Many rainfall-runoff models consider only the infiltration excess runoff generation process. The use of green roofs, bio-retention areas, and pervious pavements for urban storm water management purposes requires the modeling of surfaces where both infiltration and saturation excess runoff generation mechanisms need to be considered. Expanded from previous results of probabilistic rainfall-runoff transformation, analytical equations transforming the input rainfall frequency distribution to output runoff frequency distribution are derived to incorporate both runoff generation processes. These analytical equations can be used to calculate the average annual runoff volume and runoff event volume return period. Results from deterministic continuous simulation of various urban surfaces were compared to those from the analytical equations and satisfactory agreement was obtained. The analytical equations are therefore proposed as a complement to continuous simulation models for the modeling of urban catchments where both runoff generation processes occur.
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 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.
 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].
 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  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.
Rivera et al.  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  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  improved the method of Guo and Dai  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  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.
 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.
 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.
 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.
2. Statistical Analysis of Rainfall Data
 A continuous rainfall series is viewed as composed of individual rainfall events and interevent dry periods. To separate consecutive rainfall events, a minimum time period without rainfall needs to be selected as the minimum interevent time (MIET). Rainfall periods separated by a time interval longer than or equal to the selected MIET are considered as separate rainfall events. The choice of the MIET should be governed by the intended application. The results of this research may be applied to urban catchments with a response time (such as the time of concentration) in the order of half an hour to several hours. To ensure a one-to-one correspondence between rainfall and runoff events on these catchments, the MIET for rainfall event separation should be greater than the catchment response time. However, if the MIET is too long, meteorologically separate rainfall events may be considered as belonging to the same event. A more objective technique for obtaining an appropriate MIET is through the examination of the relationship between the MIET and the resulting average annual number of rainfall events observed. For the Toronto, Ontario climate, Kauffman  shows that increases of MIET beyond 6 h do not result in significant changes in the number of events observed. Thus, an MIET of 6 h or slightly longer may be considered appropriate for the analysis of typical urban catchments in the Toronto region. Details about the proper choice of MIET can be found in the work of Adams and Papa , Adams et al. , Eagleson , Fraser , Howard , and Restrepo-Posada and Eagleson .
 Each rainfall event is characterized by its rainfall volume (v), rainfall duration (t), and interevent time (b). A historical rainfall record can then be viewed as composed of a time series for each of the above characteristics, which individually may be submitted to a frequency analysis. Histograms of the time series can be prepared and probability density functions can be fitted to the histograms. An average annual number of storm events can also be obtained from these statistical calculations. It has been found that exponential PDFs often fit such histograms satisfactorily [Eagleson, 1972, 1978; Howard, 1976; Adams and Bontje, 1984; Adams et al., 1986]; although some authors, in their implementation of the analytical probabilistic approach, found the Weibull distribution more appropriate than the exponential distribution for rainfall event volume under some climate conditions [Bacchi et al., 2008; Balistrocchi et al., 2009]. The development in this paper employs the following exponential distributions for rainfall characteristics:
where ζ, λ, and ψ are distribution parameters. Using the method of moments, the values of ζ, λ, and ψ for a specific location may be estimated as the inverse of, respectively, the mean of rainfall event volume, the mean of rainfall event duration, and the mean of interevent time.
 As another example and also for use in verifying the probabilistic model developed in this paper, histograms of rainfall event characteristics were prepared from the 56-yr historical rainfall record (1939–1998, with 1940, 1956–1958 missing) of the Yonge Street station in Toronto, Ontario, Canada using an MIET of 9 h. In processing the Toronto rainfall data, it was found that >30% of the rainfall events have a volume ≤2.0 mm. To ensure that runoff-producing events are properly represented by the fitted exponential distributions, for the Toronto climate, storms with a rainfall volume ≤2.0 mm are omitted in the frequency analyses for the determination of the statistics of rainfall event characteristics. The omission of such small events can be justified given that a rainfall volume of 2.0 mm or less results in negligible amount of runoff on most urban catchments.
Figures 1, 2, and 3 compare the histograms and fitted exponential PDFs when events with a rainfall volume ≤2.0 mm are deleted. These figures indicate that exponential distributions again fit well with observed relative frequency data. It is worth noting that the threshold of 2.0 mm may need to be modified for other regions.
 The following derivations also require joint PDFs of the characteristics of rainfall events. In cases where the random variables are statistically independent, the joint PDF is the product of the marginal PDFs of the random variables. As shown by Adams and Papa , for most Canadian locations, only weak correlations exist between rainfall event volume and duration. Therefore, in this development, the three rainfall event characteristics are treated as statistically independent.
3. Event-Based Rainfall-Runoff Transformation
 In the work of Guo and Adams [1998a], the impervious areas which are directly connected to storm sewers and pervious areas of an urban catchment are considered separately first and then combined to form an event-by-event (or simply event-based) rainfall-runoff model. For directly connected impervious areas (hereafter referred to as impervious areas), the only potential loss results from its depression storage Sdi (mm). For pervious areas, potential losses include depression storage Sdp (mm), the initial soil-wetting infiltration loss Siw (mm), and infiltration loss at a constant rate of fc (mm h−1), where fc equals the ultimate infiltration capacity of the soil as defined by the Horton infiltration model. The initial soil wetting infiltration as defined by Guo and Adams [1998a] takes into account the additional infiltration loss resulting from the infiltration rates that are greater than fc at the beginning of a rainfall event. It was shown there that the value of Siw can be estimated by considering the recovery of the soil's infiltration capacity during interevent times. An approximate expression relating Siw to the Horton infiltration parameters of the soil and the distribution parameters of the probabilistic models of rainfall event characteristics was derived by Guo and Adams [1998a].
