Abstract:
A navigation system for a vehicle having a receiver operable to receive a plurality of signals from a plurality of transmitters includes a processor and a memory device. The memory device has stored thereon machine-readable instructions that, when executed by the processor, enable the processor to determine a set of error estimates corresponding to pseudo-range measurements derived from the plurality of signals, determine an error covariance matrix for a main navigation solution using ionospheric-delay data, and, using a solution separation technique, determine at least one protection level value based on the error covariance matrix.

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority from U.S. Provisional Appl. No. 61/012,303 entitled “RAIM WITH SPECIALLY CORRELATED IONOSPHERIC ERRORS” filed Dec. 7, 2007, which is incorporated by reference as if fully set forth herein. 
    
    
     BACKGROUND OF THE INVENTION 
     Conventional RAIM algorithms may be based on either a weighted or un-weighted least squares solution where the errors in each satellite&#39;s pseudo-range measurement are uncorrelated with the errors in the other satellites&#39; pseudo-range measurements. 
     However, the ionospheric error (which can be the dominant error source) in each satellite&#39;s pseudo-range is, in fact, highly correlated with that of each of the other satellites. By ignoring this correlation, the computed Horizontal Protection Limit (HPL) which bounds the horizontal position error is much larger than necessary. As a result the availability of GPS to do a low Required Navigation Performance (RNP) approach suffers. 
     SUMMARY OF THE INVENTION 
     In an embodiment of the invention, a navigation system for a vehicle having a receiver operable to receive a plurality of signals from a plurality of transmitters includes a processor and a memory device. The memory device has stored thereon machine-readable instructions that, when executed by the processor, enable the processor to determine a set of error estimates corresponding to pseudo-range measurements derived from the plurality of signals, determine an error covariance matrix for a main navigation solution using ionospheric-delay data, and, using a solution separation technique, determine at least one protection level value based on the error covariance matrix. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Preferred and alternative embodiments of the present invention are described in detail below with reference to the following drawings. 
         FIG. 1  shows a navigation system incorporating embodiments of the present invention; 
         FIG. 2  shows a graphical illustration of HPL determination according to an embodiment of the present invention; and 
         FIG. 3  shows a process according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       FIG. 1  shows a radio navigation system incorporating the teachings of an embodiment of the present invention. The system includes several transmitters  1 -N and user set  12 . Transmitters  1 -N, in the preferred embodiment, are a subset of the NAVSTAR GPS constellation of satellite transmitters, with each transmitter visible from the antenna of user set  12 . Transmitters  1 -N broadcast N respective signals indicating respective transmitter positions and signal transmission times to user set  12 . 
     User set  12 , mounted to an aircraft (not shown), includes receiver  14 , processor  16 , and a memory device, such as processor memory  18 . Receiver  14 , preferably NAVSTAR GPS compatible, receives the signals, extracts the position and time data, and provides pseudorange measurements to processor  16 . From the pseudorange measurements, processor  16  derives a position solution for the user set  12 . Although the satellites transmit their positions in World Geodetic System of 1984 (WGS-84) coordinates, a Cartesian earth-centered earth-fixed system, the preferred embodiment determines the position solution in a local reference frame L, which is level with the north-east coordinate plane and tangential to the Earth. This frame choice, however, is not critical, since it is well-understood how to transform coordinates from one frame to another. 
     Processor  16  also uses the pseudorange measurements to detect satellite transmitter failures and to determine a worst-case error, or protection limit, both of which it outputs with the position solution to flight management system  20 . Flight management system  20  compares the protection limit to an alarm limit corresponding to a particular aircraft flight phase. For example, during a pre-landing flight phase, such as nonprecision approach, the alarm limit (or allowable radial error) is 0.3 nautical miles, but during a less-demanding oceanic flight phase, the alarm limit is 2-10 nautical miles. (For more details on these limits, see RTCA publication DO-208, which is incorporated herein by reference.) If the protection limit exceeds the alarm limit, the flight management system, or its equivalent, announces or signals an integrity failure to a navigational display in the cockpit of the aircraft. The processor also signals whether it has detected any satellite transmitter failures. 
     An embodiment of the invention models the correlation of the ionospheric errors between each pair of satellites as a function of the distance between their ionospheric pierce points. The closer the pierce points, the higher the correlation. The root-mean-square (RMS) uncertainty (or sigma) of each satellite&#39;s pseudo-range measurement is computed using the ionospheric variance model defined in DO-229D, Appendix J. Using the computed correlation coefficients and the sigma for each satellite, the ionospheric measurement error covariance matrix is formed. The remaining errors (satellite clock and ephemeris, tropospheric, multi-path and receiver noise) are assumed to be uncorrelated. Thus, the combined measurement error covariance matrix for these error sources is diagonal. These two matrices are added to form the total measurement error covariance matrix. This matrix is then inverted to form the weighting matrix for the least squares solution. Fault detection and exclusion can then be performed and the various protection levels such as the horizontal protection level (HPL), vertical protection level (VPL), horizontal exclusion level (HEL), and vertical exclusion level (VEL) computed based on the methods of solution separation previously described in U.S. Pat. Nos. 5,760,737 and 6,639,549, each of which are incorporated by reference as if fully set forth herein. 
       FIG. 3  illustrates a process  300 , according to an embodiment of the invention, that can be implemented in the radio navigation system illustrated in  FIG. 1 . The process  300  is illustrated as a set of operations or steps shown as discrete blocks. The process  300  may be implemented in any suitable hardware, software, firmware, or combination thereof. As such the process  300  may be implemented in computer-executable instructions that can be transferred from one electronic device to a second electronic device via a communications medium. The order in which the operations are described is not to be necessarily construed as a limitation. 
     Referring to  FIG. 3 , at a step  310 , the processor  16  computes the sigma (error) values on pseudo-range and measurements. 
     At a step  320 , the processor  16  determines the measurement matrix. The true vector of pseudo-range residuals Δρ is related to the incremental position/time solution vector Δx (distance from the position linearization point) as follows:
 
Δρ=ρ−{circumflex over (ρ)}= HΔx   (1)
 
     where H is the measurement matrix and is given by: 
     
       
         
           
             
                 
             
             ⁢ 
             
               H 
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                         
                           1 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             - 
                             
                               LOS 
                               Nx 
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               Ny 
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               Nz 
                             
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                     
 
                   
                   [ 
                   
                     
                       
                         
                           LOS 
                           ix 
                         
                       
                     
                     
                       
                         
                           LOS 
                           iy 
                         
                       
                     
                     
                       
                         
                           LOS 
                           iz 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                     
                 
                 ⁢ 
                 
                   Line 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   of 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   sight 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   unit 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   vector 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   pointing 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   from 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   the 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   user 
                   ⁢ 
                   
                     
                         
                     
                     ⁢ 
                     
                         
                     
                   
                   ⁢ 
                   to 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   satellite 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   i 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 x 
               
               = 
               
                 
                   x 
                   - 
                   
                     x 
                     ^ 
                   
                 
                 = 
                 
                   true 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   position 
                   ⁢ 
                   
                     / 
                   
                   ⁢ 
                   clock 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   bias 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   linerarization 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   point 
                 
               
             
           
         
       
     
     At a step  330 , the processor  16  computes the Error Covariance Matrix. The vector of measured pseudo-range residuals Δ{tilde over (ρ)} is the true pseudo-range residual vector plus the vector of residual errors δρ and is thus: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             p 
                             ~ 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           p 
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             p 
                           
