Visibility - Derivation

Derivation

To define visibility, we examine the case of a perfectly black object being viewed against a perfectly white background. The visual contrast, C_V(x), at a distance x from the black object is defined as the relative difference between the light intensity of the background and the object

${C_V}\mathrm{\left({x}\right)}\mathrm{{=}}\frac{{F_B}\left({x}\right)\mathrm{{-}}{F}\left({x}\right)}{{F_B}\left({x}\right)}$

where F_B(x) and F(x) are the intensities of the background and the object, respectively. Because the object is assumed to be perfectly black, it must absorb all of the light incident on it. Thus when x=0 (at the object), F(0) = 0 and C_V(0) = 1. Between the object and the observer, F(x) is affected by additional light that is scattered into the observer's line of sight and the absorption of light by gases and particles. Light scattered by particles outside of a particular beam may ultimately contribute to the irradiance at the target, a phenomenon known as multiple scattering. Unlike absorbed light, scattered light is not lost from a system. Rather, it can change directions and contribute to other directions. It is only lost from the original beam traveling in one particular direction. The multiple scattering's contribution to the irradiance at x is modified by the individual particle scattering coefficient, the number concentration of particles, and the depth of the beam. The intensity change dF is the result of these effects over a distance dx. Because dx is a measure of the amount of suspended gases and particles, the fraction of F that is diminished is assumed to be proportional to the distance, dx. The fractional reduction in F is

${dF}\mathrm{{=}}\mathrm{{-}}{b_{ext}}\;{F}\;{dx}$

where b_ext is the attenuation coefficient. The scattering of background light into the observer's line of sight can increase F over the distance dx. This increase is defined as b' F_B(x) dx, where b' is a constant. The overall change in intensity is expressed as

${dF}\left({x}\right)\mathrm{{=}}{\mathrm{}}{dx}$

Since F_B represents the background intensity, it is independent of x by definition. Therefore,

${dF_B}\left({x}\right)\mathrm{{=}}{0}\mathrm{{=}}{\mathrm{}}{dx}$

It is clear from this expression that b' must be equal to b_ext. Thus, the visual contrast, C_V(x), obeys the Beer–Lambert law

$\frac{{dC_V}\left({x}\right)}{dx}\mathrm{{=}}\mathrm{{-}}{b_{ext}}\;{C_V}\left({x}\right)$

which means that the visual contrast decreases exponentially with the distance from the object:

${C_V}\left({x}\right)\mathrm{{=}}\exp\left({\mathrm{{-}}{b_{ext}}\;{x}}\right)$

Lab experiments have determined that contrast ratios between 0.018 and 0.03 are perceptible under typical daylight viewing conditions. A contrast ratio of 2% (C_V = 0.02) is usually used to calculate visual range. Plugging this value into the above equation and solving for x produces the following visual range expression (the Koschmeider equation):

${x_v}\mathrm{{=}}\mathrm{\frac{3.912}{b_{ext}}}$

with x_v in units of meters. At sea level, the Rayleigh atmosphere has an extinction coefficient of approximately 13.2 X 10−6 m−1 at a wavelength of 520 nm. This means that in the cleanest possible atmosphere, visibility is limited to about 296 km.