PATENT ABSTRACT
A method and device for field detection are provided, in which one or more gradiometers ( 31, 32, 33 ) are positioned in the field and rotated about their axes. Rotation of a single gradiometer allows an output signal of the gradiometer to be analyzed in the Fourier domain, which allows particular field components to be obtained, and also separates the field signal from noise, in the frequency domain. Use of three such rotating gradiometers with non-parallel axes allows a complete magnetic field gradient tensor to be obtained with data redundancy, and can reduce crosstalk.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This application is a 35 U.S.C. 371 national stage of International Application No. PCT/AU2003/000999 filed Aug. 7, 2003, which claims priority to Australian Application No. AU 2002950624 filed Aug. 7, 2002. 
   TECHNICAL FIELD 
   The present invention relates to a method and apparatus for measurement of a magnetic field, and in particular relates to rotating a magnetic sensor relative to the magnetic field to be measured. 
   BACKGROUND ART 
   A gradiometer is a device for measuring the difference between two signals, with an aim to facilitate rejection of common mode noise signals and improved reduction in errors that arise from misorientation of the measuring instrument. 
   The ability of a gradiometer to measure small magnetic field gradients in the presence of a large background magnetic field is determined by its ability to reject large common mode signals, and the intrinsic noise and drift performance of the field detectors employed. The first of these characteristics is called the common mode rejection ratio (CMRR). In conventional gradiometers the CMRR is determined by the precision with which the field sensors comprising the gradiometer are balanced. That is, the CMRR is determined by the relative sensitivities of the field sensors, and the accuracy to which the field sensors can be positioned in precisely parallel planes. This fundamental nexus between the precision with which conventional gradiometers can be manufactured and the ultimate CMRR performance has until now limited the CMRR to figures of the order of 10 −4  in practical instruments. 
   Further, the DC and low frequency accuracy of state of the art gradiometers is limited by the DC and drift performance of the field detectors employed, and is also limited by the low frequency noise performance of the field detectors. Furthermore, the achievable noise performance can be dependent upon the magnitude of the background homogeneous field; being determined by microphonics which arise from vibrations causing randomly varying misalignment of the axes of symmetry of the two field sensors. For example, in the case of magnetic gradiometry two appropriate magnetic sensors are the flux gate and the superconducting quantum interference device (SQUID). Flux gates have considerable drift and noise in comparison to SQUIDs, but state of the art SQUID systems are incapable of providing data about the absolute value of the magnetic field or gradient. 
   The inability of state of the art SQUIDs to measure the absolute value of the magnetic field or gradient arises for the following reasons. The radio frequency (rf) SQUID is essentially a ring of superconducting material, the ring being interrupted by a Josephson Junction. When the superconducting ring is energised by an inductively coupled resonant rf-oscillator, tunnelling of electrons takes place at the junction and a periodic signal, being a function of flux through the ring, can be detected across the junction. The periodic signal is substantially a triangular waveform, usually having a period (ΔB) in the order of a nanotesla. Therefore, in order to yield a sensitivity in the femtotesla range, the SQUID is operated in a nulling bridge mode, or flux locked loop (FLL) mode. In this mode, magnetic flux is fed back to the SQUID so as to cause the output voltage to remain relatively constant. The feedback voltage, being proportional to the difference between the applied flux and the quiescent flux level, gives a highly accurate measurement of relative magnetic flux. The feedback voltage V can therefore be written as
 
 V=M ( A   eff   B+u )  (1)
 
where
 
   M is a constant in a specific SQUID system; 
   A eff  is the effective area of the SQUID; 
   B is the applied magnetic field; and 
   u is the quiescent flux. 
   However, the quiescent flux u is unknown. Thus, conventional rf-SQUIDs provide only relative measurements of magnetic field, up to an offset (which can usually be held constant for the duration of the measurement procedure) which is an integer multiple of a field value determined by the sensitivity of the SQUID and the fundamental flux quantum. The SQUID, therefore, is not an absolute field detector and state of the art gradiometers cannot provide absolute value measurements of the gradient components. 
   A further problem arises in tensor gradiometers. The gradient of a field in 3 dimensions is a second rank tensor, and accordingly has 9 components. In the case of magnetic fields Maxwell&#39;s equations impose restrictions which imply that only 5 of the gradient components are independent. To determine the gradient tensor of a magnetic field it is therefore necessary to measure all 5 of these components. State of the art tensor gradiometers achieve this by deploying a suite of at least 5 field sensors or planar gradiometers, or a combination of field sensors and planar gradiometers. 
   However, a problem which arises from the use of a plurality of field sensors or gradiometers to measure the gradient components, is that the sensors or gradiometers distort the field in their neighbourhood. Consequently, each sensor or gradiometer adversely affects the measurement of the field by the other sensors or gradiometers which are deployed to measure other gradient components. Such a “cross-talk” effect limits the accuracy of the gradient measurement. 
   Any discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is solely for the purpose of providing a context for the present invention. It is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present invention as it existed before the priority date of each claim of this application. 
   Throughout this specification the word “comprise”, or variations such as “comprises” or “comprising”, will be understood to imply the inclusion of a stated element, integer or step, or group of elements, integers or steps, but not the exclusion of any other element, integer or step, or group of elements, integers or steps. 
   Throughout this specification, the terms ‘superconducting material’, ‘superconducting device’ and the like are used to refer to a material or device which, in a certain state and at a certain temperature, is capable of exhibiting superconductivity. The use of such terms does not imply that the material or device exhibits superconductivity in all states or at all temperatures. 
   SUMMARY OF THE INVENTION 
   According to a first aspect, the present invention provides a method of field detection comprising: 
   providing a gradiometer in a field, the gradiometer having at least first and second field vector sensors connected in a differencing arrangement; and 
   controllably altering a position of the at least first and second field vector sensors relative to the field during operation of the gradiometer. 
   According to a second aspect the present invention provides a field detection device comprising: 
   a gradiometer, the gradiometer having at least first and second field vector sensors connected in a differencing arrangement; and 
   means for controllably altering the position of the at least first and second field vector sensors relative to a field during operation of the gradiometer. 
   In preferred embodiments of the present invention, the at least first and second field vector sensors of the gradiometer are substantially axially aligned, thus forming an axial gradiometer. That is, the at least first and second field vector sensors are preferably positioned relative to each other such that first and second field vectors sensed by the respective first and second field vector sensors are substantially co-axial. However, the present invention may also be applied in respect of other devices such as planar gradiometers. 
   In particularly preferred embodiments of the present invention, the position of the at least first and second field vector sensors is controllably altered by rotating the at least first and second field vectors about an axis of rotation. Such rotation may be continuous, as in the case of a motor driving the gradiometer at a substantially constant revolutional velocity about the axis of rotation. Alternatively, the rotation may be piecewise about the axis of rotation, as in the case of manual rotation of the gradiometer through a given angle about the axis of rotation. Measurements may be obtained from the gradiometer before, during, and/or after such a manual rotation. 
   In preferred embodiments of the invention, the axis of rotation is positioned substantially perpendicular to the co-axial first and second field vectors. The axis of rotation preferably passes substantially between the first and second field vector sensors, and substantially equidistant from the first and second field vector sensors. 
   In accordance with preferred embodiments of the present invention, the gradiometer is made to rotate so that the direction of the sensitive direction of the field vector sensors or their sensitivities is changed in a known way. 
   In accordance with preferred embodiments of the invention, the gradiometer may be an axial magnetic gradiometer, and may be formed from a pair of SQUIDs, flux gates or superconducting pick-up loops with their sensitivity vectors lying substantially in a nominal x-y plane. In preferred embodiments, such an axial gradiometer is rotated about a nominal z-axis to form the rotating gradiometer. 
   While the use of an axial gradiometer may be favoured in implementations of the present invention because of its intrinsic noise immunity, the advantages of rotating a gradiometer are not limited to the axial configuration. 
   It has been realised by the present inventors that a rotating axial magnetic gradiometer returns information about the value of the g xy  component and linear combinations of the values of the g xx  and g yy  components of a field gradient tensor, in addition to information about the B x  and B y  field components. In preferred embodiments of the invention, such information is retrieved by transforming the voltage waveform from the field vector sensors into the Fourier domain. 
   Furthermore, it has been realised that it is possible to obtain the complete magnetic gradient tensor by deploying three such rotating gradiometers, configured in such a way that their respective axes of rotation are not parallel. 
   Accordingly, in a third aspect, the present invention provides a method of obtaining a complete magnetic gradient tensor of a magnetic field, the method comprising: 
   providing at least three axial gradiometers in the magnetic field such that an axis of each axial gradiometer is not parallel to an axis of any other one of the at least three gradiometers; and 
   controllably altering a position of each of the at least three gradiometers relative to the magnetic field during operation of the gradiometer. 
   Preferably the step of controllably altering comprises rotating each of the at least three axial gradiometers about a respective axis of rotation of that gradiometer. The axis of rotation of each gradiometer is preferably not parallel to an axis of rotation of any other one of the at least three gradiometers. 
   Similarly, in a fourth aspect, the present invention provides a device for obtaining a complete magnetic gradient tensor of a magnetic field, the device comprising: 
   at least three axial gradiometers positioned such that an axis of each axial gradiometer is not parallel to an axis of any other one of the at least three gradiometers; 
   means for controllably altering a position of each of the at least three gradiometers relative to the magnetic field during operation of the gradiometer. 
   Preferably the means for controllably altering comprises means for rotating each of the at least three axial gradiometers about a respective axis of rotation of that gradiometer. The axis of rotation of each gradiometer is preferably not parallel to an axis of rotation of any other one of the at least three gradiometers. 
   In preferred embodiments of the third and fourth aspects of the invention, the at least three gradiometers are rotated at a substantially constant velocity about their respective axes of rotation. Such embodiments produce data suitable for conversion to the Fourier domain. 
   Embodiments of the third and fourth aspects of the invention effectively provide a single measurement of the true value of each of the three off-diagonal components of the gradient tensor, three measurements of the diagonal components and two measurements of each of the field components. Because more than the minimum of five tensor components are obtained, the use of just three such rotating gradiometers in fact provides a redundant measurement. This redundancy permits a least-squares best fit solution, further improving the quality of data obtained. Any DC offsets such as low frequency drift in flux gates or the fixed offset of a SQUID, for example, can also be determined and monitored to provide information about the operating conditions of the detectors. Such offsets can also be separated from gradient data in the Fourier domain. 
   In preferred embodiments of the third and fourth aspects of the invention, data obtained is converted to the Fourier domain. In the Fourier domain the gradient information is distinguishable from the field information because the signals due to the gradient fields are centred at twice the rotation frequency while those due to the fields are centred at the rotation frequency. 
   Further, information about the g xy  component of the gradient tensor may be distinguished from that due the diagonal gradient components, even at the same frequency, because one involves taking the Fourier transform of a cosine waveform, which is real and even, while the other involves the Fourier transform of a sine wave, which is odd and imaginary. The Fourier transformation may be implemented by means of a conventional fast Fourier transform algorithm or even simple discrete implementation via a coarse sampling of say 8 data points per cycle of the rotation. 
   To first-order any mismatches in the sensitivities of the field sensors or their relative alignment gives rise to contributions only at the fundamental frequency (discussed in more detail in the following) and therefore such mismatches do not determine the ultimate CMRR of the gradiometer. The final CMRR performance is determined by the resolution and precision of the data handling and the ultimate noise floor of the detectors. Accordingly, preferred embodiments of the present invention may provide for a CMRR improved by several orders of magnitude over that of state of the art instruments. 
   Further, since the external gradient and field are modulated by the rotation, but DC offsets, drift or very low frequency noise in the detectors are not, the rotation provided by the present invention facilitates separation of the these effects in the Fourier domain so that the DC and low frequency performance of the instrument is not limited by the performance of the detectors in these respects. 
   Furthermore, in the case of magnetic gradiometry, if the gradiometer is to be employed in a moving platform such as an aircraft, for example, a suspension system is generally used to isolate the instrument from vibrations which would otherwise induce significant noise by moving the instrument in the earth&#39;s magnetic field. However, in previous systems, such suspension systems have proven to be extremely difficult to engineer as significant isolation is required at sub 1 Hz frequencies. However, the rotating gradiometer provided by present invention modulates low frequency signals with the rotation frequency, thus facilitating the use of suspension systems which then only need to provide effective vibration isolation near the fundamental and first overtone of the rotation frequency. Such suspension systems are typically significantly easier to engineer than sub-1 Hz suspension systems. 
   In preferred embodiments of the third and fourth aspects of the invention, the at least three gradiometers are rotated at differing frequencies. Such embodiments of the invention facilitate separation of the frequency of operation of each gradiometer, and further facilitate a reduction in “cross-talk” between the respective gradiometers, of the type set out in the preceding discussion. As such cross-talk effects will be occurring at differing frequencies, the effect of one gradiometer operation on another can be filtered out or removed in the frequency domain. 
   Accordingly, preferred embodiments of the present invention may yield significant advantages over previous field detection devices and methods in a number of ways. First, rotation of the gradiometer breaks the nexus between engineering tolerances (in positioning two sensors in precisely parallel planes) and the ultimate CMRR performance. Further, providing three rotating gradiometers in accordance with embodiments of the third and fourth aspects of the present invention yields measurements of all five components of the gradient tensor and all three the components of the field, with redundancy in the data, all by using only 3 field detectors. Additionally, the rotation of the gradiometer breaks the dependence of the DC performance of the instrument on the DC performance of the detectors. Embodiments of the present invention also facilitate absolute value measurements of both the gradient and field components using SQUIDs. Embodiments of the present invention may also provide real time information of the condition of the SQUID in operation, for example as to whether or not the SQUID has remained in lock. As noted in the preceding discussions, previous SQUID devices have only been capable of making relative field measurements, not absolute field measurements. Furthermore, embodiments of the present invention may improve the sensitivity of field detection by shifting the working point of the detectors to a higher frequency, where the noise performance of the detectors and the vibration isolation afforded by suspension systems is superior. Yet further, certain embodiments of the present invention can serve to reduce the effects of cross-talk by use of distinguishable rotation frequencies. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     By way of example only, preferred embodiments of the invention will be described with reference to the accompanying drawings, in which: 
       FIG. 1   a  illustrates a low temperature superconducting gradiometer in accordance with the present invention; 
       FIG. 1   b  illustrates a high temperature superconducting gradiometer in accordance with the present invention; and 
       FIG. 2  illustrates a three-gradiometer magnetic field detection device in accordance with the present invention. 
   

   DESCRIPTION OF THE INVENTION 
     FIG. 1   a  illustrates a low temperature superconducting gradiometer  10  in accordance with the present invention. Gradiometer  10  comprises a first field vector sensor in the form of superconducting wire loop  11 . Gradiometer  10  further comprises a second field vector sensor in the form of superconducting wire loop  12 . Sensors  11  and  12  are positioned substantially coaxially so as to sense field vectors along axis  13 . Further, sensors  11  and  12  are connected in a differencing arrangement, whereby a signal produced at terminals  14  stems from the difference between the magnetic field vectors sensed by sensors  11  and  12 . In accordance with the present invention, gradiometer  10  is configured to rotate about axis  15 , which is at a normal to the page as viewed in  FIG. 1   a . Rotation of the gradiometer is indicated by direction arrows  16 . 
     FIG. 1   b  illustrates a high temperature superconducting gradiometer  20  in accordance with the present invention. Gradiometer  20  comprises a first field vector sensor in the form of a SQUID  21 . Gradiometer  10  further comprises a second field vector sensor in the form of a SQUID  22 . SQUIDs  21  and  22  are mounted on a suitable magnetically transparent mount. Sensors  21  and  22  are positioned substantially coaxially so as to sense field vectors along axis  23 . Further, sensors  21  and  22  are connected in a differencing arrangement (not shown), whereby a signal thus produced stems from the difference between the magnetic field vectors sensed by sensors  21  and  22 . In accordance with the present invention, gradiometer  20  is configured to rotate about axis  25 . 
     FIG. 2  illustrates a three-gradiometer magnetic field detection device  30  in accordance with the present invention. The gradiometers are represented by cylinders  31 ,  32 ,  33 . Cylinders  31 ,  32 ,  33  are respectively positioned on axes  41 ,  42 ,  43 , and each of axes  41 ,  42  and  43  is not parallel with either of the other two axes. Each of gradiometers  31 ,  32 ,  33  is rotatable about unique non-parallel axes in accordance with the present invention. Operation of such a device is set out in more detail in the following. 
   This calculation models the sensitivity of the gradiometer instrument to the magnetic field and gradient tensor in the presence of 6 pairs of misalignment angles introduced by tilting the SQUIDs through arbitrary angles. The effects of precession are not considered explicitly but can probably be modelled to sufficient accuracy within the theory. It is demonstrated that the misalignment errors lead to contamination of the gradient and B-field components by terms that depend on the misalignments, but the effects of misalignment can be removed by the theory by introducing appropriate calibrations constants. A final solution is derived in terms of instrument coordinates and the transformations to geographic coordinates is also described.
 
The resulting instrument provides a measurement of the true-value of the gradient and field components which is relatively robust with respect to changes in the calibration constants.
 
   I. Overview 
   The present embodiment comprises of a set of three rotating axial gradiometers or drum gradiometers. In the 2-SQUID implementation the output voltages from a pair of SQUIDs operated in flux-locked loops are differenced to provide information about the magnetic field density and the first-order magnetic gradient tensor. If the SQUIDs are perfectly aligned with identical sensitivities then the difference voltage provides no information about the magnetic field density but this information is restored if the sensitivities of the SQUIDs are mismatched by a known amount. The difference voltage also provides a measurement of the first-order axial gradient provided that the contributions from higher-order gradients are sufficiently small to be ignored. This is easily seen by expanding the B-field as a Taylor series about the origin which is take to be centre point of the axial gradiometer. 
   A. Taylor Series for Functions of Three Variables. 
   The Taylor series for a scalar function B (x, y, z) about the origin is 
   
     
       
         
           
             
               
                 
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   In the case of a 3-dimensional vector field the previous equation applies to each of the vector components, so the α-component of B at the point r, B α (r), and at the origin, B α (0), are related by 
                     B   α     ⁡     (   r   )       =         B   α     ⁡     (   0   )       +       (       x   ⁢     ∂     ∂   x         +     y   ⁢     ∂     ∂   y         +     z   ⁢     ∂     ∂   z           )     ⁢     B   α       +       1     2   !       ⁢     (       x   ⁢     ∂     ∂   x         +     y   ⁢     ∂     ∂   y         +     z   ⁢     ∂     ∂   z           )     ⁢     B   α       +       1     3   !       ⁢       (       x   ⁢     ∂     ∂   x         +     y   ⁢     ∂     ∂   y         +     z   ⁢     ∂     ∂   z           )     3     ⁢     B   α       +       1     4   !       ⁢       (       x   ⁢     ∂     ∂   x         +     y   ⁢     ∂     ∂   y         +     z   ⁢     ∂     ∂   z           )     4     ⁢     B   α       +   ⋯             (   2   )               
If the second- and higher-order gradients of B are, small then 2 may be approximated by
 
                 =         B   α     ⁡     (   0   )       +       (       x   ⁢     ∂     ∂   x         +     y   ⁢     ∂     ∂   y         +     z   ⁢     ∂     ∂   z           )     ⁢     B   α                 (   3   )               
provided that r is small.
 
