rmmult and AVO

Introduction

Usually when processing only for structural information, applying rmmult to multiple ridden data has been pretty much pro forma. This is a brief note to point out some problems some users have been encountering doing AVO analysis after running rmmult. It was prompted in part by a discussion at the recent ILIS/USP meeting lead by Hans Sugianto who very nicely illustrated the problem. It was also prompted by the growing appreciation for the subtle yet profound effects processing can have on AVO and ultimately on interpretation itself. The golden rule of AVO might be stated as "do nothing to the data unless you absolutely have to". Alas, nature and the wave propagation spirits too often intrude and we are forced to do things to the data.

Using a very simple synthetic we will show what this particular problem is and how to get around it.

The Problem

As most users know, rmmult is run on data that has been approximately primary NMO corrected. In simple terms this results in reduced moveout on all the events and reduced curvature and enables rmmult to fit its family of parabolas (or hyperbolas) in radon space more accurately. The problem comes in when the primary NMO corrected events are almost exactly flat. Consider the input data of Figure 1. Two key features of the data are (1)the primary at 900ms is exactly flat and (2)the AVO of the primary is exactly flat. The moveout differentials of the multiples as measured at the far offset of 5050 vary from 300ms to 650ms. Some quick parameter testing yields the following rmmult run:
rmmult -N input -O input_rm -fl5 -fh55 -mute -H -mmin -500 -mmax 1500 -rmin 200 -rmax 1500

where as in the case of real data, we try to account for reverse moveout of the primaries with the mmin parameter and expand mmax to include all possible coherent events in the data down to the lowest velocities (they are in terms of moveout differentials in ms). rmin and rmax in ms define the reject zone. The results are shown in Figure 2. As usual in the shallow portion of the record along the mute line, rmmult leaves some artifacts but near the primary they are low amplitude. Using avoir and picking along the primary we extract the amplitude profile of Figure 3. The primary should have a flat AVO but that is not what we see. Overall we see the general decay with offset that has been reported by other users. Whether this has structure on it other than a linear trend will depend on the particular rmmult parameters.

The Explanation

The best way to show how this happens is to look at the problem in the domain in which rmmult does its thing: the radon domain. Running a rmmult is the equivalent of running radonf | polymute | radonr, where the polymute mutes all the samples in each trace between the ray parameters defined by rmin and rmax (the reject zone). Figure 4 shows the results of running the command line
radonf -N input -O input_radf -f27 -f350 -f455 -H -mute -mmin -500 -mmax 1500 -xmax 5050

The flat event transforms into a zone of energy that is mostly localized in a small zone in ray parameter and time. What makes rmmult work is if the localization is compact enough to be separate from the other transformed events. In other words is there a mute line which can be drawn vertically separating signal from noise (rmin)?

As we can see in Figure 4 there is considerable "streaking" of the transformed primary and first multiple energy horizontally such that there is no mute line that will not pick up some primary and multiple energy together. Further, since flat events in X-T generate infinite horizontal responses in radon space (parabolic or hyperbolic transforms), there can be no mmin or mmax large enough to adequately capture all the transformed data - some will inevitably be lost. Then when rmmult internally does the inverse radon transform and reconstructs the data in X-T space, the primary will have lost some energy between rmin and rmax but will also have some first multiple energy mixed in between mmin and rmin. Because the primary energy has been lost on the higher ray parameter side of the transform (toward mmax), the effect in X-T is to attenuate energy more and more as distance increases. Conversely, the energy mixed into the primary from the first multiple is from the smaller ray parameters (toward mmin) which correspond to the near offsets. This effect then tends to boost the transformed primary energy on the near offsets upon transform back to X-T. Lastly, being unable to capture the infinite horizontal transformed signatures of flat events in X-T means that overall energy will be lost from these events at all offsets upon inverse transform back to X-T. This is the explanation for the rmmult induced amplitude effects some users have seen.

The Solution

Do we have to live with this? No, we sure don't. The trick to avoiding this is (1) for un-NMO corrected CDPs, choose a velocity that over corrects the primaries and still leaves the multiple undercorrected; or (2) for flattened CDPs (say from a contractor), apply a second NMO correction that forces overcorrection on the primaries leaving the multiples undercorrected. In our example in Figure 1 we built a flat file that consisted of the three lines shown in Table 1

Table 1: Contents of File "junk"

0	8000	1
3000	8000	1
-1	8000	1

Where the first column is the 2-way time, the second is the rms velocity, and the third is the record number to which this function belongs, and where -1 flags the function complete. You could also use a TDFN format. We then ran

velin -N input -O vel -vjunk -flat -S

which gives us a velocity trace. We apply the moveout

bdnmo -N input -O input_nmo -vvel

with the results shown in Figure 5. The primary now has about 250 ms overcorrection on the far offsets. Running exactly the same radonf on the data of Figure 5 gives the result shown in Figure 6.

The only significant difference between Figures 4 and 6 is that in the latter there is much less horizontal "streaking" of the transformed energy. This means the primary is now much better isolated from the multiple events in this domain. Our mute line will no longer intercept significant primary energy. Figure 7 shows the result of running rmmult on the data of Figure 6 then piping into a bdnmo to remove the overcorrection velocity.

Clearly this result is cleaner than the one shown in Figure 2 which is gratifying indeed and indicates there was bad separation of the primary and multiples on the data with exactly flat primaries. In our case the event was exactly flat and this may have caused some numerical problems in rmmult.

Figure 8 shows the AVO analysis on the primary of Figure 7. While it is true that the curve still is not a flat response, it is much better than the result of Figure 3. What makes the Figure 3 result disturbing is that rmmult seems to have induced a systematic AVO response. Figure 8 at least appears approximately flat with no significant systematic component to it. Further, both Figure 3 and Figure 8 are plotted on the same scale and the latter shows a higher overall amplitude. Again this is because since there were no perfectly flat events on the input data the internal rmmult radon transform was able to contain all the energy from the input between the mmin and mmax derived ray parameters so that all the energy could be recovered on transform back to X-T.

Conclusions

In situations where multiples interfere with primaries, it is still possible to clean up the data with rmmult and do a good AVO analysis (there are other techniques which in special circumstances might work better, but we are not going to cover them here). By doing a little extra work the pathologies of flat events can be avoided.

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