                           - 
                           
                             p 
                             ^ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             p 
                             ~ 
                           
                           - 
                           
                             p 
                             ^ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                           + 
                           δp 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The processor  16  designates the post-update estimate of Δx as Δ{circumflex over (x)}. Then, the processor  16  can define the vector of post-update measurement residuals as:
 
ξ=Δ{tilde over (ρ)}− HΔ{circumflex over (x)}   (3)
 
     Each post-update measurement residual is the difference between the measured pseudo-range residual and the predicted pseudo-range residual based on the post-update estimate Δ{circumflex over (x)}. 
     A “weighted least-squares solution” can be determined by the processor  16  by finding the value of Δ{circumflex over (x)} which minimizes the weighted sum of squared residuals. Thus, the processor  16  may minimize:
 
ξ T   W ξ=(Δ{tilde over (ρ)}− HΔ{circumflex over (x)} ) T   W (Δ{tilde over (ρ)}− HΔ{circumflex over (x)} )  (4)
 
     where W is an appropriate weighting matrix. The weighting matrix generally chosen is one which normalizes the residuals based on the uncertainty of each pseudo-range measurement. Thus, the processor  16  yields: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 
                                   σ 
                                   1 
                                   2 
                                 
                               
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 1 
                                 
                                   σ 
                                   2 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               0 
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               
                                 1 
                                 
                                   σ 
                                   N 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
                         
                           
                             assuming 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             uncorrelated 
                           
                         
                       
                       
                         
                           measurements 
                         
                       
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     which represents the inverse of the pseudo-range measurement error covariance matrix assuming each pseudo-range error is uncorrelated with the others. 
     However, the vertical ionospheric delay component of each pseudo-range error is highly correlated with the others. If this correlation is known, then the processor  16  can take advantage of that knowledge by using the true pseudo-range measurement error covariance matrix R. The weighting matrix then becomes 
     
       
         
           
             
               
                 
                   W 
                   = 
                   
                     
                       R 
                       
                         - 
                         1 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 σ 
                                 1 
                                 2 
                               
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       1 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       1 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       1 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 σ 
                                 2 
                                 2 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       1 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     
                                       δρ 
                                       2 
                                     
                                     ⁢ 
                                     
                                       δρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 σ 
                                 N 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The value of Δ{circumflex over (x)} that minimizes (4) is determined by taking the derivative, setting it equal to zero, and solving for Δ{circumflex over (x)}. This yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   x 
                                   ^ 
                                 
                               
                               = 
                               
                                 ( 
                                 
                                   
                                     H 
                                     T 
                                   
                                   ⁢ 
                                   WH 
                                 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 S 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   p 
                                   ~ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     H 
                     T 
                   
                   ⁢ 
                   W 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     p 
                     ~ 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where the processor  16  has defined the weighted least-squares solution matrix S as:
 
 S =( H   T   WH ) −1   H   T   W   (8)
 
     Altitude Aiding 
     Barometric altitude can be used by the processor  16  to augment the GPS pseudo-range measurements. If it is used, the measurement matrix is then augmented as follows 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 1 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 x 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 y 
                               
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               
                                 2 
                                 ⁢ 
                                 z 
                               
                             
                           
                         
                         
                           1 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             - 
                             
                               LOS 
                               Nx 
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               Ny 
                             
                           
                         
                         
                           
                             - 
                             
                               LOS 
                               Nz 
                             
                           
                         
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             - 
                             1 
                           
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     This measurement matrix assumes that the incremental position vector (the first 3 elements) within Δx are given in local-level coordinates (with the z axis down). The line-of-sight (LOS) elements then must also be expressed in the local-level coordinates. The weighting matrix is also augmented as follows 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         = 
                           
                         ⁢ 
                         
                           R 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     σ 
                                     1 
                                     2 
                                   
                                 
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           1 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           2 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           1 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           N 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           1 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           2 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     σ 
                                     2 
                                     2 
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           2 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           N 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           1 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           N 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   
                                     E 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         
                                           δρ 
                                           2 
                                         
                                         ⁢ 
                                         
                                           δρ 
                                           N 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     σ 
                                     N 
                                     2 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     baro 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Computing the Measurement Covariance Matrix 
     There are multiple methods that can be employed to determine the measurement error covariance matrix. In the case of a Kalman filter application, the temporal behavior (time-correlation) of the ionospheric delays may be modeled. The spatially correlated ionospheric error for satellite i can be modeled as a weighted sum of three independent normalized (sigma=1.0) Gaussian random errors scaled by the nominal iono sigma value for that satellite as follows:
 
δρ iono     —     i =σ iono     —     i k iono     —     i   T x ref   (11)
 
     where x ref  is a 3×1 vector of independent Gaussian random errors with zero mean and a variance of 1. The weighting vector k iono     —     i   T  is determined by the processor  16  by first defining three grid points on the thin shell model of the ionosphere (at a height of 350 km) equally spaced in azimuth at a great circle distance of 1500 km from the user. The processor  16  may then define a 3×1 vector x grid  of normalized delays at these points. The delays at gridpoints i and j may be spatially correlated with each other based on the great circle distances between them according to:
 
 E[x   grid     —     i   x   grid     —     j   T ]=1−(1−e −     grid       —       i,grid       —       j     /d     iono   ) 2   (12)
 
where
 
d grid     —     i,grid     —     j =great circle distance between grid point i and grid point j
 
d iono =correlation distance of the ionospheric delay=4000 km  (13)
 
     Using that relationship the processor  16  can form a 3×3 covariance matrix P grid  which describes the correlations between each of the grid points:
 
P grid =E[x grid x grid   T ]
 
     If the delay processes that exist at these grid points are a certain linear combination of the reference independent Gaussian random errors, then they will have the desired spatial and temporal correlation. The processor  16  may assume that the desired linear combination is obtained by using a 3×3 upper-triangular mapping matrix U grid  as follows:
 
x grid =U grid x ref   (14)
 
     The grid covariance matrix is then: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           grid 
                         
                         = 
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   x 
                                   grid 
                                 
                                 ⁢ 
                                 
                                   x 
                                   grid 
                                   T 
                                 
                               
                               ] 
                             
                           
                           = 
                           
                             
                               U 
                               grid 
                             
                             ⁢ 
                             
                               E 
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     x 
                                     ref 
                                   
                                   ⁢ 
                                   
                                     x 
                                     ref 
                                     T 
                                   
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               U 
                               grid 
                               T 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                             T 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Therefore, the mapping matrix U grid  can be formed by the processor  16  simply by factoring the covariance matrix P grid . Since the geometry of the three gridpoints is fixed, the covariance matrix P grid  is constant and can thus be pre-computed by the processor  16 . Now the processor  16  can choose a linear combination of the three grid-point delays that yields a normalized delay at the pierce-point of the satellite i such that the proper spatial correlation with the three grid points (and thus, presumably, each of the other satellites) is achieved as follows:
 
δρ norm     —     iono     —     i =k sat     —     i     —     grid x grid   (16)
 
     where 
     k sat     —     i     —     grid =3-vector of weighting factors 
     δρ norm     —     iono     —     i =normalized delay at the satellite pierce-point 
     The satellite pseudo-range delay may be correlated to the delay at the k th  grid point according to: 
                     E   ⁡     [       δρ     norm_iono   ⁢   _i       ⁢     x   grid_k   T       ]       =     1   -       (     1   -     ⅇ     -     d     sat_i   ⁢   _grid   ⁢     _k     /     d   iono                   )     2               (   17   )               
where
 
d sat     —     i,grid     —     k =great circle distance between the satellite pierce point and the grid point
 
d iono =correlation distance of the nominal ionospheric delay  (18)
 
