Abstract:
A method includes measuring an acceleration along an input axis of an accelerometer mounted to a gradiometer disc, the accelerometer having a coordinate axis that is parallel to a spin axis of the disc, and includes calculating a gravity tensor element as a function of the measured acceleration and a component of the measured acceleration caused by an acceleration along the coordinate axis. Consequently, this technique typically yields a more accurate calculation of the gravitational field by accounting for undesired accelerations picked up by accelerometers having input axes that are not parallel to the gradiometer disc.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application claims priority to U.S. Provisional Application Serial No.  60 / 361 , 699 , filed on Mar.  6 ,  2002 , which is incorporated by reference. 
     
    
     
       BACKGROUND  
         [0002]    Geologists have discovered that one can identify certain physical characteristics of a geological formation from the gravitational potential field (denoted with the symbol G in this application) near the formation. For example, the gravitational field G can often indicate the presence and yield the identity of a mineral such as coal that is located beneath the surface of the formation. Therefore, measuring and analyzing the gravitational field G of a formation, can often yield such physical characteristics of the formation more easily and less expensively than an invasive technique such as drilling. Relevant characteristics of the field are typically determined not by measuring the gravitational potential G directly, but either by measuring components of the gravitational acceleration vector g resulting from this field or by measuring spatial derivatives of these acceleration vector components. The three components of the acceleration vector can each be spatially differentiated along three different axes providing a set of nine different signals mathematically related to the underlying gravitational potential G. These nine signals are the gravitation tensor elements (sometimes called the gravity Gradients), and much effort has gone into developing techniques for accurately measuring these tensor elements.  
           [0003]    Referring to FIG. 1, one can use a gravity gradiometer  10  to measure the gravitational potential field G near a geological formation (not shown). The notation used in this patent to the refer to the nine gravitational field tensor elements in matrix form is:  
             Γ   =     [           Γ   xx           Γ   xy           Γ   xz               Γ   yx           Γ   yy           Γ   yz               Γ   zx           Γ   zy           Γ   zz           ]             (   1   )                               
 
           [0004]    where the matrix members represent the respective gravity tensors along each of the three X, Y, and Z “body” axes, which typically intersect at the centroid  12  of the gradiometer  10 . For example, the tensor element Γ xx  (which can be expressed in equivalent units of (meters/seconds 2 )/meter, 1/seconds 2 , or Eötvös units where  109  Eötvös=1/seconds 2 ) is the spatial partial derivative along the X axis of the X component of the gravitational acceleration vector g, Γ xy  is the spatial partial derivative along the Y axis of the X component of g, Γ xz  is the spatial partial derivative along the Z axis of the X component of g, Γ yx  is the spatial partial derivative along the X axis of the Y component of g, etc. Furthermore, although the tensors elements Γ may vary over time, for many formations the tensor elements Γare constant over time, or vary so slowly that they can be treated as being constant over time. Moreover, in some applications the gradiometer  10  may make measurements sufficient to calculate only the desired elements of the full tensor Γ. To measure the gravitational potential field G of a geological formation (not shown in FIG. 1), one mounts the gradiometer  10  in a vehicle (not shown) such as a helicopter that sweeps the gradiometer over the formation. For maximum accuracy, it is desired that the gradiometer  10  not rotate at a high rate about any of the X, Y, and Z body axes as it sweeps over the formation. But unfortunately, the vehicle often generates vibrations (e.g., engine) or is subject to vibrations (e.g., wind) that causes such rotations about the body axes. Therefore, the gradiometer  10  is often rotationally isolated from the vehicle by a gimballing system (not shown) which allows the gradiometer to remain non-rotating even as the vehicle experiences varying orientations typical of its operation. The gimballing system carrying the gradiometer  10  typically includes a rotational sensor assembly  18 , such as a gyroscope assembly for measuring rotational activity (typically rotational rate ω or rotational displacement) about the X, Y, and Z body axes. Control signals derived form these measurements are fed back to the motors attached to the gimbal axes to reduce the rotations experienced by the gradiometer  10 . But although the gimballing system typically reduces the