Geophys. J. Int. (2011) 186, 711–720 doi: 10.1111/j.1365-246X.2011.05043.x

GJI

Sei

smol

ogy

Introduction of uncertainty of Green’s function into waveforminversion for seismic source processes

Yuji Yagi1 and Yukitoshi Fukahata2

1Graduate School of Life and Environmental Sciences, University of Tsukuba, Japan. E-mail: yagi-y@geol.tsukuba.ac.jp2Disaster Prevention Research Institute, Kyoto University, Japan

Accepted 2011 April 21. Received 2011 March 25; in original form 2010 November 15

S U M M A R YIn principle, we can never know the true Green’s function, which is a major error source inseismic waveform inversion. So far, many studies have devoted their efforts to obtain a Green’sfunction as precise as possible. In this study, we propose a new strategy to cope with thisproblem. That is to say, we introduce uncertainty of Green’s function into waveform inversionanalyses. Due to the propagation law of errors, the uncertainty of Green’s function results in adata covariance matrix with significant off-diagonal components, which naturally reduce theweight of observed data in later phases. Because the data covariance matrix depends on themodel parameters that express slip distribution, the inverse problem to be solved becomes non-linear. Applying the developed inverse method to the teleseismic P-wave data of the 2006 Java,Indonesia, tsunami earthquake, we obtained a reasonable slip-rate distribution and moment-ratefunction without the non-negative slip constraint. The solution was independent of the initialvalues of the model parameters. If we neglect the modelling errors due to Green’s functionas in the conventional formulation, the total slip distribution is much rougher with significantopposite slip components, whereas the moment-rate function is much smoother. If we use astronger smoothing constraint, more plausible slip distribution can be obtained, but then themoment-rate function becomes even smoother. By comparing the observed waveforms withthe synthetic waveforms, we found that high-frequency components were well reproduced onlyby the new formulation. The modelling errors are essentially important in waveform inversionanalyses, although they have been commonly neglected.

Key words: Time series analysis; Inverse theory; Earthquake dynamics.

1 I N T RO D U C T I O N

Error e in the observation equation, d = Ha + e, plays an essentialrole in inversion analyses, where d and a are data and model param-eter vectors, respectively, and H is a coefficient matrix. For exam-ple, the least-squares criterion, (d − Ha)t E−1 (d − Ha) → min, isderived from the assumption that the error e follows a Gaussian dis-tribution with zero mean and covariance E. The off-diagonal termsof the matrix E represent the covariance components between data.

The error e generally consists of observation errors and modellingerrors. In some geodetic data inversion analyses (e.g. Langbein &Johnson 1997; Segall et al. 2000; Lohman & Simons 2005; Fukahata& Wright 2008), covariance components have already been takeninto account. In these studies, however, the origin of covariance hasalways been ascribed to observation errors. Covariance originatedfrom modelling errors, which can be a key in inversion analyses(Yagi & Fukahata 2008), has been neglected. However, the effectof covariance due to modelling errors is getting more important,because we observe various data nominally continuously with high

accuracy and we can now invert such data with a very high samplingrate owing to enhanced technology of computers.

In the paper of Yagi & Fukahata (2008), we focused on thediscretization error as a source of the covariance, mainly becausethe discretization error was easier to be treated theoretically. Asmodelling errors, however, terms that originated from uncertaintyof Green’s function seem more important. It is well recognizedthat errors of Green’s function can severely bias the result in theseismic waveform inversion (e.g. Graves & Wald 2001; Wald &Graves 2001). In principle, however, we can never know the trueGreen’s function. So far, many studies have devoted their effortsto obtain a Green’s function as precise as possible, for example,by using a 3-D underground structure model (e.g. Liu & Archuleta2004) or by using observed waveforms for small earthquakes (e.g.Fukuyama & Irikura 1986; Mori & Harzel 1990). These efforts areindispensible. In this study, however, we propose a new strategy tocope with this problem. That is to say, we introduce uncertainty ofGreen’s function into inversion analyses. On the basis of the premisethat our estimates of Green’s function always include some errors,

C© 2011 The Authors 711Geophysical Journal International C© 2011 RAS

Geophysical Journal International

712 Y. Yagi and Y. Fukahata

we formulate the waveform inverse problem for seismic sourceprocesses.

Since the pioneer studies by Olson & Apsel (1982) and Hartzell& Heaton (1983), waveform inversion for seismic source processeshas been done by many researchers (e.g. Beroza & Spudich 1988;Ide et al. 1996; Delouis et al. 2002; Ji et al. 2002; Yagi et al.2004; Emolo & Zollo 2005; Hartzell et al. 2007; Piatanesi et al.2007; Uchide & Ide 2007). Seismic source models are essentiallyimportant to reveal the characteristics of earthquakes. Now, thewaveform inversion becomes a popular tool to construct seismicsource models. As pointed out by some researchers (e.g. Beresnev2003; Clevede et al. 2004; Vallee & Bouchon 2004), however, seis-mic source models for the same earthquake are often quite differentfrom one another, even if we use the same observed data (Mai et al.2007). This suggests that the existing formulations of the inverseproblem have some crucial flaws.

