Abstract:
An instrument for determining the second and/or third-order components of the gravity tensor includes a group of six accelerometers arranged at an equal radius from a spin axis and positioned at 60 degree intervals about the spin axis with the sensing axis of each accelerometer aligned tangentially to the circle subscribed by the accelerometers as they rotate about the spin axis. A gyro-stabilized platform maintains the accelerometer arrangement at a preferred alignment relative to the local gravity vector. As the accelerometers orbit about the spin axis, each accelerometer outputs a sinusoidal signal that is offset by 60 degrees from its immediately adjacent leading and trailing accelerometers with the outputs thereof processed to provide the second-order component and the third-order tensor component. In another arrangement, a group of eight accelerometers arranged at an equal radius from a spin axis and positioned at 45 degree intervals about the spin axis can provide second, third, and fourth-order tensor components. The higher-order tensor components are of use in “de-cluttering” the lower-order tensor components.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims the benefit of U.S. Provisional Patent Application No. 61/178,665 filed by the inventor herein on May 15, 2009 and in common assignment herewith. 
     
    
     BACKGROUND 
       [0002]    The present invention relates to an improved gravity gradiometer instrument (GGI) and, more particularly, to gravity gradiometer instruments that are responsive to one or more higher-order gravity-gradient characteristics. 
         [0003]    Various instruments have been developed to measure gravity gradients, these instruments include gradiometers that are designed to measure the differential curvature or ellipticity of gravity equipotential surfaces, the rate of change of the increase of gravity in the horizontal direction, and/or the rate of increase of gravity in the vertical direction. 
         [0004]    Gradiometers have been used as navigational aids in sub-surface sea-going vessels, gravity field surveys in which one or more gradiometers are carried in a vehicle (i.e., aircraft, surface or sub-surface sea-going vessel, land vehicle, etc.) and, more specifically, as an aid in identifying the boundaries of sub-surface liquid hydrocarbon deposits. 
         [0005]    A representative or example gradiometer is shown in  FIG. 5  and is sold by the Lockheed Martin Corporation (Niagara Falls N.Y. USA) and is described in more detail in U.S. Pat. No. 5,357,802 issued Oct. 25, 1994 to Hofineyer and Affleck and entitled “Rotating Accelerometer Gradiometer,” the disclosure of which is incorporated herein by reference. 
         [0006]    As shown in  FIG. 5 , the exemplary gravity gradiometer instrument GGI includes eight accelerometers  100  mounted at a common radius and equi-spaced about the periphery of a rotor assembly  102  that is rotated at a constant and controlled angular velocity about a spin axis SA x . The rotor assembly  102  includes the rotor  104  carried on a support shaft  106  for rotation therewith. The rotor assembly  102  is rotatably mounted in ball bearings  108  and, in turn, carried in a flex-mount assembly  110  and carried in a gyro-stabilized gimbal mount. Processing electronics  112  are mounted on the rotor  104  adjacent each accelerometer  100  for processing the respective accelerometer output signal. An inner housing  114  contains the rotor assembly  102  and is designed to rotate with the rotor assembly  102 . An outer housing  116  contains the interior components and includes one or more heaters  118  designed to operate the instrument at some controlled temperature above ambient and also includes a magnetic-field shield  120 . A slip-ring assembly  122  at the upper end of the mounting shaft  106  provides the electrical/signal interface with the rotor assembly  102  and the active devices thereon. A shaft encoder  124  at the lower end of the mounting shaft  106  cooperates with an encoder pick-off  126  to provide rotary position information. The output of the encoder pick-off  126  is provided to a soft/firmware-controlled computer or microcomputer and speed controller, which, in turn, controls a drive motor  128  at the upper end of the unit to provide a controlled rotary velocity. 
         [0007]    The gradiometer includes an internal linear servo controlled actuator that imparts a 2 Hz sinusoidal acceleration to each accelerometer pair to enable biasing and compensation of various errors including the g 2  rectification error. In addition, the gravity gradiometer GGI is mounted on an external vibration isolation system that assists in attenuating higher frequency vibration. 
         [0008]    Each accelerometer  100  is of the force-rebalance type and provides a substantially sinusoidally varying analog output that is a function of the acceleration experienced by each accelerometer as the accelerometer orbits the spin axis SA. For a gradiometer having its spin axis SA aligned along the field lines in an ideally uniform and unperturbed gravity field, each accelerometer experiences the same acceleration forces as its proceeds along its orbital path. However, when the local gravity field is perturbed by the presence of one or more masses and/or the spin axis SA is tilted relative to the local vertical field lines, each accelerometer will experience different accelerations throughout its respective orbit about the spin axis SA. 
         [0009]    Gradiometers have typically been positioned with their spin axis vertical (VSA—Vertical Spin Axis), their spin axis horizontal (HSA—Horizontal Spin Axis), and in a three-GGI cluster at an ‘umbrella’ angle in which the spin axis is tilted 35 degrees from the local vertical, though any orientation is possible. The quantitative output of each rotating accelerometer pair, when summed and differenced, can be used to provide information related to the local gravity gradient field. 
         [0010]    Gradiometers measure the second-order variation of gravitational potential and currently do not directly measure or otherwise determine third, fourth, or higher-order effects. Knowledge of the second-order effects can be used, for example, in verifying the veracity of the primary gradient measurement in submarine navigation systems, especially in those cases were the submarine is navigating along an iso-gradient line (wherein the second-order data would be zero, i.e., the partial derivative of the first-order gradient in the direction of movement would be zero). Additionally, knowledge of the second-order characteristics can be useful for edge detection of buried objects or bodies and fluid boundary detection, e.g., in resource exploration. Third-order gravity tensor components provide a natural filtering or upward continuation that may be useful for profiling objects close in proximity to the measuring gradiometer device. This is advantageous because background objects now only influence output data as inverse distance to the fourth power. Likewise, fourth-order gravity components filter even more background “clutter” by signal naturally rolling off proportional to inverse distance to the fifth power. 
       SUMMARY 
       [0011]    A gradiometer instrument for determining the second and/or third-order components of the gravity tensor includes, in a preferred embodiment, a group of six accelerometers A 1 , A 2 , A 3 , A 4 , A 5 , and A 6  arranged at an equal radius from a spin axis and positioned at an equiangular spacing about the spin axis with the sensing axis of each accelerometer aligned tangentially to the circle subscribed by the accelerometers as they rotate about the spin axis. A gyro-stabilized gimbal-type platform maintains the accelerometer arrangement at a preferred alignment relative to the local gravity vector. In a steady field and without moving the instrument and as the accelerometers orbit the spin axis, an accelerometer lagging another by an angular offset will produce the identical signal but with a phase shift equivalent to that offset. A non-steady field and/or a moving gradiometer instrument shows this effect, but the equality is not strictly upheld. As the accelerometers orbit about the spin axis, each accelerometer outputs a generally sinusoidal signal that is offset by 60 degrees from its immediately adjacent leading and trailing accelerometers with the outputs thereof processed to provide the second-order component in accordance with 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     A 
                     1 
                   
