Radon Transforms II: Spacial Resampling Seismic Data

Introduction

Recent work by Kurt Marfurt at APR on radon transforms has resulted in two new USP programs - radonf and radonr, respectively the forward and reverse generalized radon transforms. These codes implement the radon transform by modeling the data in the radon domain using parameterized curves in a way which minimizes the least squares error between the transformed data and the original input. This differs radically from most radon or tau-p programs which are deterministic transforms involving fast fourier transforms followed by delay-and-sum operations. There are several advantages to the least squares modeling approach:
1. artifacts minimized - one of the most cited complaints about slant stack or tau-p programs is the obvious contamination with transform noise and their effect on processing in the transform domain.
2. effects of aliasing minimized - failure to meet stringent spatial sampling requirements can have disastrous effects on current slant stack or tau-p programs often rendering the transform domain useless as a processing venue.

The thrust of this note is simple: the use of radon processing to resample irregularly sampled data on to a regularly sampled spatial grid.

Resampling in the Radon Domain

Let's take a look at one example and work it through. Consider the normal moveout corrected cdp gather shown in Figure 1. This was courtesy of Norm Allegar (OBU) who posed the resampling problem to me several months ago.

The problem is highlighted in Figure 2 which shows a portion of the file generated by the scan command

scan -N cdp1 > out0

It is clear that the trace distances on the rightmost column vary considerably and in no way could be considered regular. Looking at the range of trace distances the new near offset was chosen to be 1,200ft and the far offset 17,000ft with a group of 200ft (this was somewhat of an arbitrary decision about the new spread geometry and perhaps Norm might have chosen differently).

The next step is to run the forward radon transform on the input data with the current trace distances:

radonf -N cdp1 -O cdp1.fwd -x max17000 -m min -300 -m max300 -L -n xtaper3 -f 15 -f 28 -f 350 -f 460 -mute

where the maximum possible offset was chosen to be our new far offset. The minimum and maximum moveouts - -300ms and 300ms - were determined

1. by realizing that the moveouts of primaries inherent in nmo-corrected data will be quite small (ideally zero), and
2. by allowing enough range in ray parameters (i.e. spread of curves to be least squares fitted to the data) to capture the real data characteristics present in the input. This is a little bit of an art form but the user should know that constricting the range of ray parameters too much can result in railroad track output data which strictly speaking is not wrong but is merely severely spatially bandpass filtered to include only the very smallest wavenumbers.

The plane wave option of radonr ( -L ) was chosen for no other reason other than I'm familiar with what plane waves look like and so am comfortable interpreting what I see. For instance I know that if I choose and xmax of 17,000' and mmax is 300ms the lowest velocities transformed will be about 56,000ft/s (remember this is phase velocity not material speed) which is an event that is almost but not quite flat. However, there is no theoretical or practical reason why the user could not choose the generalized parabolic or hyperbolic transform. The -mute is set to preserve early mute.

Now that this issue has been totally muddied let's look at the results of the forward radon transform in Figure 3. The horizontal axis of this plot is ray parameter extending from most negative on the left, through zero (events with the fastest phase velocity) in the middle, to most positive on the right. The vertical axis is time in samples. Since this is nmo-corrected data most of the signal should be confined to the small ray parameters.The events at about sample 400 on the input record are circled in the forward radon (plane wave or tau-p) plot. Even though this cdp has been NMO corrected there is enough residual NMO to cause the radon events to streak horizontally across the plot. If they were absolutely flat they would appear as dots in the middle of the plot at their proper times. This is why we chose 300ms instead of a much smaller value.

The next stage is to do the inverse radon transform:

radonr -N cdp1.fwd -d min1200 -d max17000 -d x200 -R | mute -O cdp1.rsmp -R

where we have piped the results of the inverse radon transform into a mute (with the restore mute option turned on) before outputting to disk. The -R on the radonr command line tells the inverse transform to build the spread geometry based on the minimum and maximum trace distances given on the command line and the trace separation.

The results of the inverse radon transform are shown Figure 4 on the following page. Comparing this with the input in Figure 1 one notes first that the ambient noise has been greatly attenuated as a result of the radonf/r filtering process. This is a natural result of the spatial filtering inherent in any process that describes the data using a set of wavenumber band limited functions. This should generally be regarded as a good outcome (and indeed this effect of radon processing has been applied to post-stack data with very impressive results and will be the subject of a future USP note).

The other effect is that the data have been resampled spatially. Reflection events have been clearly reconfigured according to the new equally sampled grid. Figure 5 below shows the new trace distances.

More importantly the reflection events have been stretched and squeezed horizontally to conform to the new regular spatial sampling. Users concerned about AVO should know that the radonf/r process is fairly benign. Those who have confidence in rmmult should have similar confidence in radonf/r.

The downside to the resampling is that in this case with the exception of the trace distances and the depth indexing (and by default the record and trace numbers) the group and shot indexing is now wrong and the data must be reindexed either by re-prepping or (horrors!) running a ufh script. Probably in the case of land data re-prepping is mandatory.

Conclusions

A very powerful method for resampling spatially irregularly sampled is the use of radon transforms. The results are optimum in the sense that the specific radon implementation is a least squares process. One should note, however, when resampling already regularly sampled data it is better to use a tool like spacetrp which uses faster cubic spline interpolation.

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