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

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5th IASPEI / IAEE International Symposium: Effects of Surface Geology on Seismic Motion Taipei, Taiwan, August 15-17, 2016

Outline

Microtremor H/V (MHVSR)  Site Characterization Multiple Diffraction  Diffuse Fields  Coda ~ Noise Correlations  Green’s Function  Equipartition & Isotropy Auto-Correlations  Energy Densities  Im G11(x, x, ω) H/V from Ambient Seismic Noise is modeled as 2Im𝐻11/Im𝐻33 Elastic Im G11(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.

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Mic icrotemor H/V /V Spectral Ratio MHVSR Site Characterization  f0  Site effects

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Amplification and Duration

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• Nogoshi & Igarashi (1971):

Microtremors  Surface Waves

• Others Authors

H/V ≡ Rayleigh Ellipticity

• Nakamura (1989)

H/V ≡ Transfer Function SH wave

• Sánchez-Sesma et al. (2011)

H/V ≡ All Waves (Diffuse Field Theory)

Ellipticity 2.32 HZ SH TF 2.48 Hz H/V 2.38 HZ

Microtremor H/V /V Spectral Ratio M MHVSR

m

V H V H 

2 2 2 m NSm EWm

V H H V H  

Average

• f Ratios

Ratio of Averages

• Arai & Tokimatsu (2004)

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

temps

time

Propagation Regimes & Energy Density Decay Radiative Transfer

Difussion (long times)

Coda

Few bounces (short times)

Noise

Kanai (1932)

Mult ltiple scattering C Coda N Noise Dif iffu fuse Fie ield

Directional Energy Density (DED) is proportional to the imaginary part of Green’s function at the source itself.

)] , , ( Im[ 2 ) ( ) ( ) , (

11 1 * 1 1 2 1

     x x x x x G k u u

S 

  

10

This is clear from the full-space solution (Stokes,1849):

         

3 3 11

2 1 12 )] , , ( Im[      x x G

Directional Energy Densities = Auto-Correlations Im[G(x,x,ω)] 3 3 3 ) 2 1 ( 6

3 3 3 1

           

S P S

R

     

3 3 3 3 3

2 1 2 1 2 1 1 R R R R R

SV SH P

     

Weaver (1985) JASA Sánchez-Sesma & Campillo (2006) BSSA

              3 1 3 2 3 1 3 1 2 1 6 1 3 1 3 1 2 1 6 1 3 1

3 2 1

          

SV P SH SV P SH SV P

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A Theory for H/V

With Directional Energy Densities the H/V ratio is:

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

) ; ( ) ; ( ) ; ( ) ; ]( / [

3 2 1

    x x x x E E E V H   )] ; , ( Im[ )] ; , ( Im[ )] ; , ( Im[ ) ; ]( / [

33 22 11

    x x x x x x x G G G V H  

12

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The imaginary parts of the Green's functions, using Harkrider (1964) notation can be written as :

 

 

   

      



  33

2 Im Im dk kr J k f i ; r G

V SV P

 

                   

               

   

 

                         

SV P H SV P SH SH

dk kr J kr J k f i dk kr J kr J k f i ; r G

2 2 11

4 4 Im Im   

 

 

 

 

  ; r G ; r G

11 22

Im Im 

r z

Green’s function calculation

Two bli lind tests (Synthetic Noise & DFA)

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200 400 600 800 1000 1200 1400 1600 1800

• 300
• 250
• 200
• 150
• 100
• 50

 (m/s), (m/s),  (Ton/m3) 100

Depth (m) Model N101

1000 2000 3000 4000 5000 6000

• 400
• 350
• 300
• 250
• 200
• 150
• 100
• 50

 (m/s), (m/s),  (Ton/m3) 100

Depth (m) Model N104

Frequency [Hz]

Frequency [Hz] H/V 0.47Hz

10

• 2

10

• 1

10 10

10

• 1

10 10

10

H/V =√2 Im[G11]/Im[G33]

10

• 2

10

• 1

10 10

1

10

• 1

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The Texcoco Experiment

1 2 3 4 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 freq [Hz] Im[Green functions]

Im[G22] =Im[G11] Im[G33] Frequency [Hz]

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Simplified model in 3D

2 2 2

) ( c G k G         x

h

x z

 G

   z G

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Frequency and time domains 3D

) , , ( Im  G  ) , , ( t PRS          



 

) sgn( ) ( ) 1 ( ) ( 2 1 2 1

2

t n t n dt t d h c

n n

    

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.

