time-reversal method and cross-correlation techniques by normal mode theory: a three-point problem

submitted to Geophys. J. Int.

Time Reversal Method and Cross-Correlation techniques by

normal mode theory: a 3-point problem. DRAFT

J.-P. Montagner1, Y. Capdeville1, M. Fink2, C. Larmat3, H. Nguyen1, B. Romanowicz4

1 Seismology Laboratory, Institut de Physique du Globe UMR-CNRS 7154, 4 Place Jussieu 75252 Paris Cedex 05, France

2 L.O.A., E.S.P.C.I., Rue Vauquelin, 75005 Paris, France

3 Seismology Lab., U.C. Berkeley, Berkeley, CA, U.S.A.

4 L.A.N.L., Los Alamos, NM, U.S.A.

SUMMARY

Long-period time-reversal experiments in the Earth produce excellent focusing of seismic

waves at the earthquake location. Normal mode theory enables for a finite body such as the

Earth, to investigate why time–reversal technique works when applied to seismic waves

in an elastic medium. It is demonstrated how the informations given by the 3-components

of elastic waves can be optimally incorporated for the localization in time and space of

earthquake, for recovering the whole seismic moment tensor and the source time function.

A 3 point problem, with 2 fixed points and one variable point, can be defined. The similar-

ities (and differences) between time reversal imaging and cross-correlation techniques are

investigated. For a random surface distribution of seismic sources, the cross-correlation

of seismograms in 2 stations is not exactly the Green function between these 2 stations

We demonstrate the importance of a correct weighting of stations by Voronoi tessellation

on a global scale.

Key words: Time reversal, normal modes, cross-correlation, adjoint technique

2 J.-P. Montagner, Y. Capdeville, M. Fink, C. Larmat, H. Nguyen, B. Romanowicz

1 INTRODUCTION: TO BE COMPLETED

The normal mode approach is well suited for investigating the propagation of seismic waves in the

spherical Earth. So far, normal mode theory has been only applied to scalar acoustic waves (Draeger

& Fink (1997, 1999), Weaver & Lobkis (2002)). By using the normal mode formalism developed

by Capdeville (2005), Larmat et al. (2006) showed that it is very easy to back-propagate seismic

waves at long periods by using seismograms synthetized by normal mode summation to the Sumatra-

Andaman earthquake of 26 Dec. 2004. She showed that the focusing of seismic waves at the epicenter

is as good as with the Coupled Spectral Element Technique (Capdeville et al. (2002) where synthetic

seismograms are calculated in 3D-heterogeneous Earth. The time-reversal technique as implemented

by Larmat et al. (2006) has many similarities with the adjoint method recently detailed in Tromp et al.

(2005).

We show in this paper how to generalize the 1D-normal mode approach of Draeger & Fink (1999),

Weaver & Lobkis (2002)) to 3D-elastic medium. We also demonstrate why time reversal technique

works when applied to the Earth for seismic waves, and answer some questions such as how the infor-

mations given by the 3-components of elastic waves can be optimally incorporated for the localization

of earthquake, for recovering the whole seismic moment tensor and the source time function?

The potential applications of this technique are ongoing. In addition to automatically locate earth-

quakes and retrieve their seismic moment tensor, time-reversal techniques using normal modes can

be used for locate sources excited in the long-period range (period larger than 100s) where the effect

of lateral heterogeneities is not large. Larmat et al. (2008) applied time-reversal technique to locate

Glacial earthquakes. Other potential applications are the location of the source of seismic Hum (Suda

et al. (1998), Fukao et al. (2002)).

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Time Reversal Method and Cross-Correlation techniques by normal mode theory: a 3-point problem 3

2 TIME REVERSAL WITH NORMAL MODE THEORY

2.1 Forward problem

The elasto-dynamics (Navier-Stokes) equation for a continuum medium can be written:

ρd2uidt2

=∑j

σij,j + fi + ρgi

fi et ρgi are external applied forces. E is the source, R receiver for the forward problem, M the

observation point of the time reversed field. Generally, when neglecting the advection term, this local

equation can be written under the form:

(ρ∂tt +H0)u(r, t) = F(r, t) (1)

whereH0 is an integro-differential operator calculated in a reference Earth model and F expresses

the ensemble of external forces applied in r at time t. Gravity forces are neglected and it is assumed

that F is zero for t < 0. Bra-ket notations from quantum mechanics (Cohen-Tannoudji et al., 1973)

are used for displacement eigenfunction uK(r, t):

uK = D|q,n, `,m〉 =n D`|K〉 = (nU`(r)er +n V`(r)∇1 +nW`(r)(−er ×∇1)Y ml (θ, φ) = (2)

(nU`er + 1√`(`+1)

nV`(eθ ∂∂θ +eφ1

sinθ

∂

∂φ) +

1√`(`+ 1)

nW`(eθ1

sinθ

∂

∂φ− eφ

∂

∂θ)Y ml (θ, φ)

uK = |K〉 and ωK are respectively the set of eigenfunctions and eigenfrequencies, solutions of

the equation without 2nd member (−ρω2K + H0)|K〉 = 0 where the indices K = (q, n, `,m) and

k = (q, n, `); q can take 2 values, one for spheroidal modes and two for the toroidal modes (q is

often emitted when there is no ambiguity); n (radial order), ` (angular order), m (azimuthal order) are

quantum numbers. For a spherically symmetric elastic non rotating earth model, there is a degeneracy

in m, the eigenfrequency ωK is independent of m, and can be noted ωk, and the eigenfunctions nU` ,

nV` , nW` are only functions of radius r. It is possible to expand any displacement u(r, t) on the set

of eigenfunctions uK(r, t) = |K〉e−iωKt:

u(r, t) =∑K

aK |K〉e−iωKt =∑K

aKuK(r)e−iωKt (3)

|K〉 are orthogonal, normalized functions, fulfilling the orthonormalization relationship:

〈K′|K〉 =∫Vρu∗K′(r)uK(r)dV = δKK′

Let us first introduce some useful notations, such as the dot product , u.v = ui.vi


and the inner product

〈u|v〉 =∫Vρu∗(r)v(r)dV (4)

A very useful theorem is the addition theorem (also named closure relationship):∑

k |K(r)〉〈K(r′)| =

ΣkuK(r)u∗K(r′) = Iδ(r− r′) where I is the identity matrix.

