Smooth Bias Estimation for Multipath Mitigation Using Sparse Estimation

Abstract

Multipath remains the main source of error when using global navigation satellite systems (GNSS) in constrained environment, leading to biased measurements and thus to inaccurate estimated positions. This paper formulates the GNSS navigation problem as the resolution of an overdetermined system, which depends nonlinearly on the receiver position and linearly on the clock bias and drift, and possible biases affecting GNSS measurements. The extended Kalman filter is used to linearize the navigation problem whereas sparse estimation is considered to estimate multipath biases. We assume that only a part of the satellites are affected by multipath, i.e., that the unknown bias vector is sparse in the sense that several of its components are equal to zero. The natural way of enforcing sparsity is to introduce an 1 regularization associated with the bias vector. This leads to a least absolute shrinkage and selection operator (LASSO) problem that is solved using a reweighted-1 algorithm. The weighting matrix of this algorithm is designed carefully as functions of the satellite carrier to noise density ratio and the satellite elevations. The smooth variations of multipath biases versus time are enforced using a regularization based on total variation. An experiment conducted on real data allows the performance of the proposed method to be appreciated.

Robust Covariance Matrix Estimation and Sparse Bias Estimation for Multipath Mitigation

“…In this paper, we have used the "shooting algorithm" (detailed for instance in [22] and [23]). We have shown in [13] that enforcing smoothness to the MP bias amplitudes provides better local results, as presented in the next subsection.…”

“…in order to estimate the bias vector m k , and feed it to the proposed EKF in (13). Regarding the weighting matrix W k , we propose to consider the strategy of [12], leading to…”

“…Based on the fused LASSO described in [24], we proposed in [13] to introduce a penalty associated with the temporal variations of the different biases leading to the following problem arg min…”

“…The point of view considered in this work is to model the effect of MP on GNSS measurements as sparse additive biases with temporal smoothing as in [12,13]) These additive biases are then estimated and subtracted from the GNSS measurements to mitigate MP effects. However, we have observed that this method can be sensitive to the imperfect knowledge of the noise covariance matrix.…”

“…This section also recalls the Kalman filtering steps that will be used to track the receiver position. Section 2 presents the methods proposed in [12,13] to estimate MP biases using sparse estimation, formulating the positioning problem as a penalized least squares problem with a weighted 1 regularization and 1 smoothing. Section 3 describes the Danish method allowing the noise covariance matrix to be estimated, thanks to an iterative reweighted least squares (IRLS) algorithm.…”