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.

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

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

∂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?

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

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

1-Event rupture

2-Seismogram recording

3- Time reversal

4- Focusing?

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

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

PERU 23 June 2001 - 8.4 Fault Plane C. Larmat

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

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)

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)

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

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, …)

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

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)

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

Weighting of stations: Voronoi cells

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 S1 S2 E5 E3 E1

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.

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

Weighting of stations: Voronoi cells

Sumatra Normal mode Time reversal Real Data

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?

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

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

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 Different Mechanism? (Larmat et al., 2008)

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