Considering uncertainty in the collapse fragility of New Zealand buildings for risk‐targeted seismic design

The risk‐targeted seismic design framework is used to set design intensities, based on achieving a target risk level (e.g., collapse or fatality risk) with respect to an assumed building collapse response. This paper assesses the distribution of fatality risk associated with the risk‐targeted design intensity, considering uncertainty in both the hazard and the variety of buildings that can satisfy the minimum design requirement. The randomness among buildings and their response is due to other design decisions (aleatory variability with respect to the design intensity) and can be represented by a distribution of fragility curves that quantify the collapse probabilities of as‐built, code‐conforming buildings. First, a single design intensity is calculated based on a “design fragility,” the mean hazard curve, and a risk target. The design fragility is taken as a conservative estimate from the distribution of collapse fragilities and the selected risk target approximates the risk associated with New Zealand's previous (non‐risk‐targeted) criteria for design intensities. Then the risk distribution is assessed, considering the aleatory variability in as‐built fragilities and the epistemic uncertainty in the new National Seismic Hazard Model. Accounting for variability in design decisions and uncertainty in the hazard model produces a risk distribution that more fully represents the potential risk associated with a given design intensity. This distribution can be compared to guidance on tolerable risk ranges, which suggest that risk can be variable among buildings but should fall within acceptable bounds. Sensitivity studies consider epistemic uncertainty in the assumed model for the distribution of as‐built fragilities. This inclusion of uncertainty to assess the risk distribution offers a powerful extension to the risk‐targeted framework. While this extension would not affect engineering practice (as the output is still a single design intensity), it allows building code developers to better understand and consider the risk implications associated with the selected design intensity.


INTRODUCTION
The risk-targeted design framework 1 is a tool for setting seismic design intensities that result in buildings with similar risk of failure, for example, collapse or fatality risk.As such, the framework is more relevant for the decision-makers who set the design intensities in a building standard than for the engineers who use those intensities as one of many design constraints.This paper adds a new dimension to the framework, allowing decision-makers to more formally assess the potential distribution of risk among a variety of buildings that all satisfy the same design intensity.The risk-targeted framework was introduced as an alternative to the more common uniform-hazard approach, which selects the design intensity based on a given annual probability of exceedance (APoE). 2,3An APoE is a single point on a hazard curve from a probabilistic seismic hazard analysis (PSHA), which considers the randomness associated with rupture scenarios and the probabilities of ground motion shaking intensities given those scenarios.Gkimprixis et al. compiled a comprehensive discussion of both the motivation and various methodologies for shifting from APoE to risk-targeted design intensities. 1 In short, a building's response to ground shaking adds another layer of uncertainty that must be assessed over a range of shaking intensities, not just a single APoE point.The risk-targeted framework adds the uncertainty in building response via a fragility curve to characterize the probability of failure at a range of shaking intensities.When paired with the hazard curve, these fragilities can quantify the risk of failure, given uncertainty in both the shaking intensity and the building response.Risk-targeted methodologies calibrate the design intensities by comparing this risk quantification to a target risk, or a tolerable risk limit.The framework is typically based on the response of undamaged buildings and does not include the increased hazard or additional damage associated with foreshocks and aftershocks.
It is important to note that the fragility curve used for selecting the design intensity is a generic representation of response, while individual buildings may have different responses. 4,5The generic curve offers a way to achieve similar (but not equal) risk levels among buildings, without introducing building-and site-specific assessments that would complicate the design process.The variety of fragilities (and associated risk levels) among individual, as-built structures can be seen as aleatory variability with respect to the design intensity, due to the impact of other design decisions (e.g., irregularities, design ductility, member detailing, etc.).While this variability is accepted as an implicit feature of the risk-targeted framework, this paper adds a secondary risk assessment to explicitly consider the potential risk distribution associated with the aleatory variability among as-built fragilities.This secondary assessment can also include epistemic uncertainty in the hazard model, as well as sensitivity studies for the epistemic uncertainty in the assumptions regarding the distribution of fragilities.This risk distribution can be compared to tolerable risk ranges, providing additional support for decision-makers as they consider implementing the risk-targeted framework in building standards.
What follows is a more detailed overview of the risk-targeted framework, as implemented by the Luco et al. methodology. 6The building response is typically characterized by a lognormal fragility curve, which defines the probability of exceeding the collapse limit state, , over a range of shaking intensities, (|) (black line in Figure 1A).The shaking intensity is quantified by an intensity measure, IM, defined here as the psuedo-spectral acceleration, (), for the fundamental building period, T. This IM was preferred for its direct link to New Zealand's National Seismic Hazard Model (NSHM), even though other intensity measures (e.g., average spectral acceleration or a vector IM) are known to be more representative of the collapse response. 