Wide-azimuth angle gathers for wave-equation migration

Examples

We illustrate the method discussed in the preceding section with common-image-point-gathers constructed using the wide-azimuth SEAM data . Figure 7 shows the velocity model in the area used for imaging. For demonstration, we consider $16$ shots located at the locations of the thick dots in Figure 9(d). The thin dots represent all the $357$ shots available in one of the SEAM data subsets. The solid lines in Figures 9(a)-9(b) depict the decimated receiver lines for each of the $3$ shots shown. In all panels 9(a)-9(d), the large dot indicates the surface projection of the CIP used for illustration, located at coordinates $\{x,y,z\}=\{23.450,11.425,2.38\}$  km. For this example we consider the azimuth reference vector oriented in the $x$ direction, i.e. ${\bf v}=\{1,0,0\}$ .

velo Figure 7. A subset of the SEAM velocity model used for the imaging example in Figures 8-11(d).

cstk Figure 8. Conventional image obtained using wavefield extrapolation with the $16$ shots shown in Figure 9(d).

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Figure 9. Geometry of SEAM imaging experiment. Panels (a)-(c) show the position of one shot and the associated receiver lines (decimated by a factor of $30$ in the $y$ direction. Panel (d) shows the locations of the $16$ shots used for creating the image shown in Figure 8.

Figures 10(a)-10(c) show the extended image obtained at the CIP location indicated earlier using migration by downward continuation. The extended image cubes use $41$ grid points in the ${h_x}$ and ${h_y}$ directions sampled on the image grid, i.e. at every $30$  m, and $31$ grid points in the $\tau$ direction sampled on the data grid, i.e. at every $8$  ms. The vertical lag ${h_z}$ is not computed in this example, since the analyzed reflector is nearly-horizontal. This lag is computed in the decomposition process from the horizontal lag and from the known information about the normal to the reflector at the given position. Figure 10(d) shows the extended image obtained for all $16$ shots used for imaging. Although here we show the extended image cubes for independent shots, in practice these cubes need not be computed separately - the decomposition separates the information corresponds to different angles of incidence, as shown in this simple example.

Finally, Figures 11(a)-11(d) show the angle-domain decomposition of the extended image cubes shown in Figures 10(a)-10(d), respectively. In these plots, the circles indicating the reflection angles are drawn at every $5^\circ$ and the radial lines indicating the azimuth directions are drawn at every $15^\circ$ . Given the sparse shot sampling, the CIP is sparsely illuminated, but at the correct reflection and azimuth angles.

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Figure 10. Extended image cubes for the SEAM imaging experiment. Panels (a)-(c) show extended image cubes at the same location for $3$ different shots, and panel (d) shows the extended image obtained for all $16$ shots considered in this experiment.

cang-037449,cang-041729,cang-043873,cang
Figure 11. Reflectivity as a function of reflection and azimuth angles for the SEAM imaging experiment. Panels (a)-(c) show the angle-domain CIPs at the same location for $3$ different shots, and panel (d) shows the angle-domain CIP obtained for all $16$ shots considered in this experiment. The angles $\phi$ and $\theta$ are indexed along the contours using the trigonometric convention and along the radial lines increasing from the center.

Wide-azimuth angle gathers for wave-equation migration