A simple method for earthquake location by surface-wave time reversal

12 The scalar 2-D Helmholtz’ equation (i.e., “membrane waves”) can be used to model surface13 wave propagation in a laterally smooth, lossless half space. Building on this known result, we 14 develop an algorithm to localize earthquake sources based on surface-wave data, via numerical 15 time-reversal on a membrane, where monochromatic waves propagate with the phase velocity 16 of Rayleigh or Love waves at the same frequency. Importantly, membrane-wave modeling is 17 computationally much less expensive than three-dimensional surface-wave modeling. We 18 first explain rigorously the relationship between surface waves and membrane waves. Our 19 mathematical treatment is slightly different from those found in the literature, in that it does 20 not invoke variational principles. We next implement our time-reversal algorithm via spectral 21 elements as well as simple ray tracing. Both implementations account for the effects of lateral 22 variations in phase velocity. We validate the two resulting tools by means of several numerical 23 experiments. This includes synthetic tests, as well as the localization of a virtual source based 24 on a data set of real ambient-noise cross correlations, and the localization of the epicenter of a 25 real earthquake from real, raw data. In this study, applications are limited to Northern Italy 26 and the Alpine arc, where we have access to recent, high resolution phase velocity maps, 27 ambient noise cross correlations and data from a recent, relatively large earthquake. The 28 accuracy of epicenter location despite the non-uniform station coverage encourages further 29 applications of our method, in particular to the task of mapping large-earthquake rupture in 30 space and time. 31