Measuring image focusing for velocity analysis

Introduction

The effects of migration velocity on the focusing and unfocusing of seismic images is obvious when observing depth migrated seismic images obtained with different migration velocities. Quantitative measures of image focusing could provide valuable information to velocity estimation. This information is particularly abundant in areas where reflectors have strong curvature or are discontinuous; such as in presence of faults, heavily folded geology, buried channels, uncomformities or rough salt/sediment interfaces. Figure 1 shows three images obtained by migrating the same prestack data set: the top panel (a) shows the image obtained with too low migration velocity, the middle panel (b) shows the image obtained with approximately the correct velocity, and the bottom panel (c) shows the image obtained with too high velocity. An interpreter could easily spot clear signs of undermigration in Figure 1a and of overmigration in Figure 1c. However, the definition of objective quantitative criteria to measure image focusing is challenging. Consequently, current practical methods for exploiting image-focusing information are based on subjective interpretation criteria instead of quantitative measurements (Wang et al., 2006; Sava et al., 2005).

If we were able to extract reliably quantitative focusing-velocity information from migrated images it could supplement the velocity information that we routinely extract by analyzing residual moveout along offsets (after common-offset migration) or aperture-angles (after angle-domain migration) axes. Velocity estimation would be enhanced by increasing resolution and decreasing 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 high-velocity geological bodies in the overburden (e.g. salt bodies) that deflect the propagating waves. In practice, velocity analysis based on image focusing is unlikely to replace conventional velocity analysis, but only to supplement it. Therefore, a method that measures image focusing should provide velocity estimates that are consistent with conventional methods.

ResMig-stack-overn Figure 1. Three images obtained by prestack residual migration applied to the same prestack migration: the top panel (a) is undermigrated ( $\rho =0.8975$), the middle panel (b) s approximately well focused ($\rho =1.01$), and the bottom panel (c) is overmigrated ( $\rho =1.2725$). [CR]

Near-off-overn Figure 2. Near-offset section of the data set used to generate the images shown in Figure 1. [CR]

Figure 1 illustrates some of the challenges of defining quantitative criteria to measure image focusing. The main challenge is related to the ambiguity between reflectors' curvature and their apparent focusing velocity. The section migrated with approximately the correct velocity (Figure 1b) shows several convex reflectors with strong curvature. These reflectors collapse into high-amplitude foci in the overmigrated section (Figure 1c). Criteria that have been previously proposed to measure image focusing, such as maximization of the power of the stack or minimization of image entropy (Fomel et al., 2007; De Vries and Berkhout, 1984; Stinson et al., 2005; Harlan et al., 1984), would wrongly rank the overmigrated image higher than the more accurate image. When in the subsurface we have high-curvature reflectors, but not infinite curvature reflectors, the minimum-entropy criterion would fail because it assumes the presence of point scatterers in the subsurface.

Fomel et al. (2007) propose to separate in the data space the diffractions originated from point scatterers before performing minimum-entropy velocity analysis. However, in complex geology this separation can be unreliable, mostly because reflections from curved reflectors may appear as diffractions. This potential source of errors is also well illustrated by the field-data example. Figure 2 shows the near-offset section of the data set used to generate the images shown in Figure 1. The diffraction-like hyperbolic events visible in this section were generated by the high-curvature reflectors discussed above. An application aimed to separate diffractions from other events could easily classify these events for diffractions and lead to biased velocity estimates.

This paper aims to overcome the shortcomings of current methods used to measure image focusing. It presents a new method that has two important characteristics: 1) it is unbiased by reflectors' curvature, and 2) it provides velocity information from image focusing that is consistent with the velocity information that we routinely extract from migrated images by analyzing their coherency along the reflection-aperture angle axes. The method is based on the image-focusing semblance functional I introduced in Biondi (2008b), where I generalized the conventional semblance functional used to measure image coherency along the aperture-angle axes by defining an image-focusing semblance functional that simultaneously measures image coherency along the structural-dip axes and the aperture-angle axes.

To remove the bias caused by reflectors' curvature, I explicitly take into account curvature by correcting its effects on image coherency along structural dips. Making curvature an explicit parameter of the velocity estimation does not necessarily resolves the fundamental problem of the ambiguity between the determination of reflectors' curvature and migration velocity. However, I show that it enables a consistent and unbiased velocity estimation that optimally uses the information contained in the data. In the last section of the paper, I present examples of image-focusing velocity analysis applied to two synthetic zero-offset data sets. These examples indicate that image-focusing analysis could automatically extract useful velocity information from zero-offset data even when the reflectivity model contains curved reflectors with finite curvature.

The simultaneous image-coherency measurement along both the structural-dip axes and the aperture-angle axes of the curvature-corrected images, assures the consistency of the velocity information provided by the method. This consistency facilitates the interpretation of the results. Furthermore, it may improve the robustness of velocity estimation with respect to conventional angle-domain methods by automatically averaging the coherency computation along reflectors. At each point on a reflector, image coherency is measured for several dips in addition to the stationary dip. The inclusion of non-stationary dips is equivalent to averaging coherency measurements along the reflector, following both its dip and its curvature.

In this paper, I present results of the proposed method applied to 2D data. The computation of the image-focusing semblance functional could be easily generalized from 2D to 3D. In 2D, semblance is computed 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 curvature correction is also easily generalizable from 2D to 3D. However, three parameters are required to define curvature in 3D: the two main curvatures along the principal axes, and the rotation of the principal axes with respect to the coordinate axes (Al-Dossary and Marfurt, 2006). I expect the nature of the ambiguity between velocity and curvature to be different between 2D and 3D. In both cases velocity is defined by one scalar parameter, whereas curvature is defined by three parameters in 3D.

Measuring image focusing for velocity analysis