JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B07414, doi: /2004jb003378, 2007

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2004jb003378, 2007 Estimation of slip distribution using an inverse method based on spectral decomposition of Green s function utilizing Global Positioning System (GPS) data Honglin Jin, 1,2 Teruyuki Kato, 1 and Muneo Hori 1 Received 11 August 2004; revised 20 April 2007; accepted 10 May 2007; published 31 July [1] An inverse method based on the spectral decomposition of the Green s function was employed for estimating a slip distribution. We conducted numerical simulations along the Philippine Sea plate (PH) boundary in southwest Japan using this method to examine how to determine the essential parameters which are the number of deformation function modes and their coefficients. Japanese GPS Earth Observation Network (GEONET) Global Positioning System (GPS) data were used for three years covering to estimate interseismic back slip distribution in this region. The estimated maximum back slip rate is about 7 cm/yr, which is consistent with the Philippine Sea plate convergence rate. Areas of strong coupling are confined between depths of 10 and 30 km and three areas of strong coupling were delineated. These results are consistent with other studies that have estimated locations of coupling distribution. Citation: Jin, H., T. Kato, and M. Hori (2007), Estimation of slip distribution using an inverse method based on spectral decomposition of Green s function utilizing Global Positioning System (GPS) data, J. Geophys. Res., 112,, doi: /2004jb Introduction [2] Crustal deformation in the Japanese islands is caused by earthquakes and volcanic eruption as well as plate coupling interactions. Knowledge of the slip distribution on faults due to earthquakes and plate coupling interactions is required to understand earthquake mechanics and to make seismic hazard assessments. The data from a dense Global Positioning System (GPS) array in the Japanese islands, as well as other geodetic survey measurements, enable us to clarify crustal deformation in much detail [Sagiya et al., 2000]. These data are important for estimating seismic and aseismic slip distributions in and around the Japanese islands. [3] The representation theorem in elastostatics relates surface displacements to slip distribution on a fault surface through a linear integral equation that uses the Green s function [e.g., Maruyama, 1964; Okada, 1985]. Therefore we can set up an inverse problem for reconstructing the static image of a seismic or aseismic source from observed geodetic data. Menke [1984] described a range of linear inverse methods such as the least squares analysis method with prior information and the general Singular Value Decomposition (SVD) analysis method. Many previous studies in Japan have used inverse methods to investigate slip distribution over a fault and slip deficit distribution at a converging plate interface [Yoshioka et al., 1993; Ito et al., 1 Earthquake Research Institute, University of Tokyo, Tokyo, Japan. 2 Institute of Earthquake Science, China Earthquake Administration, Beijing, China. Copyright 2007 by the American Geophysical Union /07/2004JB003378$ ; Sagiya, 1999; Ito et al., 2000; Miyazaki and Heki, 2001; Miura et al., 2004a, 2004b]. Yabuki and Matsu ura [1992] developed an inverse method using the Bayesian Information Criterion for estimating the spatial distribution of fault slip. They combined prior information about the smoothness of fault slip distribution with information coming from observed data to find optimal model parameters using Akaike s Bayesian Information Criterion (ABIC) by means of the maximum likelihood method. [4] In the present study, we use a different inverse method that uses a spectral decomposition of the Green s function, which is similar to the SVD analysis method. Early introduction of this method is, for example, by Parker [1997] or Matthews and Segall [1993] or Kirsch [1996]. Here we use a method introduced by Hori [2001]. While Hori demonstrated the usefulness of this inverse method, he did not address the problem of how to determine the number of modes to be used or the coefficients of each mode for the surface functions for obtaining appropriate solutions or an actual application. [5] The present study first reviews the inverse analysis method and then conducts numerical simulations on the subducting plate boundary in southwest Japan to find a way to determine parameters for the inversion analysis. Then the inverse method is applied to the GPS data of the Japanese nationwide array to estimate the slip deficit distribution in this region, and its tectonic interpretation is given. 2. Formulation of the Inverse Method Based on a Spectral Decomposition of Green s Function [6] In this section, we briefly review the formulation of the inverse analysis method based on the spectral decomposition 1of14

2 of Green s function according to Hori [2001]. We assume the medium to be an elastic half space. Let S be the surface plane and F the fault plane, and we denote by vectors x and y for the points on S and F, respectively. According to the representation theorem in elastostatics, the static surface displacements u(x) are expressed in terms of source function p(y) (slip or back slip) and Green s function: Z ux ðþ¼ F gx; ð yþpy ðþdy [7] Since p(y) is mapped to u(x) in equation (1), the inverse problem of equation (1) is regarded to be the backward mapping from u(x)top(y). It suffices to construct an inverse operator, denoted by h(x, y), which satisfies Z py ðþ¼ S hx; ð yþux ðþdx [8] When the space (surface and fault domain) of functions u(x) and p(y) is specified, the Green s function admits the spectral decomposition [Hori, 2001]: gx; ð yþ ¼ X1 l a 8 a ðxþy a ðyþ a¼1 hx; ð yþ ¼ X1 1 l a 8a ðþy x a ðþ y a¼1 Here l a is the ath eigenvalue, and 8 a (x) and y a (y) are the associated eigenfunctions of the surface and the fault, respectively. The ath eigenvalue has positive values that converge to zero as a increases, and 8 a (x) and y a (y) form orthonormal basis functions on S and F, respectively. Note that the spectral decomposition of the Green s function depends on the domains S and F as well. That is, the set of eigenvalues and eigenfunctions change for different S or F even if the same function g(x, y) is used. [9] The eigenvalues and eigenfunctions are found by solving the following two self-adjoint operators that are constructed from g. Z rx; ð x 0 Þ ¼ F ð1þ ð2þ ð3þ ð4þ gx; ð yþgx ð 0 ; yþdy ð5þ Z ly; ð y 0 Þ ¼ gx; ð yþgx; ð y 0 Þdx ð6þ S Here r or l is an operator that maps a function on S or F to another function on S or F, respectively. Due to the selfadjointness, the eigenvalues and eigenfunctions of g are given as solutions: Z l 2 8ðxÞ ¼ rx; ð x 0 Þ8ðx 0 Þdx 0 S Z l 2 yðyþ ¼ ly; ð y 0 Þyðy 0 Þdy 0 F ð7þ ð8þ It is easily seen that the operator r or l admit the following spectral decomposition: r = P a (la ) 2 8 a 8 a and l = P a (la ) 2 y a y a. The set of eigenfunctions 8(x) and y(y) are calculated from g by these solutions. [10] We expect the response function to be expressed as a linear combination of the measurable modes, {8 1 (x),8 2 (x) 8 k (x)}, which are associated with k eigenvalues. ux ð Þ ¼ Xk u a 8 a ðþþnoise x ð Þ ð9þ a¼1 where {u a } are coefficients of 8 a (x), and k is a measurable number of the eigenvalues (or modes). The second term contains modes given by 8 a (x) fora > k, but they cannot be distinguished from error or noises due to the limited accuracy. Therefore even though a response function cannot be measured, it is possible to accurately estimate a continuous function from a set of discrete data by using a set of eigenfunctions. [11] Then p(y) can be written as the linear combination of source eigenfunctions as p(y) = P p a y a (y) where p a are coefficients of y a (y), and p a has the same order of magnitude as u a. Then, the resulting u(x) = P u a 8 a (x) has vanishing u a (= l a p a ) since l a s go to zero as a increases. Therefore when the relative accuracy of the measurements is given, we can select the number of measurable modes (k) of the displacement field from 8 a (x), which is associated with larger l a. The number of such measurable modes, k, is determined as the largest eigenvalue that satisfies l k > relative accuracy 1 l ð Þ ð10þ [12] However, it is difficult to determine the largest k when the relative accuracy of the data is not known. The next section introduces a method to find an appropriate k through numerical experiments. [13] Once the spectral decomposition of g(x, y) is computed, we obtain a set of eigenfunctions 8 a (x) and y a (y). The remaining work is to determine the coefficients {u a }by numerical computation, using data {U i }={u(x i )ji =1,2,... P}, which are actually measured at position {x i }. Here P indicates the number of observations. When the number of k modes is smaller than P, a method, generally the Least Squares Method, to determine {u a } is applied to find the set of {u a } that matches the measured data. That is, we find the set that minimizes the error E. Eðu a Þ ¼ XP i¼1 U i Xk a¼1 2 u a 8 a ðx i Þ! ð11þ [14] Once {u a } is determined, the source function is expressed in terms of the eigenvalues, {l a }, and eigenfunctions on the fault F, y a (y). Substituting equation (4) and equation (9) into equation (2), we can derive the following expression for the source function: py ðþ¼ Xk u a l a ya ðyþ a¼1 ð12þ 2of14

3 Figure 1. Tectonic settings of the Japanese islands; AM, PH, PA, and NA denote Amurian, Philippine Sea, Pacific, and North American plate, respectively. MTL denotes the Median Tectonic Line. Southwest Japan is overriding the Philippine Sea plate at the Nankai Trough. The grids of surface domains (red) and source domains (green) used for the computation are shown. The geometry of the Philippine Sea plate boundary in southwest Japan shown in green is taken from the work of Hashimoto et al. [2004]. [15] That is, {u 1, u k } is determined from equation (11) for the given data, then p(y) is obtained from equation (12). As {8 1 (x),,8 k (x)} are measurable, {y 1 (y),, y k (y)} are predictable for the measurement of the limited accuracy. This equation omits the effects of y a (y) fora > k in the response, which is inevitable since the effects cannot be measured. Omitting these y a (y) does not mean that p(y) does not have components associated with it. We need other information besides the measured data to determine mode coefficients for 8 a (x) ofa > k. However, such information cannot change the coefficients corresponding to u a for a = 1, 2,... k. On the other hand, we should point out that equation (12) means that, for a fast decrease of eigenvalues, the resolution of source function is low, since inversion can only find a small number of modes that correspond to eigenfunctions associated with sufficiently large eigenvalues. [16] Finally, the current method is not valid when the fault slip reaches the surface. In this case, it may fail because of the discontinuity in the deformation field, which negates the assumption of a smooth response function. 3. Simulation for the Inverse Analysis Method [17] In the preceding section, we introduced the inverse analysis method based on a spectral decomposition of the Green s function. This section examines the validity of this method through numerical experiments, investigates the effects of data conditions such as data noises, station distribution, and its density, and provides a way to determine k through numerical experiments. [18] First, we discuss a hypothetical slip distribution on the Philippine Sea plate (PH) boundary of southwest Japan, and then we compute surface displacements at GPS observation stations in this region by a forward method. Then a synthetic data set is calculated. Other data sets are prepared by adding errors, which are GPS velocity error estimates for white and flicker noise models [Mao et al., 1999], to examine the effects of noise in the data. Then we estimate 3of14

4 Table 1. Eigenvalues of l 1 l 3 for Different (N,M) a N,M l 1 l 2 l , , , , a (N,M) indicate a decomposed segment number on the surface and plate boundary for numerical computation. slip from (1) the error-free data, (2) data contaminated with GPS data error, and (3) cases when the number of observation points is forced to decrease or their distribution is changed. Finally, we introduce a methodology to find the most suitable k in cases (1) and (2) Surface and Plate Boundary Domain [19] First, we determine the domains of the surface and the Philippine Sea plate boundary in southwest Japan in order to apply the new inverse method. The three-dimensional geometry of the plate boundary (Figure 1) is calculated using the Crustal Activity Modeling Program model that was constructed from the topography of ocean floors and hypocenter distribution [Hashimoto et al., 2004]. Referring to previous research, such as the work of Miyazaki and Heki [2001], we take the depth range of the source domain to be 5 50 km to fully cover the possible coupling region. [20] The surface S and plate boundary F are decomposed into N and M points. The convergence check by increasing N and M, which corresponds to the discretization for surface and fault, is important to examine the numerical validity of the spectral decomposition. We just test the following four cases of discretization: {(N, M)} = {(1400, 1860)(2570, 2980) (3588, 4048)(4596, 5128)}. Table 1 shows the estimated eigenvalues as test. As shown, eigenvalues converge to within 10 4 when (N, M) are more than or equal to (3588,4048). This suggests that the spectral decomposition of Green s function can be computed accurately if (N 3588, M 4048). We note that Green s function must be discretized using the (N, M) points by employing a set of smooth functions when the spectral decomposition is obtained numerically. [21] A smoothed plate boundary is plotted on a grid into 3588 elements for numerical computations. In the present study, we use the point dislocations embedded in the smoothed curved surface, but do not use multirectangular fault planes as was used in the previous studies. Figure 2. Calculated orthonormal eigenfunctions for surface (top four inserts) and plate boundary (lower four inserts) using the spectral decomposition of Green s function for given domains of the surface and plate boundary. Only a selected four modes (a are 1, 4, 8, and 12) for E-W and dip direction are shown. The patterns are more complex for bigger a. 4of14

5 Figure 3. (a) Assumed slip distribution on the plate boundary; calculated surface displacement (b1) at 465 data position and (b2) at 365 GPS station position from the assumed slip (a). [22] The Earth s surface (Figure 1) is represented by a curved plane of km along the Nankai Trough and the surface was gridded into 4048 elements for numerical computations. Using the selected set of discrete points to represent the surface and plate boundary domains, we calculate the Green s function, g(x, y), and conduct the spectral decomposition of g(x, y) to separate it into a set of surface and fault eigenfunctions. Figure 2 shows some E-W direction s surface eigenfunctions, 8 a (x), and dip direction s fault eigenfunctions, y a (y), for a =1, 4, 8, and 12 as example, respectively. Note that the surface deformation caused by plate coupling can be expressed by a linear combination of surface eigenfunctions. The back slip distribution is expressed by a linear combination of fault eigenfunctions Slip Estimation and a Way to Find the Number of Modes [23] This section first looks into finding the largest number of modes for surface displacement and then goes on to investigate the effects of data noise and data distribution upon the estimated slip distribution. For this purpose, a theoretical slip distribution is hypothesized as is shown in Figure 3. The coefficients of the modes given in this slip distribution are u a 6¼ 0 when a 5 and u a = 0 when a >5, which means that slip distribution can be represented by five modes. Horizontal (Figure 3) displacements at 465 data position and 365 GPS station position are computed from this slip distribution. [24] Here it should be noted that the synthetic data are accurate; they include only numerical noise. Therefore we also created synthetic data that include error from real GPS velocity error estimates for a white plus flicker noise model [Mao et al., 1999] added to the theoretical surface displacements How to Determine the Largest Number of Modes? [25] Before estimating coefficients u a, we have to determine the number of modes, k. Here we consider two cases: (1) evenly spaced data points at the surface (Figure 3) locations of GPS sites (Figure 3). Thus the maximum numbers of k are 465 for the former case and 365 for the latter, respectively. But in fact we take k as a number smaller than 465 and 365 to avoid a numerically unreliable computation error, which will affect the accuracy of estimated coefficients and lead to an erroneous estimation of slip distribution. Estimation error increases as the number of modes k increases, because eigenvalues converge to zero as a increases (equation (12)). Therefore it is essential to find a suitable k first. [26] For the inverse method based on the spectral decomposition of the Green s function, the number of modes k is determined from the measured data. Since the coefficients for measurable modes u a (a =1,2,... k) are estimated by using the measured data, the coefficients u a should be stable as far as the number of modes k is limited within the measurable modes. However, if k exceeds the limit of the measurable mode, which means that surface function includes unmeasurable modes, then the coefficients u a will be contaminated by noises and thus become unstable (see equation (9)). [27] To estimate a suitable k, we study the stability of the coefficients u a as the number of modes k increases. Here we examine the estimated coefficients u* 1 u* 5. Figure 4 shows the variation of estimated coefficients u* 1 u* 5 with respect to k by error-free data at 365 GPS station position. This figure shows a comparison for the estimated coeffi- 5of14

6 Figure 4. Variability of estimated coefficients with respect to k. Five color lines are coefficient error comparisons with true values u 1 u 5 by 365 GPS station position error-free data. cients u* 1 u* 5 with true u 1 u 5, respectively. It is shown that coefficients become unstable as k exceeds about 270. When we take k to be larger than 270, the estimated coefficients will include a computational error and lead it to an unrealistic slip distribution. In this case, it is interesting to note that l 270 /l 1 is approximate to 10 2 because the estimated slip error is in the order of 10 2 when k = 270. Therefore we predict that the numerical error in case (2) is of the order of [28] In the previous section, we suggested that we can truncate k such that the relative accuracy l 270 /l 1 is nearly the order of data accuracy. In fact, we cannot use this criterion because we do not know the relative accuracy of the data. Alternatively, from the above discussion, we find that the method of studying the stability of coefficients u a by increasing the number of modes k may be used to seek the suitable number of mode k. [29] Determination of k depends on the signal-to-noise ratio in the data. Parker [1994] advocates a procedure of fitting within tolerance to determine the number of modes, k, that is to find the estimate that minimizes the model norm while keeping the data misfit within tolerance. On the other hand, Matthews and Segall [1993] use a Cross- Validation approach that does not rely on the knowledge of the data error. However, this method requires knowledge of the correlation structure in the data, and this can be a problem with GPS data. Yabuki and Matsu ura [1992] used a maximum likelihood method to determine the appropriate amount of damping. The bottom line of the ultimate way of determining the smoothing parameter is to compare various methods and not to interpret results that are dependent on a specific choice of damping Effects of Data Distribution [30] It may be naturally understood that the number and distribution of data are responsible for estimating surface response function. If the number of data is sufficient and if data are distributed fully over the deformation area, we will be able to make a highly accurate estimate of the surface response function. However, this is not always the case; in reality, data are sparse and not uniformly distributed in the surface. Therefore it is important to examine how such data distribution and density affect the estimation of the slip distribution. [31] Figure 3 show that the displacement for the number of data is reduced from 465 which are fully distributed on the surface to the 365 actual GPS station positions. Figure 5 gives an estimated slip error with respect to k, and shows that the estimated slip vectors that depart from the true value becomes much greater as the number of data points decreases. When the number of data is 465, the estimated slip is consistent with the true slip within errors (s 2 = )uptok = 400. However, if the number of data is 356, the estimated slip is consistent with the true slip within errors (s 2 = 0.02) only up to k = 270. If we take k > 270, the estimated slip is significantly contaminated by errors. Thus the number of data should be responsible for determining the number of modes in the estimated slip distribution Effects of Data Error [32] In order to investigate the effects of data error on the number of modes and the estimated slip distribution, we invert for the slip distribution on the fault from synthesized data that contain errors. Our simulations assume that the GPS data include white plus flicker noises [Mao et al., 1999]. [33] First, we determine the number of modes, k, from data with error by studying the stability of the coefficients u a with increasing k. In order to superimpose error onto the synthesized data, we used the algorithm of Mao et al. [1999] who compared white, flicker, and random walk Figure 5. Estimated slip errors (s 2 = P s 2 i ) for dip and strike direction with respect to true slip distribution, in the cases of two theoretical data distributions for (a) k = 365 and (b) k = of14

7 present study proposes that k can be determined by studying the stability of the coefficients while increasing the number of modes estimated from the measured data. Figure 6. Variability of estimated coefficients u 1 u 5 with respect to k. Five color lines are coefficient error comparisons with true values u 1 u 5 by 365 GPS station position data with error. noises in the three years of globally distributed 23 GPS sites and concluded that white and flicker noises are responsible for the GPS time series data. They estimated the error covariance of these noises using the maximum likelihood method. We applied Mao et al. s method to three years of 365 GPS Earth Observation Network (GEONET) stations that will be used in the later sections and estimated velocity errors for white plus flicker noise model. Estimated errors are 4.8 and 6.4 mm/yr for horizontal and vertical components, respectively. We added these error components to the error-free data to make a set of synthesized data with errors. [34] Using our synthetic data, we applied the technique from section for determining a suitable k. Figure 6 shows the variability of coefficients u 1 u 5 with respect to k and Figure 7 shows the estimated slip error for dip and strike directions with respect to k. These figures show that the suitable k may be 24. The estimated slip distribution is consistent with true slip when k = 24, where the variance is s 2 = P s i 2 = 0.7. If we take a larger k such as k = 30, which is in the region of unstable a, then the variance is s 2 = 2.5, the estimated slip distribution is contaminated by significant errors. From this test, we predict that k should not exceed 30 when using actual GEONET data. [35] This numerical simulation indicates that data error affects the appropriate choice of k. Therefore again, we should note that the criterion of l k /l 1 (relative accuracy of measurement) can only be used when we do not consider the data error, data number, and data distribution, and when the relative accuracy of the measurements is known. Alternatively, as shown in this section, the stability of u a may be used for finding a suitable k. [36] Summing up this section, we may derive some conclusions through the above numerical simulations as follows: The number of modes for k depends on data conditions such as data error, data density, and data distribution. So using the criteria l k /l 1 (relative accuracy of measurement) is not an appropriate way to estimate k. The 4. Application to the Philippine Sea Plate Boundary in Southwest Japan [37] Interseismic crustal deformation in southwest Japan (Figure 1) is dominated by its interaction with the Philippine Sea plate (PH) that subducts at the Nankai Trough, off Shikoku Island and the Kii Peninsula. Terrestrial geodetic surveys have been performed by the Geographical Survey Institute (GSI) of Japan for more than a century, and such data have revealed interseismic crustal deformation [Hashimoto, 1990; Hasimoto and Jackson, 1993]. These data have been used to estimate the slip deficit (or coupling) at the Philippine Sea plate boundary in southwest Japan [Ozawa et al., 1999; Ito et al., 1999; Miyazaki and Heki, 2001]. [38] The most significant feature of deformation in southwest Japan is crustal shortening in the direction of plate convergence. This shows interseismic strain accumulation at the Nankai Trough, which is the elastic deformation caused by the subduction of the PH and the strong coupling (slip deficit) at the plate interface [Ozawa et al., 1999]. Past geodetic and seismological studies constrain the downdip extent of the seismogenic zone in the Nankai Trough, so it is appropriate to select this area for the application of the new inverse method for inferring the distribution of the fault slip deficit. [39] When the plate interface is partially locked, interseismic deformation in the island arc can be modeled by imaginary backward slip at the locked fault surface [Savage, 1983]. Since the slab dip angle (7 15 ) is relatively low at the Nankai Trough, viscous flow in the asthenospheric wedge beneath the arc should not play an important role, and an elastic half-space assumption is reasonable for the interseismic timescale [Miyazaki and Heki, 2001]. If the observed crustal deformation in southwest Japan was due solely to the subduction at the Nankai Trough, the back slip Figure 7. Estimated slip error comparison with given slip for dip and strike direction by increasing k. 7of14

8 Figure 8. (a) Stability of coefficients u* 1 u* 4 and (b) estimated displacements error (s 2 = P s i 2 ) with respect to k. distribution could be estimated at the surface of the PH slab by the inversion of geodetic data. Here we will estimate the slip deficit at the Nankai Trough using the inverse analysis method based on the spectral decomposition of Green s function and GPS observation data without any prior information. Then we will discuss the tectonic implications of the results GEONET GPS Data [40] The Japanese Geographical Survey Institute (GSI) has been operating its nationwide GPS Earth Observation Network (GEONET) system since It is one of the largest permanent GPS arrays in the world with more than 1200 permanent GPS stations. GEONET covers the Japanese islands with an average baseline distance of about km. This network provides us with an ideal data set to quantitatively investigate the plate subduction process and the mechanism of earthquake generation in Japan. [41] The observation data are processed by GPS analysis software BERNESE Ver4.2 [Rothacher and Mervart, 1996] with GSI s combined GPS satellite orbits and IGS final orbits. Geocentric coordinates (x,y,z) are estimated in the reference frame of International Terrestrial Reference Frame 2000 (ITRF00) and they are converted to the daily geodetic positions in latitude, longitude, and height based on the reference ellipsoid of World Geodetic System 1984 (WGS84) for each station[miyazaki et al., 1997]. In the present study, we focus on southwest Japan and the daily coordinate data from 365 GPS stations between January 1997 and December [42] In order to discuss tectonic deformation for southwest Japan, we estimate the secular displacement rates at GPS stations from the daily coordinate solutions. We used the following function to model the daily coordinate time series data (S n i ) of the three components for the 365 stations. Sn i ¼ ai n þ bi n t þ ci n þ fn i sin ð 2wt cos ð wt Þþdi n Þþgi n Ht ð t sin ð wt Þþei n cos ð 2wt Þ Þþei n ð i ¼ 1; 2; 3; n ¼ 1; M Þ ð13þ ð13þ [43] Here the first two terms correspond to a linear trend, the next four terms with trigonometric functions are annual and semiannual variations, the next function H(t t) 8of14

9 Figure 9. Surface displacement vectors for horizontal components. Black vectors indicate observed GPS velocities, while red vectors indicate estimated velocities from GPS data using the present inverse method. represents coseismic jumps associated with earthquakes i during the periods analyzed, and the last term e n is the residual. M indicates the station number. In this study, we only discuss the linear trends, which represent interseismic crustal deformation in southwest Japan. We estimated displacement rate components for 365 stations by solving the least squares problem for equation (13). Estimated standard variances of the data are 2.5 and 8 mm for horizontal and vertical components, respectively. The velocity error estimates for white plus flicker noise based on Mao et al. [1999] are 4.8 and 6.4 mm/yr for horizontal and vertical components, respectively. [44] Recent studies suggest that southwest Japan can be considered as a part of the Amurian (AM) plate [e.g., Miyazaki and Heki, 2001; Heki et al., 1999]. In order to isolate signals of crustal deformation caused by the subduction of the Philippine Sea plate, we fixed the kinematics reference frame to the Amurian (AM) plate using the following procedure. The original velocity field is given in the ITRF00. The velocities relative to AM are obtained by subtracting the absolute rotation of AM-ITRF00 given by Linette and Bock [2004]. [45] The Median Tectonic Line (MTL) is the longest arc-parallel fault system in southwest Japan where the right-lateral strike-slip movement is related to the oblique subduction of the Philippine Sea plate (PH). Average slip rate along the MTL in the late quaternary is estimated to be 5 10 mm/yr based on geological and geomorphological observations. Miyazaki and Heki [2001] have pointed out that the southern block moves to the southwest at a rate of 2 5 mm/yr after removing the effect of PH subduction. Tabei et al. [2002] estimated this block motion to be about 5 mm/yr. However, the location of block boundaries and the transition of block motion from south to north were difficult to identify in their study because the station distribution of GSI s continuous array was too sparse. Tabei et al. suggest that the statistical test indicated that MTL is a boundary of block motion, although slip deficits on the upper fault zone were not significant throughout all segments because formal uncertainties exceeded the estimates. Consequently, we removed a block motion rate of 5 mm/yr parallel to MTL from each GPS position velocity in the block south of MTL. [46] Thus GPS velocities obtained for horizontal components in southwest Japan are shown in Figure 9. The velocity fields clearly manifest the interseismic elastic loading due to the coupling of the Philippine Sea plate slab and continental plate at the Nankai Trough. We use these velocities to estimate the back slip distribution of the PH by applying the new inverse method Estimation of Back Slip Distribution [47] From the GPS velocity distribution on the surface shown in Figure 9, the surface displacement function can be estimated from the surface eigenfunctions and coefficients u a given in equation (9). First, in order to seek suitable k,we 9of14

