MODELING AND IN INVERSION OF THE MIC ICROTREMOR H/V /V SPECTRAL - PowerPoint PPT Presentation

MODELING AND IN INVERSION OF THE MIC ICROTREMOR H/V /V SPECTRAL RATIO: TH THE PHYSICAL BASI SIS BEHIND TH THE DIF IFFUSE FIE IELD APPROACH Francisco J. Sánchez-Sesma Instituto de Ingeniería, Universidad Nacional Autónoma de México Coyoacán, 04510 CDMX, Mexico Email: sesma@unam.mx 5th IASPEI / IAEE International Symposium: Effects of Surface Geology on Seismic Motion Taipei, Taiwan, August 15-17, 2016 1

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 11 ( x , x , ω ) H/V from Ambient Seismic Noise is modeled as 2Im𝐻 11 /Im𝐻 33 Elastic Im G 11 (0,0, ω ) behavior suggests simple 3D 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. 2

Mic icrotemor H/V /V Spectral Ratio MHVSR Site Characterization   f 0 Site effects Amplification and Duration 3

Microtremor H/V /V Spectral Ratio M MHVSR Ellipticity SH TF H/V • Nogoshi & Igarashi (1971 ): 2.32 HZ 2.48 Hz 2.38 HZ Microtremors  Surface Waves • Others Authors H  H H/V ≡ Rayleigh Ellipticity V V m Average • Nakamura (1989) of Ratios H/V ≡ Transfer Function SH wave  2 2 H H H • Arai & Tokimatsu (2004)  EWm NSm Ratio of Averages 2 V V m • Sánchez-Sesma et al. (2011) H/V ≡ All Waves (Diffuse Field Theory) 4

Weaver & Lobkis (2002) Ultrasonics

Campillo & Paul (2003) Science

Shapiro et al (2005) show waveforms emerging from cross- correlations of ambient seismic noise and compared them with Rayleigh waves excited by earthquakes 30 30 days of ambie ient noise ise. . USArray

Mult ltiple scattering C Coda N Noise Dif iffu fuse Fie ield Kanai (1932) Propagation Few bounces (short times) Regimes & Radiative Transfer Energy Density Decay Difussion (long times) temps Noise Coda time

Directional Energy Densities = Auto-Correlations  Im[ G ( x , x , ω )]          2 * 1 ( x , ) u ( x ) u ( x ) 2 k Im[ G ( x , x , )] 1 1 1 S 11 Directional Energy Density (DED) is proportional to the imaginary part of Green’s function at the source itself. This is clear from the full-space solution (Stokes,1849):     1 2       x x Im[ G ( , , )]    11 3 3   12     3        3 P S ( 1 2 R )  1 S 3 6 3 3 3 10

1 1 1 1 1              P 1 P SV SH 3 1 2 R 3 6 2 3 3 1 1 1 1 R              2 P SV SH SH 3 1 2 R 3 6 2 3 3 1 2 1 R            3 P SV SV 3 3 3 3 1 2 R Weaver (1985) JASA Sánchez-Sesma & Campillo (2006) BSSA 11

A Theory for H/V With Directional Energy Densities the H/V ratio is:    E ( x ; ) E ( x ; )   1 2 [ H / V ]( x ; )  E ( x ; ) 3    Im[ G ( x , x ; )] Im[ G ( x , x ; )]   11 22 [ H / V ]( x ; )  Im[ G ( x , x ; )] 33 measurements  system properties Sánchez-Sesma et al. (2011) Kawase et al. (2011) 3D problem (BW & SW) 1D problem (BW) Matsushima et al. (2014) 2.5D Lontsi et al. (2015) 1D case (Lateral heterogeneity) H/V ( z , ω ) Data at depth 12

Green’s function calculation The imaginary parts of the Green's functions, using Harkrider (1964) notation can be written as :    r   i          V   Im G r ; Im f k J kr dk   33 P SV 0  2  0         z    Im G r ; Im G r ; 22 11         i   i                         H  Im G r ; Im f k J kr J kr dk f k J kr J kr dk    11 SH 0 2 P SV 0 2 4 4                             0 0      SH P SV 13

Two bli lind tests (Synthetic Noise & DFA) Model N101 0 -50 -100 Depth (m) -150 -200 -250 -300 0 200 400 600 800 1000 1200 1400 1600 1800  (m/s),  (m/s),  (Ton/m 3 )  100 Model N104 0 -50 -100 Depth (m) -150 -200 -250 -300 -350 -400 0 1000 2000 3000 4000 5000 6000  (m/s),  (m/s),  (Ton/m 3 )  100 Frequency [Hz] 14

The Texcoco Experiment 2 10 H/V = √ 2 Im[G 11 ] /Im[G 33 ] 1 10 0 10 H/V 0.47Hz -1 -1 10 10 -2 -1 0 1 -2 -1 0 1 10 10 10 10 10 10 10 10 Frequency [Hz] Frequency [Hz] 0.45 0.4 Im[G 22 ] =Im[G 11 ] 0.35 Im[Green functions] 0.3 Im[G 33 ] 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 freq [Hz] 15

Simplified model in 3D  G  0  z    ( x ) x     2 2 G k G h  2 c  G 0 z 16

Frequency and time domains 3D   Im G ( 0 , 0 , ) PRS ( 0 , 0 , t )      n 1 1 d ( t )  ( 1 )         ( t n ) sgn( t )  2   2 c 2 h dt n  n 0 18

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 I II     Im 0 Im 0   N N     Im 0 Im 0   N N           N N N N III k Simple poles IV Branch points     Im ( ) 0 ; Re ( ) 0   Augustin-Louis Cauchy N N Integration countour     Im ( ) 0 ; Re ( ) 0   Branch cut N N

Position of poles Dispersion curves of Integrand of Im G 33 (0,0; f =5 Hz, k ) Rayleigh waves 6 1000 Phase Velocity (m/s) Body Waves Amplitude 4 Surface Waves Branch points ( ω / α N ) 800 Branch points ( ω / β N ) 2 600 0 400 0.5 1 1.5 2 2.5 0 2 4 6 8 10 k / ( ω / β N ) f ( Hz )

Green function calculation The imaginary parts of the Green's function is computed as (García-Jerez et al., 2013):             N   1 1               2 H   Im G A A Re f k f k dk  th th  11 m Rm Lm P SV SH 4 4 4   4   m Rayleigh m Love                             0 Body Waves Surface Waves            Im G Im G 22 11       N   1 1         V Im G A Re f k dk  th  33 Rm P SV 4 2 2     m Rayleigh              0 Surface Waves Body Waves

Several views of H/V Profile Velocity Foward Problem

Inverse Problem Inverse Problem   2     Th . E xp.        H V / H V /       1 i i  E     2 n i i

Inversion (Simulated Annealing)

Inversion example of H/V

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

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

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

Cost function map for H/V & dispersion curves joint inversion Functional H/V & Dispersion curves Log Error 1 V S 2 V S Log Error 1 V S 2 V S

Example of joint inversion

Application to site effect characterization at Almería, Río Andarax, Spain (1/2) 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)

Application to site effect characterization at Almería, Río Andarax, Spain (2/2)