**3.3 Uncertainty quantification**

Both linguistic and numerical uncertainty quantifications approaches are used to analyze and assess the uncertainty in a given system. The quantification methods depend on the propagation of uncertainties in the system model and then assess the model output response due to this uncertainty propagation. The mathematical representations of the uncertainty are based on the use of probability, imprecise probability and possibility theories. For deterministic risk assessment, the uncertainty might be quantified either using a one factor at a time (OFAT) or multi- variant techniques. OFAT allows the change of one uncertain factor or parameter within a specified range with keeping the rest of the factors or parameters fixed [27]. This allows the examination of the effect of the factor variability/randomness/presence on a single process output or multi outputs. **Figure 3** illustrates the application of OFAT in assessing the risk of a system, where a single valued specified factor is propagated through the system, and the model outputs are quantified (**Figure 3a** and **b**). To quantify the uncertainty in the risk estimate of that system, discrete values or probabilistic uncertainty information of the uncertain parameter are propagated through the system which generates statistical information in the risk values for the uncertain parameter (**Figure 3a** and **c**). Different sampling methods could be used to represent the probabilistic information in that parameter, i.e. Latin hypercube sampling. OFAT does not allow the investigation of the interaction between uncertain parameters and their effect on the system output (s), nor allowing the determination of the outputs dependence [24, 26]. To overcome the latter, the parameters are often selected based on their ability to produce a conservative decision.

**Figure 2.** *Uncertainty classification according to Skinner et al. [19].*

**Figure 3.**

*Application of the OFAT approach in uncertainty quantification, (A) uncertainty propagation in the model, (b, c) modeling outputs for single parameter value (b) and uncertain parameter (c).*

The use of the multi-variant approach is adopted by varying the factors or parameters simultaneously and investigating their individual and combined effects on the process output. This approach is applied using statistical experimental design, (e.g. response surface methodology, Taguchi) which allows the development of regression models that correlate between the multi variant inputs and the process outcomes either for multi-variant – single objective or multi-variant – multi objective problem based [25, 27, 30–33]. Integrated tools were developed to quantify and assess the uncertainty in RA, an example of these tools is the Quantifying Margin and Uncertainty, which used to support the certification of the reliability and safety for a physical system and quantify the performance thresholds and their margins and the associated uncertainty in their evaluation. This tool widely used to quantify uncertainties that are dominated by lack of knowledge in risk-informed decision analysis [34].

#### **3.4 Sensitivity analysis to support the uncertainty management**

Sensitivity analyses are used as tools to reduce the uncertainty, where it is used to prioritize the research efforts to reduce uncertainty associated with the scenario, conceptual model, input data, modeling process, and the designed system [35]. Differential and probabilistic sensitivity analyses are used to support the uncertainty quantification and reduction. Differential sensitivity analysis is used when exact risk formula exists, this technique is computationally efficient; however, it is only valid in vicinity of the base case and might require intensive efforts to drive the sensitivity coefficients [35]. Probabilistic sensitivity analyses are conducted by assign probability density functions to each input parameter, generate an input matrix using suitable sampling method, calculate the outputs, and assess the influences and relative importance of each input/output relationship [36]. In probabilistic risk analysis, the marginal distributions of the studied parameters and the dependence between them need to be specified [36]. In this case, interval probability, Dempster-Shafe structure, and probability boxes are widely used approaches.

### **4. Conclusion**

In this chapter the approaches to manage uncertainty within the risk assessment framework to support the decision making process for pollution prevention *Introductory Chapter: Uncertainty Management to Support Pollution Prevention and Control… DOI: http://dx.doi.org/10.5772/intechopen.98465*

and control systems are introduced. In this respect, the risk assessment, its need, and approaches were introduced and discussed. The classification of sources and reducibility of uncertainty is presented. The approaches to quantify the uncertainty were overviewed with special reference to the role of the sensitivity analysis in uncertainty management.
