## 1 Introduction
### Artificial Non-Stationary Seismic Excitation Using the Kanai-Tajimi Filter  
During the design process, earthquake excitations are employed to perform nonlinear dynamic time history analyses. In this context, the Kanai-Tajimi [31; 32] filter is used to model the movement of the ground in terms of a single degree of freedom oscillator, which is subjected to a random white noise excitation. In doing so, site dependencies are included by using the natural frequency and the damping ratio of the ground. However, the conventional Kanai-Tajimi filter models earthquakes as stationary random processes and does not account for temporal changes in the properties of the excitation history. To this end, non-stationary Kanai-Tajimi models have been proposed to create more realistic excitations [33; 34]. In this way, the time-dependent frequency content \(\omega_{g}(t)\) and the time-dependent intensity parameter \(e(t)\) are extracted from a measured benchmark record. We adopt this procedure to provide many realistic and site-dependent artificial ground motions based on one recorded accelerogram chosen from the region around the building site. Exemplary, we use the Northridge earthquake accelerogram, recorded in California in 1994. In particular, the chosen time history is the 360\({}^{\circ}\) component from a Station of USGS (Sepulveda VA Hospital, CA: VA Hospital Bldg 40) on the ground level [35]. A time window that moves from \(t_{0}\) to \(t_{end}\) of the accelerogram is defined to capture the time-varying features, as shown in Figure 1.  
In this study, we chose a time window of size \(t_{w}=1.5\) s, which results in a smooth extraction of the parameters, while covering the frequency oscillations. Decreasing the time windows increases the deviation from window to window. By contrast, large time windows flatten the curves of the extracted parameters. The intensity of the acceleration at time \(t\) within the time-window \(t-\frac{t_{w}}{2}<t<t+\frac{t_{w}}{2}\) is calculated by:  
\[\hat{e}(t)=\int_{t-\frac{t_{w}}{2}}^{t+\frac{t_{w}}{2}}\left(\hat{x}_{g}^{(r) }\right)^{2}\text{d}t. \tag{5}\]  
By counting the number of zero crossings in the interval \([t-\frac{t_{w}}{2},t+\frac{t_{w}}{2}]\), the time dependent frequency at time \(t\) is evaluated:  
\[\hat{\omega}_{g}(t)=\frac{\hat{n}(t)}{t_{w}}2\pi. \tag{6}\]  
Figure 1: Parameter extraction from a record of the Northridge earthquake in 1994 [35] using a moving time window. We fit polynomial functions to the extracted properties, as shown in Figure 2. Therefore, next to the size of the scanning time, the order of the polynomial function must be chosen carefully to subsequently generate the nonstationary excitation. Next to these time-varying parameters, other ground properties must be chosen, such as damping of the ground. This parameter may be found through soil tests or from trial and error methods. We choose a damping ratio of \(\zeta_{g}=0.3\) based on numerical fitting methods [36; 37] and visual inspection. The nonstationary Kanai-Tajimi filter is represented as the solution to the following nonlinear single degree of freedom system [33]:  
\[\ddot{x}_{f}^{(k)}+2\zeta_{g}\omega_{g}(t)\dot{x}_{f}^{(k)}+\omega_{g}^{2}(t)x _{f}^{(k)}=n(t). \tag{7}\]  
To obtain the filter response, \(x_{f}\) and \(\dot{x}_{f}\), the single degree of freedom system is solved by the explicit central difference integration scheme [38; 39]. The filter acceleration is written as:  
\[\ddot{x}_{g}^{(k)}=-2\zeta_{g}\omega_{g}(t)\dot{x}_{f}^{(k)}-\omega_{g}^{2}(t) x_{f}^{(k)}. \tag{8}\]  
The extracted envelope function \(e(t)\) magnifies this response based on the extracted intensity parameter over the excitation time history.