Ambient Seismic Vibrations in Seismology
and Earthquake Engineering
Francisco J. Sánchez‐Sesma
Instituto de Ingeniería, National University of Mexico (UNAM)

Uji, Kyoto, 23 February 20151

Outline of Presentation
Introduction
Diffuse Fields  Multiple Diffraction
Equipartition of Energy in Dynamic Elasticity
Full Space, Half‐Space  Experimental Verification
Correlation Type Representation Theorem
Green’s Function from the Average of Correlations
Dispersion Curves for Tomography of Alluvial Valleys
Energy Densities from Average of Auto‐Correlations
Deterministic Partition of Energy
Ambient Seismic Noise and H/V Spectral Ratios, Fast Calculation
and Inversion (1D, 3D)
Conclusions
2

Introduction
The last decade witnessed the specular emergence of ambient seismic
noise as a powerful tool for imaging Earth structure at many differentscales.
All this lead to improvements of methods that utilize ambient noise. They
are now applied to more data sets with better theoretical understanding
and are useful for a wide range of applications in Seismology includingtime‐dependent imaging for diverse physical processes like helioseis‐
mology, underwater acoustics, and structural health monitoring, to cite
only a few.
These advances powered also significant applications in Earthquake
Engineering as well. In addition to improved performance of noise‐based
imaging, there are innovative applications to establish dominant periodsof sites and inverting the soil structure in order to compute seismic
3response.

Pocosrebotes(tiemposcortos)
Few bouces(shorttimes)

Propagation
regimes &Energy
densitydecay

Transferencia
Radiativa
Radiative Transfer

Difussion (longtimes) largos)
Difusión(tiempostempstime4

Radiative Transfer Theory
Originally in astrophysics by Chandrasheckar et al. (40´s & 50’s).
Introduced to sesmology by R.S. Wu (1985) and developend by K.
Aki and Y. Zeng, H. Sato, K. Mayeda, M. Campillo, L. Margerin, A.
Gusev and others.
Sato H. & M. Fehler (1998). Wave propagation and scattering in theheterogeneous Earth, Academic Press, Cambridge, Mass.
Dmowska R., H. Sato & M. Fehler (2008) (Eds) Vol. 50 of Advances inGeophysics, Academic Press, Cambridge, Mass.
Predictions for
Elastic Diffuse
FieldsMultiple Scattering  Energy Equipartition

5

Energy Equipartition Principle
Infinite Space, Volume modesLAssume stationary P waves inside a finite region (e.g. A cube ofside L >> the wave lenght) and Dirichlet boundary conditions:

Lnz

Then we can admit a modal solution of the form,withL

ModalSphere

According to the Principle of Equipartition of Energy, theenergy associated to every state is proportional to the densityof modes at a given frequency.

nynx

6

3

11 4  L   2 2 2 3 2N P   Volume of Modal Sphere     ( nx  nx  n x )88 3    6dNP  (1/ 2 2 ) 2 3Vd

dN S  2(1 / 2 2 ) 2  3Vd ,In a diffuse field the energy associated to elastic waves in a
En un campo difuso la energía asociada a las ondas elásticas en elvolumeis arranged
accordingto contheelPrincipleofEquipartición.
Equipartition.
volumenestá repartidade acuerdo
Principio deenergíaofdeeachcada stateestado ises proportional
proporcional altonúmero
de modos.
The Laenergythe densityof Más
modes
precisamente, a la densidad de modos en una banda de frecuencias. El
at a cociente
given frequency
band.
de las energíasde las ondas de S y P está dado por:

The ratio of the S and P wave energy is:
ES q  dNS
3
 2 3  2R3
EP q  dNPWeaver (1982)
7

1
P 

31  2RR3
SH 
31  2R
R3
SV 
31  2R

1111
1   P  SV  SH  
362
3
1
111
2   P  SV  SH  36231213   P   SV 
33
3Weaver (1982)

Sánchez‐Sesma & Campillo (2006)

8Half-Space, Rayleigh’s Modes
Assume now stationary Rayleigh waves associated to the freesurface (for instance, a square with side L >> wave length).
1  2w
 2w  2w
 2
c R t 2
y 2
x 2

LL1Dirichlet 2 w boundary conditions (w=0 en |x|=|y|=0,L ) 2w  2w
Consideralso
 2
22xyc R t 2
2n y ynx xLw  sen
 sen
 sen t  n x2  n y2  LLcR 
21 2 A  L  N R  (n  n )  
44   cR  4 cR2

