As mentioned before, the estimates used in a risk management framework are sensitive to the epistemic uncertainties associated with the hazard input. For example, Fig. 12a and Fig. 12b show the 84th and 50th percentile hazard maps from the current Italian national seismic hazard assessment produced by INGV (Gruppo di Lavoro MPS 2004; Meletti and Montaldo 2007). The uncertainty results from multiple seismic source zonations and many ground-motion models within the logic tree of the seismic hazard assessment (see, e.g., Douglas et al. 2014 for a discussion of these uncertainties and their sizes in recent seismic hazard assessments). The difference in the PGAs for the 84th and 50th percentiles for a given site ranges between 0.005 g and 0.063 g, according to Stucchi et al. (2011), who report higher differences when comparing the 84th and 16th percentiles (up to 0.12 g).
Differences can be observed between the results from different studies for the same area, too, as it can be inferred by comparing the results of Fig. 12b with those of Fig. 12c, showing the 50th percentile (median) PGAs obtained from the ESHM13. A comparison between the uncertainties for PGAs and response pseudo-spectral accelerations at 1 s obtained from different hazard studies and for different percentiles can be found in Douglas et al. (2014). The impact of the hazard uncertainty on the safety of the designed structures and the cost implications are investigated in the following sections by considering various case studies.
Epistemic uncertainty from different seismic hazard models: case study for Italy
Four different locations are considered in Italy (Fig. 13a), for which the hazard curves (50th percentiles) in terms of MAF of exceedance of the PGA (Fig. 13b) are obtained from the INGV and ESHM13 projects. The latter gives higher values of PGAs for the 475-year return period, in the considered areas, as can also be noticed in Fig. 12. That is not the case for the whole range of the PGA levels though, because the slopes of the INGV curves are generally steeper than those from the ESHM13 project and hence the INGV curves often show higher PGAs at higher MAFs.
In the selected locations, the values of the PGAs corresponding to a 475-year return period are found to be 40% to 95% higher for the ESHM13 compared to INGV, as shown in Table 2. These differences in the hazard estimates result in different design accelerations for the reference building, which is translated into different construction costs. The structural performance is also changed, and this affects the risk and loss levels from the expected future seismic activity. These are quantified following the procedure outlined in Sect. 2 and the results are reported in Table 2. EALr and E[LCCr] refer to the expected value of the annual losses and of the life-cycle cost when only the repair/replacement costs are considered. When the additional losses are also included in the estimates, the notation is changed to EALr,a and E[LCCr,a]. It is clarified that class B is assumed for the soil conditions in every location; thus the hazard curves are multiplied by the soil factor 1.2, expressing the ratio between PGAs on site classes B and A in EC8.
Table 2 Comparison of the results for the hazard curves according to the INGV and ESHM13 hazard models First, the initial construction cost is calculated considering as design acceleration, PGAd, the PGAUH values with a 10%-in-50-years exceedance probability. Because the initial construction costs are not very sensitive to the design acceleration, the relative differences for the two hazard models are less than 2.2% for the various sites. It is also important to mention that the variation of the initial cost in the vicinity of 0.1 g is very low, as discussed in Gkimprixis et al. (2020). This explains why the impact of the considered hazard uncertainty on the construction cost is negligible for Milan (low hazard) and much larger for Cosenza (high hazard).
The MAF of collapse, λC., evaluated via Eq. (1), is very sensitive to the hazard model choice, and the variations can be of two orders of magnitude from the estimates obtained considering the hazard from one project rather than the other. These results give a measure of the importance of considering hazard uncertainty in risk assessment.
The life-cycle cost derives from the sum of the initial costs and the expected future losses evaluated from Eq. (4) for a 50-year time period and a discount rate of 3% per year. It is noticed that the life-cycle cost is less sensitive to the hazard uncertainty than the risk. This is mainly because the construction cost (which is not very sensitive to the PGAd) contributes more than the future losses to the E[LCC]. Nevertheless, the relative variation of E[LCCr] (up to 4%) is higher than that of the construction cost. Higher variations in the life-cycle cost are observed if the additional losses are included.
Sensitivity analysis for Europe based on a simplified approach
This section investigates, through a sensitivity study, the effect of the uncertainty in the hazard input on the design and consequently on the risk and loss levels attained for the same reference building considered in the previous sections. To demonstrate the approach, five locations of different seismicity are first studied in detail, and then the procedure is repeated for all locations across Europe.
First, the same locations selected in Sect. 3.2 (Fig. 5) are considered here. A simple approach is followed to investigate the effect of hazard uncertainty, by assuming that the hazard curves of Fig. 5b, considered as the ‘reference’ ones, underestimate or overestimate the PGA levels by 50% at all the MAFs of exceedance. This corresponds to a translation of the hazard curves towards the left or the right. It is clarified that the variation in the hazard curves is described by the variations of the PGA for a given MAF, rather than by variations of the MAF for a given PGA, which would not lead to exactly the same results but the overall conclusions would likely be similar.
The varied curves are assumed to describe the actual hazard of the site, which is different than that described by the ‘reference’ one, and thus will be referred to hereafter as the ‘true’ curves. Having defined the ‘reference’ and the ‘true’ curves, the values of λc, C0, EAL, E[LCC] are calculated for three different cases (based on the consideration of the uncertainty in the design and assessment stages):
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‘Case 1’ (R/T): the building is designed with the ‘reference’ (R) curve, and the performance is assessed by evaluating the output parameters of interest using the ‘true’ (T) ones (i.e. varied by 50%, higher or lower than the ‘reference’ one), which could refer to a case where the building has already been designed for a hazard level, but at the assessment stage the hazard input is updated;
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‘Case 2’ (T/T): the ‘true’ (varied) hazard curves are used both for the design and the assessment of the various output parameters, corresponding to the situation where the true hazard at a site is known at both the stages;
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‘Case 3’ (R/R): the design and assessment of the building are both performed using the ‘reference’ curve, which corresponds to a situation of reduced knowledge of the true hazard.
Then, sensitivity analyses are performed, by comparing the results obtained from the above three cases to each other. Figure 14 summarizes the various analyses performed, using the results from the above cases. The local sensitivity of the generic parameter \(p\) (e.g. initial cost, risk, loss, life-cycle cost) is defined as follows:
$$\Delta p\left[\%\right]=({p}_{B}-{p}_{A})\cdot 100/{p}_{A}$$
(8)
where \({p}_{A}\) denotes the value of the parameters obtained from analysis A (for either ‘Case 3’ or ‘Case 1’), and \({p}_{B}\) denotes the value of the parameters obtained from analysis 2 (for either ‘Case 1’ or ‘Case 2’), as defined in Fig. 14. The values of \(\Delta p\) provide information on the sensitivity of costs, losses, and safety to the hazard, for different sites, thus allowing quantification of the importance of a more accurate hazard assessment.
Scenario I: ‘Case 3’ versus ‘Case 1’
In the first scenario, a comparison is made between the results obtained from ‘Case 3’ versus those obtained from ‘Case 1’. Figure 15 shows the results obtained based on Eq. (8), for the five considered locations. The red and blue points refer to the cases of respectively higher and lower ‘true’ hazards compared to the ‘reference’ one (i.e. hazard underestimated and overestimated by the ‘reference’ curve, respectively). Given that the design in both cases is performed with the ‘reference’ hazard curve, the initial construction costs are the same. The rest of the output parameters are all increased or decreased when the ‘true’ curve is higher or lower, respectively, than the ‘reference’ one. It is interesting to observe that the estimated values are not symmetrical around zero, although the design is the same and the hazard is increased and decreased by the same percentage. This is attributed to the fact that the fragility and vulnerability functions are not constant, but change with the IM level, resulting in higher sensitivity when the hazard is increased rather than decreased.
It can also be observed that the parameter that is most sensitive to the hazard uncertainty is the risk (differences of up to 200%). Of course, it is important to have in mind that the risk is very small in absolute terms. The EALs are also very sensitive to the hazard (differences of up to 150%), though the actual numbers are quite small compared to the initial cost. The variation of the E[LCC] is lower, given that it is affected mainly by the change in the initial cost, which is zero in this scenario. It is interesting that the seismic hazard of the site does not play an important role in the sensitivity of the considered parameters, with the exception of the E[LCC].
This type of investigation can be used, for example, to incorporate uncertainties to the insurance model suggested above. For example, the minimum premium (equal to the EALr in the case of full insurance) for Aigio should be increased by a percentage higher than the assumed hazard uncertainty, i.e. by 75% and 105% when the hazard is decreased or increased, respectively, by 50%.
Scenario II: ‘Case 3’ versus ‘Case 2’
A second sensitivity analysis is performed, where a comparison is carried out between the results for ‘Case 3’ and ‘Case 2’. The sensitivities of all the parameters in the assessed locations are calculated according to Eq. (8), and the results are presented in Fig. 16. This investigation highlights the effects of updating the design, previously done with the ‘reference’ hazard, based on the ‘true’ hazard. In contrast to the previous scenario, now the construction cost is changed to meet the true hazard needs. Because of the updated design, the variation of the rest of the parameters is reduced, in comparison with the previous scenario. Also, the variation of the parameters is more dependent on the seismic hazard of the location in contrast to the previous scenario.
Scenario III: ‘Case 1’ versus ‘Case 2’
A final sensitivity analysis is performed by comparing the results obtained from ‘Case 1’ versus those obtained from ‘Case 2’. The results of this sensitivity analysis for the five locations are presented in Fig. 17. As expected, if the design is carried out assuming a ‘true’ hazard that is higher than the ‘reference’ (red points), then the initial construction cost increases, but on the other hand there is a reduction of the risk and losses. The opposite applies in the case of a ‘true’ hazard that is lower than the ‘reference’ one (blue points). The life-cycle cost variations are influenced by changes in both the initial cost and the future losses, and thus they do not follow a constant trend. It can also be observed that the parameter that is most sensitive to the hazard uncertainty is the risk. Of course, it is important to have in mind that the risk values are very small numbers. In Aigio, for example, if the ‘true’ hazard is lower than the ‘reference’ one used for the design, the risk is 2.58·10–5. This is significantly lower than the risk corresponding to the design performed using the ‘true’ hazard curve, which is 1.00·10–4 (i.e. 289% higher). For the rest of the parameters, it is clear that the differences are in general greater in higher hazard areas than in lower hazard areas (e.g. Cologne). This suggests that ± 50% differences in hazard are relatively unimportant in low hazard areas and hence there is less need to refine hazard assessments there. It should be noted, however, that only a single building type and geometry is considered here and hence more analyses are required before drawing wider conclusions.
The EALs are very sensitive to the hazard, with a maximum difference of -30% and 98% in the case of underestimation and overestimation of the hazard, respectively. On the other hand, the sensitivity of the initial cost is lower than 4%, given that the initial cost does not present a significant variation for the different design levels (see Fig. 18a). Although the EALs are significantly affected by the hazard variations, the actual numbers are quite small compared to the initial cost. Thus, the E[LCC] exhibits a reduced sensitivity, being affected mainly by the change of the initial cost.
The graph for the initial cost is almost symmetrical, meaning that under- and over-estimating the hazard has roughly the same effect in absolute terms. That is not the case for the other studied parameters. For example, the impact of overestimating the hazard (‘true’ hazard curve 50% smaller than the ‘reference’ one) on the EAL is higher than when the hazard is underestimated.
To better understand these trends, the location of Catania is selected for further investigation. Figure 18a shows the relation between the initial cost C0 (normalized to the one for PGAd = 0, i.e. 655,840€) and the design acceleration. This relationship is almost linear for PGAd values higher than 0.1 g. This is why the 50% increased or decreased hazard results in roughly 2% change of C0 in both cases. Figure 18b shows the results in terms of EAL, normalized to those corresponding to PGAd = 0 for each case (i.e., 729€, 2,357€, and 4,362€ for -50%, ‘reference’, and + 50%, respectively). For reference, a black curve is also plotted which corresponds to considering the ‘reference’ hazard curve both for design and assessment (‘Case 3’ above). The EALs are also evaluated again considering the ‘reference’ hazard curve only for design while the assessment is done with the curve increased by 50% (red line) or decreased by 50% (blue line). Similarly, the colour of the dots shows which hazard curve was used for the design.
The same sensitivity study is performed for the E[LCC] and the results are presented in Fig. 19a, b, with the additional losses disregarded or considered along with the repair/replacement cost, respectively. In this case, the hazard uncertainty does not affect the life-cycle costs in a consistent direction, as mentioned before. This is attributed to the fact that the relation of LCC with the PGAd is not monotonic, given that the initial cost and the future losses are affected in opposite ways by a change in the PGAd.
Investigation across Europe
In this section, the effect of the hazard uncertainty on the cost and performance of the structure is investigated for all locations across Europe. As explained in Sect. 3.2, first, the PGA values that correspond to various exceedance probabilities for the different locations are obtained from the ESHM13 project and the second-order polynomial function in log-space (Vamvatsikos 2013) is used to extrapolate the hazard data to a wider range of PGA, if required. As the seismic designs were undertaken using the EC8 spectrum for site class B, the PGAs for site class A from the ESHM13 are multiplied by the soil factor 1.2, expressing the ratio between PGAs on site classes B and A in EC8, to construct the ‘reference’ hazard curves. These PGAs are further increased or decreased by 50% to perform the sensitivity analyses. In order to avoid biasing the conclusions by extrapolating the vulnerability curves to very high values of PGA (e.g. 0.60 g would become 0.90 g for a 50% increased hazard) the results presented here refer to PGAs up to 0.5 g.
The first column of Fig. 20 presents the values of C0 (normalized to the value for PGAd = 0, i.e. 655,840€), EAL, E[LCC] (both normalized to the C0 value of each location) and λc, for all the considered locations across Europe. The parameters are evaluated for ‘Case 3’ (defined above), and the resulting values are summarized in the first column of Fig. 20 versus the value of the 475-year-return-period PGAUH. Each point refers to a given location across Europe, for example the figure for the EALr summarizes Figs. 7a and 6 (see explanation in Fig. 8). Similarly, the remaining graphs of the column summarize the values of the parameters for all locations across Europe. These results have already been discussed (and presented as maps) in Gkimprixis et al. (2020), but are summarized here to help with the interpretation of the sensitivity analyses.
The variations in C0, EAL, E[LCC] and λc for all the considered locations across Europe are then calculated based on Eq. (8). In particular, in the second column of Fig. 20, the sensitivity of the parameters is evaluated for ‘Case 3’ and ‘Case 1’ and in the third column by comparing ‘Case 3’ and ‘Case 2’. The resulting variations show how much the values of the first column will be affected, considering the hazard uncertainty described by Scenarios I and II.
The trends are similar in both scenarios, for all the considered parameters, in the sense that the results of both cases present the same trend (increased or decreased) as the hazard variation. Overall, it can be observed that overestimating hazard (‘true’ lower than ‘reference’, i.e. blue points) introduces lower levels of uncertainty in the estimates, when compared to the case of underestimation of hazard. The most sensitive parameters are the EAL and the risk, mainly in low seismicity regions and for the case of underestimating hazard (red points). In the second scenario the levels of uncertainty in the high seismicity regions are smaller and close to level of the hazard uncertainty. The variation of E[LCC] is mainly affected by the variation in C0, and are both lower than the hazard uncertainty.
For the second scenario (third column of Fig. 20) and for PGAUH values higher than 0.1 g, the variations of C0 due to increased or decreased hazards are almost symmetrical. This is a consequence of the relation between C0 and the design acceleration, which is almost linear for PGAUH values higher than 0.1 g (see plot for C0 in first column of Fig. 20). In the vicinity of 0.1 g the change in C0 when the hazard is reduced by 50% is smaller than when the hazard is increased (see also Fig. 18a). This is attributed to the minimum requirements of the design code and the fact that for low levels of PGAd the design is mainly influenced by the gravity loads. This leads to an overdesign in the case of PGAd = 0, influencing the slopes of the C0 and EAL curves (Fig. 19a, b).
As previously discussed, the EALr variation graphs can be used to account for the effects of hazard uncertainty on insurance premiums. If the ‘true’ hazard is different to the ‘reference’, then the premiums obtained from ‘Case 3’ (Fig. 7a or the equivalent graph of EALr in the first column of Fig. 20) should be modified by the percentages reported in the second and the third column, if the design is performed with the ‘reference’ or the updated/ ‘true’ hazard model, respectively. It is worth noting that in many cases the level of uncertainty introduced in the insurance model is more than twice that of the hazard uncertainty (assumed 50%) for Scenario I, which is reduced to roughly equal the hazard uncertainty when the design is updated to match the ‘true’ hazard (Scenario II).
The variations in C0, EAL, E[LCC] and λc for all the considered locations across Europe are also calculated for Scenario III, using Eq. (8), and the results are summarized in Fig. 21. It is noted that, in contrast to the two previous scenarios, there is no connection of these results to the first column of Fig. 20 (i.e. with ‘Case 3’), since the variations refer to the difference in the estimates resulting from ‘Case 1’ versus ‘Case 2’.
Overall, it can be observed that the effects of the hazard variations change significantly from location to location. For example, the changes in the observed parameters are negligible for low hazard sites with PGAUH less than 0.1 g. It should be clarified that this observation is based on analyses for a single example of a code-compliant structure and may not hold for other types of structure or infrastructure. The parameter that is most sensitive to the hazard uncertainty is the risk, for the case of overestimating the hazard. It is observed that, in high seismicity regions, overestimating the actual hazard of the site can lead to underestimation of the risk by up to 500%. Significant also are the variations of the EAL, which as discussed above for the case of Catania, increase monotonically with PGAUH, while the E[LCC] does not exhibit a clear trend in its variation.
As previously commented, the results can provide useful information for the insurance sector. For example, the results for the EALr (presented in Fig. 20 and in Fig. 21) can be interpreted as the sensitivity of the minimum premium (no profit) in the extreme case of a full insurance contract (no deductible or limit point). Similarly, the sensitivity of the premium in the case of partial insurance can be determined. As an example, the analyses for the three defined scenarios are repeated for the insurance model of Fig. 2a (D = 10% and L = 50%), which significantly changes the insurance premiums, as seen in Fig. 9a.
The results for Scenarios I and II are presented in Fig. 22a, b, respectively. It is very interesting to observe that the effect of the hazard uncertainty on the partial insurance premiums is increased, compared to the case of full insurance investigated above. This effect is larger in areas of low to moderate seismicity (e.g. with a PGAUH lower than 0.3 g) for Scenarios I and II and a ‘true’ hazard curve 50% higher than the ‘reference’ one. It is recalled though that the obtained EALs are quite low in low seismicity regions (see Fig. 9a). For Scenario III on the other hand, similar results to the case of full insurance in Fig. 21 are observed in Fig. 22 (observe EALr results of the two figures), with some increased impacts in high seismicity regions and for the case of a ‘true’ hazard curve 50% lower than the ‘reference’ one.
Overall, from these results it is concluded that the necessary precision in the definition of the hazard curve depends on the parameter of interest. In other words, the need to invest in the accuracy of the hazard model depends on the stakeholder. For example, a study on the risk sensitivity can be of interest when updating design guidelines and defining the design seismic input and the target reliability levels to be achieved. On the other hand, the information on the sensitivity of the EALs would be more useful for insurance companies that cover losses from future earthquakes, since the uncertainties can seriously affect the insurance premiums.