     The 1×3 covariance matrix P sat     —     i     —     grid , which defines the correlations between satellite i and each of the grid points, is 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           
                             sat_i 
                             ⁢ 
                             _grid 
                           
                         
                         = 
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   norm_i 
                                 
                                 ⁢ 
                                 
                                   x 
                                   grid 
                                   T 
                                 
                               
                               ] 
                             
                           
                           = 
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   k 
                                   
                                     sat_i 
                                     ⁢ 
                                     _grid 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   x 
                                   grid 
                                 
                                 ⁢ 
                                 
                                   x 
                                   grid 
                                   T 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             k 
                             
                               sat_i 
                               ⁢ 
                               _grid 
                             
                             T 
                           
                           ⁢ 
                           
                             P 
                             grid 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Therefore the weighting vector k sat     —     grid  can be found by the processor  16  as follows:
 
 k   sat     —     i     —     grid   T   =P   sat     —     i     —     grid   P   grid   −1   (20)
 
     Combining (14) and (16), the processor  16  can obtain the normalized vertical delays directly from the three independent reference delays as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             
                               norm_iono 
                               ⁢ 
                               _i 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             k 
                             
                               sat_i 
                               ⁢ 
                               _grid 
                             
                             T 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             k 
                             iono_i 
                             T 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Thus, the weighting vector is:
 
 k   iono     —     i   T =k sat     —     i     —     grid   T U grid   (22)
 
     The processor  16  can form a vector of N normalized pseudo-range iono delays from (21) as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δρ 
                           
                             norm_ion 
                             ⁢ 
                             o 
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     k 
                                     
                                       sat_ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _grid 
                                     
                                     T 
                                   
                                 
                               
                               
                                 
                                   
                                     k 
                                     
                                       sat_ 
                                       ⁢ 
                                       2 
                                       ⁢ 
                                       _grid 
                                     
                                     T 
                                   
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     k 
                                     
                                       sat_N 
                                       ⁢ 
                                       _grid 
                                     
                                     T 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             K 
                             sat_grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The actual (non-normalized) delay along the line of sight can be obtained by the processor  16  by scaling the normalized delay by the sigma value for that satellite based on the geomagnetic latitude of the pierce-point and obliquity factor as defined in DO-229. In vector form, the processor  16  yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δρ 
                           iono 
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     iono_N 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             δρ 
                             norm_iono 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     iono_N 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           Γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             x 
                             ref 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     The ionospheric delay error covariance matrix may be defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           iono 
                         
                         = 
                         
                           E 
                           ⁡ 
                           
                             [ 
                             
                               
                                 δρ 
                                 iono 
                               
                               ⁢ 
                               
                                 δρ 
                                 iono 
                                 T 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ΓK 
                             sat_grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   x 
                                   ref 
                                 
                                 ⁢ 
                                 
                                   x 
                                   ref 
                                   T 
                                 
                               
                               ] 
                             
                           
                           ⁢ 
                           
                             U 
                             grid 
                             T 
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                             T 
                           
                           ⁢ 
                           
                             Γ 
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ΓK 
                             sat_grid 
                           
                           ⁢ 
                           
                             P 
                             grid 
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                             T 
                           
                           ⁢ 
                           
                             Γ 
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           R 
                           iono 
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   σ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     1 
                                   
                                   2 
                                 
                               
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         iono_N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   σ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     2 
                                   
                                   2 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         iono_N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         iono_N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   E 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         δρ 
                                         
                                           
                                             iono 
                                             ⁢ 
                                             _ 
                                           
                                           ⁢ 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         δρ 
                                         iono_N 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   σ 
                                   iono_N 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The rest of the pseudo-range measurement errors are assumed to be uncorrelated with a composite one-sigma value denoted by σ other     —     i  for satellite i. For simplicity, the processor  16  can assume that the one-sigma value for each satellite is a constant six meters. The total measurement error covariance matrix is then: 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       W 
                       
                         - 
                         1 
                       
                     
                     = 
                     
                       
                         R 
                         iono 
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 σ 
                                 
                                   
                                     other 
                                     ⁢ 
                                     _ 
                                   
                                   ⁢ 
                                   1 
                                 
                                 2 
                               
                             
                             
                               0 
                             
                             
                               … 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 σ 
                                 
                                   
                                     other 
                                     ⁢ 
                                     _ 
                                   
                                   ⁢ 
                                   2 
                                 
                                 2 
                               
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋱ 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               … 
                             
                             
                               0 
                             
                             
                               
                                 σ 
                                 other_N 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     In a snapshot RAIM approach, the correlations between satellites are computed directly without the use of a grid. Computing the correlations between satellites directly may be both simpler and slightly more accurate. 
     Specifically, ionospheric error covariance may be modeled as a function of the great circle distance between the pierce points along the ionospheric shell (350 km above the earth&#39;s surface):
 
 E[ρ   iono     —     i δρ iono     —     j ]=σ iono     —     i τ iono     —     j [1−(1 −e   −d     ij     /d     iono   ) 2 ]
 
     where: 
     d ij =great circle distance between pierce points for sats i and j 
     d iono =de-correlation distance=4000 km 
     Ionospheric errors are highly correlated. As such:
 
 W=R   −1 =( R   iono   +R   other ) −1  
 
     where: 
     
       
         
           
             
               
                 
                   
                     R 
                     iono 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             σ 
                             
                               
                                 iono 
                                 ⁢ 
                                 _ 
                               
                               ⁢ 
                               1 
                             
                             2 
                           
                         
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   iono_N 
                                 
                               
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             σ 
                             
                               
                                 iono 
                                 ⁢ 
                                 _ 
                               
                               ⁢ 
                               2 
                             
                             2 
                           
                         
                         
                           … 
                         
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   iono_N 
                                 
                               
                               ] 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   iono_N 
                                 
                               
                               ] 
                             
                           
                         
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   δρ 
                                   
                                     
                                       iono 
                                       ⁢ 
                                       _ 
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   δρ 
                                   iono_N 
                                 
                               
                               ] 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             σ 
                             iono_N 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 
                   
                     R 
                     other 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             `σ 
                             
                               
                                 other 
                                 ⁢ 
                                 _ 
                               
                               ⁢ 
                               1 
                             
                             2 
                           
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             σ 
                             
                               
                                 other 
                                 ⁢ 
                                 _ 
                               
                               ⁢ 
                               2 
                             
                             2 
                           
                         
                         
                           … 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                           
                             σ 
                             other_N 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Error Covariance for the Weighted Least Squares Solution 
     At a step  340 , the processor  16  computes a weighted least-squares solution. The error in the post-updated solution is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                             x 
                             ^ 
                           
                         
                         = 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               ^ 
                             
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     H 
                                     T 
                                   
                                   ⁢ 
                                   WH 
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               H 
                               T 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                               ρ 
                               ~ 
                             
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Substituting (2) into (27) yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                             
                               
                                   
                               
                               ⁢ 
                               x 
                             
                             ^ 
                           
                         
                         = 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     H 
                                     T 
                                   
                                   ⁢ 
                                   WH 
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               H 
                               T 
                             
                             ⁢ 
                             
                               W 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     x 
                                   
                                   + 
                                   δρ 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     H 
                                     T 
                                   