                   + 
                   
                     A 
                     5 
                   
                 
                 ) 
               
               - 
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       A 
                       2 
                     
                     + 
                     
                       A 
                       3 
                     
                     + 
                     
                       A 
                       4 
                     
                     + 
                     
                       A 
                       6 
                     
                   
                   ) 
                 
               
             
             = 
             
               3 
                
               
                 R 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             W 
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the third-order component in accordance with 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         + 
                         
                           A 
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         A 
                         3 
                       
                     
                   
                 
                 ) 
               
               - 
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           4 
                         
                         + 
                         
                           A 
                           5 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         A 
                         6 
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 3 
                 4 
               
                
               
                 
                   R 
                   2 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         x 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                         
                                           ( 
                                           
                                             W 
                                             xy 
                                           
                                           ) 
                                         
                                         y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             3 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       y 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           W 
                                           xy 
                                         
                                         ) 
                                       
                                       x 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           3 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0012]    For a gradiometer instrument having eight equi-angular spaced accelerometers A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7  and A 8 , simultaneous isolation and extraction of second, third, and fourth-order gravity tensor components is possible. The second-order components extracted at twice the spin rate are obtained by summing accelerometer outputs pursuant to 
         [0000]    
       
         
           
             
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           + 
                           
                             A 
                             3 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             5 
                           
                           + 
                           
                             A 
                             7 
                           
                         
                       
                     
                   
                   ) 
                 
               
               - 
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             2 
                           
                           + 
                           
                             A 
                             4 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             6 
                           
                           + 
                           
                             A 
                             8 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               4 
                
               
                   
               
                
               
                 R 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             W 
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0013]    Simultaneously, the third-order gravity tensor components are extracted at three times the disk spin rate by summing accelerometer outputs pursuant to 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     A 
                     1 
                   
                   - 
                   
                     A 
                     3 
                   
                 
                 ) 
               
