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.

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

Basic Principle of TRI (Time Reversal Imaging) in acoustics:

Acoustic Source -> receivers

Existence of transducers being at the same time recorders and emitters

Time Reversal- Adjoint Tomography:

-Source focusing (Green function known)

-Adjoint Tomography => Structure (source known)

S R

Direct field Residual timereversed field

pp

Time reversal Concept Time reversal Concept

Elasto-dynamics equation, for seismic displacement Elasto-dynamics equation, for seismic displacement field field uu(r,t)(r,t)

∂∂22u/u/∂∂tt22 = H.u = H.u

In the absence of attenuation, rotation, In the absence of attenuation, rotation, time invariance and spatial reciprocitytime invariance and spatial reciprocity

if u(t) is a solution, u(-t) is also a solution.if u(t) is a solution, u(-t) is also a solution.

We can send back waves with reversed time:We can send back waves with reversed time:how to get a good focusing?how to get a good focusing?

Seismic Source Imaging by time reversalSeismic Source Imaging by time reversal

Method Principle:Method Principle:

- Acoustic Source -> receiversAcoustic Source -> receivers- Existence of transducers at the same time recorders and emitters Existence of transducers at the same time recorders and emitters

sending back signal in the sending back signal in the samesame medium medium

How to apply this concept to seismic waves within How to apply this concept to seismic waves within the Earth?the Earth?

1C (scalar) ->3C (elastic case)?1C (scalar) ->3C (elastic case)?Limited number of receivers?Limited number of receivers?Realistic Propagating Medium? 1D-3D EarthRealistic Propagating Medium? 1D-3D Earth

Time reversal Time reversal Seismic displacement field Seismic displacement field uu(r,t) can be calculated everywhere (r,t) can be calculated everywhere by the by the SEM-NMSEM-NM method (Capdeville et al., 2003) method (Capdeville et al., 2003)

It is possible to numerically backpropagate It is possible to numerically backpropagate u(-t)u(-t)•Very long periods T> 150s Very long periods T> 150s •Vertical componentVertical component

•1D PREM1D PREM

•3D models3D models

Larmat et al., 2006Larmat et al., 2006

1-Event rupture

2-Seismogram recording

3- Timereversal

4- Focusing?

1- Event rupture 2-Seismogram recording

3-Time reversal experiment 4- Focusing

DATA: Peru Earthquake (23-06-2001) Mw= 8.4DATA: Peru Earthquake (23-06-2001) Mw= 8.4

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PERU 23 June 2001 - 8.4

Fault Plane

C. Larmat

Normal Mode Approach

In acoustics, for chaotic cavities,

- Draeger and Fink, 1999- Weaver and Lobkis, 2002

WHY normal modes?

Complete basis of functionsAnalytical solutions

WHY does Time Reversal work when WHY does Time Reversal work when applied to seismic waves ?applied to seismic waves ?

1D- Reference Earth Model

Seismic Source∂ttu + H0u = Fs

Synthetic Seismograms by normal Synthetic Seismograms by normal mode summation (k={n,l,m}).mode summation (k={n,l,m}).

Source Term (uk .F)E = (M:)E

M Seismic moment tensor, deformation tensor

Green tensor G(rE,r,t,0)

Displacement at point r at time t due to a force system F at point source rE

u(r,t)= k -(uk.F)E uk(r)cos kt /k2exp(-kt/2Qk)

PREM (Dziewonski & Anderson (1981)

Why does time reversal works when applied to seismic waves?

rS station, rE source location, rM observation point

u(rS,t) = k -(F.uk)E cos kt /k2 uk(rS)

u(rS,t) = FE(t) *GES(t)

( * convolution)

Time reversed seismogram in rS : FE(-t) *GES(-t)

in rM: v(rM,t) = FE(-t) *GES(-t) * GSM(t)

SE

M

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(t+)GSE()d

- If M not in E, cross-correlation:

v(rM,t) = GES(-t)*GSM(t) =∫GES(t+)GSM()d

v(rM,t) = k k’ uk’(rS)uk(rE) FSuk(rS) uk’ (rM) ∫b(t,)d

uk(rS)= nDl(rS) Ylm()

