5 GPS Data Errors 5.1 SELECTIVE AVAILABILITY ERRORS Prior to May 1, 2000, Selective Availability (SA) was a mechanism adopted by the Department of Defense (DoD) to control the achievable navigation accuracy by nonmilitary GPS receivers. In the GPS SPS mode, the SA errors were speci®ed to degrade navigation solution accuracy to 100 m (2D RMS) horizontally and 156 m (RMS) vertically. In a press release on May 1, 2000, the President of the United States announced the decision to discontinue this intentional degradation of GPS signals available to the public. The decision to discontinue SA was coupled with continuing efforts to upgrade the military utility of systems using GPS and supported by threat assess- ments which concluded that setting SA to zero would have minimal impact on United States national security. The decision was part of an ongoing effort to make GPS more responsive to civil and commercial users worldwide. The transition as seen from Colorado Springs, CO., U.S.A. at the GPS Support Center is shown in Figure 5.1. The ®gure shows the horizontal and vertical errors with SA, and after SA was suspended, midnight GMT (8 PM EDT), May 1, 2000. Figure 5.2 shows mean errors with and without SA, with satellite PRN numbers. Aviation applications will probably be the most visible user group to bene®t from the discontinuance of SA. However, precision approach will still require some form of augmentation to ensure that integrity requirements are met. Even though setting SA to zero reduces measurement errors, it does not reduce the need for and design of WAAS and LAAS ground systems and avionics. 103 Global Positioning Systems, Inertial Navigation, and Integration, Mohinder S. Grewal, Lawrence R. Weill, Angus P. Andrews Copyright # 2001 John Wiley & Sons, Inc. Print ISBN 0-471-35032-X Electronic ISBN 0-471-20071-9 Time and frequency users may see greater effects in the long term via commu- nication systems that can realize signi®cant future increases in effective bandwidth use due to tighter synchronization tolerances. The effect on vehicle tracking applications will vary. Tracking in the trucking industry requires accuracy only good enough to locate in which city the truck is, whereas public safety applications can require the precise location of the vehicle. Maritime applications have the potential for signi®cant bene®ts. The personal navigation consumer will bene®t from the availability of simpler and less expensive products, resulting in more extensive use of GPS worldwide. Because SA could be resumed at any time, for example, in time of military alert, one needs to be aware of how to minimize these errors. There are at least two mechanisms to implement SA. Mechanisms involve the manipulation of GPS ephemeris data and dithering the satellite clock (carrier frequency). The ®rst is referred to as epsilon-SA (e-SA), and the second as clock- dither SA. The clock-dither SA may be implemented by physically dithering the frequency of the GPS signal carrier or by manipulating the satellite clock correction data or both. Though the mechanisms to implement SA and the true SA waveform are classi®ed, a variety of SA models exist in the literature (e.g., [4, 15, 21, 134]). These references show various models. One proposed by Braasch [15] appears to be the most promising and suitable. Another used with some success for predicting SA is a Levinson predictor [6]. Fig. 5.1 Pseudorange residuals with SA. 104 GPS DATA ERRORS 11 21 15 25 05 0.9 150m 100m 50m 60ϒ 30ϒ Horizon 150m 100m 50m 60ϒ 30ϒ Horizon Weighted Mean Weighted Mean HDOP < =1.0 HDOP < =2.0 HDOP > 2.0 Display All Logging All HDOP < =1.0 HDOP < =2.0 HDOP > 2.0 Display All Logging All (SA)AWatch SA Watch 23 05 29 30 25 21 11 15 22 23 30 S (no SA) Fig. 5.2 Pseudorange residuals with and without SA. 5.1 SELECTIVE AVAILABILITYERRORS 105 The Braasch model assumes that all SA waveforms are driven by normal white noise through linear system (autoregressive moving-average, ARMA) models (see Chapter 3 of [46]). Using the standard techniques developed in system and parameter identi®cation theory, it is then possible to determine the structure and parameters of the optimal linear system that best describes the statistical character- istics of SA. The problem of modeling SA is estimating the model of a random process (SA waveform) based on the input=output data. The technique to ®nd an SA model involves three basic elements:  the observed SA,  a model structure, and  a criterion to determine the best model from the set of candidate models. There are three choices of model structures: 1. an ARMA model of order ( p; q), which is represented as ARMA( p; q); 2. an ARMA model of order ( p; 0 known as the moving-average MA( p) model; and 3. an ARMA model of order q; 0, the auto regression AR(q) model. Selection from these three models is performed with physical laws and past experience. 