The first experiments I ran were to simply test the result of the regularized inversion (fitting goals (3)) and the preconditioned inversion (fitting goals (5)). I am concerned with the behavior in early iterations, so I just ran 6 iterations of each. These results are in Figure .
The regularized result is high frequency, but it has barely begun to fill in the areas affected by the null space. This is exactly the behavior we expect at early iterations in a regularized inversion. The preconditioned result has completely filled in the areas affected by the null space, meaning that it has defined solutions at every point (although not the ideal solution), but it is very low frequency. Once again, this is expected and has been seen in earlier works with imaging operators Prucha et al. (2000); Prucha and Biondi (2000, 2002).
The previous example helps to demonstrate two important points made by Claerbout (1999). First, both regularized inversion and preconditioned inversion take a great many iterations to converge. While this is not a problem for a toy problem like the one presented in this paper, it is impossible for a real geophysical problem like imaging in complex areas. The operators used in such a problem are infinitely more complex than those used in this simple interpolation problem, so it is vital that we minimize the number of iterations needed Biondi and Vlad (2001). Secondly, when we limit ourselves to a small number of iterations, we encounter several problems with both regularization and preconditioning. These problems include:
Clearly, in order to obtain a high frequency result with defined solutions at every point in a small number of iterations, we need some combination of the regularized and preconditioned inversions. I chose to run a small number of preconditioned iterations then use that result as an initial model for a small number of regularized iterations. I chose to test two different combinations, one with 3 iterations of preconditioned inversion and 3 iterations of regularized inversion and one with 5 iterations of preconditioned inversion and 1 iteration of regularized inversion. These results are in Figure .
Both of the CIPR results contain higher frequencies than the purely preconditioned result (right panel Figure ) and fill the areas affected by the null space better than the purely regularized result (left panel Figure ). Determining which CIPR result is ``better'' is fairly subjective, but I chose to compare them by looking at their frequency spectrums. This can be seen in Figure . The frequencies shown in this figure are the average over all of the traces.
Figure 4 Comparison of the frequency content of the resulting models seen in Figures and along with the frequency content of the correct model (right panel of Figure ).
Figure shows the frequency spectra of the results in Figure and Figure along with the frequency spectrum of the ``ideal'' model in Figure . As expected, the frequency content of the regularized inversion is close to that of the ideal model and the frequency content of the preconditioned inversion is much lower than the ideal model. It is more interesting to compare the frequency contents of the two different CIPR results. This shows us that the inversion using 3 iterations of preconditioning with 3 iterations of regularization has a frequency content closer to the ideal model than that of the inversion using 5 preconditioned iterations and 1 regularized iteration. This is particularly interesting because it indicates that both preconditioning and regularization are important to get the most improvement.
In this paper, I will consider the CIPR result using 3 iterations of preconditioned inversion and 3 iterations of regularized inversion to be my ``best'' result. Given this result, I felt it would be instructional to see how many iterations of just preconditioned inversion (fitting goals (5)) it would take to get an equivalent frequency content. It took 30 iterations of preconditioned inversion to get the same frequency content as the ``best'' result. The frequency content of the result can be seen in Figure . One again, the frequencies shown here are the average over all of the traces.
Figure 5 Comparison of the frequency content of the results of 3 iterations of preconditioned inversion with 3 iterations of regularized inversion and 30 iterations of just preconditioned inversion.
Figure displays the models resulting from the ``best'' solution and the solution using 30 iterations of preconditioned inversion. The model resulting from 30 iterations has done a better job of filling the areas affected by the null space, as we would expect for an inversion process that used 5 times as many iterations. I have also included a model that has filled the areas affected by the null space equally well as that used only regularized inversion (fitting goals (3)). This result took 50 iterations.