magnitude of the vibration-induced rotations of the gradiometer  10  about the body axes, it is typically impossible to eliminate such rotations altogether. Even with an ideal gradiometer, the tensor measurement would, of physical necessity, be additively corrupted by the presence of gradient signals caused by these rotations. These additional non-gravitational gradients are simple deterministic functions of the rotational rates (e.g. rotational Γ=ω x , ω y  where ω j  refers to the rotational rate around the j body axis in radians/sec). Consequently, the measurements from the gradiometer  10  typically have these corrupting signals subtracted by a processor  20  to increase the accuracy of the gradiometer&#39;s measurement of the gravitational field G as discussed below in conjunction with FIG. 3. Although shown as being disposed within the housing  16 , the processor  20  may be disposed outside of the housing for processing of the measurement data from the gradiometer  10  in real time or after the gradiometer measures the gravitational field G. In the latter case, the gradiometer  10  typically includes a memory  22  for storage of the measurement data for later download to the external processor  20 . Alternatively, the gradiometer  10  may include a transmitter (not shown) for transmitting the measurement data to the external processor  20  and/or an external memory  22 . Moreover, the processor  20  or memory  22  includes a sample-and-hold circuit (not shown) and an analog-to-digital converter (ADC) (not shown) to digitize the gradiometer measurement data and any other measured signals required for optimal operation.  
           [0005]    Referring to FIG. 2, the gravity gradiometer  10  of FIG. 1 includes one or more disc assemblies—here three disc assemblies  24 ,  26 , and  28 —each for measuring a subset of the full set of tensors F for the gravitational field G of a geological formation  36 .  
           [0006]    Each disc assembly  24 ,  26 , and  28  includes a respective disc  30 ,  32 , and  34  that is mounted in a respective plane that is coincident or parallel with one of the three body-axis planes such that the spin axis of the disc is either coincident with or parallel to the body axis that is normal to the mounting plane. Furthermore, each disc includes orthogonal disc axes that lie in but rotate relative to the mounting plane. For example, the disc  30  lies in the X-Y body-axis plane, has a spin axis Z S  that is parallel to the Z body axis—that is, the X-Y coordinates of Z S  are (X=C1, Y=C2) where C1 and C2 are constants—and includes orthogonal disc axes X D  and Y D . As the disc  30  rotates—here in a counterclockwise direction—the X D  and Y D  disc axes rotate relative to the non-rotating X and Y body axes. At the instant of time represented in FIG. 2, the X D  and Y D  disc axes of the disc  30  are respectively coincident with the X and Y body axes. In addition, the disc  32  lies in a plane that is parallel to the Y-Z body-axis plane and has a spin axis X S  that is parallel to the X body axis. At the instant of time represented in FIG. 2, the X D  and Y D  disc axes of the disc  32  are respectively parallel with the Y and Z body axes.  
           [0007]    To measure the gravitational field G, the disc assemblies  24 ,  26 , and  28  each include at least one respective pair of accelerometers that are mounted π radians apart on the discs  30 ,  32 , and  34 , respectively. For clarity of explanation, only the disc assembly  24  is discussed, it being understood that the other disc assemblies  26  and  28  are similar. Here, the disc assembly  28  includes two pairs of accelerometers  38   a ,  38   b  and  38   c ,  38   d . Each accelerometer  38   a ,  38   b ,  38   c , and  38   d  includes a respective input axis  40   a ,  40   b ,  40   c , and  40   d  along which the accelerometer measures a respective acceleration magnitude Aa, Ab, Ac, and Ad, and each accelerometer is mounted to the disc  30  such that its input axis is a radius R from the spin axis Z S  and is perpendicular to R. The accelerometers  38   a  and  38   b  of the first pair are mounted π radians apart on the X D  disc axis, and the accelerometers  38   c  and  38   d  are mounted π radians apart on the Y D  disc axis. Although ideally described as being perpendicular to the radius R, the input axes  40   a ,  40   b ,  40   c , and  40   d  may actually be oriented at other angles relative to R intentionally or due to manufacturing imperfections. Furthermore, the disc assembly  24  may include additional pairs of accelerometers that are mounted on the disc  30  between the accelerometers  38   a ,  38   b ,  38   c , and  38   d . For example, the disc assembly  24  may include additional accelerometers  38   e ,  38   f ,  38   g , and  38   h , which are respectively spaced π/4 radians from the accelerometers  38   a ,  38   b ,  38   c , and  38   d . As is known, these additional accelerometers allow, through redundant measurement, an increase in the signal-to-noise ratio (SNR) of the gravitational-field measurement.  
