# Inter-Story Drift Ratio Detection of High-Rise Buildings Based on Ambient Noise Recordings
## 3 Data and Processing Methods
### Polarization Analysis
#### 3.2.1 Calculating the Polarization Parameters
Conversely, when subjected to vertical loads or when lacking horizontal loads, \(\theta_{1}\) becomes large. Therefore, \(\theta_{1}\) is often negatively correlated with horizontal deformation. In Figure 2a, the three semi-principal axes of the polarized ellipsoid are defined as follows: \(\left|\overset{\rightarrow}{a}_{1}\right|>\left|\overset{\rightarrow}{a}_{2} \right|>\left|\overset{\rightarrow}{a}_{3}\right|\), \(\overset{\rightarrow}{a}_{1}\) is a semi-major axis, and \(\overset{\rightarrow}{a}_{3}\) is a semi-minor axis. In Figure 2b, \(\lambda_{X}\) is the length of the semi-principal axis, \(\varphi_{X}\) is the back azimuth clockwise from the north, and \(\theta_{X}\) is the dip angle from the horizontal plane. Additionally, \(X\) is the serial number sorted by the length of the axes from the largest to the smallest. According to [PERSON] [26], the required polarization data can be obtained using the polarity analysis method to strengthen the polarization characteristics of seismic stations. Because the effects of wind are complex and multidirectional, accurate \(\theta_{1}\) data are often not of analytical significance. A data set made up of \(\theta_{1}\) data can better reflect the response characteristics of buildings under wind loads. Therefore, a probability density function can be used to determine the representative \(\theta_{1}\) scope. Additionally, the probability density function P(\(\theta_{1}\)) is used to describe the probability density of that the angle between the longest axis of the polarized vibration ellipse of the collected Rayleigh wave and the horizontal plane meets \(\theta_{1}\). If P(\(\theta_{1}\)) is large, the wave whose angle between the longest axis of its polarized ellipse and the horizontal plane is in the range of \((0,\theta_{1})\) dominates in the collected Rayleigh wave. Then, 0.01 is taken as the calculation precision to calculate \(\theta_{1}\) and P(\(\theta_{1}\)). The target cumulative probability density is the probability density that starts with \(\theta_{1}=0\) and accumulates to the target \(\theta_{1}\). For certain target cumulative probability densities, there will always be a unique \(\theta_{1}\) parametric curve that represents the parameter combination with the maximum response. As shown above, when wind speeds increase, \(\theta_{1}\) decreases. Therefore, an inverted diagram of the wind speeds can be used to more intuitively compare the wind speeds  
Figure 2: (**a**) Diagram of the polarized ellipsoid formed by Rayleigh waves’ particle motion in the quadrant. (**b**) Definition of the polar coordinates (\(\lambda_{X}\), \(\varphi_{X}\), \(\theta_{X}\) ) of a semi-principal axis below the reference plane. and \(\theta_{1}\). By calculating the related coefficient between each target cumulative probability density's \(\theta_{1}\) parametric curve and the diagram of the inverted diagram of the wind speed and taking the maximum of the sum of the related coefficient, the optimal target cumulative probability density can be determined as 98%. Based on the above, the \(\theta_{1}\) parameter curve was calculated, and the median \(\theta_{1}\) was taken as the parameter when the building is under normal use. Additionally, the \(\theta_{1}\) occurring with maximum wind speed was taken as the parameter when the building is subjected to strong winds. Finally, the difference value between these two parameters was taken as the change value of \(\theta_{1}\) for the analysis.