The spectral inversion (Satoh, 2016) estimated the source spectra of the foreshocks, mainshock, and moderate aftershocks of the Kumamoto Earthquake. The hypocenters of these earthquakes are shown in Fig. 9. There were 18 moderate aftershocks ranging in magnitude from 4 to 6. These source spectra were used to calculate the empirical site amplification for every earthquake at EEB.
An empirical site amplification G is calculated using equations (1), (2), and (3). In these equations, X, and f are hypocentral distance, and frequency. Q0 is the Q value obtained by the spectral inversion analysis. ρ and β were density (2.7ton/m3) and S-wave velocity (3.5km/s) in the source layer. Rθφ was the radiation coefficient. Ans and Aew are acceleration Fourier spectra of observed ground motions in NS and EW directions. S is acceleration source spectra.
The empirical site amplification during weak ground motions at EEB is the average of the empirical site amplifications of the moderate aftershocks. Prior to ESG6, the ESG local organizing committee distributed ground motion records at KUMA with magnitudes ranging from 3.3 to 5.9. One of the moderate aftershocks was included. The empirical site amplifications were calculated for this single earthquake and the targeted aftershocks in BP2, for which record at KUMA were distributed after ESG6. In KUMA, the empirical site amplifications during weak motions at KUMA is the average of the empirical site amplification of these two earthquakes. We utilized empirical site amplifications during weak motions, the foreshock, and the mainshock at EEB and KUMA to consider prediction errors. Figure 10 depicts the empirical site amplifications at EEB. The empirical site amplification during the main shock is lower than the weak motions above 2 Hz. Nonlinear soil response is thought to be a contributing factor. The empirical site amplification during the foreshock is much higher than that of the weak motions below ~ 1 Hz, which is probably caused by fixing the radiative coefficient in Eq. 4 to 0.55 (Boore et al., 1984), the mean radiation coefficient of near-earthquakes. The strain levels based on the equivalent linearization method shown later indicate that the soil is highly nonlinearized at the EEB during the foreshock. However, since the empirical site amplification below 6 Hz of the foreshock at EEB is the same as or greater than that in weak motion time, it is considered that the radiation coefficient strongly influences the empirical site amplification below 6 Hz. Therefore, only the spectral shape of the empirical site amplification of the EEB is considered. Similarly, Fig. 11 shows the empirical site amplifications at KUMA. Like EEB, the empirical site amplification during the foreshock is high at low frequencies but up to just under 2 Hz.
$$P\left(f\right)=exp\left(-\frac{\pi fX}{{Q}_{s}\left(f\right)\beta }\right)$$
1
$$C=\frac{1}{\sqrt{2}}\frac{{R}_{\theta \phi }}{4\pi \rho {\beta }^{3}}$$
2
$${G}_{ns,ew}\left(f\right)={A}_{ns,ew}\left(f\right)/\left(C\bullet P\left(f\right)\bullet S\left(f\right)\right)$$
3
BP2
The Point Source
An aftershock with a point source was modeled. A centroid and a CMT solution were estimated. The grid points were set at 3 km intervals in the NS and EW directions and 5, 7, 9, and 13 km in depth around the epicenter by the JMA. Using the least-squares method, the moment tensor was inverted at each grid to fit the velocity waveforms with a period of greater than 4 s at the observation stations depicted in Fig. 12. In this inversion, we used a source time function for far-field S-wave (Brune, 1970). The source time function STF is represented by equations (4) and (5). M0 is a seismic moment (Nm) and fs is a corner frequency. The fc was fixed at 0.44 Hz which was estimated by the spectral inversion analysis (Satoh, 2016). The centroid was determined to be the point on the grid where the difference between the calculated and observed waves was the smallest, and the moment tensor at the grid was used. Figure 12 shows the CMT.
The Prediction
The ground surface records at EEB were used to calculate the short-period incident waves in the engineering bedrock (the 7th layer) using the 1D multiple reflection theory. To determine the incident waves in the engineering bedrock (the 11th layer) at KUMA, the incident waves at EEB were multiplied by the ratio of the hypocentral distance of KUMA to EEB.
