We present a 3-D automatic migration velocity analysis (MVA) algorithm based on common-azimuth migration (CAM) in the source-geophone implementation, and using differential semblance optimization (DSO). We have successfully tested the algorithm on the 3-D SEG/EAGE overthrust model and a 3D field dataset using a simple 1D velocity model as a starting point in each case. We conclude that wave-equation-based MVA may be a good alternative to ray-based tomography for velocity model-building in complex regions of the earth.
An accurate velocity model is a prerequisite for satisfactory depth migration. Although ray-based tomography (e.g. Bishop et al., 1985, Sexton and Williamson, 1998) is still the workhorse for velocity model-building in the industry, interest has been growing in band-limited methods which more accurately represent the underlying physics, such as one-way wave equation-based MVA (Sava and Biondi, 2004; Shen et at 2003; Xie and Yang 2008; Fliedner et al, 2007; Albertin et al, 2006), and full waveform inversion (Tarantola, 1984: Pratt, 1999; Hadj-Ali et al., 2007). These methods offer improved stability in complex geology compared to ray-based ones (Sava and Biondo, 2004), and may also provide better resolution by taking advantage of the different information at different frequencies. The one-way wave equation-based MVA formulation is independent of the choice of propagator and data organization (Shen et al, 2003; Biondi and Sava, 2004; Khoury et al, 2006; Sava and Vlad, 2008). However, while shot-profile migration is now routine for depth imaging, the iterative nature of MVA makes a 3D shot-profile based inversion tremendously computationally intensive. On the other hand, CAM (Biondi and Palacharla, 1996), while potentially of limited applicability because of its assumptions, offers a very quick wave-equation based migration. For areas of moderate geological complexity, CAM provides as accurate an image as shot profile migration, but much more economically. So CAM provides an attractive framework for wave-equation based MVA. Khoury et al., 2006, presented a 2D DSRbased MVA tool in which the wave equation propagator is a GSP implemented in the mid-point offset domain. Hua et al., 2007, proposed a modified CAM, implementing the ffd wide-angle term in the source-geophone domain, which provides much better images in complex areas.
In this paper, we present a 3D implementation of DSO-based MVA based on the CAM of Hua et al, 2007; the adjoint state method is used to calculate the velocity gradient. The method is applied to 3-D synthetic and field data, validating the feasibility of the algorithm.
In this part, we briefly review the theory of FFD source geophone implementation of CAM, DSO principle, and adjoint state method.
The FFD source-geophone implementation of CAM was proposed by Hua, et al 2007. The common-azimuth DSR dispersion relation can be written as (Biondi and Palacharla, 1996):
equation (1) amounts to a 2-D inline prestack depth migration using equation (2), followed by a 2-D crossline poststack migration. The wide-angle term in the mid-point offset implementation of CAM uses the standard Padé approximation for the square roots in equation (2), but discards the relatively computationally expensive operator arising from the cross-term between Kmx and Khx.