Transcript Why learn about radiative transfer

Training Course Module DA.
Data assimilation and use of satellite data.
Introduction to infrared radiative transfer.
Marco Matricardi, ECMWF, March 2006.
Why learn about radiative transfer
►The minimisation procedure involved in 4D-Var requires the computation of
the gradient of the cost function with respect to the atmospheric profile.
►As a consequence, a prerequisite for exploiting radiance data from satellite
sounders is the availability of a radiative transfer model (usually called the
observation operator) to predict a first guess radiance from the NWP model
fields corresponding to every measured radiance.
►The radiative transfer model and its adjoint are therefore a key
component to enable the assimilation of satellite radiance in a NWP
system.
Radiance
►A fundamental quantity associated to a radiation field is the intensity of
the radiation field or radiance.
Radiance is defined as:
dE
L 
cos( ) dA dt d d
W
[ 2
]
1
m sr s
(1)
►Radiance is the amount of energy crossing, in a time interval dt and in
the frequency interval υ to υ +d υ, a differential area dA at an angle θ to
the normal to dA, the beam being confined to a solid angle dΩ.
Radiance can also be defined for a unit wavelength, λ, or wave number,  ,
interval, the relation among these quantities being:
c 1
 
(2)
 
Radiance
►Satellite radiometers make measurements over a finite spectral interval
(channel). They respond to radiation in a non-uniform way as a function of
frequency.
►To represent the outgoing radiance as viewed by a satellite, the
spectrum on monochromatic radiance must be convolved with the
appropriate instrument response function
2
Lˆ *   L f ( *  )d
(3)
1
Where f ( *  ) is the normalised instrument response function and ^ over the
symbol denotes convolution. Here  * is the central frequency of the channel.
Black-body radiation
►To explain the spectral distribution of radiance emitted by solid bodies,
Planck found that the radiance inside an enclosure (black body radiance) at
constant temperature T is expressed by:
B (T ) 
2h 3
c 2 (e
(
h
)
kT
 1)
where h is the Planck’s constant and k is the Boltzman constant.
►The spectral distribution of radiance emitted by the cavity
depends only on its temperature, whatever spectral distribution of
radiance is entering the cavity.
(4)
Black-body radiation
If we define c1  2 hc 2 and c2  hc / k as the first and second radiation
constant, we can give an equivalent formula in terms of radiance per unit
wavelength:
B (T ) 
c15
 (e
(
c2
)
T
(5)
 1)
2kT 2
►As   0 or   , B (T )  2 . This is known as Rayleigh-Jeans
c
distribution.
 hv
2h 3 kT
►As    or   0, B (T )  2 e .
c
distribution.
This is known as Wien
Brightness temperature
►Radiance can be expressed in terms of the temperature that a
perfect black body would have to emit the same radiance at the
same wave number. This quantity is referred to as brightness
temperature Tb(K).
Transmittance and optical depth
►When radiation is transported in a medium, the intensity of the radiation field
decreases and we have extinction of radiation.
►The process of extinction is governed by the Lambert’s law. It
states that the change of intensity along a path is proportional to the
amount of matter in the path.
Transmittance and optical depth
Whenever a beam of monochromatic radiation whose radiance is L(υ) enters a
medium, the fractional decrease experienced is:
dL  L ke ( ) dx
Here ρ=ρ(x) is the density of the medium at x and ke(υ) is a proportionality
factor called the mass extinction coefficient.
(6)
Transmittance and optical depth
►The
extinction coefficient can be expressed as the sum of an
Absorption coefficient,ka, and a scattering coefficient, ks, and we
can say that a radiation field transported by a medium will
experience a reduction of intensity due to absorption and scattering.
►When absorption occurs, radiant energy is transformed to kinetic energy.
►When scattering occurs, there is no change to another form of internal energy
and the radiant energy is re-emitted by the volume.
►In
a planetary atmosphere we are often confronted by the case
when there is only absorption. However, scattering can occur in
presence of aerosols and clouds.
Transmittance and opical depth
Integration of Eq.(6) between 0 and s yields:
0
L ( s)  L (0)exp( ke ( )  dx
s
where Lυ(0) is the radiance entering the medium at x=0.
►This is known as Beer-Bouguer-Lambert law. The ratio
Lυ(s)/Lυ(0) is called the spectral transmittance of the medium.
(7)
Transmittance and opical depth
optical depth τυ(s,0) of the medium between points s and 0 is
defined as:
►The
0
 (s,0)   ke  dx
(8)
s
►The
spectral transmittance can be written as:
 (s, 0)  exp(v (s, 0))
(9)
The equation of radiative transfer for a
plane-parallel atmosphere
►In a plane-parallel atmosphere variations in the intensity and
atmospheric parameters depends only on the vertical direction.
Distances can be measured along the normal Z to the plane of
stratification of the atmosphere.
Z
s
θ
z
►A beam of radiation travelling along the direction s will experience a
reduction of intensity due to extinction by the material and a increase of
intensity by emission from the material plus scattering from all other directions.
The equation of radiative transfer for a
plane-parallel atmosphere
►The equation of radiative transfer for a plane parallel
atmosphere can then be written as:
 dL ( z, )   L ( z, )ke  dz  J ( z, )  dz
The quantity Jυ is called the source function within dz and μ=cos(θ).
(10)
The equation of radiative transfer for a
plane-parallel atmosphere
In the most general case where the presence of solar radiation is considered,
the source function Jυ is written as:
J ( z ,  )  k s
2 1
ks

