A new strategy to compare inverted rupture models exploiting the eigen-structure of the inverse problem

Abstract:

Slip inversion methods differ in how the rupture model is parameterized and which regularizations or constraints are applied. However, there is still no consensus about which of the slip inversion methods are preferable and how reliable the inferred source models are due to the non-uniqueness or ill-posedness of the inverse problem. The ‘Source Inversion Validation’ (SIV) initiative aims to characterize and understand the performance of slip inversion methods (http://equake-rc.info/SIV/). Up to now, four benchmark test cases have been proposed, some of which were even conducted as blind tests. The next step is performing quantitative comparisons of the inverted rupture models. To this aim, we introduce a new comparison technique based on a Singular Value Decomposition (SVD) of the design matrix of the continuum inverse problem. We separate the range and null sub-spaces (representing resolved and unresolved features, respectively) by a selected ‘cut-off’ singular value, and compare different inverted models to the target (exact) model after projecting them on the range sub-space. This procedure effectively quantifies the ability of an inversion result to reproduce the resolvable features of the source. We find that even with perfect Green’s functions the quality of an inverted model deteriorates with decreasing cut-off singular value due to applied regularization (smoothing and positivity constraints). Applying this approach to the inversion results of the SIV2a benchmark from various authors shows that the inferred source images are very similar to the target model when we consider a cut-off at ~1/10 of the largest singular value. Although the truncated model captures the overall rupture propagation, the final slip is biased significantly, showing distinct peaks below the stations lying above the rupture. We also show synthetic experiments to assess the role of station coverage, crustal velocity model, etc. on the conditioning of the slip inversion.