geodetic data inversion using abic to estimate slip ...fukahata/papers/fukahata04gji.pdf ·...

Geophys. J. Int. (2004) 156, 140–153 doi: 10.1111/j.1365-246X.2004.02122.xG

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Geodetic data inversion using ABIC to estimate slip history duringone earthquake cycle with viscoelastic slip-response functions

Yukitoshi Fukahata,� Akira Nishitani and Mitsuhiro Matsu’uraDepartment of Earth and Planetary Science, University of Tokyo, Tokyo 113-0033, Japan. E-mail: [email protected]

Accepted 2003 September 11. Received 2003 June 26; in original form 2002 February 8

S U M M A R YWe developed a new method of geodetic data inversion to estimate slip history at a plate interfaceby using Akaike’s Bayesian Information Criterion (ABIC). In this method we considered theeffects of viscoelastic stress relaxation in the asthenosphere, which cannot be neglected toestimate slip history at a plate interface during one earthquake cycle. We also introduceda proper formulation to incorporate two sorts of partially dependent prior information intoobserved data by Bayes’ rule. By applying the new inversion method to levelling data for 1893–1983 in Shikoku, southwestern Japan, we reconstructed the pattern of space–time variationin slip motion during one earthquake cycle, including the 1946 Nankai earthquake, at theinterface between the Eurasian and the Philippine Sea plates. The result shows that a steadyslip motion at a plate convergence rate (40 mm yr−1) proceeds in the shallow and the deepregions through the entire earthquake cycle. In the intermediate depth range (10–30 km), onthe other hand, an instantaneous slip of approximately 4 m occurs at the time of the Nankaiearthquake. After that, this portion keeps in stationary contact until the occurrence of the nextNankai earthquake. If we neglect the effects of viscoelastic stress relaxation, the inversionanalysis gives geophysically unrealistic results.

Key words: earthquake cycle, inversion, Nankai earthquake, slip history, viscoelasticity.

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

In subduction zones oceanic plates bend and descend beneath continental plates at a constant rate on a long-term average. On a shorttimescale, however, this steady descending motion is disturbed by the repetition of stick and slip in a seismogenic zone at the plateinterface. The purpose of this study is to develop a new method of inversion analysis for estimating space–time variation in slip mo-tion at a plate interface during one earthquake cycle by taking the effects of viscoelastic stress relaxation in the asthenosphere intoaccount.

The elastic dislocation theory quantitatively relates a fault slip distribution with surface displacements (e.g. Maruyama 1964). There-fore, as demonstrated by Matsu’ura (1977a,b) and Yabuki & Matsu’ura (1992), we can estimate a coseismic fault slip distribution fromobserved geodetic data with an inversion method. The method of geodetic data inversion developed by them has later been applied to theproblem of estimating slip deficits during an interseismic period (Matsu’ura et al. 1986; Yoshioka et al. 1993; Sagiya 1999; Nishimuraet al. 2000). In these studies, however, elastic slip-response functions have been used to invert the observed surface displacement data.It is reasonable to use elastic slip-response functions, as far as we analyse short-term crustal movements such as coseismic deforma-tion. For long-term crustal movements, however, the effect of viscoelastic stress relaxation in the asthenosphere cannot be neglected(Thatcher & Rundle 1984; Matsu’ura & Sato 1989). To estimate space–time variation in slip motion at a plate interface during oneearthquake cycle, we must use viscoelastic slip-response functions, instead of elastic slip-response functions. The general expressionsof surface displacements caused by a unit step slip in an elastic-viscoelastic layered medium have been obtained by Matsu’ura et al.(1981) and Matsu’ura & Sato (1989). We can use their expressions as the viscoelastic slip-response functions to invert long-term crustalmovements.

In the mathematical formulation of geodetic data inversion we use a Bayesian information criterion (ABIC) proposed by Akaike (1980)on the basis of the entropy maximization principle (Akaike 1977). With this criterion we can objectively determine the optimal relative

�Corresponding author: Department of Earth Sciences, University of Oxford, Parks Road, Oxford OX1 3PR, UK. E-mail: [email protected]

140 C© 2004 RAS

Inversion using ABIC with slip response 141

weights of information from observed data and prior constraints. Yabuki & Matsu’ura (1992) have developed an inversion method usingABIC, and demonstrated its usefulness through the analysis of geodetic data associated with the 1946 Nankai earthquake. In their analysis,prior constraints on smoothness in spatial distribution of fault slip has been imposed. In the present study, to estimate space–time variation inslip motion, we need to impose prior constraints on smoothness not only in space but also in time. Yoshida (1989) and Ide et al. (1996), forexample, have treated the inverse problem of earthquake rupture processes with the two different sorts of prior constraints. In their formulationthe smoothness constraints in space and time have been assumed to be independent, but actually they are partially dependent.