5.1.1 Time Domain Description Given observed SA data, the identi®cation process repeatedly selects a model structure and then calculates its parameters. The process is terminated when a satisfactory model, according to a certain criterion, is found. We start with the general ARMA model. Both the AR and MA models can be viewed as special cases of an ARMA model. An ARMA p; q model is mathe- matically described by a 1 y k  a 2 y kÀ1 ÁÁÁa q y kÀq1  b 1 x k  b 2 x kÀ1 ÁÁÁb p x kÀp1  e k 5:1 or in a concise form by  q i1 a i y kÀi1   p j1 b j x kÀj1  e k ; 5:2 where a i ; i  1; 2; .; q, and b j ; j  1; 2; .; p, are the sets of parameters that describe the model structure, x k and y k are the input and output to the model at any time k for k  1; 2; ., and e k is the noise value at time k. Without loss of generality, it is always assumed that a l  1. 106 GPS DATA ERRORS Once the model parameters a i and b j are known, the calculation of y k for an arbitrary k can be accomplished by y k À  q i2 a i y kÀi1   p j1 b j x kÀj1  e k : 5:3 It is noted that when all of the a i in Eq. 5.3 take the value of 0, the model is reduced to the MA  p; 0 model or simply MA( p). When all of the b j take the value of 0, the model is reduced to the AR(0; q) model or AR(q). In the latter case, y k is calculated by y k À  q i2 a i y kÀi1  e k : 5:4 5.1.1.1 Model Structure Selection Criteria Two techniques, known as Akaike's ®nal prediction error (FPE) criterion and the closely related Akaike's information theoretic criterion (AIC), may be used to aid in the selection of model structure. According to Akaike's theory, in the set of candidate models, the one with the smallest values of FPE or AIC should be chosen. The FPE is calculated as FPE  1  n=N 1 À n=N V ; 5:5 where n is the total number of parameters of the model to be estimated, N is the length of the data record, and V is the loss function for the model under consideration. Here, V is de®ned as V   n i1 e 2 i ; 5:6 where e is as de®ned in Eq. 5.2. The AIC is calculated as AIC  log1  2n=N V 5:7 In the following, an AR(12) model was chosen to characterize SA. This selection was based primarily on Braasch's recommendation [14]. As such, the resulting model should be used with caution before the validity of this model structure assumption is further studied using the above criteria. 5.1.1.2 Frequency Domain Description The ARMA models can be equivalently described in the frequency domain, which provides further insight 5.1 SELECTIVE AVAILABILITYERRORS 107 into model behavior. Introducing a one-step delay operator Z À1 , Eq. 5.2 can be rewritten as AZ À1 y k  BZ Àl x k  e k ; 5:8 where AZ À1   q i1 a i Z Àil ; 5:9 BZ À1   p i1 b i Z i1 ; 5:10 and Z À1 y k  y kÀ1 : 5:11 It is noted that AZ À1  and BZ À1  are polynomials of the time-shift operator Z À1 and normal arithmetic operations may be carried out under certain conditions. De®ning a new function HZ À1  as BZ À1  divided by AZ À1  and expanding the resulting HZ À1  in terms of operator Z À1 ,wehave HZ À1  BZ À1  AZ À1    I i1 h i Z i1 : 5:12 The numbers of {h i } are the impulse responses of the model. It can be shown that h i is the output of the ARMA model at time i  1; 2; .when the model input x i takes the value of zero at all times except for i  1. The function HZ À1  is called the frequency function of the system. By evaluating its value for Z À1  e jo , the frequency response of the model can be calculated directly. Note that this process is a direct application of the de®nition of the discrete Fourier transform (DFT) of h i . 