PATENT 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 parity space technique, determine at least one protection level value based on the error covariance matrix.

PATENT DESCRIPTION
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims priority from U.S. Provisional Appl. No. 61/012,303 entitled “RAIM WITH SPATIALLY 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 parity space 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 process according to an embodiment of the invention; 
         FIG. 3  depicts a noise scatter that would occur if there was a bias on the most difficult to detect satellite in the presence of the expected noise; and 
         FIG. 4  shows an embodiment to rotate the parity space such that the parity error due to a bias failure on satellite k is entirely along axis  1  of the parity space. 
     
    
    
     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. 2  illustrates a process  200 , according to an embodiment of the invention, that can be implemented in the radio navigation system illustrated in  FIG. 1 . The process  200  is illustrated as a set of operations or steps shown as discrete blocks. The process  200  may be implemented in any suitable hardware, software, firmware, or combination thereof. As such the process  200  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. 2 , at a step  210 , the processor  16  computes the sigma (error) values on pseudo-range and measurements. 
     At a step  220 , 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 
                             
                               N 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                           
                         
                       
                       
                         
                           - 
                           
                             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 
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   linearization 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   point 
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     At a step  230 , 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 
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                         = 
                         
                           ρ 
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ρ 
                           
                           - 
                           
                             ρ 
                             ^ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ρ 
                             ~ 
                           
                           - 
                           
                             ρ 
                             ^ 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             H 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             x 
                           
                           + 
                           
                             δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ρ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   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 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 σ 
                                 2 
                                 2 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       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 
                                 
                                 ) 
                               
                               
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               H 
                               T 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ρ 
                               ~ 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           S 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             ~ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   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 
                                     
                                       N 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       x 
                                     
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     LOS 
                                     Ny 
                                   
                                 
                               
                             
                             
                               
                                 0 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     LOS 
                                     Nz 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 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 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 σ 
                                 2 
                                 2 
                               
                             
                             
                               … 
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       1 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       N 
                                     
                                   
                                   ] 
                                 
                               
                             
                             
                               
                                 E 
                                 ⁡ 
                                 
                                   [ 
                                   
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       2 
                                     
                                     ⁢ 
                                     δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ρ 
                                       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     —     i   T ]=1−(1 −e   −d grid/diono) 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   T   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     —     1   x   grid     —     k ]=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 
                             ⁡ 
                             
                               [ 
                               
                                 δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ρ 
                                   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 U grid   (22)
 
     The processor  16  can form a vector of N normalized pseudo-range iono delays from (21) as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             
                               norm_iono 
                               ⁢ 
                               _i 
                             
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     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: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             iono 
                           
                         
                         = 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       1 
                                     
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     
                                       iono_ 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   … 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     σ 
                                     iono_N 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ρ 
                             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 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           Γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                           
                           ⁢ 
                           
                             U 
                             grid 
                           
                           ⁢ 
                           
                             E 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   x 
                                   ref 
                                 
                                 ⁢ 
                                 
                                   x 
                                   ref 
                                   T 
                                 
                               
                               ] 
                             
                           
                           ⁢ 
                           
                             U 
                             grid 
                             T 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                             T 
                           
                           ⁢ 
                           
                             Γ 
                             T 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           Γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                           
                           ⁢ 
                           
                             P 
                             grid 
                           
                           ⁢ 
                           
                             K 
                             sat_grid 
                             T 
                           
                           ⁢ 
                           
                             Γ 
                             T 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     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 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     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[δρ   iono     —     i δρ iono     —     j ]=σ iono     —     i σ 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  240 , 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))}
 
     Parity Space RAIM 
     At a step  250 , the processor  16  computes at least one protection level value. In doing so, the processor  16  can employ a parity space technique. Recall that the weighted least squares solution may be expressed as: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       x 
                       ^ 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           
                             H 
                             T 
                           
                           ⁢ 
                           WH 
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       H 
                       T 
                     