 As summarized briefly in the previous paragraph, only Hortonian overland flow generation over pervious areas is considered in the work of Guo and Adams [1998a]. The same definitions adopted and processes considered by Guo and Adams [1998a] will be employed and considered here for the Hortonian portion of overland flow generation. In order to incorporate saturation excess flow, the maximum infiltration volume possible (Ss in mm) is added here as one additional characteristic of the pervious area. The value of Ss can be estimated as the difference between a soil's porosity and its wilting point times the depth of the infiltration horizon [Rossman, 2010]. The infiltration horizon can be a permeable top layer on top of an impermeable or much less permeable layer, or directly on the water table.
 Similar to what is presented in the work of Guo and Adams [1998a], the rainfall-runoff transformation described herein for the development of probabilistic models treats each individual rainfall event as a whole (i.e., event-based), and does not follow a time step-by-time step procedure as in numerical simulation models. Nevertheless, the same physical processes (i.e., interception, depression storage, and infiltration) are considered. At the beginning of a rainfall event, soils are dry and the remaining maximum possible infiltration volume is typically close to Ss. At the end of a rainfall event, the soils may become saturated or remain unsaturated. Direct surface runoff from pervious areas may be generated whether or not the soils become saturated during a rainfall event. Two scenarios (i.e., scenario 1: soils remain unsaturated throughout a rainfall event, and scenario 2: soils become saturated during a rainfall event) will be considered separately in establishing the relationship between event rainfall and runoff. Direct surface runoff from impervious areas will be generated when the input rainfall event volume v is greater than Sdi. Under both scenarios, runoff generation from impervious areas and a combination of runoff from impervious and pervious areas are calculated the same way.
3.1. Runoff Generation Solely From Infiltration Excess
 Under scenario 1, soils of the catchment remain unsaturated throughout the rainfall event and runoff generated over pervious areas is solely from infiltration excess. The event-based rainfall-runoff relationship established by Guo and Adams [1998a] would still be valid if the input rainfall event's characteristics (i.e., v and t) are such that the condition that the soils remain unsaturated is satisfied. In order for the soils to remain unsaturated, either the rainfall volume actually infiltrated must be less than or equal to the maximum possible infiltration volume [i.e., (fct + Siw) ≤ Ss or equivalently fct ≤ (Ss – Siw)], or the input rainfall volume v must be less than or equal to the sum of depression storage and maximum possible infiltration volume [i.e., v ≤ (Sdp + Ss)]. The cases represented by fct ≤ (Ss – Siw) are those in which the soils have a relatively low fc, water infiltrates into the soil so slowly that the cumulative infiltrated volume is not enough to fill the maximum possible infiltration volume, regardless of the magnitude of v. The cases represented by v ≤ (Sdp + Ss) are those in which, regardless of the time available for infiltration and the soils' infiltration capacity, the input rainfall volume is not enough to fill the maximum possible infiltration volume.
 Let Sm = Ss – Siw, which is the remaining maximum possible infiltration volume after satisfying the initial soil wetting infiltration requirement. In the APSWM, it is assumed that Siw must be filled before any runoff occurs and this is treated as a part of the initial losses. Thus, Sm can be viewed as the maximum possible infiltration volume after all initial losses are satisfied. The conditions for soils to remain unsaturated after a rainfall event (v, t) can be expressed as, condition (1), t ≤ Sm/fc; or condition (2), v ≤ (Sil + Sm), where Sil = Sdp + Siw, and is referred to as pervious area initial losses. Conditions (1) and (2) are mutually independent since t and v are considered independent random variables. If either condition (1) or (2) is satisfied, the rainfall-runoff relationship as established by Guo and Adams [1998a] and as summarized below would be valid.
 Runoff volume generated from impervious areas (vri in mm) is,
Runoff volume from pervious areas (vrp in mm) is,
The overall runoff generation of an urban catchment is the area-weighted combination of the runoff from the pervious and impervious portions of the catchment. In an urban catchment, Sdi is typically less than Sil. Therefore, if v is less than Sdi, then v must be less than (Sil + fct). If the fraction of impervious areas of the urban catchment is h (dimensionless), then combining equations (4) and (5) gives the following equation:
where Sd = hSdi + (1 − h)Sil is the area-weighted depression storage of the impervious areas and the initial losses of the pervious areas of the urban catchment. It is noted here that, in this development, no differentiation between directly or indirectly connected impervious areas is made. Equation (6) together with conditions (1) and (2) will be used to form part of a new event-based rainfall-runoff relationship by which both Hortonian and saturated overland flows are considered.