Yagi & Fukahata (2008) solved the problem that the solutiondepends on the sampling rate. However, it seems that the incon-sistency between the results cannot be solved only by this effect.In this study, by introducing uncertainty of Green’s function, weaim to contribute to constructing a standard formulation of seismicwaveform inversion.

2 M AT H E M AT I C A L F O R M U L AT I O N

2.1 Observation equation

In general, observed seismic waveform for the far-field term at astation j due to a shear dislocation source on a fault plane S is givenby (e.g. Olson & Apsel 1982)

u j (t) =2∑

q=1

∫s

G0q j (t, ξ ) ∗ D0

q (t, ξ ) dξ + ebj (t) , (1)

where G0q j is the true Green’s function, D0

q is a true spatio-temporalslip-rate distribution of qth slip component and ebj is a back-ground and instrumental noise. ξ represents a position on the faultplane. The superscript 0 denotes a true function and ∗ denotes theconvolution operator in time domain. The Green’s function G0

q j

for teleseismic body wave can be generally decomposed into twofactors:

G0q j (t, ξ ) = F (t) ∗ g0

q j (t, ξ ) , (2)

where F represents the effect of inelastic attenuation of the Earthand g0

q j is a sequence of spikes which contains a radiation pattern, ageometrical spreading factor and reflection phases. Here, we neglectthe modelling error of F for simplicity.

To formulate the inverse problem, we represent the spatio-temporal slip-rate distribution Dq by a linear combination of afinite number of basis functions:

D0q (t, ξ ) ∼=

K∑k=1

L∑l=1

aqkl Xk (ξ ) Tl (t − tk), (3)

where Xk and Tl are the basis functions for space and time, re-spectively, and aqkl are the expansion coefficients. Our problem isto estimate aqkl from observed waveform data. tk is the time whenrupture can start on a space grid k. Since this study is targeted onthe modelling error originated from Green’s function, we neglectthe discretization error introduced by Yagi & Fukahata (2008).

Substituting eqs (2) and (3) into (1), we obtain

u j (t) =2∑

q=1

K∑k=1

L∑l=1

aqkl Tl (t − tk) ∗ F (t) ∗ g0qk j (t) + ebj (t) , (4)

with

g0qk j (t) =

∫s

Xk (ξ ) g0q j (t, ξ ) dξ, (5)

where g0qk j is the spike function of qth slip component at kth space

knot. The most difficult point in estimating the Green’s functionis the amplitude and timing of reflection phases in heterogeneousmedia. Therefore, we here introduce the error of Green’s functionδgqk j as

g0qk j (t) = gqk j (t) + δgqk j (t) . (6)

Substituting eq. (6) into (4), we rewrite the expression of wave-form:

u j (t) =2∑

q=1

K∑k=1

×[

L∑l=1

Hqkl j (t) aqkl + Pqk

(t, aqkl

) ∗ δgqk j (t)

]+ebj (t) , (7)

with

Hqkl j (t) = Tl (t − tk) ∗ F (t) ∗ gqk j (t) (8)

and

Pqk

(t, aqkl

) =L∑

l=1

aqkl Tl (t − tk) ∗ F (t). (9)

Here, it should be noted that Pqk is a function of the modelparameters aqkl .

We commonly apply a low-pass filter to observed raw wave-forms to mitigate the effects of aliasing and the heterogene-ity of underground structure. With the low-pass filter B, thedata d j for the inversion analysis is related to the observed rawwaveform u j :

d j (t) = B (t) ∗ u j (t) . (10)

Substituting eq. (7) into (10), we obtain the observation equationfor the data d j of the jth station:

d j (t) =2∑

q=1

K∑k=1

L∑l=1

Hqkl j (t) aqkl

+2∑

q=1

K∑k=1

Pqk

(t, aqkl

) ∗ δgqk j (t) + B (t) ∗ ebj (t) , (11)

where

Hqkl j (t) = B (t) ∗ Hqkl j (t) (12)

Pqk (t) = B (t) ∗ Pqk (t) . (13)

We can rewrite eq. (11) in the following simple vector form:

d j = H j a + e j (a) , (14)

with

e j (a) =2∑

q=1

K∑k=1

Pqk j (a) δgqk j + B j ebj , (15)

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Geophysical Journal International C© 2011 RAS

Uncertainty of Green’s function in inversion 713

where Pqkj and Bj are Nj (number of data point at jth station) × Nj

dimensional matrices and δgqk j and ebj are Nj-dimensional vectors.We assume the modelling error δgqk j and the observation errorebj to be Gaussian with zero mean and covariance (σg Sqk j )2I j and(σb Obj )2I j , respectively, where I j is a unit matrix with a size ofNj × Nj, Sqkj is the maximum amplitude of the calculated Green’sfunction of qth slip component at kth space knot and Obj is themaximum amplitude of the observed background noise at a stationj. σg and σb are unknown scaling factors of the modelling error andbackground noise, respectively. Based on the propagation law oferrors and assuming no correlation between the two error terms, thedata covariance matrix for the data vector d j is expressed as

Cd j

(σ 2

g , σ 2b , a

) = σ 2g

2∑q=1

K∑k=1

S2qk j Pqk j (a) Pt

qk j (a) + σ 2b O2

bj B j Btj .

