Long-wavelength FWI updates beyond cycle skipping

Abstract

Full Waveform Inversion (FWI) success depends on producing seamless updates of the short-and longwavelength features missing in the starting velocity model while avoiding cycle skipping. The use of cross-correlation gradients in FWI can lead to updates with the reflectivity imprint (high-wavenumbers) before the long wavelength updates have been constructed. In addition, the use of L2norm to measure the data misfit is prone to cycle skipping. This may conduct to a local-minimum if the data lacks of low frequency information and/or the initial model is far from the true earth model. We offer a solution to these two FWI fundamental problems that combines a robust implementation of the velocity sensitivity kernel and the optimal transport norm to measure the data misfit. The new scheme can retrieve the long wavelength updates and reduce the cycle skipping problem. The velocity kernel eliminates the migration isochrones emphasizing the longwavelength updates produced by the diving waves and the "rabbit ears" provided by reflections. The optimal transport norm accentuates those long-wavelength updates while minimizing the cycle skipping. We demonstrate the advantages of our implementation on synthetic and field data examples.

Optimal Transport Based Seismic Inversion:Beyond Cycle Skipping

“…There are also modeling and numerical errors in the wave simulation as well as noisy data. What makes us confident of the practical value of the techniques discussed here is the emerging popularity in the industry and successful application to real field data, which have been reported [41,43,47]. We include one numerical example to show the robustness of W 2 -based inversion in Section 6 by using a perturbed synthetic source and adding correlated data noise.…”

Optimal Transport Based Seismic Inversion: Beyond Cycle Skipping

“…Instead of solving a 2D or 3D optimal transport problem which can be computationally challenging, the trace-by-trace approach utilizes the explicit solution to the 1D optimal transport problem. Since it is fast and accurate with a minimum numerical error, the cost-effective trace-by-trace approach is particularly appreciated by the industry [49,37]. Nevertheless, benefits have been observed regarding the lateral coherency of the data [34,25] by solving a 2D or 3D optimal transport problem to compute the W 2 metric.…”

“…There are also modeling and numerical errors in the wave simulation as well as noisy data. What makes us confident of the practical value of the techniques discussed here is the emerging popularity in the industry and successful application to real field data, which have been reported [34,37,32]. We include, however, one numerical example to show the robustness of W 2 -based inversion in Section 3.1 by using a perturbed synthetic source and adding correlated data noise.…”

Analysis and Application of Optimal Transport For Challenging Seismic Inverse Problems

“…The current challenges of FWI motivate us to replace the traditional L 2 norm with a new metric of better convexity and stability for seismic inverse problems. Soon after [16], there have been fruitful activities in the past four years in developing the idea of using optimal transport based objective functions for FWI from both academia [11,47,17,37,38,64,2,42] and industry [48,51,45] with several field data applications. By using the W 2 distance to measure the two signals in Figure 1a, we obtain a globally convex optimization landscape as illustrated in Figure 1c.…”

“…Figure 13 shows the residual spectrum of both L 2 -FWI and W 2 -FWI at different iterations. The residual is computed as the difference between synthetic data f (x r , t; m) generated by the model m at current iteration and the true data g(x r , t): (51) residual = f (x r , t; m) − g(x r , t).…”