10 Figure 10. Residual velocities determined by subtracting estimated velocities from observed ones. study the stability of the coefficient and displacement errors comparison with GPS data. By increasing k for coefficients u k and error of surface displacement (Figures 8a and 8b), we found that the suitable point of k is 13. Once k is determined, we can estimate the surface function, as shown in Figure 9. The available GPS data can extract slip modes up to 13 for the given plate boundary geometry. [48] One might wonder what happens in the estimation of k if the true slip is rougher than the simulated slip or vice versa. If the true slip is rougher and surface data are accurate enough (higher S/N) to reflect such roughness, we will have a larger k. And the estimated slip may be closer to the truth. However, if the data are not accurate enough (lower S/N), then the obtained k is not high enough to resolve such rough slip and the estimated slip distribution would be much smoother than the true slip distribution, and vice versa. Thus, again, the data noise is essential in how better we could estimate the slip or back slip on the fault. [49] Figure 9 shows a comparison between the observed and estimated velocities at GPS stations. By subtracting the estimated velocities from the observed ones, we obtain the residual velocity field shown in Figure 10. Generally speaking, the observed velocities in Chugoku and Shikoku districts are consistent with the estimated velocities (for k = 13) both in direction and their magnitudes. Kinki and Chubu districts, on the other hand, show large residual velocities and systematic errors. Residual vectors in these districts tend to show westward velocities suggesting the existence of crustal deformation caused by the collision between the continental lithospheres of southwest and northeast Japan. [50] The estimated back slip distribution along the subducting PH slab for k = 13 is shown in Figure 11. Direction and magnitudes of back slip vectors are generally consistent with those of the convergence of the Philippine Sea plate. It seems that large coupling (>50%) is occurring in the depth range from 10 to 30 km. There are three areas of nearly 100% coupling: western Tosa Bay, south of the Kii Channel, and Enshu-Nada to Suruga Bay areas. In Shikoku and the Kii peninsula areas, the result is similar to the result of Miyazaki and Heki [2001] who used prior guess of back slip in the inversion. Ozawa et al. [1999] estimated interplate coupling rate as 51% 100% at a depth of km, which is consistent with our result. However, the maximum back slip rate 6 cm/yr according to their study is smaller than the maximum back slip rate of 7 cm/yr in this study. This is perhaps owing to a different fault structure and range. The plate boundary structure data that we used might be a little more distant from the true plate boundary geometry. [51] We also examined the case when 5 mm/yr of westward block motion south of MTL is not considered. The result simply showed larger westward back slip and departed from the direction of the PH motion. Thus we judge that the block motion of 5 mm/yr toward the west in the south of MTL may be significant Comparison With the Method by Yabuki and Matsu ura [1992] [52] We also applied a conventional technique of geodetic inversion for comparison. Figure 12 is the back slip distribution estimated using the program of Yabuki and Matsu ura 10 of 14

11 Figure 11. Estimated back slip distribution and slip vectors along the subducting PH slab for model number k = 13 from GPS data. The green vector is the PH-AM convergence rate. [1992], which uses a Bayesian criterion called ABIC (Akaike s Bayesian Information Criterion) approach for estimating the necessary fault parameters. Their inversion program has widely been used for estimating fault slips and back slips [e.g., Yoshioka et al., 1993; Ito et al., 1999; Sagiya, 1999; Ito et al., 2000; Miyazaki and Heki, 2001; Miura et al., 2004a, 2004b]. This technique requires significant a priori information about the smoothness of slip along the fault. Slip directions are also constrained as a priori information in many cases. Given these a priori constraints, the fault parameters are estimated so as to minimize the ABIC that consists of a data error covariance and a hyperparameter that controls roughness of fault slip distribution (see the work of Yabuki and Matsu ura [1992] for details). [53] The fault slip distribution shown in Figure 11 is basically similar to that obtained by using Yabuki and Matsu ura s [1992] method (Figure 12). Three regions of high coupling rates are common in both of the results: western Shikoku, Kii Channel region, and the area of Tonankai earthquake. Also, the depth extent of the coupling area is at depths between 10 and 30 km in both of the results. [54] However, a little more detailed comparisons of these figures indicate significant differences between them; for example, a large slip deficit in the eastern end of the source domain and a few patchy slip deficits at the deeper plate interface in Figure 12. These differences might be due to different treatments on smoothness (or roughness) in the method. While our method discards rougher patterns in higher modes that might stem from noise in the data, they are still used in the method of Yabuki and Matsu ura [1992] to some extent in terms of minimization of the hyperparameter. Thus, ultimately, the present method might result in a little smoother pattern as compared with that of Yabuki and Matsu ura [1992]. [55] It would be difficult to judge which method provides us with a better (or closer to the truth) estimate of the fault slips. If some local surface noise contaminates the data, such erroneous surface patterns may be mapped as fictitious slips on the fault if we use the method of Yabuki and Matsu ura [1992], though such local noise may be smoothed out by the present method. On the other hand, if there is a local slip on the fault as a truth, such local slip might be smoothed out in the present method if the resultant presumably local signal on the surface is not resolved in a mode of surface function Tectonic Implication [56] We plotted the hypocenter distribution of a magnitude larger than two earthquakes along the plate boundary for the period and compared it to coupling distribution as shown in Figure 13. It can be seen that high seismic areas are concordant with two areas of full coupling areas south of the Kii Channel and Tokai areas. On the other 11 of 14