               + 
               
                 
                   1 
                   
                     2 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             6 
                           
                           + 
                           
                             A 
                             7 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             5 
                           
                           - 
                           
                             A 
                             8 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 1 
                 2 
               
                
               
                 
                   R 
                   2 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         x 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                         
                                           ( 
                                           
                                             W 
                                             xy 
                                           
                                           ) 
                                         
                                         y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             3 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       y 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           W 
                                           xy 
                                         
                                         ) 
                                       
                                       x 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           3 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0014]    Additionally and simultaneously, the fourth-order gravity tensor components are extracted at four times the spin rate by summing accelerometer outputs pursuant to 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         + 
                         
                           A 
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           A 
                           3 
                         
                         + 
                         
                           A 
                           4 
                         
                       
                     
                   
                 
                 ) 
               
               - 
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           5 
                         
                         + 
                         
                           A 
                           6 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           A 
                           7 
                         
                         + 
                         
                           A 
                           8 
                         
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 1 
                 3 
               
                
               
                 
                   R 
                   3 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         yy 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       xx 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             4 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           2 
                            
                           
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0015]    More generally, given 2N equi-angular spaced accelerometers, the N th  order tensor components in the plane of the accelerometers can be isolated (at integer multiples of the spin rate) and extracted. The tensor components available are not exhaustive of all the possible components that can be defined in that plane. 
         [0016]    For exploration applications, e.g., searching for hydrocarbons or minerals, generally the second-order tensor components are most useful due to the physical size or baseline of the object sought and its correspondingly long-wavelength signal induced and measured in the gravity data. In these applications, any higher-order tensor effects induced by near-field bodies is detrimental to the sought signal and must be removed by Post Mission Compensation (PMC) techniques. To date, these PMC techniques have relied exclusively on forward modeling signal effects induced by relative motion of a GGI in its hosting stable platform (i.e., gimbal arrangement) and vehicle; this can be a costly and time consuming calibration procedure. 
         [0017]    Direct measurement of higher-order tensor components as disclosed herein can alleviate this costly and time consuming calibration procedure. In this approach, where the sought information resides in the low-order tensor data, the higher-order tensor “noise” or “clutter” is directly removed, i.e., subtracted, from the desired low-order tensor signal. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
         [0018]      FIG. 1  is a perspective view of a six accelerometer mounting for obtaining second and third-order and higher-order tensor data, for example, in or as part of a gravity gradiometer instrument of the type shown in  FIG. 5 ; 
           [0019]      FIG. 2  is a vector diagram of the six accelerometer arrangement shown in  FIG. 1  with the equations associated with each accelerometer indicative of the respective output thereof; 
           [0020]      FIG. 2A  is a functional block diagram showing the functional processing for isolating and extracting second-order gravity tensor components at 2Ω with a six-accelerometer gradiometer instrument; 
           [0021]      FIG. 2B  is a functional block diagram showing the functional processing for isolating and extracting third-order gravity tensor components at 3Ω with a six-accelerometer gradiometer instrument; 
           [0022]      FIG. 3  is a vector diagram of an eight accelerometer arrangement with the equations associated with each accelerometer indicative of the respective output thereof; 
           [0023]      FIG. 3A  is a functional block diagram showing the functional processing for isolating and extracting second-order gravity tensor components at 2Ω with an eight accelerometer gradiometer instrument; 
           [0024]      FIG. 3B  is a functional block diagram showing the functional processing for isolating and extracting third-order gravity tensor components at 3Ω with an eight accelerometer gradiometer instrument; 
           [0025]      FIG. 3C  is a functional block diagram showing the functional processing for isolating and extracting fourth-order gravity tensor components at 4Ω with an eight accelerometer gradiometer instrument; 
           [0026]      FIG. 4  is a functional block diagram of a representative arrangement for compensating a GGI instrument output for fourth-order tensor components in order to de-noise the GGI instrument output; and 
           [0027]      FIG. 5  is an isometric view of an exemplary prior art gravity gradiometer suitable for incorporation therein of the above-described structure with selected portions thereof broken away for reasons of clarity. 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0028]    An accelerometer arrangement in accordance with the preferred arrangement is shown in schematic form in  FIG. 1  and is designated therein by the reference character  10 . As shown, the accelerometer arrangement  10  include six accelerometers A 1 , A 2 , A 3 , A 4 , A 5 , and A 6  spaced at a common radius from an axis of rotation A z  and positioned at 60 degree intervals on a disc-like or discoidal mounting structure  12 . As can be appreciated, each accelerometer orbits the axis of rotation as the discoidal mounting structure rotates about the axis A z . The exterior configuration of the accelerometers shown in  FIG. 1  and their mounting structure  12  are exemplary. Each accelerometer A n  may take the form of force re-balance accelerometers of the type shown in  FIG. 5  and described in the above incorporated patent and manufactured by the Lockheed Martin Corp. under the part number Model VII-g designation. In general, any accelerometer configuration with performance characteristics (e.g., stability, accuracy, resolution, scale factor balance) sufficient relative to sought gradient extraction accuracy may be so arranged. Since accelerometers of type described include “capture” loops that maintain the pendulum-supported proof mass at a selected position, the electronics associated with the capture loop or loops can be integrated into the accelerometer or can be external to the accelerometer. The representation of  FIG. 1  can be incorporated within the stabilized gimbal system of the instrument shown in  FIG. 5 . 
         [0029]      FIG. 2  is an idealized acceleration vector diagram of the representative physical arrangement of  FIG. 1  in which the sensitive axis of each accelerometer is shown as an arrow that is aligned tangentially to the common radius R circle with the X G  direction and the Y G  direction within the gravity field shown with the accelerometers A n  rotating about the axis A z  at some steady spin rate Ω. If the disk is stationary, each accelerometer provides a harmonically varying analog output that is a function of the acceleration experienced by each accelerometer as the accelerometer orbits the spin axis A z . Since the various accelerometers are angularly offset from one another by 60 degrees, the respective harmonic outputs will likewise be displaced 60 degree in phase, e.g., the output of the accelerometer A 1  will lead that of accelerometer A 6  by 60 degrees while trailing that of accelerometer A 4  by 60 degrees. 
         [0030]    Gradients are derived from the scalar gravitational potential field of the earth defined at a point (x,y,z) above ground in an earth-fixed reference frame as 
         [0000]    
       
