# An alternative pulse classification algorithm based on multiple wavelet analysis

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## Abstract

Near fault ground motions may contain impulse behavior on velocity records. Such signals have a particular indicator which makes it possible to distinguish them from non-impulsive signals. These signals have significant effects on structures; therefore, they have been investigated for more than 20 years. In this study, we used Ricker and Morlet wavelets in order to analyze the wavelet power spectrum of the strong motion signals to investigate the impulsiveness. Both the area around the PGV and the area that exceeds the minimum threshold for the energy function are used in order to determine the position of the pulse. On both of these cases, particular criteria are used in order to characterize the signal. Then, we calculate the pulse period of the pulse region. Ricker and Morlet wavelets are also used to mimic the pulse signal. This method provides advanced information about the position of the maximum energy of the pulse part of the signal. We found that the impulsive part of the signal is frequently at the position where PGV occurs and the Ricker wavelet is better than the Morlet wavelet on mimicking the pulse part of the waveform. Spectral responses of strong motion waveform and the wavelets have strong correlation at around pulse period. Results show consistency with previous studies; hence, it can be used as a solid alternative on pulse shape signal investigations.

## Keywords

Near-fault ground motion Pulse-like ground motion Velocity pulse Wavelet analysis Pulse shaped signal extraction## 1 Introduction

The increasing number of seismic Stations has allowed the study of near-fault earthquake ground motions in seismology. Occasionally, Stations recorded earthquake signals with unexpected patterns. These ground motions, called henceforward pulse-shape signals, can be seen at the beginning of the earthquake signal in velocity records.

Pulse-shape signals are researched in both classical and engineering seismology. In engineering seismology, it is vital to identify pulse-shape signals, since they can create high demands on structures around the period of the pulse signal (Bertero et al. 1978; Anderson and Bertero 1987; Hall et al. 1995; Iwan 1997; Alavi and Krawinkler 2001; Menun and Fu 2002; Makris and Black 2004; Mavroeidis et al. 2004; Akkar et al. 2005; Luco and Cornell 2007; Kalkan and Kunnath 2006). However, some of the probabilistic seismic hazard analysis (PHSA) models (Lanzano et al. 2016) and building design codes do not consider their effects on PHSA models. Although it is a rare phenomenon, it is vital to investigate pulse-shape signals due to their hazardous effects.

Pulse-shape signals can appear in some earthquake scenarios, like those considering forward directivity, which occurs when receivers are located in the forward direction of the fault rupture (Somerville et al. 1997; Somerville 2003, 2005; Spudich and Chiou 2008), the fling step effect, which is a permanent displacement of the ground resulting from fault rupture (Mavroeidis and Papageorgiou 2002) and when rupture velocity and shear-wave velocity of the bedrock of the site of interest are similar.

The seldom occurrence of pulse-shape signals depends mostly on the above conditions. However, because of their scarcity, velocity pulses are not taken into account in most of the ground motion prediction equations (GMPE) (Abrahamson et al. 2016; Boore et al. 2014). Yet, standard and median deviations of GMPE can be calibrated with certain factors to involve the effect of a pulse-shape signal (Akkar and Cheng 2016).

- 1.
Signals with long and large amplitudes (Somerville et al. 1997),

- 2.
A high PGV/PGA ratio (Bray and Rodriguez-Marek 2004),

- 3.
Earthquake energy concentrated on one (or a few) pulse(s) (Somerville et al. 1997),

- 4.
Unexpectedly high response values at the pulse period on response spectra (Yang and Wang 2012).

Various methods have been created for identifying the pulse-shape signals. Mavroeidis and Papageorgiou (2003) proposed a wavelet analysis to construct a mathematical representation of the pulse, which depends on amplitude, period, duration, and phase shift. Shahi and Baker (2014) used a 4th-order Daubechies wavelet to determine pulse-shape signals. The method has some constraints such as a minimum PGV amplitude, a pulse arrival located at the beginning of the signal and arbitrary thresholds for the energy function. Mena and Mai (2011) used windowed Fourier transform analysis for the pulse shape signal and its position with certain energy thresholds. Chang et al. (2016) used the energy function with certain thresholds to determine the pulse-shape signal position and period. Ghaffarzadeh (2016) used the S-transform to identify the pulses. Kardoutsou et al. (2017) used a cross-correlation between the potential pulse-shape signal and the wavelet functions to determine the pulse shape. Methods of Shahi and Baker (2014) and Chang, Chang et al. (2016) are explained in detail in Section 3.

The goal of this study is to create an alternative pulse identification algorithm. Main considerations are determining the time location of the pulse and mimicking impulsive part of the signal with known wavelets. Ricker and Morlet wavelets are used both for spectrum analysis to determine the pulse period and the region with maximum energy and mimicking impulsive part of the signal. Waveform that are identified as pulse shaped are compared with the wavelets which are created with calculated pulse period by checking spectral responses. If the wavelets correspond to the features of the long period part of the earthquake signal, the algorithm is considered as successful.

