A Solution to Determine the Optimal Number of Discrete Prolate Spheroidal Sequences in Nonstationary Earthquakes Ground-Motion Model

One of the most important issues in structural earthquake engineering is the proper definition of the time history of ground motions. It has been shown in various studies that uncertainty in the frequency content had a significant effect on the response of linear and non-linear structures. Here, it is necessary to state a series of relationships first to reach the article’s main topic. This article investigates the nonstationary stochastic model of the family of oscillatory sigma processes. The nonlinear random model of earthquakes is defined as a Gaussian oscillatory sigma process consisting of independent and uniformly modulated Gaussian processes with zero means. First, the estimation of the PSD is obtained by Thomson's short-term multiple window estimation method; then, the model parameters are used to estimate the model's PSD in such a way that the objective function takes its minimum value. Thomson’s short-term multiple window estimation method depends on describing stretched spherical wave functions. Discrete prolate spheroidal wave functions (DPSWFs), as well as discrete prolate spheroidal sequences (DPSS), are introduced by Slepian. Thomson's multiple short-term window estimations is a method to estimate non-stationary time series. Here the earthquake process is a continuous process. Sampling on the time interval is done by considering equal time intervals. Local time series are depicted on the orthogonal vectors of discrete stretched spherical wave sequences. First, using the real earthquake data of El Center and 1940 to extract the local time series, considering the Hanning window as the moving time window. Assuming that the number of DPSS is equal to 2. For each local time series of two special spectra, parameters of the oscillatory sigma process model consisting of 20 independent components of modulated processes have been estimated by the adaptive nonlinear least squares algorithm method. Our main problem is choosing the number of DPSS and knowing that is very important in obtaining the desired simulation. Therefore, by taking this approach, by drawing a graph of the number of DPSSs against the error value, its behavior can be checked. as the number of DPSSs increases from one place to the next, the error value becomes constant and tends to zero. Therefore, we seek to select the lowest value of DPSSs. It should be noted that the number of selected DPSSs is unique and depends on the length of the data under review. Here, the obtained results indicate that the presented model shows well the characteristics of the real records of the 1940 El Centro earthquake and records the time and frequency changes in the earthquake records.