Abstract:
The invention is a method and apparatus for gravity vector compensation in an inertial navigation system (INS). The INS obtains its position and determines a gravity-model vector at its position using a predetermined gravity model. The deflection of vertical (DOV), the error in the direction of the gravity-model vector, is expressed in terms of a North-South (N-S) DOV component and an East-West (E-W) DOV component. Matrices of N-S DOV and E-W DOV data are stored in memory in a plurality of files for a plurality of discrete points distributed over the surface of the earth at a plurality of altitudes. The elements of a matrix of N-S DOV data or E-W DOV data have a one-to-one correspondence with a two-dimensional array of discrete contiguous points at a specified altitude above the earth&#39;s surface. Adjacent elements in a matrix correspond to (1) adjacent points on a parallel of latitude or (2) adjacent points on a half-circle meridian, a half-circle meridian terminating at the two poles of the earth. The method consists of obtaining the INS position, separately translating the N-S DOV data and the E-W DOV data from a plurality of matrices into a plurality of supermatrices, determining the N-S DOV and the E-W DOV at the INS position utilizing data from the plurality of supermatrices, and finally compensating the direction of the gravity-model vector for the N-S DOV and the E-W DOV.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     (Not applicable) 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT 
     (Not applicable) 
     BACKGROUND OF THE INVENTION 
     This invention relates generally to inertial navigation systems which require gravity vector data in their operational regions and more specifically to methods for determining the deflection of the gravity vector at a point above the earth&#39;s surface given gravity vector deflection data for a three-dimensional grid of points above the earth&#39;s surface. 
     World wide deflection of vertical (DOV) data provided by the National Imagery and Mapping Agency (NIMA) can be used to compensate the mismodeled gravity vector error, thereby reducing velocity and position errors from an inertial navigation system. 
     NIMA provides DOV data on 32 CD-ROMs with each CD/ROM containing a 45°×45° sector of data. DOV data (East-West, North-South, as well as associated accuracies) are stored in files corresponding to grid points that are 2 arcminutes apart in 1°×1° cells. 
     The future NIMA DOV database will consist of 180×360+1=64,801 files at one altitude. There will be seven reference altitudes at 0, 10K, 20K, 30K, 50K, 70K, and 90K feet. This represents the NIMA-supplied data for the F-117 aircraft. 
     Each of the 64,801 files (except for the one discussed in the next paragraph) contains DOV data and accuracy data (all in arcseconds) of 900 grid points at 2 arcminutes spacing in a 1°×1° angular cell as shown in FIG.  1 . The files are labeled from N 01000 , N 01001 , . . . , N 01359 , N 02000 , . . . , to N 90359  and S 0000 , S 00001 , . . . , S 00359 , S 01000 , . . . , to S 89359 . Each file is 29,025 bytes in size. 
     The remaining file is named S 90000 . It contains the DOV and accuracy data for the south pole repeated 900 times to maintain the uniformity of data files. 
     BRIEF SUMMARY OF THE INVENTION 
     The invention is a method and apparatus for gravity vector compensation in an inertial navigation system (INS). The INS obtains its position, in terms of latitude, longitude and altitude, and determines a gravity-model vector at its position using a predetermined gravity model. There is an error between the gravity-model vector and the true gravity vector. 
     The error in the direction of the gravity-model vector is referred to as deflection of vertical (DOV), the DOV being expressed in terms of a North-South (N-S) DOV component and an East-West (E-W) DOV component. Matrices of N-S DOV and E-W DOV data are stored in memory in a plurality of files for a plurality of discrete points distributed over the surface of the earth at a plurality of altitudes. 
     The elements of a matrix of N-S DOV data or E-W DOV data have a one-to-one correspondence with a two-dimensional array of discrete contiguous points at a specified altitude above the earth&#39;s surface, the array of points having (1) latitude values in a specified range of latitude values, (2) longitude values in a specified range of longitude values, and (3) an altitude value in a specified range of altitude values. Adjacent elements in a matrix correspond to (1) adjacent points on a parallel of latitude or (2) adjacent points on a half-circle meridian, a half-circle meridian terminating at the two poles of the earth. 
     The first step of the method consists of obtaining the INS position. 
     The second step of the method consists of separately translating the N-S DOV data and the E-W DOV data from a plurality of matrices into a plurality of supermatrices. The INS latitude and longitude values are within the specified ranges of latitude and longitude and the INS altitude is greater than the altitude value of at least one matrix. The INS latitude and longitude values are within the specified ranges of latitude and longitude, and the INS altitude is less than the altitude value of at least one matrix. 
     The adjacent elements of a supermatrix correspond to (1) adjacent points on a parallel of latitude or (2) adjacent points on a whole-circle meridian. The elements of a supermatrix all correspond to the same altitude value. 
