Patent Publication Number: US-7720642-B2

Title: Measurement fault detection

Description:
The present application is a continuation of application Ser. No. 10/737,702, filed on Dec. 15, 2003, now U.S. Pat. No. 7,356,445, which is a continuation of application Ser. No. 09/942,060, filed on Aug. 28, 2001, now U.S. Pat. No. 6,691,066 which claims benefit of U.S. application 60/228,535 filed on Aug. 28, 2000. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates in general to the use of variable-gain filters, such as Kalman filters to estimate the state of a system, and in particular to improved measurement fault detection in such filters. 
     2. Description of the Related Art 
     The discrete time Kalman filter was introduced by Rudolf Kalman in a “A New Approach to Linear Filtering and Prediction Problems,” in the Transactions of the ASME, Series D: Journal of Basic Engineering, Vol. 82, pp. 35-45 in 1960. The Kalman filter (KF) produces the state estimate with the smallest mean-square error for a linear time-variant system with white, Gaussian state and measurement noise. The Kalman filter extended the concept of least-squares estimation to dynamic systems. The concept of least-squares estimation dates back to Gauss, at least. The Kalman filter has found innumerable applications in navigation, tracking, process control, parameter estimation, and other areas. It has been extended to include nonlinear state models, nonlinear measurement models, non-white state and measurement noise, and correlation between state and measurement noise. 
     A succinct description of the Kalman filter can be found in  Applied Optimal Estimation  by A. Gelb (ed.) from The MIT Press, 1974 pp. 1-179 which is herein incorporated by reference. A summary of the basic filter and some notes on the notation follow from Gelb. The notation to be used in the description is given in Table 1. A superscripted “T” after a matrix is the matrix transpose, and a superscripted “−1” after a matrix is it&#39;s inverse. Again following from Gelb, the discrete KF processing steps are given in Table 2. These steps advance the state estimate and it&#39;s associated error covariance matrix from one time to another. 
     
       
         
           
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Notation for Kalman Filter Description 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 X t     1     |t     2     
                 State vector estimate for the system at time t 1  using 
               
               
                   
                   
                 measurements up to and including those from time t 2 . 
               
               
                   
                 Φ 
                 State transition matrix 
               
               
                   
                 Q 
                 Process Noise Matrix 
               
               
                   
                 R 
                 Measurement Noise Covariance Matrix 
               
               
                   
                 P t     1     |t     2     
                 State error covariance matrix at time t 1  using 
               
               
                   
                   
                 measurements up to and including time t 2   
               
               
                   
                 Z 
                 Measurement vector, a set of vector measurements for 
               
               
                   
                   
                 a given time. 
               
               
                   
                 H 
                 The measurement model matrix. This is the partial 
               
               
                   
                   
                 derivative of the measurement vector with respect 
               
               
                   
                   
                 to the state vector. 
               
               
                   
                 I 
                 The identity matrix 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Five Discrete Kalman Filtering Processing Steps 
               
            
           
           
               
               
               
            
               
                 Step 
                 Step Name 
                 Step Processing 
               
               
                   
               
               
                 1 
                 State Propagation 
                 X t+Δt|t  = ΦX t|t   
               
               
                 2 
                 Covariance Propagation 
                 P t+Δt|t  = ΦP t|t Φ T  + Q 
               
               
                 3 
                 Gain Calculation 
                 K = P t+Δt|t H T (HP t+Δt|t H T  + R) −1   
               
               
                 4 
                 State Update 
                 X t+Δt|t+Δt  = X t+Δt|t  + K(Z − HX t+Δt|t ) 
               
               
                 5 
                 Covariance Update 
                 P t+Δt|t+Δt  = (I − KH)P t+Δt|t   
               
               
                   
               
            
           
         
       
     
