Summary
Time-domain Full-Waveform Inversion (FWI) aims to image the subsurface of the earth accurately from field recorded data and can be solved via the reduced adjoint-state method. However, this method requires access to the forward and adjoint wavefields that are meet when computing gradient updates. The challenge here is that the adjoint wavefield is computed in reverse order during time stepping and therefore requires storage or other type of mitigation because storing the full time history of the forward wavefield is too expensive in realistic 3D settings. To overcome this challenge, we propose an approximate adjoint-state method where the wavefields are subsampled randomly, which drastically the amount of storage needed. By using techniques from stochastic optimization, we control the errors induced by the subsampling. Examples of the proposed technique on a synthetic but realistic 2D model show that the subsampling-related artifacts can be reduced significantly by changing the sampling for each source after each model update. Combination of this gradient approximation with a quasi-Newton method shows virtually artifact free inversion results requiring only 5% of storage compared to saving the history at Nyquist. In addition, we avoid having to recompute the wavefields as is required by checkpointing.
Introduction
Time-domain Full-Waveform Inversion (FWI) is a well known and widely used method for acoustic inversion as its marching in time structure allows easy and memory-efficient implementation and fast solutions. The requirement to access the forward wavefield in a reverse order in time led to numerous techniques including checkpointing and optimal checkpointing strategies (Symes, 2007,Griewank and Walther (2000)), recomputing the wavefield when needed from a partial save of the time history, or boundary methods that use integrals to compute the wavefield from its full history saved at the boundary of the domain (Plessi, 2006; Clapp, 2009).
All approaches that aim to address the above problem are balancing memory requirements and computational cost of recovering the wavefield for the unknown part of the history. The method we are proposing in addition will also balance memory requirements with the accuracy of the gradient calculations. We motivate this strategy from recent insights in stochastic optimization that prove that gradients need not be accurate especially in the beginning of an iterative inversion procedure (van Leeuwen and Herrmann, 2014). However, our approach differs slightly because we do not approximate the objective. Instead, we approximate the gradient by randomly subsampling the time history of the wavefield prior to correlation and applying the imaging conditions. To limit the imprint of this random subsamplings, we draw independent subsamplings for each source and after each model update. As a consequence, the subsampling-related artifacts average out as the inversion procedure progresses.