 L ,
4

exp(  ) P(  ,  ) 

I ( z ,  / ) P(  ,  / ) / 4 d  / d / 
(11)
0 1
JATM ( z ,  )
Here L ,  is the radiance of the solar beam at the top of the atmosphere,  is
the angular diameter of the sun and, P, the phase function, describes the
angular distribution of the scattered energy.
The equation of radiative transfer for a
plane-parallel atmosphere
►The first term in the source function represents the increase in
intensity from the single scattering of the un-scattered direct solar
beam from the direction   to  .
►The second term represent the increase in intensity from the
multiple scattering of the diffuse intensity from the directions
 / to  .
►The third term represent the increase in intensity from emission
within the atmosphere in the direction μ.
The equation of radiative transfer for a
plane-parallel atmosphere
►If the Earth’s atmosphere is in thermodynamic equilibrium (this
happens below 40 to 60 km where a single kinetic temperature
characterise, to a good approximation, the gas), a volume of gas
behaves approximately as a black cavity (Kirchoff’s law).
►The emission from the volume of gas is then dependent only on its
temperature.
The term
JATM ( z ,  ) in the source function can then be written as:
JATM ( z,  )  ka B (T ( z ))
(12)
The equation of radiative transfer for a
plane-parallel atmosphere
►In presence of multiple scattering, the radiative transfer equation
cannot be solved analytically.
►An exact solution can only be obtained using numerical techniques
(e.g. discrete-ordinates method, doubling-adding method).
►An analytical solution can however be obtained if approximate
methods are used (e.g. two/four-stream approximation and Eddington
approximation).
The equation of radiative transfer for a
plane-parallel atmosphere
►If we consider only absorption and emission, the radiative
transfer equation for a plane-parallel atmosphere in local
thermodynamic equilibrium can be written as:
 dL ( z,  )   L ( z,  )ka  dz  B (T ( z ))ka  dz (13)
In terms of the coordinate τ , the equation can be rewritten as:
 dL ( z,  )  L ( z,  )d  B (T ( ))d
(14)
The equation of radiative transfer for a
plane-parallel atmosphere
The clear sky radiance at the top of the atmosphere can then be written as:
0
LCl ( )  B (Ts ) ( ) s   B ( / )
s
d
/
d


/
d
 0   B ( / )  d 

/
Sun
s 
[1   ( )] s   
d


L

 /

2
d

 s 



(15)
Where Ts is the surface temperature,  ( ) is the surface emissivity,  s is
sun
L
the transmittance from the surface to the top of the atmosphere and  is the
solar contribution.
The equation of radiative transfer for a
plane-parallel atmosphere
►The first term in L ( ) is the surface emission attenuated to the top of
the atmosphere.
Cl
►The second term is the emission contribution from the atmosphere.
►The third term is the downward radiance from the atmosphere reflected
back upward and then attenuated to the top of the atmosphere.
►The fourth term is the solar radiance attenuated to the surface, reflected
back upward and then attenuated to the top of the atmosphere. In the
infrared, solar radiance is important for wavelengths shorter that 5 micron.
The equation of radiative transfer for a
plane-parallel atmosphere
►Eq.(15) can be integrated numerically by dividing the atmosphere
into a number of homogeneous layers.
►For the assumption of homogeneity to be valid, the atmosphere has
to be divided up into a sufficiently large number of layers.
►The layers are defined by a number of pressure levels that usually
range from 0.005 hPa (top of the atmosphere) to 1013.5 hPa (surface).
The equation of radiative transfer for a
plane-parallel atmosphere
We can rewrite Eq.(15) in discrete layer notation for N atmospheric layers
(the layers are numbered from space, layer 1, to the surface, layer N) and for
a single angle to simplify notation:
 N

