Introduction

Transmission wavefields contain important information on subsurface velocity profiles. This is evident from the many tomography-based techniques that invert for perturbations on an assumed background velocity using the transmission wavefield response. One of the more successful approaches is waveform tomography Pratt and Worthington (1989); Woodward (1992), an approach to non-linear velocity inversion problem that iteratively obtains an estimate of velocity perturbations by minimizing the difference between forward-modeled waveforms and acquired data through residual back-projection. A commonly used and accurate way to forward model synthetic data is using two-way finite-differences. Inversion of the correspondingly large matrices required for 2-D waveform inversion is typically done using a memory-intensive LU decomposition approach Stekl and Pratt (1998). Current computer memory limitations preclude the use of this approach on typical 3-D seismic volumes Operto et al. (2006).

Wave-equation migration velocity analysis (WEMVA) is another velocity inversion technique Sava and Biondi (2004). This procedure back-projects wavefield perturbations derived from variations in migrated image volume (i.e. angle-gathers) to image velocity perturbations. Unlike typical waveform inversion approaches, this procedure is often implemented with one-way phase-only wavefield extrapolation for forward modeling, and is applied to the back-scattered reflection response. However, nothing precludes using a WEMVA-like formalism in inverting transmission wavefields for velocity perturbations. One potential benefit is that because the phase-only extrapolation operator is stated explicitly, one can represent scattering as a matrix operation that provides a direct link between a velocity perturbation and the gradient field.

In this paper, I derive a WEMVA-like framework for modeling transmission wavefields. I then use the waveform inversion objective function Pratt and Worthington (1989) to develop the equations appropriate for transmission wavefield waveform inversion using one-way extrapolation operators. Finally, I demonstrate that forward modeling in generalized coordinate systems Sava and Fomel (2005) does not pose any theoretical difficulties for the inversion process.