Applies the Koschmieder–Berek law to estimate the maximum horizontal
sighting distance in water:
\(V = -\ln(C_t) / c\)
where \(C_t\) is the inherent contrast threshold and \(c\) is the
beam attenuation coefficient. Because \(c\) is rarely measured directly,
a conversion factor kd_to_c maps diffuse attenuation Kd to \(c\).
Arguments
- Kd
Diffuse attenuation coefficient in 1/m (scalar or vector).
- contrast_threshold
Inherent contrast threshold of the visual system or camera (dimensionless, typically 0.01–0.05). Default 0.02.
- kd_to_c
Ratio \(c / K_d\). Typical values: 1.2–1.5 for clear oceanic water, 2–4 for coastal/turbid water. Default 1.5. Ignored when
beam_cis supplied.- beam_c
Optional measured beam attenuation coefficient \(c\) (1/m), e.g. from
beam_attenuation(). When supplied it is used directly instead of theKd * kd_to_capproximation.
Details
This is a contrast-threshold scenario estimate, not an empirically validated prediction of detection by a particular observer or camera.
The coefficient that governs horizontal sighting distance is the beam
attenuation coefficient \(c\), not the diffuse attenuation \(K_d\)
(Johnsen 2012, ch. 5: as you move away from an object, veiling light
replaces it at a rate equal to \(c\)). If beam_c is not given, this
function approximates \(c = K_d \times\) kd_to_c. Note that
\(c > K_d\) always, so treating \(K_d\) as \(c\) (kd_to_c = 1)
is an upper bound that overestimates range. The model assumes
horizontal viewing; for an upward-looking observer the effective
coefficient is \(c - K_L\) (smaller, so ranges are longer), which is not
modelled here.
References
Duntley SQ (1963) Light in the sea. J. Opt. Soc. Am. 53:214–233.
Zaneveld JRV, Pegau WS (2003) Robust underwater visibility parameter. Opt. Express 11:2997–3009.
Johnsen S (2012) The Optics of Life: A Biologist's Guide to Light in Nature. Princeton University Press. (Ch. 5: sighting distance and \(c\).)