Document Type : Research
One way of gridding two dimensional vector data would be gridding each component separately. Alternatively, using Green’s functions we can grid two components simultaneously in a way that couples them through elastic deformation theory. This is particularly suited, though not exclusive, to data that represent elastic/semi-elastic deformation, like horizontal GPS velocity fields. Measurements made on the surface of the Earth are often sparse and unevenly distributed. For example, GPS displacement measurements are limited by the availability of ground stations and airborne geophysical measurements are highly sampled along flight lines but there is often a large gap between lines. Many data processing methods require data distributed on a uniform regular grid, particularly methods involving the Fourier transform or the computation of directional derivatives. Hence, the interpolation of sparse measurements onto a regular grid (known as gridding) is a prominent problem in the Earth Sciences.
In this research, sparse two-dimensional vector data of the horizontal GPS velocity field are interpolated using Green’s functions derived from elastic constraints. The method is based on the Green’s functions of an elastic body subjected to in-plane forces. This approach ensures elastic coupling between the two components of the interpolation. Users may adjust the coupling by varying Poisson’s ratio. Smoothing can be achieved by ignoring the smallest eigenvalues in the matrix solution for the strengths of the unknown body forces. The study area is the oblique collision zone of Arabia-Eurasia tectonic plates, which has a GPS velocity field with sparse distribution.
Since the Green’s functions developed for the half-space environment, the Mercator map projection used to create the half-space for interpolation and gridding. Data split into a training and testing set. We'll fit the gridder on the training set and use the testing set to evaluate how well the gridder is performing. The vector gridding was done using the Poisson's ratio 0.5 to couple the two horizontal components. Then score on the testing data. The best possible score is 1, meaning a perfect prediction of the test data. By calculating the mean square deviation ratio (MSDR) to evaluate the gridding accuracy, the score of 0.86 obtained for this statistic.
While this method is not new, it provides some insight into the behavior of the coupled interpolation for a wide range of Poisson’s ratio. This approach provides improved interpolation of sparse vector data when the physics of the deforming material follows elasticity equations.
We interpolated our horizontal GPS velocities onto a regular geographic grid with 1 arc second spacing and masked the data that are far from the observation points and finally the residuals between the predictions and the original input data were calculated. Interpolation of horizontal GPS velocity fields of local geodynamic networks are proposed to obtain an estimate for Poisson's ratio values in the best case for gridding validation.
In this study, two dimensional GPS data were interpolated. Three dimensional GPS data gridding can also be done using the Green’s functions provided by Uieda et al., (2018). It is also recommended to use different Green’s functions to grid different types of spatial data.