DRAGON ADVANCED TRAINING COURSE IN ATMOSPHERE … · Day 3 Lecture 1 Retrieval techniques - Erkki...

Day 3 Lecture 1 Retrieval techniques - Erkki Kyrölä 1

DRAGON ADVANCED TRAINING COURSE IN ATMOSPHERE REMOTE SENSING

Inversion basics

Erkki KyröläFinnish Meteorological Institute

y = Kx + εˆ x = (KTK)−1KT y



Contents

1. Introduction: Measurements, models, inversion

2. Classical inversion: LSQ3. Bayes theory4. Monte Carlo Markov Chain5. References



Introduction

instrumenttarget

Inverse model

inverse model = generalized inverse of the approximate forward model

data

Forward model



Measurements

• Measurements either count or scale• Remote and in situ measurements• Direct and indirect measurements• Classical or quantum measurements



Atmospheric remote measurements

• Atmosphere is continuously changing in time and space.No repeated measurements of the same quantity.

• Radiation field measurements are direct, all othermeasurements are indirect

• Measurements probe large atmospheric volume. Large averaging. Validation by in situ measurements is difficult.



Forward models

The “true” nature

z=all other pertinent variables

G(x,z) + ε

Gknown (x,zknown = z fix ) + ε

The best forward model available.Uninteresting variables fixed.

Gapp (x,zknown = z fix ) + ε

Model used in simulation Ginv (x,zknown = z fix ) + ε

Model used for inversion



Inverse problem

x = Parameters to be determined from measurements

K = Forward model y = Measurements (data)ε = Noise

y = K(x) + ε

Find best x when y is measured.Define “best” first.



Least squares solution (LSQ)

S = (yp − K p (x))2∑Minimize

If we have a linear problem

y = Kxwe get simply

ˆ x = (KTK)−1KT y

Distance between dataand the model prediction



A systematic basis for inversion theory is given by

the Bayesian approach

• Model parameters are random variables• Probability distribution of model parameters is retrieved• A priori information is needed. This has led to many

controversies about the Bayesian approach.



• Mean• Minimum variance

• Maximum probability

Various point estimators for x

Mean = Minimum variance estimator

Mean = Maximum probability if pdf symmetric



MCMCmethod

Maximumlikelihood

LSQ

Gaussian errors

max of

max of MAP

Linearmodel

LMmethod

Closedsolution

Whole distribution Point estimation

P(x | y) = P(y | x)P(x)



Example: Linear problem

y = K true x true + ε ≈ Kx + ε

Assume Gaussian noise and prior distribution:

P(y | x) = const × e−

12

(y−Kx )T CD−1 (y−Kx )

const = ((2π )N det(CD ))−

12

P(x) = const × e−

12

(x−xa )T Ca−1 (x−xa )



ˆ x = (KTCD−1K + Ca

−1)−1(KTCD−1y + Ca

−1xa )

Cx = (KTCD−1K + Ca

−1)−1

P(x | y) = const × e−

12

(x− ˆ x )T Cx−1 (x− ˆ x )

ˆ x = xa + CaKT (KCaK

T + CD−1)−1(y − Kxa )

Maximum a posteriori

The solution can be written also as

This can be viewed as an update to a priori Assimilation

Interpretation: weighted mean between data and a priori

Posterior distribution

Model covariance



Properties of linear solution

Averaging kernel

ˆ x = (KTCD−1K + Ca

−1)−1(KTCD−1y + Ca

−1xa )= CxK

TCD−1K true x true + CxK

TCD−1ε + CxCa

−1xa

A = CxKTCD

−1K true

ˆ x − x = (A − I)(x − xa ) + CxKTCD

−1εsmoothing error random error



Bias ˆ x = CxKTCD

−1K true x true + CxKTCD

−1 ε + CxCa−1xa

Ca → ∞If and K = K true no bias

Quality of retrieval

The bias in retrieval can usually be checked only by computer simulations. Intercomparisons of real measurements can also be used to detect bias.

Residual Investigate the difference d = yobs − ymod = yobs − Kˆ x

Chi2 χ 2 =1

N − m( (yobs − Kˆ x )∑ T

CD−1(yobs − Kˆ x ) + ( ˆ x − xa )T Ca

−1( ˆ x − xa ))

linear



Special cases

1. No apriori Ca → ∞ and K = K true

A=1 and we obtain WLSQ

2. Connection to elementary data analysis. Take case 1 andonly one parameter. Then the MAP estimator is mean and

Cx =σ 2

N i.e. the standard error of the mean.



Possible solutions for linear equations

Number of data = N, number of unknowns = M

1. N = M Exact inversion possible

2. N > M Overdetermined problem. Additional information may be used to constrain the solution.

3. N < M Underdetermined problem. Needs additional information or constraints



Non-linear problems

• Gaussian errors and linear model give a quadratic optimisation problem. This leads to linear (in data) estimates.

• All other cases lead to non-linear problems.

• Gaussian errors and non-linear models can be approached by the Levenberg-Marquardt algorithm

• Sometimes a model can also be linearised

• Sometimes we can transform the problem to a new linear problem. Note: Error statistics will also change

• With very noisy data and/or complicated models several maxima of pdfcan exist. Global methods, like simulated annealing, may help but it is better to try MCMC.



Ultimate estimators: Markov chain Monte Carlo Blind Mr. Levenberg: That’s it!

Mr. Markov: Hold your horses

Twin peaks drama

Top guy: Yes! Mean guy: <Sorry but...>

Flatness dullness



Markov chain Monte Carlo

<xi>= Σ zti

Estimators from MCMC

1N t

N



Marginal posterior distributions at 30 km for different gases

Brightstar

Weakstar

MCMC examples (GOMOS)



A priori information

• Discrete grid: Assume that profile has only a finite number of free parameters

• Smoothness: Tikhonov constraint

• A priori profile

• Positivity constraint or similar



Literature and a referenceTarantola: Inverse problem theory, Methods for datafitting and model parameter estimation, Elsevier, 1987

Rodgers: Inverse Methods for Atmospheric Sounding: Theory and Practice, World Scientific, 2000

Menke: Geophysical data analysis: discrete inverse theory, Academic Press, 1984

Tamminen and Kyrölä, JGR, 106, 14377, 2001

Tamminen: Ph.D. thesis, FMI contributions 47, 2004.

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