7The fragility curve is a cumulative distribution defined by two parameters: (1) median, the median intensity at which the limit state would be exceeded and (2) beta, the dispersion (uncertainty) around the median, defined as a lognormal standard deviation.The median is the shaking intensity for (|) = 0.5 (black dot), and beta is reflected in the slope of the fragility curve passing through the median.For any given fragility curve's median and beta values, the annual individual fatality risk, AIFR, can be evaluated by the equation: where ( > im) is the hazard curve or the annual rate of exceeding a given value of the intensity measure, defined over a range of im values.The colored lines in Figure 1A are the hazard curves for three different locations, all of which have the same APoE of 1/500.The second term is the derivative of the collapse fragility curve, (|).As the functional form of the fragility curve is the cumulative distribution function (CDF) of a lognormal distribution with parameters median and beta, its derivative is the probability density function (PDF) of the same distribution.The final term converts the annual rate of collapse into annual individual fatality risk (AIFR) using a rate of fatality given collapse, (|).This term was added after the Luco et al. methodology was introduced and will be discussed at the end of this section. 8Finally, the integration over the risk integrand (i.e., the shaded area under the curves) is the total annual risk, considering the full range of shaking intensities (im values) that may cause collapse.The colored hazard curves in Figure 1A produce different risks for the same fragility curve, as demonstrated by the corresponding shaded areas.The shape of the integrand, termed the "risk disaggregation," provides insight into the relative contribution of each im to the total risk.(As the risk disaggregation focuses on the relative contributions, rather than the absolute value, the height of the risk disaggregations in Figure 1 is based on a scaling factor chosen for visualization purposes and does not correspond to either y-axis.) The adjustment for the risk target is an iterative process, shifting the fragility curve's median shaking intensity until the AIFR is equal to the risk target.This process is depicted in Figure 1B for the Wellington hazard curve from Figure 1A.The fragility's beta value is held fixed as the median is shifted.The result of the risk adjustment process is an optimal placement of the fragility curve (i.e., the shifted purple dot for the median), given the selected beta.
The final step is to extract the design intensity associated with the optimized response fragility.This is based on the design point, or the assumed probability of collapse for the design intensity.For example, in the United States the design point is (|  ) = 10% for the Risk-targeted Maximum Considered Earthquake,   , based on FEMA P-695. 6,9ASCE 7-22 pairs this MCE  design point with  = 0.6, while taking the Design Basis Earthquake, DBE, as 2/3 MCE  . 10his is equivalent to a design point of (|  ) = 2.5% for DBE.
In summary, the three input parameters that drive the design intensity are (a) beta, (b) the risk target, and (c) the design point.The first two parameters determine the median of the optimized fragility and the third extracts the design intensity from this fragility.The input parameter values can be tuned based on the study's objectives.For example, when considering a shift from the uniform hazard approach to the risk-targeted approach, one objective may be to maintain a degree of consistency with the previous design intensities.The Luco et al. implementation paired existing MCE values (based on uniform hazard with an APoE of 1/2475) with  = 0.8 to get an average annual risk of collapse across sites in the United States.As the average collapse risk was roughly 1% in 50 years, this value was selected as the risk target for setting MCE  .By selecting parameter values based on APoE of 1/2475, each individual site's MCE  probability of exceedance fluctuated above or below 1/2475 as required to match the 1% in 50-year target.(When this was introduced in ASCE7, the beta was updated to 0.6 with similar results.) Other studies have inverted the parameter selection process by starting with a risk target and a beta value. 4,8,11These two values are paired with each site's hazard curve to find the optimized fragilities.The design point is then selected to roughly approximate the traditional APoE values across all the sites, similar to the objective for ASCE7.The preference for this inverted selection process is two-fold.Firstly, international engineering communities may not consider the FEMA P-695 study's (|  ) = 10% design point to be relevant due to differences with the design intensity levels used locally.This is consistent with FEMA P-695's caution 9 that the design point was selected based on judgment and may not be appropriate for all governing jurisdictions.Secondly, Silva et al. and Horspool et al. select the risk target based on an acceptable AIFR.][14] Therefore, AIFR is considered more meaningful than collapse risk from a policy perspective, as it can be related to fatality risks in other sectors to allow for effective risk management.It is also more closely linked to the New Zealand Building Code objective to "safeguard people from injury caused by structural failure."The term "injuries" is interpreted to include both fatal (deaths) and non-fatal injuries.Since the majority of deaths and serious injuries in recent New Zealand earthquakes have been caused by structural collapse, 15 the fatality risk can be linked to the collapse risk.While this conversion to fatality risk could be handled probabilistically, most studies (e.g., literature 4,8 ) use a simple factor for the rate of fatality given collapse, (|) = 0.1, as in the last term of the Equation (1).In this manner, the AIFR risk target can be informed by comparisons with other sectors, then used to select the design point (|  ).
This overview of the risk-targeted framework provides the background for the current study, which supported the Seismic Risk Work Group's (SRWG) deliberation process for the building standards update following the release of New Zealand's new NSHM.This paper introduces key ideas that were considered during those deliberations.Section 2 outlines the relationship between the design assumptions (the three input parameters) and the assessment assumptions for the range of fragilities that represent the variability among as-built structures.Section 3 assesses the distribution of risk, considering the uncertainty in the as-built fragilities, as well as epistemic uncertainty in the hazard.Section 4 addresses epistemic uncertainty in the fragilities via sensitivity studies that perturb the assumed models for sampling and creating the as-built fragilities.

