    Shaping regularization in geophysical estimation problems  Next: Velocity estimation Up: Examples Previous: Examples

## 1-D inverse data interpolation

I start with a simple 1-D example: a benchmark data regularization test used previously by Fomel and Claerbout (2003).  both2,data2
Figure 4.
The input data (b) are irregular samples from a sinusoid (a).     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. if
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 is the first finite difference. The preconditioning operator is causal integration. The number of iterations is indicated in the plot labels.    sz
Figure 6.
The first iterations and the final result of inverse interpolation with shaping regularization using equation 13. Left: the shaping operator is lowpass filtering with a Gaussian smoother. Right: the shaping operator is bandpass filtering with a shifted Gaussian. Shaping by bandpass filtering recovers the sinusoidal shape of the estimated model. The number of iterations is indicated in the plot labels.    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.   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).    Shaping regularization in geophysical estimation problems  Next: Velocity estimation Up: Examples Previous: Examples

2013-07-26