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Strongmotion simulation for the 1944 Tonankai earthquake based on the statistical green’s function method and stochastic representation of complex source process
Earth, Planets and Space volume 75, Article number: 97 (2023)
Abstract
Precisely evaluating the source, path, and site terms in a broadband frequency range is indispensable for quantitatively predicting strong motions. We conducted a generalized spectral inversion of strong motion in Japan to delineate both the spectral amplitude and phase characteristics for statistical Green’s functions. To predict ground motion from future megathrust earthquakes, we also need to model a kinematic source with a stochastic representation of the slip and rupture velocities on the fault surface. In this fundamental study, we first reported the basic features of statistical Green’s functions used for summation. We then demonstrated the construction of a kinematic source with distinctive strongmotiongeneration areas with spatially random slip and rupture velocity variations. After the summation of statistical Green’s functions following the constructed kinematic source, we found that the peak ground accelerations and peak ground velocities of synthetics for the Mw8.2 1944 Tonankai earthquake are in good agreement with those of the wellestablished empirical formula in Japan because the medians of the synthetic values are well within the average plus/minus one standard deviation of the formula. Based on the simulation results of conducted parametric studies, we observed relatively small but meaningful effects of random slip distribution, as well as relatively small effects of random rupture velocity perturbation. In conclusion, our implementation of the complex source model and the empirically obtained statistical Green’s function used as an element source can be a viable combination for broadband (0.1 to 20 Hz) strongmotion simulations, realistic in terms of the amplitude and duration, without any hybrid scheme.
Graphical Abstract
Introduction
Quantitative prediction of strong motion with source and sitespecific schemes is crucial for mitigating earthquake disasters and for the seismic design of important structures. This is especially true in Japan, where large megathrust earthquakes are expected to occur within the next 30 years along the Nankai Trough at the subduction interface between the Japanese Islands and the Philippine Sea Plate (The Headquarters for Earthquake Research Promotion 2022). Strongmotion prediction in a broadband frequency range from a large fault is difficult because the complex nature of the source, path, and site must be represented with the same degree of accuracy.
There are several methods for simulating strong motions as waveforms on a surface at a target site located at an arbitrary position. One is a theoretical method in which wave generation at a source, propagation from the source to a site, and local site amplification near the site are represented by numerical modeling of the medium. In this method, a viable physical model of the medium is required to represent the wavepropagation phenomena in the entire path from the source to the target site. In other words, Green’s function must be calculated quantitatively for a point source on the fault surface. Because of the limited computational resources and accuracy of the geophysical model that can be used, the frequency range of such a theoretical method is usually limited to a lower frequency range below 1 Hz, although several attempts have been made to expand the limit to 5 Hz or higher (e.g., Rodgers et al., 2019). Another method is an empirical method in which we use the observed ground motions of a small earthquake or a set of earthquakes as a substitute for Green’s function and sum all the contributions from the element sources on the fault surface. This is called the empirical Green’s function (EGF) method. The EGF has been proven to be very effective in reproducing the observed mainshocks if we use proper aftershock records (e.g., Kamae and Irikura 1998; Kamae and Kawabe 2004).
Suppose that there are no appropriate small earthquake records to be used as empirical Green’s functions, as is the case for groundmotion prediction for future events. In this case, we first generate synthetic waveforms representing small earthquakes and then sum them to represent a large event, as in the EGF. This is called either the stochastic or statistical Green’s function method (SGF); the difference between them is explained later. SGF is very useful for predicting strong motions for events that have not yet occurred or that have not yet produced any weak motions.
Because the frequency range for the fully theoretical method with a coherent nature is limited to the lower end, usually below 1 Hz or lower, whereas the valid range for EGF or SGF with the inherent nature of stochasticity is higher, a hybrid scheme combining a theoretical method with EGF or SGF is naturally used. This hybrid method has been used as a standard method in current national projects for strongmotion prediction using specific sources in Japan (JSHIS 2020).
However, after the deployment of dense national strongmotion observation networks, namely KNET and KiKnet by the National Institute of Earth Science and Disaster Resilience (NIED), and the Seismic Intensity seismometer (Shindokei) network by the Japan Meteorological Agency (JMA), a significant amount of data has been accumulated, which can be used to construct a model of SGF in a broadband frequency range that can reflect the average characteristics of observed ground motions. As long as we can generate the SGF for an arbitrary size of a small earthquake at an arbitrary location in the frequency range of interest (in this study, from 0.1 Hz to 20 Hz), we need not use a hybrid scheme combined with a theoretical representation of the lowfrequency component for a broadband simulation. The effectiveness of such a broadband use of the statistical Green’s function for groundmotion prediction of notyetoccurring inland earthquakes is shown in Itoh et al. (2001) and Ho and Kawase (2007).
However, if the true complex crustal structure is known, a hybrid scheme can reflect the path or sitedependent characteristics of wave propagation inside such a structure. In contrast, statistical Green’s function modeling can reflect medium or eventspecific characteristics only as statistical deviations from the average. If they are emerging only in a limited area or event, then their effects cannot be seen because their contributions could be too small to alter the average.
For SGF modeling, we used statistical Green’s functions based on the generalized spectral inversion technique (GIT, Andrews 1986; Iwata and Irikura 1988) to observe strongmotion data in Japan. The GIT provides the statistical properties of three major terms, namely the source, path, and site terms, as reported by Kawase and Matsuo (2004), Nakano et al. (2015), Nakano et al. (2019), and Nakano and Kawase (2021). The novelty of the approach in these GIT studies is that we use the extracted seismological bedrock spectra at one site, YMGH01, as a reference to calculate the site amplification factors at all other observed sites. We performed GIT for the observed Fourier amplitudes and phase spectra. Such separation of the observed spectra into three major terms in both amplitude and phase is sufficient to generate SGF at these sites using inverse Fourier transforms.
Here, we would like to explain the reason why we call our Green’s function as the “statistical” one, rather than the “stochastic” one. The stochastic Green's function, which was initially proposed by Boore (1983) based on a predefined ω^{−2} source spectrum and a specific time window shape, was applied to artificially generated white noise. Since then, many studies have been published using different implementations of the stochastic Green’s function. A recent addition to these studies was conducted by Ji et al. (2022). The word "stochastic" means that the Green's function is somehow randomly generated without referring to a specific characteristic derived from the observation. It is acceptable as long as Green’s function is used only in the reproduction of the highfrequency range, where the stochastic nature is dominant. In contrast, in the statistical Green's function construction described in this study, we use the Green's function directly obtained from the statistical properties of the observed Fourier amplitude and phase information based on the observed ground motions using the generalized inversion technique (GIT). The merit of using the statistical Green's function is its reality as a substitute for the empirical Green's function in the broadband frequency range because of the direct connection of the characteristics to the observed small earthquakes of the same size as the element source used for summation. Another merit is the option to consider the effects of variations in the source, path, and site factors derived directly from the statistics of the observed ground motions, if we want to consider such variations. Before this implementation, the statistical Green’s function method was applied to the subduction plateboundary earthquakes by Baoyintu et al. (2013, 2021), where they only used amplitude information from GIT. Instead, they used an envelope function with coefficients obtained by the regression for magnitude and distance.