When using common spherical harmonics, the classical addition theorem expression is found:

m=∑m=−`

Y m` (θE , φE)Y m

` (θR, φR) =4π

2`+ 1P 0` (cos∆ER) (5)

where ∆ER is the angular distance between E and R. There are some other useful relationships, such

as generalization of classical addition theorem, applied to generalized spherical harmonics (Edmonds

(1960); Li & Romanowicz (1996); Capdeville et al. (2000)

m=∑m=−`

Y Nm` (θE , φE)Y Nm

` (θR, φR) =4π

2`+ 1eiN

′γERPNN′

` (cosβER)eiN′αER (6)

where PNN′

` (cosβER) are the generalized Legendre functions. The index E is related to the source

location, and the index R to the receiver location. The angle−αER is the back-azimuth at the receiver,

π− γER is the azimuth at the source, and βER is the epicentral distance between the source E and the

receiver R.

Bra and ket can be applied to the operator H0 such that:

〈K′(r)|H0|K(r′)〉 =∫VdV

∫V ′dV ′ u∗K′(r)H0uK(r′) = ω2

KδKK′ (7)

It is easy to calculate the excitation coefficient aK (Gilbert (1971)) by taking either the Fourier trans-

form or the Laplace transform of equation (2). At the receiver point R and for an earthquake at point

E, the displacement can be written:

u(rR, ω) =∑K

∫V ′dV ′ u∗K(rE)F(rE, ω)

uK(rR)−ω2 + ω2

K

=∑K

|KR〉〈KE|FE(ω)1

−ω2 + ω2K

(8)

where for sake of simplicity, we note |KE〉 = |K(rE)〉, |KR〉 = |K(rR)〉. The same notation will be

used later on for other points.

An inverse Fourier transform can be performed in order to come back in the time domain. For the

source of the earthquake, we can consider 2 simple cases. In the first one, the force system F(r, t) is a

delta function in time and space F(r, t) = FEδ(r− rE)δ(t). And in the second one, the force system

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F(r, t) is a time Heaviside function H(t): F(r, t) = H(t)F0(r), which is much more realistic for an

earthquake.

For a delta function in time:

u(rR, t) =∑K

(∫VE

dVEu∗K(rE)FE(rE))uK(rR)H(t)sin(ωKt)ωK

=∑K

|KR〉〈KE|FE〉H(t)sin(ωKt)ωK(9)

Let us define by index γ the coordinates of vectors at point E, β at point R, and later on α for any

other point M in a Cartesian coordinate system. The Green function can be written:

Gγβ(rE , rR, t; 0) =∑K

u∗γK (rE)uβK(rR)H(t)sin(ωKt)ωK

=∑K

|KβR〉〈K

γE|w(t) (10)

where w(t) = H(t) sin(ωKt)ωK

. It is also useful to define the Green function of velocity u(rR, t) for

t > 0 as will be seen in section 3.2

G(rE, rR, t; 0) =∑K

|KR〉〈KE|H(t)sin(ωKt) (11)

For acoustic waves (scalar waves), the Green function is a scalar function, whereas for elastic

waves, it is a second order tensor. The spatial reciprocity GγβER = Gβγ

RE can be directly seen, but it

must be noted that it is more complex than for scalar waves.

For a source Heaviside Function in time, the general expression of displacement is given by:

u(rR, t) =∑K

(∫VE

dVE u∗K(rE)F0(rE))H(t)1− cos(ωKt)

ω2K

uK(rR) (12)

It is a remarkable expression, since the displacement is the result of the summation of all modes

calculated at time t. We can come back to the previous expression (10), for t > 0, by taking the time

derivative of equation (12). It can be shown that the contribution of eigenmodes is usually zero except

when there is a constructive interference corresponding to a stationnary phase (Romanowicz (1987)).

For t > 0, the corresponding acceleration Γ(rR, t) has a very simple expression:

Γ(rR, t) =∑K

uK(rR)H(t)cosωK(t)∫VE

dVEu∗K(rE)F0(rE)

For a point source in time and space, it is usual to introduce the seismic moment tensor Mmγ

such that F γ0 (rE) = −Mγm∂∂ξm

δ(ξ − rE).

u(rR, t) =∑K

ε∗Kγm(rE)MγmH(t)

1− cosωK(t)ω2K

uK(rR) =∑K

|KR〉〈KE|EMw′(t) (13)

This expression corresponds to Born approximation at zero order. Instead of the Green Function

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in displacement for a point source in time and space, we must use now (with respect to Mγm)):

Gβγ,mER (t) =∑K

uβK(rR)ε∗Kγm(rE)H(t)

1− cosωKtω2K

If the static term is discarded, and if the instrument vector v is introduced, the expressions derived

by Woodhouse & Girnius (1982) for a spherically non rotating earth model are retrieved (see table 1),

v.u(t) =∑k

(−cos(ωkt)

ω2k

∑m

Rmk (rR, θR, φR)Smk (rE , θE , φE)

)with

Rmk (rR, θR, φR) =N=1∑N=−1

RkN (rR)Y Nm` (θR, φR)

and

Smk (rE , θE , φE) =N=1∑N=−1

SkN (rE)Y Nm` (θE , φE)

For example, the vertical component is expressed as:

ur(rR, t) = vr∑n,`,m

nU`(rR)Y m` (θR, φR)(∂rU(rE)Mrr +

12F (rE)(Mθθ +Mφφ))Y m

` (θE , φE)eiωkt

If the addition theorem (6) is taken into account:

ur(rR, t) = vr∑n,`

nU`(rR)−2`+ 1

4πP 0` (cos(∆ER)(∂rU(rE)Mrr +

12F (rE)(Mθθ +Mφφ))eiωkt


2.2 Time reversal Application

Now u(rR, t) is the displacement field applied by time reversal recorded in different seismic stations

in rR to the Earth surface. The equivalent dynamic force is Γ(r) = −ω2u(rR, t). A force proportional

to displacement λ.u(rR, t) (or acceleration λ′.Γ(rR, t) ) can be applied.

