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, March 2006.

2. 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.

5. Radiance ►A fundamental quantity associated to a radiation field is the intensity of the radiation field or radiance. 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)

7. 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 (3) Where is the normalised instrument response function and ^ over the symbol denotes convolution. Here is the central frequency of the channel.

9. 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: (4) 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.

10. 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. ►As This is known as Wien distribution.

12. 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).

13. 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.

14. 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.

15. 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.

16. Transmittance and opical depth Integration of Eq.(6) between 0 and s yields: (7) whereLυ(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 thespectral transmittance of the medium.

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

18. The equation of radiative transfer for a plane-parallel atmosphere Z s ►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. ►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. θ z

19. 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: The quantity Jυ is called the source function within dz and μ=cos(θ).

20. 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 Jυis 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.

21. 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 . ►The second term represent the increase in intensity from the multiple scattering of the diffuse intensity from the directions . ►The third term represent the increase in intensity from emission within the atmosphere in the direction μ.

23. 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:

24. 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).

25. 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 can be rewritten as:

26. 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: 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.

27. 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 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.

29. 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).

30. 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.

32. 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.

33. 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 line. ►The basic relation holds: where and are the two energy levels involved in the transition and is the centre frequency of the absorption/emission line. ►In general

34. 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.

35. 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 2)Collision broadening 3)Doppler broadening

36. 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.

37. 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.

38. 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 psand Ts are reference values.

40. 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.

41. 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.

42. 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

43. 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).

47. 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.

48. 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.

49. 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.

50. 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.