1. A pseudo-unitary implementation of the radial trace transform
Morgan P. Brown* and Jon F. Claerbout
Stanford Exploration Project
Stanford University

2. The Punch Line
data
“signal”

3. Outline Radial trace transform (RTT) defined. Two RTT Implementations
Overcoming Spatial Aliasing
Signal/Noise Separation on Real Data

4. Radial Trace Transform (RTT)
Simple resampling of (t,x) data onto radial coordinates (t,v).

5. RTT Schematic

6. Shot Gather Example
2-D Shot Gather
Radial Trace Transform

7. Motivation for Ground Roll Suppression
Radial events map to vertical events.
Nearly flat events stay nearly flat.
Better separation of ground roll and reflections in frequency.

8. Outline Radial trace transform (RTT) defined. Two RTT Implementations
Overcoming Spatial Aliasing
Signal/Noise Separation on Real Data

9. Resampling = Interpolation
Main Issue: How to interpolate from one grid to another?

10. Implementation #1 - “x-interpolation”
Used by Henley (1999 SEG).
Loop over (t,v) bins.
Interpolate (average) between adjacent (t,x) bins.

11. “x-interpolation” - Pros and Cons
Intuitive.
RT panel well sampled.
Smoothing of high kx events at normal receiver spacing.
Data aliasing limits interpolation options.

12. Implementation #2 - “v-interpolation”
Loop over (t,x) bins.
Interpolate between adjacent (t,v) bins.

13. “v-interpolation” - Pros and Cons
Can increase sampling density to mitigate smoothing errors.
Operator effectively unitary.
RT panel has “holes”.
Holes: bad for filtering.

14. RT Panels Compared
Shot gather
v-interpolation RTT
x-interpolation RTT

15. Interpolation Errors Compared
Shot gather
v-interpolation RTT
x-interpolation RTT

16. What have we learned? x-interpolation RTT v-interpolation RTT Unitary.
RT panel not suitable for highpass filtering.
Non-unitary.
RT panel suitable for highpass filtering.

17. The Ideal Implementation
x-interpolation RTT
v-interpolation RTT
Unitary.
RT panel not suitable for highpass filtering.
Non-unitary.
RT panel suitable for highpass filtering.

18. Where are we going? Use v-interpolation approach.
Interpolate RT panel holes to stabilize bandpass filter. Prefer filling with horizontal and vertical events.
Justification: Holes in nullspace of RTT adjoint.

19. Least Squares RT Panel “hole interpolation”
Hold “known” points constant.
Regularize undetermined model points w/assumed model covariance.

20. Least Squares RT Panel “hole interpolation”
m = unknown model.
d = original RT panel.

21. Least Squares RT Panel “hole interpolation”
K = known data mask.
A = regularization operator.

22. Regularization Operator1 -1 = 1 -1 * 1 -1 A

23. Results of Missing RT Data Interpolation
v-interpolation
v-interpolation + infill
x-interpolation

24. Outline Radial trace transform (RTT) defined. Two RTT Implementations
Overcoming Spatial Aliasing
Signal/Noise Separation on Real Data

25. RT Panels Before Decimation
Raw Data
v-interpolation+infill
x-interpolation

26. RT Panels After Decimation
Decimated Data
v-interpolation+infill
x-interpolation

27. Aliasing = Disappointment
Problem: Aliased ground roll causes poor vertical coherency in RT panels.
Solution: Modify RT panel hole interpolation to emphasize vertical coherency.

28. Human eye as interpolator
Coherency is easy to see, but hard to get if ground roll aliased.
Holes are anisotropic.
Solution #1: Redesign regularization filter.
Solution #2: Precondition to encourage simple models at early iterations.

29. Recall Regularization Operator1 -1 = 1 -1 * 1 -1 A

30. New Regularization Operator1 -1 = 1
-.5 -1 .5 * 1
-.5 A

31. Preconditioning: Am=p
Before
After
Large-scale vertical stripes appear in early iterations.

32. Preconditioning: Am=p
A is minimum phase.
Stable 1-D decon.
Large-scale vertical stripes appear in early iterations.

33. Results of Improved Missing Data Interpolation
No infill
Old infill
New infill

34. Outline Radial trace transform (RTT) defined. Two RTT Implementations
Overcoming Spatial Aliasing
Signal/Noise Separation on Real Data

35. Signal/Noise Separation Processing Flowx t
Data v t
RTT t v
6.5 Hz
Highpass x t
RTT
Adjoint
Signal x t
Noise
Subtract
from data

36. Results of Signal/Noise Separation
v-interpolation –
estimated signal
x-interpolation –
estimated signal
Raw Data

37. Results of Signal/Noise Separation
v-interpolation –
estimated noise
x-interpolation –
estimated noise
Raw Data

38. Conclusions A new implementation of the RTT. Unitary.
Good suppression of spatially aliased ground roll.
With preconditioning, cost comparable to x-interpolation (5:1).

39. Acknowledgements
SEP sponsors
Antoine Guitton