Why learn about radiative transfer

1. Training Course Module DA.Data assimilation and use of satellite data.Introduction to infrared radiative transfer.Marco Matricardi, ECMWF 6 June - 7 June 2013.

2. 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. Why learn about radiative transfer The radiative transfer model (and its adjoint) are therefore a key component in the assimilation of satellite radiance in a NWP system.

3. Electromagnetic radiation at the top of the atmosphere The forward model simulates the observed radiances using the equation of radiative transfer

4. Spectrum of electromagnetic radiation Near Infrared Far infrared After Liou (2002)

5. Radiance 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.

6. Satellite sensors measure radiance over a finite spectral interval. They respond to radiation in a non-uniform way as a function of wave number. AIRS: Atmospheric Infrared Sounder Radiance Monochromatic radiance To represent the outgoing radiance as viewed by a satellite sensor, the spectrum of monochromatic radiance must be convolved with the appropriate instrument response function. This yields what we call 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 wave number of the channel. (3)

7. (4) Blackbody radiation Blackbody radiation is the radiation emitted by a medium (e.g. a gas) in thermodynamic equilibrium. The spectral distribution of blackbody radiation is given by the Planck radiation law: where Bν (T) is the radiance emitted by the medium, h is the Planck’s constant, k is the Boltzmann’s constant and T is the temperature of the medium. Bν (T) is also known as Planck’s function. In general, blackbody radiation is used to refer to an object or system that absorbs all radiation incident on it (i.e. a blackbody). The best practical realization of an ideal blackbody is a small hole in a large cavity maintained at constant temperature. If the wavelength of the incident radiation is shorter than the diameter of the hole, most of the incident radiation is absorbed by the cavity and the spectrum of the radiation emitted by the hole is given the Planck’s function. Large cavity with absorbing walls Hole in the cavity Note that the radiance inside the cavity is also given by the Planck’s function as long as the wavelength of the radiation is smaller than the size of the cavity.

8. Blackbody radiation Limiting cases of low and high frequencies If we define c1=2πhc2andc2=hc/kas the first and second radiation constant, we can give an equivalent black body function in terms of radiance per unit wavelength: (5) for for This is known as Rayleigh-Jeans Distribution This is known an Wien distribution

9. Blackbody radiation curves for different temperatures The peak of the blackbody curve shifts to shorter wavelengths as the temperature of the blackbody increases. The wavelength of the peak is given by Wien’s displacement law: λ=2.7 103/T [µm] After Valley (1965)

10. Brightness temperature In many applications, the radiance, L(ν), is expressed in units of equivalent brightness temperature, Tb(ν). The brightness temperature is computed by applying to the measured or simulated radiance the inverse of the Planck’s function : (6) In practice, we express the radiance in terms of the temperature of a blackbody emitting the same radiance at the same wavelength.

11. Transmittance and optical depth 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 away from its original direction of propagation, i.e. scattering redistributes the energy of the incident beam to all directions. 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). 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. Absorbing medium Scattering medium • Scattering particles • After McCartney (1983)

12. 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: (7) s ds Mass extinction coefficient [m2Kg-1] Density of the medium [Kg m-3] The quantity is known as volume extinction coefficient [m-1]. In general, the extinction coefficient is written as the sum of the absorption coefficient, and the scattering coefficient, :

13. The integration of Eq. (7) between points s1 and s2=s1+s yields : (8) Transmittance and opical depth The optical depth of the medium between points s1 and s2 is defined as: (9) The transmittance of the medium between points s1 and s2 is defined as: (10)

14. The equation of radiative transfer The change of intensity resulting from the interaction between radiation and matter is the sum of the contribution due to extinction and emission of radiation. If we assume that the emission process is linear in the amount of matter, the change of intensity can be written as: (11) Eq. (11) is known as the Schwarzchild’s equation. is called the source function. It consists, in general, of two parts: (12)

15. Scattering serves as a source of energy. The scattering source function characterises the energy that is redirected from all directions into the direction of the incoming beam. The equation of radiative transfer: the scattering source function Scattering particles • • In absence of solar radiation, the scattering source function can be written as: (13) is known as the phase function. It describes the angular distribution of the scattered energy.

16. The thermal source function characterises the thermal energy emitted by the medium. The equation of radiative transfer: the thermal source function Although the earth’s atmosphere as a whole is not in thermodynamic equilibrium, we find that it is in Local Thermodynamic Equilibrium (LTE) (at least below the altitude of 70 km). In LTE conditions, the atmosphere locally behaves like a blackbody and the thermal source function is given by the Planck’s function: (14) Where T(s) is the local kinetic temperature.