                     ⁢ 
                     W 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       ρ 
                       ~ 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     S 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Δ 
                     ⁢ 
                     
                       ρ 
                       ~ 
                     
                   
                 
               
             
             
               
                 
                   W 
                   = 
                     
                   ⁢ 
                   
                     
                       R 
                       
                         - 
                         1 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           E 
                           ⁡ 
                           
                             [ 
                             
                               δρδρ 
                               T 
                             
                             ] 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
             
           
         
       
     
     The correlated measurement set can be transformed by the processor  16  into a set that is uncorrelated by factoring, using a matrix factorization method known in the art, the weighting matrix W:
 
W=L T L
 
     Where L is the lower triangular square root of W. This results in: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       x 
                       ^ 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               H 
                               T 
                             
                             ⁢ 
                             
                               L 
                               T 
                             
                             ⁢ 
                             LH 
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         H 
                         T 
                       
                       ⁢ 
                       
                         L 
                         T 
                       
                       ⁢ 
                       L 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         ρ 
                         ~ 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 ( 
                                 LH 
                                 ) 
                               
                               T 
                             
                             ⁢ 
                             LH 
                           
                           ] 
                         
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           LH 
                           ) 
                         
                         T 
                       
                       ⁢ 
                       L 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         ρ 
                         ~ 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 H 
                                 _ 
                               
                               T 
                             
                             ⁢ 
                             
                               H 
                               _ 
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           H 
                           _ 
                         
                         T 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         ρ 
                         _ 
                       
                     
                     = 
                     
                       
                         S 
                         _ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                         ρ 
                         _ 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   H 
                   _ 
                 
               
               = 
               LH 
             
             , 
             
               
                 Δ 
                 ⁢ 
                 
                   ρ 
                   _ 
                 
               
               = 
               
                 L 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                   ρ 
                   ~ 
                 
               
             
             , 
             
               
                 S 
                 _ 
               
               = 
               
                 
                   
                     ( 
                     
                       
                         
                           H 
                           _ 
                         
                         T 
                       
                       ⁢ 
                       
                         H 
                         _ 
                       
                     
                     ) 
                   
                   
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   H 
                   _ 
                 
               
             
           
         
       
     
     The covariance of the transformed measurement errors is: 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁡ 
                     
                       [ 
                       
                         δ 
                         ⁢ 
                         
                           ρ 
                           _ 
                         
                         ⁢ 
                         δ 
                         ⁢ 
                         
                           
                             ρ 
                             _ 
                           
                           T 
                         
                       
                       ] 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     E 
                     ⁡ 
                     
                       [ 
                       
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           δρδρ 
                           T 
                         
                         ⁢ 
                         
                           L 
                           T 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         LW 
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         L 
                         T 
                       
                     
                     = 
                     
                       
                         
                           L 
                           ⁡ 
                           
                             ( 
                             
                               
                                 L 
                                 T 
                               
                               ⁢ 
                               L 
                             
                             ) 
                           
                         
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         L 
                         T 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       L 
                       ⁡ 
                       
                         ( 
                         
                           
                             L 
                             
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             L 
                             
                               - 
                               T 
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       L 
                       T 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     I 
                     N 
                   
                 
               
             
           
         
       
     
     Thus, it is seen that the transformed measurements are uncorrelated, each with unity variance. Multiplying our measurement equation by L the processor  16  gets:
 
 LΔ{tilde over (ρ)}=LHΔx+Lδρ 
 
Δ  ρ =   H Δx+δ  ρ   
 
     An N×N orthogonal matrix Q can be found by the processor  16 , such that: 
     
       
         
           
             
               Q 
               ⁢ 
               
                 H 
                 _ 
               
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           q 
                           11 
                         
                       
                       
                         
                           q 
                           12 
                         
                       
                       
                         … 
                       
                       
                         
                           q 
                           
                             1 
                             ⁢ 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           q 
                           21 
                         
