Optical depth

This article deals with a measure of optical transmittance of a medium; for a comparison between a distance in a medium and in vacuum, see optical path length.

The optical thickness τ \tau , also optical depth, is a dimensionless measure of how well a physical medium allows electromagnetic waves to pass:

  • when passing through a layer of matter (e.g. the atmosphere) of optical thickness τ \tau = 1, the radiance drops to 1/e times (≈ 37 %).
  • for the case τ \tau ≫1 we speak of optically thick
  • for the case τ \tau ≪1 of optically thin.

The optical thickness of a material is fdifferent for different frequencies It is calculated by integrating the absorption coefficient aover the light path dthat the radiation must travel:

\tau (f)=\int _{0}^{d}a(x,f){\mathrm {d}}x

In a medium assumed to be homogeneous, the whole thing simplifies to a multiplication:

{\displaystyle \tau =C_{i}\cdot \sigma \cdot d}

with

Optical thickness of the atmosphere

Determination

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

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

with

  • the intensity Iof the solar radiation in the considered wavelength on the ground
  • of the exatmospheric solar radiation I_{0}(solar constant)
  • the atmospheric mass {\displaystyle 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:

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

Thereby describe

  • the gas optical thickness τ {\displaystyle \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):

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

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


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