   Let 
             g   αβ     ⁢     =   Δ     ⁢       ∂     ∂   β       ⁢     B   α             
denote components of the the first-order gradients of B. Then eq. ?? becomes simply
   B   α ( r )≠ B   α (0)+ xg   αx   +yg   αy   zg   αz   (4) 
or in coordinate independent form
   B ( r )≠ B (0)+ ĝr   (5) 
where ĝ is the gradient matrix and r is the displacement vector to the point at which the B field is measured.
 
   Using 5 if r 1  and r are two points the difference in the B-field at these points is
 
 B ( r   1 )− B ( r )≠ B (0)+ ĝr   1 −( B (0)+ ĝr )={circumflex over ( g )}( r′   1   −r′ )
 
which is proportional to the product of the gradient matrix and the displacement between the points.
 
   II. Instrument and Drum Coordinates. 
   The drum gradiometers rotate about axes with orientations denoted by the vectors U, V and W in an umbrella configuration to be described in detail below. The axial gradiometers within the drums are located at the end points of U, V and W, and facilitate absolute value measurements of linear combinations of some of the components of the magnetic gradient tensor and magnetic field with respect to a drum coordinate system, C′, which is fixed to the axis of rotation of each of the three drums. The systems of equations in drum coordinates are transformed to a common instrument coordinate system, C, and solved to yield absolute value measurements of the magnetic gradient tensor and magnetic field in instrument coordinates. 
   For use in geophysical exploration the final measurements have to be expressed in terms of the geographic coordinate system, C G . This is easily done if the orientation of C with respect to C G  is known but in general this will require additional information from position and orientation sensors. It is possible that, in some applications, the real-time measurements of the magnetic field provided by the instrument itself, might be used to provide information about the orientation of the instrument with respect to the magnetic field of the earth. Transformations of the final measurements between the instrument coordinates and geographic coordinates are described below in section VI. 
   A. Instrument Coordinate System, C. 
   The instrument coordinates system, C, is a right-handed Cartesian coordinate system defined as follows. The x-y plane is defined by the base of the instrument. The z-direction is normal to the x-y plane and forms an axis of symmetry for the instrument in the sense that the three drum gradiometers are distributed symmetrically about the z-axis. Unit vectors in the directions of the x-, y- and z-axes will be denoted i, j and k, respectively. 
   As mentioned above the axial gradiometers are located at the end points of U, V and W and rotate about the axes defined by these vectors. The direction of U is the vector obtained by revolving a unit vector i with its base at the origin of C about the −y axis through an angle λ. The directions of V and W are obtained by revolving U about the z-axis through angles of 120° and 240° respectively. U, V and W then form a type of umbrella configuration being disposed symmetrically about the z-axis and tilted above the x-y plane through an umbrella angle angle λ. Hence 
   
     
       
         
           
             
               
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   in which L u , L v  and L w  denote the distances of the axial gradiometers from the origin of C. 
   1. Properties of U, V and W.
         Note that these unit vectors form an orthogonal system only if       

   
     
       
         
           
             
               
                 
                   
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                               2 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           , 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           L 
                           v 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 4 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       1 
                                       4 
                                     
                                     ⁢ 
                                     
                                       6 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                     
                                       1 
                                       _ 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 11 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       W 
                       = 
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 1 
                                 2 
                               
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           , 
                           
                             
                               - 
                               
                                 
                                   3 
                                 
                                 2 
                               
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                           , 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               π 
                               4 
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       = 
                       
                         
                           L 
                           w 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                 
                               
                               
                                 
                                   2 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
         
       
     
   
   2. Unit Drum Vectors 
   It will prove useful to define the unit drum vectors, u 0 , v 0  and w 0 , that are normal to U, V and W respectively. Vector u 0  is the unit vector obtained by revolving a unit vector i with its base at the origin of C about the +y axis through an angle (90−λ) degrees. Vectors v 0  and w 0  are obtained by revolving u 0  about the z-axis through angles of 120° and 240° respectively, In instrument coordinates 
                   u   0     =     (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ           )             (   13   )                 v   0     =     (             -     1   2       ⁢   sin   ⁢           ⁢   λ               3     2     ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ           )             (   14   )                   w   0     =     (             -     1   2       ⁢   sin   ⁢           ⁢   λ             -       3     2       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ           )       ⁢     
     ⁢     N   .   B   .     
     ⁢               u   0     ·   U     =       (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ           )     ·       L   u     ⁡     (       cos   ⁢           ⁢   λ     ,   0   ,     sin   ⁢           ⁢   λ       )                     =   0             ⁢     
     ⁢               v   0     ·   V     =       ⁢       (             -     1   2       ⁢   sin   ⁢           ⁢   λ               3     2     ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ           )     ·                     ⁢       L   v     ⁡     (         -     1   2       ⁢   cos   ⁢           ⁢   λ     ,         3     2     ⁢   cos   ⁢           ⁢   λ     ,     sin   ⁢           ⁢   λ       )                   =       ⁢   0           ⁢     
     ⁢               w   0     ·   W     =       ⁢       (             -     1   2       ⁢   sin   ⁢           ⁢   λ             -       3     2       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ           )     ·                     ⁢       L   w     ⁡     (         -     1   2       ⁢   cos   ⁢           ⁢   λ     ,       -       3     2       ⁢   cos   ⁢           ⁢   λ     ,     sin   ⁢           ⁢   λ       )                   =       ⁢   0                   (   15   )               
as required.
 
   B. Drum Coordinate Systems C′ 
   As mentioned earlier we introduce 3 drum coordinate systems, C′, C″ and C′″ associated with the each of the three drums rotating about U, V and W respectively. Let x′, y′ and z′ denote unit vectors in the directions of the x′, y′ and z′ axes of the drum coordinate systems. C′ is defined so that x′=u 0 , y′=j, z′=U/L u  and the origins of C′ U  and C coincide. With these definitions C′ U  is a right-handed coordinate system which would become equivalent to C if the umbrella angle was set to 90°. 
   C″ and C′″ are then defined by rotating C′ about the z-axis through angles 120° and 240° so that in C″, x″=v 0  and z″=V/L v  while in C′″, x′″=w 0  and z′″=W/L w . 
   1. Unit Gradiometer Vectors. 
   The instantaneous direction of the rotating axial gradiometers will be defined, by unit gradiometer vectors u, v and w, which remain normal to U, V and W and undergo continuous rotation about the end points of U, V and W with angular frequencies w u , w v  and w w  respectively. θ u , θ v  and θ w  will denote respectively the angular displacements of the vectors u, v and w at time t, from the positions of u 0 , v 0  and w 0  defined above, so that θ u =W u t, θ v =w v t and θ w =w w t. In terms of the drum coordinates C′ U , C′ V  and C′ W  the unit gradiometer vectors u′, v′ and w′ assume the same form
 
 u ′=(cos θ u , sin θ u , 0)  (16)
 
 v ′=(cos θ v , sin θ v , 0)  (17)
 
 w ′=(cos θ w , sin θ w , 0)  (18)
 
   C. Coordinate Transformations Between C and the C′ Systems. 
   In general if P and P′ denote the components of a vector in the frames C and C′ respectively and if C is mapped to C′ by a positive (right-handed) rotation through an angle κ about an axis k (k=x, y, z) then
 
 P′={circumflex over (R)}   k (κ) P   (19)
 
where {circumflex over (R)} k (κ) denotes the standard rotation matrices
 
   
     
       
         
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     x 
                   
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   ( 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     y 
                   
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         0 
                       
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         0 
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       R 
                       ^ 
                     
                     z 
                   
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 22 
                 ) 
               
             
           
         
       
     
   
   Also if ĝ and ĝ′ are the components of a Cartesian tensor in the frames C and C′ respectively then
 
{circumflex over ( g )}′={circumflex over ( R )} k (κ){circumflex over ( g )}{circumflex over ( R   T )} k (κ).  (23)
 
   For example a rotation about the z-axis through angle κ is 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     g 
                                     11 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     1 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     13 
                                     ′ 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     1 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     3 
                                     ′ 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     13 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     3 
                                     ′ 
                                   
                                 
                                 
                                   
                                     g 
                                     33 
                                     ′ 
                                   
                                 
                               
                             
                             ) 
                           
                           = 
                             
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     g 
                                     11 
                                   
                                 
                                 
                                   
                                     g 
                                     1 
                                   
                                 
                                 
                                   
                                     g 
                                     13 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     1 
                                   
                                 
                                 
                                   g 
                                 
                                 
                                   
                                     g 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     13 
                                   
                                 
                                 
                                   
                                     g 
                                     3 
                                   
                                 
                                 
                                   
                                     g 
                                     33 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       sin 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     and 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     hence 
                   
                   ⁢ 
                   
                      
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             g 
                             11 
                             ′ 
                           
                           = 
                           
                             
                               
                                   
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     g 
                                     11 
                                   
                                   + 
                                   g 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                     
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       g 
                                       11 
                                     
                                     - 
                                     g 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               k 
                             
                             + 
                             
                               
                                 g 
                                 1 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             ′ 
                           
                           = 
                           
                             
                               
                                   
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     g 
                                     11 
                                   
                                   + 
                                   g 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 
                                     
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       g 
                                       11 
                                     
                                     - 
                                     g 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   k 
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 g 
                                 1 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             33 
                             ′ 
                           
                           = 
                           
                             g 
                             33 
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                             ′ 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   
                                       
                                     
                                       1 
                                       _ 
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         g 
                                         11 
                                       
                                       - 
                                       g 
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   k 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 g 
                                 1 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             3 
                             ′ 
                           
                           = 
                           
                             
                               
                                 g 
                                 3 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                             
                             - 
                             
                               
                                 g 
                                 13 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             13 
                             ′ 
                           
                           = 
                           
                             
                               
                                 g 
                                 13 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                             
                             + 
                             
                               
                                 g 
                                 3 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                   
                 
                 } 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
         
       
     
   
   1. Transformations C⇄C′. 
   When x′=u 0 , y′=j and z′=U/L u , C is mapped onto C′ by a rotation about y through an angle (π/2−λ). Hence if B and B′ denote the components of a vector in the frames C and C′ respectively then 
                   B   ′     =       (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ             0       1       0             cos   ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢   B             (   25   )               B   =       (           sin   ⁢           ⁢   λ         0         cos   ⁢           ⁢   λ             0       1       0                     ⁢       -   cos     ⁢           ⁢   λ           0         sin   ⁢           ⁢   λ           )     ⁢     B   ′               (   26   )               
N.B. If λ=π/4 then
 
                   B   ′     =       (           2             ⁢     1   _             0         -     2             ⁢     1   _                   0       1       0             2             ⁢     1   _             0         2             ⁢     1   _               )     ⁢   B             (   27   )               
while if λ=π/2 25 and 26 give that P′=P. i.e. the coordinate systems C and C′ U  coincide, as required. Also, using equation 6,
 
                   U   ′     =       (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ             0       1       0             cos   ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢   U                 =       (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ             0       1       0             cos   ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     (             L   u     ⁢   cos   ⁢           ⁢   λ             0               L   u     ⁢   sin   ⁢           ⁢   λ           )                   =       L   u     ⁡     (         0           0           1         )                   
as required.
 
   Using 26 the unit gradiometer vector 
                     u   ′     =     (           cos   ⁢           ⁢     θ   u                 sin   ⁢           ⁢     θ   u               0         )       ⁢     
     ⁢   becomes   ⁢     
     ⁢     u   =         (           sin   ⁢           ⁢   λ         0         cos   ⁢           ⁢   λ             0       1       0               -   cos     ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     (           cos   ⁢           ⁢     θ   u                 sin   ⁢           ⁢     θ   u               0         )       =     (           cos   ⁢           ⁢     θ   u     ⁢   sin   ⁢           ⁢   λ               sin   ⁢           ⁢     θ   u                   -   cos     ⁢           ⁢     θ   u     ⁢   cos   ⁢           ⁢   λ           )                 (   28   )               
in C. Observe
 
             u   ·   U     =         (           cos   ⁢           ⁢     θ   u     ⁢   sin   ⁢           ⁢   λ               sin   ⁢           ⁢     θ   u                   -   cos     ⁢           ⁢     θ   u     ⁢   cos   ⁢           ⁢   λ           )     ·     (             L   u     ⁢   cos   ⁢           ⁢   λ             0               L   u     ⁢   sin   ⁢           ⁢   λ           )       =   0           
so u⊥U as required.
 
   a. Gradient Transformation. The transformations of the gradient tensor from C to C′ follows from the use of the transformation matrix in 25, with the general transformation equation, 23, i.e. 
                   (           g   11   ′           g   1   ′           g   13   ′               g   1   ′           g   ′           g   3   ′               g   13   ′           g   3   ′           g   33   ′           )     =       (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ             0       1       0             cos   ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     (           g   11           g   1           g   13               g   1         g         g   3               g   13           g   3           g   33           )     ⁢     (           sin   ⁢           ⁢   λ         0         cos   ⁢           ⁢   λ             0       1       0               -   cos     ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )               (   29   )                         g             ⁢   11               ⁢   ′       =                     ⁢     1   _         ⁢     (         g   ⁢             11     +     g             ⁢   33       -       g             ⁢   11       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       g             ⁢   33       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -     2   ⁢     g             ⁢   13       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ       )                     g   ′     =   g                       ⁢       ⇒     g   33   ′       =             1   _       ⁢     (       g   11     +     g   33     +       g   11     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   33     ⁢   cos   ⁢           ⁢   2   ⁢   λ     +     2   ⁢     g   13     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ       )                       g   1   ′     =       2             ⁢     1   _         ⁢     (       g   1     -     g   3       )                     g   13   ′     =     -                     ⁢     1   _         ⁢     (       2   ⁢     g   13     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   11     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +       g   33     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ       )                       g   3   ′     =       2             ⁢     1   _         ⁢     (       g   1     +     g   3       )               }           (   30   )               
In the special case λ=π/4
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               g 
                               11 
                               ′ 
                             
                             = 
                             
                               
                                 g 
                                 11 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               - 
                               
                                 g 
                                 13 
                               
                               + 
                               
                                 g 
                                 33 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             ′ 
                           
                           = 
                           g 
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               g 
                               33 
                               ′ 
                             
                             = 
                             
                               
                                 g 
                                 11 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               + 
                               
                                 g 
                                 13 
                               
                               + 
                               
                                 g 
                                 33 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               g 
                               1 
                               ′ 
                             
                             = 
                             
                               
                                 2 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     g 
                                     1 
                                   
                                   - 
                                   
                                     g 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               g 
                               13 
                               ′ 
                             
                             = 
                             
                               
                                 g 
                                 11 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               - 
                               
                                 g 
                                 33 
                                   
                                   
                                   
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               g 
                               3 
                               ′ 
                             
                             = 
                             
                               
                                 2 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     g 
                                     1 
                                   
                                   + 
                                   
                                     g 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
                 . 
               
             
             
               
                 ( 
                 31 
                 ) 
               
             
           
         
       
     
   
   2. Transformations C⇄C″. 
   When x″=v 0  and z″=V/L v , C is mapped onto C″ by a rotation about z through an angle 2π/3 followed by a rotation about y through an angle (π/2−λ) 
                   B   ″     =       (             -                     ⁢     1   _         ⁢   sin       ⁢           ⁢   λ             3             ⁢     1   _         ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               -     3             ⁢     1   _                   -           1   _         ⁢                 0               -                     ⁢     1   _         ⁢   cos       ⁢           ⁢   λ             3             ⁢     1   _         ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢   B             (   32   )               B   =       (             -                     ⁢     1   _         ⁢   sin       ⁢           ⁢   λ           -     3             ⁢     1   _                   -                     ⁢     1   _         ⁢   cos       ⁢           ⁢   λ                 3             ⁢     1   _         ⁢     (     sin   ⁢           ⁢   λ     )               -           1   _         ⁢                     3             ⁢     1   _         ⁢     (     cos   ⁢           ⁢   λ     )                   -         cos       ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     B   ″               (   33   )               
If λ=π/4
 
             B   ″     =       (             -     1   4       ⁢     2               1   4     ⁢     6             -     2             ⁢     1   _                     -     3             ⁢     1   _                 -           1   _             0               -     1   4       ⁢     2               1   4     ⁢     6             2             ⁢     1   _               )     ⁢     B   .             
Using 33 the unit gradiometer vector
 
                     v             ⁢   ″       =     (           cos   ⁢           ⁢     θ             ⁢   υ                   sin   ⁢           ⁢     θ             ⁢   υ                 0         )       ⁢     
     ⁢   becomes   ⁢     
     ⁢     v   =         (             -           ⁢                     ⁢     1   _         ⁢           ⁢   sin       ⁢           ⁢   λ           -           ⁢     3             ⁢     1   _                   -           ⁢                     ⁢     1   _         ⁢           ⁢   cos       ⁢           ⁢   λ                       ⁢       3             ⁢     1   _         ⁢           ⁢     (     sin   ⁢           ⁢   λ     )                 -           ⁢                   ⁢     1   _           ⁢                           ⁢       3             ⁢     1   _         ⁢           ⁢     (     cos   ⁢           ⁢   λ     )                     -         cos       ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     (           cos   ⁢           ⁢     θ             ⁢   υ                   sin   ⁢           ⁢     θ             ⁢   υ                 0         )       =     (               -           ⁢                     ⁢     1   _         ⁢           ⁢   sin       ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ       -       3             ⁢     1   _         ⁢           ⁢     (     sin   ⁢           ⁢     θ   υ       )                       -           ⁢                     ⁢     1   _         ⁢           ⁢   sin       ⁢           ⁢     θ   υ       +       3             ⁢     1   _         ⁢           ⁢     (     sin   ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ       )                     -   cos     ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ             )                 (   34   )               
in C.
 