In this 3 point problem, E(θE , φE), R(θR, φR),M(θM , φM ), the time-reversed signal will be

successively calculated at any point at the surface of the earth M and then, at the particular case of

point E (where M = E). Since the problem is linear, the time-reversed field at any point M due a

distribution of receivers will be also consider . Finally, the effect of a random distribution of sources

will be calculated. At the receiver point rR, the signal is found to be the result of the convolution of

the Green function with the source time function f(t) ∗GER(t). The time reversed seismogram in rR

is f(−t) ∗GER(−t) and in point rM the wavefield S(rM , t) will be:

S(rM, t) = f(−t) ∗GER(−t) ∗GRM (t) (14)

or in terms of deformation:

Ξ(rM, t) = M(−t) ∗EER(−t) ∗ERM (t)

where EER = 1/2(∇+∇T )GER

So the time-reversed signal is the cross-correlation of 2 Green functions. As already noted by

Draeger & Fink (1999) the signal in point rE is the autocorrelation function of GER which is known

to have its maximum at t = 0. At other points M, this signal is the cross-correlation of the 2 Green

Functions GER and GRM .

GER(−t) ∗GRM (t) =∫dτGER(τ + t)GRM (τ)

This result can be used as well to understand the cross-correlation of noise signals in 2 different

stations (see section 3.2). This correspondence between cross-correlation and time reversal was al-

ready noted in several papers since Draeger & Fink (1999) (see for example Derode et al. (2003)). But

this property has been primarily used for getting the Green function between 2 points, not for studying

the source, as done for example in seismology by Gajewski & Tessmer (2005) for local studies and by

Larmat et al. (2006), Larmat et al. (2008) at regional and global scale.

For the elastic case, we can send back the expression of the displacement found in section 2.1 and

following the approach of Capdeville (2005), all expressions are written in the Fourier domain. For

a point source in time F0(rE, ω) = eFδ(rE), the displacement field in the Fourier domain resulting

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from the forward propagation is according to equation (8):

uR(ω) =∑K

|KR〉〈KE|FE1

−ω2 + ω2k

(15)

And in the time domain (delta function in time): uR(t) =∑

K |KR〉〈KE|FEsinωktωk

For a point force Heaviside function H(t):

uR(ω) =∑K

|KR〉〈KE|FE1

iω(−ω2 + ω2k)

In point M , the time-reversed field for a delta-function in time is:

SM (ω) =∑J

|JM〉〈JR|uR(ω)1

(−ω2 + ω2j )

Usinjg the expression of uR(ω), it is found for a source delta function that:

SM (ω) =∑J

∑K

|JM〉〈JR|KR〉〈KE|FE1

(−ω2 + ω2k)

1(−ω2 + ω2

j )

It is necessary to bear in mind that in this expression, 〈JR|KR〉 does not correspond to the inner

product as defined in equation (5) but only to the scalar product of 2 eigenfunctions at point R. Be

careful since the convolution F ∗GER(−t) ∗GRM (t) is written from left (source) to right (point M)

whereas equation (15) is written in the opposite sense, which might induce some confusion. In terms

of cartesian coordinates, this equation can be written:

SαM (ω) =∑

J

∑K Σβγ

|JαM〉〈JβR|K

βR〉〈K

γE|F

γE

(−ω2+ω2k)(−ω2+ω2

j )

=∑J

Σβ|JαM〉〈JR

β|(−ω2 + ω2

j )︸︷︷︸BαβRM (ω)

∑K

Σγ|Kβ

R〉〈KγE|F

γE

(−ω2 + ω2k)︸︷︷︸

CβER(ω)

For sake of simplicity, let us continue the calculation by assuming that the source function in E is

a delta function. The case of a source Heaviside function in E is detailed in Appendix A. Sums over

J and K can be separated:

SM (ω) =∑J

|JM〉〈JR|(−ω2 + ω2

j )︸︷︷︸BRM (ω)

∑K

|KR〉〈KE|FE

(−ω2 + ω2k)︸︷︷︸

CER(ω)

(16)

Since the spectrum of the time-reversed field SR is the product of the spectra of 2 functions BRM

and CER, the field in the time domain is the convolution of these 2 fonctions. Since the time-reversed

of uR(t) is sent back, the cross-correlation between the 2 functions must be calculated (the cross-


correlation is the convolution of the time-reversed signal) . The Fourier transform of a time reversed

function f(−t) is F (−ω). Consequently,

BRM (t) =∑J

|JM〉〈JR|sinωjt

ωjH(t)

CER(t) =∑K

|KR〉〈KE|FEsinωkt

ωkH(t)

And finally, if the time-reversed field is calculated for time duration ∆T = [t1; t2] between times

−t2, and −t1, we introduce the time variable tR so that refocusing occurs at tR = 0.