         
           
             
               U 
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 - 
                 G 
               
                
               
                 
                   ∫ 
                   ξ 
                 
                  
                 
                   
                     ∫ 
                     η 
                   
                    
                   
                     
                       ∫ 
                       ζ 
                     
                      
                     
                       
                         
                           ρ 
                            
                           
                             ( 
                             
                               ξ 
                               , 
                               η 
                               , 
                               ζ 
                             
                             ) 
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         x 
                                         - 
                                         ξ 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       ( 
                                       
                                         y 
                                         - 
                                         η 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         z 
                                         - 
                                         ζ 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                        
                       
                          
                         ξ 
                       
                        
                       
                          
                         η 
                       
                        
                       
                          
                         ζ 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    where G is the universal gravitational constant (6.6720E-11 m 3 /(kg·s 2 ), and p is density of an infinitesimal particle (of the earth) located at coordinates (ξ,η,ζ). The triple integral is computed over all of the earth&#39;s mass and location. 
         [0031]    The vector of gravitational force, or merely the gravity vector, comprises the three first-order spatial derivatives of the scalar potential in each of respective x-, y-, and z-directions. These three components comprise a vector field that describes how the scalar potential varies spatially, and are written 
         [0000]    
       
         
           
             
               
                 g 
                 x 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
                
               
                 U 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 g 
                 y 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   y 
                 
               
                
               
                 U 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 g 
                 z 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
                
               
                 U 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
     
         [0032]    The second-order derivatives of the scalar potential are referred to as the second-order tensor components of gravity and comprise a tensor field. The components are identical to the first derivative of the gravity vector above, and thus describe how the gravity vector components vary spatially, i.e., describes how each of the three gravity vector components varies in each of the three coordinate directions. The total number of second-order tensor components is nine, but by virtue of the conservative nature of the scalar potential field only five of these are independent and the order of differentiation is not relevant. The second-order tensor components are written 
         [0000]    
       
         
           
             
               
                 W 
                 xx 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
                
               
                 
                   g 
                   x 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 xy 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   y 
                 
               
                
               
                 
                   g 
                   x 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 xz 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
                
               
                 
                   g 
                   x 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 yx 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 W 
                 xy 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 W 
                 yy 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   y 
                 
               
                
               
                 
                   g 
                   y 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 yz 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   z 
                 
               
                
               
                 
                   g 
                   y 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 W 
                 zx 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 W 
                 xz 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 W 
                 zy 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 W 
                 yz 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 W 
                 zz 
               
                
               
                 ( 
                 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   ∂ 
                   
                     ∂ 
                     z 
                   
                 
                  
                 
                   
                     g 
                     z 
                   
                    
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               = 
               
                 - 
                 
                   ( 
                   
                     
                       