After constructing the Green’s function, which can be used under generic conditions, the next step comprises a complex source representation of the fault for realistic synthetics of broadband ground motions.
From the beginning of kinematic source modeling in strongmotion seismology in the late’70 s and the early’80 s, it is apparent that we need to introduce spatial heterogeneity into slip distributions (Das and Aki 1977; Aki 1979; Papageorgiou and Aki, 1983 for Barrier model and Lay and Kanamori 1981 for Asperity model). In the’80 s and’90 s source process, inversion studies were conducted on major earthquakes worldwide. We can now refer to the SRCMOD database collected by Mai and his group (http://equakerc.info/srcmod/) to see those events analyzed before 2000.
For inland crustal earthquakes, Mai and Beroza (2002) referred to the kinematic source inversion results of 44 previous seismic events collected at that time to obtain a stochastically randomized source model with a fractal structure by considering the correlation distance with respect to the corner wave number, kc. Guatteri et al. (2003) performed a strong groundmotion simulation from rupture scenarios constructed based on the stochastic source model of Mai and Beroza (2002) and showed its validity by comparing the simulated results with empirical predictions using the groundmotion prediction equation (GMPE). Graves and Pitarka (2004, 2010, 2016) performed stronggroundmotion simulations of the past four earthquakes in the U.S. and validated the resultant strong ground motions by comparing them with the observed the peak ground accelerations (PGAs), the peak ground velocities (PGVs), and acceleration response spectra.
However, a stochastically randomized source model with smallscale variations cannot reproduce largeamplitude velocity pulses owing to the forward rupture directivity from a distinctive patch or asperity (i.e., asperity pulses), which is characteristic of strong ground motions in the nearfault region directly connected to structural damage in the epicentral regions of crustal earthquakes (Kamae and Irikura 1995; Pitarka and Irikura 1996; Kawase et al. 2000). Because it was found that the waveform of the nearfault region of the 1995 Hyogoken Nanbu earthquake was primarily composed of these asperity pulses (Kamae and Irikura 1995; Kawase et al. 2000), a method for extracting several large slip areas of the source and modeling them according to the scaling law of the outer fault parameters (e.g., Somerville et al. 1999) has been developed in Japan. The resulting source model is called the characterized source model (Irikura and Miyake 2001; Irikura et al. 2004).
The validity of the strong ground motions reproduced based on this characterized source model was demonstrated by comparing the observed records of several past earthquakes (Miyakoshi et al. 2000; Miyake et al. 2003; Miyakoshi et al. 2004). To systematize the method, Irikura and Miyake (2001) proposed a procedure (the socalled “recipe” for strongmotion prediction) to construct a characterized source model with distinct strongmotiongeneration areas (SMGAs) by analyzing observed records of several past earthquakes. However, such a distinctive SMGA representation for source complexity is problematic because the theoretical Green’s function emphasizes only the wave motions from the SMGAs. Therefore, the reality of the envelope form over the entire observed duration would not be well reproduced.
As part of the activities in the broadband strongmotion prediction platform project within the Southern California Earthquake Center (SCEC), Pitarka et al. (2020) validated the stochastically randomized model by Graves and Pitarka (2016), characterized source model by Irikura and Miyake (2011), and a hybrid model that combines the two models by comparing the synthetic strong ground motions from those models with those observed during the 2016 Kumamoto earthquake, as well as the model bias with respect to GMPEs. As a result, both the hybrid model and each of the two models show that they can generally reproduce the acceleration response spectra of the observations equally well.
However, when we look at subductionzone earthquakes macroscopic source rupture processes have also been studied since the early’80 s using recorded longperiod seismic waveforms from distant locations (e.g., Kanamori 1981; Lay and Kanamori 1980, 1981; Kikuchi and Kanamori 1982, 1986, 1991). Longperiod source inversions have revealed that the slip distribution on a fault plane is heterogeneous, as observed in crustal earthquakes. The large slip areas (asperities) determined in these studies exhibited slight variations, reflecting the resolution limit of the seismic waves used.
Irikura et al. (2004) and Miyake et al. (2006) proposed a method (recipe) for constructing a source model that can reproduce up to short periods based on the results of an analysis of past subductionzone earthquakes as well as inland crustal earthquakes, and strongmotion predictions have been made using this method (e.g., Kamae and Kawabe, 2004). However, the number of subductionzone earthquakes that have occurred in the past around Japan is much smaller than that of crustal earthquakes. Consequently, it is not easy to sufficiently verify the recipe.
Sekiguchi and Yoshimi (2011) proposed a method for constructing a heterogeneous source model based on the method of Mai and Beroza (2002) to add stochastic randomness to the distinct SMGAs proposed by the Central Disaster Prevention Council (2003). The basic concept of the stochastic slip distribution is similar to that of the hybrid model of Pitarka et al. (2020). Based on these reports on the complex nature of the fault rupture process, it is important to introduce both a coherent nature for asperities or SMGAs and a stochastic nature for the entire fault surface to represent realistic features observed in the simulated waveforms. In this regard, we are not trying to improve the implementation of complex source modeling in the scheme of Sekiguchi and Yoshimi (2011). Rather, we are trying to observe the effects of complex source modeling on the simulated waveforms in terms of spectra and maximum amplitudes when we combine it with the statistical Green’s functions mentioned above.
Then, the M 9.0 off the Pacific coast of Tohoku Earthquake occurred on March 11, 2011. The results of source inversion of the earthquake showed that a large slip region related to tsunami generation was located near the trench axis. In contrast, strongmotion generation areas (SMGAs) with a period of down to 2 s, which are directly related to the cause of building damage, are located on the western side of the ruptured area (e.g., Lay et al. 2012; Asano and Iwata, 2012; Kurahashi and Irikura, 2013). This finding is consistent with the depth dependence of the Brune’s stress parameter (the deeper, the larger) reported by Nakano et al. (2015).
Basic strategy
To construct a more realistic source model, it is necessary to construct a heterogeneous source model based on knowledge obtained from the latest megathrust earthquake. Therefore, in this study, we constructed a heterogeneous source model by arranging the size and locations of SMGAs based on the recipe and findings from the 2011 Tohoku earthquake, together with random perturbations in both the SMGAs and background regions using the method of Sekiguchi and Yoshimi (2011). As will be mentioned later, we introduce a partially stochastic nature to the phase characteristics of all statistical Green’s functions. Therefore, it is interesting to observe the effects of heterogeneity introduced in the rupture process of the hypothesized source under this partly stochastic SGF representation.
