Mountain roads are particularly prone to natural hazards, and consequently, risk assessment for road infrastructure focused on a range of different hazard processes, such as landslides (Benn, 2005; Schlögl et al., 2019), rockfall (Bunce et al., 1997; Hungr and Beckie, 1998; Roberds, 2005; Ferlisi et al., 2012; Michoud et al., 2012; Unterrader et al., 2018) and snow avalanches (Schaerer, 1989; Kristensen et al., 2003; Margreth et al., 2003; Zischg et al., 2005; Hendrikx and Owens, 2008; Rheinberger et al., 2009; Wastl et al., 2011). These studies have in common that they exclusively address the interaction of individual hazards with values at risk of the built environment and/or of society and use qualitative, semi-quantitative and/or quantitative approaches. However, there is still a gap in multi-hazard risk assessments for road infrastructure. The article provides a comparison of a standard (deterministic) risk assessment approach for road infrastructure exposed to a multi-hazard environment with a probabilistic risk analysis method to show the potential bias in the results. The multi-hazard scope of the study is based on a spatially oriented approach to include all relevant hazards within our study area. Using this approach, we address the consequences of multi-hazard impact on road infrastructure and compare the monetary loss of the different hazard types. The standard framework from ASTRA (2012) for road risk assessment is based on a deterministic approach and computes road risk based on a variety of input variables. Data are generally addressed with single values without considering potential input data uncertainty. We used this standardized framework for operational risk assessment for roads and transportation networks and supplemented this well-established deterministic method with a probabilistic framework for risk calculation (Fig. 1). A probabilistic approach enables the quantification of epistemic uncertainty and uses probability distributions to characterize data uncertainty of the input variables, while a deterministic computation uses single values with discrete values without uncertainty representation. While the former calculates risk with constant or discrete values, ignoring the epistemic uncertainty of the variables, the latter enables the consideration of the potential range of parameter values by using different distributions to characterize the input data uncertainty. Our study focuses on the epistemic uncertainty of the risk terms (exposure situations, vulnerability factors and monetary values), ignoring potential sources of variation within hazard analysis. Thus, the probability of occurrence of the hazard event was not assessed in a probabilistic way. Since deriving the likelihood of occurrence as part of the hazard analysis is crucial for risk analysis, a large source of uncertainty is attributed to this factor (Schaub and Bründl, 2010).

The study area is located in the Eastern European Alps, within the federal state of Salzburg, Austria (Fig. 2). The case study is a road segment of the federal highway B99 with an overall length of 2 km ranging from km 52.8 to km 54.8 and is endangered by multiple types of natural hazards. The road segment was chosen to demonstrate the advantages of using probabilistic risk approaches in comparison to traditional deterministic methods. The mountain road under examination is part of a north–south traverse over the main ridge of the Eastern European Alps and is therefore an important regional transit route. Furthermore, the road provides access to the ski resort of Obertauern.

As shown in Fig. 2, the road segment is affected by three avalanche paths, four torrent catchments and one rockfall area. The four torrent catchments have steep alluvial fans on the valley basin. The road segment is located at the base of these fans or the road is slightly notched in the torrential cone and passes the channels either with bridges or with culverts. The rockfall area is situated in the western part of the road segment. Approximately two-thirds of the study area is affected from rockfall processes either as single blocks or by multiple blocks.

The road is frequently used for individual traffic from both sides of the alpine pass. Hence, a mean daily traffic (MDT) of 3600 cars is observed. This constant frequency represents the standard situation for the potentially exposed elements at risk. However, especially in the winter months the average daily traffic can considerably increase up to an amount of about 7000 cars. Thus, the traffic data underlie short-term daily and longer-term seasonal fluctuations with peaks up to double of the mean value. The importance of dynamic risk computation needed for traffic corridors was also discussed earlier by Zischg et al. (2005) and Fuchs et al. (2013) with respect to the spatiotemporal shifts in elements at risk. Besides its use as a regional transit route, the road is also a central bypass for one of the main transit routes through the Eastern European Alps. Hence, any closure of this main transit route (A10 Tauern motorway) results in a significant increase of daily traffic frequency up to a total of 19 650 cars. The evaluation of the dataset in terms of the bandwidth of the traffic data is shown in Table A6.

