Atmospheric Optical Depth

Measurement of optical depth

Beer-Lambert law applied to the atmosphere

`I (lambda) = I_0 (lambda) * e^[(-m *(tau_a + tau_g + tau_[NO_2] + tau_w + tau_[O_3] + tau_r))]`                 [1]


`I_0` : intensity of sunlight outside the atmosphere

`I` : light received on the ground

`lambda` is the light wavelength

`tau_a` : aerosols transparency coefficient

`tau_g` : gas transparency coefficient (CO2 et O2 )

`tau_[NO_2]` : nitrogen dioxide transparency coefficient  (pollution)

`tau_w` : water steam transparency coefficient

`tau_[o_3]`: ozone transparency coefficient

`tau_r` : Rayleigh scattering coefficient

`m` : air mass coefficient crossed through by light  (optical path)

`m = 1 / sin(theta)`    with `theta` angle between the position of the sun and the horizon line


In the case of aerosol measurements, the equation will be simplified by considering that the total atmospheric optical thickness depends solely on the dissipation of light by the molecules (Rayleigh) and by the aerosols. We will distinguish the contribution "natural" (molecular) and "contaminant" (aerosols + others).

Contributions due to ozone (and possibly other absorbent gases under certain conditions) and aerosols may be separated after measurement, either by using climatological data and latitude-dependent average ozone values , Or by using actual total measurements of the air column with time and place of data collection. Satellite-mounted instruments such as the Total Ozone Mapping Spectrometer (TOMS) provide this type of data.


Equation [1] becomes :         `I(lambda) = I_0(lambda) * e^[(-m(tau_a + tau_r  + tau_[O_3]))]`

  • We seek to determine `tau_a`.
  • `tau_r` coefficient is proportional to the ratio of atmospheric pressure measured at the point of observation to that measured at the surface of the sea :`p / p_0` and thus : `tau_r = a_R * p / p_0`
  • `tau_[O_3]` coefficient id supplied by LOA in the red and green wavelengths. In the blue wavelength, this coefficient is nul.


Our photometer gives to us a value directly proportional to the light intensity. We will name it : `N`.

`N_0` is the value that our photometer would give to us, for a measurement of the light intensity out of atmosphere at 1 AU of the sun.

(AU :  Astronomical unit. It is equal to the average Earth-Sun distance and is worth 150 million kilometers.)


`N = N_0 * e^[(-m(tau_a + a_R * p / p_0 + tau_[O_3]))]`


We will introduce a corrective term taking into account the Earth-Sun distance which varies according to the day of the year.


`N = N_0 * [ r_0 / r]^2 *  e^[(-m(tau_a + a_R * p / p_0 + tau_[O_3]))]`


`r_0` is the 1 AU and `r` The Earth-Sun distance at the date of measurement (in AU).

`r= [1-e^2]/[1+e * cos(2pi * n / 365)]` with `e = 0,0167`


We will now express `tau_a`, the aerosols optical depth, according to the other terms.

`ln(N) - ln(N_0 * [r_0 / r]^2) = -m * (tau_a + a_R * p / p_0 + tau_[O_3])`

`tau_a = [ - [ln(N)-ln(N_0.[r_0 / r]^2) ]] / m - a_R * p / p_0 - tau_[O_3]`                         [2]                


The optical thickness of the atmosphere (Atmospheric Optical Thickness) is denoted AOT.

The part of this thickness due to aerosols is called Aerosol Optical Depth (AOD).

See paragraph 3 for the parameters `N_0` et `a_R`


Banizoumbou : march 8th 2007 -  april 2nd 2007 . Photo : LOA



LOA (Service d'Observation PHOTONS)