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Next: Conclusion Up: Guitton and Claerbout: Galilee Previous: A new fitting goal

Approximating the model covariance with a PEF

It is well known that the ideal regularization operator (squared) is the model covariance matrix Tarantola (1987). Estimating this matrix is not straightforward. We often approximate the model covariance matrix with roughening operators like the Laplacian or the derivative. Claerbout and Fomel (2002) advocate that in principle, an ``ideal'' regularization operator is a PEF estimated from an a-priori model. In this section, we test the idea of using a PEF instead of the Helix derivative for the regularization operator.

Starting from equation (9), we simply replace the Helix derivative ${\bf H}$ with a PEF ${\bf A_m}$ estimated from a model of the lake bathymetry. Since we do not know a-priori the exact $\bold{h}$,a PEF is computed from the depth map ${\bf h}$ estimated in equation (9) with the Helix derivative. We estimate a 3 by 4 filter. This procedure can be interpreted as a bootstrapping of the model covariance matrix. We bootstrapped the PEF estimation six times before converging to a satisfying result, meaning that we use the last depth map to estimate a PEF and reiterate with the new filter. Including the PEF ${\bf A_m}$ into equation (9), we have the new fitting goals  
 \bold 0 &\approx& \bold r_d &=& \bold...
 ...prox& \epsilon_2 \bold r_q &=& \epsilon_2 \bold q.
 \end{array}\end{displaymath} (12)
Again, we set $\epsilon_1=\epsilon_2=0$ and estimate ${\bf p}$ and ${\bf q}$ with conjugate gradients.

To our great surprise, after our first attempt, the bootstrapping technique did not converge, meaning that we were unable to estimate a correct PEF for the inversion. The final results were extremely low frequency. We were able to stabilize the inversion by adding some random noise to the model $\bold{h}$ before the PEF estimation. However, the noise level needed was extremely high, around one meter. The filter we estimate from this noisy image is then far from the PEF we are looking for, to our disappointment.

In Figure [*]b, we show the estimated ${\bf p}$ with the PEF as a preconditioning operator after six iterations of bootstrapping with noise added to the model. To increase the contrast inside the lake, we apply a weighting function on Figure [*]b that boosts up the low values of ${\bf p}$ in the middle of the sea and deemphasizes the sea shores. Figure [*]c displays the final result. We see that the structure inside the lake is more visible. We can almost follow this pattern up to the north side of the lake (we tried a similar weighting function on the result with the Helix derivative but with no improvement in the middle of the lake.)

Although encouraging, this result has major shortcomings. First, we have lost resolution on the sea shores. The result with the Helix derivative in Figure [*]a is much better in this area. Second, it is surprising that the PEF is not able to eliminate the low frequency trend visible throughout the lake in Figures [*]b and [*]c. Finally, we still do not fully understand why we could not find a PEF without adding noise to the model that would allow us to converge to a satisfying depth map. Maybe we should estimate non-stationary PEFs as opposed to one PEF for the whole model space. Maybe we should try to estimate the PEF and the model at the same time, as done in the missing data problem Claerbout (1992). Maybe we should include a starting guess in our inversion to insure better convergence because bootstrapping is essentially a non-linear process. All these suggestions are possible niches of improvement with the Sea of Galilee dataset. Our feeling is that more work needs to be done for the PEF estimation.

Figure 10
(a) Estimated ${\bf p}$ with the Helix derivative.(b) Estimated ${\bf p}$ with the PEF. (c) Scaled version of (b) to increase the contrast inside the lake.
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Next: Conclusion Up: Guitton and Claerbout: Galilee Previous: A new fitting goal
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