As reported elsewhere (Ader et al, 2018, Ioannou, et al, 2015), fragility curves derived from data in which the PGA is uncertain, for example in the USGS Shakemap values used here, have larger variances (σ) than the “true” curve which would be derived from the actual PGA at the location of the survey. This problem has been addressed by Ader et al (2018), who have presented a process for removing this bias and estimating an unbiased fragility curve. The Ader process has been adopted here to estimate the parameters of the unbiased fragility curve.
The parameters used are defined as follows:
σest is the estimate of the variance of the lognormal fragility distribution derived from the original data points
σ is the unbiased estimate of the variance of the lognormal fragility distribution
σPGA is the uncertainty in the ln(PGA), based on the ground motion prediction equation used
σD is the standard deviation of the ln(PGA) of the data points projected onto the original estimated fragility curve by minimising the distance between the data point and the curve.
The value of σ can be estimated from the following relationship:
where the parameter λ2 is found (Ader et al., 2018) to have the approximate value of 1.66.
As Ader et al. (2018) point out, to be rigorous σD should be estimated by projecting the data points onto the true fragility curve rather than the original biased fragility curve. Thus, a better estimate of σ can be found by iteratively updating the value of σD and the resulting unbiased fragility curve. In this study only two iterations were used as it was found that further iterations made no significant difference given the data uncertainties.
The value of σPGA to be used derives from data given by Worden et al (2012) describing the Shakemap archive. Its value depends not just on the ground motion prediction equation used, but also on whether any observed ground motion data was available. Although for some locations in the Shakemap archive σPGA varies from 0.4 to 0.7, for the vast majority of the Shakemap dataset σPGA = 0.5, and this value has been adopted here.
In this study the value of σest was found by a least squares regression process as described earlier. The projected points were found by finding the distance from the data point to the nearest point on the regression curve using Excel’s Solver routine.
The calculation of the unbiased σ as described above was carried out for the DS1 fragility curves for the 5 building classes, and the σ values for DS1 were then used for the fragility curves for all damage states, for the reasons explained in Section 4.2 above. The values of µ for each damage state were then determined using least squares regression based on the revised σ.
For each of the five regions, the values of the unbiased σ, and associated values of µ for each building class and each damage state are given in Table 5. The resulting lognormal fragility curves for building class RCFL in each region are shown in Figure 3. It is important to note that only in Europe were all the 5 classes represented in the available data.
Table 5: Parameters (𝞵 and 𝞼) of the unbiasing fragility curves for the five regions and all CEQID in the different building classes
Region
|
µDS1(g)
|
µDS2(g)
|
µDS3(g)
|
µDS4(g)
|
µDS5(g)
|
σ
|
Building Class
|
ASIA
|
|
|
|
|
|
|
WM
|
0.15
|
0.22
|
0.25
|
0.32
|
0.53
|
0.6
|
URM
|
0.12
|
0.22
|
0.34
|
0.50
|
0.90
|
0.6
|
RCFL
|
0.16
|
0.39
|
0.68
|
1.46
|
2.20
|
0.5
|
AUSTRALASIA
|
|
|
|
|
|
|
URM
|
0.38
|
0.48
|
0.52
|
0.59
|
1.40
|
0.6
|
RCFL
|
0.50
|
0.67
|
0.88
|
1.11
|
1.12
|
0.5
|
RCFM
|
0.38
|
0.51
|
0.54
|
0.56
|
3.07
|
0.5
|
EUROPE
|
|
|
|
|
|
|
WM
|
0.07
|
0.22
|
0.38
|
0.59
|
1.52
|
0.6
|
URM
|
0.13
|
0.59
|
0.75
|
0.96
|
2.97
|
0.6
|
RCFL
|
0.30
|
0.72
|
1.01
|
1.92
|
3.10
|
0.5
|
RCFM
|
0.33
|
0.64
|
0.70
|
0.76
|
1.49
|
0.5
|
RFCH
|
0.31
|
0.62
|
1.20
|
1.40
|
1.70
|
0.5
|
MIDDLE EAST
|
|
|
|
|
|
|
WM
|
0.16
|
0.30
|
0.33
|
0.38
|
0.49
|
0.6
|
URM
|
0.17
|
0.36
|
0.41
|
0.47
|
0.79
|
0.6
|
RCFL
|
0.08
|
0.39
|
0.51
|
0.83
|
0.75
|
0.5
|
LATIN AMERICA
|
|
|
|
|
|
|
RCFL
|
0.40
|
0.58
|
0.64
|
0.78
|
0.82
|
0.5
|
CEQID
|
|
|
|
|
|
|
WM
|
0.08
|
0.27
|
0.32
|
0.42
|
0.72
|
0.6
|
URM
|
0.16
|
0.39
|
0.45
|
0.57
|
1.06
|
0.6
|
RCFL
|
0.27
|
0.49
|
0.64
|
1.01
|
1.45
|
0.5
|
RCFM
|
0.32
|
0.51
|
0.59
|
0.62
|
1.49
|
0.5
|
RCFH
|
0.31
|
0.62
|
1.20
|
1.40
|
1.70
|
0.5
|