Frequency Domain

To develop an algorithm in the frequency domain, we basically have to take a Fourier transform in the horizontal direction of the data in the left panel of Figure 1. The algorithm is therefore:

1.

Design the filters in the frequency domain, as before.

2.

Take a Fourier transform in the horizontal direction (that is, a Fourier transform for each row of the matrix on the left panel of Figure 1) and form the corresponding frequency-domain convolutional matrix. Figure 2 shows the resultant matrix (amplitude spectrum only). This matrix is called the frequency connection matrix. On the left is a horizontally-shifted version of the matrix. The center ``trace'' corresponds to the stationary response and the ``traces'' away from it represent the departure from stationarity. Only a few ``traces'' are shown. On the right panel we have the complete dataset shifted so that the ``stationary trace'' is along the diagonal, which means that the off-diagonal energy represents again the departure from stationarity.

3.

Take the Fourier transform of the input trace.

4.

Multiply the frequency connection matrix (right panel of Figure 2) with the Fourier transform of the input trace to get the filtered trace in the frequency domain.

5.

Take the inverse Fourier transform of the filtered trace to get it in the time domain.