In Table 3 the results for each risk group (RP, RA and RD) as well as for the total multi-hazard risk RC calculated with the standard deterministic risk approach are shown and compared to those obtained by the two probabilistic setups using two different probability distributions (TPD and BPD). The results associated with the two distribution functions are displayed as a median value of the PDF to show their deviation to the outcome of the standard approach. Based on our case study, the road risk over all hazard types and scenarios (multi-hazard risk) with the deterministic approach results in EUR 76.0 thousand per year. The results with the probabilistic approach referring to the median of the PDFs amounts to a monetary risk of EUR 105.6 thousand per year (TPD) and EUR 90.9 thousand per year (BPD), respectively. Compared to the standard approach the median of the PDFs equals an increase of 38 % (BPD) and 19 % (TPD), depending on the choice of probability distribution to model the uncertainties of the input variables. Focusing on the 95th percentile (P95) of the results – non-exceedance probability of 95 %, shown in Fig. 3 – an increase of 79 % (TPD) and 46 % (BPD) to the deterministic result can be observed. Figure 3 illustrates, based on the Lorenz curves for the two distributions (TPD and BPD), the scale of deviation of the total multi-hazard risk RC within the probabilistic risk modelling and compared to the standard outcome. The graphs show the potential uncertainties of the risk computation, which can be covered by a suitable choice of a value-at-risk (VaR) level. For example, with a benchmark of the 95th percentile (P95), 95 % of the potential uncertainties within the risk calculation can be covered by using a probabilistic risk assessment approach. However, a suitable VaR level depends on the general safety requirement of the system as well as on the degree of uncertainty of the input variables.

Geological hazards (rockfall) contribute with a fraction of 7.8 % to the total risk (or, in absolute numbers, EUR 5.9 thousand, see Table 3) based on the deterministic model, which can be attributed to the relatively small importance in comparison to the other hazard types in the study area. Hydrological hazards pose the highest risk (50.5 %, or, in absolute numbers, EUR 38.4 thousand per year) previous to avalanche hazards (41.7 %, or, in absolute numbers, EUR 31.7 thousand per year). Overall, RP (44.9 %; EUR 34.1 thousand per year) has the highest share on the total multi-hazard risk narrowly followed by RA (38.9 %; EUR 29.6 thousand per year), both associated with direct damage. The hydrological hazards (predominantly debris flow processes) with a portion of 76.5 % or EUR 26.1 thousand per year have a disproportionately high share on RP due to the high-intensity hazard impact. Similarly, the semi-empirical lethality factors shown in Table A7 have high values (λD=0.8) just like the impact of rockfall on cars with a probability of death of λR=1.0. Thus, these event types yield in high monetary losses in contrast to snow avalanches with a lethality factor for a high intensity of λA=0.2. By modelling the hazard-specific lethality with probability functions a wider scatter can be achieved, but the effect still remains due to the heavy weight around the most likely value m. The indirect losses related to RD with a fraction of 16.3 %, or, in absolute numbers, EUR 12.4 thousand per year have a minor portion because this risk group is only relevant for snow avalanches.