     The third step of the method consists of determining the N-S DOV and the E-W DOV at the INS position utilizing data from the plurality of supermatrices. 
     The fourth step of the method consists of compensating the direction of the gravity-model vector for the N-S DOV and the E-W DOV. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows the 900 grid points in a DOV cell corresponding to N 36251 . 
     FIG. 2 shows a flow diagram for the method of the invention. 
     FIG. 3 shows a block-diagram embodiment of apparatus for gravity vector compensation in an inertial navigation system. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As a vehicle navigates over the earth, an inertial navigation system (INS) utilizes an 8×8 array consisting of 64 points at a reference altitude above the vehicle and a corresponding array of 64 points at a reference altitude below the vehicle to determine the North-South and East-West components of the DOV at the vehicle location by means of 64 linear interpolations between the altitudes and 9 natural cubic spline interpolations between the horizontal data points. 
     A cubic spline interpolation is a piecewise third-order polynomial approximation that divides an interval into a collection of subintervals and constructs an approximating cubic polynomial on each subinterval. A natural cubic spline interpolation is one where the second derivative of the cubic spline interpolation function at the interval boundaries are set equal to zero. 
     The natural cubic spline interpolation is fully described in Daniel Zwillinger, Editor-in Chief,  STANDARD MATHEMATICAL TABLES AND FORMULAE,  30th ed., CRC Press, Inc., Boca Raton, Fla., 1996, p. 680. 
     The flow diagram governing the operations of a processor in determining the DOV of a vehicle is shown in FIG.  2 . The first step  11  of the process begins with an interrupt from the INS indicating the availability of new position information. The program causes the processor to enter into RAM the latitude, longitude, and altitude coordinates (φ v , λ v , h v ) of the vehicle supplied by the INS. 
     The second step  13  consists of determining whether the 90×90 North-South and East-West supermatrices of DOV data presently in RAM are adequate. A supermatrix is constructed from the elements of a 30×30 900-element primary matrix that corresponds at the time of data retrieval to a region containing the vehicle together with eight 30×30 900-element secondary matrices that surround the primary matrix. The elements of the North-South and East-West matrices contain respectively the North-South and East-West components of the DOV and are identified within a matrix by the latitude φ m  and longitude λ m . 
     The nine matrices that constitute a supermatrix are so arranged that for most of the earth&#39;s surface, the elements of the supermatrix can also be identified by the latitude φ m  and longitude λ m  but with φ m  and λ m  each taking on 90 contiguous values for most of the earth&#39;s surface. When a supermatrix includes elements that surround either the North or South pole, there are discontinuities in φ m  and λ m . However, adjoining elements of a supermatrix always correspond to adjoining points above the earth&#39;s surface. 
     There are North-South and East-West matrices and supermatrices for each of seven reference altitudes h m . 
     It is necessary at some point in the process to extract interpolation matrices from supermatrices at two reference altitudes for the purpose of obtaining the DOV components at the position of the vehicle. If the interpolation matrices are not within the latitude, longitude, and altitude bounds of the supermatrices, then new supermatrices must be retrieved from memory and stored in RAM. If the answer to the question posed in step  13  is yes, the program jumps to step  21 . If the answer is no, files containing the primary and secondary matrices are identified and designated for retrieval in step  15 . The primary matrices are those for which the bounding latitudes and longitudes also bound the vehicle position. The designated files containing the primary and secondary matrices are retrieved into RAM in step  17 . 
     The four required supermatrices are constructed in step  19 . Two superrnatrices contain the North-South and East-West components of the DOV for the reference altitude just below the vehicle altitude, and the other two supermatrices contain the North-South and East-West components of the DOV for the reference altitude just above the vehicle altitude. In discussing the construction of the supermatrices, the k′th 30×30 matrix that contains the North-South DOV components will be denoted by [ξ] k . Similarly, the k′th 30×30 matrix that contains the East-West DOV components will be denoted by [η] k . 
     Let us consider a vehicle position in the northern hemisphere away from the north pole and the equator. For purposes of data retrieval, if the vehicle position is given by (100  v ,λ v )=(35.4°×251.3°), then the primary matrix for each reference altitude is contained in the file with the label N 36251 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 250° &lt; λ ≦ 251° 
                 251° &lt; λ ≦ 252° 
                 252° &lt; λ ≦ 253° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 36° &lt; φ ≦ 37° 
                 N37250 
                 N37251 
                 N37252 
               
