The Sigsbee data set was modeled by simulating the geological setting found on the Sigsbee escarpment in the deep-water Gulf of Mexico. The model exhibits the illumination problems due to the complex salt shape, with a rugose salt top (Figure ) found in this area. We choose a target zone (indicated with the "target" box in Figure ) to see the effects of illumination on the Hessian close to the salt.

Figure 4

Figures a and b show the Hessian matrix for the Sigsbee model. Notice the banded nature of the matrix (Figure b), with most of the energy on the main diagonal (Figure a). As opposed to the case with the constant-velocity, the energy decreases considerably in some areas of the diagonal due to illumination problems cause by the salt body. But, similar to the constant velocity case, amplitudes become dimmer at the extremes of the diagonal due to the acquisition geometry.

Figure 5

We fixed seven points, all of them at the same depth, to see the corresponding lines of the Hessian. Figures and show the Hessian and the envelope of the Hessian, respectively. The envelope of the Hessian (Figure ) shows clearly the main features of interest.

Figure 6

Figure a shows a close-up of the velocity model in the area where the target-oriented Hessian was computed (salt body to the right and sediments to the left). Figure b shows point 1, with coordinates ; since this point in the model is well illuminated, the resulting ellipse looks similar to the constant-velocity ellipses. Figures c-f show points 2 to 5, with coordinates , , , and , respectively. As the points enter a shadow zone, the ellipses lose energy and splits. A diagonal matrix approximation of the Hessian would not be appropriate to describe this behavior, since there is considerable energy away from the point where the ellipse should be centered. Figure g shows point 6, with coordinates ; out of the shadow zone, the ellipse gains energy.

Figure 7

Finally, Figure h shows point 7, with coordinates . As the point gets closer to the salt boundary it enters a new shadow zone. This point behaves differently, the Hessian not only losses energy but the ellipse center is away from where it should be. This behavior might suggest that not enough reference velocities where used in the split-step computation of the Green functions. Or more fundamentally, that the physics of wave propagation is not well modeled by the acoustic one-way wave-equation close to the salt. This subject deserves more attention in the future.

10/23/2004