SYNTHETIC EXAMPLE

In this section, I will show the application of the algorithm previously explained to the inversion of synthetic traveltimes from a cross-well geometry. 4992 traveltimes were generated through the model shown in Figure  for a geometry where sources were located within a range of $\pm 45$ degrees with respect to the horizontal at each receiver position. The ray tracing algorithm described in Michelena (1992) was used to compute the synthetic traveltimes and to trace the rays needed in the nonlinear inversion.

 

syn-model
Figure 4 Synthetic model used to test the inversion. The third interface is dipping 15 degrees and the fourth one - 30. The inclination of the axis of symmetry in the second layer is -15 degrees and 40 degrees in the fourth layer. The ratio $\frac{V_{\perp}}{V_{\parallel}}$ at the fourth layer is 1.26. The gray scale shows variations in velocity. ``V+'' stands for $V_{\perp}$,``V||'' for $V_{\parallel}$ and ``gamma'' for $\gamma$.

The starting model used for the inversion was homogeneous isotropic described by 17 horizontal layers of equal thickness. The inclination used for the axis of symmetry was $\gamma = 0$ in all layers. Figure  shows the initial positions of the boundaries in the starting model. By starting the iterations with this model I wanted to test how the interfaces arrange themselves to create a dipping layer not present in the initial model.

 

initial-model-syn
Figure 5 Initial position of the interfaces in the starting model (homogeneous isotropic). The velocity is equal to 12000 ft/sec and the inclination of the axes of symmetry with respect to the vertical is zero for all layers.

The inversion was constrained by not allowing parallel layers (within $\pm 5$ degrees) to be thinner than 15 ft. When this condition was met, the corresponding layer was eliminated, reducing the number of model parameters. No smoothing was applied to the model after each iteration.

Figure  shows the result of the inversion after tracing rays 35 times, with few conjugate gradient iterations between each ray tracing. Notice how boundaries have changed positions with respect to their initial values. Notice also that the initial 17 layers were reduced to 10 to allow the positioning of the dipping ones at the correct depths with the correct dips. The inclination of the axes of symmetry estimated by the algorithm is also correct. Figure  compares a profile at x=0 of the real model (Figure ) and the estimated one (Figure ). The agreement is almost perfect.

 

final-model-syn
Figure 6 Result of the inversion. Notice how the interfaces have changed their initial positions. The position of each arrow's head shows the interval that correspond to each $\gamma$.The gray scale shows variations in velocity. ``V+'' stands for $V_{\perp}$,``V||'' for $V_{\parallel}$ and ``gamma'' for $\gamma$.

In this example, it was possible to estimate correctly both small and moderate dips (between 0 and 30 degrees), but the question of the maximum and minimum dips that can be retrieved from the data will depend in general upon the aperture of the recording geometry, the interval between sources and receivers and the frequency.