Regression coefficients
Table 2 lists the attenuation coefficients and the standard deviation values obtained from the regression for PGA, PGV, SA, and AvgSA. For simplicity, Table 2 shows the coefficients for 41 and 30 structural periods for SA and AvgSA, respectively. The complete table of the regression coefficients is provided as an electronic supplement. Figures 3a and 3b compare the prior and posterior period-dependent coefficients estimated for SA and AvgSA, respectively. In these figures, it is noted that the prior and posterior coefficients follow a similar shape with significant differences in their amplitude, especially in the coefficients c0, c2, and c4, where the updated values are so far from those computed theoretically, which means that our prior knowledge of these coefficients is vague and wrong. Regarding the total standard deviation, the computed SA values are like those of AvgSA at the period between 0.1s and 4.0s. Therefore, it is thought that both models have a similar predictive capacity.
Covariance matrixes and comparison of variances
The covariance matrixes needed to compute the forecasted variability, see Eq. (5), are shown in Table 3 for some IMs such as PGA, PGV, SA (0.2, 0.5, 1.0, 1.5, 2.0) and AvgSA (0.1, 0.2, 0.5, 1.0, 1.5, 2.0). As stated before, with these matrixes and the parameters to be forecasted, the epistemic uncertainty due to the sampling of the dataset could be considered in the estimation of the hazard. The supplementary material shows an extended version of Table 3 for all IMs reported in Table 2.
Moreover, to see the impact of adding the epistemic uncertainty due to the dataset sampling, a ratio between the standard deviation from the total forecasted and the one coming from the regression analysis, \(s={\sigma }_{p}/{\sigma }_{e}\), is computed for some intensity measures of SA and AvgSA, and it is shown in Figs. 4a and 4b, respectively. The dash lines correspond to the contour of s values, while the dots are the observations of the dataset (see Table 1). As it can be seen, s values close to 1 occur in the vicinity of the well-sampled region of the dataset, while higher s values (which means higher values of total forecasted variability) occur when there is a lack of data. As will be seen later, this uncertainty increase directly impacts the hazard assessment estimated for the site.
Residuals Analysis
The residuals as a function of magnitude (Mw) and distance (Rrup) are shown in Figs. 5a and 5b, respectively, for PGA, PGV, SA, and AvgSA. For the last two IMs, the residuals are shown for four different periods of vibration that correspond to 0.5s, 1s, 1.5s, and 2s. The plotted residuals represent the difference between the natural logarithms of the observed data and the predicted SA (circles) and AvgSA (triangles) values. The solid and dotted lines correspond to the linear fit between the residuals computed and the Mw and Rrup values, respectively. As can be seen, the mean residuals obtained for Mw and Rrup are almost negligible for PGA, PGV, and SA, which means that the estimated coefficients of the GMPMs are adequate and provide a reasonable estimation of the GMI. Similar trends are obtained for the rest of the periods of vibration considered in this study for SA. For the case of AvgSA, the slope decreases slightly as Mw increases, especially for T > 1s. This behavior may be related to the number of events used in the regression process to obtain the coefficients for long periods since, as mentioned earlier, the number of events in the dataset is reduced by approximately 23% to estimate AvgSA at T ≥ 1.8s.