3.2. Runoff Generation When Soil Saturation Occurs
 Under scenario 2, soils of the catchment become saturated during an input rainfall event. Runoff generated from pervious areas under this scenario may be from infiltration excess at the early part of the storm and then saturation excess during the later part of the storm. The conditions for saturation to occur are: fct > Sm, referred to as condition (3), and v > (Sil + Sm), referred to as condition (4). Condition (3) specifies that the speed of infiltration must be fast enough so that the total volume infiltrated during the rainfall event could exceed the maximum possible infiltration volume. Condition (4) specifies that there must be enough incoming rainfall to fill both Sil and Sm. Violation of either condition (3) or (4) would result in unsaturation of the soils. Therefore, conditions (3) and (4) must be satisfied simultaneously in order for saturation to occur. Under this scenario, the runoff volume generated from the pervious area is (v – Sil – Sm) and the runoff volume generated from the impervious area is (v – Sdi). Combining the two parts, the event runoff volume vr from an urban catchment with an imperviousness fraction of h can be calculated as,
Equation (7) and conditions (3) and (4) form the other part of the event-based rainfall-runoff relationship by which both Hortonian and saturated overland flows are considered. Conditions defining the two parts of the relationship are mutually exclusive and collectively exhaustive, therefore combining the two parts, a complete event-based rainfall-runoff relationship as summarized below is obtained:
4. Probability Distribution of Runoff Event Volume
Equations (8) and (9) describe the functional relationship between the dependent random variable vr and independent random variables v and t. With a given value vo (mm), the probability that vr ≤ vo (denoted as P[vr ≤ vo]) can be determined as the sum of the probability that vr ≤ vo, with v and t satisfying the conditions of scenario 1 (denoted as P1[vr ≤ vo]) and the probability that vr ≤ vo, with v and t satisfying the conditions of scenario 2 (denoted as P2[vr ≤ vo]), since the conditions delineating the two scenarios (i.e., the two sets of v and t values) are mutually exclusive. In the following, P1[vr ≤ vo] and P2[vr ≤ vo] are derived separately for the determination of P[vr ≤ vo].
4.1. Derivation for the Scenario With Saturation Excess Runoff
 P2[vr ≤ vo] is derived first as the functional relationship between vr and (v, t) under this scenario is simpler. According to equation (9), in order for vr ≤ vo, v must be less than or equal to [vo + Sd + (1 – h)Sm]; at the same time, in order to satisfy condition (4), v must be greater than (Sil + Sm). Thus, if the given value vo is such that (Sil + Sm) ≥ [vo + Sd + (1 – h)Sm], i.e., vo ≤ [Sil + Sm − Sd − (1 – h)Sm], which can be further simplified to vo ≤ [h(Sil – Sdi) + hSm], no v value would satisfy the above-described two conditions simultaneously, therefore P2[vr ≤ vo] = 0.
 To simplify notation, let Sdd = (Sil – Sdi), which is the difference between pervious area initial losses and impervious area depression storage. When vo > (hSdd + hSm), in order for vr ≤ vo, v must be less than or equal to [vo + Sd + (1 − h)Sm], at the same time v must be greater than (Sil + Sm) (i.e., condition 4) and t > Sm/fc (i.e., condition 3), therefore
Summarizing the above results, P2[vr ≤ vo] can be expressed as,
4.2. Derivation for the Scenario With Only Infiltration Excess Runoff
 P1[vr ≤ vo] may be determined following a procedure similar to the above for the determination of P2[vr ≤ vo]. However, the process of integration would be much more complex, as can be seen in the work of Guo and Adams [1998a], where without the conditions delineating scenario 1, the process of integration is already quite complex. To avoid undue complexity, an approximate solution of P1[vr ≤ vo] is sought in the following derivations. Let A be defined as the event that vr ≤ vo with v and t constrained only by what is required according to equation (6), i.e., either with or without saturation excess runoff; let B be defined as the event that either t ≤ Sm/fc or v ≤ (Sil + Sm), i.e., saturation of the soil does not occur. Thus, the intersection of events A and B (denoted as A∩B) describes cases in which vr ≤ vo and runoff generation is solely from infiltration excess. P1[vr ≤ vo] is therefore equal to the probability of occurrence of A∩B, denoted as P[A∩B]. If A and B are statistically independent, then P[A∩B] = P[A] × P[B], where P[A] and P[B] are, respectively, the probability of occurrence of events A and B.
 Events A and B are dependent upon each other to some degree since they are both dependent on the values that random variables t and v might take. To simplify the derivation for P1[vr ≤ vo], it may be assumed that A and B are statistically independent for the majority of the cases. This assumption may be reasonable because, first of all, event A is not affected by Sm but event B is mainly controlled by Sm; and second, event A depends largely on vo whereas event B is not influenced by vo at all. Nevertheless, an occurrence of event A may be accompanied by t ≤ Sm/fc and/or v ≤ (Sil + Sm), i.e., may be accompanied by the occurrence of B, and therefore events A and B are not completely independent. When Sm takes on comparatively small values, an occurrence of A for a given value of vo would be more likely accompanied by v ≤ (Sil + Sm), thus resulting in the occurrence of B. Therefore, the assumption of independence between A and B may result in a poor estimate of P[A∩B] for smaller Sm values. However, P1[vr ≤ vo] would be smaller as compared to P2[vr ≤ vo] when Sm is small. As a result, the sum of P1[vr ≤ vo] and P2[vr ≤ vo] could still be reasonably accurately estimated even for small Sm values since P2[vr ≤ vo] can always be determined exactly using equation (11). Still, the accuracy of the simplifying assumptions will be examined by comparing with continuous simulation results.