L  B (Ts )  s    B (T j )( , j 1   , j )  
 j 1

Cl
 N
 2s ( , j 1   , j ) 
(1   s )   B (T j )



 j 1

 , j  , j 1
(1   ) L,   s (  ) s (  )
(16)

Here  , j is the transmittance from a given level to space and T ( j ) is the
average temperature of the layer.
Mechanism for gaseous absorption
►Emission spectra of the Earth and atmosphere show large variations in
energy emitted upwards. These variations are due to complex interactions
taking place within the atmosphere between molecules and
electromagnetic fields.
►For interaction to take place, a force must act on a molecule in the
presence of an external electromagnetic field. For such a force to exist,
the molecule must possess an electric or magnetic dipole moment.
►In general, only asymmetric molecules as CO, N20, H2O and O3 possess a
permanent dipole moment. Symmetric molecules as N2, O2, CO2 and CH4
do not.
►However, as a molecule like CO2 vibrates, an oscillating electrical dipole
moment is generated and an interaction can take place.
Mechanism for gaseous absorption
►Interaction between the molecule and the external field take place
whenever a quantum of energy h o is extracted from (absorption
process) or added (emission) to the external field. When this process
occurs, we say we are in presence of an absorption line.
►The basic relation holds:
E /  E //  h o
(17)
where E / and E // are the two energy levels involved in the transition and
 o is the centre frequency of the absorption/emission line.
►In general
E /  E //  Eelec  Evib  Erot
(18)
Mechanism for gaseous absorption
► Eelec are changes in the molecule electron energy levels and result in
absorption/emission at U.V. and visible wavelengths
► Evib are changes in the molecule vibrational energy levels and result
in absorption/emission at near-infrared wavelengths. They are generally
accompanied by rotational transitions and one observes a group of lines that
constitutes a vibration-rotation band.
► Erot are changes in the molecule rotational energy levels and result
in absorption/emission at microwave and far-infrared wavelengths.
Gaseous absorption:line shape and absorption coefficient
►For a strictly monochromatic absorption and emission to occur at  o ,
the energy involved should be exactly E  h o implying that the energy
levels are exactly known.
►The molecular absorption coefficient
kaM  S (  o )
k aM can then be expressed as:
(19)
where is the delta Dirac function.
►However three physical phenomena occur in the atmosphere, which produce
broadening of the line:
1)Natural broadening
2)Collision broadening
3)Doppler broadening
Gaseous absorption:line shape and absorption coefficient
Natural broadening:
It is caused by smearing of the energy levels involved in the transition. In
quantum mechanical terms it 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:
n
S
k 
 (  o )2   n2
M
a
,

S   k ( )d

(20)
Where S is the line strength and  n is the line half width. The line half
width is independent of frequency and its value is of the order of 10 5 nm.
Gaseous absorption:line shape and absorption coefficient
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 emitted and absorbed radiance. The
absorption coefficient is:
k aM 
S
d

exp[
(  o ) 2
 d  3.58 10  o
7
Where
Ma
is the molecular mass.

