# 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))]                 

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  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]                         

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