An Overview of Adjustment Methods for Mixed Additive and Multiplicative Random Error Models

Abstract

Geodetic adjustment theory has been developed on the basis of a linear or nonlinear Gauss-Markov model, in which the random errors of measurements are always assumed to be independent of the true values of measurements themselves and naturally added to the functional model. However, modern geodetic instruments and geodetic imaging systems have clearly shown that the random errors of such measurements consist of two parts: one is of local nature and has nothing to do with the quantity under observation, and the other is proportional to the true value of measurement. From the statistical point of view, these two types of errors are called additive and multiplicative errors, respectively. Obviously, the conventional geodetic adjustment theory and methods for Gauss-Markov models with additive errors cannot theoretically meet the need of processing measurements contaminated by mixed additive and multiplicative random errors. This paper presents an overview of parameter estimation methods for processing mixed additive and multiplicative random errors. More specifically, we discuss two types of methods to estimate parameters in a mixed additive and multiplicative error model, namely, quasi-likelihood and least-squares-based methods. From this point of view, we extend the conventional adjustment theory and methods and give a solid theoretical foundation to process geodetic measurements contaminated by mixed additive and multiplicative random errors. Finally, we further discuss parameter estimation with prior information.