Probabilistic modeling of vulnerability of road infrastructures to floods

Factors which contribute to the vulnerability of physical elements such as road infrastructures to a natural hazard such as a flood event are pervaded by uncertainty due to the complexity of the hazard, of the vulnerable infrastructure and of their physical interaction. In the context of risk management efforts, it is conceptually correct to explicitly address this uncertainty and to parameterize the criticality of the vulnerable element and, consequently, an explicit target degree of conservatism and reliability in risk assessment and mitigation strategies. This paper illustrates the results of the probabilistic characterization of the vulnerability of road infrastructures to flood events for two areas in South-Eastern Norway. Flood intensity and road vulnerability serve as inputs to an analytical model, which expresses the latter as a function of the former with respect to a user-set level of probability of exceedance. Deterministic and probabilistic vulnerability estimates are compared quantitatively, and the results are assessed and analyzed critically.


Introduction
The quantitative modeling of vulnerability is a central module in quantitative risk assessment of critical infrastructures to extreme weather events.Vulnerability is most often estimated deterministically, i.e., with no explicit modelling, processing and quantification of uncertainty.It is well known, however, that the factors which contribute to the vulnerability of physical elements such as road infrastructures to a natural hazard such as a flood event are pervaded by uncertainty due to the inherent complexity of the hazard, that of the vulnerable infrastructure and of their physical interaction.The deterministic approach to the quantification of vulnerability, in which uncertainties are not explicitly specified, processed and reported, thus hinders a comprehensive assessment of the fragility of the vulnerable element.Hence, such an approach may not provide optimal support for informed risk management purposes.
This paper illustrates the results of the probabilistic characterization of the vulnerability of road infrastructures to flood events for two counties in South-Eastern Norway.Damage surveys from flood events of 2011 are used jointly with peak 24-hour rainfall data to parameterize loss and hazard intensity, respectively.Vulnerability is defined quantitatively hereinafter as the ratio of measured economic loss to the total cost of reconstruction of the road infrastructures.Such definition is also valid for other definitions of vulnerability representing degree of loss.Intensity and vulnerability serve as inputs to a reference analytical vulnerability model, which expresses quantitatively the relationship between the two.
Model parameters are first calibrated deterministically using generalized least-squares regression.Subsequently, vulnerability is characterized probabilistically through quantile regression.Quantile regression (e.g.Yu et al. 2003) allows the calibration of model parameters with reference to a user-defined probability of non- exceedance.Through this approach, it is possible to calibrate model parameters corresponding to a desired level of conservatism in vulnerability estimates, thereby obtaining "characteristic" vulnerability functions.Such functions reflect the user-assigned qualitative degree of criticality of the vulnerable infrastructure, thereby allowing a full probabilistic flood risk assessment.