For checking the 3D velocity model, we simulated long-period ground motions at KMM006 and EEB during the targeted aftershock in BP2 by the 3D finite difference method. A comparison of synthesized and observed waveforms is shown in Fig. 13. The observed waveforms and the synthesized waveforms at both locations are in agreement. At EEB, synthesized waveform using original JIVSM is also shown. The two synthesized waveforms are essentially identical. No significant improvement was recognized from the modification of JIVSM. In the engineering bedrock at KUMA, the synthetic wave was cut in half to become the incident wave and was referred to as the long-period incident wave.
Figure 14 shows the predicted and observed waveforms during the aftershock. The amplitude levels and envelope shapes for both acceleration and velocity are almost exactly reproduced by the predictions. Figure 15 shows the Fourier spectra and the spectral ratios of the prediction to the observation. To avoid overlapping with the spectral ratios, the Fourier spectra were plotted five times. The Fourier spectra were smoothed by the parent window with the bandwidth of 0.1 Hz, except for the average and the average +/− a standard deviation of the Fourier spectra by BP2 participants. A parent window within the bandwidth of 0.4 Hz was used to smooth the Fourier spectrum before calculating the spectral ratio. Thereafter, the Fourier spectra and the Fourier spectrum ratios were similarly smoothed and then plotted. Between 3 and 5 Hz, predictions are too frequent. Particularly, NS components significantly outperform at 5 Hz.
Spectral ratios of upgoing waves of S-waves on the ground surface to that on an upper boundary of a layer were calculated by the multiple reflection theory (hereafter, theoretical S-wave amplification). Theoretical S-wave amplifications from the engineering bedrock at EEB and KUMA (the 7th and 11th layers, respectively) are depicted in Fig. 16. and the ratios between the theoretical KUMA and EEB S-wave amplification. and the ratios of the theoretical S-wave amplification at EEB to KUMA. Short-period predicted ground motions are represented by the convolutions of these spectral ratios and the Fourier spectra of the ground surface records at EEB. The spectral ratios are predominant at 3 to 4 Hz where the predictions are overpredicted. The top and bottom of theoretical S-wave amplifications at EEB and KUMA, respectively, are what led to this dominance. Therefore, to check the validity of the frequency characteristics of the theoretical S‐wave amplifications at EEB and KUMA, it was compared with the empirical site amplifications at each site.
The theoretical S-wave amplification from the seismic bedrock (the 18th layer) and the empirical site amplification at EEB is shown in Fig. 17. The empirical site amplifications are averagely large in a frequency range of just below 4 to 5 Hz. On the other hand, the theoretical S-wave amplifications of the seismic bedrock are also predominant in the same frequency range, but they are a little higher frequency. Theoretical S-wave amplifications from each layer between the engineering bedrock and the seismic bedrock are shown in Fig. 18 for EEB. It is clear from the transitions of the theoretical S-wave amplifications from the seismic bedrock to the engineering bedrock that the shallower layer than the engineering bedrock is primarily responsible for the mean large amplification of 4 to 5 Hz of the theoretical S-wave amplification from the seismic bedrock. As a result, the S-wave velocities above the engineering bedrock in the velocity model at EEB are considered to be a little higher than the actual one.
The seismic bedrock (the 14th layer) at KUMA is amplified theoretically and empirically by the site in Fig. 19. The empirical site amplifications are predominant around 4 Hz. Theoretical S-wave amplifications, on the other hand, tend to dominate between 3–5 Hz on average, although they do occasionally drop below 4 Hz. Unlike EEB, no obvious difference in the predominant frequencies is recognized.
The overprediction in the 3 to 6 Hz range may be due to modeling errors in layers shallower than the engineering bedrock at EEB, according to the information presented above. With this modeling error, the incident wave in the engineering bedrock may have been overestimated. Figure 20 shows the Fourier spectra of the ground surface records at EEB and KUMA. The records of the ground surface at KUMA are the targets of BP2 and BP3. NS component at EEB is particularly large at 5 Hz. This peak is noticeably more prominent than the peak of the theoretical S-wave amplification and the empirical site amplification, which both peak at about 5 Hz. And this peak is not recognized at KUMA. This peak is thought to be the result of the frequency response of the propagation path that is localized in the EEB based on these. In the NS component, in addition to modeling errors in the surface layers at EEB, this peak is also a major cause of overprediction. The local frequency response of the propagation path cannot be taken into account when estimating the incident waves in the engineering bedrock from those at a different point using only hypocentral distance correction.