           [0008]    Referring to FIG. 3, the operation of the disc assembly  24  is discussed, it being understood that the operation of the disc assemblies  26  and  28  of FIG. 2 is similar.  
           [0009]    [0009]FIG. 3 is a top view of the disc assembly  24  where the spin axis Zs extends out of the page from the center  50  of the disc  30 . For purposes of explanation, the following ideal conditions are assumed. First, the disc  30  spins in a counterclockwise direction at a constant rate of C), which has units of radians/second. Second, the input axes  40  are each aligned perfectly with either the X D  or Y D  disc axes and as a result lie in or parallel to the X-Y plane. Third, the accelerometers are all the same radial distance R from the Z S  spin axis. And fourth, there are no rotations of the disc  30  about the X or Y body axes.  
           [0010]    At a time t=0, the X D  and Y D  disc axes of the disc  30  are respectively coincident with the body axes X and Y. As the disc  30  spins, the X D  disc axis forms an angle Ωt with the X body axis. To illustrate this rotation, the position of the X D  and Y D  axes and the accelerometer  38   a  are shown in dashed line at ωt=π/4 radians. Although not shown in dashed line, the other accelerometers  38   b ,  38   c , and  38   d  are also π/4 radians from their illustrated (Ωt=0) positions when Ωt=π/4 radians. Consequently, an equation that represents the acceleration Aa in terms of the gravity tensor elements Γ xx , Γ xy , Γ yx , and Γ yy  can be derived as follows, where a x  and a y  are the gravitational-field induced accelerations at the center  50  in the X and Y directions, respectively. Specifically, Aa equals the component of acceleration along the input axis  40   a  caused by an acceleration in the Y direction minus the component of acceleration along the input axis caused by an acceleration in the X direction. Therefore,  
             Aa =( a   y +Γ yx   R  cos Ω t+Γ   yy   R  sin Ω t )cos Ω t −( a   x   +Γ   xx   R  cos Ω t+Γ   xy   R  sin Ω t )sin Ω t   (2)  
           [0011]    Expanding the terms of equation (2) gives:  
             Aa=a   y  cos Ω t+Γ   yx   R  cos 2   t+Γ   yy   R  sin Ω t  cos Ω t−a   x  sin Ω t−Γ   xx   R  cos Ω t  sin Ω t−Γ   xy   R  sin 2   Ωt   (3)  
           [0012]    Using trigonometric identities for cos 2 Ωt and cos Ωt sin Ωt, and realizing that Γ xy =Γ yx  for any gravitational field G, one obtains:  
             Aa   =         a   y        cos                 Ω                 t     -       a   x        sin                 Ω                 t     +       Γ   xy          R        (       1   2     +       1   2        cos                 2      Ω                 t       )         +       Γ   yy          R   2        sin                 2      Ω                 t     -       Γ   xx          R   2        sin                 2      Ω                 t     -       Γ   xy          R        (       1   2     -       1   2        cos                 2      Ω                 t       )                   (   4   )                               
 
           [0013]    And combining terms of equation (4) gives:  
             Aa   =         a   y        cos                 Ω                 t     -       a   x        sin                 Ω                 t     +       Γ   xy        R                 cos                 2      Ω                 t     +       R   2        sin                 2      Ω                   t        (       Γ   yy     -     Γ   xx       )                   (   5   )                               
 
           [0014]    Because the accelerometer  38   b  is always π radians from the accelerometer  38   a , one can easily derive the following equation for the acceleration magnitude Ab by replacing “Ωt” with “Ωt+π” in equations (2)-(5):  
             Ab   =         -     a   y          cos                 Ω                 t     +       a   x        sin                 Ω                 t     +     R                 cos                 2      Ω                 t                   Γ   xy       +       R   2        sin                 2      Ω                   t        (       Γ   yy     -     Γ   xx       )                   (   6   )                               
 