               
                 35° &lt; φ ≦ 36° 
                 N36250 
                 N36251 
                 N36252 
               
               
                 34° &lt; φ ≦ 35° 
                 N35250 
                 N35251 
                 N35252 
               
               
                   
               
             
          
         
       
     
     The supermatrix that contains all 90×90 North-South DOV components of these nine matrices is                [   ξ   ]     =     [             [   ξ   ]     9             [   ξ   ]     2             [   ξ   ]     3                 [   ξ   ]     8             [   ξ   ]     1             [   ξ   ]     4                 [   ξ   ]     7             [   ξ   ]     6             [   ξ   ]     5           ]             (   1   )                                
     The supermatrix that contains all 90×90 East-West DOV components of these nine matrices is                [   ξ   ]     =     [             [   η   ]     9             [   η   ]     2             [   η   ]     3                 [   η   ]     8             [   η   ]     1             [   η   ]     4                 [   η   ]     7             [   η   ]     6             [   η   ]     5           ]             (   2   )                                
     Let us now consider a vehicle position in the southern hemisphere away from the south pole and the equator. If the vehicle position is given by (φ v ,λ v )=(−35.4°,251.3°), then the primary matrix for each reference altitude is contained in the file with the label S 35251 . The eight adjoining secondary matrices are contained in the files indicated below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 250° &lt; 
                   
                   
               
               
                   
                 λ ≦ 251° 
                 251° &lt; λ ≦ 252° 
                 252° &lt; λ ≦ 253° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 −35° &lt; φ ≦ −34° 
                 S34250 
                 S34251 
                 S34252 
               
               
                 −36° &lt; φ ≦ −35° 
                 S35250 
                 S35251 
                 S35252 
               
               
                 −37° &lt; φ ≦ −36° 
                 S36250 
                 S36251 
                 S36252 
               
               
                   
               
             
          
         
       
     
     The supermatrices that contain all 90×90 North-South DOV components and all 90×90 East-West DOV components for these nine matrices are similar to those given in equations (1) and (2). 
     Now consider a point in the northern hemisphere near the equator. If the point is given by (φ v ,λ v )=(0.4°,121.3°), then the primary matrix for each reference altitude is contained in the file with the label NO 1121 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 120° &lt; λ ≦ 121° 
                 121° &lt; λ ≦ 122° 
                 122° &lt; λ ≦ 123° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 1° &lt; φ ≦ 2° 
                 N02120 
                 N02121 
                 N02122 
               
               
                 0° &lt; φ ≦ 1° 
                 N01120 
                 N01121 
                 N01122 
               
               
                 −1° &lt; φ ≦ 0°   
                 S00120 
                 S00121 
                 S00122 
               
               
                   
               
             
          
         
       
     
