Anisotropic Covariance Model Applied for the Computation of Surface Deformation using Least Squares Collocation, Case Study: Kenai Peninsula

Least squares collocation (LSC) is one of the usual methods for the interpolation of the GPS displacement field and computation of the surface deformations. The goal of this research is to improve the quality of prediction by LSC using anisotropic covariance model in which the covariance depends on azimuth and the distance of points. The Restricted Maximum Likelihood method (REML) is used for the precise determination of the parameters of anisotropic covariance model. For implementation of the REML method we have applied the Fisher scoring technique. The required mathematical relations for computation of the elements of strain tensor with LSC using anisotropic covariance model are also derived. We interpolated the displacement vectors on a regular grid of points with 15 minutes spacing in Kenai Peninsula in South central Alaska by the LSC method using the anisotropic covariance function. We employed a trend, signal noise LSC model in which de-trended data are used for estimation of covariance parameters. The input data is displacement vectors of a GPS network derived by processing the two campaigns of GPS observations in 1996 and 1998. The east-west and north-south components of the displacement vector were predicted independently from each other. In addition, the displacement vectors were interpolated employing an isotropic covariance function. In order to estimate the isotropic covariance parameters, we applied the method of model fitting to empirical covariances. The LSC computed results using the proposed method are compared with the isotropic covariance model. For this purpose, we applied the usual measures for precision and accuracy assessment of LSC predictions which are LSC cross validation errors and LSC prediction errors, respectively. Employing the proposed method, the root mean square of cross validation errors for east-west and north-south components of the displacement vector are reduced about 42% and 45%, respectively in comparison with the method employing isotropic covariance model. Meanwhile, the root mean square of LSC prediction errors reduced about 25% and 10% for east-west and north-south components. The strain tensor is also computed for the regular grid of points using anisotropic covariance model and the respective formula developed in this work. Principle strain components and dilatation parameter were computed from the estimated tensors and the results are illustrated graphically. Then, the pattern of deformation derived by the proposed method is compared with the known pattern of deformation in the region which is represented by other researchers. The figure of principle strains computed on a grid shows two components of contraction from north-west and south-east directions in the area in which the maximum computed contraction is located at the center of the northern area of Kenai Peninsula. Map of computed dilatation in this area also shows a dome type pattern of deformation in the region in which the maximum compression (negative dilatation) occurs at the same point in the middle of Peninsula. This is in agreement with the results derived from the leveling observations in the region which showed the maximum uplift at the same point. Shape of uplift in the region is also like a dome. Therefore, the proposed method and developed formulae were validated in two ways: One with the LSC prediction quality measures and the other by the independent leveling observations. Finally, we conclude that using the proposed method for this case study, one can get reliable LSC estimations even with the data points sparsely located in the region.