Figure 21 depicts the average and the average +/− a standard deviation of the Fourier spectra predicted by the participants compared to this study. Except for the 5Hz of NS components, our predictions are nearly within the participant's average +/− standard deviation.
BP3
The Nonlinear Soil Properties
The ESG local organizing committee has had five soil samples from the borehole at KUMA tested in a lab (Matsushima, 2022). Nonlinear properties T1, T2, Tr-3, Tr-4, and Tr-5 were obtained and modeled by H-D models. Each layer of the 1D models at EEB and KUMA received the H-D-model assignment. The unknown is the geology of EEB. T-1 obtained in the layer with an S-wave velocity of approximately 100 m/s was assigned to layers with the same degree, and Tr-5 was assigned to other layers because it was the most average. To ensure that geology and depth correspond, the nonlinear property was given to each layer of the 1D model at KUMA. Tr-3 was not used because it deviated from the H-D model in a large strain range. Tables 1 and 2 display the nonlinear characteristics given to each layer of 1D models at KUMA and EEB, respectively. In these tables, Li is a case in which nonlinear is not considered.
The equivalent nonlinear method was used to calculate short-period incident waves in the engineering bedrock at EEB from the aftershock and the mainshock records on the ground surface. We used damping that decreases at high frequencies because the equivalent linear method frequently overestimates damping at those frequencies (Satoh, 1997). The damping factor used in the nonlinear method. is represented by Eq. (6). γ is strain and h(γ) is damping factor of H-D models. Q0 and Qf are written in Table 1 and Table 2.
$$h{\prime }\left(\gamma \right)=\left(h\left(\gamma \right)+1/\left(2{Q}_{0}\right)\right){f}^{-{Q}_{f}}$$
6
DAYNEQ (Yoshida, 2008) was used to perform calculations included in the above method.
Figure 22 depicts the S-wave velocity structures and maximum strain at EEB and KUMA during the foreshock. Each layer was divided to precisely calculate the strain. The maximum strains in almost all layers at EEB and KUMA are less than 1% where the equivalent linear analysis is valid. The maximum strain in the layer at a depth of just less than 5 m at KUMA is 3% in the NS component. Due to the thinness of this layer, the effect is thought to be minimal although the maximum strain slightly exceeds the strain at which the equivalent linear analysis is valid. Similarly, Fig. 23 illustrates the maximum strain during the mainshock. The maximum strains during the main shock and the foreshock are similar.
The Prediction
Figure 24 illustrates the predicted and observed waveforms during the foreshock. The prediction closely matches the envelope shapes and amplitude levels of acceleration and velocity. Figure 25 shows their Fourier spectra. Below 0.5 Hz and above 10 Hz, the predictions are overpredicted, and between 1.5 Hz to 3 Hz, they are underpredicted.
The layers between the engineering bedrock and the seismic bedrock have a significant influence on long-period ground motions below 0.5 Hz. Large estimation errors are unlikely because the average S-wave velocity estimated by participants in BP1 is nearly identical to that of us. The underprediction below 0.5 Hz is mainly attributed to the source model.
The ratios of the theoretical S-wave amplification of EEB to KUMA are shown in Fig. 26 along with the theoretical S-wave amplifications from the engineering bedrock at EEB and KUMA. The frequency at which the theoretical S-wave amplifications predominates is lower than that of aftershocks because of the nonlinear soil responses. Around 2 Hz, where the predictions are exaggerated, the spectral ratios predominate. This predominance is responsible for the over predictions of 3 Hz from 1.5 Hz. The spectral ratios are lower than 1 above 5 Hz. This small spectral ratio is a direct cause of underprediction. The under- and overestimation of the theoretical S-wave amplifications by the nonlinear soil responses in EEB and KUMA, respectively, are thought to be the cause of the small spectral ratios.
