Abstract

Numerous problems in the field of seismology require the determination of parameters of a physical model that are compatible with a set of observations and prior assumptions. This type of problem is generally termed inverse problem. While, in many cases, we are able to predict observations, given a particular set
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of parameters with high accuracy and precision using numerical and analytical methods, often we are unable to directly infer model parameters from observations. This has lead to the development of a large number of inverse methodologies. For many applications it is necessary not only to obtain a set of best fitting parameters, but also an estimate of the attached uncertainties, which arise from the fact that any observation is subject to measurement uncertainties and from other sources of uncertainty that may be present due to numerical limitations or simplifying assumptions. Often, the situation is further complicated by the non-linear nature of many inverse problems, leading to non-Gaussian and possibly non-unique posterior distributions. An important seismological inverse problem is the determination of earthquake source parameters. Source parameters are on one hand needed as an input for subsequent investigations, such as tomographic imaging or finite fault studies. On the other hand source parameter estimates are required in order to provide information to the public as quickly as possible after a new earthquake has initiated - a field known as earthquake early warning. We have developed a methodology for rapid probabilistic moment tensor point source inversions able to cope with a wide variety of data types. Accurate and computationally expensive synthetic forward modelling can be incorporated without significantly altering the time required for the inversions. Our methodology is based on finding an approximation to the conditional posterior probability of source models given observations by smoothly interpolating a set of prior samples. The interpolation is obtained using ensembles of Mixture Density Networks (MDNs) - a class of artificial neural networks, whose output are the parameters of a Gaussian mixture model. Once an ensemble has been constructed, new observations can be inverted within the fraction of a second on a standard desktop computer. The method is therefore well suited for earthquake early warning, where new observations must be inverted routinely and rapidly. We apply the method to two scenarios in Southern California using both synthetic and observed datasets. We invert regional static and dynamic GPS displacement data for the 2010 M 7.2 El Mayor Cucapah earthquake in Baja California to obtain estimates of magnitude, centroid location and depth, and focal mechanism. Moreover, we show how accurate 3-D spectral element simulations can be incorporated into the rapid inversion scheme by means of a model scenario focussing on the 2008 M 5.4 Chino Hills earthquake. We discuss general implications, advantages and limitations of the method and suggest directions for potential future research.
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