In this study, as a primary step toward understanding the source rupture process of the 1944 Tonankai earthquake, we combined the statistical Green’s function proposed by Nakano and Kawase (2021) and the heterogeneous source model proposed by Sekiguchi and Yoshimi (2011) to reproduce the strong motion of an earthquake. It is worthy of note that the 1944 Tonankai earthquake was an M8 class subduction plateboundary earthquake (e.g., Ando 1975) whose entire source has not been understood in detail because of the lack of observed waveforms in the nearfault region. The validity of the heterogeneous source model and the effects of introducing heterogeneity will be confirmed. The remainder of this paper is organized as follows. First, we briefly explain the GIT approach proposed by Nakano and Kawase (2021) and the resultant separate site amplification factors necessary for calculating Green’s function. Then we show the proposed concept of basininduced surface waves and coda waves that can be represented by the wholewave to Swave spectral ratio (WSR). We also demonstrate how to represent the group delay time, as proposed by Nakano and Kawase (2021). The amplitude and phase spectra of an incident wave at an arbitrary site from an element source on a fault surface are necessary to calculate the synthetic Green’s function. We then explain the method of constructing the heterogeneous source model from the SMGA distribution, based on the model of the Cabinet Office (2015) and the method of Sekiguchi and Yoshimi (2011). Based on this method, we constructed models with (i) various slip values and a constant rupture velocity (CASEVScr); (ii) a constant slip value and a constant rupture velocity (CASECScr); and (iii) both various slip and rupture velocity values (CASEVSvr). To compare the validity of these models and the Green’s functions used, we show the PGA and PGV distributions as a function of the distance and spectral amplitudes of selected sites for each case, together with a couple of representative acceleration time histories. The effects of heterogeneity on the PGA and PGV were studied by comparing CaseVScr with CaseCScr and CaseVScr with CaseVSvr. In addition, the effect of nonlinearity in the resultant synthetics, based on the approximated method proposed by Nakano and Kawase (2021), was investigated as CaseVScrNL. Through these analyses, we demonstrate that the proposed method is reasonable for predicting strong ground motions from a megathrust event. Please note that we previously published part of this study in the DPRI Annuals, the institutional collection of research reports from DPRI researchers (Ito et al. 2020a).
Statistical green’s function
Outline of GIT
In this section, we briefly introduce the observed horizontal site amplification factor (HSAF) and vertical one (VSAF) derived from GIT (Nakano et al. 2015; Nakano and Kawase 2021). Here, we only introduce their basic aspects because we used their results as a starting point.
Based on the ordinary nonparametric GIT concept, the Swave Fourier spectrum of the horizontal motion, F_{S_ij}, of earthquake i observed at site j is decomposed into the common logarithmic sum of the source term S_{S_i}, path term P_{S_ij}, and the horizontal site amplification factor at site j, H_{S_j}, as shown in the following equations:
Similarly, the Swave Fourier spectra of the vertical motion G_{S_ij} is decomposed into the following equation:
From these equations, the separation of variables through the GIT analysis should be performed simultaneously for both the horizontal and vertical components to obtain the common source and path terms. Otherwise, we may have slightly different source and path terms for the horizontal and vertical components.
These equations assume that ground motion from the source is propagated as Swaves until it reaches the seismological bedrock immediately below the observation site, after which a part of the energy of the Swaves is converted to Pwaves, which are observed as a vertical motion on the ground surface after amplification through the layers above the seismological bedrock. Here, V_{S_j} is the vertical site amplification factor for site j, and the third term on the righthand side of Eq. (2), V_{B}H_{B}R, is a coefficient accounting for the ratio of the vertical amplitude of an Swave part V_{B} to the horizontal amplitude of the same Swave part H_{B} under the equipartition condition of DFC. This term corresponds to the inverse of the horizontaltovertical amplitude ratio of the incident wave at the seismological bedrock, which is theoretically equal to the square root of the ratio of the Pwave velocity to the Swave velocity of the seismological bedrock in the diffuse field regime (e.g., Kawase et al. 2011; Nagashima et al. 2014; Ito et al., 2020b). Therefore, this coefficient should be virtually frequencyindependent. However, we observed some small fluctuations around the theoretical value at the reference site, YMGH01, as shown in Ito et al. (2020b). This coefficient is required because the main portion of the wave energy consists of Swaves propagated and scattered through the medium from the hypocenter to the point on the seismological bedrock immediately below the site, whereas V_{S_j} represents the vertical (Pwave) amplification factor of the vertical component from the seismological bedrock to the surface. HSAF and VSAF are the general terms referring to the site amplifications in the horizontal and vertical directions, while H_{S_j} and V_{S_j} are the specific terms derived from GIT at site j. Therefore, we refer to the horizontal and vertical site amplification factors as HSAF and VSAF hereafter for simplicity.
We restricted events and sites with JMA magnitude M_{JMA} ≥ 4.5; source depth ≤ 60 km; hypocentral distance ≤ 200 km; peak ground acceleration ≤ 2 m/s^{2}; and number of observation sites triggered simultaneously for one event ≥ 3. These selection criteria resulted in 77,213 eventstation pairs at 2,105 sites for 967 events from the KNET, KiKnet, and JMA Shindokei networks. Only a relatively short duration of acceleration record from the onset of the Swave is analyzed (5 s if 4.5 < M_{JMA} ≤ 6; 10 s if 6 < M_{JMA} ≤ 7; 15 s if 7 < M_{JMA} ≤ 8) to focus on the Swave part. The contributions of basininduced and basintransduced surface waves as well as coda waves were evaluated separately, as explained in the next section. A Parzen window of 0.1 Hz was used as the minimum level of smoothing. As mentioned above, the most important feature of the GIT used here is that it determines the Swave velocity structure at the reference site (YMGH01) using the transfer function (the spectral ratio and the phase difference) between the surface and the borehole 200 m below, and deconvolves the observed Fourier spectra on the surface to obtain the hypothesized outcrop spectra on the seismological bedrock with an Swave velocity of 3,450 m/s. Nakano et al. (2015), Nakano et al. (2019), and Nakano and Kawase (2021) successfully separated the source spectra and path terms as evidenced by their correspondence to the ω^{−2} source spectra shapes and regional Q values (transformed from the path terms) similar to previous studies in Japan.
Here, we emphasize the importance of the reference spectral conditions used in the GIT. It is acceptable to use any site or sites with any site amplification factors as a reference site or sites. However, if we use a site with site amplification as a reference, all the site amplifications will be carried over to the source factors for all events used. This is a direct consequence of Eqs. (1) and (2): If we set one site as a reference site, it is equivalent to assume that H_{S_j} at the reference site is one (i.e., log H_{S_j} is zero); therefore, the actual amplification at that site, if it exists, should be included in the source spectrum S_{S_i}. This is because the path term is strongly constrained by all paths between the sourcetosite pairs. However, the source and site terms are interconnected simply as a sum in the logarithmic space. If several sites are selected as a group of reference sites, their average amplification factor is carried over to the source spectrum. This situation is true not only for GITs, but also for any GMPEs. Therefore, the modeled source spectra in any GMPEs with a reference site condition not specifically defined as the outcrop of the seismological bedrock with a sufficiently high surface Swave velocity (Vs≧3 km) would be higher than the true source spectra by the amount of the actual site amplification included in the reference spectra.