                       
                       
                         
                           q 
                           22 
                         
                       
                       
                         … 
                       
                       
                         
                           q 
                           
                             2 
                             ⁢ 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           q 
                           31 
                         
                       
                       
                         
                           q 
                           32 
                         
                       
                       
                         … 
                       
                       
                         
                           q 
                           
                             3 
                             ⁢ 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           q 
                           41 
                         
                       
                       
                         
                           q 
                           42 
                         
                       
                       
                         … 
                       
                       
                         
                           q 
                           
                             4 
                             ⁢ 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           q 
                           51 
                         
                       
                       
                         
                           q 
                           52 
                         
                       
                       
                         … 
                       
                       
                         
                           q 
                           
                             5 
                             ⁢ 
                             N 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           q 
                           
                             N 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                       
                         
                           q 
                           
                             N 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       
                         
                           q 
                           
                             N 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             3 
                           
                         
                       
                       
                         
                           q 
                           NN 
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   H 
                   _ 
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           A 
                         
                       
                       
                         
                           B 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                     H 
                     _ 
                   
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             u 
                             11 
                           
                         
                         
                           
                             u 
                             12 
                           
                         
                         
                           
                             u 
                             13 
                           
                         
                         
                           
                             u 
                             14 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             u 
                             22 
                           
                         
                         
                           
                             u 
                             23 
                           
                         
                         
                           
                             u 
                             24 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             u 
                             33 
                           
                         
                         
                           
                             u 
                             34 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             u 
                             44 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           U 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Thus, if the processor  16  multiplies the measurement equation by Q, the result is: 
     
       
         
           
             
               Q 
               ⁢ 
               
                   
               
               ⁢ 
               Δ 
               ⁢ 
               
                 ρ 
                 _ 
               
             
             = 
             
               
                 
                   Q 
                   ⁢ 
                   
                     H 
                     _ 
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 + 
                 
                   Q 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   δ 
                   ⁢ 
                   
                     
                       ρ 
                       _ 
                     
                     ⁢ 
                     
                       
 
                     
                     [ 
                     
                       
                         
                           A 
                         
                       
                       
                         
                           B 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                     ρ 
                     _ 
                   
                 
               
               = 
               
                 
                   
                     [ 
                     
                       
                         
                           U 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 + 
                 
                   
                     [ 
                     
                       
                         
                           A 
                         
                       
                       
                         
                           B 
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   δ 
                   ⁢ 
                   
                     ρ 
                     _ 
                   
                 
               
             
           
         
       
     
     Where A is the upper 4×N portion of Q and B is the lower (N−4)×N portion. The following two equations result:
 
 AΔ  ρ =UΔx+Aδ  ρ   
 
BΔ  ρ =Bδ  ρ =p
 
     The first equation can be used to solve for the estimated least squares solution by setting the pseudo-range error to zero:
 
AΔ  ρ =UΔ{circumflex over (x)}           Δ{circumflex over (x)}=U −1 AΔ  ρ 

     This equation is a more efficient than the previously given one since it only requires inverting an upper-triangular matrix. 
     The second equation gives us the parity vector p which is non-zero only when there are pseudo-range errors. In the absence of a failure, the parity covariance is: 
     
       
         
           
             
               
                 
                   
                     E 
                     ⁡ 
                     
                       [ 
                       
                         pp 
                         T 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         BE 
                         ⁡ 
                         
                           [ 
                           
                             δ 
                             ⁢ 
                             
                               ρ 
                               _ 
                             
                             ⁢ 
                             δ 
                             ⁢ 
                             
                               
                                 ρ 
                                 _ 
                               
                               T 
                             
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         B 
                         T 
                       
                     
                     = 
                     
                       BB 
                       T 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     I 
                     
                       N 
                       - 
                       4 
                     
                   
                 
               
             
           
         
       
     
