Spreading Loss
Spreading loss (also called geometrical spreading or geometrical divergence) is the reduction in seismic wave amplitude that occurs as the wavefront expands outward from the source and its energy is distributed over an increasingly large area, independent of any absorption or scattering of energy in the medium; in a homogeneous medium with constant velocity, a seismic wave propagates as a spherical wavefront whose surface area increases as the square of the travel distance, so the energy per unit area (intensity) decreases as one over the square of distance, and the amplitude (proportional to the square root of intensity) decreases as one over the distance traveled; this one-over-distance amplitude decay is referred to as spherical divergence or geometric spreading and is the dominant cause of amplitude reduction at short to moderate travel distances in typical seismic surveys, far exceeding the effects of anelastic attenuation (Q) except at very long travel distances or in highly attenuating formations; in seismic data processing, the spherical divergence gain correction (also called the geometrical spreading correction or t-squared gain) multiplies each seismic trace by a factor proportional to the travel time squared to compensate for the amplitude decay and equalize the amplitudes of reflections from different depths before further processing steps such as deconvolution, stack, or amplitude-versus-offset (AVO) analysis; applying an accurate spreading loss correction is essential for AVO analysis because the relative amplitude variation between near-offset and far-offset traces is the diagnostic signal for lithology and fluid content, and an incorrect divergence correction will introduce offset-dependent amplitude errors that masquerade as AVO anomalies or suppress genuine ones.
Key Takeaways
- The distinction between spherical spreading loss and cylindrical spreading loss matters in different seismic wave types: body waves (P-waves and S-waves that travel through the bulk of the rock) experience spherical spreading because their wavefronts are approximately spherical surfaces expanding in three dimensions, giving amplitude decay proportional to one over distance; surface waves (Rayleigh waves and Love waves, which propagate along the earth's surface and are the primary source of coherent noise on land seismic records) experience cylindrical spreading because they are confined to the surface and their energy spreads in only two dimensions, giving amplitude decay proportional to one over the square root of distance; the slower amplitude decay of surface waves with distance is one reason they dominate the noise field on long-offset land seismic records, outcompeting the body wave reflections from deep targets that have experienced much more severe spherical spreading; noise attenuation methods that target surface waves (F-K filters, radon transforms, surface-consistent deconvolution) must account for the cylindrical spreading geometry when designing amplitude-preserving noise suppression.
- The precise mathematical form of spreading loss in a real earth with vertical velocity gradients and layering differs from the simple one-over-distance relationship of a homogeneous medium: when velocity increases with depth (as is typical in compacting sedimentary sequences), the ray paths curve upward and the effective spreading distance for a reflected ray is not the geometric source-to-reflector-to-receiver path length but a modified distance that accounts for the ray curvature; the standard processing correction uses the root-mean-square velocity to compute an effective spreading correction that approximates the true spreading loss in the layered earth; more accurate spreading corrections require ray tracing through the actual velocity model (computing the geometric spreading factor, the Jacobian of the ray coordinate transformation) and are applied in the context of migration algorithms that handle the full propagation geometry; in areas with complex velocity variations such as gas chimneys, salt flanks, or zones of rapid lateral velocity change, the simple travel-time-based spreading correction is inadequate and migration-based amplitude corrections must be used.
- The interplay between geometrical spreading and anelastic attenuation (Q) in real formations determines the overall amplitude decay with depth in a seismic section: geometrical spreading dominates at early travel times (shallow targets), causing amplitude to decay roughly as one over time; anelastic attenuation, which absorbs energy at a rate proportional to frequency and travel distance, becomes increasingly important at greater depths and removes progressively more high-frequency energy from the wavelet; the net result is that deep reflections arrive with both reduced amplitude (from spreading) and altered wavelet shape (from frequency-dependent Q attenuation, which preferentially removes high frequencies and broadens the wavelet); correcting for both effects simultaneously requires applying the geometrical spreading gain followed by a Q compensation or inverse-Q filter that amplifies the high-frequency content of deeper reflections to restore the original source wavelet bandwidth; over-application of Q compensation amplifies noise along with signal, requiring careful noise regularization to prevent the correction from boosting noise to levels that dominate the corrected image.
- Amplitude-preserving processing for quantitative interpretation requires accounting for spreading loss with greater precision than is needed for purely structural imaging: structural seismic interpretation uses the migrated image for picking horizons and faults, tolerating amplitude errors that do not affect the geometry of the events; quantitative interpretation (AVO analysis, impedance inversion, rock physics analysis) uses the absolute and relative amplitudes to infer rock and fluid properties, requiring that the amplitude corrections applied in processing accurately remove all non-geological amplitude effects so that the remaining amplitude variations reflect only subsurface geology; a systematic over-correction of geometrical spreading at far offsets (which decreases the effective spreading loss correction at near offsets relative to far) will inflate far-offset amplitudes and produce false positive AVO gradients that look like gas sands; the precision required for quantitative amplitude work is significantly higher than for conventional structural mapping, and the processing sequence must be audited specifically for amplitude fidelity at each step.
