SYNTHETIC EXAMPLE

Scheduled tests begin with a simple 2D isotropic example, to test the convergence of the ray and velocity optimization. Non-uniqueness here will most likely remain with the additional complications of 3D and anisotropic models.

I define a 2D velocity model (in km/s) over the ranges $0 \leq x \leq 10$ (km) and $0 \leq z \leq 2$ (km), with

$$\begin{eqnarray} v(x,z) = 2 + z - 0.6 &\exp&\{- \pi [(x-3)^2 + (z-1)^2 ]\} \nonumber \\ + &\exp&\{- \pi [(x-7)^2 + (z-1)^2 ]\} .\end{eqnarray}$$ (18)

This velocity model (Figure ) has a background that increases linearly from 2 km/s to 4 km/s from z=0 km to z=2 km. Two smooth velocity anomalies, Gaussian with a width of 1, one slow and one fast, are placed at depths of z=1 km, at x=3 km and x=7 km.

 

fig2
Figure 2 This is the assumed velocity model.

Sources and receivers are placed at 0.5 km intervals of x within $0 \leq x \leq 10$, with a minimum separation (offset) of 0.5 km and a maximum of 7 km, for a total of 189 data pairs. Figure shows the resulting coverage of the velocity model by raypaths. If not for the velocity anomalies, a maximum offset of 7 km would produce a ray reaching 2 km depth. Rays through the faster part of the model cross at shallower depths. Rays also attempt to pass around the slow anomaly, leaving a small hole in coverage.

 

fig3
Figure 3 Rays are modeled from sources and receivers placed every 0.5 km, with a maximum offset of 7 km.

The most robust initialization of this velocity model would first optimize a velocity that changed only with depth. Only two parameters are necessary to describe a velocity that changes linearly with depth, so I initialize with the model in Figure : the velocity increases from 2 km/s at the surface to 4 km/s at 2 km depth.

 

fig4
Figure 4 An initial model assumes velocity increases linearly from 2 km/s at the surface to 4 km/s at 2 km depth.

The rays are laterally invariant for a given offset. The ray coverage does not agree with that of the original model in Figure .

Eight large iterations were performed. In the first iteration isotropic velocities were described with overlapping Gaussian basis functions with a width of 1 km vertically and 4 km horizontally. During each large iteration, the raypaths were estimated once, then a linearized perturbation of velocities was estimated for these reference raypaths with a conjugate gradient algorithm. The perturbation was added to the reference velocities only after a line-search to find the proper scale factor. Rays were then reëstimated for the next large iteration and velocity perturbations were allowed to change more rapidly, with geometrically decreasing widths, until the final two iterations, where basis functions have widths of 0.1 km.

To control the rate of convergence for a general dataset, I always begin with velocity function that is smooth over most of the vertical and horizontal span. The final maximum resolution is limited by the density of rays and the spatial wavelengths. The widths of intermediate basis functions are reduced by a constant factor for each iteration. The final two iterations are at maximum resolution. Users are allowed to increase the total number of iterations for a more robust convergence.

After four iterations, the very smooth velocity model in Figure has begun to show the velocity anomalies in outline. After two more iterations, the shape of the estimated anomalies in Figure has approached the scale of the true anomalies, but still with flattened magnitudes. After eight large iterations, the velocity model in Figure shows minor unreliable detail on the scale of the sampling of raypaths, but the anomalies do not much better resemble the original model. The raypaths in Figure have converged to roughly the same coverage as in the original model in Figure .

 

fig5
Figure 5 After four iterations, very smooth variations in interval velocity have been optimized.

 

fig6
Figure 6 After six iterations, velocities are allowed to change more rapidly.

 

fig7
Figure 7 After eight iterations, velocities are allowed to change most arbitrarily.

 

fig8
Figure 8 The final estimation of raypaths for the model in Figure has converged to a coverage similar to that in Figure .

The traveltimes plotted in Figure were integrated from the raypaths in the original model in Figure . Traveltimes from the estimated raypaths in Figure were subtracted from the correct traveltimes for the errors in Figure . These errors ranged from -6 ms to +10 ms, with a root-mean-square average of 2.3 ms. Those errors which remain do not appear to have any systematic correlation. Such time errors are much less than would be found at these scales in times picked from recorded surface seismic data, where a seismic temporal wavelength often exceeds 20 ms. Thus, the original and estimated velocity models in Figures and produce effectively indistinguishable traveltimes for this survey.

 

fig9
Figure 9 These traveltimes were modeled from the source and receiver positions in Figure .

 

fig10
Figure 10 Traveltimes from the original model in Figure minus traveltimes from the estimated model in Figures and . The root-mean-square average of these errors was 2.3 ms, well below any possible accuracy in picking.

Figure shows the difference of the original velocities in Figure minus the estimated velocities in Figure . The magnitudes of both anomalies has been underestimated. Moreover, sidelobes appear about the anomalies. The center of the estimated slow anomaly is too fast and the edges of the slow anomaly are too slow. The center of the estimated fast anomaly is too slow and the edges of the fast anomaly are too fast. Both anomalies have lost resolution in the reconstruction. The deeper portion of the fast anomaly was missed altogether because the original rays did not reach this depth.

 

fig11
Figure 11 The original velocities in Figure minus the estimated velocities in figure .

The most important application of diving wave tomography is the correction of structural distortions in underlying reflections. To see the effect of errors in the velocity model, I calculate the vertical two-way traveltime from the surface to a horizontal reflector at 2 km depth, for all surface positions. The true velocity model gives two distinct peaks in traveltime in Figure , but the estimated velocity model gives much smoother peaks. The positioning of peaks is correct, but lost resolution has the effect of reducing the magnitude of the relative changes in vertical time.

Smearing of the velocity anomalies clearly occurs parallel to the raypaths. Resolution is greatest perpendicular to the raypaths. The angular coverage of a particular anomaly is limited. Broader velocities anomalies are easier to invert than this example, but we must acknowledge this loss of resolution will affect the accuracy of our near-surface velocities. Other information, such as known shallow faults, might allow us to identify and introduce sharper edges on lateral velocity changes.

 

fig12
Figure 12 Vertical two-way times integrated from the surface down to a flat reflector at 2 km depth, for the original model in Figure 2 and for the estimated model in Figure . The estimated curve has broader and shallower peaks (time sags) than the actual curve.