     An implementation issue with the KF is the matrix inverse called for in step 3, the gain calculation, from table 2. Computing the inverse is often a substantial burden, particularly in a real-time system. A common practice, therefore, has been to divide up the processing of the vector measurement (steps 3, 4, and 5 from Table 2) into a sequence of scalar measurement processing. If the measurement vector Z, has one element, in other words when we have a scalar measurement, the measurement derivative with respect to the state, H, is a row vector. In this case the gain calculation, step 3 from Table 2 above reduces to a division rather than a matrix inversion. 
             K   =       PH   T         HPH   T     +   R             
If a measurement vector, Z, contains multiple elements, it can be partitioned into a set of scalar measurements. In the case of a diagonal measurement noise covariance matrix R, partitioning into scalar measurements is no more complicated than picking out the diagonal term from the matrix R, and processing it with the corresponding row of H and the corresponding element of the vector Z. This is generally referred to as sequential measurement processing. We can expand our notation to include sequential measurement processing such that Z A , Z B , . . . are elements of the original vector measurement Z; H A , H B , . . . are the corresponding rows of H; R A , R B , . . . are the corresponding diagonal elements of the noise covariance matrix. The sequential measurement processing KF then follows these steps. The gain and residual calculations have been broken out for discussion of the Chi-square test in subsequent paragraphs. The state and covariance notation has been augmented for sequential processing, such that X t+Δt|t,A  is the state vector estimate for time t+Δt given measurements up to time t and measurement A. This state, updated for measurement A, is the input state for steps 10 and 13 of measurement B processing, and so on.
 
                     TABLE 3                  Discrete Sequential Kalman Filter Processing Steps                                 Repeated               Step   Steps   Step Name   Step Processing               1       State Propagation   X t+Δt|t  = ΦX t|t         2       Covariance Propagation   P t+Δt|t  = ΦP t|t Φ T  + Q       Loop for each   3, 9, 15 ...   Residual Variance   Γ A  = H A PH A   T  + R A         measurement:   4, 10, 16 ...   Residual   δ A  = Z A  − H A X t+Δt|t                 A, B, C ...   5, 11, 17 ...   Chi-Square test statistic               χ   A     2     =         δ   A     2       Γ   A                               6, 12, 18 ...   Gain Calculation   K A  = P t+Δt|t H A   T /Γ A             7, 13, 19 ...   State Update   X t+Δt|t, A  = X t+Δt|t  +K A δ A             8, 14, 20 ...   Covariance Update   P t+Δt|t, A  = (I − KH)P t+Δt|t                      
In the case where R is not diagonal, in other words the measurement noises are correlated, techniques may be used which transform the R matrix to diagonal form for easy partitioning. The necessary transformations are defined by G. Bierman in  Factorization Methods for Discrete Sequential Estimation , Academic Press 1977, pp. 47-55 which is herein incorporated by reference.
 
     It is possible to prove that the order in which a set of sequential measurements is processed in a linear system using the steps of Table 3 is irrelevant to the final state estimate. A simple one-dimensional example of this fact is given in Table 4. In Table 4, the two measurements are taken to be measurements of the state vector directly, thus H is a two-by-two identity matrix. The example is for the sequential measurement processing, not the state propagation, so it corresponds to steps 3 through 14 inclusive of Table 3. The Chi-square test in the example of Table 4, will be described subsequently. There are two processing columns in Table 4, one for measurement order A,B, and the other for B,A. While the intermediate steps are different, the final state and covariance estimates are the same. Thus, in the prior art, the order in which measurements from the same time have been processed in a sequential filter has been a matter of convenience, the measurements are processed in whatever order they are gathered from the measuring device or stored in memory, or some other mechanism. 
     