           [0015]    Summing equations (5) and (6) gives the following equation for the expected idealized output of the sum of these two accelerometers:  
             Aa+Ab= 2Γ xy   R  cos 2Ω t+R  sin 2Ω t (Γ yy −Γ xx )  (7)  
           [0016]    To increase the accuracy of the measurement (in view of potential errors as discussed below), one can derive equations that represent Ac and Ad in terms of the gravity tensor elements Γ xx , Γ xy =Γ yx , and Γ yy  by respectively replacing “Ωt” with “Ωt+π/2” and “Ωt+3π/2” in equations (2)-(6) to arrive at the following equation:  
             Ac+Ad=− 2Γ xy   R  cos 2Ω t−R  sin 2Ω t (Γ yy −Γ xx )  (8)  
           [0017]    Subtracting these two accelerometer sums (reflected in equations 7 and 8) provides the following equation, which is the basic element of measurement for gradiometers of this design:  
                   (     Aa   +   Ab     )     -     (     Ac   +   Ad     )       2     =       2        Γ   xy        R                 cos                 2      Ω                 t                +     R                 sin                 2      Ω                   t        (       Γ   yy     -     Γ   xx       )                   (   9   )                               
 
           [0018]    This combination signal, which is normally called the bandpass signal, is typically bandpass filtered and digitized, and is then synchronously demodulated by the processor  20  at sin 2Ωt and cos 2Ωt to recover Γ xy =Γ yx  and (Γ yy −Γ xx )/2.  
           [0019]    Still referring to FIG. 3, the conditions that were assumed to be ideal for the derivation of equations (2)-(9) are, unfortunately, seldom ideal. Consequently, these non-ideal conditions introduce additional acceleration terms into these equations, and these terms can reduce the accuracy of the calculated gravity tensor elements if unaccounted for. But fortunately, the processor  20  can account for many of these additional terms as discussed below.  
           [0020]    For example, still referring to FIG. 3, the motor (not shown) that spins the disc  30  may be unable to maintain a constant rotation rate Ω. Such uneven rotation may cause the pairs of accelerometers to sense reinforcing accelerations that swamp out the accelerations caused by the gravitational field. Consequently, the gradiometer  10  (FIG. 1) may include a sensor (not shown) that measures the rotation rate Ω as a function of time, and the processor  20  can use this measurement to conventionally include the acceleration term introduced by the uneven rotation in the equations (2)-(9).  
           [0021]    Furthermore, as discussed above in conjunction with FIG. 1, vibrations of the vehicle (not shown) or other forces may cause the gradiometer  10  to rotate about the X or Y body axes. Such rotations may cause the pairs of accelerometers to sense reinforcing accelerations that swamp out the accelerations caused by the gravitational field. For example, assume that the gradiometer  10  rotates about the Y body axis with a rotational rate (in units of radians/second) ω y . This rotation causes the accelerometer  38   a  to sense a centripetal acceleration toward the Y axis along a moment arm  52  according to the following equation, where AaY is the acceleration term added to equation (2) due to this centripetal acceleration:  
             Aay =(ω y ) 2   R  cos Ω t  sin Ω t   (10)  
           [0022]    The accelerometer  38   b  senses an identical centripetal acceleration AbY, and the corresponding centripetal accelerations AcY and AdY sensed by the accelerometers  38   c  and  38   d  are given by an equation similar to equation (10). Consequently, the processor  20  can use the signals (from the gyroscope assembly  18  of FIG. 1) that represent the rotational rates ω x , ω y , and ω z  to include AaY, AbY, AcY, and AdY in equation (9), and thus to compensate the gradient measurements for centripetally induced errors introduced by rotations about the X, Y, and Z body axes.  