DESIGN ASSUMPTIONS FOR THE RISK-TARGETED INPUT PARAMETERS
A potential shift from APoE to risk-targeted design intensities starts with selecting the input parameters, similar to those studies described above.Here, the parameters for the design fragility are selected first, then used to identify a risk target that is roughly equivalent to the risk associated with the traditional APoE design intensities.

Selecting a design fragility
The design fragility is a generic representation of building response, which should generally reflect the behavior of buildings that comply with code criteria at the design intensity.The following discussion considers the range of behavior of New Zealand's code-conforming buildings, before selecting a single design fragility.This range of behavior will also inform the distribution of as-built fragilities used in the subsequent risk assessment (Section 3).
There is limited data on the collapse response of New Zealand buildings and particularly on the design point, (|  ).An alternative to defining a design point is to define a collapse margin ratio as the ratio between the median of the collapse fragility and the design intensity,  = ∕  .A range of 3 ≤ CMR ≤ 9 was adopted, representing the variability among as-built structures.The lower bound was roughly based on experimental and analytical research studies, which are often considered to provide conservative estimates of collapse performance, as compared to field observations.This perceived bias is due to a variety of factors, including conservative definitions of analytical collapse (e.g., 5% peak story drift), the presence of (unmodeled) gravity-load resisting systems that improve performance, and the effects of soil-structureinteraction, among others.In light of these factors, numerical research based on non-linear dynamic analyses informed the lower bound for the CMR values.Data from three studies [16][17][18] indicated that new code-compliant buildings are likely to have median collapse capacities ranging from 2.5 to 4.8 times the design intensity.These studies considered 4-and 12-story buildings in Christchurch and Wellington, including steel moment-resisting frame, eccentrically braced frame, and reinforced concrete wall systems.The upper CMR bound would represent buildings where seismic loads were not found to be critical and/or where the inherent strength, stiffness, ductility and deformation capacity are significantly underestimated by current code provisions.Finally, the distribution of CMR values was characterized as a truncated normal distribution, with a mean of 6 and a standard deviation of 1.5, and resampling for any values below the minimum of 3 or above the The second parameter that defines the collapse fragility is beta, or the uncertainty in the shaking intensity that would cause collapse.Other studies, such as the seminal FEMA P-695, discuss several sources of uncertainty, including variability in the response to ground motion, and variability in design requirements, test data, and modeling. 9The latter three pertain to knowledge of the structural system of interest, such as how comprehensive the prescriptive design requirements are, how much empirical evidence exists from system testing, and how well the generic structural system is represented by the suite of archetypes used to analytically interrogate the performance.It is also worth noting that even the response fragilities derived from non-linear analysis for individual buildings are estimates, due to ground motion selection and the numerical model that represents the building response. 19In FEMA P-695, these various sources of uncertainty are implicitly considered by combining them into a single, inflated beta value.However, this study posits that any given building will have an as-built (though unknown) response behavior, for which the primary source of uncertainty is in the ground motion (record-to-record variability, RTR), rather than uncertainty in building design, materials, or compliance during construction.Furthermore, the uncertainty in the median and beta estimates for individual buildings can be accounted for within a suite of fragilities, wherein the as-built response of any given structure will be represented, even if the exact values are unknown.This study adopted a range of 0.35 ≤  ≤ 0.45, based on FEMA P-695's suggestion for RTR variability in most building types that experience period elongation (i.e., not those with brittle, non-ductile behaviour or certain base-isolated systems).Figure 2B shows 100 beta values uniformly sampled from this range.
The sampled CMR and beta values were randomly paired with no correlation to produce 100 as-built fragility curves, shown in grey in Figure 2C.This represents a range of as-built fragilities that could result from the design process.Fox et al. suggested that the fragility will also depend on the location within New Zealand, not just the design, due to the regional variation in ground motion characteristics. 20While this study assumes that the distribution of fragilities does not vary by location, this regional effect could be included once there are simplified recommendations on how to account for it.The dashed black line represents an aggregated fragility curve across the as-built designs.The aggregated fragility was assessed by extracting 100 collapse intensities from each sampled as-built fragility curve, with the extracted values cumulatively representing the 1-99 th percentiles of the fragility.These 100 extracted intensities were compiled into 1000 Comparison of the adopted design fragility (blue) versus the ASCE7 fragility (orange).The dots at probability of collapse (|) = 0.5 are the CMR values, or the ratio of the median collapse capacity to the design intensity, IM  (thick dashed line).The slope through the CMR value is defined by beta.The design point is the probability of collapse at the design intensity, (|  ).The ASCE7 fragility's design point is defined as (|  ) = 10% but was converted to (|  ) = 2.5% based on the relationship between MCE  and DBE.DBE, Design Basis Earthquake; MCE  , Risk-targeted Maximum Considered Earthquake.
collapse intensities across the 100 sampled fragility curves.Then a lognormal distribution was fitted to the combined collapse intensities, characterized by a median and beta value for the aggregated fragility.
In the absence of other information about the design of a building, the aggregated fragility is the central estimate of its collapse fragility.The median CMR of the aggregated fragility is 6, or the mean of the distribution from which the CMRs were sampled.The aggregated fragility's beta increases to 0.44, near the upper end of the underlying beta distribution.This is because the beta also accounts for the spread in the medians.This increased beta is conceptually similar to the way that FEMA P-695 increases the beta to account for uncertainty in building design, materials, or compliance during construction, which are now represented within the distribution of fragilities.(Note that 0.44 may suggest an unwarranted level of precision.Rather than rounding to 0.45, the more precise value is reported for comparison with other CMR distributions that will be explored in Section 4.) This aggregated fragility curve is a central estimate of the as-built performance, recognizing that not every potential design will exhibit this behavior.Using the aggregated fragility curve would be similar to other risk-targeted studies, as visually depicted by Silva et al., where a mean fragility represents the individual building fragilities for mid-rise reinforced concrete buildings. 4However, the SRWG preferred to use a more conservative fragility with  = 4, based on a philosophy that building standards are intended to constrain minimum conforming design.The implications of this decision on the risk target selection will be discussed at the end of Section 2.2.
The design fragility's CMR was also paired with a conservative beta value.Increasing the beta effectively pivots the slope of the fragility through the CMR value, increasing the modeled probability of collapse for intensities below the median and reducing it above the median.The steep negative slope of hazard curves (such that the probabilities of exceedance are always viewed in logscale) mean that larger beta values will produce higher risk.Therefore,  = 0.45 was selected as the upper bound of the beta range.
Based on these decisions,  = 4 and  = 0.45 were adopted for the design fragility, in support of the SRWG study.This means the design intensity can be extracted from a risk-optimized fragility using   =   ∕4 or the equivalent design point, (|  ) = 1 × 10 −3 .This fragility is compared to the ASCE7 fragility in Figure 3. Recall that ASCE7's (|  ) = 10% and  = 0.6 is equivalent to (|) = 2.5%.These assumptions are roughly equivalent to  = 3.2.