                                   ⁢ 
                                   WH 
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               H 
                               T 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             δρ 
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   T 
                                 
                                 ⁢ 
                                 WH 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             H 
                             T 
                           
                           ⁢ 
                           W 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           δρ 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           S 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           δρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     Thus, the solution matrix S maps the pseudo-range errors into the post-updated solution error vector. The solution error covariance matrix may be defined as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         = 
                         
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   x 
                                   ^ 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   T 
                                 
                               
                               ] 
                             
                           
                           = 
                           
                             
                               SE 
                               ⁡ 
                               
                                 [ 
                                 
                                   δρδρ 
                                   T 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               S 
                               T 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             SW 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             S 
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   T 
                                 
                                 ⁢ 
                                 WH 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             H 
                             T 
                           
                           ⁢ 
                           
                             WW 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               WH 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     T 
                                   
                                   ⁢ 
                                   WH 
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 H 
                                 T 
                               
                               ⁢ 
                               WH 
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     The x and y horizontal position errors are statistically described by the upper 2×2 portion of P. The major and minor axes of the horizontal position error ellipse are equal to the square roots of the maximum and minimum eigenvalues of this 2×2 matrix and represent the one-sigma errors in the corresponding directions. Thus, the one-sigma error in the worst-case direction is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           horz_max 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             λ 
                             max 
                             
                               P 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     1 
                                     : 
                                     2 
                                   
                                   , 
                                   
                                     1 
                                     : 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 
                                   p 
                                   11 
                                 
                                 + 
                                 
                                   p 
                                   22 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           p 
                                           11 
                                         
                                         - 
                                         
                                           p 
                                           22 
                                         
                                       
                                       2 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       p 
                                       12 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The one-sigma error in the vertical position is given by:
 
σ vert =√{square root over ( p   33 )}  (31)
 
     Horizontal Figure of Merit is a conservative 95% fault-free error bound and may be computed by the processor  16  as the 2D RMS error from the error covariance matrix
 
HFOM=2√{square root over ( P (1,1)+ P (2,2))}{square root over ( P (1,1)+ P (2,2))}
 
     Similarly, the Vertical Figure of Merit may be computed by the processor  16  as the 2-sigma vertical error from the error covariance matrix
 
VFOM=2√{square root over ( P (3,3))}
 
     Snapshot Solution Separation RAIM 
     At a step  350 , the processor  16  computes at least one protection level value. In doing so, the processor  16  can employ a snapshot solution separation algorithm. Snapshot solution separation RAIM is analogous to a hybrid Kalman filter solution separation FDE. A main snapshot least-squares solution may be computed by the processor  16  using all N satellites in view (plus barometric altitude, if desired). In addition, N sub-solutions may be computed using each combination of N−1 satellites (plus altitude, if desired). Satellite failure detection occurs when the separation between the main position solution and one of the sub-solutions exceeds a threshold based on the expected statistical separation. Likewise, N−1 sub-sub-solutions may be computed by the processor  16  for each sub-solution using each combination of N−2 satellites which excludes the satellite excluded from the parent sub-solution plus one other. (Again altitude may also be used as an additional measurement). Isolation and exclusion of the failed satellite is accomplished by the processor  16  by comparing the separation of each sub-solution from each of its sub-sub-solutions against a threshold based on the expected statistical separation. In order to meet the time-to-detect requirement of eight seconds, fault detection may be performed and the RAIM HPL may be calculated by the processor  16  at least every four seconds. 
     As long as there are at least five satellites being tracked, the n th  sub-solution can be computed by the processor  16  by zeroing out the n th  row of the measurement matrix. If the processor  16  designates the measurement matrix of the sub-solution as H 0n , then the resulting least-squares sub-solution is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               x 
                               ^ 
                             
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   W 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                                 ⁢ 
                                 
                                   H 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             H 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                           ⁢ 
                           
                             W 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             S 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     where 
     H 0n =geometly matrix with n th  row zeroed 
     R 0n =covariance matrix R with n th  row and column deleted 
     W 0n =R 0n   −1  with n th  zero row and column inserted 
     and where the weighted least-squares sub-solution matrix S 0n  is:
 
 S   0n =( H   0n   T   WH   0n ) −1   H   0n   T   W   (33)
 
     Note that the n th  column of S 0n  will contain all zeros. If the processor  16  designates the main solution with the subscript 00, then the separation between the main solution and sub-solution 0n is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           dx 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         = 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               00 
                             
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                               ρ 
                               ~ 
                             
                           
                           - 
                           
                             
                               S 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                               ρ 
                               ~ 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 S 
                                 00 
                               
                               - 
                               
                                 S 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             dS 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Where dS 0n  is referred to as the separation solution matrix. 
     RAIM Fault Detection 
     In order to detect a satellite failure, the horizontal position solution separation (i.e., the discriminator) between each sub-solution and the main solution may be compared by the processor  16  to a detection threshold. The discriminator for sub-solution 0j is given by:
 
 d   0n   =|d{circumflex over (x)}   0n (1:2)|=√{square root over ([ d{circumflex over (x)}   0n (1)] 2   +[d{circumflex over (x)}   0n (2)] 2 )}  (35)
 
     The detection threshold is set by the processor  16  to a multiple of the one-sigma separation in the worst case direction in order to give a false detection rate of 10 −5 /hr. Using (34) the processor  16  can express the solution separation in terms of the normalized pseudo-range errors as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           dx 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         = 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               00 
                             
                           
                           - 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 ^ 
                               
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   00 
                                 
                               
                             
                             ) 
                           
                           - 
                           
                             ( 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 x 
                               
                               + 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     ^ 
                                   
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               00 
                             
                           
                           - 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               x 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             δρ 
                           
                           - 
                           
                             
                               S 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                             ⁢ 
                             δρ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 S 
                                 00 
                               
                               - 
                               
                                 S 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           δρ 
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             dS 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           δρ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The covariance of this solution separation is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           dP 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         = 
                         
                           E 
                           ⁡ 
                           
                             [ 
                             
                               
                                 dx 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                               · 
                               
                                 dx 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                                 T 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             dS 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 δρδρ 
                                 T 
                               
                               ] 
                             
                           
                           ⁢ 
                           
                             dS 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             dS 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           
                             RdS 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 S 
                                 00 
                               
                               - 
                               
                                 S 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     S 
                                     00 
                                   
                                   - 
                                   
                                     S 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           E 
                           ⁡ 
                           
                             [ 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       00 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                       
                                         ρ 
                                         _ 
                                       
                                       00 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       
                                         0 
                                         ⁢ 
                                         n 
                                       
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                       
                                         ρ 
                                         _ 
                                       
                                       
                                         0 
                                         ⁢ 
                                         n 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           S 
                                           _ 
                                         
                                         00 
                                       
                                       ⁢ 
                                       δ 
                                       ⁢ 
                                       
                                         
                                           ρ 
                                           _ 
                                         
                                         00 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           S 
                                           _ 
                                         
                                         
                                           0 
                                           ⁢ 
                                           n 
                                         
                                       
                                       ⁢ 
                                       δ 
                                       ⁢ 
                                       
                                         
                                           ρ 
                                           _ 
                                         
                                         
                                           0 
                                           ⁢ 
                                           n 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             
                               RS 
                               00 
                               T 
                             
                           
                           + 
                           
                             
                               S 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                             ⁢ 
                             
                               RS 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                               T 
                             
                           
                           - 
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             
                               RS 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                               T 
                             
                           
                           - 
                           
                             
                               S 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                             ⁢ 
                             