In Section 2 we formulate a method of geodetic data inversion with the two sorts of partially dependent prior constraints to estimate a sliphistory at a plate interface during one earthquake cycle. In Section 3 we apply this inversion method to a set of levelling data for 1893–1983in Shikoku, southwestern Japan, and demonstrate its validity.

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

2.1 Observation equations for viscoelastic crustal movements

In general, the viscoelastic surface displacements w caused by a slip motion u along a plate interface � can be written in the following formof hereditary integral:

w(x, t) =∫ t

−∞

∫�

G(x, t ; ξ, τ )u̇(ξ, τ ) dξ dτ. (1)

Here, we consider a vertical component of the surface displacements, and so G(x , t ;ξ , τ ) indicates a vertical viscoelastic displacement at atime t and a point x on the Earth’s surface caused by a unit step slip at a time τ and a point ξ on the plate interface. The dot means partialdifferentiation with respect to time. The concrete expression of the viscoelastic slip-response function G(x , t ; ξ , τ ) is given in Matsu’ura& Sato (1989) for a thrust-type point dislocation source and in Sato & Matsu’ura (1993) for a thrust-type line dislocation source. If weare only interested in instantaneous coseismic crustal deformation we can, of course, use an elastic slip-response function G(x , τ+; ξ , τ )instead of the viscoelastic slip-response function G(x , t ; ξ , τ ). In the case of the analysis of long-term crustal movement, however, wecannot neglect the effect of viscoelastic stress relaxation in the asthenosphere, because the effective relaxation time is 10–100 yr in a realisticsituation.

Now we decompose the slip velocity u̇ into the uniform steady slip at a plate convergence rate vpl and its perturbation u̇(ξ, τ ):

u̇(ξ, τ ) = vpl + u̇(ξ, τ ). (2)

The viscoelastic surface displacements due to slip perturbation at a time τ become substantially constant after τ + τ e, where τ e is theeffective relaxation time of the lithosphere-asthenosphere system. Therefore, taking t = t0 as a reference time, we can rewrite the observationeq. (1) as

w(x, t) = vplU�(x)(t − t0) +∫ t

t0−τe

∫�

G(x, t − τ ; ξ, 0)u̇(ξ, τ ) dξ dτ + C(x ; t0) (3)

with

U�(x) =∫

�

G(x, ∞; ξ, 0) dξ, (4)

where C(x;t0) represents some viscoelastic effect that is independent of time t. We parametrize the slip velocity perturbation u̇ by thesuperposition of basis functions X k in space and Tl in time as

u̇(ξ, τ ) =K∑

k=1

L∑l=1

akl Xk(ξ )Tl (τ ), (5)

where akl (k = 1, . . ., K ; l = 1, . . ., L) are the expansion coefficients to be determined from observed geodetic data. The concrete expressionsof X k(ξ ) and Tl(τ ) are given in the next section.

The levelling data give change in the height difference between adjacent points (x = xi and x = xi−1) for a certain period of time ( t= t j − t j−1), and so it can be written as

di j = w(xi , t j ) − w(xi−1, t j ) − w(xi , t j−1) + w(xi−1, t j−1) + ei j , (6)

where eij denote measurement errors (Fukahata et al. 1996). Substituting eqs (3) and (5) into eq. (6), we can obtain the following observationequations:

di j − dsi j =

K∑k=1

L∑l=1

Hi jklakl + ei j , (7)

where

dsi j = vpl(t j − t j−1)[U�(xi ) − U�(xi−1)] (8)

Hi jkl = �kl (xi , t j ) − �kl (xi−1, t j ) − �kl (xi , t j−1) + �kl (xi−1, t j−1) (9)

C© 2004 RAS, GJI, 156, 140–153

142 Y. Fukahata, A. Nishitani and M. Matsu’ura

with

�kl (x, t) =∫ t

t0−τe

∫�

G(x, t − τ ; ξ, 0)Xk(ξ )Tl (τ ) dξ dτ. (10)

Here, dsij indicates the response to the uniform steady slip over the plate interface �, which is treated as a correction term in the following

inversion analysis. Rewriting the observation eq. (7) in a vector form, we obtain

d − ds = Ha + e, (11)

where d, ds and e are N-dimensional vectors, a is an M(= K × L)–dimensional vector, and H is an N × M dimensional matrix. Our problemis to solve the ill-conditioned linear system (11) for the model parameters a.

