Lecture 19 Continuous Problems: Backus-Gilbert Theory and Radon’s Problem

1. Lecture 19 Continuous Problems:Backus-Gilbert TheoryandRadon’s Problem

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 • Extend Backus-Gilbert theory to continuous problems • Discuss the conversion of • continuous inverse problems to discrete problems • Solve Radon’s Problem • the simplest tomography problem

4. Part 1 Backus-Gilbert Theory

5. Continuous Inverse Theorythe data are discretebutthe model parameter is a continuous function

6. One or several dimensions

7. One or several dimensions data model function

8. hopeless to try to determine estimates of model function at a particular depthm(z0) = ? localized average is the only way to go

9. hopeless to try to determine estimates of model function at a particular depthm(z0) = ? the problem is that an integral, such as the data kernel integral, does not depend upon the value of m(z) at a “single point” z0 localized average is the only way to go continuous version of resolution matrix

10. let’s retain the idea that the“solution”depends linearly on the data

11. let’s retain the idea that the“solution”depends linearly on the data continuous version of generalized inverse

12. implies a formula for R

13. comparison to discrete case <m>=G-gd d=Gm <m>=Rm • R=G-gG

14. implies a formula for R

15. Now define the spread of resolution as

16. fine generalized inversethat minimizes the spread Jwith the constraint that = 1

18. J has exactly the same form as the discrete caseonly the definition of S is different

19. Hence the solution is thesame as in the discrete case where

20. furthermore, just as we did in the discrete case, we can add the size of the covariance where

21. as beforethis just changes the definition of S and leads to a trade-off of resolution and variance α=1 size of variance α=0 spread of resolution

22. Part 2 Approximating aContinuous Problemas a Discrete Problem

23. approximation using finite number of known functions

24. approximation using finite number of known functions known functions continuous function unknown coefficients = discrete model parameters

25. posssiblefj(x)’s voxels (and their lower dimension equivalents) polynomials splines Fourier (and similar) series and many others

26. does the choice of fj(x) matter? Yes! The choice implements prior information about the properties of the solution The solution will be different depending upon the choice

27. conversion to discrete Gm=d

28. special case of voxels 1 if x inside Vi • fi(x) = 0 otherwise size controlled by the scale of variation of m(x) integral over voxel j

29. approximation when Gi(x) slowly varying center of voxel j size controlled by the scale of variation of Gi(x) more stringent condition than scale of variation of m(x)

30. Part 3 Tomography

31. Greek Root tomos a cut, cutting, slice, section

32. “tomography”as it is used in geophysics data are line integrals of the model function curve i

33. you can force this into the form if you want Gi(x) but the Dirac delta function is not square-integrable, which leads to problems

34. Radon’s Problemstraight line raysdata d treated as a continuous variable

35. (u,θ) coordinate system forRadon Transform y s u θ x integrate over this line

36. Radon Transformm(x,y) → d(u,θ)

37. (A) m(x,y) (B) d(u,θ) x u y θ

38. Inverse Problem findm(x,y) given d(u,θ)

39. Solution via Fourier Transforms x→kx • kx→ x

40. now Fourier transform u→ku now change variables • (u,θ) →(x,y)

41. now Fourier transform u→ku now change variables • (s,u) →(x,y) J=1, by the way Fourier transform of d(u,θ) Fourier transform of m(x,y) evaluated on a line of slopeθ

42. y ky ^ ^ m(x,y) m(kx,ky) d(u,θ0) u θ0 θ0 x kx FT

43. Learned two things • Proof that solution exists and unique, based on “well-known” properties of Fourier Transform • Recipe how to invert a Radon transform using Fourier transforms

44. (A) (B) (C) x u x y θ y