Patent Abstract:
An electromagnetic vector sensor (EMVS) system, having a plurality of EMVS devices consisting of a plurality of loop antenna elements spatiatally orthogonally integrated with and electrically isolated from a plurality of dipole antenna elements, mounted on a rotatably adjustable platform having a true north orientation, including active circuitry residing in antenna housings, and external executing software programs causing the active circuitry in cooperation with the EMVS device and receivers to determine angle of arrival and resolution of incoming wave vectors and polarization of incoming signals and to perform accurate high frequency geolocation signal processing; the programs which perform calibration and antenna element placement determination operations, also cause the system to collect data of known transmitted high frequency skywave signals, and estimate direction of arrival of unknown signals by detecting, resolving and measuring components of an electric field and a magnetic field at a single point.

Full Description:
RELATED APPLICATIONS 
     This U.S. patent application is related to U.S. Provisional Application for Patent No. 61/788,650 for Electromagnetic Vector Sensors (EMVS) Apparatus Method and System, filed on Mar. 15, 2013 (Mar. 15, 2013), having the same inventor or joint inventor as the instant application, and where the afore mentioned provisional application for patent is incorporated by reference and relied upon herein in its entirety. 
     FIELD OF THE INVENTION 
     The present invention relates in general to radio communications and antenna devices which can be used in direction finding applications. More particularly, this invention provides a simple means to integrate a series of dipole elements with a series of loop elements such that electromagnetic interactions between the dipole and loop elements are minimized. 
     BACKGROUND OF THE INVENTION 
     The ability to discriminate polarization states of an incoming wavefront has led to the development of various sensor designs which can be utilized to extract polarization information from an incoming signal. With advances in detection processing techniques, these sensors can also be used to provide increased accuracy in determining the direction-of-arrival of unknown signals. The development of electromagnetic vector sensors, as disclosed herein, enables new processing techniques to be utilized in real life situations. 
     Current direction finding devices, used in the field having mutual orthogonal axial elements, include vector sensors. There have been several multi-element sensing antennas designed and utilized in the field that provide HF radio wave direction finding capabilities. In addition, these multi-element sensing devices utilize various processing algorithms to accurately determine angle of arrival and thus require the multiple elements to be spatially orthogonal to each other, which in turn requires a high degree of isolation between the multiple loop and dipole antenna elements. However, current devices exhibit poor isolation characteristics between the multiple antenna elements (i.e., the dipole and loop elements) and inadequately resolve incoming wave vectors in elevation and azimuth directions, because of the requirement that the multiple elements be spatially and electrically orthogonal. 
     Therefore, the need exists for an electromagnetic vector sensor device which facilitates data collection of known transmitted high frequency skywave signals for purposes of achieving high frequency geolocation signal processing, using electromagnetic vector sensors (EMVS) direction-of-arrival estimation of unknown signals, by measuring three complete components of the electric field and three components of the magnetic field at a single point. 
     Furthermore, the need exists for an electromagnetic vector sensor device having a series of loops and a series of dipoles configured spatially orthogonal to each other in such a manner as to maintain a high degree of isolation between the dipole and loop elements. 
     Further, the need exists for the electromagnetic vector sensor device to utilize various processing algorithms to accurately determine angle of arrival and to be able to clearly resolve, to a high degree of accuracy, incoming wave vectors and polarization of incoming signals, by minimizing interactions between dipole and loop elements. 
     SUMMARY OF THE INVENTION 
