MODELING AND INVERSION OF THE MICROTREMOR H/V SPECTRAL RATIO:

Transcription

1 MODELING AND INVERSION OF THE MICROTREMOR H/V SPECTRAL RATIO: THE PHYSICAL BASIS BEHIND THE DIFFUSE FIELD APPROACH Francisco J. Sánchez-Sesma Instituto de Ingeniería, Universidad Nacional Autónoma de México Coyoacán, 0450 CDMX, Mexico 5th IASPEI / IAEE International Symposium: Effects of Surface Geology on Seismic Motion Taipei, Taiwan, August 5-7, 06

2 Outline Microtremor H/V (MHVSR) Site Characterization Multiple Diffraction Diffuse Fields Coda ~ Noise Correlations Green s Function Equipartition & Isotropy Auto-Correlations Energy Densities Im G (x, x, ω) H/V from Ambient Seismic Noise is modeled as ImG /ImG Elastic Im G (0,0,ω) behavior suggests simple D models Inversion of H/V Cauchy s theorem Soil Information Joint inversion of H/V and DC mitigates non-unicity New software allows modeling and inversion of MHVSR.

3 Microtemor H/V Spectral Ratio MHVSR Site Characterization f 0 Site effects Amplification and Duration

4 Microtremor H/V Spectral Ratio MHVSR Nogoshi & Igarashi (97): Ellipticity. HZ SH TF.48 Hz H/V.8 HZ Microtremors Surface Waves Others Authors H/V Rayleigh Ellipticity Nakamura (989) H/V Transfer Function SH wave H V H V m Average of Ratios Arai & Tokimatsu (004) H V H EWm V m H NSm Ratio of Averages Sánchez-Sesma et al. (0) H/V All Waves (Diffuse Field Theory) 4

5 Weaver & Lobkis (00) Ultrasonics

6 Campillo & Paul (00) Science

7 Shapiro et al (005) show waveforms emerging from crosscorrelations of ambient seismic noise and compared them with Rayleigh waves excited by earthquakes 0 days of ambient noise. USArray

8 Multiple scattering Coda Noise Diffuse Field Propagation Regimes & Energy Density Decay Few bounces (short times) Radiative Transfer Kanai (9) Difussion (long times) temps Noise time Coda

9 Directional Energy Density (DED) is proportional to the imaginary part of Green s function at the source itself. )],, ( Im[ ) ( ) ( ), ( * x x x x x G k u u S 0 This is clear from the full-space solution (Stokes,849): )],, ( Im[ x x G Directional Energy Densities = Auto-Correlations Im[G(x,x,ω)] ) ( 6 S P S R

10 R R R R R SV SH P Weaver (985) JASA Sánchez-Sesma & Campillo (006) BSSA 6 6 SV P SH SV P SH SV P

11 A Theory for H/V With Directional Energy Densities the H/V ratio is: [ H / V ]( x; ) E ( x; ) E( x; ) E ( x; ) [ H / V ]( x; ) Im[ G ( x, x; )] Im[ G( x, x; )] Im[ G ( x, x; )] measurements system properties Sánchez-Sesma et al. (0) Kawase et al. (0) D problem (BW & SW) D problem (BW) Matsushima et al. (04).5D case (Lateral heterogeneity) Lontsi et al. (05) D H/V (z, ω) Data at depth

12 Green s function calculation The imaginary parts of the Green's functions, using Harkrider (964) notation can be written as : r Im i V G r; Im f kj krdk 0 PSV 0 z Im r; G r; G Im Im i SH SH i PSV 0 dk 4 0 PSV H G r; Im f kj kr J krdk f kj kr J kr

13 Two blind tests (Synthetic Noise & DFA) 0 Model N Depth (m) (m/s), (m/s), (Ton/m ) 00 0 Model N Depth (m) (m/s), (m/s), (Ton/m ) 00 Frequency [Hz] 4

