Radon demultiple

Part 4 — Multiple Attenuation

Learning objectives

Define the parabolic Radon transform and read a (τ, p) panel
Map primaries to p≈0 and multiples to p>0 after NMO
Apply a Radon mute and inverse-transform to kill multiples
Identify when Radon works (strong velocity differential, enough offset) and when it fails

SRME is thorough and expensive. Radon is cheap, fast, and — for the right kind of multiple — just as effective. The key idea: after NMO, primaries at a given t₀ are flat, multiples at the same t₀ are NOT (because they were over-corrected with a velocity that was right for primaries, too high for multiples). So primaries and multiples occupy different positions in a space that measures RESIDUAL curvature. That space is the Radon domain.

1. The parabolic Radon transform

For a CMP gather d(t, x), the parabolic Radon transform is

R(\tau, p) = \sum_{x} d(\tau + p\,x^{2}, x)

— for each (τ, p), sum the data along the parabola t = τ + p·x². An event that actually lies on that parabola stacks coherently and produces a bright spot at (τ, p). An event that lies on a different parabola stacks incoherently and produces no bright spot.

A correctly NMO-corrected primary has t = t₀ regardless of x — flat, which is a parabola with p = 0. It appears at (t₀, 0) in the Radon panel. A multiple that was under-corrected has t = t₀ + p·x² for some p > 0. It appears at (t₀, p>0).

Left: the post-NMO CMP gather, with a flat primary at 0.8 s and a curved-up multiple at 1.1 s. Middle: the Radon panel. Two bright spots — one at (0.8 s, 0) for the primary, one at (1.1 s, 0.17) for the multiple. The red dashed line is the user’s p-cutoff; everything to the right of it (red) gets muted in the Radon domain. Right: the inverse Radon using only the unmuted bins — multiple is gone.

Start the cutoff high (0.25), and the multiple slips through because the mute does not catch it. Drop the cutoff to 0.10 and the multiple is fully muted. Drop it further to 0.05 and you risk clipping near-zero-p real-data noise.

3. Other Radon variants

Linear (τ–p). Sum along straight lines t = τ + p·x. Suited to linear events — ground roll, direct arrivals, refractions. Not as good for reflections which are hyperbolic.
Parabolic. The workhorse. Exact for mild residual move-out after NMO, which is exactly what you want for multiples vs primaries.
Hyperbolic. Sum along true hyperbolae t = √(τ² + x²/V²). More accurate for events that have not been NMO-corrected; more expensive.

4. What Radon gets right

Primary/multiple velocity differential. If the multiple has been under-corrected by ±10–20 % in velocity, its p is clearly different from the primary’s. Mute the wrong p, keep the right one.
Cost. Radon is O(N_τ · N_p · N_x) — a fraction of SRME.
Locality. Works on a single CMP. No need for dense neighborhood sampling.

5. What Radon gets wrong

Small velocity differential. If primary and multiple have nearly the same velocity (shallow water + nearby seafloor multiple), they map to nearly the same p and the mute takes both or neither.
Short offsets. The p·x² term is small — the Radon transform has poor resolution in p — primaries and multiples smear together.
Smearing / leakage. The discrete Radon transform has artifacts that can leak signal into nearby p bins. A high-resolution Radon (sparse-regularized inversion) fixes this at extra cost.
Aliased Radon. Coarse trace spacing aliases the Radon transform. Interpolate traces first if needed.

6. How Radon fits in the multiple-attenuation flow

Apply NMO with best primary velocity.
Forward-Radon transform the gather.
Design a mute function in (τ, p) — typically a velocity-dependent curve that widens with τ.
Inverse-Radon transform the non-muted bins.
Optional: compute the MUTED part as a “predicted multiple” and feed adaptive subtraction (§4.4) for a final clean-up.

Radon is often used AFTER SRME — SRME removes the bulk of surface-related multiples, and Radon cleans up the residual. Or without SRME, on cheaper projects where full SRME is budget-prohibitive.

**The one sentence to remember**

Radon sums the gather along parabolas, sorting flat primaries to p=0 and residually-curved multiples to p>0; mute p≥cutoff and inverse-transform to kill multiples — cheap, effective, and the go-to for residual post-SRME multiples.

Where this goes next

§4.4 makes both SRME and Radon better. Adaptive subtraction takes a predicted multiple train (from either method) and fits a short time-varying filter that minimizes the residual energy under the assumption that the prediction is approximately right. Half the multiple-attenuation literature since 2000 is about getting the adaptive filter right.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Verschuur, D. J., Berkhout, A. J., Wapenaar, C. P. A. (1992). Adaptive surface-related multiple elimination. Geophysics, 57, 1166.
Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.