GNSS and InSAR have their own strengths and weaknesses, and they provide highly complementary measurements of surface displacements (Wei et al., 2010). In this context, my colleagues and I developed a new joint inversion algorithm of GNSS and InSAR for continuous surface 3D motions and associated horizontal strain rate field on the surface of a sphere.

Target Area:

Southern California, where San Jacinto Fault splits off the San Andreas Fault

This area is covered by many GNSS stations and two Sentinel-1 tracks (Kim, 2022)

The inversion algorithm solves for the best linear combination of basis functions, which represent the Earth's responses to body-force equivalents embedded within uniform isotropic (or anisotropic) sheet, providing an optimal match to both GNSS and InSAR data.

Basis functions for the Nth model grid cell. The background indicates thin elastic-sheet responses to four different body-force equivalents. In our current workflow, we utilize a total of 1600 linearly independent basis functions (Kim 2022).

How does the inversion algorithm work?

How do we choose an "optimal" solution?

We determine optimal regularization parameters and the relative weighting of InSAR with respect to GNSS by analyzing various misfit metrics. Additionally, we adapted a 10-fold cross-validation technique. We tested these methods using various synthetic datasets. 

The "L-curve" and three different methods to determine the corner (Kim, 2022)

Data Predictions and Modeling Results

Using the best model coefficients of the basis functions, we obtain a continuous 3D velocity field on the surface and a tectonic  horizontal strain rate field, along with horizontal gradients of vertical velocity! 

For testing purposes, we utilized MIDAS GNSS velocities and InSAR LOS velocities, provided by Blewitt et al. (2016) and Xu et al. (2021), respectively (Kim, 2022). This figure illustrates our results: showcases the input data sets (Top), and presents the data predictions (Bottom) that were deemed satisfactory, achieved using the best coefficients from our inversion model. 

Predicted horizontal strain rate fields from the joint inversion (InSAR + GNSS) and the interpolation of GNSS data (Kim, 2022). The background shows the second invariant of strain rates. Red and gray bars are the principal axes of the strain rates. Red for the maximum stretching direction; Gray for the maximum shortening direction.

We compared predicted horizontal strain rate fields using two methodologies: a joint inversion approach that integrates both InSAR and GNSS data, and a separate analysis involving only the interpolation of GNSS data. The comparison, as detailed in Kim (2022), highlights our preference for the joint inversion solution. This preference is based on the observation that the joint inversion more accurately concentrates strain rates along major faults compared to the GNSS interpolation technique.

The Ridge (L2) and LASSO (L1) Regularizations

The joint inversion algorithm can employ one of two different regularization method: L2 or L1. Utilizing the LASSO technique (L1) for inversion results in most of the model coefficients being reduced to zero! This sparsest solution still produces a good deformation model. However, obtaining a solution with the L1 method tends to require more time. 

We're applying our joint inversion algorithm across southern California. Check out our most recent poster below for insights! In addition, my colleagues and I are diligently working on two research papers: one focusin g on the methodology (Kim, Vashishtha, and Holt) and the other on the application of this algorithm (Vashishtha, Kim, and Holt).