Seismic Tomography: Example of a geophysical inverse problem

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1 Seismic Tomography: Example of a geophysical inverse problem Dr. Karin Sigloch (karin.sigloch@lmu.de) Theresienstr. 41, Zi What is isan inverse problem? A real-world example: tomography of the Earth s mantle under North America How does it itwork? Seismic tomography Folie 1

2 Q: What is an inverse problem? A: An indirect measurement. We Wewant to to measure some important EARTH_PROPERTY (e.g., seismic velocity v(x)), and and have no no tools to to do do it. it. Insteadweknowhowto measuresomeotherpropertycalled DATA (e.g., traveltime delays dt) dt) And And we weknow some phys./math. relationship MAPPING_FCT, so so that: DATA = MAPPING_FCT(EARTH_PROPERTY) If Ifwe weare areable ableto to find find an an inverse function MAPPING_FCT -1-1 so so that that EARTH_PROPERTY = MAPPING_FCT -1-1 (DATA), then the theproblem is issolved. Seismic tomography Folie 2

3 Inverse problems are common Seismology EARTH_PROPERTY: as as a function of of space (x,y,z), e.g., P-velocity or or intrinsic attenuation, or or rock rock composition DATA: Seismograms (and (and data dreived from them, like liketraveltimes, amplitudes...) at at discrete points at at the thesurface MAPPING_FCT: wave equation (or (orsome approximation to to it, it, like likerays raysfrom Snell s law) law) Seismic tomography Folie 3

4 Inverse problems are common Medical Imaging: Computed Tomography EARTH_PROPERTY: structureof tissuein thehuman body DATA: X-ray imaging in in multiple plane by by how how much do do x-rays get getattenuated? MAPPING_FCT: wave propagation and and attenuation (optics, geometrical ray ray approximation) Seismic tomography Folie 4

5 Inverse problems are common (also outside geophysics) Medical Imaging: Computed Tomography Seismic tomography Folie 5

6 Inverse problems are common Planetary Science:Composition of of a Jupiter moon EARTH_PROPERTY: density as as function of of x,y,z DATA: gravity measurements: deflection of of a satellite upon upon its its fly-by MAPPING_FCT: Newton s law lawof of gravity Common theme: measure interior properties from the the outside but not not all all inverse problems are are like like that Seismic tomography Folie 6

7 Inverse problems are common Borehole seismics: EARTH_PROPERTY: shallow earth properties (velocity, density, attenuation, ) as as a function of of depth DATA: hydrophone recordings inside the the borehoel MAPPING_FCT: wave equation: reflections Seismic tomography Folie 7

8 Inverse problems are common Environmental remediation/hydrology: EARTH_PROPERTY: source location(s) and and quantity of of contaminants DATA: contaminant sensors in in several deep holes around a chemical factory MAPPING_FCT: diffusion equation/transport in in porous media Seismic tomography Folie 8

9 Summary: What is an inverse problem? Weareunableto directlymeasurean interesting EARTH_PROPERTY. Instead we wemeasure some other DATA, because we weknow how how to to derive/compute a physical relationship MAPPING_FCT so so that: DATA = MAPPING_FCT(EARTH_PROPERTY) We We try try to to find find the the inverse MAPPING_FCT -1-1, so so that that EARTH_PROPERTY = MAPPING_FCT -1-1 (DATA) Seismic tomography Folie 9

10 A realistic experiment: Seismic tomography of the Earth s mantle Geophysicists mission: Discover new things about the Earth. If it itis isa good problem then many other people (geologists, geodynamicists, economists, etc.) will be interested in in the results. Seismic tomography Folie 10

11 Imaging the subducted Farallon plate under North America Seismic tomography Folie 11

12 The Farallon plate 140 Myr ago F Engebretson et al

13 and 80 Myr ago F F Engebretson et al

14 and today. F Engebretson et al

15 150 million years of textbook-like subduction? A single large plate has been subducting beneath the North American west coast for 150+ million years. No significant interference from other plates. Ren et al 2007, after Engebretson