       
         
           
               
             
               
                 TABLE 4 
               
               
                   
               
               
                 Single State KF Sequential Measurement Processing Example 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 State 
                 Truth: X = 0 Estimate: X t|t−Δt  = 1 
               
               
                 State Variance 
                 P t|t−Δt  = 1 (state estimate correctly modeled) 
               
               
                 Measurement Model 
                 H A  = H B  = H = 1 
               
               
                   
                 R A  = R B  = R = 1 
               
               
                 Measurements 
                 Z A  = −1 (One sigma measurement, correctly modeled) 
               
               
                   
                 Z B  = 2 (Two sigma measurement, correctly modeled) 
               
            
           
           
               
               
               
            
               
                 Measurement Processing 
                 A,B 
                 B,A 
               
               
                 Order 
               
               
                 First Measurement Residual 
                 Γ A  = 1 + 1 = 2 
                 Γ B  = 1 + 1 = 2 
               
               
                 Variance 
               
               
                 First Measurement Residual 
                 δ A  = Z A  − X t|t−Δt  = −2 
                 δ B  = Z B  − X t|t−Δt  = 1 
               
               
                 First Measurement Three 
                 (−2) 2  &lt; (3 2  · 2) ? 
                 (1) 2  &lt; (3 2  · 2) ? 
               
               
                 Sigma Chi-Square Test 
                 Passed 
                 Passed 
               
               
                 First Measurement Gain 
                 K A  = P t|t−Δt /Γ A  = 1/2 
                 K B  = P t|t−Δt /Γ B  = 1/2 
               
               
                 First Measurement State 
                 X t|t−Δt, A  = X t|t−Δt  + K A δ A  = 0 
                 X t|t−Δt, B  = X t|t−Δt  + K B δ B  = 3/2 
               
               
                 Update 
               
               
                 First Measurement State 
                 P t|t−Δt, A  = (1 − K A )P t|t−Δt  = 1/2 
                 P t|t−Δt, B  = (1 − K B )P t|t−Δt  = 1/2 
               
               
                 Variance Update 
               
               
                   
               
               
                 Second Measurement Residual Variance 
                 
                   
                     
                       
                         
                           Γ 
                           B 
                         
                         = 
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             1 
                           
                           = 
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Γ 
                           A 
                         
                         = 
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             1 
                           
                           = 
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Second Measurement 
                 δ B  = Z B  − X t|t−Δt,|A  = 2 
                 δ A  = Z A  − X t|t−Δt, B  = −5/2 
               
               
                 Residual 
               
               
                   
               
               
                 Second Measurement Three Sigma Chi-Square Test 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                                 2 
                               
                               &lt; 
                               
                                 
                                   ( 
                                   
                                     
                                       3 
                                       2 
                                     
                                     · 
                                     
                                       3 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ? 
                               
                             
                           
                         
                         
                           
                             Passed 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     
                                       5 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               &lt; 
                               
                                 
                                   ( 
                                   
                                     
                                       3 
                                       2 
                                     
                                     · 
                                     
                                       3 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ? 
                               
                             
                           
                         
                         
                           
                             Passed 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Second Measurement Gain 
                 K B  = P t|t−Δt, A /Γ B  = 1/3 
                 K A  = P t|t−Δt, B /Γ A  = 1/3 
               
               
                 Second Measurement State 
                 X t|t  = X t|t−Δt, A  + K B δ B  = 2/3 
                 X t|t  = X t|t−Δt, B  + K A δ A  = 2/3 
               
               
                 Update 
               
               
                 State Measurement State 
                 P t|t  = (1 − K B )P t|t−Δt, A  = 1/3 
                 P t|t  = (1 − K A )P t|t−Δt, A  = 1/3 
               
               
                 Variance Update 
               
               
                   
               
            
           
         
       
     
     One of the problems apparent to Kalman filter practitioners is that measurements are often generated by physical mechanisms which can result in measurement faults. These faults, by definition, are components of the measurement values that are not included in the measurement model. So, for example, if the measurement is a pseudorange measurement to a GPS satellite, and the direct ray to the satellite has been blocked but a reflected ray is tracked, the resulting measurement has a fault, errors that fall outside the model used to process the measurement. This condition is shown in  FIG. 2 . Measurement faults have at least two adverse affects; they corrupt the resultant estimate of the state, and their effects are not considered in the update of the error covariance matrix. After processing a measurement with a fault, the state error covariance matrix may no longer be an accurate representation of the state error. 
     This problem led to the development of Chi-square measurement fault detection, sometimes referred to as Chi-square residual testing or innovations variance testing. The purpose is to reject measurements that we believe have been corrupted by a fault, and therefore fall outside of our filter model. The test is a statistical hypothesis test for a random variable. The application of a Chi-square test to the measurement residual dates back, at least, to Mehra and Peschon [5] in a 1971 paper titled “An Innovations Approach to Fault Detection and Diagnosis in Dynamic Systems” in Automatica, Vol. 7, pp. 637-640, 1971. 
     The chi-square residual test is a test on the hypothesis that the measurement we are about to process is actually a member of the measurement set we have defined. We define the residual as:
 