Table 2
Coefficients obtained (see Eq. 2)
T(s)
|
SA(T)a
|
|
AvgSA(T)a
|
c0
|
c1
|
c2
|
c4
|
σ
|
|
c0
|
c1
|
c2
|
c4
|
σ
|
PGV
|
2.7281
|
1.4706
|
-0.0434
|
-0.0021
|
0.4286
|
|
|
|
|
|
|
PGA
|
5.1671
|
0.9599
|
0.0925
|
-0.0043
|
0.3248
|
|
|
|
|
|
|
0.05
|
5.2099
|
0.9543
|
0.0944
|
-0.0044
|
0.3225
|
|
|
|
|
|
|
0.10
|
5.4459
|
0.9146
|
0.0946
|
-0.0048
|
0.3335
|
|
5.2567
|
1.2978
|
-0.0800
|
-0.0046
|
0.3170
|
0.20
|
6.1631
|
0.8212
|
0.0727
|
-0.0057
|
0.3115
|
|
5.6687
|
1.2175
|
-0.0706
|
-0.0052
|
0.2892
|
0.30
|
6.1508
|
0.8243
|
0.1053
|
-0.0055
|
0.2426
|
|
5.7540
|
1.2128
|
-0.0639
|
-0.0051
|
0.2793
|
0.40
|
6.0079
|
0.8925
|
0.0730
|
-0.0050
|
0.2759
|
|
5.7877
|
1.1997
|
-0.0590
|
-0.0049
|
0.2806
|
0.50
|
6.0645
|
0.8560
|
0.0831
|
-0.0046
|
0.3089
|
|
5.7642
|
1.1942
|
-0.0429
|
-0.0047
|
0.2806
|
0.60
|
5.8783
|
0.8368
|
0.1315
|
-0.0041
|
0.3310
|
|
5.7265
|
1.1821
|
-0.0204
|
-0.0045
|
0.2763
|
0.70
|
5.8213
|
0.8693
|
0.1874
|
-0.0040
|
0.3063
|
|
5.6818
|
1.1806
|
-0.0093
|
-0.0042
|
0.2781
|
0.80
|
5.7247
|
0.8557
|
0.1884
|
-0.0035
|
0.2885
|
|
5.6589
|
1.1871
|
-0.0070
|
-0.0041
|
0.2771
|
0.90
|
5.6159
|
0.8464
|
0.2132
|
-0.0031
|
0.2856
|
|
5.6316
|
1.1921
|
-0.0020
|
-0.0040
|
0.2864
|
1.00
|
5.5138
|
0.9118
|
0.1594
|
-0.0028
|
0.3198
|
|
5.6186
|
1.2039
|
-0.0050
|
-0.0040
|
0.2869
|
1.10
|
5.5682
|
0.9517
|
0.1434
|
-0.0032
|
0.3162
|
|
5.6002
|
1.2242
|
-0.0068
|
-0.0039
|
0.2933
|
1.20
|
5.6461
|
0.9630
|
0.1513
|
-0.0036
|
0.3139
|
|
5.5799
|
1.2460
|
-0.0070
|
-0.0039
|
0.3033
|
1.30
|
5.5169
|
0.9728
|
0.1441
|
-0.0033
|
0.3589
|
|
5.5510
|
1.2627
|
-0.0126
|
-0.0038
|
0.3092
|
1.40
|
5.5904
|
0.9919
|
0.1222
|
-0.0034
|
0.3613
|
|
5.4831
|
1.2873
|
-0.0142
|
-0.0037
|
0.3171
|
1.50
|
5.6398
|
1.0678
|
0.1012
|
-0.0036
|
0.3819
|
|
5.4274
|
1.3130
|
-0.0173
|
-0.0036
|
0.3255
|
1.60
|
5.5405
|
1.1104
|
0.0993
|
-0.0033
|
0.4162
|
|
5.3988
|
1.3365
|
-0.0231
|
-0.0036
|
0.3340
|
1.70
|
5.5055
|
1.1516
|
0.1010
|
-0.0033
|
0.4340
|
|
5.3445
|
1.3629
|
-0.0270
|
-0.0036
|
0.3360
|
1.80
|
5.4324
|
1.1951
|
0.0851
|
-0.0033
|
0.4481
|
|
5.3166
|
1.3812
|
-0.0304
|
-0.0036
|
0.3393
|
1.90
|
5.2698
