# Theory of 3-D angle gathers in wave-equation seismic imaging

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## Abstract

I present two methods for constructing angle gathers in 3-D seismic imaging by downward extrapolation. Angles in angle gathers refer to the scattering angle at the reflector and provide a natural access to analyzing migration velocity and amplitudes. In the first method, angle gathers are extracted at each downward-continuation step by mapping transformations in constant-depth frequency slices. In the second method, one extracts angle gathers after applying the imaging condition by transforming local offset gathers in the depth domain. The second approach generalizes previously published algorithms for angle-gather construction in 2-D and common-azimuth imaging.

### Keywords

Geophysics Seismic imaging Velocity analysis Amplitude analysis## Introduction

Wave extrapolation provides an accurate method for seismic imaging in structurally complex areas (Biondi 2006; Etgen et al. 2009). Wave extrapolation methods have several known advantages in comparison with direct methods such as Kirchhoff migration thanks to their ability to handle multi-pathing, strong velocity heterogeneities, and finite-bandwidth wave-propagation effects (Gray et al. 2001). However, velocity and amplitude analysis in the prestack domain are not immediately available for wave extrapolation methods. To overcome this limitation, several authors (de Bruin et al. 1990; Prucha et al. 1999; Mosher and Foster 2000; Rickett and Sava 2002; Xie and Wu 2002; Soubaras 2003; Sava and Fomel 2003, 2005, 2006) suggested methods for constructing angle gathers from downward-continued wavefields. Angles in angle gathers are generally understood as the reflection (scattering) angles at reflecting interfaces (Xu et al. 2001; Brandsberg-Dahl et al. 2003). Angle gathers facilitate velocity analysis (Liu et al. 2001; Stork et al. 2002) and can be used in principle for extracting angle-dependent reflectivity information directly at the target reflectors (Sava et al. 2001). Stolk and de Hoop (2002) assert that angle gathers generated with wavefield extrapolation are genuinely free of artifacts documented for Kirchhoff-generated angle gathers (Stolk and Symes 2002, 2004).

There are two possible approaches to angle-gather construction with wavefield continuation. In the first approach, one generates gathers at each depth level converting offset-space-frequency planes into angle-space planes simultaneously with applying the imaging condition. The offset in this case refers to the local offset between source and receiver parts of the downward continued prestack data. Such a construction was suggested, for example, by Prucha et al. (1999). This approach is attractive because of its localization in depth. However, the method of Prucha et al. (1999) produces gathers in the offset ray parameter as opposed to angle. As a result, the angle-domain information becomes structure-dependent: the output depends not only on the scattering angle but also on the structural dip.

In the second approach, one converts migrated images in offset-depth domain to angle-depth gathers after imaging of all depth levels is completed. Sava and Fomel (2003) suggested a simple Radon-transform procedure for extracting angle gathers from migrated images. The transformation is independent of velocity and structure. Rickett and Sava (2002) adopted it for constructing angle gathers in the shot-gather migration. Biondi and Symes (2004) demonstrate that the method of Sava and Fomel (2003) is strictly valid in the 3-D case only in the absence of cross-line structural dips. They present an extension of this method for the common-azimuth approximation (Biondi and Palacharla 1996).

In this paper, I present a more complete analysis of the angle-gather construction in 3-D imaging by wavefield continuation. First, I show how to remove the structural dependence in the depth-slice approach. The improved mapping retains the velocity dependence but removes the effect of the structure. Additionally, I extend the second, post-migration approach to a complete 3-D wide-azimuth situation. Under the common-azimuth approximation, this formulation reduces to the result of Biondi et al. (2003) and, in the absence of cross-line structure, it is equivalent to the Radon construction of Sava and Fomel (2003).

## Traveltime derivatives and dispersion relationships for a 3-D dipping reflector

*x*,

*y*,

*z*} coordinates by the general equation

*α*,

*β*, and

*γ*satisfy

*D*is the length of the normal to the reflector from the midpoint (distance \(MM\prime\) in Fig. 2)

*m*_{x} and *m*_{y} are the midpoint coordinates, *h*_{x} and *h*_{y} are the half-offset coordinates, and *v* is the local propagation velocity.

In the frequency-wavenumber domain, Eq. 11 serves as the basis for 3-D shot-geophone downward-continuation imaging. In the Fourier domain, each *t*_{x} derivative translates into −*k*_{x}/*ω* ratio, where *k*_{x} is the wavenumber corresponding to *x* and ω is the temporal frequency.

*θ*and each frequency

*ω*, Eq. 13 specifies the locations on the four-dimensional (\(k_{m_x}, k_{m_y}, k_{h_x}, k_{h_y}\)) wavenumber hyperplane that contribute to the common-angle gather. In the 2-D case, Eq. 13 simplifies by setting \(k_{h_y}\) and

*k*

_{y}to zero. Using the notation \(k_{m_x}=k_m\) and \(k_{h_x}=k_h,\) the 2-D equation takes the form

*k*

_{h}resulting in the convenient 2-D dispersion relationship

In the next section, I show that a similar simplification is also valid under the common-azimuth approximation. Equations (13) and (15) describe an effective migration of the downward-continued data to the appropriate positions on midpoint-offset planes to remove the structural dependence from the local image gathers.

*v*from Eqs. 11 and 12. Expressing

*v*

^{2}from Eq. 12 and substituting the result in Eq. 12, we arrive (after a number of algebraical transformations) to the frequency-independent equation

where \(k_{h_z}\) refers to the vertical offset between source and receiver wavefields (Biondi and Shan 2002).

which is the equation suggested by Sava and Fomel (2003). Equation (18) appeared previously in the theory of migration-inversion (Stolt and Weglein 1985).

