Analysis and Assessment of Ground Motion Characteristics and Similarity Using Dynamic Time Warping Distance

Abstract

Due to the complex nonstationarity of ground motion in time-frequency domain, the traditional methods of comparing and evaluating earthquake waveforms have not enough ability and accuracy to distinguish the details and changing features of the similar waves, which makes the similarity evaluation of waveform is difficult to be quantified accurately. The similarity degree of different signals can be calculated precisely according to Dynamic time warping (DTW) algorithm, so it can be used for waveform comparison and similarity evaluation. In order to improve the traditional method, a method based on DTW distance is proposed to identify the earthquake waveform and analyze the ground motion characteristics. Based on the statistical analysis of a great quantity of earthquake waves, the changes law of DTW distance considering amplitude, time lag, noise signal ratio, site type and the comprehensive effect is obtained. DTW distance is proved to be used as a compatible evaluation standard for waveform refinement. It is verified that DTW distance and vector norm are essentially equivalent. In the analysis of ground motion, DTW distance is implicated in the equivalent amplitude and energy of earthquake waves. The physical connotation of DTW distance is demonstrated by analyzing the data of the station array, and the results show that the distribution of DTW distance can accurately imply the time-space variation effect of the earthquake in the region. The reasonable reference range of DTW distance is defined by statistical method, and the corresponding evaluation standard of synthetic multi-point ground motion with real characteristics is proposed. In the synthetic accuracy evaluation of artificial ground motion with spatial variation effect, the combination of ground motions with more real characteristics can be obtained by evaluating and optimizing the waveforms according to the variation rule and range of DTW distance.

1. Introduction

The ground motion is a complex time-space stochastic process. Due to the influence of the focal mechanism, propagation path, dispersion effect and site conditions, the frequency distribution and amplitude intensity of a certain earthquake wave have obvious time-varying characteristics and nonstationary characteristics (Faravelli, 1988; Kubo and Penzien, 1979), which result in the structural seismic response are nonstationary both in time domain and frequency domain (Hasgur, 1995). In addition, due to the strong randomness and inhomogeneity of the ground motion in space, the wave characteristics of different measuring position under the same earthquake are variable, which results in the difference in seismic excitation and response at different locations for long-span structures or multipoint support structures (Lin et al., 1997). Many studies have shown that intensity or amplitude, spectrum distribution and duration are three elements of the basic characteristics of ground motion. The current seismic design and structural analysis mainly represent the spectrum characteristics and spectral intensity of the ground motion in terms of the site classification and the amplitude of response spectrum, and this method is relatively macroscopic and rough.

With the development of earthquake engineering (Massa et al, 2006 and Saad et al, 2019), whether accurately describing the seismic time-frequency evolution characteristics and the dissimilarity among different ground motions or evaluating the artificial earthquake wave or the simulated wave generated by shaking table meet the requirements of design specifications and engineering requirements, it is necessary to adopt more accurate quantitative analyses for intensity, frequency distribution and duration. Thus, it is of great research and engineering significance to introduce a comprehensive evaluation index in order to reveal the slight differences in waveform and characteristics of different earthquake waves.

In fact, there are similar demands to discriminate different time-domain signals in other research fields, such as mechanical engineering, clinic diagnosis and signal analysis, the relevant methods can be called waveform similarity evaluation. The index of similarity evaluation for different earthquake waves mainly include peak deviation, wave correlation coefficient and waveform distortion. It is undoubtedly inaccurate to distinguish different waveforms only by the peak deviation. The waveform correlation coefficient method uses the maximum of convolutions of two signal to measure the degree of waveform correlation, which has a large amount of calculation and cannot display the local difference of the signals. The distortion of time domain can be defined as the ratio of the amplitude difference of the contrast wave and the fundamental wave to the amplitude of the fundamental wave. The frequency domain distortion of the signal can be defined as the energy of each subharmonic of the contrast signal and the ratio of the total energy to the fundamental wave. The waveform distortion is generally considered the first choice to measure the similarity of earthquake waves, and the distortion degrees of the displacement waveform and acceleration waveform are mainly used to evaluate the waveform distortion. Although the physical meaning of waveform distortion is clear and easy to calculate, it cannot entirely and accurately reveal the coincidence degree between the contrast waveform and the standard waveform, especially to evaluate the local stochastic characteristics. Therefore, it is necessary to introduce a more accurate method and an index of waveform similarity evaluation.

