A lens example

The presence of a lens anomaly in a velocity model results in a variety of ray paths, the most interesting of which is a development of a triplication in the wavefront. This multi-arrival traveltime phenomenon typically occurs when a negative velocity anomaly is present. The intriguing issue is that triplication can also occur when we have positive $\eta$anomalies.

Figure 5 shows rays and corresponding wavefronts that were obtained using conventional raytracing in the depth domain (black curves), and using the equivalent raytracing in the $(x-\tau)$-domain (gray curves) through a VTI model with $\eta$=0.1. The velocity model is shown in the background with a negative velocity anomaly that has a peak of -1.0 km/s. The result is a noticeable triplication that develops soon after the rays pass the anomaly. Despite the triplication, the results of raytracing in the two domains (depth and time) are similar.

 

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Figure 5 Raypaths (solid curves) and corresponding wavefronts (dashed curves) for an inhomogeneous VTI model, with $\eta=0.1$. The rays are superimposed on the velocity model, given in km/s, of a negative velocity anomaly. The black curves are obtained through conventional raytracing in the depth domain, and the gray curves are obtained using the equivalent $(x-\tau)$-domain raytracing, where the results are later converted to depth. The curves nearly overlap even in the presence of triplication.

Figure 6 also shows raypaths through an anomaly. The anomaly now is in $\eta$, and it is positive. Therefore, the background is an $\eta$ model, with $\eta$=0 everywhere other than in the anomaly. Again, the black curves correspond to solutions of raytracing in the depth domain, while the gray curves correspond to raytracing in the $(x-\tau)$-domain. Triplication, smaller than that associated with the velocity perturbation, occurs in the wavefront. Velocity-wise this medium is homogeneous; it is $\eta$ that is causing the severe bending of the rays! The rays with larger propagation angles from the vertical are the most influenced by the $\eta$ anomaly.

 

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Figure 6 Raypaths and wavefronts for an inhomogeneous VTI model, with v=3.5 km/s. The rays are superimposed on the $\eta$ distribution, which includes a positive $\eta$ anomaly. The black curves are obtained through conventional raytracing in the depth domain, and the gray curves are obtained using the equivalent $(x-\tau)$-domain raytracing, where the results are later converted to depth. The curves nearly overlap even in the presence of $\eta$-induced triplication.