## Common-azimuth approximation

Common-azimuth migration (Biondi and Palacharla 1996) is a downward continuation imaging method tailored for narrow-azimuth streamer surveys that can be transformed to a single common azimuth with the help of azimuth moveout (Biondi et al. 1998). Employing the common-azimuth approximation, one assumes the reflection plane stays confined in the acquisition azimuth. Although this assumption is strictly valid only in the case of constant velocity (Vaillant and Biondi 2000), the modest azimuth variation in realistic situations justifies the use of the method (Biondi 2003).

*h*

_{y}to zero assuming the

*x*coordinate is oriented along the acquisition azimuth. In particular, from Eqs. 8, 9, we obtain

which shows that, under the common-azimuth approximation and in a laterally homogeneous medium, 3-D seismic migration amounts to a cascade of a 2-D prestack migrations in the in-line direction and a 2-D zero-offset migration in the cross-line direction (Canning and Gardner 1996).

which is identical to the 2-D Eq. 14. This proves that under this approximation, one can perform the structural correction independently for each cross-line wavenumber.

obtained previously by Biondi et al. (2003). In the absence of cross-line structural dips \((k_{m_y}=0),\) it is equivalent to the 2-D Eq. 18.

## Algorithm I: Angle gathers during downward continuation

*z*:

- 1.
Generate local offset gathers and transform them to the wavenumber domain. In the double-square-root migration, the local offset wavenumbers are immediately available. In the shot gather migration, local offsets are generated by cross-correlation of the source and receiver wavefields (Rickett and Sava 2002).

- 2.
For each frequency

*ω*, transform the local offset wavenumbers \({k_{h_x},k_{h_y}}\) into the angle coordinates \(\sin{\theta}/v\) according to Eq. 13. The angle coordinates depend on velocity but do not depend on the local structural dip. In the 2-D case, each frequency slice is simply the \({k_m,k_h}\) plane, and each angle coordinate corresponds to a circle in that plane centered at the origin and described by Eq. 14. Figure 3 shows an example of a 2-D frequency slice transformed to angles. - 3.
Accumulate contributions from all frequencies to apply the imaging condition in time.

This algorithm is applicable for targets localized in depth. The local offset gathers need to be computed for all lateral locations, but there is no need to store them in memory, because conversion to angles happens on the fly. The algorithm outputs not angles directly, but velocity-dependent parameters \(\sin{\theta}/v.\) Alkhalifah and Fomel (2009, 2011) have recently extended this algorithm to transversally isotropic media.

## Algorithm II: Post-migration angle gathers

- 1.
Generate and store local offset gathers. In the double-square-root migration, the local offsets are immediately available. In the shot gather migration, local offsets are generated by cross-correlation of the source and receiver wavefields.

- 2.
Estimate the dominant local structural dips at the common image point by using one of the available dip estimation methods: local slant stack, plane-wave destruction, etc.

- 3.
After the imaging has completed, transform local-offset gathers into the slant-stack domain either by slant-stacking in the \(\{ {z,h_{x} ,h_{y} } \}\) physical domain or by radial-trace construction in the \(\{{k_z,k_{h_x},k_{h_y}}\}\) Fourier domain (Sava and Fomel 2003).

- 4.
Using estimated dips, convert slant stacks into angles by applying Eq. 16. The mapping from offset-depth slopes to angles is illustrated in Fig. 4.

The last two steps can be combined into one. It is sufficient to compute the effective offset \(\hat{h} = \sqrt{h_x^2+h_y^2+(h_x z_y -h_y z_x)^2}\) and apply the basic 2-D angle extraction algorithm to the effective offset gather.

The second method is applicable to selected common-image gathers, which can be spread on a sparse grid. The local offset gathers need to be computed and stored at all depths. The method works independent of the velocity. The main disadvantage is the need to estimate local structural dips. In the common-azimuth approximation, only the cross-line dip is required (Biondi et al. 2003). In the 2-D case (zero cross-line dip), the method is dip-independent (Sava and Fomel 2003).

## Discussion

Since the first presentation of the 3-D angle-gather theory (Fomel 2004), many new research results have appeared in the literature. By the end of 2000s, prestack 3-D reverse-time migration has become a standard tool for depth imaging in structurally-complex areas, and it is becoming feasible to generate 3-D angle gathers as part of routine processing (Luo et al. 2010; Vyas et al. 2010; Xu et al. 2010). The most important new theoretical developments are the ability to extract angle information from time-shift angle gathers (Sava and Fomel 2006; Vyas et al. 2010), the ability to extract not only reflection-angle but also azimuth information (Xu et al. 2010), and the extension of the angle-gather theory to anisotropy (Biondi 2007; Alkhalifah and Fomel 2009, 2011).

## Conclusions

Angle gathers present a natural tool for analyzing velocities and amplitudes in wave-equation imaging. I have discussed two approaches for angle-gather construction. In the first approach, angle gathers are constructed on the fly at different depth steps of the wave extrapolation process. In the second approach, angle gathers are extracted from the local-offset gathers after imaging has completed. The second method was previously presented for the 2-D case and for the case of a common-azimuth approximation. Both approaches have advantages and disadvantages. The preference depends on the application and the input data configuration.

## Notes

### Acknowledgments

I am grateful to Nanxun Dai, John Etgen, Sergey Goldin, and Paul Sava for enlightening discussions.

This publication is authorized by the Director, Bureau of Economic Geology, The University of Texas at Austin.

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