The 2-dimensional t-x technique predicts linear events with a 2-dimensional time-space domain filter. This filter is calculated to minimize the energy inside a design window using a conjugate-gradient routine [Claerbout 1992a]. The filters used in the examples in this article have the form:
As an example of a calculated filter, a flat event in a noiseless window produces a filter with all zeros except on the row containing the 1, where each of the a01,a02,a03, and a04 coefficients in the filter above becomes -0.25. This filter minimizes the energy of the output by exactly predicting the flat event. Anything left after filtering is considered noise and removed from the input. Although making any one of the a01,a02,a03, or a04 coefficients -1 produces a filter that exactly predicts an event, I set up the problem so that the coefficients tend to be equal for the best noise attenuation.
The choice of the filter size depends on the size of the design window, the maximum dip in the data, the number of dips within the window, and the desired strength of the prediction effect. For t-x prediction, the choice of the filter's length in space is similar to the choice of the filter length in f-x prediction. Enough traces need to be included to create a good estimate of the signal. I tend to use 5 to 7 traces on 2-dimensional data, while for 3-dimensional data, this number can be somewhat smaller in each direction. The filter length in time for the t-x filter does not seem to be an especially sensitive parameter unless very steep events exist. In the examples shown here, I used 3 to 5 samples. Events that have a moveout greater than the filter length in time are easily predicted, provided the bandwidth of the data is not too high. To keep the application of the calculated filter symmetrical, the filter is applied in both forward and reverse directions in space, with the results averaged.
The process of applying t-x prediction, as well as the f-x prediction discussed next, is applied to windows small enough for events of interest to appear linear. After filtering, these windows are merged to produce the output image.