SM (t) = BRM (t) ∗CER(−t) =∑J

∑K

|JM〉〈JR|KR〉〈KE|FEωk

1ωj

∫ t2

t1

dτsinωjτsinωk(tR + τ)︸︷︷︸=jk

(17)

The case with a Heaviside source time function can be done in a similar way (see Appendix A)

2.2.1 Calculation of =jk

=jk =∫ t2

t1

dτsin(ωk(tR + τ))sin(ωjτ)

=jk = −12sinωktR

∫ t2

t1

dτ [sin(ωk+ωj)τ+sin(ωk−ωj)τ ]+12cosωktR

∫ t2

t1

dτ [cos(ωk+ωj)τ−cos(ωk−ωj)τ ]

The terms with (ωj +ωk)τ are negligible since 1/(ωj +ωk)� ∆T , provided that a long enough time

series is used.=jk = 1

2∆TcosωktR if ωk = ωj

=jk = 12sinωktRωk−ωj (cos(ωk − ωj)t2 − cos(ωk − ωj)t1) if ωk 6= ωj

+12cosωktRωk−ωj (sin(ωk − ωj)t2 − sin(ωk − ωj)t1)

The first term is dominant provided that ∆T is long enough. The second term (when ωk 6= ωj) is

only important when this difference is small. The time 1∆ω corresponds to the Heisenberg time. At very

long period, the effect of heterogeneities (and rotation) is small, and their effect can be neglected. For

the earth normal modes, 1∆ω ≈ 2.103s for eigenperiods around 200s, which means that if δT � 1

∆ω ,

the non-diagonal term of Ijk will tend to interfere destructively and the contribution of non-diagonal

terms will be small compared to the diagonal contribution. However, that is only a zeroth-order ap-

proximation valid at very long-period. These expressions make it possible to incorporate the coupling

between different modes, but this study is beyond the scope of this paper (see Capdeville et al. (2000)

for a complete discussion of such a coupling). Therefore, if ∆T � 1∆ω , =jk ≈ 1

2δkj∆TcosωktR.


S(rM , tR) =∑J

1ωj|JM〉〈JR|

∑K

|KR〉〈KE|F (rE)ωk

12δkj∆TcosωktR (18)

2.2.2 The observation point is at the source point E: Cavity Equation

The time reversal of acoustic waves (scalar case) has been extensively investigated by Draeger & Fink

(1999). If FEδ(t) = F , the convolution F.GER(−t) ∗GRE(t) if J = K, according to equation (18),

is:

SE(tR) =∑K

1ω2k

|KE〉〈KR|KR〉〈KE|12

∆TcosωktR

For acoustic waves, the bras and kets are scalar functions. It can be easily seen that this expression

is the same as the convolution of GEE(−t) by GRR(t) if J = K. As a matter of fact, F.GEE(t) =∑J

1ωk|KE〉〈KE|F sinωkt

ωk, GRR(t) =

∑J

1ωj|JR〉〈JR|

sinωjtωk

, and

F.GEE(−t) ∗GRR(t) =∑J

1ωj|JR〉〈JR|

∑K

1ωk|KE〉〈KE|F

1ωk=jk

Since there is no difference for scalar eigenfunctions between bra and ket, for J = K, Draeger & Fink

(1999) obtained the remarkable cavity equation for acoustic waves:

F.GER(−t) ∗GRE(t) = F.GEE(−t) ∗GRR(t) =∑K

1ωk2|KR〉〈KR|KE〉〈KE|

12

∆TcosωktR

In the elastic case, the equivalent of the cavity equation is more complex, since it is necessary to

take account of the tensorial character of Green functions and of the difference between bra and ket.

Considering the equation (18) and if point M is located in E,

SαE(tR) =∑K

∑J

1ωkωj

Σβγ |JαE〉〈JβR|K

βR〉〈K

γE|F

γE =jk

If only the dominant diagonal elements of =jk is taken into account, the sum over J can be sup-

pressed and by reordering the different terms:

SαE(tR) =∑K

Σβγ1ω2k

〈KβR|K

βR〉|K

αE〉〈K

γE|F

γE =kk (19)

If only the convolution product CC1 = GER(−t) ∗GRE(t) is considered:

CCαγ1 = GER(−t) ∗GRE(t)(tR) =∑K

Σβ1ω2k

〈KβR|K

βR〉|K

αE〉〈K

γE| =kk (20)

So, the effect of the source is separated from the effect of the receiver and the 2 Green functions

GEE(t) and GRR(t) can be introduced:


GαγEE(t) =∑K

|KαE〉〈K

γE|H(t)

sinωkt

ωk

GβζRR(t) =∑J

|JβR〉〈JζR|H(t)

sinωjt

ωj

The cross-correlation CC2 of GEE(−t) ∗ GRR(t) involves the same integral =kj(t) as in equation

(16)

CCαβγζ2 = GαγEE(−t) ∗GβζRR(t) =∑K

∑J

|JβR〉〈JζR|

ωj

|KαE〉〈K

γE|

ωk=kj(t) (21)

This expression of CCαβγζ2 can be contracted by taking ζ = α in order to have a cross-correlation

similar to CCαγ1 , and by summing over β,

CCβγ2 = GαγEE(−t) ∗GβαRR(t) =∑K

∑J

Σα|JβR〉〈JαR|

ωj

|KαE〉〈K

γE|

ωk=kj(t)

If J = K

CCβγ2 = GαγEE(−t) ∗GβαRR(t) =∑K

Σα1ω2k

|KβR〉〈K

αR|Kα

E〉〈KγE|=kk(t)

In order to facilitate the comparison of CC1 and CC2, we interchange the role of E and R, α and β,

we obtain:

CCαγ2 = GβγRR(−t) ∗GαβEE(t) =∑K

Σβ1ω2k

|KαE〉〈K

βE|K

βR〉〈K

γR|=kk(t) (22)

The expression of CCαγ2 in equation (22) looks similar to the one in equation (20), but it is

not. Actually, the scalar product 〈KβR|K

βR〉 appears in equation (20) whereas it is the scalar product

〈KβR|K

βE〉 which comes up in equation (22). However, if we apply to CCαγ1 and CCαγ2 a vector force

proportional to |KR〉, then the same expression is obtained:

CCαγ1 |KγR〉 = CCαγ2 |K

γR〉 =

∑K

Σβγ1ω2k

|KαE〉〈K

βR|K

βR〉〈K

γE|K

γR〉=kk(t)

If we reverse the roles of E and R, we have to apply a vector force 〈KR|.

