A multilayer level-set method for eikonal-based traveltime tomography

TL;DR

This paper introduces a multilayer level-set method (MLSM) to address first-arrival traveltime tomography in media with multiple discontinuous phases. By representing several interfaces with a single level-set function through a sequence of $i_n$-level-sets, MLSM captures complex multilayer structures while preserving an Eulerian formulation for the eikonal equation and using adjoint-state methods for efficient gradient computation. Regularization strategies including multilayer reinitialization, arc-length penalization, and Sobolev smoothing stabilize the inversion, and an illumination-based error measure accounts for nonuniform ray coverage. Numerical experiments demonstrate that MLSM accurately recovers both domain interfaces and discontinuous slowness parameters, even in challenging topologies, highlighting its potential for multiphase traveltime tomography and related inverse problems.

Abstract

We present a novel multilayer level-set method (MLSM) for eikonal-based first-arrival traveltime tomography. Unlike classical level-set approaches that rely solely on the zero-level set, the MLSM represents multiple phases through a sequence of $i_n$-level sets ($n = 0, 1, 2, \cdots$). Near each $i_n$-level set, the function is designed to behave like a local signed-distance function, enabling a single level-set formulation to capture arbitrarily many interfaces and subregions. Within this Eulerian framework, first-arrival traveltimes are computed as viscosity solutions of the eikonal equation, and Fréchet derivatives of the misfit are obtained via the adjoint state method. To stabilize the inversion, we incorporate several regularization strategies, including multilayer reinitialization, arc-length penalization, and Sobolev smoothing of model parameters. In addition, we introduce an illumination-based error measure to assess reconstruction quality. Numerical experiments demonstrate that the proposed MLSM efficiently recovers complex discontinuous slowness models with multiple phases and interfaces.

Figures (17)

• Figure 1: Two examples of the multilayer level-set function. (a) $\phi(x,y)$ from Section \ref{['subsubsec_EX1']}; (b) $\phi(x,y)$ from Section \ref{['subsubsec_EX2']}. The white line indicates the 0-level-set, and the black line indicates the 1-level-set.
• Figure 2: (Section \ref{['SubSubSec:NomalMotion']} Test 1) The inner circle expands while the outer circle shrinks. The average distance between the two circles at the final time is computed using different mesh resolutions. The red line shows the least-squares fit, given by $1.8(\Delta x)^{0.83}$.
• Figure 3: (Section \ref{['SubSubSec:NomalMotion']} Test 2) The upper-right curve evolves according to motion in the normal direction, while the lower-left circle remains stationary. From left to right, the panels show the evolution of the solution using a mesh with $n = 101^2$ points.
• Figure 4: (Section \ref{['SubSubSec:NomalMotion']} Test 2) The upper-right curve evolves according to motion in the normal direction, while the lower-left circle remains stationary. From left to right, the final solutions are shown using meshes with $n = 101^2$, $201^2$, $401^2$, and $801^2$ points, respectively.
• Figure 5: (Section \ref{['SubSubSec:MeanCurvature']}, Test 1) Time evolution of the mean radii of two circles, $\Gamma_1$ and $\Gamma_2$, with initial radii of 2 and 4, respectively. (a) In the first case, both circles evolve with normal velocity $v_n = -\kappa$ using meshes with $401^2$ points. (b) In the second case, the normal velocity of the outer circle $\Gamma_2$ is increased to $v_n = -5\kappa$, again using meshes with $401^2$ points.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• proof