(16)

As shown in eq. (16), the data covariance matrix Cd j is a functionof the model parameters a. This means that the inverse problem tobe solved is non-linear.

Based on eqs (14) and (16), we can describe a stochastic model,which relates the data d for all the stations with the model parametersa as

p(d|a; σ 2

g , σ 2b

) = (2π)−N/2∣∣Cd

(σ 2

g , σ 2b , a

)∣∣−1/2

× exp

[−1

2(d − Ha)t C−1

d

(σ 2

g , σ 2b , a

)(d − Ha)

], (17)

where Cd is the direct sum of Cd j (j = 1, 2, 3, . . .), | · | denotes thedeterminant of the matrix and N is the total number of data.

2 . 2 A k a i k e ’ s B a y e s i a n I n f o r m a t i o nC r i t e r i o n a n d n o n - l i n e a r i n v e r s i o n

In inversion analyses, we can use information from prior constraintsas well as from observed data. The explicit use of prior constraintsenables us to naturally compromise reciprocal requirements formodel resolution and estimation errors (Jackson 1979; Tarantola1987). It is considered that the slip rate on seismic faults does notprefer sudden change in some degree (e.g. Yabuki & Matsu’ura1992). So, as prior constraints, we use smoothness of slip rate inspace and time.

∇2 D (t, ξ ) + es = 0, (18)

∂2

∂t2D (t, ξ ) + et = 0, (19)

where es and et are Gaussian with zero mean and covariance ρ21 I and

ρ22 I, respectively. As shown in eq. (18), we apply the smoothness

constraint to the instantaneous spatial distribution of slip rate, sincethe smoothness constraint on the total slip distribution, which hasbeen used by many previous studies (e.g. Ide et al. 1996), tends tobe a cause of unstable results. Eqs (18) and (19) can be rewritten inthe following simple vector form, respectively:

S1a + es = 0, (20)

S2a + et = 0. (21)

Here S1 and S2 are M × M-dimensional matrices and es and et areM-dimensional vectors, where M (=2KL) is the number of modelparameters. Following Fukahata et al. (2003, 2004), we combine

the two prior constraints, eqs (20) and (21), into one probabilitydensity function

p(a; ρ2

1 , ρ22

) = (2π)−M/2

∣∣∣∣ 1

ρ21

G1 + 1

ρ22

G2

∣∣∣∣1/2

× exp

[−at

(1

2ρ21

G1 + 1

2ρ22

G2

)a

], (22)

with

G1 = St1S1, G2 = St

2S2, (23)

where ρ21 and ρ2

2 are unknown hyperparameters that control thedegree of smoothness. Following the concept of Sekiguchi et al.(2000), we introduce a resolution ratio χ 2 = ρ2

1/ρ22 for simplicity,

and set this ratio to the square of the shear wave velocity. Then, eq.(22) becomes

p(a; ρ2

1

) = (2π )−M/2

∣∣∣∣ 1

ρ21

(G1 + χ 2G2

)∣∣∣∣1/2

× exp

[− at

2ρ21

(G1 + χ 2G2

)a

], (24)

We incorporate the information from prior constraints (24) andobserved data (17) by Bayes’ theorem.

p(a; σ 2

g , σ 2b , ρ2

1 |d) = cp

(d|a; σ 2

g , σ 2b

)p

(a; ρ2

1

), (25)

where c is a normalizing factor. Eq. (25) includes unknown hyper-parameters σ 2

g , σ 2b and ρ2

1 together with unknown model parametersa.

As pointed out in Section 2.1, the inverse problem to be solvedis non-linear because the data covariance matrix Cd is a function ofthe model parameters a. To cope with this problem, we introducean initial model parameter vector ai to evaluate the data covariancematrix, and iteratively solve the inverse problem by improving theestimate for the model parameters a. In each step of the iteration,the selection of the optimal values of the hyperparameters can beobjectively done by minimizing ABIC (Akaike 1980; Yabuki &Matsu’ura 1992). ABIC is defined by

ABIC = −2 log

[∫p

(d|a; σ 2

g , σ 2b

)p

(a; ρ2

1

)da

]+ C. (26)

Carrying out the integration and introducing new hyperparame-ters α2 (= σ 2

g

/ρ2

1 ) and γ 2 (= σ 2g

/σ 2

b ) instead of ρ21 and σ 2

b , respec-tively, we obtain the expression of ABIC for a given initial modelparameter vector ai (Yabuki & Matsu’ura 1992; Fukahata et al.2004):

ABIC(α2, γ 2

) = N log s (a∗) − log α2∣∣G1 + χ 2G2

∣∣ + log∣∣Ht C−1d

(γ 2, ai

)H + α2

(G1 + χ 2G2

)∣∣ + log∣∣Cd

(γ 2, ai

)∣∣ + C ′,(27)

where

s (a) = (d − Ha)t C−1d

(γ 2, ai

)(d − Ha) + α2at

(G1 + χ 2G2

)a,

(28)

a∗ = [Ht C−1

d

(γ 2, ai

)H + α2

(G1 + χ 2G2

)]−1Ht C−1

d

(γ 2, ai

)d.

