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Convexification for the 3D Problem of Travel Time Tomography

Michael V. Klibanov, Jingzhi Li, Vladimir G. Romanov, Zhipeng Yang

Abstract

The travel time tomography problem is a coefficient inverse problem for the eikonal equation. This problem has well known applications in seismic. The eikonal equation is considered here in the circular cylinder, where point sources run along its axis and measurements of travel times are conductes on the whole surface of this cylinder. A new version of the globally convergent convexification numerical method for this problem is developed. Results of numerical studies are presented.

Convexification for the 3D Problem of Travel Time Tomography

Abstract

The travel time tomography problem is a coefficient inverse problem for the eikonal equation. This problem has well known applications in seismic. The eikonal equation is considered here in the circular cylinder, where point sources run along its axis and measurements of travel times are conductes on the whole surface of this cylinder. A new version of the globally convergent convexification numerical method for this problem is developed. Results of numerical studies are presented.
Paper Structure (13 sections, 82 equations, 8 figures)

This paper contains 13 sections, 82 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Statement of the Problem
  3. The Monotonicity Property of the Function $\tau \left( r, \varphi ,z,z_{0}\right)$
  4. Integral Differential Equation
  5. A Special Orthonormal Basis in $L_{2}\left( 0,B\right)$
  6. Boundary Value Problem 2
  7. Partial Finite Differences
  8. Some specifics of partial finite differences
  9. Completion of the approximate mathematical model
  10. Convexification
  11. The cost functional of the convexification
  12. The global convergence
  13. Numerical Studies

Figures (8)

  • Figure 1: A schematic diagram of the source/detectors of this paper. The large circle is the outer boundary of the domain $\Omega$ defined in (\ref{['1.3']}), the small circle is the inner boundary of domain $\Omega_{\varepsilon}$ defined in (\ref{['1.31']} ), and the red line with red star is position of the source $L_{\text{source}}$ defined in (\ref{['1.32']}).
  • Figure 1: Test 1: The reconstructions of $n( \mathbf{x} )$ of different $\lambda$ with $N=8$, when the shape of the inclusion in (\ref{['9.02']}) is horizontal letter '$B$' with $c_{a}=1.5$. The value of $\lambda$ can be seen on the top side of each square. The images have a low quality for $\lambda =0,1,2$. The quality is improved with $\lambda =3,4,5$. Then it starts to deteriorate at $\lambda =6$ and becomes unsatisfactory at $\lambda =10$. Thus, we select $\lambda=3$ as the optimal value of the parameter $\lambda$.
  • Figure 2: Test 1: Reconstructions of $n( \mathbf{x} )$ of $N=2, 4, 8$ at $\lambda=3$, when the shape of the inclusion in (\ref{['9.02']}) is horizontal letter '$B$' with $c_{a}=1.5$. The reconstruction for $N=2$ has a low quality, and the reconstructions for $N=4, 8$ are almost same. Thus, keeping also in mind (\ref{['9.09']}) and (\ref{['900']}) and also to reduce the computational cost, we choose in Tests 2-5 $N=4$ and $\lambda =3$.
  • Figure 3: Test 2: The exact (left) and reconstructed (right) function $n( \mathbf{x})$, when the shape of the inclusion in (\ref{['9.02']}) is vertically oriented letter '$B$' with $c_{a}=1.5$ in it. Here $\lambda=3,N=4$ as in (\ref{['900']}). The reconstruction is accurate.
  • Figure 4: Test 3: The exact (left) and reconstructed (right) function $n( \mathbf{x})$, when the shape of the inclusion in (\ref{['9.02']}) is vertically oriented letter '$B$' with $c_{a}=3$ in it. Here $\lambda =3,N=4$ as in (\ref{['900']}). The inclusion/background contrast in (\ref{['9.03']}) is $3:1$. The computed inclusion/background contrast in (\ref{['9.04']}) is accurate.
  • ...and 3 more figures