Simultaneous and Integrated Strain Tensor Estimation From Geodetic and Satellite Deformation Measurements to Obtain Three-Dimensional Displacement Maps

We propose a new technique, named SISTEM, based on the elastic theory, to efficiently estimate 3-D displacements for producing deformation maps by integrating sparse Global Positioning System (GPS) measurements of deformations and differential interferometric synthetic aperture radar (DInSAR) maps of movements of the Earth´s surface. Previous approaches in the literature to combine GPS and DInSAR data require two steps: a first step in which sparse GPS measurements are interpolated in order to fill in GPS displacements in the DInSAR grid and a second step to estimate the 3-D surface displacement maps by using a suitable optimization technique. One of the advantages of the proposed approach, compared to previous ones, is that it does not require the preliminary interpolation of the observed deformation pattern. Indeed, we propose a linear matrix equation which accounts for both the GPS and DInSAR data whose solution simultaneously provides the strain tensor, the displacement field, and the rigid body rotation tensor. The mentioned linear matrix equation is solved by using the weighted least square (WLS), thus assuring both numerical robustness and high computation efficiency. The methodology was tested on both synthetic and experimental data, these last from GPS and DInSAR measurements carried out on Mount Etna during the 2003-2004 period. In order to appreciate the accuracy of the results, the estimated standard errors computed by the WLS are provided. These tests also allow optimizing the choice of specific parameters of this algorithm. This method can be further exploited to account for other available data sets, such as additional interferograms or other geodetic data (e.g., leveling, tilt, etc.), in order to achieve higher accuracy.