## 2 Data

The analyzed ground motions are selected from NGA-West2 (Ancheta et al. 2012), GeoNet, Itaca (Pacor et al. 2011; Luzi et al. 2016), and K-Net databases, which contain data from crustal earthquakes. Earthquake signals that are recorded due to *M**w* ≥ 5.5 earthquakes with a maximum distance range of 150 km from the epicenter are selected. In order to study pulse-shape signals, East and North components are rotated to radial and transverse components. Acceleration waveform has been bandpass filtered between 0.05 and 10 Hz and integrated to obtain velocity waveform. In total, our database contains 2785 waveform.

## 3 Previous methods

Two aforementioned algorithms, Shahi and Baker (2014) and Chang et al. (2016), are used for comparison with our new method. These algorithms are chosen since both are well known and widely used when dealing with this topic.

### 3.1 Shahi and Baker (2014)

Shahi and Baker (2014) classification algorithm uses wavelet-based signal processing to detect pulse-shape signals in the area with the largest value of velocity (PGV). The algorithm can differentiate early and late arrival pulses by analyzing the arrival of PGV. Early arrivals of PGV generally indicate directivity effects.

Classification algorithm uses two criteria to determine whether the signal has impulsive or non-impulsive behavior. First criterion is the hazardousness of the signal. If PGV is less than 30 cm/s, it is considered as non-hazardous signal. Second criterion is that the pulse indicator (PI) values should be bigger than 0. Calculation of PI has two stages. In the first stage, principal component (PC) is found as explained in Eq. 1:

*P*

*G*

*V*

*r*

*a*

*t*

*i*

*o*indicates the ratio of the PGV value of the residual signal and the PGV value of the original signal, which is calculated by subtracting the original signal from that produced by the 4th-order Daubechies wavelet signal.

*e*

*n*

*e*

*r*

*g*

*y*

*r*

*a*

*t*

*i*

*o*is the ratio calculated by dividing the power of the residual signal by the power of the original signal. In the second stage, PI is calculated as Eq. 2:

^{th}order Daubechies wavelet is fitted to entire waveform. It is hard to determine where the impulsive part of the signal starts and ends by using Shahi and Baker (2014) classification algorithm.

### 3.2 Chang et al. (2016)

Chang et al. (2016) use an energy-based classification algorithm. The algorithm determines a region around the PGV and determines the energy ratio between the pulse region and the total energy of the signal by taking the squared values on both signals. The region around PGV is calculated by using a least-square fitting for various pulse periods; then, the one with the smallest residual is used for the pulse region. The energy ratio is then calculated as Eq. 3:

*t*

_{s}and

*t*

_{e}represent the starting and ending point of the impulse part in time axis and

*v*

^{2}represents the velocity time history of the signal. If the ratio between the pulse region energy, the numerator part of Eq. 3, and the total energy, the denominator of Eq. 3, exceeds 0.34, the signal is considered as a pulse-shaped signal.

If the signal is considered as pulse shaped signal, then a waveform is fitted to impulsive part of signal. Contrary to Shahi and Baker (2014), one can identify the impulsive part of the seismic signal by using the algorithm of Chang et al. (2016).

## 4 The new method

Previous attempts to determine pulse shape signals were concentrated on determining if the signal has an impulsive or non-impulse behavior. Another common goal is to determine the period of the pulse, since it can have significant effects on structures. Therefore, the pulse period has been focused on when mimicking the pulse signal. One of the main assumptions on pulse shape signals is that the impulsive part is where PGV is located.

In this method, we took the minimum threshold of PGV as 30 cm/s, if the pulse occurs where PGV has occurred. As proposed by Shahi and Baker (2014), we used wavelet analysis to determine pulse shape signals. Unlike previous studies, apart from the PGV time interval, we also considered the possibility that the pulse occurs at other time intervals of the earthquake signal.

Mavroeidis and Papageorgiou (2003) and Chang et al. (2016) focused on the impulsive part of the signal for analysis whereas Shahi and Baker (2014) fitted almost all of the signal. We focused only on the impulsive part of signal, similar to Mavroeidis and Papageorgiou (2003) and Chang et al. (2016). Main goal of this study is to create a robust alternative to identify the pulse shaped signals. We implemented threshold for PGV and wavelet analysis for signal process to have similarities with previous studies while adding new features that these studies did not consider.

### 4.1 Wavelet analysis

The resolution of the wavelet function depends on the width of real space and width in Fourier space. A broad function will give a poor time resolution but a good frequency resolution, and vice versa. The width of the wavelet is proportional to the sampling rate of the signal.