The dynamic time warping (DTW) algorithm is firstly introduced in speech signal processing and classification (Itakura, 1975). Its core idea is to elongate or shorten the length of the signal until it fits the length of the reference signal. In this process, the time axis of the contrast signal will be distorted or bent, so that its characteristics and standard mode correspond to the contrast and analysis. The DTW can be used to calculate the similarity degree of two signal sequences and provide the point-to-point match, which can minimize the two sequence distances, and the matching and recognition effect is good. The DTW algorithm has been widely used in several fields such as the fault diagnosis of dynamic multivariable industrial processes, biological gene sequencing, and electrocardiogram signal classification (Zhen et al., 2013; Han et al., 2016). Bagnall et al.(2017) have implemented 18 recently proposed algorithms in a common framework and compared them against two standard benchmark classifiers (and each other) by performing 100 resampling experiments on each of the 85 datasets. The results show that the recognition method based on DTW distance has good discrimination capacity.

Currently, researchers attempt to apply the DTW algorithm to seismic analysis and comparison. Joswig and Schultetheis(1993) and Schultetheis and Joswig(1993) take the DTW distance between earthquake waves as the criterion to determine the correlation of weak local earthquakes and use the DTW distance to compare and cluster the locations of induced earthquakes in the Ruhr basin. Orozco-Alzate et al.(2015) proposed a dissimilarity space based on the DTW distance measure to classify seismic volcanic patterns either waveforms or spectrograms as alternative to both the nearest neighbor rule directly applied to the dissimilarities and classification in a dissimilarity space based on the Euclidean distance between pairs of seismic spectra. Li et al.(2010) studied the application of similarity measurement in the similarity analysis of simulated multi-point ground motions, and the analysis results show that the DTW distance method not only can quantitatively represent the similarity of simulated ground motion, but also offers advantages in clustering analysis and singularity recognition of actual multi-point ground motions. However, the variation of DTW distance and the mechanism relationship between DTW distance and ground motion characteristics have not been deeply explored. Hu et al.(2018) proposed a method integrating dynamic time warping and local outlier factor to identify anomalies of time series on various time scales, and a case study is presented focus on the measured foundation uplift beneath the local riverbed blocks of Xixi reservoir dam. Results show that this method can accurately identify abnormal time.

In summary, the above research usually apply DTW distance as the only index of ground motion similarity or convergence, and lack of further excavation of the basic performance and laws of DTW distance, There is also no demonstration of the physical connotation between the non-stationary and distribution characteristics of seismic data in time and space and DTW distance.

In view of this, this paper further studies the application of DTW in the discrepancy of ground motion and introduces an evaluation method of ground motion waveform identification and ground motion characteristics analysis based on the DTW distance. The method considers the change in DTW distance under the effects of the time, amplitude, signal-to-noise ratio and spatial distribution of the factors, and take it as the evaluation standard of artificially generated earthquake wave. The results show that the DTW distance can accurately characterize these factors in detail and identify the stationary and nonstationary characteristics of different waveforms, which becomes an effective basis for the wave similarity evaluation and selection.

2. Dynamic Time Warping (Dtw) Algorithm

When solving the correlation degree of time series data, it can be regarded as the distance between multi-dimensional vectors. Euclidean distance is the most frequently used distance measure in data mining algorithm, but it can only calculate the distance sum of the same sequence position. For two time series with similar shape but staggered in time axis, the distance calculated by Euclidean distance is very large, so the two time series are misjudged as different signals.

The dynamic time warping (DTW) algorithm is a nonlinear regularity algorithm that combines time warping and distance measurement. Its core idea is to find a valid path through the search to accurately calculate and measure the distance between two waves at each time and determine the similarity based on the distance. A smaller DTW distance corresponds to a higher similarity degree (Gollmer and Posten 1996; Forestier et al., 2012). Suppose there are two earthquake waves T and R with N and M data points, respectively, and N ≠ M. n and m are the data point number corresponding to each wave. As shown in Fig. 1, data points T and R along the horizontal and vertical coordinates are listed, the relation between each data point can form a grid, any cross point in the grid (n, m) means an intersection of T(n) and R(m), and the distortion degree of the intersection point is d[T(n), R(m)], i.e., the Euclidean distance between T(n) and R(m). Based on DTW algorithm, the best path from the starting point to the termination point through each intersection point can be found and the total degree of distortion of all intersection points on the path is minimum.