Quantitative vulnerability model
A vulnerability model is proposed in the following functional form: in which: -V ub is the inherent upper-bound vulnerability, i.e., the maximum value which vulnerability can take as a consequence of its definition.For example, when considering direct physical damage to critical infrastructure (CI) and loss is measured as repair cost, vulnerability can be given by the ratio of repair cost to replacement cost.In such case, V ub =1; -I is the intensity of the extrme weather event (EW E), efficiently parameterized in terms of an EWE's physical attribute (e.g., wind speed, rainfall intensity, etc.) or of the event's presumed return period.
-A is the location parameter, describing the value of intensity corresponding to the maximum vulnerability gradient.The location parameter also corresponds to the abscissa of the flex of the vulnerability function; -B is the brittleness parameter, describing the rapidity with which vulnerability increases with increasing EWE intensity.High values of B correspond to rapidly increasing vulnerability, whereas low values of B correspond to a gradual increase in vulnerability.
The vulnerability model in Eq. ( 1) allows for considerable flexibility in the intensity-vulnerability relationship, and is thus able to accommodate different behaviors through variations in the model parameters A and B. Fig. 1 illustrates comparatively two very different intensity-vulnerability relationships: vulnerability curve V1 displays a rapid increase for low intensity values, and a progressive decrease in gradient with increasing intensity.Vulnerability curve V2 corresponds to a scenario in which vulnerability remains low for low intensity values, then increases significantly for intermediate to high intensity values.
Though vulnerability can be conceptually defined as the expected degree of loss (damage), with respect to the maximum possible degree, suffered by one or more vulnerable elements as the consequence of the impact of a hazardous event with a given intensity level, the operational definition of vulnerability will vary depending on type of CI and the vulnerable element under investigation.Some examples of impact variables could be: x Down time: Down time of the CI could for example be normalised by the reconstruction time from scratch of the CI component that was impacted x Number of people losing CI service (for a longer duration than a threshold time period): Number of impacted people could be normalised by the number of people being serviced by the CI in question x Repair cost: Repair cost could be normalised by the cost of a full reconstruction of the CI in question x Degree of service loss of the CI.An example could be a road that has reduced capacity due to an EWE.
Figure 2a and Figure 2b illustrate qualitatively the effects of variations of the model parameters A (location) and B (brittleness), respectively, on vulnerability with respect to prior states A 0 and B 0 .
While increases in the location parameter A are univocally beneficial in terms of decreasing vulnerability for any given intensity level, a qualitative reasoning on the brittleness parameter B suggests that an increase in the brittleness parameter leads to a reduction in vulnerability for intensity values lower than the location parameters, while vulnerability increases for intensity values higher than the location parameter (see Figure 3).This non-uniform effect, in planning risk mitigation actions, to assess whether mitigation is envisaged primarily for low or high intensity ranges.This in turn relates to return periods, with lower intensities corresponding to lower return periods and higher intensities to higher return periods.Risk mitigation could thus be pursued quantitatively, with the aid of this model, using performance-based criteria for vulnerability mitigation and referring to specific life cycle durations for a CI.Target design vulnerability values could be set for given levels of expected intensity in order to pursue a rational cost-benefit analysis.The counties of Oppland and Hedmark in the eastern part of Norway experienced two very similar flood events in 2011 and 2013.They were both caused by snow melt in the mountains combined with intense rainfall.The descriptions of the two events are accounts taken mainly from [1][2][3][4].Figure 4 shows flood reports from the peak of the two events, indicating a large number of flood and landslide damages.

The June 2011 events
During the Spring of 2011, the Norwegian Water Resources and Energy Directorate (NVE) assessed the likelihood of a large spring floods as small because of relatively little snow in the mountains and early snow melt in the first half of April.But at the end of May there was still snow in the mountains, and in addition heavy rainfall saturated the ground and filled groundwater reservoirs.In the period 6th -12th June, significant rainfall occurred in several places simultaneously as high temperature led to strong snowmelt in the mountains.NVE issued the first flood warning on Monday 6 June, and upgraded the warning to "major flood" level on June 10.Several main rivers reached 100-year flood level, however it was the flood level in side tributaries and the large number of landslides that caused the most significant impacts.The main road through Østerdalen, Highway 3, was closed on the night of June 10.Subsequently, an increasing number of roads closed out overnight and on the morning of June 10 both in the counties of Hedmark and Oppland.Early on June 10 the police received the first reports of flood damage, closed roads and people who evacuated themselves.Throughout the day on June 10, it became clear that conditions stabilized in Hedmark.However, in Oppland and especially in the region of Gudbrandsdalen, flood levels increased and damages caused by floods and landslides were reported through the day.The European main road through Gudbrandsdalen, E6, was closed in one location due to debris flows.Eventually, it also closed at several places during the period 10-14 June.In the county of Oppland more than 30 roads were closed due to flooding and landslides.In Hedmark, up to seven roads were closed at the same time due to the flood.In addition, the north south railroad, Dovrebanen, was closed.More than 270 people were evacuated over from their homes, mainly in Oppland.In addition, an unknown number of people evacuated and accommodated themselves privately .Overall damage costs were estimated to 800 million Norwegian Kroner (NOK).

The May 2013 events
The winter and spring of 2013 were cold and relatively dry with mostly less snow than normal in Eastern Norway.On May 3, a flood alert was issued by the Norwegian Water Resources and Energy Directorate (NVE).Significantly warmer weather was forecasted   Other railroad lines were also closed for several days.The main road, E6, suffered damage along with 24 county roads and more than 20 minor roads, which had to be closed for a period.As was the case in 2011, the two main roads connecting north and south, Highway 3 and European road E6, were closed at several locations.The damage costs are estimated at a total of 1 200 million Norwegian Kroner.Peak 24 hour-rainfall was for both events in the order of 100 mm.A detailed account of the road damages after the 2011 event is given in [5].The total estimate is roughly 240 million NOK for the two counties of Oppland and Hedmark.The costs are given for individual road segments, as shown in Table 1.The roads segments included in Table 1 are typically 2-lane county roads (Fylkesveier) with width of 6.5 meters.To obtain dimensionless damages estimates, the damages are normalised by typical construction costs for Norwegian roads.Flood return period for adjacent rivers are extracted from [6].Examples of the road damages are shown in Figure 5.

Example case study
This Section details the result of the application of deterministic and probabilistic calibration of the vulnerability model presented in Section 2 to the case study described in Section 3 .

Definition and parameterization of intensity
In the present case study, intensity is parameterized as the base-10 logarithm of the estimated return period of the observed EWE: (2) in which T R is the return period of estimated river discharge (in yrs).The above model, albeit arbitrarily defined, is convenient in that it yields a non-negative, linear intensity scale (as long as T R >1 yr, which is most usual).Ouput intensity values are reported in Table 2.

Definition and parameterization of vulnerability
In the present case study, vulnerability is defined as the ratio between the cost of the damage induced by the EWE and the cost of reconstruction of the CI.Costs of road development for communal roads can be quantified tentatively at 50,000-90,000 NOK/m.For simplicity, an average deterministic value of 70,000 NOK/m is assumed.The total lengths of road segments were obtained from a GIS-based analysis.Reconstruction costs were obtained by multiplying the lengths of the road segments by the deterministic cost per kilometer.Vulnerability was then calculated as the ratio of damage to reconstruction cost.The parameter V ub was set equal to unity since the cost of complete reconstruction is an upper-bound value to the cost or repair, which quantifies damage in this example.Table 3 details the assignment of vulnerability values for the examined cases.The definition of vulnerability as given herein is arbirary and case-specific, and reflects the scope of the analysis and the goal to assess the magnitude of post-event reconstruction costs.

Deterministic model calibration
Deterministic calibration of the vulnerability model can be achieved by generalized least squares (GLS) regression.The regressed coefficients are estimates of central tendency, i.e., of a "mean" curve.Deterministic calibration yielded the deterministic model parameters A det =5.9 and B det =5.96. Figure 6 shows the deterministic vulnerability function and the source data.It is evident that deterministic calibration does not yield a conservative vulnerability function, as most source data points lie above the output vulnerability curve.