Figure 1 shows examples of separated HSAF and VSAF at four representative sites in the Mie Prefecture, directly derived from Eqs. (1) and (2) as H_{S_j} and V_{B}H_{B}R*V_{S_j}. The choice of these sites was somewhat arbitrary; they were selected from a region relatively close to the epicenter. Among them, three were from the hilly zones of the Shima District with relatively stiff soil layers, and the other (JMAE43) was located close to the edge of the Ise Basin. We observed significant differences in HSAF from site to site. The amplitude and fluctuation of the VSAF were much smaller than those of the HSAF, especially below 3–7 Hz. Therefore, the earthquake horizontaltovertical spectral ratio, eHVSR, tends to be like HSAF until the fundamental peak frequency of VSAF is reached. However, to obtain HSAF from eHVSR, we need to correct VSAF, as recently proposed by Ito et al. (2020b). The VSAFs shown in Fig. 1 are the site amplification factors of the vertical component with respect to the horizontal component of the seismological bedrock, referred to as VSAF* by Ito et al. (2020b). VSAF* corresponds to VSAF multiplied by V_{B}H_{B}R; thus, VSAF* is slightly (theoretically 72%) smaller than VSAF. Hereafter, we use the abbreviation “VSAF” simply to refer to V_{B}H_{B}R*VSAF without any confusion, because we specifically predict the Swave part of the spectra radiated from the source. Therefore, we need to use the relative amplification factor of the vertical component to the Swave input on the seismological bedrock (H_{B}). Please note that we have not yet included a Pwave portion in our groundmotion simulations.
WSR to account for basin effects
One of the novelties of our SGF based on the GIT is the back calculation of the site factors for the entire duration of motion. When our target sites are located within a sedimentary basin, we observe additional amplification due to basininduced or basintransduced surface waves generated at the edge of a 2D or 3D basin (e.g., Bard and Bouchon 1980; Kawase and Aki 1989; Kawase 2003), in addition to the contribution of the Swave coda. They were significant in the lower frequency range (< 1 Hz), especially inside soft and large basins. However, if we directly used the entire duration of motion as the target of GIT analysis, the resultant attenuation would be distorted. When we use the entire duration, the Fourier spectra inside a basin but away from the source become larger than those spectra at sites closer to the source but outside of a basin. As a result, we cannot observe any attenuation in proportion to the distance; hence, we will have negative Q values.
To account for the effects of basininduced and basintransduced surface waves inside sedimentary basins, Nakano and Kawase (2019) and Nakano and Kawase (2021) proposed the use of an empirical ratio called the wholewavetoSwave ratio (WSR), where the spectral ratios of the whole duration with respect to the Swave portion with a relatively short duration (5–15 s) are averaged over all the observed events at a site. Although we have never performed another GIT for the entire duration, conceptually, we are doing the same as the GIT analysis for the wholewave Fourier spectra of the horizontal motion, F_{W_ij}, based on the following equation:
where H_{W_j} is the wholewave horizontal site amplification factor at site j. We fixed the source and path terms as those obtained for the Swave part. Similarly, the wholewave Fourier spectra of the vertical motion G_{W_ij} were decomposed into the following equation:
where V_{W_j} is the wholewave vertical site amplification factor at site j. WSR is the ratio of either H_{W_j}/H_{S_j} or V_{W_j}/V_{S_j}. Surprisingly, we found that the WSR for the horizontal and vertical components were quite similar. Therefore, we applied the same WSR for both the horizontal and vertical components after spatial interpolation, as described below.
We found that the WSR tends to be close to 1 irrespective of the frequency of a site on hard rock, whereas it can easily exceed 10 in the lower frequency range for a site inside a soft sedimentary basin. Even for such a site, the WSR converges to 1 (or close to 1) toward a higher frequency range, usually higher than 1 or 2 Hz. In Fig. 2, the averaged HSAFs for only the Swave part (black line) and those for the whole duration (red line) with their variation (red dotted lines) are compared. Thin gray lines represent individual HSAFs for the whole duration. MIE009, MIE016, and MIEH16 show the normal characteristics of sites outside a sedimentary basin, whereas JMAE43 shows those inside a deep (> 1 km) sedimentary basin close to the shoreline. The WSR is the spectral ratio between the red and black lines in this figure.
The spatial variation of WSR at a specific frequency is highly correlated with that of the basin depths, as reported by Nakano and Kawase (2019). Nakano and Kawase (2021) proposed a scheme to interpolate WSRs to make it possible to calculate a scenariotype hazard map with a much higher spatial density (in a 250 m grid) than those of strongmotion observation sites. The interpolation scheme utilizes the surface function of the GMT (Smith and Wessel 1990), together with the code developed by Renka (1999). This WSR correction is a simple empirical method that accounts for additional amplifications caused by soft sedimentary basins. In Fig. 3, we plot the original WSRs for HSAF across Japan and those interpolated based on the scheme proposed by Nakano and Kawase (2021) at 0.3174 Hz. It is evident that the WSR corrections are significant in the lowerfrequency range inside the large sedimentary basins, especially in Eastern Japan, east of the Itoigawa–Shizuoka tectonic line.
Tgr modeling
To calculate the broadband synthetic waveforms for smallto moderatesized earthquakes as SGFs, we must specify their phase information or envelope functions to constrain the timevarying characteristics of the synthetic SGFs as a function of frequency. It is desirable to use phase spectral information because it can account for the frequency dependence of the envelope shape (Sato et al 2002; Satoh 2004). Thus, we followed the procedure of Nakano and Kawase (2021) to model the group delay time t_{gr} as
where,
From Eqs. (5) and (6), following Satoh (2004), we use the following equation:
where R(ω) and I(ω) are the real and imaginary part of the spectra and \(func{\left(\omega \right)}^{^{\prime}}\) means the differentiation of \(func\left(\omega \right)\) with respect to ω. Then, we can model the average and variance characteristics of t_{gr} for the whole duration of observed record μ(f) and σ(f) as
for the ith source observed at the jth site. The derivation of these equations is the same as that of the GIT for spectral amplitudes, that is, Eqs. (1) and (2), except for the fixed geometrical spreading factor of 1.9, based on a study by Satoh (2004). X_{ijk} is the apparent section of the entire path in the kth region (k = 1, 6). Please refer to Nakano et al. (2015) for information on separate regions of path effects. The reference site for t_{gr} was also YMGH01, and t_{gr} at that site was set to zero. In the actual SGF calculation, we introduced a random number for each frequency to consider σ(f) as the perturbation from the average t_{gr}, μ(f).
In Fig. 4, we showed examples of the site terms in Eq. (8), that is, \({\mu }^{Site\_j}\left(f\right)\), for the same sites shown in Figs. 1 and 2. The physical meaning of the average site terms, \({\mu }^{Site\_j}\left(f\right)\) is the difference of the peakout time relative to that at YMGH01. Although we omitted any examples here, the physical meaning of the variance, \({\sigma }^{Site\_j}\left(f\right)\) is the spreading characteristics in time (i.e., the width of the main duration). Among the four sites, only JMAE43 site shows a significantly longer duration, especially in the lowerfrequency range, owing to the basininduced surface waves of the Ise Basin. As for the vertical component, we used the same \({\mu }^{Site\_j}\left(f\right)\) and \({\sigma }^{Site\_j}\left(f\right)\) as the horizontal component based on the comparison of t_{gr} for the horizontal and vertical components.
Stochastic source model
A broadband heterogeneous source model was generated for a plate boundary earthquake along the Nankai Trough by introducing fractal heterogeneity into a characterized source model consisting of background and strongmotion generation regions.
