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)
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