Time Domain

The algorithm in the time-domain is:

1.

Design the filters in the frequency domain. These may be trapezoidal tapered filters or some other suitable bandpass filter. We can consider these filters as making up a matrix whose horizontal coordinate is time $\tau$ (that is, the sample time of application of every filter) and whose vertical component is frequency as shown in the left-hand side of Figure 1.

2.

Get the filter impulse responses in time domain. This essentially means taking an inverse Fourier transform of each column of the left panel in Figure 1.

3.

Form the non-stationary impulse response matrix in time domain. The right panel in Figure 1 shows an example of this matrix for the case of three different filters to be applied in three windows of data. The wavelet is zero phase and the impulse responses are shifted so that the wavelet is centered along the diagonal.

4.

Apply the non-stationary convolution. This is done by matrix multiplication between the matrix in the right panel of Figure 1 and the seismic trace to be filtered.

 

tvf_td1
Figure 1 Filter design in the time-frequency domain. On the left, filter spectra as a function of time. On the right, impulse responses on time

 

tvf_fd1
Figure 2 Frequency domain convolutional matrix. On the left, the filter spectra (amplitude only). The center ``trace'' represents the stationary response, the traces to the right positive frequencies and the traces to the left negative frequencies (only a few ``traces'' are shown). On the right the complete matrix shifted so that the stationary ``trace'' is on the diagonal of the matrix