|
1.2280
|
0.0707
|
-0.0030
|
0.4566
|
|
5.1532
|
1.4037
|
-0.0317
|
-0.0033
|
0.3477
|
2.00
|
5.0881
|
1.2747
|
0.0629
|
-0.0028
|
0.4676
|
|
5.1154
|
1.4284
|
-0.0401
|
-0.0033
|
0.3498
|
2.20
|
4.8606
|
1.3411
|
0.0635
|
-0.0027
|
0.5058
|
|
5.0254
|
1.4749
|
-0.0527
|
-0.0032
|
0.3561
|
2.40
|
4.8331
|
1.4113
|
-0.0015
|
-0.0029
|
0.4919
|
|
4.9666
|
1.4991
|
-0.0572
|
-0.0032
|
0.3588
|
2.60
|
4.8047
|
1.4658
|
0.0149
|
-0.0033
|
0.4726
|
|
4.9109
|
1.5246
|
-0.0669
|
-0.0031
|
0.3651
|
2.80
|
4.5538
|
1.5624
|
-0.0117
|
-0.0030
|
0.4802
|
|
4.8426
|
1.5496
|
-0.0786
|
-0.0031
|
0.3684
|
3.00
|
4.3871
|
1.6209
|
-0.0452
|
-0.0028
|
0.4840
|
|
4.7690
|
1.5684
|
-0.0835
|
-0.0030
|
0.3755
|
3.20
|
4.2437
|
1.6869
|
-0.0850
|
-0.0027
|
0.4641
|
|
4.3509
|
1.7058
|
-0.1216
|
-0.0026
|
0.3934
|
3.40
|
3.9932
|
1.7560
|
-0.1353
|
-0.0023
|
0.4927
|
|
4.2715
|
1.7273
|
-0.1271
|
-0.0025
|
0.3959
|
3.60
|
3.7148
|
1.8239
|
-0.1515
|
-0.0021
|
0.5414
|
|
4.1992
|
1.7507
|
-0.1340
|
-0.0024
|
0.3999
|
3.80
|
3.5121
|
1.8791
|
-0.1664
|
-0.0020
|
0.5350
|
|
4.1349
|
1.7721
|
-0.1437
|
-0.0024
|
0.3998
|
4.00
|
3.3245
|
1.9383
|
-0.1759
|
-0.0019
|
0.5326
|
|
4.0692
|
1.8000
|
-0.1532
|
-0.0023
|
0.4049
|
4.20
|
3.1290
|
1.9937
|
-0.2012
|
-0.0018
|
0.5110
|
|
|
|
|
|
|
4.40
|
2.9587
|
2.0580
|
-0.2130
|
-0.0017
|
0.4908
|
|
|
|
|
|
|
4.60
|
2.8559
|
2.1169
|
-0.2451
|
-0.0017
|
0.4852
|
|
|
|
|
|
|
4.80
|
2.7152
|
2.1657
|
-0.2475
|
-0.0016
|
0.4944
|
|
|
|
|
|
|
5.00
|
2.4967
|
2.2212
|
-0.2546
|
-0.0012
|
0.4964
|
|
|
|
|
|
|
5.20
|
2.3061
|
2.2762
|
-0.2666
|
-0.0009
|
0.4896
|
|
|
|
|
|
|
5.40
|
2.1444
|
2.3345
|
-0.2942
|
-0.0006
|
0.5040
|
|
|
|
|
|
|
5.60
|
2.0383
|
2.3891
|
-0.3098
|
-0.0006
|
0.5074
|
|
|
|
|
|
|
5.80
|
1.9543
|
2.4432
|
-0.3053
|
-0.0007
|
0.5182
|
|
|
|
|
|
|
6.00
|
1.8801
|
2.4877
|
-0.2985
|
-0.0008
|
0.5080
|
|
|
|
|
|
|
a) The period-independent coefficient used in the regression analysis for all IMs: c3 = -0.5.
Note that the median values obtained from these coefficients are expressed in acceleration units (cm/s2) for PGA, SA, and AvgSA. For PGV, the results are expressed in velocity units (cm/s).
