Bayesian Inversion of Wrapped Satellite Interferometric Phase to Estimate Fault and Volcano Surface Ground Deformation Models

Abstract

<p>Phase unwrapping is the process of recovering the absolute phase from unambiguous wrapped phase values that are measured modulo 2pi rad. From a mathematical point of view, phase unwrapping is an inverse problem, however, it is ill-posed and notoriously difficult to solve in the presence of noise. Meanwhile, phase unwrapping errors severely impact the estimation of earthquake and volcano source parameters using interferometric observations, therefore avoiding phase unwrapping completely is desirable. </p><p>A potential solution to avoid the unwrapping error issue completely, is to carry out a geophysical inversion directly on the wrapped phase observations. To overcome the need for phase unwrapping, we propose a novel approach that we can invert directly the interferometric wrapped phase, circumventing the ill-posed phase unwrapping processing step. This approach includes (1) a downsampling algorithm, (2) a method to estimate the covariance function of the wrapped phase, (3) an appropriate misfit function between the observed and the simulated wrapped phase. We also assess the uncertainties of source parameters within a Bayesian approach, and finally we test the robustness of the inversion methodology in multiple simulations including variable decorrelation and atmospheric noise simulations.</p><p>We demonstrate the proposed methodology on synthetic cases with variable noise and one real earthquake case. We show that the method is robust in challenging noise scenarios. We also show an improvement with the Bayesian approach in performance with respect to similar previous methods, resulting in avoiding any influence of seed starting models, and escaping local minima. We study the impact of a small percentage of incorrectly unwrapped phase observations in current state-of-the-art methods, and show that the presence of a small fraction of unwrapping errors affect strongly the estimation process. We conclude that in the cases where phase unwrapping is difficult or even impossible, the proposed inversion methodology with wrapped phase will provide an alternative approach to assess earthquake and volcano source model parameters.</p>

Mitigating bias in inversion of InSAR data resulting from radar viewing geometries

“…In order to give equal weights to each data set during an inversion, we investigated several methods: one involves adjusting the number of subsampled points for each data set; the second, hereafter referred to as the Covariance Weighting inversion method, involves weighting the data through the covariance matrix; and the last, hereafter referred to as the Wrapped Inversion method, involves removing the influence of data magnitude by carrying out inversions on wrapped interferograms (Feigl & Thurber 2009;Jiang & González 2020).…”

“…Another approach consists of carrying out inversions on wrapped data. This method has been proposed by Feigl & Thurber (2009); Ali & Feigl (2012) and improved by Jiang & González (2020) in order to overcome errors related to the phase-unwrapping step. Here, we use it in order to cancel the influence of data magnitudes.…”

Aseismic Fault Slip During a Shallow Normal‐Faulting Seismic Swarm Constrained Using a Physically Informed Geodetic Inversion Method

“…Then, the fault slip distribution is forward modeled to estimate surface displacement. Following Jiang and González (2020), a misfit function is constructed based on the wrapped phase residuals and the weighting matrix. The misfit function is then regarded as the likelihood function fed into the Bayesian process to retrieve the posterior distribution of crack model parameters.…”

“…To develop the kinematic fault model, we first constructed the fault geometry derived from a nonlinear fault inversion of InSAR wrapped phase observations, solving for uniform distribution on rectangular faults (Jiang & González, 2020). A geodetic inversion directly using the interferometric wrapped phase avoids any potential phase unwrapping error (Figure S6 in Supporting Information S1).…”

“… Fault geometry for the 2011 Hawthorne seismic swarm. Image (a) indicates the fault plane with uniform slip retrieved by WGBIS (Jiang & González, 2020) from the wrapped interferograms, and the modeled phase and phase residuals are shown in Figure S8 in Supporting Information S1. In image (a), the green rectangle indicates the southern subfault which is active during the pre‐M4.6 stage, retrieved from RADARSAT‐2 interferogram 22 March 2011–15 April 2011; yellow rectangle indicates the northern subfault which is active during the co‐ and post‐M4.6 stages, retrieved from the RADARSAT‐2 interferogram 15 April 2011–26 June 2011, and the yellow triangle indicates the joint fault connecting two rectangle subfaults.…”