     An electromagnetic vector sensor (EMVS) system, as disclosed herein, comprises a plurality of EMVS devices each consisting of a plurality of loop antenna elements spatially and orthogonally integrated with a plurality of dipole antenna elements, while maintaining electrical isolation from the plurality of dipole antenna elements. This EMVS configuration of spatially orthogonally integrated antenna elements is mounted on a rotatably adjustable platform having a true north orientation; further having active circuitry residing in antenna holding housings to couple the RF signals from the loops/dipoles to various receivers. Also having program code and executable instructions in a plurality of computer processors communicatively coupled with and causing the active circuitry in cooperation with the EMVS device to determine angle of arrival and resolution of incoming wave vectors and polarization of incoming signals and to perform accurate high frequency geolocation through signal processing. Furthermore, the program code and executable instructions which perform calibration and antenna element placement determination also cause the system to collect data of known transmitted high frequency skywave signals, and estimate direction of arrival of unknown signals by detecting, resolving and measuring components of an electric field and a magnetic field at a single point. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an overall isometric view of a fully assembled electromagnetic vector sensor  100 . 
         FIG. 2  illustrates a completed electromagnetic vector sensor  100  assembly showing spatially and orthogonally integrated dipole antenna and loop antenna elements. 
         FIG. 3  illustrates a close view of a crossover region of the dipole antenna and loop antenna elements and the mechanical configuration which isolates the dipole and loop antenna elements from each other (i.e., Loop/Dipole Cross Point Insulator  208 ), having a dielectric block as the Loop/Dipole Cross Point Insulator  208  and loop notches  304  shown in the loop antenna blades. 
         FIG. 4  illustrates loop antenna element feed points connected to a coupling section between the loop antenna elements and a loop electronics section, where the loop electronics section houses the loop antenna electronics; these elements are all contained in a portion of a base plate (platform). 
         FIG. 5  illustrates two coupling sections  502  which provide RF signals to the loop antenna circuitry mounted in the loop electronics  402  section, which contains electronics circuitry (i.e., such as active balun circuits  804 ) for the loop antenna elements (i.e., loop elements  204 ); these elements are also housed in portions of the base plate  702  platform. 
         FIG. 6  illustrates a dipole electronics assembly, which consists of an upper holder  602  and a bottom holder  606 ; and dipole electronic circuitry  604  is housed in the dipole electronics assembly. 
         FIG. 7  illustrates a base plate  702  (platform) of the electromagnetic vector sensor  100  apparatus. The base plate  702  serves as a mounting platform in which the loop antenna element(s)  204  are configured. Also, the base plate  702  serves as the platform upon which the electromagnetic vector sensor  100  assembly is mounted. 
         FIG. 8  illustrates loop antenna RF connections (i.e., loop coupling section to electronics  502 ) configured in portions of the base plate platform. In addition active circuitry  840  is also included in  FIG. 8 . 
         FIG. 8  illustrates loop RF connections  800  for the loop element(s)  204 , as contained in a portion of the base plate  702 , where portions of the base plate  702  platform include a loop holder  802 , the loop coupling section to electronics  502  and active circuitry  840 . 
         FIG. 9  illustrates loop antenna active circuitry and feed point(s)  210 , as configured in portions of the base plate platform, where active circuitry  840  can include active loop balun circuit(s)  841  and active loop matching networks  842  (low impedance—see  FIG. 31A  and  FIG. 31B ). 
         FIG. 10  illustrates configuration and orientation of a first loop antenna element, such as loop element  204 , as mounted in the base plate  702 . The angle of the loop pairs (i.e., loop element  204 ) relative to the base plate  702  mounting platform is 54.74 degrees (i.e., the tilt angle of each loop pair is 54.74 degrees from the base plate of the loop holder assembly). 
         FIG. 11  illustrates configuration and orientation of the first loop antenna and a second loop antenna Loop Element(s)  204  of a plurality of antennae, as mounted in the base plate  702 . 
         FIG. 12  illustrates configuration and orientation of the first loop antenna, the second loop antenna and a third loop antenna, as mounted in the base plate  702 . Also, Loop/Dipole Cross Point Insulator(s)  208  are mounted, on the loop antenna elements. 
         FIG. 13  illustrates a dipole antenna support column  1302  vertically positioned in the bottom center of the base plate of the electromagnetic vector sensor  100  apparatus and provides support for the dipole antenna section. 
         FIG. 14  illustrates a bottom holder  606  (also called a bottom or lower dipole cap) section for mounting dipole antenna element(s)  204 . 
         FIG. 15  illustrates balun circuitry  1502  for dipole antenna elements. 