   Observe that
 
 v·V= 0
 
confirming that v⊥V as required. Also using 6
 
                         V   ′′     =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               -             1   _       ⁢     3               -     1   _           0               -             1   _       ⁢   cos       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢   V                 =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               -             1   _       ⁢     3               -     1   _           0               -             1   _       ⁢   cos       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢     (             -     L   υ                 1   _         ⁢   cos   ⁢           ⁢   λ                 L   υ                   3     _       ⁢   cos   ⁢           ⁢   λ                 L   υ     ⁢   sin   ⁢           ⁢   λ           )                   =       L   υ     ⁡     (         0           0           1         )                     
as required.
 
   a. Gradient Transformation. The transformations of the gradient tensor from C to C′ follows from the use of the transformation matrix in 32, with the general transformation equation, 23, i.e. 
                     (           g   11   ′′           g   1   ′′           g   13   ′′               g   1   ′′           g   ′′           g   3   ′′               g   13   ′′           g   3   ′′           g   33   ′′           )     =     (             -             1   _       ⁢   sin       ⁢           ⁢   λ                     3     _       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               -       3     _             -     1   _           0               -             1   _       ⁢   cos       ⁢           ⁢   λ                     3     _       ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )       ⁢     
     ⁢       (           g   11           g   1           g   13               g   1         g         g   3               g   13           g   3           g   33           )     ⁢     (                     -     1   _       ⁢   sin       ⁢           ⁢   λ           -       3     _               -             1   _       ⁢   cos       ⁢           ⁢   λ                         3     _       ⁢     (     sin   ⁢           ⁢   λ     )             -     1   _                       3     _       ⁢     (     cos   ⁢           ⁢   λ     )                   -   cos     ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )               (   35   )                         g   11   ′′     =         1   8     ⁢     g   11       +       3   8     ⁢   g     +     g   33                 1   _       -       1   8     ⁢     g   11     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       3   8     ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   λ     +                     g   33                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       g   13                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     -       1   4     ⁢     g   1     ⁢     3       +       1   4     ⁢     g   1     ⁢     3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   3                 1   _       ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                   g   ′′     =         1   4     ⁢   g     +       1   4     ⁢     g   1     ⁢     3       +       3     ⁢     (         1   4     ⁢     g   1       +       1   4     ⁢     g   11     ⁢     3         )                       g   33   ′′     =         1   8     ⁢     g   11       +       3   8     ⁢   g     +     g   33                 1   _       -       1   8     ⁢     g   11     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       3   8     ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   λ     -                     g   33                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   13                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     -       1   4     ⁢     g   1     ⁢     3       -       1   4     ⁢     g   1     ⁢     3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       g   3                 1   _       ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                 ⇒                         g   1   ′′     =                 1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢     g   3       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   1       +               1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢     g   13     ⁢     3     ⁢           +     1   4         ⁢     (     sin   ⁢           ⁢   λ     )                         g   11     ⁢     3       -       1   4     ⁢     (     sin   ⁢           ⁢   λ     )     ⁢   g   ⁢     3                     g   13   ′′     =         g   13                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       1   8     ⁢     g   11     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +       3   8     ⁢   g   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   33                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     -     g   3                 1   _                         3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       1   4     ⁢     g   1     ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                   g   3   ′′     =         -             1   _       ⁢     (     cos   ⁢           ⁢   λ     )         ⁢     g   1       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   3       +       1   4     ⁢     (     cos   ⁢           ⁢   λ     )     ⁢     g   11     ⁢     3       -       1   4     ⁢     (     cos   ⁢           ⁢   λ     )     ⁢   g                     3     -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   13     ⁢     3               }           (   36   )               
In the special case λ=π/4
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             11 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               g 
                               13 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             + 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             - 
                             
                               
                                 g 
                                 3 
                                   
                                   
                                   
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                   
                                 
                                   3 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 3 
                                 4 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               g 
                             
                             + 
                             
                               
                                 g 
                                 1 
                                   
                                   
                                   
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 3 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             33 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             - 
                             
                               g 
                               13 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             + 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             + 
                             
                               
                                 3 
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                             + 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 13 
                               
                             
                             - 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               g 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             13 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             - 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             3 
                             ′′ 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                             + 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 13 
                               
                             
                             - 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               g 
                             
                           
                         
                       
                     
                   
                   } 
                 
                   
               
             
             
               
                 ( 
                 37 
                 ) 
               
             
           
         
       
     
   
   3. Transformations C⇄C′″. 
   When x′″=w 0  and z′″=W/L w , C is mapped onto C′″ by a rotation about z through an angle 4π/3 followed by a rotation about y through an angle (π/2−λ). Hence 
                   B   ′′′     =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ             -     3     1   _         ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               3     1   _             -     1   _           0               -             1   _       ⁢   cos       ⁢           ⁢   λ             -     3     1   _         ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢   B             (   38   )               B   =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ           3     1   _               -             1   _       ⁢   cos       ⁢           ⁢   λ                 -     3     1   _         ⁢     (     sin   ⁢           ⁢   λ     )             -     1   _               -     3     1   _         ⁢     (     cos   ⁢           ⁢   λ     )                   -         cos       ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     B   ′′′               (   39   )               
If λ=π/4
 
                   B   ′′′     =       (             -     1   4       ⁢     2               -     1   4       ⁢     6             -     2     1   _                   3     1   _             -     1   _           0               -     1   4       ⁢     2               -     1   4       ⁢     6             2     1   _             )     ⁢     B   .               (   40   )               
Using 39 the unit gradiometer vector
 
                     w   ′′′     =     (           cos   ⁢           ⁢     θ   w                 sin   ⁢           ⁢     θ   w               0         )       ⁢     
     ⁢   becomes   ⁢     
     ⁢     w   =         (             -             1   _       ⁢   sin       ⁢           ⁢   λ           3     1   _               -             1   _       ⁢   cos       ⁢           ⁢   λ                 -     3     1   _         ⁢     (     sin   ⁢           ⁢   λ     )             -     1   _               -     3     1   _         ⁢     (     cos   ⁢           ⁢   λ     )                   -         cos       ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢     (           cos   ⁢           ⁢     θ   υ                 sin   ⁢           ⁢     θ   υ               0         )       =     
     ⁢     (               -             1   _       ⁢   sin       ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ       +       3     1   _       ⁢     (     sin   ⁢           ⁢     θ   υ       )                       -             1   _       ⁢   sin       ⁢           ⁢     θ   υ       -       3     1   _       ⁢     (     sin   ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ       )                     -   cos     ⁢           ⁢   λ   ⁢           ⁢   cos   ⁢           ⁢     θ   υ             )                 (   41   )               
in C.
 
   Observe that
 
 w·W= 0
 
confirming that w⊥W as required. Also using equation 8
 
                   W   ′′′     =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ             -     3     1   _         ⁢     (     sin   ⁢           ⁢   λ     )               -         cos       ⁢           ⁢   λ               3     1   _             -     1   _           0             -                   1   _         ⁢     (     cos   ⁢           ⁢   λ     )                 -     3     1   _         ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢   W                 =       (             -             1   _       ⁢   sin       ⁢           ⁢   λ             -     3     1   _         ⁢     (     sin   ⁢           ⁢   λ     )               -         cos       ⁢           ⁢   λ               3     1   _             -     1   _           0             -                   1   _         ⁢     (     cos   ⁢           ⁢   λ     )                 -     3     1   _         ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢     (             -     L   w                 1   _         ⁢   cos   ⁢           ⁢   λ                 -     L   w                   3     _         ⁢   cos   ⁢           ⁢   λ                 L   w     ⁢   sin   ⁢           ⁢   λ           )                   =       L   w     ⁡     (         0           0           1         )                   
as required.
 
   a. Gradient Transformation. The transformations of the gradient tensor from C to C′″ follows from the use of the transformation matrix in 38, with the general transformation equation, 23, i.e. 
                     (           g   11   ′′′           g   1   ′′′           g   13   ′′′               g   1   ′′′           g   ′′′           g   3   ′′′               g   13   ′′′           g   3   ′′′           g   33   ′′′           )     =     (             -             1   _       ⁢   sin       ⁢           ⁢   λ           -               3     _       ⁢     (     sin   ⁢           ⁢   λ     )                 -   cos     ⁢           ⁢   λ                       ⁢       3     _             -     1   _           0               -             1   _       ⁢   cos       ⁢           ⁢   λ           -               3     _       ⁢     (     cos   ⁢           ⁢   λ     )               sin   ⁢           ⁢   λ           )       ⁢     
     ⁢       (           g   11           g   1           g   13               g   1         g         g   3               g   13           g   3           g   33           )     ⁢     (                     -     1   _       ⁢   sin       ⁢           ⁢   λ             3     1   _       _             -             1   _       ⁢   cos       ⁢           ⁢   λ               -               3     _       ⁢     (     sin   ⁢           ⁢   λ     )               -     1   _             -               3     _       ⁢     (     cos   ⁢           ⁢   λ     )                     -   cos     ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )               (   42   )                         g   11   ′′′     =         1   8     ⁢     g   11       +       3   8     ⁢   g     +     g   33                 1   _       -       1   8     ⁢     g   11     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       3   8     ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   λ     +                     g   33                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       g   13                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +       1   4     ⁢     g   1     ⁢     3       -       1   4     ⁢     g   1     ⁢     3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       g   3                 1   _       ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                   g   ′′′     =         3   4     ⁢     g   11       +       1   4     ⁢   g     -       g   1                 1   _       ⁢     3                       g   33   ′′′     =         1   8     ⁢     g   11       +       3   8     ⁢   g     +     g   33                 1   _       +       1   8     ⁢     g   11     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       3   8     ⁢   g   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   λ     -                     g   33                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   13                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +       1   4     ⁢     g   1     ⁢     3       +       1   4     ⁢     g   1     ⁢     3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   3                 1   _       ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                 ⇒                         g   1   ′′′     =                 1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢     g   3       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   1       -               1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢     g   13     ⁢     3     ⁢           -     1   4         ⁢     (     sin   ⁢           ⁢   λ     )                         g   11     ⁢     3       +       1   4     ⁢     (     sin   ⁢           ⁢   λ     )     ⁢   g   ⁢     3                     g   13   ′′′     =         g   13                 1   _       ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       1   8     ⁢     g   11     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +       3   8     ⁢   g   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     -       g   33                 1   _       ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ     +     g   3                 1   _                         3     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢   λ     +       1   4     ⁢     g   1     ⁢     3     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢   λ                   g   3   ′′′     =         -             1   _       ⁢     (     cos   ⁢           ⁢   λ     )         ⁢     g   1       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   3       -       1   4     ⁢     (     cos   ⁢           ⁢   λ     )     ⁢     g   11     ⁢     3       +       1   4     ⁢     (     cos   ⁢           ⁢   λ     )     ⁢   g                     3     +               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢     g   13     ⁢     3               }           (   43   )               
In the special case λ=π/4
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             g 
                             11 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               g 
                               13 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             + 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             + 
                             
                               
                                 3 
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 3 
                                 4 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               g 
                             
                             - 
                             
                               
                                   
                                 
                                   3 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             33 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             - 
                             
                               g 
                               13 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             + 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             - 
                             
                               
                                 3 
                                 
                                   1 
                                   _ 
                                 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                             - 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 13 
                               
                             
                             + 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               g 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             13 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 3 
                                 8 
                               
                               ⁢ 
                               g 
                             
                             - 
                             
                               g 
                               33 
                                 
                                 
                                 
                               
                                 1 
                                 _ 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 3 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             3 
                             ′′′ 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 1 
                               
                             
                             - 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 2 
                               
                               ⁢ 
                               
                                 g 
                                 3 
                               
                             
                             - 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 11 
                               
                             
                             + 
                             
                               
                                 1 
                                 4 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               
                                 g 
                                 13 
                               
                             
                             + 
                             
                               
                                 1 
                                 8 
                               
                               ⁢ 
                               
                                 6 
                               
                               ⁢ 
                               g 
                             
                           
                         
                       
                     
                   
                   } 
                 
                   
               
             
             
               
                 ( 
                 44 
                 ) 
               
             
           
         
       
     
   
   D. SQUID Configuration 
   The SQUIDs composing the axial gradiometer rotating about U are located at the endpoints of the vectors r=U±ρu, where ρ=d/2 and d is the total distance between the SQUIDs. In drum coordinates 
                   r   +   ′     =         U   ′     +     ρ   ⁢           ⁢     u   ′         =         (         0           0             L   u           )     +     (           ρ   ⁢           ⁢   cos   ⁢           ⁢     θ   u                 ρ   ⁢           ⁢   sin   ⁢           ⁢     θ   u               0         )       =     (           ρ   ⁢           ⁢   cos   ⁢           ⁢     θ   u                 ρ   ⁢           ⁢   sin   ⁢           ⁢     θ   u                 L   u           )                 (   45   )                 r   -   ′     =         U   ′     -     ρ   ⁢           ⁢     u   ′         =         (         0           0             L   u           )     -     (           ρ   ⁢           ⁢   cos   ⁢           ⁢     θ   u                 ρ   ⁢           ⁢   sin   ⁢           ⁢     θ   u               0         )       =     (             -   ρ     ⁢           ⁢   cos   ⁢           ⁢     θ   u                   -   ρ     ⁢           ⁢   sin   ⁢           ⁢     θ   u                 L   u           )                 (   46   )               
Using eqs. 45 and 46 in equation 5 the magnetic field at the SQUIDs may be written
 
                           B   +   ′     ⁡     (     r   +   ′     )       =       (           B   x   ′               B   y   ′               B   z   ′           )     +       (           g   xx   ′           g   xy   ′           g   xz   ′               g   yx   ′           g   yy   ′           g   yz   ′               g   zx   ′           g   yz   ′           g   zz   ′           )     ⁢     (           ρcosθ   u               ρsinθ   u               L   u           )                     =     (             B   x   ′     +       L   u     ⁢     g   xz   ′       +     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xy   ′       +     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xx   ′                     B   y   ′     +       L   u     ⁢     g   yz   ′       +     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xy   ′       +     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   yy   ′                     B   z   ′     +     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xz   ′       +     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   yz   ′       +       L   u     ⁢     g   zz   ′               )                   (   47   )                         B   -   ′     ⁡     (     r   -   ′     )       =       (           B   x   ′               B   y   ′               B   z   ′           )     +       (           g   xx   ′           g   xy   ′           g   xz   ′               g   yx   ′           g   yy   ′           g   yz   ′               g   zx   ′           g   zy   ′           g   zz   ′           )     ⁢     (           -     ρcosθ   u                 -     ρsinθ   u                 L   u           )                     =     (             B   x   ′     +       L   u     ⁢     g   xz   ′       -     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xy   ′       -     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xx   ′                     B   y   ′     +       L   u     ⁢     g   yz   ′       -     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xy   ′       -     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   yy   ′                     B   z   ′     -     ρ   ⁢           ⁢     (     cos   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   xz   ′       -     ρ   ⁢           ⁢     (     sin   ⁢           ⁢     θ   u       )     ⁢           ⁢     g   yz   ′       +       L   u     ⁢     g   zz   ′               )                   (   48   )               in   ⁢           ⁢   which                             B   ′     ⁢     =   △     ⁢     (           B   x   ′               B   y   ′               B   z   ′           )             (   49   )               
is the magnetic field at the origin of C′.
 
   E. SQUID outputs. 
   In the instrument the gradiometer signal is produced by forming the sum
 
 U=U   +   +U   −   (50)
 
where U +  and U −  are the responses of the SQUIDs when operated in a flux-locked loop. In response to the magnetic fields in equations 47 and 48 U +  and U −  may be written
 
 U   +   =S′   1   ·B′   + ( r′   + )+ a   +   (51)
 
 U   −   =S′·B′   − ( r′   − )+ a   −   (52)
 
where the S′ 1  and S′ denote the SQUID sensitivity vectors. These are normal to the SQUIDs with lengths proportional to the sensitivity of the SQUID system (V/T) and the a are arbitrary offset voltages which can change value upon each reset of the SQUID system electronics. In a perfectly aligned system S′ 1 =S 1 u′ and S′=−S u′ but in any real system the SQUID vectors will be tilted with respect to u′ due to alignment errors. To characterize the alignment we introduce 2 pairs of alignment error angles. Let σ 1 (σ) denote the error angle between the projection of the S′ 1 (S′) vector onto the x′-y′ plane and the unit vectors u′(−u′), and τ 1 (τ) is the angle of inclination of S′ 1 (S′) above the x′-y′ plane. Then
 
                     S   1   ′     =     (             S   1     ⁢   cos   ⁢           ⁢     τ   1     ⁢   cos   ⁢           ⁢     θ   1                   S   1     ⁢   cos   ⁢           ⁢     τ   1     ⁢   sin   ⁢           ⁢     θ   1                   S   1     ⁢   sin   ⁢           ⁢     τ   1             )       ,       θ   1     ⁢     =   △     ⁢     (       θ   u     +     σ   1       )               (   53   )                   S   ′     =     (             -   S     ⁢           ⁢   cos   ⁢           ⁢   τ   ⁢           ⁢   cos   ⁢           ⁢   θ                 -   S     ⁢           ⁢   cos   ⁢           ⁢   τ   ⁢           ⁢   sin   ⁢           ⁢   θ               S   ⁢           ⁢   sin   ⁢           ⁢   τ           )       ,     θ   ⁢     =   △     ⁢     (       θ   u     +   σ     )               (   54   )               
If σ 1 =σ=τ 1 =τ=0 then S′ 1 =S 1 u′ and S′=−S u′ as required.
 
