Modelling and Simulation in Engineering

Volume 2017, Article ID 2614769, 9 pages

https://doi.org/10.1155/2017/2614769

## Synthesis of Spatially Correlated Earthquake Ground Motions Based on Hilbert Transform

^{1}School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China^{2}Hubei Key Laboratory of Roadway Bridge & Structure Engineer, Wuhan University of Technology, Wuhan 430074, China

Correspondence should be addressed to Yu Miao; nc.ude.tsuh@uyoaim

Received 14 March 2017; Accepted 4 June 2017; Published 6 July 2017

Academic Editor: Hongyi Li

Copyright © 2017 Erlei Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simplified method for synthesizing spatially correlated earthquake ground motions is developed based on Hilbert transform and a reference earthquake record. In this method, one reference earthquake record is treated as the original ground motion, based on a series of generated ground motions. This procedure uses the instantaneous amplitude and the instantaneous phase of the record obtained using Hilbert transform to achieve the nonstationarity of ground motion. To establish the coherency between generated ground motions, an incoherence model is employed to describe the relation between the instantaneous phase at the present station and the instantaneous phases at previous stations. This type of phase is defined as the instantaneous coherence phase. In addition, time lag is included in the instantaneous coherence phase to prescribe the wave passage effect. The proposed Hilbert-transform-based method is efficient and avoids cumbersome parameter estimations as well as other drawbacks involved in some traditional synthesizing methods. Applications of this method demonstrate that the generated ground motions are statistically analogous with the reference record.

#### 1. Introduction

In the past decades, the methods of generating spatially variable ground motions were studied thoroughly. A number of studies demonstrate that the effect of spatially variable ground motions on the responses of long structures is not negligible [1–7]. Nowadays, the synthesizing methods [8–11] based on stochastic process theory are popular and are implemented as a tool in most of the finite-element packages [12]. Stochastic procedures usually include power spectral matrix/incoherence matrix decomposition [13, 14] or spectral factorization [15], which involves massive calculation, thus decreasing the synthesizing efficiency. In addition, the nonstationarity of ground motion is a significant factor in the process of generating ground motions, which includes intensity nonstationarity and frequency nonstationarity [16]. Intensity nonstationarity is usually achieved by multiplying the generated stationary process using an envelope function. However, this method may not be applicable for simulations based on a recorded accelerogram, because the applied envelop function may not completely present the intensity nonstationarity of the original ground motion. Moreover, studies by Ohsaki [17] showed that the seismic waveform was governed by the distribution of the phase difference spectrum; thus, phase difference spectrum could be used to achieve intensity nonstationarity. Zhu and Feng [18, 19] studied the distribution characteristic of the phase difference spectrum and proposed a method of generating a random phase. This method can achieve full nonstationarity; however, the effect of the randomness of the generated phase on the ground motions is not confirmed. Several methods have been proposed for frequency nonstationarity [20–26]. These methods involve complex processes and massive calculation. Conditional simulation is an alternative and several contributions have been made to the study of ground motion simulation [27, 28].

In the present paper, a simplified conditional simulation method of synthesizing spatially correlated earthquake ground motions is proposed based on Hilbert transform and earthquake record. In this method, one reference earthquake record is treated as the original ground motion, based on which a series of ground motions are generated. By performing Hilbert transform on the known earthquake record, the instantaneous amplitude and the instantaneous phase of the earthquake record can be obtained. The instantaneous amplitude is utilized as an envelope function to achieve the intensity nonstationarity of each simulated ground motion. To establish the coherency between generated ground motions, a coherence model is employed to describe the relation between the instantaneous phase at the present station and the instantaneous phases at previous stations. The instantaneous coherence phases at different stations are all statistically analogous with that of the known record. Thus every generated ground motion can present similar frequency nonstationarity. This type of phase is defined as an instantaneous coherence phase. In addition, time lag is included in the instantaneous coherence phase to prescribe the wave passage effect. The proposed Hilbert-transform-based method can efficiently achieve intensity nonstationarity and frequency nonstationarity, and this approach avoids cumbersome parameter estimations, as well as other drawbacks involved in some traditional synthesizing methods. The application of this method demonstrates its validity and practical value.

#### 2. Hilbert Transform

For a real-valued function , its Hilbert transform is defined aswhere denotes the Cauchy principal value of the integral. Thus, by using the Hilbert transform, an analytic signal which is a complex-valued function can be obtained as follows [29]:which can be further expressed as where .

Thus, the original function can be expressed as where the independent variable denotes time and time functions and are called the instantaneous amplitude and the instantaneous phase function or, by Bendat and Piersol [30], the envelop signal and the instantaneous phase signal of , respectively. In addition, represents the amplitude modulation, and represents the frequency modulation mechanisms contained in the original signal and is between –*π* and *π*. In other words, governs the intensity nonstationarity (or the temporal variation of amplitude), whereas dominates the frequency nonstationarity (or the temporal variation of frequency content) of the signal .

#### 3. Proposed Method

In this study, the spatial variation of ground motion is prescribed in terms of wave scattering as well as wave passage effects. The soil condition and geology of the field of interest were assumed to be uniform. The well-known north-south component of the natural ground motion recorded at the El Centro station during the 1940 earthquake in Imperial Valley, California, which had a magnitude of 6.95 Mw, is chosen as the reference accelerogram, as shown in Figure 1(a). The recorded ground motion exhibits a sampling rate of 50 Hz and the anterior 40.94 s time history of the record is extracted to be studied with a total of 2048 values. First, the instantaneous amplitude and the instantaneous phase functions can be obtained by performing the Hilbert transform on the known earthquake record, as indicated in Figures 1(b) and 1(c), respectively. The instantaneous amplitude is treated by the method as an envelop function, with a value that will be preserved into the synthetic samples.