Tomographic full waveform inversion and linear modeling of multiple scattering

Introduction

In a recent report Biondi and Almomin (2012) and Almomin and Biondi (2012) presented waveform-inversion methods with robust convergence characteristics even when the initial velocity model is far away from the correct one. These methods are based on an extension of velocity and reflectivity along the subsurface offset axes. This extension enables the kinematics of reflected to be correctly modeled by a linear operator even when the velocity errors are large. However, the extension also explodes the null space of the inverse problem. To ensure convergence towards desirable models a tomographic term is added to the inversion objective function that penalizes velocity models with energy at non-zero subsurface offsets.

In this paper I introduce a tomographic full waveform inversion (TFWI) that is based on an extension of the velocity model along the time axis instead of the subsurface offset axes. This time extension has the theoretical advantage that it can be directly linked to the modeling of multiple scattering phenomena; therefore, overcoming the limitations of conventional full waveform inversion (FWI), whose gradient is based on a first-order scattering approximation. Furthermore, the velocity extension along the time axis should enable robust convergence from transmitted, or refracted events, in addition to reflected events. This versatility can be beneficial when inverting long offset data that contain overturned and refracted events as well conventional reflections. A one-dimensional extension along time is also computationally more efficient than a two-dimensional extension along subsurface offsets. This is an important practical advantage since the computational cost of modeling wave propagation in extended velocity models is substantially larger than in conventional velocity models (Almomin, 2012).

Throughout this paper I illustrate the theory with simple 1D examples. Waves are propagated in 1D, and model parameter, both background and perturbations, are averaged over the whole propagation interval. In addition to be fast to compute by using Matlab, the 1D examples have the advantage of reducing the dimensionality of the model space and thus making the analysis of behavior of objective functions and gradients illustrative of the more general conceptual contributions of the paper. The numerical examples describes a transmission tomography problem to illustrate the capability of the proposed method to effectively use transmitted events, in addition to reflected ones.

Tomographic full waveform inversion and linear modeling of multiple scattering