Gradient-constrained model parametrization in 3-D compact full waveform inversion

Abstract

SUMMARY Seismic full waveform inversion (FWI) can produce high-resolution subsurface models by using the complete information of the observed data, but its computational cost can be prohibitively large, particularly for realistically sized 3-D problems. Due to its relatively fast convergence rate, it would be beneficial when Gauss–Newton algorithms could be employed for such problems, but the approximate Hessian matrix required for Gauss–Newton schemes would be too large to be kept in computer memory. Therefore, compact FWI (CFWI) was introduced, with which the number of inversion model parameters can be reduced substantially, and thus the amenable properties of Gauss–Newton inversion schemes can be exploited. Here, we extend the CFWI technology to 3-D problems. Since the spatial coverage of sources and receivers is generally sparser in 3-D (compared with 2-D problems), the total number of model parameters can become very large, and adequate model parametrization is particularly important for 3-D problems. Furthermore, we introduce gradient constrained CFWI (GC-CFWI). This is a novel development that allows the number of model parameters to be further reduced significantly. CFWI employs hierarchical model parametrizations that can be, for example, based on spatial Fourier transforms or wavelet transforms. Only those parameters of such a hierarchical parametrization are retained that exhibit a sufficiently high formal resolution. With GC-CFWI, it is further checked which of these parameters are expected to be changed significantly during a single CFWI iteration. Only parameters with a potentially significant adjustment are retained in the inversion parameter space. We have performed numerical experiments to analyse the performance of CFWI and GC-CFWI for 3-D acoustic FWI problems. For that purpose, we have considered a crosshole geometry including four boreholes and a surface deployment of sources and receivers. As parametrizations, we have considered the Fourier-based Hartley transform and the Haar wavelet transform. For both set-ups, the number of inversion model parameters could be reduced to about 20 per cent for the crosshole model and to about 10 per cent for the surface-based acquisition using CFWI. With GC-CFWI, a further reduction of about 50 per cent for both experimental set-ups could be achieved. The different results for the crosshole and surface-based set-ups indicate that an optimal model parametrization is tightly coupled to the experimental layout.