 The cumulative distribution function (CDF) of vr derived by Guo and Adams [1998a] with vr replaced by vo is equivalent to P[A] here. It can be expressed as,
As shown in Figure 4, the region of (v, t) representing event B can be viewed as composed of subregion 1, where 0 < t ≤ Sm/fc and 0 < v < ∞, and subregion 2, where Sm/fc < t < ∞ and 0 < v < (Sil + Sm). Thus, P[B] can be determined by integrating the joint PDF of v and t over the two subregions:
For a specific catchment under a given climate, P[B] is a constant equaling the probability that the catchment would generate only infiltration excess runoff under a random input rainfall event. To simplify notation, let C1 = P[B], C2 = λ/(λ + ζ fc – ζ fch), C3 = ζ fc(1 – h)/(λ + ζ fc − ζ fch), C4 = − ζSdi + λSdd/fc. P1[vr ≤ vo] can be determined as,
4.3. Combination of Two Scenarios to Obtain the Probability Distribution of Runoff Event Volume
 P[vr ≤ vo], i.e., the probability per rainfall event that the generated runoff volume is less than or equal to a given value vo can be determined as the sum of P1[vr ≤ vo] and P2[vr ≤ vo], i.e.,
In equation (15), C5 = λSm/fc + ζSd + ζSm − ζhSm. Note that C1 – C5 are all dimensionless constants dependent on the local climate and catchment characteristics.
Equation (13) shows that when Sm → ∞, C1 = P[B] = 1. With C1 = 1 and Sm → ∞, equation (15) reduces to equation (12), i.e., the equation by Guo and Adams [1998a] for cases where only Hortonian runoff is considered. Under a specific climate, as Sm increases, C1 increases if Sil and fc remain constant, indicating that the probability increases per rainfall event that runoff is generated solely from infiltration excess, or that the probability of saturation of the soils decreases. Physically, with a specific type of soil, an increase in Sm requires a thicker infiltration horizon. A thicker infiltration horizon takes more water to saturate. When Sm → ∞, the infiltration horizon will never be saturated. Therefore, the characteristics of the derived equations conform to the physical behavior of pervious areas.
 The CDF of random variable vr, F(vr), can be obtained by replacing vo in the right-hand side of equation (15) with vr. The PDF of vr, f(vr), can then be obtained by taking the first order derivative of its CDF with respect to vr. The final result is,
 Based on the PDF of vr, the expected value of vr per rainfall event can be found as follows:
The expected annual runoff volume is simply the product of E(vr) and θ, where θ is the average number of rainfall events per year. Equation (17) is not an exact solution because the PDF of vr is an approximation. However, when Sm → ∞, C1 = 1, and equation (17) reduces to
which is an exact solution, the same one as presented by Guo and Adams [1998a]. When Sm = 0 and Sil = 0, which is the other extreme, C1 = 0 and from equation (17), E(vr) = (1/ζ – hSdi). This is also an exact solution because the pervious areas behave like impervious areas with zero depression storage and 1/ζ is the expected volume of an input rainfall event. These features are indications of correct derivation and reasonableness of assumptions.
 The probability per rainfall event that the generated runoff volume is greater than vr is denoted as G(vr). This probability is referred to as the exceedance probability of vr and can be calculated from F(vr) as,
The return period of vr, T(vr) (yr), can be determined as,
The above-derived equations complement those already included in the APSWM, all together they will still be referred to as the APSWM.
5. Comparison With Hydrologic Engineering Center-Hydrologic Modeling System (HEC-HMS) Continuous Simulation Results
 The closed-form mathematical expressions derived in section 4.3 provide an efficient means of estimating the average annual runoff volume and return periods of runoff event volumes from urban catchments where pervious areas have limited capacities to absorb infiltrated rain water. To verify the reasonableness of the simplifying assumptions made in establishing the event-based rainfall-runoff relationships and in the derivation of the probability distribution of runoff event volume, results calculated using the analytical equations are compared to those determined from continuous simulation-generated flow series. HEC-HMS [U. S. Army Corps of Engineers (USACE), 2009] is chosen for continuous simulation because it is widely used in North America and has a flexible module for rainfall loss calculations. For the simulation of catchment rainfall-runoff transformations, the HEC-HMS model does not require the same simplifying assumptions invoked in the development of the analytical equations. Thus, results from continuous simulation using long-term historical rainfall records can be regarded as accurate values.
5.1. Correspondence Between HEC-HMS and APSWM Input Parameters
 The input parameters required by HEC-HMS and APSWM are not identical due to their difference in representing urban catchments. To ensure that a particular catchment as modeled by HEC-HMS is the same as that represented by APSWM, catchment parameter values must be the same if the parameters are essentially the same, or properly related if the definitions of the parameters are not exactly the same but the parameters are related. To establish these parameter relationships, the method that HEC-HMS uses to model a catchment is briefly reviewed below.
 A catchment or watershed in HEC-HMS is composed of impervious surface areas directly connected to storm sewers and pervious surface areas. Directly connected impervious surface has no infiltration, evaporation, or other losses, all precipitation falling onto it runs off. For pervious areas, HEC-HMS is equipped with various loss model options. The soil moisture accounting model (SMA), capable of detailed simulation of both the wet and dry weather behavior of pervious surfaces, is the most suitable for continuous simulation and was selected in this study. The SMA model represents the pervious area with a series of storage layers and simulates the movement of water through these layers. Storage layers include canopy-interception storage, surface-depression storage, soil-profile storage, and groundwater storage. The SMA model computes flow into, out of, and between the storage layers on a time step-by-time step basis. The main flows calculated are infiltration, percolation, surface runoff, groundwater flow, and evapotranspiration (ET). Infiltration is water that enters the soil profile from the ground surface. Percolation is the movement of water downward from the soil profile, through the groundwater layers, and into a deep aquifer.
