Angle-domain common-image gathers in generalized coordinates

Test 2: BP velocity model

The second test compares elliptic and Cartesian coordinate ADCIG volumes computed for the BP synthetic velocity model. Images computed in elliptic coordinates used only one coordinate system per shot. For the one-sided data set, I used a (surface) migration aperture of 12km and located the source and farthest offset receiver (at 8km) points 2km in from the edges of the computational mesh. (Note that the migration aperture effectively expands during wavefield extrapolation because the coordinate mesh expands outward.) I found this initial migration geometry to produce the best results for the BP synthetic model through iterative testing. Generally, the optimal elliptic coordinate migration geometry is controlled by the velocity model.

Figure 7 shows slices all clipped at the 99$^{th}$ percentile from the corresponding elliptic and Cartesian ADCIG image volumes.

GOOD
Figure 7. Vertical elliptic and Cartesian ADCIGs slices using the correct migration velocity model. a) Elliptic coordinate image with three vertical lines showing the locations of ADCIG gathers from left to right in panels b-d. e) Cartesian coordinate image with three vertical lines showing the locations of ADCIG gathers from left to right in panels f-h.

Panel 7a shows an elliptic coordinate image with three vertical lines indicating the angle-gather locations from left to right in Figures 7b-d. The three panels show predominantly one-sided reflectivity, which is to be expected because the input migration data were not in a split-spread geometry. (This statement holds for all subsequent images calculated using this data set.) The only significant exceptions occur in panel 7b within salt where energy is exhibited for both positive and negative reflection angles. I attribute this to the reversal of source and receiver wavefield orientations within the salt.

The image in panel 7d has a wide reflection zone between 3.75-4.25 km in depth, which occurs because the shown angle gather is a vertical slice through the nearly vertical salt flank. This creates the appears of low-frequency noise, which is the appropriate response for a near-vertical reflector. Panel 7e shows the Cartesian image for the same location as panel 7a, while panels 7f-h are extracted from the same locations as panels 7b-d. The Cartesian angle gathers look similar to those in elliptic coordinates, except for the salt flanks to the right-hand-side of panel 7h.

A final observation from Figure 7 is that ADCIGs calculated via subsurface correlations will generate artifacts at locations near salt-sediment interfaces - whether in an elliptic or a Cartesian coordinate system. This geologic setting leads to situations where a wavefield sample inside a salt body is correlated with another sample located in the sediment with a significantly different velocity. This velocity difference violates one of the theoretical ADCIG assumptions, namely that the velocity remains constant across the correlation window. Hence, one must be careful not to interpret ADCIG artifacts as signal useful for migration velocity analysis.

Figure 8 shows horizontal slices that better resolve the vertical salt flank. Panel 8a presents the elliptic coordinate image, with three horizontal lines showing the ADCIG slice locations from top to bottom. The right-hand sides of panels 8b-d display the well-focused vertical salt-flank reflector. This demonstrates the robustness of the ADCIG calculation in elliptic coordinates.

GOODZ
Figure 8. Horizontal elliptic and Cartesian ADCIGs slices using the correct migration velocity model. a) Elliptic coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers from top to bottom in panels b-d. e) Cartesian coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers from top to bottom in panels f-h.

Panel 8e shows the Cartesian coordinate image with three horizontal lines showing the locations of the ADCIG slices. The right-hand salt-flank reflector in panel 8f is similarly well-resolved, largely because the structural dip is relatively low. However, the salt-flank images in panels 8g-h are somewhat blurred out. I attribute this to the combined effects of inaccurate large-angle extrapolation and insensitivity of the ADCIG calculation to steep structural dip.

An additional test examines how the ADCIG volumes change when introducing an incorrect migration velocity profile. Figure 9 presents ADCIG volumes similar to those shown in Figure 7 after using a migration velocity profile rescaled by 98%.

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Figure 9. Vertical elliptic and Cartesian ADCIGs slices using an incorrect migration velocity model. a) Elliptic coordinate image with three vertical lines showing the locations of vertical ADCIG gathers from left to right in panels b-d. e) Cartesian coordinate image with three vertical lines showing the locations of vertical ADCIG gathers from left to right in panels f-h.

Both images are poorly focused and have residual curvature indicating an incorrect migration velocity. Because the reflectors are near vertical, though, the sensitivity of horizontal gathers is weak. This low sensitivity is greatly improved when examining the horizontal slices in Figure 10 taken at the same locations as in Figure 8.

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Figure 10. Horizontal elliptic and Cartesian ADCIGs slices using an incorrect migration velocity model. a) Elliptic coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers from top to bottom in panels b-d. e) Cartesian coordinate image with three horizontal lines showing the locations of horizontal ADCIG gathers from top to bottom in panels f-h.

The elliptic angle gathers in panels 10b-d, and especially to the right-hand side in panel 10d, show much greater residual curvature. This indicates that the elliptic coordinate horizontal ADCIGs have greater sensitivity to velocity error for near-vertical structures than Cartesian coordinate horizontal ADCIGs. The imaging enhancements afforded by elliptic coordinates should improve any migration velocity analysis approach that uses residual curvature in steeply dipping reflectors to compute velocity model updates.

Angle-domain common-image gathers in generalized coordinates