2.2 Inversion algorithm using ABIC

In geophysical observations, unlike the case of laboratory experiments, data are always inaccurate and insufficient, and so, as pointed outby Backus & Gilbert (1970), the essential problem in geophysical data inversion is how to compromise reciprocal requirements for modelresolution and estimation errors in a natural way. In the present study, to address this problem, we combine two sorts of prior information onthe smoothness of slip motion in space and time with information coming from observed data by using Bayes’ theorem, and construct a highlyflexible model with hyperparameters, called a Bayesian model (Yabuki & Matsu’ura 1992). The Bayesian model consists of a family of usualparametric models. The selection of a specific model from among the family of parametric models can be objectively done by using a Bayesianinformation criterion (ABIC) proposed by Akaike (1980). Once a specific parametric model is selected, we can use the maximum-likelihoodmethod to determine the optimal values of model parameters. In the following part of this subsection we present the concrete expressions ofinversion algorithm.

Assuming a Gaussian distribution with zero mean and unknown variance σ 2 for the data errors e in eq. (11), we can describe a stochasticmodel which relates the data d with the model parameters a as

p(d|a; σ 2) = (2πσ 2)−N/2 exp

[− 1

2σ 2(d − ds − Ha)T (d − ds − Ha)

]. (12)

In addition to the above information, coming from observed data, we have another sort of information concerning the slip velocity perturbationu̇ such that variations of u̇ must be smooth in some degree both in space and time, except for an instantaneous coseismic slip. As a measureof roughness of the slip velocity perturbation we introduce the following quantities;

r1 =∫

T

∫�

[∂2u̇(ξ, τ )/∂ξ 2]2 dξ dτ (13)

r2 =∫

T

∫�

[∂u̇(ξ, τ )/∂τ ]2 dξ dτ, (14)

where the integration is done over the model space–time region. Then, substituting the parametric expansion of u̇ in eq. (5) into eqs (13)and (14), we obtain

r1 =K∑

k=1

L∑l=1

K∑p=1

L∑q=1

akl G1klpqapq (15)

r2 =K∑

k=1

L∑l=1

K∑p=1

L∑q=1

akl G2klpqapq (16)

with

G1klpq =

∫�

∂2 Xk(ξ )

∂ξ 2

∂2 X p(ξ )

∂ξ 2dξ

∫T

Tl (τ )Tq (τ ) dτ (17)

G2klpq =

∫�

Xk(ξ )X p(ξ ) dξ

∫T

∂Tl (τ )

∂τ

∂Tq (τ )

∂τdτ (18)

or, in vector form,

r1 = aT G1a (19)

r2 = aT G2a. (20)

Since the roughness r1 and r2 have a positive-definite quadratic form of the model parameters a, using these quantities, we may express priorconstraints on the roughness of the slip velocity perturbation in the form of a probability density function (pdf) with hyperparameters ρ2

1 andρ2

2 as

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

], (21)

C© 2004 RAS, GJI, 1


where ‖G1/ρ21 + G2/ρ

22‖ represents the absolute value of the determinant of (G1/ρ

21 + G2/ρ

22), which is usually a full-rank M × M

matrix.In the inversion analyses of waveform data using ABIC (e.g. Yoshida 1989; Ide et al. 1996), the prior constraints on the roughness of

slip velocity in space and time have been expressed in the following form of pdf:

p′ (a; ρ21 , ρ

22

) = (2πρ21 )−P1/2(2πρ2

2 )−P2/2‖�1‖1/2‖�2‖1/2 exp

[−aT

(1

2ρ21

G1 + 1

2ρ22

G2

)a

], (22)

where P1 and P2 are the rank of G1 and G2, respectively, and ‖�1‖ and ‖�2‖ represent the absolute value of the product of non-zero eigenvaluesof G1 and G2, respectively. However, eq. (22) is improper, because the constraints on spatial variation and the constraints on temporal variationare partially dependent in usual cases. Actually, if P1 + P2 > M , the two sorts of prior constraints must be partially dependent. In such acase, the prior pdf defined by eq. (22) cannot be normalized correctly:∫

p′(a; ρ21 , ρ

22 ) da �= 1. (23)

Now we incorporate the prior distribution in eq. (21) with the data distribution in eq. (12) by using Bayes’ theorem, and construct ahighly flexible model with the hyperparameters, σ 2, ρ2

1 and ρ22, called a Bayesian model:

p(a; σ 2, ρ2

1 , ρ22 |d ) = cp(d|a; σ 2)p

(a; ρ2

1 , ρ22

), (24)

where c is a normalizing factor independent of the model parameters a and the hyperparameters σ 2, ρ21 and ρ2

2. It should be noted that theprior information on the hyperparameters are assumed to be non-informative. Substituting eqs (12) and (21) into eq. (24), and introducingnew hyperparameters α2(= σ 2/ρ2

1) and β2(= σ 2/ρ22) instead of ρ2

1 and ρ22, we obtain

p(a; σ 2, α2, β2|d) = c(2πσ 2)−(N+M)/2∥∥α2G1 + β2G2

∥∥1/2exp

[− 1

2σ 2s(a)

