Ignoring density in waveform inversion

Introduction

Velocity models for processing seismic reflection data are usually derived from traveltime tomography or other methods that depend on detection of moveout in picked reflection events. Picking is time consuming and prone to error and makes use of only a subset of the information available in a dataset. Waveform inversion provides an alternative approach for deriving velocity models. As an automatic algorithm, waveform inversion is less dependent on human input. The goal of the inversion is to match both data phase and amplitude, so it is theoretically possible to recover subtle local variations that are too small to lead to measurable moveouts in gathers.

Though the method is conceptually appealing, several barriers have prevented waveform inversion from becoming viable for real data: it is only able to recover anomalies that are either very small in magnitude or that have similar spatial wavelengths as the seismic data; it is computationally expensive, especially if based on time-domain modeling; and when the physics of wave-propagation is simplified to reduce cost and complexity, the inversion may not converge to a useful solution. In this report we investigate the last issue.

Early formulations (Tarantola, 1984; Lailly, 1984) describe the method as simultaneous inversions for the source function, the density field, and the bulk modulus field. Woodward (1990) and Luo and Schuster (1991) choose to invert only for a velocity field. Due to the limited geometries of seismic reflection surveys, there is an ambiguity between velocity and density: a velocity anomaly, a density anomaly, or a combination of the two can all create reflections, and near-vertical-incidence waves do not contain much information to distinguish between these cases.

Here we first present a simplified formulation of waveform inversion, based on Tarantola (1984), and show the results of inverting a small synthetic dataset modeled with the same physics--the constant-density acoustic wave equation--as used in the inversion. Then we show results from a velocity-only inversion of a dataset modeled with both velocity and density contrasts.

Ignoring density in waveform inversion