The selection of the DTW path is not arbitrary. First, considering the time sequence of the ground motion waveform, the order of every part cannot be reversed, so the choice of the route must start from the lower left corner and terminate at the upper right corner. Second, to make the path not excessively lean, the maximum and minimum of the average slope of each path must be limited, so that with the symmetrical distribution of slope on both sides of the diagonal, the maximum and minimum of the slope can be constrained in the range of 0.5 ~ 2. Then, the path selection must consider all data points and cannot skip any point. Finally, the former point when one arrives at any point (n_i, m_i) can only be a point among three points (n_i−1, m_i), (n_i−1, m_i−1), and (n_i, m_i−1). At this point, the DTW distance is D(n, m) and it can be expressed as a recursive formula as follows:

With the above recursive formula, the best path from the starting point to the end point and the DTW distance between the two waveforms can be obtained. The above analysis shows that it is feasible to use the DTW distance to test the similarity degree of the ground motion and evaluate the characteristics of the waveform. To test the accuracy of the DTW algorithm, two typical earthquake waves are selected, the time interval is 0.02 s and the first 10 seconds are mainly analyzed. Figure 2 shows the dynamic warping path graph of the two waves, where the white curve indicates the selection process of the best path in the calculation, and the DTW distance is 12.97, so the algorithm eliminates the adverse effects of the bending phenomenon on the distance measure, can effectively handle the local time displacement, and finally the reasonable matching results are obtained. If the local amplitude values of one earthquake wave are shifted, as shown in Fig. 3, the distance of the DTW path is 7.15, which is relatively small. When the local waveforms of the two waves coincide, the DTW path is a straight line, whereas the path of the alignment part becomes a curve. The result shows the calculation process of the DTW algorithm is dynamic when selecting the best path, and the DTW distance and path are sensitive to the change in waveform. The above research shows that DTW distance is relatively accurate and objective as an index for evaluating the similarity of ground motion waves.

3. Characteristic Evaluation Of The Earthquake Wave Based On Dtw

The characteristics of ground motion can generally be indicated by the three basic elements: amplitude, spectrum and time history. For different ground motions, it is relatively easy to identify and evaluate the characteristics when the three elements are only different on the extreme or overall boundary. However, when all three elements changes or only local subtle differences occur, the difficulty of the similarity evaluation dramatically increases. For natural ground motion, the earthquake waves acquired under analogous site conditions contain similar mechanisms and change regulation, but their randomness and nonstationary often obscure the inherent similarity, so we must analyze the data in detail. On the other hand, for the synthesis of artificial ground motion, because the algorithm cannot fully represent the real vibration characteristics, besides the introduced random noise, it is necessary to introduce a precise and stable method to evaluate the accuracy of different synthesis methods or the similarity of different waves generated by the same synthesis method. In addition, when one uses the shaking table structural dynamic test, the error control system causes an output wave with different amplitudes of the original wave and a mix of noise and output time-lag, the determination of whether the wave generated by the shaking table satisfies the accuracy requirements needs an accurate waveform similarity evaluation method (Shortreed et al., 2002; Phillips et al., 2014).

As previously mentioned, the waveform similarity evaluation based on DTW has promising prospects, and it can comprehensively compare the amplitude ratio, traveling wave effect (time lag), noise level and local variation in detail, and it is the ideal waveform similarity identification index. Hence, it is necessary to explore the effect of these factors on the characteristics of ground motion based on the DTW distance. To obtain the relevant mechanism, the DTW statistical analysis of natural ground motion data is conducted in this study. First, the variation rule of the DTW distance under the influence of each factor is studied; then, the ground motion waveform similarity evaluation and feature recognition standard considering the comprehensive effectiveness are proposed. In the specific analysis, 40 typical acceleration waves are selected from the PEER ground motion database (Harichandran and Vanmarcke, 1986) as samples. Ten earthquake waves are collected in four site types, and all the time intervals are 0.02 s. The variation rule of the DTW distance under various factors is statistically analyzed.

