An image-focusing semblance functional for velocity analysis

Introduction

Even a superficial analysis of depth migrated seismic images obtained with different migration velocities clearly shows that velocity information could be extracted by measuring image focusing along the spatial dimensions (i.e. horizontal axes and depth). This information is particularly abundant in areas where complex structure and discontinuous reflectors reveal lack of focusing caused by velocity errors; such as in presence of faults, point diffractors, buried channels, uncomformities or rough salt/sediment interfaces.

If we were able to extract this focusing-velocity information reliably from migrated images it could supplement the velocity information that we routinely extract by analyzing residual moveout along the offsets or aperture-angles axes, and thus enhance velocity estimation by increasing resolution and decrease uncertainties. It would be particularly useful to improve the interpretability of the final image and the accuracy of time-to-depth conversion in areas where the reflection aperture range is narrow either because of unfavorable depth/offset ratio or because of the presence of fast body in the overburden (e.g. salt bodies) that deflect the propagating waves. Today, the most common application of image focusing is to migration-velocity scans for subsalt imaging (Wang et al., 2006). However, current practical applications exploit the image-focusing information by using subjective interpretation criteria instead of quantitative measurements (Sava et al., 2005). This limitation makes almost impossible to automate the process and potentially reduces its reliability, and thus it is a serious obstacle to its extensive application.

Minimum entropy has been often proposed as a quantitative measure of image focusing, starting with Harlan et al. (1984), De Vries and Berkhout (1984), and more recently by Stinson et al. (2005) and Fomel et al. (2007). Minimizing the ``spatial entropy'' measured on image windows privileges images that consist of isolated spikes. If the reflectivity function consists of isolated diffractors, minimum entropy is a good indicator of image focusing. However, field data are usually a combination of diffracted events, specular reflections from planar reflectors, and reflections for high-curvature reflectors. In these cases minimum entropy may yield bias estimates unless the diffractions are successfully separated from the other events before performing the analysis (Fomel et al., 2007). In complex geology, this separation can be unreliable when performed in the data space, and even more challenging when performed in the image space because it is biased by the initial migration velocity. In the following section I show that in presence of reflector curvature (e.g. a sinusoidal reflector) measuring focusing by minimum entropy leads to under-migrated images of convex reflectors (e.g. an anticline,) and over-migrated images of concave reflectors (e.g. a syncline.)

I aim to overcome these shortcomings by generalizing the conventional concept of semblance commonly used in velocity analysis. In addition to measuring semblance along the reflection-aperture angle (or offset for Kirchhoff migration,) as is routinely done, I propose to measure semblance along the structural-dip axes. In this paper I work with 2D data, and thus I compute semblance on 2D patches (structural dip and aperture angle.) With 3D full-azimuth data, semblance would be computed on 4D patches (indexed by two structural dips, reflection aperture and reflection azimuth.)

The proposed method can be applied to locally select the best-focused image among an ensemble of images obtained with different migration velocities. I use residual prestack depth migration in the angle domain (Biondi, 2008) to generate this ensemble of images starting from prestack depth-migrated image in the angle domain. Stolt prestack depth migration could be used as well to perform residual prestack migration (Sava, 2003). With either choice of residual migration, the image decomposition according to structural dip is easily performed within the residual prestack migration process, since both migrations require the image to be transformed into the spatial Fourier domain. The final goal, not addressed by this paper, is to use the image focusing information to enhance interval-velocity estimation for depth migration. In particular, I plan to update the interval-velocity model by using the wave-equation migration velocity analysis method starting from a spatially-varying field of optimal-focusing parameters (Sava and Biondi, 2004b,a; Sava, 2004; Biondi and Sava, 1999).

An image-focusing semblance functional for velocity analysis