δ= Z−HX   t+Δt|t  
 
The hypothesis is tested by forming the ratio of the observed residual to it&#39;s expected value, thus, for a scalar measurement.
 
               χ   2     =       δ   2         HPH   T     +   R             
The test would be passed if the computed value of chi-squared is less than a threshold value. A typical threshold of nine, for example, would be a three sigma test on the measurement, meaning we would expect 99.7% of normally distributed measurements to pass the rest. So if the value χ 2  computed above is greater than nine, we have detected a failure at the three sigma level. This test is well-known and in common practice.
 
     Since the residual variance, HPH T +R, is also part of the scalar gain calculation, the chi-square test is often conducted in the sequential processing of the scalar measurements. Referring back to Table 3, the residual variance has been computed in step 3, and the Chi-square test statistic was formulated in step 5. Thus the test is computed in-line with the sequential measurement processing. The example in Table 4 shows the application of the test, the ratio is computed and checked before the gain computation and the state update are completed for each measurement. A flowchart of this process is given in  FIG. 1 . 
     The term Satellite Positioning System (SATPS) is used to define a satellite-based system transmitting ranging signals for use in navigation. Examples of SATPS include the Global Positioning System (GPS), the Russian GLONASS system, and the proposed European Galileo system. In all of these systems the receiver develops pseudorange measurements to the transmitting satellites by synchronizing a code replica with the transmitted satellite code. Doppler measurements, and integrated doppler measurements, may be made as well, with integrated doppler measurements commonly being referred to as delta-pseudorange measurements. Since the inception of the GPS, Kalman filtering has been a popular method for estimating the navigation state from the pseudorange and delta-pseudorange measurements. The paper “Aircraft Navigation with the United Operational Phase of the NAVSTAR Global Positioning System” by L. R. Kruczynski in Global Positioning System, Volume 1, pp. 154-165, 1980 from the Institute of Navigation, from his 1977 conference paper, describes a Kalman filter used to process GPS measurements. In satellite positioning system measurement processing there is some prior art on measurement fault detection methodology. Much of this art is concerned with ad hoc attempts to define a measurement edit or weighting strategy to reduce the effects of multipath. In U.S. Pat. No. 5,883,595 by Colley, for example, a mechanism for de-weighting suspect measurements is disclosed based on the size of their residuals and an undisclosed reliability factor. In U.S. Pat. No. 5,590,043 by McBurney some adaptation of filter parameters on the basis of multipath presence is disclosed, but with no mechanism apparent for detection of the multipath. The existence of these methods indicates the relative lack of success of unaltered Chi-square residual test at least in the field of satellite navigation, and calls out the need for improved measurement fault detection and editing. 
     It can be seen then, that there is a need in the art for improved measurement fault detection in state estimation. It can also be seen that there is a need in the art for improved multipath detection in satellite positioning system receivers. 
     SUMMARY OF THE INVENTION 
     To minimize the limitations in the prior art, and to minimize other limitations that will become apparent upon reading and understanding the present specification, the present invention discloses a method and apparatus for improving measurement fault detection in a sequential measurement processing estimator. 
     An object of the present invention is to provide for improved measurement fault detection in state estimation. Another object of the present invention is to provide for improved multipath detection in satellite positioning system receivers. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a flow-chart of a generic sequential measurement processing system for a Kalman filter with Chi-square residual testing; and 
         FIG. 2  is a diagram of a constellation of visible satellite positioning system spacecraft being tracked by a terrestrial used with a measurement corrupted by multipath in the so-called urban canyon environment. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS 
     In the following description of the preferred embodiment, reference is made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration a specific embodiment in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
     Overview 
     The invention improves measurement fault detection by ordering the processing of measurements in order of increasing probability of a measurement fault, where we have some means to assess the relative likelihood of a fault on the measurements to be processed, other than the measurement residual itself. For example, suppose a set of pseudorange measurements is to be processed in a satellite positioning system receiver to obtain an estimate of the current position, as in  FIG. 2 . It appears in this Figure, and it is quite true that satellites at a low elevation angle as seen from the receiver are much more likely to have their transmitted signals affected by multipath and blockage for a terrestrial user. Thus we can say, without quantifying the exact probability of failure, that the lower elevation satellite has a higher probability of a multipath-induced fault on its pseudorange measurement than a higher elevation satellite. 
     Similarly, multipath corrupted measurements are typically observed to have a lower carrier to noise density ratio than uncorrupted measurements, and carrier tracking of a corrupted measurement may result in larger phase-errors at the output of the carrier tracking loop. Thus these two additional parameters, the C/N 0  of the measurement and the number and magnitude of the carrier tracking errors observed over some measurement interval are other indicators that may be used to order measurements for sequential processing with increasing measurement fault probability. These ordering elements could be used individually or in combination to determine the best processing order. 
     The key to the invention is to order the sequence of the measurements presented to the filter for processing so that those with a lower fault probability are processed first. This processing reduces the trace of the state error covariance matrix and increases the probability of fault-detection on subsequent measurements. This has not been recognized in the prior art, Chi-square measurement fault detection induced order dependence into sequential Kalman filter measurement processing. Once we have a measurement with a fault the measurement processing sequence is no longer irrelevant. This is demonstrated in another one-dimensional example in Table 5. Here we have a measurement with a fault and the outcome of the processing cycle is highly dependent on the order in which the measurements are processed. In this case, processing the measurements in the wrong order (BA) results in two estimation system failures, first, the faulty measurement is not detected, then, following the processing of the faulty measurement, a good measurement is rejected because of the damage done to the state estimate. 
     