T
Ma
2
d
]
(21)
(22)
Gaseous absorption:line shape and absorption coefficient
Collisional broadening:
It is due to the modification of molecular potentials, and hence the energy levels,
which take place during each emission (absorption) process, and is caused by
inelastic as well as elastic collision between the molecule and the surrounding ones.
The shape of the line is Lorentzian, as for natural broadening, but the half width is
several order of magnitudes greater, and is inversely proportional to the mean free
path between collisions, which indicates that the half width will vary depending on
pressure p and temperature T of the gas.
When the partial pressure of the absorbing gas is a small fraction of the total gas
pressure we can write:
 c   c,s
p
ps
Ts
T
Where ps and Ts are reference values.
(23)
Gaseous absorption:line shape and absorption coefficient
►Collisions are the major cause of broadening in the troposphere while Doppler
broadening is the dominant effect in the stratosphere.
►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 are independent, the collision broadened line shape can be
shifted by the Doppler shift and averaged over the Maxwell distribution to obtain
the Voigt line shape.
►The Voigt line shape cannot be evaluated analytically. For its computation, fast
numerical algorithms are available.
Gaseous absorption:departure from Voigt line shape
►Comparison of accurate calculations with measurements taken by
high spectral resolution instruments have shown the importance of the
finer details of line shape. For some molecules the simple Voigt line
shape is inadequate.
►In particular, the continuum type absorption must be considered.
Gaseous absorption:continuum absorption
►The continuum absorption is not accounted for by line shapes based on simple
collision broadening theory and having Lorentzian wings.
Mechanism: the exact mechanism continues to be a matter of debate. There are
two main theories, (1) the continuum is due to inadequate description of line shape
away from line centres, (2) the continuum is due to molecular polymers (e.g. water
vapour dimers).
Formulation:empirical algorithms based on laboratory and field measurements are
available that provide an estimate of the continuum absorption for any atmospheric
path. The problem is that most of the measurements are for warm paths (300 K)
whereas most atmospheric paths are colder than this.
Gases: H2O, CO2, N2,O2
Gaseous absorption:computation of the optical depth
due to line absorption
►The optical depth due to line absorption for an atmospheric layer
comprising J single gas is computed by performing the sum of the optical
depth evaluated for each single gas and each single absorption line.
 aM ( )  all lines i all gases j [Si , j g ( , i ) j ]
where
(24)
Si , j
is the strength of the line i adjusted to the conditions of the gas,
g ( , i ) is the normalized line shape function for line i, and  j is the gas
amount in the layer.
►The models used to compute the gaseous optical depth due to line
absorption are called line-by-line models (GENLN2, LBLRTM,
HARTCODE).
Gaseous absorption:computation of the optical depth
►Line-by-line models are computationally expensive both in CPU and
disk space.
►Efforts to alleviate this have lead to the development of radiative transfer
models (4A, K-carta) that use absorption coefficients stored in a look-uptable.
►Because the monochromatic absorption coefficient vary slowly with
temperature and is directly proportional to the absorber amount, the
monochromatic optical depths stored in the look-up table can be
interpolated in temperature and modified for changes in absorber amount to
give the most appropriate optical depths for a given profile.
Computation of the optical depth due to scattering
►Particulates contained in the Earth’s atmosphere vary from aerosols, to
water droplets and ice crystals.
►The range of shapes for aerosols vary from quasi-spherical to highly
irregular with a size typically less than 1 μm.
►Small water droplets are by their nature spherical in shape with a size
typically less than 10 μm.
►Ice crystals are mainly present in cirrus clouds. The shape of ice crystals
vary greatly with a size typically less than 100 μm. Although their shape
include solid and hollow columns, prisms, plates, aggregates and branched
particles, an hexagonal column shape is typically assumed for ice crystals.
Computation of the optical depth due to scattering
►The computation of the absorption/scattering coefficient (and phase
function) for particles with a spherical shape can be performed by using the
exact Lorentz-Mie theory for any practical size. This is the approach usually
followed for aerosols and water droplets.
►For nonspherical ice crystals, an exact solution that covers the whole
range of shapes and sizes observed in the Earth’s atmophere is not
available in practice.
Computation of the optical depth due to scattering by
ice crystals
►If the size of an ice crystal is much larger than the wavelength of the