Selecting a risk target
Having set the first two parameters (beta and the design point), the risk target is a free parameter, as described previously for the original implementation in ASCE7.As in that study, the risk target is informed by the risk for the APoE value associated with the traditional design intensities.Typically, the free parameter is selected to match a metric of interest for one period and one soil condition.For example, in a previous study 8 by the same authors (in which the risk target was selected first) the design point was optimized to match the APoE of 1/500 for SA(0.5) and Site Class C, the underlying parameters that define New Zealand's current design spectrum. 21(The current site classes are no longer included in the new National Seismic Hazard Model (NSHM), which now defines the soil conditions based on the average shear wave velocity over the top 30 m,  30 . 22) In contrast, this study selects the risk target based on analysis across multiple periods and  30 .This approach accounts for variable risk across these parameters, which change the shape of the hazard curve and increase the risk for hazard curves with long tails.For example, Auckland is a lower hazard region that is far away from New Zealand's most active faults.This means that short period ground motions from distant subduction earthquakes are dissipated before they reach Auckland.At longer periods, however, the energy from distant earthquakes can reach Auckland, increasing the low-probability tail of the hazard curve and the associated risk.Similarly, soft soils with low  30 values cannot sustain high ground motion intensities without nonlinear behavior, which decreases the low-probability tail of the hazard curve as compared to stiffer soils.This effect of nonlinearity in soft soils is compounded for short period ground motions.Furthermore, the lognormally distributed uncertainty in the Ground Motion Models (GMM), sigma, also increases with period and Vs30, which inherently increases the hazard curve.The NSHM uses GMM that include these effects. 22This study harnesses the NSHM to ensure that the risk target selection considers a range of periods and site conditions, not just a reference value.Therefore, the risk, AIFR, was evaluated at a variety of periods,  30 , and locations across New Zealand, using the mean APoE of 1/500 as the design intensity,   .
The periods ranged from 0.5 to 5 s, with the intent of focusing on the velocity and displacement-controlled portions of the design spectrum, that is, the descending branch.This preference for matching the risk over the descending branch was due to short period truncations that the SRWG was also considering, as is typical in other international codes (e.g., ASCE7).The  30 values were  30 = 200, 250, 300, 350, 400, 450, 750 m/s.The selected sites were Auckland, Blenheim, Christchurch, Dunedin, Gisborne, Greymouth, Masterton, Napier, Nelson, Queenstown, Tauranga, and Wellington.The results are shown in Figures 4 and 5.
The figures demonstrate that the uniform-hazard approach results in variability in risk across sites, as expected due to the use of a single point on the hazard curve, rather than the full range of shaking intensities.There is also variability across periods and  30 values.The black markers show the average for each site, across all the periods and  30 values listed in the legends.The black line shows the average of the black markers across the 12 sites.The distance between the black markers and the corresponding black line shows the between-site variability.The colors in Figure 4 delineate between periods, showing the risk for the period, averaged over the  30 values.The corresponding line (e.g.purple for T = 2 s) is the average across the 12 sites.Figure 5 is the equivalent, showing the risk for each  30 value, averaged across the periods.Note that sites with an overall risk that is higher than the average tend to have higher risk for all periods and  30 values as well.That is, a site with a black marker that falls above the black line will also have a purple marker that falls above the purple line.The order of the colored markers in Figures 4 and 5 is relatively consistent, as the APoE of 1/500 risk increases with both  and  30 , due to the impact of these parameters on the shape of the hazard curve.Specifying design intensities based on a risk target results in uniform risk across the country (assuming the design fragility).For example, the risk could be set at the average value for the APoE of 1/500 design intensities (the black line in Figures 4 and 5).This selection would mean that for any site with a APoE of 1/500 risk that is higher than the selected value, the risk would be reduced via an increased risk-targeted design spectrum (and the equivalent APoE value will be lower).The reverse would be true for sites with a APoE of 1/500 risk that is lower than the selected value.Furthermore, the risk would also become uniform across periods and  30 values.These adjustments would increase the risk-targeted spectra at long periods, while decreasing it at short periods (pulling each colored line towards the black lines in Figure 4).Similarly, the risk-targeted spectra would increase for high  30 values and decrease for low  30 values.
Based on this preliminary risk assessment,  = 1.5 × 10 −5 was selected as comparable to the risk for an APoE of 1/500.(This is equivalent to an annual collapse rate of 1.5 × 10 −4 or ∼ 0.7% in 50 years, just under ASCE7's 1% target.)Three other risk targets (AIFR = 0.1, 0.5, and 5 ×10 −5 ) were also selected for comparison with the uniform-hazard approach.Figure 6 shows the resulting spectra for four major urban centers, with  30 = 250 and 400 m/s on the top and bottom rows, respectively.The 1.5 × 10 −5 risk target (green line) tends to match the APoE of 1/500 uniform hazard spectrum at ∼1.5-3 s.The design intensities for shorter periods are lower than the APoE of 1/500, due to the relative shape of the hazard curves, as discussed above.
The spectra shown in Figure 6 are all based on the mean hazard curve as the central estimate of the NSHM.This is consistent with recommendations, 23 though some have raised concerns about using the mean. 24A previous study by the authors 8 addressed this concern by using the full distribution of hazard curves to produce a distribution of risk-targeted design intensities, from which a single intensity could be sampled based on a metric of interest (e.g., the mean or a percentile).There is value in such an approach, particularly for selecting a non-central metric based on risk tolerance.However, this study includes the NSHM's epistemic uncertainty in the subsequent risk assessment (Section 3), rather than in the initial calculation of design intensities.
It is worth noting the high AIFR risk target (1.5 × 10 −5 ) as compared to a previous study by the authors, 8 which took the approach by Douglas et al. and Silva et al. with regards to the parameter selection process.In that study, the risk target was set to 1 × 10 −6 , for consistency with wider studies on acceptable levels of fatality risk (e.g., Tsang et al. 25 ).The beta value was taken as 0.6, in keeping with other risk-targeted studies and the preference for a higher beta value to account for the uncertainty in design requirements, test data, and modeling espoused in FEMA P-695.Having fixed the risk target and beta value, the design point was taken as (|  ) = 1 × 10 −4 to produce a design spectrum similar to an APoE of 1/500.When paired with  = 0.6, this design point is equivalent to  ≈ 9.As this CMR is at the upper bound of expected building response, the current study took the alternate approach of setting the risk target based on the design fragility, rather than the reverse.
Regardless of the process for selecting a design fragility and a risk target, the resulting design intensities for both studies were constrained by the objective of roughly matching the APoE of 1/500.For example, if the aggregated fragility had been selected ( = 6 vs.  = 4), the risk target would be 5 × 10 −6 , with roughly similar design intensities.One benefit of the higher risk target is that the risk-optimized fragility is less sensitive to the low probability tail of the hazard curve, with the bulk of the risk disaggregation occurring between APoE of 1/500 and 1/10,000 (see Figure 1).As will be discussed in the next section, the specified risk target is less important than the resulting risk distribution, as the distribution can be compared to guidance on tolerable risk ranges.

