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The transmittance T quantifies the amount of energy transmitted by an atmospheric column compared to that incident at the top of this column. Let FincUicm and Ftransmiej be respectively the incident flux at the top of the atmosphere and the flux received on the ground. The transmittance T is defined as the ratio of these two fluxes:

where г is the optical depth (vertical path) of the atmosphere column and m is the air mass. The transmittance is dimensionless. It varies between 0 (totally opaque medium) and 1 (fully transparent medium). It is the transmittance that is plotted in Figure 4.2, which I then called transmitted fraction, since transmittance was not defined yet.

Since the optical depths add up along the radiation beam, the transmittances multiply. Given an atmosphere column containing absorbing gases, scattering air molecules, and scattering aerosols, the transmittances Tgases, Tmoiecu/es, and Taerosois may be defined for absorbing gases, scattering molecules, and scattering aerosols, respectively:

Each transmittance can itself be broken down into the product of transmittances. For example, the transmittance of gases is the product of that of ozone, that of dioxygen, etc.

I draw your attention to the fact that transmittance is linked to the optical depth and to the air mass by an exponential. A small variation in one or other of the variables results in a greater variation in transmittance. Let me take a few examples. Assume a large solar zenithal angle в$ equal to 75°, and let me use the approximation m = 1/ cos(#s). If the optical depth is 0.1, which is very small, then the transmittance is 0.68. If the optical depth is slightly greater and equal to 0.5, the transmittance is 0.14, i.e., much smaller. In another example, the optical depth is now set to 1. When Q$ is respectively 30° and 75°, T is 0.32 and 0.02. These few examples demonstrate the importance of the optical depth and the solar zenithal angle on the transmittance.

Linke turbidity factor

The Linke turbidity factor[1] synthesizes the atmospheric attenuation under cloudless sky conditions. It is often noted 7/, and is dimensionless. It takes into account absorption and scattering by water vapor and aerosols relative to that of a clean and dry atmosphere. TL is 1 for this atmosphere. Given an air mass, the Linke turbidity factor is the number of clean and dry atmospheres that should theoretically be superimposed to obtain an attenuation equivalent to that observed for the atmosphere considered. The greater the attenuation by the atmosphere, the greater TL.

Unlike the attenuation, the air mass, the transmittance, or the optical depth, which can be defined for a wavelength, a range of wavelengths, or the entire spectrum, the Linke turbidity factor applies to the total radiation though it could be spectrally defined in theory.

The Linke turbidity factor is often given for an air mass equal to 2, i.e., #s=60°. A value of 3 is typical for Africa and Europe. The Linke turbidity factor can reach 7 or more, in the case of polluted areas, such as urban areas with heavy road traffic. It is a very convenient quantity for synthesizing the attenuation of the cloudless atmosphere and is often used by engineers and other practitioners in various fields of solar radiation.

If Tmixed_gases is the optical depth due to the absorption by mixed gases (mainly CO2, O2) in the atmosphere, rozone that due to ozone, and tscaUerinz_moecuies that due to the scattering by the air molecules, the optical depth of a clean and dry atmosphere iciean_dry is given by:

Let rwaterj/apor be the optical depth due to the absorption by water vapor and raerosois that due to the scattering by aerosols. The optical depth г of this atmosphere is then:

and it comes:

For illustration purposes, Figure 4.6 exhibits typical monthly values of the Linke turbidity factor at several locations. The site of Barrow in Alaska in the United States (latitude: 71.32°; longitude: -156.61°; elevation: 10m) is north of the northern polar circle. It experiences very clear skies in boreal winter, and TL is small and close to 2 from November to January. TL increases up to around 4 in summer as the content in water vapor increases in the atmosphere. Bondville (latitude: 40.11°; longitude: -88.37°; elevation: 220 m) is a small city in rural landscape in Illinois in the United States that experiences a temperate climate. Similarly to Barrow, TL reaches its minimum in

Typical monthly values of the Linke turbidity factor at several locations

Figure 4.6 Typical monthly values of the Linke turbidity factor at several locations.

(Source: SoDa Service (

winter (Tl~2.5) and its maximum (7^»4.5) in summer. Carpentras (latitude: 44.06°; longitude: 5.05°; elevation: 100 m) is also a small city in rural landscape with a temperate climate in Provence, which is the sunniest part of France. Like the previous two sites, the minimum is reached in winter and the maximum in summer, but unlike the other two, its variation throughout the year is small. TL ranges between 2.6 and 3.9. Actually, Provence is known for its often very clear sky. Alice Springs (latitude: -23.80°; longitude: 133.89°; elevation: 550 m) is a small city situated roughly in the center of Australia in an arid environment. It experiences a subtropical hot desert climate. Expectedly, the skies are often clear all year round, and TL ranges from 2.9 to 4. The minimum is reached in austral winter (June-August) and the maximum is observed in summer when the atmospheric content in water vapor increases and dust storms occur.

The two other sites are megacities and are air-polluted areas. TL at Mexico City, the capital of Mexico (latitude: 19.43°; longitude: -99.13°; elevation greater than 2200m), ranges between 4 and 5. A greater TL could be expected due to air pollution, but it is partly offset by the high altitude of the city. Cairo, the capital of Egypt (latitude: 30.05°; longitude: 31.24°; elevation: 20m), has almost 20 million inhabitants and millions of vehicles and thousands of factories. It has an arid climate with increased air moist in summer. 7/, is about 4 in January and increases till May, partly because of the occurrence of dust storms. After a fall in June, it increases again up to 6 in August, partly because of the increase in air moist, and partly because at the end of the growing season, farmers used to burn rice straw, thus adding to the air pollution in summer and thus increasing the depletion of the solar radiation reaching the ground. The values given here for illustration are indicative. They may evolve with changing urban policies and other constraints.


Unlike the Linke turbidity factor, visibility is not just about cloudless skies. It quantifies the horizontal attenuation of the atmosphere, cloudy or not, at ground level.

Visibility is the greatest horizontal distance at which an observer can distinguish a black object of sufficient size against the background of the sky. It is usually expressed in km. The greater the attenuation at ground level, the smaller the visibility. Given its definition, the visibility is a quantity related to the human visual perception and is not spectrally defined.

Visibility is very much linked to the particle content in the atmosphere but also to the presence of fogs, which are defined as clouds in contact with the ground. For example, it can be very small, less than 1 km, in the case of fogs. The combination of suspended desert dust and high water vapor content in the coastal area means that visibility can be less than a few km around the Arabian Gulf, also known as the Persian Gulf or Gulf of Iran. On the contrary, from Sophia Antipolis, in the southeast of France, you can distinguish the Corsican Mountains 200 km apart when the atmosphere is clear, particularly early in the morning in winter.

Visibility is a quantity measured at all airports. It is very often reported in meteorological bulletins and related databases. It is a convenient way to get a first idea of how the atmosphere attenuates the radiation in a given place and time. Certain numerical models modeling the radiative transfer in the atmosphere allow the visibility as input to quantify the extinction of the atmosphere without cloud. For a beginner, it is indeed an easier quantity to grasp than the optical depth of aerosols at 500 nm, or another wavelength, or the water vapor content of the atmospheric column. For example, a visibility of 50 km may be input to the model for a clear sky, or 10 km for a turbid sky loaded with aerosols and water vapor.

  • [1] Franz (exactly Karl Wilhelm Franz) Linke was a German geophysicist and meteorologist (1878-1944).He defined the turbidity factor (Triinbungsfaktor) in 1932. As Linke is a German name, the final e mustbe pronounced.
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