                               RS 
                               00 
                               T 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Note that: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             
                               RS 
                               00 
                               T 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         H 
                                         00 
                                         T 
                                       
                                       ⁢ 
                                       
                                         WH 
                                         00 
                                       
                                     
                                     ) 
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 W 
                               
                               ] 
                             
                             ⁢ 
                             
                               
                                 W 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   
                                     
                                       WH 
                                       00 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           00 
                                           T 
                                         
                                         ⁢ 
                                         
                                           WH 
                                           00 
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 ( 
                                 
                                   
                                     H 
                                     00 
                                     T 
                                   
                                   ⁢ 
                                   
                                     WH 
                                     00 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 
                                   WH 
                                   00 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     H 
                                     00 
                                     T 
                                   
                                   ⁢ 
                                   
                                     WH 
                                     00 
                                   
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 
                                   WH 
                                   00 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             00 
                           
                           ⁢ 
                           
                             RS 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       H 
                                       00 
                                       T 
                                     
                                     ⁢ 
                                     
                                       W 
                                       00 
                                     
                                     ⁢ 
                                     
                                       H 
                                       00 
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 H 
                                 00 
                                 T 
                               
                               ⁢ 
                               
                                 W 
                                 00 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
                               W 
                               00 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   W 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       H 
                                       
                                         0 
                                         ⁢ 
                                         n 
                                       
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           
                                             0 
                                             ⁢ 
                                             n 
                                           
                                           T 
                                         
                                         ⁢ 
                                         
                                           W 
                                           
                                             0 
                                             ⁢ 
                                             n 
                                           
                                         
                                         ⁢ 
                                         
                                           H 
                                           
                                             0 
                                             ⁢ 
                                             n 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 
                                   W 
                                   00 
                                 
                                 ⁢ 
                                 
                                   H 
                                   00 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             H 
                             00 
                             T 
                           
                           ⁢ 
                           
                             W 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               
                                 H 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     W 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                   
                                   ⁢ 
                                   
                                     H 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     But, it can be shown that
 
H 00   T W 0n H 0n =H 0n   T W 0n H 0n   (40)
 
     Substituting into (39) yields: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             00 
                           
                           ⁢ 
                           
                             RS 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 
                                   W 
                                   00 
                                 
                                 ⁢ 
                                 
                                   H 
                                   00 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             H 
                             
                               0 
                               ⁢ 
                               n 
                             
                             T 
                           
                           ⁢ 
                           
                             W 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               
                                 H 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                     T 
                                   
                                   ⁢ 
                                   
                                     W 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                   
                                   ⁢ 
                                   
                                     H 
                                     
                                       0 
                                       ⁢ 
                                       n 
                                     
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ( 
                             
                               
                                 H 
                                 00 
                                 T 
                               
                               ⁢ 
                               
                                 W 
                                 00 
                               
                               ⁢ 
                               
                                 H 
                                 00 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             S 
                             00 
                           
                           ⁢ 
                           
                             RS 
                             00 
                             T 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     Taking the transpose of each side, the processor  16  gets:
 
S 0n RS 00   T =S 00 RS 00   T   (42)
 
     Substituting (41) and (42) into (37), the processor  16  gets: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           dP 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         = 
                         
                           
                             
                               S 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                             
                             ⁢ 
                             
                               RS 
                               
                                 0 
                                 ⁢ 
                                 n 
                               
                               T 
                             
                           
                           - 
                           
                             
                               S 
                               00 
                             
                             ⁢ 
                             
                               RS 
                               00 
                               T 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   H 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   W 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                                 ⁢ 
                                 
                                   H 
                                   
                                     0 
                                     ⁢ 
                                     n 
                                   
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                           - 
                           
                             
                               ( 
                               
                                 
                                   H 
                                   00 
                                   T 
                                 
                                 ⁢ 
                                 
                                   W 
                                   00 
                                 
                                 ⁢ 
                                 
                                   H 
                                   00 
                                 
                               
                               ) 
                             
                             
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             P 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                           - 
                           
                             P 
                             00 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     The x and y horizontal separations are statistically described by the upper 2×2 portion of dP. The major and minor axes of the horizontal position separation ellipse are equal to the square roots of the maximum and minimum eigenvalues of this 2×2 matrix and represent the one-sigma separations in the corresponding directions. Thus, the one-sigma separation in the worst-case direction is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           σ 
                           
                             d 
                             
                               0 
                               ⁢ 
                               n 
                             
                           
                         
                         = 
                         
                           
                             λ 
                             max 
                             
                               
                                 dP 
                                 
                                   0 
                                   ⁢ 
                                   n 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     1 
                                     : 
                                     2 
                                   
                                   , 
                                   
                                     1 
                                     : 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 
                                   dp 
                                   11 
                                 
                                 + 
                                 
                                   dp 
                                   22 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           dp 
                                           11 
                                         
                                         - 
                                         
                                           dp 
                                           22 
                                         
                                       
                                       2 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       dp 
                                       12 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     The processor  16  can assume that the separation is dominant along this worst case direction and thus the distribution can be considered Gaussian. 
     The detection threshold is computed by the processor  16  using the allowed false alarm probability and a Normal distribution assumption as follows: 
     
       
         
           
             
               
                 
                   
                     D 
                     
                       0 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       
                         σ 
                         
                           d 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                       
                       ⁢ 
                       
                         
                           Q 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               p 
                               fa 
                             
                             
                               2 
                               ⁢ 
                               N 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         σ 
                         
                           d 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                       
                       · 
                       
                         
                           K 
                           fa 
                         
                         ⁡ 
                         
                           ( 
                           N 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     where: 
     p fa =probability of false alert per independent sample 
     N=Number of sub-solutions (i.e., number of satellites being tracked) 
     and Q −1  is the inverse of: 
     
       
         
           
             
               
                 
                   
                     Q 
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         1 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             π 
                           
                         
                       
                       ⁢ 
                       
                         
                           ∫ 
                           z 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   u 
                                   2 
                                 
                               
                               / 
                               2 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             u 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           
                             1 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                           ⁢ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               z 
                             
                             ⁢ 
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     
                                       u 
                                       2 
                                     
                                   
                                   / 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 u 
                               
                             
                           
                         
                       
                       = 
                       
                         1 
                         - 
                         
                           F 
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     The function F(z) is the well known standard normal distribution function. 
     Horizontal failure detection may be declared by the processor  16  when a discriminator exceeds its threshold, according to the following: 
     For each active sub-solution 0n 
     If |d 0n |&gt;D on , then 
     DETECTION=TRUE 
     End If 
     End For 
     RAIM HPL 
     The RAIM horizontal protection level (HPL) is the largest horizontal position error in the least-squares position solution that can occur and that meets the required probability of missed detection of 0.001. For a given sub-solution 0n, this occurs when the separation between the sub-solution and the main solution is just under the detection threshold and the sub-solution position error, which may be called a 0n , is at its 99.9% performance bound. Thus, the HPL for sub-solution 0n is:
 
 a   0n =σ δr     0n     Q   −1 ( p   md )=σ δr   0n   k   md   (47)
 
     where Q −1  is as defined in (46) and:
 
σ δr     0n   =√{square root over (λ P     0n     (1:2,1:2) )}  (48)
 
     and λ P     0n     (1:2,1:2)  is the maximum eigenvalue of the 2×2 horizontal position elements of the sub-solution covariance matrix P 0n  computed by the processor  16  from the normalized geometry matrix  H   0n  as follows:
 
