REMOTE SENSING & THE INVERSE PROBLEM “Remote sensing is the science and art of obtaining information about an object, area, or phenomenon through the analysis of data acquired by a device that is not in contact with the object, area, or phenomenon under investigation“ (Lillesand and Kiefer 1987)

ACTIVE VS. PASSIVE REMOTE SENSING Passive Remote Sensing: Natural sources of radiation, the attenuated, reflected, scattered, or emitted radiation is analysed  Usually UV, visible, IR  Sources include: sun, moon, Earth Active Remote Sensing: Artificial source of radiation, the reflected or scattered signal is analysed  Usually lasers (LIDAR) or RADAR  Active source, usually co-located with receiver

WAVELENGTHS FOR REMOTE SENSING UV:some absorptions + profile information aerosols vis:surface information (vegetation) some absorptions aerosol information IR:temperature information cloud information water / ice distinction many absorptions / emissions + profile information MW:no problems with clouds ice / water contrast surfaces some emissions + profile information

WHAT WE MEASURE VS. WHAT WE WANT! MEASURED: absolute intensities in dedicated wavelength intervals intensities relative to the intensity of a reference source ratios of intensities at different wavelengths variations of intensities degree of polarisation RETRIEVED: Surface properties (albedo, vegetation, fires, wind speed, ice cover, etc.) Meteorological properties (pressure, temperature, cloud cover, lightning, etc.) Chemical composition (trace gases, aerosol burden & type) The “magic” of remote sensing

THE FORWARD MODEL Atmospheric Composition (x true ) Radiation Spectra (y) Radiative Transfer Model (F) Observations (e.g. radiation) State of interest (e.g. concentrations) Complementary parameters Errors If linear (or linearize first): Jacobians (forward model)

THE INVERSE PROBLEM Given observed spectra, what is the atmospheric state that produced it? Problems: 1.Non-uniqueness of solution (need to apply a priori information) 2.Discreteness of measurements of a smoothly varying function 3.Instability of the solution due to errors in the observations ? Retrieval Classes: 1.Physical Retrievals: use a first guess and the forward model to try to match observed radiances (computationally expensive) 2.Statistical Retrievals: Use a training set of observed relationships between radiance and atmospheric state (requires an exhaustive dataset for completeness) 3.Hybrid Retrievals: Like physical retrievals, but apply approximations (e.g. linearize the RTE)  Optimal estimation is part of this class (we will focus on this here)

BAYES’ THEOREM: FOUNDATION FOR MAP SOLUTION (ALSO KNOWN AS OPTIMAL ESTIMATION) P(x) = probability distribution function (pdf) of x P(y|x) = pdf of y given x A priori pdfObservation pdf Normalizing factor (unimportant) A posteriori pdf Maximum a posteriori (MAP) is the solution to

AN ASIDE: EXAMPLE OF BAYES’ THEOREM… Imagine we are determining the likelihood of a patient’s illness… What is the probability of having the flu (A) given that the patient has a high temperature (B)?  what is P(A|B)? In a flu outbreak, say 20% of the population has the flu  P(A). 90% of people who have the flu have a high temperature  (P(B|A). In the general public 10% of the non-flu public has a high temperature for other reasons (cold, other illness, etc.) So, we want to solve: P(A|B) = P(B|A) P(A) / P(B) P(B)=probability of having a high temperature = 0.80(0.10)+0.20(0.90)=0.26 Thus, P(A|B) = (0.90) (0.20) / (0.26) = 69% Quite likely that a high temperature is an indicator of flu during the outbreak…

SIMPLE LINEAR INVERSE PROBLEM FOR A SCALAR Bayes’ theorem: Max of P(x|y) is given by minimum of cost function SOLUTION: where g is a gain factor Let x be the true value: where a is an averaging kernel a priori “bottom-up” estimate x a ±  a Observation y=kx ±  y Errors from: fwd model and instrument If errors are Gaussian:

GENERALIZATION: CONSTRAINING n STATES WITH m OBSERVATIONS Forward model: A cost function defined as is generally not adequate because it does not account for correlation between states or between observations. Need to go to vector-matrix formalism: with Jacobian matrix K (elements k ij) ) and error covariance matrices leading to formulation of cost function: S y = Var(i,i) Var(j,j) S y (i,j)

THE COST FUNCTION a priori term J a Observation term J O Bouttier and Courtier, 1999 J a (x) J(x)=J a (x)+J o (x) xa S a = covariance matrix of the background error S  = covariance matrix of the observation error + covariance matrix of representativeness error (interpolation, discretization) K = linearized forward model Minimization of the cost function:

VECTOR-MATRIX REPRESENTATION OF LINEAR INVERSE PROBLEM Scalar problemVector-matrix problem Optimal a posteriori solution (retrieval): Gain factor: A posteriori error: Averaging kernel: Jacobian matrixsensitivity of observations to true state (fwd model) Gain matrix sensitivity of retrieval to observations Averaging kernel matrix sensitivity of retrieval to true state If the forward model is non-linear, formulation may need to be applied iteratively