Present-day crustal deformation in Alborz region based on GPS velocity field and spline functions

GPS observations over the Earth surface occur at the location of stations. So, they have a discrete form and are often sparse and irregularly distributed. Most of the data processing approaches require the data over a regular grid. As such, the interpolation of sparse observations onto a regular grid, known as gridding, is a challenging step in geosciences. To overcome this problem, interpolation with spline functions can be used.
In this method, the geographic coordinates and velocity components of GPS stations are inputs, and the two-dimensional velocity field components on a regular grid are outputs. The sparse vector data of horizontal displacements of 86 GPS stations with non-uniform distribution in the Alborz region located in the north of Iran has been analyzed. Due to ample GPS stations and a tectonically active area, this region has been selected for study.
Interpolation is done for each component separately and in coupled form. Both cases are based on the Green functions of an elastic body subjected to in-plane forces. The second approach ensures elastic coupling between the two components. Coupling could be adjusted by varying Poisson’s ratio. Vector gridding is done using the Poisson's ratio 0.5 to couple the two horizontal components.
Since the used Green functions developed for the half-space, an arbitrary map projection e.g., Mercator used to create it. Trend analysis was done on the input data first. Then residuals were calculated by subtracting predicted trend from the input data. The output is passed onto block reduction to generate reduced data. Then, the spline is fitted to the residuals of trend analysis and block reduction.
To validate the model, ten percent of the data are selected for testing and the rest for training. The coefficient of determination for eastern and northern components are as 0.25 and 0.88, respectively. To reduce the correlation effect of data with close distances in the validation results, data reduction is done with 50 km blocks. In this case, the coefficient of determination for eastern and northern components are -0.18 and 0.47. For the coupled interpolation of two components, the coefficient of determination is 0.88 and by reducing the data with 50 km blocks, it is 0.84. Validations show that the coupled interpolation of velocity components using elasticity constraint leads to improved interpolation of sparse vector data.
The results of velocity field interpolation show that the spatial distribution of the crustal deformation in the Alborz region is irregular and has partitioning characteristics. The range of the northern components is larger than the eastern components. The direction of the northern components is always towards Eurasia, but the direction of the eastern components changes in the region in such a way that the amplitude of the eastern component in the inner region is around zero. The amplitude of the northern components is decreasing from south to north and from west to east. Estimated strain rate from interpolated velocity field shows the convergence deformation in the study area and confirms the compressing of high-elevation region of Alborz.