Angle-azimuth decomposition of converted waves using extended images

In this paper, we discuss an algorithm to decompose converted shear waves seismic data into angle gathers. This work is an extension to the work published by Sava and Vlad (2010), which presents an equivalent method for pure compressional waves. This method can be applied using a variety of solutions for the wave-equation and wave extrapolation, e.g. finitedifference, kirchhoff, and frequency-domain solutions. However, the power of this method is due to its applicability with finite-difference solution for the wave-equation and a consistent imaging condition. Hence, this method is able to deal with issues due to complex geology and fits perfectly the RTM migration. Introduction In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earths interior. The quality of the final image greatly depends on the quality of the velocity model and on the quality of the technique used for wavefield reconstruction in the subsurface (Gray et al., 2001). However, structural imaging is not the only objective of wave-equation imaging. It is often desirable to construct images depicting reflectivity as a function of reflection angles. Such images not only highlight the subsurface illumination patterns, but could potentially be used for image postprocessing for amplitude variation with angle analysis. Furthermore, angle domain images can be used for tomographic velocity updates. Decomposition of migrated images into angle and azimuth components is useful for several purposes, for example for illumination studies or AVAZ analysis. An algorithm for angle decomposition of pure acoustic waves using extended imaging condition has been recently presented by Sava and Vlad (2010). Here we show what changes have to be introduced in the algorithm so that the angle decomposition can be applied for converted waves. Our algorithm uses seismic wavefields reconstructed with the complete wave equation, thus it inherits all attributes of this methodology for imaging complex geologic structures. We use conventional modifications of wave-equation migration algorithms, for example by using different velocities for the the source and receiver wavefields, i.e. P-wave velocity for the source and S-wave velocity for the receiver wavefields. The angle decomposition is based on extended common image-point gathers. Such gathers are advantageous for wave-equation migration because they can be constructed at sparse locations in the image, thus reducing computational cost. Moreover, the common image point gathers are not biased toward the vertical direction, as is the case for conventional common-imagegathers, and they also avoid calculations in areas that are not useful for velocity analysis. These advantages are discussed in details by Sava and Vasconcelos (2011). Extended Imaging Condition The extended imaging condition differs from the conventional imaging condition in that it is a correlation between the source and receiver wavefields shifted in the space and in the time directions (Claerbout, 1971 and Sava and Vasconcelos, 2011). R(~x, t,~λ ,τ) = ∑ t Ws(~x−~λ , t + τ)Wr(~x+~λ , t − τ) (1) The equations defining the spatial extent of the incident and reflected wavefronts at a time t, considering the origin at (xs, ts) and (xr, tr), are: ns · (~x−~xs) = vs(t − ts) (2) nr · (~x−~xr) = vr(t − tr) (3) The variables with index s are related to the source wavefield, and variables with index r are related to the receiver wavefield. If symmetrical space shifts (−λ and +λ ) are applied to each of these wavefronts, then a shift in time τ is necessary to bring them to a position where once again they cross the original point of reflection (Sava and Vlad, 2010). The relation between the spatial and temporal shifts are given by the equations ns · (~x−~xs −~λ ) = vs(t − ts − τ) (4) nr · (~x−~xr +~λ ) = vr(t − tr + τ) (5) We can use equation 2 in equation 4 and equation 3 in equation 5 to cancel out the spatial and temporal variables, so that we end up with an expression which depends only with the lag variables: