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

Geophys. J. Int. () 15, 1 153 doi: 1.1111/j.135-X..1.x Geodetic data inversion using ABIC to estimate slip history during one earthquake cycle with viscoelastic slip-response functions Yukitoshi Fukahata, Akira Nishitani and Mitsuhiro Matsu ura Department of Earth and Planetary Science, University of Tokyo, Tokyo 113-33, Japan. E-mail: fukahata@eps.s.u-tokyo.ac.jp Accepted 3 September 11. Received 3 June ; in original form February GJI Tectonics and geodynamics 1 INTRODUCTION SUMMARY We developed a new method of geodetic data inversion to estimate slip history at a plate interface by using Akaike s Bayesian Information Criterion (ABIC). In this method we considered the effects of viscoelastic stress relaxation in the asthenosphere, which cannot be neglected to estimate slip history at a plate interface during one earthquake cycle. We also introduced a proper formulation to incorporate two sorts of partially dependent prior information into observed data by Bayes rule. By applying the new inversion method to levelling data for 193 193 in Shikoku, southwestern Japan, we reconstructed the pattern of space time variation in slip motion during one earthquake cycle, including the 19 Nankai earthquake, at the interface between the Eurasian and the Philippine Sea plates. The result shows that a steady slip motion at a plate convergence rate ( mm yr 1 ) proceeds in the shallow and the deep regions through the entire earthquake cycle. In the intermediate depth range (1 3 km), on the other hand, an instantaneous slip of approximately m occurs at the time of the Nankai earthquake. After that, this portion keeps in stationary contact until the occurrence of the next Nankai earthquake. If we neglect the effects of viscoelastic stress relaxation, the inversion analysis gives geophysically unrealistic results. Key words: earthquake cycle, inversion, Nankai earthquake, slip history, viscoelasticity. In subduction zones oceanic plates bend and descend beneath continental plates at a constant rate on a long-term average. On a short timescale, however, this steady descending motion is disturbed by the repetition of stick and slip in a seismogenic zone at the plate interface. The purpose of this study is to develop a new method of inversion analysis for estimating space time variation in slip motion at a plate interface during one earthquake cycle by taking the effects of viscoelastic stress relaxation in the asthenosphere into account. The elastic dislocation theory quantitatively relates a fault slip distribution with surface displacements (e.g. Maruyama 19). Therefore, as demonstrated by Matsu ura (1977a,b) and Yabuki & Matsu ura (199), we can estimate a coseismic fault slip distribution from observed geodetic data with an inversion method. The method of geodetic data inversion developed by them has later been applied to the problem of estimating slip deficits during an interseismic period (Matsu ura et al. 19; Yoshioka et al. 1993; Sagiya 1999; Nishimura et al. ). 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 deformation. For long-term crustal movements, however, the effect of viscoelastic stress relaxation in the asthenosphere cannot be neglected (Thatcher & Rundle 19; Matsu ura & Sato 199). To estimate space time variation in slip motion at a plate interface during one earthquake cycle, we must use viscoelastic slip-response functions, instead of elastic slip-response functions. The general expressions of surface displacements caused by a unit step slip in an elastic-viscoelastic layered medium have been obtained by Matsu ura et al. (191) and Matsu ura & Sato (199). We can use their expressions as the viscoelastic slip-response functions to invert long-term crustal movements. In the mathematical formulation of geodetic data inversion we use a Bayesian information criterion (ABIC) proposed by Akaike (19) 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: fukahata@earth.ox.ac.uk 1 C RAS

Inversion using ABIC with slip response 11 weights of information from observed data and prior constraints. Yabuki & Matsu ura (199) have developed an inversion method using ABIC, and demonstrated its usefulness through the analysis of geodetic data associated with the 19 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 in slip motion, we need to impose prior constraints on smoothness not only in space but also in time. Yoshida (199) and Ide et al. (199), for example, have treated the inverse problem of earthquake rupture processes with the two different sorts of prior constraints. In their formulation the smoothness constraints in space and time have been assumed to be independent, but actually they are partially dependent. In Section we formulate a method of geodetic data inversion with the two sorts of partially dependent prior constraints to estimate a slip history at a plate interface during one earthquake cycle. In Section 3 we apply this inversion method to a set of levelling data for 193 193 in Shikoku, southwestern Japan, and demonstrate its validity. MATHEMATICAL FORMULATION.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 form of hereditary integral: t w(x, t) = G(x, t; ξ,τ) u(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 a time 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 partial differentiation with respect to time. The concrete expression of the viscoelastic slip-response function G(x, t; ξ, τ) is given in Matsu ura & Sato (199) for a thrust-type point dislocation source and in Sato & Matsu ura (1993) for a thrust-type line dislocation source. If we are 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, we cannot neglect the effect of viscoelastic stress relaxation in the asthenosphere, because the effective relaxation time is 1 1 yr in a realistic situation. Now we decompose the slip velocity u into the uniform steady slip at a plate convergence rate v pl and its perturbation u(ξ,τ): u(ξ,τ) = v pl + u(ξ,τ). () The viscoelastic surface displacements due to slip perturbation at a time τ become substantially constant after τ + τ e, where τ e is the effective relaxation time of the lithosphere-asthenosphere system. Therefore, taking t = t as a reference time, we can rewrite the observation eq. (1) as t w(x, t) = v pl U (x)(t t ) + G(x, t τ; ξ,) u(ξ,τ) dξ dτ + C(x; t ) (3) with U (x) = G(x, ; ξ,) dξ, t τ e where C(x;t ) represents some viscoelastic effect that is independent of time t. We parametrize the slip velocity perturbation u by the superposition of basis functions X k in space and T l in time as K L u(ξ,τ) = a kl X k (ξ)t l (τ), (5) k=1 l=1 where a kl (k = 1,..., K ; l = 1,..., L) are the expansion coefficients to be determined from observed geodetic data. The concrete expressions of X k (ξ) and T l (τ) are given in the next section. The levelling data give change in the height difference between adjacent points (x = x i and x = x i 1 ) for a certain period of time ( t = t j t j 1 ), and so it can be written as d ij = w(x i, t j ) w(x i 1, t j ) w(x i, t j 1 ) + w(x i 1, t j 1 ) + e ij, () where e ij denote measurement errors (Fukahata et al. 199). Substituting eqs (3) and (5) into eq. (), we can obtain the following observation equations: K L d ij d s ij = H ijkl a kl + e ij, (7) k=1 l=1 where d s ij = v pl (t j t j 1 )[U (x i ) U (x i 1 )] () H ijkl = kl (x i, t j ) kl (x i 1, t j ) kl (x i, t j 1 ) + kl (x i 1, t j 1 ) (9) C RAS, GJI, 15, 1 153 ()

1 Y. Fukahata, A. Nishitani and M. Matsu ura with kl (x, t) = t t τ e G(x, t τ; ξ,)x k (ξ)t l (τ) dξ dτ. (1) Here, d s ij 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 d s = Ha + e, (11) where d, d s and e are N-dimensional vectors, a is an M(= K L) dimensional vector, and H is an N M dimensional matrix. Our problem is to solve the ill-conditioned linear system (11) for the model parameters a.. Inversion algorithm using ABIC In geophysical observations, unlike the case of laboratory experiments, data are always inaccurate and insufficient, and so, as pointed out by Backus & Gilbert (197), the essential problem in geophysical data inversion is how to compromise reciprocal requirements for model resolution and estimation errors in a natural way. In the present study, to address this problem, we combine two sorts of prior information on the smoothness of slip motion in space and time with information coming from observed data by using Bayes theorem, and construct a highly flexible model with hyperparameters,called a Bayesian model (Yabuki &Matsu ura 199). The Bayesian model consists of a family of usual parametric models. The selection of a specific model from among the family of parametric models can be objectively done by using a Bayesian information criterion (ABIC) proposed by Akaike (19). Once a specific parametric model is selected, we can use the maximum-likelihood method to determine the optimal values of model parameters. In the following part of this subsection we present the concrete expressions of inversion algorithm. Assuming a Gaussian distribution with zero mean and unknown variance σ for the data errors e in eq. (11), we can describe a stochastic model which relates the data d with the model parameters a as [ p(d a; σ ) = (πσ ) N/ exp 1 ] σ (d ds Ha) T (d d s Ha). (1) In addition to the above information, coming from observed data, we have another sort of information concerning the slip velocity perturbation u such that variations of u must be smooth in some degree both in space and time, except for an instantaneous coseismic slip. As a measure of roughness of the slip velocity perturbation we introduce the following quantities; r 1 = [ u(ξ,τ)/ ξ ] dξ dτ (13) T T r = [ u(ξ,τ)/ τ] dξ dτ, where the integration is done over the model space time region. Then, substituting the parametric expansion of u in eq. (5) into eqs (13) and (1), we obtain K L K L r 1 = a kl G 1 klpq a pq (15) k=1 l=1 p=1 q=1 (1) r = K L K k=1 l=1 p=1 q=1 with G 1 klpq = G klpq = L a kl G klpq a pq X k (ξ) X p (ξ) dξ T ξ ξ l (τ)t q (τ) dτ T X k (ξ)x p (ξ) dξ T T l (τ) T q (τ) dτ τ τ (1) (17) (1) or, in vector form, r 1 = a T G 1 a (19) r = a T G a. Since the roughness r 1 and r have a positive-definite quadratic form of the model parameters a, using these quantities, we may express prior constraints on the roughness of the slip velocity perturbation in the form of a probability density function (pdf) with hyperparameters ρ 1 and ρ as p ( ) a; ρ 1,ρ = (π) M/ 1 G ρ1 1 + 1 1/ [ ( 1 G ρ exp a T G ρ1 1 + 1 ) ] G ρ a, (1) () C RAS, GJI, 1

Inversion using ABIC with slip response 13 where G 1 /ρ 1 + G /ρ represents the absolute value of the determinant of (G 1/ρ 1 + G /ρ ), which is usually a full-rank M M matrix. In the inversion analyses of waveform data using ABIC (e.g. Yoshida 199; Ide et al. 199), the prior constraints on the roughness of slip velocity in space and time have been expressed in the following form of pdf: p ( a; ρ 1,ρ ) = (πρ 1 ) P 1/ (πρ ) P / 1 1/ 1/ exp [ ( 1 a T G ρ1 1 + 1 ) G ρ a ], () where P 1 and P are the rank of G 1 and G, respectively, and 1 and represent the absolute value of the product of non-zero eigenvalues of G 1 and G, respectively. However, eq. () is improper, because the constraints on spatial variation and the constraints on temporal variation are partially dependent in usual cases. Actually, if P 1 + P > M, the two sorts of prior constraints must be partially dependent. In such a case, the prior pdf defined by eq. () cannot be normalized correctly: p (a; ρ 1,ρ ) da 1. (3) Now we incorporate the prior distribution in eq. (1) with the data distribution in eq. (1) by using Bayes theorem, and construct a highly flexible model with the hyperparameters, σ, ρ 1 and ρ, called a Bayesian model: p ( a; σ,ρ 1,ρ d ) = cp(d a; σ )p ( a; ρ 1,ρ ), where c is a normalizing factor independent of the model parameters a and the hyperparameters σ, ρ 1 and ρ. It should be noted that the prior information on the hyperparameters are assumed to be non-informative. Substituting eqs (1) and (1) into eq. (), and introducing new hyperparameters α (= σ /ρ 1 ) and β (= σ /ρ ) instead of ρ 1 and ρ, we obtain p(a; σ,α,β d) = c(πσ ) [ (N+M)/ α G 1 + β G 1/ exp 1 ] σ s(a) (5) with s(a) = (d d s Ha) T (d d s Ha) + a ( ) T α G 1 + β G a. () () Our problem is to find the values of a, σ, α and β which maximize the posterior pdf in eq. (5) for given data d. Ifwefix the hyperparameters σ, α and β to certain values, then the maximum of the posterior pdf is realized by minimizing s(a) in eq. (). The best estimates of the model parameters a and also the covariance matrix C for the fixed σ, α and β can be obtained by a = [ ] H T H + α G 1 + β 1 G H T (d d s ) (7) C = σ [ ] H T H + α G 1 + β 1 G. () Here, we used the following relation (Yabuki & Matsu ura 199): s(a) = s(a ) + (a a ) ( ) T H T H + α G 1 + β G (a a ). (9) To determine the best estimates of the hyperparameters σ, α and β, we can use a Bayesian information criterion (ABIC) proposed by Akaike (19). In the present case, where the number of adjustable hyperparameters is definite, ABIC is defined by ABIC = logl(σ,α,β d) + C (3) with L(σ,α,β d) = p(a; σ,α,β d) da (31) and the values of σ, α and β which minimize the ABIC are chosen as the best estimate of the hyperparameters. Here, L(σ, α, β d) is called the marginal likelihood of σ, α and β for given data d. Carrying out the integration in eq. (31) with respect to a, we obtain L(σ,α,β d) = c(πσ ) N/ α G 1 + β G 1/ [ H T H + α G 1 + β G 1/ exp 1 ] σ s(a ). (3) The minimum of ABIC is realized by maximizing L(σ, α, β d). Thus the necessary conditions for the minimum of ABIC are L σ = L α = L β =. From the first condition we can analytically obtain σ = s(a )/N. We substitute eq. (3) into eq. (3). Then, following the definition, we may write the ABIC in the form of ABIC(α,β ) = N log s(a ) log α G 1 + β G + log H T H + α G 1 + β G + C, (35) C RAS, GJI, 15, 1 153 (33) (3)

1 Y. Fukahata, A. Nishitani and M. Matsu ura where C is independent of α and β. The search for the values of α and β which minimize the ABIC can be carried out numerically. Once the values of α and β minimizing the ABIC has been found, the best estimates of the model parameters a are directly obtained from eq. (7) by substituting those values. If we use the improper prior pdf in eq. (), instead of the proper prior pdf in eq. (1), the expression of ABIC becomes ABIC(α,β ) = (N + P 1 + P M)logs(a ) log α P 1 β P + log H T H + α G 1 + β G +C. (3) This expression has a fatal defect that ABIC decreases infinitely as α and β approach infinity. 3 APPLICATION TO LEVELLING DATA IN SHIKOKU In the preceding section we developed an inversion algorithm to estimate a slip history at a plate interface from a set of coseismic and interseismic geodetic data. In this section we apply the inversion algorithm to levelling data for 193 193 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 of approximately mm yr 1 (Seno et al. 1993). A location map of Shikoku and its surrounding area is shown in Fig. 1. Large thrust-type earthquakes have periodically occurred along this plate boundary with a recurrence time of approximately 1 yr (Ando 1975). The last and penultimate great events occurred in 19 (Showa Nankai earthquake, M =.1) and in 15 (Ansei Nankai earthquake, M =.), respectively. As shown in Fig. 1, we take the x axis in the direction perpendicular to the strike (N7 E) of the Nankai trough and project the locations of levelling points on it. The distances of these levelling points are measured from the trough axis. In the present inversion analysis we use a set of level change data for 193 193 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. ), the first levelling in Shikoku was done in the early 19s. Since then, the levelling along the Muroto Sakaide route has been repeated twice before the 19 Nankai earthquake and five times after the event in most places. From a set of these levelling data Thatcher (19) and Fukahata et al. (199) have reconstructed the space time pattern of crustal movements during one earthquake cycle including the 19 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 the Nankai 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 in the direction perpendicular to the strike (N7 E) of the Nankai trough. The ξ-axis, along which the horizontal distance of the plate interface from the Nankai trough is measured, is identical to the x axis. C RAS, GJI, 1

Inversion using ABIC with slip response 15 19 19 Time [year] 19 19 19 M K S 1 1 1 Distance from Nankai Trough [km] Figure. A diagram showing the time and section of levelling repeated for 193 193. The broken line represents the time of Showa Nankai earthquake. M, K and S denote Muroto, Kochi and Sakaide, respectively. Depth [km] Distance from Nankai Trough [km] 5 1 15 5 1 3 5 Lithosphere Asthenosphere Σ x (ξ) Figure 3. The structure model used for the inversion analysis. The structure consists of a 3 km thick elastic surface layer and a viscoelastic half-space under gravity. 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.,. km s 1 and 3. 1 3 kg m 3, respectively; and those of the asthenosphere as.,.5 km s 1 and 3. 1 3 kg m 3, respectively. The viscosity of the asthenosphere is taken as 5 1 1 Pa s. X k (ξ) 3 X 1 X X 3 X K ξ 1 ξ ξ 3 ξ ξ K (ξ 1 ξ) Figure. A diagram showing the basis functions for space. (ξ K + ξ) ξ The structure model used for inverting the observed levelling data is shown in Fig. 3; that is, the crust and mantle structure is modelled by a 3 km thick elastic surface layer overlying a viscoelastic half-space under gravity. The rheological property of the asthenosphere is a Maxwell fluid in shear and an elastic solid in bulk and the viscosity of the asthenosphere is taken to be 5 1 1 Pa s (Matsu ura & Iwasaki 193). Mizoue et al. (193) have estimated the depth to the upper boundary of the descending Philippine Sea plate from the hypocentre distributions of microearthquakes in this region. On the basis of their results, we determine the configuration of the plate interfaces as shown in Fig. 3. Along this plate interface we take a model space region extending from ξ = to km in horizontal distance from the Nankai trough. The depth of the plate interface at the northernmost point (ξ = km) is approximately 5 km. As the basis functions to parametrize the slip velocity perturbation along the plate interface, we choose the normalized cubic B-splines X k (k = 1,... K ) (Fig. ): C RAS, GJI, 15, 1 153

1 Y. Fukahata, A. Nishitani and M. Matsu ura T 1 T 1 1 T T 3 T T 5 T 3 T (T L+1 ) T (T L+1 ) τ τ τ 1 τ τ 3 τ τ L τ L 1 τ 1 τ τ L 1 (3) τ (19) (15) Figure 5. A diagram showing the basis functions for time. = (ξ ξ k + ξ) 3 (ξ k ξ<ξ<ξ k ξ) 3(ξ ξ k ) 3 ξ(ξ ξ k ) + ξ 3 (ξ k ξ<ξ<ξ k ) X k (ξ) = 1 ξ 3(ξ ξ k ) 3 ξ(ξ ξ k ) + ξ 3 (ξ k <ξ<ξ k + ξ) 3 (ξ ξ k ξ) 3 (ξ k + ξ<ξ<ξ k + ξ) (ξ<ξ k ξ, ξ > ξ k + ξ) with ξ = ξ k ξ k 1 ξ 1 ξ = ξ K + ξ =. (37) (3) In the present case, we divide the model space region into subsections (K = 17), and so ξis 1 km. As to time, we model a period covering the recent two cycles of Nankai earthquakes (15 3). It should be noted that the levelling data for 193 193 not only include the coseismic elastic and post-seismic viscoelastic crustal deformation associated with the 19 event, but also the post-seismic viscoelastic deformation associated with the 15 event. This is the reason why it is necessary to take such a long period as the model time region in the analysis. In practice, however, we do not have sufficient data to determine the slip history over the two earthquake cycles. Fortunately, it is known that the 19 and 15 Nankai earthquakes were quite similar in many respects, including the spatial pattern and amplitude of crustal deformation (Kawasumi & Sato 199). In the present analysis, for simplicity, we assume that the slip histories 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 T 1 and the linear B-splines T l (l =,..., L + 1) (Fig. 5): T 1 (τ) = T δ(τ τ ) (39) (τ τ l + τ) (τ l 3 <τ<τ l ) T l (τ) = 1 ( τ + τ l + τ) (τ l <τ<τ l 1 ) τ (τ<τ l 3,τ >τ l 1 ) () with τ = τ l τ l 1 τ = 15 or 19 τ L 1 = τ + T. The delta function T 1 is needed for representing instantaneous coseismic slip. In the present case, we take the recurrence interval T of earthquakes to be 9 yr and L to be 1, and so τ is approximately 1 yr. Since we assume the completely cyclic slip history, the first linear B-spline T is equal to the last linear B-spline T L+1. The delta function T 1, which represents instantaneous coseismic slip motion, has a special property, and so the careful treatment of it is needed in the expression of prior constraints. First, the delta function T 1 should be free from the smoothness constraints in time, since it represents instantaneous coseismic motion: T 1 (τ) T l (τ) dτ =, (l = 1,...,L). () T τ τ Here and in the following equations, it is important not to confuse the subscripts 1 (one) and l (el). Secondly, we may assume that the instantaneous coseismic slip motion T 1 and other interseismic gradual slip motion T l (l =,..., L) have no correlation in time: T 1 (τ)t l (τ) dτ =, (l =,...,L). (3) T (1) C RAS, GJI, 1

