Coordinate transformation is a fundamental issue in the related studies of measurement. However, existing methodologies often need to pay more attention to the available spatial information, leading to suboptimal results. This paper addresses this issue by incorporating geometric constraints into the symmetric coordinate transformation. We propose the so-called geo-constrained transformation method based on the joint adjustment of the coordinate transformation in conjunction with the geometric constraints over a set of points. By formulating the geometric constraints as the conditional model, we analyze the effects of geometric constraints on the estimated transformation parameters and point locations. By removing such effects during the symmetric transformation algorithm, the results show better statistical performance and satisfy geometric constraints. Two numerical examples are given to demonstrate the expected improvement in the statistical accuracy. It is shown that the improvement of the point determination accuracy can go beyond 50%.

The reliability theory has been an important element of the classical geodetic adjustment theory and methods in the linear Gauss-Markov model. Although errors-in-variables (EIV) models have been intensively investigated, little has been done about reliability theory for EIV models. This paper first investigates the effect of a random coefficient matrix A on the conventional geodetic reliability measures as if the coefficient matrix were deterministic. The effects of such geodetic internal and external reliability measures due to the randomness of the coefficient matrix are worked out, which are shown to depend not only on the noise level of the random elements of A but also on the values of parameters. An alternative, linear approximate reliability theory is accordingly developed for use in EIV models. Both the EIV-affected reliability measures and the corresponding linear approximate measures fully account for the random errors of both the coefficient matrix and the observations, though formulated in a slightly different way. Numerical experiments have been carried to demonstrate the effects of errors-in-variables on reliability measures and compared with the conventional Baarda's reliability measures. The simulations have confirmed our theoretical results that the EIV-reliability measures depend on both the noise level of A and the parameter values. The larger the noise level of A, the larger the EIV-affected internal and external reliability measures; the larger the parameters, the larger the EIV-affected internal and external reliability measures.