Addition theorem: m Ylm()Yl

m()= Pl0(cos(r1,r2))

k multiplet: {n,l,m}

v(rM,t)=n,l n’,l’ n’Dl’Pl’0(cos(rS,rM))FS nDlPl

0(cos(rR,rE)) ∫ b(t,)d

M (rS,rM)

E S

(rE,rS)

=>Max if =0 or and M=E(Stationnary phase approximation :

Romanowicz, Snieder, …)

Why does time reversal works when applied to seismic waves?

A 3-POINT PROBLEM

TR-field: v(rM,t)= f(rE,rS,rM) B(t)Max if =0 or and if M=0=> Focus at the Source with 1 receiver point=> Focus at the Source with 1 receiver point

(but imperfect)(but imperfect)

M (rS,rM)

E S

(rE,rS)

Linear Problem

Why does time reversal works when applied to seismic waves?

TR-field: v(rM,t)= f(rE,rS,rM) (t)

Linear Problem:

-several sources Ei

v(rM,t) = (GE1S(-t) + GE2S(-t) + GE3S(-t)+ …)*GSM(t)

-several stations Si

v(rM,t) = (GS1E(-t) + GS2E(-t) + GS3E(-t)+ …)*GEM(t)

Why does time reversal works when applied to seismic waves?

A 3-POINT PROBLEM

v(rM,t)= Si k k’FE uk(rE) uk’(rSi

) uk(rSi) uk’(rM) ∫b(t,)d

S2

E S1

S3

S4

• NETWORK OF STATIONS Si => Source study

M (rS,rM)

E S

(rE,rS)

= ∫ dStations

Weighting of stationsFocusing in E at t=0

Weighting of stations: Voronoi cells

A 3-POINT PROBLEM

v(rM,t)= Ei k k’ uk(rS) uk’(rEi

)FEi2uk(rEi

) uk’(rM) ∫b(t,)d

E2

E4

S1 S2 E5

E3

E1

M (rS,rM)

E S

(rE,rS)

• DISTRIBUTION OF SOURCES Ei => Cross-correlation

= ∫ dsourcesWeighting of sources

Random distributionof point sources

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.

Normal Mode Approach

Same formula apply for time reversal imaging and cross-correlation techniques

Sumatra-AndamanEarthquake(26/12/04)

FDSN stations

Weighting of stations: Voronoi cells

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Normal modeTime reversal

Real Data

Source Rupture Imaging

u(r,t) = k uk (r) cos kt /k2 exp(-kt/2Qk) (uk.F)S

u(r, ) = G (r,rS, ) S(rS,

G (r,rS, ) Green Function S(rS, Source Function

=> Reference source: delta function?

Glacial Earthquakes

(Ekstrom et al., 2003, 2006)

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Greenland - 28 dec 2001- M=5.0

(Larmat et al., 2008)

(SEM, Komatitsch & Tromp, 2002)

Greenland - 28 dec 2001- M=5.0

(Larmat et al., 2008)

Greenland - 21 dec 2001- M=4.8

(Larmat et al., 2008)

Greenland - 21/12/2001

(Larmat et al., 2008)

Different Mechanism?

TIME REVERSALTIME REVERSAL

Normal mode theory enables to Normal mode theory enables to understand why, how TR works.understand why, how TR works.

Similarities between time reversal Similarities between time reversal imaging and cross-correlation techniquesimaging and cross-correlation techniques

Application to real seismograms ofApplication to real seismograms ofbroadband FDSN stationsbroadband FDSN stations

Good localization in time and in spaceGood localization in time and in space of earthquakes, and ice-quakes of earthquakes, and ice-quakes Spatio-temporal Imaging of seismic sourceSpatio-temporal Imaging of seismic source Applications to seismic Tomography- Adjoint methodApplications to seismic Tomography- Adjoint method

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