(29)

Here, we used

σ 2g = s(a∗)

/N , (30)

which is derived from the necessary condition that the partial deriva-tive of the ABIC with respect to σ 2

g must be zero (Akaike 1980).

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Geophysical Journal International C© 2011 RAS

714 Y. Yagi and Y. Fukahata

Once the values of α2 and γ 2 minimizing ABIC has been found, wecan obtain the best estimates of the model parameters a∗ from eq.(29) with the covariance matrix for the model parameters:

Ca = σ 2g

[Ht C−1

d

(γ 2, ai

)H + α2

(G1 + χ 2G2

)]−1. (31)

As mentioned earlier, we must apply an iterative method to obtainthe optimal model parameters a that satisfy a∗ ∼= ai . At the firstiteration, we give initial model parameters ai

1, and obtain the optimalvalues of the hyperparameters α2

1 and γ 21 , and the model parameters

a∗1. In the nth iteration, we set initial model parameters as

ain = ai

n−1 + w(a∗

n−1 − ain−1

)(0 < w < 1) , (32)

and obtain α2n , γ 2

n and a∗n . This iterative procedure is continued

until the normalized L2 norm ‖a∗n − ai

n‖/∥∥ai

n

∥∥ becomes acceptablysmall.

3 A P P L I C AT I O N

To examine the validity of the inverse method developed in theprevious section, we perform seismic source inversion for the 2006July 17 Java tsunami earthquake. Because the source area of thisearthquake extended to near the Java trench (e.g. Okamoto &Takenaka 2009), the true Green’s function must include the effectof complicated multipaths due to the 3-D structure of the seafloor(e.g. Wiens 1989; Okamoto & Miyatake 1989). However, the effectof multipaths is very difficult to be accurately evaluated in the cal-culation of theoretical Green’s function. Therefore, by consideringerrors in the calculated Green’s function, we invert the waveformdata.

3.1 Data and fault model

The 2006 Java earthquake occurred far off the Java, Indonesia, andgenerated devastating tsunamis along the southern coast of centralJava. From the analyses of tsunamis and surface waveforms, thisearthquake is classified as a tsunami earthquake (e.g. Ammon et al.2006; Fujii & Satake 2006). The moment tensor solution and theaftershock distribution indicated that the earthquake was a thrustfaulting dipping NNE with about 10◦. Seismic source models havebeen obtained by body and surface waves (Ammon et al. 2006), bybody waves (Bilek & Engdahl 2007; Okamoto & Takenaka 2009)and by near-source waveforms (Nakano et al. 2008). These modelsshow that the rupture propagated southeastwards in a unilateralmanner with the rupture velocity of 1.0–1.5 km s–1 along the fault,which is about 200 km in length.

Teleseismic P-wave data recorded at FDSN (International Fed-eration of Digital Seismograph Networks) network stations andGlobal Seismograph Network stations were retrieved from IRIS-dmc (Data Management Center, Incorporated Research Institutionsfor Seismology). 58 stations were selected from the viewpoint ofdata quality. The locations of the teleseismic stations are shown inFig. 1. The teleseismic body waves were converted into ground dis-placements with a sampling rate of 1.5 s. For mitigation of aliasing,we applied the Butterworth low-pass filter before the resampling.

We calculated the theoretical Green’s functions for teleseismicbody waves using the method of Kikuchi & Kanamori (1991) witha sampling rate of 0.1 s. As a structure model near the source, weused the 1-D crustal model interpolated form CRUST2.0 (Bassinet al. 2000). The attenuation time constant t∗ for the P wave wastaken to be 1.0 s.

We assumed that the faulting occurred on a single flat plane.We adopted the epicentre (9.28◦S, 107.42◦E) determined by USGS,

Figure 1. Teleseismic station distribution shown as a map view. The trian-gles and star indicate the teleseismic stations and epicentre, respectively.

and the fault mechanism (strike, dip) = (290◦, 10◦), which wereslightly modified form the moment tensor solution to be consistentwith amplitudes of the P waves. We took a fault area of 360 km ×132 km, which was expanded into bilinear B-splines with an intervalof about 10 km. On each space knot we also took a slip-rate durationof 60 s, which was expanded into linear B-splines with an interval of1.5 s. A long time window of 60 s allows a flexible slip model. Thestarting time tk at each space knot was controlled by the rupture frontvelocity Vm and the hypocentral distance, which is the function of theepicentral location and hypocentral depth d. Based on preliminaryanalyses, we determined Vm and d to be 1.5 km s–1 and 15 km,respectively.

So far, the non-negative constraint has been commonly applied inthe waveform inversion for source processes to obtain a plausible so-lution (e.g. Harzell & Heaton 1983; Yoshida & Koketsu 1990; Das &Schadolc 1995). However, the inversion with the non-negative con-straint forcefully gives a non-negative slip distribution, even if weuse incorrect fault parameters and/or a structure model. Of course,such slip distribution may significantly differ from the true one, butthe non-negative constraint apparently conceals the defects causedby the use of an incorrect model. When we use an inappropriate in-verse method, we can say the same thing. Apparently not bad resultscan be easily obtained by imposing the non-negative constraint, butthere is no guarantee that the obtained result is actually not bad.In other words, if we cannot obtain a reasonable result without thenon-negative constraint, this implies that something of the modeland/or the method is inappropriate. Therefore, in the following in-version analyses, we do not apply the non-negative constraint tomore clearly see the validity of the new formulation.

