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Principles of remote sensing
of atmospheric parameters from space
February 1998
By R. Rizzi and updated by
R. Saunders
European Centre for Medium-Range Weather
Forecasts, Shinfield Park, Reading RG2 9AX, U.K.
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8 . Line shapes and the absorption coefficient.
For a strictly monochromatic absorption and emission to occur at
, the energy involved for each molecule of gas should
be exactly 
, implying
that the energy levels are exactly known. The mathematical description of
the absorption line would be 
where 
is the line strength and 
is the Dirac delta function centred at 
. However three
physical phenomena occur in the atmosphere (and elsewhere) which produce
broadening of the line: (i) natural broadening, (ii) collision broadening
and (iii) doppler broadening, which will be briefly discussed.8.1 Natural broadening.
It is caused by smearing of the energy levels involved in the transition. In quantum mechanical terms this is due to the uncertainty principle and depends on the finite duration of each transition. It can be shown that the appropriate line shape to describe natural broadening is the Lorentz line shape
Figure 14 . Emission spectrum of the atmosphere measured at 10 km height with a resolution of approximately half wavenumber. Most of the lines are due to the water vapour vibro-rotational band.
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where
is the line strength
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and
is the line half width, which is the distance from
the line centre 
to 
where 
has decreased to half of its maximum power. The shape
of a Lorentzian line is shown in Fig. 16 . It can be shown that 
is independent of wave number and its value is of the order of 10-5
nm
Figure 15 .Transmittances of the millimetre wave region calculated for a path between the surface and space for molecular oxygen, water vapour and their product (taken from Grody, 1976).
8.2 Collisional broadening.
It is due to the modification of molecular potentials, and hence to the energy levels, which take place during each emission (absorption) process, and is caused by inelastic as well as elastic collisions between the molecule and the surrounding ones. The shape of the line is still Lorentzian, as for natural broadening, but the half width
is several orders of magnitude greater,
and is inversely proportional to the mean free path between collisions,
which indicatesthat
will vary depending on pressure 
and temperature
of the gas. When the partial pressure
of the absorbing gas is a small fraction of the total gas pressure we can
write:
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where
and 
are reference values. As an example the collisional
half width for the CO molecule in the vicinity of the vibrational 
vibrational transition, at standard pressure and 
K is 
nm8.3 Doppler broadening.
Molecules in a volume of air possess a Maxwell velocity distribution, hence the velocity components along any direction of observation produce a Doppler effect which induces a shift in frequency in the emitted and absorbed radiance. The line shape is
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where
is the molar mass. The Doppler half width for the
CO transition of the previous paragraph is 
nm which is about 1/8 of the collisional half width. Collisions
are the major cause of broadening in the troposphere, since their effect
is proportional to pressure, and pressure variations are larger than temperature
variations; while Doppler broadening is the dominant effect in the stratosphere,
due to the larger mean free path and high temperatures, the latter producing
larger standard deviations for the molecular velocity distribution. There
is however an intermediate region where neither of the two shapes is satisfactory
since both processes are active at once. Assuming the collisional and doppler
broadening can be assumed to be independent we can combine both line shapes
to give the Voigt shape. This is often used to allow for both tropospheric
and stratospheric broadening of an absorption/emission line in one calculation.Actually the level of our knowledge of spectroscopic phenomena, that are at the very core of our understanding of the radiative processes, is still insufficient for many purposes, and a large effort is taking place to perform more accurate measurements of the key parameters, in order to avoid, whenever possible, the use of empirical tuning to reduce discrepancies between measurements and model results.
Figure 16 . Spectral line shape produced by: (a) Doppler broadening and (b) natural and collision broadening (taken from Levi (1968).
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