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Applications of Kalman filter in WAAS
Estimation of Ionospheric Delay and Instrumental Differential Group Delay Biases
The ionospheric TEC at any observation direction can be modeled as a function F describing the vertical TEC on the subionospheric point of observation, mapped to the line of sight by means of a slant function S: Iji(t) = S(eji) F(dji, t) (9) where t is the observation time, eij is the elevation of the satellite i as seen from the receiver j , dij is the geocentric direction of the subionospheric point. The slant function S, given by Sovers and Fanselow [1987] can be used here. 
where e is the satellite elevation, h1 and h2 are the lower and the upper height of the ionosphere and RE is the mean radius of the earth.  Figure. 2: Local ionosphere around the zenith of the station j. P is subionospheric point of satellite i as seen from station j, and Z is subionospheric point of the zenith of station j. We consider the geocentric reference system as shown in Figure 1 with axis z towards the solar center. The biased ionospheric term can be modeled as (refer to Figure 2): Iji (t) = S(eji) [Aj(t)+Bj(t)dYji+Cj(t)dcji]+kj+ki (11) where dYji=Yji-Yj, dcji=cij-ci and Aj(t), Bj(t), Cj(t) represent the coefficients of the local linear approximation to the global function F. Aj(t) is the vertical TEC for the zenith direction of station j. The parameters for which we should solve are the differential instrumental biases and Aj(t), Bj(t), Cj(t) at each station j and for each time t. Eqn. (11) does not allow an unambiguous solution for kj and ki, so one way out is to choose an arbitrary receiver kr. Therefore, the estimated satellite and receiver are kri=ki+kr and kj,r=kj–kr . The parameters A, B, C, kj,r, kri are estimated using a Kalman filter. The state equations that are used are: Aj(t2) = Aj(t1) + Bj(t1)[ Yj(t2) - Yj(t1)] + Cj(t1)[cj(t2) - cj(t1)] Bj(t2) = Bj(t1) Cj(t2) = Cj(t1) kj,r(t2) = kj,r(t1) kri(t2) = kri(t1) Time synchronization Implementation In US WAAS, GPS Common View Time Transfer Technique is used to estimate the difference between the Reference station clocks and the Master clock located at the MCC. The basic equation used to estimate the clock offset is the following pseudorange residual equation: {(Dk,m = Drk.1k,m + Dbm-DBk+ vk,m)k=1}m=1,M (1) In eqn. (1), as stated above, Dr k is the vector that connects the true location of the kth satellite, to its location according to the broadcast navigation message. 1k,m denotes the unit vector from the kth satellite to the mth reference station. Additionally, Dbm and DBk are the offsets in the Reference station and the satellite clocks with respect to the GPS time. The measurement noise is given by vk,m. K is the no. of satellites in view of the mth Reference station and M is the total no. of Reference stations. As seen from eqn. (1), considering only one Reference station, i.e. m=1, we can say that there are 5 unknowns. They are, the three components of Drk namely, the along track error, the cross track error and the radial track error in the ephemeris of the kth satellite, and Dbm as well as DBk . Since, in the Common View Time Transfer Technique, a single satellite is tracked by all M Reference stations, in subsequent equations i.e. m=2, ….., M, the number of unknowns is incremented by one each time, namely, the corresponding Dbm . Thus we see that the total number of unknowns is 5+M-1 and the number of equations are only M. Moreover, the observations of the clock offsets between the k th satellite and the mth Reference station are embedded in noise vk,m . The clock offset estimation problem can be formulated in terms of the well known system equation and the noisy observation equation of the Kalman filter. Hence, this problem can be easily solved by using the Kalman filtering technique. Conclusions In this paper, we have briefly outlined the applications of the Kalman filtering technique in Wide Area Augmentation Systems (WAAS). This technique has the potential of efficiently solving the problems of estimation of ionospheric Total Electron Content as well as the Time Synchronization of the atomic clocks at the RIMS. References
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