](25)

with

s(a) = (d − ds − Ha)T (d − ds − Ha) + aT(α2G1 + β2G2

)a. (26)

Our problem is to find the values of a, σ 2, α2 and β2 which maximize the posterior pdf in eq. (25) for given data d. If we fix thehyperparameters σ 2, α2 and β2 to certain values, then the maximum of the posterior pdf is realized by minimizing s(a) in eq. (26). The bestestimates of the model parameters a∗ and also the covariance matrix C for the fixed σ 2, α2 and β2 can be obtained by

a∗ = [HT H + α2G1 + β2G2

]−1HT (d − ds) (27)

C = σ 2[HT H + α2G1 + β2G2

]−1. (28)

Here, we used the following relation (Yabuki & Matsu’ura 1992):

s(a) = s(a∗) + (a − a∗)T(HT H + α2G1 + β2G2

)(a − a∗). (29)

To determine the best estimates of the hyperparameters σ 2, α2 and β2, we can use a Bayesian information criterion (ABIC) proposed byAkaike (1980). In the present case, where the number of adjustable hyperparameters is definite, ABIC is defined by

ABIC = −2 log L(σ 2, α2, β2|d) + C (30)

with

L(σ 2, α2, β2|d) =∫

p(a; σ 2, α2, β2|d) da (31)

and the values of σ 2, α2 and β2 which minimize the ABIC are chosen as the best estimate of the hyperparameters. Here, L(σ 2, α2, β2| d) iscalled the marginal likelihood of σ 2, α2 and β2 for given data d.

Carrying out the integration in eq. (31) with respect to a, we obtain

L(σ 2, α2, β2|d) = c(2πσ 2)−N/2∥∥α2G1 + β2G2

∥∥1/2 ∥∥HT H + α2G1 + β2G2

∥∥−1/2exp

[− 1

2σ 2s(a∗)

]. (32)

The minimum of ABIC is realized by maximizing L(σ 2, α2, β2| d). Thus the necessary conditions for the minimum of ABIC are

∂L

∂σ 2= ∂L

∂α2= ∂L

∂β2= 0. (33)

From the first condition we can analytically obtain

σ 2 = s(a∗)/N . (34)

We substitute eq. (34) into eq. (32). Then, following the definition, we may write the ABIC in the form of

ABIC(α2, β2) = N log s(a∗) − log∥∥α2G1 + β2G2

∥∥ + log∥∥HT H + α2G1 + β2G2

∥∥ + C ′, (35)

C© 2004 RAS, GJI, 156, 140–153


where C′ is independent of α2 and β2. The search for the values of α2 and β2 which minimize the ABIC can be carried out numerically. Oncethe values of α2 and β2 minimizing the ABIC has been found, the best estimates of the model parameters a∗ are directly obtained from eq.(27) by substituting those values.

If we use the improper prior pdf in eq. (22), instead of the proper prior pdf in eq. (21), the expression of ABIC becomes

ABIC(α2, β2) = (N + P1 + P2 − M) log s(a∗) − log α2P1β2P2 + log ‖HT H + α2G1 + β2G2‖ + C ′′. (36)

This expression has a fatal defect that ABIC decreases infinitely as α2 and β2 approach infinity.

3 A P P L I C AT I O N T O L E V E L L I N G DATA I N S H I KO K U

In the preceding section we developed an inversion algorithm to estimate a slip history at a plate interface from a set of coseismic andinterseismic geodetic data. In this section we apply the inversion algorithm to levelling data for 1893–1983 in Shikoku, southwestern Japan,and demonstrate its validity.

3.1 Observed data and modelling

In southwestern Japan the Philippine Sea plate is descending beneath the Eurasian plate along the Nankai trough at the convergence rate ofapproximately 40 mm yr−1 (Seno et al. 1993). A location map of Shikoku and its surrounding area is shown in Fig. 1. Large thrust-typeearthquakes have periodically occurred along this plate boundary with a recurrence time of approximately 100 yr (Ando 1975). The lastand penultimate great events occurred in 1946 (Showa Nankai earthquake, M = 8.1) and in 1854 (Ansei Nankai earthquake, M = 8.4),respectively.

As shown in Fig. 1, we take the x–axis in the direction perpendicular to the strike (N67◦E) of the Nankai trough and project the locationsof levelling points on it. The distances of these levelling points are measured from the trough axis. In the present inversion analysis we use aset of level change data for 1893–1983 along the levelling route from Muroto to Sakaide via Kochi, reported by Geographical Survey Institute(GSI) of Japan. As shown in a space–time diagram (Fig. 2), the first levelling in Shikoku was done in the early 1890s. Since then, the levellingalong the Muroto–Sakaide route has been repeated twice before the 1946 Nankai earthquake and five times after the event in most places.From a set of these levelling data Thatcher (1984) and Fukahata et al. (1996) have reconstructed the space–time pattern of crustal movementsduring one earthquake cycle including the 1946 event.

