Parameter estimation methods for nonlinear mixed additive and multiplicative random error model

Objectives: In the field of geodesy, with the continuous development of modern observation technology, so that the measurement data not only contain additive errors, there are also multiplicative errors, purely considering the processing of additive errors can no longer meet the requirements. Existing methods for dealing with mixed additive and multiplicative errors models are mainly based on the fact that the unknown parameters and observations are in linear form, and few studies have been conducted on the fact that the unknown parameters and observations are in nonlinear form. Methods: In order to extend the parameter estimation method of the mixed additive and multiplicative errors model, this paper determines the reasonable weight matrix of nonlinear mixed additive and multiplicative errors model based on the law of error propagation and the principle of least squares and applies the idea of Taylor's formula expansion, and derives the least squares, Gauss-Newton method, and weighted least squares of the nonlinear mixed additive and multiplicative errors model. Due to the nonlinear nature, it makes the weighted least squares solution biased, so it needs to be analyzed for its bias. The bias-corrected weighted least squares method is derived by deviation analysis and proof. Results: It can be seen through the calculation and comparative analysis of the arithmetic examples that a reasonable weighting method is conducive to improving the correctness of the model parameter estimation results, and when the model nonlinearity is high, the bias-corrected weighted least squares method can obtain better parameter estimation results. Conclusions: The feasibility and validity of four methods for parameter estimation of the nonlinear mixed additive and multiplicative errors model are demonstrated, and the bias-corrected weighted least squares method is more suitable for processing geodetic data from such nonlinear models with a mixture of mixed additive and multiplicative errors.