|
Table 3
Covariance matrixes obtained (see Eq. 5)
SA(T)
|
|
AvgSA(T)
|
PGV
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
|
|
|
|
|
|
c0
|
556.488
|
-4.220
|
2.006
|
-114.572
|
0.325
|
|
|
|
|
|
|
|
c1
|
-4.220
|
0.277
|
-0.145
|
0.848
|
-0.002
|
|
|
|
|
|
|
|
c2
|
2.006
|
-0.145
|
0.083
|
-0.403
|
0.001
|
|
|
|
|
|
|
|
c3
|
-114.572
|
0.848
|
-0.403
|
23.599
|
-0.067
|
|
|
|
|
|
|
|
c4
|
0.325
|
-0.002
|
0.001
|
-0.067
|
0.00019
|
|
|
|
|
|
|
|
PGA
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 0.1
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
319.506
|
-2.423
|
1.152
|
-65.781
|
0.186
|
|
c0
|
305.722
|
-2.318
|
1.102
|
-62.943
|
0.178
|
c1
|
-2.423
|
0.159
|
-0.083
|
0.487
|
-0.001
|
|
c1
|
-2.318
|
0.152
|
-0.080
|
0.466
|
-0.001
|
c2
|
1.152
|
-0.083
|
0.048
|
-0.231
|
0.001
|
|
c2
|
1.102
|
-0.080
|
0.046
|
-0.221
|
0.001
|
c3
|
-65.781
|
0.487
|
-0.231
|
13.549
|
-0.038
|
|
c3
|
-62.943
|
0.466
|
-0.221
|
12.965
|
-0.037
|
c4
|
0.186
|
-0.001
|
0.001
|
-0.038
|
0.00011
|
|
c4
|
0.178
|
-0.001
|
0.001
|
-0.037
|
0.00011
|
T = 0.2
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 0.2
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
293.782
|
-2.228
|
1.059
|
-60.485
|
0.171
|
|
c0
|
253.350
|
-1.921
|
0.913
|
-52.161
|
0.148
|
c1
|
-2.228
|
0.146
|
-0.077
|
0.448
|
-0.001
|
|
c1
|
-1.921
|
0.126
|
-0.066
|
0.386
|
-0.001
|
c2
|
1.059
|
-0.077
|
0.044
|
-0.213
|
0.001
|
|
c2
|
0.913
|
-0.066
|
0.038
|
-0.183
|
0.00049
|
c3
|
-60.485
|
0.448
|
-0.213
|
12.458
|
-0.035
|
|
c3
|
-52.161
|
0.386
|
-0.183
|
10.744
|
-0.031
|
c4
|
0.171
|
-0.001
|
0.001
|
-0.035
|
0.00010
|
|
c4
|
0.148
|
-0.001
|
0.00049
|
-0.031
|
0.00009
|
T = 0.5
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 0.5
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
288.499
|
-2.188
|
1.040
|
-59.397
|
0.168
|
|
c0
|
236.724
|
-1.795
|
0.853
|
-48.738
|
0.138
|
c1
|
-2.188
|
0.144
|
-0.075
|
0.440
|
-0.001
|
|
c1
|
-1.795
|
0.118
|
-0.062
|
0.361
|
-0.001
|
c2
|
1.040
|
-0.075
|
0.043
|
-0.209
|
0.001
|
|
c2
|
0.853
|
-0.062
|
0.035
|
-0.171
|
0.00046
|
c3
|
-59.397
|
0.440
|
-0.209
|
12.234
|
-0.035
|
|
c3
|
-48.738
|
0.361
|
-0.171
|
10.039
|
-0.029
|
c4
|
0.168
|
-0.001
|
0.001
|
-0.035
|
0.00010
|
|
c4
|
0.138
|
-0.001
|
0.00046
|
-0.029
|
0.00008
|
T = 1.0
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 1.0
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
309.823
|
-2.350
|
1.117
|
-63.788
|
0.181
|
|
c0
|
247.918
|
-1.880
|
0.894
|
-51.043
|
0.145
|
c1
|
-2.350
|
0.154
|
-0.081
|
0.472
|
-0.001
|
|
c1
|
-1.880
|
0.123
|
-0.065
|
0.378
|
-0.001
|
c2
|
1.117
|
-0.081
|
0.046
|
-0.224
|
0.001
|
|
c2
|
0.894
|
-0.065
|
0.037
|
-0.180
|
0.00048
|
c3
|
-63.788
|
0.472
|
-0.224
|
13.139
|
-0.037
|
|
c3
|
-51.043
|
0.378
|
-0.180
|
10.513
|
-0.030
|
c4
|
0.181
|
-0.001
|
0.001
|
-0.037
|
0.00011
|
|
c4
|
0.145
|
-0.001
|
0.00048
|
-0.030
|
0.00009
|
T = 1.5
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 1.5
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
443.511
|
-3.363
|
1.599
|
-91.312
|
0.259
|
|
c0
|
319.290
|
-2.421
|
1.151
|
-65.737
|
0.186
|
c1
|
-3.363
|
0.221
|
-0.116
|
0.676
|
-0.002
|
|
c1
|
-2.421
|
0.159
|
-0.083
|
0.487
|
-0.001
|
c2
|
1.599
|
-0.116
|
0.066
|
-0.321
|
0.001
|
|
c2
|
1.151
|
-0.083
|
0.048
|
-0.231
|
0.001
|
c3
|
-91.312