3.2 A synthetic test

We first made a synthetic test to compare the new formulation withthe conventional formulation. We gave a source model with twomajor slip areas as shown in Fig. 2(a), and computed the synthetic

C© 2011 The Authors, GJI, 186, 711–720

Geophysical Journal International C© 2011 RAS

Uncertainty of Green’s function in inversion 715

Figure 2. Results of synthetic test. (a) Slip distribution of the assumed source model. (b) Slip distributions inverted form the same synthetic data using thenew formulation (left-hand panel) and the conventional formulation (right-hand panel). In the new formulation covariance components due to uncertainty ofGreen’s function are taken into account.

waveform at each station shown in Fig. 1. In the computation ofthe synthetic waveform, we added background noise and an errorof Green’s function as in eq. (6). The error of Green’s functionand background noise were random Gaussian noise with zero meanand standard deviation of 3 per cent of the maximum amplitudeof the calculated Green’s function and 3 μm s–1, respectively. Thesynthetic data were calculated every 0.05 s, and then resampled witha sampling rate of 1.5 s with the Butterworth low-pass filter.

The difference between the new and the conventional formula-tions only consists in the data covariance matrix Cd j . In the newformulation Cd j is defined by eq. (16), while in the conventionalformulation Cd j is a diagonal matrix defined by

Cconventionald j

= σ 2b I j . (33)

Fig. 2(b) shows the inverted source models obtained by the newand the conventional formulations. The broad pattern with two ma-jor slip areas was reproduced by both formulations. Especially, theslip distribution by the new formulation is very similar to the givenslip distribution in Fig. 2(a). However, the slip distribution by theconventional formulation is significantly rougher than the given slipdistribution. Right-lateral strike-slips are estimated in the lower leftof the fault plane, whereas left-lateral slips are estimated in theshallower side. The two contradictory slips are ruptured almost atthe same timing, which is unrealistic. The discrepancy between thegiven and the estimated models tends to increase with the distancefrom the hypocentre. The covariance components of waveform datadue to modelling error of Green’s function generally increase withtime at each station, because the modelling error of the Green’sfunction is the convolution of the slip-rate function at each spaceknot and the random error, as shown in eqs (7) and (9). The synthetic

test shows that the neglect of the covariance components results ina biased and unrealistic seismic source model.

3.3 Application to real waveform data

Now we invert the real observed waveform data of the 2006 July17 Java tsunami earthquake. To examine the dependence of theinverted result on initial model parameters, we applied two kinds ofinitial slip models: (1) uniform slip model and (2) vertical-stripedslip model. Fig. 3 shows the comparison between the total slipdistributions for the cases of (1) and (2). Six and five iterationswere enough to get acceptably small values of the normalized L2norm, where we set w in eq. (32) to be 0.5. For the synthetic testin the previous subsection, we obtained similar results. As seenfrom Fig. 3, exactly the same total slip distribution was obtained forboth of the initial slip models. We also confirmed that the estimatedvalues of the model parameters were the same for both models.This means that the final result after iterations is independent of theinitial values of the model parameters.

Fig. 4 shows the snapshots of the inverted slip distribution foreach time interval on a map view. As can be seen, the source ruptureprocess is divided into two episodes. The first is rupture propagationaround the epicentre during the initial 30 s. The rupture swept overthe fault area about 60 km along both the fault-strike and fault-dip.The second phase of the rupture started at 30 s later after the initialbreak and propagated unilaterally to the east. The rupture reachedabout 200 km far from the hypocentre. The rupture propagatedintermittently, and the average velocity was about 1.2 km s–1. Thegloss feature of the seismic source process is consistent with otherstudies (e.g. Ammon et al. 2006; Okamoto & Takenaka 2009). Thetotal seismic moment M0 is 6.6 × 1020 Nm (Mw = 7.8), which

C© 2011 The Authors, GJI, 186, 711–720

Geophysical Journal International C© 2011 RAS

716 Y. Yagi and Y. Fukahata

Figure 3. Independence of inversion result to initial slip models. We gave a uniform slip (a) and a vertical-striped slip (b) as initial slip distribution. Eachdiagram shows the inverted slip distribution. The number of iteration is shown at the right bottom.

is slightly larger than that of the Global centroid-moment-tensor(CMT) solution (http://www.globalcmt.org/), 4.61 × 1020 Nm.

We also performed waveform inversion with the conventionalformulation. In Fig. 5 we compare the results inverted by the newand the conventional formulations. The total slip distribution by theconventional formulation is much rougher than the one by the newformulation, whereas the moment-rate function is clearly smootherin the conventional formulation. The discrepancy between the to-tal slip distributions increases with distance from the hypocentre,as expected in the synthetic test. The total slip model by the con-ventional formulation contains large normal slip components in theshallow left area of the fault plane, which is contradictory to thegeneral feature of the fault slip and the Global CMT solution. Asshown in Fig. 6, if we use a stronger smoothing constraint than theoptimal value determined by ABIC, such unrealistic slips disap-pear. In this case, however, the moment-rate function becomes evensmoother.