16 A simple story? Yes, but. Extensive mountain building and volcanism far inland (since ~70 Myr). Not a conventional volcanic arc. Why is the North American Cordillera so wide and stands so high? 16

area graphics: Blakely (online) Laramide orogeny (70-50 Myr):

17 The Laramide orogeny : Rapid uplift, far inland at ~70 Myr ago 75 Myr ago 65 Myr ago A shallow inland sea covers the Rocky Mountain area graphics: Blakely (online) Laramide orogeny (70-50 Myr): basement uplift by thrust faulting, volcanic arc along trench has shut off. 17

18 Geologists explanation: Laramide thrust faulting was caused by anomalously flat subduction Extremely flat slab scrapes along bottom of continental crust 18

19 but Western North America has stood high ever since. WY UT AZ CO NM NASA satellite photo of Western U.S.; mountains are from Laramide times19

P-wave seismograms from large earthquakes (magnitude >= 5.

20 Our tomographic experiment 637 earthquake sources 1125 broadband receivers (seismometers) We use teleseismic P-wave seismograms from large earthquakes (magnitude >= 5.8, ) Many new USArray stations in Western U.S since

21 Tomography step by step observed seismogram deconvolve source time function extract scalar observables dt(f), da(f) dt da = solve * dv dq result: earth model predicted seismogram compute sensitivity kernels parameterize 21

DATA observed seismogram Tomography step

earth model predicted seismogram compute

22 DATA observed seismogram Tomography step by step deconvolve source time function extract scalar observables dt(f), da(f) dt da = solve * EARTH PROPERTY dv dq result: earth model predicted seismogram compute sensitivity kernels parameterize MAPPING_FCT 22

23 Result: 3-D model of P-wave velocities under North America 23

24 Locations of interesting cross-sections F 49ºN D C B A 42ºN PA 24

25 The big picture: Not one, but two episodes of whole mantle subduction S1 C F1 42ºN S2 F2 F 49ºN D C B A 42ºN dv p /V p in % PA 25

26 Image of the subducted Farallon slab in the mantle depth/km Seismically fast material is contoured (fast means cold). Color signifies depth. We can confidently image ~1500 km deep. Crust and lithosphere not rendered. 26

27 Interpretation: a frontal plate break ended the Laramide era at ~50 Myr N1 N2 S2 F2 S1 S2 S2 F1 F2 W F1 F2 depth/km F1 must have been part of the Laramide flat slab. It still fills the transition zone. Lower end of S2/N2 subducted ~55 Myr ago. 27

28 Interpretation: a frontal plate break ended the Laramide era at ~50 Myr 55 Myr ago 40 Myr ago Today All velocities in a hotspot frame. 28

29 How does this work? Some intuitive examples Seismic tomography Folie 29

30 Tomography: Intuitive example 1 Surface waves (of (of a certain frequency) have sampled the the shallow mantle of of North America, along the theshown raypaths. Your prior guess is is that that traveltime ~ length of of ray, ray, meaning v(x,y)=v 0 = 0 const. everywhere. In In reality you youobserve anomalies in in the thetraveltime DATA: Red Red ray raymeans traveltime was was longer than expected. Blue Blue ray raymeans traveltime was was as as expected. Red Red rays raysmust have traversed some slow slowmaterial. Where exactly is is is islocated? Seismic tomography Folie 30

31 Tomography: Intuitive example 1 Seismic tomography Folie 31

32 Tomography: Intuitive example 1 Idea: Consider ray crossings Seismic tomography Folie 32

33 Tomography: Intuitive example 1 Idea: Consider ray crossings Seismic tomography Folie 33

34 Tomography: Intuitive example 1 It Itworked pretty well. recovered area true area (was used to generate the colored rays) Why is isthe reconstruction not perfect? Seismic tomography Folie 34

