Displacement calculations from geodetic data and the testing of geophysical deformation models

The inverse problem of estimating displacements on the Earth's surface from repeated geodetic measurements is usually mixed-determined. Absolute displacements are underdetermined if the network is not tied to an external reference frame, but relative displacements may be overdetermined if the network is geometrically redundant. Least squares analysis of underdetermined linear systems may be carried out by optimizing some criterion of solution desirability over the class of solutions that fits the data. We review solution by constrained optimization, its connection with generalized matrix inverses, and currently used solution methods, such as the “inner coordinate” and “outer coordinate” solutions. In the context of fitting deformation models to geodetic data, we introduce a “model coordinate” solution that fixes the indeterminate components of the displacement field by minimizing the difference between the computed displacements and those predicted by a geophysical model. This gives an appropriate solution for model assessment. We also propose a means to exploit the overdetermination of some geodetic data by decomposing the model residuals from a geometrically redundant network into two orthogonal components; the “pure error,” which contains only observation errors, and the “lack of fit,” which may contain both observation errors and model misfit. Like the model coordinate displacements, this decomposition is useful in model assessment. Under certain statistical assumptions, the relative sizes of the pure error and lack of fit may be compared to test for significant model misfit.