Toward understanding slip inversion uncertainty and artifacts: 2. Singular value analysis

Abstract

Seismic slip inversion is studied by means of singular value decomposition (SVD), with emphasis on the role of singular vectors and regularization of the solution. Because the stable part of the slip inversion result is given in terms of a linear combination of the leading singular vectors (representing directions in the model space most sensitive to data), the performance of the inversion depends simply on how well the real slip model can be expanded into these vectors. The analysis is demonstrated using a synthetic model of a symmetric bilateral rupture with two asperities. Inverting the data yields a persisting false asperity in the middle of the fault, regardless of the inversion scheme used, because such a model cannot be well represented by only the leading singular vectors. The parallel study of the singular vectors from the individual stations and the whole network is suggested in order to understand how to achieve balance between overregularized solutions (with possible slip inversion artifacts) and underregularized solutions (vulnerable to data errors). The ideas are applied to the Mw 6.3 Movri Mountain earthquake, 2008, Greece, showing a weak indication of an asymmetric bilateral rupture. We suggest that inversion methods, not working explicitly with singular vectors, may yield artifacts, the origin of which can be also often explained by the SVD technique. Therefore, practical earthquake studies should be broadly complemented by the analysis of the leading singular vectors, a simple tool to evaluate the efficiency and potential drawbacks of the slip inversion.

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

“…Detailed presentations of the role of the SVD in understanding the properties of linear-inverse problems can be found in textbooks on geophysical inverse problems (Menke, 2012), or, for example, Olson and Apsel (1982) and Gallovič and Zahradník (2011) in the context of the slip inversions. The SVD approach implicitly assumes as the objective function the L2 norm of data residuals relative to model predictions.…”

“…in which K is the index of the smallest singular value larger than λ c (Jackson, 1972;Wiggins, 1972;Gallovič and Zahradník, 2011). After truncating both the inverted and the target model, we evaluate the MVR using the following variant of equation (7), which we refer to as the truncated model variance reduction (TMVR): E Q -T A R G E T ; t e m p : i n t r a l i n k -; d f 9 ; 5 2 ; 7 2 1 TMVR 1 − jjm T −m T jj=jjm T jj:…”

“…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.…”

Resolution of Rise Time in Earthquake Slip Inversions: Effect of Station Spacing and Rupture Velocity

“…In principle, the model discretization in time is similar to the classical multi-timewindow method introduced by Hartzell and Heaton (1983), but the implementations of that approach are typically limited to about 10 windows (Hartzell and Langer, 1993). Our unrestricted multi-time-window approach allows for time windows to cover the whole duration of the rupture, at all fault locations, unlike other multi-time-window approaches used (Olson and Anderson, 1988;Das and Kostrov, 1990;Gallovič et al, 2009;Gallovič and Zahradník, 2011). Our approach does assume a known fault geometry, as typical in inversions (Hartzell and Heaton, 1983;Olson and Anderson, 1988;Graves and Wald, 2001;Ji et al, 2002).…”

“…Saraò et al (1998) found that stations on the hanging wall facilitate source inversion on dip-slip faults. The case of single-station inversion was considered in Gallovič and Zahradník (2011) to understand the individual contribution of each station. The present study is the first to consider systematically the effect of network spacing in regular 2D networks, including dense networks with a large number of stations that were prohibitively expensive for earlier studies and for certain inversion methods.…”