         FIG. 16  illustrates an upper holder  602  section (also called an upper dipole cap) attached to the bottom holder  606  section, where both the upper holder  602  section and the bottom holder  606  section can be configured to hold a plurality of dipole antenna elements mounted in the electromagnetic vector sensor apparatus. 
         FIG. 17  illustrates at least three dipole antenna elements, such as dipole element(s)  202  of a plurality of dipole element(s)  202  mounted in the upper holder  602  section. 
         FIG. 18  illustrates a completed electromagnetic vector apparatus assembled with a plurality of loop and dipole antenna elements. 
         FIG. 19  illustrates the completed electromagnetic vector sensor  100  apparatus assembled and fielded, showing the length of a dipole element  202  being four feet (4′) long and the diameter of a loop element  204  being three feet (3′). 
         FIG. 20  illustrates the completed electromagnetic vector sensor  100  apparatus assembled, fielded, mounted on a three legged platform, and directionally situated having a true north indicator  2002  orientation. 
         FIG. 21  illustrates a close-in view of the completed dipole antenna holder  2102  where the upper holder  602  has been connected to and on top of the bottom holder  606  by use of a collar assembly, and where the completed dipole antenna holder  2102  contains the plurality of dipole antenna elements, such as dipole element(s)  202 , as preferably mounted in the electromagnetic vector sensor  100  apparatus; and contains dipole active circuitry  604 . 
         FIG. 22  illustrates a close-in view of the dipole antenna support column  1302  vertically positioned in the bottom center of the base plate  702  of the electromagnetic vector sensor  100  apparatus, where these portions of the base plate  702  contain active loop circuitry  840 . 
         FIG. 23A  illustrates one way EM propagation. 
         FIG. 23B  illustrates a Range Doppler Plot with range separated ground wave, one-hop O and X, and two hop O and X modes. 
         FIG. 24  illustrates a description of the polarization state as the rotation of the E-field vector in the plane orthogonal to the direction of propagation where polarization is elliptical and is described by an ellipticity angle alpha (α) in radians and orientation angle beta (β) in radians. 
         FIG. 25  illustrates the two coordinate system for a two dimensional array manifold vector with azimuth defined from North to West and elevation defined from zenith to horizon. 
         FIG. 26A  illustrates an isometric view of the completed electromagnetic vector sensor  100  assembly with orthogonal E-field dipoles and H-filed loops labeled. 
         FIG. 26B  illustrates a top-down view of the completed electromagnetic vector sensor  100  assembly. 
         FIG. 26C  illustrates EMVS E-filed dipole coordinate rotations applied about the Z axis, relative to the normal x, y, z axis, at 45 degrees. 
         FIG. 26D  illustrates EMVS E-filed dipole coordinate rotations applied about the Y axis, relative to a varying x, y, z axis, at an Ez angle of 54.73 degrees. 
         FIG. 26E  illustrates E-field rotation representations, upon realignment of E-field dipole antenna elements relative to original x, y, z cartization coordinate system. 
         FIG. 27A  illustrates an estimated Az/El obtained with a single EMVS. 
         FIG. 27B  illustrates an estimated Az/El obtained with a 2D array of three-EMVS. 
         FIG. 27C  illustrates an estimated orientation/ellipticity of a single EMVS. 
         FIG. 27D  illustrates an estimated orientation and ellipticity of three EMVS. 
         FIG. 28A  illustrates estimated Az/El angle of arrival of O-mode signal of interest. 
         FIG. 28B  illustrates signal Hop estimated left circular orientation/ellipticity of O-mode signal of interest. 
         FIG. 28C  illustrates signal Hop estimated right circular orientation/ellipticity of X-mode signal of interest. 
         FIG. 28D  illustrates signal Hop estimated Az/El of X-mode of signal of interest. 
         FIG. 29A  illustrates a plot of estimated slant range, as a function kilometers verses UTC time. 
         FIG. 29B  illustrates a plot of polarization ellipticity, as a function of radians verses UTC time. 
         FIG. 29C  illustrates an azimuth plot, as a function of degrees verses UTC time. 
         FIG. 30  illustrates a plot of elevation, as a function of degrees and UTC time. 
         FIG. 31A  illustrates an overhead view of single active loop matching networks (low impedance). 
         FIG. 31B  illustrates a schematic view of single active loop matching networks (low impedance). 
         FIG. 31C  illustrates an overhead view of 3 complete active dipole matching networks (high impedance). 
         FIG. 31D  illustrates a schematic view of 3 active dipole matching networks (low impedance). 
         FIG. 32  illustrates In-Situ Calibration Processing flow chart. 
         FIG. 33  illustrates Quick Look Processing flow chart. 
         FIG. 34  illustrates active circuitry  840  and a network environment of an EMVS system. 
     
