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In this paper, we present a new ridge estimation method for solving rank-deficient least squares problems, in which a rank-deficient matrix is regarded as an almost rank-deficient. First, we give an algebraic derivation that the optimal solution can in fact be obtained by solving a related regularized problem on the optimal worst-case residual. Second, we give a new iterative algorithm to solve ridge parameter and prove its convergence. Finally, examples are given to demonstrate the efficiency of our new method. It is shown that the proposed algorithm can not only assess the stability of solution but also use additional prior information to guarantee the uniqueness of solutions to the problem of rank-deficient free-network adjustment. [ABSTRACT FROM AUTHOR]

With the rapid development of BeiDou system (BDS) and steady progress of Galileo system, the current GNSS (Global Navigation Satellite System) constellations consist of GPS, GLONASS, BeiDou and Galileo. The real signals from the four constellations have been available, which allows us to analyse and compare their measurement noises and multipath effects. In this study, a zero-baseline test is conducted using two ‘Trimble NetR9’ receivers to assess and compare the noises and multipath of measurements on multiple frequencies from the four satellite systems. The zero-baseline double difference approach is utilised to analyse the receiver noises. The code multipath combination and triple-frequency carrier phase combination approaches are exploited to analyse a comprehensive effect of the multipath and noises on the code and carrier phase measurements, respectively. Based on the analysis of the zero-baseline dataset, the results indicate that the code measurement noise levels range from 5 to 25 cm while the carrier phase noise levels vary within 0.9–1.5 mm for different frequencies and constellations. The code multipath and noise (CMN) level for GLONASS is the largest with a root mean square (RMS) value of 39 cm on both G1 and G2 frequencies whereas the Galileo code measurements exhibit a smallest level on the E5 frequency with a RMS value of only 10 cm. The RMS of the carrier phase multipath and noises (PMN) ranges from 1.3 to 2.6 mm for BeiDou and Galileo satellites. By contrast, the triple-frequency carrier phase combinations from the GPS Block IIF satellites demonstrate a much larger RMS value of 5.6 mm owing to an effect of inter-frequency clock biases. [ABSTRACT FROM PUBLISHER]