incident radiation, the geometric optics approach can then be used.
►The geometric optics approach is based on the assumption that a light
beam can be considered to be made of a bundle of separate parallel rays that
undergo reflection and refraction outside and inside the crystal. This is the
only practical method to compute optical parameters for large non-spherical
particles.
►For smaller sizes, other techniques have to be employed such as the
Finite-Difference Time Domain Method, the T-Matrix method and the
Direct Dipole Approximation Method.
Computation of the total optical depth
►The total extinction optical depth for an atmospheric layer where
absorption and scattering take place, can be written as:
 etot ( )   aM ( )   acont ( )   sscatt ( )
 aM ( )  Line absorption
 acont ( )  Continuum type absorption
 sscatt ( )  Scattering (aerosols and clouds)
Fast radiative transfer model for use in the NWP model
►A prerequisite for exploiting satellite radiance data in the NWP
model using the variational analysis scheme is the prediction of
radiances given first guess model fields.
►Line-by-line (and fast line-by-line) models are too slow to be used
operationally in NWP.
►To cope with the processing of observations in near real-time,
hyper fast radiative transfer models have been developed. An hyper
fast radiative transfer model has to be accurate and computationally
efficient.
Fast radiative transfer model for use in the NWP model
►There are several types of fast radiative transfer models in use or
under development, which are relevant to infrared radiance
assimilation.
The various models can be categorised into:
1) Regression based fast models
2) Physical models
3)Neural network based models
RTTOV, the regression model on pressure levels used at
ECMWF
►In the RTTOV model, the computation of the channel averaged
optical depth involves a polynomial with terms that are functions of
temperature, absorber amount, pressure and angle.
For a given homogeneous layer j we can write:
M
el
ˆmod
  a j ,k , * X k , j
j , *
(25)
k 1
where M is the number of predictors and the functions X k , j constitute the
profile-dependent predictors of the fast transmittance model.
►To compute the expansion coefficients a j ,k , , a line-by-line
model is used to compute accurate channel averaged optical depths
for a number of temperature, humidity and ozone atmospheric
profiles.
*
RTTOV, the regression model on pressure levels used at
ECMWF
►These atmospheric profiles are chosen to be representative of widely
differing atmospheric situations.
►The line-by-line optical depths ˆ j , * are then used to compute the
expansion coefficients by linear regression of X k , j against the predictor
values calculated from the profile variables for each profile at each angle.
lbl
►The expansion coefficients can then be used by the model to compute
optical depths given any other input profile.
RTTOV, the regression model on pressure levels
used at ECMWF
►The functional dependence of the predictors used to parameterise the
optical depth depends mainly on factors such as the absorbing gas, the
spectral response function and the spectral region although also the layer
tickhness can be important.
►The basic predictors are defined from the layer temperature and the
absorber amount of the gas.
►Since we are predicting channel averaged optical depths (polychromatic
regime) a number of predictors have to be included that in general depend
on pressure-weighted quantities above the layer.
HIRS: High resolution Infrared Radiation Sounder
IASI: Infrared Atmospheric Sounding Interferometer
RTTOV, the regression model on pressure levels used at
ECMWF
►The error introduced by the parameterisation of the optical depths can
be assessed by comparing fast model and line-by-line computed radiances.
►The largest errors are usually associated with water vapour and ozone but
in general, for all the instruments simulated by RTTOV, the radiance rms
error is usually below the instrument noise.
►Note that RTTOV comes with associated routines to compute Jacobians
with respect to input profile variables. This is a prerequisite for the model
to be used in NWP assimilation.
HIRS:High Resolution Sounder
AIRS: Atmospheric Infrared Sounder
The parametrization of scattering in a fast radiative
transfer model
►The computational efficiency of a fast radiative transfer model can be
seriously degraded if explicitly calculations of multiple scattering are to be
introduced.
►However, a parameterization of multiple scattering is possible that allows
to write the radiative transfer equation in a form identical to that in clear
sky conditions.
The parametrization of scattering in a fast
radiative transfer model
►This parameterization (scaling approximation) rests on the hypothesis that the
diffuse radiance field is isotropic and can be approximated by the Planck
function.
►In the scaling approximation the absorption optical depth, a , is replaced by an
