Page updated: April 12, 2020
Author: Curtis Mobley

# Gershun's Law

The radiative transfer equation is a statement of energy conservation in the sense that it accounts for all the losses and gains to a swarm of photons moving through the water along a path in a ﬁxed direction. We now derive a useful conservation statement that holds at a ﬁxed point in the water, through which photons are moving in all directions.

The desired result is obtained by integrating the 1-D, time-independent, source-free RTE

over all directions. Dropping the wavelength argument for brevity and writing the diﬀerential element of solid angle $sin𝜃d𝜃d\varphi$ as $d\Omega \left(𝜃,\varphi \right)$, the left hand side of Eq. (1) yields

$\begin{array}{llll}\hfill {\int }_{0}^{2\pi }{\int }_{0}^{\pi }cos𝜃\frac{dL\left(z,𝜃,\varphi \right)}{dz}d\Omega \left(𝜃,\varphi \right)\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}\frac{d}{dz}{\int }_{0}^{2\pi }{\int }_{0}^{\pi }L\left(z,𝜃,\varphi \right)cos𝜃d\Omega \left(𝜃,\varphi \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}\frac{\phantom{\rule{1em}{0ex}}d}{dz}\left[{E}_{d}\left(z\right)-{E}_{u}\left(z\right)\right]\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\end{array}$ after noting that $cos𝜃<0$ for $\pi ∕2<𝜃\le \pi$ and recalling the deﬁnitions of the upwelling and downwelling plane irradiances as integrals of the radiance. The $-cL$ term becomes

$\begin{array}{llll}\hfill \iint -c\left(z\right)L\left(z,𝜃,\varphi \right)d\Omega \left(𝜃,\varphi \right)\phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}-c\left(z\right)\iint L\left(z,𝜃,\varphi \right)d\Omega \left(𝜃,\varphi \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}-c\left(z\right){E}_{o}\left(z\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\end{array}$ where the double integration over all directions is the same as shown in Eq. (2), and ${E}_{o}\left(z\right)$ is the scalar irradiance. The elastic scatter path function gives

$\begin{array}{llll}\hfill .& \phantom{\rule{1em}{0ex}}\iint \left[\iint L\left(z,{𝜃}^{\prime },{\varphi }^{\prime }\right)\phantom{\rule{0.3em}{0ex}}\beta \left(z,{𝜃}^{\prime },{\varphi }^{\prime }\to 𝜃,\varphi \right)d\Omega \left({𝜃}^{\prime },{\varphi }^{\prime }\right)\right]d\Omega \left(𝜃,\varphi \right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{1em}{0ex}}=& \phantom{\rule{1em}{0ex}}\iint L\left(z,{𝜃}^{\prime },{\varphi }^{\prime }\right)\left[\iint \beta \left(z,{𝜃}^{\prime },{\varphi }^{\prime }\to 𝜃,\varphi \right)d\Omega \left(𝜃,\varphi \right)\right]d\Omega \left({𝜃}^{\prime },{\varphi }^{\prime }\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}b\left(z\right)\iint L\left(z,{𝜃}^{\prime },{\varphi }^{\prime }\right)d\Omega \left({𝜃}^{\prime },{\varphi }^{\prime }\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill =& \phantom{\rule{1em}{0ex}}b\left(z\right){E}_{o}\left(z\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\end{array}$ Here we recall that the integral of the volume scattering function over all directions is the scattering coeﬃcient.

Collecting terms (2)-(4) resulting from the directional integration of the RTE, we have

$\frac{d}{dz}\phantom{\rule{0.3em}{0ex}}\left[{E}_{d}-{E}_{u}\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}-c{E}_{o}+b{E}_{o}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},$

or

 $\frac{d}{dz}\left[{E}_{d}\left(z,\lambda \right)-{E}_{u}\left(z,\lambda \right)\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}-a\left(z,\lambda \right){E}_{o}\left(z,\lambda \right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(W\phantom{\rule{1em}{0ex}}{m}^{-3}\phantom{\rule{1em}{0ex}}n{m}^{-1}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}},$ (5)

which is the desired result. This equation is known as Gershun’s law (Gershun (1936), Gershun (1939)).

The physical signiﬁcance of Eq. (5) is that it relates the depth rate of change of the net irradiance ${E}_{d}-{E}_{u}$ to the absorption coeﬃcient a and the scalar irradiance ${E}_{o}$. If inelastic scattering (ﬂuorescence and Raman scattering) and internal sources (such as bioluminescence) are negligible at the wavelength of interest, then Eq. (5) can be used to obtain the absorption coeﬃcient a from in situ measurements of the irradiances ${E}_{d}$, ${E}_{u}$, and ${E}_{o}$. This is an example of an inverse model – a model that retrieves an inherent optical property from measurements of the light ﬁeld.

Voss (1989) used Gershun’s law (5) to recover a values to within an estimated error of order 20%. Inelastic scattering and internal source eﬀects were reasonably assumed to be negligible in his study. The needed irradiances were all computed from a measured radiance distribution, so that no intercalibration of instruments was required.

A more general development can be made to account for internal sources or inelastic scatter and for 3-D and time-dependent light ﬁelds, as shown in Light and Water section 5.10. The result is known as the divergence law for irradiance.

Maﬃone et al. (1993) determined absorption values by writing the source-free form of the 3-D divergence law in spherical coordinates and applying the result to irradiance measurements made using an underwater, artiﬁcial, isotropic light source. The artiﬁcial light source allowed measurements to be made at night, thus there was no inelastic scattering from other wavelengths. Their instrument did not require absolute radiometric calibration.

Note, however, Gershun’s law will give incorrect absorption values if naively applied to waters and wavelengths where inelastic processes such as Raman scattering or ﬂuorescence are signiﬁcant. For this reason, and because of calibration diﬃculties if diﬀerent instruments are used to measure ${E}_{d}$, ${E}_{u}$, and ${E}_{o}$, Gershun’s law is seldom used as a way to measure absorption. Nevertheless, it is sometimes a useful check on the internal consistency of numerical models or measured data, and it leads to a convenient way of calculating radiant heating rates.