Scintillation Index
Rytov variance, aperture averaging, and fade probability for atmospheric turbulence.
Rytov variance σ_R²
0.338
Moderate fluctuations
Scintillation index σ_I² (point)
0.312
σ_I² with D = 0.1 m
0.0279
aperture averaging factor 0.0895
Fresnel zone √(L/k)
1.57 cm
sets the aperture-averaging scale
Fried parameter r₀
5.71 cm
plane wave
Fade probability
2.25e-5 / 1.40e-43
P(I < ⟨I⟩ − 3 dB) / P(I < ⟨I⟩ − 10 dB), log-normal
Fade probability (log-normal)
Scintillation vs distance
Model
σ_R² = 1.23 Cn² k^(7/6) L^(11/6), d² = kD²/(4L)
σ_I²(D) = exp[0.49σ_R²/(1 + 0.65d² + 1.11σ_R^(12/5))^(7/6) + 0.51σ_R²(1 + 0.69σ_R^(12/5))^(−5/6)/(1 + 0.90d² + 0.62d²σ_R^(12/5))] − 1
Plane wave, Kolmogorov spectrum with zero inner scale (Andrews & Phillips, Laser Beam Propagation through Random Media, 2nd ed., 2005). The expression covers weak to strong fluctuations: σ_I² ≈ σ_R² when weak, peaks above 1 in the focusing regime and saturates at 1. A receiver larger than the Fresnel zone averages the scintillation down.
The fade probabilities assume log-normal irradiance with σ_ln² = ln(1 + σ_I²), P(I < F⟨I⟩) = Φ((ln F + σ_ln²/2)/σ_ln). That holds in weak fluctuations; in moderate to strong turbulence the gamma-gamma distribution has heavier tails, so these values are optimistic there.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Plane wave on a horizontal path with constant Cn² and a Kolmogorov spectrum without inner scale. Fade probabilities assume log-normal irradiance, which is optimistic in moderate to strong turbulence.
- Andrews L. C., Phillips R. L. (2005). Laser Beam Propagation through Random Media, 2nd ed. SPIE. Plane-wave Rytov variance, Fried parameter and aperture-averaged scintillation index.