2-D inversion

To test the performance of the algorithm in inverting data generated in a 2-D medium, we computed synthetic traveltimes through the isotropic model shown in Figure . The separation between contiguous sources and receivers is 10 ft and for each receiver gather, only sources located at $\pm 50$ degrees are used. With a geometry like this, we pretend to simulate the geometry of the real data example to be analyzed later. As in the 1-D example, the slowness contrast between the anomaly and the background is small ($5 \%$), and therefore, straight rays can be used again.

 

s-model-test Figure 7 Isotropic slowness model. The radius of the circular anomaly is r=50 and is centered at (100,700). The background slowness is 1.0 and the slowness of the disc is 1.05.

 

sx2d-test
Figure 8 Reconstructed horizontal and vertical component of the slowness. The ratio of the two components is shown at the right.

The unknown model was discretized in 241 x 46 pixels (5 ft²) and therefore, the inversion has to estimate 241 x 46 x 2 parameters from 2200 synthetic traveltimes. Figure  shows the results of the inversion. The slowness of the isotropic circular anomaly (1.05) is better estimated by the horizontal than by the vertical component of the slowness. Remember that this is not the case in the 1-D inversion, where both slowness components can be perfectly recovered even thought the vertical component of the slowness is not properly sampled. The extra information introduced in that problem by assuming that the model is layered compensates for the limited view of the measurements. In the 2-D inversion, where the unknown is less constrained, the better sampling of the horizontal component translates into a better recovery of that component and as a result, some artificial anisotropy is introduced by the reconstruction. In this noise-free example such an anisotropy is not greater than $3 \%$ as shown in Figure  by the ratio $\frac{S_z}{S_x}$.Doing more CG-iterations does not help to reduce this artificial anisotropy to zero, like in the 1-D inversion (Figures 5 and 6). In the present case the images didn't change after 120-CG iterations.

The artifacts in both slowness components are similar to the well known truncation artifacts in isotropic inversion although they are different from one component to the other. The estimated S_x is smeared along the horizontal direction whereas S_z is not. This is because the estimation of S_z is not affected by rays that travel horizontally. The different character of the artifacts for each slowness component can limit our ability to recover variations in anisotropy at the same scale of variations in velocity when data from only one geometry is used. This will be clearly observed later in the application to field data.