                         W 
                         x 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                     + 
                     
                       
                         W 
                         yy 
                       
                        
                       
                         ( 
                         
                           x 
                           , 
                           y 
                           , 
                           z 
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0033]    The last equation reflects the fact that everywhere the trace of the second-order tensor is zero (i.e., which satisfies the Laplacian constraint) 
         [0000]        W   xx ( x,y,z )+ W   yy ( x,y,z )+ W   zz ( x,y,z )=0 
         [0034]    Higher-order partial derivatives of the scalar potential are likewise equivalent to respective derivatives of the second-order tensor components. For example, third-order gravity tensor components are equivalent to first-order partials of second-order components, and, likewise, fourth-order gravity tensor components are equivalent to second-order derivatives of second-order components and first-order derivatives of third-order components. 
         [0035]    The convention used here for defining higher-order tensor components is the latter approach described above, namely, first-order spatial differentiations of next-highest order tensor components, i.e., recursively. Thus, third-order tensor components are written 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     W 
                     xx 
                   
                   - 
                   
                     W 
                     yy 
                   
                 
                 ) 
               
               x 
             
             = 
             
               
                 
                   ∂ 
                   
                     ∂ 
                     x 
                   
                 
                  
                 
                   
                     W 
                     xx 
                   
                    
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ∂ 
                   
                     ∂ 
                     x 
                   
                 
                  
                 
                   
                     W 
                     yy 
                   
                    
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   W 
                   xy 
                 
                 ) 
               
               x 
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
                
               
                 
                   W 
                   xy 
                 
                  
                 
                   ( 
                   
                     x 
                     , 
                     y 
                     , 
                     z 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     W 
                     xx 
                   
                   - 
                   
                     W 
                     yy 
                   
                 
                 ) 
               
               y 
             
             = 
             
               
                 
                   ∂ 
                   
                     ∂ 
                     y 
                   
                 
                  
                 
                   
                     W 
                     xx 
                   
                    
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
               - 
               
                 
                   ∂ 
                   
                     ∂ 
                     y 
                   
                 
                  
                 
                   
                     W 
                     yy 
                   
                    
                   
                     ( 
                     
                       x 
                       , 
                       y 
                       , 
                       z 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    and fourth-order components are written 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     W 
                     xx 
                   
                   - 
                   
                     W 
                     yy 
                   
                 
                 ) 
               
               xx 
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
                
               
                 
                   ( 
                   
                     
                       W 
                       xx 
                     
                     - 
                     
                       W 
                       yy 
                     
                   
                   ) 
                 
                 x 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     W 
                     xx 
                   
                   - 
                   
                     W 
                     yy 
                   
                 
                 ) 
               
               xy 
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   y 
                 
               
                
               
                 
                   ( 
                   
                     
                       W 
                       xx 
                     
                     - 
                     
                       W 
                       yy 
                     
                   
                   ) 
                 
                 x 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     W 
                     xx 
                   
                   - 
                   
                     W 
                     yy 
                   
                 
                 ) 
               
               yy 
             
             = 
             
               
                 ∂ 
                 
                   ∂ 
                   y 
                 
               
                
               
                 
                   ( 
                   
                     
                       W 
                       xx 
                     
                     - 
                     
                       W 
                       yy 
                     
                   
                   ) 
                 
                 y 
               
             
           
         
       
     
         [0000]    where only the relevant components in the plane of the gradiometer disk are shown. 
         [0036]    In a manner consistent with the explanation in the preceding description, the second-order tensor components (in the XY plane) can be obtained as follows: 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     A 
                     1 
                   
                   + 
                   
                     A 
                     5 
                   
                 
                 ) 
               
               - 
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       A 
                       2 
                     
                     + 
                     
                       A 
                       3 
                     
                     + 
                     
                       A 
                       4 
                     
                     + 
                     
                       A 
                       6 
                     
                   
                   ) 
                 
               
             
             = 
             
               3 
                
               
                 R 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             W 
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0037]    Additionally and also in a manner consistent with the explanation in the preceding description, the third-order tensor components (in the XY plane) can be obtained as follows: 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         + 
                         
                           A 
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         A 
                         3 
                       
                     
                   
                 
                 ) 
               
               - 
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           4 
                         
                         + 
                         
                           A 
                           5 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         A 
                         6 
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 3 
                 4 
               
                
               
                 
                   R 
                   2 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         x 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                         
                                           ( 
                                           
                                             W 
                                             xy 
                                           
                                           ) 
                                         
