Plane Wave

A plane wave is a wave whose wavefronts are flat (planar) surfaces, meaning that at any instant in time, all points on a given wavefront are at the same phase of the wave cycle. This is the simplest possible wave geometry and contrasts with a spherical wave, which radiates outward from a point source in an expanding sphere. In reality, all seismic sources generate spherical waves. However, at distances far from the source compared to the wavelength, or within a limited aperture around the receiver array, the curvature of the spherical wavefront is so slight that the wave behaves as a plane wave to an excellent approximation. This plane-wave assumption underlies most seismic processing theory, including Snell's law for refraction and reflection, migration algorithms, and full-waveform inversion.

Key Takeaways

A plane wave traveling in a uniform medium has a wavefront that is a flat plane perpendicular to the direction of propagation. All points on the wavefront move together with the same amplitude and phase. The wave's energy is spread uniformly across the wavefront and does not decrease with distance (unlike a spherical wave, whose energy per unit area decreases as 1/r²).
In seismic processing, the far-field approximation treats seismic waves as plane waves when the receiver-to-source distance is large compared to the wavelength. For a 40 Hz wave at 3,000 m/s velocity (wavelength 75 metres), a source-receiver distance of 500 metres is more than 6 wavelengths away, and the plane-wave approximation is reasonable.
Snell's law for seismic refraction and reflection (n₁ sin θ₁ = n₂ sin θ₂, where n is the seismic slowness) is derived under the plane-wave assumption. It correctly predicts the angle at which a seismic wave bends when it crosses a velocity interface, and this calculation is the basis for all seismic ray-tracing software used in velocity model building and depth imaging.
Plane-wave decomposition is a seismic processing technique that decomposes the recorded wavefield into a set of plane waves traveling at different angles (the plane-wave components). This representation is used in plane-wave common-image gathers (CIGs) for velocity analysis and migration, and in plane-wave modeling for full-waveform inversion.
Electromagnetic waves (including those used in ground-penetrating radar and marine CSEM) are also treated as plane waves at sufficient distances from the source. The plane-wave assumption is what makes Snell's law applicable to electromagnetic refraction at interfaces, which is the basis for understanding how EM signals penetrate the seafloor in CSEM surveys.

What Is a Plane Wave?

Drop a stone into a still pond. Circular ripples expand outward from the point where the stone hit. Each ripple is a wavefront: a circle of water at the same height (phase) of the wave cycle. As the ripple expands, the circle grows larger and the curvature of the wavefront decreases. Far from the stone, any short arc of the ripple looks nearly straight. In the limit, at infinite distance from the stone, the ripple is so large that it looks completely flat from any local vantage point. That flat wavefront is a plane wave.

In three dimensions, a plane wave has a flat plane as its wavefront rather than a flat line. A plane wave generated by a seismic source would look like a flat sheet of pressure rippling through the earth in one direction. No real seismic source generates a perfect plane wave (they all create spherical waves), but the plane-wave model is an approximation that works extremely well for most seismic processing applications because the receivers are far from the source compared to the scale of the structures being imaged.

The mathematical simplicity of plane waves is what makes them so useful in theory. A plane wave in a uniform medium has a simple sinusoidal form that can be described by amplitude, frequency, wavelength, and propagation direction. More complex wave patterns (curved wavefronts, reflections from complex interfaces) can be decomposed into sums of plane waves traveling in different directions, which is the key idea behind the Fourier transform applied to spatial data and behind plane-wave decomposition in seismic processing.

Fast Facts

The concept of plane-wave imaging (also called controlled-beam illumination or plane-wave common-image gathers) was commercialized in seismic processing in the early 2000s. It involves recording data from a seismic source that fires at multiple times with controlled time delays between shots, so that the combined wavefield from the delayed shots sums constructively in a specific direction, creating an effective plane wave propagating at a controlled angle. Processing these plane-wave data into subsurface images is computationally cheaper than migrating each shot independently and gives similar image quality. Ion Geophysical (now ION), CGG, and PGS have all developed commercial implementations used on marine and land surveys globally.

