Radiative Transfer of Seismic Waves

Transcription

1 Radiative Transfer of Seismic Waves L. Margerin CEREGE, CNRS, Aix en Provence, France Atelier sur les ondes élastiques, Col de Porte, 13 janvier 2009 En collaboration avec: N. Le Bihan, M. Campillo, E. Larose, G. Nolet, C. Sens-Schönfelder, N. Shapiro, B. van Tiggelen,

2 Content 1 Introduction to Radiative Transfer 2 3

3 Radiative Transfer in Seismology: Why? PYLO PYTB PYBB PYAD PYPU PYPC PYAT PYLU PYPP (x10) PYBE (x10) PYPR (x10) Observations Duration of signal travel time of ballistic waves Rapid attenuation of direct waves Cigar -shaped envelope Pronounced Coda Ti = 18:19:42.77 Temps apres Ti (s) Role of Scattering lat = deg 17 Novembre 2006 t0 = 18 h 19 mn 00 s (TU) lon = deg Argeles!Gazost (65) prof = 8 km M=5.0 Crustal earthquake in the Pyrenees

4 What is Radiative Transfer? Definition Radiative Transfer is a theory aimed at predicting the spatio-temporal distribution of energy in a scattering medium The physical concepts of Radiative Transfer Local Energy Balance Angularly resolved energy flux: Specific intensity Scattering strength: mean free path

5 Some references in seismology Introduced by Wu (1985) and developed by Aki, Zeng, Sato Monte Carlo simulations: Gusev and Abubakirov, Hoshiba Sato and Fehler (Wave propagation and scattering in the heterogeneous Earth, Academic Press, 1998) Dmowska, Sato and Fehler, Eds, Vol. 50 of Advances in Geophysics, Academic Press, 2008 IASPEI Task group: scattering and heterogeneity in the earth:

6 Content 1 Introduction to Radiative Transfer 2 3

7 The specific intensity d 2ˆk ds θ Definition I(ω, t, r, ˆk) ds cos(θ) dt dω = Amount of energy within the frequency band [ω, ω + dω] flowing through ds around direction ˆk during time dt Angularly-resolved energy flux through a surface

8 Relation to the wavefield Wigner-Ville Spectrum Consider a time-dependent field ψ(t): + W ψ (t, ω) = ψ(t τ/2)ψ(t + τ/2) e iωτ dτ 2 Π Ω Enveloppe t : Ensemble Average Separation of Time Scales W ψ : Instantaneous Energy Spectrum

9 Wave content of the specific intensity d 2ˆk Spatial Wigner-Ville θ W ψ (x, k) = ψ(x r/2)ψ(x + r/2) e ik r d 3 r ds Angular Energy Spectrum Slowly-modulated wave packet Specific intensity contains information on the correlation properties of the wavefield

10 The Equation of Radiative Transfer Scalar Case ( ) 1 c t + ˆk x I(t, x, ˆk) = ( 1 l s + 1 ) l a I(t, x, ˆk) + 1l s l s : scattering mean free path l a : absorption length p(ˆk, ˆk )I(t, x, ˆk )d 2ˆk c: wave speed p(ˆk, ˆk ): scattering anisotropy δi = I out I in Loss Gain I in I out cdt

11 Assumptions in radiative transfer Averaging the products of 2 Green functions Example of scattering path Visit the same scatterers Pairing of trajectories From Akkermans & Montambaux (2005)

12 The case of vector waves The correlation tensor Γ ij (x, k) = ψ i (x r/2)ψ j (x + r/2) e ik r d 3 r Rotate from the frame (ˆx 1, ˆx 2, ˆx 3 ) onto the frame (ˆk 1 = ˆk, ˆk 2, ˆk 3 ): Γ(x, k) = Γˆk 1ˆk 1 Γˆk 1ˆk 2 Γˆk 1ˆk 3 Γˆk 2ˆk 1 Γˆk 2ˆk 2 Γˆk 2ˆk 3 Γˆk 3ˆk 1 Γˆk 3ˆk 2 Γˆk 3ˆk 3 Interpretation Γˆk 1ˆk 1 : P-wave intensity Γˆk 2ˆk 2, Γˆk 3ˆk 3 : S-wave intensity along ˆk 2, ˆk 3 Γˆk 2ˆk 3, Γˆk 3ˆk 2 : Stokes-like parameters polarization Remaining terms: correlation between P and S waves

13 The equation of radiative transfer for elastic waves (c 1 t + ˆk x ) I(t, x, ˆk) = (l 1 + l a 1) I(t, x, ˆk) + l 1 p(ˆk, ˆk )I(t, x, ˆk )d 2ˆk l = Diag(l p, l s, l s, l s, l s ) c = Diag(v p, v s, v s, v s, v s ) l a = Diag(lp, a ls a, ls a, ls a, ls a ) I = (I p, I x, I y, U, V ) Interpretation of U and V Linear +45 y Linear 45 y Circular Left y Circular Right y x x x x

14 Content 1 Introduction to Radiative Transfer 2 3

15 Numerical Tests Models of random media 2-D Gaussian, exponential media Finite-Difference calculations P wave source Ensemble average Elastic R.T.E. p(ˆk, ˆk ) from Born approx. Monte-Carlo simulations Przybilla et al., J. Geophys. Res, 111, B04305, 2006