To simplify these introduce
 
                             M   1     ⁢     =   △     ⁢       S   1     ⁢   cos   ⁢           ⁢     τ   1         ,       N   1     ⁢     =   △     ⁢       S   1     ⁢   sin   ⁢           ⁢     τ   1                       M   ⁢     =   △     ⁢     S   ⁢           ⁢   cos   ⁢           ⁢   τ       ,     N   ⁢     =   △     ⁢     S   ⁢           ⁢   sin   ⁢           ⁢   τ               }           (   55   )               
so that 53 and 54 become
 
   
     
       
         
           
             
               
                 
                   S 
                   1 
                   ′ 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             M 
                             1 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             M 
                             1 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           N 
                           1 
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 56 
                 ) 
               
             
           
           
             
               
                 
                   S 
                   ′ 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             - 
                             M 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                     
                       
                         
                           
                             - 
                             M 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                     
                       
                         N 
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 57 
                 ) 
               
             
           
         
       
     
   
   Using 47 and 56 in equation 51 gives 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         U 
                         + 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     M 
                                     1 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     M 
                                     1 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         · 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     B 
                                     x 
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       xz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xy 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xx 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     y 
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       yz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xy 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       yy 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     z 
                                     ′ 
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       yz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       zz 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           a 
                           + 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 58 
                 ) 
               
             
           
           
             
               
                 = 
                 
                   
                     ( 
                     
                       
                         M 
                         1 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         1 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         B 
                         x 
                         ′ 
                       
                       + 
                       
                         
                           L 
                           u 
                         
                         ⁢ 
                         
                           g 
                           xz 
                           ′ 
                         
                       
                       + 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xy 
                           ′ 
                         
                       
                       + 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xx 
                           ′ 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 59 
                 ) 
               
             
           
           
             
               
                 
                   + 
                   
                     ( 
                     
                       
                         M 
                         1 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         1 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       B 
                       y 
                       ′ 
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         yz 
                         ′ 
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         xy 
                         ′ 
                       
                     
                     + 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         yy 
                         ′ 
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 60 
                 ) 
               
             
           
           
             
               
                 
                   
                     + 
                     
                       ( 
                       
                         N 
                         1 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         B 
                         z 
                         ′ 
                       
                       + 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xz 
                           ′ 
                         
                       
                       + 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           yz 
                           ′ 
                         
                       
                       + 
                       
                         
                           L 
                           u 
                         
                         ⁢ 
                         
                           g 
                           zz 
                           ′ 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   a 
                   + 
                 
               
             
             
               
                 ( 
                 61 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             B 
                             x 
                             ′ 
                           
                           ⁢ 
                           
                             M 
                             1 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                         + 
                         
                           
                             L 
                             u 
                           
                           ⁢ 
                           
                             g 
                             xz 
                             ′ 
                           
                           ⁢ 
                           
                             M 
                             1 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                         + 
                         
                           
                             g 
                             xx 
                             ′ 
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               M 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   u 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 1 
                               
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           g 
                           xy 
                           ′ 
                         
                         ⁢ 
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             M 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             + 
                             
                               B 
                               y 
                               ′ 
                             
                           
                           ⁢ 
                           
                             M 
                             1 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                         + 
                         
                           
                             g 
                             yz 
                             ′ 
                           
                           ⁢ 
                           
                             L 
                             u 
                           
                           ⁢ 
                           
                             M 
                             1 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             1 
                           
                         
                         + 
                         
                           
                             g 
                             yy 
                             ′ 
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               M 
                               1 
                             
                             ⁡ 
                             
                               ( 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   u 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 1 
                               
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           g 
                           xy 
                           ′ 
                         
                         ⁢ 
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             M 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             + 
                             
                               B 
                               z 
                               ′ 
                             
                           
                           ⁢ 
                           
                             N 
                             1 
                           
                         
                         + 
                         
                           
                             g 
                             zz 
                             ′ 
                           
                           ⁢ 
                           
                             N 
                             1 
                           
                           ⁢ 
                           
                             L 
                             u 
                           
                         
                         + 
                         
                           
                             g 
                             xz 
                             ′ 
                           
                           ⁢ 
                           
                             N 
                             1 
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         + 
                         
                             
                         
                         ⁢ 
                         
                           
                             g 
                             yz 
                             ′ 
                           
                           ⁢ 
                           
                             N 
                             1 
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           a 
                           + 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 62 
                 ) 
               
             
           
         
       
     
   
   where we have used that g xy =g yx , g xz =g zx  and g yz =g xy  from our knowledge of the properties of the magnetic gradient tensor. 
   Similarly using 48 and 57 in equation 52 gives 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         U 
                         - 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     - 
                                     M 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     M 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   θ 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         · 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     B 
                                     x 
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       xz 
                                       ′ 
                                     
                                   
                                   - 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xy 
                                       ′ 
                                     
                                   
                                   - 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xx 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     y 
                                     ′ 
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       yz 
                                       ′ 
                                     
                                   
                                   - 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xy 
                                       ′ 
                                     
                                   
                                   - 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       yy 
                                       ′ 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     B 
                                     z 
                                     ′ 
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         cos 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       xz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     ρ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           θ 
                                           u 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       g 
                                       yz 
                                       ′ 
                                     
                                   
                                   + 
                                   
                                     
                                       L 
                                       u 
                                     
                                     ⁢ 
                                     
                                       g 
                                       zz 
                                       ′ 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           a 
                           - 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 63 
                 ) 
               
             
           
           
             
               
                 = 
                 
                   
                     ( 
                     
                       
                         - 
                         M 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         B 
                         x 
                         ′ 
                       
                       + 
                       
                         
                           L 
                           u 
                         
                         ⁢ 
                         
                           g 
                           xz 
                           ′ 
                         
                       
                       - 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xy 
                           ′ 
                         
                       
                       - 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xx 
                           ′ 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 64 
                 ) 
               
             
           
           
             
               
                 
                   + 
                   
                     ( 
                     
                       
                         - 
                         M 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       B 
                       y 
                       ′ 
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         yz 
                         ′ 
                       
                     
                     - 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         xy 
                         ′ 
                       
                     
                     - 
                     
                       ρ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         g 
                         yy 
                         ′ 
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 65 
                 ) 
               
             
           
           
             
               
                 
                   
                     + 
                     
                       ( 
                       N 
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         B 
                         z 
                         ′ 
                       
                       - 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           xz 
                           ′ 
                         
                       
                       - 
                       
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           g 
                           yz 
                           ′ 
                         
                       
                       + 
                       
                         
                           L 
                           u 
                         
                         ⁢ 
                         
                           g 
                           zz 
                           ′ 
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   a 
                   - 
                 
               
             
             
               
                 ( 
                 66 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             - 
                             
                               B 
                               x 
                               ′ 
                             
                           
                           ⁢ 
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           
                             L 
                             u 
                           
                           ⁢ 
                           
                             g 
                             xz 
                             ′ 
                           
                           ⁢ 
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         + 
                         
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               g 
                               xx 
                               ′ 
                             
                             ⁡ 
                             
                               ( 
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   u 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             g 
                             xy 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             - 
                             M 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               B 
                               y 
                               ′ 
                             
                             ⁡ 
                             
                               ( 
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             g 
                             yz 
                             ′ 
                           
                           ⁢ 
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               L 
                               u 
                             
                             ( 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             g 
                             yy 
                             ′ 
                           
                           ⁢ 
                           M 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             ) 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           g 
                           xy 
                           ′ 
                         
                         ⁢ 
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ρ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             + 
                             
                               B 
                               z 
                               ′ 
                             
                           
                           ⁢ 
                           N 
                         
                         + 
                         
                           
                             g 
                             zz 
                             ′ 
                           
                           ⁢ 
                           N 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             L 
                             u 
                           
                         
                         - 
                         
                           
                             g 
                             xz 
                             ′ 
                           
                           ⁢ 
                           N 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         - 
                         
                             
                         
                         ⁢ 
                         
                           
                             g 
                             yz 
                             ′ 
                           
                           ⁢ 
                           N 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ρ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 u 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           a 
                           - 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 67 
                 ) 
               
             
           
         
       
     
   
   Substituting  62  and  67  into 50 gives the gradiometer output 
   
     
       
         
           
             
               
                 U 
                 = 
                   
                 ⁢ 
                 
                   
                     
                       B 
                       x 
                       ′ 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       1 
                     
                   
                   + 
                   
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       1 
                     
                   
                   + 
                   
                     
                       g 
                       xx 
                       ′ 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     g 
                     xy 
                     ′ 
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       M 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         1 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       - 
                       
                         B 
                         x 
                         ′ 
                       
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   - 
                   
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         g 
                         xx 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   M 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         B 
                         y 
                         ′ 
                       
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       1 
                     
                   
                   + 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       1 
                     
                   
                   + 
                   
                     
                       g 
                       yy 
                       ′ 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     g 
                     xy 
                     ′ 
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       M 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         1 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       - 
                       M 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         B 
                         y 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         ) 
                       
                     
                   
                   - 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         L 
                         u 
                       
                       ( 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     g 
                     yy 
                     ′ 
                   
                   ⁢ 
                   M 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         B 
                         z 
                         ′ 
                       
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                   
                   + 
                   
                     
                       g 
                       zz 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       L 
                       u 
                     
                   
                   + 
                   
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   + 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     a 
                     + 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         B 
                         z 
                         ′ 
                       
                     
                     ⁢ 
                     N 
                   
                   + 
                   
                     
                       g 
                       zz 
                       ′ 
                     
                     ⁢ 
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       L 
                       u 
                     
                   
                   - 
                   
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   - 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     a 
                     - 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       B 
                       x 
                       ′ 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           θ 
                           u 
                         
                         + 
                         
                           σ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           θ 
                           u 
                         
                         + 
                         
                           σ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     g 
                     xx 
                     ′ 
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       M 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             u 
                           
                           + 
                           
                             σ 
                             1 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         g 
                         xy 
                         ′ 
                       
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             
                               σ 
                               1 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       B 
                       x 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           θ 
                           u 
                         
                         + 
                         σ 
                       
                       ) 
                     
                   
                   - 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     L 
                     u 
                   
                   ⁢ 
                   
                     g 
                     xz 
                     ′ 
                   
                   ⁢ 
                   M 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   cos 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         θ 
                         u 
                       
                       + 
                       σ 
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       M 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ρg 
                         xy 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             σ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ρg 
                         xx 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             σ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         B 
                         y 
                         ′ 
                       
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           θ 
                           u 
                         
                         + 
                         
                           σ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                     ⁢ 
                     sin 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           θ 
                           u 
                         
                         + 
                         
                           σ 
                           1 
                         
                       
                       ) 
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     g 
                     yy 
                     ′ 
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       M 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             θ 
                             u 
                           
                           + 
                           
                             σ 
                             1 
                           
                         
                         ) 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         g 
                         xy 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         ρM 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             
                               σ 
                               1 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       MB 
                       y 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             σ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       - 
                       
                         g 
                         yz 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         ML 
                         u 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 θ 
                                 u 
                               
                               + 
                               σ 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             σ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         g 
                         yy 
                         ′ 
                       
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               θ 
                               u 
                             
                             + 
                             σ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       B 
                       z 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                   
                   + 
                   
                     
                       g 
                       zz 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     
                       L 
                       u 
                     
                   
                   + 
                   
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         g 
                         yz 
                         ′ 
                       
                     
                     ⁢ 
                     
                       N 
                       1 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     a 
                     + 
                   
                   + 
                   
                     
                       B 
                       z 
                       ′ 
                     
                     ⁢ 
                     N 
                   
                   + 
                   
                     
                       g 
                       zz 
                       ′ 
                     
                     ⁢ 
                     
                       NL 
                       u 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       - 
                       
                         g 
                         xz 
                         ′ 
                       
                     
                     ⁢ 
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       g 
                       yz 
                       ′ 
                     
                     ⁢ 
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     a 
                     - 
                   
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       B 
                       x 
                       ′ 
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           - 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           - 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       
                         g 
                         xx 
                         ′ 
                       
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         M 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           - 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                         
                         ) 
                       
                     
                   
                   + 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁢ 
                     ρ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       M 
                       1 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   ( 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         1 
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     - 
                     
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         1 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                   
                   ) 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       - 
                       
                         B 
                         x 
                         ′ 
                       
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           σ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         - 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           σ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                       
                       ) 
                     
                   
                   - 
                   
                     
                       L 
                       u 
                     
                     ⁢ 
                     
                       g 
                       xz 
                       ′ 
                     
                     ⁢ 
                     M 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           σ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         - 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           σ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                     
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     
                       + 
                       M 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ρg 
                         xy 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                           - 
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               u 
                             
                           
                         
                         ) 
                       
                     
                   
                   + 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   M 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ρg 
                       xx 
                       ′ 
                     
                     ⁡ 
                     
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                 = 
                   
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                 ⁢ 
                 
                   
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             N 
                             1 
                           
                           + 
                           N 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     g 
                     zz 
                     ′ 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   + 
                   
                     
                       B 
                       z 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           N 
                           1 
                         
                         + 
                         N 
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   + 
                   a 
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       x 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       y 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     - 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     - 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       x 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     ρ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       y 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     2 
                   
                   ⁢ 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           M 
                           1 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           1 
                         
                       
                       + 
                       
                         M 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         σ 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       g 
                       xy 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           u 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         θ 
                         u 
                       
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     
                       
                         ρg 
                         xz 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         N 
                         1 
                       
                       - 
                       N 
                     
                     ) 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     + 
                     
                       
                         ρg 
                         yz 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             u 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         N 
                         1 
                       
                       - 
                       N 
                     
                     ) 
                   
                 
               
             
           
         
       
     
   
   where a           a + +a − . Expanding the trigonometric functions within each term and collecting harmonics gives
                 U   =         (         L   u     ⁢     g   zz   ′       +     B   z   ′       )     ⁢     (       N   1     +   N     )       +   a   +       1   2     ⁢   ρ   ⁢           ⁢     (         M   1     ⁢   cos   ⁢           ⁢     σ   1       +     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )     ⁢     (       g   xx   ′     +     g   yy   ′       )       +       (         M   1     ⁢   cos   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )     ⁢     (       B   x   ′     +       L   u     ⁢     g   xz   ′         )     ⁢           ⁢   cos   ⁢           ⁢     θ   u       +       (         M   1     ⁢   sin   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )     ⁢     (       B   y   ′     +       L   u     ⁢     g   yz   ′         )     ⁢           ⁢   cos   ⁢           ⁢     θ   u       +     ρ   ⁢           ⁢     (       N   1     -   N     )     ⁢           ⁢     g   xz   ′     ⁢   cos   ⁢           ⁢     θ   u       -       (         M   1     ⁢   sin   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )     ⁢     (       B   x   ′     +       L   u     ⁢     g   xz   ′         )     ⁢           ⁢   sin   ⁢           ⁢     θ   u       +       (         M   1     ⁢   cos   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )     ⁢     (       B   y   ′     +       L   u     ⁢     g   yz   ′         )     ⁢           ⁢   sin   ⁢           ⁢     θ   u       +     ρ   ⁢           ⁢     (       N   1     -   N     )     ⁢           ⁢     g   yz   ′     ⁢   sin   ⁢           ⁢     θ   u       +       1   2     ⁢   ρ   ⁢           ⁢     (         M   1     ⁢   cos   ⁢           ⁢     σ   1       +     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )     ⁢     (       g   xx   ′     +     g   yy   ′       )     ⁢           ⁢   cos   ⁢           ⁢   2   ⁢           ⁢     θ   u       +     ρ   ⁢           ⁢     (         M   1     ⁢   sin   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )     ⁢           ⁢     g   xy   ′     ⁢   cos   ⁢           ⁢   2   ⁢     θ   u       +       1   2     ⁢   ρ   ⁢           ⁢     (         M   1     ⁢   sin   ⁢           ⁢     σ   1       +     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )     ⁢     (       g   yy   ′     -     g   xx   ′       )     ⁢           ⁢   sin   ⁢           ⁢   2   ⁢           ⁢     θ   u       +     ρ   ⁢           ⁢     (         M   1     ⁢   cos   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )     ⁢           ⁢     g   xy   ′     ⁢   sin   ⁢           ⁢   2   ⁢     θ   u                 (   68   )               Hence                           U   =         μ   u     ⁡     (         L   u     ⁢     g   zz   ′       +     B   z   ′       )       +   a   +       ξ   u     ⁡     (       g   xx   ′     +     g   yy   ′       )       +         ς   u     ⁡     (       B   x   ′     +       L   u     ⁢     g   xz   ′         )       ⁢           ⁢   cos   ⁢           ⁢     θ   u       +         η   u     ⁡     (       B   y   ′     +       L   u     ⁢     g   yz   ′         )       ⁢           ⁢   cos   ⁢           ⁢     θ   u       +       υ   u     ⁢     g   xz   ′     ⁢   cos   ⁢           ⁢     θ   u       -         η   u     ⁡     (       B   x   ′     +       L   u     ⁢     g   xz   ′         )       ⁢           ⁢   sin   ⁢           ⁢     θ   u       +         ς   u     ⁡     (       B   y   ′     +       L   u     ⁢     g   yz   ′         )       ⁢           ⁢   sin   ⁢           ⁢     θ   u       +       υ   u     ⁢     g   yz   ′     ⁢   sin   ⁢           ⁢     θ   u       +         ξ   u     ⁡     (       g   xx   ′     +     g   yy   ′       )       ⁢           ⁢   cos   ⁢           ⁢   2   ⁢     θ   u       +     2   ⁢           ⁢     (     χ   u     )     ⁢           ⁢     g   xy   ′     ⁢   cos   ⁢           ⁢   2   ⁢     θ   u       +         χ   u     ⁡     (       g   yy   ′     -     g   xx   ′       )       ⁢           ⁢   sin   ⁢           ⁢   2   ⁢     θ   u       +     2   ⁢           ⁢     (     ξ   u     )     ⁢           ⁢     g   xy   ′     ⁢   sin   ⁢           ⁢   2   ⁢     θ   u                 (   69   )               where                                                             ξ   u     ⁢     =   △     ⁢               1   _       ⁢   ρ     ⁢           ⁢     (         M   1     ⁢   cos   ⁢           ⁢     σ   1       +     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )                     χ   u     ⁢     =   △     ⁢               1   _       ⁢   ρ     ⁢           ⁢     (         M   1     ⁢   sin   ⁢           ⁢     σ   1       +     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )                           υ   u     ⁢     =   △     ⁢     ρ   ⁢           ⁢     (       N   1     -   N     )                           ς   u     ⁢     =   △     ⁢     (         M   1     ⁢   cos   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   cos   ⁢           ⁢   σ       )                         η   u     ⁢     =   △     ⁢     (         M   1     ⁢   sin   ⁢           ⁢     σ   1       -     M   ⁢           ⁢   sin   ⁢           ⁢   σ       )                         μ   u     ⁢     =   △     ⁢     (       N   1     +   N     )             }           (   70   )               
N.B. In the absence of alignment errors, i.e. σ 1 =σ=0, τ 1 =τ=0 from equation 55
 
