Multiple Regression Results of Ground Motion Prediction Equations, GMPE
The calculated regression parameters of Equations (1) and (2) for the logarithm of the peak ground acceleration, PGA, are presented in
Table 2.
Figure 6 shows the recorded values of the logarithm of the peak ground acceleration at the surface stations and the theoretical values determined by the GMPE model (1). This regression procedure includes all the gathered seismic data records.
A large discrepancy between theoretical and observed peak ground acceleration values can be readily noticed for extreme, i.e., low and high, PGA values in
Figure 6. For low peak ground acceleration values, a significant overestimation of the GMPE model is visible, whereas for high peak ground acceleration values, the empirical values are significantly higher than the theoretical values determined by the GMPE model. As mentioned earlier, the solution to this problem can be to estimate the GMPE model parameters of peak ground acceleration models (1) and (2) only on the basis of seismic PGA records exceeding a certain predetermined threshold. Therefore, in further analysis, it was assumed that seismic records with PGA values not lower than 150 mm/s
2 would be included in calculations. This 150 mm/s
2 threshold value corresponds to the lower limit of the first degree of the mining seismic intensity scale, GSIS-2017 [
3], and makes it possible to include seismic PGA records felt by local communities on the surface. This threshold value resulted in limiting the sample to 132 observations and also shows that only about 5% of all seismic PGA records caused noticeable effects on the surface infrastructure according to the GSIS-2017 mining seismic intensity scale.
Figure 7a shows the recorded values of the logarithm of peak ground acceleration at the surface stations and the theoretical values determined by the GMPE model (1) for all seismic records.
Figure 7b shows the recorded values of the logarithm of peak ground acceleration at the surface stations and the theoretical values determined by the GMPE model (1) for the seismic records with PGA values not lower than 150 mm/s
2. It can be easily noticed that the discrepancy between the theoretical and observed peak ground acceleration values is much smaller for the GMPE model (1) which includes only seismic records with PGA values not lower than 150 mm/s
2 (
Figure 7b). This partially validates the concept of using PGA values not lower than 150 mm/s
2 in the regression procedure. The calculated regression parameters of Equation (1) for the logarithm of peak ground acceleration with PGA values not lower than 150 mm/s
2 are presented in
Table 3.
The regression GMPE model parameters in
Table 3 are statistically significant. Nevertheless, it is not enough to use this model for the prediction of peak ground acceleration values (PGA). The prediction accuracy and the correct construction of the confidence intervals require us to conduct the verification of the estimated GMPE model in terms of the normality and homoscedasticity of the random component. For the analyzed GMPE model shown in
Table 3, the
p-values were calculated for the Kolmogorov–Smirnov and Anderson–Darling statistical tests verifying the normality of the residual component. The obtained
p-values are 0.0175 and 0.0197, respectively, indicating that the distribution of the residual component of this model does not form the normal distribution at the significance α = 0.05. Additionally, the Breusch–Pagan statistical test indicates the presence of heteroscedasticity in the analyzed model. Therefore, there are no grounds at the level of α = 0.05 to reject the hypothesis of the homoscedasticity of the random component.
Therefore, a bootstrap approach was used to estimate the model parameters and to determine the confidence intervals of the forecasts. This approach does not require the normality and homoscedasticity of the random component.
Bootstrap model of ground motion prediction equations (GMPE) obtained from seismic data records with PGA values exceeding 150 mm/s2.
In order to determine the bootstrap model of ground motion prediction equations (GMPE), we have also constrained our seismic records to PGA values exceeding 150 mm/s
2. We have utilized the bootstrap procedure described in
Section 2.2.2 and assumed that the number of replications r is equal to 1000. The number of bootstrap replications needed depends on the precision required for the estimation and the complexity of the model being analyzed. In general, the more bootstrap samples are used, the more accurate the estimation of the parameter of interest or the sampling distribution of a statistic will be. However, as the number of bootstrap samples increases, so does the computational cost of the analysis. There is no fixed number of bootstrap replications that can be universally recommended, as the appropriate number depends on the specific analysis and research question. A common rule of thumb is to use at least 1000 bootstrap samples to obtain stable and reliable estimates [
27].