35 Tomography: Intuitive example 2 Image reconstruction original image Seismic tomography Folie 35 We smear the image in horizontal direction (like an x-ray integrates over different body tissues along its path)

36 Tomography: Intuitive example 2 Image reconstruction: Generating DATA We smear over more directions to simulate more x-ray data : what rays see from all these different angles Seismic tomography Folie 36

37 Tomography: Intuitive example 2 Now we wetry to to reconstruct the image (principle of of destructive/constructive interference): Addition of two directions of the data Addition of all 8 directions of the data Seismic tomography Folie 37

38 Tomography: Intuitive example 2 Reconstruction result Original Reconstruction How could we weimprove on on this? Seismic tomography Folie 38

39 Basic modeling Acoustic tomography: A few bricks are standing next to to each other. To To first order they all all have the same, known P- P- velocity v 0 (or slowness u 0 =1/v 0 ), ), except for small variations: u i = i u 0 + u i, i, where u i << i << u 0 We want to to estimate the small anomalies u ii u 0 + u 1 u 0 + u 2 u 0 + u 3 Seismic tomography Folie 39

40 Basic modeling Acoustic tomography: Blocks are x wide and y high. Traveltime: t 11 =u =u 11 s u +u 2 s u +u 3 s Traveltime anomaly: t 11 = u 11 s u 2 s u 3 s The s ij ij can be becomputed from the given geometry (general case = Snell s law!) x y u 0 + u 1 u 0 + u 2 u 0 + u 3 s 11 s 12 s 13 t 1, t 1 Seismic tomography Folie 40

41 Basic modeling Acoustic tomography: Linear system: two equations, three unknowns Matrix notation: x t 2 y s 21 u 0 + u 1 u 0 + u 2 s 11 s 12 u 0 + u 3 s 13 t 1 Seismic tomography Folie 41

42 Basic modeling Acoustic tomography: For full rank: two equations, two unknowns. Full rank means: Matrix notation (and its itsinverse): y x t 2 s 21 2x u 0 + u 1 u 0 + u 2 s 11 s 12 t 1 Seismic tomography Folie 42

43 Basic modeling Acoustic tomography: Does this system have full rank? How many measurements M need to to be bemade for the matrix to to have an an inverse? x t 2 t 3 y s 21 u 0 + u 1 u 0 + u 2 s 11 s 12 u 0 + u 3 t 1 s 13 Seismic tomography Folie 43

44 Basic modeling Acoustic tomography: How about these geometries? Ideally, each measurement should contribute as as much new information as as possible ( independent measurements --> --> experiment design) x x y u 0 + u 1 u 0 + u 2 y u 0 + u 1 u 0 + u 2 Seismic tomography Folie 44

45 Real-world experiments Parameterizations of of the unknowns (grid) u 1 u 2 Coarse parameterization in blocks; few unknowns vs. Complex parameterization (irregular tetrahedra, 10 5 unknowns Seismic tomography Folie 45

46 Real-world experiments Source and receiver geometry Optimally designed sourcereceiver geometry vs. Take what you can get --> Seismic tomography Folie 46

47 Real-world experiments Signals and wave propagation modeling Sharp pulses modelled as optical rays vs. Realistic wavelets with broad Fresnel zones --> Seismic tomography Folie 47

48 Real-world experiments System to to solve Small sytem, well conditioned, exactly determined vs. DATA EARTH_PROPERTY Huge system, ill conditioned, both underdetermined and overdetermined = MAPPING_FCT * ART OF TOMOGRAPHY: Finding smart ways to solve this anyway. Seismic tomography Folie 48

49 Seismic tomography Folie 49

Earth Science ENR Plate Boundaries Notes

Name Earth Science ENR Plate Boundaries Notes Per Tchr Plate Boundary Types: 1) Divergent Plate Boundary (Seafloor Spreading Centers) 2) Divergent Plate Boundary (Continental Rift Valley) 3) Transform