Inversion using ABIC with slip response 17 log(β ) log(β ) log(β ) log(β ) (a) (b) (c) (d) log(α ) Figure. Contour maps of ABIC(α, β ) calculated with viscoelastic slip-response functions for given γ : (a) γ =.1, (b) γ = 1, (c) γ = 1 and (d) γ = 1. The contour intervals are taken to be 3 in the values of ABIC. For the autocorrelation of T 1 we need a special treatment, because it formally diverges to infinity. Thus, we introduce a new parameter γ, T 1 (τ)t 1 (τ) dτ = γ T l (τ)t l (τ) dτ, (l =,...,L), () T T which controls the relative weight of coseismic slip variation to interseismic slip variation in space. If we take larger γ, the smoothness constraint on coseismic slip in space becomes more dominant. For smaller γ, on the other hand, the smoothness constraint on interseismic slip in space becomes more dominant. In the following inversion analysis, we treat γ as an independent model control parameter. 3. Inverted slip history at the plate interface For given γ (.1, 1, 1 and 1), we compute the values of ABIC(α, β ) from eq. (35) with viscoelastic slip-response functions, and plot them in contour maps (Fig. ). The point of the ABIC minimum in each diagram gives the best estimates of the hyperparameters α and β for a given γ. With these values of α and β we can directly compute the optimal values of the model parameters a from eq. (7). In the left-hand column of Fig. 7 we show the slip histories u(ξ, τ) during one earthquake cycle at the plate interface, reconstructed from eqs () and C RAS, GJI, 15, 1 153

1 Y. Fukahata, A. Nishitani and M. Matsu ura 1 3 1 3 1 3 5 1 15 5 1 15 5 1 15 (a) (b) (c) 1 1 - - 1 1 1 1 - - 1 1 1 1 - - 1 1 1 3 Time (yr) 5 1 15 (d) Distance from Trench (km) 1 1 - - 1 1 Distance from Trench [km] Figure 7. Slip histories u(ξ, τ) during one earthquake cycle at the plate interface (left-hand column), and the spatial distributions of instantaneous coseismic slip (solid lines) with those of total slip deficits (broken lines) just before the occurrence of the next Nankai earthquake (right-hand column). (a) (d) represent the results for γ =.1, 1, 1 and 1, respectively. Viscoelastic slip-response functions are used. The time is measured from just after the occurrence of the Nankai 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 instantaneous coseismic 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,asin the case of (a), the smoothness constraint on interseismic slip becomes too strong in comparison with that on coseismic slip. In this case the inverted result gives a very smooth interseismic slip distribution both in space and time. On the contrary, if we choose a too large γ,asinthe case of (d), the smoothness constraint on interseismic slip becomes too weak in comparison with that on coseismic slip. In this case the inverted result gives a very rough interseismic slip distribution both in space and time. These qualitative criteria are useful for excluding geophysically unrealistic models, but not for selecting the most likely model from among many likely models. As a criterion for the model selection, we may use a prior constraint on earthquake cycles imposed by plate tectonics; that is, the total slip deficits just before the occurrence of the next earthquake should be almost the same as the amount of instantaneous coseismic slip (Fukuyama et al. ; Hashimoto & Matsu ura ). Applying this criterion to the inverted results of coseismic slip and total slip deficits in the right-hand column of Fig. 7, we can objectively select the case (c) with γ = 1 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 m occurs at the time of the Nankai earthquake in the intermediate depth range (1 3 km). After that, this portion keeps in stationary contact until the occurrence of the next Nankai earthquake. In the shallow (<5 km) and the deep regions (> km), on the other hand, steady slip at the plate convergence rate ( mm yr 1 ) proceeds through the entire earthquake cycle. Given such a slip history, we can calculate the absolute crustal movements during one earthquake cycle in Shikoku C RAS, GJI, 1

Inversion using ABIC with slip response 19 Absolute height change (mm) - - 1 15 5 Distance from Nankai Trough [km] 19 19 [year] Figure. Crustal movements in Shikoku during one earthquake cycle (19 3) calculated from the inverted slip history in Fig. 7(c) with viscoelastic slip-response functions. The absolute height of the Earth s surface just before the occurrence of the 19 Nankai earthquake is taken as a reference height. from eq. (3) as shown in Fig.. In this 3-D diagram the absolute height of the Earth s surface just before the occurrence of the 19 Nankai earthquake is taken as a reference height. Fukahata et al. (199) have obtained a similar space time pattern of crustal movements in Shikoku during the earthquake cycle, but from a direct inversion of levelling data. From the 3-D diagram in Fig. we can calculate the difference in absolute height between two arbitrary