5.1.1.3 AR Model Parameter Estimation The parameters of an AR model with structure AZ À1 y k  e k 5:13 may be estimated using the least-squares (LS) method. If we rewrite Eq. 5.13 in matrix format for k  q; q  1; .; n, we get y n y nÀl ÁÁÁ y nÀq1 y nÀl y nÀ2 ÁÁÁ y nÀq . . . . . . . . . . . . y q y qÀ1 ÁÁÁ y 1 P T T T R Q U U U S a 1 a 2 . . . a q P T T T R Q U U U S  e n e nÀ1 . . . e q P T T T R Q U U U S 5:14 108 GPS DATA ERRORS or H Á A  E; 5:15 where H  y n y nÀl ÁÁÁ y nÀq1 y nÀl y nÀ2 ÁÁÁ y nÀq . . . . . . . . . . . . y q y qÀ1 ÁÁÁ y 1 P T T T T T R Q U U U U U S ; 5:16 A a 1 a 2 a 3 ÁÁÁ a q  T ; 5:17 and E e n e nÀ1 e nÀ2 ÁÁÁ e q  T : 5:18 The LS estimation of the parameter matrix A can then be obtained by A H T H À1 H T E: 5:19 5.1.2 Collection of SA Data To build effective SA models, samples of true SA data must be available. This requirement cannot be met directly as the mechanism of SA generation and the actual SA waveform are classi®ed. The approach we take is to extract SA from ¯ight test data. National Satellite Test Bed (NSTB) ¯ight tests recorded the pseudorange measurements at all 10 RMS (Reference Monitoring Station) locations. These pseudorange measurements contain various clock, propagation, and receiver measurement errors, and they can, in general, be described as PR M  r  DT sat  DT rcvr  DT iono  DT trop  DT multipath  SA  Dt noise 5:20 where r is the true distance between the GPS satellite and the RMS receiver; DT sat and DT rcvr are the satellite and receiver clock errors; DT iono and DT trop are the ionosphere and troposphere propagation delays, DT multipath is the multipath error; SA is the SA error; and Dt noise is the receiver measurement noise. To best extract SA from PR M , values of the other terms were estimated. The true distance r is calculated by knowing the RMS receiver location and the precise orbit data available from the National Geodetic Survey (NGS) bulletin board. GIPSY= OASIS analysis (GOA) was used for this calculation, which re-created the precise orbit and converted all relevant data into the same coordinate system. Models for propagation and satellite clock errors have been built into GOA, and these were used 5.1 SELECTIVE AVAILABILITYERRORS 109 to estimate DT sat ; DT iono , and DT trop . The receiver clock errors were estimated by the NSTB algorithm using data generated from GOA for the given ¯ight test conditions. From these, a simulated pseudorange PR sim was formed.: PR sim  r sim  DT sat sim  DT rcvr sim  DT iono sim  DT trop sim 5:21 where DT sat sim ; DT rcvr sim ; DT iono sim , and DT trop sim are, respectively, the estimated values of DT sat ; DT rcvr ; DT iono , and DT trop in the simulation. From Eqs. 5.20 and 5.21, pseudorange residuals are calculated: DPR  PR M À PR sim  SA  DT multipath  Dt noise  DT models ; 5:22 where DT models stands for the total modeling error, given by DT models  r À r sim ÀÁ  DT sat À DT sat sim   DT rcvr À DT rcvr sim   DT iono À DT iono sim   DT trop À DT trop sim  5:23 It is noted that the terms DT multipath and Dt noise should be signi®cantly smaller than SA, though it is not possible to estimate their values precisely. The term DT models should also be negligible compared to SA. It is, therefore, reasonable to use DPR as an approximation to the actual SA term to estimate SA models. Examination of all available data show that their values vary between Æ80 m. These are consistent with previous reports on observed SA and with the DoD's speci®cation of SPS accuracy. 5.2 IONOSPHERIC PROPAGATION ERRORS The ionosphere, which extends from approximately 50 to 1000 km above the surface of the earth, consists of gases that have been ionized by solar radiation. The ionization produces clouds of free electrons that act as a dispersive medium for GPS signals in which propagation velocity is a function of frequency. A particular location within the ionosphere is alternately illuminated by the sun and shadowed from the sun by the earth in a daily cycle; consequently the characteristics of the ionosphere exhibit a diurnal variation in which the ionization is usually maximum late in midafternoon and minimum a few hours after midnight. Additional variations result from changes in solar activity. The primary effect of the ionosphere on GPS signals is to change the signal propagation speed as compared to that of free space. A curious fact is that the signal modulation (the code and data stream) is delayed, while the carrier phase is advanced by the same amount. Thus the measured pseudorange using the code is larger than the correct value, while that using the carrier phase is equally smaller. The magnitude of either error is directly proportional to the total electron count (TEC) in a tube of 1 m 2 cross section along the propagation path. The TEC varies spatially 110 GPS DATA ERRORS due to spatial