Wide-azimuth angle gathers for wave-equation migration |
Conventional seismic imaging is based on the concept of single scattering. Under this assumption, waves propagate from seismic sources, interact with discontinuities and return to the surface as reflected seismic waves. We commonly speak about a ``source'' wavefield, originating at the seismic source and propagating in the medium prior to any interaction with discontinuities, and a ``receiver'' wavefield, originating at discontinuities and propagating in the medium to the receivers (Berkhout, 1982; Clærbout, 1985). The two wavefields kinematically coincide at discontinuities.
We can formulate imaging as a process involving two steps: the wavefield reconstruction and the imaging condition. The key elements in this imaging procedure are the source and receiver wavefields, and which are -dimensional objects as a function of space and time , or as a function of space and frequency . For imaging, we need to analyze if the wavefields match kinematically in time and then extract the reflectivity information using an imaging condition operating along the space and time axes.
A conventional cross-correlation imaging condition (cIC) based on the reconstructed wavefields can be formulated in the time or frequency domain as the zero lag of the cross-correlation between the source and receiver wavefields (Clærbout, 1985):
An extended imaging condition preserves in the output image certain acquisition (e.g. source or receiver coordinates) or illumination (e.g. reflection angle) parameters (Weglein and Stolt, 1999; Clayton and Stolt, 1981; Clærbout, 1985; Stolt and Weglein, 1985). In shot-record migration, the source and receiver wavefields are reconstructed on the same computational grid at all locations in space and all times or frequencies, therefore there is no a-priori wavefield separation that can be transferred to the output image. In this situation, the separation can be constructed by correlation of the wavefields from symmetric locations relative to the image point, measured either in space (Rickett and Sava, 2002; Sava and Fomel, 2005) or in time (Sava and Fomel, 2006). This separation essentially represents local cross-correlation lags between the source and receiver wavefields. Thus, an extended cross-correlation imaging condition (eIC) defines the image as a function of space and cross-correlation lags in space and time. This imaging condition can also be formulated in the time and frequency domains (Sava and Vasconcelos, 2011):
Typically, angle decomposition with extended images uses common image gathers, i.e. representations of (a subset of) the extensions as a function of a space axis, typically the depth axis. As pointed out by Sava and Vasconcelos (2011), this approach suffers from major drawbacks. Common-image-gathers are appropriate for nearly horizontal structures and they are computationally wasteful since they require un-necessary calculations, e.g. inside massive salt bodies. In contrast, the common-image-point-gathers advocated by Sava and Vasconcelos (2011) are constructed at selected points in the image, thus eliminating unnecessary calculations, and they can accomodate arbitrary orientations of the reflectors.
In this paper, we use extended common-image-point-gathers to extract angle-dependent reflectivity at individual points in the image. The method described in the following section is appropriate for 3D wide-azimuth wave-equation imaging. The problem we are solving is to decompose extended CIPs as a function of azimuth and reflection angles at selected points in the image. In general, the input for such decomposition are gathers in the domain, and the output are gathers in the domain:
(5) |
In the remainder of the paper, we show that all the necessary information for this decomposition is available after wave-equation migration, regardless of its implementation, e.g by depth or time extrapolation. A pre-requisite for this decomposition is the moveout function characterizing individual shots, as discussed in the following section. We then show how the moveout information can be used for angle decomposition and illustrate the method with simple and complex synthetic examples.
Wide-azimuth angle gathers for wave-equation migration |