1. Methods for isolating coherent noise in the Radon domain
Shauna Oppert and R. James Brown
University of Calgary

2. Coherent energy involved in high-velocity layers
The Radon transform
Some new Radon variants
Outline: I will briefly go over the specific type of coherent noise I am trying to remove, then give a quick review of the Radon transform (although I assume this audience is already familiar with the standard process). The bulk of this presentation will discuss some new variations I am adapting in the Radon transform and the results of these methods applied to synthetic data. I will end with drawing some conclusions based on these preliminary tests and proposing potential areas of interest in this work to push further in.
Testing on synthetic data

3. Mode-converted reflectionsP
PSPP
PPSP
PSSP
PPPP
Surface
High-velocity
salt
In 1992, Guy Purnell showed how efficient coupling between P- and S-waves at salt interfaces can introduce mode-converted reflections on typical P-wave surveys. The moveout of these reflections are similar to that of Primary energy, allowing it to be stacked into data with typical processing, affecting the overall appearance of the data and the interpretation of it. This energy can be treated as coherent noise, similar to multiples, and through my work I show that the Radon transform may be the best method for removal.

4. Transform at slowness p and zero-offset time 
Radon Transform
One-sided CMP gather
Radon domain x p • • •  t
The Radon transform utilizes the differences in moveout of coherent reflections by summing over curves in the x-t domain and outputting the energy that falls along the curves in the Radon domain. An efficient approximation of the reflections results in a high energy point in the Radon Domain. Mathematically it is simply written as this expression, where p is slowness, tau is the intercept time, and x is offset. The results I will show are from a least-squares approximation algorithm, although it is important to note that a higher resolution transform first described by Sacchi and Porsani in 1995 is available (and currently in the works!). Slide 4: you may want to mention that this depicts a hyperbolic Radon transform
[or, in a cartoon manner, generally, any quasi-hyperbolic, like the various
parabolic transforms, that try to map the t-x reflection curves to points] and
NOT the conventional (-p transform or slant stack that maps lines to points.
Until fairly recently this linear Radon transform was the only kind generally
known. See e.g. Sheriff's dictionary entry "tau-p mapping" (p 294).
Transform at slowness p and zero-offset time 
CMP data

5. t2-stretched parabolic transform
Input =
Hyperbolic multiple transform
The simplest summation curves I will be discussing involve parabolic summation curves as shown here. In 1989, Oz Yilmaz introduced the t-square-stretched parabolic transform using these summation curves. This transform makes hyperbolic reflections in the x-t domain approximately parabolic in the x-t-squared domain, allowing for more efficient summation. In this case, the input is t-squared, and the summation are along these parabolic curves in the x-t-squared domain. In 1994, Foster and Mosher developed a method to remove specific multiple energy by designing a summation curve using an input depth variable, z. The input is the typical x-t data, however, a focusing depth is chosen, optimizing the focusing of the multiple energy in the Radon domain.

6. Fourth-order non-hyperbolic NMO equation:
Castle (1994)
Dix hyperbola
Shifted hyperbola
Fourth-order non-hyperbolic NMO equation:
The last Radon transform I will discuss involves a fourth-order NMO equation. To explain this, it is helpful to look at the work done by Castle in 1994, where he discussed a variety of NMO equations and concluded that the shifted hyperbola equation was the most robust equation for NMO applications. These two plots compare a Dix equation hyperbola and the Shifted Hyperbola to that of a ray-traced true response (bold lines). As depth becomes larger than offset, both approximations stray from the true response, however, the shifted-hyperbola equation is much more accurate than the Dix equation. In the present work, I use a variation of the sixth order shifted hyperbola equation, where the last term is dropped and it becomes a fourth order equation, as shown on this slide, where mu2 is the slowness, and mu4 and t0 must be “tuned” for specific events (in the same way that depth is tuned for in the foster and mosher equation).
Slide 6: I've suggested a slight change in your notes in describing shifted
hyperbola cf. Dix.

7. The CMP gather model Primary Mode-converted reflection Base-of-salt
This is the synthetic CMP gather used for testing of these transforms. It is a representation of a HVL with a BOS primary reflection, BOS multiple reflection, and a PSSP mode-converted reflection. The mode-converted energy involves a 90-degree phase shift at the critical angle and generally increases amplitude with offset in contrast to the primary and multiple energies. Typical processing can involve stacking in the precritical energy from the mode-converted reflection, creating artifacts in the data.
Base-of-salt
multiple

8. Parabolic Radon transform
Amplitude scale (x10-3)
Far-offset
smear
 (s)
Near-offset
smear
The first Radon transform that was applied to the gather is shown here with the respective amplitude scale on the side. This parabolic RT does an adequate job of focussing each event (Point out each event). The smearing is partly due to the lower resolution least squares algorithm used and some smearing due to the near (horizontal smear) and far offsets (diagonal smear). A typical mute would look something like this—as you can see, with adequate separation in the Radon domain, other events are not generally removed by this technique. FAR OFFSET TAPER CAN REDUCE DIAGONAL SMEAR.
p (x10-8 s/m)

9. Data weighting prior to a parabolic Radon transform
Amplitude scale (x104)
 (s)
One of the variations I tried was data weighting prior to a radon transform. This example is the result of using the same parabolic transform on offset-squared-weighted data. In this case, the multiple and primary energy was more diminished, while the mode-converted energy was enhanced. This effect is directly related to the nature of the mode-converted energy and how it increases amplitude with offset. This technique might be very helpful especially on more complex CMP gathers, where mode-converted energy is difficult to separate from other types of energy. The results indicate that data weighting might provide a method for discrimination of different types of events in the Radon domain, although further work in this area is necessary to determine the effectiveness.
p (x10-8 s/m)