16 Comparison between RT and FD simulations Left: t s = 11.4s, weak anisotropy, weak scattering Middle: t s = 0.5s Przybilla et al., J. Geophys. Res, 111, B04305, 2006 Right: t s = 0.1s, strong anisotropy, strong scattering Excellent agreement even for strong forward scattering Uniform energy distribution at long time

17 Comparison between RT and FD simulations Excellent prediction of energy partition on radial and transverse components Equipartition at large time Energy ratios stabilize: E ( ) s vp 2 = E v s p Przybilla et al., J. Geophys. Res, 111, B04305, 2006

18 Content 1 Introduction to Radiative Transfer 2 3

19 Imaging of volcanic heterogeneity Experiment at Osama Volcano (Japan) 5 Dynamite shots Vertical sensors 2Hz Station spacing: m Data normalized with late coda Shot 3 in the 8-16 Hz frequency band Courtesy of M. Yamamoto

20 Spatial Distribution of Energy Observe the two slopes Shear waves are mode converted Spatial homogenization of energy at late time Courtesy of M. Yamamoto

21 Role of the scattering parameters Reference: v p = 2.7km/s, v p /v s = 3, l pp = 3 km, l pp /l ps = 2, l pp /l ss = 3 Smaller l ps : l pp /l ps = 4 Smaller l ss : l pp /l ss = 6 Courtesy of M. Yamamoto 3-D elastic radiative transfer Isotropic scattering Explosion: isotropic P source Reciprocity: l sp = 2 ( v p v s ) 2 l ps Vary the ratios l pp /l ps and l pp /l ss

22 Constraint on medium heterogeneity Impressive agreement! Yamamoto and Sato, to appear in J.G.R. Inversion results v p = 2.7 km/s l pp = 3.8 km, l pp = 2.4l ps, l pp = 3l ss Absorption: Q i = 100 Extremely strong scattering: l p l s 1km Coupling parameters: rich information on the nature of heterogeneity

23 Modeling of Lg blockage through the Pyrenees Wavepaths through the Pyrenees 51 o N 48 o N 45 o N 42 o N 39 o N 36 o N 2 3 MFF 1 SJPF PAB 6 o W 4 o W 2 o W 0 o 2o E 4 o E 6 o E Extinction of Lg waves western Pyrenees eastern Pyrenees v [km/s] Blockage in a localized zone Not explained by Moho jump or faults Monte Carlo simulations of RT 3-D anistropic scattering and mode conversion Depth varying velocity and scattering properties

24 Introduction to Radiative Transfer Snapshot of numerical simulation Undisturbed zone v [km/s] 2 Through anomaly v [km/s] Sens-Scho nfelder et al., J. Geophys. Res., 114, B07309, 2009 L. Margerin Radiative Transfer of Seismic Waves 2

25 Applications at the global scale Multiple scattering of Rayleigh waves Scattering in the lower mantle Texture Fluctuation Spectrum Spatial Extension Precursors Scattered Waves Shadow Zone Sato and Nishino, J. Geophys. Res., 107, 2343, 2001 Margerin and Nolet, J. Geophys. Res., 108, 2514, 2003

26 Application to source studies The Kursk Explosion Comparison between observed and theoretical explosion Sèbe et al., Geophys. Res. Lett., 32, L14308, 2005

27 More radiative transfer Asymptotic solution of Bethe-Salpeter equation Wigner distribution of two Green functions: Γ(x, r; t, τ) = G(x + r/2, t + τ/2)g(x r/2, t τ/2) Fourier transform over τ: C(x, r; t, ω) = Γ(x, r; t, τ)e iωτ dω Asymptotic result t C(x, r; t, ω) e ωt/q i Im G(r, ω) (Dt) 3/2

28 Consequence 1: Equipartition Full-Space Elastic Green Function Spectral domain: ImĜ ij (k, ω) (δ ij ˆk i ˆk j )δ(ω 2 v 2 s k 2 ) + ˆk i ˆk j δ(ω 2 v 2 p k 2 ) Total Energy: [ Trace [ImG ik (0, ω)] = Trace ] ImĜ ij (k, ω)d 3 k Shear + Long {}}{{}}{ 2 ω2 ω 2 vs 3 + vp 3

29 Application of Equipartition to Site Effect Studies Velocity Model h 1 = 4m α 1 = 300m/s β 1 = 150m/s ρ 1 = 2200kg/m 3 h 2 = 11m α 2 = 900m/s β 2 = 500m/s ρ 2 = 2200kg/m 3 h 3 = 50m α 3 = 3100m/s β 3 = 1600 m/s ρ 3 = 2700kg/m 3 α = 5400m/s β = 3000m/s ρ = 2700kg/m 3 V 2 H 2 Energy partition in the coda Frequency Hz Margerin et al., Geophys. J. Int., 177, , 2009

30 Consequence 2: Green function reconstruction Temporary experiment in Alaska Correlation of Coda waves Paul et al., J. Geophys. Res., 110, B08302, 2005

31 Introduction to Radiative Transfer Beyond radiative transfer: Weak localization Interference of reciprocal paths Ultrasound propagation in sand Ballistic waves Log(Energy) Diffuse halo Weak Localization Courtesy A. Derode, L.O.A. L. Margerin Radiative Transfer of Seismic Waves