## 1. Introduction

[2] Sewer flooding caused by overloaded urban drainage systems is an unwelcome reality in many parts of the world, producing significant adverse economic, social, and environmental impacts. In the UK, for example, sewer flooding is considered to be the second most serious issue (after drinking water quality) facing water companies, with an estimated cost of 270 million GBP a year in England and Wales alone [*Parliamentary Office of Science and Technology (POST)*, 2007]. Further, there is an increasing probability of sewer flooding due to the expansion of urban areas and the likely adverse impacts of global climate change [*Ryu*, 2008].

[3] Most sewer systems have been designed on the basis of simple deterministic methods, such as the rational method or time-area method [*Butler and Davies*, 2004; *Thorndahl and Willems*, 2008]. These methods commonly use a design storm with a typical return period from 1 to 10 years to determine the maximum (minor) system capacity. However, the return period of sewer flooding is certainly not equivalent to that of the design storm, as the system capacity is increased so that sewers can accommodate a considerable surcharge before surface flooding occurs [*Butler and Davies*, 2004]. For existing sewer systems, flood frequency estimation can be further complicated with the issues of pipe deterioration and network expansion to new developments. Thus, the estimation of sewer flood frequency statistics for an urban catchment is of great interest in practice, as it provides direct assessment of hydraulic performance of the sewer system and supports decision making for sewer flood risk management [*Schmitt et al.*, 2004; *Ryu*, 2008].

[4] Handling uncertainty is a major issue in modeling water systems, including sewer systems, given the complexity and extent of uncertainty sources involved. This uncertainty has received increasing attention in recent years [e.g., *Guo and Adams*, 1998; *Adams and Papa*, 2000; *Matott et al.*, 2009]. Uncertainty can be broadly classified as stochastic or epistemic. Stochastic uncertainty refers to the randomness observed in nature, which is normally irreducible due to the inherent variation of physical systems. Epistemic uncertainty arises from incomplete knowledge about a physical system, which can be reduced with improved understanding of the system. Various approaches to characterize uncertainty are available, such as probability distributions, fuzzy sets, and random sets. Selecting an appropriate characterizing approach is essentially subjective as, in general, it is difficult to recommend one over another.

[5] Classical probability theory (Bayesian methods) has most often been used to quantify uncertainties in, for example, rainfall [*Guo and Adams*, 1998; *Thorndahl and Willems*, 2008], model parameters [*Lei and Schilling*, 1994], model structures [*Freni et al.*, 2009], and system dimensions such as storage volume and runoff basins [*Korving et al.*, 2002]. However, it is increasingly recognized that the concept of uncertainty is too broad to be captured by probability measures alone [*Ross et al.*, 2009]. So, for example, the theory of fuzzy sets [*Zadeh*, 1965] has also been increasingly applied, albeit in an attempt to describe imprecision and vagueness arising from the modeling process. Numerous applications can be found in hydrologic and hydraulic engineering [e.g., *Revelli and Ridolfi*, 2002; *Jacquin and Shamseldin*, 2007], including applications in the decision-making context of urban water management [e.g., *Makropoulos et al.*, 2003].

[6] It is not uncommon to have to address different uncertainty types simultaneously in the modeling and decision making processes. For example, some uncertainties are represented by probability distributions when sufficient data is available, while others are better represented by fuzzy sets to capture linguistic expert knowledge (qualitative data). Effort has been made to accommodate both probabilistic and fuzzy uncertainties in a unified framework of uncertainty analysis. The straightforward way is to transform one type of uncertainty into another, for example, probability distributions can be transformed into fuzzy sets or vice versa with little difficulty [*Zhang et al.*, 2009]. *Guyonnet et al.* [2003] proposed a hybrid method by embedding the -cut propagation method for fuzzy variables within each sample simulation of the Monte Carlo (MC) technique for random variables, and as a result a large number of fuzzy sets were obtained for output variables. However, these methods cannot effectively handle imprecise probabilities in which only probability bounds (rather than one precise probability) can be defined as a result of scarce, vague, or conflicting information [*Walley*, 1991].

[7] Random set theory [*Kendall*, 1974; *Matheron*, 1975; *Dubois and Prade*, 1991] has attracted increasing attention in recent years, as it can cope with varying levels of precision regarding information, and uncertainty can be represented directly using original uncertainty forms without further assumptions. It is a theory of set-valued stochastic processes, observations of which are intervals or sets rather than precise point values, and thus it can be viewed as a natural generalization of probability and statistics on random variables [*Nguyen*, 2006]. Most importantly, it can serve as a bridge between different uncertainty representations, thus allowing them to be handled simultaneously in a single modeling framework [*Hall*, 2003].

[8] The aim of this work is to present a new methodology for imprecise probabilistic evaluation of sewer flooding using random set theory. In this methodology, temporal uncertainty in rainfall data is considered (spatial distribution and measurement uncertainties are neglected) and represented using imprecise probability distributions of rainfall depth and duration. Synthetic rainfall events of uniform shape are used both because they are simple and typically used in practice [*Butler and Davies*, 2004], and because they are generally assumed when there is a complete lack of evidence on the appropriateness of other shapes. Model parameter uncertainty is characterized by fuzzy numbers with assumed shapes only. The most commonly used discretization method is used initially to propagate the two different types of uncertainties, and a MC based method is then developed to improve computational efficiency. What results from the method are the lower and upper cumulative distribution functions (CDFs) for model outputs (flood depth), constructed using the propagated random set. This methodology can potentially handle different uncertainty characterizations simultaneously, and thus allows for a more complete, and arguably accurate representation of uncertainty in data and models in terms of the most appropriate form wherever they originally appear, i.e., without further assumptions that might reduce the information or lead to inappropriate conclusions.