Figure 1. Location map of Shikoku, southwestern Japan. The open arrows give the relative motion of the Philippine Sea plate to the Eurasian plate at theNankai trough. The thick line with solid squares indicates the levelling route. The x–axis, on which the locations of levelling points are projected, is taken inthe direction perpendicular to the strike (N67◦E) of the Nankai trough. The ξ -axis, along which the horizontal distance of the plate interface from the Nankaitrough is measured, is identical to the x–axis.

C© 2004 RAS, GJI, 1


1900

1920

1940

1960

1980

2000

Tim

e [y

ear]

140 160 180 200 220 240Distance from Nankai Trough [km]

M K S

Figure 2. A diagram showing the time and section of levelling repeated for 1893–1983. The broken line represents the time of Showa Nankai earthquake. M,K and S denote Muroto, Kochi and Sakaide, respectively.

0

10

20

30

40

50

Dep

th [

km]

0 50 100 150 200 250Distance from Nankai Trough [km]

Lithosphere

Asthenosphere

Σ

(ξ)x

Figure 3. The structure model used for the inversion analysis. The structure consists of a 30 km thick elastic surface layer and a viscoelastic half-space undergravity. The plate interface � is indicated by the thick solid line. The P–wave velocity, S–wave velocity and density of the lithosphere are taken as 7.0, 4.0km s−1 and 3.0 × 103 kg m−3, respectively; and those of the asthenosphere as 8.0, 4.5 km s−1 and 3.4 × 103 kg m−3, respectively. The viscosity of theasthenosphere is taken as 5 × 1018 Pa s.

23

0 240

X1 X2 X3

ξ

(ξ1−2∆ξ)

Xk (ξ)

∆ξξ1 ξ2 ξ3

XK

ξK(ξK +2∆ξ)

Figure 4. A diagram showing the basis functions for space.

The structure model used for inverting the observed levelling data is shown in Fig. 3; that is, the crust and mantle structure is modelledby a 30 km thick elastic surface layer overlying a viscoelastic half-space under gravity. The rheological property of the asthenosphere is aMaxwell fluid in shear and an elastic solid in bulk and the viscosity of the asthenosphere is taken to be 5 × 1018 Pa s (Matsu’ura & Iwasaki1983). Mizoue et al. (1983) have estimated the depth to the upper boundary of the descending Philippine Sea plate from the hypocentredistributions of microearthquakes in this region. On the basis of their results, we determine the configuration of the plate interfaces as shownin Fig. 3. Along this plate interface we take a model space region extending from ξ = 0 to 240 km in horizontal distance from the Nankaitrough. The depth of the plate interface at the northernmost point (ξ = 240 km) is approximately 50 km.

As the basis functions to parametrize the slip velocity perturbation along the plate interface, we choose the normalized cubic B-splinesX k (k = 1, . . . K ) (Fig. 4):

C© 2004 RAS, GJI, 156, 140–153


(1946)(2038)(1854)

T1

=

T1

T2T3 T4 T3T51 T2 (TL+1) T2 (TL+1)

τ 0 τ1 τ1τ L−1 τ L−1τ 2 τ 3 τ L−2 τ 2

ττ 4

τ 0

Figure 5. A diagram showing the basis functions for time.

Xk(ξ ) = 1

6ξ 3×

(ξ − ξk + 2ξ )3 (ξk − 2ξ < ξ < ξk − ξ )

−3(ξ − ξk)3 − 6ξ (ξ − ξk)2 + 4ξ 3 (ξk − ξ < ξ < ξk)

3(ξ − ξk)3 − 6ξ (ξ − ξk)2 + 4ξ 3 (ξk < ξ < ξk + ξ )

−(ξ − ξk − 2ξ )3 (ξk + ξ < ξ < ξk + 2ξ )

0 (ξ < ξk − 2ξ, ξ > ξk + 2ξ )

(37)

with

ξ = ξk − ξk−1

ξ1 − 2ξ = 0

ξK + 2ξ = 240.

(38)

In the present case, we divide the model space region into 20 subsections (K = 17), and so ξ is 12 km.As to time, we model a period covering the recent two cycles of Nankai earthquakes (1854–2038). It should be noted that the levelling

data for 1893–1983 not only include the coseismic elastic and post-seismic viscoelastic crustal deformation associated with the 1946 event,but also the post-seismic viscoelastic deformation associated with the 1854 event. This is the reason why it is necessary to take such a longperiod as the model time region in the analysis. In practice, however, we do not have sufficient data to determine the slip history over the twoearthquake cycles. Fortunately, it is known that the 1946 and 1854 Nankai earthquakes were quite similar in many respects, including thespatial pattern and amplitude of crustal deformation (Kawasumi & Sato 1949). In the present analysis, for simplicity, we assume that the sliphistories of the last and penultimate earthquake cycles are completely the same.

