with a focus on seismic tomography CIDER2012 KITP Santa Barbara Seismic travel time tomography 1 In the background reference model Travel time T along a ray g v 0 s velocity at point s on ID: 1025092
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1. Geophysical Inverse Problemswith a focus on seismic tomographyCIDER2012- KITP- Santa Barbara
2. Seismic travel time tomography
3. 1) In the background, “reference” model: Travel time T along a ray g:v0(s) velocity at point s onthe rayu= 1/v is the “slowness”Principles of travel time tomographyThe ray path g is determined by the velocity structure using Snell’s law. Ray theory.2) Suppose the slowness u is perturbed by an amount du small enoughthat the ray path g is not changed. The travel time is changed by:
4. lij is the distance travelled by ray i in block jv0j is the reference velocity (“starting model”) in block jSolving the problem: “Given a set of travel time perturbations dTi on an ensemble of rays {i=1…N}, determine the perturbations (dv/v0)j in a 3Dmodel parametrized in blocks (j=1…M}” is solving an inverse problem ofthe form:d= data vector= travel time pertubations dTm= model vector = perturbations in velocity
5. G has dimensions M x NUsually N (number of rays) > M (number of blocks):“over determined system”We write:GTG is a square matrix of dimensions MxMIf it is invertible, we can write the solution as:where (GTG)-1 is the inverse of GTGIn the sense that (GTG)-1(GTG) = 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 GTG, in general, is singular:“””least squares solution”Minimizes ||d-Gm||2Some Gij are null ( lij=0)-> infinite elements in the inverse matrix
7. How to choose a solution?Special solution that maximizes or minimizes some desireable property through a normFor example:Model with the smallest size (norm): mTm=||m||2=(m12+m22+m32+…mM2)1/2Closest possible solution to a preconceived model <m>: minimize ||m-<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-GmTikhonov 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:<m> reference model
10. Weighted damped least squaresMore generally, the solution has the form:For more rigorous and complete treatment (incl. non-linear):See Tarantola (1985) Inverse problem theoryTarantola 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 SquaresFor Damped Least SquaresNote : Generally G-g ≠G-1
12. As you increase the damping parameter e, more priority is given to model-norm part of functional.Increases Prediction ErrorDecreases model structure Model will be biased toward smooth solutionHow to choose e so that model is not overly biased?Leads to idea of trade-off analysis.η“L curve”
13. Model Resolution MatrixHow 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 testR contains theoretical assumptionson wave propagation, parametrizationAnd assumes the problem is linearAfter Masters, CIDER 2010
16. Ingredients of an inversionImportance of sampling/coveragemixture of data typesParametrizationPhysical (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. PSSurface wavesSS50 mnP, PPS, SSArrivals well separated on the seismogram, suitable for traveltime measurementsGenerally:Ray theoryIterative back projection techniques- Parametrization in blocks
18. Van der Hilst et al., 1998Slabs……...and plumesMontelli et al., 2004P velocity tomography
19.
20. Vasco and Johnson,1998P TravelTimeTomography:RayDensitymaps
21. Karason andvan der Hilst,2000Checkerboard tests
22. Honshu410660±1.5 %151305060708091112141513northern Bonin±1.5 %4106601000Fukao andObayashi2011
23. ±1.5%TongaKermadec06070809101112131415±1.5%4106601000Fukao andObayashi2011
24. PRI-S05Montelli et al., 2005EPRSouth Pacific superswellTongaFukao andObayashi,20116601000400S40RTSRitsema et al., 2011
25. Rayleigh waveovertonesBy 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 subsets120 km600 km1600 km2800 kmAfter Ritsema et al., 2004
27. SdiffScS2The travel time dataset in this model includes:Multiple ScS: ScSn
28. Coverage of S and PAfter Masters, CIDER 2010
29. PSSurface wavesSS
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 GUC B e r k e l e y
31. PAVANACTSSSdiffLi and Romanowicz , 1995
32. PAVANACT
33. 2800 km depthfrom Kustowski, 2006Waveforms only, T>32 s!20,000 wavepacketsNACT
34. To et al, 2005
35. Indian Ocean Paths - SdiffractedCorner frequencies: 2sec, 5sec, 18 secTo 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 syntheticscomputed using the Spectral Element Method (SEM)DataSynthetics
38. SEMum (Lekic and Romanowicz, 2011)S20RTS (Ritsema et al. 2004)70 km125 km180 km250 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.
40.
41. Courtesy of Scott French
42. SEMum2S40RTSRitsema et al., 2011French et al., 2012EPRSouth Pacific superswellTongaSamoaEaster IslandMacdonaldFukao andObayashi, 2011
43. Summary: what’s important in global mantle tomographySampling: 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 functionsError estimation