Nonparametric Methods for Detecting Structure and Dynamics of Earth s Deep Interior. The Wahba Conference June 6, 2014

Transcription

1 Nonparametric Methods for Detecting Structure and Dynamics of Earth s Deep Interior The Wahba Conference June 6, 2014

2 Connection with Grace Ph.D with Chong Gu (First generation) Postdoc with Jun S Liu (Second generation) Happy Birthday Earth 2

3 Earth 3

4 Earth 4

5 Earth 5

6 EARTH S STRUCTURE Earth 6

7 Earth Structure Earth 7

8 Earth 8

9 Keith E. Bullen s Earth Model Bullen (1940, 1942) divided Earth into seven major layers and labeled them as A, B, C, D, E, F, G Sub-layers near major layers D were labeled as D and D. Earth 9

10 Earth 10

11 Earthquake Waves Van der Hilst et al (2007) Science Earth 11

12 USArray - Earthscope Rapid deployment of large number of high-quality seismometers world-wide Earth 12

13 Big Data Science Earth 13

14 Exploring the Inner Earth using Seismic Waves The limitations of forward modeling (1) prior knowledge of earth structure (2) signal has sufficient amplitude (3) only practical for a small dataset Inverse modeling methods that are geophysical sound, mathematical elegant, statistical rigorous are still of rarity. Earth 14

15 Isochrons and Image Point Gathers r s Van der Hilst et al (2007) Science Earth 15

16 Generalized Radon Transform in Nutshell s r Waveform measurement u t x x at time and s with source coordinate x, receiver coordinate x Travel time function s r T( x, x, y) with source s coordinate x, reflected at coordinate y, recorded at x Objective: get image at a depth with coordinate GRT consists of integrating curves r (,, ) t over a set of s r s r s r G( y) ( t T( x, x, y)) u( t, x, x ) dtdx dx s r u( t, x, x ) r y Earth 16

17 Earth 17

18 Generalized Radon Transform in Action Factors need to be taken into account (1) source-receiver geometry: slowness vectors (2) properties of Earth s mantle and crust mass density s s s r r r p ( y), p ( x ), p ( y), p ( x ) s s s s s V ( y) 1/ p ( y), ( y) p ( y)/ p ( y) m s r p ( y) p ( y) p ( y) ( x) stiffness cx ( ) Earth 18

19 Generalized Radon Transform in Action Start with inhomogeneous elastic wave equation of body force f ( x, t) 2 1 u( x, t) 1 u( x, t) f ( x, t) 2 2 ( x) c ( x) t ( x) The solution is the Green s function Approximations are applied to get computable formulae. Earth 19

20 Common Image Point Gathers s r s r r r s r u( x, x, y) W ( x, x, y) h ( x ) u( x, x, y) p 2[ ( x ) V ( x ) V ( y) V ( y) ( x ) V ( x )] [det Q( x, y)det Q( y, x )] r r r r s s s 1/2 s r 1/2 G( y;, ) E v m s r m 3 u( x, x, y) p ( y) s r s r W ( x, x, y) w( x, x, y) dv m Earth 20

21 Generalized Radon Transform in Action u 1 u 2 s r r s y y 2 s = source ; r = receiver ; y = image point ; u = data Earth 21

22 Common Image Point Gathers u 1 u 2 s r r s Image points are predefined Earth 22

23 Common Image Point Gathers u 1 u 2 s r r s Earth 23

24 Interpretation of Common Image Point Gathers Geophysical view: the stack of isochrons. Mathematical view: weight average of wave functions. Statistical view: the strength of the reflectors. Earth 24

25 ScS study: data 1,500 earthquakes 1,200 seismic stations 80,000 ScS (SH) Earth 25

26 Simple stacking does not wok (Wang et al., JGR, 2006) Earth 26

27 2-D image Juxtaposition (Ma et al., JGR, 2007) Earth 27

28 Angle Dilation Depth Harmonic Model The image point gathers at angle is G ij i and depth x j G g( x ) a cos( w ) a ij i j ik ik i ik ij k ik ij 2 ~ N(0, i ) 2 ~ N(0, ) Gu & Ma (05) Ann Stat Earth 28

29 (Wang et al., JGR, 2006) Earth 29

30 D layer (Ma et al., JGR, 2007) Earth 30

31 D layer Earth 31

32 Routine 3-D exploration of lowermost mantle now possible!! along with calculation of formal uncertainties (Ma et al. JGR, 2006) (van der Hilst et al., Science, 2007) Earth 32

