Numerical examples

Using the ray-tracing system of equations derived earlier, we can compute traveltimes numerically. Unlike numerical solutions of the eikonal equation, raytracing provides multi-arrival traveltimes and amplitudes. We want to confirm numerically the following two aspects of implementing raytracing in the new coordinate system:

• The traveltime solution when transformed to depth agrees with results from conventional depth-domain raytracing.
• The traveltime solution in the $(x-\tau)$-domain is independent of the vertical P-wave velocity for media that are factorized laterally [$\alpha(z)$].

Figure 1 shows sixteen rays originating from a source on the surface at position x=0 through the same depth velocity model of v_v(x,z)=1.5+0.225 z+0.15 x, v(x,z)=2.0+0.3 z+0.2 x, and $\eta(x,z)=0.1+0.05 z+0.05 x$ using conventional raytracing in the depth domain (black curves), and the new raytracing in the $(x-\tau)$-domain (gray curve). We achieved the $(x-\tau)$-domain ray tracing results by mapping the depth velocity model to time using equation (3), and then mapping the ray solutions back to depth using equation (30). The sixteen rays have ray parameters ranging from zero to the maximum value of 1/V_h (V_h is the horizontal velocity), with a fixed ray-parameter spacing of 1/(15 V_h). The rays terminate at the same time of 8 s, and the wavefronts (given by the dashed curves) are plotted at about 1.6-s intervals. The wavefronts that correspond to the different raytracings are virtually coincident, a result that agrees with our analytical findings.

 

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Figure 1 Raypaths (solid curves) and corresponding wavefronts (dashed curves) for an inhomogeneous VTI model with v(x,z)=2.0+0.3 z+0.2 x km/s, v_v(x,z)=1.5+0.225 z+0.15 x km/s, and $\eta(x,z)=0.1+0.05 z+0.05 x$. 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 ultimately converted to depth. In this case, the curves nearly overlap; they are only barely distinguishable, which agrees with the theoretical results. The small difference is numerical noise resulting from the different schemes used to solve the ordinary differential equations (Runge-Kutta versus Euler schemes).

In Figure 2, we check for another aspect of the theory, that is, the independence of raytracing from the vertical velocity for laterally factorized VTI media. Again, sixteen rays were ray traced through a VTI model with v(x,z)=2+0.2 x km/s, and $\eta(x,z)=0.1+0.05 z+0.05 x$. The raytracing was done in the $(x-\tau)$-domain coordinates, and, as a result, the rays and corresponding wavefronts appear in the $(x-\tau)$-domain. The vertical velocity varies considerably between the two sets of curves (black and gray), and yet the two curves coincide exactly. That is because in both models $\alpha$, which is the ratio of the vertical to NMO P-wave velocity, does not vary laterally-- a condition for the independence of raytracing from vertical velocity in the $(x-\tau)$-domain. Therefore, under this condition, raytracing is dependent on only v and $\eta$.

 

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Figure 2 Raypaths and corresponding wavefronts in the $(x-\tau)$-domain for an inhomogeneous VTI model with v(x,z)=2+0.2 x km/s, and $\eta(x,z)=0.1+0.05 z+0.05 x$.The black curves correspond to v_v(x,z)=1.5+0.15 x km/s ($\alpha=0.75$), and the gray curves correspond to v_v(x,z)=1.5+0.15 x+0.75 z +0.075xz km/s ($\alpha(z)=0.75+0.375 z$). In both cases, $\alpha$is laterally invariant, and as a result the two curves overlap.

However, when $\alpha$ varies laterally, raytracing does depend on the vertical velocity. The amount of its dependence is controlled by the size of the lateral variation in $\alpha$. Figure 3 shows rays penetrating in the same model as Figure 2, but with the gray curves corresponding to a laterally varying $\alpha$that satisfies

$$\alpha(x,z)=\frac{(1.5+0.1x)(1+0.5z)}{2+0.2x}.$$

On the other hand, for the black curves, $\alpha=0.75$. For the laterally varying $\alpha$ model, at x=0 and z=5 km, $\alpha$=2.625, while at x=5 and z=5 km, $\alpha$=2.333. This big difference corresponds to a large variation in the ratio of the vertical to NMO P-wave velocity, a lot more than would be expected in practice. Yet the differences in traveltimes between the two models is moderate. This fact implies that, despite the apparent influence of vertical velocity on raytracing in the $(x-\tau)$-domain coordinates, when $\alpha$ varies laterally, such influence is overall small.

 

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Figure 3 Raypaths and corresponding wavefronts in the $(x-\tau)$-domain for an inhomogeneous VTI model with v(x,z)=2+0.2 x km/s, and $\eta(x,z)=0.1+0.05 z+0.05 x$.The black curves correspond to v_v(x,z)=1.5+0.15 x km/s ($\alpha=0.75$), and the gray curves correspond to v_v(x,z)=1.5+0.1 x+0.75 z +0.05xz km/s ($\alpha(x,z)=\frac{(1.5+0.1x)(1+0.5z)}{2+0.2x}$). While for the black curves $\alpha$ is laterally invariant, for the gray curves $\alpha$ varies laterally, and as a result, the black and gray curves no longer coincide.

Considering that $\delta$, the parameter that relates the vertical and NMO velocity, ranges typically between -0.1 and 0.4 , Figure 4 shows a more practical $\alpha$ variation, in which the curves given by the two models are extremely close.

 

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Figure 4 Raypaths and corresponding wavefronts in the $(x-\tau)$-domain for an inhomogeneous VTI model with v(x,z)=2+0.2 x km/s, and $\eta(x,z)=0.1+0.05 z+0.05 x$.The black curves correspond to v_v(x,z)=1.5+0.15 x km/s ($\alpha=0.75$), and the gray curves correspond to v_v(x,z)=1.5+0.13 x+0.75 z +0.065xz km/s ($\alpha(x,z)=\frac{(1.5+0.13x)(1+0.5z)}{2+0.2x}$). Despite the fact that for the gray curves $\alpha$ varies laterally, the two curves are extremely close.

The slightness of the variation suggests that for practical applications of the $(x-\tau)$-domain coordinate processing, we can simply ignore the vertical velocity, and rely on the NMO velocity and $\eta$.