nonhomogeneity of the ionosphere. Temporal variations are caused not only by ionospheric dynamics, but also by rapid changes in the propagation path due to satellite motion. The path delay for a satellite at zenith typically varies from about 1 m at night to 5±15 m during late afternoon. At low elevation angles the propagation path through the ionosphere is much longer, so the corresponding delays can increase to several meters at night and as much as 50 m during the day. Since ionospheric error is usually greater at low elevation angles, the impact of these errors could be reduced by not using measurements from satellites below a certain elevation mask angle. However, in dif®cult signal environments, including blockage of some satellites by obstacles, the user may be forced to use low-elevation satellites. Mask angles of 5  7:5  offer a good compromise between the loss of measurements and the likelihood of large ionospheric errors. The L 1 -only receivers in nondifferential operation can reduce ionospheric pseudorange error by using a model of the ionosphere broadcast by the satellites, which reduces the uncompensated ionospheric delay by about 50% on the average. During the day errors as large as 10 m at midlatitudes can still exist after compensation with this model and can be much worse with increased solar activity. Other recently developed models offer somewhat better performance. However, they still do not handle adequately the daily variability of the TEC, which can depart from the modeled value by 25% or more. The L 1 =L 2 receivers in nondifferential operation can take advantage of the dependency of delay on frequency to remove most of the ionospheric error. A relatively simple analysis shows that the group delay varies inversely as the square of the carrier frequency. This can be seen from the following model of the code pseudorange measurements at the L 1 and L 2 frequencies: r i  r Æ k f 2 i ; 5:24 where r is the error-free pseudorange, r i is the measured pseudorange, and k is a constant that depends on the TEC along the propagation path. The subscript i  1, 2 identi®es the measurement at the L 1 or L 2 frequencies, respectively, and the plus or minus sign is identi®ed with respective code and carrier phase pseudorange measurements. The two equations can be solved for both r and k. The solution for r for code pseudorange measurements is r  f 2 1 f 2 1 À f 2 2 r 1 À f 2 2 f 2 1 À f 2 2 r 2 ; 5:25 where f 1 and f 2 are the L 1 and L 2 carrier frequencies, respectively, and r 1 and r 2 are the corresponding pseudorange measurements. An equation similar to the above can be obtained for carrier phase pseudorange measurements. However, in nondifferential operation the residual carrier phase 5.2 IONOSPHERIC PROPAGATION ERRORS 111 pseudorange error can be greater than either an L 1 or L 2 carrier wavelength, making ambiguity resolution dif®cult. With differential operation ionospheric errors can be nearly eliminated in many applications, because ionospheric errors tend to be highly correlated when the base and roving stations are in suf®ciently close proximity. With two L 1 -only receivers separated by 25 km, the unmodeled differential ionospheric error is typically at the 10±20-cm level. At 100 km separation this can increase to as much as a meter. Additional error reduction using an ionospheric model can further reduce these errors by 25±50%. 5.2.1 Ionospheric Delay Model J. A. Klobuchar's model [37,76] for ionospheric delay in seconds is given by T g  DC  A 1 À x 2 2  x 4 24 ! for jxj p 2 5:26 where x  2pt À T p  P (rad) DC  5 ns (constant offset) T p  phase  50,400 s A  amplitude P  period t  local time of the earth subpoint of the signal intersection with mean ionospheric height (s) The algorithm assumes this latter height to be 350 km. The DC and phasing T p are held constant at 5 ns and 14 h (50,400 s) local time. Amplitude (A) and period (P) are modeled as third-order polynomials: A   3 n0 a n f n m s; P   3 n0 b n f n m s; where f m is the geomagnetic latitude of the ionospheric subpoint and a n ; b n are coef®cients selected (from 370 such sets of constants) by the GPS master control station and placed in the satellite navigation upload message for downlink to the user. For Southbury, Connecticut, a n 0:8382  10 À8 ; À0:745  10 À8 ; À0:596  10 À7 ; 0:596  10 À7 ; b n 0:8806  10 5 ; À0:3277  10 5 ; À0:1966  10 6 ; 0:1966  10 6  The parameter f m is calculated as follows: 112 GPS DATA ERRORS [...]