 P   0n =( H   0n   T   WH   0n ) −1   (49)
 
     Note that the processor  16  may only consider one side of the distribution, since the failure biases the distribution to one side. Evaluating the K md  for 99.9% Gaussian bound the processor  16  gets:
 
K md =3.1  (50)
 
     The HPL for each active sub-solution 0n, is then computed by the processor  16  as:
 
HPL RAIM ( n )= D   0n   +a   0n   (51)
 
     The overall HPL and the hardest-to-detect satellite ID (needed for fault insertion test purposes) are computed by the processor  16  as follows:
 
HPL RAIM =0
 
     For each active sub-solution 0n
 
HPL RAIM =max(HPL RAIM ,HPL RAIM ( n ))  (52)
 
If HPL RAIM =HPL RAIM (n), then
 
 i   SVID     —     hard     —     det     —     horz   f   ix ( n )  (53)
 
End If
 
End for
 
     where f ix (n) is an array that maps the sub-solution number n to the satellite ID. A graphical illustration of HPL determination is provided in  FIG. 2 . 
     RAIM HUL 
     The Horizontal Uncertainty Level (HUL) is an estimate of horizontal position that bounds the true horizontal position error with a probability greater than or equal to 0.999. A conservative 0.999 bound on the horizontal position error of the primary solution can be computed by the processor  16  by simply adding the sub-solution horizontal position 0.999 error bound a 0n  to the horizontal separation between the primary solution and the sub-solution. This will bound the error even in the case of a failure on the satellite excluded by that sub-solution. To bound the error for any satellite failure, the processor  16  may compute the HUL for all sub-solutions and take the worst case. 
     The HUL is thus computed by the processor  16  as follows:
 
HUL RAIM =max( d   0n   +a   0n ), n= 1,N  (54)
 
     where a 0n  is as defined in (47). 
     The HUL can be substituted for the HPL when a failure is detected. In this way, the failure continues to be bounded until the faulty satellite is isolated and excluded. 
     Vertical Position Error Detection 
     For protection of vertical position, the discriminator is:
 
 d   0n   vert   =|dx   0n (3)|
 
     For vertical error detection, the sigma of the vertical position error is computed by the processor  16  from the third diagonal of the covariance:
 
σ d     0n     vert =√{square root over ( dP   0n (3,3))}
 
     The threshold set to meet allowed false detection probability is: 
     
       
         
           
             
               D 
               
                 d 
                 
                   0 
                   ⁢ 
                   n 
                 
               
               vert 
             
             = 
             
               
                 
                   σ 
                   
                     d 
                     
                       0 
                       ⁢ 
                       n 
                     
                   
                   vert 
                 
                 ⁢ 
                 
                   
                     Q 
                     
                       - 
                       1 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         p 
                         fa 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         N 
                       
                     
                     ) 
                   
                 
               
               = 
               
                 
                   σ 
                   
                     d 
                     
                       0 
                       ⁢ 
                       n 
                     
                   
                   vert 
                 
                 · 
                 
                   
                     K 
                     fd 
                   
                   ⁡ 
                   
                     ( 
                     N 
                     ) 
                   
                 
               
             
           
         
       
     
     Vertical Protection Level (VPL) 
     The error bound of the sub-solution vertical position may be determined by the processor  16  from the sub-solution error covariance matrix:
 
σ δr     0n     vert =√{square root over ( P   0n (3,3))}
 
 a   0n   vert =σ δr     0n     vert   Q   −1 ( p   md )=σ δr     0n     vert   K   md  
 
     VPL may be computed by the processor  16  by adding this bound to the threshold for each sub-solution and taking the worst case:
 
VPL=max( D   0n   vert   +a   0n   vert )
 
     RAIM Fault Isolation and Exclusion 
     Fault isolation for RAIM may be performed by the processor  16  by performing fault detection on the separations between each sub-solution and its corresponding sub-sub-solutions. 
     Sub-sub-solution nm which excludes satellites n and m is: 
                     Δ   ⁢           ⁢       x   ^       n   ⁢           ⁢   m         =       ⁢         (       H   nm   T     ⁢     W     n   ⁢           ⁢   m       ⁢     H   nm       )       -   1       ⁢     H   nm   T     ⁢     W   nm     ⁢   Δ   ⁢           ⁢     ρ   ~                   =       ⁢       S   nm     ⁢   Δ   ⁢           ⁢     ρ   ~                   
where:
         H nm =geometry matrix with n th  and m th  row zeroed   R nm =R with n th  and m th  rows and columns deleted   W nm =R 0n   −1  with n th  and m th  zero row and column inserted       

     Solution separation between sub and sub-sub-solution is: 
     
       
         
           
             
               
                 
                   
                     dx 
                     nm 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                       
                       - 
                       
                         Δ 
                         ⁢ 
                         
                           
                             x 
                             ^ 
                           
                           nm 
                         
                       
                     
                     = 
                     
                       
                         
                           S 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           ρ 
                           ~ 
                         
                       
                       - 
                       
                         
                           S 
                           nm 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           ρ 
                           ~ 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           S 
                           
                             0 
                             ⁢ 
                             n 
                           
                         
                         - 
                         
                           S 
                           nm 
                         
                       
                       ) 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ρ 
                       ~ 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       dS 
                       nm 
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       ρ 
                       ~ 
                     
                   
                 
               
             
           
         
       
     
     The horizontal position solution separation vector for each active sub-solutions 0n and each of its corresponding active sub-sub-solutions (nm for m&gt;n or mn for n&gt;m) may be calculated by the processor  16  as:
 
 dx   nm ( i )=Δ x   0n ( i )×Δ x   nm ( i );  i= 1, 2 ; m&gt;n   (55)
 
 dx   nm ( i )=Δ x   0n ( i )−Δ x   mn ( i );  i= 1, 2 ; n&gt;m   (56)
 
     The horizontal discriminator (the horizontal solution separation distance) for each active sub-solution may be calculated by the processor  16  using the x and y position separation states:
 
 d   nm =√{square root over ([ dx   nm (1)] 2   +[dx   nm (2)] 2 )}{square root over ([ dx   nm (1)] 2   +[dx   nm (2)] 2 )}  (57)
 
     As for detection, the threshold is based on the separation covariance matrix. For each active sub-solution 0n and each of its corresponding active sub-sub-solutions (nm for m&gt;n or mn for n&gt;m), the separation covariance matrix is computed by the processor  16  as:
 
 dP   nm   =P   nm   −P   0n   ; m&gt;n   (58)
 
 dP   nm   =P   mn   −P   0n   ; n&gt;m   (59)
 
     Again, the horizontal position error is an elliptical distribution in the x-y plane. The error along any one axis is normally distributed. For the purpose of setting a threshold to meet the required false alarm rate and to calculate the protection level, the processor  16  makes the worst case assumption that the error is entirely along the semi-major axis of this ellipse. The variance in this worst case direction is given by the maximum eigenvalue λ dP     max     (1:2,1:2)  of the 2×2 matrix formed from the horizontal position elements of the separation covaniance. Thus, the horizontal position separation uncertainty in the worst case direction for each sub-solution is computed by the processor  16  as follows:
 
σ d     nm   =√{square root over (λ max   dP     nm     (1:2,1:2) )}  (60)
 
     Each threshold is computed by the processor  16  as follows: 
     
       
         
           
             
               
                 
                   
                     D 
                     nm 
                   
                   = 
                   
                     
                       
