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![]() | Shaping regularization in geophysical estimation problems | ![]() |
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both2,data2
Figure 4. The input data (b) are irregular samples from a sinusoid (a). |
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The input synthetic data are irregular samples from a sinusoidal
signal (Figure 4). The task of data regularization
is to reconstruct the data on a regular grid. The forward operator
in this case is forward interpolation from a regular grid
using linear (two-point) interpolation.
Figure 5 shows some of the first iterations and the final
results of inverse interpolation with Tikhonov's regularization
using equation 1 and with model preconditioning
using equation 3. The regularization operator
in equation 1 is the first finite
difference, and the preconditioning operator
in 3 is the inverse of
or causal
integration. The preconditioned iteration converges faster but its
very first iterations produce unreasonable results. This type of
behavior can be dangerous in real large-scale problems, when only few
iterations are affordable.
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Figure 5. The first iterations and the final result of inverse interpolation with Tikhonov's regularization using equation 1 (left) and with model preconditioning using equation 3 (right). The regularization operator ![]() ![]() |
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sz
Figure 6. The first iterations and the final result of inverse interpolation with shaping regularization using equation 13. Left: the shaping operator ![]() ![]() |
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spec
Figure 7. Spectrum of the estimated model (solid curve) fitted to a shifted Gaussian (dashed curve). The Gaussian band-limited filter defines a shaping operator for recovering a band-limited signal. |
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The left side of Figure 6 shows some of the first
iterations and the final result of inverse interpolation with shaping
regularization, where the shaping operator was chosen to
be Gaussian smoothing with the impulse response width of about 10
samples. The final result is smoother, and the iteration is both
fast-converging and producing reasonable results at the very first
iterations. Thanks to the fact that the smoothing operation is
applied at each iteration, the estimated model is
guaranteed to have the prescribed shape.
Examining the spectrum of the final result (Figure 7), one can immediately notice the peak at the dominant frequency of the initial sinusoid. Fitting a Gaussian shape to the peak defines a data-adaptive shaping operator as a bandpass filter implemented in the frequency domain (dashed curve in Figure 7). Inverse interpolation with the estimated shaping operator recovers the original sinusoid (right side of Figure 6). Analogous ideas in the model preconditioning context were proposed by Liu and Sacchi (2001).
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![]() | Shaping regularization in geophysical estimation problems | ![]() |
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