In Fig. 7 we compare the observed waveforms with the syntheticwaveforms obtained by the new and the conventional formulations,in which the values determined by the ABIC minimum are usedeven in the conventional formulation. The fitting to the waveformsis basically good for both cases. However, high-frequency compo-

nents of the observed waveforms are well reproduced only by thenew formulation, which is consistent with the difference in smooth-ness of the moment-rate functions (Fig 5). In brief, the smoothnessconstraint in the conventional formulation is too strong to repro-duce the high-frequency components of the observed waveforms,whereas the smoothness constraint is too weak to obtain a reasonableslip distribution that does not contain substantial slips of oppositedirections. Here, we can see the meaning to use a better model inexpressing errors.

4 D I S C U S S I O N A N D C O N C LU S I O N S

By introducing uncertainty of Green’s function, which seems to bethe most dominant source of modelling errors, we have developeda new method of inversion analyses of waveform data for seismicsource processes. The uncertainty of Green’s function results insignificant covariance components due to the law of propagation oferrors (eq. 16). As shown through the synthetic test and actual wave-form data analysis, it is crucially important to take the covariancecomponents into account in obtaining reasonable inversion results.

The source process of the Java tsunami earthquake has beenestimated by some studies (e.g. Ammon et al. 2006; Okamoto &

C© 2011 The Authors, GJI, 186, 711–720

Geophysical Journal International C© 2011 RAS

Uncertainty of Green’s function in inversion 717

Figure 4. Snapshots of the inverted slip distribution for each time interval on a map view. The star and the rectangle indicate the epicentre and the fault plane,respectively. The solid line of the rectangle represents the intersection with the Earth’s surface.

Takenaka 2009), but the effect of covariance components has alwaysbeen neglected. For example, Okamoto & Takenaka (2009) invertedbody waves using more realistic Green’s function than our study,but the obtained total slip distribution contains many small patchesof relatively large slips. Ammon et al. (2006) used surface wavesas well as body waves. The off-diagonal components of the datacovariance matrix is considered to be larger in the surface wavethan in the body wave because of longer periods of the surfacewave. In fact, their slip model contains an extremely large slip areawith the maximum slip of 16 m, which is about 10 times as large asour and other studies (e.g. Okamoto & Takenaka 2009).

In addition, the smoothing parameter was manually adjusted inthese studies. Because a wide range of slip distribution can be repro-duced according to the selected value of the smoothing parameter,it should be objectively determined by using a statistical criterion,such as ABIC. If the value of the smoothing parameter is subjec-tively selected, the solution may strongly reflect the thought of theanalyst. As Jackson (1979) insisted, if we have some information orknowledge for the solution, such information or knowledge shouldbe incorporated at the beginning rather than implicitly referred atthe end of the analysis.

It should also be noted that the non-negative constraint was notimposed in this study. As mentioned in Section 3.1, the non-negativeconstraint contributes to making a solution apparently plausible.However, if the solution obtained without the non-negative con-

straint has negative slips that are significantly larger than the es-timation error, we should suspect something in the model and/orthe method to be inappropriate. In fact, we admit that we have alsoneeded the non-negative constraint in previous analyses to obtainreasonable slip distribution even in Yagi & Fukahata (2008), inwhich covariance components of modelling errors were included ina simpler way. By introducing uncertainty of Green’s function, thenon-negative constraint has become unnecessary.

It seems to be a common agreement that waveform fittings justafter the first motion of P or S wave are important indicators inevaluating the result, because the seismogram just after the firstmotion is not contaminated by later phases created by a complexunderground structure. However, previous inversion studies havecommonly put an equal weight on observed data regardless of time.In fact, a reasonable way to control the weight has not been found. Inour new formulation the modelling error due to Green’s function isdescribed by the convolution of the slip-rate function at each spaceknot and the random error, and so the covariance components ofwaveform data generally increase with time. This means that theweight of observed data in later phases is naturally reduced in thenew formulation. Because of that, it is considered that a reason-able result was obtained without the non-negative constraint. Thiseffect is more important for the analysis of large earthquakes. Incor-porating the uncertainty of Green’s function would be essentiallyimportant not only for source process inversion but also for other

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718 Y. Yagi and Y. Fukahata

Figure 5. Moment-rate function and total slip distribution inverted with the new formulation (a) and the conventional formulation (b). The contour interval ofthe total slip distribution is 0.2 m.

Figure 6. Moment-rate function and total slip distribution obtained by the conventional formulation, in which 10 times stronger smoothness constraint thanthe optimal value determined by ABIC is used.

seismic waveform analyses such as moment tensor analysis andtomography.

In this study, we assumed the error of Green’s function to beGaussian with zero mean, for simplicity. This assumption workswell when we use reasonable fault parameters and Green’s function.In other words, improvements of the fault parameter model and theGreen’s function are always important to estimate a seismic sourcemodel more precisely. However, we can never know the true Green’sfunction in principle. In estimating the seismic source process as

precise as possible, we must encounter the problem of uncertaintyof Green’s function.