     Notice the mixture of Nyyxxx and Syyxxx files. The supermatrices that contain all 90×90 North-South DOV components and all 90×90 East-West DOV components for these nine matrices are similar to those given in equations (1) and (2). 
     Now let us consider a vehicle location in the southern hemisphere near the equator. If the vehicle location is given by (φ v ,λ v )=(−0.4°,121.3°), then the primary matrix for each reference altitude is contained in the file with the label S 00121 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 120° &lt; λ ≦ 121° 
                 121° &lt; λ ≦ 122° 
                 122° &lt; λ ≦ 123° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   0° &lt; φ ≦ 1° 
                 N01120 
                 N01121 
                 N01122 
               
               
                 −1° &lt; φ ≦ 0° 
                 S00120 
                 S00121 
                 S00122 
               
               
                 −2° &lt; φ ≦ −1° 
                 S01120 
                 S01121 
                 S01122 
               
               
                   
               
             
          
         
       
     
     Notice the mixture of Nyyxxx and Syyxxx files. The supermatrices that contain all 90×90 North-South DOV components for these nine cells are similar to those given in equations (1) and (2). 
     Let us now consider a point near the north pole. If the point is given by (φ v ,λ v )=(89.4°, 121.3°), then it is located in the file with the label N 90121 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
           
               
                   
               
             
             
               
                   
                 302° &lt; λ ≦ 303° 
                 301° &lt; λ ≦ 302° 
                 300° &lt; λ ≦ 301° 
               
               
                   
               
               
                 89° &lt; φ ≦ 90° 
                 N90302 
                 N90301 
                 N90300 
               
               
                   
               
               
                   
                 120° &lt; λ ≦ 121° 
                 121° &lt; λ ≦ 122° 
                 122° &lt; λ ≦ 123° 
               
               
                   
               
               
                 89° &lt; φ ≦ 90° 
                 N90120 
                 N90121 
                 N90122 
               
               
                 88° &lt; φ ≦ 89° 
                 N89120 
                 N89121 
                 N89122 
               
               
                   
               
             
          
         
       
     
     The matrix that contains all 90×90 North-South DOV components for these nine cells is                [   ξ   ]     =     [             [   ξ   ]     9   *             [   ξ   ]     2   *             [   ξ   ]     3   *                 [   ξ   ]     8             [   ξ   ]     1             [   ξ   ]     4                 [   ξ   ]     7             [   ξ   ]     6             [   ξ   ]     5           ]             (   3   )                                
     The matrix that contains all 90×90 East-West DOV components for these nine cells is                [   η   ]     =     [             [   η   ]     9   *             [   η   ]     2   *             [   η   ]     3   *                 [   η   ]     8             [   η   ]     1             [   η   ]     4                 [   η   ]     7             [   η   ]     6             [   η   ]     5           ]             (   4   )                                
     The asterisks in the top row of these two matrices indicate that the following four operations are performed before the matrices are patched together. 
     (1) Delete the first row from [ξ] 9 , [ξ] 2 , [ξ] 3 , [η] 9 , [η] 2 , [η] 3 ; 
     (2) Pad these six matrices with a row of 30 zeros at the bottom so that each matrix still has 30 rows and 30 columns; 
     (3) Flip each of these matrices left to right (i.e. column 1 becomes column 30, column 2 becomes column 29, etc.); 
     (4) Flip each of these matrices upside down (i.e. row 1 becomes row 30, row 2 becomes row 29, etc.). 
     The reasons for these operations are: 
     (1) All files with the label N 90 xxx have the North pole as the common upper (northern) boundary, hence there are duplications; 
     (2) We need to preserve relative physical location of the grid points in the two augmented 90×90 matrices [ξ], [η]. 
     Let us now consider a vehicle position near the south pole. If the vehicle position is given by (φ v ,λ v )=(−89.4°, 121.3°), then it is located in the file with the label S 89121 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
           
               
                   
               
             
             
               
                   
                 120° &lt; 
                   
                   
               
               
                   
                 λ ≦ 121° 
                 121° &lt; λ ≦ 122° 
                 122° &lt; λ ≦ 123° 
               
               
                   
               
               
                 −89° &lt; φ ≦ −88° 
                 S88120 
                 S88121 
                 S88122 
               
               
                 −90° &lt; φ ≦ −89° 
                 S89120 
                 S89121 
                 S89122 
               