                         σ 
                         
                           d 
                           nm 
                         
                       
                       ⁢ 
                       
                         
                           Q 
                           
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               p 
                               fa 
                             
                             
                               2 
                               ⁢ 
                               
                                 ( 
                                 
                                   N 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         σ 
                         
                           d 
                           nm 
                         
                       
                       · 
                       
                         
                           K 
                           fa 
                         
                         ⁡ 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
     
     where p fa , N, Q −1 , and K fa  are as previously defined for detection. 
     Exclusion proceeds in an analogous manner to detection with the sub-solution taking the role of the main solution and the sub-sub-solution taking the role of the sub-solution. For each sub-solution, the separation from each of its sub-sub-solutions is determined by the processor  16  and tested against its threshold. 
     The exclusion logic is as follows: 
     If sub-solution n is separated from at least one of its sub-sub-solutions by more than its threshold, then satellite n cannot be the failed satellite (i.e. the failed satellite is still included in sub-solution n). On the other hand, if the separation of sub-solution r from each of its sub-sub-solutions is less than the exclusion threshold, then satellite r is the failed satellite (providing each of the other sub-solutions is separated from at least one of its sub-sub-solutions by more than its threshold). 
     This is equivalent to saying that satellite r is isolated as the failed satellite if and only if:
 
d rm &lt;D rm ; for all m≠r
 
and
 
d nm ≧D nm ; for at least one m≠n for all n≠r
 
     Exclusion of a vertical position failure may be accomplished by the processor  16  in a similar manner. 
     The following algorithm executed by the processor  16  will accomplish the isolation as described above: 
     
       
         
               
             
           
               
                   
               
             
             
               
                 If (DETECTION_HORZ) 
               
               
                  ISOLATION_HORZ = FALSE 
               
               
                  PREVIOUS_ISOLATION_HORZ = FALSE 
               
               
                  For each active sub-solution 0n 
               
               
                   If |d nm   horz | &lt; D nm   horz  for all m (active sub-sub-solutions only), then 
               
               
                    If (PREVIOUS_ISOLATION_HORZ), then 
               
               
                     ISOLATION_HORZ = FALSE 
               
               
                    Else 
               
               
                     ISOLATION_HORZ = TRUE 
               
               
                     PREVIOUS_ISOLATION_HORZ = TRUE 
               
               
                     k failed     —     svid  = f ix  (n) 
               
               
                    End if 
               
               
                   End if 
               
               
                  End for 
               
               
                 End if 
               
               
                 If (ISOLATION_HORZ), then 
               
               
                  SAT_STATUS = ISOLATION 
               
               
                 End if 
               
               
                   
               
             
          
         
       
     
     When the failure is isolated, the satellite is marked by the processor  16  as failed, all the solutions excluding satellite are dropped, and the satellite will be retried in one hour. 
     RAIM HEL 
     RAIM HEL is computed by the processor  16  in a fashion analogous to that of the Kalman filter solution separation. At exclusion, the horizontal position solution of sub-solution 0n is separated by the processor  16  from sub-sub-solution nm by D nm  (see equation (61)). The sub-solution position error with respect to the true position is thus D nm  plus the sub-sub-solution position error (assuming, in the worst case, that they are in opposite directions). The sub-solution position error bound a nm  can be determined by the processor  16  from the maximum eigenvalue λ P     nm     (1:2,1:2)  of the 2×2 matrix formed from the horizontal position error elements of the sub-sub-solution error covariance matrix. That is:
 
 a   nm =σ Υr     nm     Q   −1 ( p   md )=σ δr     nm     K   md   (62)
 
     where Q −1  is as previously defined and
 
σ δr     nm   =√{square root over (λ P     nm     (1:2,1:2) )}  (63)
 
     Again, the processor  16  may only consider one side of the distribution, since the failure biases the distribution to one side. The allowed probability of missed detection is 1.0 e-3  and is assumed by the processor  16  to be the same as the allowable probability of failed exclusion. The sigma multiplier K md  is as previously defined for the RAIM HPL calculation. The RAIM HEL for each active sub-sub-solution nm of sub-solution 0n is then computed by the processor  16  as:
 
HEL RAIM ( nm )= D   nm   +a   nm   (64)
 
     Note that the HEL does not consider the failure-free rare normal condition, since exclusion assumes that a failure is present. The overall HEL and the hardest-to-exclude satellite ID (needed for fault insertion test purposes) may be computed by the processor  16  as follows:
 
HEL RAIM ( nm )=0
 
     For each active sub-solution 0n 
     For each active sub-sub-solution nm of sub-solution 0n
 
HEL RAIM =max(HEL RAIM ,HEL RAIM ( nm ))  (65)
 
     If HEL RAIM =HEL RAIM (nm), then
 
 i   SVID     —     hard     —     excl     —     horz   =f   ix ( n )  (66)
 
     End If 
     End for 
     End for 
     As such:
 
HEL=max(max( D   nm   +a   nm )), n=1, . . . N, m=1, . . . N
 
Similarly:
 
VEL=max(max( D   nm   vert   +a   nm   vert )), n=1, . . . N, m=1, . . . N
 
     Ionospheric Error Model Calculations 
     Determination of Ionospheric Grid Points and Pierce Point Coordinates 
     For a Kalman filter approach, and in order to utilize ( 17 ), the processor  16  may first determine the coordinates of each gridpoint and the coordinates of the satellite&#39;s ionospheric pierce point. Then, using those two sets of coordinates, the great circle distance between the pierce point and the grid point can be calculated by the processor  16 . For either a Kalman filter or snapshot RAIM approach, knowing the coordinates of a point i (e.g., the system illustrated in  FIG. 1 , or “user”) and the distance and azimuth from point i to a point j (e.g. the gridpoint), the coordinates of point j can be determined by the processor  16  as follows: 
     
       
         
           
             
               
                 
                   
                     λ 
                     j 
                   
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               λ 
                               i 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ψ 
                               ij 
                             
                           
                           + 
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               λ 
                               i 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ψ 
                               ij 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               ij 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       λ 
                                       i 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ψ 
                                       ij 
                                     
                                   
                                   - 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       λ 
                                       i 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ψ 
                                       ij 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       A 
                                       ij 
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               
                                 ψ 
                                 ij 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               
                                 A 
                                 ij 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .1 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Λ 
                     j 
                   
                   = 
                   
                     
                       Λ 
                       i 
                     
                     + 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ψ 
                               ij 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               A 
                               ij 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 i 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ψ 
                                 ij 
                               
                             
                             - 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 i 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ψ 
                                 ij 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 A 
                                 ij 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .2 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           λ 
                           i 
                         
                         = 
                           
                         ⁢ 
                         
                           Geodetic 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           latitude 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                       
                     
                   
                   
                     
                       
                         
                           λ 
                           j 
                         
                         = 
                           
                         ⁢ 
                         
                           Geodetic 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           latitude 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           Λ 
                           i 
                         
                         = 
                           
                         ⁢ 
                         
                           Geodetic 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           longitude 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                       
                     
                   
                   
                     
                       
                         
                           Λ 
                           j 
                         
                         = 
                           
                         ⁢ 
                         
                           Geodetic 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           longitude 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             A 
                             ij 
                           
                           = 
                             
                           ⁢ 
                           
                             Azimuth 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             angle 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               bearing 
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             from 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             point 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             to 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             point 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             j 
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             ψ 
                             ij 
                           
                           = 
                             
                           ⁢ 
                           
                             Angular 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             distance 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   earth 
                                   &#39; 
                                 