|
0.676
|
-0.321
|
18.808
|
-0.053
|
|
c3
|
-65.737
|
0.487
|
-0.231
|
13.540
|
-0.038
|
c4
|
0.259
|
-0.002
|
0.001
|
-0.053
|
0.00015
|
|
c4
|
0.186
|
-0.001
|
0.001
|
-0.038
|
0.00011
|
T = 2.0
|
c0
|
c1
|
c2
|
c3
|
c4
|
|
T = 2.0
|
c0
|
c1
|
c2
|
c3
|
c4
|
c0
|
666.811
|
-5.057
|
2.404
|
-137.286
|
0.389
|
|
c0
|
727.929
|
-6.446
|
2.582
|
-149.233
|
0.416
|
c1
|
-5.057
|
0.332
|
-0.174
|
1.016
|
-0.003
|
|
c1
|
-6.446
|
0.271
|
-0.136
|
1.308
|
-0.004
|
c2
|
2.404
|
-0.174
|
0.099
|
-0.483
|
0.001
|
|
c2
|
2.582
|
-0.136
|
0.075
|
-0.524
|
0.001
|
c3
|
-137.286
|
1.016
|
-0.483
|
28.277
|
-0.080
|
|
c3
|
-149.233
|
1.308
|
-0.524
|
30.604
|
-0.085
|
c4
|
0.389
|
-0.003
|
0.001
|
-0.080
|
0.00023
|
|
c4
|
0.416
|
-0.004
|
0.001
|
-0.085
|
0.00024
|
Median predictions
Figure 6 compares the observations against the predicted values of PGV, PGA, SA, and AvgSA for two different magnitudes (Mw) as a function of the source-to-site distance (Rrup). For SA and AvgSA, the comparison is carried out for four different periods of vibration that correspond to 0.5s, 1s, 1.5s, and 2s. In this figure, the black and red solid lines represent the median prediction (50th percentile) for a magnitude value (Mw) of 6.5 and 7.5, respectively, and the dashed lines correspond to the 16th and 84th percentiles. Likewise, the filled circles correspond to the 6.3 ≤ Mw<6.7 observations and the filled triangles for 7.3 ≤ Mw<7.7. Significant data variability for the evaluated magnitude ranges can be observed, especially for 6.3 ≤ Mw < 6.7, whose dispersion increases as the period increases. This dispersion is also observed in PGV, whose intensity measure is more sensitive to the low-frequency amplitudes of the ground motion. This dispersion increase over long periods is likely related to the characteristics of the seismic recording equipment operated at the station CU since most have been short-period accelerometers with low sensitivity at frequencies below 1 Hz. Despite this, it is observed that the empirical data is relatively well represented by the GMPM proposed.
Figure 7 compares the response spectra observed (solid black line) with the median estimated (dotted red line) at the station CU for SA and AvgSA from the GMPM derived. Only twelve of the most intense earthquakes in the dataset are compared. As expected, the suitability of the fit varies from event to event, showing a good agreement for most of them. In the case of the events: 5 (Mw 7.7), 12 (Mw 6.9), 15 (Mw 7.3), and 26 (Mw 7.25), there is a significant difference between the observed and predicted values for both SA and AvgSA; however, the fitting is reasonable given the uncertainty of the GMPM. On the other hand, it can be noted that the spectral shape of SA is strongly influenced by the site effects present at station CU, which yield a large spectral amplification at the period range from 1s to 3s and, consequently, widen the spectral shape of AvgSA to long periods. This condition is particularly atypical at sites classified as firm soils, where it is expected that the peak amplitude has a place at short periods below 1s. However, in the firm zone of Mexico City, this amplification is caused by the rather shallow (< 1 km) sediments that lie below the volcanic rocks that cover the hill zone of the Valley of Mexico (Ordaz and Singh 1992).