The results related to our case study (Table 3 and Fig. 4) show that due to the shape and the mathematical definition of the distribution the TPD leads to the highest variation in the monetary losses. The boxplots in Fig. 4 display the results from the probabilistic simulation for the three risk categories (RP, RA and RD) and for the total hazard-specific risk (RC) relating to the three hazard types (Fig. 3a–c) and for the total multi-hazard collective risk (Fig. 4d) with respect to the measures of the central tendency of the PDF. The boxplot diagrams are thereby plotted against the deterministic value to show its position. The wide range of the distribution in RC is markedly caused by RP, which exhibits a broad bandwidth and a right-skewed distribution. Hence, unlike RA and RD, the physical injuries expressed as the economic losses of persons (RP) are responsible for the highest divergence to the standard approach and show a considerable scatter. The main causes for the striking deviations can be associated with the relatively high monetary value of persons which was modelled as discrete point value in combination with the fluctuations of the MDT and the variations of the hazard-specific lethality. The monetized costs for a statistical human life equal EUR 3 million (Table A7) and are based on a statistical survey of the economic expenses for a road accident in Austria (BMVIT, 2014). Although we ascribe this value to a high degree of uncertainty the valuation of the expenses for a statistical human life was not attributed to a probability distribution due to the case-study-specific fixed governmental requirements in Austria. The discussion of a monetarily evaluation of a human life is still ongoing across scientific disciplines using different economic approaches (e.g. Hood, 2017). Furthermore, the lethality factors also correspond to the high variation of RP which are seen as very sensitive parameters. Therefore, we encourage further research on hazard-specific lethality functions for road risk management either based on comprehensive empirical datasets or on representative hazard impact modelling. Due to the strong effect of RP on RC the results have to be carefully interpreted, as they are sensitive to the input variables. Therefore, the values in our case study especially the cost for human life cannot be directly transferred to other application without a detailed validation and verification of national regulations.

Apart from RP where the deterministic result is located below or near the 5th percentile of both PDFs, RA and RD are mostly within the interquartile range between the 25th quartile and the median compared to the standard approach (Fig. 4). In this context, RA for snow avalanche exceeds the median and is situated between the median and the 75th quartile. The effect can be traced back to the left-skewed distribution of the vulnerability factor vB,A for medium avalanche hazard intensities regarding the object class structures (bridges and culverts) in Table A8. In general, due to the shape and the mathematical characteristics of the distribution, the BPD leads to a stronger compaction around the median than the TPD which can be well explained by the properties of the BPD which has, in comparison to the TPD, a larger weight around the most likely value m.

In Fig. 5, the PDF and the cumulative distribution function (CDF) are shown for RC with the two probabilistic model results and the deterministic result. In both cases (TPD and BPD), the deterministic result is situated at the lower edge of the PDF near or under the 5th percentile. Thus, the deterministic result of our case study covers approximately less than 5 % of the potential bandwidth of the probability distribution. The TPD has a wide range, whereas the BPD is considerably flattened on the boundary of the amplitude. The results of the two distributions have in common that they are allocated right skewed. In contrast to the location of the median, the deterministic result is on the far-left side of both distributions and is exceeded by more than 95 % of the potential outcome.

The results based on our case study provide evidence that the monetary risk calculated with a standard deterministic method following the conventional guidelines is lower than applying a probabilistic approach. Thus, without consideration of uncertainty of the input variables risk might be underestimated using the operational standard risk assessment approach for road infrastructure. The mathematical product of the frequency of occurrence and the potential consequences with single values and, in a narrower sense, the multiplication of the partial risk factors in the second part of the risk equation may lead to a bias in the risk magnitude because the multiplication of the ancillary calculations generates a theoretical value ignoring the full scope of the total risk.

The far-left position of the deterministic value within the PDF of the probabilistic result in our study can be traced back to fact that the multiplication of two positive symmetrical distributions results in a right-skewed distribution because the product of the small numbers at the lower ends of the bandwidths results in much smaller numbers than the product of the high numbers at the upper ends of the bandwidths. When right-skewed distributions are used as an input and aggregated, the effect of skewness shifts the deterministic value (represented by the most likely value) to the left side of the resulting distribution. Even if conservative risk values are used in a deterministic setup, a potential scatter (upper and lower bounds) remains, which leads within a probabilistic calculation through aggregation of the partial risk elements and sub-results to a right-skewed distribution according to the skewness of input variables. Since risk values of our study are in most cases asymmetric with primarily positive skews, the deterministic result migrates during aggregation to the left side of the PDF in Fig. 5. The deterministic risk value is usually expressed either as a theoretical mean value or as the most likely value, neglecting the potential distribution functions of the input data. Thus, the compression of the input values to a single deterministic risk value with total determination prevents an actual prognosis of reliability that would have been achieved by specifying bandwidths (Sander, 2012). Furthermore, the simple summation of the scenario related and the object-based risk to receive the cumulative risk level instead of using probabilistic risk aggregation leads to an underestimation of the final risk. Hence, the full spectrum of risk cannot be represented with deterministic risk assessment, which may further lead to biased decisions on risk mitigation.