     Thus the parity elements are also uncorrelated zero mean Gaussian random variables with unity variance. 
     In an embodiment, the processor  16  employs a chi-square method using the concept of pbias. In such an embodiment, the processor  16  uses the square of the parity magnitude as the discriminator (test statistic) d as follows:
 
 d=p   T   p=p   1   2   +p   2   2   + . . . +p   N−4   2  
 
     The discriminator will then have a central chi-square distribution with N−4 degrees of freedom. The processor  16  places a threshold on this discriminator above which a failure is declared. This threshold D is computed by the processor  16  from the chi-square probability density function to yield an allowable false alarm probability. 
     Once the threshold has been set, the question becomes, with a failure on a single satellite that results in the discriminator just reaching the threshold, how large the resulting position error can be while meeting the probability of missed detection. A bias failure ε on the kth satellite will result in a solution error and a parity magnitude error of: 
     
       
         
           
             
               δ 
               ⁢ 
               
                 x 
                 ^ 
               
             
             = 
             
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 δρ 
               
               = 
               
                 
                   S 
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           ɛ 
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         s 
                                         
                                           1 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           k 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         s 
                                         
                                           2 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           k 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   s 
                                   
                                     3 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     k 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             s 
                             
                               4 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                             
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   ɛ 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                      
                     p 
                      
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         p 
                         T 
                       
                       ⁢ 
                       p 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         δρ 
                         T 
                       
                       ⁢ 
                       
                         L 
                         T 
                       
                       ⁢ 
                       
                         B 
                         T 
                       
                       ⁢ 
                       BL 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       δρ 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     ɛ 
                     ⁢ 
                     
                       
                         Λ 
                         kk 
                       
                     
                   
                 
               
             
             
               
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Λ 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         
                           B 
                           _ 
                         
                         T 
                       
                       ⁢ 
                       
                         B 
                         _ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         B 
                         _ 
                       
                     
                     = 
                     BL 
                   
                 
               
             
           
         
       
     
     Thus the parity bias is related to the horizontal position error through the following slope: 
     
       
         
           
             
               slope 
               ⁡ 
               
                 ( 
                 k 
                 ) 
               
             
             = 
             
               
                 
                   
                     s 
                     
                       1 
                       ⁢ 
                       k 
                     
                     2 
                   
                   + 
                   
                     s 
                     
                       2 
                       ⁢ 
                       k 
                     
                     2 
                   
                 
               
               
                 
                   Λ 
                   kk 
                 
               
             
           
         
       
     
     The satellite with the largest slope is the most difficult to detect. This slope is referred to as Slope max .  FIG. 3  depicts a noise scatter that would occur if there was a bias on the most difficult to detect satellite in the presence of the expected noise. The specific bias that results in the percentage of data to the left of the detection threshold D equal to the missed detection probability is of particular interest. The parity magnitude associated with this bias is called “pbias.” 
     With the bias present, the discriminator (square of the parity magnitude) has a non-central chi-square distribution with N−4 degrees of freedom. It can be shown that the non-centrality parameter λ of the chi-square distribution is:
 
λ=pbias 2  
 
     Thus, using the non-central chi-square probability density function, the processor  16  can determine the value for pbias which meets the required probability of missed detection. 
     The Horizontal Protection Level (HPL) is then:
 
HPL=Slope max   ·p bias
 
     Gaussian Method by Rotation of the Parity Space 
     With reference to  FIG. 4 , through additional orthogonal transformations, an embodiment can rotate the parity space such that the parity error due to a bias failure on satellite k is entirely along axis  1  of the parity space. 
     