                                 ⁢ 
                                 s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 central 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 angle 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           from 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           to 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             d 
                             ij 
                           
                           
                             
                               R 
                               e 
                             
                             + 
                             
                               h 
                               l 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           d 
                           ij 
                         
                         = 
                           
                         ⁢ 
                         
                           Great 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           circle 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           distance 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           from 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           to 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           point 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           R 
                           e 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             Radius 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             of 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             the 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             earth 
                           
                           = 
                           
                             6378 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             km 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           h 
                           l 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             Height 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             of 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             the 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ionosphere 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             thin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             shell 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             model 
                           
                           = 
                           
                             350 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             km 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The coordinates of the ionospheric pierce point of the satellite can also be calculated using (A.1) and (A.2). In this case, ψ ij  represents the central angle from the user location to the pierce point and may be calculated by the processor  16  as follows: 
     
       
         
           
             
               
                 
                   
                     ψ 
                     ij 
                   
                   = 
                   
                     
                       π 
                       2 
                     
                     - 
                     E 
                     - 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               R 
                               e 
                             
                             
                               
                                 R 
                                 e 
                               
                               + 
                               
                                 h 
                                 l 
                               
                             
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           E 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .3 
                   
                   ) 
                 
               
             
           
         
       
     
     where E is the elevation angle of the satellite from the user location with respect to the local tangent plane. 
     Computing Elevation and Azimuth Angles of Satellite 
     The elevation angle E of a satellite is defined as the angle the line-of-sight vector makes with the user&#39;s local tangent (horizontal) plane. The azimuth angle A of the satellite is the angle of the line-of-sight vector with respect to true north as measured in the horizontal plane. Thus, we have the following 
     
       
         
           
             
               
                 
                   E 
                   = 
                     
                   ⁢ 
                   
                     ATAN 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           - 
                           
                             u 
                             LOS_z 
                           
                         
                         , 
                         
                           
                             
                               u 
                               LOS_x 
                               2 
                             
                             + 
                             
                               u 
                               LOS_y 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .4 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     0 
                   
                   = 
                   
                     
                       ATAN 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                       ⁢ 
                       
                         ( 
                         
                           
                             u 
                             LOS_y 
                           
                           , 
                           
                             u 
                             LOS_x 
                           
                         
                         ) 
                       
                     
                     + 
                     α 
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .5 
                   
                   ) 
                 
               
             
             
               
                 
                   A 
                   = 
                   
                     { 
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 A 
                                 0 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               - 
                               π 
                             
                             ≤ 
                             
                               A 
                               0 
                             
                             &lt; 
                             π 
                           
                         
                       
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               A 
                               0 
                             
                             ≥ 
                             π 
                           
                         
                       
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   A 
                                   0 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               A 
                               0 
                             
                             &lt; 
                             π 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .6 
                   
                   ) 
                 
               
             
           
         
       
     
     where 
     u LOS     —     x , u LOS     —     y , u LOS     —     z =x, y, and z components of the line-of-sight vector u LOS   L    
     α=wander angle (angle in azimuth from north to the x local-level frame axis) 
     Note that the azimuth angle is adjusted by ±2π so that the result is between −π and +π. 
     Determination of Great Circle Distance 
     The great circle distance along the ionospheric thin shell model from a point i (e.g. satellite pierce point) to another point j (e.g. grid point) may be calculated by the processor  16  as follows: 
     
       
         
           
             
               
                 
                   
                     d 
                     ij 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           R 
                           e 
                         
                         + 
                         
                           h 
                           I 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                 ⁢ 
                                 
                                   λ 
                                   j 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   2 
                                 
                                 ⁢ 
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   Λ 
                                   ij 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         λ 
                                         i 
                                       
                                       ⁢ 
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         λ 
                                         j 
                                       
                                     
                                     - 
                                     
                                       sin 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         λ 
                                         i 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         λ 
                                         j 
                                       
                                       ⁢ 
                                       cos 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         Λ 
                                         ij 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 i 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 j 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ΔΛ 
                                 ij 
                               
                             
                             + 
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 i 
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 λ 
                                 j 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .7 
                   
                   ) 
                 
               
             
           
         
       
     
     where
 
ΔΛ ij =Λ j −Λ i,  
 
     Ionospheric Variance Model 
     The algorithm that may be executed by the processor  16  for calculation of the ionospheric model error variance may be from ICD-GPS-200C and DO-229D J.2.3. Note that the symbols in this section are unique to this section. 
     Using the satellite&#39;s elevation angle E, form the earth&#39;s central angle between the user position and the earth projections of ionospheric pierce point ψ pp  using equation (A.3). 
     Next, using the satellite&#39;s elevation angle E, the azimuth angle A, the earth&#39;s central angle ψ pp  and the user geodetic latitude λ u  and longitude Λ u  determine the pierce point geodetic latitude φ pp  and longitude λ pp  using equations (A.1) and (A.2). 
     Form the absolute value of the geomagnetic latitude of the ionospheric pierce point.
 
|λ m ∥=λ pp +0.064π cos(Λ pp −1.617π)|radians  (A.8)
 
     Form an estimate of the vertical delay error based on geomagnetic latitude 
     
       
         
           
             
               
                 
                   
                     τ 
                     vert 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 9 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 meters 
                               
                               , 
                             
                           
                         
                         
                           
                               
                             ⁢ 
                             
                               
                                  
                                 
                                   λ 
                                   m 
                                 
                                  
                               
                               ≤ 
                               
                                 20 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 degrees 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 4.5 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 meters 
                               
                               , 
                             
                           
                         
                         
                           
                               
                             ⁢ 
                             
                               
                                 22.5 
                                 ⁢ 
                                 ° 
                               
                               &lt; 
                               
                                  
                                 
                                   λ 
                                   m 
                                 
                                  
                               
                               ≤ 
                               
                                 55 
                                 ⁢ 
                                 ° 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 6 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 meters 
                               
                               , 
                             
                           
                         
                         
                           
                               
                             ⁢ 
                             
                               
                                  
                                 
                                   λ 
                                   m 
                                 
                                  
                               
                               &gt; 
                               
                                 55 
                                 ⁢ 
                                 ° 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .9 
                   
                   ) 
                 
               
             
           
         
       
     
     Using the elevation angle E, calculate the square of the obliquity factor. 
     
       
         
           
             
               
                 
                   
                     F 
                     pp 
                     2 
                   
                   = 
                   
                     1 
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             
                               
                                 R 
                                 e 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   E 
                                   ) 
                                 
                               
                             
                             
                               
                                 R 
                                 e 
                               
                               + 
                               
                                 h 
                                 l 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .10 
                   
                   ) 
                 
               
             
           
         
       
     
     Form the modeled estimated variance of the ionospheric delay.
 
σ model   2 =F pp   2 τ vert   2   (A.11)
 
     Form the estimated variance using the compensation that is applied if available from the receiver. (If not, assume zero). 
     
       
         
           
             
               
                 
                   
                     σ 
                     comp 
                     2 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           CT 
                           IONO 
                         
                         5 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .12 
                   
                   ) 
                 
               
             
           
         
       
     
     Form the estimated variance of the ionospheric delay.
 
τ iono   2 =max(τ model   2 ,τ comp   2 )  (A.13)
 
     While the preferred embodiment of the invention has been illustrated and described, as noted above, many changes can be made without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is not limited by the disclosure of the preferred embodiment. Instead, the invention should be determined entirely by reference to the claims that follow.