4. Spatial Variation Effect Of Ground Motion Based On The Dtw Distance

In the above studies, each earthquake wave is independently selected from different earthquake records, and the calculated DTW distance is not directly correlative. The corresponding results are applicable to the evaluation of the usual earthquake waveform. For a certain earthquake, the ground motion in a small-scale area has similar propagation paths, site conditions, and attenuation characteristics. Therefore, there is both the relation and different between the waveforms recorded by different points on the surface, which reveals the effect of the time-space variation. This effect has great influence on long-span structures and bridges. Therefore, the time-space evolutionary characteristics and laws of ground motions must be revealed in real time for the seismic analysis of long-span structures and bridges. To verify whether the DTW distance can represent the time-space variation characteristics of ground motion and its capability as the evaluation index of the multipoint ground motion and the variation regularity of earthquake motion field, we must analyze the strong motion observation data of the actual array.

China's Taiwan SMART-1 (Strong Motion Array in Taiwan-1) array is a high-density array established in 1979, which includes a central station measuring point (C-00) and 36 stations which are evenly distributed at the station survey point and the extension line at three radii of 200 m, 1000 m and 2000 m on concentric circles (Forestier et al., 2012; Shortreed et al., 2002). The distribution of the SMART-1 array points is shown in Fig. 10. Except two stations on the extension line, all stations are located on alluvial soil. Thus, all the points except the two extension points are selected as the effective points.

In this study, the representative earthquake data (north-south direction) in the Event40 of the array records are selected. The point O07, which is the nearest point to the focus is set as the reference point, and the distance between the measuring point and the reference point of the DTW distance are calculated. The DTW of acceleration signals from the outer ring (the initial is O), the central ring (the initial is M) and the inner ring (the initial is I) are plot in the polar plot, respectively, and the results are shown in Fig. 11. According to the similar method, the DTW distance of the corresponding displacement is drawn in the polar coordinate Fig. 12.

The DTW distance near each focal point can exhibit the decay characteristics of different earthquake waves. The acceleration DTW distance distribution on each ring is irregular. In addition to the attenuation in the northwest direction, the randomness in other directions is more obvious. Furthermore, the distribution of DTW distance of displacement on each ring is more regular.

Next, we further discuss the physical connotation of the DTW distance. According to the principle of the DTW algorithm, we know that the DTW distance is the cumulative sum of the best path distance from the starting point to the termination point, and the norm in mathematics is a length measure of the current vector in space. Therefore, we can assume that the DTW distance has a similar mathematical meaning to the vector norm. Moreover, the DTW distance is equal or greater than 0 and linearly related to the dynamic amplitude of the earthquake, so it can be considered positive definite and homogeneous in the perspective of the vector norm. The sum of the absolute value of the signal is the 1-norm, and the square root of the sum of the signal energy is the 2-norm. Thus, according to the norm equivalence principle, the DTW distance should be closely related to these two norms, i.e., the equivalent amplitude and energy of the signal. To verify this assumption, the sums of the absolute values of the displacement wave, velocity wave and acceleration wave are calculated and compared. The results of the outer ring, middle ring and inner ring are shown in Fig. 13-Fig. 15.

By comparing the results of Fig. 11-Fig. 15, we find that the distribution characteristics of the acceleration DTW distance of each measuring point are very similar to the sum of the absolute values of the acceleration of earthquake waves at each measuring point, and it also has some characteristics of the sum of the absolute value of displacement or velocity. The distribution of the displacement DTW distance of each measurement point is also very similar to the sum distribution of the absolute value of the displacement history at each test point.

Hence, the DTW distance of each physical quantity can represent the equivalent amplitude and energy of the corresponding time domain signal of the corresponding physical quantity. In addition, it can show the comprehensive effect of different earthquake wave propagation characteristics and time difference, and the distribution of DTW distance can generally represent the time-space characteristics of ground motion. Therefore, it is reasonable to take the DTW distance as an effective evaluation index of the time-space evolutions of the multi-point ground motion and seismic dynamic field.

