Fog Attenuation
Optical attenuation in fog from visibility with the Kim, Kruse or Al Naboulsi (advection, radiation fog) model: specific attenuation in dB/km, path loss and transmittance.
Specific attenuation
33.9 dB/km
extinction 7.81 km⁻¹ (Np/km)
Path loss
33.9 dB
over 1 km
Transmittance
0.0405 %
Exponent q
0
no wavelength dependence
Specific attenuation at 1550 nm
Model
Kim, Kruse: σ = σ_R(λ) + [3.912/V − σ_R(550 nm)] · (λ/550 nm)^(−q)
Al Naboulsi: σ = (0.11478λ + 3.8367)/V (advection), (0.18126λ² + 0.13709λ + 3.7502)/V (radiation)
σ is the extinction in km⁻¹ (e-folds of power per km), with λ in µm and V in km; the attenuation in dB/km is 10 log₁₀(e) ≈ 4.343 times larger. The visibility V (Koschmieder, 2 % contrast at 550 nm) fixes σ at 550 nm. Kim's exponent q is 1.6 (V > 50 km), 1.3 (6–50 km), 0.16V + 0.34 (1–6 km), V − 0.5 (0.5–1 km) and 0 in fog below 0.5 km, where droplets are much larger than the wavelength. Kruse's older law keeps q = 0.585V^(1/3) below 6 km, so it predicts less loss at longer wavelengths even in dense fog; Kim et al. measured no such difference between 785 and 1550 nm there. Al Naboulsi fitted FASCOD results for advection fog (sea fog, droplet diameters near 20 µm) and radiation fog (inland, about 4 µm) over 690–1550 nm and V = 50–1000 m.
Visibility laws give a mean loss for a typical droplet size distribution; measured fog attenuation scatters widely around them. Not modelled: absorption bands, rain and snow, turbulence, geometric loss.
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
Empirical visibility laws: Kim or Kruse aerosol exponents, or Al Naboulsi's advection and radiation fog fits (690–1550 nm, V = 50–1000 m). They give a mean loss; measured fog loss scatters widely around it.
- Kim I. I., McArthur B., Korevaar E. (2001). Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications. Proc. SPIE 4214, 26–37. Visibility-based aerosol exponent q. doi:10.1117/12.417512
- Al Naboulsi M., Sizun H., de Fornel F. (2004). Fog attenuation prediction for optical and infrared waves. Opt. Eng. 43(2), 319–329. Advection and radiation fog laws. doi:10.1117/1.1637611
- Nadeem F., Leitgeb E. (2010). Dense maritime fog attenuation prediction from measured visibility data. Radioengineering 19(2), 223–227. Kruse, Kim and Al Naboulsi laws, eqs. (1)–(6). radioeng.cz
- Kruse P. W., McGlauchlin L. D., McQuistan R. B. (1962). Elements of Infrared Technology: Generation, Transmission and Detection. Wiley. Kruse visibility law.