Probabilistic model calibration
The above result confirms that deterministic calibration is very often conceptually not adequate for engineering analysis and design purposes.As stated previously, deterministic regression yields estimates of mean functions, i.e., functions which tend to provide central estimates of relationships between independent and dependent parameters.In engineering analysis, it is most often necessary to operate with reference to a target degree of conservatism.Conservatism is related to safety and performance of engineering systems, and is an especially relevant concept in the context of critical infrastructures, for which reduced serviceability or ± worse ± collapse, are likely to result in unacceptable consequences and risk.The target degree of conservatism is expected to increase with the criticality of the vulnerable infrastructure under investigation, as the threshold of tolerable/acceptable risk decreases with increasing criticality.
Regression models are almost invariably simplifications and approximations of relationships between parameters.In other words, they are approximate, simplified models of the phenomena which they parameterize.The scatter of data points around regression models represent the indetermination in the phenomena and, in the case of the present analysis, of the domains in which intensity and vulnerability are parameterized.In quantitative terms, indetermination, vagueness and complexity are parameterized by uncertainty.A detailed treatment of uncertainty in the geosciences is not provided here.Readers are referred, for instance, to [7].In very general terms, uncertainties can be categorized (see e.g.[7]) into aleatory (representing the "true" variability in the parameters and phenomena under investigation) and epistemic (representing, among other things: the imperfections in measurements and estimates of the parameters; the imperfections in the models; the uncertainty in statistical estimates of model parameters resulting from the limited size of data sets).Aleatory uncertainty can possibly be reduced by varying the scale of investigation of the analysis.Epistemic uncertainty can be reduced, but never completely eliminated, by increasing the size and quality of data sets and by improving models).With reference to critical infrastructures, a tentative list of sources of uncertainty can be attempted for the various dimensions of vulnerability.Some examples are shown in Table 4.
Neglecting uncertainties leads to gross simplification of the systems under investigation, and hinders a comprehensive understanding of the sensitivity of a system to the (effectively existing) indetermination in its parameters and models.Moreover, a purely deterministic analysis impedes the rational assessment of risk and target level of reliability of a system.Hence, the uncertainty-based analysis of engineering systems is receiving increasingly focused attention, as attested by the development of reliability-based design codes and by the rise of systematic risk analysis approaches.
Uncertainty-based modelling of vulnerability is achieved here using quantile regression.Quantile regression is a type of regression analysis often used in statistics and econometrics.Whereas the method of least squares results in estimates that approximate the conditional mean of the response variable given certain values of the predictor variables, quantile regression aims at estimating either the conditional median or other quantiles of the response variable [8].Quantile regression is a most convenient and useful tool if confident estimates of conditional quantile functions are of interest.This is often the case in risk management, since conservatism and safety are conceptually related to high probabilities of non-exceedance, i.e., to high quantiles of vulnerability and risk as output variables of quantitative estimation frameworks.It is thus possible to define "characteristic vulnerability", analogously to characteristic values in modern design codes such as the Eurocodes, as values of a parameter or model explicitly related to a target probability of exceedance or nonexceedance.The selection of a characteristic value or function thus expresses the analyst's target degree of  conservatism, which is expected to increase with the criticality of the vulnerable infrastructure under investigation.
Quantile regression thus allows a direct estimate of characteristic vulnerability.Besides the full consistency with the fundamental concept of conservatism in engineering, quantile regression is beneficial in that it allows a more comprehensive analysis of the relationship between variables.Furthermore, quantile regression estimates are more robust against outliers in the response measurements relative to the ordinary least squares regression.
Consider the following general response model in which T is an 2-sized vector denoting the set of model coefficients [A,B], I is an N-sized vector of intensity values (independent variable) and V is an N-sized vector of vulnerability values (dependent variable).The pth regression quantile (0<p<1) is defined as any solution T p to the quantile regression minimization problem where the loss function for the i-th observation (i «N) is given by 2 1 2 and is the difference between the i-th observed vulnerability value and model-predicted value for the i-th intensity value I i .Quantile regression implements the minimization algorithm and yields model parameters which define the characteristic vulnerability curve for the p-th regression  Quantile regression was performed on the data for a characteristic quantile of 0.95. Figure 7 shows comparatively the deterministic and characteristic vulnerability curves, the latter referring to a quantile (probability of non-exceedance) of 0.95.The values of model parameters corresponding to the characteristic vulnerability curve are A QR =58.5, B QR =1.26.The vulnerability models are not static, and can be updated as new data become available, provided that data are homogeneous in terms of at least the operational definitions of intensity and vulnerability, and the reference area in which they are collected.This paper will hopefully serve as an example which may contribute to stimulate and facilitate a more systematic collection and organization of quantitative data regarding damage to critical infrastructures from hazardous events such as flooding, in the broader perspective of an increasingly objective and quantitative approach to risk analysis.

Figure 1 .
Figure 1.Comparative plot of two vulnerability functions with V ub =1 I max I min

Figure 2 .
Figure 2. Qualitative explanation of the effects of model parameters A and B on the vulnerability model

Figure 4 .
Figure 4. Flood reports from the peak of the two flood events in 2011 and 2013.The counties of Oppland and Hedmark were most impacted during both events.Source: www.varsom.no (a) Veikledalen in the municipality of Nord Fron, Gudbrandsdalen (b) Overview of Europe road E6 through Gudbrandsdalen.
The counties of Oppland and Hedmark were most impacted during both events.Source: www.varsom.nob A: Ditch; B: Landslide; C: Erosion; D: FLOODrisk 2016 -3 rd European Conference on Flood Risk Management