               
                   
               
               
                   
                 302° &lt; 
               
               
                   
                 λ ≦ 303° 
                 301° &lt; λ ≦ 302° 
                 300° &lt; λ ≦ 301° 
               
               
                   
               
               
                 −90° &lt; φ ≦ −89° 
                 S89302 
                 S89301 
                 S89300 
               
               
                   
               
             
          
         
       
     
     The matrix that contains all 90×90 North-South DOV components for these nine cells is                [   ξ   ]     =     [             [   ξ   ]     9             [   ξ   ]     2             [   ξ   ]     3                 [   ξ   ]     8             [   ξ   ]     1             [   ξ   ]     4                 [   ξ   ]     7   *             [   ξ   ]     6   *             [   ξ   ]     5   *           ]             (   5   )                                
     The matrix that contains all 90×90 East-West DOV components for these nine cells is                [   η   ]     =     [             [   η   ]     9             [   η   ]     2             [   η   ]     3                 [   η   ]     8             [   η   ]     1             [   η   ]     4                 [   η   ]     7   *             [   η   ]     6   *             [   η   ]     5   *           ]             (   6   )                                
     The asterisks in the bottom row of these two matrices indicate that the following five operations are performed before the matrices are patched together. 
     (1) Delete the first row from [ξ] 5 , [ξ] 6 , [ξ] 7 , [η] 5 , [η] 6 , [η] 7 ; 
     (2) Extract the North-South DOV component ξ sp  and East-West DOV component η sp  from S 90000  (south pole DOV components) at two levels; 
     (3) Pad these six matrices with a row of 30 ξ sp &#39;s at the bottom of [ξ] 5 , [ξ] 6 , [ξ] 7  and a row of 30 η sp &#39;s at the bottom of [η] 5 , [η] 6 , [η] 7  so that each matrix still has 30 rows and 30 columns; 
     (4) Flip each of these matrices left to right (i.e. column 1 becomes column 30, column 2 becomes column 29, etc.); 
     (5) Flip each of these matrices upside down (i.e. row 1 becomes row 30, row 2 becomes row 29, etc.). 
     The reasons for these operations are: 
     (1) All files with the label S 89 xxx do not have south pole DOV&#39;s which must therefore be appended; 
     (2) We need to preserve relative physical location of the grid points in the two augmented 90×90 matices [ξ], [η]. 
     Let us now consider a vehicle position in the northern hemisphere near the Greenwich meridian. If the vehicle position is given by (φ v ,λ v )=(14.3°, 0.30°), then it is located in the file with the label N 15000 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 359° &lt; λ ≦ 360° 
                 0° &lt; λ ≦ 1° 
                 1° &lt; λ ≦ 2° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 15° &lt; φ ≦ 16° 
                 N16359 
                 N16000 
                 N16001 
               
               
                 14° &lt; φ ≦ 15° 
                 N15359 
                 N15000 
                 N15001 
               
               
                 13° &lt; φ ≦ 14° 
                 N14359 
                 N14000 
                 N14001 
               
               
                   
               
             
          
         
       
     
     The matrices that contain all 90×90 all North-South and East-West DOV components for these nine cells are similar to those given in equations (1) and (2). 
     Let us now consider a vehicle position in the southern hemisphere near the Greenwich meridian. If the vehicle position is given by (φ v ,λ v )=(−14.3°, 0.3°), then it is located in the file with the label S 14000 . The files containing the eight secondary matrices are identified below. 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 359° &lt; λ ≦ 360° 
                 0° &lt; λ ≦ 1° 
                 1° &lt; λ ≦ 2° 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 −14° &lt; φ ≦ −13° 
                 S13359 
                 S13000 
                 S13001 
               
               
                 −15° &lt; φ ≦ −14° 
                 S14359 
                 S14000 
                 S14001 
               
               
                 −16° &lt; φ ≦ −15° 
                 S15359 
                 S15000 
                 S15001 
               
               
                   
               
             
          
         
       
     
