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.
SUMMARYA boundary integral formulation is presented and applied to model the ground motion on alluvial valleys under incident P, S and Rayleigh waves. It is based on integral representations for the diffracted and the refracted elastic waves using single-layer boundary sources. This approach is called indirect BEM in the literature as the sources' strengths should be obtained as an intermediate step. Boundary conditions lead to a system of integral equations for boundary sources. A discretization scheme based on the numerical and analytical integration of exact Green's functions for displacements and tractions is used. Various examples are given for two-dimensional problems of diffraction of elastic waves by soft elastic inclusion models of alluvial deposits in an elastic half-space. Results are displayed in both frequency and time domains. These results show the significant influence of locally generated surface waves in seismic response and suggest approximations of practical interest. For shallow alluvial valleys the response and its resonant frequencies are controlled by a coupling mechanism that involves both the simple one-dimensional shear beam model and the propagation of surface waves.
We describe a new solution for recovering the long-wavelength features of a velocity model in gradient-based full-waveform inversion (FWI). The method uses reflected and transmitted wave modes to recover high-resolution velocity models. The new FWI gradient enables reliable velocity updates deeper than the maximum penetration depth of diving waves and reduces the FWI dependency on recording ultralong offsets. We also discuss a new FWI regularization scheme that overcomes the limitations of the inversion in the presence of high-contrast geobodies and cycle skipping. The solution utilizes a priori information about the earth model in the regularization as an extra term in the objective function. The implementation makes use of the split Bregman method, making it efficient and accurate. Results from applying the new FWI gradient to field data show that we can combine both transmitted and reflected energy in a global FWI scheme to obtain high-resolution velocity models without imprint of the reflectivity on the velocity updates. We illustrate the new regularization method's potential on the BP 2004 velocity benchmark model where our regularized FWI solution is capable of using a simple starting velocity model to deliver a high-quality reconstruction of salt bodies.
Splitting of zero‐offset reflected shear‐waves is measured directly from three‐component finite‐difference synthetic seismograms for media with intersecting vertical crack systems. Splitting is simulated numerically (by finite differencing) as a function of crack density, aspect ratio, fluid content, bulk density, and the angle between the crack systems. The type of anisotropy symmetry in media containing two intersecting vertical crack systems depends on the angular relation between the cracks and their relative crack densities, and it may be horizontal transverse isotropy (HTI), tetragonal, orthorhombic, or monoclinic. The transition from one symmetry to another is visible in the splitting behavior. The polarities of the reflected quasi‐shear waves polarized perpendicular and parallel to the source particle motion distinguish between HTI and orthorhombic media. The dependence of the measured amount of splitting on crack density for HTI symmetry is consistent with that predicted theoretically by the shear‐wave splitting factor. In orthorhombic media (with two orthogonal crack systems), a linear increase is observed in splitting when the difference between crack densities of the two orthogonal crack systems increases. Splitting decreases nonlinearly with the intersection angle between the two crack systems from 0° to 90°. Surface and VSP seismograms are simulated for a model with several flat homogeneous layers, each containing vertical cracks with the same and with different orientations. When the crack orientation varies with depth, previously split shear waves are split again at each interface, leading to complicated records, even for simple models. Isotropic and anisotropic three‐component S-wave zero‐offset sections are synthesized for a zero‐offset survey line over a 2.5-D model of a carbonate reservoir with a complicated geometry and two intersecting, dipping crack sets. The polarization direction of the fast shear wave, propagating obliquely through the cracked reservoir, is predicted by theoretical approximations for effective properties of anisotropic media with two nonorthogonal intersecting crack sets.
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