33 Verification of D Consistent with previous studies! km km (Thomas et al., JGR, 2004) Earth 33

34 Compare with geochemical evidence perovskite -> post-perovskite Ed Garnero (ASU) Earth 34

35 Earth 35

36 Earth 36

37 Earth 37

38 Earth 38

39 EARTH S DYNAMICS Earth 39

40 Earth Structure Earth 40

41 The Core Core (Oldham 1906) studied S-wave. Inner Core: I. Lehmann (1936): Studied the P-wave. Those detected waves are reflected from the inner core boundary. Birch (1952): The inner core is made up of crystalline iron (depth km). Dziewonski and Gilbert (1971): Existence confirmed via normal mode analysis. Outer Core: Birch (1952): The outer core is composed of a liquid iron alloy (depth km). Today: liquid iron alloy + some light elements. Earth 41

42 Earth 42

43 PKP(AB) PKP(BC) PKP(DF) SUPERROTATION It has been observed that differential travel times thought the inner core along certain paths have changed over the past four decades. Alaskan Stations Cola, Alaska Source: Song and Richards [8], Nature (1996) Possible Explanation: The inner core is not completely homogenous, it possesses regions with distinct seismic velocities. The inner core rotates different than the rest of the Earth (super rotation). Goal of the research: to resolve the structure in the inner core and its velocity simultaneously. Waveform Doublets South Sandwich Islands Earth 43 Source: Zhang et al. [9], Science (2005)

44 Seismic Network Stations Locations of the Alaskan Seismic Network station (black ) and the new ARCTIC stations (red) used in this study. Earth 44

45 LATERAL COVERAGE Map of pathways from South Sandwich Islands earthquakes to the Alaska Seismic Network (red) and ARCTIC stations (purple). Earth 45

46 Single Station travel time difference Earth 46

47 TEMPORAL COVERAGE We use a total number of 1165 differential time measurements covering a time span of 56 years that are obtained from earthquakes that originate in the South Sandwich Island and are recorded at Alaskan stations. Earth 47

48 Turning point Earth 48

49 Travel time difference = Mantle Contribution + Mantle Contribution Model: Model o Earthquake epicenter: South Sandwich Islands o The seismic network stations: Northern Alaska. The path through the mantle depends mainly on the position of the receiving station and hence denote the mantle contribution can by f Mantle (s1,s2). Core Contribution o The turning point longitude of the ray in the inner core, x2. o The event time t o The travel time a of the ray in the inner core Core Contribution Northern Alaska o The structure we search is a coherent entity and can be described by a moving wave with speed v. The core contribution denoted by f Core (x2-v t, a). y ' y =f Mantle (s1,s2) + f Core (x2-v t,a) Alaska Stations (s1,s2) x2 a South Sandwich Islands Earth 49

50 STEP 1: SIMULTANEOUS ESTIMATION Both contributions are fitted simultaneously without any constraint: o Core: f 1 (x2,t,a) o Mantle: f 2 (s1,s2) Earth 50

51 STEP 2: SEPARATING THE INNER CORE STRUCTURE FROM ITS TIME EVOLUTION The core contribution is constrained to be of the functional form γ = f 1 (x2-v t,a) representing a coherent object moving with velocity v. The velocity, v, is determined via a grid search {v1, v2,, vn}: o For each velocity vi the core contribution is refitted as γ = f 1 (x2-vi t,a) = f 1 (zi,a). Earth 51

52 Mantle Contribution Travel Time Residuals: Earth 52

53 v=0.39 Core Contribution v 0.24 or v 0.56 A clear spatial gradient of the inner core with some fine structure is apparent. Earth 53

54 Validation Earth 54

55 Robustness Different partition of observation period: t jump = 1975 t jump = 1995 Different Data Subsets: All data above t = 1962 All data above t = 1972 All data above t = 1990 A data with: -81.3< x2< All tests yield: An average velocity 0.4 per year An increase in velocity from about from 0.3/0.4 to 0.5 per year. Earth 55

56 Rotation Rate Average rotation rate of 0.39 per year to the East. In agreement with previous studies. Both westward and no rotation can be rejected. Possible change in the rotation rate from 0.24 to 0.56 per year within the last 55 years. This yields a minimum angular acceleration α s -2 that has acted on the inner core within the past 36 years due to: The deceleration of the rest of the planet. A real net torque that acts on the inner core. Earth 56

57 Acknowledgement Rob van der Hilst, Ping Wang (MIT) Maarten de Hoop (Purdue) Luis Tenorio (Mines) a Earth 57

58 Acknowledgement Xiaodong Song (UIUC) Daniela Lindner (UIUC) Earth 58