Microlocal analysis of borehole seismic data

Raluca Felea; Romina Gaburro; Allan Greenleaf; Clifford Nolan

doi:10.3934/ipi.2022026

Article Contents

2022, Volume 16, Issue 6: 1543-1570. Doi: 10.3934/ipi.2022026

This issue Previous Article Global unique continuation from the boundary for a system of viscoelasticity with analytic coefficients and a memory term Next Article A uniqueness theorem for inverse problems in quasilinear anisotropic media

Microlocal analysis of borehole seismic data

1.
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY, 14623, USA
2.
Department of Mathematics and Statistics, CONFIRM, HRI, University of Limerick, Limerick, V94 T9PX, Ireland
3.
Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

^* Corresponding author: Allan Greenleaf, allan@math.rochester.edu
^* Corresponding author: Allan Greenleaf, allan@math.rochester.edu

Dedicated to the memory of Victor Isakov, for his contributions to the field and for his friendship

Received: October 2021

Revised: March 2022

Early access: May 2022

Published: December 2022

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Abstract

Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or another well. Using microlocal analysis, we study possible approximate reconstruction, via linearized, filtered backprojection, of an isotropic sound speed in the subsurface for three types of data sets. The sources may form a dense array on the surface, or be located along a line on the surface (walkaway geometry) or in another borehole (crosswell). We show that for the dense array, reconstruction is feasible, with no artifacts in the absence of caustics in the background ray geometry, and mild artifacts in the presence of fold caustics in a sense that we define. In contrast, the walkaway and crosswell data sets both give rise to strong, nonremovable artifacts.

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Mathematics Subject Classification: Primary: 86A22, 35R30, 35S30.

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Figure 1. Illustration of data acquisition geometry and filtering: contributions to the data from unbroken rays such as that illustrated here are filtered out by removing data associated to those rays arriving from nearby directions, as indicated by the gray cone

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Figure 2. Construction of the $ y $-coordinates, with the $ y_1 $-direction being tangent to the ray connecting $ y $ to a source $ (s, 0)\in\Sigma_S $

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