                         M   1     =     S   1       ,                 M   =   S     ,           ⁢             N   1     =   0               N   =   0             }         
and from equations 70 χ u , η u , ν u  and μ u  vanish, and
 
                           ξ   u     =               1   _       ⁢   ρ     ⁢           ⁢     (       S   1     +   S     )                     ς   u     =     (       S   1     -   S     )             }           (   71   )               
Finally if S 1             S say, then            u  also vanishes and the only non-zero constant is
 ξ u   =ρS.   (72)
   III. Data Extraction 
   One method to extract the data is to take a series of measurements over a number of angular increments of the pick-up loop structure and solve the resulting system of equations algebraically. A series of voltage measurements is now taken at angles θ u =−π/4, 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2 to yield the following set of equations: 
   
     
       
         
           
             
               
                 
                   U 
                   0 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           
                             - 
                             π 
                           
                           / 
                           4 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       ϛ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       u 
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         g 
                         13 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         g 
                         3 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         
                           η 
                           u 
                         
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 73 
                 ) 
               
             
           
           
             
               
                 U 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           π 
                           / 
                           4 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         g 
                         13 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         g 
                         3 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   4 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           3 
                           ⁢ 
                           
                             π 
                             / 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         g 
                         13 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         g 
                         3 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   6 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           5 
                           ⁢ 
                           
                             π 
                             / 
                             4 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         B 
                           
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         g 
                         13 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         g 
                         3 
                         ′ 
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                           
                           
                           
                         
                           1 
                           _ 
                         
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   1 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   3 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           π 
                           / 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     + 
                     
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   5 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         π 
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     - 
                     
                       
                         B 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     - 
                     
                       
                         B 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         11 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
           
             
               
                 
                   U 
                   7 
                 
                 ⁢ 
                 
                   = 
                   △ 
                 
                 ⁢ 
                 
                   
                     U 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         θ 
                         = 
                         
                           3 
                           ⁢ 
                           
                             π 
                             / 
                             2 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     a 
                     + 
                     
                       
                         B 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     - 
                     
                       
                         B 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         B 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                     - 
                     
                       2 
                       ⁢ 
                       
                         g 
                         1 
                         ′ 
                       
                       ⁢ 
                       
                         χ 
                         u 
                       
                     
                     - 
                     
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         υ 
                         u 
                       
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         g 
                         ′ 
                       
                       ⁢ 
                       
                         ξ 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         13 
                         ′ 
                       
                       ⁢ 
                       
                         η 
                         u 
                       
                     
                     - 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         3 
                         ′ 
                       
                       ⁢ 
                       
                         ς 
                         u 
                       
                     
                     + 
                     
                       
                         L 
                         u 
                       
                       ⁢ 
                       
                         g 
                         33 
                         ′ 
                       
                       ⁢ 
                       
                         μ 
                         u 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   Note that even with all of these misalignment errors considered the following L.C. vanishes if the SQUID stays in lock throughout the measurement series:
 
 U   0   −U   1   +U−U   3   +U   4   −U   5   +U   6   −U   7 =0  (74)
 
So this is still a good check to test the SQUID lock and the integrity of the data with respect to random pointing errors which might occur during the measurement sequence.
 
   A. Solving for Gradient Components. 
   Assuming that ξ u , χ u ,              u , η u , ν u , μ u  are all known by calibration, the 8 equations in 73 contain 10 unknown independent variables, namely B′ x , B′ y , B′ z , g′ xx , g′ yy , g′ zz , g′ xy , g′ yz , g′ zx , a, since voltages U 0  . . . U 7  are determined by measurement. The set of equations in 73 is clearly underdetermined and additional data from the other two drums is necessary to determine a complete solution. Nevertheless it is possible to solve for the off-diagonal gradient component, g′ xy , and the difference between the diagonal components, g′ xx −g′ yy , independently of the measurements from the other two drums. This is done by forming the following linear combinations of eqs. 73 to eliminate the dependence on the SQUID offset voltage a and on the field components,
                     U   0     -   U   +     U   4     -     U   6       =       4   ⁢     χ   u     ⁢           ⁢     Δ   ′       -     8   ⁢           ⁢     ξ   u     ⁢     g   xy   ′                 (   75   )                       U   1     -     U   3     +     U   5     -     U   7       =       4   ⁢     ξ   u     ⁢     Δ   ′       +     8   ⁢           ⁢     χ   u     ⁢     g   xy   ′           ,     
     ⁢   where     ⁢          ⁢       Δ   ′     ⁢     =   Δ     ⁢       g   xx   ′     -     g   yy   ′         ⁢     
     ⁢   Then           (   76   )                   (           g   xy   ′               Δ   ′           )     =     (                 χ   u     ⁡     (       U   1     -     U   3     +     U   5     -     U   7       )       -       ξ   u     ⁡     (       U   0     -   U   +     U   4     -     U   6       )           8   ⁢     (       ξ   u     +     χ   u       )                         χ   u     ⁡     (       U   0     -   U   +     U   4     -     U   6       )       +       ξ   u     ⁡     (       U   1     -     U   3     +     U   5     -     U   7       )           4   ⁢     (       ξ   u     +     χ   u       )               )       ⁢     
     ⁢   provided           (   77   )                 (       ξ   u     +     χ   u       )     ≠   0           (   78   )               
which is always satisfied because both ξ u  and χ u  are real and ξ u &gt;0.
 
   The results from these additional gradiometers give solutions with exactly the same form in their respective coordinate systems, i.e. 
   
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             g 
                             xy 
                             ′′ 
                           
                         
                       
                       
                         
                           
                             Δ 
                             ′′ 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 
                                   χ 
                                   υ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       1 
                                     
                                     - 
                                     
                                       V 
                                       3 
                                     
                                     + 
                                     
                                       V 
                                       5 
                                     
                                     - 
                                     
                                       V 
                                       7 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   ξ 
                                   υ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       0 
                                     
                                     - 
                                     V 
                                     + 
                                     
                                       V 
                                       4 
                                     
                                     - 
                                     
                                       V 
                                       6 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               8 
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     ξ 
                                     υ 
                                   
                                   + 
                                   
                                     χ 
                                     υ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   χ 
                                   υ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       0 
                                     
                                     - 
                                     V 
                                     + 
                                     
                                       V 
                                       4 
                                     
                                     - 
                                     
                                       V 
                                       6 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   ξ 
                                   υ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       1 
                                     
                                     - 
                                     
                                       V 
                                       3 
                                     
                                     + 
                                     
                                       V 
                                       5 
                                     
                                     - 
                                     
                                       V 
                                       7 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               4 
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     ξ 
                                     υ 
                                   
                                   + 
                                   
                                     χ 
                                     υ 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 and 
               
             
             
               
                 ( 
                 79 
                 ) 
               
             
           
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           g 
                           xy 
                           ′′′ 
                         
                       
                     
                     
                       
                         
                           Δ 
                           ′′′ 
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               
                                 χ 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     W 
                                     1 
                                   
                                   - 
                                   
                                     W 
                                     3 
                                   
                                   + 
                                   
                                     W 
                                     5 
                                   
                                   - 
                                   
                                     W 
                                     7 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ξ 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     W 
                                     0 
                                   
                                   - 
                                   W 
                                   + 
                                   
                                     W 
                                     4 
                                   
                                   - 
                                   
                                     W 
                                     6 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             8 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   w 
                                 
                                 + 
                                 
                                   χ 
                                   w 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 χ 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     W 
                                     0 
                                   
                                   - 
                                   W 
                                   + 
                                   
                                     W 
                                     4 
                                   
                                   - 
                                   
                                     W 
                                     6 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ξ 
                                 w 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     W 
                                     1 
                                   
                                   - 
                                   
                                     W 
                                     3 
                                   
                                   + 
                                   
                                     W 
                                     5 
                                   
                                   - 
                                   
                                     W 
                                     7 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             4 
                             ⁢ 
                             
                               ( 
                               
                                 
                                   ξ 
                                   w 
                                 
                                 + 
                                 
                                   χ 
                                   w 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 80 
                 ) 
               
             
           
         
       
     
   
   Equations 77, 79 and 80 are now transformed to the instrument coordinate system and the resulting system of equations may be solved to yield the gradient measurements. 
   1. Transformations for the “Proof of Concept” Instrument. 
   In the case of the present embodiment λ=π/4, and the transformations for the xy, xx and yy gradient components of the C′, C″ and C′″ coordinate systems are those given by the relevant equations in 36, 37 and 44, which are brought together here for convenience: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′ 
                                 
                               
                               = 
                               
                                 
                                   2 
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       g 
                                       1 
                                     
                                     - 
                                     
                                       g 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         4 
                                         ⁢ 
                                         
                                             
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       3 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       
                                         
                                             
                                         
                                         ⁢ 
                                         4 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       3 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       13 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′ 
                                 
                               
                               ⁢ 
                               
                                 = 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     1 
                                     _ 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     11 
                                   
                                 
                                 - 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     13 
                                   
                                 
                                 ⁢ 
                                 
                                   + 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       1 
                                       _ 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     33 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′ 
                                 
                               
                               = 
                               g 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   + 
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     13 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 + 
                                 
                                   
                                     3 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                                 ⁢ 
                                 
                                   + 
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     33 
                                   
                                 
                                 - 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     3 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     3 
                                     
                                       1 
                                       _ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     3 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 + 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   + 
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     3 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 + 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     13 
                                   
                                     
                                     
                                     
                                   
                                     1 
                                     _ 
                                   
                                 
                                 + 
                                 
                                   
                                     3 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       8 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   + 
                                   
                                     1 
                                     _ 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   
                                     
                                         
                                     
                                     ⁢ 
                                     33 
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     3 
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     3 
                                     
                                       1 
                                       _ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   
                                       
                                   
                                   ⁢ 
                                   ′′′ 
                                 
                               
                               = 
                               
                                 
                                   
                                     3 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       11 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       4 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   g 
                                 
                                 - 
                                 
                                   
                                     3 
                                     
                                       1 
                                       _ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     g 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 } 
               
             
             
               
                 ( 
                 81 
                 ) 
               
             
           
         
       
     
   
   Using the last six of these equations we can derive the transformation equations for the diagonal component differences Δ′, Δ″ and Δ′″, i.e. 
                         Δ   ′     =       g   11   ′     -     g   ′                   =         1   2     ⁢     g   11       -     g   13     -   g   +       1   2     ⁢     g   33                     =         1   2     ⁢     Δ   1       -       1   2     ⁢   Δ     -     g   13                     (   82   )                       Δ   ′′     =       g   11   ′′     -     g   ′′                   =         -     5   8       ⁢     g   11       +       1   8     ⁢   g     +       1   2     ⁢     g   33       -       3   4     ⁢     g   1     ⁢     3       -       1   2     ⁢   g   ⁢           3     ⁢     3         +       1   2     ⁢     g   13                     =         -     1   8       ⁢     Δ   1       +       1   2     ⁢     Δ   3       -       3   4     ⁢     g   1     ⁢     3       -       1   2     ⁢   g   ⁢           3     ⁢     3         +       1   2     ⁢     g   13                       (   83   )                         Δ   ′′′     =       g   11   ′′′     -     g   ′′′                   =         -     5   8       ⁢     g   11       +       1   8     ⁢   g     +       1   2     ⁢     g   33       +       3   4     ⁢     g   1     ⁢     3       +       1   2     ⁢   g   ⁢           3     ⁢     3         +       1   2     ⁢     g   13                     =         -     1   8       ⁢     Δ   1       +       1   2     ⁢     Δ   3       +       3   4     ⁢     g   1     ⁢     3       +       1   2     ⁢   g   ⁢           3     ⁢     3         +       1   2     ⁢     g   13                 ⁢     
     ⁢   where           (   84   )                         Δ   1     ⁢     =   Δ     ⁢       g   11     -   g                 Δ   ⁢     =   Δ     ⁢     g   -     g   33                     Δ   3     ⁢     =   Δ     ⁢       g   33     -     g   11               }           (   85   )               
Combining these with the first three equation of 81 yields the following set of equations
 
                           g   1   ′     =         2     1   _       ⁢     g   1       -       2     1   _       ⁢     g   3                       g   1   ′′     =         -     1   4       ⁢     2     ⁢     g   1       +       1   4     ⁢     2     ⁢     g   3       +       1   4     ⁢     6     ⁢     g   13       +       1   8     ⁢     6     ⁢     Δ   1                       g   1   ′′′     =         -     1   4       ⁢     2     ⁢     g   1       +       1   4     ⁢     2     ⁢     g   3       -       1   4     ⁢     6     ⁢     g   13       -       1   8     ⁢     6     ⁢     Δ   1                       Δ   ′     =       Δ   1                 1   _       -             1   _       ⁢   Δ     -     g   13                     Δ   ′′     =         -     1   8       ⁢     Δ   1       +     Δ   3                 1   _       -       3   4     ⁢     3     ⁢     g   1       -       3     1   _       ⁢     g   3       +     g   13                 1   _                       Δ   ′′′     =         -     1   8       ⁢     Δ   1       +     Δ   3                 1   _       -       3   4     ⁢     3     ⁢     g   1       +       3     1   _       ⁢     g   3       +     g   13                 1   _                 }           (   86   )               
Although the corresponding matrix of coefficients, with respect to the variables g 1 , g 3 , g 13 , Δ 1 , Δ, Δ 3 ,
 
               (           -     2     1   _               2     1   _           0       0       0       0               1   4     ⁢     2               -     1   4       ⁢     2               -     1   4       ⁢     6               -     1   8       ⁢     6           0       0               1   4     ⁢     2               -     1   4       ⁢     2               1   4     ⁢     6               1   8     ⁢     6           0       0           0       0       1         -     1   _             1   _         0               3   4     ⁢     3             3     1   _             -     1   _             1   8         0         -     1   _                   -     3   4       ⁢     3             -     3     1   _               -     1   _             1   8         0         -     1   _             )     ⁢     (           g   1               g   3               g   13               Δ   1             Δ             Δ   3           )       =     (           -     g   1   ′                 -     g   1   ′                 -     g   1   ′                 -     Δ   1   ′                 -     Δ   ′                 -     Δ   3   ′             )           
is singular a solution can by obtained after observing that, by definition,
 Δ 1 +Δ+Δ 3   =g   11   −g+g−g   33   +g   33   −g   11 =0 
and use this equation to augment  86 , i.e.
 
                                       ⁢       g             ⁢   1               ⁢   ′       ⁢           =           ⁢         2             ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   1         ⁢           -           ⁢       2             ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   3                                 ⁢       g             ⁢   1               ⁢   ′′       ⁢           =           ⁢         -           ⁢     1             ⁢   4         ⁢           ⁢     2     ⁢           ⁢     g             ⁢   1         ⁢           +           ⁢       1             ⁢   4       ⁢           ⁢     2     ⁢           ⁢     g             ⁢   3         ⁢           +           ⁢       1             ⁢   4       ⁢           ⁢     6     ⁢           ⁢     g             ⁢   13         ⁢           +           ⁢       1             ⁢   8       ⁢           ⁢     6     ⁢           ⁢     Δ             ⁢   1                                 ⁢       g             ⁢   1               ⁢   ′′′       ⁢           =           ⁢         -           ⁢     1             ⁢   4         ⁢           ⁢     2     ⁢           ⁢     g             ⁢   1         ⁢           +           ⁢       1             ⁢   4       ⁢           ⁢     2     ⁢           ⁢     g             ⁢   3         ⁢           -           ⁢       1             ⁢   4       ⁢           ⁢     6     ⁢           ⁢     g             ⁢   13         ⁢           -           ⁢       1             ⁢   8       ⁢           ⁢     6     ⁢           ⁢     Δ             ⁢   1                                             ⁢       Δ             ⁢   ′       ⁢           =           ⁢                       ⁢     1             ⁢   _           ⁢           ⁢     Δ             ⁢   1         ⁢           -           ⁢                     ⁢     1             ⁢   _           ⁢           ⁢   Δ     ⁢           -           ⁢     g             ⁢   13                               ⁢       Δ             ⁢   ′′       ⁢           =           ⁢         -           ⁢     1             ⁢   8         ⁢           ⁢     Δ             ⁢   1         ⁢           +           ⁢                     ⁢     1             ⁢   _           ⁢           ⁢     Δ             ⁢   3         ⁢           -           ⁢       3             ⁢   4       ⁢           ⁢     3     ⁢           ⁢     g             ⁢   1         ⁢           -           ⁢       3             ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   3         ⁢           +           ⁢                     ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   13                                 ⁢       Δ             ⁢   ′′′       ⁢           =           ⁢         -           ⁢     1             ⁢   8         ⁢           ⁢     Δ             ⁢   1         ⁢           +           ⁢                     ⁢     1             ⁢   _           ⁢           ⁢     Δ             ⁢   3         ⁢           -           ⁢       3             ⁢   4       ⁢           ⁢     3     ⁢           ⁢     g             ⁢   1         ⁢           -           ⁢       3             ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   3         ⁢           +           ⁢                     ⁢     1             ⁢   _           ⁢           ⁢     g             ⁢   13                         0   ⁢           =           ⁢       Δ             ⁢   1       ⁢           +           ⁢   Δ   ⁢           +           ⁢     Δ             ⁢   3                       }               
The corresponding matrix form
 
               (           -     2     1   _               2     1   _           0       0       0       0               1   4     ⁢     2               -     1   4       ⁢     2               -     1   4       ⁢     6               -     1   8       ⁢     6           0       0               1   4     ⁢     2               -     1   4       ⁢     2               1   4     ⁢     6               1   8     ⁢     6           0       0           0       0       1         -     1   _             1   _         0               3   4     ⁢     3             3     1   _             -     1   _             1   8         0         -     1   _                   -     3   4       ⁢     3             -     3     1   _               -     1   _             1   8         0         -     1   _               0       0       0         -   1           -   1           -   1           )     ⁢     (           g   1               g   3               g   13               Δ   1             Δ             Δ   3           )       =     (           -     g   1   ′                 -     g   1   ′′                 -     g   1   ′′′                 -     Δ   ′                 -     Δ   ′′                 -     Δ   ′′′               0         )           
is transformed to a square matrix by multiplying on the left by the transpose of the coefficient matrix to give
 