5. Spatial Variation Effect And Accuracy Evaluation Of Artificial Ground Motion Based On Dtw Distance

In the field of ground motion synthesis, the researchers have conducted numerous studies and proposed synthetic methods (Harichandran and Vanmarcke, 1986; Hao et al., 1989; Abrahamson et al., 1991; Yang and Chen, 2000; Bogdanoff et al., 1961). Because the main purpose of ground motion synthesis is to provide basic data for structural seismic design and analysis, most of the current methods attempt to synthesize the earthquake waves that match both the design response spectra and the power spectra (Shinozuka, 1972; Jennings et al., 1968; Saragoni and Hart, 1973). In addition, the original waveform is multiplied by the envelope function in time domain to ensure the nonstationarity of earthquake waves. Recent studies have shown that the stochastic characteristics of ground motion are jointly determined by the amplitude spectra and phase difference spectra. The synthetic ground motion based on the phase difference spectrum can more realistically represent the nonstationarity of ground motion (Niu, 1991; Clough and Penzien, 1975). In addition, considering the spatial variation effects of the seismic field (including effect of traveling wave, partial coherence, and local field), synthesizing the ground motion at multiple locations based on a theoretical model of the coherence function is necessary. Synthesized ground motion derived from stochastic simulation is usually unable to fully and accurately reveal the nonstationarity of real earthquakes in time-frequency domain. When considered with the inherent characteristics and limitations of various theories and methods, this leads to different accuracy in various ground motion synthesis methods. Therefore, whether the accuracy of different methods or the similarity of different waveforms randomly generated by the same method is evaluated, the accurate and stable evaluation criteria are required to select the most realistic and reliable synthesized ground motion. Waveform evaluation based on DTW distance provides a relatively good solution for the aforementioned problems.

6. Conclusion

• Based on the dynamic time warping (DTW) algorithm, we can calculate the similarity of two signals and give the point-to-point matching value, which can minimize the two sequence distances. Therefore, we can use the DTW path and DTW distance to achieve a waveform comparison and similarity evaluation. Since the DTW distance can accurately and comprehensively represent the difference in waveform from different ground motions, a method based on the DTW distance to identify the ground motion waveform and the characteristics of ground motion is proposed. Through the calculation and analysis of many typical earthquake waves, we obtain the DTW distance variation rule considering only the amplitude, time difference, noise-signal ratio and synthetically considering time difference and noise- signal ratio. This finding proves that the DTW distance can be used as the standard for the fine evaluation of waveforms. A statistical analysis of Event40 ground motion data in Smart-1 array records shows that the DTW distance distribution can accurately reflect the time-space effect of regional vibration.

• Based on the physical meaning of the DTW distance and the mathematical meaning of vector norm, it is expounded that the DTW distance is equivalent to a vector norm in essence. Furthermore, it is considered that the DTW distance is closely related to the signal's equivalent amplitude and energy in ground motion analysis, and its distribution rule is similar. The above views are verified through the analysis of the array data.

• Based on the DTW distance, the accuracy and spatial variation effect of artificial ground motion are evaluated. By calculating the DTW distance and statistical parameters between a large numbers of earthquake waves, the reasonable DTW distance range and the acceptable DTW distance range are determined, which are used as the accuracy evaluation standard of synthetic ground motion. The analysis results indicate that the DTW distance can be used as an effective evaluation index for the evolution law of multi point ground motion and seismic dynamic field. In addition, through the synthetic and comparative analysis of different synthetic methods, coherence functions, envelope functions and power spectrum models of synthetic ground motion, the selected artificial earthquake waves can be more close to the real ground motion in time domain and frequency domain.

• For the combination of earthquake waves that can reveal the spatial variation effect of basic ground motion, the DTW distance should increase with the increase in distance. Because the random selection of earthquake waves cannot fully represent the above laws, we can recombine and evaluate multiple earthquake waves based on the variation rule of the DTW distance to achieve the synthetic accuracy evaluation and selection of artificial ground motion with spatial variation effect. The relevant methods proposed in this study can also provide reference for signal analysis and waveform similarity evaluation in other fields.

Declarations

Acknowledgments

We thank the Editor and anonymous reviewers.

Author contribution

Haoxiang He carried out the whole essay idea and article writing. Haoding Sun carried out part of the partial article and picture writing. Yifei Chen carried out the table and part of the photo production project. All the authors contributed equally in writing the manuscript. All authors reviewed the manuscript.

Funding information

This work is partially supported by the National Key R&D Program of China under Grant No. 2017YFC1500604 and 2017YFC1500603 and the National Natural Science Foundation of China under Grant No. 51878017.

Competing interests The authors declare no completing interests.

Ethics approval and consent to participate Not applicable.

Consent for publication Not applicable.

Conflict of interest The authors declare no competing interests.

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