Spherical Divergence: Seismic Amplitude Decay, Geometric Spreading, and Gain Correction in WCSB Imaging
Spherical divergence is the geometric decay of wave energy as a wavefront expands outward from its source, and it is one of the first corrections any processor applies to raw seismic data. When a seismic source like a vibroseis truck or a dynamite charge fires near surface, it radiates energy as an expanding spherical wavefront. The total energy on that wavefront is conserved, but it is spread over a surface area that grows with the square of the radius, so the energy density at any point falls as one over distance squared and the wave amplitude, which is proportional to the square root of energy, falls as one over distance. This is the same inverse-square law that governs the apparent dimming of a gravitational or magnetic field with distance, and in geophysics it is also called geometric spreading. Because seismic travel paths in the Western Canadian Sedimentary Basin can run from a few hundred metres to many kilometres, a reflection from a deep Duvernay or Leduc reef target arrives at the geophone vastly weaker than a shallow reflection, purely because of how far the wavefront has stretched, not because the deep rock reflects poorly. If left uncorrected, this decay would make every seismic section fade to nothing with depth, hiding the deep targets that justify the survey. Processors therefore apply a spherical divergence correction, also called a gain recovery or t-squared gain, that multiplies each sample by a factor increasing with travel time to restore the true relative reflection strength. The correction is complicated by the fact that real earth velocity increases with depth, so the spreading is not purely geometric; a common formulation scales amplitude by velocity squared times travel time, using the velocity function from velocity analysis. Getting this correction right is essential for any quantitative interpretation, especially amplitude-versus-offset (AVO) work used to detect gas sands in Cretaceous Mannville or Cardium plays, because AVO depends on comparing true reflection amplitudes across offsets. Over-correcting boosts deep noise and creates false bright spots; under-correcting leaves real deep reflectors invisible. Spherical divergence is purely a geometric effect and is distinct from the other two great amplitude killers in seismic data, intrinsic attenuation, which is frequency-dependent absorption of energy into heat, and transmission loss at each interface. Modern processing separates these effects so that the recovered amplitudes reflect genuine subsurface reflectivity. In magnetics and gravity surveys the same square-of-distance falloff sets how rapidly an anomaly weakens as the sensor moves away from a buried salt body or basement fault, which is why airborne survey line spacing and flight height are chosen to keep targets within detectable range.
Key Takeaways
- Energy Falls As Distance Squared: A spherical wavefront spreads its fixed energy over a surface area of 4 pi r squared, so energy density drops as one over r squared and amplitude drops as one over r. A reflector at 3,000 m returns far less energy than one at 500 m for purely geometric reasons, independent of how reflective the rock actually is.
- Gain Correction Restores Deep Reflectors: Processors apply a spherical divergence or t-squared gain that multiplies each sample by a time-increasing factor, recovering the relative strength of deep events. Without it, every WCSB seismic section would fade to noise below a few hundred milliseconds, hiding deep Leduc and Nisku carbonate targets.
- Velocity Complicates The Simple Law: Because earth velocity rises with depth, the practical correction scales amplitude by roughly velocity squared times travel time rather than a pure one-over-r factor. The velocity function comes from velocity analysis, linking divergence correction directly to the same picks used for normal moveout.
- AVO Depends On Getting It Right: Amplitude-versus-offset gas detection in Cardium and Mannview sands compares true reflection amplitudes across offsets. An incorrect divergence correction distorts the AVO gradient, producing false bright spots or masking real ones, which can mislead a multimillion-dollar drilling decision.
- It Is Geometric, Not Absorptive: Spherical divergence is a frequency-independent geometric effect, separate from intrinsic attenuation, which preferentially removes high frequencies as heat, and from transmission loss at interfaces. Correct processing treats the three losses separately so recovered amplitude reflects genuine subsurface reflectivity.
Why Deep Duvernay Targets Need Strong Gain Recovery
A Duvernay survey in the Kaybob area images source-rock targets near 3,400 m, with seismic travel times past 2.6 seconds two-way. By that time a reflection has spread across a wavefront kilometres in radius, and its raw amplitude is a tiny fraction of a shallow Belly River reflection. The processor applies a spherical divergence correction tied to the stacking velocity function, typically rising from about 2,400 m/s near surface to 4,500 m/s at target. The recovered section shows the Duvernay and underlying Swan Hills carbonate at usable amplitude. Skipping or weakening this gain would leave the interpreter unable to map the very interval the survey was shot to evaluate, wasting a CAD 1.5 million 3D acquisition program.
Spherical Divergence In Gravity And Magnetic Surveys
The same inverse-square principle governs potential-field surveys used for regional WCSB framework mapping. A buried Precambrian basement fault or a Prairie Evaporite salt-dissolution feature produces a magnetic or gravity anomaly whose strength falls off with the square of the distance from sensor to source. Airborne crews set flight-line spacing and terrain clearance, often 200 to 500 m line spacing at 80 to 120 m height, so the target stays within detectable amplitude before divergence buries it in the regional field. This is why a deep, broad salt body shows a smooth low-amplitude gravity low while a shallow density contrast produces a sharp, strong anomaly, the difference being entirely a function of distance-driven spherical divergence.
Fast Facts
The inverse-square spreading law that underlies spherical divergence was understood in optics and acoustics centuries before seismology adopted it, descending directly from the geometry Kepler and Newton used to describe how light and gravity weaken with distance. Early reflection crews in the 1930s applied crude exponential gain by hand, turning a knob to brighten deep traces until the section looked balanced. The shift to physics-based, velocity-weighted divergence correction in the 1970s was a precondition for quantitative AVO, which now routinely separates gas from brine in WCSB sands on the strength of amplitude alone.
Related Terms
Spherical divergence correction depends on the velocity field derived during velocity analysis, since the gain factor is velocity-weighted, and an unaccounted velocity change can mimic or distort the expected geometric decay. It is one of several amplitude effects removed alongside attenuation, the frequency-selective absorption of energy into heat. Accurate recovery feeds directly into amplitude versus offset analysis, the gas-detection technique that lives or dies on true relative amplitudes, and supports the broader goal of trustworthy seismic interpretation across the basin.
Real-World WCSB Scenario: Gain Calibration On A Cardium AVO Project
A Calgary processing shop handles a Cardium 3D for an operator near Pembina, where the goal is to map gas-charged sandstone using AVO. The interpreter flags that early bright spots near 1.4 seconds look too strong relative to deeper events. Diagnosis points to an over-aggressive spherical divergence correction that boosted shallow noise. The processor re-derives the gain using a refined velocity function, balancing amplitudes across the section. Reprocessing costs roughly CAD 40,000 in analyst time.
With corrected divergence recovery, the AVO anomaly that survives is a genuine class III response over the Cardium sand, and the operator drills it with confidence. The recalibration paid for itself many times over by preventing a CAD 6 million horizontal well from chasing a processing artifact instead of real gas pay.