Using Effective Media Theory to Better Constrain Seismic Full Waveform Inversion
Abstract:In seismology, effective media theory exploits the fact that bandlimited elastic waves are sensitive to small-scale heterogeneity in only an ‘effective’ manner. From effective media theories follow modern homogenization techniques, which seek to upscale a fine-scale medium into an effective medium that produces an identical long-period solution to the elastic wave equation. These techniques are currently finding heavy use in seismic waveform modelling, which has immediate applications to seismic full waveform inversion (FWI).
FWI is usually formulated as a non-linear gradient-based inverse problem. As with most problems of this class, the solution is strongly dependent on the starting model: if a given starting model predicts 'cycle-skipped' synthetic data, the method is unlikely to converge to the global minimum. As low frequencies are less prone to this ‘cycle-skipping’, the issue is commonly solved by following a multi-scale approach. While this is often successful, it requires that each new frequency band satisfies the half-cycle criterion before being used. We propose to use effective media theory as a communicator between scales and frequency bands; to allow us to broaden the investigated frequency band earlier in the inversion workflow.
To accomplish this, we take advantage of the intuitive notion that the homogenized gradient should vanish at scales much larger than that probed by waves within a given frequency band. Practically, this involves first calculating a broadband waveform sensitivity. Then, within a narrow frequency band which includes the lowest frequency data, we calculate the gradient with respect to a modified misfit function. This modification comes in the form of a Lagrange multiplier constraint which forces the homogenized gradient to zero. Then, we cascade upwards through frequencies, adding constraints which enforce the homogenized gradient in any frequency band to equal the constrained gradient at the next lowest frequency band. By investigating the behaviour of the gradient across different scales of effective media, we hope to place additional physical constraints on the descent direction of a gradient-based FWI problem, to mitigate cycle-skipping issues, and to improve the quality of the solution.