12 Figure 12. Estimated back slip distribution and slip vectors along the subducting PH slab by Yabuki and Mastu ura s [1992] inverse method. hand, low seismic activity but full coupling is seen west of Tosa Bay. Different characteristics of coupling and seismic activity between these areas show that, in eastern Shikoku, many microearthquakes occur right beneath the fully coupled plate interface, whereas in western Shikoku, microseismic activity is low though the plate interface is fully coupled. Why does this difference occur? While an area of high coupling aligns with low seismic activity in the subducting slab, the other areas show an opposite tendency of high coupling with high seismic activity. All of the studied areas experienced large interplate earthquakes of the 1854 Ansei, 1944 Tokai, and 1946 Nankai earthquakes. Thus if the ruptured fault surface recovered strong coupling soon after the earthquake, the shearing stress may have accumulated enough around such asperity to generate microearthquakes around these areas. Western Shikoku is a somewhat interesting area where old geodetic data have suggested little coupling after the Nankai earthquake, contrary to the present study. [57] Kato and Tsumura [1979] analyzed sea level data to estimate vertical movements of the tide station at Muroto and Tosa Shimizu, in eastern and western Shikoku, respectively. At Muroto, the subsidence that shows clear evidence of coupling seems to have occurred before the beginning of observations in This suggests that coupling has continued for more than 30 years. On the contrary, vertical movement at Tosa Shimizu has shown no clear evidence of subsidence until This leads us to the idea that strong coupling in western Shikoku began only very recently. The short history of strong coupling might lead to insufficient stress accumulation around the coupling area to trigger microseismicity. This evidence indicates that long-term coupling at the Muroto Misaki area has accumulated enough strain energy to cause many of the microearthquakes, but the short history of coupling at Tosa Simizu area has not accumulated enough strain energy to cause microearthquakes yet. This hypothesis may have to be examined together with other kinds of data such as seismic and geomagnetic structure, heat flow data, etc. 5. Conclusions [58] A number of numerical simulations were conducted to examine how parameters (k, u a ) are to be estimated in the inversion method based on the spectral decomposition of Green s function. Results suggest that, first, finer gridding is necessary for accurate estimations of eigenvalues (l a ) and eigenfunctions (8 a (x), y a (y)). The number of modes, k, depends on data condition, data quality, and data distributions. Best estimates of k can be obtained by testing the stability of mode coefficients. 12 of 14

13 Figure 13. Comparison between estimated slip distribution from GPS and hypocenter distribution for the same period of [59] The inverse method was applied to estimate back slip distribution at the subducting Philippine Sea plate boundary below southwest Japan using GEONET GPS data without any prior conditions. Results show that estimated back slip distribution is consistent with the PH motion and strong coupling occurs in the depth range of km and there are three areas of strong coupling. Two of the areas are concordant with areas of high seismic activity in the subducting slab, while the western most area of strong coupling has little seismic activity in the slab. This difference may be due to the difference of history of coupling. [60] The inverse method introduced in this study may provide a powerful tool for estimating fault slip or back slip distribution in the areas where dense geodetic networks are available such as the Japanese Islands. [61] It should be noted, however, that, because the GPS data used are limited in the land area which is only on one side of the plate boundary and far from coupling areas, it would limit the number of modes and hamper the accurate estimation of back slips. A problem shows up where part of the highly coupled areas shows estimated slip deficit rates larger than a plate motion of 4 5 cm/yr probably due to the effect of such biased site distribution. [62] Acknowledgments. We would like to express our gratitude to Chihiro Hashimoto for providing us plate boundary data. We thank Shin ichi Miyazaki of Earthquake Research Institute, University of Tokyo, for discussing the theory of inversion analysis. Comments and annotation by reviewers and Jeff McGuire of Woods Hole Oceanographic Institution were helpful for improving the manuscript. This research was supported by the Special Project for Earthquake Disaster Mitigation in Urban Areas ( ). References Hashimoto, C., K. Fukui, and M. Matsu ura (2004), 3-D modeling of plate interfaces and numerical simulation of long-term crustal deformation in and around Japan, Pure Appl. Geophys., 161, Hashimoto, M. (1990), Horizontal strain rates in the Japanese islands during interseismic period deduced from geodetic surveys: Part I. Honshu, Shikoku and Kyushu, ZISIN, 43, Hasimoto, M., and D. D. Jackson (1993), Plate tectonics and crustal deformation around the Japanese islands, J. Geophys. Res., 98(B9), 16,149 16,166. Heki, K., S. Miyazaki, and H. Takahashi (1999), The Amurian Plate motion and current plate kinematics in eastern Asia, J. Geophys. Res., 104(B12), 29,147 29,155. Hori, M. (2001), Inversion analysis method using spectral decomposition of Green s function, Geophys. J. Int., 147, Ito, T., S. Yoshioka, and S. Miyazaki (1999), Interplate coupling in southwest Japan deduced from inversion analysis of GPS data, Phys. Earth Planet. Inter., 115, Ito, T., S. Yoshioka, and S. Miyazaki (2000), Interplate coupling in northeast Japan deduced from inversion analysis of GPS data, Phys. Earth Planet. Inter., 176, Kato, T., and K. Tsumura (1979), Vertical land movement in Japan as deduced from tidal records ( ) (in Japanese), Bull. Earthq. Res. Inst. Univ. Tokyo, 54, Kirsch, A. (1996), An Introduction to the Mathematical Theory of Inverse Problems, vol. 12, 284 pp., Springer, New York. 13 of 14

