Multidimensional recursive filter preconditioning in geophysical estimation problems

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# One-dimensional synthetic examples

data
Figure 1.
The input data (right) are irregularly spaced samples of a sinusoid (left).

In the first example, the input data were randomly subsampled (with decreasing density) from a sinusoid (Figure 1). The forward operator in this case is linear interpolation. In other words, we seek a regularly sampled model on 200 grid points that could predict the data with a forward linear interpolation. Sparse irregular distribution of the input data makes the regularization necessary. We applied convolution with the simple difference filter as the operator that forces model continuity (the first-order spline). An appropriate preconditioner in this case is the inverse of , which amounts to recursive causal integration (Claerbout, 1999). Figures 2 and 3 show the results of inverse interpolation after exhaustive 300 iterations of the conjugate-direction method. The results from the model-space and data-space regularization look similar except for the boundary conditions outside the data range. As a result of using the causal integration for preconditioning, the rightmost part of the model in the data-space case stays at a constant level instead of decreasing to zero. If we specifically wanted a zero-value boundary condition, we could easily implement it by adding a zero-value data point at the boundary.

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Figure 2.
Estimation of a continuous function by the model-space regularization. The difference operator is the derivative operator (convolution with ).

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Figure 3.
Estimation of a continuous function by the data-space regularization. The preconditioning operator is causal integration.

The model preconditioning approach provides a much faster rate of convergence. We measured the rate of convergence using the power of the model residual, defined as the least-squares distance from the current model to the final solution. Figure 4 shows that the preconditioning (data regularization) method converged to the final solution in about 6 times fewer iterations than the model regularization. Since the cost of each iteration for both methods is roughly equal, the computational economy is evident. Figure 5 shows the final solution, and the estimates from model- and data-space regularization after only 5 iterations of conjugate directions. The data-space estimate appears much closer to the final solution.

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Figure 4.
Convergence of the iterative optimization, measured in terms of the model residual. The d'' points stand for data-space regularization; the m'' points for model-space regularization.

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Figure 5.
The top figure is the exact solution found in 250 iterations. The middle is with data-space regularization after 5 iterations. The bottom is with model-space regularization after 5 iterations.

Changing the preconditioning operator changes the regularization result. Figure 6 shows the result of data-space regularization after a triangle smoother is applied as the model preconditioner. Triangle smoother is a filter with the -transform (Claerbout, 1992). We chose the filter length .

fm6
Figure 6.
Estimation of a smooth function by the data-space regularization. The preconditioning operator is a triangle smoother.

If, instead of looking for a smooth interpolation, we want to limit the number of frequency components, then the optimal choice for the model-space regularization operator is a prediction-error filter (PEF). To obtain a mono-frequency output, we can use a three-point PEF, which has the -transform representation . In this case, the corresponding preconditioner can be the three-point recursive filter . To test this idea, we estimated the PEF from the output of inverse linear interpolation (Figure 3), and ran the data-space regularized estimation again, substituting the recursive filter in place of the causal integration. We repeated this two-step procedure three times to get a better estimate for the PEF. The result, shown in Figure 7, exhibits the desired mono-frequency output. One can accommodate more frequency components in the model using a longer filter.

pm1
Figure 7.
Estimation of a mono-frequency function by the data-space regularization. The preconditioning operator is a recursive filter (the inverse of PEF).

Subsections
 Multidimensional recursive filter preconditioning in geophysical estimation problems

Next: Regularization after binning: missing Up: Multidimensional recursive filter preconditioning Previous: Data-space regularization (model preconditioning)

2013-03-03