           [0023]    Similarly, the processor  20  can often account for errors introduced by the input axes  40  of the accelerometer pairs not making the same angle with the respective X D  or Y D  disc axis or not being the same radial distance R from the disc center  50 . In these cases, the precise amount of misalignment or radial distance error is typically unknown (though it may be relatively constant for a given gradiometer instrument and may be imperfectly known), and thus the error introduced into the gravitational-field measurement may not be exactly known. However if the functional relationship between the causative error and the resulting signal corruption is known, then this information can be included in an estimation procedure that allows test measurements to be processed, an optimal fit to be made between the measurements identified as being corrupt, and finally a compensation to be applied using these optimal fit estimates. Most often there is a linear (or linearizable) relationship between the error parameters and resultant signal corruptions and a standard least-squares fit is made between the corrupted measurements and arbitrarily scaled calculations of the expected signal corruptions. These expected functions are called regressors, and the fit procedure calculates the extent to which these regressors appear in the raw measurements.  
           [0024]    Unfortunately, no set of regressors fits all the acceleration and rotationally induced error found in a gradiometer system. An important part of improving gradiometer instrument performance is the identification of error sources, estimation and compensation of the error effect found in particular instruments, and perhaps adjustment of the instrument build/setup to reduce the physical effects leading to the errors.  
         SUMMARY  
         [0025]    Embodiments of the invention as discussed below concern the discovery of one such error mechanism, the calculation of the error effect resulting from this (thus allowing for compensation and improved measurement performance), and identification of instrument adjustments for reducing the magnitude of the raw effect of the error  
           [0026]    In one aspect of the invention, a method includes measuring an acceleration along an input axis of an accelerometer mounted to a gradiometer disc, the accelerometer having a coordinate axis that is parallel to a spin axis of the disc, and includes calculating a gravity tensor element as a function of the measured acceleration and a component of the measured acceleration caused by an acceleration along the coordinate axis.  
           [0027]    This technique typically yields a more accurate calculation of the gravitational field by accounting for undesired accelerations picked up by accelerometers having input axes that are not parallel to the gradiometer disc. Furthermore, this technique is applicable to systems that measure the full gravitational tensor as well as to those that measure a subset of the full tensor. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0028]    [0028]FIG. 1 is a view of a conventional gravity gradiometer.  
         [0029]    [0029]FIG. 2 is a view of the conventional gradiometer disc assemblies inside the gravity gradiometer of FIG. 1.  
         [0030]    [0030]FIG. 3 is a plan view of one of the gradiometer disc assemblies of FIG. 2.  
         [0031]    [0031]FIGS. 4A and 4B are side views of respective first and second accelerometers of an accelerometer pair according to an embodiment of the invention.  
         [0032]    [0032]FIG. 5 is a side view of a gradiometer disc assembly that is rotating about a non-spin axis according to an embodiment of the invention. 
     
    
     DESCRIPTION OF THE INVENTION  
       [0033]    The following discussion is presented to enable a person skilled in the art to make and use the invention. Various modifications to the embodiments will be readily apparent to those skilled in the art, and the generic principles herein may be applied to other embodiments and applications without departing from the spirit and scope of the present invention as defined by the appended claims. Thus, the present invention is not intended to be limited to the embodiments shown, but is to be accorded the widest scope consistent with the principles and features disclosed herein.  