                                         y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             3 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       y 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           W 
                                           xy 
                                         
                                         ) 
                                       
                                       x 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           3 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0038]      FIGS. 2A and 2B  illustrate, respectively, exemplary functional block diagrams presenting data-channel processing for simultaneously isolating and obtaining gravity second- and third-order tensor components in the sensing plane of a 6-accelerometer gradiometer. 
         [0039]    In  FIG. 2A , the signals from accelerometers A 1  and A 5  are provided to a summation block  10  and, concurrently, the signals from accelerometers A 2 , A 3 , A 4 , and A 6  are provided to a summation block  12  with the output thereof subsequently divided by two at functional block  14 . The outputs of summation block  10  and divider  12  are differenced at  16  with the output provided to a first demodulator  18  that receives its reference signal sin 2 Ωt from reference signal source  20  and a second demodulator  22  that receives its reference signal cos 2 Ωt from another reference signal source  24  to provide the second-order gravity tensor components outputs shown. 
         [0040]    In  FIG. 2B , the signals from accelerometers A 1 , A 2 , and A 3  are provided to a summation block  50  and, concurrently, the signals from accelerometers A 4 , A 5 , and A 6  are to a summation block  52 . The outputs of summation blocks  50  and  52  differenced at  56  with the output provided to a first demodulator  58  that receives its reference signal sin 3 Ωt from reference signal source  60  and a second demodulator  62  that receives its reference signal cos 3 Ωt from another reference signal source  64  to provide the third-order gravity tensor components outputs shown. 
         [0041]    In a similar manner and for a gradiometer having eight equi-angular spaced accelerometers, simultaneous isolation and extraction of second, third, and fourth-order gravity tensor components is possible. The second-order components extracted at twice disk spin rate are obtained by summing accelerometer outputs per 
         [0000]    
       
         
           
             
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             1 
                           
                           + 
                           
                             A 
                             3 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             5 
                           
                           + 
                           
                             A 
                             7 
                           
                         
                       
                     
                   
                   ) 
                 
               
               - 
               
                 
                   1 
                   2 
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             2 
                           
                           + 
                           
                             A 
                             4 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             6 
                           
                           + 
                           
                             A 
                             8 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               4 
                
               
                   
               
                
               
                 R 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               1 
                               2 
                             
                              
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             2 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             W 
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           2 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0042]    Simultaneously, the third-order gravity tensor components are extracted at three times the disk spin rate by summing accelerometer outputs per 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     A 
                     1 
                   
                   - 
                   
                     A 
                     3 
                   
                 
                 ) 
               
               + 
               
                 
                   1 
                   
                     2 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             A 
                             6 
                           
                           + 
                           
                             A 
                             7 
                           
                           - 
                         
                       
                     
                     
                       
                         
                           
                             A 
                             5 
                           
                           - 
                           
                             A 
                             8 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 1 
                 2 
               
                
               
                 
                   R 
                   2 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         x 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                        
                                       
                                         
                                           ( 
                                           
                                             W 
                                             xy 
                                           
                                           ) 
                                         
                                         y 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             3 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       y 
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         ( 
                                         
                                           W 
                                           xy 
                                         
                                         ) 
                                       
                                       x 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                            
                           sin 
                            
                           
                               
                           
                            
                           3 
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0043]    Additionally and simultaneously, the fourth-order gravity tensor components are extracted at four times the disk spin rate by summing accelerometer outputs per 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           1 
                         
                         + 
                         
                           A 
                           2 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           A 
                           3 
                         
                         + 
                         
                           A 
                           4 
                         
                       
                     
                   
                 
                 ) 
               
               - 
               
                 ( 
                 
                   
                     
                       
                         
                           A 
                           5 
                         
                         + 
                         
                           A 
                           6 
                         
                         + 
                       
                     
                   
                   
                     
                       
                         
                           A 
                           7 
                         
                         + 
                         
                           A 
                           8 
                         
                       
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 1 
                 3 
               
                
               
                 
                   R 
                   3 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               W 
                                               xx 
                                             
                                             - 
                                             
                                               W 
                                               yy 
                                             
                                           
                                           ) 
                                         
                                         yy 
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             W 
                                             xx 
                                           
                                           - 
                                           
                                             W 
                                             yy 
                                           
                                         
                                         ) 
                                       
                                       xx 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             4 
                              
                             Ω 
                              
                             
                                 
                             
                              
                             t 
                           
                           + 
                         
                       
                     
                     
                       
                         
                           2 
                            
                           
                             
