Prediction of seismic signals in the t-x domain

The two-dimensional t-x technique predicts linear events with a two-dimensional time-space domain filter. This filter is calculated by minimizing the energy inside a window with a conjugate-gradient routine (Claerbout [1992a]). The filters used in the examples in this article have this form:  

$$\begin{array} {ccccc} 0 & a_{-2,1} & a_{-2,2} & a_{-2,3} & ... ...1,4} \\ 0 & a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \end{array}.$$ (1)

The vertical axis is the time axis, and the horizontal axis is the space axis. The output position is under the 1 coefficient on the left side of the filter. This filter has no free coefficients in the column corresponding to the output trace, which forces whatever predictions are made to be lateral.

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, we 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 window, the maximum dip in the data, the number of dips within the window, and the desired strength of the prediction effect. 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.