1. Integral Equations, 27.09.2007 Inverse methods for the remote sensing of atmospheric parameters Rolando Rizzi Atmospheric Dynamics Group, Department of Physics University of Bologna

2. Integral Equations, 27.09.2007 Purpose Show an example of the intimate relationship between important physical processes and the mathematical tools required for the their quantitative description. It is a presentation of the physical interpretation behind an integral equation that describes modern remote sensing techniques to retrieve properties of the atmospheres, and of current methodologies applied to solve this problem. Therefore it is not intended to be a presentation of up-to-date personal research work.

3. Integral Equations, 27.09.2007 Radiances from Earth Over the Antarctic Over the Sahara Desert Measured by IRIS-D on Nimbus-4

4. Integral Equations, 27.09.2007 Radiances from Mars Over lat. 21 ° South Measured by IRIS-D on Mariner 9 Near lat. 66 ° North

5. Integral Equations, 27.09.2007 How can we interpret these spectra? Qualitative: knowledge about spectroscopy of gaseous substances, that is how they interact with light; Quantitative: an equation that predicts the absorption and emission of light by atmospheric (Earth, Mars,…) gases

6. Integral Equations, 27.09.2007 Relationship bewteen radiance and matter There is a relationship between the energy flowing in a certain direction per unit time in a specific wavelength interval and certain properties of the surface and of the atmosphere: Simplifications: plane parallel atmosphere horizontal homogeneity dependence on time implicit No particulate matter/clouds

7. Integral Equations, 27.09.2007 Relationship bewteen radiance and matter Simplification; we consider energy flowing at zenith at TOA; It is also understood that radiance L, transmittance τ and absorption coefficient k all depend on T s, T, q i,…

8. Integral Equations, 27.09.2007 The meaning of the Schwarzchild equation III

9. Integral Equations, 27.09.2007 Limiting cases Case 1: k very small: τ(0,z TOA ) ~1, term I >> term II Radiance represents surface properties Case 2: k very large: τ(0,z TOA )~ 0, term I << term II Radiance represents some atmospheric properties

10. Integral Equations, 27.09.2007 Some atmospheric properties?

11. Integral Equations, 27.09.2007 Transmittance and Weighting function:example

12. Integral Equations, 27.09.2007 Transmittance and Weighting function

13. Integral Equations, 27.09.2007 Application to remote sensing? We have seen that by increasing the absorption coefficient k of a particular gas we can measure radiance that comes from broad atmospheric layers closer to the measuring device. How can we change the absorption coefficient? By changing the wavelength! Spectrometers are designed to measure spectral radiance. A sequence of spectral radiances contain information on how some atmospheric parameters ( T,q i ) change with height.

14. Integral Equations, 27.09.2007 Some preliminary steps From radiance measured at wavelengths where k is very small we can derive the magnitude of the first term: We can extrapolate from these measurements the properties of the surface (the product of emissivity ε and Planck radiance B ) also at wavelengths where k is very large (emissivity ε varies slowly with wavelength and we know the wavelength dependence of B )

15. Integral Equations, 27.09.2007 The integral equation Finally we rewrite the integral equation with the dependence on T(z) and q(z) and…taking into consideration the ever present measurement errors! The temperature T can be derived from analytical inversion once the Planck function B(z,v) is known. The other unknown ( q ) is embedded into the kernel. It looks as the problem of retrieving q is not the simplest problem.

16. Integral Equations, 27.09.2007 The integral equation To retrieve the Temperature(z) we would ideally exploit the properties of a gas whose concentration is precisely known. Until ten years ago CO 2 concentration was considered well known… Assume that M channels within the CO 2 15  m band have been chosen to recover the temperature profile. Thus, a set of M observations give M integral equations as: Index i identifies the selected wavelengths. Note that the spectral dependence of B has been factorised into the kernel function.

17. Integral Equations, 27.09.2007 This Freedholm integral equation of first kind is clearly an ill-posed problem, since the unknown is a continuous function of height but there are only a finite number of observations. It is clearly both mathematically and physically incorrect to solve an under- constrained set of equations. There is an infinite manifold of solutions which satisfy them exactly, and an even larger number of solutions which satisfy them within experimental error.

18. Integral Equations, 27.09.2007 An example of an initial error Most linearisation schemes (such as expansion on a complete basis) can be expressed as: The straightforward least squares solution (that neglects experimental errors) provides an unstable solution. matrix A T A has a small determinant, due to the structure of the weighting functions for the temperature (and humidity) and is very sensitive to very small measurement errors. Therefore the matrix inverse operation amplifies these small differences.