Based on these 1000 resamplings with replacement replications, we have estimated the mean values of the parameters of the GMPE model (1) and the corresponding lower and upper limits of 95% confidence intervals. These values are presented in
Table 4.
Figure 8 shows the GMPE model parameter distributions for our bootstrap procedure. The dashed red line shows the estimated mean values of the GMPE parameters, whereas the dashed blue lines show the lower and upper 95% confidence intervals.
A comparison of the GMPE bootstrap model with the GMPE linear regression model from
Table 3 is shown in
Figure 9. It is clearly seen that theoretical values determined from the bootstrap GMPE model better correspond to the observed PGA values, i.e., the differences between theoretical and recorded PGA values are smaller for most of the observations for the second level of mining seismic intensity scale GSIS-2017 and in all observations of the third and fourth levels of the mining seismic intensity scale GSIS-2017.
Table 5 summarizes these findings for the PGA values exceeding 600 mm/s
2. The differences between the theoretical values of the GMPE bootstrap model and the GMPE linear regression model from
Table 3 reach 136 mm/s
2. Therefore, the bootstrap analysis of recorded peak ground acceleration values of high-energy mining tremors can provide important information regarding the level of seismic hazard on the surface infrastructure by estimating the level of the mining seismic intensity scale,
Figure 9. By estimating bootstrap PGA values caused by high-energy mining tremors, it is possible to assess the seismic hazard on the surface infrastructure related to the level of the mining seismic intensity scale. For example, if the probability of the level of the mining seismic intensity scale is high, then the surface infrastructure may be at a higher risk of damage or failure due to the ground motions caused by the tremors. Thus, the proposed tool may be directly applicable for preventing damage to buildings and protecting local populations by identifying areas in coal mines that are prone to high-energy seismic activity and strong ground motions.
Based on the 1000 resamplings with replacement replications for 132 seismic records exceeding 150 mm/s
2, we have estimated the mean PGA values. The results of our analysis are presented in
Figure 10a, which displays the estimated mean PGA values, and
Figure 10b, which shows the upper limits of the 95% confidence intervals. In addition to estimating the mean PGA values, our analysis also involved performing a comparison test. This test involved plotting the mean and upper 95% confidence intervals for a linear regression model from
Table 3, as shown in
Figure 10. This allowed for a comparison between the estimated mean PGA values and the predictions from the regression model, which was previously fitted to the data. One can clearly observe that both the mean PGA values and the upper limits of the 95% confidence intervals of our bootstrapping method yield higher estimated values compared to the ordinary least square linear model (OLS) in the right part of
Figure 10a,b and correspondingly smaller values for the left part of
Figure 10a,b. This means we have obtained larger predicted values for samples 100–132, i.e., samples with the highest recorded PGA values, and smaller predicted values for samples 1–50, i.e., samples with the lowest recorded PGA. Overall, our analysis provides more valuable insights into the characteristics of the seismic records under consideration and helps to inform decisions related to seismic hazard assessment and risk management.
Table 6 shows the upper limits of 95% confidence for both considered models for recordings exceeding the value of vibration acceleration equal to 0.6 m/s
2. It is clearly seen that the bootstrap GMPE model reflects the observed PGA values much better and therefore provides more accurate estimators compared to the GMPE model from
Table 3. This finding has particular importance for the largest PGA values related directly to high seismic hazards and the highest impact on the surface infrastructure. The accuracy of the analyzed bootstrap model depends on the quality and representativeness of the observed seismic data. When these factors are properly accounted for, our bootstrap ground motion prediction model can provide a reliable and accurate estimate of the distribution of PGA values.
The proposed bootstrap tool for determining peak ground acceleration estimators and identifying areas in coal mines that are prone to high-energy seismic activity has several significant implications. First, it can improve safety in coal mines by identifying areas in coal mines that are prone to high seismic ground motion vibrations, and second, by providing more accurate estimations of peak ground acceleration values, the tool can help companies assess the risk of seismic events more effectively and implement appropriate risk management strategies.