Weighted Total Least Squares with Singular Covariance Matrices Subject to Weighted and Hard Constraints

Abstract

Weighted total least squares (WTLS) has been widely used as a standard method to optimally adjust an errors-in-variables (EIV) model containing random errors both in the observation vector and in the coefficient matrix. An earlier work provided a simple and flexible formulation for WTLS based on the standard least-squares (SLS) theory. The formulation allows one to directly apply the available SLS theory to the EIV models. Among such applications, this contribution formulates the WTLS problem subject to weighted or hard linear(ized) equality constraints on unknown parameters. The constraints are to be properly incorporated into the system of equations in an EIV model of which a general structure for the (singular) covariance matrix QA of the coefficient matrix is used. The formulation can easily take into consideration any number of weighted linear and nonlinear constraints. Hard constraints turn out to be a special case of the general formulation of the weighted constraints. Because the formulation is based on the SLS theory, the method automatically approximates the covariance matrix of the estimates from which the precision of the constrained estimates can be obtained. Three numerical examples with different scenarios are used to demonstrate the efficacy of the proposed algorithm for geodetic applications.