    
    
     DETAILED DESCRIPTION 
     Preferred exemplary embodiments of the present invention are now described with reference to the figures, in which like reference numerals are generally used to indicate identical or functionally similar elements. While specific details of the preferred exemplary embodiments are discussed, it should be understood that this is done for illustrative purposes only. A person skilled in the relevant art will recognize that other configurations and arrangements can be used without departing from the spirit and scope of the preferred exemplary embodiments. It will also be apparent to a person skilled in the relevant art that the exemplary embodiments can also be employed in other applications. Further, the terms “a”, “an”, “first”, “second” and “third” etc. used herein do not denote limitations of quantity, but rather denote the presence of one or more of the referenced items(s). 
     Referring to  FIG. 1 ,  FIG. 2 , and  FIG. 3 , it can be seen that  FIG. 1  illustrates an isometric view of an overall view of a completed electromagnetic vector sensor  100  assembly.  FIG. 2  illustrates the completed electromagnetic vector sensor  100  assembly showing dipole antenna and loop antenna elements, including dipole elements  202 , loop elements  204 , loop/dipole cross point insulator(s)  208 , loop feed point(s)  210 , and dipole feed point(s)  212 .  FIG. 3  illustrates a close view of a crossover region of the dipole antenna and loop antenna elements and the mechanical configuration which provides orthogonal integration of the dipole elements  202  and the loop elements  204 , and which isolates the dipole and loop antenna elements from each other (i.e., Loop/Dipole Cross Point Insulator  208 , which includes a dielectric block as the Loop/Dipole Cross Point Insulator  208 . Maintaining orthogonal integration of the antenna elements and maintaining isolation of the antenna elements are essential aspects of the exemplary embodiments. 
     Again referring to  FIG. 3 , the Loop/Dipole Cross Point Insulator  208  illustrated in  FIG. 3  consists of a square piece of dielectric material which is used to support the dipole element(s)  202  and loop element(s)  204  sections. This piece of dielectric material (i.e., the dielectric material used as the Loop/Dipole Cross Point Insulator  208 ), which is used to support the dipole element(s)  202  and the lop element(s)  204 , comprises the main focus of the instant invention. Loop antenna elements i.e., loop element(s)  204  consist of double and/or twin blade like circular sections of 50 mil thick aluminum. The blade portions are 2 inches in width. As illustrated in  FIG. 3 , the loop element(s)  204  are designed such that a pair of the blades forming loop element(s)  204  can be collocated side-by-side, such that the blades are isolated from each other. The distance between the loop blades is 0.95 inches apart. The Loop/Dipole Cross Point Insulator  208  provides the isolation, decoupling and mechanical support between the loop blades, as well as provides the structural layout and symmetry of the device. To maintain the decoupling of the double loop element(s)  204  from each other, a series of notches are cut into the loop elements, such that when the loops are integrated together the loops are not in electrical contact, while having sufficient mechanical support between the loop element(s)  204  and the dipole element(s)  202 . Also, the Loop/Dipole Cross Point Insulator  208  provides the orthognallity between the loop element(s)  204  and the dipole element(s)  202 , where the dipole element(s)  202  are inserted through the center of the dielectric material of the Loop/Dipole Cross Point Insulator  208  (see  FIG. 3 ). Thus, the main advantages of this design, includes the ability to decouple both the dipole element(s)  202  from the loop element(s)  204  by using flat loop element(s)  204  in which small cut-outs and/or notches are used, such that when all of the loops are integrated together, the loops are not electrically touching each other or touching the dipoles; these design techniques provide the ability to maintain symmetry around the dipole element and maintain decoupling between the dipole element(s)  202  and the loop element(s)  204 . 
     Referring to  FIG. 2 ,  FIG. 11 ,  FIG. 12  and  FIG. 18 , integration of the dipole element(s)  202  with the loop element(s)  204  having dual loop design (also known as “loop pair”) provides increased operational bandwidth (BW), because orthogonallity of the antenna elements is achieved and maintained; then stable isolation between both the dipole element(s)  202  and loop element(s)  204  is maintained. Other advantages of exemplary embodiments disclosed herein include: (1) a smaller footprint than conventional array antenna sensors, (2) the ability to provide azimuth and elevation as well as polarization estimates, and (3) decoupling of antenna array elements, associated with dipole element(s)  202  and loop element(s)  204  integration. In addition, multiple loop element(s)  204  can be not only integrated with dipole element(s)  202 , but can also be integrated with a plurality of loop element(s)  204  (see  FIG. 11 ,  FIG. 12  and  FIG. 18 ). 
