# Joint Interpolation of 3-component GPS Velocities Constrained by Elasticity

Leonardo Uieda, David Sandwell, Paul Wessel

## Info

This talk is about some early results from a collaboration me and Paul have with David Sandwell. The project was started by Paul and Dave as a follow up to their 2016 paper about interpolating 2D vector data. We're expanding it into 3D and ironing out some kinks in the methodology. I picked it up at the beginning of the year and have been slowly trying things out. It was a great chance to play with some tools from machine learning since this is a supervised prediction problem. Unlike most geophysical inversion, we're not really interested in the estimated parameters themselves. They are only a means to predict new values (on a regular grid in the case of gridding). I started implementing the tools I would need in a Python library called Verde, which served as the basis for the results shown in the presentation.

I made the slides using Google Docs with the Barlow font for the most part and EB Garamond for some of the math symbols. I used an online Latex equation editor to generate the equations as high resolution PNGs. Not ideal but it's a price I will pay for avoiding LibreOffice Impress.

## Abstract

Vertical ground motion at fault systems can be difficult to detect due to their small amplitude and contamination from non-tectonic sources, such as ground water loading. However, it may play an important role in our understanding of the earthquake cycle and the associated seismic hazards. Ground motion measurements from GPS are often sparse and must be interpolated onto a regular grid (e.g., for computing strain rate), ideally taking into account the varying degrees of uncertainty of the data. Traditionally, each vector component is interpolated separately using minimum curvature or biharmonic spline methods. Recently, a joint interpolation of the two horizontal components has been developed using the Green's functions for a point force deforming a thin elastic sheet. The elasticity constraints provide a coupling between the two vector components and lead to improved results because the underlying physics of the method approximately matches that of the GPS observations. We propose an expansion of this method into 3D in order to incorporate vertical GPS velocity measurements. To smooth the model and avoid singularities, we formulate the interpolation as a weighted least-squares inverse problem with damping regularization. Optimal values of the regularization parameter and the Poisson's ratio of the elastic medium are determined through K-fold cross-validation, a technique often used in machine learning for model selection. Additionally, the cross-validation provides a measure of the accuracy of model predictions and eliminates the need for manual configuration. The computational load of the inversion is lessened by imposing a cutoff distance to the Green's functions computations, which makes the sensitivity matrix sparse. We will present preliminary results from an application to EarthScope GPS data from the San Andreas Fault system. In the future, we aim to develop a joint inversion of 3D GPS and InSAR line-of-sight velocities to improve data coverage.