Virtual Observation Iteration Solution and A-Optimal Design Method for Ill-Posed Mixed Additive and Multiplicative Random Error Model in Geodetic Measurement

Abstract

The mixed additive and multiplicative random error model is a combination of the additive error random model and multiplicative random error model. Weakness is an attribute of the mixed additive and multiplicative random error model, and the ill-posed problem of the coefficient matrix is ignored in the existing parameter estimation methods for addressing the model, which will result in an unstable or nonconvergent solution. Aiming at solving this problem, this paper first derives the virtual observation iterative solution (VOIS) formula for the ill-posed mixed additive and multiplicative random error model by combining the observation equation of the mixed additive and multiplicative random error model and the virtual observation equation. Furthermore, based on the principle of the A-optimal design, the A-optimal design method is proposed to determine the regularization parameter of the ill-posed model. Finally, the VOIS method is applied in simulated and actual data for verification and analysis and is compared with existing methods. The experimental results show that the A-optimal design method can determine reasonable regularization parameters and that the VOIS method can obtain more accurate parameter estimations than existing methods and has strong feasibility and applicability.