 In the SMA model, the rate of infiltration during a time step is a function of the volume of water available for infiltration, the storage capacity of the soil profile still remaining at the time step, and the maximum infiltration rate of the soil. The two input parameters required for infiltration calculations are the maximum infiltration rate of the soil and the maximum volume of the soil storage. The soil storage is divided into the upper zone storage and tension storage. The upper zone storage loses water to both ET and percolation, while the tension storage loses water to ET only. The potential ET during a time step is computed from input monthly pan evaporation depths, multiplied by monthly varying pan-correction coefficients.
 Groundwater storage in the SMA model is used to simulate the behavior of the groundwater reservoirs underneath a watershed and their contributions to streamflow in the form of interflow and base flow. Since APSWM is developed for small urban catchments where interflow and base flow are nonexistent or insignificant, the groundwater storage layers of the SMA model do not need to be used and the soil profile is assumed to be directly above an impermeable layer. Thus, the rate of percolation and volume of groundwater storage in the SMA model are set to zero to eliminate percolation, interflow, and base flow calculations in this study.
 Without percolation, groundwater storage, and groundwater flow calculations, the input parameters to the SMA model include the initial conditions of the storage layers, maximum depth of canopy storage, maximum depth of surface depression storage, maximum infiltration rate of the soil profile ( fm, mm h−1), percent imperviousness, total soil profile storage, and tension storage. The initial conditions of the storage layers only affect a few initial flow values generated. In evaluating runoff event volume statistics, those initial flow values were removed. There are no related parameters, nor is there a need for such parameters in APSWM.
 The definition of imperviousness in HEC-HMS is the same as the directly connected imperviousness used in APSWM; for a particular catchment, the value of this parameter is kept the same in the corresponding HEC-HMS and APSWM models. For impervious areas of a catchment, HEC-HMS does not account for any depression storage. In APSWM models, Sdi was therefore specified as zero to ensure equivalence in this comparison study. In other studies, by using Sdi, APSWM gives additional capability in modeling the behavior of the impervious areas of an urban catchment. This is advantageous considering that the main application of APSWM is for small urban catchments where Sdi may be significant or needs to be accounted for due to the use of various storm water-management measures.
 The pervious area depression storage Sdp in APSWM is made equivalent to the sum of the maximum depths of canopy and surface storages in a SMA model. Since the rate of percolation is set to zero to eliminate groundwater storage and groundwater flow calculations, water stored in soil storage depletes only through ET; thus, there is no need to differentiate between upper zone storage and tension storage (these are simplified versions of real-world catchments and are only used here for model comparison purposes). Therefore, tension storage input to HEC-HMS is set to zero, and the maximum infiltration volume Ss input to APSWM is made equal to the total soil profile storage input to HEC-HMS. The value of fm input to HEC-HMS depends on the soil type, and so does the value of fc used in APSWM. Thus, the value of fc is related to fm from the known type of soil.
 The only input parameter of APSWM that cannot be determined directly from those of HEC-HMS is Siw, the initial soil wetting infiltration loss. Siw accounts for the infiltration volume due to soil wetting during the initial period of a storm when infiltration capacity is higher than fc. For an individual storm event, it can be calculated as [Guo and Adams, 1998a],
where fo is the initial infiltration capacity at the beginning of the storm event, t is the duration of the storm event, and k (in h−1) is the infiltration capacity decay coefficient employed in the Horton infiltration model. The value of fo changes from storm to storm and depends on how the infiltration capacity of the soil is recovered during the dry period preceding a storm event. The recovery of soil's infiltration capacity is a result of ET after a rainfall event and the continuing downward movement of water if there is unsaturated space in the soil profile. Both processes deplete soil moisture at the surface layer of the soil horizon. APSWM requires an input of a long-term average Siw which can be calculated if a long-term average of fo can be obtained.
 Assuming that ET would not deplete soil moisture before it completely depletes water accumulating in surface depressions, and that surface depressions are always full of water at the beginning of a dry period, the maximum ET depletion of soil moisture during an interevent time b can be estimated as (RETb – Sdp), where RET is the potential rate of ET calculated as the annual average of the HEC-HMS input monthly potential ET rates times monthly correction coefficients, scaled to appropriate units. Since the average of b is 1/ψ, the long-term average of the potential maximum ET depletion of soil moisture during a dry period preceding a rainfall event is (RET/ψ – Sdp).
 The actual amount of soil moisture depletion resulting from ET and the extent of downward movement of water during dry periods are also dependent on the soil moisture available at the beginning of the dry period and the maximum soil profile storage volume (i.e., Ss). The average soil moisture available at the beginning of a dry period is generally less than or equal to the average rainfall event volume (1/ζ). Therefore, when (RET/ψ – Sdp) is greater than Ss or when (RET/ψ – Sdp) is greater than 1/ζ, ET alone will likely deplete all of the soil moisture during an average dry period, thus fo will recover to its maximum fm. When (RET/ψ – Sdp) is less than Ss or when (RET/ψ – Sdp) is less than 1/ζ, fo will likely not recover to its maximum due to ET alone. However, when Ss is greater than 1/ζ, fo will still recover to fm regardless of the magnitude of (RET/ψ – Sdp) because the continuing downward movement of soil moisture will dry up the surface layer of the soil profile.
