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The algorithm for tomographic inversion of focusing operators with data dependent parameterization is applied to a synthetic 2D case. The model contains a salt dome, fault structures beneath the dome, a complex turbidite velocity structure with low velocities, and lateral and vertical velocity gradients within the layers (Fig.7a).
This is the "real" velocity model of the subsurface, in which (in this synthetic case) the focusing operators are generated (Fig.7b). The positions of the focal points are represented as black dots in Fig.7a. An initial velocity model and an initial estimate of the focal point positions are needed to start the updating procedure (Fig.7c). From these initial locations focusing operators are modeled in the initial velocity model (Fig.7d). Note that for display purposes, a wavelet has been assigned to each operator, and the resulting wave fields have been shown. In the inversion algorithm, only the traveltimes are used. By the traveltime difference between the modeled operators and the real operators, the velocity model and focal point positions are updated. In each iteration, the parameterization will be adjusted and the velocity model and focal point positions will be updated.
The final result after 6 iterations is shown in Figure 8 (displayed in terms of squared-slowness). Note that at locations with high data information (i.e. originally low variances), the density of the grid-points is high. The corresponding velocity model of the final result is also shown in Figure 8. The positions of the focal points are similar to the real focal point positions in Figure 7a. The velocity model also resembles the real model, altough it is a smooth solution of the real case. However, the upper two layers (water layer, and layer below) are not well resolved, which is indicated by the higher velocity, and the downward shifted focus points. This artifact is caused by the severe ray distortion originated in the salt dome.
When this velocity model is used for migration (Fig.8), most reflectors are clearly visible. Nevertheless, the inaccuracy of the upper two layers (water layer, and layer below) is also visible in this migration. In particular above the salt dome, where the reflectors are shifted downward.

**synth
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Figure 7 Velocity models of a synthetic case. Black dots represent focal points. (a) Real velocity model and focal points
(b) Real focusing operators (modeled in (a), using a wavelet for display);
(c) Initial model (squared slowness) and initial focal points;
(d) Focusing operators modeled in (c)

**res
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Figure 8 *Top*: Final (sixth) update of model (squared slowness) and focal points; open dots: real focus point locations, black dots: updated focus point location; *Middle*: Final update of model (velocity) and focal points. *Bottom*: Migration with velocity model displayed in figure above.

** Next:** Approximate covariance calculation
** Up:** Data dependent parameterization
** Previous:** A posteriori covariance
Stanford Exploration Project

9/18/2001