  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. Our approach achieves a
reduction of the unknowns by a factor of approximately 1000 for various
numerical experiments for a 2D subsurface model of Groningen, the Netherlands,
even if the wave speeds of the subsurface are relatively varied. Moreover,
using parallel computing, the reduced model 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 by considering the Laplace-transformed
problem in frequency domain. 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.
