# 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 26 April - 27 April 2011.

The exploitation of radiance data from satellite sounders requires 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 in the assimilation of satellite radiance in a NWP system.

at the top of the atmosphere The forward model simulates the observed radiances using the equation of radiative transfer

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, Lν, is defined as the amount of energy crossing, in a time interval dt and in the frequency interval to , a differential area dA at an angle θ to the normal to dA, the pencil of radiation being confined to a solid angle dΩ. (1) Normal to dA Pencil of radiation The radiance can also be defined for a unit wavelength, λ, or wave number, , interval. If, c, is the speed of light in vacuum, the relation between these quantities is: After Liou (2002) (2) Differential area, dA Wavelengths are usually expressed in units of microns (1µm=10-6 m) whereas wave numbers are expressed in units of cm-1.

Radiance Satellite radiometers make measurements over a finite spectral interval. They respond to radiation in a non-uniform way as a function of frequency (or wave number). AIRS: Atmospheric Infrared Sounder Monochromatic radiance 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: Where is the normalised instrument response function and ^ over the symbol denotes convolution. Here is the central frequency of the channel. (3)

If we consider a gas in thermodynamic equilibrium inside an isothermal enclosure at temperature T, the radiance inside the enclosure is expressed by the Planck’s function: where h is the Planck’s constant and k is the Boltzman constant. Note that depends only on the temperature of the enclosure (for a fixed frequency). The radiance inside an isothermal enclosure is the same as the radiance emitted by a blackbody at the same temperature. A blackbody is a body which absorbs all the radiation incident on it. Bν(T) is also known an blackbody function and the radiation inside the enclosure is called blackbody radiation. (4) Note that for a finite-size cavity, the radiance inside the cavity can be expressed by the Planck’s function only if the wavelength of the radiation is smaller than the size of the cavity. A small hole in a large cavity is a very close approximation to an ideal blackbody if the wavelength of the incident radiation is shorter than the diameter of the hole. Large cavity with absorbing walls

If we define c1=2πhc2 and c2=hc/k as the first and second radiation constant, we can give an equivalent black body function in terms of radiance per unit wavelength: (5) for This is known as Rayleigh-Jeans Distribution This is known an Wien distribution

The peak of the blackbody curve shifts to shorter wavelengths
as the temperature of the blackbody increases. The peak is given by Wien’s displacement law: λ= /T [µm] Blackbody radiation curves for different temperatures After Valley (1965)

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

Transmittance and optical depth
Electromagnetic radiation in the atmosphere interacts with molecules and aerosol/cloud particles. This interaction can increase or decrease the intensity of the radiation field. The decrease of the intensity is called extinction whereas the increase of the intensity is called emission. In general, the extinction of radiation is the sum of two different mechanisms: absorption and scattering. In the infrared, atmospheric scattering occurs in presence of aerosol and cloud particles. When we have absorption, the intensity of the radiation field decreases because radiant energy is converted into other forms of energy (e.g. kinetic energy of the medium). When we have scattering, there is no conversion of the radiant energy to other forms of energy. The intensity of the radiation field decreases because radiant energy is redirected from its original direction. Absorbing medium Scattering medium Scattering particles After McCartney (1983)

Transmittance and optical depth
The process of extinction is governed by the Beer-Bouguer-Lambert law. It states that extinction is linear in the amount of matter and in the intensity of radiation. The attenuation of intensity along a path ds is thus: (6) s ds Mass extinction coefficient [m2Kg-1] Density of the medium [Kg m-3] The extinction coefficient can be written as the sum of the absorption coefficient, , and the scattering coefficient,

Transmittance and opical depth
The integration of Eq. (6) between points s1 and s2=s1+s yields : (7) The optical depth of the medium between points s1 and s2 is defined as: (8) The transmittance of the medium between points s1 and s2 is defined as: (9)

The change of intensity resulting from the interaction between radiation and matter is the sum of the contribution due to extinction and emission. If we assume that the emission process is linear in the amount of matter, the change of intensity can be written as: (10) Eq. (10) is the equation of radiative transfer (Schwarzchild’s equation). is called the source function. It consists, in general, of two parts: (11)

The equation of radiative transfer: the scattering source function
Scattering is a source of emission in a given direction because it can redirect radiant energy from other directions into the actual direction. Scattering particles In absence of solar radiation, the scattering source function can be written as: (12) is known as the phase function. It describes the angular distribution of the scattered energy.

The equation of radiative transfer: the thermal source function
Below the altitude of ~70 km, emissions from localized volumes of the Earth’s atmosphere are said to be in Local Thermodynamic Equilibrium (LTE). Under LTE conditions, a volume of gas behaves like a blackbody. In LTE conditions, the thermal source function of the emission from a local volume of the atmosphere is equal to the Planck’s function computed at the local kinetic temperature: (13)

In presence of scattering, the radiative transfer equation cannot be solved analytically. An “exact” solution for the scattering radiative transfer equation can only be obtained using numerical techniques (e.g. discrete-ordinate, doubling-adding, Monte-Carlo). An analytical solution, however, can be sought if approximate methods are used (e.g. two/four-stream approximation, Eddington/Delta-Eddington approximation, single scattering approximation, etc.)

