The laws of extinction

flux of radiation falls on a point A1 at the

top of the atmosphere.
Radiation flux at point B’ is
and at point B is
The value is extinction of the radiation flux as it passes from B’ to B.
According to Bouguer’s law

Mass extinction index

A2

A

A1

ho

B’

B

To obtain the value of the radiation flux at point A, the last formula is to be integrated from the top (A1) to the ground (A).

vertical air column
Absorption and diffusion is different at different altitudes

due to the fact that the composition of the air is not homogeneous.

Ozone
Oxygen

CO2
Water vapor
aerosols

Molecular

Molecular + aerosol

Aerosol

Absorption

Scattering

optical mass of the atmosphere (m) and altitude of the

Sun (ho)

Let’s inspect the area A1A2A

At high altitude of the Sun ho, this area can be approximated as rectangular triangle. A1A is hypotenuse, A1A2 and A2A are legs.

Optical mass of the atmosphere is equal to cosecant of the Sun altitude.

curvature of the Earth’s surface and Sun beam refraction must

be accounted for at ho>30°

Optical mass of the atmosphere is quick to rise with decrease of the Sun altitude

m=1

m=35,4

ho=0°

ho=90°

Formulas of solar radiation extinction in the atmosphere
Dimensionless magnitude
Solar

radiation extinction index or OPTICAL DEPTH
of the atmosphere

It means that value Iλ is quick to increase. Before

sunset value m is quick to increase. It means that value Iλ is quick to decrease.
Near the noon hours the solar radiation flux is slow to change its value.

In case the air density is constant (the beam spreads horizontally and a short distance)

Volume extinction index

times
We obtain that
Optical depth of the atmosphere (

) is the magnitude inverse to that optical mass, which attenuates the radiation flux in e times.

one more notion transmission coefficient (коэф.прозрач.)of the atmosphere (Pλ).

Adopting

m=1, i. e. the Sun is in zenith

Transmission coefficient is the fraction of the solar radiation flux, which reaches the Earth surface as the Sun is in zenith.

Since

in the atmosphere, the more the value of for a

given wavelength, and the less the transmission coefficient.

Important: Transmission coefficient of monochromatic flux depends on physical state of the vertical air column and does not depend on altitude of the Sun.
Transmission coefficient is also a function of wavelength. For the ideal atmosphere (no water vapor, no admixtures), this coefficient increases with the increase of the wavelength since the scattering of the shorter wavelengths is more intensive than the longer ones.

to know total (integrated) flux of SR but monochromatic.


Due to

very intricate dependence of the transmission coefficient on wavelength, it is not easy to take this integral. The only way to make necessary calculation is to use average values of P and


We may do it because in this case transmission coefficient P also shows the fraction of SR that reaches Earth’s surface when the Sun is in zenith:
However, in this case P value depends on the optical mass m.

only being attenuated, but it also change its spectral composition
Maximum

emittance is shifted to the longer wave side as the Sun altitude decreases.

The shorter wave beams suffer the largest extinction. Thus, passing through every new layer, the SR becomes more and more enriched with longer wave radiation.

presented as a sum of three items.

is the optical depth

of the ideal atmosphere.
is the optical depth formed by variable constituents (CO2, H2O )
is the optical depth formed by aerosols.
is turbidity factor.

As we know

Comparing the formulas suggests how many masses of ideal atmosphere are needed to get the SR extinction produced by one mass of the real atmosphere.

the transmission coefficient does.
ATF does not depend on m

value as much as the transmission coefficient does.

ATF depends on physical properties of air masses

Air mass is a huge air body characterized by homogeneous distribution of the air properties such as temperature, humidity, transparency etc.

point as a bundle of parallel rays is called DSR.
Fluxes

of I and depends on the following factors:
Solar constant.
Distance between the Earth and the Sun.
Physical state of the atmosphere over the point.
Altitude of the Sun.
Values of I and I’ have well-defined diurnal and annual variations. Maximal values is observed at the local noon. They are also influenced by turbidity of the atmosphere. They increase with increasing altitude of a locality (in this case optical mass decreases). It is why in mountain areas these quantities are larger than over planes.

condition it completely blocks the DSR.
The DSR fluxes falling on

slanted surfaces are different of those falling on horizontal surfaces

unit of area in a unit of time is named

SCATTERED RADIATION FLUX (i).
It depends on
Altitude of the Sun
Transparency of the atmosphere
Cloudiness

The DR flux reaches its maximum value at medium and high level clouds. At some cases it can be 2-3 times more intensive than the clear sky does.
The maximal value of DR is observed at local noon when the Sun altitude is the highest for the given day.

In a certain condition, contrary to the DSR, cloudiness makes DR stronger. However, some interior clouds (St, Sc at ho<15°)can not do that.

atmosphere
Empirical coefficient
For ideal atmosphere
For real atmosphere
There are some other formulas
“c”

is parameter describing the atmosphere transparency.

From these formulas it follows:
At c=const , the DR flux is proportional to I . The Sun altitude increases (m value decreases), DR grows up.
The ratio i/I depends upon c value only. For ho=10..75
The ratio i/I grows up when the Sun altitude and c value decrease

in the sky, amount of DR grows up.
Snow cover also

makes some contribution into increase of DR.

DSR is reflected by the snow

Atmosphere scatters the reflected DSR

A part of back scattered radiation comes back to the surface

The maximal energy of the DR falls on wavelength