Attenuation – extinction
In this section, I present the attenuation or extinction as well as several quantities associated with this notion of attenuation and often used in the fields of radiative transfer or solar radiation. All of these quantities can be defined for a wavelength, a range of wavelengths, or the entire spectrum.
The two phenomena - absorption and scattering - remove energy from the incident solar radiation in its downwelling path to the ground. This decrease in radiation is called attenuation, or extinction. Like absorption and scattering, attenuation depends on the wavelength. An attenuation coefficient, c, also called extinction coefficient is defined, such that if Fis the incident flux, the flux F' transmitted over a length / in an absorbing-scattering medium is:
with c=a + b.
The dimension of c is the inverse of a length.
Quantities related to the attenuation
The extinction of a radiation beam depends on the number and sizes of particles and molecules on the path, on the volumetric density of the air, and on the path length across the attenuating medium. The relative air mass characterizes this extinction for any incidence relative to that at zenith, i.e., for a vertical path. It is often abbreviated as air mass and noted m. It is dimensionless. This term is not to be confused to the air mass used in meteorology. The air mass can be defined for a given atmospheric layer, or for a given constituent, e.g., the ozone air mass, or for the entire atmospheric column from sea level to the top. By default, the latter prevails.
At normal incidence, i.e., when the solar zenithal angle is zero, the air mass is equal to 1 by definition. The closer the sun to the horizon, the longer the path through the atmosphere, and the more important the extinction of the beam radiation. When 6s is 60°, or the solar elevation angle is 30°, the air mass is 2. It reaches approximately 40 when the sun is on the horizon. The simplest equation of the air mass is:
The results are quite acceptable in most cases when is less than 70° in the case of total radiation. More complex formulations have been proposed, some of them taking into account changes in density in the atmospheric column and depending on the spectral range under concern. For the total radiation, the Kasten and Young formula gives accurate enough results even when the sun is on the horizon:
where в$ is in degrees.
The geometrical thickness of the atmosphere depends on the elevation, or altitude, of the site above the mean sea level. The air mass is given for the mean sea level, i.e., for an atmospheric pressure at surface P0 equal to 101.324 hPa. At a given site, the atmospheric pressure at surface P is related to the elevation of the site. Since the elevation and the air pressure are correlated, a usual correction of the air mass for difference in elevation is given by:
Optical depth – optical thickness
The optical depth r is a measure of the attenuation of the radiation during its vertical path though the atmosphere, i.e., when % is zero. It is the natural logarithm (In) of the ratio of the incident flux Fmcuiem at the top of an atmosphere layer to the flux Ftransmitted transmitted by this layer:
The atmosphere layer can be composed of molecules of air, aerosols, or clouds. The greater the attenuation in this layer, the greater the optical depth. It is positive and dimensionless. The optical depth of aerosols typically varies between 0 and 5 as it will be seen later. The range of variation of the optical depth of clouds is much larger, from 0 to more than 100. An observer on the ground can hardly see the sun when the optical depth of the cloud is around 3.
Using the previous equations, a relationship is obtained between the optical depth r and the attenuation coefficient c on a vertical path:
where таь50грц0П and тscattering are the optical depths due respectively to absorption and scattering. In other words, the optical depths add up along the radiation beam. In the case of an atmosphere column containing absorbing gases, scattering air molecules, and scattering aerosols, Beer-Lambert’s law links the incident flux Finci<jeM and the transmitted flux
where xgases, xmoiecuies> and гaerosols are respectively the optical depths of the absorbing gases, air molecules, and aerosols. Each optical depth can itself be broken down into the sum of optical depths. For example, the optical depth of gases is the sum of that for ozone, that for dioxygen, etc.
Some authors use the terms optical depth and optical thickness interchangeably. This is not the general case. For example, the World Meteorological Organization limits the use of optical thickness to oblique paths. If т^ерц, and Tthickness denote respectively the optical depth and thickness, the following relationship holds:
-  Kasten F., Young A. T., 1989. Revised optical air mass tables and approximation formula. AppliedOptics, 28,4735-4738. doi:10.1364/A0.28.004735.
-  Guide to Instruments and Methods of Observation. WMO-No. 8, 2018 edition, 2018, WorldMeteorological Organization, Geneva, Switzerland. [Vol 1 Measurement of Meteorological Variables,Chapter 7, Measurement of Radiation.]