10. Hyperbolic multiple transform
Amplitude scale (x10-3)
 (s)
The second type of transform I looked at was the Hyperbolic Multiple, from Foster & Mosher’s work, with a focusing depth of the BOS. The “smiles” evident on the radon domain result indicate an inaccurate approximation of events, specifically of the far offsets. However, one interesting feature of this transform is the focusing of separate near and far-offset energy on the mode-converted reflection. Now remember that it is usually the near offsets that cause problems in typical processing. Hence, this suggests that this method may be used to remove only the near-offset energy of mode-converted events, and be less damaging to the data.
p (x10-4 s/m)

11. t2-stretched parabolic transform
Amplitude scale
 (s)
This slide shows the result of applying Yilmaz’s method of t-squared stretching the data prior to a parabolic transform. There is very good focusing in this result, and in fact it may be the most robust method I am discussing (right now) because it does not rely on a focussing depth or otherwise tuning factor.
p (x10-4 s/m)

12. Fourth-order non-hyperbolic transform
Amplitude scale
 (s)
In comparison, the Fourth-order transform tuned for the converted reflection also had excellent focusing capabilities and may even include less smearing of events, although it is difficult to determine with this low-resolution algorithm.
p (x10-4 s/m)
Tuned for mode-converted reflection

13. Fourth-order non-hyperbolic transform
Amplitude scale
 (s)
In contrast, the fourth order tuned for primary or multiple reflection caused drastic smearing of other events-and may not be applicable to real data. One interesting point is that the multiple energy is focused in this case and may be removed using this method.
p (x10-4 s/m)
Tuned for primary reflection

14. Fourth-order non-hyperbolic transform
Near offsets
Far offsets
 (s)
 (s)
The last test involved separating CMP’s based on the critical angle of converted reflection and then performing a fourth-order Radon transform. The results are superior due to less smearing and better focusing, and it may provide a method involving less damage to the original data set. However, doing both transforms would require twice the number of Radon transforms, increasing the expense of the algorithm.
p (x10-4 s/m)
p (x10-4 s/m)
Tuned for mode-converted reflection

15. Some things to think about…
Stretching and unstretching the data (t2) can
cause aliasing
Tuned Radon transforms become expensive
when applied to real data
The t-squared stretched approach appeared to be the most robust algorithm, however by stretching and unstretching the data, it is possible to introduce aliasing (or further aliasing) into the data set. So, this method might not be as advantageous as first thought. Also, by using the fourth-order or higher equations or the Hyperbolic multiple equation, focusing parameters must be chosen and used for specific reflections. This can be very expensive to apply to entire data sets. An alternative that I continue to research involves removing this dependency on tuning by introducing some kind of fudge factor that relates known parameters to the unknown variables. Successful implementation of this type of method could drastically improve computational time and results.
Introducing a ‘fudge’ factor for 4 and t0 may
make the fourth-order algorithm more robust

16. Conclusions Data weighting prior to a Radon transform
may allow for discrimination of specific events
Aid in designing substantially more accurate
mutes for unwanted energy
Focusing of all events was maximized when
using a t2-stretched parabolic or a fourth-
order transform tuned for converted waves
I hope that throughout this talk I have impressed upon you that work in the field of Radon transforms is still relatively untouched. Faster algorithms can improve the cost and efficiency of this process, and combinations of techniques can be used to better isolate energy in the Radon domain. The results from offset weighting of the data indicate that this type of technique may be helpful for discrimination of different types of energy on CMP data and may aid in designing better mutes for noise. The results indicate the t-squared stretched parabolic or fourth order transform tuned for converted waves produce the best results and should be used with long offsets and HVL’s.
Slide 16: I've suggested minor changes to notes.

17. Conclusions The removal of near-offset mode-converted
energy is possible using the hyperbolic
multiple or fourth-order non-hyperbolic
transform on near-offset data.
Mode-converted reflections can be diminished
with near-offset mutes and stacking
Furthermore, the results showed methods for removing near-offset energy separately, a technique that may prove useful in dealing with mode-converted energy with typical processing.
Slide 17: I've suggested "separately" instead of "only" so it sounds more like
increased versatility than some sort of limitation.

18. Future Work Improving a data-weighting function to account
for AVO affects in all types of reflections
Develop a robust method for implementation
of a fourth-order or larger NMO-equation
transform
In this work, I’ve only touched upon some valuable Radon algorithms, however there are many other variations that have yet to be included (such as applying these transforms after NMO or PSDM). Additionally, it would be beneficial to look into data weighting within the Radon transform to account for different types of AVO affects. I hope to continue the work with the higher order NMO equations and develop and effective method for handling long offsets and even anisotropy. Of course first, I will need to code the high-resolution transform for better comparison of each transform variation.
Test all transforms with the high-resolution
Radon algorithm

19. Acknowledgements Sponsors of Dr. Mauricio Sacchi Marco Perez
Xinxiang Li

20. Fourth-order non-hyperbolic transform
Amplitude scale
 (s)
REMOVE THIS SLIDE???
p (x10-4 s/m)
Tuned for multiple reflection