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

The Punch Line data “signal”

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

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

RTT Schematic

Shot Gather Example 2-D Shot Gather Radial Trace Transform

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.

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

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

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

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

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

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

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

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

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.

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.

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.

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

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

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

Regularization Operator 1 -1 = 1 -1 * 1 -1 A

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

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

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

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

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

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.

Recall Regularization Operator 1 -1 = 1 -1 * 1 -1 A

New Regularization Operator 1 -1 = 1 -.5 -1 .5 * 1 -.5 A

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

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

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

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

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

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

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

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

Acknowledgements SEP sponsors Antoine Guitton