Joint Minimization of the Mean and Information Entropy of the Matching Filter Distribution for a Robust Misfit Function in Full-Waveform Inversion

Abstract

Full-Waveform Inversion (FWI) is a highly nonlinear inversion methodology. FWI tends to converge to a local minimum rather than a global one. We refer to this phenomena as "cycle skipping" in FWI. A cost-effective solution for resolving this issue is to design a more convex misfit function for the optimization problem. A global comparison based on using a matching filter admits more robust misfit functions. In this case, we would compute a matching filter first by deconvolving of the predicted data from the measured ones. When the velocity model is accurate, the predicted data resemble the measured ones, and the resulting matching filter would be an approximated Dirac delta function. If the velocity produces data that are different from the observed ones, a misfit function can be formulated by penalizing the energy away from the zero-lag time (the center). Here, we develop a general mechanism for an evolution of the matching filter to our objective in FWI. Specifically from the statistics point of view, rather than using a penalty, we propose a novel misfit by minimization the mean and information entropy of the matching filter distribution. We show that the resulting misfit function can mitigate the "cycle skipping" as well as reduce the mean and variance of the resulting matching filter distribution. We use a modified Marmousi example to demonstrate the features of the proposed misfit. We also evaluate the robustness of the proposed method using inaccurate (rotation in phase) source wavelets and measured data with different levels of Gaussian random noise.

Full-waveform inversion based on generalized Rényi entropy using patched Green’s function techniques

“…The PGF method is widely used in condensed matter physics for modeling transport of electronic waves [49,50] through the calculation of the complete Green's function G of a system using the so-called Dyson equation. By considering that the Green's function in Eq (35) is invertible, we may apply the Dyson equation to compute G −1 [67][68][69]. In this regard, the total Green's function G is associated to a self-energy g by employing a connection potential term V. We highlight that the connection potential is a very important term in the Dyson equation, as it is responsible for connecting the elements necessary for the calculation of the recursive process, which will be described later.…”

“…[52] reformulated this technique for propagating acoustic waves in disordered (non-homogeneous) media. In this regard, the solution of the wave equation in (35) is efficiently solved by computing the following Green's functions [53]:…”

“…1 and q = 2 limits, respectively. Indeed, generalizations of Gauss' law of error based on the foundations of statistical physics have been successfully applied to perform robust physical parameters' estimation in non-linear geophysical problems, such as misfit functions based on Student's t distribution [18][19][20], deformed Gaussian distributions [21][22][23][24][25], generalized maximum likelihood approaches [26][27][28], non-parametric methods [29], normalized-based objective functions [30,31], Wasserstein metric from the optimal transport distance [32][33][34], matching filter techniques [35], as well as in the Re ´nyi framework [36].…”

Wavefield Reconstruction Inversion via Physics-Informed Neural Networks

“…We can also seek a solution to cycle skipping from another perspective by formulating new objective functions instead of the conventional l 2 norm. In adaptive waveform inversion (AWI), [15], [16] proposed to transform the data using a convolutional operators and force Wiener filters to zero-lag delta function instead of the conventional data comparison. Wavefield reconstruction inversion (WRI) relaxes the wave equation accuracy to allow more data fitting and mitigate the nonlinearity of FWI [17], [18].…”

Potential of Phase-Amplitude-Based Multi-Scale Full Waveform Inversion with Total-Variation Regularization for Seismic Imaging of Deep-Seated Ores

“…In addition, correlation-based FWI misfit functions focus on the phase similarity and relax the requirement for amplitude matching between synthetic and observed data [36][37][38][39][40]. Furthermore, adaptive matching filter-based FWI misfit functions are also a kind of phase adjustment approach, which can effectively invert subsurface structures [41][42][43][44].…”