# Numerical Study of the Seismic Response of an Instrumented Building with Underground Stories
## 3 Numerical Model
### Numerical Models vs. Recorded Measurements Comparisons
#### 4.2.2 Comparisons Using Statistical Parameters  
Among the several alternatives to compare time-history records (i.e., representative parameters of the entire record or averaged difference following point-by-point), this study used error formulas to compare on-site values with those of the models used in this work. Specifically, comparisons used absolute error of the analyzed recording window across representative parameters: maximum acceleration, number of crossover distortions, dominant frequency, and duration of the strongest signal, as detailed below. In total, 46 models were run for the 11 events. The parameters used for the analysis were: the average absolute error, calculated using Equation (10); the relative error of the maximum acceleration calculated by Equation (11); the relative error of the number of crossover distortions, given by Equation (12); the relative error of the number of crossover distortions for the strongest signal of the recording, given by Equation (13); the relative error of the duration of the strongest signal, calculated through Equation (14); and the relative error of the dominant frequency, given by Equation (15). The definitions of the parameters and the formulas are listed below. 1. _The average relative error_ is used to analyze the differences between the on-site records and the model results in a point-by-point approach. The average relative error for the accelerations of the event \(j\), in the accelerometer \(k\) for the model \(m\) is calculated as: \[RE_{reg(j,k,m)}=\frac{\sum_{i=1}^{n_{j}}\frac{\left|\frac{a_{mod(j,k,m)}-a_{sit (j,k)}}{\left|a_{sit(j,k)}\right|}\right.}{n_{j}}}{\] (10)
2. _Maximum acceleration relative error_ compares the peak acceleration, essential to design, across different models for the same record or different records for the same model. In this study, the relative error across models is analyzed. \[RE_{amax(j,k,m)}=\frac{\left|amax_{mod(j,k,m)}-amax_{sit(j,k)}\right|}{amax_{sit (j,k)}}.100\] (11)
3. _The number of crossover distortions_ can be used to define whether two datasets are synchronized or not. In this case, the relative error of the number of crossover distortions is calculated using: \[RE_{cross(j,k,m)}=\frac{\left|Ncross_{mod(j,k,m)}-Ncross_{sit(j,k)}\right|}{ Ncross_{sit(j,k)}}.100\] (12)
4. The comparison of the _number of crossover distortions during the strongest part of the signal_ shows the significant changes between datasets. A lower number of this parameter comparing the times the strongest signal crossed the \(x\)-axis implies a better fit. This parameter is also useful, as it disregards small loads that may alter the data (wind, environmental vibrations). The relative error number of crossover distortions during the strongest part of the signal is calculated using:\[RE_{crossSP(j,k,m)}=\frac{\left|NcrossSP_{mod(j,k,m)}-NcrossSP_{sit(j,k)}\right|}{ NcrossSP_{sit(j,k)}}.100 \tag{13}\]
5. The _relative error for the duration of the strongest signal of the event_ is also used. The duration of the empirical window (10% maximum acceleration before and after peak) and that of the model are compared to each other to observe if the latter may reproduce the same window. \[RE_{DSP(j,k,m)}=\frac{\left|DSP_{mod(j,k,m)}-DSP_{sit(j,k)}\right|}{DSP_{sit(j,k)}}.100\] (14)
6. The similarity in the dominant frequency between the model response and the recorded data would imply that the mechanical properties of the model closely match reality. The dominant frequency of the recording is obtained by Fourier transform followed by obtaining the maximum value of the real parts of the data. Then, the _relative error of the dominant frequency of the recording_ is calculated: \[RE_{DF(j,k,m)}=\frac{\left|DF_{mod(j,k,m)}-DF_{sit(j,k)}\right|}{DF_{sit(j,k)} }.100\] (15)  
Only the predominant period of the records is analyzed (related to the first vibration mode), considering that the contribution of the higher modes in the dynamic response of the structure is negligible in the linear-elastic range.