                               ( 
                               
                                 
                                   W 
                                   xx 
                                 
                                 - 
                                 
                                   W 
                                   yy 
                                 
                               
                               ) 
                             
                             xy 
                           
                            
                           cos 
                            
                           
                               
                           
                            
                           4 
                            
                           
                               
                           
                            
                           Ω 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0044]    In a manner similar to  FIG. 2 ,  FIG. 3  is an idealized acceleration vector diagram for an eight accelerometer A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 , and A 8  arrangement in which the sensitive axis of each accelerometer is shown as an arrow that is aligned tangentially to a common radius R circle with the accelerometers A n  orbiting about the axis A z  at some steady spin rate Ω. Since the various accelerometers are angularly offset from one another by 45 degrees, the respective harmonic outputs will likewise be displaced 45 degrees in phase, e.g., the output of the accelerometer A 1  will lead that of accelerometer A 8  by 45 degrees while trailing that of accelerometer A 5  by 45 degrees. 
         [0045]      FIGS. 3A ,  3 B, and  3 C illustrate, respectively, an exemplary functional block diagram illustrating data channel processing for simultaneously isolating and obtaining gravity second-, third-, and fourth-order tensor components in the sensing plane of an eight-accelerometer gradiometer instrument. 
         [0046]    In  FIG. 3A , the signals from accelerometers A 1 , A 3 , A 5 , and A 7  are provided to a summation block  100  and, concurrently, the signals from accelerometers A 2 , A 4 , A 6 , and A 8  are provided to a summation block  102  with the respective outputs thereof each divided by two at functional blocks  104  and  106 . The so-divided outputs of summation blocks  100  and  102  are differenced at  108  with the output thereof provided to a first demodulator  110  that receives its reference signal sin 2 Ωt from reference signal source  112  and a second demodulator  114  that receives its reference signal cos 2 Ωt from another reference signal source  116  to provide the second-order gravity tensor components outputs shown. 
         [0047]    In  FIG. 3B , the signals from accelerometers A 1  and A 3  are differenced in functional block  150  while the signals from accelerometers A 6  and A 7  are provided to a functional block  152  for summation while the signals from accelerometers A 5  and A 8  are decremented therefrom in functional block  152  with the output thereof divided by 1/√2 in functional block  156 . The outputs from functional blocks  150  and  156  and summed at functional block  158  and the output thereof provided to a first demodulator  160  that receives its reference signal sin 3 Ωt from reference signal source  162  and a second demodulator  164  that receives its reference signal cos 3 Ωt from another reference signal source  166  to provide the third-order gravity tensor component outputs shown. 
         [0048]    In  FIG. 3C , the signals from accelerometers A 1 , A 2 , A 3 , and A 4  are summed in summation block  200  and the signals from accelerometers A 5 , A 6 , A 7 , and A 8  are provided to a summation block  202  for summation. The outputs from functional blocks  200  and  202  are differenced at functional block  208  and the output thereof provided to a first demodulator  210  that receives its reference signal sin 4 Ωt from reference signal source  212  and to a second demodulator  214  that receives its reference signal cos 4 Ωt from another reference signal source  216  to provide the fourth-order gravity tensor component outputs shown. 
         [0049]      FIG. 4  illustrates one manner by which the 4Ω higher-order harmonics may be removed from the desired lower-order signals in a GGI instrument. In  FIG. 4 , the processing path of a conventional GGI instrument (for example, of the type disclosed in the aforementioned incorporated patent) is represented at  300 . As shown, the outputs of accelerometers A 1 , A 3 , A 5 , and A 7  are summed in summation block  302  and the signals from accelerometers A 2 , A 4 , A 6 , and A 8  are summed in summation block  304 . The outputs of summations blocks  302  and  304  are divided by some value (i.e., ½) and then the difference taken at  310  with the output provided to a first demodulator  312  that receives its reference signal sin 2 Ωt from reference signal source  314  and to a second demodulator  316  that receives its reference signal cos 4 Ωt from another reference signal source  318  to provide the lower-order gravity tensor component outputs. In addition to the desired lower-order components, the output signals also include the 4Ω higher-order harmonics; thus, the outputs of demodulators  312  and  316  can be characterized, respectively, as follows: 
         [0000]      2R(W xx −W yy )+(4Ω) terms 
         [0000]      4RW xy +(4Ω) terms 
         [0050]    The signal path carrying the first of these signals from demodulator  312  includes first and second differential units  320  and  322  (i.e., subtractors) and the signal line carrying the second of these signals from demodulator  316  also includes third and fourth differential units  324  and  326 . 
         [0051]    The functional block diagram of  FIG. 3C  described above has been presented in the lower portion of  FIG. 4  and provides the fourth-order gravity tensor component outputs, respectively, from demodulators  210  and  214 : 
         [0000]    
       
