Journal of Surveying Engineering

The total least-squares (TLS) method and its variations have recently received increasing research attention. However, little attention has been given to the weighted TLS adjustment method with condition equations. In this paper, a weighted TLS method designed for condition equations (WTLSC) is presented with the assumption that both the observation vector and design matrix contain errors. The covariance matrices are estimated for both the observation vector and design matrix after the adjustment, and the biases are corrected for the adjusted observation vector, design matrix, and corresponding covariance matrices in the WTLSC method. The proposed approach was used in an adjustment problem of an object point photographed by three terrestrial cameras. The results show that the proposed method resolves the condition equations with errors in the design matrix without linearization in the case study. The proposed WTLSC method generates stable error vector and matrix for the observation vector and design matrix, which satisfy the condition equation in the repeated simulation experiments. The results also show that there are biases in the adjusted observation vector, design matrix, and corresponding covariance matrices, although the biases are small in the case study.

Global positioning system (GPS) technology has been frequently used to monitor geological hazards associated with ground deformations, such as long-term landslides and subsidence. When GPS data are processed, they yield ellipsoid heights, which are the distances above a smooth ellipsoid surface. However, orthometric heights are often used in practical surveying and engineering applications. Orthometric height is a physical quantity that refers to the surface of the geoid. In this study, a more practical alternative definition of orthometric height is used. This approximation is the one commonly implemented in practical surveying and engineering applications to compute relative orthometric height values. This well-known procedure computes orthometric heights by combining GPS-measured ellipsoid height and a geoid model. Any type of orthometric height is a physically based quantity. GPS alone, which is a geometric technique, cannot directly measure orthometric heights. This study investigates the vertical displacements (subsidence or uplift) derived independently from ellipsoid heights on one hand and modeled orthometric heights (computed from GPS and a geoid model) on the other hand and compares the results. Long-term GPS observations at a subsidence site in Houston, Texas, and a landslide site in Ponce, Puerto Rico, are investigated as examples. The major conclusion derived from this study is that, in practice, directly GPS-obtained ellipsoid heights and GPS-derived orthometric heights determined using GPS and a geoid model will result in the same subsidence measurements. Hence, ellipsoid heights derived from GPS observations, which are geometric quantities, could be directly used to measure long-term subsidence without the need of performing leveling techniques. It was further concluded that the choice of the software packages for GPS data postprocessing [Precise Point Positioning (PPP) and/or Online Positioning User Service (OPUS)] was not critical for tracking long-term subsidence rates. However, users should avoid mixing the ellipsoid heights calculated by different software packages or by different versions of the same software package processed at different times.