The motivation for utilizing the radon domains (e.g. plane wave or tau-p) is strong. In this domain predictive decon works optimally and noise trains often separate more usefully than in say f-k. It is the latter we will explore in this note.
We will consider a very simple synthetic data example to illustrate our points. The input data is shown below in Figure 1 (an xsd display). It consists of 100 traces with a group interval of 25ft and a trace length of 250 samples (1,000ms @ 4ms sample interval). There are 5 positively dipping events and 5 correspondingly negatively dipping events. These events have velocities varying from 5,000ft/s to 9,000ft/s. For clarity the events are all linear.
Figures 2 and 3 show respectively the forward f-k amplitude spectrum and the forward radonf tau-p spectrum (i.e. radonf run in plane wave mode with a -L on the command line). The k-axis of the f-k spectrum is the horizontal axis and runs from negative spatial nyquist on the left to positive spatial nyquist on the right. The temporal frequency extends vertically from zero at the bottom of the plot to nyquist at the top. The right hand side or positive wave numbered events correspond to the events in x-t space that dip upward from left to right.
The f-k spectrum was generated by the command line:
fft2da -N dips.f -O junkfft
where dips.f and junkfft are the names of the input x-t data record of Figure 1 and the output f-k data set respectively.
The tau-p spectrum has a horizontal ray parameter axis that extends from most negative on the left to most positive on the right. In fact these data were generated by running the command line
where 2,600 ft is the maximum absolute range in the data and where the minimum and maximum moveouts (of the parameterized curves) are -1,000ms and +1,000ms (in fact these values were chosen to be the actual trace length and the xmax range was chosen to be the maximum trace distance. The vertical axis of Figure 3 is of course time (in samples).
In Figure 3 the grouping of events in the lower left part of the record correspond to the events in Figure 1 which dip upward from left to right. The center traces are the smallest absolute ray parameters, i.e. highest absolute horizontal phase velocities. This means events closest to the vertical center of the record are fastest.
A cursory comparison of Figures 2 and 3 leads to the notion of the tau-p domain as the one exhibiting superior separation of events. In f-k the separability gets even worse for the lower frequencies. The next step is to actually attempt to excise certain events.
We will attempt to remove the 8,000ft/s and 9,000ft/s events that dip upward from left to right in Figure 1. Polygonal mute zones will be constructed in both axis space and in tau-p space.
In f-k space the mute zone is shown in Figure 4 which is a zoomed version of the scale of Figure 2. The mute zone is actually a trapezoid only the two major sides of which are shown. The upper phase velocity cutoff is infinite; the lower phase velocity cutoff is about 7,500ft/s. The picks were made using xsd. The muting operation and the transform back to x-t space is done with the command line
where fft.pik is the name of the pick file xsd wrote, -Min means mute inside the polygon, and where -R means inverse of the previous forward fft2da transform (back to x-t). The results are shown in Figure 5.
It is apparent that the effectiveness is somewhat mixed. The 8,000 and 9,000 events have been attenuated but the 7,000 and even the 6,000ft/s events have been affected. Also the edges of the record have suffered somewhat. In fact we have applied an 8 trace linear taper to the edges of the record in order to reduce edge effects by using the command line
where the first ramp tapers up from trace 1 to trace 8, and where the second ramp tapers down from trace 92 to trace 100 (see man page for ramp explanation). Without the taper fairly strong residual energy associated with the target velocities would appear at the record edges.
For velocity filtering though, this is an excellent result, especially compared to more traditional dip or velocity filtering programs like dipf.
In Figure 6 the mute zone in the tau-p domain is defined and written to an xsd pick file.
The mute and the reverse radon transform are applied using the command line
where taup.pik is the name of the pick file writen by xsd.
The output is shown in Figure 7. The differences between this and the f-k method seem to be
The taper on the edges of the record is a natural side effect of the processing. As the velocity or dip mute zones become more restrictive the advantages of processing with radonf/r become even more apparent.
Velocity or dip filtering in the tau-p domain using the radonf and radonr programs can have distinct advantages of more traditional methods. In particular reduced artifacts and increased effectiveness of the filtering become increasingly important as the velocity or dip fan becomes tighter.
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