1. COST 723 Training School - Cargese 4 - 14 October 2005 OBS 2 Radiative transfer for thermal radiation. Observations Bruno Carli

2. COST 723 Training School - Cargese 4 - 14 October 2005 Table of Contents Geometry of observation: zenith, nadir and limb sounding. Thermal radiation and types of measurements. Spectroscopy. The forward model for thermal radiation. The inverse problem of atmospheric constituent retrieval. Definition of variance-covariance matrix and of averaging kernel matrix.

3. COST 723 Training School - Cargese 4 - 14 October 2005 Geometry of observation Zenith sounding Nadir sounding Limb sounding

4. COST 723 Training School - Cargese 4 - 14 October 2005 Geometry of observation Limb sounding

5. COST 723 Training School - Cargese 4 - 14 October 2005 The vertical and horizontal resolution depend on the geometry of observation. Typical resolution of nadir measurements is 10 km horizontal and 10 km vertical. Typical resolution of limb measurement is 2 km vertical and 700 km horizontal. Geometry of observation

6. COST 723 Training School - Cargese 4 - 14 October 2005 Thermal radiation Sun/Star Moon Earth/planet Atmosphere Sun Earth/atmosphere

7. COST 723 Training School - Cargese 4 - 14 October 2005 Thermal radiation Thermal radiation Sources Sun: solar occultation Atmosphere: emission sounding

8. COST 723 Training School - Cargese 4 - 14 October 2005 Spectroscopy Rotational spectra Vibrational spectra Electronic spectra

9. COST 723 Training School - Cargese 4 - 14 October 2005 Spectroscopy Main spectroscopic constituents for the atmosphere: water vapor (  ) ozone (  ) carbon dioxide (  ) methane (  ) nitrous oxide (  ) nitric acid (  ) Rotational spectra Vibrational spectra

10. COST 723 Training School - Cargese 4 - 14 October 2005 Spectroscopy Wavenumber cm -1 Altitude km 40 10 30 80 100 90 70 60 50 20 40 10 30 80 100 90 70 60 50 20 110100100010000 Temperature broadening: Gaussian line shape Pressure broadening: Lorentzian line shape Voight line shape equal convolution between Gaussian and Lorentzian distributions.

11. COST 723 Training School - Cargese 4 - 14 October 2005 Integral equation of Radiative Transfer Absorption term Emission term Intensity of the background source “Transmittance” between 0 and L Spectral intensity observed at L “Transmittance” between l and L “Optical depth”

12. COST 723 Training School - Cargese 4 - 14 October 2005 Numerical integral of Radiative Transfer This integral is numerically performed as:

13. COST 723 Training School - Cargese 4 - 14 October 2005 Tree of operations for Radiative Transfer calculation segmentation Spectral intensity observed at L The optical path is divided in a set of contiguous segments in which the path is straight and the atmosphere has constant properties (segmentation). In each segment the absorption coefficient is also calculated. Segments of the optical path

14. COST 723 Training School - Cargese 4 - 14 October 2005 Tree of operations for Radiative Transfer calculation absorption coefficient Spectral intensity observed at L “Absorption coefficient” “number density” “Cross section” “Volume Mixing Ratio” “Line shape”“Line strength”

15. COST 723 Training School - Cargese 4 - 14 October 2005 The forward model calculates the spectrum measured by the instrument. This is equal to the atmospheric spectral intensity I ,  (L), obtained with the radiative transfer model, convoluted with instrument effects. Forward Model of the measurements Where AILS is the “apodized instrument line shape” and FOV is the “field of view “ of the instrument.

16. COST 723 Training School - Cargese 4 - 14 October 2005 The inversion problem Forward model Inverse model

17. COST 723 Training School - Cargese 4 - 14 October 2005 The forward and the inverse problem The measured signal is determined for several values of the frequency  and of the limb angle  : S( ,  ), and depends on the properties of the atmosphere that, in the case of a stratified atmosphere, are only a function of the altitude z : S( ,  ) = F(p(z), T(z), VMR i (z) ) where p(z) is the pressure, T(z) is the temperature and VMR i (z) is the volume mixing ratio of the atmospheric species i.

