A New Strategy to Compare Inverted Rupture Models Exploiting the Eigenstructure of the Inverse Problem

Finite-fault-slip inversions provide crucial information on earthquake rupture phenomena. Many slip-inversion methods exist and differ in how the rupture model is parameterized and which regularizations or constraints are applied (e.g., Ide, 2007, and references therein). Some methods are utilized even routinely for large earthquakes and published online (e.g., the U.S. Geological Survey website http://earthquake.usgs.gov/, last accessed August 2015). However, the slip-inversion results obtained by various authors for the same event may differ (e.g., Clévédé et al., 2004). There is currently no consensus about which slip-inversion method is preferable, and there are concerns about the reliability of the inferred source models due to the nonuniqueness or ill conditioning of the inverse problem (Hartzell et al., 2007; Zahradník and Gallovič, 2010; Gallovič and Zahradník, 2011; Shao and Ji, 2012). Therefore, slip inversion is still a subject of active research. A requisite to understand the variability of slip-inversion results across different methods is the characterization of their similarities and differences. Methods to compare spatial distributions of final slip have been previously developed and applied to synthetic and real cases (Clévédé et al., 2004; Razafindrakoto et al., 2015; Zhang et al., 2015). Here, we propose an approach to compare the complete space–time evolution of rupture models. The basic ideas behind our comparison technique are as follows. If the fault geometry is assumed, the forward problem of the slip inversion is a linear mapping from the model space (the spatial–temporal distribution of slip) to the data space (the seismograms) by means of the representation theorem (e.g., Aki and Richards, 2002). The spectral decomposition of the forward operator and its discrete counterpart, the singular value decomposition (SVD), provide a natural set of basis functions (singular vectors) in the model space. Any source model then can be decomposed into two parts made of linear combinations of singular vectors lying in the coimage and in the null space, which are associated with large and small (or even zero) singular values, respectively. Because only the coimage components provide significant signal, any slip-inversion method should resolve them correctly. The inversion results obtained by different methods may then differ in their null-space contributions, which are implicitly determined by the particular choices of source model parameterization and regularization of each method. In the subsequent section, we introduce the principles of our comparison method, including an objective method to determine the boundary between the coimage and null spaces. We then briefly describe the Source Inversion Validation (SIV2a) benchmark problem conducted under the SIV (http:// equake‐rc.info/SIV/, last accessed August 2015) project. We use the SIV2a setup to illustrate the practical application of our method and compare SIV2a-inversion results obtained by various authors. We also assess how the conditioning of the slip inversion is affected by frequency band, station weighting, network coverage, and crustal model. Finally, we discuss the origin of slip-inversion bias and the implications of our results for the comparison of performance of source-inversion methods.