The gross ruptured area (i.e., the assumed fault plane) of the fault map of the Central Disaster Management Council (CDMC, 2003) was projected onto the upper surface of the Philippine Sea plate of the Japan Integrated Velocity Structure Model (JIVSM; Koketsu et al. 2012) and adjusted to be compatible with the fault size of the CDMC, and included the strongmotion generation areas (SMGAs) of the Cabinet Office (2015). For the assumed SMGAs, their areas, moments, stress parameters, rise times, and center positions were adopted from the values provided by the Cabinet Office (2015) as the basic model (Table 1). However, the shape of the SMGA is assumed to be circular, unlike those of the Cabinet Office (2015). We assumed as such because we want to use an automatic generation system for a large number of parametric studies with various locations and areas of SMGAs in the future. We also believe that circular and halfcircular shapes may work as representative shapes for SMGAs based on the spatial distribution of largeslip areas from past kinematic sourceprocess inversions. The parameters of the background region of the basic model were based on CDMC (2003). The amount of average slip was calculated from the total moment and fault area, similar to the method used by the CDMC (2003). The average area of each fault element was approximately 25 km^{2}, with some variations. The averaged seismic moment of each element is 4.3 × 10^{18} N·m.
Broadband source heterogeneity was calculated by randomly adding the heterogeneity of slip distributions from approximately half of the SMGA areas to several km^{2} in size to the basic source parameter distribution consisting of the background and SMGAs, as described by Sekiguchi and Yoshimi (2011). The amplitude of the fluctuation was proportional to the size of heterogeneity. The power of the falloff in the spatial distribution of the slip heterogeneity was adjusted and is equal to the value obtained by Mai and Beroza (2002), i.e., − 1.75. The stress parameter distribution is proportional to the variation in the slip distribution. In the end, the level of introduced slip heterogeneity in terms of the standard deviation was turned out to be approximately 1.3 m for the entire fault area.
The rupture velocity distribution was started from a uniform distribution of 2.7 km/s and a constant amount of variation regardless of the area of the inhomogeneous patch was added and finally adjusted so that the standard deviation of the spatial variation was set to ± 20% (Sekiguchi and Yoshimi 2011). This is equivalent to the level obtained by Miyakoshi and Petukhin (2005) in their analysis from the sourceinversion models. We set the maximum allowable rupture velocity to be the Swave velocity of the seismogenic zone. The distribution of rupture arrival times at an arbitrary point on the fault was obtained through wavepropagation analysis using an ordinary twodimensional finite difference method with a perturbed (or constant) rupture velocity on the fault surface.
Figure 5 shows the assumed shape of the whole ruptured area and the SMGAs, together with the locations and names of the prefectures that were damaged by the 1944 Tonankai earthquake. Randomized slip and randomized rupture velocity on the stochastic fault model used for strongmotion simulations are shown in Fig. 6 and 7, respectively, and the results are shown in the next section.
Results of parametric studies
In this chapter, to observe the effect of the variable slip and rupture velocity proposed by Sekiguchi and Yoshimi (2011), we tested three scenarios: a model with variable slip values and a constant rupture velocity (CASEVScr), a model with a constant slip value and a constant rupture velocity (CASECScr), and a model with variable slip values and rupture velocities (CASEVSvr). We calculated the PGAs and PGVs at the KNET, KiKnet, and JMA Shindokei networks in the Mie, Aichi, and Shizuoka Prefectures, where severe building damage and tsunamis were observed during the 1944 Tonankai earthquake (Takemura and Toraya, 2015). All target sites used in this study are listed in Table 2 with their altitudes. Table 2 also includes Vs30 (the socalled timeaveraged Swave velocity of top 30 m) from the PS logging data as in Wang et al. (2023) for KNET and KiKnet sites or from the estimated shallow velocity profiles of Senna et al. (2019) and Wakai et al. (2019) for JMA sites, as well as Z1.4 (the depth to the layer with the Swave velocity higher than 1.4 km/s) from the estimated deep velocity profiles of Senna et al. (2019) and Wakai et al. (2019). These are the site parameters in the empirical formula of Morikawa and Fujiwara (2013).
Model with variable slip and constant rupture velocity (CASEVScr)
Figure 8 shows the attenuation relationships of the PGA and PGV in CASEVScr with respect to the shortest distance to the fault plane. The orange and blue circles show the PGA and PGV at each site, respectively, the black line shows the empirical relationship from Model 2 of Morikawa and Fujiwara (2013), and the black dotted lines show the average plus/minus one standard deviation. CASEVScr yields strongmotion synthetics with PGAs that correspond to slightly higher values than the empirical relationship of Morikawa and Fujiwara (2013), whereas PGVs tend to correspond to the average values. We used the baseline attenuation formulas in Morikawa and Fujiwara (2013), where Vs30 is assumed to be 350 m/s and Z1.4 to be 0.1 km. These reference values would be appropriate for sites outside large basins, such as MIE016. However, these references would be too large for Vs30 and too small for Z1.4, for sites inside large basins, such as JMAE43. If we consider the range of effects in these site parameters for typical values inside large basins, such as Vs30 = 200 m/s or Z1.4 = 1 km, the deviation from the baseline shown here would yield a 1.3 to 1.5 times increase (See Morikawa and Fujiwara (2013) for detailed information). To see the effects of Vs30 and Z1.4 in the simulated PGA and PGV, we distinguished the basin sites with Vs30 < 250 m/s and Z1.4 > 500 m (14 in total) with small solid circles inside. In Fig. 8, as well as the other figures with different source parameters, namely, Figs. 13 and 15, we can see clearly the effects of soft and deep sedimentary basins as increasing amplitudes on both PGA and PGV. Relative amplifications are slightly larger in PGV than those in PGA, which is consistent with their difference in coefficients shown in Morikawa and Fujiwara (2013).
Figures 9 and 10 show the acceleration waveforms and the acceleration Fourier and pseudovelocity response spectra at JMAE43, and Figs. 11 and 12 show those at MIE016 in the same manner. At JMAE43, located inside the Ise Plain, the peaks of the response spectra exist in the period range of 0.5–2 s. At MIE016, which is located inside a hilly zone, the response spectra were much smaller than those at JMAE43, except for the period shorter than 0.3 s. Please note that the east–west (EW) component in the low frequency range is larger than the north–south (NS) component because of the frequencydependent radiation pattern.
Model with constant slip and constant rupture velocity (CASECScr)
Figure 13 shows the attenuation relationships of the PGAs and PGVs in CASECScr. The orange and blue circles show the PGA and PGV at each site, respectively. The black line shows the empirical relationship from Morikawa and Fujiwara (2013), and the black dotted lines show the average plus/minus one standard deviation. This model, with a constant slip value and a constant rupture velocity, yields strongmotion synthetics with PGAs that correspond to the empirical relationship, but the PGVs are slightly lower than the empirical relationship at sites near the hypocenter. At MIE016, the Fourier spectra are about half of the one in CASEVScr within 0.3–0.4 Hz, where the difference between them is the most significant. The response spectra were slightly smaller for CASECScr in almost every period band, with the smallest being approximately onehalf to twothirds that of CASEVScr at 0.4 to 0.5 s. Because we omitted the acceleration waveform plots for CASECScr and CASEVSvr owing to the difficulty in observing the difference, we added a list of PGA values as Table 3.
Figure 14 shows the Fourier and response spectra at MIE016 for CASE–CScr (solid lines) and CASEVScr (dotted lines). We do not show the results for JMAE43. The systematic differences between CASEVScr and CASECScr were observed; the amplitudes of CASECScr are almost always smaller than those of CASEVScr in the frequency range below 2 Hz (above 0.5 s in period). Direct comparisons of the PGAs and PGVs are presented later in the discussion section.
Model with variable slip and variable rupture velocity (CASEVSvr)