         [0034]    [0034]FIGS. 4A and 4B are side views of respective first and second accelerometers  60   a  and  60   b  of an accelerometer pair according to an embodiment of the invention. Referring to FIG. 3, the accelerometers  60   a  and  60   b  are mounted to a disc such as the disc  30  and are spaced π radians apart like the accelerometers  38   a  and  38   b . But unlike the ideally oriented accelerometers  38   a  and  38   b , the accelerometers  60   a  and  60   b  have respective input axes  62   a  and  62   b  that, perhaps through manufacturing irregularities, are not parallel to the disc, and thus may introduce additional acceleration terms into equations (2)-(9). Each accelerometer  60   a  and  60   b  has a coordinate system with an origin  64   a  and  64   b , respectively. Referring to the accelerometer  60   a , the Z A  axis is parallel to the Z S  spin axis of the disc, the Y A  axis is parallel to the disc and is orthogonal to the radius of the disc at the origin  64   a , and the X A  axis is coincident with the radius of the disc that intersects the origin  64   a , and thus is normal to the drawing page at the origin  64   a . Likewise, referring to the accelerometer  60   b , the Z B  axis is parallel to the Z S  spin axis of the disc, the Y B  axis is parallel to the disc and is orthogonal to the radius of the disc at the origin  64   b , and the X B  axis is coincident with the radius of the disc that intersects the origin  64   b , and thus is normal to the drawing page at the origin  64   b.    
         [0035]    Referring to FIG. 4A, the accelerometer  60   a  measures a component of accelerations that occur along the Z A  axis, and thus will add to the acceleration terms in equations (2)-(9). Unless these accelerations are removed from the acceleration measurements, they will introduce errors into the calculation of the gravity tensor elements. Specifically, the input axis  62   a  of the accelerometer  60   a  makes a nonzero angle β a  with the Y A  axis—unlike the input axis  40   a  of the accelerometer  38   a  (FIG. 3), which makes a zero angle (β a =0) with its Y A  axis (not shown in FIG. 3). Therefore, because the input axis  62   a  has a projection along the Z A  axis, the accelerometer  62   a  will measure an acceleration term Aaz of Aa in response to an acceleration AZ A  along the Z A  axis according to the following equation:  
           Aaz=AZ   A  sin β a   (11)  
         [0036]    Therefore, to accurately reflect the effect of axial misalignment, β a , in the calculation of the gravitational field, the term Aaz should be included in the right-hand sides of equations (2)-(5). Similarly, referring to FIG. 4B, the accelerometer  60   b  will measure an acceleration term Abz of Ab in response to an acceleration AZ B  (that is, an acceleration along the Z B  axis at location  64   b ) according to the following equation:  
           Abz=AZ   B  sin(β b )  (12)  
         [0037]    And to accurately reflect the effect of β b , the term Abz should be included in the right-hand side of equation (6).  