Equation (20) has the interesting property that it displays the contribution of one receiver and it

can be easily generalized to a complete global distribution of earthquakes as it will be seen in section

3.


2.2.3 Degenerate modes in azimuthal number m

In order to go further, the expression of the eigenfunction provided in equation (2) can be taken into

account. Some particular cases will be detailed when the eigenfunctions are degenerate.∑

k is now

written explicitly∑

k =∑

n,`,m as well as the eigenvector |kR〉 = nD`(r)Y m` (θ, φ)

In the general case, the equation (18) can be rewritten

SM (tR) =∑

n′,`′,m′n′D`′Y

m′`′ (rM)n′D∗`′Y

m′∗`′ (rR)

∑n,`,m

nD`Ym` (rR)nD∗`Y

m∗` (rE)FE

=jknω`n′ω`′

(23)

For example, the vertical component of the displacement, assuming also that the force is also a

vertical force uK(r, θ, φ) = nU`(r)Y m` (θ, φ). In that case, the addition theorem (or Closure relation-

ship) for spherical harmonics (equation 5) can be used:

SM (tR) =∑

n′,`′,m′n′U`′Y

m′`′ (rM)n′U`′Y

m′∗`′ (rR)

∑n,`,m

nU`Ym` (rR)nU`Y m∗

` (rE)FE=jk

nω`n′ω`′

=∑n′,`′

n′U`′(rM )P 0`′(cos∆RM )n′U`′(rR)

∑n,`

nU`(rR)P 0` (cos∆ER)nU`(rE)FE

=jknω`n′ω`′

For the 3- components of displacement, the same kind of calculation can be performed, but the

vectorial operators nD` impose to use the more general addition theorem as defined by equation

(6) (Capdeville et al. (2000)), which introduces generalized Legendre functions PNN′

` (cos∆) with

Nε[−2,+2] The source term nD∗`Ym∗` (rE)FE can be transformed if the source force is the gradient

of the moment tensor and is equal to nE`M

SM (tR) =∑n′,`′

n′D`′(rM)P 0`′(cos∆RM )n′D∗`′(rR)

∑n,`

nD`(rR)nE∗`M(rE)P 0` (cos∆ER)

=jknω`n′ω`′

(24)

For sake of simplicity, the notation of Woodhouse & Girnius (1982) can be used.

BARBARA

According to Romanowicz (1987), it is possible to apply a phase stationary approximation. The

multiplication of Legendre polynomials P 0`′(cos∆RM ) by P 0

` (cos∆ER) is maximum for ∆ = 0 or π.

It means that points E, R and M must be on the same great circle in order to provide a maximum

contribution to the time-reversed displacement (see Figure 1). In addition, since in equation (24), the

contributions of each eigenfrequency will tend to cancel out and all sinusoids will only be in phase

(and to interfere constructively) when tR = 0.

There will be also some non null contributions to time-reversed seismograms for some stationary

phases, corresponding to particular group velocities. In that case some ghosts wavetrain will come up


E R

M

ΔER

ΔEM ΔRM

N

Figure 1. Earthquake epicenter E- receiver R geometry. M is the observation point of the time-reversed field

as demonstrated by Capdeville et al. (2000). A simple way to get rid of these ghost trains is to send

back as many seismograms as possible.


3 FROM A 3-POINT PROBLEM TO A MULTIPLE POINT-PROBLEM

According to equation (18) and (24), it is very easy to sum the contributions of many receivers (or

many sources). A global coverage of the earth by seismic stations or seismic sources will be considered

3.1 Time-reversal Imaging: 1 source, n receivers

Let us consider that the earth is covered by a global seismic network at points rRi, i = (1, nR). Due

to the fact that the Earth covered by 2 thirds of oceans and that most broadband seismic stations are

installed in northern hemishere, the spatial coverage is obviously uneven. How can we take account

of the non-uniform spatial coverage of the Earth? It is necessary to weight the contribution of each

station to the time-reversed field. The simplest choice consists in weighting each station by a surface

ai such that∑nR

i=1 ai = Searth, Searth being the surface of the Earth.

The time-reversed field at any point M at the surface of the Earth is then:

SM (tR) =∑i

aiS∆T (tR)i =∑i

ai∑k

∑j

1ωkωj

|JM〉〈JR|KR〉〈KE|FE =jk

If we assume that the spatial coverage by receivers is good enough, the sum∑

i ai can be replaced

by the integral∫earth dS. Since |KR〉 = nD`(r=0)Y m

` (θ, φ), we can integrate over the whole surface

of the Earth:nR∑i=1

ai〈JR|KR〉 =∫dS nD`(r=0)Y m

` (θ, φ)n′D`′(r=0)Y m′l′ (θ, φ)nD`(r=0)n′D`′(r=0)δ``′δmm′

The eigenfunctions of receiver stations are considered at the surface of the Earth r = 0.

Finally, for M = E:

SE(tR) =∑k

1ω2k

|KE〉〈KE|FEnD`2(0)

∆TcosωktR2ω2

k

This expression is maximum for tR = 0 since it is the only time where all cosine terms add

constructively. Alternative demonstration by using equation (24), where point E is at the pole:∫dSP 0

`′(cos∆RE)P 0` (cos∆ER) =

12`+ 1

δ``′

SE(tR) =∑k

1nω`2

nD`(rE)(nD`(rE).FE)nD`2(0)

12

∆Tcosnω`tR

At any point M at the surface of the Earth, the calculation done in section 2.1 is still valid, except

that the eigenfunctions in point rM must be introduced.

