1. 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.

2. Basic Principle of TRI (Time Reversal Imaging) in acoustics:
Acoustic Source -> 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

3. Time Reversal- Adjoint Tomography:
Source focusing (Green function known)
Adjoint Tomography => Structure
(source known) S R
Residual time
reversed field dp
Direct field

4. ∂2u/∂t2 = H.u 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?

5. Seismic Source Imaging by time reversal
Method Principle:
Acoustic Source -> 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) ->3C (elastic case)?
Limited number of receivers?
Realistic Propagating Medium? 1D-3D Earth

6. Time reversal 1D PREM 3D models Larmat et al., 2006
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 T> 150s
Vertical component
1D PREM
3D models
Larmat et al., 2006

7. 1-Event rupture

8. 2-Seismogram
recording

9. 3- Time
reversal

10. 4- Focusing?

11. 2-Seismogram recording
1- Event rupture
3-Time reversal experiment
4- Focusing

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

13. PERU 23 June
Fault Plane
C. Larmat

14. 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

15. 1D- Reference Earth Model
Seismic Source
r∂ttu + H0u = Fs
Synthetic Seismograms by normal mode summation (k={n,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 = (M:e)E
M Seismic moment tensor, e deformation tensor
Green tensor G(rE,r,t,0)

16. 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)

17. 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+t)GSE(t)dt
If M not in E, cross-correlation:
v(rM,t) = GES(-t)*GSM(t) =∫GES(t+t)GSM(t)dt

18. 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’ n’Dl’Pl’0(cosD(rS,rM))FS nDlPl0(cosD(rR,rE)) ∫ b(t,t)dt M
D(rS,rM) f
E S
D(rE,rS)
=>Max if f =0 or  and M=E
(Stationnary phase approximation :
Romanowicz, Snieder, …)

19. 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 M=0
=> Focus at the Source with 1 receiver point
(but imperfect)
Linear Problem

20. 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)

21. NETWORK OF STATIONS Si => Source study
A 3-POINT PROBLEM M
D(rS,rM) f
E S
D(rE,rS)
NETWORK OF STATIONS Si => Source study
v(rM,t)= SSi Sk Sk’FE 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 t=0

22. Weighting of stations: Voronoi cells

23. S = ∫W dW A 3-POINT PROBLEM M D(rS,rM) f E S D(rE,rS)
DISTRIBUTION OF SOURCES Ei => 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
S S E5 E3 E1

24. Same formula apply for time reversal imaging
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.

26. Sumatra-Andaman
Earthquake
(26/12/04)
FDSN stations

27. Weighting of stations: Voronoi cells

28. Sumatra
Normal mode
Time reversal
Real Data

29. 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
=> Reference source: delta function?

32. Glacial Earthquakes
(Ekstrom et al., 2003, 2006)

33. Greenland - 28 dec 2001- M=5.0 (SEM, Komatitsch & Tromp, 2002)
(Larmat et al., 2008)

34. Greenland - 28 dec M=5.0
(Larmat et al., 2008)

35. Greenland - 21 dec M=4.8
(Larmat et al., 2008)

36. Greenland - 21/12/2001
Different Mechanism?
(Larmat et al., 2008)

37. 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