     Referring to  FIG. 1 .  FIG. 2 ,  FIG. 8 ,  FIG. 10 ,  FIG. 18  and  FIG. 19 , in exemplary embodiments, a three element, six axis electromagnetic vector sensor  100  array assembly has operational characteristics which include an operational frequency range from about 3 MHz up to about 15 MHz, but ideally from about 3 MHz up to about 10 MHz with an optimum NF (noise figure) at 7 MHz and estimates direction of arrival and polarization for unknown signals. And, where, each loop element(s)  204  having dual loop design (also known as “loop pair”) is positioned with a tilt angle of each loop pair is about 54.74 degrees from the base of the loop holder  802  assembly. 
     Referring to  FIG. 9 ,  FIG. 31A ,  FIG. 31B ,  FIG. 31C ,  FIG. 31D  and  FIG. 34 , active matching networks include low impedance matching networks, as well as high impedance matching networks and utilize high third-order intercept point (IP3) or (TOI) characteristics which provide a measure for nonlinear systems and devices and such IP3 characteristics concomitantly reduce the effects of interfering signals. The active circuitry  840  includes active loop balun circuit(s)  841 , active loop matching networks  842  (low impedance—see  FIG. 31A  and  FIG. 31B ) or active dipole networks  843  (high impedance—see  FIG. 31C  and  FIG. 31D ). The system utilizes an algorithm unit  1230  having algorithms A 1 , A 2 , A 3  . . . An, where A 1  determines angle of arrival, A 2  estimates direction of arrival of unknown signals, A 3  determines polarization and computer program code (for process and/or method  2000  (see  FIG. 34 )) executed on computer processor(s)  1206 , having memory  1208  residing in the computer processor (s)  1206 , where the algorithms and code are required to discriminate polarization states of an incoming wavefront, provide increased accuracy in determining the direction of arrival of unknown signals, facilitate data collection of known transmitted high frequency skywave signals for achieving high frequency geolocation signal processing process and/or method  2000 , to determining angle of arrival, and resolve incoming wave vectors (in elevation and azimuth directions) and polarization by measuring three complete components of the electric field and three components of the magnetic field of incoming signals, at a single point, by minimizing interactions between dipole and loop elements. 
     Referring to  FIG. 34 , further the external equipment consists of a display  1202  having a user interface  1204 , memory  1208  having a dynamic repository  1210  having data repositories R 90  through R 94  up to Rn, where R 90  can contain known skywave data  1211 , R 91  can contain electric field components data  1212 , R 92  can contain magnetic field components data  1214 , R 93  can contain unknown signal data  1216 , R 94  can contain elevation data  1218  and up to Rn, which can contain azimuth data  1220 , all callable and executable by program code instructions (such as instructions from a method  2000  of geolocation processing code, which can reside in program unit  1240 . Also, the external system may contain network interface  1270  modules, memory controller  1260  modules, I/O controller  1250  modules output devices  1254 , input devices  1252 , and can be connected in a network  1272  environment. 
     The completed electromagnetic vector sensor  100  can include at least three active loop matching networks  842  and at least three active dipole matching networks  843 . For Active Network Calibration, each of the 3 dipole antenna (i.e., dipole element(s)  202 , the bottom dipole end of each dipole element(s)  202  is physically positioned toward the center calibration whip using a counterclockwise 30 degree offset rotation, and frequency sweeps performed between 3 MHz to 15 MHz. The E-field dipole antenna, i.e., dipole element(s)  202  are then realigned to true north. Final antenna calibration is facilitated using a high fidelity model (based on method of moments) to determine antenna patterns for field site configuration. The high fidelity model includes active load matching and associated cables, for measurements collected. Because of active network variations/drift, each set of loop element(s)  204  and dipole element(s) requires both phase and amplitude alignment (i.e., matching calibration) to adjust for cable phase/amplitude variations and system receiver phase variations. 
     Referring to  FIG. 32 , regarding In-Situ Calibration processing, radar data waveform patterns are used to select a known 18×1 directional signal; then directional weighting using array manifold parameters of known directional transmit (tx) location, where
 
