Homogenization of seismic point and extended sources

Abstract

SUMMARY Seismic sources are mostly modelled as point sources: moment tensors associated with the gradient of a Dirac distribution. Such sources contain an infinite range of scales and induce a discontinuity in the displacement wavefield. This makes the near-source wavefield expensive to model and the event location complex to invert, in particular for large events for which many point sources are required. In this work, we propose to apply the non-periodic two-scale homogenization method to the wave equation source term for both force and couple-sources. We show it is possible to replace the Dirac point source with a smooth source term, valid in a given seismic signal frequency band. The discontinuous wavefield near-source wavefield can be recovered using a corrector that needs to be added to the solution obtained solving the wave equation with the smooth source term. We show that, compared to classical applications of the two-scale homogenization method to heterogeneous media, the source term homogenization has some interesting particularities: for couple-sources, the leading term of the homogenization asymptotic expansion is dependent on the fine spatial scale, depending on the source type, only one or two first terms of the expansion are non-zero and there is no periodic case equivalent (the source term cannot be made spatially periodic). For heterogeneous media, two options are developed. In the first one, only the source is homogenized while the medium itself remains the same, including its discontinuities. In the second one, both the source and the medium are homogenized successively: first the medium and then the source. We present a set of tests in 1-D and 2-D, showing accurate results both in the far-source and near-source wavefields, before discussing the interest of this work in the forward and inverse problem contexts.

Interpretation of time reversal focal spots based on point-source homogenization

“…In theory, this discontinuity is even sharper than what we observed in Fig. 5 because the simulation is limited by the mesh resolution (Capdeville 2021).…”

“…It is not trivial to prove that such a combination can generate a continuous wavefield. In the following part, we show that it can be demonstrated thanks to the theory of point-source homogenization developed by Capdeville (2021).…”

“…where λ 0 is the wavelength under which the scales are considered as small (defined by the user) and λ min the minimum wavelength to be propagated in the model. Capdeville (2021) recently applied homogenization to a point source. The idea is to filter the high-wavenumber content of a point source in order to compute a smooth effective source.…”

Novel Hybrid Numerical Simulation of the Wave Equation by Combining Physical and Numerical Representation Theorems and a Review of Hybrid Methodologies

“…This is because the wavefields in the elements containing the sources are not accurate for SEM (Nissen‐Meyer et al., 2007). In the framework of normal numerical simulations using SEM, recent effective sources can be used to address this problem (Capdeville, 2021). In the hybrid simulation framework, an additional layer of elements can be used to bypass its influence (Lin et al., 2019).…”

“…41625016, 41888101, and 42004045), the China Scholarship Council (File No. 201804910289), and the French National Research Agency (ANR-16-CE31-0022-01, 2016-2021. The authors note that there are no data-sharing issues because all of the numerical parameter information is provided in the figures produced by solving the global and local hybrid numerical simulations in this study.…”