AS-BUILT RISK ASSESSMENT
The as-built risk assessment extends the traditional risk-targeted framework to consider the impact of uncertainties in the characterization of building response fragilities and in the hazard curves.This is in contrast to other risk-targeted studies, in which the risk variability is implicitly accepted (in the form of a generic fragility), but is not explicitly quantified.This assessment step allows decision makers to consider the distribution of risk as part of a decision to adopt the risk-targeted design intensities.The potential as-built risk is assumed to have a range of collapse fragilities, with CMR and beta values sampled from the distributions shown in Figure 2.This represents aleatory variability, or the randomness associated with other design decisions that will result in a variety of building responses for the same design intensity,   .The epistemic uncertainty associated with the choice of the CMR and beta distributions and even the lognormal functional form of the fragility curve will be discussed in Section 4. A significant motivation for assessing a risk distribution is the New Zealand engineering community's growing focus on the importance of designing for uncertainty. 26The context for this shift has been the long-anticipated release of the new NSHM, which was published in October 2022. 22The new NSHM is similar to the United States's UCERF3 model 27 in using an "inversion recipe" of existing datasets for earthquake geology, geodesy, and earthquake catalogs to develop a range of possible models for the seismic sources that could rupture and cause shaking across the country. 28This range of models accounts for the scientific community's limited knowledge of the underlying earthquake phenomenon (epistemic uncertainty).Similarly, the NSHM includes a range of GMM that are used to quantify the probability of shaking intensities given the rupture of a seismic source. 29Taken in combination, these models result in 900k+ possible hazard curves, ( > im) (see, Equation 1).This addition of uncertainty in the hazard model has generated discussions about other sources of uncertainty that are relevant for design. 26The secondary risk assessment proposed here extends the risk-targeted framework to fit within these broader discussions among the New Zealand engineering community.
The procedure for setting the design intensities relied on a single conservative design fragility and a central estimate (the mean) of the hazard curve.The first step towards understanding the as-built risk is to re-assess the central estimate of the risk, using the aggregated fragility instead of the design fragility.Recall that in the absence of other information about the design of an as-built structure, the aggregated fragility ( = 6 and  = 0.44) is the central estimate of its collapse fragility.Using the aggregated fragility, the central estimate of the risk for any given building is ∼ 0.5 × 10 −5 , or one third of the risk target (black dot and dashed line, respectively, in Figure 7A).(Equivalently, a ∼ 0.  The next step is to incorporate the distribution of as-built fragilities.These fragilities are based on 5000 CMRs sampled from a normal distribution with a mean of 6 and a standard deviation of 1.5, then resampled if any are above or below the 3 ≤  ≤ 9 range, similar to the 100 samples shown in Figure 2A.These CMRs are randomly paired with 5000 beta values sampled from a uniform distribution ranging from 0.35 ≤  ≤ 0.45 (Figure 2B).(The number 5000 was chosen to be consistent with the number selected for the NSHM uncertainty, described below.)These 5000 as-built fragilities are convolved with the mean hazard curve to produce the risk distribution shown in Figure 7B's gray histogram.The mean (circle marker) is similar to the central estimate using the aggregated fragility.This assessment quantifies the range of risk around that central estimate, due to variability in design.The median risk (50 th percentile, diamond marker) is just below the mean, while the 90 th percentile (triangular marker) is roughly twice the mean.The curve plotted over the histogram is a PDF, spanning the 1 st and 99 th percentiles (i.e., ranging over the CDF for 1% ≤  ≤ 99%).The darker red at higher probabilities allows the shape of the PDF to still be perceived as a horizontal line in Figure 7A.Other studies have suggested characterizing the risk as a beta distribution. 30In this study, the addition of the NSHM's epistemic uncertainty in Figures 7C,D resulted in a more concave distribution, which the beta distribution did not fit well.No single distribution type fit the data well in all cases so the figures show a smoothed empirical PDF, using a Savitzky-Golay filter with a 1-D polynomial and a 20 point filter window. 31ather than adding the hazard uncertainty directly, Figure 7C isolates the uncertainty due to the NSHM, allowing the resulting distribution to be compared with that of the fragility uncertainty.Therefore, this assessment uses only the aggregated fragility.There is global debate over the nature of epistemic uncertainty in a hazard model, particularly whether the mean hazard is the "true" hazard or whether there is value in considering the broader distribution via a suite of F I G U R E 8 An uncertainty matrix for the distribution of the as-built risk for structures with 0.5, 1.5, and 3.0 s periods.The grouped vertical lines at each period of interest show the 1  to the 99 ℎ percentile of the risk distribution, equivalent to the horizontal lines in Figure 7A.The left red lines are the distribution for the sampled as-built fragilities, evaluated for the mean hazard.The center blue lines are the distribution for the aggregated fragility, evaluated for the sampled hazard curves.The right purple lines are the distribution for the sampled as-built fragilities, evaluated at the sampled hazard curves.The diamonds, circles, and triangles mark the median, mean, and 90 ℎ percentiles, respectively.The upper row (A-D) shows results for  30 = 250 m/s while the lower row (E-H) shows  30 = 400 m/s.The columns show each of four major urban centers: Auckland, Christchurch, Dunedin, and Wellington.hazard curves representing the potential variability in the model. 23,32As mentioned regarding the engineering community's growing focus on uncertainty, there is strong incentive in the New Zealand context to consider the broader distribution.Conceptually, the assessment could use every branch of the NSHM logic tree (900k+ model combinations).However, due to resource constraints, 5000 branches are randomly sampled based on the relative weights of each branch.This sampling method produces a good representation of the underlying distribution, without needing to account for branch weights in later steps.An alternative is to sample randomly but keep the weights of the individual samples, which are then used to adjust the sample to represent the underlying weighted distribution.The two sampling methods were tested for accuracy in representing the underlying distribution's mean hazard curve, recognizing that this metric would be less sensitive than the percentiles.The sample sizes included 50, 100, 500, 1000, 5000, and 10,000, with two sampling attempts for each size.The sampling method based on weights produced a stable, accurate representation of the mean with as few as 100 samples.The sampling method with subsequent reweighting was less stable, requiring 5000 samples before the two sampling attempts produced identical means.(Note that these tests were performed for Wellington, with (1.5)and  30 = 250 m/s.More work is needed to confirm the results for other locations, periods, and site conditions.)Having selected the weighted sampling method, the sample size was tested for stability in the standard deviation of the resulting risk distribution.By 5000 samples, the standard deviation was stable to within two significant digits.Figure 7C shows the resulting risk distribution, with a similar mean and 90 th percentile as compared to the distribution for the fragility uncertainty in Figure 7B.However, the median is reduced significantly, due to more results with  < 0.1 × 10 −5 .
Finally, both sources of uncertainty are combined in Figure 7D, randomly pairing the 5000 sampled fragilities and hazard curves.Once again, the mean hovers around 0.5 × 10 −5 , though the spread increases.The median risk is similar to that of the distribution due to the hazard uncertainty in Figure 7C, again due to a significant number of results with less than 0.1 × 10 −5 .On the other hand, there are also more results with high risk, as seen by the 90 th percentile reaching the risk target of 1.5 × 10 −5 .
These plots for variability due to fragilities versus hazard curves were visually reconfigured in Figure 8 to show results for Auckland, Christchurch, Dunedin, and Wellington.The rows are for  30 = 250 and 400 m/s, respectively.The x-axis of each plot reflects the period, with each grouping of vertical lines at  = 0.5, 1.5, and 3.0 s equivalent to the set of horizontal lines in Figure 7A.In all cases, the risk distribution for uncertainty in both the as-built fragilities and the NSHM epistemic uncertainty (purple lines) has the widest spread, though the mean value remains similar across all three combinations.Moreover, the mean value is around 0.5 × 10 −5 for all sets of location,  30 and period.
The risk distributions in Figure 8 provide more context for evaluating whether the risk is acceptable.Although New Zealand has no regulatory framework for risk tolerance, there are a number of existing risk tolerability guidelines in use that can be compared to the risk distributions.Because there is no national framework, various sectors and authorities in New Zealand have defined their own risk tolerance levels.This include land-use planning guidelines by the Otago Regional Council 33 and the Bay of Plenty Regional Council. 34These guidelines both consider a range, where the risk is tolerable considering the cost of further reductions.The Otago guidelines for new development considers less than 1 × 10 −6 "acceptable" while up to 1 × 10 −5 is "tolerable".The Bay of Plenty guidelines suggest 1 × 10 −5 and 1 × 10 −4 for the tolerable range.
The concept of a tolerable range is reminiscent of the assessed distribution of risk, where the intent is to shift the majority of the distribution toward the lower bound, even if higher risk is still tolerable.The risk assessment provides insight into the potential spread of risk, which supports decisions around setting the design intensities.The mean risk (circle markers in Figure 8) is close to 0.5 × 10 −5 while the 90 ℎ percentile is around 1.5 × 10 −5 , or the risk target for the conservatively selected design fragility.This risk distribution falls within the tolerable risk ranges from land-use planning guidelines in New Zealand, as described above.However, the risk distribution could be shifted (e.g., to keep the upper tail within Otago's tolerable range of 1 × 10 −6 to 1 × 10 −5 ) by setting the risk target based on public policy-related selection criteria, rather than the objective of matching the APoE of 1/500 design spectrum.These criteria could be informed by societal expectations 35,36 and consider aggregated risk in dense urban areas. 8,37

EPISTEMIC UNCERTAINTY IN AS-BUILT RESPONSE FRAGILITIES
The previous section highlights the value of assessing risk with respect to the variability in as-built fragilities.However, only the aleatory variability was considered, that is, the randomness that is due to the wide variety of designs that could be based on the same design intensity.It did not consider epistemic uncertainty in how to model that aleatory variability.So far, the assumptions were based on a CMR range with a truncated normal distribution.The CMR values sampled from this distribution became the medians of lognormal fragility curves.This section will explore the risk distribution's sensitivity to these assumptions via perturbations to the shape of the distribution and to the functional form of the fragility curve.

Epistemic uncertainty in the distribution of collapse margin ratios
The assumed range of CMR values is one of the fundamental decisions affecting the risk distribution range for as-built fragilities shown in Figure 7B.The results shown in that figure were sampled from a truncated normal distribution with a mean of 6, standard deviation of 1.5, and resampling for any values below 3 or above 9, as described in the previous section.There are a variety of perturbations that could be applied to consider the sensitivity to these assumptions, which were loosely adopted to represent the minimal data for New Zealand buildings.For example, the range could be amended, especially considering the fact that ASCE7's equivalent fragility is roughly  = 3.2 and that the CMR discussion did not explicitly consider the impact of long duration shaking, variability in structural ductility, or design/construction errors.However, in the absence of alternate recommendations for an appropriate range, this sensitivity study focuses on the shape of the distribution, while maintaining the same 3 ≤  ≤ 9 range.The shape of the original distribution is plotted in red in Figure 9A, along with two alternate distributions.A uniform distribution (green) is the simplest option for a distribution ranging between two end points.A triangular distribution (orange) could represent the as-built distribution if structures tend to converge to the minimum conforming design.
The results for the truncated normal distribution are shown again in Figure 9B.The alternate risk distributions are show in Figures 9C,D .The color coded dashed lines show the risk for the aggregated fragilities that were reassessed for each CMR distribution:  = 5.9 and  = 0.48 for the uniform distribution and  = 4.6 and  = 0.50 for the triangular distribution.As noted previously, this level of precision for beta may not be warranted and is only shown for comparison of the alternate distributions.The aggregated fragility for the truncated normal and uniform distributions results in a risk value between the median and mean values, suggesting that they can represent the central tendency of the distribution.However, the triangular distribution's aggregated fragility results in a risk value higher than the mean.This is likely due to the combined impact of the aggregated fragility's low median and high beta value, which increased with respect to the truncated normal distribution to account for the disparity between the bulk of the fragilities with low CMR values versus the few fragilities with high CMR values.This sensitivity study shows the significant impact of the assumed shape of the distribution for CMR values.Changes to the upper and lower bounds would have a similar impact.This result invites further New Zealand-specific research into the variable building performance among designs that comply with the same design intensity.

Epistemic uncertainty in the functional form
The lognormal functional form is widely accepted for a variety of uses in earthquake engineering, including FEMA P-695 for quantifying the risk of collapse for a variety of structural systems and FEMA P-58 for assessing building and component level performance. 9,38This two parameter form is particularly convenient when shifting the design fragility to achieve the risk target.In the context of the risk assessment, however, it is possible to test the sensitivity to this assumption by introducing other functional forms.This flexibility provided a convenient way to address a potential concern regarding the high intensity, low probability portion of the hazard curves, where the uncertainty is the largest.The optimized fragilities stretch across very high intensities, which can give the impression that the risk-targeted design intensities are sensitive to this portion of the hazard curve.To address this concern, the risk distribution was reassessed using truncated lognormal fragility curves to demonstrate the fact that the risk is driven by the low probability portion of the fragility, that is, the higher probability portion of the hazard curve.
A truncated lognormal fragility was introduced, with a maximum value of +1 standard deviation.The comparison between the fragilities' PDFs is shown in Figure 10A.The original form (red) is assymptotic, resulting in a CDF(Figure 10B) that extends infinitely.The blue PDF is truncated at +1 standard deviation and scaled such that the area under the curve still reaches 1.0 at the end of the distribution.This effectively means that an intensity that originally had an 84% probability of collapse now has a 100% probability (dashed lines in upper right of Figure 10B).In order to achieve this new distribution, however, the median and beta values must be adjusted.This is done via a first order approximation, starting with finding an adjusted input median such that the truncated median (where the blue CDF crosses (|) = 0.5) is still equivalent to the original median (red).Then the beta is adjusted to fit the less-than-median portion of the CDF (0%-50%) as closely as possible.
The red and blue risk disaggregations in Figure 10B show the impact of this truncation for the design fragility, assuming a design intensity for Wellington, (1.5)and  30 = 250 m/s.As expected, there is little difference in the shape of the disaggregation.In fact, the difference is more apparent in the 0%-50% portion of the CDF than in the truncated portion.This is due to the exponential nature of the hazard curve, which can only be viewed on a log-scale (left y-axis of Figure 10B).
The truncation's impact on the full risk distribution is shown in Figure 10C.Again, there is little difference in the shape of the distribution or in the median, mean, and 90 th percentile values.This demonstrates the that insensitivity to the upper tail of the fragility curve is consistent across all the sampled fragilities, regardless of whether the fragility is steep (e.g., low CMR with low beta) or shallow.
A truncation on the lower tail of the fragilities would have a greater impact.As can be seen in Figure 10B, the lower tail of the risk disaggregation dips well below the design intensity.This is driven by the assumed lognormal functional form, rather than empirical evidence of collapse probabilities for such low intensities.Cook et al. demonstrated that using a truncated lower tail better approximated the number of collapsed buildings following the 1999 Northridge earthquake. 39hile that study used a single assumption for the truncation, a risk distribution assessment could consider the effect of various truncation levels over the full range of as-built fragilities.Such a study would indicate how much the assumption of a lognormal fragility form affects the perceived risk distribution.

CONCLUSIONS
The risk-targeted framework has proven to be a valuable tool for decision-makers as they seek to establish design intensities that produce more uniform risk across a country.The existing procedure assumes a generic building response, as a necessary simplification to reduce the burden of building-and site-specific assessment in the design process.Yet this simplification also limits the decision-makers' ability to assess the potential distribution of risk associated with the selected design intensity, due variability among the responses of as-built structures.The proposed extension described herein considers this risk distribution, allowing decision-makers to take a broader view, including comparisons with tolerable risk ranges instead of a single risk target value.This extension to consider a risk distribution in the decision-making process is particularly suited to New Zealand, where the engineering community is embracing the concept of designing for uncertainty.This shift is partly in response the new NSHM, which now includes epistemic uncertainty to explicitly account for the lack of knowledge in the models for potential rupture scenarios and the resulting probabilities of shaking intensities.This addition to the hazard model has also led to conversations about other sources of uncertainty.This study caters to those conversations by assessing the risk distribution that could be expected from risk-targeted design, considering variability among building performance and uncertainty within the hazard model.
One source of uncertainty is the performance across a variety of buildings that comply with the same design intensity.This uncertainty was represented by a distribution of collapse fragilities.In this study, the median of the collapse fragilities were defined by a collapse margin ratio with respect to the design intensity,  = ∕  .These values were sampled from a truncated normal distribution over the range ≤ CMR ≤ 9.Each fragility's uncertainty around the median (the logstandard deviation, beta) was sampled from a uniform distribution over the range 0.35 ≤ beta ≤ 0.45.While this distribution of fragilities represented the variability of potential building performance of as-built structures,  = 4 and  = 0.45 was selected as a single, generic design fragility for the risk-targeted procedure.
The risk target was selected to match the current design approach's uniform hazard spectrum with an APoE of 1/500.This selection process considered a variety of periods and site conditions, both of which affect the risk due to their influence on the low-probability tail of a hazard curve.An AIFR target of 1.5 × 10 −5 was adopted for producing the design intensities, assuming the design fragility and the mean hazard curve.The risk distribution was then assessed for aleatory variability in the as-built fragilities and epistemic uncertainty in the hazard curves.This distribution ranged from less than 0.1 × 10 −5 to over 3.0 × 10 −5 , with the mean value hovering around 0.5 × 10 −5 and the 90 th percentile around 1.5 × 10 −5 .This risk distribution falls within the assorted tolerable risk ranges from land-use planning guidelines in New Zealand.
Finally, the epistemic uncertainty of the models that represented the as-built fragilities was addressed via sensitivity analysis.Given that the as-built fragility concept is new to this study, the truncated normal distribution for the CMR values serves as a placeholder.Two alternative distributions (a uniform and a triangular distributions) were also assessed to test the sensitivity to these assumptions and highlight the need for more formal studies on the expected range of performance for New Zealand buildings.The sensitivity to the functional form of the fragility curve can also be tested, as demonstrated with a truncated fragility.
The assumptions used in this study represent a practical application of the proposed extension to the risk-targeted framework.These assumptions, which can be refined with further study, include the CMR of the design fragility, the approach for selecting the risk target (matching the APoE of 1/500), the range of CMR values for the as-built fragilities, the shape of the CMR distribution, and the functional form of the fragility curve.The extension to the risk-targeted framework presented in this paper supported the Seismic Risk Working Group's decision-making process for establishing design intensities by framing the discussion around risk distributions and tolerable ranges.