effective extinction optical depth,  e , defined as
 e   a  b s
(26)
The parametrization of scattering in a fast
radiative transfer model
Here  a is the scattering optical depth and b is the integrated fraction of
energy scattered backward for radiation incident either from above or from
below.
If P (  ,  ) is the azimuthally averaged phase function, the scaling factor b can
be written in the form
/
1
0
1
/
/
b   d   P (  ,  )d 
20
1
(27)
Fast radiative transfer model:regression model on layers of
equal absorber amount
►This approach (known as the Optical Path Transmittance (OPTRAN)
method) is similar to the one used in the fast models that use fixed pressure
levels but uses layers of equal absorber amount instead.
►In OPTRAN the atmosphere is sliced into layers according to layer-tospace absorber amount rather than atmospheric pressure.
This can be advantageous for gases like water vapour where the path
absorber amounts are not simple functions of pressure.
In fact in the OPTRAN approach the layer absorber amount is constant
across the layer and pressure becomes a predictor.
Fast radiative transfer model:physical models
►This approach averages the spectroscopic parameters for each
channel and uses these to compute layer optical depths.
The advantages of this approach are:
1)More accurate computations for some gases
2)Any vertical co-ordinate grid can be used
3)It is to modify if the spectroscopic parameters change
►However to date these models are a factor 2-5 slower than the regression
based ones which is significant for assimilation purposes.
Fast radiative transfer model:neural networks
►Models using neural networks have been developed and may
provide even faster means to compute radiances.
►However, the gradient versions of the model to compute
Jacobians are proving difficult to develop and more work is needed
before operational centres can consider using these techniques for
radiance assimilation.
Bibliography
Many of the aspects of the subjects treated in this lectures are covered in the
following books:
Goody, R.M. and Yung,Y.L., 1995: Atmospheric Radiation:Theoretical
Basis . Oxford University.
Chandrasekhar, S., 1950:Radiative transfer. Dover.
Liou, K.N., 2002:An introduction to atmospheric radiation.Academic Press.
Bibliography
Mechanism for absorption and scattering
Armstrong, B.H., 1967:Spectrum line profiles:the Voigt function. J. Quant.
Spectrosc. Rad. Transfer, 52, pp. 281-294.
Clough, S.A., Kneizys, F.X. and Davis, R.W., 1989:Line shape and the
water vapour continuum, Atmos. Research, 23, pp. 228-241.
Van de Hulst, H.C. 1957:Light scattering by Small Particles. Wiley.
Mishchenko, M.I., Hovenier, J.W. and Travis, L.D., 2000:Light scattering
by Nonspherical particles. Academic Press.
Bibliography
Line-by-line models:
Edwards, D.P., 1992: GENLN2. A general line-by-line atmospheric
transmittance and radiance model. NCAR Technical note NCAR/TN367+STR (National Center for Atmospheric Research, Boulder, Co., 1992)
Clough, S.A., Jacono, M.J. and Moncet, J.L., 1992: Line-by-line
calculations of atmospheric fluxes and cooling rates: application to water
vapour. J. Geophys. Res., 97, pp. 15761-15785.
Miskolczi, F., Rizzi,R., Guzzi, R. and Bonzagni, M.M., 1998: A new high
resolution atmospheric transmittance code and its application in the field of
remote sensing. In Proceedings of IRS88: Current problems in atmospheric
radiation, Lille, France, 18-24 August 1988, pp. 388-391.
Bibliography
Line-by-line models based on look-up tables:
Strow,L.L., Motteler,H.E., Benson,R.G.,Hannon, S.E. and De SouzaMachado,S., 1998: Fast computation of monochromatic infrared
atmospheric transmittances using compressed look-up-tables. J. Quant.
Spectrosc. Rad. Transfer, 59, pp. 481-493.
Scott, N.A. and Chedin,A., 1981: A fast line-by-line method for
atmospheric absorption computation: the Automatized Atmospheric
Absorption Atlas, J. Appl. Meteor., 20, pp. 802-812.
Bibliography
Fast models on fixed pressure levels:
Mc Millin L.M., Fleming, H.E. and Hill, M.L., 1979:Atmospheric
transmittance of an absorbing gas. 3: A computationally fast and accurate
transmittance model for absorbing gases with variable mixing ratios.
Applied Optics, 18, pp. 1600-1606.
Eyre, J.R. 1991: A fast radiative transfer model for satellite sounding
systems. ECMWF Research Department Technical Memorandum 176
(available from the librarian at ECMWF).
Matricardi, M. and Saunders, R., 1999: A fast radiative transfer model for
simulation of IASI radiances. Applied Optics, 38, pp. 5679-5691.
Bibliography
Regression based fast models on levels of fixed absorber amount:
Mc. Millin, L.M., Crone, L.J. and Kleespies, T.J., 1995:Atmospheric
transmittances of an absorbing gas. 5. Improvements to the OPTRAN
approach. Applied Optics, 24, pp. 8396-8399.
Physical models:
Garand, L., Turner, C., Chouinard, C. and Halle J., 1999: A physical
formulation of atmospheric transmittances for the massive assimilation of
satellite infrared radiances. J. Appl. Meteorol., 38, pp. 541-554.