Inverse Theory. COST WaVaCS Winterschool Venice, February Stefan Buehler Luleå University of Technology Kiruna

Transcription

1 Inverse Theory COST WaVaCS Winterschool Venice, February 2011 Stefan Buehler Luleå University of Technology Kiruna

2 Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) 2

3 Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) 3

4 Remote Sensing Systems Provide Indirect Measurements Why is this called an inverse problem? Figure: University of Bremen

5 Inverse Problem Notation y = F(x) + ε y: Measurement vector x: State vector F: Forward model ε: Noise Often: y = y 0 + Kx K: Jacobi matrix (df/dx) 5

6 Measurement Vector y and State Vector x x y Figure: University of Bremen, stolen from last weeks lecture by Uwe.

7 The Jacobi Matrix K y = y 0 + Kx Figure from Clive D. Rodgers, Inverse Methods for Atmospheric Sounding, World Scientific,

8 Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) 8

9 A Linear Equation System y = y 0 + K x For simplicity: y = K x If K is a n y *n x matrix, this defines a system of n y equations for the n x unknown elements of x. Mathematically, this system can be overdetermined (more equations than unknowns) or underdetermined (fewer equations than unknowns). If we have more equations than unknowns, the different equations will contradict each other, due to the noise, so that there is no solution to the equation system. 9

10 Matrix Inversion Approach y = K x We could get the idea to choose n x = n y. Then: y = K -1 x Problem solved? 10

11 Matrix Inversion Result Figure from Clive D. Rodgers, Inverse Methods for Atmospheric Sounding, World Scientific,

12 Matrix Inversion Result Small errors (noise) in y can lead to large differences in x. (Noise amplification.) The problem is that the information provided by the different equations is not independent. (They are linearly dependent, or almost so.) Mathematically, we say that the inversion problem is ill-posed, and that the K matrix is illconditioned. 12

13 Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) 13

14 The Least Squares Method Least-squares fitting is a technique that can be used if there are more equations than unknowns. Idea: Find the x that minimizes the disagreement between F(x) and y. Disagreement measured by a chi-square test: χ 2 = (y F(x))'S y 1 (y F(x)) The measurement error covariance matrix S y needs more explanation. 14

15 Describing Errors for Vectors Scalar: deviation) Vector: y = y ± σ y S y = 2 σ 11 (mean and standard 2 σ 1ny 2 σ ny 1 2 σ ny n y σ ij 2 = s (y s [i] y[i])(y s [ j] y[ j]) The covariance matrix is the generalization of the variance σ 2 for vectors. 15

16 Covariance Matrix Diagonal elements are variances for the individual vector elements. Off-diagonal elements describe correlations between the elements. S y = 2 σ 11 2 σ 1ny 2 σ ny 1 2 σ ny n y x[2] x[1] 16

17 Empirical Orthogonal Functions (EOF) Also called principal components. Can be obtained by finding a coordinate transformation that diagonalizes S. PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. Source: Ben FrantzDale (Wikipedia) 17

18 Back to the Least Squares Fits Idea: Find the x that minimizes the disagreement between F(x) and y. Disagreement measured by a chi-square test: χ 2 = (y F(x))'S 1 y (y F(x)) S y describes the measurement error (noise behavior of the instrument). The test measures the discrepancy between measurement and forward model for each element of y. Weighted by the measurement error for that element. 18

19 Multilinear Regression If the forward model is linear, then the least square solution for x given y, y 0 and K can be found by simple linear algebra. This is called (multi)linear regression. y = F(x) = y 0 + Kx Very easy to do in Matlab with the backslash operator. Works even if there are more equations than unknowns. Also suffers from noise amplification, like the direct matrix inversion method. 19

20 Singular Value Decomposition (SVD) Helps us to understand the noise amplification problem. K = UDV' U and V are orthogonal matrices, D is a diagonal matrix. If one has the SVD, then one gets the inverse of K for free: K 1 = VD 1 U' The elements of D are called singular values. 20

21 Singular Values Example for D -1 for n y =4, n x =3: If d ii are small, there will be numerical problems (division by zero). If all are small, we can simply scale the whole matrix, the problem is, if some are large and some are small. Condition number c = 1 d d d largest singular value smallest singular value 21

22 Ill-Conditioned Matrices The larger c, the more numerically unstable the matrix inversion will be. We say, matrix K is ill-conditioned, if c 1. If K is ill-conditioned, then the inverse problem y=kx is ill-posed. There is no stable solution, unless we make additional assumptions. We call these assumptions a-priori information. 22

23 Overview Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) 23

24 Bayesian Methods Are methods that are consistent with probability theory. Bayes theorem states how different pieces of information can be combined. It can be used to derive the OEM method, but I will show you only the result, not the derivation. The OEM method is based on the assumption that both measurement error and climatology of x have Gaußian statistics. 24

25 The Optimal Estimation Method (OEM) Make the a-priori assumption, that the solution x should be close to a climatological mean value. For the least square approach, the costfunction becomes: (y F(x))'S 1 y (y F(x)) κ (x) = (x x 0 )'S 1 0 (x x 0 ) The x that minimizes this cost function is the OEM solution. Can be found by simple algebra for the linear problem. + 25

26 OEM Solution for Linear Problem Is given by: ˆx = x 0 + (K'S 1 y K + S 1 0 ) 1 K'S 1 y (y Kx 0 ) or ˆx = x 0 + S 0 K'(KS 0 K'+ S y ) 1 (y Kx 0 ) Looks tough, but all quantities are known. Easy to calculate with Matlab. Expensive part: One matrix inversion. 26

27 Averaging Kernel Matrix Jacobian: K = dy dx Contribution function matrix: Averaging kernel matrix: Connects true state to retrieved state: D = dˆx dy A = DK A OEM = (K'S 1 y K + S 1 0 ) 1 K'S 1 y K ˆx x 0 = A(x x 0 ) + Dε 27

28 Example: Groundbased Microwave H2O Radiometer 28

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34 Dealing with Non-Linearity Completely linear: y=kx Almost completely linear: y-y 0 =K(x-x 0 ) Moderately nonlinear (typical trace gas retrieval) Iterative methods, particularly Levenberg-Marquardt iteration (recalculate K(x i ) as x i is iteratively improved) Strongly nonlinear: Neural Nets, 34

35 Summary Inversion 1 The Inverse Problem 2 Simple Minded Approach (Matrix Inversion) 3 Less Simple Minded Approach (Linear Regression, SVD, Condition Number) 4 Optimal Estimation (Bayesian Methods) Perhaps a bit abstract, we will try out the different concepts in the practical! 35

36 Questions? 36