The equation of radiative transfer for a plane-parallel atmosphere
We can solve the radiative transfer equation assuming that in localized portions the atmosphere is plane-parallel. In a plane-parallel atmosphere variations in the radiance and atmospheric parameters depend only on the vertical direction, and distances can be measured along the normal Z to the plane of stratification of the atmosphere. Localized portion of the Atmosphere Z s z θ

The equation of radiative transfer for a plane-parallel atmosphere
In absence of scattering, the radiative transfer equation for a plane-parallel atmosphere can be written as: (14) Eq. (14) can be rewritten in terms of the optical depth coordinate: (15)

The equation of radiative transfer for a plane-parallel atmosphere
Eq. (15) can be solved for the upward and downward radiances in a homogeneous atmospheric layer applying the appropriate boundary conditions at the top and at the bottom of the layer. If we divide the atmosphere into N homogeneous layers (N=1 is the top layer) bounded by N+1 pressure levels, the upward radiance at the top of the atmosphere and the and downward radiance at the surface can be written respectively as: (16) (17) Average temperature in the layer Radiance emitted by the surface Transmittance from the surface to the top of the atmosphere Transmittance from pressure level j to the top of the atmosphere Radiance coming from the top of the atmosphere. In the infrared this term can be omitted.

The equation of radiative transfer for a plane-parallel atmosphere
The radiance emitted by the surface can be written as: (18) Spectral emissivity of the surface Surface temperature The downward radiance is reflected upward by the surface The second term of Eq.(18) is valid under the assumption that the reflection from the surface is quasi-specular. Emissivity of sea water: nadir view and zero wind speed A blackbody as an emissivity of 1 at any frequency Wave number (cm-1)

Top of the atmosphere radiance spectrum
After Hanel (1971)

Mechanism for gaseous absorption
Radiance spectra measured at the top of the atmosphere show large variations in energy emitted upwards. These variations are due to complex interactions between molecules and the atmospheric radiation field. 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. Asymmetric molecules as CO, N20, H2O and O3 possess a permanent dipole moment. Symmetric molecules as N2, O2, CO2 and CH4 do not. For symmetric molecules, however, molecular vibrations and interactions with other molecules can generate a temporary dipole moment and an interaction can take place.

Mechanism for gaseous absorption
When a molecule interacts with electromagnetic radiation having a frequency , a quantum of energy is extracted from (absorption process) or added (emission process) to the external field and the molecule undergoes a transition between two different energy levels. When either process occurs, we say we are in presence of an absorption/emission line. The basic relation holds: (19) where and are the final and initial energy level of the molecule. In general: (20)

Mechanism for gaseous absorption
involves changes in the molecule electron energy levels and result in absorption/emission at U.V. and visible wavelengths. involves changes in the molecule vibrational energy levels and result in absorption/emission at near-infrared wavelengths. involves changes in the molecule rotational energy levels and result in absorption/emission at microwave and far-infrared wavelengths. Vibrational transitions are generally accompanied by rotational transitions. The associated absorption/emission lines form what is known as a vibration-rotation band.

Gaseous absorption: line shape and absorption coefficient
For an ideal monochromatic absorption line, the molecular absorption coefficient can be expressed as: (21) where S is known as the line strength and is the delta Dirac function. S Delta Dirac function νo In a real atmosphere, however, the absorption line is broadened by three physical phenomena: 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: (22) 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 10-5 nm.

Gaseous absorption: line shape and absorption coefficient
Doppler broadening: It is caused by a Doppler shift in the frequency of the emitted and absorbed radiation. The Doppler shift is due to the molecular velocity component along the direction of propagation of the radiation. The absorption coefficient is: (23) (24) 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. The modification of the molecular potentials is caused by inelastic as well as elastic collision between the molecules. 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: (25) where ps and Ts are reference values.

Doppler (a) and collisional (b) line shape
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 Doppler shifted and averaged over a Maxwell velocity distribution to obtain what is known as the Voigt line shape. The Voigt line shape includes the Gaussian and Lorentzian line shapes as limiting cases. It cannot be evaluated analytically. For its computation, fast numerical algorithms are available.

Gaseous absorption: continuum type absorption
The comparison of accurate calculations and measurements taken by high spectral resolution instruments have shown the importance of the finer details of line shape. For water vapour in particular, measurements show that there is a component of the absorption that varies slowly with frequency. This component cannot be explained by a spectrum computed using Lorentzian line shapes based on collision broadening theory. To reconcile measurements and simulations, in addition to spectral lines we introduce a continuum type absorption.

Gaseous absorption: continuum type absorption
Mechanism: the exact mechanism for the continuum absorption 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 have to extrapolated to atmospheric paths that are colder than the warm paths used in the laboratory.

Scattering and absorption by aerosol and cloud particles
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. Their shape include solid and hollow columns, prisms, plates, aggregates and branched particles, and, hexagonal columns.

Scattering and absorption by aerosol and cloud particles
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 atmosphere is not available in practice.