S(rM , tR) =∑k

|KM〉〈KE|F (rE)ω2k

D`n(0)D`

n(0)12

∆TcosωktR (25)


Barbara: it must be shown now that the sum of the products |K(rM)〉〈K(rE)|ω2k

is in average close to

zero except at point E. May be, it is obvious.

3.2 Cross-correlations: n sources, 2 receivers

- Random distribution of seismic noise sources

Due to time reversal invariance and spatial reciprocity, it can be imagined that instead of a distribu-

tion of stations in rR, we have a distribution of random sources. In equations of the forward problem,

the roles of point E and R are symmetric, we can consider noise sources at point rR, with a random

distribution. Due to the linearity of the problem it is possible to stack all contributions from these dif-

ferent sources, and the equation (23) is very similar to the Green function defined at the end of section

(1). The 2 Green functions GER(t) and GRM (t) are given by equation (10) and the corresponding

seismograms of displacement due to a force system F γE are:

uβR(t) = ΣγFγEG

γβER(t) =

∑K Σγ |Kβ

R〉〈KγE|F

γEH(t) sinωktωk

uαM (t) = ΣγFγEG

γαEM (t) =

∑J Σγ |JαM〉〈J

γE|F

γEH(t) sinωjtωj

or in terms of velocity for t > 0: uβR(t) = ΣγFγEG

γβER(t) =

∑K Σγ |Kβ

R〉〈KγE|F

γEH(t)sinωkt

uαM (t) = ΣγFγEG

γαEM (t) =

∑J Σγ |JαM〉〈J

γE|F

γEH(t)sinωjt

The cross-correlation CC of GER(t) ∗ GEM (t) involves the same integral =kj(t) as in equation (17)

CCβα(τ) =∫ t2

t1

uβ?R (t)uαM (t+ τ)dt =∑J

Σζ |JαM〉〈JζE|F

ζE

∑K

ΣγFγE |K

γE〉〈K

βR|=jk(τ)

Let us consider the simple case where only one component for F γE is excited (for example the

vertical component) and equal to 1, the sum over γ disappear:

CCβα(τ) = F γ2E

∑K

∑J

|JαM〉〈JγE|K

γE〉〈K

βR|=jk(τ) (26)

And for the dominant term, when J = K:

CCβα(τ) = F γ2E

∑K

|KαM〉〈K

γE|K

γE〉〈K

βR|=kk(τ) (27)

The cross-correlation CCβα(τ) is the Green function of velocity as defined in equation (11) if and

only if 〈KγE|K

γE〉 = 1. If sources are only located at the surface of the Earth, this condition is not

fulfilled.

The expression in equation (26) is very similar to equation (18). The role of point R is played by

point E, but we have now to consider a 2x2 tensor instead of a vector. We can apply twice the addition


theorem and an expression similar to equation (24) is found implying P 0` cos∆ME and P 0

` cos∆RE .

By using the same arguments as in section 2.2.3, it can be demonstrated that the most important

contributions to CCβα(τ) are provided by sources located along the great-circle between R and M .

The same kind of calculation as in section 3.1 can be performed as well. If a homogeneous dis-

tribution of sources is assumed, with a density distribution n(θ, φ) associated with a surface an(θ, φ)

such that∫S n(θ, φ)sinθdθdφ = 1, and if only the case J = K is considered, then it is possible to

integrate:

CCβα(τ) = F γ2E

∑n,`,m

∑n′,`′,m′

〈(n, `,m)βR|(n′, `′,m′)αM〉(

∫Sn(θ, φ)sinθdθdφ〈(n′, `′,m′)γE(|n, `,m)γE〉)=jk(τ)

By using orthonormalization of eigenfunctions:

CCβα(τ) = F γ2E

∑n,`,m

∑n′,`′,m′

〈(n, `,m)βR|(n′, `′,m′)αM〉nD

γ` (rE)n′Dγ

`′(rE)=jk(τ)

and the addition theorem for J = K:

CCβα(τ) = F γ2E

∑n,`

nDβ` (rR)nDα

` (rM)P 0` (cos∆RM )nD

γ2` (rE)=kk(τ)

It is directly proportional to the Green function of velocity between points M and R. In isotropic

media if β 6= α CCβα(τ) = 0

These calculations demonstrate that time reversal (adjoint field) approach, and cross-correlations

of seismograms are directly related and that the same kind of calculations apply in both cases.


4 DISCUSSION AND APPLICATIONS

4.1 Voronoi cells

We develop in previous sections the whole theory of time-reversal method by using the normal mode

formalism. An important result is that the focusing at the epicenter is largely dependent on the weight-

ing of stations as defined in section 3.1. If the contribution of the different stations is not correctly

taken into account, the focus point will not be correctly located in time and space. This weighting is

particularly important when the station distribution is non-uniform, which is the case for the coverage

of Earth. Most broadband stations are located in continents and primarily in the northern hemisphere.

A simple way to take account of the uneven distribution is to use a Voronoi tessellation. This kind of

mapping at the surface of the Earth was implemented for example for global tomography by Debayle

& Sambridge (2004) and for time-reversal earthquake localization by Larmat et al. (2006). An exam-

ple of Voronoi cell distribution is shown in Figure 2 for a global distribution of 115 stations. As it can

be seen the surface distribution is very heterogeneous and some stations in the Pacific, Atlantic and

Indian oceans may have a very large distribution.

Figure 3 shows an example of focusing of time-reversed seismograms (an example of seismogram

is given on figure 3a) for an isotropic source located close to Sumatra obtained by using the 115

stations of Figure 2. Figure 3c shows a perfect refocusing when using the Voronoi cell tessellation

and Figure 3e when no weighting is applied. In this case the localization in space is very poor and the

radiation pattern (a circle for an isotropic source) is deformed.

