1. NCAS Atmospheric Measurement Summer School, September 2010 Page 1/12 Deriving useful information from satellite data (a remote sensing application) Satellite measurements do not measure atmospheric quantities directly (e.g. radiances for passive IR sounding). What is the relationship between the atmospheric profile and the satellite measurements? What is the procedure of extracting information from satellite data? How should we Interpret the result? Reminder The forward problem The inverse problem Warning Ross Bannister, High Resolution Atmospheric Assimilation Group, NERC National Centre for Earth Observation, University of Reading, UK

2. NCAS Atmospheric Measurement Summer School, September 2010 Page 2/12 Passive IR sounding Passive IR sounding relies on the following basic physics of the atmosphere. All bodies above absolute zero (-273 C) emit thermal radiation (black body radiation described by the Planck function). Radiation from an air parcel suffers absorption and scattering as it travels through the atmosphere towards the detector. From: K. N. Liou, 2002, Fig. 4.1

3. NCAS Atmospheric Measurement Summer School, September 2010 Page 3/12 A radiative transfer problem in the atmosphere (the forward problem) Thermal radiation in the atmosphere is affected by a number of processes Emission (a function of the temperature of the emitter, its density and its absorption cross-section). Absorption. Scattering. We consider the first two only (for simplicity). I λ () monochromatic radiance emitted to space I λ (s) monochromatic radiance emitted from layer B λ (T) Planck function τ λ (s) transmittance of the atmosphere from surface to space surface Temperature T thickness ds density ρ cross-section k λ

4. NCAS Atmospheric Measurement Summer School, September 2010 Page 4/12 The temperature weighting functions Suppose that we know the composition of the atmosphere (density, water vapour, O 3 ). Hence we know the absorption characteristics, i.e. k λ, τ λ. What information can we deduce about the temperature of the atmosphere? Introduce the temperature weighting functions – the sensitivity of the emitted radiation at wavelength λ to temperature at height s.

5. NCAS Atmospheric Measurement Summer School, September 2010 Page 5/12 Linearization of the radiative transfer equation The radiative transfer equation is a non-linear function of T(s). The problem is simplified by linearizing it (first discretize).

6. NCAS Atmospheric Measurement Summer School, September 2010 Page 6/12 The inverse problem T(s i ) I λ () This is an inexact inverse problem (all observations are subject to observation error) Solving the inverse problem is the basis of the remote sensing problem Fundamental: also used in medical imaging, astrophysics, geology, oil prospecting, etc.

7. NCAS Atmospheric Measurement Summer School, September 2010 Page 7/12 Solving the inverse problem N vertical levels of T(s i ) M measured radiances, I λj meas () M simulated observations, I λj (; T 0:N ) σ λj error standard deviation of instrument Reminder of the linear forward problem Cost function (least squares) measurements simulated measurements

8. NCAS Atmospheric Measurement Summer School, September 2010 Page 8/12 Solving the inverse problem Introduce a vector/matrix notation Problem in terms of vectors/matrices measurements simulated measurements atmospheric T profile matrix of weighting functions observation error oovariance matrix T = T B + δT

9. NCAS Atmospheric Measurement Summer School, September 2010 Page 9/12 Solving the inverse problem This problem is ill-posed when W T R -1 W is a singular matrix (like dividing by zero). This happens when M < N (and often when M N). The solution, T A is then not unique. Need to regularize the problem (e.g. choose minimum with smallest δT).

10. NCAS Atmospheric Measurement Summer School, September 2010 Page 10/12 Interpretation of the result Imagine we know the true temperature profile of the atmosphere, T true. Write as a perturbation from the background Simulate the observations using this truth Assimilate the observations Interpret the result (the averaging kernel matrix)

11. NCAS Atmospheric Measurement Summer School, September 2010 Page 11/12 Interpretation of the result Courtesy S. Ceccherini (IFAC-CNR) The averaging kernels tell us about the resolution of the solution to the inverse problem True T profile Averaging kernel for Real averaging kernels perfect resolution

12. NCAS Atmospheric Measurement Summer School, September 2010 Page 12/12 Summary and references Satellites do not measure atmospheric quantities directly. An inverse problem needs to be. All measurements are subject to errors. The inverse problem requires solution of the forward problem. The method of least squares is the basis of many inverse. Usually background information is required to regularize the problem. Care should be taken when interpreting the solution of the inverse problem. Similar techniques are used in operational weather forecasting to determine the initial conditions of the Numerical Weather Prediction model (N ~ 10 7 - 10 8, M ~ 10 6 -10 7 ). Brugge & Stuttard, From Sputnik to EnviSat, and beyond Weather 58 (March 2003), 107-112; Weather 58 (April 2003), 140-143, Weather 58 (May 2003), 182-186. Rodgers C.D., Inverse methods for atmospheric sounding, theory and practice, World Scientific, Singapore, 2000. Kalnay E., Atmospheric modelling, data assimilation and predictability, Cambridge University Press, Cambridge, 2003. More at www.met.reading.ac.uk/~ross/measurements/index.html