       
         
           
             
               
                 
                   
                     p 
                     k 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       Q 
                       k 
                     
                     ⁢ 
                     
                       B 
                       _ 
                     
                     ⁢ 
                     δρ 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         B 
                         _ 
                       
                       k 
                     
                     ⁢ 
                     δρ 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 b 
                                 _ 
                               
                               1 
                               k 
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 b 
                                 _ 
                               
                               k 
                               k 
                             
                           
                           
                             … 
                           
                           
                             
                               
                                 b 
                                 _ 
                               
                               N 
                               k 
                             
                           
                         
                         
                           
                             X 
                           
                           
                             ⋮ 
                           
                           
                             0 
                           
                           
                             ⋮ 
                           
                           
                             X 
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             X 
                           
                           
                             ⋮ 
                           
                           
                             0 
                           
                           
                             ⋮ 
                           
                           
                             X 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             ɛ 
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     b 
                                     _ 
                                   
                                   k 
                                 
                                 ) 
                               
                               T 
                             
                           
                         
                         
                           
                             X 
                           
                         
                         
                           
                             X 
                           
                         
                         
                           
                             X 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             ɛ 
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         b 
                         _ 
                       
                       k 
                       k 
                     
                     ⁢ 
                     ɛ 
                   
                 
               
             
           
         
       
     
     Since the bias only shows up on axis  1 , the result is a scalar, and the discriminator, in general, can be determined by the processor  16  using the following:
 
 d   k   =p   1   k =(   b     k ) T δρ
 
     With no failure, only correlated random errors w are present:
 
d k =(  b   k ) T  w=(b k ) T  Lw=(b k ) T   w 
         where B k =Q k B=  B   k L −1 ,  w =Lw, E[  w   w   T ]=I N      Note: σ d     k     2 =E[(d k ) 2 ]=E[(b k ) T   w   w b k ]=E[(b k ) T b k ]=1       

     With a bias failure on satellite k plus the correlated random errors w on each of the satellites, the discriminator can be determined by the processor  16  using the following:
 
 d   k   =  b     k   k ε+( b   k ) T     w   
 
     The impact of the noise in addition to failure on the horizontal position is: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       h 
                     
                   
                   = 
                   
                     
                       
                         S 
                         h 
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ρ 
                     
                     = 
                     
                       
                         
                           
                             s 
                             k 
                             h 
                           
                           ⁢ 
                           ɛ 
                         
                         + 
                         
                           
                             S 
                             h 
                           
                           ⁢ 
                           w 
                         
                       
                       = 
                       
                         
                           
                             s 
                             k 
                             h 
                           
                           ⁢ 
                           ɛ 
                         
                         + 
                         
                           
                             
                               S 
                               _ 
                             
                             h 
                           
                           ⁢ 
                           Lw 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         s 
                         k 
                         h 
                       
                       ⁢ 
                       ɛ 
                     
                     + 
                     
                       
                         
                           S 
                           _ 
                         
                         h 
                       
                       ⁢ 
                       
                         w 
                         _ 
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             where S h ,  S   h =1 st  two rows of S and  S , and s k   h =column k of S h    
           
         
       
    
     Using the Gaussian probability density function, the threshold D, which meets the probability of false alarm, can be determined by the processor  16 . At detection, the discriminator magnitude is equal to the threshold:
 
| d   k   |=|  b     k   k ε+( b   k ) T     w |=D 
 
     Assuming the failure is positive and much larger than the noise: 
     
       
         
           
             
               
                 
                    
                   
                     
                       b 
                       _ 
                     
                     k 
                     k 
                   
                    
                 
                 ⁢ 
                 ɛ 
               
               + 
               
                 
                   
                     ( 
                     
                       b 
                       k 
                     
                     ) 
                   
                   T 
                 
                 ⁢ 
                 
                   w 
                   _ 
                 
               
             
             = 
             
               
                 D 
                 ⇒ 
                 ɛ 
               
               = 
               
                 
                   D 
                   
                      
                     
                       
                         b 
                         _ 
                       
                       k 
                       k 
                     
                      
                   
                 
                 - 
                 
                   
                     
                       
                         ( 
                         
                           b 
                           k 
                         
                         ) 
                       
                       T 
                     
                     ⁢ 
                     
                       w 
                       _ 
                     
                   
                   
                      
                     
