## 1. Introduction

[2] Deterministic hydrologic models are widely used to estimate flood peaks for flood control and floodplain delineation purposes. These models can be simplified as single-event models or continuous simulation models capable of modeling major hydrological processes occurring on a watershed over a long period of time [*American Society of Civil Engineers and Water Environment Federation* (*ASCE WEF*), 1992; *Bicknell et al.*, 1997]. Single-event modeling usually follows the design storm approach, whereby a design storm of a specific return period is used as the input rainfall to a watershed; the peak discharge of the resulting runoff is assumed to be the watershed's flood peak having the same return period as the input design storm. The total volume of a flood event is traditionally not the main concern and cannot be estimated using the design storm approach.

[3] Continuous simulation models use long-term precipitation and other meteorological records as inputs to generate continuous series of flows. Frequency analyses on the generated flow series can then be conducted to estimate flood peaks of different return periods. Separating the generated flow series into individual runoff events, flood event volumes of different return periods can also be estimated from continuous simulation results. The extensive data and computation requirements limit the practical application of the continuous simulation approach for routine design purposes. However, when runoff event volume is the major concern, as in the case of storm water quality management, continuous simulation has to be conducted to estimate the frequency distribution of runoff event volumes [*Reese*, 2009].

[4] Analytical probabilistic storm water models were developed specifically for storm water-management planning and design purposes [e.g., *Guo and Adams* 1998a, 1998b, 1999a, 1999b; *Li and Adams*, 2000; *Bacchi et al.*, 2008; *Balistrocchi et al.*, 2009]. These models are composed of closed-form analytical equations which can be used to directly calculate the values of not only flood peaks but also runoff volumes of different return periods from small urban watersheds [e.g., *Guo and Adams* 1998a, 1998b]. Some of the analytical equations can also be used to calculate the values of the performance indexes of storm water management systems [e.g., *Bacchi et al.*, 2008; *Balistrocchi et al.*, 2009; *Li and Adams*, 2000; *Guo and Adams* 1999a, 1999b]. To expand the utility of the models developed by *Guo and Adams* [1998a, 1998b, 1999a, 1999b], *Guo and Zhuge* [2008] derived additional equations for the probabilistic routing of floods through channel reaches and detention ponds. The simple storage-based channel flow routing by *Guo and Zhuge* [2008] was made equivalent to the Muskingum-Cunge routing method by *Guo et al.* [2009]. For ease of reference, the analytical equations derived by *Guo and Adams* [1998a, 1998b, 1999a, 1999b], *Guo and Zhuge* [2008], and *Guo et al.* [2009] are collectively referred to as the analytical probabilistic storm water model (APSWM). *Guo and Dai* [2009] further expanded the capabilities of APSWM to consider rainfall areal reduction for large watersheds and to allow the use of either the Horton model or the runoff curve-number procedure for infiltration calculations.

[5] The accuracy of the APSWM was initially investigated by *Guo and Adams* [1998a, 1998b, 1999a, 1999b] by comparing the APSWM results with continuous simulation results for hypothetical catchments of various imperviousness, slopes, and soil characteristics. Rainfall data from the Pearson International Airport in Ontario, Canada were used in these earlier studies. *Guo* [2001] provided a comparison between the analytical probabilistic, design storm, and continuous simulation approaches for hypothetical test cases in Chicago, Illinois, United States. It was found that the three approaches could generate similar results for the prediction of peak discharges of various return periods. It was also demonstrated that appropriate design storm durations and hyetographs must be chosen in order for the design storm approach to provide results similar to continuous simulation and the APSWM.

[6] *Rivera et al.* [2005] applied the APSWM for two locations: Fort Collins, Colorado, United States and Santiago, Chile. The APSWM results were compared to continuous storm water management model (SWMM) simulation results. Using 50 yr of rainfall records from Fort Collins, it was found that rainfall event volume and duration are statistically independent, satisfying the APSWM's assumptions about rainfall characteristics. While using 44 yr of historical rainfall records from Santiago, it was found that there is a positive correlation between rainfall event volume and duration. Their studies showed that the APSWM results agree well with continuous simulation results for Fort Collins. However, for Santiago, the comparisons are not as good as those for Fort Collins. The authors suggested ways to improve the APSWM so that locations where rainfall event volume and duration are positively correlated can also be dealt with using the analytical probabilistic approach. *Quader and Guo* [2006] applied the APSWM for the estimation of peak discharge rates in an actual design case with multiple subcatchments in Kingston, Ontario, Canada and compared results with the design storm approach. Peak discharge estimates from the two approaches were found to be generally comparable. *Guo and Markus* [2011] improved the method of *Guo and Dai* [2009] for incorporating the curve-number procedure and developed a method for incorporating Clark's unit hydrograph for runoff routing calculations into the APSWM framework. *Guo and Markus* [2011] also applied APSWM for 12 urbanizing watersheds in the Chicago metropolitan area with calibrated parameters representing two degrees of urbanization. The APSWM results were found to be similar to those from single-event design storm modeling.

[7] The APSWM is computationally efficient, and if implemented in a spreadsheet or computer program, it is easier to use than either the design storm or the continuous simulation approach. One shortcoming of the APSWM and many other earlier probabilistic models [e.g., *Eagleson*, 1972; *Chan and Bras*, 1979; *Loganathan and Delleur*, 1984; *Bierkens and Puente*, 1990] is that only the infiltration-excess runoff generation process is considered. In those earlier probabilistic models, surface runoff is considered to be produced at the ground surface when rainfall intensity exceeds the soil's infiltration capacity. This process, known as Hortonian overland flow, occurs frequently on the majority of urban catchments. Another storm water runoff process, known as saturated overland flow, can occur when the surface horizon of the soil becomes saturated as a result of either the build-up of a saturated zone above a soil horizon of lower hydraulic conductivity or the rise of a shallow water table to the surface.

[8] The use of low-impact development practices (LIDs) such as infiltration trenches, rain gardens, pervious pavement, and green roofs results in urban areas where the maximum infiltration volume provided by the LIDs may sometimes be less than the total runoff generated from all of the contributing areas. This phenomenon necessitates the modeling of saturated overland flow generated from urban catchments where many LIDs are implemented. To properly model these new types of urban storm water management facilities, in its latest version (version 5), SWMM incorporated the maximum infiltration volume possible as an additional input parameter to model saturation excess runoff [*Rossman*, 2010]. The earlier versions of the analytical probabilistic models also need to be improved so that both Hortonian and saturated overland flow mechanisms can be considered, especially when runoff volume becomes the main concern for storm water management. In this paper, an analytical probabilistic storm water runoff model incorporating both Hortonian and saturation excess overland flows is developed and verified.

[9] The probability density functions (PDFs) of rainfall event characteristics and the mathematical representation of the rainfall-runoff transformation on an urban catchment are the basis for deriving the PDFs of runoff characteristics. The derivation process for the determination of the desired PDFs follows the principle of derived probability distribution theory [*Benjamin and Cornell*, 1970; *Eagleson*, 1972]. In this paper, historical rainfall data are examined to further verify the exponential PDF assumptions about rainfall event characteristics, a new functional relationship between input rainfall and output runoff is established, and the probability distribution of the output runoff event volume is then derived and compared to continuous simulation results.