Comparison of standard deviations
Figure 8a compares the total standard deviation of PGA and SA (σlnIM) obtained from the GMPM proposed with those obtained from two previously developed models for the station CU, corresponding to R99 and J06. This figure shows that the σlnSA computed in this study is less than estimated by R99 and J06 models at all periods of vibration. Note that the PGA value is associated with T = 0s. On the other hand, Fig. 8b clearly shows the percentage reduction achieved for σlnIM regarding the values reported at the two GMPMs evaluated. This percentage varies from 5–60% for R99 and from 0.3–58% for J06. For both models, the lowest value occurs in the period range from 1s to 3s. In general, a greater reduction of the dispersion of PGA and SA is obtained for the R99 model compared to that of J06. These differences are related to the number of earthquakes used in the regression approach since CU21 contains 18 more events than R99 and 13 more than J06. These results indicate a more significant predictive capacity of the proposed GMPM than previous models.
Comparison with previous models
To evaluate the performance of the proposed GMPM concerning previous models developed for the station CU, Fig. 9 presents the differences in natural logarithm between the predictions of GMPM–SA proposed and the R99–SA model. Similarly, Fig. 10 shows the differences regarding the J06–SA model. In both figures, the results are presented through surface plots that allow to quickly evaluate the behavior of the residuals obtained for a wide range of magnitude values (6.0 ≤ Mw ≤ 8.5) and periods of vibration (0 ≤ T ≤ 6), associated with a set of four different distances. These distances correspond to 250, 300, 350, and 450 km. Likewise, a color scheme ranging from blue to red facilitates interpreting the results; the blue tones indicate that the predicted values are lower regarding the evaluated model, and the red tones indicate the opposite. In both figures, it can be observed that for T < 3s, the GMPM proposed predicts intensities below R99 and J06 for the magnitude range of 6 to 8 and distances less than 350 km. In this zone, the differences are less than 0.2 log units. For T > 3s, the model predicts intensities above R99 and J06 for the entire range of magnitude and distance. The differences are greater than 1 log unit in this area, especially at T > 5s. In general, a smaller difference is obtained regarding the J06 model than that of J06. These differences are mainly attributed to the dataset size since CU21 includes thirteen earthquakes not considered in J06 and eighteen more than R99, which provide essential information about the GMI for events with magnitudes (Mw) between 6.0 and 7.4, and distances (Rrup) between 260 and 400 km.
Comparison of hazard curves and uniform hazard spectra
Figure 11 compares the empirical and theoretical hazard curves computed from a probabilistic seismic hazard analysis (PSHA) for PGA and SA at five structural periods that correspond to 0.2s, 0.5s, 1s, 1.5s, and 2s. The empirical curve is obtained by counting the number of times per year a given IM value has been exceeded, dividing by the observation period that, in this case, corresponds to 57 years for PGA, SA(T ≤ 2.7), and AvgSA(T ≤ 1.8), and 37 years for PGV, SA(T > 2.7) and AvgSA(T > 1.8). For these IMs, the observation period decreases because only twenty-four events recorded from 1985 to 2021 were used to derive the GMPM. The PSHA analysis is performed just for interface earthquakes along the Mexican Pacific Coast. The characteristics of the seismogenic zones (i.e., geometry and seismicity) are taken from the current version of Mexico’s Seismic Design Code of the Federal Electricity Commission (CFE 2015). The attenuation models used correspond to R99, J06, and the one proposed in this study. The computations were made using R-CRISIS (Ordaz et al. 2021), based on the classic Esteva–Cornell approach (McGuire 2008).