- In marine seismic acquisition, the sea-surface ghost (the downward reflection of the upgoing wavefield from the water surface) interacts with spreading loss in a way that affects amplitude differently at different frequencies and offsets: the ghost reflection arrives shortly after the primary upgoing wave (the delay time determined by the streamer depth and the incidence angle), and its amplitude is close to unity (the reflection coefficient of the air-water interface is approximately -1); the interference between the primary upgoing wavelet and the ghost reflection creates a frequency-dependent amplitude pattern (the ghost notch) that removes energy at specific frequencies and doubles it at others; this frequency-dependent amplitude modification is applied before spreading loss corrections in the processing sequence and must be corrected (deghosting) before meaningful spreading loss analysis can be performed, because the ghost changes the apparent spreading loss differently at different frequencies in a way that corrupts the uniform gain correction needed for subsequent processing.
Fast Facts
The inverse square law for intensity decay (and the associated one-over-distance amplitude decay) is the same physical principle that explains why a person standing 10 meters from a loudspeaker hears the music at one-hundredth the intensity of someone standing 1 meter away, why the apparent brightness of a star decreases as the square of its distance from Earth, and why seismic waves from deep earthquakes arrive at distant seismograph stations with amplitudes far smaller than their surface-proximate counterparts. This universality — the same law governing acoustics, optics, gravitational attraction, and seismic propagation — reflects the fundamental geometry of three-dimensional space, in which the surface area of an expanding sphere increases as the square of its radius regardless of the nature of the wave or field filling it.
What Is Spreading Loss?
Spreading loss is the fundamental reason that seismic reflections from deep formations arrive at the surface with much smaller amplitudes than reflections from shallow ones, even when the reflection coefficients are identical. As the seismic wavefront expands outward from the source, the same amount of energy is distributed over an ever-larger spherical surface. The energy per unit area — which determines the wave amplitude — drops in proportion to the square of the distance traveled. Nothing is absorbing the energy; it is simply diluted over space. This geometric reality must be corrected in seismic data processing before any amplitude-based interpretation is attempted, because without the correction, deep reflections appear artificially weak, shallow reflections appear strong, and any attempt to compare amplitudes at different depths is comparing geometric artifacts rather than geological properties. The spreading loss correction is one of the earliest and most fundamental steps in the seismic processing sequence, and getting it right is a prerequisite for everything that follows in quantitative seismic interpretation.
Synonyms and Related Terminology
Spreading loss is most commonly called geometrical spreading or spherical divergence in seismic contexts. The correction applied in processing is called the divergence correction, spherical divergence gain, or t-squared gain. Related terms include anelastic attenuation (Q attenuation, the amplitude decay caused by energy absorption in real rocks, which supplements geometrical spreading as a mechanism of amplitude loss with depth and preferentially removes high-frequency energy), AVO (amplitude versus offset, the analysis technique for hydrocarbon detection that requires accurate correction of geometrical spreading before the offset-dependent amplitude variations related to geology can be isolated), inverse-Q filtering (the processing step that compensates for frequency-dependent attenuation to restore the high-frequency content absorbed by anelastic processes, applied after the geometrical spreading correction), ghost (the sea-surface reflection of the upgoing wavefield in marine seismic acquisition, which causes frequency-dependent amplitude distortion that interacts with spreading loss corrections), and true amplitude (the objective of amplitude-preserving seismic processing, in which all non-geological amplitude effects including spreading loss, attenuation, and acquisition effects are removed to leave amplitudes that reflect only subsurface reflectivity).
Why Correcting Geometry Before Interpreting Geology Is Non-Negotiable
Every bright spot anomaly on a seismic section, every AVO gradient that suggests a gas-saturated sand, every impedance inversion that claims to resolve rock properties between wells — all of these interpretations rest on the assumption that the amplitudes in the processed seismic data reflect geological reflectivity rather than acquisition and propagation effects. Geometrical spreading is the largest single acquisition and propagation effect that must be removed before that assumption is valid. An uncorrected seismic dataset shows a progressive amplitude decay with depth that has nothing to do with the geology and everything to do with the geometry of wave propagation. Mistaking that geometric effect for a geological one would be as fundamental an error as mistaking a camera's depth-of-field effect for real blurring in the photographed scene. The spreading correction does not produce truth — it removes a known artifact so that truth has a chance to be visible in the data that remains.