Lectures 12 & 13: Geophysical Inverse Theory I, II

1. Lectures 12 & 13: Geophysical Inverse Theory I, II H. SAIBI & SASAKI Kyushu University Dec. 24, 2015 & Jan. 7, 2016

2. Geophysical inverse problems Inferring seismic properties of the Earth’s interior from surface observations

3. Inverse problems are everywhere When data only indirectly constrain quantities of interest

4. Thinking backwards Most people, if you describe a train of events to them will tell you what the result will be. There are few people, however that if you told them a result, would be able to evolve from their own inner consciousness what the steps were that led to that result. This power is what I mean when I talk of reasoning backward. Sherlock Holmes, A Study in Scarlet, SirArthur Conan Doyle (1887)

5. Reversing a forward problem

6. Inverse problems=quest for information What is that ? What can we tell about Who/whatever made it ? Collect data: • Measure size, depth properties of the ground Can we expect to reconstruct the whatever made it from the evidence ? Use our prior knowledge: • Who lives around here ? Make guesses ?

7. Anatomy of an inverse problem Hunting for gold at the beach with a gravimeter X X X X ? Gravimeter

8. Forward modelling example: Treasure Hunt We have observed some values: 10, 23, 35, 45, 56  gals How can we relate the observed gravity values to the subsurface properties? We know how to do the forward problem: X X X X X Gravimeter ? This equation relates the (observed) gravitational potential to the subsurface density. -> given a density model we can predict the gravity field at the surface!

9. Treasure Hunt: Trial and error What else do we know? Density sand: 2.2 g/cm3 Density gold: 19.3 g/cm3 Do we know these values exactly? Where is the box with gold? X X X X X Gravimeter ? One approach is trial and (t)error forward modelling Use the forward solution to calculate many models for a rectangular box situated somewhere in the ground and compare the theoretical (synthetic) data to the observations.

10. Treasure Hunt: model space But ... ... we have to define plausible models for the beach. We have to somehow describe the model geometrically. We introduce simplifying approximations - divide the subsurface into rectangles with variable density - Let us assume a flat surface X X X X X Gravimeter ? x x x x x surface sand gold

11. Treasure Hunt: Non-uniqueness • Could we compute all possible models • and compare the synthetic data with the • observations? • at every rectangle two possibilities • (sand or gold) • 250 ~ 1015 possible models • Too many models! X X X X X Gravimeter • We have 1015 possible models but only 5 observations! • It is likely that two or more models will fit the data (maybe exactly) • Non-uniqueness is likely

12. Treasure hunt: a priori information Is there anything we know about the treasure? How large is the box? Is it still intact? Has it possibly disintegrated? What was the shape of the box? X X X X X Gravimeter This is called a priori (or prior) information. It will allow us to define plausible, possible, and unlikely models: plausible possible unlikely

13. Treasure hunt: data uncertainties Things to consider in formulating the inverse problem • Do we have errors in the data ? • Did the instruments work correctly ? • Do we have to correct for anything? • (e.g. topography, tides, ...) X X X X X Gravimeter • Are we using the right theory ? • Is a 2-D approximation adequate ? • Are there other materials present other than gold and sand ? • Are there adjacent masses which could influence observations ? Answering these questions often requires introducing more simplifying assumptions and guesses. All inferences are dependent on these assumptions. (GIGO)

14. X X X X X Gravimeter What we have learned from one example Inverse problems = inference about physical systems from data • Data usually contain errors (data uncertainties) • Physical theories require approximations • Infinitely many models will fit the data (non-uniqueness) • Our physical theory may be inaccurate (theoretical uncertainties) • Our forward problem may be highly nonlinear • We always have a finite amount of data Detailed questions are: How accurate are our data? How well can we solve the forward problem? What independent information do we have on the model space (a priori information) ?

15. Estimation and Appraisal

16. Objective of geophysical survey to obtain information about geologic structure or buried object (such as the existence of the object and its size and shape) by generating the distribution of physical properties Data(Earth response) Distribution of physical property

17. Forward & inverse problems Forward modeling Distribution of physical property (Earth model) Earth response (Data) Inversion

18. Forward problem “Convolved” (or averaging) process Earth model Earth response Governingequation We have a unique solution! e.g., finite difference modeling

19. Governing equation in forward problem Density Laplace’s equation Gravity Magnetic Laplace’s equation Magnetic fieldsusceptibility Electrical Poisson’s equation Electric potentialresistivity Electrical Maxwell equation EM fieldresistivity (diffusion or wave) - - - - -

20. Inverse problem “Deconvolved” process Earth model Earth response Governingequation We may have many (non-unique) solutions! --

21. Why is the inverse solution non-unique?(Why is the inverse problem ill-posed?) • Data are incomplete; data include measurement errors and various errors. • Data are insufficient; data coverage is limited in space (and in frequency).