14 Linette, P., and Y. Bock (2004), Instantaneous global plate motion model from 12 years of continuous GPS observations, J. Geophys. Res., 109, B08405, doi: /2003jb Mao, A., G. C. A. Harrison, and T. H. Dixon (1999), Noise in GPS coordinate time series, J. Geophys. Res., 104(B2), Maruyama, T. (1964), Statistical elastic dislocation in and infinite and semiinfinite medium, Bull. Earthq. Res. Inst. Univ. Tokyo, 42, Matthews, M. K., and P. Segall (1993), Estimation of depth-dependent fault slip from measured surface deformation with application to the 1906 San Francisco earthquake, J. Geophys. Res., 98(B7), 12,153 12,163. Menke, W. (1984), Geophysical Data Analysis: Discrete inverse Theory, 289 pp., Elsevier, New York. Miura, S., Y. Suwa, T. Sato, K. Tachibana, and A. Hasegawa (2004a), Slip distribution of the 2003 northern Miyagi earthquake (M6.4) deduced from geodetic inversion, Earth Planets Space, 56, Miura, S., Y. Suwa, A. Hasegawa, and T. Nishimura (2004b), The 2003 M8.0 Tokachi-Oki earthquake How much has the great event paid back slip debts?, Geophys. Res. Lett., 31, L05613, doi: / 2003GL Miyazaki, S., T. Saito, M. Sasaki, Y. Hatanaka, and Y. Iimura (1997), Expansion of GSI s nationwide GPS array, Bull. Geogr. Surv. Inst., 43, Miyazaki, S., and K. Heki (2001), Crustal velocity field of southwest Japan: Subduction and arc-arc collision, J. Geophys. Res., 106(B3), Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space, Bull. Seismol. Soc. Am., 75(4), Ozawa, T., T. Tabei, and S. Miyazaki (1999), Interpolate coupling along the Nankai Trough off southwest Japan derived form GPS measurements, Geophys. Res. Lett., 26, Parker, R. L. (1994), Geophysical Inverse Theory, 386 pp., Princeton Univ. Press, Princeton, N. J. Parker, R. L. (1997), Understanding inverse theory, Annu. Rev. Earth Planet. Sci., 5, Rothacher, M., and L. Mervart (1996), Bernese GPS Software Version 4.0, 418 pp., Astronomical Institute University of Berne, Switzerland. Sagiya, T. (1999), Interplate coupling in the Tokai District, Central Japan, deduced from continuous GPS data, Geophys. Res. Lett., 26(15), Sagiya, T., S. Miyazaki, and T. Tada (2000), Continuous GPS array and present-day crustal deformation of Japan, Pure Appl. Geophys., 157, Sagiya, T., and W. Thatcher (1999), Coseismic slip resolution along a plate boundary mega-thrust: The Nankai Trough, southwest Japan, J. Geophys. Res., 104(B1), Savage, J. C. (1983), A dislocation model of strain accumulation and release at a subduction zone, J. Geophys. Res., 88(B6), Tabei, T., M. Hashimoto, and S. Miyazaki (2002), Surface structure and faulting of the Median Tectonic Line, Southwest Japan inferred from GPS velocity field, Earth Planets Space, 54, Yabuki, T., and M. Matsu ura (1992), Geodetic data inversion using a Bayesian information criterion for spatial distribution of fault slip, Geophys. J. Int., 109, Yoshioka, S., T. Yabuki, T. Sagiya, T. Tada, and M. Matsu ura (1993), Interplate coupling and relative plate motion in the Tokai district, central Japan, deduced from geodetic data inversion using ASIC, Geophys. J. Int., 113, M. Hori and T. Kato, Earthquake Research Institute, University of Tokyo, Tokyo, Japan. (hori@eri.u-tokyo.ac.jp; teru@eri.u-tokyo.ac.jp) H. Jin, Institute of Earthquake Science, China Earthquake Administration, Beijing , China. (jhl@seis.ac.cn) 14 of 14

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