... Standard C/A reference code waveform t Tc Cross-correlation with received C/A-code – Tc 0 τ Tc Multipath-mitigating reference code waveform t Tc Cross-correlation with received C/A-code Tc – Tc 0 Fig 5.5 Multipath-mitigating reference code waveform τ 5.6 5.6.3 METHODS OF MULTIPATH MITIGATION 123 Performance of Time Domain Methods Ranging with the C=A-Code Typical C=A-code ranging performance curves for several... considered as zero-mean random processes that can be combined to form a single UERE This is accomplished by forming the root-sum-square of the UERE errors from all sources: s n € UERE ˆ …UERE†2 : i iˆ1 …5:36† Figure 5.10 depicts the various GPS UERE errors and their combined effect for both C=A-code and P(Y)-code navigation at the 1-s level When SA is on, the UERE for the C=A-code user... Narrow-Correlator Technology (1990±1993) The ®rst signi®cant means to reduce GPS multipath effects by receiver processing made its debut in the early 1990s Until that time, most receivers had been designed with a 2-MHz precorrelation bandwidth that encompassed most, but not all, of the GPS spread-spectrum signal power These receivers also used one-chip spacing between the early and late reference C=A-codes... direct-path delay estimate Even in the simple two-path model (Eq 5.32) there are six signal parameters, so that a very large number of correlation function shapes must be handled An example of a heuristically developed shape-based approach called the early±late slope (ELS) method can be found in [119], while a method based on maximum-likelihood estimation called the multipath-estimating delay-lock... ®gure The result of using an 8-MHz bandwidth is shown in Fig 5.4, where it can be noted that the sharper peak of the direct-path cross-correlation function is less easily shifted by the secondary-path component It can also be shown that at larger bandwidths the sharper peak is more resistant to disturbance by receiver thermal noise, even though the precorrelation signal-to-noise ratio is increased Another... ˆ x1 ‡ w2 ; where w1 …t† and w2 …t† are independent zero-mean white-noise processes with known variances The equivalent discrete-time model has the state vector xˆ x1 x2 ! and the stochastic process model 1 xk ˆ Dt ! ! w1;kÀ1 0 x ‡ ; w2;kÀ1 1 kÀ1 …5:35† where Dt is the discrete-time step and fw1;kÀ1 g; fw2;kÀ1 g are independent zero-mean white-noise sequences with known variances 5.11 ERROR BUDGETS... the multipath-induced distortions near the peak of the correlation function Leading-Edge Techniques Because the direct-path signal always precedes secondary-path signals, the leading (left-hand) portion of the correlation function is uncontaminated by multipath, as is illustrated in Fig 5.4 Therefore, if one could measure the location of just the leading part, it appears that the direct-path delay could... direct-path GPS L1 C=A-code signal arrives with a phase such that all of the signal power lies in the baseband I-channel, so that the baseband signal is purely real Further assume an in®nite signal bandwidth so that the cross-correlation of the baseband signal with an ideal C=A reference code waveform will be an isosceles triangle 600 m wide at the base (a) Suppose that in addition to the direct-path... function is the superposition of the cross-correlation functions of the direct- and secondary-path signals (b) Repeat the calculations of part (a) but with a secondary-path relative time delay of precisely 250 1 carrier cycles Note that in this case the secondary2 path phase is 180 out of phase with the direct-path signal, but still lies entirely in the baseband I-channel 5.3 (a) Using the discrete matrix... MULTIPATH PROBLEM Multipath propagation of the GPS signal is a dominant source of error in differential positioning Objects in the vicinity of a receiver antenna (notably the ground) can 116 GPS DATA ERRORS easily re¯ect GPS signals, resulting in one or more secondary propagation paths These secondary-path signals, which are superimposed on the desired direct-path signal, always have a longer propagation . Inc. Print ISBN 0-4 7 1-3 5032-X Electronic ISBN 0-4 7 1-2 007 1-9 Time and frequency users may see greater effects in the long term via commu- nication systems. with a 2-MHz precorrela- tion bandwidth that encompassed most, but not all, of the GPS spread-spectrum signal power. These receivers also used one-chip spacing