         
           
             
               1 
               3 
             
              
             
               
                 R 
                 3 
               
                
               
                 ( 
                 
                   
                     
                       ( 
                       
                         
                           W 
                           xx 
                         
                         - 
                         
                           W 
                           yy 
                         
                       
                       ) 
                     
                     yy 
                   
                   - 
                   
                     
                       ( 
                       
                         
                           W 
                           xx 
                         
                         - 
                         
                           W 
                           yy 
                         
                       
                       ) 
                     
                     xx 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               2 
               3 
             
              
             
               
                 
                   R 
                   3 
                 
                  
                 
                   ( 
                   
                     
                       W 
                       xx 
                     
                     - 
                     
                       W 
                       yy 
                     
                   
                   ) 
                 
               
               xy 
             
           
         
       
     
         [0052]    The output of demodulator  210  is provided as inputs to scaling/compensation amplifiers  328  and  330 , which, in turn, provide their respective outputs to differential units  324  and  320 . In a similar manner, the output of demodulator  214  is provided as inputs to scaling/compensation amplifiers  332  and  334 , which, in turn, provide their respective outputs to differential units  326  and  322 . The amplifiers  328 / 330  and  332 / 334  thus function to ‘condition’ the outputs of the demodulators  210  and  214  for mixing with the outputs of the demodulators  312  and  316 . Additionally, a compensation functional block  336  provide control signals to each of the scaling/compensation amplifiers to adjust gain or gain response based upon a system model, measured values, and/or a measured value history, or the estimated influence of the fourth-order components within the lower-order signals from the demodulators  312  and  316 . 
         [0053]    The outputs of the various amplifiers  338 ,  332 ,  330 , and  334  are provided to their respective differential units  324 ,  326 ,  320 , and  322  to effectively remove the undesired fourth-order components. Nominally, the fourth-order tensor components are rectified at 4-times the rotor spin rate, i.e., at 4Ω. In practice, however, and as a consequence to various tiny residual misalignments and dynamic perturbations, the fourth-order tensor components are also rectified at twice the rotor spin rate (2Ω). Fortunately, their influence at 2Ω is scaled by tiny misalignments and residual dynamic perturbations so the net effect is usually a low-level noise or cluttering of the sought second-order component data. Having directly measured fourth-order components as described herein, the second-order data output stream can be de-noised or de-cluttered by scaling the fourth-order effects by otherwise measured, estimated, or calibrated misalignments and such, then subtracted from the data stream subsequently, resulting in purer  20  output data. 
         [0054]    As can be appreciated, the arrangement of  FIG. 4  processes a signal or signals from the GGI  300  containing a sought-after lower-order signal that also includes a higher-order component or harmonic therein and generates another signal or signals representative of that higher-order component or harmonic and controllably uses that higher-order component or harmonic signal to attenuate or remove the higher-order component or harmonic from the output of the GGI to effectively “de-clutter” or de-noise” the output of the GGI. 
         [0055]    The functional block diagrams of  FIGS. 2A-2B ,  3 A- 3 C, and  4  and the equations therein can be implemented in analog or digital form (or a combination thereof) and can take the form of discrete devices or, more preferably, as one or more firmware- or software-controlled microprocessors or microcomputers (as well as special-purpose processors, including RISC processors), application specific integrated circuits (ASIC), programmable logic arrays (PLA), discrete logic or analog circuits, and/or combinations thereof. If desired, multi-processor parallel processing can be utilized. 
         [0056]    The present invention can be implemented in a preferred embodiment by modifying existing gravity gradiometer design to incorporate a second rotating disc structure (in a manner consistent with  FIG. 1 ) carrying the accelerometers A 1 -A 6  (or A 1 -A 8 ) described above or by the addition of the accelerometers described above on the existing disc that carries the accelerometer pairs for the conventional gradient measurements. 
         [0057]    As can be appreciated, the above described six and eight accelerometer embodiments can extended to ten accelerometer embodiment; in general, n-accelerometer variants can be configured such that each of the n accelerometers is spaced 360/n degrees from its neighbor around the circumference of a disk or instrument block. 
         [0058]    As will be apparent to those skilled in the art, various changes and modifications may be made to the illustrated embodiment of the present invention without departing from the spirit and scope of the invention as determined in the appended claims and their legal equivalent.