TWO-DIMENSIONAL PREDICTION ERROR FILTERING

If we attempt to calculate an operator that will annihilate a hyperbola, this operator should eliminate any predictable part of this hyperbola. The result of filtering the hyperbola with this operator should be our desired output, a whitened hyperbola.

The operator we calculate was expected to have a form like:

$$\begin{array} {ccccccc} \cdot &\cdot &\cdot &1 &a &a &a \\ ... ... &a \\ a &a &a &a &a &a &a \\ a &a &a &a &a &a &a \end{array}$$ (1)

A dot indicates a zero and an a indicates the individual filter coefficients. This prediction-error filtering is done with a spatial prediction-error filter similar to that of pe2 from Claerbout1992, page 193, that calls the conjugate-gradient routine in Claerbout1992, page 137.

The first attempt with the filter above produced hyperbolas that were asymmetric, a result that was unexpected. To make the operator symmetric, we zeroed out the last half of the first row.

$$\begin{array} {ccccccc} \cdot &\cdot &\cdot &1 &\cdot &\cdo... ... &a \\ a &a &a &a &a &a &a \\ a &a &a &a &a &a &a \end{array}$$ (2)

By calculating a two-dimensional filter that removes the predictable part of the hyperbola, the filter will be whitened as desired. The evanescent zone is likely to create a problem if we cannot suppress the whitening by providing a small amount of white light.

Since the evanescent zone produces problems shown later in this paper, an operator of the following form could be used.

$$\begin{array} {ccccccc} \cdot &\cdot &\cdot &1 &\cdot &\cdo... ...dot \\ a &a &a &a &a &a &a \\ a &a &a &a &a &a &a \end{array}$$ (3)

The challenge is zeroing the proper coefficients to suppress the noise created in the evanescent zone. This idea needs more development before it is implemented.