Applications of Kalman filter in WAAS


M. R. Sivaraman



Rajiv Ranjan Sahay
Satcom & IT Applications
Area,Space Applications Centre
Ahmedabad –380015, India.
sivaraman_55@yahoo.com

Abstract
The Kalman filter being a multi-input, multi-output, recursive digital filter that produces estimates of states of a system, which are optimal in the mean square sense, finds application in two ways in GPS based navigation systems like WAAS. The first application is estimating the ionospheric TEC from the GPS observables and the other is in the Time Synchronization of the atomic clocks placed in the RIMS.

Introduction
The Kalman filter is a multi-input, multi-output, recursive digital filter that can optimally estimate, in real time, the states of the system based on its noisy outputs. The estimates are statistically optimal in the sense that they minimize the mean square estimation error. This filter finds application in GPS based navigation systems, (like Wide Area Augmentation System for Category I precision approach of aircrafts), in two ways, namely, estimation of the ionospheric Total Electron Content (TEC) and Time Synchronization of the Range and Integrity Monitoring Stations (RIMS). A brief approach of using the Kalman filter for estimation of TEC and synchronization of the atomic clocks at the RIMS is given below. Firstly, we look at the problem of estimation of ionospheric TEC in section 2 and 2.1, then we focus on the time synchronization of the various RIMS in section 3.1.

Estimation of Ionospheric TEC
The ionospheric Total Electron Content (TEC) can be estimated from the GPS observables, namely pseudorange and carrier-phase measurements. However, in the estimation of the ionospheric TEC form the GPS observables several instrumental systematic effects such as the biases in the GPS satellites and the GPS receivers on the ground must be modeled. The Kalman filtering technique can be adopted for estimating the instrumental biases as well as the TEC at each GPS station using dual GPS data.

There will be a bias for each of the two GPS frequencies ( f₁ =1575.42 and f₂ =1227.60 MHz) and their difference, the differential instrumental bias, will produce systematic errors in the estimates of the ionospheric delays. For accurate estimates of the TEC these differential instrumental biases have to be removed.

The GPS carrier phases Lⁱ_kj and pseudoranges Pⁱ_kj , in range units, can be modeled using the following formulae [Sardon et al]:
L¹_kj = rⁱ_j + c(dtⁱ– dt_j) + dⁱ_{trop j} – dⁱ_{ion 1j} - l₁bⁱ_1j
L²_kj = rⁱ_j + c(dtⁱ– dt_j) + dⁱ_{trop j} – dⁱ_{ion 2j} - l₁bⁱ_2j         (1)
Pⁱ_1j = rⁱ_j + c(dtⁱ– dt_j) + dⁱ_{trop j} + dⁱ_{ion 1j} + d_q1j + d_q1ⁱ
Pⁱ_2j = rⁱ_j + c(dtⁱ– dt_j) + dⁱ_{trop j} + dⁱ_{ion 2j} + d_q2j + d_q2ⁱ
Here, the subscripts k=1,2 refer to frequency f_k.

r_jⁱ the distance from the receiver to the satellite

c the speed of light

dt_j, dtⁱ receiver and satellite clock offsets, respectively

dⁱ_{trop j} tropospheric delay

l_k is the carrier wavelength at frequency f_k

bⁱ_kj initialization constant at frequency f_k , lumping together the carrier phase ambiguity and the satellite and receiver instrumental phase delay biases.

dq_kⁱ , dq_kj satellite and receiver instrumental group delay biases at frequency f_k , respectively.

dⁱ_{ion kj}ionospheric delay at frequency f_k which can be modeled as [Sovers and Fanselow,1987 ]

Figure 1. Coordinates Y and c of the point P in the chosen reference system

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