Model Order Reduction for Seismic Applications

In this paper, we propose a model order reduction approach to speed up the computation of seismograms, i.e., the solution of the seismic wave equation evaluated at a receiver location, for different model parameters. This is highly relevant for seismic applications, such as full waveform inversion, seismic tomography, or monitoring tools of seismicity, that are computationally challenging,
as the discretized (forward) model often has a huge number of unknowns and needs to be solved many times for different model parameters. Our approach achieves a reduction of the unknowns by a factor of approximately 1000 for various numerical experiments for a two-dimensional subsurface model of Groningen, the Netherlands (a region known for its seismic activity), even if the (known) wave speeds of the subsurface are relatively varied. Moreover, when using multiple cores to construct
the reduced model, we can approximate the (time domain) seismogram in a lower wall-clock time than an implicit Newmark-beta method. To realize this reduction, we exploit the fact that seismograms are low-pass filtered for the observed seismic events. We thus consider the Laplace-transformed problem in the frequency domain and implicitly restrict ourselves to the frequency range of interest by adjusting the parameters of a source function, which is a popular model in computational seismology for the temporal response of an earthquake. Therefore, we can avoid the high frequencies that would require many reduced basis functions to reach the desired accuracy and generally make the reduced order approximation of wave problems challenging. Instead, we can prove for our ansatz that for a fixed subsurface model, the reduced order approximation converges exponentially fast in the
frequency range of interest in the Laplace domain. We build the reduced model from solutions of the Laplace-transformed problem via a (proper orthogonal decomposition-)greedy algorithm targeting the construction of the reduced model to the time domain seismograms; the latter is achieved by using an a posteriori error estimator that does not require computing any time domain counterparts.
Finally, we show that we obtain a stable reduced model, thus overcoming the challenge that standard model reduction approaches do not necessarily yield a stable reduced model for wave problems.