14 The Texcoco Experiment 0 0 H/V = Im[G]/Im[G] H/V 0.47Hz Frequency [Hz] Frequency [Hz] Im[Green functions] Im[G ] =Im[G ] Im[G] freq [Hz] 5

15 Simplified model in D h G z 0 x ( x ) G k G c z G 0 6

16 Frequency and time domains D ImG(0,0, ) PRS( 0,0, t) n d ( t) ( ) ( t n )sgn( t) c h dt n0 n 8

17 Green s function calculation Cauchy s Residue theorem The integrand of the Green s function has simple poles isolated on the real axis k and branch points ω/β y ω/α. ik Im Im II N N 0 0 Im Im I N N 0 0 N N N N III Simple poles IV k Branch points Integration countour Branch cut Im( Im( N N ) ) 0; Re( 0; Re( N N ) ) 0 0 Augustin-Louis Cauchy

18 Amplitude Body Waves Branch points (ω/α N ) Branch points (ω/β N ) Surface Waves Phase Velocity (m/s) Position of poles 6 Integrand of ImG (0,0; f=5 Hz, k) 000 Dispersion curves of Rayleigh waves k /(ω/β N ) f (Hz)

19 The imaginary parts of the Green's function is computed as (García-Jerez et al., 0): Im Green function calculation Re m Rm Lm 4 m Rayleigh m Love th f k th PSV SH dk 0 N H G A A f k Surface Waves Body Waves Im Im G G Rm mrayleigh Im Re PSV th dk 4 0 N V G A f k Surface Waves Body Waves

20 Several views of H/V Profile Velocity Foward Problem

21 Inverse Problem Inverse Problem Th. E xp. H / V / i H V i E n i i

22 Inversion (Simulated Annealing)

23 Inversion example of H/V

24 Application to site effect characterization at Texcoco, México D.F.

25 Non-uniqueness of H/V, dispersion curves Velocity Profile Rayleigh Love Rayleigh Love Rayleigh Love H/V H/V H/V

26 Cost function map for H/V & dispersion curves joint inversion Functional H/V & Dispersion curves

27 Log Error Cost function map for H/V & dispersion curves joint inversion Functional H/V & Dispersion curves V S V S Log Error V S V S

28 Example of joint inversion

29 Application to site effect characterization at Almería, Río Andarax, Spain (/) SPAC - Pentagonal Array Rmax 450m At each station, we computed H/V Also local dispersion curves are computed using SPAC technique Stations distribution (from Dr. E. Carmona)

30 Application to site effect characterization at Almería, Río Andarax, Spain (/)

31 Conclusions Green s function (GF) can be retrieved from correlations within a diffuse field. Directional Energy Densities from autocorrelations are related with GF. E x, ω ~ < u (x, ω) > ~ImG x, x, ω. Assuming noise is diffusive: H V x, ω = E x, ω + E x, ω E x, ω = ImG x, x, ω + ImG x, x, ω ImG x, x, ω This expression relates Field Measurements and System s Properties, and allows extraction of soil information hidden in ambient seismic noise. Even if noise is not fully diffusive, residual coherency may allow to retrieve Green s functions. Appropriate data processing H/V may allow the inversion of soil profile and then its effects in strong ground motion can be explored. 6

32 Two References A García-Jerez et al. (06), A computer code for forward calculation and inversion of the H/V spectral ratio under the diffuse field assumption, Computers and Geosciences, in press. J Piña-Flores et al. (06), The inversion of spectral ratio H/V in a layered media using the diffuse field assumption, Geophysical Journal International, Submitted. 7

33 Thank you! 8

34 Im[ G ( x, x, )] 4

35 4

36 Im[G] =Im[G] 0.5 Im[Green functions] Im[G] H/V = Im[G]/Im[G] freq [Hz] H/V 0.47Hz freq [Hz] freq [Hz] 4

37 x u (4) u () x ξ ξ ξ u () u () u () u () u () ξ ξ ξ 44

38 x x ξ + ξ = ξ ξ + ξ = ξ ξ + ξ = ξ 45