Surrogate models for seismological inverse problems: Constructing fast approximations to misfit functions

Generally, solving an inverse problem involves finding the global minimum of some 'misfit function', which provides a measure of how well any given set of model parameters explain available data. The misfit function also encapsulates information about the resolution properties and uncertainties associated with any solution, and accessing and understanding this is necessary if results are to be properly interpreted. In seismology -- where we are typically interested in adjusting earth or source models to bring synthetic waveforms (or measurements made thereon) into agreement with recorded data -- a variety of tools have been developed to enable misfit to be evaluated at any point in model-space. However, these calculations are computationally demanding, making it impossible to find best-fitting solutions via brute-force search. One way around this is to linearise the inverse problem, evaluate the gradient of the misfit function at a given location, and then use this information to iteratively step towards a minimum. However, unless the misfit function has a simple form, linearised algorithms may fail to converge to the global minimum. A second class of approaches involve directed random search, using various strategies to preferentially sample low-misfit regions of model space. This is computationally expensive, and may become infeasible as the dimension of the model space increases. We show that it is possible to construct an approximation to the misfit function using a learning algorithm. This assimilates information obtained by evaluating the forward problem, and interpolates between these samples. It is possible to progressively refine the approximation, by using its current state to direct the generation of new samples. Evaluating the approximation at any point in model space is computationally cheap, and it has a well-defined (and differentiable) functional form. The approximation may therefore be substituted for a full evaluation of the misfit in a wide range of calculations, and may be regarded as a 'surrogate model' for the misfit. Furthermore, we show that the process of learning the approximation can be accelerated if information about the gradient of misfit is also provided. At present, we focus on seismological applications, but the approach may be of interest throughout geophysics.