     The matrices that contain all 90×90 all North-South and East-West DOV components or these nine cells are similar to those given in equations (1) and (2). 
     Four interpolation matrices are extracted respectively from the four supermatrices in step  21 . The interpolation matrices for the preferred embodiment of the invention are 8×8 64-element matrices. The interpolation matrices are chosen such that the latitudes of the fourth and fifth rows of the matrices bound the latitude of the vehicle and the longitudes of the fourth and fifth columns of the matrices bound the longitude of the vehicle. 
     Interpolated values for the North-South and East-West components of the DOV corresponding to the vehicle position (φ v ,λ v ) are obtained in step  23 . An interpolation matrix containing the North-South DOV components and corresponding to the vehicle&#39;s altitude λ v  is obtained by performing linear interpolations between the values contained in the two interpolation matrices containing the North-South DOV components and corresponding to reference altitudes above and below the vehicle&#39;s altitude h v . An interpolation matrix containing the East-West DOV components and corresponding to the vehicle&#39;s altitude h v  is obtained in a similar manner. 
     Interpolated values for the North-South DOV components corresponding to the vehicle&#39;s latitude and eight reference longitudes are obtained by natural 8-point cubic spline interpolations using the altitutde-interpolated interpolation matrix data. An interpolated value for the North-South DOV component corresponding to the vehicle&#39;s latitude and longitude is obtained by a natural 8-point cubic spline interpolation using the interpolated values corresponding to the vehicle&#39;s latitude. Interpolated values for the East-West DOV components are obtained in a similar fashion. 
     The preferred embodiment of the invention retrieves at any one time for a particular DOV component and a particular reference altitude the files containing the primary matrix and eight secondary matrices. Other embodiments could retrieve at various times (1) the primary matrix by itself, (2) the primary matrix and one secondary matrix, and (3) the primary matrix and three secondary matrices. 
     If the INS position (φ v , λ v , h v ) is in the central portion of the primary matrices, the 8×8 arrays that provide the basis for determining the DOV for the INS position can be extracted from the primary matrices, and it is necessary to retrieve only the primary cells from memory. 
     If, however, the INS position is near one of the bounding parallels of latitude or meridians (but not the intersection of a bounding parallel of latitude and a bounding meridian), it becomes necessary to also retrieve for each reference altitude the secondary matrix that is nearest to the INS position and adjacent to the primary matrix. 
     If the INS position is near the intersection of a bounding parallel of latitude and a bounding meridian, then it becomes necessary to retrieve for each reference altitude the three secondary matrices adjacent to the primary matrix and closest to the INS position. 
     To avoid the processing necessary to determine for each reference altitude of interest whether one, two, or four secondary matrices must be retrieved from memory, it may be desirable to always retrieve the primary matrix and three adjacent secondary matrices. The secondary matrices to be retrieved can be identified by determining the quadrant of the primary matrix in which the INS position is located. 
     It may even be more cost-effective to simply retrieve the eight secondary matrices surrounding a primary matrix since the only processing required in this case is to identify the primary cell for the INS position. 
     However many cells are retrieved at one time, a new set of cells must be retrieved whenever the DOV data available is insufficient to construct a DOV interpolation matrix surrounding the point INS position. 
     We have chosen for the preferred embodiment of the invention 30×30 900-point matrices, 90×90 8100-point supermatrices (one primary matrix and eight secondary matrices), and 8×8 interpolation matrices. Other embodiments may utilize matrices, supermatrices, and interpolation matrices of other sizes. The matrices, supermatrices, and interpolation matrices may be rectangular rather than square. A supermatrix may encompass any number of matrices. 
     Estimates of DOV components in the preferred embodiment for a particular altitude involves a linear interpolation between the elements of two interpolation matrices, one corresponding to a reference attitude less than the INS altitude and the other corresponding to a reference attitude greater than the INS altitude. The altitude interpolation could involve interpolation matrices for more than two reference altitudes. 
     An 8-point natural cubic spline interpolation is used in the preferred embodiment. However, other embodiments could use other interpolation methods based on other numbers of points. 
     The latitude, longitude, and altitude interpolations may be carried out in any order.