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[SCG55-P02] Application of spectral analysis with autoregressive model for frequency response of ground amplification
Keywords:Autoregressive model, Spectral analysis, Ground amplification, Frequency response, Real-time ground motion estimation, Long-period ground motion
1. Introduction
In order to achieve more accurate “real-time strong ground motion estimation” and “long-period ground motion estimation”, it is effective to take into account differences in frequency for site amplification factors (e.g., Hoshiba (2013) and Ogiso et al. (2016)). Generally, the Fast Fourier Transform (FFT) is used to calculate the site amplification factor, but when the FFT is used to obtain the spectrum in the low-frequency (long-period) region, it is considered to be difficult to obtain the correct spectrum from a short-time data. In addition, since the FFT provides equally spaced analysis results on the frequency axis, the low frequency side of the spectrum data is sparse. On the other hand, the autoregressive (AR) model, which is used to predict future data based on past data, obtains a spectrum during the estimation of the model, and the spectrum does not have the above limitations of FFT. Therefore, we tried to evaluate the frequency characteristics of the ground amplification factor using the spectrum by the AR model, and examined its effectiveness.
2. Analysis method
(1) Autoregressive (AR) model
The AR model is a model that estimates present and future values from past values, and the output at a certain point in time is expressed as equation (1) in Figure 1. The spectrum is represented as equation (2) in Figure 1. The autoregressive coefficient φk was calculated by the Yule-Walker method. The order p of the autoregressive model was chosen as the highest possible order (= number of sample data minus 2) because the Akaike Information Criterion (AIC) tends to be smaller (= better fit between prediction and observation) when p is high, and spectra from low-order AR models are too smoothed especially in the low-frequency region. We confirmed that the spectrum obtained by the AR model with higher order p has a similar shape to the spectrum obtained by FFT.
(2) Data and analysis procedure
The data used in this study are shown in Table 1 and Figure 2.
The spectra of P and S waves of NS, EW, and UD components of the surface and subsurface seismographs were calculated by FFT and AR models, and the amplitude spectrum ratios of the surface and subsurface were calculated. In order to check for the similarity of the “amplitude spectrum ratios” and “amplitude ratios” of the surface/subsurface in the low frequency region, the amplitude spectrum ratios obtained from the FFT and AR model spectra were compared with the amplitude ratios obtained from the original waveforms applied a bandpass filter from 0.2 Hz to 0.8 Hz.
3. Results
As a result of the analysis, we found the followings.
(1) The spectrum of the AR model has a similar shape to the spectrum calculated by FFT.
(2) The variation of the amplitude spectral ratio for each event tends to be smaller in the spectrum calculated by FFT than in the AR model.
(3) In the low frequency region, the amplitude spectral ratios calculated by FFT tend to be closer to the amplitude ratios of actual seismic waveforms than those calculated by AR model.
Examples of the results are shown in Figure 3.
4. Conclusion
It is considered that spectra calculated by FFT is more appropriate than AR model to use for evaluation of the frequency characteristics of ground amplification factor, even in the low frequency region.
[Acknowledgments]
We appreciate the National Research Institute for Earth Science and Disaster Resilience (NIED) for allowing us to use data from KiK-net (NIED, 2019).
[References]
Hoshiba M (2013) Real-time correction of frequency-dependent site amplification factors for application to earthquake early warning. Bull Seismol Soc Am 103:3179-3188. doi:10.1785/0120130060.
Ogiso M, Aoki S, Hoshiba M (2016) Real-time seismic intensity prediction using frequency-dependent site amplification factors. Earth, Planets and Space 68:83. doi:10.1186/s40623-016-0467-4.
NIED (2019) NIED K-NET, KiK-net. doi: 10.17598/nied.0004.
In order to achieve more accurate “real-time strong ground motion estimation” and “long-period ground motion estimation”, it is effective to take into account differences in frequency for site amplification factors (e.g., Hoshiba (2013) and Ogiso et al. (2016)). Generally, the Fast Fourier Transform (FFT) is used to calculate the site amplification factor, but when the FFT is used to obtain the spectrum in the low-frequency (long-period) region, it is considered to be difficult to obtain the correct spectrum from a short-time data. In addition, since the FFT provides equally spaced analysis results on the frequency axis, the low frequency side of the spectrum data is sparse. On the other hand, the autoregressive (AR) model, which is used to predict future data based on past data, obtains a spectrum during the estimation of the model, and the spectrum does not have the above limitations of FFT. Therefore, we tried to evaluate the frequency characteristics of the ground amplification factor using the spectrum by the AR model, and examined its effectiveness.
2. Analysis method
(1) Autoregressive (AR) model
The AR model is a model that estimates present and future values from past values, and the output at a certain point in time is expressed as equation (1) in Figure 1. The spectrum is represented as equation (2) in Figure 1. The autoregressive coefficient φk was calculated by the Yule-Walker method. The order p of the autoregressive model was chosen as the highest possible order (= number of sample data minus 2) because the Akaike Information Criterion (AIC) tends to be smaller (= better fit between prediction and observation) when p is high, and spectra from low-order AR models are too smoothed especially in the low-frequency region. We confirmed that the spectrum obtained by the AR model with higher order p has a similar shape to the spectrum obtained by FFT.
(2) Data and analysis procedure
The data used in this study are shown in Table 1 and Figure 2.
The spectra of P and S waves of NS, EW, and UD components of the surface and subsurface seismographs were calculated by FFT and AR models, and the amplitude spectrum ratios of the surface and subsurface were calculated. In order to check for the similarity of the “amplitude spectrum ratios” and “amplitude ratios” of the surface/subsurface in the low frequency region, the amplitude spectrum ratios obtained from the FFT and AR model spectra were compared with the amplitude ratios obtained from the original waveforms applied a bandpass filter from 0.2 Hz to 0.8 Hz.
3. Results
As a result of the analysis, we found the followings.
(1) The spectrum of the AR model has a similar shape to the spectrum calculated by FFT.
(2) The variation of the amplitude spectral ratio for each event tends to be smaller in the spectrum calculated by FFT than in the AR model.
(3) In the low frequency region, the amplitude spectral ratios calculated by FFT tend to be closer to the amplitude ratios of actual seismic waveforms than those calculated by AR model.
Examples of the results are shown in Figure 3.
4. Conclusion
It is considered that spectra calculated by FFT is more appropriate than AR model to use for evaluation of the frequency characteristics of ground amplification factor, even in the low frequency region.
[Acknowledgments]
We appreciate the National Research Institute for Earth Science and Disaster Resilience (NIED) for allowing us to use data from KiK-net (NIED, 2019).
[References]
Hoshiba M (2013) Real-time correction of frequency-dependent site amplification factors for application to earthquake early warning. Bull Seismol Soc Am 103:3179-3188. doi:10.1785/0120130060.
Ogiso M, Aoki S, Hoshiba M (2016) Real-time seismic intensity prediction using frequency-dependent site amplification factors. Earth, Planets and Space 68:83. doi:10.1186/s40623-016-0467-4.
NIED (2019) NIED K-NET, KiK-net. doi: 10.17598/nied.0004.