Figure 15 shows the attenuation relationships of the PGAs and PGVs in CASEVSvr. The symbols and lines are the same as those in Fig. 13. Figure 16 shows the Fourier and response spectra at MIE016 in Fig. 14 for both CASEVSvr (solid lines) and CASEVScr (dotted lines). We do not show the results for JMAE43. A model with variable slip and rupture velocity (CASEVSvr) yields strongmotion synthetics with PGAs and PGVs that correspond well with the empirical relationship. In the Fourier spectra, although there were some differences in the frequency range lower than 1 Hz, they were at the same level on average. The response spectra also differed in the longperiod range. However, the difference between CASEVSvr and CASEVScr is not systematic; at one period in one component CASEVSvr is larger and at another period in another component CASEVScr is larger. The difference in the shorter period range is marginal. Therefore, it can be said that the overall levels of Fourier and response spectra were similar to each other. Direct comparisons of the PGAs and PGVs are presented later in the Discussion section.
Discussion
Effects of slip variation on PGA and PGV
As shown in previous sections, there is a relatively small difference between the simulated synthetics from CASEVScr and those from CASECScr at a selected site (MIE016) near the hypocenter. This difference reflects the effect of the slip variation on the fault. We need to check whether this is a universal observation at all the analyzed sites.
Figure 17 shows comparisons of the PGAs and PGVs at all sites in the Tokai area. The PGAs from the model with spatial slip variations (CASEVScr) tended to be slightly higher than the PGAs from the model without spatial slip variations (CASECScr). For a smooth slip model (CASECScr), we observed a 23% reduction in the PGA compared with a variable slip model (CASEVScr) based on the regression coefficient of their ratios, 0.77. However, the PGVs from the model with spatial slip variations (CASEVScr) tended to be higher than the PGVs from the model without spatial slip variations (CASECScr). For a smoothslip model (CASECScr), we observed approximately 29% reduction in PGVs in comparison with a variableslip model (CASEVScr) based on the regression coefficient of their ratios, 0.71.
These observations may be attributed to the fact that the PGA is the result of random superposition of acceleration waves from each element source, so that it is difficult to see the effects by the slip variations in individual elements with the size of 5 km × 5 km, whereas the PGV is controlled in the intermediate frequency component (0.5 to 5 Hz) so that coherent interferences of waves from different element sources are taking place. In our simulations, we did not assume any spatial fluctuation in the stress drop in the element sources (except for the SMGAs), which controlled the highfrequency radiation level without slip fluctuation. In the future, it may be necessary to consider the spatial fluctuation of the stress drop to understand how it affects the highfrequency range of the synthetic waveforms.
Effects of rupture velocity variation on PGA and PGV
As shown in previous sections, we can observe a minor effect between the simulated synthetics from CASEVScr and those from CASEVSvr at a selected site near the hypocenter. This difference reflects the effect of rupture velocity variation on the fault. We need to check whether this is a universal observation at all sites analyzed.
Figure 18 shows the comparisons of PGAs and PGVs for CASEVScr and CASEVSvr at all sites in the Tokai area. Clearly, both the PGAs and PGVs show an almost 1:1 correspondence. The stability of the results in PGAs and PGVs between CASEVScr and CASEVSvr could be because of the random phase component introduced in the element source waveforms, which may prevent the possibility of coherent summation of waves, even for the case with a constant rupture velocity.
These values inevitably fluctuate from site to site, even for the same parameter case, because we included sitespecific amplification factors that differ significantly from site to site, as shown in Figs. 1 and 2.
Effects of nonlinearity on PGA and PGV
As per the simulation results for CASEVScr (Figs. 9 and 10 for JMAE43), there are high PGV areas inside the Ise Plain, where strong site amplification is observed in the lower frequency range. When a strong input of seismic motion impinges on thick, soft sediments, soil nonlinearity occurs, and we will have a smaller amplitude in the site amplification factor than the linear one (i.e., amplitude degradation) and a peak frequency shift. Nakano and Kawase (2021) proposed a method using only amplitude degradation, without considering the peak frequency shift. In this report, we consider only amplitude degradation for simplicity.
The amplitude degradation curve proposed by Nakano and Kawase (2021) was based on 174 high PGA and PGV records, together with the Vs30 values at their observation sites. The method follows the one proposed by Yamaguchi and Midorikawa (2014), in which the pseudoeffective shear strain γ’_{eff} is defined as:
Then the spectral amplitude degradation G'(f) is modeled as:
in which a(f) is the regression coefficient for each frequency using a linear function with respect to the excess strain between γ’_{eff} and γ’_{eff_0}, where γ’_{eff_0} = 3.0 × 10^{–4} is the threshold strain level of the linear regime assumed a priori. Two examples of the initial regression for a(f) are shown in the top panels of Fig. 19, whereas the final regression curve a(f) as a function of frequency is shown in the bottom panel of Fig. 19. The regression curve was plotted here not as a function of frequency but period T = 1/f for comparison with Yamaguchi and Midorikawa (2014). Nonlinearity emerged as a stronger amplitude degradation in the frequency range 1–10 Hz.
Figure 20 shows the comparisons of the PGAs and PGVs for CASEVScr and CASEVScrNL calculated using the spectral nonlinearity shown in the above formula in addition to the synthetics calculated under the same assumption as CASEVScr at all sites in the Tokai area. In this figure, the PGAs and PGVs at sites where the PGVs were over 40 cm/s are shown as red circles. From this figure, it can be deduced that the degree of nonlinearity is proportional to the PGV value, which is also evident from Eq. (8) for the same Vs30. The threshold value in terms of the PGV is 40 cm/s. Therefore, the PGAs at sites where the PGVs are over 40 cm/s are clearly affected by nonlinearity, while the PGAs at sites where the PGVs are under 40 cm/s remain the same as in the linear regime. It is also obvious that the effect of nonlinearity on the PGAs for sites with PGV > 40 cm/s tends to be larger than the effect of nonlinearity on the PGVs. This is because PGVs are primarily controlled by the intermediate frequency range from 0.5 Hz to 5 Hz, while PGAs are controlled by the whole frequency range of accelerograms.
It is worth mentioning that the sites with high PGAs near the hypocenter did not show any significant reduction in their Fourier spectra, as shown in Fig. 21, for MIE016 as an example. This is because the effective shear strains in Eq. (8) at these sites were not very large owing to their smaller PGVs and larger Vs30.
Considering these fundamental characteristics of the resultant soil nonlinearity, we believe that the empirical approach for soil nonlinearity correction adopted in the current implementation will be useful for the largescale prediction of strong motions as a firstorder approximation. Please note that the discrepancy between the synthetics and GMPE will increase inevitably by considering the nonlinearity, which is natural as the current synthetic PGA matches to the GMPE. Further validation of the empirical method to consider nonlinearity is needed.
Conclusions
We planned to clarify the source processes of past megathrust earthquakes, the ground motions of which have not been recorded, by adapting the simulated strongmotion waveforms to the results of the damage survey. For realistic strongmotion synthetics, we introduced random spatial heterogeneity in both the slip and rupture propagation velocities in addition to the presumed strongmotion generation areas. We have developed an effective method to predict strong ground motions over a wide frequency range from 0.1 to 20 Hz by combining a randomized source model that can represent complex rupture process on the fault and a statistical Green’s function that reflects the statistical properties of small to mediumsized earthquakes extracted from a large number of strongmotion observation records in Japan. The method was applied to the 1944 Tonankai earthquake, and a parametric analysis was performed to investigate the effects of fluctuations in slip and rupture propagation velocities. The main results are summarized as follows.