 Only when (RET/ψ – Sdp) is less than Ss and Ss is less than 1/ζ, fo may not recover to fm. This is because, on average, individual rainfall events would fill the entire storage volume of the soil horizon, so there will be almost no downward movement of soil moisture after the rainfall event, and ET during the dry period cannot deplete the soil moisture stored from the previous rainfall event. Under this circumstance, the total moisture stored inside the soil profile at the beginning and end of an average dry period is Ss and (Ss − RET/ψ + Sdp), respectively. In SMA model flow calculations, the rate of infiltration is assumed to be an inverse function of soil profile storage volume. This function is linear with an infiltration rate of fm corresponding to zero soil profile storage and an infiltration rate approaching zero when the soil profile storage reaches its maximum Ss. Based on this, the relationship between fo and fm when (RET/ψ – Sdp) < Ss and Ss < 1/ζ is
Summarizing the above, the average fo in equation (21) can be expressed as,
 The expected value of the initial soil wetting infiltration loss, E(Siw), of a storm event can be determined as,
Using equation (25) to evaluate E(Siw), the result may be less than zero for extremely humid areas or areas with high Sdp. For those cases, E(Siw) should be set to zero to indicate that, on average, there is no recovery of infiltration capacity between storm events. The value of k should be determined in accordance with the known soil type, together with the determination of the value of fc. In the development of APSWM, the value of Siw for all storm events is treated as a constant equaling the expected value E(Siw), and the symbol Siw was used to denote E(Siw) to simplify notation [Guo and Adams, 1998a]. Equation (25) is derived here mainly for comparison and verification purposes. In the actual application of APSWM, Siw may be estimated through other means, depending on available data.
 It can be seen from the above that some of the input parameters of HEC-HMS are essentially the same as those used in APSWM and some others are closely related. Establishment of their relationships based on physical grounds ensures that an HEC-HMS modeled catchment is physically the same as that represented by APSWM for the purpose of comparing results from the two different modeling approaches. A calibration of parameter values is not required, rather APSWM results are directly compared to those from HEC-HMS to verify the APSWM simplifying assumptions.
5.2. Average Annual Runoff Volume
 The emphasis of comparison and verification was placed on the effect of the maximum infiltration volume possible since the effects of other input parameters were already examined extensively by Guo and Adams [1998a]. A set of continuous HEC-HMS simulation runs were performed for catchments with various soil types and different maximum infiltration volumes. The 56-yr historical rainfall record of the Toronto's Yonge Street station was used as the rainfall input to the continuous simulations.
 Test catchments with different soil types and depths were simulated. The corresponding maximum infiltration volumes possible were estimated based on the soil depth and the soil's porosity and wilting point. The levels of imperviousness simulated include 20%, 45%, and 70%, representing different intensities of urban development. Three soil types, i.e., clay, silt, and sand, with porosities of 0.48, 0.50, and 0.47, respectively, and wilting points of 0.27, 0.14, and 0.02, respectively, were evaluated. Three to five levels of Ss (resulting from different soil depths) were considered for each type of soil. The continuous HEC-HMS simulation results of flow rates at the outlet of the catchment were separated into individual runoff events, according to a selected minimum interevent time (MIET). As runoff continues for an additional time (this additional time equals the time of concentration of the catchment) after the end of an input rainfall event, to avoid the aggregation of runoff events generated from two consecutive rainfall events, the MIET for runoff events is reduced by the time of concentration of the catchment from the MIET used to separate rainfall events. The times of concentration of the test catchments are input to the HEC-HMS models according to the “SCS unit hydrograph” option of the HEC-HMS and range from half an hour to ∼4 h. The separated runoff events from the continuous HEC-HMS simulations were then subjected to frequency analyses.
 The average annual runoff volumes determined from continuous HEC-HMS simulations are compared with those estimated from APSWM. The average annual rainfall volume determined from the 56-yr rainfall record of the Toronto's Yonge Street station is ∼475 mm including events with rainfall volumes ≤2.0 mm. With an MIET of 9 h, the average rainfall event volume, average rainfall event duration, average interevent time, and average annual number of rainfall events were determined to be 11.72 mm, 10.04 h, 121.6 h, and 39 h, respectively. For each year, the rainfall data contain precipitation records for only the rainfall year (April–October).
 The values of the major catchment hydrologic parameters in the HEC-HMS models are listed in Table 1 together with the average annual runoff volumes from the continuous HEC-HMS simulations. Table 2 presents the input parameter values for the analytical models together with analytically calculated (using equation (17)) average annual runoff volumes. As discussed in section 5.1, the values of the input parameters in Table 2 are either the corresponding HEC-HMS input parameter values or calculated according to the related HEC-HMS input parameter values based on physical grounds.
Table 1. HEC-HMS Input Parameter Values and Resultsa
Canopy Storage (mm)
Surface Storage (Sdp, mm)
Maximum Infiltration ( fm, mm h−1)
Soil Storage (Ss, mm)
Average Annual Runoff (mm)
As discussed in section 5.1, tension storage, soil percolation, groundwater storage, and percolation are all set to zero in all cases.
Table 2. Analytical Model Input Parameter Values and Results
This is the relative difference between analytical model and HEC-HMS continuous simulation results on average annual runoff.
 The relative differences between the analytically calculated average annual runoff volumes and those computed from the HEC-HMS simulations are also included in Table 2. It shows that the relative differences are all <20%. The close agreement on average annual runoff volumes indicates that the event-based catchment rainfall-runoff transformation utilized in the analytical probabilistic models closely approximates the time step-based rainfall-runoff transformation employed in the HEC-HMS models, and that the simplifying assumptions made in establishing the event-based rainfall-runoff transformation are generally acceptable.
