Geophys. J. Int. (2008) 175, 686–688

doi: 10.1111/j.1365-246X.2008.03926.x

Time-reversal seismic-source imaging and moment-tensor inversion
Hitoshi Kawakatsu and Jean-Paul Montagner∗
Earthquake Research Institute,
The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032 Japan. E-mail: hitosi@eri.u-tokyo.ac.jp

Accepted 2008 July 23. Received 2008 June 18; in original form 2008 April 25

SUMMARY
The time reversal operation in seismic source estimation is considered. We show that thetime reversal operation, equally the adjoint operation, for seismic source imaging gives anapproximate solution to more conventional seismic source inverse problem through the ‘happy
approximation’ underlined by Claerbout. Practical applications of such methods in a longperiod range to monitor earth’s activities in realtime are also discussed.
Key words: Theoretical seismology.

and in vector form,

GJI Seismology

1 I N T RO D U C T I O N
Availability of high-performance computing and numerical techniques have brought a new class of analysis methods in seismology,
and so-called adjoint or time reversal operations are getting greatinterests in global seismology (Tromp et al. 2005; Larmat et al.
2006). For example, Tromp et al. (2005) drew connections among
different techniques, such as seismic tomography, adjoint methods,and ‘banana-doughnut’ kernels, and demonstrated how waveform
tomography for 3-D earth structure may be performed based on
numerically simulated waveform synthetic techniques (Komatitsch& Tromp 2002; Tsuboi et al. 2003). By analyzing broadband seismic recording of the global network for the 2004 great Sumatra
earthquake, Larmat et al. (2006) showed the time reversal source
imaging may be feasible on global scale. This short note is motivated by the work of Larmat et al. (2006), and draws connections
between the time reversal source imaging and more conventional
moment tensor inversion techniques. For the purpose, we focus on
the distinction between ‘inverse’ and ‘adjoint’ methods introducedby Claerbout (2001).
2 S E I S M I C S O U R C E E S T I M AT I O N
Green’s function
For simplicity, we start with the excitation problem by a single force.
The displacement at x excited by a single force point source
located at ξ may be given as
u n (x, t) = G ni (x, t; ξ , 0) ∗ f i (t)(1)

where G ni and f i are the Green’s function and single force source
time function respectively, and the multiplication * denotes convolution (Aki & Richards 1980).
Equivalently in frequency domain, we haveu n (x, ω) = G ni (x, ω; ξ ) f i (ω)∗ On leave from Institut de Physique du Globe de Paris, Paris, France.

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(2)

u=G· f(3)

Time reversal operation
For the observation waveform at a point x, d i (t), the time reversal(TR) operation for ‘wavefield’ at ξ may be written as
TRn (t) = G ni (ξ , t; x, 0) ∗ di (t0 − t),(4)where TR n (t) denotes the n-th component of the time-reversed seismogram at ξ and t 0 is an arbitrary reference time; in frequency
domain it becomesTRn (ω) = G ni (ξ , ω; x)d̄i (ω)eiωt0 ,(5)

where d̄ denotes complex conjugate of d, and the sign conventionof the Fourier transform is the same as in Aki & Richards (1980).

Source inversion
In source inversion, if the source location is assumed to be known,
we search f (t) such that the difference between predicted waveforms and observed ones are minimized. If we use the ordinary
least-squares criterion, the solution f̂ in frequency domain may be
given asf̂ (ω) = (G ∗ G)−1 G ∗ d(6)where G ∗ is the conjugate transpose (or adjoint transpose) of G.
Also as the intermediate equation, the normal equation, we haveG ∗ G · f (ω) = G ∗ d(7)

The right hand side of this equation may be written as
rhs = Ḡ in (x, ω; ξ )di (ω) = Ḡ ni (ξ , ω; x)di (ω)(8)

where we used the spatial reciprocity of Green’s function. Comparing with (5), this is a phase-shifted complex conjugate of TR n (ω) in(5). In other words, the time reversal operation is nothing different
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Time-reversal seismic-source imaging
from the phase-shifted cross-correlation between the observed displacement and the Green’s function (e.g. Draeger & Fink 1997). As
we employ only the spatial reciprocity of Green’s function to draw
the connection between (5) and (8), it should hold for anelastic caseas well (Dahlen & Tromp 1998).
Imaging vs. inversion(9)

where α is some real proportionality constant. In this case, theleast-squares solution (6) can be also approximated asf̂ T R (ω) ≈ α −1 G ∗ d.

(10)

Here following the idea of Claerbout, we define this type of solution as ‘imaging’, while the conventional least squares solution as‘inversion’.
Multi-station caseWe can easily extend the above argument for observations with multiple stations. In many seismological source estimation problems,(9) appears to be a good approximation (e.g. Kawakatsu 1996, forthe moment tensor of deep earthquakes with a sufficient station coverage), and ‘source imaging’ using (10) may be possible. As (10)
is just a phase-shifted complex conjugate of the time reversal, now
we understand why the time reversal appears to work (e.g. Larmat
et al. 2006).
In case the point source approximation is appropriate, the convergence (goodness of fit) can be measured by the ratio of the crosscorrelation of observed (d) and predicted (G f̂ ) waveforms to theobserved waveform energy (d 2 ),
R=

(11)
d2
which is equal to the variance reduction for the least-squares solution. For source imaging, this may be approximated byf̂ 2T R,(12)d2
and a TR-image which gives the largest energy may be consideredas the best one. For a spatially extended source such as the 2004
Sumatra event, more sophisticated measure may be used; nevertheless, (12) should still give some idea of the spatial extent of theseismic source (e.g. Larmat et al. 2006).
Moment tensor caseIn case of a more general seismic source represented by the moment
tensor M i j (t), the displacement is now given as
u n (x, t) = G ni, j (x, t; ξ , 0) ∗ Mi j (t),(13)

but the whole argument above can be repeated if we use G ni, jinstead of the Green’s function itself and if we treat six independent
components of symmetric M i j as a vector. So G ni, j should be usedrather than G ni to do TR source imaging. It should be noted thatthe spatial derivative , j is applied at the source location ξ ; i.e. inC 2008 The Author, GJI, 175, 686–688
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TRi j (t) = E inj (ξ , t; x, 0) ∗ dn (t0 − t)(14)

TRi j (ω) = E inj (ξ , ω; x) d̄n (ω) eiωt0 .

