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.

## 5.1 Artificial ground motion accuracy evaluation based on DTW distance

For the schemes evaluation of the synthetic ground motion, we can determine whether the DTW distance accuracy of different synthetic schemes satisfies the requirements, so we must determine the DTW distance accuracy evaluation standard. This paper selects 40 earthquake waves calculated by the DTW distance, obtains 780 different results from the DTW. The statistics of the mean and standard deviation are used to determine the standard DTW distance accuracy evaluation. The standard deviation of ± 0.5 times of the mean value is taken as the reasonable DTW range of the synthetic seismic wave, and the standard deviation of ± 1 times of the mean value is taken as the acceptable DTW range. Through statistical calculation and analysis, the reasonable range of DTW distance of synthetic earthquake waves is [134.9, 287.1], and the corresponding range of DTW distance is [58.9, 363.2].

After the accuracy evaluation standard is determined, multiple earthquake waves are generated through the various synthetic methods randomly, and the DTW distance between the seismic waves obtained by each synthesis method is obtained, and its mean value and variance are calculated. Compared with the above DTW distance range standard, the rationality and accuracy of each synthesis method are evaluated.

In this paper, various theoretical models and synthetic methods in the ground motion simulation are used to produce artificial ground motion, and the relevant references are marked in Table 1.

Table 1

Relation between each method or model and acronym symbol for nonstationary wave generation

| A | B | C | D | E | N |

Synthesis method | Based on envelope function and power spectrum | Based on phase difference spectrum model | Based on known seismic records | Fitting based on response spectrum with envelope function | Fitting based on phase difference spectrum | - |

Coherence function model | Harichandran[19] | Hao[20] | Abrahamson [21] | Yang[22] | - | - |

Envelope function model | Bogdanoff[23] | Shinozuk[24] | Jennings[25] | Saragoni[26] | - | - |

Power spectrum model | Kanai-Tajimi[27] | Markov[28] | Clough-Pizen[29] | | - | - |

The synthetic earthquake wave is the corresponding seismic acceleration for different field sites. Four different methods or models such as the synthesis method, coherence function, envelope function and power spectrum are represented by letters A, B, C, D, E respectively to form different combinations to represent various earthquake generation schemes. N means no corresponding method is adopted. Table 1 shows the relationship among the methods or models and the letters. For example, ABBC indicates the method that synthesize spatially correlated nonstationary seismic waves generated by the envelope function and power spectrum. Specifically, Hao model is considered a coherent function model, the envelope function is selected as Shinozuk model, and the power spectrum model is Clough-Pizen model (Hao et al., 1989; Shinozuk, 1972; Clough and Penzien, 1975).

In the synthetic process of artificial ground motion, the site type is assumed as class C, and the spatial variation effect of ground motion is considered. Ten earthquake waves are synthesized by each generation scheme, the DTW distance between 10 earthquake waves in each scheme is calculated, and the statistical mean is calculated. The results are shown in Table 2. The analysis of the result table shows that the correlation function between types of artificial earthquake waves from the DTW effect is similar and negligible. Therefore, we only need to analyze the last column in Table 2 of the data integration scheme and compare the results in Fig. 16 with the above statistical earthquake wave DTW obtained from the mean DTW, the reasonable range is set as [*µ* + 0.5*σ*, µ-0.5*σ*], and the acceptable range is set as [*µ* + *σ*, *µ*-*σ*].

Table 2

Mean value of the DTW distance for each generation scheme

Scheme 1 | Scheme 2 | Scheme 3 | Mean of integration scheme |

ABBC = 129.1 | AABC = 109.5 | ADBC = 106.9 | A-BC = 115.2 |

ABCC = 275.1 | AACC = 275.4 | ADCC = 250.3 | A-CC = 275.3 |

BBNC = 482.9 | BANC = 425.7 | BDNC = 527.8 | B-NC = 482.4 |

DBBC = 216.3 | DABC = 246.3 | DDBC = 259.0 | D-BC = 246.5 |

DBCC = 378.3 | DACC = 371.8 | DDCC = 297.8 | D-CC = 378.2 |

EBNC = 463.8 | EANC = 434.3 | EDNC = 443.1 | E-NC = 443.3 |

Figure 16 shows that schemes D-BC and A-CC are in the reasonable DTW range of [*µ* + 0.5*σ*, *µ*-0.5*σ*], and A-BC is in the acceptable DTW range of [*µ* + *σ*, *µ*-*σ*], whereas the D-CC, B-NC and E-NC data significantly deviate in an unreasonable range. In summary, the simulation precision of ground motions are generally required to choose Clough-Pizen model as the power spectrum model generation program, including the following methods: (1) Using envelope function and power spectrum to generate spatially correlated nonstationary seismic field, and Jennings model is selected for envelope function model; (2) The response spectrum fitting based on envelope function is selected for the spatial correlation non-stationary seismic field, and Shinozuk model is selected for the envelope function model. The acceptable generation scheme include that the envelope function and power spectrum are used to generate the spatial correlation non-stationary seismic field, and the Shinozuk model is selected as the envelope function model. The other ground motion schemes may be unstable and should be carefully chosen.

## 5.2 Evaluation of spatial variation effect of synthetic ground motion

After determining the reasonable generation scheme, we must further evaluate different waveform characteristics generated by the same plan to select more typical ground motion. When multiple ground motions are required, it is necessary to ensure that the ground motions contain the basic rules of the spatial variation effect. Considering the effect of spatial variation, it is generally assumed that if two earthquake waves are relatively close to each other, the difference between them is smaller, and the corresponding DTW distance is smaller. If the two waves are far away from each other, the difference between them and the corresponding DTW distance are larger. According to the previous study, especially the results of Fig. 11 and Fig. 12, we can think that when the multipoint ground motion is generated, with the increase in distance of each point, the DTW distance also gradually increases, which can represent the general rule of the time and space effect of ground motion. Therefore, when selecting earthquake waves from the results of several multipoint synthetic earthquake waves, we should at least ensure that one set of seismic accelerations, DTW distance and displacement DTW distance can satisfy the above conditions.

The requirements are further illustrated by an example. Five points that must generate ground motion on the same line have the coordinates of (0, 0), (500, 0), (1000, 0), (1500, 0) and (2000, 0), and the unit is m. The selected site type is class C considering the spatial variation effect, and the apparent wave velocity is 1000 m/s. Three sets of earthquake wave data are synthesized using a reasonable DTW distance generation scheme 2 with 5 random earthquake waves in each group. The DTW distance between the points and the origin is calculated, and the corresponding results are shown in subgraphs (a), (b) and (c) of Fig. 17 and Fig. 18.

It is obvious that the DTW distance of the 3 sets of earthquake waves does not always increase with increasing distance, so it cannot completely represent the basic law of the spatial variation effect. To solve this problem, we combine the earthquake waves and calculate the DTW distance among 5 new earthquake waves. After the adjustment, the results satisfy the requirements, as shown in Fig. 17 (d) and Fig. 18 (d). The acceleration time history and displacement time history of corresponding earthquake waves are shown in Fig. 19 and Fig. 20, respectively.

For this group of earthquake waves, with the increase in distance among the points, the displacement DTW distance and acceleration DTW distance increase, so they can better represent the spatial variation effect of ground motion. Hence, the synthetic accuracy evaluation and selection of artificial ground motion based on the DTW distance are more accurate and effective, which can select more authentic and effective ground motion, and provide more reliable excitation source data for the structural dynamic analysis.