17. In presence of scattering, the radiative transfer equation cannot be solved analytically. The equation of radiative transfer 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.)

18. We can solve the radiative transfer equation making what we call the plane-parallel approximation. The equation of radiative transfer for a non-scattering plane-parallel atmosphere In the atmosphere, the horizontal variations of parameters like temperature, density, etc., are small compared to the vertical variations of the same parameters. Hence, we can assume that all radiative quantities depend only on the vertical direction. Distances can then be measured along the normal Z to the plane of stratification of the atmosphere. Z s z θ

19. In absence of scattering, the radiative transfer equation for a plane-parallel atmosphere can be written as: (15) The equation of radiative transfer for a non-scattering plane-parallel atmosphere Eq. (15) can be rewritten in terms of the optical depth coordinate: (16)

20. The equation of radiative transfer for a non-scattering plane-parallel atmosphere The solution of Eq.(16) yields the radiance at the top of the atmosphere. If we neglect the contribution of solar radiation and apply appropriate boundary conditions at the top and at the bottom of the atmosphere, after some mathematical manipulation we can write: (17) Note that : the first term on the right hand side of Eq. (17) is the spectral radiance emitted by the Earth’s surface and attenuated by the atmosphere, the second term is the spectral radiance emitted to space directly by the atmosphere the third term is the spectral radiance emitted to space by the atmosphere by reflection from the Earth surface.

21. The practical calculation of the radiance at the top of the atmosphere is carried out by dividing the atmosphere in to a number of L homogeneous layers bounded by L+1 fixed pressure levels. We can then rewrite Eq. (17) in discrete layer notation noting that layer L represents the surface layer: The equation of radiative transfer for a non-scattering plane-parallel atmosphere (18) Spectral emissivity of the surface Transmittance from the surface to the top of the atmosphere Average temperature in the layer Transmittance from pressure level j to the top of the atmosphere Surface temperature The third term of Eq.(18) is valid under the assumption that the reflection from the surface is quasi-specular. Spectral reflectivity of the surface

22. The equation of radiative transfer for a non-scattering plane-parallel atmosphere In Eq. (18), the radiance emitted to space directly by the atmosphere is obtained by weighting the Planck’s function by the quantity The function varies with altitude. The region around its maximum makes the bulk of the contribution to the radiance emitted to space. Note that the location of the maximum varies with frequency. Radiance measurements made at different frequencies receive contributions from different regions in the atmosphere and can thus provide information on the temperature and absorbing gaseous concentration in those regions.

23. The equation of radiative transfer for a non-scattering plane-parallel atmosphere Emissivity of sea water: nadir view and zero wind speed Wave number (cm-1) A blackbody as an emissivity equal to unity at any frequency

24. Radiance spectrum at the top of the atmosphere over a warm water surface Radiance spectrum at the top of the atmosphere over a sea ice surface

25. 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.

26. 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)

27. 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.

28. 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. Gaseous absorption: line shape and absorption coefficient 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

29. 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.

30. 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.

31. 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 psand Ts are reference values.

32. Doppler (a) and collisional (b) line shape After Levi (1968)

33. 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.

34. The comparison of accurate calculations and measurements taken by high spectral resolution instruments have shown the importance of the finer details of line shape. Gaseous absorption: continuum type absorption 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 absorption due to spectral lines we introduce a continuum type absorption.

35. 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.

36. The models used to compute the gaseous absorption coefficients are called line-by-line models (e.g. GENLN2, LBLRTM, HARTCODE, RFM). For an homogeneous atmospheric layer bounded by z1 and z2, the absorption coefficient for a gas j is computed by performing the sum of the absorption contributions from all lines i: Line-by-line models (26) Line strength adjusted to the conditions in the layer (e.g. the line strength is temperature dependent) Normalized line shape function The total optical depth for a layer comprising J single gas is then written as: (27)

37. 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).

39. Absorption by different molecules N2O Atmospheric window CO2 Atmospheric window CH4 CO O3 H2O CO2 N2O

40. 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.

41. 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).

42. 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

43. 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 L homogeneous layers bounded by L+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.

44. 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. The RTTOV fast radiative transfer model 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.

45. 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.

46. 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

47. 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.

48. 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. AIRS: Atmospheric Infrared Sounder HIRS: High Resolution Sounder

49. The parametrization of scattering in RTTOV In RTTOV we have introduced a computationally efficient parameterization of scattering that allows to write the radiative transfer equation in a form that is identical to that in clear sky conditions. 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.

50. The parametrization of scattering in RTTOV 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)