5.3. Relative Frequency Distribution of Runoff Event Volume
 Using the separated runoff events from the continuous simulations, the relative frequency distribution histogram of runoff event volumes can be constructed for each simulated case. Using equation (16), the probability density of the runoff event volume can be analytically calculated for the same case. An example of a comparison of the analytically calculated probability density distribution and continuous simulation-determined relative frequency distribution is presented in Figure 5 for sandy soil, 10 cm deep, in a catchment with an imperviousness of 70%. Satisfactory comparison is shown in Figure 5. Comparisons for other cases are similar.
5.4. Exceedance Probability Distribution of Runoff Event Volume
 The runoff event series obtained from the continuous HEC-HMS simulations are sorted and ranked in descending order of magnitude of runoff event volume to calculate runoff event volume exceedance probabilities. The exceedance probability is equated to the Weibull plotting position as follows:
where P is the Weibull plotting position of the runoff event; M is the rank of the runoff event according to a descending series of runoff event volumes; and N is the total number of rainfall events contained in the historical rainfall series input to the HEC-HMS model. Use of the total number of input rainfall events for N ensures that the runoff event series analyzed include those events with zero runoff volume, so that the plotting position P is equivalent to the runoff event volume exceedance probability per rainfall event.
Equation (19) can be used to analytically calculate runoff event volume exceedance probabilities per rainfall event. An example of a comparison of exceedance probability is shown in Figure 6 for the cases of silty soils, 12 and 24 cm deep and an imperviousness of 45%. Figure 6 shows that the HEC-HMS modeled exceedance probability distributions for the two cases (12 and 24 cm soil depths) are almost identical; this is the same as shown by the analytical probabilistic model results. It is probably because when the soil is 12 cm deep, the void spaces between soil particles are large enough to accommodate infiltration from the vast majority of storm events. When the soil depth increases to 24 cm, the additional void spaces in the lower horizon are seldom filled by infiltrated water during rainfall events. In other words, the maximum infiltration volume provided by a 12-cm deep silt soil layer is large enough to accommodate infiltration from the vast majority of storm events at the Toronto location.
Figure 6 reveals that analytically determined exceedance probability curves closely resemble those obtained from continuous HEC-HMS simulations. Similar comparisons are obtained for other cases; to save space, these comparison figures are not included here. The minor deviation of exceedance probabilities for the extreme low values of runoff event volume is partly a result of the fact that rainfall events with a volume v ≤ 2.0 mm are included in the HEC-HMS simulations but excluded in the statistical analysis of the rainfall record for the determination of the PDFs of rainfall characteristics.
5.5. Runoff Event Volume Return Period
 To further verify the event-based rainfall-runoff transformation models and the resulting analytical expressions developed herein, the analytically calculated runoff event volume return periods are compared with those determined from continuous HEC-HMS simulation results. Two example comparisons are presented in Figure 7. For the two cases compared, the agreement between the analytical and HEC-HMS results for lower return periods are fairly good, but for longer return periods, the differences get larger. Overall, the difference between the two cases as reflected in the HEC-HMS results are well reproduced by the APSWM results. The larger difference for longer return periods is absent in Figure 6. This is because the inversion of exceedance probability values much less than unity are involved in the conversion from exceedance probabilities to return periods for runoff event volumes of longer return periods. Thus, indiscernible differences in the lower tails of the curves in Figure 6 are magnified in the upper tails of the curves in Figure 7.
 The above comparisons demonstrate that the analytical models for runoff volume derived from the event-based rainfall-runoff transformation and the fitted exponential distributions of rainfall event characteristics can generate results comparable to those from continuous HEC-HMS simulations. It also indicates again that the simplifying assumptions made in developing the probabilistic equations are acceptable.
6. Example Results From the Analytical Probabilistic Model
 Another set of calculations using the probabilistic model alone were performed to test the sensitivity of the model results to input parameters. The results of these example calculations are summarized in Figures 8, 9, and 10. In Figure 8, runoff event volume versus return period for catchments with an imperviousness of 45% and silt soils are presented. Soils in various cases have 2.4-, 4.8-, 12-, 24-cm, and infinite depths. It can be seen that as the soil depths increase, the runoff event volumes of the same return period decrease. This is due to the fact that, while allowing the same infiltration rate, shallow soil depths can get saturated faster than deep soils and therefore generate more runoff as a result of saturation. Figure 8 also shows that the curve for 24-cm soil is almost identical to the one for infinite soil depth. This can be explained by the fact that when the silty soil is 24-cm deep, its maximum possible infiltration can accommodate infiltration from the largest storm that occurred, therefore, any additional depth would not affect the rainfall-runoff transformation under almost all storms.
 Probabilistic model results comparing three soil types are presented in Figure 9. The cases in Figure 9 all have a soil depth of 5 cm and imperviousness of 30%. Figure 9 shows that, for the same return period, the runoff event volume from catchments with sandy soils is the lowest, whereas the runoff event volume from catchments with clayey soils is the highest. This is due to the combined effect of the soil's infiltration capacity and the maximum infiltration volume possible.
 In Figure 10, probabilistic model results comparing different imperviousness values are presented. All of the cases in Figure 10 have 15-cm sand soil and their imperviousness ranges from 5% to 100%. As expected, Figure 10 shows that, for the same return period, runoff event volume from catchments with lower imperviousness is lower than those from catchments with higher imperviousness.
