Interpretation of the focal spot obtained with time reversal

Zoé Renat and Paul Cupillard and Yann Capdeville. ( 2021 )
in: 2021 RING Meeting, ASGA

Abstract

The time reversal method is based on the back-propagation of seismic waveforms recorded at a set of receivers which form a closed surface. When the elastic properties of the medium are correct, the seismic energy focuses at the source location, creating a focal spot. Such a spot is smooth in space, whereas the original wavefield shows a displacement discontinuity at the source, hence the question: what is the link between the focal spot and the original source. The goal of the present paper is to answer this question in a proper way, based on the concept of homogenized point-source recently proposed by Capdeville (2021). We show that the back-propagated wavefield is equivalent to the sum of two low-wavenumber fields resulting from the homogenization of the original point-source. In other words, the homogenized point-source is the equivalent force for producing the focal spot. Here we present the mathematical demonstration of it in one dimension and some numerical examples in two dimensions.

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BibTeX Reference

@INPROCEEDINGS{RENAT_RM2021,
    author = { Renat, Zoé and Cupillard, Paul and Capdeville, Yann },
     title = { Interpretation of the focal spot obtained with time reversal },
 booktitle = { 2021 RING Meeting },
      year = { 2021 },
 publisher = { ASGA },
  abstract = { The time reversal method is based on the back-propagation of seismic waveforms recorded at a set of receivers which form a closed surface. When the elastic properties of the medium are correct, the seismic energy focuses at the source location, creating a focal spot. Such a spot is smooth in space, whereas the original wavefield shows a displacement discontinuity at the source, hence the question: what is the link between the focal spot and the original source. The goal of the present paper is to answer this question in a proper way, based on the concept of homogenized point-source recently proposed by Capdeville (2021). We show that the back-propagated wavefield is equivalent to the sum of two low-wavenumber fields resulting from the homogenization of the original point-source. In other words, the homogenized point-source is the equivalent force for producing the focal spot. Here we present the mathematical demonstration of it in one dimension and some numerical examples in two dimensions. }
}