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Microlocal analysis of borehole seismic data

Raluca Felea, Romina Gaburro, Allan Greenleaf, Clifford Nolan

Abstract

Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or in 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.

Microlocal analysis of borehole seismic data

Abstract

Borehole seismic data is obtained by receivers located in a well, with sources located on the surface or in 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.

Paper Structure

This paper contains 16 sections, 5 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering model, microlocal analysis and singularity theory
  3. Scattering model and normal operator
  4. Microlocal analysis
  5. Dense array: constant $c_0$
  6. Dense Array: No Caustics
  7. Dense Array: Fold Caustics
  8. Fold caustics: single receiver
  9. Fold caustics: borehole data
  10. Proof of Thm. \ref{['thm folded cc']}
  11. Crosswell and walkaway geometries
  12. Crosswell geometry
  13. Walkaway geometry
  14. Artifacts for crosswell and walkaway
  15. Appendix: Singularity classes
  16. ...and 1 more sections

Key Result

Theorem 3.1

Suppose, in addition to Assumptions separation and nsop, the ray geometry of a smooth background sound speed $c_0(x)$ satisfies Assumptions simple_geom and no_grazing below. Then the linearized scattering operator for the dense array data set, $F:\mathcal{E}'(\mathbb R^3_+)\to \mathcal{D}'(\mathbb D

Figures (2)

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

Theorems & Definitions (16)

  • Theorem 3.1
  • proof : Proof of Thm. 3.1
  • Remark 4.0.1
  • Definition 5.1
  • Theorem 5.1
  • Definition 5.2
  • Remark 5.0.1
  • Theorem 5.2
  • Theorem 6.1
  • proof : Proof of Thm. 6.1
  • ...and 6 more