7. Summary and Conclusions
 In recent years, the use of LIDs such as green roofs, bio-retention areas, and pervious pavements has increased significantly to improve the management of urban storm water. One common feature of these storm water-management measures is that their maximum infiltration volume possible is limited and often lower than natural areas. This is because, while facilitating infiltration of rainfall fallen on their own surface areas, some of the LIDs also allow infiltration of runoff from their runoff contributing areas. As a result, the area-averaged maximum infiltration volume possible over all of the service areas of some LIDs is still relatively low compared to natural areas. For the proper design of these facilities, engineers need to estimate their long-term average performance as well as performance under extreme weather conditions. Developed here are closed-form mathematical expressions for the determination of the probability distribution of runoff event volume and average annual runoff volume from areas where infiltration is controlled by both the infiltration capacity and the maximum possible infiltration volume. The exponential PDFs of rainfall event characteristics, tested and accepted in previous research for various locations, are used as the starting point for the mathematical derivations.
 Continuous HEC-HMS simulations are conducted for catchments with various types of soils, different soil depths, and different maximum possible infiltration volumes. The HEC-HMS-simulated average annual runoff volumes, runoff event volume frequency distributions, exceedance probabilities, and return periods are compared with those calculated from the analytical probabilistic models. Close agreement is obtained between the analytical models and the simulation models. The input parameter values for the analytical models in these comparative studies are taken directly from those input to the HEC-HMS model, or directly calculated from related HEC-HMS input parameter values.
 The close agreement between the analytical and simulation model results indicates that the event-based rainfall-runoff transformation proposed herein generates runoff volumes comparable with those from time step-by-time step rainfall-runoff transformations. The close agreement of results also illustrates that the exponential distributions used for rainfall event characteristics and the simplifying assumptions made for the derivation of the probabilistic models are reasonably acceptable for the locations tested.
 The analytical probabilistic models, i.e., equations (16), (17), (19), and (20), developed herein can be used as a complement to continuous simulation for the analysis of runoff volume from urban catchments where both Hortonian and saturation runoff generation processes need to be considered. For example, for the optimum design of facilities or systems, the analytical probabilistic models can be used first to gain a quick appreciation of the overall characteristics of the system; based on that, detailed continuous simulation runs can be conducted for system configurations that are closer to the optimum setting. The analytical probabilistic models can be implemented easily in a computer spreadsheet. Once the statistical analysis of the rainfall record of a location is completed, considerable time savings can be realized by using the derived probabilistic models in place of, or together with, numerical hydrologic models.
 The storm water management effects (e.g., runoff volume reduction rates) of some of the LIDs can be modeled by lumping them with their runoff contributing areas and treating the lumped unit as a pervious area. The lumped pervious area's depression storage and infiltration related parameter values can be estimated as the area average of the individual contributing area's parameter values. An impervious contributing area may be treated as pervious for lumping purposes with its infiltration capacities and maximum infiltration volume possible equal to zero. The analytical probabilistic models developed here can then be directly applied to the lumped areas. Additional equations may be derived to consider the runoff routing and storage effects of some of the LIDs.
 The following symbols are used in this paper:
the event that vr ≤ vo with or without saturation excess runoff,
the event that saturation of the soil does not occur,
rainfall interevent time (h),
C1, … , C5
dimensionless constants defined in text (all are functions of the local climate and catchment characteristics),
the expected value of vr (mm),
the expected value of the initial soil wetting infiltration loss (mm),
evapotranspiration (mm h−1),
the probability density function of interevent time,
ultimate infiltration capacity of soil (mm h−1),
the maximum infiltration capacity of soil (mm h−1),
initial infiltration capacity of soil at the beginning of a storm (mm h−1),
the probability density function of rainfall event duration,
the probability density function of rainfall event volume,
the cumulative distribution function of runoff event volume,
the probability density function of runoff event volume,
the probability per rainfall event that the generated runoff volume is greater than vr,
the fraction of directly-connected impervious areas of a catchment,
infiltration capacity decay coefficient (h−1),
the rank of a runoff event in a descending series of runoff event volumes,
the total number of rainfall events contained in the input rainfall series;
the Weibull plotting position of a runoff event,
the probability of what is defined in the brackets,
the probability of what is defined in text,
the probability of what is defined in text,
the potential rate of evapotranspiration (mm h−1),
area-weighted depression storage of the directly connected impervious area and initial losses of the pervious area of a catchment (mm),
the difference between Sil and Sdi (mm),
depression storage of the directly-connected impervious areas (mm),
depression storage of the pervious areas (mm),
pervious area initial losses (mm),
initial soil wetting infiltration volume at the beginning of a rainfall event (mm),
Ss − Siw, i.e., the remaining maximum possible infiltration volume after satisfying the initial soil wetting infiltration requirement (mm),
maximum infiltration volume possible of pervious areas (mm),
rainfall event duration (h),
the return period of vr (yr),
rainfall event volume (mm),
a given value of the volume of runoff resulting from a rainfall event (mm),
volume of runoff resulting from a rainfall event (mm),
volume of runoff generated from impervious areas (mm),
volume of runoff generated from pervious areas (mm),
distribution parameter of point rainfall event volume (1 mm−1),
average number of rainfall events per year,
distribution parameter of rainfall event duration (1 h−1), and
distribution parameter of interevent time (1 h−1).
 Financial support provided by the Natural Sciences and Engineering Research Council of Canada, the Ministry of Water Resources and the Ministry of Science and Technology of China, and the Kwang-Hua Fund for College of Civil Engineering, Tongji University is gratefully acknowledged.