                       
                         b 
                         _ 
                       
                       k 
                       k 
                     
                      
                   
                 
               
             
           
         
       
     
     The resulting horizontal position error can be determined by the processor  16  using the following: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       h 
                     
                   
                   = 
                   
                     
                       
                         
                           s 
                           k 
                           h 
                         
                         ⁢ 
                         ɛ 
                       
                       + 
                       
                         
                           
                             S 
                             _ 
                           
                           h 
                         
                         ⁢ 
                         
                           w 
                           _ 
                         
                       
                     
                     = 
                     
                       
                         
                           s 
                           k 
                           h 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               D 
                               
                                  
                                 
                                   
                                     b 
                                     _ 
                                   
                                   k 
                                   k 
                                 
                                  
                               
                             
                             - 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       b 
                                       k 
                                     
                                     ) 
                                   
                                   T 
                                 
                                 ⁢ 
                                 
                                   w 
                                   _ 
                                 
                               
                               
                                  
                                 
                                   
                                     b 
                                     _ 
                                   
                                   k 
                                   k 
                                 
                                  
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             S 
                             _ 
                           
                           h 
                         
                         ⁢ 
                         
                           w 
                           _ 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         s 
                         k 
                         h 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             D 
                             
                                
                               
                                 
                                   b 
                                   _ 
                                 
                                 k 
                                 k 
                               
                                
                             
                           
                           - 
                           
                             
                               1 
                               
                                  
                                 
                                   
                                     b 
                                     _ 
                                   
                                   k 
                                   k 
                                 
                                  
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                 
                                   b 
                                   i 
                                   k 
                                 
                                 ⁢ 
                                 
                                   
                                     w 
                                     _ 
                                   
                                   i 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             s 
                             _ 
                           
                           i 
                           h 
                         
                         ⁢ 
                         
                           
                             w 
                             _ 
                           
                           i 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The position error magnitude in the direction of the failure is: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       k 
                       h 
                     
                   
                   = 
                   
                     
                       
                         
                           s 
                           k 
                           h 
                         
                         
                            
                           
                             s 
                             k 
                             h 
                           
                            
                         
                       
                       · 
                       δ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       h 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                          
                         
                           s 
                           k 
                           h 
                         
                          
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             D 
                             
                                
                               
                                 
                                   b 
                                   _ 
                                 
                                 k 
                                 k 
                               
                                
                             
                           
                           - 
                           
                             
                               1 
                               
                                  
                                 
                                   
                                     b 
                                     _ 
                                   
                                   k 
                                   k 
                                 
                                  
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 N 
                               
                               ⁢ 
                               
                                 
                                   b 
                                   i 
                                   k 
                                 
                                 ⁢ 
                                 
                                   
                                     w 
                                     _ 
                                   
                                   i 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             
                               s 
                               k 
                               h 
                             
                             · 
                             
                               
                                 s 
                                 _ 
                               
                               i 
                               h 
                             
                           
                           
                              
                             
                               s 
                               k 
                               h 
                             
                              
                           
                         
                         ⁢ 
                         
                           
                             w 
                             _ 
                           
                           i 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           
                              
                             
                               s 
                               k 
                               h 
                             
                              
                           
                           
                              
                             
                               
                                 b 
                                 _ 
                               
                               k 
                               k 
                             
                              
                           
                         
                         ⁢ 
                         D 
                       
                       
                         ︸ 
                         Bias 
                       
                     
                     + 
                     
                       
                         
                           1 
                           
                              
                             
                               
                                 b 
                                 _ 
                               
                               k 
                               k 
                             
                              
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         s 
                                         k 
                                         h 
                                       
                                       · 
                                       
                                         
                                           s 
                                           _ 
                                         
                                         i 
                                         h 
                                       
                                     
                                     
                                        
                                       
                                         s 
                                         k 
                                         h 
                                       
                                        
                                     
                                   
                                   ⁢ 
                                   
                                      
                                     
