Optical depth

Author: Leandro Alegsa Creation date: November 13, 2022

Optical depth is a measure of transparency. It is defined as the fraction of radiation (or light) that is scattered or absorbed on a path. One way of visualizing optical depth is to think of a fog. The fog between a person and an object that is immediately in front of them has an optical depth of zero. As the object moves away, the optical depth increases until it reaches a large value and the object is no longer visible.

Optical thickness of the atmosphere

Determination

The optical thickness τ $\tau$ the atmosphere enters as extinction coefficient into the transmissivity the atmosphere. This is calculated for a given wavelength according to Lambert-Beer's law to:

$T={\frac {I}{I_{0}}}=e^{-\tau \cdot m}$

with

the intensity of the solar radiation in the considered wavelength on the ground
of the exatmospheric solar radiation $I_{0}$ (solar constant)
the atmospheric mass $m=1/\cos \Theta _{z}$ , i.e. the distance through the atmosphere as a multiple of the shortest possible distance at zenith insolation ( $\Theta _{z}$ is the solar zenith angle).

Due to the atmospheric mass, the transmissivity depends on the position of the sun, i.e., it changes during the day, even if atmospheric conditions remain constant. In contrast, the optical thickness of the atmosphere does not depend on the position of the sun; it can be measured with a photometer.

Components

The optical thickness of the atmosphere is composed additively:

$\tau =\tau _{\text{Gas}}+\tau _{R}+\tau _{A}$

Thereby describe

the gas optical thickness τ $\tau _{\text{Gas}}$ the absorption at the atmospheric gases (mainly ozone, oxygen and water vapor), but only in the wavelength ranges λ $\lambda$ of the absorption bands of the gases. The optical thickness of the atmospheric gases (except water vapor) is quasi-constant and can be taken from tables.
the Rayleigh optical thickness τ $\tau _{R}(\lambda )=0{,}008735\cdot \lambda ^{{-4{,}085}}$ the extinction caused by Rayleigh scattering of air molecules.
the aerosol optical thickness τ $\tau _{A}$ the Mie scattering from larger particles (aerosols). It can be determined from the other components (measured or looked up):

$\Leftrightarrow \tau _{A}=\tau -\tau _{R}-\tau _{\text{Gas}}$

For a more detailed breakdown, see Lambert-Beer's Law, Remote Sensing (Atmosphere).