Lecture 7 Backus-Gilbert Generalized Inverse and the Trade Off of Resolution and Variance

1. Lecture 7 Backus-Gilbert Generalized Inverse and the Trade Off of Resolution and Variance

2. Syllabus Lecture 01 Describing Inverse ProblemsLecture 02 Probability and Measurement Error, Part 1Lecture 03 Probability and Measurement Error, Part 2 Lecture 04 The L2 Norm and Simple Least SquaresLecture 05 A Priori Information and Weighted Least SquaredLecture 06 Resolution and Generalized Inverses Lecture 07 Backus-Gilbert Inverse and the Trade Off of Resolution and VarianceLecture 08 The Principle of Maximum LikelihoodLecture 09 Inexact TheoriesLecture 10 Nonuniqueness and Localized AveragesLecture 11 Vector Spaces and Singular Value Decomposition Lecture 12 Equality and Inequality ConstraintsLecture 13 L1 , L∞ Norm Problems and Linear ProgrammingLecture 14 Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15 Nonlinear Problems: Newton’s Method Lecture 16 Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals Lecture 17 Factor AnalysisLecture 18 Varimax Factors, Empircal Orthogonal FunctionsLecture 19 Backus-Gilbert Theory for Continuous Problems; Radon’s ProblemLecture 20 Linear Operators and Their AdjointsLecture 21 Fréchet DerivativesLecture 22 Exemplary Inverse Problems, incl. Filter DesignLecture 23 Exemplary Inverse Problems, incl. Earthquake LocationLecture 24 Exemplary Inverse Problems, incl. Vibrational Problems

3. Purpose of the Lecture Introduce a new way to quantify the spread of resolution Find the Generalized Inverse that minimizes this spread Include minimization of the size of variance Discuss how resolution and variance trade off

4. Part 1A new way to quantify thespread of resolution

5. criticism of Direchletspread() functionswhen m represents m(x)is that they don’t capture the sense of “being localized” very well

6. These two rows of the model resolution matrix have the same Direchlet spread … Rij Rij index, j index, j i i … but the left case is better “localized”

7. old way new way

8. old way add this function new way

9. old way Backus-Gilbert Spread Function new way

10. grows with physical distance between model parameters zero when i=j

11. if w(i,i)=0 then

12. for one spatial dimensionm is discretized version of m(x)m = [m(Δx), m(2 Δx), … m(M Δx) ]T w(i,j) = (i-j)2 would work fine

13. for two spatial dimensionm is discretized version of m(x,y)on K⨉L gridm = [m(x1, y1), m(x1, y2), … m(xK, yL) ]T w(i,j) = (xi-xj) 2 + (yi-yj) 2 would work fine

14. Part 2Constructing a Generalized Inverse

15. under-determined problem find G-g that minimizes spread(R)

16. under-determined problem find G-g that minimizes spread(R) with the constraint

17. under-determined problem find G-g that minimizes spread(R) with the constraint (since Rii is not constrained by spread function)

18. once again, solve for each row of G-g separatelyspread of kth row of resolution matrix R symmetric in i,j with

19. for the constraint [1]k= with

20. Lagrange Multiplier Equation + now set ∂Φ/∂G-gkp

21. Sg(k) – λu = 0 with g(k)Tthe kth row of G-g solve simultaneously with uTg(k) = 1

22. putting the two equations together

23. construct inverse A, b, c unknown

24. construct inverse so

25. = g(k) = uT g(k)T = kth row of G-g

26. Backus-Gilbert Generalized Inverse (analogous to Minimum Length Generalized Inverse) written in terms of components with

27. (B) (A) j i (D) (C) j i

28. Part 3Include minimization of the size of variance

29. minimize

30. new version of Jk

31. with

32. same with

33. so adding size of varianceis just a small modification to S

34. Backus-Gilbert Generalized Inverse (analogous to Damped Minimum Length) written in terms of components with

35. In MatLab GMG = zeros(M,N); u = G*ones(M,1); for k = [1:M] S = G * diag(([1:M]-k).^2) * G'; Sp = alpha*S + (1-alpha)*eye(N,N); uSpinv = u'/Sp; GMG(k,:) = uSpinv / (uSpinv*u); end

36. $20 Reward!to the first person who sends me MatLab code that computes the BG generalized inverse without a for loop(but no creation of huge 3-indexed quantities, please. Memory requirements need to be similar to my code)

37. The Direchlet analog of theBackus-Gilbert Generalized Inverseis theDamped Minimum LengthGeneralized Inverse

38. 0 special case #2 1 ε2 I G-g[GGT+ε2I]=GT G-g=GT [GGT+ε2I]-1 damped minimum length

39. Part 4the trade-off of resolution and variance

40. the value ofa localized average with small spreadis controlled by few dataand so has large variance the value ofa localized average with large spreadis controlled by many dataand so has small variance

41. (B) (A)

42. small spreadlarge variance large spreadsmall variance

43. α near 1small spreadlarge variance α near 0 large spreadsmall variance

44. Trade-Off Curves (B) (A) α=1 α=1 log10 size of covariance log10 size of covariance α=½ α=½ α=0 α=0 spread of model resolution log10 spread of model resolution