Noise attenuation suite

Part 2 — Pre-Processing Foundations

Learning objectives

Match each coherent noise type to the transform where it separates from signal
Apply an f-k filter to suppress ground roll while preserving primaries
Describe when Radon, median, and adaptive filters are the right tool
Recognize the cost — loss of low-frequency signal — and when it is acceptable

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-offset to frequency-wavenumber. A linear event in the (t, x) plane maps to a STRAIGHT LINE in (f, k) passing through the origin. The slope of that line is its apparent velocity v = ω/k. Ground roll, direct arrivals, refractions — all are linear events and all sit on their own line in f-k.

Reflections are curved in (t, x), so in f-k they spread over a BROAD region. This is the key separation: linear noise narrow and predictable; signal broad and diffuse. Mute the noise region, inverse-transform, done.

The widget above shows the gather (signal plus ground roll), the amplitude spectrum in f-k with the mute fan shaded in red, and the filtered gather. Slide the velocity cutoff: at 500 m/s the mute just covers the ground roll; at 2000 m/s you start cutting into near-offset signal too. Real production flows tune the cutoff per dataset.

2. Radon transform — for hyperbolic noise (multiples)

The Radon transform parameterises the (t, x) plane by (p, τ) pairs:

Linear Radon (τ–p): integrate along straight lines t = τ + p x. Noise events map to points; signal hyperbolae to curves.
Parabolic Radon: integrate along parabolas t = τ + p x². For mild dip, reflections at p=0 (primaries) vs p>0 (multiples with residual NMO) separate cleanly.
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 p. Mute them in Radon, inverse-transform. Covered at length in Part 4.

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 and asymmetric in what they pass. 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 — amplitude and phase are slightly wrong. Adaptive subtraction lets a short filter adjust the prediction before subtraction:

d_{\text{out}} = d - f \ast n_{\text{predicted}}

where the filter f is picked by least-squares to minimize the energy of d_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 kills low-k high-f signal at near offsets — typically the shallowest reflections. Inspect the near-offset traces before and after.
Radon muting kills reflections with incorrect picked NMO velocity — i.e., the steep-dip primaries whose velocity was underestimated.
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 around and reappear on the “other side” of the k-Nyquist. If your ground-roll fan hits the edge of k_Ny, it can alias back and look like signal. Two fixes: tighter receiver spacing, or anti-alias filter in x before 2D FFT. The spatial Nyquist condition from §0.5 still applies.

**The one sentence to remember**

Every denoiser maps your data to a domain where noise and signal separate — f-k for linear noise, Radon for multiples, median for spikes, adaptive for model-predicted noise — and every one steals a little signal in the process.

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 §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.