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Angle decomposition

In this section we discuss the steps required to transform lag-domain CIPs into angle-domain CIPs using the moveout function derived in the preceding section. We also present the algorithm used for angle decomposition and illustrate it using a simple 3D model of a horizontal reflector in a medium with constant velocity which allows us to validate analytically the procedure.

The outer loop of the algorithm is over the CIPs evaluated during migration. The angle decompositions of individual CIPs are independent of one-another, therefore the algorithm is easily parallelizable over the outer loop. At every CIP, we need to access the information about the reflector normal ( ${\bf n}$ ) and about the local velocity ( ). The reflector dip information can be extracted from the conventional image, and the velocity is the same as the one used for migration.

Prior to the angle decomposition, we also need to define a direction relative to which we measure the reflection azimuth. This direction is arbitrary and depends on the application of the angle decomposition. Typically, the azimuth is defined relative to a reference direction (e.g. North). Here, we define this azimuth direction using an arbitrary vector ${\bf v}$ . Using the reflector normal ( ${\bf n}$ ) we can build the projection of the azimuth vector ( ${\bf a}$ ) in the reflector plane as

$\displaystyle {\bf a}= \left ({\bf n}\times{\bf v}\right)\times {\bf n}\;.$

(19)

This construction assures that vector ${\bf a}$ is contained in the reflector plane (i.e. it is orthogonal on ${\bf n}$ ) and that it is co-planar with vectors ${\bf n}$ and ${\bf v}$ , Figure 1. Of course, this construction is just one of the many possible definitions of the azimuth reference. In the following, we measure the azimuth angle $\phi$ relative to vector ${\bf a}$ and the reflection angle $\theta$ relative to the normal to the reflector given by vector ${\bf n}$ .

Then, for every azimuth angle $\phi$ , using the reflector normal ( ${\bf n}$ ) and the azimuth reference ( ${\bf a}$ ), we can construct the trial vector ${\bf q}$ which lies at the intersection of the reflector and the reflection planes. We scan over all possible vectors ${\bf q}$ , although only one azimuth corresponds to the reflection from a given shot. This scan ensures that we capture the reflection information from all shots in the survey. Given the reflector normal (the axis of rotation) and the trial azimuth angle $\phi$ , we can construct the different vectors ${\bf q}$ by the application of the rotation matrix

$\displaystyle Q\left ({\bf n},\phi \right)= \left [\begin{array}{ccc} n_x^2+\le... ...)+n_x\sin\phi & n_z^2+\left (n_x^2+n_y^2 \right)\cos\phi \ \end{array} \right]$

(20)

to the azimuth reference vector ${\bf a}$ , i.e.

$\displaystyle {\bf q}= Q\left ({\bf n},\phi \right){\bf a}\;.$

(21)

In this formulation, the normal vector ${\bf n}$ of components $\{n_x,n_y,n_z\}$ can take arbitrary orientations and does not need to be normalized. Then, for every reflection angle $\theta$ , we map the lag-domain CIP to the angle-domain by summation over the surface defined by equation 17. This operation represents a planar Radon transform (a slant-stack) over an analytically-defined surface in the $\{ {\boldsymbol{\lambda}} ,\tau \}$ space. The output is the representation of the CIP in the angle-domain. In order to preserve the signal bandwidth, the slant-stack needs to use a ``rho filter'' which compensates the high frequency decay caused by the summation (Clærbout, 1976). The explicit algorithm for angle decomposition is given in Appendix A.

Consider a simple 3D model consisting of a horizontal reflector in a constant velocity medium. We simulate one shot in the center of the model at coordinates km and km, with receivers distributed uniformly on the surface on a grid spaced at every m in the and directions. We use time-domain finite-differences for modeling. Figure 4 represents the image obtained by wave-equation migration of the simulated shot using downward continuation. The illumination is limited to a narrow region around the shot due to the limited array aperture.

img-3d Figure 4. The image obtained for a horizontal reflector in constant velocity using one shot located in the center of the model.

Figures 5(a)-5(d) depict CIPs obtained by migration of the simulated shot at the reflector depth and at coordinates $\{x,y\}$ equal to $\{3.2,3.2\}$ km, $\{3.2,4.8\}$ km, $\{4.8,4.8\}$ km and $\{4.8,3.2\}$ km, respectively. For these CIPs, the reflection angle is invariant $\theta=48.5^\circ$ , but the azimuth angles relative to the axis are $-135^\circ$ , $+135^\circ$ , $+45^\circ$ and $-45^\circ$ , respectively. Figures 5(e)-5(h) show the angle decomposition in polar coordinates. Here, we use the trigonometric convention to represent the azimuth angle $\phi$ and we represent the reflection angle in every azimuth in the radial direction (with normal incidence at the center of the plot). Each radial line corresponds to $30^\circ$ and each circular contour corresponds to $15^\circ$ .


hic-A,hic-B,hic-C,hic-D,aca-A,aca-B,aca-C,aca-D Figure 5. Illustration of CIP angle decomposition for illumination at fixed reflection angle. Panels (a)-(d) show lag-domain CIPs, and panels (e)-(f) show angle-domain CIPs in polar coordinates. The angles $\phi$ and $\theta$ are indexed along the contours using the trigonometric convention and along the radial lines increasing from the center.

Similarly, Figures 6(a)-6(d) depict CIPs obtained by migration of the simulated shot the reflector depth and at coordinates $\{x,y\}$ equal to $\{2.8,2.8\}$ km, $\{3.2,3.2\}$ km, $\{3.6,3.6\}$ km and $\{4.0,4.0\}$ km, respectively. For these CIPs, the azimuth angle is invariant $\phi=-135^\circ$ , but the reflection angles relative to the reflector normal are $59.5^\circ$ , $48.5^\circ$ , $29.5^\circ$ , and $0^\circ$ respectively.


hic-F,hic-G,hic-H,hic-I,aca-F,aca-G,aca-H,aca-I Figure 6. Illustration of CIP angle decomposition for illumination at fixed azimuth angle. Panels (a)-(d) show lag-domain CIPs, and panels (e)-(h) show angle-domain CIPs in polar coordinates. The angles $\phi$ and $\theta$ are indexed along the contours using the trigonometric convention and along the radial lines increasing from the center.