Multidimensional recursive filter preconditioning in geophysical estimation problems |
One of the factors affecting the convergence of iterative data regularization is clustering of data points in the output bins. When least-squares optimization assigns equal weight to each data point, it may apply inadequate effort to fit a cluster of data points with similar values in a particular output bin. To avoid this problem, we can replace the regularized optimization with a less accurate but more efficient two-step approach: data binning followed by missing data interpolation.
Missing data interpolation is a particular case of data regularization, where the input data are already given on a regular grid, and we need to reconstruct only the missing values in empty bins. The basic principle of missing data interpolation is formulated as follows (Claerbout, 1992):
A method for restoring missing data is to ensure that the restored data, after specified filtering, has minimum energy.Mathematically, this principle can be expressed by the simple equation
We illustrate the missing data problem with a simple 1-D synthetic data test taken from Claerbout (1999). Figure 8 shows the interpolation results of the unpreconditioned technique with two different filters . For comparison with the preconditioned scheme, we changed the boundary convolution conditions from internal to truncated transient convolution. As in the previous example, the system was solved with a conjugate-gradient iterative optimization.
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Figure 8. Unpreconditioned interpolation with two different regularization filters. Left plot: the top shows the input data; the middle, the result of interpolation; the bottom, the filter impulse response. The right plot shows the convergence process for the first four iterations. |
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As depicted on the right side of the figures, the interpolation process starts with a ``complicated'' model and slowly extends it until the final result is achieved.
Preconditioned interpolation behaves differently (Figure 9). At the early iterations, the model is simple. As the iteration proceeds, new details are added into the model. After a surprisingly small number of iterations, the output closely resembles the final output. The final output of interpolation with preconditioning by recursive deconvolution is exactly the same as that of the original method.
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Figure 9. Interpolation with preconditioning. Left plot: the top shows the input data; the middle, the result of interpolation; the bottom, the filter impulse response. The right plot shows the convergence process for the first four iterations. |
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The next section extends the idea of preconditioning by inverse recursive filtering to multiple dimensions.
Multidimensional recursive filter preconditioning in geophysical estimation problems |