# Training Course Module DA. Data assimilation and use of satellite data

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Training Course Module DA. Data assimilation and use of satellite data
Training Course Module DA. Data assimilation and use of satellite data. Introduction to infrared radiative transfer. Marco Matricardi, ECMWF 5 May - 6 May 2010.

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.

Electromagnetic radiation at the top of the atmosphere

Near Infrared Far infrared After Liou (2002)

A fundamental quantity associated to a radiation field is the intensity of the radiation field or radiance. The monochromatic Radiance is defined as: (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: (2)

Normal to dA Pencil of radiation Differential area, dA
After Liou (2002)

Radiance Satellite radiometers make measurements over a finite spectral interval. They respond to radiation in a non-uniform way as a function of frequency. To represent the outgoing radiance as viewed by a radiometer, the spectrum of monochromatic radiance must be convolved with the appropriate instrument response function. This yields to so-called polychromatic or channel radiance. The channel radiance is defined as: (3) Where is the normalised instrument response function and ^ over the symbol denotes convolution. Here is the central frequency of the channel.

Central frequency of the channel

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: (4) where h is the Planck’s constant and k is the Boltzman constant. The spectral distribution of radiance emitted by the enclosure depends only on its temperature, whatever spectral distribution of radiance is entering the enclosure. Example of a blackbody source

Black-body radiation If we define and as the first and second radiation constant, we can give an equivalent formula in terms of radiance per unit wavelength: (5) As This is known as Rayleigh-Jeans Distribution This is known an Wien distribution

After Valley (1965)

Brightness temperature
The radiance, L(ν), can be expressed in units of equivalent brightness temperature, Tb(ν). The brightness temperature is the temperature (in Kelvin) that a perfect black body would need to have to emit the observed radiance at a given frequency. The brightness temperature can be computed by inverting Planck’s formula (equation (4) ).

Transmittance and optical depth
When electromagnetic radiation is transported in a medium, the radiance decreases and we have extinction of radiation. The process of extinction is governed by the Lambert’s law. It states that the change of radiance along a path is proportional to the amount of matter in the path. After McCartney (1983)

Transmittance and optical depth
Whenever a beam of monochromatic radiation whose radiance is L(υ) enters a medium, the fractional decrease experienced is: (6) Here ρ=ρ(x) is the density of the medium at x and ke(υ) is a proportionality factor called the mass extinction coefficient.

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 the radiance due to absorption and scattering by the medium. 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 optical depth
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.

Transmittance and opical depth
Integration of Eq.(6) between x=0 and x=s yields: (7) 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.

Transmittance and opical depth
The optical depth τυ(s,0) of the medium between points s and 0 is defined as: (8) The spectral transmittance can be written as: (9)

The equation of radiative transfer for a plane-parallel atmosphere
In a plane-parallel atmosphere variations in the radiance and atmospheric parameters depend only on the vertical direction (i.e. the atmosphere is horizontally homogeneous). Distances can be measured along the normal Z to the plane of stratification of the atmosphere. A beam of radiation travelling along the direction s, will experience absorption and emission processes simultaneously. Z s θ z Localized portion of the Atmosphere

The equation of radiative transfer for a plane-parallel atmosphere
atmosphere can then be written as: The quantity Jυ is called the source function within dz and μ=cos(θ). The radiance of a pencil of radiation is reduced due to absorption and scattering by the medium The radiance of a pencil of radiation is increased by emission from the medium plus multiple scattering from all other directions into the pencil.

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: Here 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 radiance due to the single scattering of the un-scattered direct solar beam from the direction The second term represent the increase in radiance due to the multiple scattering of the diffuse radiance from all the directions The third term represent the increase in radiance due to the 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 in the source function can then be written as:

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: In terms of the coordinate τ , the equation (13) can be rewritten as:

The equation of radiative transfer for a plane-parallel atmosphere
Eq. (14) can be solved for the upward and downward radiances. If we apply the appropriate boundary conditions at the surface and at the top of the atmosphere, the clear sky radiance at the top of the atmosphere can then be written as: Where Ts is the surface temperature, is the surface emissivity, is 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 is the surface emission attenuated to the top of the atmosphere. The second term is the upward radiance 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 only 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 hPa (top of the atmosphere) to 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: Here is the transmittance from a given level to space and is the average temperature of the layer.

After Hanel (1971)

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 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/emission line. The basic relation holds: where and are the two molecular energy levels involved in the transition and is the centre frequency of the absorption/emission line. A molecule can have rotational energy due to its rotation, vibrational energy due to the oscillations of the atoms and electronic energy. In general

Mechanism for gaseous absorption
are changes in the molecule electron energy levels and result in absorption/emission at U.V. and visible wavelengths 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. 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 , the energy involved should be exactly implying that the energy levels are exactly known. The molecular absorption coefficient can then be expressed as: where is the delta Dirac function. However three physical phenomena occur in the atmosphere, which produce broadening of the line: 1)Natural broadening )Collision broadening )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: Where S is the line strength and is the line half width. The line half width is independent of frequency and its value is of the order of 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: Where is the molecular mass.

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: Where ps and Ts are reference values.

After Levi (1968)

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

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:

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. where is the strength of the line i adjusted to the conditions of the gas, is the normalized line shape function for line i, and 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, RFM).

After Valley (1965)

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-up-table. 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.

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 4) Principal component 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: where M is the number of predictors and the functions constitute the profile-dependent predictors of the fast transmittance model. To compute the expansion coefficients , 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 are then used to compute the expansion coefficients by linear regression of against the predictor values calculated from the profile variables for each profile at each angle. 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 thickness 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 explicit 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, , is replaced by an effective extinction optical depth, , defined as

The parametrization of scattering in a fast radiative transfer model
Here 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 is the azimuthally averaged phase function, the scaling factor b can be written in the form

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-to-space 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:optimal spectral sampling (OSS) method
The Optimal spectral sampling method (OSS) uses a few representative monochromatic transmittances to derive narrowband or moderate-to-high spectral resolutions transmittances. In the OSS method, top of the atmosphere radiances are computed by statistically selecting top of the atmosphere monochromatic radiances. This approach has the advantage that, for instance, multiple scattering can be easily incorporated in these 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.

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.

Principal component (PC) based fast models have been developed for applications to hyperspectral remote sensing. Instead of predicting layer optical depths, PC models compute the Principle Component scores, which have much smaller dimensions as compared to the number of channels. This optimizations results in a significant decrease of the computational time without affecting the accuracy of the results.

Bibliography Many of the aspects of the subjects treated in this lecture 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 Clough, S.A., Kneizys, F.X. and Davis, R.W., 1989:Line shape and the water vapour continuum, Atmos. Research, 23, pp 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/TN-367+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 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, August 1988, pp

Bibliography Line-by-line models based on look-up tables:
Strow,L.L., Motteler,H.E., Benson,R.G.,Hannon, S.E. and De Souza-Machado,S., 1998: Fast computation of monochromatic infrared atmospheric transmittances using compressed look-up-tables. J. Quant. Spectrosc. Rad. Transfer, 59, pp 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

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 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

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 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

Bibliography Principal components model:
X. Liu, W.L. Smith, D.K. Zhou, A. Larar: “Principal component-based radiative transfer model for hyperspectral sensors: theoretical concept”, Appl. Opt, 45, (2006). M. Matricardi: “A principal component based version of the RTTOV fast tradiative transfer model”, ECMWF Tech. Memo No. 617 (2010). OSS Method: J.L. Moncet, G. Uymin, H.E. Snell, “Atmospheric radiance modelling using the optimal spectral sampling (OSS) method”, Proc. Of SPIE, 5425, (2004).

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