Radon demultiple

SRME is thorough and expensive. Radon demultiple is cheap and, for the right kind of multiple, just as effective. The key idea: after NMO with the primaries’ velocity a primary is flat, while a multiple at the same $t\_0$ is not, because it travelled slower and the velocity that is right for the primaries is too high for it. So primaries and multiples differ in their residual curvature, and the Radon domain is a space that measures residual curvature.

1. The parabolic Radon transform

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

For each $(\\tau, p)$, sum the data along the parabola $t = \\tau + p\\,x^{2}$. An event that lies on that parabola stacks coherently and makes a bright focus at $(\\tau, p)$; an event on a different parabola stacks incoherently and makes none.

A correctly NMO-corrected primary has $t = t\_0$ regardless of $x$: it is flat, which is a parabola with $p = 0$, and it focuses at $(t\_0, 0)$. A multiple corrected with the primaries’ velocity has $t \\approx t\_0 + p\\,x^{2}$ for some $p > 0$ and focuses at $(t\_0, p)$ with $p > 0$. The approximation is real. NMO applies the primaries’ velocity at the corrected time $t^{\\prime}$, not at $t\_0$, so the multiple lands at the $t^{\\prime}$ that solves $t^{\\prime 2} + x^2/V\_{\\mathrm{NMO}}(t^{\\prime})^2 = t\_0^2 + x^2/V\_{\\mathrm{m}}^2$. Its residual moveout $t^{\\prime} - t\_0$ is close to a hyperbola, which a parabola matches well only while the residual is mild (Hampson, 1986).

Summing along parabolas is the adjoint $L^{\\mathsf T}$ of the operator $L$ that models a gather from a panel, $d = L\\,m$. It is not an inverse: transforming its panel back gives a blurred copy of the gather. Hampson’s step was to solve for the panel instead, $\\min\_m \\lVert L\\,m - d \\rVert^2$, whose foci are sharp and whose forward transform rebuilds the data.

2. The figure

The gather below holds one primary at 0.80 s and one multiple at 1.10 s, 10 % slower than the primaries, after NMO with the primaries’ velocity. It opens with the cut $p\_{\\mathrm{cut}}$ at 0.040 s/km², beyond the multiple’s focus. Move the cut until the multiple leaves the kept gather (c) while the primary stays, then change the velocity gap, the cable and the transform, and find where the separation fails.

In (b) the primary focuses at $p = 0$ and the multiple at $p = 0.024$ s/km². The multiple’s residual moveout in (a) is 228 ms at 3000 m, and (e) shows that the parabola of its focus follows the picked moveout only approximately. At the opening cut of 0.040 the multiple slips through: 98 % of it is left in (c). With the cut at 0.012, between the foci, 12 % of the multiple is left and 93 % of the primary is kept; the rest of the primary is the part of its focus that smears across the cut. Push the cut down to 0 and the mute takes the primary’s own focus: only 42 % of it survives.

Switch to the adjoint with the cut at 0.012 and 21 % of the multiple is left while only 77 % of the primary is kept; even with nothing muted, the adjoint rebuilds only 89 % of the primary. The removed gather (d) is a prediction of the multiple, which is what Section 4.4 subtracts adaptively.

3. Other Radon variants

• Linear ($\\tau$-$p$). Sum along straight lines $t = \\tau + p\\,x$. Suited to linear events: ground roll, direct arrivals, refractions. Not as good for reflections, which are hyperbolic.
• Parabolic. The workhorse. A good approximation to the mild residual moveout left after NMO (Hampson, 1986), which is what separates multiples from primaries.
• Hyperbolic. Sum along true hyperbolae $t = \\sqrt{\\tau^{2} + x^{2}/V^{2}}$. More accurate for events that have not been NMO-corrected; more expensive.

4. What Radon gets right

• Primary and multiple velocity difference. If the multiple’s velocity is 10 to 20 % lower than the primary’s, its $p$ is clearly different from the primary’s. Mute the multiple’s $p$, keep the primary’s.
• Cost. The adjoint Radon costs $O(N\_\\tau N\_p N\_x)$ and the least-squares version a few such passes (or a small $N\_p \\times N\_p$ solve per frequency), still a fraction of SRME.
• Locality. Works on a single CMP. No need for dense neighborhood sampling.

5. What Radon gets wrong

• Small velocity difference. If primary and multiple have nearly the same velocity (shallow water and its water-bottom multiple), they map to nearly the same $p$ and the mute takes both or neither. In the figure a multiple only 3 % slower moves 42 ms at 2475 m, about one period of the 25 Hz wavelet, and a cut midway between the foci, at 0.003 s/km², still leaves 26 % of it while keeping only 67 % of the primary.
• Short offsets. The term $p\\,x^{2}$ is small, so the transform has poor resolution in $p$ and primaries and multiples smear together. On a 1000 m cable the multiple moves only 19 ms and 31 % of it is left.
• Smearing and leakage. The adjoint smears each focus along $p$ and leaks signal into nearby $p$; least squares focuses it, and a high-resolution (sparse) Radon (Sacchi and Ulrych, 1995) sharpens it further at extra cost.
• Aliasing. Coarse trace spacing aliases the transform. With a 15 % gap on a 4000 m cable at 100 m spacing, the far multiple steps 37 ms between neighbouring traces, more than half its period, and 16 % of it is left against 8 % at 25 m. Interpolate traces first.

6. How Radon fits in the multiple-attenuation flow

1. Apply NMO with the best primary velocity.
2. Transform the gather to the Radon panel, by least squares.
3. Design a mute in $(\\tau, p)$, typically a curve that widens with $\\tau$.
4. Transform the unmuted part back: the primaries.
5. Optional: transform the muted part back as a predicted multiple and feed adaptive subtraction (Section 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. It is also used without SRME, on projects where full SRME is too expensive.

Where this goes next

Section 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.
• Hampson, D. (1986). Inverse velocity stacking for multiple elimination. J. Can. Soc. Expl. Geophys., 22, 44-55.
• Sacchi, M. D., Ulrych, T. J. (1995). High-resolution velocity gathers and offset space reconstruction. Geophysics, 60, 1169-1177.