 w=Z   c   /a   0   (1)
 
     where w represents directional weighting; 
     where Z C  represents a known 18×1 directional signal; and 
     where a 0  represents an array manifold for a given theta (θ). 
     An 18×1 directional signal of interest is identified, as designated by Z i ; then an array response for the signal of interest is normalized by calculated weights, where
 
 Z′   i   =Z   i   /w   (2)
 
     where Z i  represents an 18×1 directional signal of interest; 
     where w represents directional weighting; and 
     where Z′ i  represents a normalized array response for the signal of interest by calculated weights. 
     Assuming a right circular (RC) polarization array manifold (a rc ) and a left circular (LC) polarization array manifold (a lc ) estimation of the electromagnetic vector sensor Beamform, a refined Joint azimuth, elevation estimate is obtained. 
     Referring to  FIG. 33  (also see  FIG. 24 ), quick look mode processing requires acquiring and inputting a plurality of signal raw data for baseband processing. After range and Doppler processing in-situ known local waveform patterns are used to obtain calibration factors that are applied to Select Training O-mode and/or X-mode (O/X) transmissions. Whereby, estimates of azimuth and elevation are obtained. 
     Referring to  FIG. 24 ,  FIG. 25 , and  FIG. 33 , Beamforming with a single electromagnetic vector signals is accomplished by defining a received signal y(t) as composed of signal and noise:
 
 y ( t )= a (Θ) s ( t )+ e ( t )  (3),
 
     where S(t) is the complex signal envelope and a(θ) is the array manifold defined for:
 
θ=[φ,ψ,α,β] T   (4),
 
     for a non-rotated x, y, z-axis, and
 
 a (Θ)= B (φ,θ) Q (β) h (α)  (5),
 
     where B(φ, ψ) is the steering matrix for azimuth φ∈[−π,π] and 
     elevation θ∈[−π/2, π/2], and 
     
       
         
           
             
               
                 
                   
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     with rotation matrix Q(β), and h(β) representing the unit-norm vector for ellipticity of polarization 
     
       
         
           
             
               
                 
                   
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     where Q(β)∈[0,π] is the polarization orientation angle and 
             α   ∈     [       -     π   4       ,     π   4       ]           
is the polarization ellipticity angle (see A. Nehorai, K. S. Ho, and T. T. G. Tan, “Minimum-Noise-Variance Beamformer with an Electromagnetic Vector Sensor,” IEEE Trans. Signal Processing, vol. 47, pp. 601-618 March 1999).
 
     Again referring to  FIG. 25  and concerning a 2D EMVS Array Manifold Vector, an EMVS positioning and/or placement can be setup in spatial 2D triangular formation with spatial positions defined by Xk, where k is defined as: 
     
       
         
           
             
               
                 
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     For 3 element 2D array, X is a 3×3 matrix representing the relative sensor positions and/or placement (see  FIG. 25 ). 
     Conventional spatial beamforming weights are then obtained as: 
     
       
         
           
             
               
                 
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                     v 
                   
                   = 
                   
                     
                       ⅇ 
                       
                         j 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           λ 
                         
                         ⁢ 
                         Xk 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     For a Single EMVS Manifold Vector, the constructed EMVS is defined with a rotation relative to the normal x, y, z-axis. Referring to  FIG. 26C , the rotation about the z-axis is defined for angle θ Z  as: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       z 
                     
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         z 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               z 
                             
                           
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               z 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               z 
                             
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               z 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Referring to  FIG. 26D  and  FIG. 26E , the rotation about the y-axis is defined for angle θ y  as: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       y 
                     
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         y 
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               y 
                             
                           
                         
                         
                           0 
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               y 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               y 
                             
                           
                         
                         
                           0 
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               y 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The 6×6 rotation matrix for the E1, E2, E3 dipole and H1, H2, H3 loop sensors is:
 
 R (θ y ,θ z )= I             ( R   z (θ z ) R   y (θ y ))  (15)

     The 6×1 array manifold vector is:
 
 a (θ y ,θ z ,φ,θ,β,α)= R (θ y ,θ z ) B (φ,θ) Q (β) h (α)  (16).
 
     While the exemplary embodiments have been particularly shown and described with reference to preferred embodiments thereof, it will be understood, by those skilled in the art that the preferred embodiments have been presented by way of example only, and not limitation; furthermore, various changes in form and details can be made therein without departing from the spirit and scope of the invention. Thus, the breadth and scope of the present exemplary embodiments should not be limited by any of the above described preferred exemplary embodiments, but should be defined only in accordance with the following claim and/or claims and their equivalents. Any and/or all references cited herein are each entirely incorporated by reference herein, including all data, tables, figures, and text presented in the cited references. Also, it is to be understood that the phraseology or terminology herein is for the purpose of description and not of limitation, such that the terminology or phraseology of the present specification is to be interpreted by the skilled artisan in light of the teachings and guidance presented herein, in combination with the knowledge of one of ordinary skill in the art. The foregoing description of the specific embodiments will so fully reveal the general nature of the invention that others can, by applying knowledge within the art, readily modify and/or adapt for various applications such specific embodiments, without undue experimentation, without departing from the general concept of the exemplary embodiments. Therefore, such adaptations and modifications are intended to be within the meaning and range of equivalents of the disclosed embodiments, based on the teaching and guidance presented herein.

Technology Classification (CPC): 6