A C K N OW L E D G M E N T S

The teleseimsic body waves used in this study are from FDSN net-work stations and Global Seismograph Network stations and pro-vided by IRIS-dmc. We acknowledge the comments by PengchengLiu and an anonymous reviewer in improving the manuscript. This

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Uncertainty of Green’s function in inversion 719

Figure 7. Comparison of the observed waveform (upper trace) to the synthetic waveform obtained by the new formulation (middle trace) and the conventionalformulation (lower trace). The arrows highlight the point where high-frequency components of the observed waveforms are well reproduced by the newformulation.

study has been supported by the Grant-in-Aid for Scientific Re-search No. 21540428 of the MEXT to YY and YF.

R E F E R E N C E S

Akaike, H., 1980. Likelihood and Bayes procedure, in Bayesian Statistics,pp. 143–166, eds Bernardo,J. M., DeGroot, M. H., Lindley, D. V. & Smith,A.F.M., University Press, Valencia, Spain.

Ammon, C.J., Kanamori, H., Lay, T. & Velasco, A.A., 2006. The 17July 2006 Java tsunami earthquake, Geophys. Res. Lett., 33, L24308,doi:10.1029/2006GL028005.

Bassin, C., Laske, G. & Masters, G., 2000. The current limits of resolutionfor surface wave tomography in North America, EOS, Trans. Am. geophys.Un., 81, F897.

Beresnev, I.A., 2003. Uncertainties in finite-fault slip inversions: to whatextent to believe? Bull. seism. Soc. Am., 93(6), 2445–2458.

Beroza, G.C. & Spudich P., 1988, Linearized inversion for fault rupturebehavior: application to the 1984 Morgan Hill, California, earthquake, J.geophys. Res., 93, 6275–6296.

Bilek, S.L. & Engdahl, E.R., 2007. Rupture characterization and aftershockrelocations for the 1994 and 2006 tsunami earthquakes in the Java subduc-tion zone, Geophys. Res. Lett., 34, L20311, doi:10.1029/2007GL031357.

Clevede, E., Bouin, M.P., Bukchin, B., Mostinskiy, A. & Patau, G., 2004.New constraints on the rupture process of the 1999 August 17 Izmitearthquake deduced from estimates of stress glut rate moments, Geophys.J. Int, 159, 931–942.

Das, S. & Suhadolc, P., 1995. On the inverse problem for earthquake rupture:the Haskell-type source model, J. geophys. Res., 101, 5725–5738.

Delouis, B., Giardini, D., Lundgren, P. & Salichon, J., 2002. Joint inversionof InSAR, GPS, teleseismic and strong motion data for the spatial andtemporal distribution of earthquake slip: application to the 1999 IzmitMainshock, Bull. seism. Soc. Am., 92, 278–299.

Emolo, A. & Zollo, A., 2005. Kinematic source parameters for the 1989Loma Prieta earthquake from the nonlinear inversion of accelerograms,Bull. seism. Soc. Am., 95, 981-994.

Fujii, Y. & Satake, S., 2006. Source of the July 2006 West Java tsunamiestimated from tide gauge records, Geophys. Res. Lett., 33, L24317,doi:10.1029/2006GL028049.

Fukahata, Y. & Wright, T.J., 2008. A non-linear geodetic data inversionusing ABIC for slip distribution of a fault with an unknown dip angle,Geophys. J. Int., 173, 353–364.

Fukahata, Y., Yagi, Y. & Matsu’ura, M., 2003. Waveform inversion forseismic source processes using ABIC with two sorts of prior constraints:comparison between proper and improper formulations, Geophys. Res.Lett., 30, 1305, doi:10.1029/2002GL016293.

Fukahata, Y., Nishitani, A. & Matsu’ura, M., 2004. Geodetic data inver-sion using ABIC to estimate slip history during one earthquake cyclewith viscoelastic slip-response functions, Geophys. J. Int., 156, 140–153.

Fukuyama, E. & Irikura, K., 1986. Rupture process of the 1983 Japan Sea(Akita-Oki) earthquake using a waveform inversion method, Bull. seism.Soc. Am., 76, 1623–1640.

Graves, R. W. & Wald, D.J., 2001. Resolution analysis of finite fault sourceinversion using one- and three-dimensional Green’s functions 1. Strongmotions, J. geophys. Res., 106, 8745–8766.

Hartzell, S.H. & Heaton, T.H., 1983. Inversion of strong ground motionand teleseismic waveform data for the fault rupture history of the 1979Imperial Valley, California earthquake, Bull. seism. Soc. Am., 73, 1553–1583.

Hartzell, S., Liu, P., Mendoza, C., Ji, C. & Larson, M.K., 2007. Sta-bility and uncertainty of finite-fault slip inversions: application to the2004 Parkfield, California, earthquake, Bull. seism. Soc. Am., 97, 1911–1934.

Ide, S., Takeo, M. & Yoshida, Y., 1996. Source process of the 1995Kobe earthquake: determination of spatio-temporal slip distribution byBayesian modeling, Bull. seism. Soc. Am., 86, 547–566.

Jackson, D.D., 1979. The use of a priori data to resolve non-uniqueness inlinear inversion, Geophys. J. R. astr. Soc., 57, 137–157.