               (           33   8           3   _         0       0       0       0             3   _           9   4         0       0       0       0           0       0         9   4           -     1   4             1   _           1   _             0       0         -     1   4             47   3           3   4           7   8             0       0         1   _           3   4           5   4         1           0       0         1   _           7   8         1         3   _           )     ⁢     (           g   1               g   3               g   13               Δ   1             Δ             Δ   3           )       =     
     ⁢     (               g   1   ′             1   _       ⁢     2       -       1   4     ⁢     g   1   ′′     ⁢     2       -       3   4     ⁢     Δ   ′′     ⁢     3       -       1   4     ⁢     g   1   ′′′     ⁢     2       +       3   4     ⁢     Δ   ′′′     ⁢     3                       -     g   1   ′             1   _         ⁢     2       +       1   4     ⁢     g   1   ′′     ⁢     2       -       Δ       ′′             1   _       ⁢     3       +       1   4     ⁢     g   1   ′′′     ⁢     2       +       Δ       ′′′             1   _       ⁢     3                     -     Δ   ′       +     Δ       ′′             1   _       +     Δ       ′′′             1   _       +       1   4     ⁢     g   1   ′′     ⁢     6       -       1   4     ⁢     g   1   ′′′     ⁢     6                     Δ       ′             1   _       -       1   8     ⁢     Δ   ′′       -       1   8     ⁢     Δ   ′′′       +       1   8     ⁢     g   1   ′′     ⁢     6       -       1   8     ⁢     g   1   ′′′     ⁢     6                   -             1   _       ⁢     Δ   ′                     Δ       ′′             1   _       +     Δ       ′′′             1   _               )           
which is now non-singular. The inverse of the coefficient matrix is
 
                 (           8   5           -     16   75           0       0       0       0             -     16   75             44   75         0       0       0       0           0       0         136   5           88   5           -     64   5             -     6   5               0       0         88   5           304   5           -     11   5             -     44   75               0       0         -     64   5             -     11   5             436   5           -     68   75               0       0           -   6     5           -     44   75             -     68   75             17   75           )           
and the solution of the system of equations, 86, is
 
                   (           g   1               g   3               g   13               Δ   1             Δ             Δ   3           )     =     (                 3     15     ⁢     (       Δ   ′′′     -     Δ   ′′       )       +       √   15     ⁢     (       2   ⁢     g   1   ′       -     g   1   ′′     -     g   1   ′′′       )                         3     15     ⁢     (       Δ   ′′′     -     Δ   ′′       )       +       √   5     ⁢     (       g   1   ′′     -     2   ⁢     g   1   ′       +     g   1   ′′′       )                       15   _     ⁢           ⁢     (       Δ   ′′     -     2   ⁢     Δ   ′       +     Δ   ′′′       )       +         6     5     ⁢     (       g   1   ′′     -     g   1   ′′′       )                       8   15     ⁢     Δ   ′       -       4   15     ⁢     Δ   ′′       -       4   15     ⁢     Δ   ′′′       +       4   15     ⁢     g   1   ′′     ⁢     6       -       4   15     ⁢     g   1   ′′′     ⁢     6                       -     14   15       ⁢     Δ   ′       -       8   15     ⁢     Δ   ′′       -       8   15     ⁢     Δ   ′′′       -       15   _     ⁢     g   1   ′′     ⁢     6       +       15   _     ⁢     g   1   ′′′     ⁢     6                       5   _     ⁢           ⁢     Δ   ′       +       4   5     ⁢     Δ   ′′       +       4   5     ⁢     Δ   ′′′       -       15   _     ⁢     g   1   ′′     ⁢     6       +       15   _     ⁢     g   1   ′′′     ⁢     6               )             (   87   )               
Note that at this point this solution has been obtained without using the tracelessness of the gradient tensor and is written in terms of the differences between the diagonal components. The diagonal components, however, follow immediately from equations 85 and the traceless of the gradient tensor, i.e. given that
 
                                   g   11     -   g     =     Δ   1                   g   -     g   33       =   Δ                   g   33     -     g   11       =     Δ   3                     g   11     +   g   +     g   33       =   0           }     ⇒       (         1         -   1         0           0       1         -   1               -   1         0       1           1       1       1         )     ⁢     (           g   11             g             g   33           )         =     (           Δ   1             Δ             Δ   3             0         )       ⁢     
     ⁢   then           (   88   )                 (           g   11             g             g   33           )     =       1   3     ⁢     (             -     Δ   3       +     Δ   1                   -     Δ   1       +   Δ                 -   Δ     +     Δ   3             )               (   89   )               
Substituting for the component differences from the last three equations in 87 gives
 
   
     
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           g 
                           11 
                         
                       
                     
                     
                       
                         g 
                       
                     
                     
                       
                         
                           g 
                           33 
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               - 
                               
                                 45 
                                 _ 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     Δ 
                                     ′ 
                                   
                                 
                                 + 
                                 
                                   8 
                                   ⁢ 
                                   
                                     Δ 
                                     ′′ 
                                   
                                 
                                 + 
                                 
                                   8 
                                   ⁢ 
                                   
                                     Δ 
                                     ′′′ 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 6 
                               
                               15 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   g 
                                   1 
                                   ′′ 
                                 
                                 - 
                                 
                                   g 
                                   1 
                                   ′′′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               - 
                               
                                 45 
                                 _ 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   11 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     Δ 
                                     ′ 
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     Δ 
                                     ′′ 
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     Δ 
                                     ′′′ 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 6 
                               
                               15 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   g 
                                   1 
                                   ′′′ 
                                 
                                 - 
                                 
                                   g 
                                   1 
                                   ′′ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             4 
                             9 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 Δ 
                                 ′ 
                               
                               + 
                               
                                 Δ 
                                 ′′ 
                               
                               + 
                               
                                 Δ 
                                 ′′′ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 90 
                 ) 
               
             
           
         
       
     
   
   The final solution for the gradient components follow after substituting for the Δ&#39;s and g&#39;s from equations 77, 79 and 80 into equation 90 and the first three equations of 87. The resulting equations axe brought together here for convenience. 
   
     
       
         
             
           
             
               
                 
                   
                     
                       g 
                       11 
                     
                     = 
                     
                       
                         
                           - 
                           
                             45 
                             _ 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               - 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               8 
                               ⁢ 
                               
                                 Δ 
                                 ′′ 
                               
                             
                             + 
                             
                               8 
                               ⁢ 
                               
                                 Δ 
                                 ′′′ 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     g 
                     = 
                     
                       
                         
                           - 
                           
                             45 
                             _ 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               11 
                               ⁢ 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 Δ 
                                 ′′ 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Δ 
                                 ′′′ 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       33 
                     
                     = 
                     
                       
                         4 
                         9 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             Δ 
                             ′ 
                           
                           + 
                           
                             Δ 
                             ′′ 
                           
                           + 
                           
                             Δ 
                             ′′′ 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                     
                     = 
                     
                       
                         
                           
                             3 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′′ 
                             
                             - 
                             
                               Δ 
                               ′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           √ 
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 g 
                                 1 
                                 ′ 
                               
                             
                             - 
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       3 
                     
                     = 
                     
                       
                         
                           
                             3 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′′ 
                             
                             - 
                             
                               Δ 
                               ′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           √ 
                           5 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 g 
                                 1 
                                 ′ 
                               
                             
                             + 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       13 
                     
                     = 
                     
                       
                         
                           15 
                           _ 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               Δ 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           5 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     in 
                     ⁢ 
                     
                       
                           
                       
                       ⁢ 
                       
                           
                       
                     
                     ⁢ 
                     which 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 U 
                                 1 
                               
                               - 
                               
                                 U 
                                 3 
                               
                               + 
                               
                                 U 
                                 5 
                               
                               - 
                               
                                 U 
                                 7 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ξ 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 U 
                                 0 
                               
                               - 
                               U 
                               + 
                               
                                 U 
                                 4 
                               
                               - 
                               
                                 U 
                                 6 
                               
                             
                             ) 
                           
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               u 
                             
                             + 
                             
                               χ 
                               u 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             υ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 1 
                               
                               - 
                               
                                 V 
                                 3 
                               
                               + 
                               
                                 V 
                                 5 
                               
                               - 
                               
                                 V 
                                 7 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ξ 
                             υ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 0 
                               
                               - 
                               V 
                               + 
                               
                                 V 
                                 4 
                               
                               - 
                               
                                 V 
                                 6 
                               
                             
                             ) 
                           
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               υ 
                             
                             + 
                             
                               χ 
                               υ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′′′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             w 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 W 
                                 1 
                               
                               - 
                               
                                 W 
                                 3 
                               
                               + 
                               
                                 W 
                                 5 
                               
                               - 
                               
                                 W 
                                 7 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             ξ 
                             w 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 W 
                                 0 
                               
                               - 
                               W 
                               + 
                               
                                 W 
                                 4 
                               
                               - 
                               
                                 W 
                                 6 
                               
                             
                             ) 
                           
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               w 
                             
                             + 
                             
                               χ 
                               w 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 U 
                                 0 
                               
                               - 
                               U 
                               + 
                               
                                 U 
                                 4 
                               
                               - 
                               
                                 U 
                                 6 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ξ 
                             u 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 U 
                                 1 
                               
                               - 
                               
                                 U 
                                 3 
                               
                               + 
                               
                                 U 
                                 5 
                               
                               - 
                               
                                 U 
                                 7 
                               
                             
                             ) 
                           
                         
                       
                       
                         4 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               u 
                             
                             + 
                             
                               χ 
                               u 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             υ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 0 
                               
                               - 
                               V 
                               + 
                               
                                 V 
                                 4 
                               
                               - 
                               
                                 V 
                                 6 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ξ 
                             υ 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 1 
                               
                               - 
                               
                                 V 
                                 3 
                               
                               + 
                               
                                 V 
                                 5 
                               
                               - 
                               
                                 V 
                                 7 
                               
                             
                             ) 
                           
                         
                       
                       
                         4 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               υ 
                             
                             + 
                             
                               χ 
                               υ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′′′ 
                     
                     = 
                     
                       
                         
                           
                             χ 
                             w 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 W 
                                 0 
                               
                               - 
                               W 
                               + 
                               
                                 W 
                                 4 
                               
                               - 
                               
                                 W 
                                 6 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ξ 
                             w 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 W 
                                 1 
                               
                               - 
                               
                                 W 
                                 3 
                               
                               + 
                               
                                 W 
                                 5 
                               
                               - 
                               
                                 W 
                                 7 
                               
                             
                             ) 
                           
                         
                       
                       
                         4 
                         ⁢ 
                         
                           ( 
                           
                             
                               ξ 
                               w 
                             
                             + 
                             
                               χ 
                               w 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       ξ 
                       u 
                     
                     ⁢ 
                     
                       = 
                       Δ 
                     
                     ⁢ 
                     
                       
                         
                             
                           
                             1 
                             _ 
                           
                         
                         ⁢ 
                         ρ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               M 
                               1 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                           
                           + 
                           
                             M 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       χ 
                       u 
                     
                     ⁢ 
                     
                       = 
                       Δ 
                     
                     ⁢ 
                     
                       
                         
                             
                           
                             1 
                             _ 
                           
                         
                         ⁢ 
                         ρ 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               M 
                               1 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               1 
                             
                           
                           + 
                           
                             M 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         M 
                         1 
                       
                       ⁢ 
                       
                         = 
                         Δ 
                       
                       ⁢ 
                       
                         
                           S 
                           1 
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           τ 
                           1 
                         
                       
                     
                     , 
                     
                       
                         N 
                         1 
                       
                       ⁢ 
                       
                         = 
                         Δ 
                       
                       ⁢ 
                       
                         
                           S 
                           1 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           τ 
                           1 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       M 
                       ⁢ 
                       
                         = 
                         Δ 
                       
                       ⁢ 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         cos 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         τ 
                       
                     
                     , 
                     
                       N 
                       ⁢ 
                       
                         = 
                         Δ 
                       
                       ⁢ 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
     
   
   2. Special Case: “Proof of Concept Instrument” with Zero Alignment Errors. 
   If all of the alignment errors vanish then 
             ξ   u     =               1   _       ⁢   ρ     ⁡     (       S   1     +   S     )                     χ   u     =   0         
and equation 91 reduces to
 
   
     
       
         
             
           
             
               
                 
                   
                     
                       g 
                       11 
                     
                     = 
                     
                       
                         
                           - 
                           
                             45 
                             _ 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               - 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               8 
                               ⁢ 
                               
                                 Δ 
                                 ′′ 
                               
                             
                             + 
                             
                               8 
                               ⁢ 
                               
                                 Δ 
                                 ′′′ 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     g 
                     = 
                     
                       
                         
                           - 
                           
                             45 
                             _ 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               11 
                               ⁢ 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 Δ 
                                 ′′ 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Δ 
                                 ′′′ 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       33 
                     
                     = 
                     
                       
                         4 
                         9 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             Δ 
                             ′ 
                           
                           + 
                           
                             Δ 
                             ′′ 
                           
                           + 
                           
                             Δ 
                             ′′′ 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                     
                     = 
                     
                       
                         
                           
                             3 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′′ 
                             
                             - 
                             
                               Δ 
                               ′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           √ 
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                 g 
                                 1 
                                 ′ 
                               
                             
                             - 
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       3 
                     
                     = 
                     
                       
                         
                           
                             3 
                           
                           15 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′′ 
                             
                             - 
                             
                               Δ 
                               ′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           √ 
                           5 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 g 
                                 1 
                                 ′ 
                               
                             
                             + 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       13 
                     
                     = 
                     
                       
                         
                           15 
                           _ 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               Δ 
                               ′′ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 Δ 
                                 ′ 
                               
                             
                             + 
                             
                               Δ 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             6 
                           
                           5 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               g 
                               1 
                               ′′ 
                             
                             - 
                             
                               g 
                               1 
                               ′′′ 
                             
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′ 
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               U 
                               0 
                             
                             - 
                             U 
                             + 
                             
                               U 
                               4 
                             
                             - 
                             
                               U 
                               6 
                             
                           
                           ) 
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ξ 
                           u 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′′ 
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               V 
                               0 
                             
                             - 
                             V 
                             + 
                             
                               V 
                               4 
                             
                             - 
                             
                               V 
                               6 
                             
                           
                           ) 
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ξ 
                           υ 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       g 
                       1 
                       ′′′ 
                     
                     = 
                     
                       
                         - 
                         
                           ( 
                           
                             
                               W 
                               0 
                             
                             - 
                             W 
                             + 
                             
                               W 
                               4 
                             
                             - 
                             
                               W 
                               6 
                             
                           
                           ) 
                         
                       
                       
                         8 
                         ⁢ 
                         
                           ξ 
                           w 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′ 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             U 
                             1 
                           
                           - 
                           
                             U 
                             3 
                           
                           + 
                           
                             U 
                             5 
                           
                           - 
                           
                             U 
                             7 
                           
                         
                         ) 
                       
                       
                         4 
                         ⁢ 
                         
                           ξ 
                           u 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′′ 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             V 
                             1 
                           
                           - 
                           
                             V 
                             3 
                           
                           + 
                           
                             V 
                             5 
                           
                           - 
                           
                             V 
                             7 
                           
                         
                         ) 
                       
                       
                         4 
                         ⁢ 
                         
                           ξ 
                           υ 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       Δ 
                       ′′′ 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             W 
                             1 
                           
                           - 
                           
                             W 
                             3 
                           
                           + 
                           
                             W 
                             5 
                           
                           - 
                           
                             W 
                             7 
                           
                         
                         ) 
                       
                       
                         4 
                         ⁢ 
                         
                           ξ 
                           w 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
           
         
       
     
   
   IV. Solving for Field Components. 
   If the gradiometer is intentionally unbalanced, i.e. S 1 ≠S then it is capable of returning information about the magnetic field in addition to the gradient. As for the gradient case considered above we form linear combinations of 73 to both eliminate the dependence on the SQUID offset voltage, a 1  and to reduce the number of gradient terms. To eliminate the SQUID offset we start with the differences between voltages corresponding to opposing positions, and then take the sums and differences of these pairs, i.e.
 