As a result of the analysis, one can determine the power spectrum values over time. The maximum power spectrum values at PGV and the biggest power spectrum value of the signal (if it does not occur at PGV) are used in the pulse identification.

### 4.2 Pulse identification

Unlike previous studies, our method can identify velocity pulses that occur away from the time interval where the PGV is located. Several decision mechanisms are used to identify pulse shape signals. The criteria for pulse shape signals differ with respect to the position of the pulse, as explained in Section 4.2.1 and in Section 4.2.2.

#### 4.2.1 Velocity pulse at PGV

- 1.
PGV ≥ 30 cm/s.

- 2.$$ \frac{\left( \frac{{\int}_{t_{\mathrm{s}}}^{t_{\mathrm{e}}} v^{2}(\tau) d\tau}{{\int}_{0}^{\infty} v^{2}(\tau) d\tau}+\frac{{\int}_{t_{\mathrm{s}}}^{t_{\mathrm{e}}} WPS(\tau) d\tau}{{\int}_{0}^{\infty} WPS(\tau) d\tau}\right)}{2} \geq 0.30 $$(4)

*t*

_{s}and

*t*

_{e}represent the starting and ending points of the pulse, respectively. These points are found by identifying the period (

*T*

_{p}) where the maximum wavelet power spectrum occurs at PGV. The pulse area is then identified as

*t*

_{PGV}∓

*T*

_{p}/2 where

*t*

_{PGV}represents the time of the PGV.

*W*

*P*

*S*indicates the wavelet power spectrum. The parameters can be seen in Fig. 2.

The left side of the numerator in Eq. 4 indicates the energy ratio between the impulsive part (velocity time history between *t*_{s} and *t*_{e} in time axis). The right side of the numerator is the ratio between wavelet spectrum energy of the waveform and impulsive part between the aforementioned area of the signal. Integrals are for summation process and infinity signs indicate the whole waveform.

#### 4.2.2 Velocity pulse outside the PGV region

- 1.
The biggest amplitude, in absolute sense, in the area where the maximum power spectrum value occurred, should be equal or bigger than 25 cm/s.

- 2.
Difference between PGV and the time where the maximum power spectrum in time axis should be larger than

*T*_{p}/4. - 3.$$ \frac{{\int}_{{{}{t}_{\mathrm{s}}}_{\text{emax}}}^{{{}{t}_{\mathrm{e}}}_{\text{emax}}} v^{2}(\tau) d\tau}{{\int}_{t_{\mathrm{s}}}^{t_{\mathrm{e}}} v^{2}(\tau) d\tau} \geq 1.1 $$(5)
- 4.$$ \frac{{\int}_{{{}{t}_{\mathrm{s}}}_{\text{emax}}}^{{{}{t}_{\mathrm{e}}}_{\text{emax}}} WPS(\tau) d\tau}{{\int}_{t_{\mathrm{s}}}^{t_{\mathrm{e}}} WPS(\tau) d\tau} \geq 1.1 $$(6)
- 5.$$ \frac{\left( \frac{{\int}_{{{}{t}_{\mathrm{s}}}_{\text{emax}}}^{{{}{t}_{\mathrm{e}}}_{\text{emax}}} v^{2}(\tau) d\tau}{{\int}_{0}^{\infty} v^{2}(\tau) d\tau}+\frac{{\int}_{{{}{t}_{\mathrm{s}}}_{\text{emax}}}^{{{}{t}_{\mathrm{e}}}_{\text{emax}}} WPS(\tau) d\tau}{{\int}_{0}^{\infty} WPS(\tau) d\tau}\right)}{2} \geq 0.30 $$(7)

*t*

_{e}

_{emax}and

*t*

_{s}

_{emax}represent the starting and ending points of the pulse in the maximum energy area in time axis. These points are found by identifying, in the area where the maximum power spectrum values are located, the maximum pulse period (

*T*

_{p,emax}) of the signal. The pulse area is then identified as

*t*

_{e}

_{emax}∓

*T*

_{p,emax}where

*t*

_{e}

_{emax}represents the time of the biggest value in the

*T*

_{p,emax}region. The parameters can be seen in Fig. 3.

Equations 5 and 6 describe the threshold for the energy ratios between the area around the PGV and the area around the maximum energy, if exists, for waveform and wavelet power spectrum, respectively. Other parameters have the same meanings that are explained in Section 4.2.1.

Energy ratio between PGV region and maximum energy region is determined by trial and error method. Time gap of *T*_{p} between PGV region and maximum energy region is implemented since Ricker wavelet power spectrum can identify discontinuities and that may cause erroneous interpretation of a single pulse into two or more separate pulses depending on the period. Amplitude threshold of the maximum power spectrum region is selected by considering the same idea behind the 30 cm/s of PGV, which is the possibility of creating damages on structures.

