Noise characterization on a shot gather

Part 1, Acquisition & the data we process

Learning objectives

Distinguish the six most common events on a shot gather by their slope, curvature, and frequency content
Recognize which noises are coherent (structured) and which are random, and why that matters for attenuation strategy
Predict where each noise attenuator (f-k filter, Radon, bandpass, SRME) fits in the processing flow
Identify noise on a real shot gather by eye before choosing which tool to apply

A raw shot gather is not pure signal with a sprinkle of noise. It is a superposition of many events, only some of which you want. The job of noise characterization is to recognize each event from its signature and choose the right attenuator. Attempt that without looking first and you will attenuate signal.

1. The classic events

A typical land or marine gather contains, at minimum:

Primary reflections. Hyperbolic in (offset, time), $t(x) = \\sqrt{t\_0^2 + x^2/V\_{\\mathrm{rms}}^2}$ , with the curvature set by the RMS velocity of the section above the reflector. This is the signal.
Direct arrival. Linear, travelling straight from the source to the receivers through the near surface, at the velocity of the top layer. On land the air wave, sound in air at about 340 m/s, is a slower and higher-frequency line of its own.
Ground roll (land). Low-frequency (5 to 15 Hz), very slow, dispersive surface waves that fan out in a cone from the source. Its marine cousin, swell noise, is low-frequency burst noise on a few neighbouring streamer traces, caused by sea-surface motion.
Refracted head waves. Linear and fast, from a refractor below the near surface, and present only beyond the critical distance, where they touch the refractor’s reflection. Essential data for statics, not for imaging.
Multiples. Reflections that have bounced more than once: hyperbolic like primaries, but at any given zero-offset time they have a lower stacking velocity, so NMO with primary velocities leaves them curving down. A free-surface multiple also flips polarity at the surface.
Ambient random noise. Uncorrelated, roughly Gaussian noise that stacking suppresses: $N$ traces raise the signal-to-noise ratio by $\\sqrt{N}$ .

In Figure 1.4 you build a synthetic land or marine gather event by event, from one layered earth, and see it in both the $t$ - $x$ and the $f$ - $k$ domains. Put the probe on an event and the figure measures its fingerprint from the traces: how fast it crosses the spread, whether it curves, and its dominant frequency. Keep the names switched off, name each event from those numbers alone, then switch the names on to check yourself.

2. What makes each signature unique

The figure opens with the probe on the first arrival at 1500 m. The line it fits crosses the spread at 3000 m/s and meets zero offset at 0.16 s, not at the source, so this is not the direct wave: it is the head wave from the 3000 m/s refractor, and its apparent velocity is the refractor velocity. Put the probe on the steep cone at 700 m and 1.25 s and the numbers change completely: 760 m/s at 7 Hz, the fingerprint of ground roll.

All the coherent events have a signature in the (time, offset) plane that maps to a distinct region of the (frequency, wavenumber) domain. That is the trick $f$ - $k$ filtering uses: it separates linear events by their slope, their apparent velocity. Separating by curvature is the job of the Radon transform (section 4.3).

Apparent velocity of a linear event on a shot gather is $v_a = \Delta x / \Delta t$ (the reciprocal of the event’s slope $\Delta t / \Delta x$ ). It is the speed at which the event moves across the receiver array, not necessarily a subsurface velocity.

Ground roll: tiny apparent velocity (300 to 1000 m/s), low frequency (5 to 15 Hz). It dips steeply in $t$ - $x$ and plots in a slow fan near the wavenumber axis in $f$ - $k$ , often aliased. Filter: $f$ - $k$ fan reject, or adaptive surface-wave attenuation if amplitude is extreme.
Direct arrival: apparent velocity equal to the near-surface velocity, broadband, on a line through the source. Filter: top mute before processing; its frequency content is fine, but its amplitude at near offset ruins everything.
Refracted head wave: high apparent velocity, equal to the refractor velocity on a flat refractor (typically 1.8 to 6 km/s), on a line that meets zero offset later than the source time. Filter: top mute, but keep an unmuted copy for refraction statics.
Multiples: hyperbolic like primaries, but their stacking velocity is lower, because all of their travel time is spent in the slower shallow section. Filter: Radon de-multiple, SRME, or adaptive subtraction (all in Part 4).
Random noise: uncorrelated, spread evenly across $f$ - $k$ . Filter: stacking, median filters, or random-noise attenuators.

3. Coherent vs random is the key distinction

Every attenuator is designed around one of two assumptions:

Noise is random: attenuate by averaging (stacking, median filtering). The signal-to-noise ratio grows as $\\sqrt{N}$ (Section 0.8).
Noise is coherent, but it has a different signature from signal in some domain ( $f$ - $k$ , $\\tau$ - $p$ , offset-dependence, angle). Attenuate by transforming into that domain and muting.

If you treat coherent noise as random, your stack hides it but leaves it lurking, producing coherent amplitude artifacts on the stacked section that look like events. If you treat random noise as coherent, you cannot find a domain where it is well-separated from signal; you simply waste effort.

4. A noise checklist for any new dataset

Display the raw shot gather with a gain (AGC or $t^2$ ) and no filter, then again with a low-cut filter to see what the ground roll was hiding.
Note each coherent event and estimate its apparent velocity.
Estimate the dominant frequency of each event (or do an $f$ - $k$ analysis).
For every coherent event, name the processing step that will attenuate it.
Estimate the amplitude of the random ambient noise; its ratio to the primary amplitude tells you how much fold you need.

5. Why this matters for processing

Every denoising step in every seismic flow takes one of two forms:

\\text{signal}\_{\\text{out}} = \\text{signal}\_{\\text{in}} + (\\text{noise}\_{\\text{in}} - \\text{noise}\_{\\text{estimated}})

\\text{signal}\_{\\text{out}} = F(\\text{signal}\_{\\text{in}} + \\text{noise}\_{\\text{in}})

where the second form is a transform-domain filter that mutes the estimated noise and leaves the signal. Either way, the algorithm starts with a noise model, and the model comes from knowing the data, which is exactly what this section trains.

The one sentence to remember

Every noise has a signature: slope, curvature, frequency content. Learn to recognize them on a gather before you reach for the attenuator, otherwise you are guessing, and denoising is an expensive way to guess.

Where this goes next

Section 1.5 closes Part 1 with survey design sanity: how source and receiver spacing, spread geometry, fold, offset range, and azimuth distribution decide which processing problems you can even attempt and which you cannot.

References

Yilmaz, Ö. (2001). Seismic Data Analysis (2 vols.). SEG.
Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge UP.
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.