times. The solid lines in Fig. 9 are the profiles of vertical displacements calculated in this way, and the solid squares are the height changes obtained from levelling data. The diagrams (a) (d) correspond to the preseismic, coseismic, post-seismic and interseismic periods, respectively. Here, it should be noted that the observed coseismic displacements (b) include some preseismic and post-seismic displacements. From comparison with the calculated vertical displacements and observed data in these diagrams we can see that the inverted slip history explains the observed crustal movements well except for the southern half of the interseismic movements (d). Thatcher (19) and Fukahata et al. (199) have pointed out the fast subsidence in the northernmost part of Shikoku from the analysis of tide-gauge data. The inverted slip history model cannot reproduce this subsidence motion, which might be caused by the 3-D effect of the descending slab (Hashimoto et al. 3). If we ignore the effects of viscoelastic stress relaxation in the asthenosphere and use elastic slip-response functions in the analysis, the inverted results will dramatically change. In Fig. 1 we show the contour maps of ABIC(α, β ) for given γ (.1, 1., 1 and 1). Even in the elastic case, we can find the clear ABIC minima for given γ. And so, with the best estimates of α and β, we compute the optimal values 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 column of Fig. 11. From comparison of the slip-history diagrams in Figs 7 and 11, we can find remarkable differences in the inverted slip pattern between the viscoelastic case and the elastic case. To check the reality of these inverted models, we compare the instantaneous coseismic slip distributions (solid lines) with the total slip deficit distributions (broken lines) in the right-hand column of Fig. 11. From these diagrams we can see that any model (even the case (c) with γ = 1) inverted with elastic slip-response functions does not satisfy the constraint that the total 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. DISCUSSION AND CONCLUSIONS We developed an algorithm of geodetic data inversion to estimate slip history during one earthquake cycle at a plate interface with viscoelastic slip-response functions. So far elastic slip-response functions have been widely used for inverting geodetic data. The use of elastic slip-response functions is reasonable, if our interest is limited to short-term crustal movements such as coseismic deformation. In the analysis of long-term crustal movements, however, the use of elastic slip-response functions definitely leads us to incorrect results as demonstrated in Section 3.. In order to correctly estimate the long-term slip history at a plate boundary, we cannot neglect the effects of viscoelastic stress relaxation in the asthenosphere. We obtained a proper formulation to incorporate two sorts of partially dependent prior information into observed data, and improved the Bayesian inversion algorithm developed by Yabuki & Matsu ura (199). In this inversion algorithm, first, we construct a highly flexible C RAS, GJI, 15, 1 153

15 Y. Fukahata, A. Nishitani and M. Matsu ura [19-199] [197-1939] 1 a -1 Vertical Displacement [mm] - 1 5-5 -1 3 1 [199-197] [1939-197] b [197-195] [197-19] c -1 3 [195-193] [19-19] 1 d -1 1 1 1 Distance from Nankai Trough [km] 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), with observed 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 data with Bayes theorem. Next, we select a specific model from among the family of parametric models by minimizing ABIC. The smoothness constraints on spatial and temporal slip variations are partially dependent in usual cases. In the old formulation, however, they have been assumed to be completely independent. If ABIC is defined on the basis of the improper formulation, as pointed out in Section., it decreases infinitely as the hyperparameters approach infinity. This problem, which is very serious in the ill-conditioned inversion analysis (Fukahata et al. 3), 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 control parameter γ, which prescribes the relative weight of coseismic slip variation to interseismic slip variation in space. In the inversion algorithm we explicitly imposed the local prior constraints that the slip motion should be smooth both in