Fast automatic wave-equation migration velocity analysis using encoded simultaneous sources

introduction

Accurate reflectivity imaging requires an accurate background velocity model. As seismic exploration moves towards structurally complex areas, wave-equation migration velocity analysis (WEMVA) that better models band-limited wave phenomena becomes necessary for high-quality velocity model building. WEMVA, however, is still expensive for industrial-scale applications (Biondi and Sava, 1999; Shen et al., 2005; Albertin et al., 2006; Fei et al., 2009), both because the method uses expensive wavefield modeling engines, and because the computation needs to be carried out for each shot, resulting in a cost proportional to the number of sources, which is huge for large surveys.

Source encoding has been used in both seismic acquisition (Berkhout, 2008; Hampson et al., 2008; Beasley et al., 1998; Beasley, 2008; Tang and Biondi, 2009) and processing (Romero et al., 2000; Whitmore, 1995; Zhang et al., 2005; Liu et al., 2006; Krebs et al., 2009) to reduce the cost. The idea is that instead of firing one impulsive source at a time, we fire all encoded impulsive sources simultaneously for acquisition or/and processing. By doing so, the acquisition or/and processing effort is reduced to just one super areal shot gather instead of many impulsive source gathers, significantly reducing the acquisition or/and processing cost. In this paper, I mainly focus on applying the source-encoding method to seismic processing, hence I assume that the data are acquired using conventional impulsive separate sources without any overlaps.

Source encoding has been widely used in migration processing, where random-phase encoding and plane-wave-phase encoding are the most popular encoding schemes. The random-phase-encoding migration, however, has had limited success. This is because the more shots randomly encoded together, the more crosstalk present in the migration image. Consequently, images obtained with many realizations need to be computed and stacked in order to attenuate the crosstalk (Romero et al., 2000). Plane-wave phase-encoding migration, on the other hand, has wider applications than random phase-encoding migration. This is because plane-wave phase-encoding function has good properties in terms of converging to a Dirac delta function (Liu et al., 2006). However, multiple plane waves need to be synthesized and migrated to remove the crosstalk artifacts. As a result, source encoding in migration can usually achieve a cost reduction by a factor of about $10$ or less.

As opposed to seismic migration processing, source encoding (especially random phase encoding) seems to be more effective in seismic inversion processing, such as least-squares migration (Dai and Schuster, 2009; Dai et al., 2010; Tang and Biondi, 2009) and full waveform inversion (Ben-Hadj-Ali et al., 2011; Tang and Lee, 2010; Krebs et al., 2009). The key element in encoded simultaneous-source inversion is the regeneration of random codes at each iteration (Ben-Hadj-Ali et al., 2011; Tang and Lee, 2010; Dai et al., 2010; Krebs et al., 2009). Different sets of random codes at each iteration enable destructive summation of the crosstalk over iterations, and consequently the residual crosstalk in the inverted model is gradually removed as inversion proceeds. Encoded simultaneous-source inversion operates on one super shot gather instead of many impulsive-source gathers at each iteration, therefore the computational cost is independent of the number of sources. Although more or less counter-intuitive, Krebs et al. (2009) have reported that encoded simultaneous-source inversion has a similar convergence rate compared to separate-source inversion. As a result, source encoding in inversion can achieve a cost reduction by a factor of the number of sources, which can be significant for large surveys.

One commonality of the above mentioned inversion processing (least-squares migration and full waveform inversion) is the minimization of a data-domain objective function, which compares the differences between the modeled and the observed data. The difference-based objective function, however, restricts the application of encoded simultaneous-source inversion to only data acquired with a fixed receiver spread, such as in land or ocean bottom cable (OBC) acquisition geometries (Krebs et al., 2009). This is because modeling using encoded simultaneous sources implicitly assumes that each receiver listens to all shots. This is obviously not the case for marine acquisition geometries, where the towed receiver spread moves along with the sources. The mismatch in acquisition is irreconcilable and will cause wrong model updates.

In this paper, I apply the source-encoding method to WEMVA, which optimizes an objective function formulated in the image domain instead of the data domain. In particular, I optimize the velocity by maximizing the image stack power (or minimizing its negative). I will show that the objective function to maximize (or minimize) is based on the crosscorrelation between the source and receiver wavefields, and that source encoding can be applied to arbitrary acquisition geometries, regardless of whether or not the receiver spread is fixed. Similar to the data-domain multi-source inversion, encoded simultaneous-source WEMVA also generates gradients contaminated by crosstalk. Therefore, regeneration of random codes at each iteration becomes necessary to mitigate the impact of crosstalk on velocity updates.

In the subsequent sections, I first review the theory of WEMVA based on image-stack-power maximization (or equivalently negative image-stack-power minimization). I prove that minimizing the negative image stack power is equivalent to the data domain Born wavefield inversion, which minimizes the difference between the modeled and observed primary reflections. I then show how source encoding can be applied to WEMVA. Finally, I apply both separate-source WEMVA and encoded simultaneous-source WEMVA to invert a truncated Marmousi model.

Fast automatic wave-equation migration velocity analysis using encoded simultaneous sources