# Coupled interpolation of three-component GPS velocities

Leonardo Uieda, Xiaohua Xu, Paul Wessel, David Sandwell

## Info

This is update on my GPS interpolation work, which is part of our EarthScope NSF grant. The implementation is entirely based on the Verde library (there is an introduction to what it can do on the blog). This was the first test of the Verde API and I'm glad to say that it passed with flying colors. Implementing the new methods was very straight forward and all the tools in Verde made the entire data processing and model selection pipeline simple and easy.

## Abstract

GPS/GNSS measurements of deformation have high accuracy and temporal resolution but are spatially sparse. Conversely, InSAR provides great spatial resolution but is limited by the satellite look angle, atmospheric noise, and the delay between repeat passes. The sparse GPS data often need to be interpolated on regular grids to be used as constraints during InSAR processing or to calculate strain rates. The interpolation is routinely done separately for each component of the velocity field using minimum curvature or specialized geostatistical algorithms. Recently, a joint interpolation of the horizontal components has been proposed. It estimates forces on a thin elastic sheet that fit the observed data and subsequently uses the estimated model to predict data on regular grids or arbitrary points. The Green’s functions for the physical model serve as a coupling between the two vector components through elasticity theory. We propose an extension of this method to 3D, using the elastic Green’s functions to couple the horizontal and vertical components. This enables the inclusion of vector data projected in arbitrary directions, such as InSAR line-of-sight velocities. The degree of coupling can be controlled through the Poisson’s ratio of the medium. We apply damping regularization to smooth the model and avoid instabilities in the inverse problem. Furthermore, we automatically select optimal values for the Poisson’s ratio and regularization parameter through cross-validation, which is common in machine learning applications. We compare the performance of the coupled model with uncoupled alternatives to grid 2- and 3-component GPS velocities and calculate derivatives through finite-differences approximations. We will present preliminary results from applications to GPS data from the Himalayas and the calibration of InSAR data products. A future goal is to integrate InSAR line-of-sight velocities in a joint interpolation with GPS velocities.

## Citation

Uieda, L., Xu, X., Wessel, P. & Sandwell, D. (2018). Coupled interpolation of three-component GPS velocities. In Eos Trans. AGU (Abstract G23B-0587).