                                       
                                         b 
                                         _ 
                                       
                                       k 
                                       k 
                                     
                                      
                                   
                                 
                                 - 
                                 
                                   
                                      
                                     
                                       s 
                                       k 
                                       h 
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     b 
                                     i 
                                     k 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 w 
                                 _ 
                               
                               i 
                             
                           
                         
                       
                       
                         ︸ 
                         Noise 
                       
                     
                   
                 
               
             
           
         
       
     
     Since the random pseudo-range error is uncorrelated with unity variance, the variance of the noise term about the mean can be determined by the processor  16  using the following: 
     
       
         
           
             
               σ 
               
                 δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   r 
                   k 
                   h 
                 
               
             
             = 
             
               
                 
                   1 
                   
                      
                     
                       
                         b 
                         _ 
                       
                       k 
                       k 
                     
                      
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     N 
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 s 
                                 k 
                                 h 
                               
                               · 
                               
                                 
                                   s 
                                   _ 
                                 
                                 i 
                                 h 
                               
                             
                             
                                
                               
                                 s 
                                 k 
                                 h 
                               
                                
                             
                           
                           ⁢ 
                           
                              
                             
                               
                                 b 
                                 _ 
                               
                               k 
                               k 
                             
                              
                           
                         
                         - 
                         
                           
                              
                             
                               s 
                               k 
                               h 
                             
                              
                           
                           ⁢ 
                           
                             b 
                             i 
                             k 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
               
             
           
         
       
     
     The Horizontal Protection Level for satellite k can be determined by the processor  16  using the following: 
     
       
         
           
             
               
                 
                   
                     HPL 
                     k 
                   
                   = 
                   
                     
                       
                         
                           s 
                           k 
                           h 
                         
                         
                            
                           
                             s 
                             k 
                             h 
                           
                            
                         
                       
                       ⁢ 
                       D 
                     
                     + 
                     
                       
                         K 
                         md 
                       
                       ⁢ 
                       
                         σ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             k 
                             h 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         
                           s 
                           k 
                           h 
                         
                         
                            
                           
                             s 
                             k 
                             h 
                           
                            
                         
                       
                       ⁢ 
                       
                         K 
                         fa 
                       
                     
                     + 
                     
                       
                         K 
                         md 
                       
                       ⁢ 
                       
                         σ 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             k 
                             h 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             where K md  and K fa  are sigma multipliers set to meet the probabilities of false alarm and missed detection 
           
         
       
    
     This process is repeated for all N satellites and the total HPL can be determined by the processor  16  using the following:
 
HPL=max(HPL k ), k=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 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           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 
                                   ′ 
                                 
                                 ⁢ 
                                 s 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 central 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 angle 
                               
                               ) 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             from 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             point 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             to 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             point 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             j 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           = 
                             
                           ⁢ 
                           
                             
                               d 
                               ij 
                             
                             
                               
                                 R 
                                 c 
                               
                               + 
                               
                                 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 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .2 
                   
                   ) 
                 
               
             
           
         
       
     
     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 
                   = 
                   
                     
                       A 
                       ⁢ 
                       TAN 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                       ( 
                       
                         
                           - 
                           
                             u 
                             LOS_z 
                           
                         
                         , 
                         
                           
                             
                               u 
                               LOS_x 
                               2 
                             
                             + 
                             
                               u 
                               LOS_y 
                               2 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     ⁢ 
                     .4 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     0 
                   
                   = 
                   
                     
                       
                         A 
                         ⁢ 
                         TAN 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       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 
                           l 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 j 
                               
                               ⁢ 
                               
                                 sin 
                                 2 
                               
                               ⁢ 
                               
                                 ΔΛ 
                                 ij 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       λ 
                                       i 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       λ 
                                       j 
                                     
                                   
                                   - 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       λ 
                                       i 
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ΔΛ 
                                       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.
 
σ iono   2 =max(σ model   2 ,σ 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.