2-D spectral signal estimation theory

To further enhance the characterization of the signal, a two-dimensional annihilation filter is used. This filter is represented in this section by

$$\sv 0 \approx \st F_s \sv s,$$ (131)

where $\sv s'$ is the scaled signal and $\st F_s$ is the matrix form of the filter. As before, I design a one-dimensional filter on each trace to annihilate the noise. This appears as

$$\sv 0 \approx \st F_n(x) \sv n.$$ (132)

$\st F_n(x)$ is a function of x because a separate filter is computed for each trace. At present, this filter is calculated on the data above the start times.

Since the 2-D filter is expected to have much different characteristics than the 1-D noise filter, the regressions are scaled to each other as follows:  

$$\sv 0 \approx \lambda_1 \st F_s \frac{1}{\sigma_s} \sv s',$$ (133)

 

$$\sv 0 \approx \lambda_2 \st F_n(x) \frac{1}{\sigma_n(x)} \sv n.$$ (134)

It is interesting to note that the regression in equation () is just a weighted version of t-x predictionAbma and Claerbout (1993).

Expanded to predict the signal, these minimizations become the single system of regressions that follows:  

$$\left( \begin{array} {c} \sv 0 \\ \lambda_2 \st F_n \frac{1... ...2 \st F_n(x) \frac{1}{\sigma_n(x)} \\ \end{array}\right) \sv s.$$ (135)

If the inversion of system () is attempted with the initial value of $\sv s$ being zero, many iterations of the solver are needed to produce a reasonable result. In Abma1995, I showed that initializing the value of $\sv s$ to the result obtained from prediction-error filtering reduced the cost of the inversion and improved the results. Although the estimated signal from prediction-error filtering is not a perfect fit, it appears close enough to reduce the number of iterations by an order of magnitude, making this process a practical production technique.