Time reversal imaging in longperiod Seismology

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Time reversal imaging in long-period Seismology

• Jean-Paul Montagner,
• Yann Capdeville, Huong Phung
• Dept. Sismologie, I.P.G., Paris, France
• Mathias Fink, LOA, ESPCI, Paris, France
• Carène Larmat, LANL, New Mexico, U.S.A.

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• Basic Principle of TRI (Time Reversal Imaging)
  in acoustics
• Acoustic Source -gt receivers
• Existence of transducers being at the same time
  recorders and emitters

Refocusing at the source location by sending back
signal (- t) through the SAME medium from a small
number of emitters
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Time Reversal- Adjoint Tomography

• Source focusing (Green function known)
• Adjoint Tomography gt Structure
• (source known)

S
R
Residual time reversed field
dp
Direct field
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Time reversal Concept

• Elasto-dynamics equation, for seismic
  displacement field u(r,t)
• ?2u/?t2 H.u
• In the absence of attenuation, rotation,
• time invariance and spatial reciprocity
• if u(t) is a solution, u(-t) is also a solution.
• We can send back waves with reversed time
• how to get a good focusing?

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Seismic Source Imaging by time reversal

• Method Principle
• Acoustic Source -gt receivers
• Existence of transducers at the same time
  recorders and emitters sending back signal in the
  same medium
• How to apply this concept to seismic waves within
  the Earth?
• 1C (scalar) -gt3C (elastic case)?
• Limited number of receivers?
• Realistic Propagating Medium? 1D-3D Earth

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Time reversal

• Seismic displacement field u(r,t) can be
  calculated everywhere by the SEM-NM method
  (Capdeville et al., 2003)
• It is possible to numerically backpropagate
  u(-t)
• Very long periods Tgt 150s
• Vertical component

• 1D PREM
• 3D models

Larmat et al., 2006
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1-Event rupture
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2-Seismogram recording
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3- Time reversal
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4- Focusing?
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2-Seismogram recording
1- Event rupture
3-Time reversal experiment
4- Focusing
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DATA Peru Earthquake (23-06-2001) Mw 8.4
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PERU 23 June 2001 - 8.4
Fault Plane
C. Larmat
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Normal Mode Approach
WHY does Time Reversal work when applied to
seismic waves ?
In acoustics, for chaotic cavities, - Draeger
and Fink, 1999 - Weaver and Lobkis, 2002
WHY normal modes? Complete basis of
functions Analytical solutions
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1D- Reference Earth Model

• Seismic Source
• r?ttu H0u Fs
• Synthetic Seismograms by normal mode summation
  (kn,l,m).

PREM (Dziewonski Anderson (1981)
Displacement at point r at time t due to a force
system F at point source rE u(r,t) Sk -(uk.F)E
uk(r)cos wkt /wk2exp(-wkt/2Qk)
Source Term (uk .F)E (Me)E M Seismic moment
tensor, e deformation tensor Green tensor
G(rE,r,t,0)
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Why does time reversal works when applied to
seismic waves?
rS station, rE source location, rM observation
point u(rS,t) Sk -(F.uk)E cos wkt /wk2 uk(rS)
u(rS,t) FE(t) GES(t) ( convolution)
M
S
E
Time reversed seismogram in rS FE(-t) GES(-t)
in rM v(rM,t) FE(-t) GES(-t) GSM(t)
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Why does time reversal works when applied to
seismic waves?
for a point source E, - If M in E,
autocorrelation v(rE,t) GES(-t)GSE(t)
?GES(tt)GSE(t)dt

• If M not in E, cross-correlation
• v(rM,t) GES(-t)GSM(t) ?GES(tt)GSM(t)dt

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Why does time reversal works when applied to
seismic waves?
v(rM,t) Sk Sk uk(rS)uk(rE) FSuk(rS) uk (rM)
?b(t,t)dt
k multiplet n,l,m
uk(rS) nDl(rS) Ylm(q,f)
Addition theorem Sm Ylm(q1,f1)Ylm(q2,f2)
Pl0(cos D(r1,r2))
v(rM,t)Sn,l Sn,l nDlPl0(cosD(rS,rM))FS
nDlPl0(cosD(rR,rE)) ? b(t,t)dt
M D(rS,rM) f E S D(rE,rS)
gtMax if f 0 or ? and ME (Stationnary phase
approximation Romanowicz, Snieder, )
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Why does time reversal works when applied to
seismic waves?
A 3-POINT PROBLEM
M D(rS,rM) f E S
D(rE,rS)
TR-field v(rM,t) f(rE,rS,rM) B(t) Max if f 0
or ? and if M0 gt Focus at the Source with 1
receiver point (but imperfect)
Linear Problem
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Why does time reversal works when applied to
seismic waves?

TR-field v(rM,t) f(rE,rS,rM) B(t) Linear
Problem
-several stations Si v(rM,t) (GS1E(-t)
GS2E(-t) GS3E(-t) )GEM(t)
-several sources Ei v(rM,t) (GE1S(-t)
GE2S(-t) GE3S(-t) )GSM(t)
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A 3-POINT PROBLEM
M D(rS,rM) f E S D(rE,rS)

• NETWORK OF STATIONS Si gt Source study

v(rM,t) SSi Sk SkFE uk(rE) uk(rSi) uk(rSi)
uk(rM) ?b(t,t)dt
S2 E
S1 S3 S4

• S ?W dW
• Stations
• Weighting of stations
• Focusing in E at t0

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Weighting of stations Voronoi cells
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A 3-POINT PROBLEM
M D(rS,rM) f E S D(rE,rS)

• DISTRIBUTION OF SOURCES Ei gt Cross-correlation

v(rM,t) SEi Sk Sk uk(rS) uk(rEi)FEi2uk(rEi)
uk(rM) ?b(t,t)dt

• S ?W dW
• sources
• Weighting of sources
• Random distribution
• of point sources

E2 E4
S1 S2
E5 E3 E1
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Normal Mode Approach
Same formula apply for time reversal imaging and
cross-correlation techniques
LIMITATIONS

• - Signal dominated by surface waves.
• Some missing modes (when station at the node of
  some
• eigenmodes excited by the source)
• Not exactly the Green functions (limited
  bandwidth, )
• Attenuation
• Improvement if several stations are available.

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Sumatra-Andaman Earthquake (26/12/04) FDSN
stations
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Weighting of stations Voronoi cells
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Sumatra Normal mode Time reversal Real Data
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Source Rupture Imaging
u(r,t) Sk uk (r) cos wkt /wk2 exp(-wkt/2Qk)
(uk.F)S
u(r, w) G (r,rS, w) S(rS, w)
G (r,rS, w) Green Function S(rS, w) Source
Function gt Reference source delta function?
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Glacial Earthquakes

• (Ekstrom et al., 2003, 2006)

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Greenland - 28 dec 2001- M5.0
(SEM, Komatitsch Tromp, 2002)
(Larmat et al., 2008)
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Greenland - 28 dec 2001- M5.0
(Larmat et al., 2008)
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Greenland - 21 dec 2001- M4.8
(Larmat et al., 2008)
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Greenland - 21/12/2001
Different Mechanism?
(Larmat et al., 2008)
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TIME REVERSAL

• Normal mode theory enables to
• understand why, how TR works.
• Similarities between time reversal
• imaging and cross-correlation techniques
• Application to real seismograms of
• broadband FDSN stations
• Good localization in time and in space
• of earthquakes, and ice-quakes
• Spatio-temporal Imaging of seismic source
• Applications to seismic Tomography- Adjoint method