Both Ricker and Morlet wavelets are fitted to the pulse region when the algorithm detects a pulse shape signal.

## 5 Results

We also created a method to check the phase of the impulsive part of the velocity waveform. Impulsive signals that can be identified with Ricker wavelet is analyzed, since Ricker wavelet dominated the representation of the impulsive part of the signals. Ricker wavelet can be fitted to the original waveform to visualize the impulsive part more easily, however it is not providing further information about the impulsive part. Method for phase determination is explained in Appendix III.

### 5.1 Comparison with previous studies

Pulses that are also occurred outside of the PGV region are partially also detected by Shahi and Baker (2014) and Chang et al. (2016) (Fig. 5). Shahi and Baker (2014) identified 18 out of 26 of the signals as impulsive signal. Since Shahi and Baker (2014) are fitting full waveform on impulsive signals, it is not clear whether the impulsive part that was detected is the same region with our algorithm or not. Chang et al. (2016) identified 20 out of 26 of the signals as impulsive signal. Pulse periods that are calculated by our algorithm and Shahi and Baker (2014) and Chang et al. (2016) algorithms are close to each other.

Numerous signals with dissimilar impulsive results

Waveform | | PI | | | | Waveform | Wavelet power |
---|---|---|---|---|---|---|---|

name | Shahi and Baker (2014) | Chang et al. (2016) | energy | spectrum energy | |||

TCU078 | 0 | − 0.71 | 1 | 0.50 | 3.60 | 74.15 | 63.79 |

Vineyard canyon 1E | 0 | − 1.63 | 1 | 0.50 | 1.27 | 47.90 | 37.02 |

Brawley airport | 0 | − 2.4 | 0 | 0.29 | 6.05 | 57.91 | 47.53 |

D08C | 3.89 | 1.90 | 0 | 0.30 | 0 | 29.74 | 23.41 |

AQK | 2.04 | 0.69 | 1.7 | 0.38 | 0 | 34.76 | 24.33 |

Pacoima dam | 0.78 | 7.69 | 0.7 | 0.38 | 0 | 39.78 | 19.19 |

KJMA | 1.09 | 5.82 | 1 | 0.35 | 0 | 38.34 | 19.24 |

Port Island | 2.7 | 5.94 | 2.1 | 0.39 | 0 | 32.35 | 18.21 |

*PI*is very close to the threshold of 0 on these examples. It is also valid for Chang et al. (2016). Threshold of 0.34 for Eq. 3 is almost exceeded at D08C Station. Brawley Airport Station is also just below the thresholds of Eq. 2 and Eq. 3, which gets a long pulse period by our study (Fig. 7). On the other hand, our method fails when the pulse period is short. In waveform energy parameter, which is the left side of the numerator of Eq. 4, the threshold is exceeded almost all non-impulsive signals, which are AQK, Pacoima Dam, KJMA, and Port Island Stations (Fig. 7). However, wavelet power spectrum energy (right side of the numerator of Eq. 4) is so small that it makes the signal not impulsive. Common feature of these signals that are identified as impulsive is the fact that they have very short impulses. One of the features of impulsive signals is their long periods (Section 1). Our method filters short period signals thanks to wavelet power spectrum energy.

## 6 Conclusion

- 1.
Ricker wavelet analysis gives a higher resolution in the time domain, which is more suitable for determining the exact timing of the pulse.

- 2.
A Ricker wavelet is better than Morlet wavelets for mimicking the pulse part of the earthquake signal based on residual analysis.

- 3.
Our method is reproducing the spectral periods of the pulses, which makes the method convincing.

- 4.
Most of the velocity pulses occurred at PGV. However, it is worth mentioning that pulses may occur also in other intervals of the signal.

- 5.
This study has correlated with previous studies while expanding the information about the pulse shaped signal such as determining the pulse that occur outside the PGV region.

## Notes

### Acknowledgments

We would like to thank Assistant Professor Dr. Zhiwang Chang from School of Civil Engineering at Southwest Jiaotong University for sharing his pulse identification algorithm with us. We are grateful to anonymous referees for their vulnerable feedback on our work. We also would like to thank the National Research Institute for Earth Science and Disaster Resilience (NIED) for their website for K-NET and KiK-net (www.kyoshin.bosai.go.jp/, last accessed February, 2019) that makes it possible to access strong-motion seismographs in Japan, and GeoNet (www.geonet.org.nz/, last accessed February, 2019) for providing New Zealand strong-motion data. Python code of the method can be found at www.github.com/dertuncay/Pulse-Identification.

### Funding information

This study received financial support from the Italian Civil Protection (DPC) and Regional Civil Protection of Friuli-Venezia Giulia (FVG).

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