1. Geophysical Inverse Problems with a focus on seismic tomography CIDER2012- KITP- Santa Barbara

2. Seismic travel time tomography

3. 1) In the background, “reference” model: Travel time T along a ray  v 0 (s) velocity at point s on the ray u= 1/v is the “slowness” Principles of travel time tomography The ray path  is determined by the velocity structure using Snell’s law. Ray theory. 2) Suppose the slowness u is perturbed by an amount  u small enough that the ray path  is not changed. The travel time is changed by:

4. l ij is the distance travelled by ray i in block j v 0 j is the reference velocity (“starting model”) in block j Solving the problem: “Given a set of travel time perturbations  T i on an ensemble of rays {i=1…N}, determine the perturbations (dv/v 0 )j in a 3D model parametrized in blocks (j=1…M}” is solving an inverse problem of the form: d= data vector= travel time pertubations  T m= model vector = perturbations in velocity

5. G has dimensions M x N Usually N (number of rays) > M (number of blocks): “over determined system” We write: G T G is a square matrix of dimensions MxM If it is invertible, we can write the solution as: where (G T G) -1 is the inverse of G T G In the sense that (G T G) -1( G T G) = I, I= identity matrix “least squares solution” – equivalent to minimizing ||d-Gm|| 2

6. -G contains assumptions/choices: -Theory of wave propagation (ray theory) -Parametrization (i.e. blocks of some size) In practice, things are more complicated because G T G, in general, is singular: “ ””least squares solution” Minimizes ||d-Gm|| 2 Some G ij are null ( l ij =0)-> infinite elements in the inverse matrix

7. How to choose a solution? Special solution that maximizes or minimizes some desireable property through a norm For example: –Model with the smallest size (norm): m T m=||m|| 2 =(m 1 2 +m 2 2 +m 3 2 +…m M 2 ) 1/2 –Closest possible solution to a preconceived model : minimize ||m- || 2  regularization

8. Minimize some combination of the misfit and the solution size: Then the solution is the “damped least squares solution”: e=d-Gm Tikhonov regularization

9. We can choose to minimize the model size, –eg ||m|| 2 =[m] T [m] - “norm damping” Generalize to other norms. –Example: minimize roughness, i.e. difference between adjacent model parameters. –Consider ||Dm|| 2 instead of ||m|| 2 and minimize: –More generally, minimize: reference model

10. Weighted damped least squares More generally, the solution has the form: For more rigorous and complete treatment (incl. non-linear): See Tarantola (1985) Inverse problem theory Tarantola and Valette (1982)

11. Concept of ‘Generalized Inverse’ Generalized inverse (G -g ) is the matrix in the linear inverse problem that multiplies the data to provide an estimate of the model parameters; –For Least Squares –For Damped Least Squares –Note : Generally G -g ≠G -1

12. As you increase the damping parameter , more priority is given to model-norm part of functional. –Increases Prediction Error –Decreases model structure –Model will be biased toward smooth solution How to choose  so that model is not overly biased? Leads to idea of trade-off analysis. η “L curve”

13. Model Resolution Matrix How accurately is the value of an inversion parameter recovered? How small of an object can be imaged ? Model resolution matrix R: –R can be thought of as a spatial filter that is applied to the true model to produce the estimated values. Often just main diagonal analyzed to determine how spatial resolution changes with position in the image. Off-diagonal elements provide the ‘filter functions’ for every parameter.

14. Masters, CIDER 2010

15. 80% Checkerboard test R contains theoretical assumptions on wave propagation, parametrization And assumes the problem is linear After Masters, CIDER 2010

16. Ingredients of an inversion Importance of sampling/coverage –mixture of data types Parametrization –Physical (Vs, Vp, ρ, anisotropy, attenuation) –Geometry (local versus global functions, size of blocks) Theory of wave propagation – e.g. for travel times: banana-donut kernels/ray theory

17. P S Surface waves SS 50 mn P, PP S, SS Arrivals well separated on the seismogram, suitable for travel time measurements Generally: -Ray theory -Iterative back projection techniques - Parametrization in blocks

18. Van der Hilst et al., 1998 Slabs……...and plumes Montelli et al., 2004 P velocity tomography

20. Vasco and Johnson,1998 P Travel Time Tomography: Ray Density maps

21. Karason and van der Hilst, 2000 Checkerboard tests

22. Honshu 410 660 ±1.5 % 15 13 05 0606 07 08 09 11 12 14 15 13 northern Bonin ±1.5 % 410 660 1000 Fukao and Obayashi 2011

23. ±1.5% Tonga Kermadec 06 07 08 09 10 11 12 13 14 15 ±1.5% 410 660 1000 Fukao and Obayashi 2011

24. PRI-S05 Montelli et al., 2005 EPR South Pacific superswell Tonga Fukao and Obayashi, 2011 660 1000 400 S40RTS Ritsema et al., 2011

25. Rayleigh wave overtones By including overtones, we can see into the transition zone and the top of the lower mantle. after Ritsema et al, 2004

26. Models from different data subsets 120 km 600 km 1600 km 2800 km After Ritsema et al., 2004

27. Sdiff ScS 2 The travel time dataset in this model includes : Multiple ScS: ScS n

28. Coverage of S and P After Masters, CIDER 2010

29. P S Surface waves SS

30. Full Waveform Tomography  Long period (30s-400s) 3- component seismic waveforms  Subdivided into wavepackets and compared in time domain to synthetics.  u(x,t) = G(m)  du = A dm  A= ∂u/∂m contains Fréchet derivatives of G UC B e r k e l e y

31. PAVA NACT SS Sdiff Li and Romanowicz, 1995

32. PAVANACT

33. 2800 km depth from Kustowski, 2006 Waveforms only, T>32 s! 20,000 wavepackets NACT

34. To et al, 2005

35. Indian Ocean Paths - Sdiffracted Corner frequencies: 2sec, 5sec, 18 sec To et al, 2005

36. To et al., EPSL, 2005

37. Full Waveform Tomography using SEM: UC B e r k e l e y Replace mode synthetics by numerical synthetics computed using the Spectral Element Method (SEM) Data Synthetics

38. SEMum (Lekic and Romanowicz, 2011) S20RTS ( Ritsema et al. 2004 ) 70 km 125 km 180 km 250 km -12% +8% -7% +9% -6% +8% -5% +5% -7% +6% -6% +8% -4% +6% -3.5% +3%

39. French et al, 2012, in prep.

41. Courtesy of Scott French

42. SEMum2 S40RTS Ritsema et al., 2011 French et al., 2012 EPR South Pacific superswell Tonga Samoa Easter Island Macdonald Fukao and Obayashi, 2011

43. Summary: what’s important in global mantle tomography Sampling: improved by inclusion of different types of data: surface waves, overtones, body waves, diffracted waves… Theory: to constrain better amplitudes of lateral variations as well as smaller scale features (especially in low velocity regions) Physical parametrization: effects of anisotropy!! Geographical parametrization: local/global basis functions Error estimation