Least-squares migration/inversion of blended data

introduction

High-quality seismic images are extremely important for subsalt exploration, but data collected from conventional narrow-azimuth towed streamers (NATS) often produce poor subsalt images due to insufficient azimuth coverage. Recently developed wide-azimuth towed streamers (WATS) (Michell et al., 2006) and multi-azimuth towed streamers (MATS) (Keggin et al., 2006; Howard and Moldoveanu, 2006) acquisition technologies have greatly improved subsalt illumination, and hence better subsalt images are obtained. However, acquiring marine WATS or MATS data is expensive. One main reason is the inefficiency of the conventional way of acquiring data, which requires waiting long enough between shots to prevent interference (Berkhout, 2008; Beasley et al., 1998; Beasley, 2008). As a consequence, the source domain is often poorly sampled to reduce the survey time.

To gain efficiency, simultaneous shooting (Hampson et al., 2008; Beasley et al., 1998; Beasley, 2008) and continuous shooting, or more generally, blended acquisition geometry (Berkhout, 2008), have been proposed to replace the conventional shooting strategy. In the blended acquisition geometry, we try to keep shooting and recording continuously, so that waiting between shots is minimized and a denser source sampling can be obtained. However, this shooting and recording strategy results in two or more shot records blending together and brings processing challenges. A common practice for processing these blended data is to first separate the blended shot gathers into individual ones in the data domain (Spitz et al., 2008; Akerberg et al., 2008), called "deblending" by Berkhout (2008). Then conventional processing flows are applied to these deblended shot gathers. The main issue with this strategy is that it can be extremely difficult to separate the blended gathers when the shot spacing is close and many shots are blended together.

In this paper, we present an alternative method of processing these blended data sets. Instead of deblending the data prior to the imaging step, we propose to directly image them without any pre-separation. The simplest way for direct imaging would be migration; however, migration of blended data generates images contaminated by crosstalk. The crosstalk is due to the introduction of the blending operator (Berkhout, 2008), which makes the corresponding combined Born modeling operator far from unitary; thus its adjoint, also known as migration, gives poor reconstruction of the reflectivity. A possible solution is to go beyond migration by formulating the imaging problem as a least-squares migration/inversion (LSI) problem, which uses the pseudo-inverse of the combined Born modeling operator to reconstruct the reflectivity of the subsurface.

We extend the LSI theory from the conventional acquisition geometry (Clapp, 2005; Tang, 2008b; Nemeth et al., 1999; Valenciano, 2008) to the blended acquisition geometry and develop inversion schemes in both data space and model space. The former minimizes a data-space defined objective function, while the latter minimizes a model-space defined objective function. By comparing the pros and cons of both inversion schemes, we show that the data-space approach is preferred over the model-space approach if the combined Born modeling operator is far from unitary; that is, its normal operator, the Hessian, has many non-negligible off-diagonal elements. Hence an approximate Hessian with a limited number of off-diagonal elements cannot capture the characteristics of the crosstalk, making it less effective in removing the crosstalk in the model space. Big Hessian filters, which sufficiently capture the information of the crosstalk, are too expensive for practical applications. Therefore, the data-space inversion approach, which does not require explicitly computing the Hessian, becomes more attractive. We demonstrate our ideas with simple synthetic examples, and we also test the data-space inversion scheme on the Marmousi model to illustrate how the crosstalk is suppressed through inverting the combined Born modeling operator. Application to 4-D (time-lapse) inversion using blended data sets is discussed in a companion paper by Ayeni et al. (2009).

This paper is organized as follows: we first describe the problem of directly imaging the blended data through migration; then we develop the theory of LSI in both data space and model space for blended data, and compare the pros and cons of the two domains for imaging blended data. Finally, we apply the data-space inversion approach to the Marmousi model to test its performance on a complex model.

Least-squares migration/inversion of blended data