A C K N O W L E D G M E N T S
The authors would like to thank the other members of the Seismic Risk Working Group (funded by the Ministry of Business, Innovation & Employment) for discussions on the design versus as-built fragilities and other aspects of the risk-targeted framework discussed herein.The SRWG deliberations were still in progress when this paper was submitted.The first author would also like to thank Adam Zsarnóczay for countless conversations about epistemic uncertainty.Chris DiCaprio and Chris Chamberlain provided the data from the New Zealand National Seismic Hazard Model.
Open access publishing facilitated by The University of Auckland, as part of the Wiley -The University of Auckland agreement via the Council of Australian University Librarians.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

F I G U R E 1
Framework for quantifying risk: (A) Hazard curves for Haast, Otira, and Wellington (blue, orange, and green respectively).The black diamond marks the design intensity for an APoE of 1/500, which is consistent for all three locations.The black fragility curve corresponds to the right vertical axis, with a black dot at the median.The upper edge of each shaded region is the risk disaggregation, that is, the integrand of the risk equation for each hazard curve.The area under these curves (i.e., AIFR) is quantified in the legend.(B) The adjustment process for achieving a risk target of 1.5 × 10 −5 .The black hazard curve is for Wellington.The red fragility curve and risk disaggregation assume the APoE of 1/500 design intensity (red diamond) and produce an AIFR of 1.1 × 10−5.The purple diamond is the risk-targeted design intensity, resulting in the purple fragility curve and risk disaggregation with an AIFR of 1.5 × 10 −5 .AIFR, individual fatality risk; APoE, annual probability of exceedance.

F I G U R E 2
Accounting for variability in the performance of as-built structures, given a design intensity.(A) A histogram of sampled collapse margin ratios (CMRs, grey) using a normal distribution with a mean of 6 and a standard deviation of 1.5 and resampling below 3 and above 9. (B) A histogram of sampled beta values, using a uniform distribution between 0.35 and 0.45.(C) Sampled fragility curves (grey), given the CMR and beta values from (A) and (B).The vertical line at 1 represents the design intensity.By definition, the median of each curve crosses the horizontal (|) = 0.5 line at the CMR value.The solid black line represents the selected fragility curve for setting the design intensity, assuming close to minimum compliance.The dashed line is the aggregated fragility and, in the absence of any other information, represents the central estimate of a design's as-built fragility.CMR, collapse margin ratio; IM, intensity measure.maximum of 9.This distribution is depicted in Figure2A, with the normal distribution (solid black line) centered around 6 (short dashed line) and the resampling limits (tall dashed lines).The grey histogram shows 100 CMR values sampled from this distribution.

F I G U R E 4
Risk assessment based on design intensities for an APoE of 1/500.The black dots for each site are the average across eight periods 0.5 ≤  ≤ 5 s and the seven  30 values in the Figure 5 legend.The black line is the average across all sites and  30 values, considering all periods.The colored dots and lines represent the same for individual periods.APoE, annual probability of exceedance.

F I G U R E 5
Risk assessment based on design intensities for an APoE of 1/500.The black dots for each site are the average across the seven  30 values listed in the legend and the eight periods in the Figure 4 legend.The black line is the average across all sites and periods, considering all  30 values.The colored dots and lines represent the same for individual  30 values.APoE, annual probability of exceedance.F I G U R E 6 Comparison of spectra using the uniform-hazard approach for APoE of 1/500 (black) versus the risk-targeted approach for four risk targets.The 1.5 × 10 −5 target (green) is the proposed risk target for closely matching the APoE of 1/500 spectrum.The top and bottom rows show  30 = 250 and 400 m/s, respectively.The columns are for each of four major urban centers: Auckland, Christchurch, Dunedin, and Wellington.APoE, annual probability of exceedance.

F I G U R E 7
An uncertainty matrix for the distribution of the as-built risk for a structure with a 1.5 s period at a Wellington site with  30 = 250 m/s, considering the separate and combined impact of two sources of uncertainty: (A) a summary of the risk distributions shown in the other subplots.The black dot is the risk for the aggregated fragility, evaluated for the mean hazard.The vertical dashed line is the risk target, assuming the more conservative design fragility.The three horizontal lines are top down views of the probability distributions shown in (B-D), with the median, mean, and 90 ℎ percentile marked by a diamond, circle, and triangle, respectively.(B) The risk distribution for the sampled as-built fragilities, evaluated for the mean hazard.The gray histogram shows the risk for all the sampled fragilities, while the red line is a smoothed empirical PDF spanning the 1  to the 99 ℎ percentile of the data.The red hue increases with the height of the PDF and is reflected in the hue of the corresponding red horizontal line in (A).(C) An equivalent histogram and empirical PDF for the aggregated fragility, evaluated for 5000 sampled hazard curves.(D) The histogram and PDF for the sampled as-built fragilities, evaluated for the sampled hazard curves.PDF, probability density function.

F
I G U R E 9 A sensitivity study for the impact of the distribution of CMR, for as-built fragilities on the resulting as-built risk distribution.(A) Three alternate distribution shapes for the same range of 3 ≤  ≤ 9. (B) The truncated normal CMR distribution, with a smoothed empirical probability density function (PDF, red) fitted to the gray histogram.The diamond, circle, and triangle mark the median, mean, and 90 ℎ percentile, respectively.The red dashed line represents the risk for the aggregated fragility, while the black dashed line is the risk target for the design fragility.(C) An equivalent histogram and PDF for the uniform CMR distribution.(D) An equivalent histogram and PDF for the triangular CMR distribution.CMR, collapse margin ratios; PDF, probability density function.

F
I G U R E 1 0 A sensitivity study for the impact of a fragility with a lognormal functional form versus a truncated lognormal functional form: (A) The PDF(the derivative of the fragility curve) of a lognormal distribution (red) with a CMR of 4 and a beta of 0.45.The PDF of a truncated lognormal distribution (blue) with an adjusted CMR and beta to match the lower half of the distribution, given the truncation at +1 standard deviation.(B) The fragility curves (CDFs) associated with the PDFs shown in (A).The black line is the hazard curve, with the design intensity marked at the vertical dashed line.The shaded areas are the risk disaggregation corresponding to the red and blue of the fragility curves.(C) The risk distribution (shown via smoothed empirical PDFs) for the two functional forms, considering the distribution of as-built fragilities.The diamonds, circles, and triangles mark the median, mean, and 90 ℎ percentiles, for each functional form.CDF, cumulative distribution function; CMR, collapse margin ratios; PDF, probability density function.