Figures 27 and 28 show the theoretical S-wave amplifications and the empirical site amplifications from the seismic bedrock during the foreshock at EEB and KUMA, respectively. As mentioned in the chapter on empirical site amplification, only the frequency characteristics are taken into account for the empirical site amplifications at EEB during the foreshock. The empirical site amplifications at EEB are predominant at 4 Hz. The theoretical S-wave amplifications from the seismic bedrock, on the other hand, are also weakly dominant at 4 Hz, though they are slightly lower in the NS component and slightly higher in the EW component. In the nonlinear response below the engineering bedrock, the equivalent S-wave velocity may be incorrect at EEB, and the amplification from 1.5 Hz to 3 Hz may not be properly assessed. Theoretical S‐wave amplifications are greater than observed site amplifications at frequencies close to 2 Hz. The overestimation of 1.5 to 3 Hz is also due to the overestimation of amplification at KUMA.
Above 5Hz, the aftershock is not underpredicted, so the underprediction of the foreshock is caused by the failure to reproduce the decrease in S-wave amplification due to the nonlinear soil response. From 5Hz to 10Hz, the theoretical S-wave amplification exceeds the empirical site amplification at both EEB and KUMA sites. The reason for the underprediction is considered to be the underestimation of the decrease in S-wave amplification at EEB. Above 10Hz, the theoretical S-wave amplification at EEB is comparable to the empirical site amplification, so the reason for the underprediction is considered to be the overestimation of the decrease in S-wave amplification at KUMA.
Figure 29 shows the average and average +/- standard deviation of the Fourier spectrum predicted by the participants. At 1 Hz, the targeted record at KUMA has a sizable amplitude. This study reproduced this large amplitude. The large amplitude cannot be predicted, though, because the participant’s average is lower than our predictions. The overpredicted 1.5 Hz to 3 Hz is greater than the participant’s average + standard deviation. Except for this frequency range, however, our predictions are almost within the participant’s average +/- standard deviation.
The predicted and observed waveforms during the mainshock are shown in Fig. 30. The prediction almost reproduces the amplitude level of acceleration and velocity and the envelope shape of acceleration. The large waves that appeared between 7 and 10 seconds in the observation are not replicated by the velocity's envelope shape. The fault model has been estimated using a large number of records near the fault (Tsuda et al., 2021). We did simulations of log-period ground motions at SEVO and KUMA by the finite difference method using the source model used in our prediction and the source model by Tsuda et al.. Figure 31 shows these source models and the observation stations used for the source inversions. A comparison of the simulated and observed ground motions is shown in Fig. 32. The source model by Tsuda et al. reproduces well the observed waveforms at two sites. It appears that predictions of the ground motions near a fault benefit from a source model estimated by a sizable number of records close to the fault.
Figure 33 shows the Fourier spectra of observations and predictions. Similar to the foreshock, predicted ground motions are overestimated between 1.5 Hz to 3 Hz and underestimated below 0.5Hz and above 5 Hz. Figure 34 shows the theoretical S-wave amplifications from the engineering bedrock at EEB and KUMA, and the ratio of the theoretical S-wave amplification of EEB to KUMA. Figures 35 and 36 show the theoretical S-wave amplification of the seismic bedrock and the empirical site amplification during the mainshock at EEB and KUMA respectively. Since these spectra are almost the same as those of the foreshock, the same considerations can be made for the foreshock. The Fourier spectra of the mainshock's observed ground motions at EEB are shown in Fig. 37. NS components are particularly large at 1.6 Hz. Compared to the theoretical S-wave amplification peak and the empirical site amplification peak, this peak is noticeably more prominent. This peak is not recognized at KUMA. From these, this peak is considered to be the frequency response of the propagation path localized in EEB. This predominance, along with the surface layer modeling error at EEB, greatly influences the overprediction of 3 Hz from 2 Hz in NS components.
Compared with our predictions, the average and the average +/− a standard deviation of the Fourier spectra predicted by the participants are shown in Fig. 38. It nearly falls within the range of the participants' average +/- standard deviation, except for the overpredicted frequency range of 1.5 Hz to 3 Hz. As with the foreshock, the targeted record at KUMA has a large amplitude at 1 Hz. This study reproduced this large amplitude. The large amplitude cannot be predicted, though, because the average participant size is lower than in this study.