As the basis functions to parametrize the slip velocity perturbation in time, we take a delta function T1 and the linear B-splines T l (l =2, . . . , L + 1) (Fig. 5):

T1(τ ) = T δ(τ − τ0) (39)

Tl (τ ) = 1

τ

(τ − τl−2 + τ ) (τl−3 < τ < τl−2)

(−τ + τl−2 + τ ) (τl−2 < τ < τl−1)

0 (τ < τl−3, τ > τl−1)

(40)

with

τ = τl − τl−1

τ0 = 1854 or 1946

τL−1 = τ0 + T .

(41)

The delta function T1 is needed for representing instantaneous coseismic slip. In the present case, we take the recurrence interval T ofearthquakes to be 92 yr and L to be 10, and so τ is approximately 10 yr. Since we assume the completely cyclic slip history, the first linearB-spline T2 is equal to the last linear B-spline TL+1.

The delta function T1, which represents instantaneous coseismic slip motion, has a special property, and so the careful treatment of itis needed in the expression of prior constraints. First, the delta function T1 should be free from the smoothness constraints in time, since itrepresents instantaneous coseismic motion:∫

T

∂T1(τ )

∂τ

∂Tl (τ )

∂τdτ = 0, (l = 1, . . . , L). (42)

Here and in the following equations, it is important not to confuse the subscripts 1 (one) and l (el). Secondly, we may assume that theinstantaneous coseismic slip motion T1 and other interseismic gradual slip motion T l (l = 2, . . ., L) have no correlation in time:∫

TT1(τ )Tl (τ ) dτ = 0, (l = 2, . . . , L). (43)

C© 2004 RAS, GJI, 1


0

2

4

6

log(

β2 )

(a)

0

2

4

6

log(

β2 )(b)

0

2

4

6

log(

β2 )

(c)

0

2

4

6

log(

β2 )

0 2 4 6log(α2)

(d)

Figure 6. Contour maps of ABIC(α2, β2) calculated with viscoelastic slip-response functions for given γ : (a) γ = 0.1, (b) γ = 1, (c) γ = 10 and (d) γ =100. The contour intervals are taken to be 3 in the values of ABIC.

For the autocorrelation of T1 we need a special treatment, because it formally diverges to infinity. Thus, we introduce a new parameter γ ,∫T

T1(τ )T1(τ ) dτ = γ

∫T

Tl (τ )Tl (τ ) dτ , (l = 2, . . . , L), (44)

which controls the relative weight of coseismic slip variation to interseismic slip variation in space. If we take larger γ , the smoothnessconstraint on coseismic slip in space becomes more dominant. For smaller γ , on the other hand, the smoothness constraint on interseismicslip in space becomes more dominant. In the following inversion analysis, we treat γ as an independent model control parameter.

3.2 Inverted slip history at the plate interface

For given γ (0.1, 1, 10 and 100), we compute the values of ABIC(α2, β2) from eq. (35) with viscoelastic slip-response functions, and plotthem in contour maps (Fig. 6). The point of the ABIC minimum in each diagram gives the best estimates of the hyperparameters α2 and β2

for a given γ . With these values of α2 and β2 we can directly compute the optimal values of the model parameters a from eq. (27). In theleft-hand column of Fig. 7 we show the slip histories u(ξ , τ ) during one earthquake cycle at the plate interface, reconstructed from eqs (2) and

C© 2004 RAS, GJI, 156, 140–153


050 100 150 200

020

4060

80

12

34

Tota

l Slip

(m

)

(a)

050 100 150 200

020

4060

80

12

34

To

tal S

lip (

m)

(b)

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Figure 7. Slip histories u(ξ , τ ) during one earthquake cycle at the plate interface (left-hand column), and the spatial distributions of instantaneous coseismicslip (solid lines) with those of total slip deficits (broken lines) just before the occurrence of the next Nankai earthquake (right-hand column). (a)–(d) representthe results for γ = 0.1, 1, 10 and 100, respectively. Viscoelastic slip-response functions are used. The time is measured from just after the occurrence of theNankai earthquake, and the distance is measured from the Nankai trough.

(5) with the optimal values of a for given γ . In each diagram, the time is measured from just after the occurrence of the Nankai earthquake,and the distance is measured from the Nankai trough. In the right-hand column of Fig. 7 we show the spatial distributions of instantaneouscoseismic slip (solid lines) with those of total slip deficits (broken lines) just before the occurrence of the next Nankai earthquake.

