Torwards time-domain extended source waveform inversion
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Full waveform inversion (FWI) updates the subsurface model from an initial model by comparing observed and synthetic seismograms. Despite the potential of FWI for high resolution seismic imaging, this technique is a nonlinear optimization and may be trapped into local minima. Wavefield reconstruction inversion (WRI) relaxes the condition that the wave equation associated with a true physical source does need to be accurately satisfied during the iteration process. Alternatively, extended source waveform inversion (ESI) introduces a non-physical source deployed into the whole computing domain for matching the observed data by synthetic data. The iterative process will move this extended source to the physical source. WRI and ESI are formulated within the same mathematical framework using Lagrangian-based adjoint-state method with a special focus on time-domain formulation using extended sources, while putting connections between classical FWI, WRI and ESI: both WRI and ESI can be viewed as weighted versions of classic FWI. Due to symmetric positive definite Hessian, the conjugate gradient is explored to efficiently solve the normal equation in a matrix free manner, while both time and frequency domain wave equation solvers are feasible. The most significant challenge comes from the huge storage demand to store time-domain wavefields through iterations. We describe two possible workaround strategies by extracting sparse frequencial wavefields or by considering time-domain data instead of wavefields for reducing such challenge. We suggest that these options should be explored more intensively for tractable workflows.
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