22. How to tackle this problem of non-uniqueness Incorporate additional ( a priori) informationinto the inverse problem Approach to solving the inverse problem as an least-squares problem with constraints (or optimization problem)

23. Regression analysis as an inverse problem In this section, we regard regression analysis as a kind of the inverse problem. Learning regression analysis help to understand the basis of inversion.

24. Linear regression y Assume that the observations can be expressed as where y is an observation, x is independent variables, a and b are unknown parameters. Determine a and bwhen we are given N data sets (x1 , y2), - - -, (xN , yN). (xi , yi) x

25. Formulation of regression analysis (1) Observation equation - - - - - - - One can solve the above observation equation in a least-squares sense.

26. Formulation of regression analysis (2) Minimization of with respect to a and b requires that

27. Formulation of regression analysis (3) Normal equation Solving the simultaneous equations, we obtain

28. Regression from perspective of inversion Model parameter: a and b Model function (response): Data (observations): y1 , y2 , - - -, yN Independent variable such as measuring location: x1 , x2 , - - -,xN

29. How to formulate least-squares problem In this section, we don’t consider non-uniqueness problem and thus incorporation of constraints. We just focus on how to find a model that matches the data in a least-squares sense. So the problem so far may be called unconstrainedleast-squares problem. We assume that we can do forward modeling.

30. Setting of inverse problem Model parameters Model response - - - - - Data (observations) 2D discretization Note that model response is nonlinear function with respect to model parameters.

31. Formulation of unconstrained nonlinear least-squares problem - - - - - - : objective function (data misfit) Gaussian noise : L2 (least-squares) norm Observation equations

32. Linearized objective function Initial model estimate: Applying Taylor expansion to model response gives Then we have : Model parameter change (unknowns) : Difference between data and model response

33. Linearized objective functionin matrix form : parameter change vector (unknowns) : data residual vector A : N x M Jacobian matrix (matrix of partial derivative with respect to model parameters)

34. Minimization of objective function Minimization of with respect to requires that Then we have the normal equation, T : transpose Note that the normal equation is M x M linear system of equations

35. There is an alternative way to obtaining the least-squares solution by forming the normal equation. It is through forming the over-determined system equations. In general, this approach is more accurate particularly when the number of the unknowns is large.

36. = A Minimization of objective function is equivalent to obtaining least-squares solution from over-determined system of equations Over-determined: the number of unknowns is smaller than the number of (observation) equations

37. Exercise 1 Assume that we have When the initial estimate m0is given, show the linearized data objective function in both summation and matrix forms. In the matrix form, show all the elements. Use the notation We define the data objective function as

38. Exercise 2 Based on the result of Exercise 1,form the over-determined system of equations.

39. A x y = How to solve least-squares problems In this section, we just focus on obtaining the least-squares solution when we are given an over-determined system of equations

40. A x y = Classic approach to least-squares problem Minimization of with respect to x requires that Then we have the normal equation, = M x M matrix

41. A a1 a2 Why is it called normal ? Error vector i=1, 2, - - -, M Inner product The error vector is normal (or orthogonal) to the column vectors of A

42. A x y = Another approach to least-squares problem:least-squares solvers • Modified Gram Schmidt method (QR decomposition ) • Householder transformation method • Singular value decomposition (SVD) method

43. A x y = Least-squares solver: QR decomposition A can be decomposed using modified Gram Schmidt method into A=QR Q: N x M orthogonal matrix R: M x M upper triangular matrix

44. How to incorporate constraints (or a priori information) into a least-squares problem In this section, we consider the most important aspect of inversion, that is, how to pose some soft restrictions on the least-squares solution so that the solution has the likely features of the reality.

45. Prerequisites for solving inverse problems • Forward problem can be solved.(Model response can be calculated) • Jacobian matrix (or sensitivity matrix) can be obtained • Incorporate a priori information (or the likely or expected earth structure)

46. Sidetrack: Occam’s inversion To find the smoothest model so that its features depart from the simplest case only as far as is necessary to fit the data (Constable et al., 1987) Occam’s razor:the explanation of any phenomenon should make as few assumptions as possible, eliminating those that make no difference in the observable predictions of the explanatory hypothesis or theory(Wikipedia).

47. Constrained least-squares problem Find a model that fit the data as well as incorporates additional information Design an objective function that include a norm (measure) for model structure besides the data misfit : model objective function (constraints) : tradeoff (regularization) parameter

48. An example of model objective function (simplest one) Data misfit Size of model structure(Constraint)

49. Selection of model objective function ? Model objectivefunction Data misfit

50. Various desirable characteristics of structure and their representations (1D) • be as smooth as possible • be as flat as possible • be as close to a reference model as possible • be as blocky as possible