A Homotopy Method for Solving the Impedance Inversion of 1‐D Wave Equation

Abstract

Based on the inversion of one-dimensional seismic wave equation, we study a homotopy method for determining the parameters of the earth. With this method, the problem of nonlinear system of equations is reduced to an initial value problem of ordinary differential equations. This algorithm is not only fast and stable but also globally convergent. Both synthetic data and field data confirm the effectiveness of the homotopy method as an inversion algorithm. It is especially appropriate for non-linear geophysical inversion problems with multi-extreme, and has widespread application prospect in non-linear geophysical inversion.

A Regularization Homotopy Method for the Inverse Problem of 2-D Wave Equation and Well Log Constraint Inversion

“…From Eq. (19), the approximation V N of the N th equation can be solved, which can be taken as the initial guess of (15). Iterations is performed repeatedly until the regularized approximation of V true is obtained.…”

“…Iterations is performed repeatedly until the regularized approximation of V true is obtained. Therefore, (15) and (18) form a stable and widely convergent method for the inversion of the 2-D acoustic wave equation. We call this method Regularization-homotopy method which integrates the virtues of the Tikhonov regularization method and homotopy method.…”

“…Jegen [12] applied the homotopy method to geophysical inversion of EM data based on the normal equations. Recently, Han et al [13] used the homotopy method to solve the 1-D inverse problem in porous media. Bao and Liu [14] suggested a homotopy-regularization method to solve inverse scattering problem with multi-experimental limited aperture data.…”

“…Feng and Han [7] combine the homotopy method with the generalied pulse-spectrum technique to solve the inverse problem of 2-D wave equation. Zhang et al [15] studied the homotopy method for obtaining the impedance profile for 1-D wave equation. Up to now, the homotopy inversion has been induced as one of the inversion methods of elastic dynamics [16] .…”

The Geophysical Inverse Method Based on Quantum Monte Carlo

“…Supposing the underground medium is isotropic and horizontally layered, the seismic wave propagates down to subsurface media as plane wave. Here we used the 1-D wave equation to compute seismic wave field with finite difference algorithm [52,53] .…”