The empirical curves were computed for 30 intensity values logarithmically separated by equal intervals between the limits 1 to 1000 cm/s2 for the SA curve (Fig. 11) and 1 to 100 cm/s2 for the AvgSA curve (Fig. 12). Therefore, their ordinates are the same in the six figures associated with a specific IM. In these figures, most of the empirical hazard curves, compared with the theoretical hazard curves computed from PSHA, tend to saturate at intensities below 5 cm/s2, which may be related to the fact that earthquakes with low ground-motion intensities were dismissed from the dataset. The same behavior occurs at intensities above 30 cm/s2 because there is only one event in the dataset that generates intensities above this threshold, so its exceedance rate corresponds to the maximum observation period, equivalent to v(a) = 0.0175 for PGA, SA(T ≤ 2.7) and AvgSA(T ≤ 1.8), and v(a) = 0.027 for PGV, SA(T > 2.7) and AvgSA(T > 1.8). For moderate intensities, the differences between both hazard curves are not so large and seem fit for the observed data. Similar observations were made by (Ordaz and Reyes 1999).
Figure 11 shows that the shape of the hazard curves estimated with the proposed model disregarding the uncertainty in the regression coefficients (herein, HDU, dotted black line), follows a similar shape to the estimated with the R99 (continuous gray line) and J06 (dotted red line) models. The three GMPMs compute similar exceedance rates, v(a), for low intensities. However, their differences increase as the intensity increases. Compared to the proposed model, the R99 model estimates higher exceedance rates for PGA and SA for all periods evaluated, except for T = 2s. On the other hand, the J06 model estimates lower exceedance rates for PGA and SA at T = 0.2s and T = 0.5s, the opposite case occurs for T = 1.0s and T = 1.5s. Like R99, the forecasts of J06 at T = 2s are the same as those of the proposed model.
Likewise, in Fig. 11, the hazard curves are also compared with the proposed model, considering the uncertainty in the regression coefficients (herein, HCU, continuous black line). As expected, for small and large intensities, the hazard level of HDU is larger than HCU. This increment at small intensities, with v(a) ≥ 0.1, is produced by the additional epistemic uncertainty for small magnitude events (Mw<6) at short distances (Rrup<240 km) whose predictive standard deviation, σp, is between 20% and 100% more than the obtain directly from the Bayesian regression, σe (see Fig. 4). The same behavior occurs for large intensities, with v(a) ≤ 0.01, since the hazard level is controlled by large events with Mw >7.5 and Rrup>280km, in which σp is between 10% and 80% more than σe. The differences become less evident at moderate intensities, with 0.1 ≥ v(a) ≥ 0.01, whose hazard level is controlled by earthquakes with magnitudes (Mw) between 6.0 to 7.5 and distances (Rrup) between 250 to 350 km. These parameters fall in the range where σe2 and σp2 are similar (see Fig. 4) and correspond to the well-sampled region of the dataset. The same effect is observed in the hazard curves computed for AvgSA (see Fig. 12), although less evident at low intensities than in SA curves.
Finally, Fig. 12 compares the empirical and theoretical hazard curve computed from PSHA for AvgSA at six periods corresponding to 0.1s, 0.2s, 0.5s, 1s, 1.5s, and 2s. In this case, the hazard curves are estimated directly from GMPM-AvgSA proposed, disregarding the uncertainty in the regression coefficient (HDU), and compared with those obtained indirectly using the existing GMPM-SAs (i.e., R99 and J06) and a SA inter-period correlation model proposed by (Rodríguez-Castellanos et al. 2021). The stepped shape of the empirical curve is due to the lack of events in the database that generate large intensities. Therefore, the exceedance rates computed for large accelerations are less reliable for comparison purposes. Despite this, it is noted that the shapes of the calculated hazard curves match reasonably well with the observed data at intensities below 20 cm/s2. The three GMPMs compute similar exceedance rates, v(a), for low intensities. However, as observed for SA, their differences increase as intensity increases. The R99 model estimates higher exceedance rates for all periods than the proposed mode. Likewise, the J06 model also estimates higher exceedance rates for most periods, except for T = 0.1s and T = 0.2s. There are no significant differences in the seismic hazard estimations computed from the proposed model compared to previous models. However, according to the available data, the GMPM suggested estimates the seismic hazard more adequately than the other models.