The slip-history diagrams in Fig. 7 indicate that the inverted results strongly depend on the choice of γ . If we choose a too small γ , as inthe case of (a), the smoothness constraint on interseismic slip becomes too strong in comparison with that on coseismic slip. In this case theinverted result gives a very smooth interseismic slip distribution both in space and time. On the contrary, if we choose a too large γ , as in thecase of (d), the smoothness constraint on interseismic slip becomes too weak in comparison with that on coseismic slip. In this case the invertedresult gives a very rough interseismic slip distribution both in space and time. These qualitative criteria are useful for excluding geophysicallyunrealistic models, but not for selecting the most likely model from among many likely models. As a criterion for the model selection, wemay use a prior constraint on earthquake cycles imposed by plate tectonics; that is, the total slip deficits just before the occurrence of the nextearthquake should be almost the same as the amount of instantaneous coseismic slip (Fukuyama et al. 2002; Hashimoto & Matsu’ura 2002).Applying this criterion to the inverted results of coseismic slip and total slip deficits in the right-hand column of Fig. 7, we can objectivelyselect the case (c) with γ = 10 as the most likely model of slip history at the Nankai subduction zone.

The most likely model in Fig. 7(c) shows that an instantaneous slip of approximately 4 m occurs at the time of the Nankai earthquake in theintermediate depth range (10–30 km). After that, this portion keeps in stationary contact until the occurrence of the next Nankai earthquake. Inthe shallow (<5 km) and the deep regions (>40 km), on the other hand, steady slip at the plate convergence rate (40 mm yr−1) proceeds throughthe entire earthquake cycle. Given such a slip history, we can calculate the absolute crustal movements during one earthquake cycle in Shikoku

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from eq. (3) as shown in Fig. 8. In this 3-D diagram the absolute height of the Earth’s surface just before the occurrence of the 1946 Nankaiearthquake is taken as a reference height. Fukahata et al. (1996) have obtained a similar space–time pattern of crustal movements in Shikokuduring the earthquake cycle, but from a direct inversion of levelling data. From the 3-D diagram in Fig. 8 we can calculate the difference inabsolute height between two arbitrary times. The solid lines in Fig. 9 are the profiles of vertical displacements calculated in this way, and thesolid squares are the height changes obtained from levelling data. The diagrams (a)–(d) correspond to the preseismic, coseismic, post-seismicand interseismic periods, respectively. Here, it should be noted that the observed coseismic displacements (b) include some preseismic andpost-seismic displacements. From comparison with the calculated vertical displacements and observed data in these diagrams we can seethat the inverted slip history explains the observed crustal movements well except for the southern half of the interseismic movements (d).Thatcher (1984) and Fukahata et al. (1996) have pointed out the fast subsidence in the northernmost part of Shikoku from the analysis oftide-gauge data. The inverted slip history model cannot reproduce this subsidence motion, which might be caused by the 3-D effect of thedescending slab (Hashimoto et al. 2003).

If we ignore the effects of viscoelastic stress relaxation in the asthenosphere and use elastic slip-response functions in the analysis, theinverted results will dramatically change. In Fig. 10 we show the contour maps of ABIC(α2, β2) for given γ (0.1, 1.0, 10 and 100). Evenin the elastic case, we can find the clear ABIC minima for given γ . And so, with the best estimates of α2 and β2, we compute the optimalvalues of the model parameters and reconstruct the slip histories u(ξ , τ ) at the plate interface for given γ as shown in the left-hand columnof Fig. 11. From comparison of the slip-history diagrams in Figs 7 and 11, we can find remarkable differences in the inverted slip patternbetween the viscoelastic case and the elastic case. To check the reality of these inverted models, we compare the instantaneous coseismic slipdistributions (solid lines) with the total slip deficit distributions (broken lines) in the right-hand column of Fig. 11. From these diagrams wecan see that any model (even the case (c) with γ = 10) inverted with elastic slip-response functions does not satisfy the constraint that thetotal slip deficits just before the occurrence of the next earthquake should be almost the same as the amount of instantaneous coseismic slip.This means that the use of viscoelastic slip-response functions is essentially important in the analysis of long-term crustal movements.

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

We developed an algorithm of geodetic data inversion to estimate slip history during one earthquake cycle at a plate interface with viscoelasticslip-response functions. So far elastic slip-response functions have been widely used for inverting geodetic data. The use of elastic slip-responsefunctions is reasonable, if our interest is limited to short-term crustal movements such as coseismic deformation. In the analysis of long-termcrustal movements, however, the use of elastic slip-response functions definitely leads us to incorrect results as demonstrated in Section 3.2.In order to correctly estimate the long-term slip history at a plate boundary, we cannot neglect the effects of viscoelastic stress relaxation inthe asthenosphere.

