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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

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Passive IR soundingPassive 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

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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).

Iλ (∞) monochromatic radiance emitted to spaceIλ (s) monochromatic radiance emitted from layerBλ (T) Planck functionτλ(s) transmittance of the atmosphere from surface to space

surface

Temperature Tthickness dsdensity ρcross-section kλ

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The temperature weighting functions

Suppose that we know the composition of the atmosphere (density, water vapour, O3).Hence we know the absorption characteristics, i.e. kλ, τλ.

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.

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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).

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The inverse problemT(si) Iλ (∞)

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.

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Solving the inverse problem

N vertical levels of T(si)M measured radiances, Iλj

meas(∞)M simulated observations, Iλj(∞; T0:N)σ λj error standard deviation of instrument

Reminder of the linear forward problem

Cost function (least squares)measurements simulated measurements

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Solving the inverse problemIntroduce 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 = TB + δT

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Solving the inverse problem

This problem is ill-posed when WTR-1W is a singular matrix (like dividing by zero).

•This happens when M < N (and often when M ≥ N).•The solution, TA is then not unique.•Need to regularize the problem (e.g. choose minimum with smallest δT).

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Interpretation of the result

Imagine we know the true temperature profile of the atmosphere, Ttrue.

Write as a perturbation from the background

Simulate the observations using this ‘truth’

Assimilate the observations

Interpret the result (the averaging kernel matrix)

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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

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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 ~ 107-108, M ~ 106-107).

• 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