Application of Theil Model to Adjustments of Network Densifications

Abstract

A new least-squares solution to the Theil model is presented. The proposed solution is particularly suitable for applications of network densifications. By assuming fictitious observations made on the constraints, the pseudoobservation equation is formed and then solved according to the least-squares principle. Instead of the noninteger degrees of freedom as appeared previously, the proposed solution has an integer degree of freedom. Generally, these two solutions are not identical, although statistically both are unbiased and minimum-variance estimates for the parameters. In addition, the proposed method results in a smaller matrix dimension for the corresponding pseudonormal equation. The unknowns to be solved for in the normal equation decrease in such a way that the more the constraints, the less the unknowns will be.