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![]() | Multidimensional recursive filter preconditioning in geophysical estimation problems | ![]() |
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data
Figure 1. The input data (right) are irregularly spaced samples of a sinusoid (left). |
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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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im1
Figure 2. Estimation of a continuous function by the model-space regularization. The difference operator ![]() ![]() |
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fm1
Figure 3. Estimation of a continuous function by the data-space regularization. The preconditioning operator ![]() |
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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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schwab1
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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early1
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. |
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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 ![]() |
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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 ![]() |
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![]() | Multidimensional recursive filter preconditioning in geophysical estimation problems | ![]() |
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