We obtained a proper formulation to incorporate two sorts of partially dependent prior information into observed data, and improvedthe Bayesian inversion algorithm developed by Yabuki & Matsu’ura (1992). In this inversion algorithm, first, we construct a highly flexible

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Figure 9. Comparison of the profiles of vertical displacements (solid lines) along the levelling route, calculated from the inverted slip history in Fig. 7(c), withobserved data (solid squares). The diagrams (a)–(d) correspond to the crustal movements for the preseismic, coseismic, post-seismic and interseismic periods,respectively.

parametric model with hyperparameters by incorporating smoothness constraints on spatial and temporal slip variations into observed datawith Bayes’ theorem. Next, we select a specific model from among the family of parametric models by minimizing ABIC. The smoothnessconstraints on spatial and temporal slip variations are partially dependent in usual cases. In the old formulation, however, they have beenassumed to be completely independent. If ABIC is defined on the basis of the improper formulation, as pointed out in Section 2.2, it decreasesinfinitely as the hyperparameters approach infinity. This problem, which is very serious in the ill-conditioned inversion analysis (Fukahataet al. 2003), was completely resolved in the new formulation.

Another problem in the inversion algorithm is how to treat the prior constraint for earthquake cycles in relation to the model controlparameter γ , which prescribes the relative weight of coseismic slip variation to interseismic slip variation in space. In the inversion algorithmwe explicitly imposed the local prior constraints that the slip motion should be smooth both in space and time. In addition to the local priorconstraints we have a global prior constraint for slip imposed by plate tectonics; that is, the spatial distribution of instantaneous coseismicslip should be almost the same as that of total slip deficits just before the occurrence of the next earthquake. If the global prior constraint wasincorporated into the inversion algorithm from the beginning, we did not need the final process to select the most likely model from amongABIC minimum models with different γ . In this case, however, we would need one more hyperparameter, in addition to σ , α, β and γ , todescribe the global prior constraint, and we would have to numerically search the best estimates of the hyperparameters in the 4-D parametric

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space, which would not be practical. It should be noted that γ controls the relative weight of the smoothness of coseismic slip and interseismicslip. Therefore, γ has an inherent property to connect the local smoothness constraints with the global constraint. Thus, externally controllingγ as an independent parameter, we can find an inverted model which satisfies the global constraint as well as the local smoothness constraints,as demonstrated in Section 3.2.

In the case with elastic slip-response functions (Fig. 11), no model could satisfy the global constraint that the spatial distribution ofinstantaneous coseismic slip should be almost the same as that of total slip deficits just before the occurrence of the next earthquake, althoughall the inverted models well fitted the data and satisfied the local smoothness constraints on slip. If the global constraint was explicitlyincorporated into the inversion algorithm, the inverted result would show the improvement on the global constraint, but the fitting of data andthe smoothness of slip would become still worse.

In the present study, applying the global prior constraint, we objectively selected the ABIC minimum solution with γ = 10 in Fig. 7(c) asthe most likely model of slip history at the Nankai subduction zone. This model indicates that instantaneous slip of approximately 4 m occursat the time of the Nankai earthquake in the intermediate depth range. After that, this portion keeps in stationary contact until the occurrenceof the next Nankai earthquake. In the shallow and deep regions, on the other hand, steady slip motion at approximately 40 mm yr−1 proceeds

C© 2004 RAS, GJI, 156, 140–153


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Figure 11. Slip histories u(ξ , τ ) at the plate interface (left-hand column), and the instantaneous coseismic slip distributions (solid lines) with the total slipdeficit distributions (broken lines) (right-hand column), inverted with elastic slip-response functions. (a)–(d) represent the results for γ = 0.1, 1, 10 and 100,respectively. The time is measured from just after the occurrence of the Nankai earthquake, and the distance is measured from the Nankai trough.

through the entire earthquake cycle. Here, it should be noted that a slip motion in the offshore region is detectable in a practical level throughthe viscoelastic response of the Earth. A slip in the offshore region causes little deformation on land as long as the Earth behaves like anelastic half-space. After the completion of viscoelastic stress relaxation in the asthenosphere, however, the Earth responds as an elastic plate.The slip response for the elastic plate has much less geometrical attenuation than that for the elastic half-space, since the plate is a body oftwo dimensions, and so a slip in the offshore region causes a substantial crustal deformation on land after the viscoelastic relaxation of theasthenosphere. The remarkable difference in coseismic slip distribution between Figs 7 and 11 is mainly derived from this reason.

At the Nankai subduction zone a similar slip history to Fig. 7(c) has been obtained by Stuart (1988) through numerical simulation of theearthquake cycle with a rate- and state-dependent friction law. For the first time we have succeeded in reconstructing the actual slip historyduring one earthquake cycle at the subduction zone from the inversion analysis of geodetic data.

A C K N O W L E D G M E N T S

We thank Yuji Yagi, Paul Segall and the anonymous reviewer for their useful comments.

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