19. Integral Equations, 27.09.2007 Given the measured radiance vector g, the statistics of the experimental error ε, the weighting functions K, and any other relevant information, what can be said, in a physically meaningful way, about the unknown f ?

20. Integral Equations, 27.09.2007 Other relevant information? Other relevant information, beyond those provided by the measurements, is needed, in the form of constraints on the solution profile, in order that the error variance of the solution be finite. If no constraints are available, or the constraints are insufficient, the only valid approach is to accept to find a smoothed version of the true profile, with a known smoothing function, by some linear combination of the measurements. The most enlightening approach to understanding the role of the added information is to regard them as virtual measurements, that is, as being of the same nature as real measurements. They say something about the unknown profile just as the measurements do, and together with the measurements they determine whether the problem is well-posed.

21. Integral Equations, 27.09.2007 Virtual measurements Many of the constraints that have been used in the literature fall into the class of linear constraints. Some of them are described below: – Statistics. The climatological mean profile and its covariance may be regarded as a measurement and its uncertainty. – Forecast profile. The forecast profile and its error covariance may similarly be regarded as a measurement and its uncertainty. However, the error covariance is often not known very well, so must be overestimated. – Twomey-Tikhonov. The virtual measurement is an a priori guessed profile. The error covariance is proportional to Twomey's constraint matrix H, often taken to be a unit matrix. – Discretisation. It is usual to express continuous functions such as the unknown profile and the weighting functions in a discrete form n order to simplify the numerical calculations, and so that the algebra of vectors and matrices may be used instead of Hilbert space. This is of course a linear representation and may be regarded as a linear constraint.

22. Integral Equations, 27.09.2007 A few examples of inversion methods Constrained linear inversion (Phillips (1962), Twomey(1963): minimization of the deviations from the mean Many versions of constrained linear inversion differing a) in the quantity to be minimised, b) for methods to compute γ and c) for the structure of matrix H. For example we can minimise difference from a climatology vector f c :

23. Integral Equations, 27.09.2007 Maximum probability solution The maximum probability solution: the inversion problem is formulated as a variational problem: a cost function which has to be minimized : J(x) = (x-x b ) T B -1 ( x-x b ) + (y m - y{x}) T Y -1 ( y m - y{x})= minimum x represents the atmospheric state (a vector of the vertical temperature profile and other atmospheric variables such as water vapour concentration and surface temperature. x b is a background profile y m is the measurement vector y{x} is the forward model B -1 is the covariance of the background information Y -1 is the covariance of the measurements and of the forward model

24. Integral Equations, 27.09.2007 Maximum probability solution The most probable (and therefore optimal) profile is found by minimizing J(x) or by solving its gradient equation: if J’(x) represents the gradient of J(x) with respect to x, then J'(x) = B -1 (x-x b ) - K(x) T Y -1 (y m -y{x}) = 0 Where K(x) is a matrix containing the Jacobian

25. Integral Equations, 27.09.2007 Maximum probability solution In general, this problem is not trivial and there is no general analytic solution to this equation; but many techniques exist for solving it numerically for problems of interest. Here we shall restrict only the linear case when K(x) = K(x b ) = K where K is taken to be a constant matrix. x = x b + (B -1 + K T Y -1 K) -1 K T Y -1 (y m -y{ x b })

26. Integral Equations, 27.09.2007 Conclusions This talk ends where your interests start! I could not possibly say anything that is new, from the mathematical point of view, of the hundreds of solutions that have been attempted to solve this difficult problem tend to be iterative versions of the constrained and maximum probability solutions to partially deal with non-linearities, especially for the retrieval of q. Atmospheric physicists with expertise in radiative transfer have succeded in convincing WMO and international meteorological centres that temperature and humidity profiles derived from satellite “sounders” are data that should be used.

27. Integral Equations, 27.09.2007 Climatology will obtain a great advantage from this kind of data. The new, higher resolution, sounders (NASA-AIRS and European IASI) represent important positive steps forward. In particular IASI is demonstrating a clear positive impact on operational numerical weather prediction (application) and is certainly a great instrument to work with.

28. Integral Equations, 27.09.2007 Molte grazie per l’attenzione.

29. Integral Equations, 27.09.2007