space and time. In addition to the local prior constraints we have a global prior constraint for slip imposed by plate tectonics; that is, the spatial distribution of instantaneous coseismic slip should be almost the same as that of total slip deficits just before the occurrence of the next earthquake. If the global prior constraint was incorporated into the inversion algorithm from the beginning, we did not need the final process to select the most likely model from among ABIC minimum models with different γ. In this case, however, we would need one more hyperparameter, in addition to σ, α, β and γ,to describe the global prior constraint, and we would have to numerically search the best estimates of the hyperparameters in the -D parametric C RAS, GJI, 1

Inversion using ABIC with slip response 151 log(β ) log(β ) log(β ) log(β ) (a) (b) (c) (d) log(α ) Figure 1. Contour maps of ABIC(α, β ) calculated with elastic slip-response functions for given γ : (a) γ =.1, (b) γ = 1, (c) γ = 1 and (d) γ = 1. The contour intervals are taken to be 5 in the values of ABIC. space, which would not be practical. It should be noted that γ controls the relative weight of the smoothness of coseismic slip and interseismic slip. Therefore, γ has an inherent property to connect the local smoothness constraints with the global constraint. Thus, externally controlling γ 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.. In the case with elastic slip-response functions (Fig. 11), no model could satisfy the global constraint that the spatial distribution of instantaneous coseismic slip should be almost the same as that of total slip deficits just before the occurrence of the next earthquake, although all the inverted models well fitted the data and satisfied the local smoothness constraints on slip. If the global constraint was explicitly incorporated into the inversion algorithm, the inverted result would show the improvement on the global constraint, but the fitting of data and the 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 = 1 in Fig. 7(c) as the most likely model of slip history at the Nankai subduction zone. This model indicates that instantaneous slip of approximately m occurs at the time of the Nankai earthquake in the intermediate depth range. After that, this portion keeps in stationary contact until the occurrence of the next Nankai earthquake. In the shallow and deep regions, on the other hand, steady slip motion at approximately mm yr 1 proceeds C RAS, GJI, 15, 1 153

15 Y. Fukahata, A. Nishitani and M. Matsu ura 1 3 1 3 1 3 5 1 15 5 1 15 5 1 15 (a) (b) (c) 1 1 - - 1 1 1 1 - - 1 1 1 1 - - 1 1 1 3 Time (yr) 5 1 15 (d) Distance from Trench (km) 1 1 - - 1 1 Distance from Trench [km] 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 slip deficit distributions (broken lines) (right-hand column), inverted with elastic slip-response functions. (a) (d) represent the results for γ =.1, 1, 1 and 1, 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 through the viscoelastic response of the Earth. A slip in the offshore region causes little deformation on land as long as the Earth behaves like an elastic 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 of two dimensions, and so a slip in the offshore region causes a substantial crustal deformation on land after the viscoelastic relaxation of the asthenosphere. 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 (19) through numerical simulation of the earthquake cycle with a rate- and state-dependent friction law. For the first time we have succeeded in reconstructing the actual slip history during one earthquake cycle at the subduction zone from the inversion analysis of geodetic data. ACKNOWLEDGMENTS We thank Yuji Yagi, Paul Segall and the anonymous reviewer for their useful comments. REFERENCES Akaike, H., 1977. On entropy maximization principle, in, Application of Statistics pp. 7 1, ed. Krishnaiah, P.R., North-Holland, Amsterdam. Akaike, H., 19. Likelihood and the Bayes procedure, in, Bayesian Statistics pp. 13 1, eds Bernardo, J.M., DeGroot, M.H., Lindley, D.V. & Smith, A.F.M., University Press, Valencia. Ando, M., 1975. Source mechanisms and tectonic significance of historical earthquakes along the Nankai trough, Japan, Tectonophysics, 7, 119 1. C RAS, GJI, 15, 1 153

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