( U   0   −U   4 )+( U−U   6 )=2η u   B′ √{square root over (2)}+2             u   B′   1 √{square root over (2)}+2ν u   g′   13 √{square root over (2)}+2 L   u η u   g′   3 √{square root over (2)}+2 L   u             u   g′   13 √{square root over (2)}
 
( U   0   −U   4 )−( U−U   6 )=2η u   B′   1 √{square root over (2)}+2           u   B′ √{square root over (2)}+2ν u   g′   3 √{square root over (2)}+2 L   u η u   g′   13 √{square root over (2)}+2 L   u             u   g′   3 √{square root over (2)}
 
( U   1   −U   5 )+( U   3   −U   7 )=2 B′η   u −2 B′   1 η u +2 B′   1             u +2 B′             u +2 g′   13 ν u +2 g′   3 ν u −2 L   u   g′   13 η u +2 L   u   g′   3 η u +2 L   u   g′   13             u +2 L   u   g′   3             u  
 
( U   1   −U   5 )−( U   3   −U   7 )=2 B′   1 η u +2 B′η   u +2 B′   1             u −2 B′             u +2 g′   13 ν u −2 g′   3 ν u +2 L   u   g′   13 η u +2 L   u   g′   3 η u 2 L   u   g′   13             u −2 L   u   g′   3             u  
 
and then simplify further by forming the following L.C. of these equations:

                         U   1     +     U   3     -     U   5     -     U   7       2     -         U   0     +   U   -     U   4     -     U   6         2   ⁢     2           =         Ϛ   u     ⁢     B   ′       -       η   u     ⁢     B   1   ′       +       υ   u     ⁢     g   3   ′       -       L   u     ⁢     η   u     ⁢     g   13   ′       +       L   u     ⁢     Ϛ   u     ⁢     g   3   ′                 (   3   )                       U   1     -     U   3     -     U   5     +     U   7       2     -         U   0     -   U   -     U   4     +     U   6         2   ⁢     2           =         η   u     ⁢     B   ′       +       Ϛ   u     ⁢     B   1   ′       +       υ   u     ⁢     g   13   ′       +       L   u     ⁢     η   u     ⁢     g   3   ′       +       L   u     ⁢     Ϛ   u     ⁢     g   13   ′                 (   94   )               
As for the gradient calculation previously the results from the other two drums have exactly the same form in their respective coordinate systems, i.e.
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         1 
                       
                       + 
                       
                         V 
                         3 
                       
                       - 
                       
                         V 
                         5 
                       
                       - 
                       
                         V 
                         7 
                       
                     
                     2 
                   
                   - 
                   
                     
                       
                         V 
                         0 
                       
                       + 
                       V 
                       - 
                       
                         V 
                         4 
                       
                       - 
                       
                         V 
                         6 
                       
                     
                     
                       2 
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ϛ 
                       υ 
                     
                     ⁢ 
                     
                       B 
                       ′′ 
                     
                   
                   - 
                   
                     
                       η 
                       υ 
                     
                     ⁢ 
                     
                       B 
                       1 
                       ′′ 
                     
                   
                   + 
                   
                     
                       υ 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′ 
                     
                   
                   - 
                   
                     
                       L 
                       υ 
                     
                     ⁢ 
                     
                       η 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       13 
                       ′′ 
                     
                   
                   + 
                   
                     
                       L 
                       υ 
                     
                     ⁢ 
                     
                       ϛ 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′ 
                     
                   
                 
               
             
             
               
                 ( 
                 95 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         V 
                         1 
                       
                       - 
                       
                         V 
                         3 
                       
                       - 
                       
                         V 
                         5 
                       
                       + 
                       
                         V 
                         7 
                       
                     
                     2 
                   
                   - 
                   
                     
                       
                         V 
                         0 
                       
                       - 
                       V 
                       - 
                       
                         V 
                         4 
                       
                       + 
                       
                         V 
                         6 
                       
                     
                     
                       2 
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       η 
                       υ 
                     
                     ⁢ 
                     
                       B 
                       ′′ 
                     
                   
                   + 
                   
                     
                       ϛ 
                       υ 
                     
                     ⁢ 
                     
                       B 
                       1 
                       ′′ 
                     
                   
                   + 
                   
                     
                       υ 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       13 
                       ′′ 
                     
                   
                   + 
                   
                     
                       L 
                       υ 
                     
                     ⁢ 
                     
                       η 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′ 
                     
                   
                   + 
                   
                     
                       L 
                       υ 
                     
                     ⁢ 
                     
                       ϛ 
                       υ 
                     
                     ⁢ 
                     
                       g 
                       13 
                       ′′ 
                     
                   
                 
               
             
             
               
                 ( 
                 96 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       + 
                       
                         W 
                         3 
                       
                       - 
                       
                         W 
                         5 
                       
                       - 
                       
                         W 
                         7 
                       
                     
                     2 
                   
                   - 
                   
                     
                       
                         W 
                         0 
                       
                       + 
                       W 
                       - 
                       
                         W 
                         4 
                       
                       - 
                       
                         W 
                         6 
                       
                     
                     
                       2 
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ϛ 
                       w 
                     
                     ⁢ 
                     
                       B 
                       ′′′ 
                     
                   
                   - 
                   
                     
                       η 
                       w 
                     
                     ⁢ 
                     
                       B 
                       1 
                       ′′′ 
                     
                   
                   + 
                   
                     
                       υ 
                       w 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′′ 
                     
                   
                   - 
                   
                     
                       L 
                       w 
                     
                     ⁢ 
                     
                       η 
                       w 
                     
                     ⁢ 
                     
                       g 
                       13 
                       ′′′ 
                     
                   
                   + 
                   
                     
                       L 
                       w 
                     
                     ⁢ 
                     
                       ϛ 
                       w 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′′ 
                     
                   
                 
               
             
             
               
                 ( 
                 97 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       - 
                       
                         W 
                         3 
                       
                       - 
                       
                         W 
                         5 
                       
                       + 
                       
                         W 
                         7 
                       
                     
                     2 
                   
                   - 
                   
                     
                       
                         W 
                         0 
                       
                       - 
                       W 
                       - 
                       
                         W 
                         4 
                       
                       + 
                       
                         W 
                         6 
                       
                     
                     
                       2 
                       ⁢ 
                       
                         2 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       η 
                       w 
                     
                     ⁢ 
                     
                       B 
                       ′′′ 
                     
                   
                   - 
                   
                     
                       ϛ 
                       w 
                     
                     ⁢ 
                     
                       B 
                       1 
                       ′′′ 
                     
                   
                   + 
                   
                     
                       υ 
                       w 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′′ 
                     
                   
                   + 
                   
                     
                       L 
                       w 
                     
                     ⁢ 
                     
                       η 
                       w 
                     
                     ⁢ 
                     
                       g 
                       3 
                       ′′′ 
                     
                   
                   + 
                   
                     
                       L 
                       w 
                     
                     ⁢ 
                     
                       ϛ 
                       w 
                     
                     ⁢ 
                     
                       g 
                       13 
                       ′′′ 
                     
                   
                 
               
             
             
               
                 ( 
                 98 
                 ) 
               
             
           
         
       
     
   
   Equations 93-?? may be simplified by making the following substitutions: 
                           U   †     ⁢     =   Δ     ⁢           U   1     +     U   3     -     U   5     -     U   7       2     -         U   0     +   U   -     U   4     -     U   6         2   ⁢     2         -       g   3   ′     ⁡     (         L   u     ⁢     ϛ   u       +     υ   u       )       +       L   u     ⁢     g   13   ′     ⁢     η   u                       U   ‡     ⁢     =   Δ     ⁢           U   1     +     U   3     -     U   5     +     U   7       2     -         U   0     -   U   -     U   4     +     U   6         2   ⁢     2         -       g   13   ′     ⁡     (         L   u     ⁢     ϛ   u       +     υ   u       )       +       L   u     ⁢     g   3   ′     ⁢     η   u                       V   †     ⁢     =   Δ     ⁢           V   1     +     V   3     -     V   5     -     V   7       2     -         V   0     +   V   -     V   4     -     V   6         2   ⁢     2         -       L   u     ⁢     g   13   ′′     ⁢     η   u       -       g   3   ′′     ⁢     υ   υ       -       L   υ     ⁢     g   3   ′     ⁢     η   u                       V   ‡     ⁢     =   Δ     ⁢           V   1     -     V   3     -     V   5     +     V   7       2     -         V   0     -   V   -     V   4     +     V   6         2   ⁢     2         -       g   13   ′′     ⁢     υ   υ       -       L   υ     ⁢     g   3   ′′     ⁢     η   υ       -       L   υ     ⁢     g   13   ′′     ⁢     ϛ   υ                       W   †     ⁢     =   Δ     ⁢           W   1     +     W   3     -     W   5     -     W   7       2     -         W   0     +   W   -     W   4     -     W   6         2   ⁢     2         -       L   w     ⁢     g   13   ′′′     ⁢     η   w       -       g   3   ′′′     ⁢     υ   w       -       L   w     ⁢     g   3   ′′′     ⁢     ϛ   w                       W   ‡     ⁢     =   Δ     ⁢           W   1     -     W   3     -     W   5     +     W   7       2     -         W   0     -   W   -     W   4     +     W   6         2   ⁢     2         -       g   3   ′′′     ⁢     υ   w       -       L   w     ⁢     g   3   ′′′     ⁢     η   w       -       L   w     ⁢     g   13   ′′′     ⁢     ϛ   w                 }           (   99   )               
Observe that all of the terms on the right hand sides of these equations are known i.e. the U, V and W voltages are returned by instrument, the gradient components axe known in the instrument frame and can be easily found using coordinare transformations, and the remaining terms are known by calibration. With these substitutions 93-?? become
 
                           U   †     =         ϛ   u     ⁢     B   ′       -       η   u     ⁢     B   1   ′                       U   ‡     =         η   u     ⁢     B   ′       +       ϛ   u     ⁢     B   1   ′                       V   †     =         ϛ   υ     ⁢     B   ′′       -       η   υ     ⁢     B   1   ′′                       V   ‡     =         η   υ     ⁢     B   ′′       +       ϛ   υ     ⁢     B   1   ′′                       W   †     =         ϛ   w     ⁢     B   ′′′       -       η   w     ⁢     B   1   ′′′                       W   ‡     =         η   w     ⁢     B   ′′′       +       ϛ   w     ⁢     B   1   ′′′                 }           (   100   )               
Equations 100, corresponding to measurements in the C′, C″ and C′″ coordinate systems, may be solved by transforming to the common instrument coordinate system using the coordinate transformations for field components derived earlier, but the resulting sets of equations rapidly become intractible. It is simpler to first solve these pairwise in their respective coordinate systems,
 
                   (           B   1   ′               B   ′           )     =       1       η   u     +     ς   u         ⁢     (             U     ς   u     ‡     -     U     η   u     †                   U     η   u     ‡     +     U     ς   u     †             )               (   101   )                 (           B   1   ′′               B   ′′           )     =       1       η   v     +     ς   v         ⁢     (             V     ς   v     ‡     -     V     η   v     †                   V     η   v     ‡     +     V     ς   v     †             )               (   102   )                 (           B   1   ′′′               B   ′′′           )     =       1       η   ω     +     ς   ω         ⁢     (             W     ς   ω     ‡     -     W     η   ω     †                   W     η   ω     ‡     +     W     ς   ω     †             )               (   103   )               
and then carry out the transformation to the instrument system. As derived above the relevant, transformation equations for the field components are
 
                   B   ′     =       ⁢         (           sin   ⁢           ⁢   λ         0           -   cos     ⁢           ⁢   λ             0       1       0             cos   ⁢           ⁢   λ         0         sin   ⁢           ⁢   λ           )     ⁢   B     ⇒     {             B   1   ′     =         -     (     cos   ⁢           ⁢   λ     )       ⁢           ⁢     B   3       +       (     sin   ⁢           ⁢   λ     )     ⁢           ⁢     B   1                       B   ′     =   B                 B   3   ′     =         (     cos   ⁢           ⁢   λ     )     ⁢           ⁢     B   1       +       (     sin   ⁢           ⁢   λ     )     ⁢           ⁢     B   3                                   B   ′′     =       ⁢         (             -             1   _       ⁢   sin       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ               -             1   _       ⁢     3               -           1   _             0               -             1   _       ⁢   cos       ⁢           ⁢   λ                     1   _       ⁢     3       ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢           ⁢   B     ⇒                     ⁢     {             B   1   ′′     =         -     (     cos   ⁢           ⁢   λ     )       ⁢           ⁢     B   3       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢           ⁢     B   1       +               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢           ⁢   B   ⁢     3                       B   ′′     =       -             1   _       ⁢   B       -       B   1                 1   _       ⁢     3                       B   3   ′′     =         -             1   _       ⁢     (     cos   ⁢           ⁢   λ     )         ⁢           ⁢     B   1       +       (     sin   ⁢           ⁢   λ     )     ⁢           ⁢     B   3       +               1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢           ⁢   B   ⁢     3                                 B   ′′′     =       ⁢         (             -             1   _       ⁢   sin       ⁢           ⁢   λ             -             1   _       ⁢     3         ⁢     (     sin   ⁢           ⁢   λ     )               -   cos     ⁢           ⁢   λ                       1   _       ⁢     3             -           1   _             0               -             1   _       ⁢   cos       ⁢           ⁢   λ             -             1   _       ⁢     3         ⁢     (     cos   ⁢           ⁢   λ     )             sin   ⁢           ⁢   λ           )     ⁢           ⁢   B     ⇒                     ⁢     {             B   1   ′′′     =         -     (     cos   ⁢           ⁢   λ     )       ⁢           ⁢     B   3       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢           ⁢     B   1       -               1   _       ⁢     (     sin   ⁢           ⁢   λ     )       ⁢           ⁢   B   ⁢     3                       B   ′′′     =       -             1   _       ⁢   B       +       B   1                 1   _       ⁢     3                       B   3   ′′′     =         -             1   _       ⁢     (     cos   ⁢           ⁢   λ     )         ⁢           ⁢     B   1       +       (     sin   ⁢           ⁢   λ     )     ⁢           ⁢     B   3       -               1   _       ⁢     (     cos   ⁢           ⁢   λ     )       ⁢           ⁢   B   ⁢     3                               
and if λ=π/4
 
                           B   1   ′     =               1   _       ⁢     2       ⁢     (       B   1     -     B   3       )                     B   ′     =   B                 B   1   ′′     =       1   4     ⁢     2     ⁢     (       -     B   1       +       3     ⁢   B     -     2   ⁢     B   3         )                     B   ′′     =     -             1   _       ⁢     (         B   1     ⁢     3       +   B     )                       B   1   ′′′     =       1   4     ⁢     2     ⁢     (       -     B   1       -       3     ⁢   B     -     2   ⁢     B   3         )                     B   ′′′     =             1   _       ⁢     (         3     ⁢     B   1       -   B     )               }           (   104   )               
and equations 101-103 become
 
                                                       U     ς   u     ‡     -     U     η   u     †           η   u     +     ς   u         =               1   _       ⁢     2       ⁢     (       B   1     -     B   3       )                         U     η   u     ‡     +     U     ς   u     †           η   u     +     ς   u         =   B                           V     ς   v     ‡     -     V     η   v     †           η   v     +     ς   v         =       1   4     ⁢     2     ⁢     (       -     B   1       +       3     ⁢   B     -     2   ⁢     B   3         )                               V     η   v     ‡     -     V     ς   v     †           η   v     +     ς   v         =     -             1   _       ⁢     (         B   1     ⁢     3       +   B     )                                 W     ς   ω     ‡     -     W     η   ω     †           η   ω     +     ς   ω         =       1   4     ⁢     2     ⁢     (       -     B   1       -       3     ⁢   B     -     2   ⁢     B   3         )                               W     η   ω     ‡     -     W     ς   ω     †           η   ω     +     ς   ω         =             1   _       ⁢     (         3     ⁢     B   1       -   B     )               }           (   105   )               
A least-squares best-fit solution may be found by first rewriting this overdetermined system of equations in matrix form in terms of the unknowns B 1 , B, B 3  to give
 
               (                   1   _       ⁢     2           0         -             1   _       ⁢     2                 0       1       0               -     1   4       ⁢     2               1   4     ⁢     6             -             1   _       ⁢     2                   -             1   _       ⁢     3               -           1   _             0               -     1   4       ⁢     2               -     1   4       ⁢     6             -             1   _       ⁢     2                           1   _       ⁢     3             -           1   _             0         )     ⁢     (           B   1             B             B   3           )       =     (               U     ς   u     ‡     -     U     η   u     †           η   u     +     ς   u                       U     η   u     ‡     +     U     ς   u     †           η   u     +     ς   u                       V     ς   v     ‡     -     V     η   v     †           η   v     +     ς   v                       V     η   v     ‡     -     V     ς   v     †           η   v     +     ς   v                       W     ς   ω     ‡     -     W     η   ω     †           η   ω     +     ς   ω                       W     η   ω     ‡     -     W     ς   ω     †           η   ω     +     ς   ω               )           
and then multiplying on the left by the transpose of the coefficient matrix to yield
 
   
     
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           
                               
                             
                               9 
                               4 
                             
                           
                           ⁢ 
                           
                             B 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             9 
                             4 
                           
                           ⁢ 
                           B 
                         
                       
                     
                     
                       
                         
                           B 
                           3 
                             
                             
                             
                           
                             3 
                             _ 
                           
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     3 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         V 
                                         
                                           η 
                                           v 
                                         
                                         ‡ 
                                       
                                       + 
                                       
                                         V 
                                         
                                           ς 
                                           v 
                                         
                                         † 
                                       
                                     
                                     
                                       
                                         η 
                                         v 
                                       
                                       + 
                                       
                                         ς 
                                         v 
                                       
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     2 
                                   
                                   ⁢ 
                                   
                                     3 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         W 
                                         
                                           η 
                                           ω 
                                         
                                         ‡ 
                                       
                                       + 
                                       
                                         W 
                                         
                                           ς 
                                           ω 
                                         
                                         † 
                                       
                                     
                                     
                                       
                                         η 
                                         ω 
                                       
                                       + 
                                       
                                         ς 
                                         ω 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               U 
                                               
                                                 ς 
                                                 u 
                                               
                                               ‡ 
                                             
                                             - 
                                             
                                               U 
                                               
                                                 η 
                                                 u 
                                               
                                               † 
                                             
                                           
                                           
                                             
                                               η 
                                               u 
                                             
                                             + 
                                             
                                               ς 
                                               u 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           1 
                                           4 
                                         
                                         ⁢ 
                                         
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               V 
                                               
                                                 ς 
                                                 v 
                                               
                                               ‡ 
                                             
                                             - 
                                             
                                               V 
                                               
                                                 η 
                                                 v 
                                               
                                               † 
                                             
                                           
                                           
                                             
                                               η 
                                               v 
                                             
                                             + 
                                             
                                               ς 
                                               v 
                                             
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         1 
                                         4 
                                       
                                       ⁢ 
                                       
                                         2 
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             W 
                                             
                                               ς 
                                               ω 
                                             
                                             ‡ 
                                           
                                           - 
                                           
                                             W 
                                             
                                               η 
                                               ω 
                                             
                                             † 
                                           
                                         
                                         
                                           
                                             η 
                                             ω 
                                           
                                           + 
                                           
                                             ς 
                                             ω 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       U 
                                       
                                         η 
                                         u 
                                       
                                       ‡ 
                                     
                                     + 
                                     
                                       U 
                                       
                                         ς 
                                         u 
                                       
                                       † 
                                     
                                   
                                   
                                     
                                       η 
                                       u 
                                     
                                     + 
                                     
                                       ς 
                                       u 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     1 
                                     4 
                                   
                                   ⁢ 
                                   
                                     6 
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         V 
                                         
                                           ς 
                                           v 
                                         
                                         ‡ 
                                       
                                       - 
                                       
                                         V 
                                         
                                           η 
                                           v 
                                         
                                         † 
                                       
                                     
                                     
                                       
                                         η 
                                         v 
                                       
                                       + 
                                       
                                         ς 
                                         v 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           1 
                                           4 
                                         
                                         ⁢ 
                                         
                                           6 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               W 
                                               
                                                 ς 
                                                 ω 
                                               
                                               ‡ 
                                             
                                             - 
                                             
                                               W 
                                               
                                                 η 
                                                 ω 
                                               
                                               † 
                                             
                                           
                                           
                                             
                                               η 
                                               ω 
                                             
                                             + 
                                             
                                               ς 
                                               ω 
                                             
                                           
                                         
                                       
                                       - 
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               V 
                                               
                                                 η 
                                                 v 
                                               
                                               ‡ 
                                             
                                             + 
                                             
                                               V 
                                               
                                                 ς 
                                                 ν 
                                               
                                               † 
                                             
                                           
                                           
                                             
                                               η 
                                               v 
                                             
                                             + 
                                             
                                               ς 
                                               v 
                                             
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           1 
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                             
                                               W 
                                               