       
         
           
               
             
               
                 TABLE 5 
               
               
                   
               
               
                 Single-State KF Sequential Measurement Processing with a Measurement Fault 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 Initial State 
                 Truth: X = 0 Estimate: X t|t−Δt  = 1 
               
               
                 Initial State Variance 
                 P t|t−Δt  = 1 (State estimate error is correctly modeled) 
               
               
                 Measurement Model 
                 H A  = H B  = H = 1 
               
               
                   
                 R A  = R B  = R = 1 
               
               
                 Measurements 
                 Z A  = −1 (One sigma measurement, correctly modeled) 
               
               
                   
                 Z B  = 5 (Five sigma measurement-faulty) 
               
            
           
           
               
               
               
            
               
                 Measurement Processing 
                 A,B 
                 B,A 
               
               
                 Order 
               
               
                 First Measurement Residual 
                 Γ A  = 1 + 1 = 2 
                 Γ B  = 1 + 1 = 2 
               
               
                 Variance 
               
               
                 First Measurement Residual 
                 δ A  = Z A  − X t|t−Δt  = −2 
                 δ B  = Z B  − X t|t−Δt  = 4 
               
               
                 First Measurement Three 
                 (−2) 2  &lt; (3 2  · 2) ? 
                 (4) 2  &lt; (3 2  · 2) ? 
               