(15)

Here
is the strain field due to an unit force directed to nth
direction, and defined as∂G in (ξ , t; x, 0) ∂G jn (ξ , t; x, 0)+E inj (ξ , t; x, 0) = ci j
, (16)∂ξ j∂ξiwhere the repeated index summation rule does not apply, and wechoose c i j = 1/2 for i = j, and c i j = 1 for the rest to be consistent
with (13). The ‘stress-glut field’ will be back-propagated in thiscase, and will converge at the source location as the moment tensor.
The time reversal imaging of the moment tensor may be obtainedasM̂TRi j (ω) ≈ α −1 Ē inj (ξ , ω; x) dn (ω)(17)

Similar to (12), the convergence may be measured by2M̂ 2i≤ j M̂ TRi j
≡
α
,
(18)
R ≈ α TRd2d2
where i≤ j denotes a summation over six independent components
2of M̂ TRi j .
It should be noted that the imaging solution (10) or (17) does notscale properly when the parameterization is changed. For example,
if we solve for M 11 = 2 × M 11 instead of M 11 while keeping otherparametrization the same, the resulting solution is not guaranteed
to be consistent with that solved for M 11 (this is not the case for the
least-squares solution). Also the convergence criteria (18) should
be modified accordingly.

3 DISCUSSIONd ∗ G f̂R≈αcase of TR imaging, it is the strain field which will be used forback-propagation: i.e.

E ijn

Claerbout (2001) introduces a clear definition of the difference
between inverse and adjoint operators as follows: In many geophysical inverse problems (e.g. Tarantola 1984), G ∗ G can be often wellapproximated by the identity matrix as
G∗ G ≈ α I

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Imaging or inversion
We have shown that the time reversal (adjoint) source imaging
gives an approximation to the seismic source inversion by way of(9). Claerbout (2001) notes that ‘the adjoint is the first step and apart of all subsequent steps in this inversion process. Surprisingly,
in practice the adjoint sometimes does a better job than the inverse!
This is because the adjoint operator tolerates imperfections in thedata and does not demand that the data provide full information.’
The question left is then whether we should use imaging or inversion
for the seismic source estimation problems.
There are cases when certain components of the moment tensor are not well resolved by data (e.g. Kanamori & Given 1981;
Kawakatsu 1996). There may also exist cases that the knowledge ofthe Green’s function is largely poor. In these cases, spurious solution
may be obtained by the inverse and the imaging may be preferred.
Other than those cases, considering the easiness of calculating
the inverse for source estimation, the conventional moment tensor
inversion may be preferred to the time-reversal source imaging. The
non-uniqueness of the solution mentioned above may also limit theusage for earthquake mechanism estimation purposes. On the other
hand, once the parameterization scheme is fixed, (12) or (18) can beused for estimating strength of the source, and the spatial extent ofseismic source may be imaged effectively.

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H. Kawakatsu and J.-P. Montagner

Realtime monitoring applications
Although (9) may give a good approximation, calculation of theactual inverse is not so difficult for the source inversion; for the
moment tensor source we need to solve for 6 × 6-matrix, andfor the single force source only for 3 × 3-matrix. So there appears no practical difficulty in solving the inverse problem. Indeed, in GRiD MT (Grid-based Realtime Determination of Moment
Tensors) algorithm (Kawakatsu 1998; Tsuruoka et al. 2008), moment tensor inversions are performed in realtime continuously forevery one second for limited source areas. GRiD MT is a gridbased earthquake analysis system that continuously monitors longperiod seismic wavefield of 20 to 50 seconds recorded by broadband
seismometers, and automatically and simultaneously determines
the origin time, location and seismic moment tensor of seismic
events within three minutes of their occurrence. This system has
been in operation since 2003 at the Earthquake Research Institute(http://wwweic.eri.utokyo.ac.jp/GRiD MT/), and has been shown
to be successful (Tsuruoka et al. 2008). The success of GRiD MT
proves that it is now practical to perform the seismic source inversion
in realtime, consequently the time reversal source imaging as well.
Either performing inversion or imaging, such a realtime operation in a long-period range is now possible as mentioned above
for regional scale monitoring, and may be for global scale as well(e.g. Larmat et al. 2006; Ekström 2006). This type of source inversion/imaging should also help monitoring glacial quakes (Ekström et al. 2003) and sources of background seismic microseismand HUM (e.g. Nishida & Fukao 2007), as well as some otherexotic seismic sources (e.g. Kumagai et al. 2001; Ito et al. 2007).
The realtime monitoring of Earth’s activity field may soon become
available.
AC K N OW L E D G M E N T S
This work is conducted while the second author was a visiting
professor at the Ocean Hemisphere Research Center (OHRC) of
ERI in 2008. He thanks the members of ERI, especially to those of
the OHRC for hospitality. Comments by Göran Ekström helped to
improve the manuscript.
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