PDF Evaluation of the Integer Ambiguity Residuals

Abstract

A parameter estimation theory is incomplete if no rigorous measures are available for validating the parameter solution. Since the classical theory of linear estimation does not apply to the integer GPS model, rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency.

In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals, which are defined as the difference between the real-valued float ambiguity estimates and the integer-valued fixed ambiguity estimates. Since the residuals and their properties depend on the integer estimation principle used, we will present the PDF of the ambiguity residuals for the whole class of admissible integer estimators. This includes the estimation principles of integer rounding, integer bootstrapping and integer least-squares. In order to get a better understanding of the various features of the joint PDF of the ambiguity residuals we will use a step-by-step construction aided by graphical means. Although the results apply for any dimension, the one-dimensional case and the two-dimensional case are highlighted.