Spherical Wave: Point Source Wavefronts, Geometric Spreading, and Seismic Amplitude Recovery
A spherical wave is an elastic disturbance that radiates outward in every direction from a point source, forming a continuously expanding spherical wavefront whose surface area grows in proportion to the square of the radius travelled. The cleanest physical example is the compressional pulse from a buried explosive charge, where a small chemical detonation in a shot hole behaves almost exactly like an idealized point source and drives a single coherent shell of energy into the surrounding rock. Conventional seismic sources used across the Western Canadian Sedimentary Basin, including Vibroseis truck arrays on land and air-gun arrays in marine work, do not start as perfect points, yet at the offsets relevant to reflection imaging their combined output is treated as a spherical wave for the purposes of ray tracing, amplitude analysis, and wavefront modelling. The defining property of any spherical wave is geometric spreading, also called spherical divergence: because a fixed amount of source energy is smeared across a spherical surface of area 4 pi r squared, the energy density falls off as one over r squared, and because wave amplitude scales with the square root of energy density, the amplitude of a spherical wave decays as one over r. A reflection arriving from the Montney at roughly 2,500 m (about 8,200 ft) therefore carries far less amplitude per unit area than the same pulse measured a few metres from the shot, and that decay must be reversed before any meaningful interpretation of reflection strength can begin. In real layered ground the situation is more severe than the simple one-over-r law, because seismic velocity generally rises with depth through the Cretaceous and Devonian section, bending rays upward and accelerating the divergence so that amplitude decay is better approximated by one over the product of velocity squared and traveltime. Processors restore this lost energy with a spherical divergence or geometric spreading correction, a time-dependent gain function applied early in the sequence so that deep Montney and Duvernay reflectors are not rendered artificially weaker than shallow Mannville events. Once a spherical wave has travelled far enough that the receiver spread is small compared with the radius, the curved wavefront is locally almost flat and can be approximated as a plane wave, an idealization that underlies most practical normal-moveout, stacking, and migration mathematics. Understanding the spherical wave is thus the starting point for amplitude-versus-offset analysis, for true-amplitude migration, and for any quantitative seismic interpretation that ties reflection strength to lithology or pore fluid.
Key Takeaways
- Point Source Origin: A spherical wave emanates from a localized point, the closest real-world analogue being a dynamite charge in a shot hole. Vibroseis and air-gun arrays are engineered point approximations: the array nulls side lobes and shapes a downgoing wavelet that, beyond the near field, expands as a single spherical shell suitable for the ray-based assumptions used in WCSB land and Scotian Shelf marine processing.
- One Over r Amplitude Law: Energy density on the wavefront falls as one over r squared because surface area grows as 4 pi r squared; amplitude, the square root of energy density, falls as one over r. A reflector ten times deeper returns roughly one tenth the amplitude from spreading alone, before any intrinsic absorption, which is why raw deep Devonian events look far weaker than shallow Cretaceous reflectors.
- Spherical Divergence Correction: Processors apply a time-dependent gain, often a function of velocity squared times time, to undo geometric spreading. Applied too early or too aggressively it boosts noise; applied correctly it equalizes amplitude with depth so that a 3,200 m (10,500 ft) Duvernay reflection can be compared honestly against a 900 m (2,950 ft) Viking event in the same survey.
- Plane Wave Far-Field Limit: At large radius the curvature of a spherical wavefront across a finite receiver spread becomes negligible, so the wave is locally treated as a plane wave. This far-field simplification is what makes normal moveout, semblance velocity analysis, and Kirchhoff migration tractable on routine 3D volumes acquired over the Deep Basin.
- Foundation For True Amplitude: Any amplitude-versus-offset or inversion workflow that infers gas saturation or lithology depends on first removing spherical divergence. If geometric spreading is mis-modelled, a genuine Montney gas anomaly can be masked or a flat spot can be mimicked by uncorrected decay, so the spherical wave model is the bedrock of quantitative interpretation, not a textbook abstraction.
Geometric Spreading Versus Intrinsic Attenuation
Amplitude loss along a spherical wave splits into two physically distinct effects that processors must separate. Geometric spreading is purely a matter of energy spreading over a larger surface and is frequency independent, decaying as one over r in a uniform medium and faster where velocity climbs with depth. Intrinsic attenuation, described by the quality factor Q, is the conversion of elastic energy into heat through friction and fluid movement and is strongly frequency dependent, preferentially stripping high frequencies from deep Montney and Duvernay reflections. A spherical divergence gain corrects only the first effect; the second requires inverse-Q filtering. Confusing the two, for example using an aggressive time-power gain to compensate for what is really Q loss, distorts the wavelet and corrupts any amplitude-versus-offset interpretation built on the data.
From Spherical Wavefronts to Plane-Wave Stacking
The practical reason geophysicists care whether a wave is spherical or plane is the mathematics of imaging. A spherical wavefront curving across a 600-channel spread produces hyperbolic moveout whose curvature encodes velocity, the basis of semblance analysis. As traveltime grows, the wavefront flattens relative to the spread and the local plane-wave assumption becomes accurate enough for normal-moveout stacking and for plane-wave decomposition methods such as tau-p and plane-wave migration. Modern WCSB processing routinely toggles between the two views: spherical divergence correction and ray tracing honour the true expanding wavefront, while stacking and many migration algorithms exploit the plane-wave limit to keep compute cost manageable on multi-terabyte 3D surveys.
Fast Facts
The one-over-r amplitude rule has a brutal consequence that surprises newcomers: spreading alone, ignoring absorption entirely, costs about 20 decibels of amplitude going from a 100 m near offset to a 1,000 m deep reflector, and roughly 40 decibels by the time energy returns from a 10,000 m total path. That is why a buried Vibroseis sweep that registers volts of ground motion near the source returns reflections measured in microvolts at the geophone, and why a single mistuned divergence-correction exponent can swing an interpreted Montney amplitude anomaly from convincing to invisible.
Related Terms
A spherical wave is intimately connected to several other glossary concepts. The plane wave is its far-field limit, the flat-front idealization that stacking and migration assume once the source is distant. The TWT two-way traveltime measures how long the spherical wave takes to reach a reflector and return, the very variable that spherical divergence corrections key on. Compressional P-wave energy is what most explosive and Vibroseis spherical sources predominantly emit, and the broader idea of geometric spreading ties the spherical wave directly to true-amplitude processing and amplitude-versus-offset workflows used to detect gas in WCSB reservoirs.
Real-World WCSB Scenario: Vibroseis 3D Over the Montney Near Dawson Creek
An operator shooting a 110 square kilometre Vibroseis 3D over a Montney development block northwest of Dawson Creek, British Columbia, budgets roughly CAD 28,000 to 38,000 per square kilometre, putting the acquisition near CAD 3.5 million before processing. Each Vibroseis point emits an upsweep that, beyond the near field, behaves as a spherical wave; the processing contractor applies a spherical divergence correction scaled by interval velocity squared times time so that the deep 2,600 m (8,530 ft) Montney doublet is amplitude-balanced against the shallow 700 m (2,300 ft) Dunvegan markers. Without that correction the target zone would appear as a faint smear beneath bright shallow events.
After divergence correction and inverse-Q filtering, the amplitude-versus-offset response over the Montney brightens with offset in the pattern expected for an organic-rich, overpressured gas siltstone, guiding the placement of a CAD 9 million pad of multi-stage horizontal wells. Here the abstract one-over-r law translated directly into a multi-million-dollar drilling decision.