               
                 Sigma Chi-Square Test 
                 Passed 
                 Passed 
               
               
                 First Measurement Gain 
                 K A  = P t|t−Δt /Γ A  = 1/2 
                 K B  = P t|t−Δt /Γ B  = 1/2 
               
               
                 First Measurement State 
                 X t|t−Δt, A  = X t|t−Δt  + K A δ A  = 0 
                 X t|t−Δt, B  = X t|t−Δt  + K B δ B  = 3 
               
               
                 Update 
               
               
                 First Measurement State 
                 P 0|A  = (1 − K A )P 0  = 1/2 
                 P 0|B  = (1 − K B )P 0  = 1/2 
               
               
                 Variance Update 
               
               
                   
               
               
                 Second Measurement Residual Variance 
                 
                   
                     
                       
                         
                           Γ 
                           B 
                         
                         = 
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             1 
                           
                           = 
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Γ 
                           A 
                         
                         = 
                         
                           
                             
                               1 
                               2 
                             
                             + 
                             1 
                           
                           = 
                           
                             3 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Second Measurement 
                 δ B  = Z B  − X t|t−Δt, A  = 5 
                 δ A  = Z A  − X t|t−Δt, B  = −4 
               
               
                 Residual 
               
               
                   
               
               
                 Second Measurement Three Sigma Chi-Square Test 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   5 
                                   ) 
                                 
                                 2 
                               
                               &lt; 
                               
                                 
                                   ( 
                                   
                                     
                                       3 
                                       2 
                                     
                                     · 
                                     
                                       3 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ? 
                               
                             
                           
                         
                         
                           
                             Failed 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     - 
                                     4 
                                   
                                   ) 
                                 
                                 2 
                               
                               &lt; 
                               
                                 
                                   ( 
                                   
                                     
                                       3 
                                       2 
                                     
                                     · 
                                     
                                       3 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ? 
                               
                             
                           
                         
                         
                           
                             Failed 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 Update State Estimate 
                 X t|t  = 0 
                 X t|t  = 3 
               
               
                 State Estimate Error 
                 0 
                 3 
               
               
                   
               
            
           
         
       
     
     An example of the invention has been given for sequential measurement processing of measurements taken at a single time, however, the method can be extended. Suppose there are multiple measurements available, at different times, but all available prior to the next required reporting time for the state estimate. We can extend the concept of ordered measurement processing to measurements taken at different times by propagating the state and state error covariance estimates backwards and forwards in time to process the measurements with the lowest probability of a fault first. If, for example, we have a scalar measurement at time t 1  and a second measurement at a later time t 2 , and we are not required to produce an updated state estimate until a third time, t 3 . We can process these non-simultaneous measurements in order by increasing fault probability even if the measurement at the later time, t 2 , is the measurement with the lowest probability of a fault. 
     The steps required to process the non-simultaneous measurements are:
         1. Record in a memory measurement  1  at time t 1 .   2. Record measurement  2  at time t 2 .   3. Sort the measurements in order of increasing fault probability, we&#39;ll assume for that the measurement with the lower fault probability is the measurement  2 .   4. Propagate the state and covariance matrix forward from t 0  to t 2      5. Process measurement  2 , updating the state and covariance at time t 2      6. Propagate the state and covariance backwards in time to t 1 , noting that we subtract the relevant process noise matrix from the state error covariance matrix when propagating backwards.   7. Process measurement  1 , updating the state and covariance at time t 1      8. Propagate the state and covariance forward form t 1  to the reporting time t 3          

     CONCLUSION 
     A method of improving measurement fault detection using Chi-square residual test by ordering measurements for sequential processing by increasing probability of a measurement fault has been disclosed. A simple one-dimensional example of the efficacy of the method has been presented, and the application to ordering of measurements for processing in a satellite positioning system receiver has been presented. 
     In summary, the present invention provides a method for improving fault detection. The method comprises recording a plurality of measurements, each measurement having an associated measurement time, sorting the plurality of measurements, wherein the sort of the plurality of measurements is performed in order of increasing fault probability, propagating a state matrix and a covariance matrix forward from the first time to a time of the measurement having a highest fault probability, processing the measurement having the highest fault probability first, continuing processing until all measurements have been processed, and propagating the state matrix and the covariance matrix forward from a last measurement time to a reporting time. The processing comprises updating the state matrix and the covariance matrix at a time of the measurement having the highest fault probability, and propagating the state matrix and the covariance matrix backwards in time to a time of a measurement with a next highest fault probability. 
     The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention not be limited by this detailed description, but by the claims appended hereto.