An introduction to the two-scale homogenization method for seismology

Yann Capdeville and Paul Cupillard and Sneha Singh. ( 2020 )
in: Advances in Geophysics, pages 217-306, Elsevier

Abstract

The Earth is a multi-scale body meaning small scales cannot be avoided in geophysics, particularly in seismology. In this paper, we present an introduction to the two-scale non-periodic homogenization method, which is designed to deal with small scales, both for the forward and the inverse problems. It is based on the classical two-scale periodic homogenization, which requires a periodic or a stochastic media, but has been extended to geological media, which are deterministic and multi-scale with no scale separation. The method is based on the minimum wavelength of the wavefield to separate the scales. It makes it possible to compute an effective medium, valid up to a given maximum source frequency, from a given fine-scale description of a medium. The effective medium is in general fully anisotropic and smooth but not constant. It can be tuned so that the wavefield computed in the effective medium is the same as the true solution up to the desired accuracy for all waves, including reflected, refracted or surface waves. For inverse problems, we will numerically check that a limited frequency band full waveform inversion can retrieve, at best, only the homogenized medium and not the true fine-scale one. We will first present the subject through a numerical experiment in 1-D. Then, we will present the method in 1-D and next in 2-D/3-D. Finally, we will present a series of examples in 2-D and 3-D in the forward modeling and inverse problem contexts.

BibTeX Reference

```@incollection{capdeville:hal-03031441,
abstract = {The Earth is a multi-scale body meaning small scales cannot be avoided in geophysics, particularly in seismology. In this paper, we present an introduction to the two-scale non-periodic homogenization method, which is designed to deal with small scales, both for the forward and the inverse problems. It is based on the classical two-scale periodic homogenization, which requires a periodic or a stochastic media, but has been extended to geological media, which are deterministic and multi-scale with no scale separation. The method is based on the minimum wavelength of the wavefield to separate the scales. It makes it possible to compute an effective medium, valid up to a given maximum source frequency, from a given fine-scale description of a medium. The effective medium is in general fully anisotropic and smooth but not constant. It can be tuned so that the wavefield computed in the effective medium is the same as the true solution up to the desired accuracy for all waves, including reflected, refracted or surface waves. For inverse problems, we will numerically check that a limited frequency band full waveform inversion can retrieve, at best, only the homogenized medium and not the true fine-scale one. We will first present the subject through a numerical experiment in 1-D. Then, we will present the method in 1-D and next in 2-D/3-D. Finally, we will present a series of examples in 2-D and 3-D in the forward modeling and inverse problem contexts.},
author = {Capdeville, Yann and Cupillard, Paul and Singh, Sneha},
doi = {10.1016/bs.agph.2020.07.001},
editor = {B. Moseley and L. Krischer},
hal_id = {hal-03031441},
hal_version = {v1},
keywords = {Acoustic Waves ; Effective Solution ; Elastic Waves ; Full Waveform Inversion ; Helmholtz Resonator ; Homogenization ; Multi-scales ; Numerical Modeling ; Rotational Seismology ; Seismology ; Synthetic seismograms ; Up-scaling},
pages = {217-306},
pdf = {https://hal.science/hal-03031441/file/Capdeville_etal_AGEO2020.pdf},
publisher = {{Elsevier}},
title = {{An introduction to the two-scale homogenization method for seismology}},
url = {https://hal.science/hal-03031441},
volume = {61},
year = {2020}
}
```