4.2 One-bit discretization

In most calculations, we have used arguments based on the stationnary phase approximation. Figure

3b shows binarized seismograms which suppress the information on the amplitude of seismograms.

Binarization of seismograms is routinely used when calculating Green functions by cross-correlation

between 2 stations (Shapiro et al. (2004)). When binarized seismograms are time-reversed with the

appropriate weighting scheme, the focusing is almost as good as the focusing obtained by time-reversal

of complete seismograms. This experiment demonstrates that the information is primarily conveyed

by the phase of seismograms. It is not really a surprise, since the location of the earthquake and the

focusing time correspond to a constructive interference of the whole parts of seismograms.


Figure 2. Voronoi cells used for time-reversal of Sumatra-Andaman earthquake for a global network of 115

stations. Each cell is associated with a station. The area of each cell reflects the global coverage of the Earth.

4.3 Inversion of the force system

According to equation (25), it is possible to obtain a linear system of equations which enables to

retrieve the vector force FFE. We only remind the results. The Green function components between

points rE and rR are equal to

Gβγ(rE, rR, t) =∑

K |KβR〉〈K

γE|

H(t)sinωktωk

and uβR(ω) =∑

K

∑γ |K

βR〉〈K

γE|

F γEωk=jk(ω)

At point source rE, by reordering the different terms, and taking J = K:

SαE(tR) =∑K

∑βγ

|KαE〉〈KR

β|KβR〉〈KE

γ |F γE=kk(tR)ω2k

If F γE is a scalar force, 〈KEγ |F γE |JαE〉 = F γEu

γK(rE)uαJ (rE), and for an uniform distribution of stations

Ri at the surface of the Earth, weighted by using for example a Voronoi tesselation, which means that

the contribution of each station is associated with an area ai,∑i ai〈JR

β|KβR〉 ≈

∑β n′Dβ∗

`′ (0)nDβ` (0)

∫S Y

m′∗`′ Y m

` sinθdθdφ =∑

β n′Dβ∗`′ (0)nD

β` (0)δ``′δmm′


[a]

0 2000 4000 6000 8000 10000Time (s)

-4e+14

-2e+14

0

2e+14

4e+14

Am

plitu

de (

m)

[b]

[c] [d]

[e: wrong weighting Huong ]

Figure 3. One-bit normalization versus Amplitude time-reversal experiment. a: real seismogram. b: one-bit

seismogram. c: real seismograms and weighting by Voronoi tessellation. d:one-bit seismograms and weighting

by Voronoi tessellation. e: real seismograms and no weighting


a) Strike-slip fault b) Inverse fault c) Normal fault

Figure 4. Sensitivity of the time-reversed field to the focal mechanism

assuming J = K

SαE(tR) =∑γβ

F γE

∑n`

nDβ2` (0)nD

γ` (rE)nDα

` (rE)||Y m` (θE , φE)||2=kk(tR)

ω2k

(28)

All oscillations are in phase for tR = 0, which explains why we get focusing of energy at that

time.

BARBARA: by using arguments of stationary phase, SαE ,∑

n` ....

It is then obtained a matrix relationship S(tR=0) = QF and it is possible to retrieve the force

system by inversion of the matrix Q. This linear system with respect to the different components FE ,

can be easily inverted. The coefficients of the matrix Q can be calculated by normal mode summation.


4.4 Time reversal of deformation- Moment tensor inversion

Similarly, it is possible to calculate the spatial derivatives of the Green function at the source point,

such that:

Gβγ,m(rE, rR, t) =∑K

|KβR〉ε

K?γm(rE)H(t)(

1− cosωktω2k

)

where the deformation tensor components are given by εγm(rE) = 12(∂|K

γE〉

∂ξm+ ∂|Km

E 〉∂ξγ

).

In terms of operators, the operator deformation is E = 12(∇+∇T ).

And uβR(ω) =∑

K

∑γ |K

βR〉

12(∂〈K

γE|

∂ξm+ ∂〈Km

E |∂ξγ

)Mγm

Similarly,Gαβ(rM, rR, ω) =∑

K |KαM〉〈K

βR|=(ω), andGαβ,m(rM, rR, ω) =

∑K |Kα

M〉εK?βm=′(ω)

If the spatial derivatives of the Green functions are time-reversed, an homogeneous expression with a

2nd-order tensor is obtained, which will give access the seismic moment tensor (Kawakatsu & Mon-

tagner (2008))

Ξ(rM , t) = M(−t) ∗G′ER(−t) ∗G′RM (t)

where G′ER and G′RM are the deformations associated with Green functions GER and GRM . In

terms of cartesian coordinates:

ΞαmM (ω) =∑J

∑K

∑βγ

|JαM〉iω(−ω2 + ω2

j )〈JβR|

|KβR〉

(−ω2 + ω2k)

12

(∂〈Kγ

E|∂ξm

+∂〈Km

E |∂ξγ

)Mγm

Actually in order to ensure the symmetry of the calculation (Kawakatsu & Montagner (2008)) such

that ΞαmM = ΞmαM , we have to consider not only Gβγ,m but also Gmβ,γ , since the contribution of Mmγ

must be the same as the contribution of Mγm

Gmβ,α(rM, rR, t) =∑K

εK?βα |KmM〉=′(ω)

and

ΞmαM (ω) =∑J

∑K

∑βγ

|JmM〉

iω(−ω2 + ω2j )〈JβR|

|KβR〉

(−ω2 + ω2k)

12

(∂〈Kγ

E|∂ξα

+∂〈Kα

E|∂ξγ

)Mγα

In the time domain:

ΞαmM tR) =∑J

∑K

∑βγ

|JαM〉〈JβR|K

βR〉

12

(∂〈Kγ

E|∂ξm

+∂〈Km

E |∂ξγ

)Mγm=′jk(tR)

The same theorems (addition theorem, orthonormalization of eigenfunctions) as for retrieving the


force system can be applied:

ΞαmM tR) =∑n`

nD`α(rE)P 0

` (cos∆ER)||nD`(0)||2P 0` (cos∆ER)nD`

γ(rE)EγmMγm

As in equation (28) for the force system, it is found a linear system with respect to the moment

tensor components. Appendix B details how to retrieve the 6 different components of and the practical

application is beyond the scope of this paper.

