Damped least squares method for nonlinear mixed additive and multiplicative errors model

Abstract

Measurement data in the field of modern geodesy contains not only additive errors but also multiplicative errors related to signal strength. The existing models for dealing with mixed additive and multiplicative errors are mainly based on the linear form of unknown parameters and observations, and there are few studies on the nonlinear form of the two. In the parameter estimation method of the nonlinear mixed additive and multiplicative errors model, the initial value of the Gauss–Newton parameter estimation method is selected by previous experience. The initial value determined by this method deviates far from the true value due to a lack of experience, which will lead to inaccurate parameter estimation results. In order to solve this problem, based on the least squares principle and the introduction of the damping factor, this paper deduces the damping least squares parameter solution formula for the parameter estimation of the nonlinear mixed additive and multiplicative errors model. The superiority of the damping least squares algorithm is reflected in the adjustment of the damping factor, taking into account the advantages of the Gauss–Newton method and the steepest descent method, and some weighted selection is obtained in the two algorithms. The calculation and comparative analysis of the simulated cases show that the damped least squares method is more suitable for handling geodetic data with this nonlinear mixed additive and multiplicative errors model when the initial value deviates far from the true value.