## 2 Materials and Methods
### Methods  
#### 2.3.1 Kanai-Yoshizawa Formula for Response Prediction  
We follow the approach of [PERSON] et al. [22] using the KY formula in the present study. The inter-story drifts and motion calculations for the two levels are carried out using records from the basement and six-story levels of the building located in the center of the Bulgarian capital, Sofia, during the 2012 Pernik earthquake (Figure 2a,b). The KY formula for predicting the response at any level of the building was derived by [PERSON] et al. [22], representing the relationship between height H and the apparent velocity of vertically propagating waves V (Figure 2c). The \(z\)-axis is oriented upwards, so that at the basement \(z=0\), and the sixth-floor level is \(z=H\). The incident motion at the base is assumed to be \(u_{0}=F(t)\), and it is transmitted into the structure at the basement and the sixth floor, respectively [22],  
\[u_{tr}(0^{+},t)=k_{tr}F(t) \tag{1}\]  
\[u_{tr}(H,t)=k_{tr}F\bigg{(}t-\frac{H}{V}\bigg{)}, \tag{2}\]  
where \(k_{tr}\) is the transmission coefficient for the waves incident from the soil into the structure, and H/V is the time delay for the transmitted wave to reach the sixth floor. For the case of the studied building, depending on the soil conditions, category C is in the range V\({}_{\text{s}}\) = 180-360 m/s, or the delay is 0.11-0.06 s [47]. Figure 2c shows a typical cantilever beam equivalent model of the analyzed building. This KY formula can be generalized to any intermediate level \(d\) (e.g., a building floor), using Equations (1) and (2).  
Let \(d\) be the distance of that level from the sixth floor (\(0\leq d\leq H\)). Then  
\[u(H-d,t)=\frac{1}{2}[u(H,t-\frac{d}{V})+\mathfrak{u}(H,t+\frac{d}{V})]. \tag{3}\]  
The derivation details can be found in [PERSON] et al. [22].  
Equation (3) shows that if the motion on the sixth floor and the apparent wave velocity V are known, the motion at any level can be computed simply as a superposition of two time shifts (one forward and one backward in time). For a cantilever uniform shear beam, V = 4 H/T\({}_{1}\), where T\({}_{1}\) is its fixed-basement fundamental period of vibration. Approximately T\({}_{\text{sys}}\geq\) T\({}_{1}\) if the soil is very stiff compared to the structure by the soil-structure system period T\({}_{\text{sys}}\), which we measured here from the Fourier spectrum of the sixth-floor record [22]. T\({}_{\text{sys}}\)-T\({}_{1}\) can generally be estimated from the sixth-floor record only, e.g., from the zero crossings. A detailed analysis of how soil conditions in Sofia influence the seismic response of the building is outside the scope of this study. The amplification effects will be assessed by incorporating the recorded accelerogram data with soil modeling. Consequently, the motion at any level in the building can be computed approximately only from the sixth-floor record, even without the Fourier spectrum being computed.  
The accelerograms were processed and analyzed using SeismoSignal v2025 and SeismoSignal 3D v2025, specialized software tools provided under a research license ([[https://seismosoft.com/](https://seismosoft.com/)]([https://seismosoft.com/](https://seismosoft.com/)), accessed on 24 January 2025). The programs are efficient and widely used for processing strong-motion data in earthquake engineering. Both tools are essential for refining strong-motion records, filtering noise, and improving seismic data analysis for structural assessments [48].