18. COST 723 Training School - Cargese 4 - 14 October 2005 The forward and the inverse problem In our case, the relationship : S( ,  ) = F(p(z), T(z), VMR i (z)) is the forward problem and its reciprocal p(z) = F 1 (S( ,  )) T(z) = F 2 (S( ,  )) VMR i (z) = F i (S( ,  )) is the inverse problem.

19. COST 723 Training School - Cargese 4 - 14 October 2005 The forward and the inverse problem The inverse problem does not always have a useful solution. In particular it can be : ill posed (either impossible solution or infinite possible solutions). This occurs when either an error is made in the definition of the problem or the observations do not provide the required information. ill conditioned (the solution exists but small errors in the observations lead to large errors in the solution). This occurs when the observations contains insufficient information.

20. COST 723 Training School - Cargese 4 - 14 October 2005 The inversion problem Ill conditioned Ill posed ?

21. COST 723 Training School - Cargese 4 - 14 October 2005 The mathematics of the retrieval The inversion In the forward problem S( ,  ) = F(p,T,VMR) ( ,  )=m are the observation variables that number the observations and (p,T,VMR)=q are the atmospheric variables that become the unknowns in the inverse problem. Using these notations : S m = F(q) In general, this equation cannot be analytically inverted and does not have an exact solution.

22. COST 723 Training School - Cargese 4 - 14 October 2005 The mathematics of the retrieval The least-squares solution We can look for a least-squares solution. The least-square solution q r is the value of q which minimises the quantity:  2 = y T (V y ) -1 y where V y is the variance covariance matrix of the measurements and y = S m -F(q) is the difference between the measurements and the forward model calculated for q.

23. COST 723 Training School - Cargese 4 - 14 October 2005 The mathematics of the retrieval The Gauss-Newton method The least-square solution is the solution of equation:  (  2 )/  q = 0. This equation can be solved with an iterative procedure that starts from an initial guess q o of the unknown. The Gauss-Newton method assumes that q o is near enough to the solution q r and adopts a linear expansion of the non linear functions.

24. COST 723 Training School - Cargese 4 - 14 October 2005 The mathematics of the retrieval The Gauss-Newton solution At each step of the iterative process a new estimate q n is obtained : q n = q n-1 +(K T (V y ) -1 K) -1 K T (V y ) -1 (S m -F(q n-1 )) where the quantity K=  (F(q))/  (q) is the Jacobian of the measurements. When the convergence criteria are satisfied the iterative process is stopped and q r =q n.. The variance covariance matrix of the solution is equal to: V q = (K T (V y ) -1 K) -1

25. COST 723 Training School - Cargese 4 - 14 October 2005 The mathematics of the retrieval The Levenberg - Marquardt Method The iterative process may be unstable (the values retrieved at each iteration oscillate around the solution). In order to limit this effect it is useful to introduce a damping in the variation of the unknown (Lavenberg - Marquardt method). In this case the solution is equal to: q r = (K T (V y ) -1 K + I) -1 K T (V y ) -1 (S m -F(q n-1 )) where I is the unit matrix.

26. COST 723 Training School - Cargese 4 - 14 October 2005 In the ideal case of absence of both random and systematic errors in the measurements and in the instrument’s forward model, for each state of the atmosphere the observing system provides a retrieved profile. Expanding up to the first order about a generic atmospheric state, we obtain: The quantity: is called averaging kernel (AK) matrix and it is a function of the state. Averaging kernels

27. COST 723 Training School - Cargese 4 - 14 October 2005 Averaging kernels The AK matrix describes how the real state of the atmosphere is distorted by the retrieval process. This information is important whenever the retrieved quantities are used together with other measurements (data assimilation, data validation).

28. COST 723 Training School - Cargese 4 - 14 October 2005 Averaging kernels The AKs are the rows of the AK matrix and provide for each retrieved value the contribution of the atmosphere at the different altitudes.

29. COST 723 Training School - Cargese 4 - 14 October 2005 variability Averaging kernels variability Ozone, all latitudes (April only)

30. COST 723 Training School - Cargese 4 - 14 October 2005 Final considerations The inversion problem is characterised by: the number of observations the number of unknowns the VCM of the observations the VCM of the retrieved unknowns the averaging kernel of the retrieval.