Image-space wave-equation tomography in the generalized source domain

introduction

Wave-equation tomography has the potential to accurately estimate the velocity model in complex geological scenarios where ray-based traveltime tomography is prone to fail. Wave-equation-based tomography uses band-limited wavefields instead of infinite-frequency rays as carriers of information, thus it is robust even in the presence of strong velocity contrasts and immune from multi-pathing issues. Generally speaking, wave-equation tomography can be classified into two different categories based on the domain where it minimizes the residual. The domain can be either the data space or the image space. The data-space approach directly compares the modeled waveform with the recorded waveform, and is widely known as waveform inversion, or data-space wave-equation tomography (Mora, 1989; Pratt, 1999; Woodward, 1992; Tarantola, 1987). The main disadvantage of the data-space approach is that in complex areas, the recorded waveforms can be very complicated and are usually of low signal-to-noise ratio (S/N), so matching the full waveform might be extremely difficult. On the other hand, the image-space approach, also known as image-space wave-equation tomography, minimizes the residual in the image domain obtained after migration. The migrated image is often much simpler than the original data, because even with a relatively inaccurate velocity, migration is able to (partially) collapse diffractions and enhance the S/N; thus the image-space wave-equation tomography has the potential to mitigate some of the difficulties that we encounter in the data-space approach. Another advantage of the image-space approach is that the more efficient one-way wave-equation extrapolator can be used. In waveform inversion, however, the one-way propagator is difficult (if not impossible) to use because of its inability to model the multiple arrivals, although some tweaks can be employed so that the one-way propagator can be applied to turning-wave tomography (Shragge, 2007).

However, despite its theoretical advantages, image-space wave-equation tomography is still computationally challenging. Each iteration of tomographic velocity updating is computationally expensive and often converges slowly. Practical applications are still rare and small in scale (Biondi and Sava, 1999; Shen et al., 2005; Albertin et al., 2006). The goal of this paper is to extend the theory of image-space wave-equation tomography from the conventional shot-profile domain (Shen, 2004; Shen et al., 2005) to the generalized source domain, where a smaller number of synthesized shot gathers make the tomographic velocity update substantially faster.

The generalized source domain can be obtained either by data-space phase encoding or image-space phase encoding. For the data-space phase encoding, the synthesized shot gathers are obtained by linear combination of the original shot gathers after some kind of phase encoding; in particular, here we mainly consider plane-wave phase encoding (Duquet and Lailly, 2006; Whitmore, 1995; Liu et al., 2006; Zhang et al., 2005) and random phase encoding (Romero et al., 2000). As the encoding process is done in the data space, we call it data-space phase encoding. For the image-space phase encoding, the synthesized gathers are obtained by prestack exploding-reflector modeling (Biondi, 2007,2006; Guerra and Biondi, 2008b), where several subsurface-offset-domain common-image gathers (SODCIGs) and several reflectors are simultaneously demigrated to generate areal source and areal receiver gathers. To attenuate the cross-talk, the SODCIGs and the reflectors have to be encoded, e.g., by random phase encoding. Because the encoding process is done in the image space, we call it image-space phase encoding. We show that in these generalized source domains, we can obtain gradients, which are used for updating the velocity model, similar to that obtained in the original shot-profile domain, but with less computational cost.

This paper is organized as follows: We first briefly review the theory of image-space wave-equation tomography. Then we discuss how to evaluate the forward tomographic operator and its adjoint in the original shot-profile domain. The latter is an important component in computing the gradient of the tomography objective functional. We then extend the theory to the generalized source domain. Finally, we show examples on a simple synthetic model.

Image-space wave-equation tomography in the generalized source domain