The value-at-risk (VaR) approach by considering a reliable percentile of the non-exceedance probability e.g. P95 as shown in Fig. 3 – depending on the desired coverage of the risk potential from society, authorities or organizations – might be an appropriate concept to tackle this challenge. In this context, a higher VaR value implies a higher safety level for the system under investigation. The final results of risk assessments are subject to uncertainties mainly due to insufficient data basis of input variables, which can be addressed using a PDF to represent uncertainties involved. For further decisions on the realization of mitigation measures a high VaR value such as P95 covers these uncertainties with a defined shortfall probability and thus supports decision-makers with more information about road risk. In turn, as a further practical improvement this benchmark can be compared to the same grade of safety for the costs of mitigation measures, since cost assessments for defence structures are also subject to considerable uncertainties. Thus, an optimal risk-based design of defence structures might encompass a balance between the same VaR level both of a probabilistic risk and a probabilistic cost assessment utilizing a cost–benefit analysis (CBA). However, within a probabilistic approach the scale of deviation is dependent on the choice of distribution for modelling the bandwidth of the variables, and the results are sensitive to the defined spectrum of input information stated in Tables A6–A9. These variables are case study specific and cannot be directly transferred to other road risk assessments without careful validation. However, probabilistic risk assessment (PRA) enables a transparent representation of potential losses due to the explicit consideration of the entire potential bandwidth of the variables contributing to risk. Since comparable results can be achieved based on predefined values (Bründl et al., 2009), we still recommend the consideration of the deterministic value as a comparative value to the probabilistic method.

Road risk assessment is usually afflicted with data scarcity; thus, risk operators and practitioners are often dependent on expert appraisals, which are subject to uncertainties. In order to improve data quality, upper and lower values and the expected value can be easily estimated for fitting a simple distribution of the input variables. Even though empirical values such as statistical data are available, a certain degree of uncertainty remains. Therefore, simple distribution functions such as TPD or BPD can adjust the shape of the distribution more conveniently than complex probability distributions, since the required additional parameters to adjust a complex distribution are simple not available. Hence, for a prognostic prediction, risk modelling with complex distributions in contrast to simple techniques cannot be justified if there is a lack of empirical data.

A limitation of our study is that the performance of the probabilistic approach cannot be verified and validated with empirical data, but the results show that the explicit inclusion of epistemic uncertainty leads to a bias in risk magnitude. The probabilistic approach allows for the quantification of uncertainty and thus enables decision-makers to better assess the quality and validity of the results from road risk assessments. This can facilitate the improvement of road safety guidelines (for example by implementing a VaR concept) and thus is of particular importance for authorities responsible for operational road safety, for design engineers and for policymakers due to a general increase of information for optimal decision-making under budget constraints. Furthermore, the paper addresses the second part of the risk concept in terms of the consequence analysis. The results of the hazard analysis serve thereby as a constant input using the physical modelling of the hazard processes without the consideration of probabilistic methods. Thus, the probability of occurrence of the hazard processes was mathematically processed as a point value within the probabilistic design, since the hazard analyses (with deterministic design events to assess the hazard intensities as a function of the return interval) was part of prior technical studies. Further considerations of a probabilistic modelling of the frequency of the events were outside of the study design and might be addressed in subsequent studies. Therefore, we expect a considerable source of epistemic uncertainty within the hazard analysis which emphasizes the necessity for the additional inclusion of probability-based hazard analyses in a holistic multi-hazard risk environment. Even though the presented methodology in this study focuses on a road segment exposed to a multi-hazard environment on a local scale, the approach can easily be transferred to other risk-oriented purposes.