1)
The standard model with random slip fluctuation without rupture velocity perturbation (CASEVScr), a model with constant slip and constant rupture velocity (CASECScr), and a model with both random slip fluctuation and variable rupture velocity (CASEVSvr) yield strongmotion synthetics with both PGAs and PGVs corresponding to the empirical relationship. The PGVs in CASECScr were underestimated compared with those in CASEVScr and CASEVSvr because CASECScr has a constant slip value and rupture velocity. The introduction of fluctuations in the slip values under the current model resulted in an increase in the PGA by 23% and PGV by 29%.

2)
The variations in the rupture propagation velocity had no significant effects on either the PGA or PGV, probably because of the stochastic nature of the statistical Green’s function method adopted here.

3)
The proposed empirically modelled nonlinearity, which considers a strong amplitude degradation in the frequency range from 1 to 10 Hz, reduces both PGAs and PGVs at sites with higher PGVs and smaller Vs30. However, the degree of nonlinearity is larger in PGAs than in PGVs. This is because the PGA is controlled over a broader frequency range than the PGV.
As for the overall characteristics of the calculated synthetics, we found that our current implementation of the complex source model used as a broadband kinematic source representation and the empirically obtained statistical Green’s function used as an element source of strong motions can be a viable combination for broadband (0.1 to 20 Hz) strongmotion simulations, realistic in terms of the amplitude and duration, without any hybrid scheme.
In the future, we will estimate building damage ratios by inputting the acceleration waveforms into the wooden building model with construction age and then search for the most realistic source rupture model through a large number of parametric analyses until the estimated damage ratio at each site matches the observed one.
Availability of data and materials
All data generated or analyzed in this study are included in the published article. Digital data are available upon request from the corresponding author.
Abbreviations
 CDMC:

