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 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.
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 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.
The inversion was constrained by not allowing parallel layers (within 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.
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.