Augustin-Louis Cauchy

Simple poles Branch points Integration countour Branch cut

N

  

N

  

N

 

N

 

Im Im  

N N

 

  Im Im  

N N

 

 

Re Im Re Im     ) ( ; ) ( ) ( ; ) (

N N N N

   

   

k ik

II I III IV

Dispersion curves of Rayleigh waves f (Hz) Phase Velocity (m/s)

600 400 800 1000 6 4 2 10 8

Surface Waves Body Waves

Integrand of ImG33(0,0; f=5 Hz, k)

Branch points (ω/αN) Branch points (ω/βN)

k /(ω/βN ) Amplitude

2 1 0.5 2.5 1.5 4 6 2

Position of poles

The imaginary parts of the Green's function is computed as

(García-Jerez et al., 2013):

 

 

 

 

 

   

                           

Waves Body 4 4 Waves Surface Love Rayleigh 2 11

Re 4 1 4 1 Im dk k f k f A A G

N th th

SH H SV P m Lm m Rm m

  

          

    

  

 

 

 

 

 

11 22

Im Im G G 

 

 

 

 

               

Waves Body 4 Waves Surface Rayleigh 33

Re 2 1 2 1 Im dk k f A G

th N

V SV P m Rm  

 

  

 

 

Green function calculation

Profile Velocity

Foward Problem

Several views of H/V

Inverse Problem

   

 

 

2 . xp. 2

/ / 1

Th E i i i i

H V H V E n                

Inverse Problem

Inversion (Simulated Annealing)

Inversion example of H/V

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

Rayleigh Love

Velocity Profile

H/V Rayleigh Love H/V Rayleigh Love H/V

Non-uniqueness of H/V, dispersion curves

Functional H/V & Dispersion curves

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

VS

2

VS

1

VS

1

VS

2

Log Error Log Error

Example of joint inversion

Stations distribution (from Dr. E. Carmona)

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

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

Conclusions

• Green’s function (GF) can be retrieved from correlations within a diffuse field.
• Directional Energy Densities from autocorrelations are related with GF.

𝑭𝟐 𝒚, 𝝏 ~ < 𝒗𝟐(𝒚, 𝝏) 𝟑 > ~𝐉𝐧𝑯𝟐𝟐 𝒚, 𝒚, 𝝏 .

• Assuming noise is diffusive:

𝑰 𝑾 𝒚, 𝝏 = 𝑭𝟐 𝒚, 𝝏 + 𝑭𝟑 𝒚, 𝝏 𝑭𝟒 𝒚, 𝝏 = 𝐉𝐧𝑯𝟐𝟐 𝒚, 𝒚, 𝝏 + 𝐉𝐧𝑯𝟑𝟑 𝒚, 𝒚, 𝝏 𝐉𝐧𝑯𝟒𝟒 𝒚, 𝒚, 𝝏

• 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.

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Two References A García-Jerez et al. (2016), 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. (2016), The inversion of spectral ratio H/V in a layered media using the diffuse field assumption, Geophysical Journal International, Submitted.

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Thank you !

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         

3 3 11

1 2 12 )] , , ( Im[      x x G

42

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1 2 3 4 5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 freq [Hz] Im[Green functions]

Im[G22] =Im[G11] Im[G33] freq [Hz] freq [Hz] H/V 0.47Hz

10

• 2

10

• 1

10 10

1

10

• 1

10 10

1

10

2

H/V =√2 Im[G11]/Im[G33]

10

• 2

10

• 1

10 10

1

10

• 1

44

x2 x1

u(1) u(2) u(3) u(4) u(1) u(3) u(2) ξ1 ξ1 ξ1 ξ2 ξ2 ξ2

45

x2 x1

ξ1 + ξ2 = ξ ξ1 + ξ2 = ξ ξ + ξ = ξ