The Central Disaster Management Council
 EGF:

The empirical Green’s function method
 eHVSR:

The earthquake horizontaltovertical spectral ratio
 GIT:

The generalized spectral inversion technique
 GMPE:

Groundmotion prediction equation
 GMT:

Generic mapping tools
 HSAF:

Horizontal site amplification factor
 JIVSM:

The Japan Integrated Velocity Structure Model
 JMA:

Japan meteorological agency
 JSHIS:

Japan seismic hazard information station
 PGA:

Peak ground acceleration
 PGV:

Peak ground velocity
 pSv:

Pseudovelocity response spectra
 SCEC:

Southern California Earthquake Center
 SGF:

Statistical Green’s function method
 SMGA:

Strongmotiongeneration areas
 2D/3D:

Two dimensional/three dimensional
 VSAF:

Vertical site amplification factor
 Vs30:

The timeaveraged Swave velocity of top 30 m
 WSR:

The wholewavetoSwave ratio
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This work was partially supported by KAKENHI 21K14388.
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Statistical Green’s function from the GIT was provided by KN, and waveform synthetic calculations were performed by EI with instructions from KN. The stochastic source model was constructed by EI using the code provided by HS. A parametric study of the source models and their analyses were performed by EI, HK, and NF through discussions with KN and HS. All the authors contributed to the editing and revision of the manuscript. All the authors have read and approved the final version of the manuscript. All authors read and approved the final manuscript.
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Ito, E., Nakano, K., Sekiguchi, H. et al. Strongmotion simulation for the 1944 Tonankai earthquake based on the statistical green’s function method and stochastic representation of complex source process. Earth Planets Space 75, 97 (2023). https://doi.org/10.1186/s4062302301847y
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DOI: https://doi.org/10.1186/s4062302301847y
Keywords
 1944 Tonankai earthquake
 Heterogeneous source model
 Statistical Green’s function