Comparing linearizations of forward modeling in $\left(\tau,\xi\right)$ and $\left(z,x\right)$

One of the motivations for performing tomography in $\left(\tau,\xi\right)$ domain is to improve the linearity of the forward modeling problem. Therefore, we compare the accuracy of the linearized forward modeling for the $\left(\tau,\xi\right)$ tomography with the linearized forward modeling for the $\left(z,x\right)$ tomography in a few significant examples. To make this comparison we define a reflector geometry, and a velocity model, that we call the ``true'' model. The true model is defined as a velocity anomaly superimposed onto a background model. Then we define a starting velocity model and a starting reflector geometry. We assume that the starting model was estimated by interpreting a migrated section, therefore the starting velocity model in the $\left(\tau,\xi\right)$ domain is equal to the background model in the $\left(\tau,\xi\right)$ domain. The starting model in the $\left(z,x\right)$ domain is defined by the starting model in the $\left(\tau,\xi\right)$ domain mapped into depth. Notice that while both the true and the starting models map into each other according to the mapping defined in equations (2) and (3), their perturbations do not. The starting reflector geometry is the results of map-migrating the true zero-offset arrivals assuming the starting velocity model.

To analyze the linearity of the forward modeling, we compare the linearization errors of the two different tomographic methods. We define the linearization errors as the differences between the reflection traveltimes modeled with the true model, and the reflection traveltimes that are predicted by linearizing the forward modeling at the starting model. To compute the traveltimes perturbations we set as velocity perturbations the difference between the true model and the starting model.

The true model for the first example is a positive Gaussian-shaped velocity anomaly with peak amplitude of 0.5 km/s superimposed onto a constant velocity background of 2 km/s. We positioned two flat reflectors. The deep reflector at 3 km is below the anomaly, while the shallow one at 2 km cuts through the anomaly. Figure 4 shows the true velocity model with superimposed both the starting (dashed lines) and the true reflectors (solid lines) along with few reflected raypaths. The raypaths on the left are traced in the $\left(\tau,\xi\right)$ domain while the raypaths on the right are traced in the $\left(z,x\right)$ domain. We can notice how both the reflectors and the rays move less in the $\left(\tau,\xi\right)$ representation.

Figure 5 shows comparison of the linearization errors in modeling traveltimes of the reflections from the deeper reflector. The errors are shown as a function of the midpoint for two different offsets. The thicker lines show the errors at zero offset, and the thinner lines show the errors at 3.2 km offset. The solid lines show the error in the $\left(z,x\right)$ domain and the dashed lines show the errors in $\left(\tau,\xi\right)$ domain. There is no significant differences in errors between the two domains. This result is not surprising since the background model is constant and the deeper reflector does not interfere with the anomaly.

Figure 6 shows comparison of the linearization errors in modeling traveltimes of the reflections from the shallower reflector. In this case the reflector interferes with the anomaly, and the errors for $\left(\tau,\xi\right)$ domain tomography are smaller. We explain these differences with the interference between the reflector movements and the velocity perturbations.

 

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Figure 4 True velocity model with superimposed both the starting (dashed) and true reflectors (solid) in both the $\left(z,x\right)$ and $\left(\tau,\xi\right)$ domains. A few reflected raypaths are superimposed onto the model.

The second example has a more complex background velocity. A fast body (e.g. salt layer) is placed at depth, below a shallow positive anomaly. The shallow anomaly distorts the time image of the fast body. Because we assume that the shallow anomaly is not known, the fast body is vertically distorted in the $\left(z,x\right)$ domain starting model. On the contrary, in the $\left(\tau,\xi\right)$ domain the fast body has the same position in both the starting and true model. Figure 7 show the true velocity model and the reflector geometry in the $\left(z,x\right)$ domain. A contour plot of the starting model is superimposed onto the true model to illustrate the vertical distortion of the fast body. A sample of zero-offset rays for both the true (solid lines) and the starting models (dashed lines) are superimposed onto the velocity model. Figure 8 shows the equivalent objects of Figure 7, but in the $\left(\tau,\xi\right)$ domain.

Figure 9 show the linearization errors for zero-offset reflections and 2 km offset reflections. The solid lines show the error in the $\left(z,x\right)$ domain and the dashed lines show the error in $\left(\tau,\xi\right)$ domain. The errors for $\left(\tau,\xi\right)$ domain tomography are smaller, although the differences are not that large.

The final example is similar to the previous one, except that the shallow anomaly is negative instead of positive. In this case, surprisingly, the linearization errors shown in Figure 10 are lower for the $\left(z,x\right)$ domain tomography. than for the $\left(\tau,\xi\right)$ domain tomography. A possible explanation of these results is that, by coincidence, the additional errors in the $\left(z,x\right)$ domain caused by the reflectors and velocity model movements have opposite sign of the errors caused by the non-linear behavior of the forward modeling.

A comparison of Figure 7 with Figure 8 shows that the deep, fast body shifts vertically in $\left(z,x\right)$ domain (Figure 7) while it is stationary in the $\left(\tau,\xi\right)$(Figure 8). Because of the depth shift of the fast body, the velocity perturbations in the $\left(z,x\right)$ domain are a dipole with a positive and negative anomaly close to each other. Usually tomographic inversions strongly penalize features like a dipole that are rapidly variant in space. They are difficult to resolve by tomography, and they can lead to divergence if not kept in check. Therefore correcting the initial distortion by a linearized inversion would be difficult; a full migration followed by reflector interpretation are probably required.