Scattering and absorption by aerosol and cloud particles
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 extinction coefficient
In the most general case where absorption and scattering take place, the total extinction coefficient can be written as: (26)

Line-by-line models For a given frequency, the line absorption coefficient for a combination of J different molecules is computed by performing the sum of the absorption coefficient evaluated for each single molecule and each single i absorption line. (27) Gas density Line strength adjusted to the path conditions (e.g. the line strength is temperature dependent) Normalized line shape function The models used to compute the gaseous line absorption coefficients are called line-by-line models (e.g. GENLN2, LBLRTM, HARTCODE, RFM).

Line-by-line models In addition to the computation of the line absorption coefficient, line-by-line models also perform the computation of the continuum absorption. They can then be used to compute monochromatic radiance at the top of the atmosphere using the radiative transfer equation. Monochromatic radiances must be convolved with the appropriate channel response functions. The accuracy of the line-by-line calculations is mainly controlled by the accuracy of line shape and line parameters (e.g. line strength and line width).

Absorption by different molecules
After Valley (1965)

Pseudo line-by-line models
Line-by-line models are computationally expensive both in CPU and disk space. Efforts to alleviate this have lead to the development of pseudo line-by-line models (e.g. 4A, K-carta) that use absorption coefficients stored in a look-up-table. Because the monochromatic absorption coefficient varies 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.

Fast radiative transfer model for use in NWP
Line-by-line and pseudo line-by-line models are too slow to be used operationally in NWP (note that operational forward models must also be capable of performing gradient computations, e.g. tangent linear and adjoint). To cope with the operational processing of observations in near real-time, fast radiative transfer models have been developed. These models are very computationally efficient and also accurate (i.e. they can reproduce line-by-line “exact” calculations very closely).

Fast radiative transfer model for use in NWP
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

The RTTOV fast radiative transfer model
The fast model used at ECMWF (and in many others NPW centres) is called RTTOV. RTTOV is a regression based fast model that computes channel optical depths using profile dependent predictors that are functions of temperature, absorber amount, pressure and viewing angle. In RTTOV, the atmosphere is divided into N homogeneous layers bounded by N+1 fixed pressure levels. The channel optical depth for layer j is written as: (28) where M is the number of predictors and the functions constitute the profile-dependent predictors of the fast transmittance model.

The RTTOV fast radiative transfer model
To compute the expansion coefficients , a line-by-line model is used to compute accurate channel averaged optical depths for a diverse set of temperature and atmospheric constituent (typically water vapour and ozone) profiles. 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 against the predictor values calculated from the profile variables for each profile at several viewing angles. The expansion coefficients can then be used by the model to compute optical depths for any other input profile.

The RTTOV fast radiative transfer model
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.

The ability of RTTOV to reproduce line-by-line optical depths
HIRS: High resolution Infrared Radiation Sounder (wheel radiometer with broad channels) IASI: Infrared Atmospheric Sounding Interferometer (hyperspectral sensor with very narrow channels) Water vapour sounding channel of the HIRS instrument Water vapour sounding channel of the IASI instrument Optical depth of the path Stars denote the fast model optical depths Water vapour amount in the path Squares denote the line-by-line optical depths

The RTTOV fast radiative transfer model
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 channels. In general, however, RTTOV can reproduce the line-by-line radiances to an accuracy typically below the instrument noise. Note that to characterize the total RTTOV error we must include the error contribution from the underlying line-by-line model. It should be stressed that RTTOV comes with associated gradient routines. This is a prerequisite for a fast model to be used in NWP assimilation.

The ability of RTTOV to reproduce line-by-line radiances
The statistic of the error has been obtained from the difference between RTTOV and line-by-line radiances computed or 82 profiles and six different viewing angles. HIRS: High Resolution Sounder AIRS: Atmospheric Infrared Sounder

The parametrization of scattering in RTTOV
The computational efficiency of a fast radiative transfer model can be seriously degraded if explicit calculations of multiple scattering are to be introduced. In RTTOV we have introduced parameterization of multiple scattering that allows to write the radiative transfer equation in a form that is identical to that in clear sky conditions.

The parametrization of scattering in RTTOV
The scattering parameterization used in RTTOV (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: (29) is the scattering optical depth 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: (30)

The red line denotes the difference between approximate (RTTOV) and
The black line denotes the difference between clear sky and exact scattering computations performed introducing either aerosol or ice crystal particles. The red line denotes the difference between approximate (RTTOV) and exact scattering computations

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 (e.g. RTTOV) 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 resolution channel transmittances. In the OSS method, top of the atmosphere channel radiances are computed by statistically selecting top of the atmosphere monochromatic radiances. The OSS approach has the advantage that, for instance, accurate scattering computations 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 easy 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.

Neural networks based fast radiative transfer models
Fast radiative transfer models using neural networks have been developed and may provide even faster means to compute radiances. A limitation of neural network based models is that accurate gradient versions of the models are proving difficult to develop.

Principal components based fast radiative transfer models
Principal component (PC) based fast models have been developed for hyperspectral (i.e. for sensors with many thousand channels) remote sensing applications. PC based fast models compute the PC scores of the radiance spectrum. The PC scores 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”, Q. J. R. Meteorol. Soc.,136,1823–1835 (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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