Plane Waves in Seismic Refraction

Seismic refraction surveys use the fact that waves traveling along an interface between two rock layers of different velocity emerge at the surface ahead of the direct wave. The geometry of these refracted arrivals is described by Snell's law applied to plane waves at each interface. The refractor (a fast layer below a slower one) can be detected and its depth calculated from the slope of the travel-time curve of the refracted arrivals on a shot record.

Near-surface refraction is used routinely in Alberta and in the Foothills to map the depth to the base of a slow, weathered surface layer. This near-surface information is converted into static corrections that shift each seismic trace in time to compensate for the variable delay introduced by the weathered layer. Without these corrections, subsurface reflections appear smeared and the structural image below is blurred. The plane-wave assumption is central to the refraction formula used to calculate the statics.

In marine seismic surveying on the Norwegian Continental Shelf, refraction data from ocean-bottom seismometers (OBS) is used to build velocity models of the deeper crust and mantle below the sedimentary section. These velocity models constrain the depth imaging of oil and gas prospects beneath the salt and sub-salt section.

Plane Waves and Full-Waveform Inversion

Full-waveform inversion (FWI) is a seismic processing technique that iteratively updates a velocity model by comparing recorded waveforms with synthetic waveforms computed from the model, and minimizing the difference. FWI can be implemented shot by shot (using each shot's data independently) or in the plane-wave domain (using combinations of shots that simulate plane waves at specific angles). Plane-wave FWI reduces the computational cost significantly because far fewer effective sources (plane waves) need to be modeled than individual shots.

Equinor, TotalEnergies, and bp have applied plane-wave FWI to improve velocity models for complex subsalt imaging in the Gulf of Mexico and the North Sea, where conventional velocity analysis fails to converge on accurate models under thick, irregular salt bodies. The accuracy of the velocity model directly controls the quality of the depth-migrated image that exploration and appraisal teams use to pick drilling locations.

Plane wave is also called a plane wave component when used in decomposition of a more complex wavefield. Related terms include spherical wave (a wave that expands outward from a point source in a sphere; the actual wave geometry generated by a seismic source; approximated as a plane wave far from the source), wavefront (the surface connecting all points of the same phase in a propagating wave; a flat wavefront is a plane wave; a curved wavefront is a spherical or cylindrical wave), Snell's law (the equation governing the angle of refraction or reflection when a plane wave crosses an interface between two media of different wave velocity; n₁ sin θ₁ = n₂ sin θ₂), plane-wave common-image gather (a seismic data organization in which images are formed for each plane-wave angle rather than each shot, used in migration velocity analysis and full-waveform inversion), and full-waveform inversion (FWI, a seismic processing technique that minimizes the difference between recorded and synthetic waveforms to update the subsurface velocity model; often implemented in the plane-wave domain to reduce computational cost).

Why a Missing Plane-Wave Assumption Cost Three Weeks of Reprocessing on a Norwegian North Sea Survey

A processing team was working on a full-waveform inversion project using an ocean-bottom cable (OBC) dataset from a complex gas field on the Norwegian Continental Shelf. The FWI algorithm being applied had been tested and validated on marine streamer data from other projects. When applied to the OBC dataset, the velocity model update after the first FWI iteration was erratic, with velocity artifacts appearing in the shallow section and the misfit function not converging as expected.

After two weeks of diagnostic work, the processing geophysicist identified the root cause: the FWI code assumed the input data was in the plane-wave domain and expected the source signatures (the time delays between shots forming each effective plane wave) to be organized in a specific way. The OBC dataset had not been pre-processed into plane-wave gathers. The code was receiving individual shot gathers and misinterpreting them as plane-wave data, producing incorrect gradient updates to the velocity model.

The fix required re-running the plane-wave encoding step (summing individual shots with the appropriate linear time shifts to form controlled-beam super-shots) before feeding the data to the FWI algorithm. The re-encoding took four days of processing time. The subsequent FWI run converged smoothly in six iterations. Total time lost to the coding assumption mismatch: three weeks. The plane-wave assumption in the FWI code had been documented in the code's technical specification, but the QC step that confirmed the input data format was in plane-wave organization had not been included in the processing workflow. It was added as a mandatory pre-FWI check after the incident.