2x2x

1 AdN R d22 cRny

1/4nx
93ES  2 3 EP  2R3 EPq AER  q  dN R   2 d2 cR

E  EP  ES  (1  2R ) EP
33

12q   P  2 2  d  E / V ,  P  EP / V , S  ES / V ,  R  ER / A
131  2R
2R3
S 31  2R

P Weaver (1985)
Perton et al. (2009)2    R3 R   3  cR  1  2R
10

Semiespacio (3D)

Perton et al (2009)This result shows two ways
the equipartition can occur:(1) à la Maxwell or (2) à la Weaver
11

Looking for an marker of the diffuse regime
The Equipartition Principle for Elastic Waves
In a diffusive regime all “modes” participate
in the same proportion

G ij ( X ,  , t )    n  ( X ) exp(  i  n t )nnwhere εn are independent random variables
Energy proportions for an infinite elastic solid
Independent of “scattering’’details !
Independent of position in afull-space with homogeneous“illumination” !

12The Principle of Equipartition ofEnergy in Elastodynamics

Experimental Verification

cR

50 m aperture
Campillo et al. (1999); Shapiro et al. (2000)

13

Experimental Verification

Hennino et al. (2001)

14

Hennino et al. (2001)

15

Hennino et al. (2001)16

Correlation type Representation Theorem

2i Im Gij (rA , rB )    Gil (rA , r )Tlj* (r, rB ) G *jl (rB , r )Tli (r, rA ) dSrBWeaver & Lobkis (2004), Wapenaar (2004),
Van Manen, Curtis & Robertson (2006)rAr 2in 2D
2ES  E P  2 3in 3D3 ¡ Equipartition !

ui (xA , )u*j (xB , )  4 ES k 3Im Gij (xA , xB , )17

SURFACE WAVE TOMOGRAPHY
Mexico Valley
70 stations
Traces163 epicenters(~280 km between Mexico Cityand the Pacific coast)

1056 km²18

3 WAYS FOR RETRIEVING THE GREEN’S FUNCTION
By definition: Impulsive Response

Plane waves: An
equipartitioned cocktail ofP,SV and Rayleigh waves

Independent distant seismic sources andlots of randomly placed difractors

19

GREEN’S FUNCTION G22 (SH CASE)

20GREEN’S FUNCTION Gij (P‐SV CASE)

21

LOVE WAVES DISPERSION CURVES
Analytical curves: layer over half‐space
Experimental curves: filtering cross‐correlation

22

RAYLEIGH WAVES DISPERSION CURVES
Analytical curves: layer over half‐space
Experimental curves: filtering cross‐correlation

23

Retrieval of Directional
Energy Densities byAveraging
Auto-correlations
24E (x)   2 um (x)um* (x)  4 ES k 1  Im[Gmm (x, x)]

Directional Energy Density (DED). It is theImaginary part of Green’s function at source

Re[G11(x, x;)  i e ]   ImG11(x, x;)
it

Proportional to the power transmited tothe medium by the unit harmonic force
25

From the Representation
Theorem

Im[Gmn (x A , x B )] exp(ikr j n j ) exp(iqr j n j )nn(nn)
m nmnm n d23316 rA     12
E1  A  Im[G11 (x, x)]  A   3  3   E1P  E1S
12   Stokes (1849)
Sánchez-Sesma et al. (2008)
26

AS A CONSEQUENCE OF THE IDENTITY

EnergyGreen’s Function

E1  E2  E3  A  Im[Gkk (x, x)]  EP  ES
E1   2 u12  Im[G11 (x, x)]
E2   2 u22  Im[G22 (x, x)]
E3   2 u32  Im[G33 (x, x)]

Directional Energy Densities (DEDs)
Perton et al. (2009)27

Deterministic Partition of Energy
Vertical force

Rayleigh

Lamb(1904)
Miller & Pursey(1955)

SVP

Weaver (1985)

SH=0 % R=67% SV=26% P=7%28

Deterministic Partition of Energy
Horizontal force

RayleighChao(1960)
Cherry (1962)

SHSVP

Weaver (1985)

SH=60% R=18% SV=16% P=6%
Sánchez-Sesma et al (2011) BSSA29Φ=0SVPSVP

Φ=π/2
SH

Sánchez-Sesma et al. (2011) BSSA
30

Seismic Noiseand H/V
31

A Theory for H/VWith Directional Energy Densities one can computethe H/V ratio as:

E1 (x;  )  E2 (x;  )[ H / V ](x;  ) E3 (x;  )measurements  system properties

Im[G11 (x, x;  )]  Im[G22 (x, x;  )][ H / V ](x;  ) Im[G33 (x, x;  )]Sánchez-Sesma et al. (2011)
3D problem (BW & SW)

Kawase et al. (2011)
1D problem (BW)32

Computation of ImG11 , and ImG33 by an integral on the radial wavenumber

33

Layer over Half-space

Im[G11(0,0, ω)], Im[G33(0,0, ω)]
3D Solution

0.45
0.4

Im[G22] =Im[G11]Im[Green functions]

0.35
0.3
0.25
0.2

Im[G33]

0.15
0.1
0.05
0012345freq [Hz]34

The Texcoco Experiment102

H/V ~ √ Im[G11]/Im[G33]

H/V

Im[Green functions]

100-1

10-1

10

freq [Hz]010

110 -21010-1100101

freq [Hz]

0.45

0.35

10.47Hz-1

10 -2
100.4

10

Sánchez‐Sesma et al. (2011)

Im[G22] =Im[G11]

0.3
0.25

Im[G33]

3D Effect !

0.2
0.15
0.1

Layer over half-space

0.05
0012

3
freq [Hz]4535

Fast computation of Im[Gij(0,0, ω)]with Cauchy Residue Theorem
An oportunity to speed up inversion
García‐Jerez et al (2013)36

Poles localization – Dispersion Curves370Velocity Profile

10Direct
Problem

Depth (m)

2030

4050
0500

1000

1500

2000

2500

3000

Velocity (m/s)
380Velocity Profile

20181016
14

Amplitude10820

Deptf (m)InverseProblem

1230

64

40

2
0.01

0.1

1

Frequency (Hz)

1050
0500

1000

1500

2000

2500

3000

Velocity (m/s)
39

Global optimization using
simulated annealing40

Inversion using Simulated Annealing (Piña et al., 2015)41

Application to site effect characterization
at Almería, Andarax River, Spain
Stability of H/VStations distribution

FTAN H/V(from Dr. E.
Carmona)
42Application to site effect characterization
at Almería, Andarax River, Spain

Depth (m)Depth (m)SPAC ‐ 5 vertex, Rmax 450m

Velocity (m/s)

Velocity (m/s)
43

H/V with lateral irregularity
Imaginary Part of Green's function

Verification of 1D results with numerical modeling‐0.07

Im G11 (Theory)
Im G33 (Theory)
Im G11 (SEM)
Im G33 (SEM)

‐0.06
‐0.05

• Imaginary Parts of Green’sfunction

‐0.04
‐0.03
‐0.02
‐0.01
06 012

3

H/V Spectral Ratio

54

H/V (Theory)

5• Good agreement between
theory and SEM for G11 yG33

• Spectral Ratio H/VH/V (SEM)• Excellent agreement
between theory and SEM

4321

Matsushima et al. (2014)

001

23Frequency (Hz)4544

H/V with lateral irregularity
61D theory

H/V Spectral Ratio

5

2D basin (NS/UD)
2D Basin (EW/UD)

4

Observed (NS/UD) 6/21
3

Observed (EW/UD) 6/21

• Spectral ratios H/V• The efects of lateral irregularity
are clear in NS/UD and EW/UD• Peak Amplitudes• NS/UD > EW/UD• Peak Frequency

2• NS/UD < EW/UD1• Qualitative Agreement

00.1

1.0

Frequency[Hz]Matsushima et al. (2014)site C45

Comments and Conclusions (1)The Principle of Equipartition of Energy allows characterizationof diffuse fields. In seismology a key issue is Multiple Scattering.
We examined the Properties of the Equipartition in a full‐space,a half‐space and we discussed the experimental verification.
We reviewed retrieval of the Green’s function from the averageof correlations in a diffuse field (coda, noise) or for earthquakes
with dominance of body waves. Generalized diffuse field.
Deterministic Gij with equipartitioned plane wave cocktails.
46

Comments and Conclusions (2)Directional Energy Densities from autocorrelation averages.
• Deterministic partition of Energy• H/V ratios• For seismic noise  fast calculation of H/V  inversion• For incoming body waves  fast calculation  inversion• Effects against depth  fast calculation  inversion• Effects in the presence of lateral irregularities
Without a doubt, these concepts will keep surprising us withnew applications.
47

Thank you..!
48