Earth Resources Laboratory :: Abstracts

Nonlinear Waveform Tomography:
Theory and Application to Crosshole Seismic Data

Delaine Rebecca Thompson

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 4, 1993 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

The acoustic inverse problem of crosshole seismology is nonlinear in the medium velocities and ill-posed because of the lack of complete data coverage surrounding the area of interest. In light of these facts, this thesis develops a new nonlinear waveform technique which inverts crosshole seismic data for acoustic velocities in the shallow subsurface.

We first describe the methods we use to treat the nonlinearity and nonuniquesness of the inverse problem. Tikhonov regularization methods are used to define a minimization condition which balances the normal misfit between observed and modeled data with a normed measure of the spatial roughness of the model. This type of regularization produces minimum structure velocity models which can vary in their degree of smoothness versus fit to the data. The measure of spatial roughness we use is a two-dimensional second derivative operator applied to the area of velocity reconstruction; the degree of smoothness in the solution is controlled by a regularization parameter set prior to the inversion. We numerically solve the Tikhonov minimization condition using a conjugate algorithm; this algorithm require a gradient direction which we derive directly from the acoustic wave equation.

To accurately calculate the components of the minimization condition and its gradient, we use frequency-domain moment method with sinc basis functions. The transformation of the acoustic wave equation into the frequency domain reduces the necessary forward modeling computations; his reduction is possible because of the spatial wavenumber redundancy in crosshole data. The moment method does not use the Born approximation or high frequency ray asymptotics to simplify the forward modeling. A two-dimensional area between the source and receiver boreholes is sampled with a grid of basis functions and a point-matching procedure discretizes the integral form of the acoustic wave equation. This discretization produces a two part matrix problem which we solve for the Green’s functions and total fields in the medium using general matrix decomposition techniques. The combination of the Tikhonov regularization technique and the frequency-domain moment method results in a inversion technique we call nonlinear waveform tomography.

We test the validity of nonlinear waveform tomography on two synthetic data sets. We show the superiority of nonlinear waveform tomography over conventional linear diffraction tomography by inverting the analytical data from a disk for a variety of experimental and model parameters. Then we invert acoustic finite difference data produced through a laterally varying medium to demonstrate the flexibility of nonlinear waveform tomography in a more geophysical realistic setting. These inversions produce guidelines for the usage of the method in more complex and potentially noisy situations.

Finally, we successfully apply nonlinear waveform tomography to two scale model data sets obtained from an ultrasonic modeling tank. The first data set comes from a mostly plane-layered epoxy-resin model. The data exhibit elastic effects and other complicated wave phenomena. We invert this data set for the lateral variations in the model using a smoothed one-dimensional starting model. Then we perturb some inversion parameters to determine the robustness of nonlinear waveform tomography. The second scale model data set is from a rubber half cylinder model submerged in water. The limited coverage of the experiment and the strong velocity contrast between rubber and water tax the ability of nonlinear waveform tomography to reconstruct the correct velocity model. In this case, a good initial model is required for accurate velocity reconstruction, but we are able to locate and describe the shape of the rubber half cylinder using only a zero-perturbation initial model.