Full-waveform inversion with an exponentially encoded optimal-transport norm

Abstract

Full waveform inversion (FWI) with 2 norm objective function often suffers from cycle skipping that causes the solution to be trapped in a local minimum, usually far from the true model. We introduce a new norm based on the optimal transport theory for measuring the data mismatch to overcome this problem. The new solution uses an exponential encoding scheme and enhances the phase information when compared with the conventional 2 norm. The adjoint source is calculated trace-wise based on the 1D Wasserstein distance. It uses an explicit solution of the optimal transport over the real line. It results in an efficient implementation with a computational complexity of the adjoint source proportional to the number of shots, receivers and the length of recording time. We demonstrate the effectiveness of our solution by using the Marmousi model. A second example, using the BP 2004 velocity benchmark model, illustrates the benefit of the combination of the new norm and Total Variation (TV) regularization.

Long-wavelength FWI updates beyond cycle skipping

“…where and ) ( t d are encoded versions of the modeled and field data. The definition of the quadratic Wasserstein distance and the resulting Frechet derivative to obtain the adjoint source are explained in Qiu et al (2017).…”

“…Moreover, in complex geological settings such as those with the presence of salt, a small error in the location of the reflectors may lead to large kinematic errors. Thus, there is an incentive to change the metrics away from the L2 norm for quantifying the data misfit (e.g., Enquist et al, 2016;Métivier et al, 2016;Qiu et al, 2017).…”

“…The new objective function was developed in the context of optimal transport theory. Our implementation (Qiu et al, 2017) uses an encoding scheme based on a logistic function that assures the mass conservation and positiveness required by the optimal transport theory.…”

Long-wavelength FWI updates in the presence of cycle skipping

“…Moreover, in complex geologic settings such as salt environments, a small error in the location of the reflectors would lead to large kinematic errors. Hence, there is an incentive to use a different metric than the L2 norm for quantifying the data misfit (e.g., Engquist et al, 2016;Métivier et al, 2016;Qiu et al, 2017).…”

“…It effectively separates the migration isochrones produced by the specular reflectivity from the components produced by transmitted arrivals. Our implementation (Qiu et al, 2017) uses an encoding scheme based on a logistic function that assures the positiveness and mass conservation conditions required by the optimal-transport theory.…”