F-K Plot: Frequency-Wavenumber Transform, Dip-Based Noise Rejection, and WCSB Seismic Processing

An f-k plot is a graphical display of seismic data transformed from the time-space domain into the frequency-wavenumber domain, where f stands for temporal frequency and k stands for spatial wavenumber. The transform, computed with a two-dimensional Fourier transform applied across both the time axis and the spatial trace axis of a seismic gather, decomposes the data into components characterized by how fast they oscillate in time, the frequency, and how rapidly they vary across space, the wavenumber. The power of the f-k plot is that a coherent linear event in the time-space domain, such as a dipping reflection, a head wave, or a train of ground roll, maps to a distinct radial trend in the f-k plane whose slope equals the apparent velocity of that event. Steeply dipping, slow-moving noise such as ground roll plots at high wavenumber and low apparent velocity, while flat or gently dipping reflections concentrate near the frequency axis at low wavenumber. This separation by dip and velocity is what makes the f-k plot a workhorse for filtering. By designing a fan-shaped or polygonal pass or reject zone in the f-k domain, a processor can attenuate energy travelling at particular apparent velocities, then inverse-transform back to time-space, suppressing coherent noise while preserving signal. The classic application is removing ground roll and other surface-wave noise from land data and suppressing coherent linear noise from marine streamers. The f-k plot is also the natural place to diagnose spatial aliasing, the wraparound artifact that appears when the trace spacing is too coarse for the wavenumbers present. Aliased energy folds back across the wavenumber axis and can corrupt both interpretation and filter performance, so processors widen reject fans or apply linear moveout corrections before f-k filtering to manage it. In the Western Canadian Sedimentary Basin, where land 3D surveys over the Montney and Deep Basin are routinely degraded by strong ground roll across muskeg, farmland, and foothills topography, f-k filtering remains a standard early step in the processing flow, often applied before deconvolution and velocity analysis. The f-k plot is more than a filter design tool; it is a diagnostic window that reveals the velocity makeup of a gather, flags acquisition problems such as inadequate group spacing, and guides decisions about whether dip filtering, frequency filtering, or more advanced methods such as tau-p or curvelet transforms are appropriate. Modern processing systems display the f-k spectrum interactively so the geophysicist can sketch a reject polygon over the noise trend and immediately preview the cleaned gather, blending mathematical rigor with hands-on interpretation.

Designing an F-K Reject Fan for Ground Roll

On a typical WCSB land shot gather, ground roll appears as high-amplitude, low-velocity, dispersive energy travelling at perhaps 300 to 800 metres per second, while primary reflections move at apparent velocities of several thousand metres per second. In the f-k plot the ground roll forms a steep, high-wavenumber wedge well separated from the near-vertical reflection trend. The processor sketches a reject fan bounding that wedge, zeroes the energy inside it, and inverse-transforms. The cleaned gather shows reflections that were previously buried under the surface-wave cone, all without the phase distortion that an aggressive bandpass filter alone would introduce.

Managing Spatial Aliasing in the Transform

Spatial aliasing is the chief limitation of f-k filtering. When receiver group spacing is large, say 30 to 40 metres on a wide foothills line, steeply dipping or high-frequency events exceed the spatial Nyquist wavenumber and wrap around, contaminating the very region a reject fan must protect. A common remedy is to apply a linear moveout correction that rotates the unwanted energy to a gentler dip before transforming, perform the f-k filter, then reverse the correction. Recognizing aliasing on the f-k plot before filtering prevents a processor from accidentally removing signal or leaving wrapped noise in the final image.

Related Terms

The f-k plot is one of several domains used to separate signal from noise, closely related to the deconvolution step that sharpens the wavelet after coherent noise is removed. It targets the same surface-wave problem addressed by careful survey design and ground roll attenuation, and the apparent velocities it reveals connect directly to the seismic velocity model built later in processing. Each tool fits into the broader sequence that turns raw field records into an interpretable seismic volume.

Real-World WCSB Scenario: Foothills 3D Over a Montney Fairway

A contractor processing a foothills 3D survey over a Montney fairway southwest of Grande Prairie faced severe ground roll from variable near-surface conditions across muskeg and till. Initial brute stacks showed reflections smeared beneath the surface-wave cone. The processing geophysicist examined f-k plots gather by gather, confirmed the ground roll wedge was well separated in wavenumber, and applied a tailored f-k reject fan with a linear-moveout step to control aliasing on the wider-spaced foothills lines.

The f-k filtering recovered usable reflection energy across the deeper Montney and underlying carbonate section, lifting the signal-to-noise ratio enough that velocity analysis and migration produced an interpretable structural image. The added processing effort, a modest fraction of the multi-million-CAD survey cost, salvaged data that would otherwise have required costly reacquisition.