                                                 η 
                                                 ω 
                                               
                                               ‡ 
                                             
                                             + 
                                             
                                               W 
                                               
                                                 ς 
                                                 ω 
                                               
                                               † 
                                             
                                           
                                           
                                             
                                               η 
                                               ω 
                                             
                                             + 
                                             
                                               ς 
                                               ω 
                                             
                                           
                                         
                                       
                                       - 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           
                             2 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     U 
                                     
                                       ς 
                                       u 
                                     
                                     ‡ 
                                   
                                   - 
                                   
                                     U 
                                     
                                       η 
                                       u 
                                     
                                     † 
                                   
                                 
                                 
                                   
                                     η 
                                     u 
                                   
                                   + 
                                   
                                     ς 
                                     u 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     V 
                                     
                                       ς 
                                       ν 
                                     
                                     ‡ 
                                   
                                   - 
                                   
                                     V 
                                     
                                       η 
                                       ν 
                                     
                                     † 
                                   
                                 
                                 
                                   
                                     η 
                                     ν 
                                   
                                   + 
                                   
                                     ς 
                                     ν 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     W 
                                     
                                       ς 
                                       ω 
                                     
                                     ‡ 
                                   
                                   - 
                                   
                                     W 
                                     
                                       η 
                                       ω 
                                     
                                     † 
                                   
                                 
                                 
                                   
                                     η 
                                     ω 
                                   
                                   + 
                                   
                                     ς 
                                     ω 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 106 
                 ) 
               
             
           
         
       
     
   
   We bring the final forms of the solutions here for convenience 
                         B   1     =       ⁢           2   ⁢     2       9     ⁢         U     ς   u     ‡     -     U     η   u     †           η   u     +     ς   u           -         2     9     ⁢         V     ς   v     ‡     -     V     η   ν     †           η   v     +     ς   v           -         2   ⁢     3       9     ⁢       (       V     η   v     ‡     +     V     ς   ν     †       )         η   v     +     ς   v           -                     ⁢           2     9     ⁢         W     ς   ω     ‡     -     W     η   ω     †           η   ω     +     ς   ω           +         2   ⁢     3       9     ⁢       (       W     η   ω     ‡     +     W     ς   ω     †       )         η   ω     +     ς   ω                           (   107   )                     B   =       ⁢         4   9     ⁢         U     η   u     ‡     +     U     ς   u     †           η   u     +     ς   u           +         6     9     ⁢         V     ς   v     ‡     -     V     η   ν     †           η   v     +     ς   v           -       2   9     ⁢         V     η   v     ‡     +     V     ς   ν     †           η   v     +     ς   v           -                     ⁢           6     9     ⁢         W     ς   ω     ‡     -     W     η   ω     †           η   ω     +     ς   ω           -       2   9     ⁢         W     η   ω     ‡     +     W     ς   ω     †           η   ω     +     ς   ω                           (   108   )                 B   3     =       -       2     3       ⁢     (           U     ς   u     ‡     -     U     η   u     †           η   u     +     ς   u         +         V     ς   v     ‡     -     V     η   ν     †           η   v     +     ς   v         +         W     ς   ω     ‡     -     W     η   ω     †           η   ω     +     ς   ω           )               (   109   )             where                                           U   †     ⁢     =   △     ⁢       ⁢           U   1     +     U   3     -     U   5     -     U   7       2     -         U   0     +   U   -     U   4     -     U   6         2   ⁢     2         +                     ⁢         L   u     ⁢     g   13   ′     ⁢     η   u       -       g   3   ′     ⁢     v   u       -       L   u     ⁢     g   3   ′     ⁢     ς   u                                   U   ‡     ⁢     =   △     ⁢       ⁢           U   1     -     U   3     -     U   5     +     U   7       2     -         U   0     -   U   -     U   4     -     U   6         2   ⁢     2         -                     ⁢         g   13   ′     ⁢     v   u       -       L   u     ⁢     g   3   ′     ⁢     η   u       -       L   u     ⁢     g   13   ′     ⁢     ς   u                                   V   †     ⁢     =   △     ⁢       ⁢           V   1     +     V   3     -     V   5     -     V   7       2     -         V   0     +   V   -     V   4     -     V   6         2   ⁢     2         +                     ⁢         L   v     ⁢     g   13   ′′     ⁢     η   v       -       g   3   ′′     ⁢     v   v       -       L   u     ⁢     g   3   ′′     ⁢     ς   v                                   V   ‡     ⁢     =   △     ⁢       ⁢           V   1     -     V   3     -     V   5     +     V   7       2     -         V   0     -   V   -     V   4     +     V   6         2   ⁢     2         -                     ⁢         g   13   ′′     ⁢     v   v       -       L   v     ⁢     g   3   ′′     ⁢     η   v       -       L   v     ⁢     g   3   ′′     ⁢     ς   v                                   W   †     ⁢     =   △     ⁢       ⁢           W   1     +     W   3     -     W   5     -     W   7       2     -         W   0     +   W   -     W   4     -     W   6         2   ⁢     2         +                     ⁢         L   ω     ⁢     g   13   ′′′     ⁢     η   ω       -       g   3   ′′′     ⁢     v   ω       -       L   ω     ⁢     g   3   ′′′     ⁢     ς   ω                                   W   ‡     ⁢     =   △     ⁢       ⁢           W   1     -     W   3     -     W   5     +     W   7       2     -         W   0     -   W   -     W   4     +     W   6         2   ⁢     2         -                     ⁢         g   13   ′′′     ⁢     v   ω       -       L   ω     ⁢     g   3   ′′′     ⁢     η   ω       -       L   ω     ⁢     g   13   ′′′     ⁢     ς   ω                       }     ,                                     g   13   ′     =       g   11                 1   _       -     g   33                 1   _           ,                   g   3   ′′     =       ⁢         -     1   4       ⁢     2     ⁢     g   1       -       1   4     ⁢     2     ⁢     g   3       +                     ⁢         1   8     ⁢     6     ⁢     g   11       -       1   4     ⁢     6     ⁢     g   13       -                     ⁢       1   8     ⁢     6     ⁢   g                           g   3   ′     =               1   _       ⁢     2       ⁢     (       g   1     +     g   3       )         ,                   g   13   ′′′     =       ⁢         1   8     ⁢     g   11       +       3   8     ⁢   g     -                     ⁢       g   33                 1   _       +       1   4     ⁢     3     ⁢     g   1                               g   13   ′′     =         1   8     ⁢     g   11       +       3   8     ⁢   g     -     g   33                 1   _       -       1   4     ⁢     3     ⁢     g   1           ,                   g   3   ′′′     =       ⁢         -     1   4       ⁢     2     ⁢     g   1       -       1   4     ⁢     2     ⁢     g   3       -                     ⁢         1   8     ⁢     6     ⁢     g   11       +       1   4     ⁢     6     ⁢     g   13       +                     ⁢       1   8     ⁢     6     ⁢   g                                         
Formally the scolding coefficients, ν u ,              u  and η u  are defined by equations 55 and 70 but would normally be determined by calibration.

   V. EXAMPLES 
   A. Precession About the z-Axis. 
   Say σ 1 =σ=0, S 1 =S=S. If the gradiometer precesses about the the axis of rotation by virtue of an error angle of φ say, then the theory developed above suffices to model this by letting, τ 1 =−τ=φ.  1 . Using equations ?? 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ξ 
                                               u 
                                             
                                             ⁢ 
                                             
                                               = 
                                               △ 
                                             
                                             ⁢ 
                                             
                                               ρ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               QS 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               cos 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               ϕ 
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               χ 
                                               u 
                                             
                                             ⁢ 
                                             
                                               = 
                                               △ 
                                             
                                             ⁢ 
                                             0 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ν 
                                         u 
                                       
                                       ⁢ 
                                       
                                         = 
                                         △ 
                                       
                                       ⁢ 
                                       
                                         2 
                                         ⁢ 
                                         ρ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         QS 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         ϕ 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ς 
                                   u 
                                 
                                 ⁢ 
                                 
                                   = 
                                   △ 
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             η 
                             u 
                           
                           ⁢ 
                           
                             = 
                             △ 
                           
                           ⁢ 
                           0 
                         
                       
                     
                   
                   } 
                 
                 ⁢ 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   α 
                                                   u 
                                                 
                                                 ⁢ 
                                                 
                                                   = 
                                                   △ 
                                                 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                       ⁢ 
                                                       QS 
                                                       ⁢ 
                                                       
                                                           
                                                       
                                                       ⁢ 
                                                       ρcosϕ 
                                                     
                                                     ) 
                                                   
                                                   
                                                     - 
                                                     1 
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   β 
                                                   u 
                                                 
                                                 ⁢ 
                                                 
                                                   = 
                                                   △ 
                                                 
                                                 ⁢ 
                                                 
                                                   
                                                     ( 
                                                     0 
                                                     ) 
                                                   
                                                   
                                                     - 
                                                     1 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             U 
                                             † 
                                           
                                           ⁢ 
                                           
                                             = 
                                             △ 
                                           
                                           ⁢ 
                                           
                                             
                                               U 
                                               0 
                                             
                                             - 
                                             U 
                                             + 
                                             
                                               U 
                                               4 
                                             
                                             - 
                                             
                                               U 
                                               6 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       U 
                                       ‡ 
                                     
                                     ⁢ 
                                     
                                       = 
                                       △ 
                                     
                                     ⁢ 
                                     
                                       
                                         U 
                                         1 
                                       
                                       - 
                                       
                                         U 
                                         3 
                                       
                                       + 
                                       
                                         U 
                                         5 
                                       
                                       - 
                                       
                                         U 
                                         7 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   M 
                                   1 
                                 
                                 ⁢ 
                                 
                                   = 
                                   △ 
                                 
                                 ⁢ 
                                 
                                   S 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ϕ 
                                 
                               
                               , 
                               
                                 
                                   N 
                                   1 
                                 
                                 ⁢ 
                                 
                                   = 
                                   △ 
                                 
                                 ⁢ 
                                 
                                   S 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ϕ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           M 
                           ⁢ 
                           
                             = 
                             △ 
                           
                           ⁢ 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                         , 
                         
                           N 
                           ⁢ 
                           
                             = 
                             △ 
                           
                           ⁢ 
                           
                             
                               - 
                               S 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 110 
                 ) 
               
             
           
         
       
     
   
   VI. Appendix 1: Geographic Coordinate System, C G . 
   The geographical coordinates system, C G , is defined as follows:
     x G -axis: The x G -axis is along the traverse in the direction of increasing station number (“grid north”).   y G -axis: The y G -axis is at 90° to the x G -axis, parallel to the ground and points right as you look along the traverse (“grid east”).   z G -axis: The z G -axis is vertically down (pointing at the centre of mass of the earth).   

   This is a right-handed Cartesian coordinate system in terms of which a unit vector with azimuth. A (the angle measured clockwise from grid north) and inclination I (angle measured positive downward from the horizontal) is
 
unit vector with azimuth  A  and inclination  I =(cos  A  cos  I , sin  A  cos  I , sin  I ).  (111)
 
   1. Transformations Between C and C G . 
   The transformation between C and C G  assumes a particularly simple form if the instrument is oriented with its base horizontal and its x-axis aligned due north. In this case C is mapped to C G  by a positive rotation through an angle π radians about the x-axis and the transformations of the field and tensor and data become simply 
   
     
       
         
           
             
               
                 
                   B 
                   G 
                 
                 = 
                 
                   
                     
                       R 
                       ^ 
                     
                     π 
                   
                   ⁢ 
                   B 
                 
               
             
             
               
                 ( 
                 112 
                 ) 
               
             
           
           
             
               
                 
                   
                     g 
                     ^ 
                   
                   G 
                 
                 = 
                 
                   
                     
                       R 
                       ^ 
                     
                     π 
                   
                   ⁢ 
                   
                     g 
                     ^ 
                   
                   ⁢ 
                   
                     
                       R 
                       ^ 
                     
                     π 
                     
                       - 
                       1 
                     
                   
                 
               
             
             
               
                 ( 
                 113 
                 ) 
               
             
           
           
             
               where 
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     R 
                     ^ 
                   
                   π 
                 
                 = 
                 
                   
                     
                       R 
                       ^ 
                     
                     π 
                     
                       - 
                       1 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 114 
                 ) 
               
             
           
           
             
               Hence 
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             B 
                             
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             B 
                             G 
                           
                         
                       
                       
                         
                           
                             B 
                             
                               G 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             B 
                             1 
                           
                         
                       
                       
                         
                           
                             - 
                             B 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               B 
                               3 
                             
                           
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 and 
               
             
             
               
                 ( 
                 115 
                 ) 
               
             
           
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             11 
                           
                         
                       
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             13 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       
                         
                           g 
                           G 
                         
                       
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             3 
                           
                         
                       
                     
                     
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             31 
                           
                         
                       
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             3 
                           
                         
                       
                       
                         
                           g 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             33 
                           
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           g 
                           11 
                         
                       
                       
                         
                           - 
                           
                             g 
                             1 
                           
                         
                       
                       
                         
                           - 
                           
                             g 
                             13 
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             g 
                             1 
                           
                         
                       
                       
                         
                           g 
                           ⁢ 
                           
                               
                           
                         
                       
                       
                         
                           g 
                           3 
                         
                       
                     
                     
                       
                         
                           - 
                           
                             g 
                             31 
                           
                         
                       
                       
                         
                           g 
                           3 
                         
                       
                       
                         
                           g 
                           33 
                         
                       
                     
                   
                   ) 
                 
               
             
             
               
                 ( 
                 116 
                 ) 
               
             
           
         
       
     
   
   More generally if the instrument has azimuth A and inclination I (in geographic coordinates) then C is mapped to C G  by first rotating about y through an angle of −I then by a rotation about the resulting z through an angle A followed lastly by a rotation about the resulting x of π. Hence 
   
     
       
         
           
             
               
                 
                   B 
                   G 
                 
                 = 
                 
                   
                     
                       
                         R 
                         ^ 
                       
                       x 
                     
                     ⁡ 
                     
                       ( 
                       π 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         R 
                         ^ 
                       
                       z 
                     
                     ⁡ 
                     
                       ( 
                       A 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         R 
                         ^ 
                       
                       y 
                     
                     ⁡ 
                     
                       ( 
                       
                         - 
                         I 
                       
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
               
             
             
               
                 ( 
                 117 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 B 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 B 
                                 G 
                               
                             
                           
                           
                             
                               
                                 B 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                 
                               
                             
                           
                         
                         ) 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 1 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       - 
                                       I 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     - 
                                     sin 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       - 
                                       I 
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 1 
                               
                               
                                 0 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       - 
                                       I 
                                     
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       - 
                                       I 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   B 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                               
                             
                             
                               
                                 
                                   B 
                                   3 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               
                                 
                                   
                                     - 
                                     cos 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                 
                               
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   A 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                               
                                 0 
                               
                               
                                 
                                   
                                     - 
                                     cos 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   I 
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   B 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                               
                             
                             
                               
                                 
                                   B 
                                   3 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 118 
                 ) 
               
             
           
           
             
               and 
             
             
               
                   
               
             
           
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   11 
                                 
                               
                             
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                           
                           
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                             
                               
                                 g 
                                 G 
                               
                             
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                 
                               
                             
                           
                           
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   13 
                                 
                               
                             
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                 
                               
                             
                             
                               
                                 g 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   33 
                                 
                               
                             
                           
                         
                         ) 
                       
                       = 
                         
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                               
                             
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
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                                   cos 
                                 
                                 ⁢ 
                                 
                                     
                                 
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                                 A 
                               
                             
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
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                                 sin 
                                 ⁢ 
                                 
                                     
                                 
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                                   cos 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 g 
                                 11 
                               
                             
                             
                               
                                 g 
                                 1 
                               
                             
                             
                               
                                 g 
                                 13 
                               
                             
                           
                           
                             
                               
                                 g 
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                                 3 
                               
                             
                           
                           
                             
                               
                                 g 
                                 13 
                               
                             
                             
                               
                                 g 
                                 3 
                               
                             
                             
                               
                                 g 
                                 33 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Acos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 A 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                             
                               
                                 
                                   - 
                                   cos 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 I 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                   
               
             
           
         
       
     
   
   In the special case A=I=0 we get so that 
             (           B     G   ⁢           ⁢   1                 B   G               B     G   ⁢           ⁢   3             )     =     (           B   1               -   B               -     B   3             )                   (           g     G   ⁢           ⁢   11             g     G   ⁢           ⁢   1             g     G   ⁢           ⁢   13                 g     G   ⁢           ⁢   1             g   G           g     G   ⁢           ⁢   3                 g     G   ⁢           ⁢   13             g     G   ⁢           ⁢   3             g     G   ⁢           ⁢   33             )     =     (           g   11           -     g   1             -     g   13                 -     g   1           g         g   3               -     g   13             g   3           g   33           )           
as above.
 
   The approximation used in obtaining equation (3) above, which is significantly simplified as compared to equation (2), involves truncating the Taylor series expansion of the magnetic field as a function of position to include the effects of the B-field and the first-order gradients only. This simplifying approximation is desirable because the Taylor series expansion has an infinite number of terms. The approximation is justified because the magnitudes of the higher-order terms decreases as the order of the terms increases, for three reasons. The higher-order terms in the Taylor series are scaled by prefactors which may be written as the product of two factors. Firstly, one of these factors is a scaling term which decreases with increasing order. The second factor is proportional to 1/r n , where r is half the distance of separation between the field detectors, and n is the order of the term. Since the separation between the field sensors is small, raising this to an increasing power rapidly decreases the magnitude of this factor. The third reason the approximation is justified is that higher-order gradients fall off faster with distance from the field source than lower-order gradients, so the signal at the gradiometer is dominated by the field and the first-order gradient components. 
   The effects of misalignments of the field detectors on the output voltages from the instrument are also scaled by these higher-order terms, but these effects are small and similarly decrease with increasing order. 
   Referring to equations (6)-(8), it is noted that the value of λ may be any value in the range of 0° to 90°, inclusive. As noted in equation (9), when λ is approximately 35.3°, an orthogonal system is provided. In the presently described embodiment, λ is 45°. 
   It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the spirit or scope of the invention as broadly described. In particular, the present invention may be applied in conjunction with the invention set out in the co-pending PCT application claiming priority from Australian Provisional Patent Application No. 2002950624 and filed simultaneously with the present application, the contents of which are herein incorporated by reference. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.