Ji, C., Wald, D. J. & Helmberger, D.V., 2002. Source description of the 1999Hector Mine, California, earthquake, part II: complexity of slip history,Bull. seism. Soc. Am., 92, 1208–1226, doi:10.1785/0120000917.

Kikuchi, M. & Kanamori, H., 1991. Inversion of complex body wave-III,Bull. seism. Soc. Am., 81, 2335–2350.

Langbein, J. & Johnson, H. 1997. Correlated errors in geodetic time se-ries: implications for time-dependent deformation, J. geophys. Res., 102,591–603.

Lohman, R.B. & Simons M., 2005. Some thoughts on the use of In-SAR data to constrain models of surface deformation: noise struc-ture and data downsampling, Geochem. Geophys. Geosyst., 6, Q01007,doi:10.1029/2004GC000841.

Liu, P. & Archuleta, R.J., 2004. A new nonlinear finite fault inver-sion with tree-dimensional Green’s functions: application to the 1989Loma Prieta, California, earthquake, J. geophys. Res., 109, B02318,doi:10.1029/2003JB002625.

Mai, P., Burjanek, J., Delouis, B., Festa, G., Francois-Holden, C., Monelli, D.,Uchide, T. & Zahradnik, J., 2007. earthquake source inversion blindtest:initial results and further developments, EOS, Trans. Am. geophys. Un.,88, Fall Meet. Suppl., Abstract S53C-08.

Mori, J. & Hartzell, S., 1990. Source inversion of the 1988 Upland, Cali-fornia, earthquake: determination of a fault plane for a small event, Bull.seism. Soc. Am., 80, 507–518.

C© 2011 The Authors, GJI, 186, 711–720

Geophysical Journal International C© 2011 RAS

720 Y. Yagi and Y. Fukahata

Nakano, M., Kumagai, H. & Inoue, H., 2008. Waveform inversion in the fre-quency domain for the simultaneous determination of earthquake sourcemechanism and moment function, Geophys. J. Int., 173, 1000–1011.

Okamoto, T., & Miyatake, T., 1989. Effects of near source seafloor topog-raphy on long-period teleseismic P waveforms, Geophys. Res. Lett., 16,1309–1312.

Okamoto, T. & Takenaka, H., 2009. Waveform inversion for slip distribu-tion of the 2006 Java tsunami earthquake by using 2.5D finite-differenceGreen’s function, Earth Planets Space, 61, e17–e20.

Olson, A.H. & Apsel, R.J., 1982. Finite faults and inverse theory withapplications to the 1979 Imperial Valley earthquake, Bull. seism. Soc.Am., 72, 1969–2001.

Piatanesi, A., Cirella, A., Spudich, P. & Cocco, M., 2007. A global searchinversion for earthquake kinematic rupture history: application to the2000 Western Tottori, Japan earthquake, J. geophys. Res., 112, B07314,doi:10.1029/2006JB004821.

Segall, P., R. Burgmann & M. Matthews, 2000. Time-dependent triggeredafterslip following the 1989 Loma Prieta earthquake, J. geophys. Res.,105, 5615–5634.

Sekiguchi, H., Irikura, K. & Iwata, T., 2000. Fault geometry at the rupturetermination of the 1995 Hyogo-ken Nanbu earthquake, Bull. seism. Soc.Am., 90, 117–133.

Tarantola, A., 1987. Inverse Problem Theory, Methods for Data Fittingand Model Parameter Estimation, pp. 37–51, Elsevier Science PublishersB. V., New York, NY.

Uchide, T. & Ide, S., 2007. Development of multiscale slip inversion methodand its application to the 2004 mid-Niigata Prefecture earthquake, J.geophys. Res., 112, B06313, doi:10.1029/2006JB004528.

Vallee, M. & Bouchon, M., 2004. Imaging coseismic rupture in far field byslip patches, Geophys. J. Int., 156, 615–630.

Wald, D.J. & Graves, R.W., 2001. Resolution analysis of finite fault sourceinversion using one- and three-dimensional Green’s functions 2. Com-bining seismic and geodetic data, J. geophys. Res., 106, 8767–8788.

Wiens, D.A., 1989. Bathymetric effects on body waveforms from shallowsub-duction zone earthquakes and application to seismic processes in theKurile Trench, J. geophys. Res., 94, 2955–2972.

Yabuki, T. & Matsu’ura, M., 1992. Geodetic data inversion using a Baysianinformation criterion for spatial distribution of fault slip, Geophys. J. Int.,109, 363–375.

Yagi, Y. & Fukahata, Y., 2008. Importance of covariance components ininversion analyses of densely sampled observed data: an application towaveform data inversion for seismic source processes, Geophys. J. Int.,175, 215–221.

Yagi, Y. Mikumo, T., Pacheco, J. & Reyes, G., 2004. Source rupture processof the Tecoman, Colima, Mexico earthquake of January 22, 2003, deter-mined by joint inversion of teleseismic body wave and near-field data,Bull. seism. Soc. Am., 94, 1795–1807, doi:10.1785/012003095.

Yoshida, S. & Koketsu, K., 1990. Simultaneous inversion of waveform andgeodetic data for the rupture process of the 1984 Naganoken-seibu, Japan,earthquake, Geophys. J. Int., 103, 355–362.

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