Noise attenuation suite

Every noise suppression algorithm follows the same three-step recipe: transform the data into a domain where noise and signal separate; mute (or shape) the region containing noise; transform back. What differs across noise types is which transform to use. This section walks the main four.

1. $f$-$k$ filter, the workhorse for linear noise

The two-dimensional Fourier transform of a shot gather maps time and offset to frequency and wavenumber. A linear event in the $(t, x)$ plane maps to a straight line through the origin of the $(f, k)$ plane: $f = v\_{\\mathrm{app}}\\,k$ with $k$ in cycles per metre, equivalently $v\_{\\mathrm{app}} = \\omega/k$ with $k$ in radians per metre. Ground roll, direct arrivals and refractions are all linear events, and each sits on its own line through the origin in $f$-$k$.

Reflections are curved in $(t, x)$, but their apparent velocities are large, so in $f$-$k$ they fill a wedge around $k = 0$. Slow linear noise lies along lines at large $|k|$. That difference in apparent velocity is the separation: mute the slow wedge with a fan that rejects every apparent velocity below a cutoff $v\_c$, inverse-transform, and the reflections survive. In the figure below, set the cutoff to clear the ground roll, then keep raising it and find the first signal it takes.

With the fan’s edge on the ground roll’s own velocity, 500 m/s, 79 % of the ground roll’s energy still gets through: the event has width in $k$ as well as a line, so the cutoff needs margin above the noise. At 900 m/s only 0.7 % gets through (−22 dB) and the shallow reflection keeps 99 % of its energy. Keep raising the cutoff and watch (c): the first signal to go is the far-offset limb of the shallow reflection, where its apparent velocity $v^2 t/x$ falls to about 2100 m/s, and not the near offsets, whose moveout is flat. By 3000 m/s the shallow reflection has lost more than half of its energy. Production flows tune the cutoff per dataset, and they look at what was removed.

2. Radon transform, for hyperbolic noise (multiples)

The Radon transform parameterises the $(t, x)$ plane by an intercept $\\tau$ and a slope $p$ (or a curvature $q$):

• Linear Radon ($\\tau$-$p$): integrate along straight lines $t = \\tau + p\\,x$. Linear noise events map to points; reflection hyperbolae map to ellipses.
• Parabolic Radon: integrate along parabolas $t = \\tau + q\\,x^2$. After NMO, primaries sit near $q = 0$ and multiples, still under-corrected, at $q > 0$, so the two separate cleanly for mild dips.
• Hyperbolic Radon: integrate along true hyperbolae. Most accurate for NMO-style separation.

Radon is the standard tool for water-bottom multiples: after NMO correction with a primary-velocity field, multiples are still under-corrected (their NMO velocity is lower), so they sit at nonzero $q$. Mute them in the Radon domain and inverse-transform; Part 4 covers this at length.

3. Median filter, for random impulsive noise

For spiky, burst-like noise (e.g., pick-up from nearby blasts), a sliding-window median filter is the right tool. Each output sample is the median of the values within a window, not the mean. Median is robust: a single giant spike barely moves the median because it ends up at one end of the sorted window.

Median filters are nonlinear: they pass steps and flat events intact but delete isolated spikes shorter than half the window. Tune the window length to match the temporal scale of the spikes; too long a window smears events.

4. Adaptive subtraction, for model-based noise

Sometimes you can predict the noise. Surface-related multiples can be predicted by autoconvolving the data with itself; water-bottom reverberations by forward-modeling through the known water layer. Subtract the predicted noise from the data. But the prediction is never exact: its amplitude and phase are slightly wrong. Adaptive subtraction lets a short filter adjust the prediction before subtraction:

where the short filter $f$ is chosen by least squares to minimize the energy of $d\_{\\text{out}}$ in windows where noise dominates. This is the SRME post-processing step; also used for inter-bed multiples.

5. What each tool costs you

No denoiser is free. Each removes signal that happens to sit in the noise region:

• $f$-$k$ muting removes any signal with low apparent velocity: the far-offset limbs of shallow reflections, steep dips, diffraction tails, and low-frequency energy near the fan’s apex. Inspect the far-offset traces and the difference panel before and after.
• Radon muting removes primaries left under-corrected by an NMO velocity that was too fast for them: they carry residual moveout like a multiple and fall inside the mute.
• Median filters smear short high-frequency events, specifically the thin-bed tuned responses.
• Adaptive subtraction can leak primary into the subtracted noise if the predictor is too aggressive, or leave residual noise if too conservative.

Always QC before/after on a raw gather, a stack, and a frequency analysis. If you cannot see what was removed, you have no idea whether signal went with it.

6. The $f$-$k$ alias problem

Coarse receiver spacing aliases steep events in the $f$-$k$ domain: they wrap past the spatial Nyquist wavenumber $k\_N = 1/(2\\Delta x)$ and reappear on the other side of $k = 0$. Ground roll of velocity $v$ aliases above $f\_a = v/(2\\Delta x)$, which is 10 Hz for 500 m/s at 25 m spacing. There its wrapped energy has a small $|k|$ and a high apparent velocity, so it passes a fan and looks like signal: in the figure at 25 m, 30 % of the ground roll’s energy lies above $f\_a$ and 9 % gets through a 900 m/s fan that left only 0.7 % at 10 m. Two fixes: tighter receiver spacing, or receiver arrays in the field that attenuate the short wavelengths before they are sampled. The spatial Nyquist condition from Section 0.5 still applies.

Where this goes next

Section 2.6 introduces deconvolution, the other major operation between loading and imaging. Spiking deconvolution attempts to sharpen the wavelet toward a delta; predictive deconvolution targets predictable multiples. Both rest on the Wiener-filter theory we set up in Section 0.6.

References

• Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
• Claerbout, J. F. (1976). Fundamentals of Geophysical Data Processing. McGraw-Hill.
• Oppenheim, A. V., Schafer, R. W. (2009). Discrete-Time Signal Processing (3rd ed.). Prentice Hall.
• Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill.