INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA

Transcription

1 Problems in Solid Mechanics A Symposium in Honor of H.D. Bui Symi, Greece, July 3-8, 6 INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA M. HORI (Earthquake Research Institute, University of Tokyo) Contents. Stress inversion method: find equilibrating stress using measured strain and partial information on stress-strain relation. Elasticity inversion method: find local elasticity using densely measured displacement

2 GPS NETWORK AND ITS DATA 4Ê 4Ê GPS stations displacement rate 5cm/yr 3Ê 3Ê 4Ê 3Ê 3Ê 4Ê

3 NEED FOR LOCAL STRESS PREDICTION Material Test of Next Generation Earthquake Prediction GPS data strain σ σ ε local ε σ relation ε regional stress 3

4 IS STRESS INVERSION POSSIBLE? 3D State Most Difficult D State Possible? 3 UNKNOWNS σ σ σ EQUATIONS equilibrium in x -direction equilibrium in x -direction CONDITION STRAIN 4

5 STRESS INVERSION σ σ σ Airy = a = a,, = a, partial information: σ + σ = f ( ε σ f ( ε ij + σ ) = κ( ε = κ( ε + ε + ε ) ) ij ) Poisson equation self-equilibrating stress G.E. B.C. B.V.P. a n, a + a, +, n = κ( ε a, = n + ε r + ) n r D bulk modulus? measured strain resultant forces 5

6 EXTENSION TO OTHER DEFORMATION STATE Dynamic State σ σ σ,, + σ + σ + σ,, = = ρ&& u = ρ&& u f ( ε) Finite Deformation State: σ j,k j,k ( X ( X + σ k k x x j j ) σ ) σ = f ( ε) j,k j,k = = ( σ = σ (X), x = x (X)) ij ij i i 6

7 NUMERICAL SIMULATION Conditions elasto-plastic material with unknown yield function prediction of stress and stress-strain relation distributed force displacement field is used as input data.. FEM computation with x48 elements 7

8 RESULTS OF INVERSION σ σ distribution of σ exact sample surface predicted principle stressplastic strain rate σ σ 3 σ σ 3 σ σ 3 σ σ 3 good agreement exact yield locus predicted 8

9 MODEL EXPERIMENT 7cm Riedel shears Torsional Shearing 8cm examples of image video recorder CCD camera Experiment Apparatus servomotor max. shear strain 9

10 OVERALL STRESS-STRAIN RELATION stress (kgf/m 3 ) time (sec) stress (kgf/m 3 ) rotation (radian) strain will be different from local relations? time (sec)

11 LOCAL STRESS-STRAIN RELATIONS σ [kpa] 4 σ [kpa] D A D A normal stress-strain ε shear stress-strain ε A: far from crack D: near crack no common relations?

12 RESULTS OF INVERSION max. shear stress (kpa) 3 yielding A B C D E max. shear strain common elasto-plastic relations?

13 APPLICATION TO GPS ARRYA DATA verification of numerical analysis method check numerical stability of solving boundary value problem check dependency of parameters application of stress inversion method geophysical interpretation of analysis results critical examination of assumption of plane state development of crust deformation monitor automatic processing of GPS array data 3

14 CONVERGENCE σ (hydrostatic stress) Δ: resolution of strain distribution (degree) 4

15 EFFECT OF REFERENCE τ (max. shear stress) 5

16 COMPARISON OF STRESS WITH STRAIN σ (hydrostatic stress) τ (max. shear stress) ε (vol. strain) γ (max. shear strain) 6

17 REGIONAL CONSTITUTIVE RELATIONS τ/γ regional stiffness (τ, γ: max. shear stress and strain) φ θ regional anisotropy (φ, γ: principle stress and strain) regional heterogeneity and anisotropy 7

18 CHANGE IN INVARIANT st invariant nd invariant 8

19 CHANGE IN REGIONAL STATE st invariant 5 ε σ nd invariant 5 5 ε σ A γ τ γ τ C 5 ε σ 5 5 B 3 5 Y A B γ τ ε σ C -5 3 γ τ 9

20 REGIONAL STRESS AND STRAIN st invariant 5 σ ε nd invariant τ γ 75 ε γ 5 γ τ A C 5 σ ε τ γ 75 B A -5-5 B σ ε t-g C

21 GPS DATA DURING GPS Data no spatial filtering to get rid of measurement noise linear interpolation between two GPS station More Sophisticated Treatment of BVP FEM with triangle element weak form regionally averaged field quantities

22 APPLICATION TO GPS NETWORK DATA BVP in Rate Form and Weak Form G.E. B.C. a&, n a& + a&,, + n a& ϕ, a&, + ϕ,a&, κ( ϕ,u& + ϕ,u& )ds = Computation of Average Quantities =κ( ε&, +ε& = n r& ) + n r& strain rate displacement rate ε& σ& = Ω = Ω Ω Ω n n u& a& dl, dl average stress rate computed by using st -order derivative

23 GPS NETWORK AND ITS DATA 4Ê 4Ê GPS stations displacement rate 5cm/yr 3Ê 3Ê 4Ê 3Ê 3Ê 4Ê 3

24 REGIONAL STRAIN RATE 6e-7 6e-7 4Ê 4e-7 4Ê e-7 4e-7 -e-7 e-7-4e-7 volumetric -6e-7 Strain max. shear Strain 3Ê 3Ê 4Ê 3Ê 3Ê 4Ê 4

25 REGIONAL STRESS RATE 6e-7 6e-7 4Ê 4e-7 4Ê e-7 4e-7 -e-7 e-7-4e-7 volumetric -6e-7 x κ [Pa] max. shear x κ [Pa] 3Ê 3Ê 4Ê 3Ê 3Ê 4Ê 5

26 COMPARISON WITH SEISMIC EVENTS? 6e-7 4Ê 4e-7 4Ê e-7 -e-7-4e-7 volumetric -6e-7 >[km], >M3 x κ [Pa] 3Ê 3Ê 4Ê 3Ê 3Ê 4Ê 6

27 REGIONAL CONSTITUTIVE RELATIONS findκ( x) s.t. τ& ( x; κ) = κ( x) γ& ( x). τ &, γ& maximum shear 4.5 κ regional stiffness. κ is originally used to relate σ & ε through σ = κ ε not too far from known geological structure

28 DRAWBACKS OF STRESS INVERSION Need to Know One Constitutive Relation bulk stress and bulk strain isotropy assumption Need to Know Boundary Traction/Resultant Force assumption of uniform stress fast decrease of non-uniform boundary traction Difficulty in Understanding Plane-Stress-State Model another analysis method needed? 8

29 DRAWBACKS OF ELASTICITY INVERSION Sensitive to Displacement Error need to make fine discretization of target body need to have some strong modes of deformation Why is it so? no mistakes in mathematics poor understanding of physics 9

30 PHYSICAL PROCESS AND MEASUREMENT GIS data data inverse analysis May not be good to pose an inverse problem from data to characteristics, because a path from characteristics to data has physical process and measurement measurement X unknown mechanical property characteristics response crust deformation physical process source unknown plate slip/ underground deformation 3

31 ESTIMATION AND INVERSION estimation data measurement inversion estimation: from data to response, i.e., estimate function for displacement from data which are measured discretely inversion: from response to characteristics, i.e., find most suitable characteristics for physical process even though source is not known characteristics response physical process source 3

32 BLOCK IN CONTINUUM nodes at which displacement is measured inner node body B x boundary node x principal of elasticity inversion block S k k displacement at all nodes given Δ Δ Δ Δ = k k no need to consider interaction of S with outside region 3

33 IDENTIFICATION OF DISPLACEMENT MODE concentrated force node block hexagonal block block block 3 material sample material sample test apply several BC s, and measure displacement at nodes of a hexagonal block.. identify displacement modes (a characteristic set of nodal displacement). identify local elasticity 33

34 IDENTIFICATION OF DISPLACEMENT MODE displacement mode elastic parameters c c.5 constraint 3 c 4 c 5 c 6 c block g. g block block unconstraint

35 ELASTICITY INVERSION METHOD body B local field variables allows Taylor series expansion block Ω nodes at which displacement is measured BASIC PROCEDURES OF INVERSION. use displacement data to determine Tayler series expansion coefficients of displacement. use equilibrium equation to estimate elastic parameters by expanding stress in Talker series uniform Poisson's ratio is assumed 35

36 DETERMINATION OF DISPLACEMENT COEFFICIENT. Taylor Expansion: { a ip } u = P i ( x) a ipf p ( x) p= {a {f ip p } = {u i } = {, x,u i,, x,u, x i,,, x u x i,,u, x i,,, L} u i,,u i,, L}. Displacement Data: u n i = P p= f pn a ip { u n i } n 3. Solution of Matrix Equation f = f ( x ) pn p a ip = A α= λ α n u n i ϕ α n ψ α p + P A β= b β i ψ β p { λ } α, ϕ α ψ α n, p : SVD of f pn fully determined undetermined 36

37 37. Elasticity Tensor Expressed in Terms of Poisson Ratio. Equation of Equilibrium and Its Taylor Expansion 3. Coefficient of Expansion: b ip = for th Order (p=) ESTIMATION OF POISSON RATIO ν ESTIMATION OF POISSON RATIO ν + ν = a a a b b M ) ( f b )) ( u (c ) ( b p p ip j, k,l ijkl i = = = x x x ijkl ijkl ijkl c c c ν + = linear equation of ν is derived

38 NUMERICAL SIMULATION (). error error..5 nd order 3rd order.5 nd order 3rd order ν ν=.5 -. ν ν=.35 ν=.5 ν=.35 frequency 8 6 measurement 4 expansion of displacement expansion of equilibrium 3 rd order th or st order ν 38

39 NUMERICAL SIMULATION (). Measurement Error: { e n i } th. P n u i = f pna ip + p= e n i e'.8.4 ν exact =.35 ν exact =.5. Find ν such that ν minimize subjected to e = (e b ( ν) = ip n i ) st e measurement expansion of displacement 3 rd order expansion of equilibrium th or st order ν exact = ν 39

40 APPLICATION OF LOCAL GPS DATA pre-wte 35.5 Velocity field Pre-Tottori Earthquake cm/year GPS array Western Tottori Earthquake (M JMA =7.3) examine change in deformation and elasticity before and after this earthquake 53 GPS observation points 8 triangular elements GPS data obtained from 997 to annual and biannual sinusoidal variations excluded post-wte Velocity field Post-Tottori Earthquake 3cm/year

41 STRAIN RATE pre-wte Dilatational strain rate post-wte Dilatational strain rate e e-7 4e-7 4e-7 dilatational 35 e-7 35 e e e-7-4e-7-4e e-7 [Strain/Year] 34-6e-7 [Strain/Year] Maximum shear strain rate Maximum shear strain rate e e-6 8e-7 8e-7 shear 35 6e e-7 4e-7 4e e-7 e-7 34 [Strain/Year] 34 [Strain/Year]

42 STRESS RATE pre-wte Dilatational stress increment post-wte Dilatational stress increment dilatational [KPa/Year] 34 - [KPa/Year] Maximum shear stress increment Maximum shear stress increment shear [KPa/Year] 34 [KPa/Year]

43 PRINICPLE AXIS pre-wte Principal axes of strain rate post-wte Principal axes of strain rate strain (extensional) (contractional).5 [Micro Strain/Year] (extensional) (contractional).5 [Micro Strain/Year] Principal axes of stress rate Principal axes of stress rate stress (extensional) (contractional) [KPa/Year] (extensional) (contractional) [KPa/Year]

44 POISSON RATIO pre-wte Poisson s ratio Pre-Tottori Earthquake Mt. Daisen post-wte Poisson s ratio Post-Tottori Earthquake Mt. Daisen Naka-Umi Naka-Umi Change of Poisson s ratio Mt. Daisen Naka-Umi difference ν has been reduced near the source faults. The comparison with other analyses are being made [%]

45 CONCLUDING REMARKS Two inverse analysis methods stress inversion find Airy s stress function by solving Poisson s equation elasticity inversion find elastic parameters by estimating displacement expansion coefficients Development of new inversion is needed for geophysics where experiments cannot be made. Application small material samples used for bio-mechanics geomaterials new image analysis with higher spatial resolution 45