         [0038]    Referring to FIGS. 1, 4A, and  4 B and assuming that the accelerometers  60   a  and  60   b  are mounted to the disc  30  in place of the accelerometers  38   a  and  38   b , one cause of accelerations AZ A  and AZ B  along the Z A  and Z B  axes is a nonrotational acceleration along the Z body axis, and thus along the spin axis Z S . For example, the vehicle carrying the gradiometer  10  may be accelerated along the Z axis by a gust of wind. In this case AZ A =AZ B =AZ S    
         [0039]    An established technique for canceling the terms Aaz and Abz introduced into the equations (2)-(9) by such a nonrotational acceleration is to mount the accelerometers  60   a  and  60   b  on the disc  30  such that β b =β a . Because the accelerations Aa and Ab are summed together per equation (7), then Aaz+Abz=AZ S  sin β a +AZ S  sin β b =AZ S  sin β a −AZ S  sin β a =0. And even if one cannot mount the accelerometers  60   a  and  60   b  such that β b  exactly equals −β a , often one can get β b  close enough to −β a  such that Aaz+Abz is negligible and Aaz and Aab can be eliminated from equations (2)-(9). But in general Aaz+Abz is not negligible and the misalignments β a  and β b  are too small to identify using conventional accelerometer calibration techniques. Therefore, one method developed for gradiometer use is to inject a common (i.e. non-rotational) acceleration along the Z S  axis and through examination of the accelerometer summation signal identify the common part of the misalignments. This acceleration can be injected by a calibration machine during pre-shipment calibration of the gradiometer. Alternatively, the gradiometer can self calibrate by using accelerations provided by the vehicle in which it is mounted. In this way the axial misalignment of one arbitrarily selected accelerometer can be adjusted to make the net effect from all accelerometers equal zero. That is sin β a +sin β b +sin β c +sin β d =0, where sin β c  and sin β d  represent the acceleration terms from another pair of accelerometers that are respectively similar to the accelerometers  60   a  and  60   b  but are mounted to the disc  30  in place of the ideal accelerometers  38   c  and  38   d  (FIG. 3). For cases where the ability to calibrate this net misalignment is better than our ability to realign the accelerometers (that is, where sin β a +sin β b +sin β c +sin β d ≠0), one can alter the measurement processing algorithms in processor  20  (FIG. 1) to include compensation for the net effect of Aaz, Abz, Acz and Adz (Acz and Adz being the accelerations from the other pair of accelerometers) in equation (9) and hence improve the resulting measurements. These established practices, although helpful in rejecting common axial acceleration (Z S ), do nothing to identify or reduce the effects of the individual axial misalignments. Therefore, as discussed below, one embodiment of the invention addresses this failure.  
         [0040]    Referring to FIGS. 1, 4A,  4 B, and  5  and again assuming that the accelerometers  60   a  and  60   b  and a corresponding pair of accelerometers are mounted to the disc  30  in place of the accelerometers  38   a ,  38   b ,  38   c , and  38   d , another cause of an acceleration AZ A  along the Z A  axis is a rotational acceleration ((α=dω/dt where ω=rotational rate as discussed above) about the X or Y body axes of the gradiometer  10 . Unfortunately, as discussed below, one cannot reduce or eliminate the acceleration terms introduced by these rotational accelerations by setting β b =−β a .  
         [0041]    [0041]FIG. 5 is an end view of the disc  30  taken along lines A-A of FIG. 3, where the accelerometers  38   a  and  38   b  are replaced by the accelerometers  60   a  and  60   b  of FIGS. 4A and 4B, the accelerometers  38   c  and  38   d  are replaced by accelerometers that are similar to the accelerometers  60   a  and  60   b  and whose input axes make respective angles β c  and β d  with the Z C  and Z D  axes, and the acceleration terms introduced by a rotational acceleration α are included in the equations (2)-(9) according to an embodiment of the invention.  
         [0042]    The normal position of the disc  30  is drawn in solid line. In the normal position when Ωt=0, the X D  and Y D  (normal to drawing page) disc axes are coincident with the X and Y (normal to drawing page) body axes, respectively, and the Z S  spin axis is parallel to the Z body axis.  
         [0043]    When a rotational acceleration occurs, for example a counterclockwise acceleration α y  about the Y body axis, then the disc  30  is accelerated toward a position that is drawn in dashed line. If β a  and β b  have opposite signs, then the acceleration component Aaz(α y ) measured by the accelerometer  60   a  is reinforced by the acceleration component Abz(α y ) measured by the accelerometer  60   b . More specifically, referring to FIG. 4A, because the input axis  62   a  of the accelerometer  60   a  has a projection on the positive Z A  axis, the accelerometer  60   a  measures a positive acceleration Aaz(α y ) in response to the rotational acceleration α y . Similarly, referring to FIG. 4B, because the input axis  62   b  of the accelerometer  60   b  has a projection on the negative Z B  axis, the accelerometer  60   b  measures a positive acceleration Abz(α y ) due to the rotational acceleration α y . Consequently, unlike the terms Aaz and Abz (equations (11) and (12)) introduced by a nonrotational acceleration as discussed above in conjunction with FIGS. 4A and 4B, the term Aaz(α y ) introduced by the rotational acceleration coy tends to be reinforced by, and not cancelled by, the term Abz(α y ) introduced by α y . This is true even if β b =−β a  exactly.  
