f-k Domain: Frequency-Wavenumber Transforms, Dip Filtering, and Coherent Noise Removal in Seismic Processing

The f-k domain is a two-dimensional representation of seismic data in which the time and space axes of a conventional shot gather or stacked section are replaced by temporal frequency (f, measured in Hz) and spatial wavenumber (k, the reciprocal of wavelength, measured in cycles per metre or cycles per kilometre). A processor reaches the f-k domain by applying a two-dimensional Fourier transform: one transform over time converts each trace into its frequency spectrum, and a second transform over the receiver-offset or distance axis converts the spatial pattern into wavenumber. The result is an f-k spectrum, often displayed as an amplitude map with frequency on the vertical axis and wavenumber on the horizontal axis. The power of this transform lies in a simple geometric fact: an event that dips linearly in the time-space (t-x) domain maps to a straight radial line through the origin in the f-k domain, and the slope of that line is the apparent velocity of the event, since apparent velocity equals frequency divided by wavenumber. Because different arrivals travel across a spread at different apparent velocities, they separate cleanly in f-k space even when they overlap badly in the raw record. Flat-lying primary reflections, which have very high apparent velocity, cluster near the frequency axis at low wavenumber, while slow, steeply dipping coherent noise such as ground roll, guided waves, and side-scatter fan out toward high wavenumber. This separation is what makes the domain so useful for dip filtering, also called fan filtering or pie-slice filtering, where the processor designs a reject zone bounded by two apparent-velocity lines and zeroes the amplitudes inside it before applying the inverse transform back to t-x. Unlike a simple bandpass filter that removes whole frequency bands and damages signal, an f-k dip filter discriminates by apparent velocity, so it can strip slow coherent noise while preserving the full frequency content of the primaries underneath it. The same transform underpins f-k migration, f-k interpolation of missing traces, and the diagnosis of spatial aliasing, which appears in the spectrum as energy wrapping around the Nyquist wavenumber set by the receiver-group interval. In Western Canadian Sedimentary Basin land processing, where source-generated ground roll from shallow glacial till and prairie weathering layers routinely buries the reflection signal on raw Vibroseis records, f-k filtering is a standard early step in the noise-attenuation flow.

Designing a Pie-Slice Reject Fan

A dip filter is specified by the apparent-velocity window the processor wants to keep or reject. On a typical WCSB Vibroseis shot gather, primaries arrive with apparent velocities above roughly 2,500 m/s while Rayleigh-wave ground roll crawls in at 300 to 900 m/s. The processor draws two fan lines through the f-k origin at those velocity bounds, zeroes amplitudes inside the slow-velocity wedge, and tapers the fan edges over several wavenumber samples to avoid Gibbs ringing on the inverse transform. A hard, untapered cut leaves diagonal smearing across the section. The reject fan must also stay clear of the aliasing wrap-around, so the design always references the spatial Nyquist set by group interval.

Ground Roll on Prairie Vibroseis Lines

Across the Alberta plains, the low-velocity weathering layer of glacial till and dry near-surface sand generates strong dispersive ground roll that can reach 80 to 100 dB above the deep Montney and Duvernay reflections on a raw 2D line. Because this noise is slow and low frequency, it occupies a compact wedge at high wavenumber and low frequency in the f-k spectrum, well separated from the steep, high-velocity primary trend. An f-k fan filter, applied shot-by-shot before stack, removes the bulk of it without touching the 30 to 80 Hz reflection band that carries Cardium and Viking detail. Processors usually pair the f-k pass with a surface-consistent deconvolution to recover residual amplitude balance.

Related Terms

The f-k domain is reached through the Fourier transform, the mathematical engine that converts time and space samples into frequency and wavenumber. Its most common job is suppressing ground roll, the dispersive surface-wave noise that dominates raw land records. Filter design is constrained by aliasing, because energy beyond the spatial Nyquist wraps into the passband and leaks artefacts. The domain also hosts migration, where Stolt f-k migration repositions dipping reflectors to their true subsurface location using the same velocity-as-slope geometry.

Real-World WCSB Scenario: Cleaning a Kaybob Duvernay 2D Line

A junior operator shooting a 22 km regional 2D Vibroseis line over the Kaybob Duvernay fairway northwest of Fox Creek, Alberta, found that raw shot gathers were dominated by ground roll reaching apparent velocities near 600 m/s, completely masking the deep 3,000 m Duvernay reflection. The processing contractor in Calgary applied an f-k fan filter rejecting everything below 1,800 m/s apparent velocity, with a tapered edge over four wavenumber samples to prevent ringing. Reprocessing the 110 shot records cost roughly CAD 18,000 in additional turnaround.

The filtered stack revealed a continuous Duvernay event and a clear Swan Hills carbonate marker beneath it that had been invisible in the brute stack. The improved section let the operator high-grade a horizontal landing zone and avoid a CAD 9 million dry-hole risk on a mispicked target, a return that dwarfed the modest reprocessing fee.