5 CONCLUSIONS

ACKNOWLEDGMENTS

A number of colleagues have helped with suggestions for the improvement of this material and I

would particularly like to thank Hitoshi Kawakatsu, Earthquake Research Insititute of Tokyo for his

criticisms and corrections.

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Table A1. The coefficients RkN and SkN

N RkN = RNk (0, 0) SkN = SNk (0, 0)

0 k0vr −k0(∂rUMrr + 12F (Mθθ +Mφφ))

±1 k1(V ± iW )(∓vθ − ivφ) k1(X ∓ iZ)(Mrθ − iMrφ)

±2 k2(V ∓ iW )((Mθθ −Mφφ)∓ iMθφ)

F = r−1E (2U − `(`+ 1)V ) kn = 1

2n

(2`+14π

(`+n)!(`−n)!

)1/2

X = ∂rV + r−1E (U − V )

Z = ∂rW + r−1E W

Mθθ,Mφφ... are components of the moment tensor, vr, vθ, vφ are components of the instrument vector.

APPENDIX A: TIME REVERSAL WITH A HEAVISIDE FUNCTION: CALCULATION OF

=′(TR)

For a source Heaviside function H(t):

uR(ω) =∑k

|kR〉iω(−ω2 + ω2

k)〈kE|FE (A.1)

SM (ω) =∑

j

∑k

|jM〉(−ω2+ω2

j )〈jR| |kR〉

iω(−ω2+ω2k)〈kE|FE for a Heaviside function in E. Sums over j

and k can be separated:

SR(ω) =∑j

|jM〉〈jR|(−ω2 + ω2

k)︸︷︷︸BRM (ω)

∑k

|kR〉〈kE|FE

iω(−ω2 + ω2k)︸︷︷︸

CER(ω)

(A.2)

For a Heaviside function, the function CER is

CER(t) =∑k

|kR〉〈kE|1− cosωkt

ω2k

H(t)

SM (ω) = BRM (t) ∗CER(−t) =∑j

∑k

|jM〉〈jR|kR〉〈kE|FEω2k

1ωj=′jk(tR) (A.3)

The integral=′jk(tR) =∫ t2t1dτ−cosωk(tR+τ)sinωjτ = 1

2sinωktR∫ t2t1dτ(cos(ωj+ωk)τ−cos(ωj−

ωk)τ)+ 12cosωktR

∫ t2t1dτ(sin(ωj +ωk)τ−sin(ωj−ωk)τ). The terms with (ωj +ωk)τ are negligible

since 1/(ωj + ωk)� ∆T

(A VERIFIER)

The integral=jk(tR) = −∫ t2t1dτsinωk(tR+τ)sinωjτ ≈ 1

2∆TsinωktRδjk. It is a similar integral

as previously calculated in the scalar case.


APPENDIX B: INVERSION OF SEISMIC MOMENT TENSOR

Coordinates x (resp y, z) corresponds to 1 (resp 2, 3). For order 2 tensors, such as moment tensor Mij ,

the classical transformation of coordinates is used 12 → 6, 13 → 5, 23 → 4. So the 6 independent

components of Mij can come up explicitly. In case a force system, the order 2 Green tensor is square.

It is no longer the case when considering the deformation tensor. Following Kawakatsu & Montagner

(2008), we can write:

u1 = G11,1M1 +G1

2,2M2 +G13,3M3+ (G1

2,3 +G13,2)M4 + (G1

1,3 +G13,1)M5 + (G1

1,2 +G12,1)M6

= G11M1 +G1

2M2 +G13M3+ G1

4M4 +G15M5 +G1

6M6

u2 = G21,1M1 +G2

2,2M2 +G23,3M3+ (G2

2,3 +G23,2)M4 + (G2

1,3 +G23,1)M5 + (G2

1,2 +G22,1)M6

= G21M1 +G2

2M2 +G23M3+ G2

4M4 +G25M5 +G2

6M6

u3 = G31,1M1 +G3

2,2M2 +G33,3M3+ (G3

2,3 +G33,2)M4 + (G3

1,3 +G33,1)M5 + (G3

1,2 +G32,1)M6

= G31M1 +G3

2M2 +G33M3+ G3

4M4 +G35M5 +G3

6M6

Conversely,

M1 = G11u1 + G2

1u2 + G31u3

G11,1 u1 + G2

1,1 u2 + G31,1 u3

M2 = G122u1 + G2

22u2 + G322u3

G12,2 u1 + G2

2,2 u2 + G32,2 u3

M3 = G13u1 + G2

3u2 + G33u3

G13,3 u1 + G2

3,3 u2 + G33,3 u3

M4 = G14u1 + G2

4u2 + G34u3

(G12,3 + G1

3,2)u1 + (G22,3 + G2

3,2)u2 + (G32,3 + G3

3,2)u3

M5 = G15u1 + G2

5u2 + G35u3

(G11,3 + G1

3,1)u1 + (G21,3 + G2

3,1)u2 + (G31,3 + G3

3,1)u3

M6 = G16u1 + G6u2 + G3

6u3

(G11,2 + G1

2,1)u1 + (G21,2 + G2

2,1)u2 + (G31,2 + G3

2,1)u3

So now, it is easy to replace the expressions of the moment tensor as defined in text.

This paper has been produced using the Blackwell Scientific Publications GJI LATEX2e class file.

time-reversal method and cross-correlation techniques by normal mode theory: a three-point problem

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