         [0044]    Referring to FIGS. 3, 4A, and  5 , the acceleration term Aaz(α y ) introduced by α y  is a function of the length of the moment arm  52  and the projection of the input axis  62   a  on the Z A  axis, and is thus given by the following equation:  
           Aaz (α y )=−α y   R  cos Ω t  sin β a   (13)  
         [0045]    And the acceleration term Aaz(α x ) introduced by a rotational acceleration α x  about the X body axis is given by the following equation:  
           Aaz (α x )=+α x   R  sin Ω t  sin β a   (14)  
         [0046]    Similarly, the error acceleration terms for the remaining accelerometer  60   b  and the other pair of accelerometers are given by the following equations:  
           Abz (α y )=+α y   R  cos Ω t  sin β b   (15)  
           Abz (α x )=−α x   R  sin Ω t  sin β b   (16)  
           Acz (α y )=+α y   R  sin Ω t  sin β c   (17)  
           Acz (α x )=+α x   R  cos Ω t  sin β c   (18)  
           Adz (α y )=−α y   R  sin Ω t  sin β d   (19)  
           Adz (α x )=−α x   R  cos Ω t  sin β d   (20)  
         [0047]    A similar analysis can be made for discs lying in or parallel to the X-Z or Y-Z planes.  
         [0048]    By including the above acceleration terms in equation (9) along with measurements from the rotational sensor assembly  18  (FIG. 1), the processor  20  can account for these measurement errors in terms of the axial misalignments of the accelerometers: β a , β b , β c , and β d . Including the terms of equations (14)-(20) in equation (9) and subtracting the ideal result (the right-hand side of equation (9)) leaves the following formulation of errors induced by rotational accelerations:  
         signal errors induced by rotational accelerations=(α x  sin Ω t−α   y  cos Ω t )(sin β a −sin β b )−(α x  cos Ω t+α   y  sin Ω t )(sin β c −sin β d )) R/ 2  (21)  
         [0049]    In a manner analogous to the error correction for non-rotational accelerations described above, this information can be used in several ways to improve the gradient measurement. If the misalignments β a , β b , β c , and β d  have been calibrated, then the resulting errors can be simply calculated and removed by the processor  20 . Conversely, a calibration of these misalignments can be implemented by obtaining an optimal fit between the error gradients obtained during a calibration procedure and the measurements obtained by the rotational sensor assembly  18  (FIG. 1). As discussed above, a rotational calibration acceleration can be injected by a calibration/test machine during pre-shipment calibration of the gradiometer. Alternatively, the gradiometer can self calibrate by using rotational accelerations provided by the vehicle in which it is mounted.  
         [0050]    Furthermore, the technique is similar for discs that lie in the X-Z or Y-Z planes. Other embodiments of the invention are contemplated. For example, the rotational sensor assembly  18  (FIG. 1) may be partially or wholly located on the rotating disc  30 . In this case the resolution of the sensed rotations into the frame of the accelerometers, implied by the sin(ωt) and cos(ωt) terms in equation (21), is unnecessary. Furthermore, equations (11)-(21) can be modified according to known mathematical principles where a disc lies in a plane that is not coincident with or parallel to one of the X-Y, X-Z, or Y-Z body-axis planes. Moreover, one can determine the angles β a  and β b  by conventional techniques such as spinning the disc  30  about the Y D  axis at a known rotational acceleration when the disc is in the Ωt=0 position.