Derived Probability Distribution Theory Research Articles

Permeable pavement hydraulic optimization by using an analytical-probabilistic model

Abstract On-source storage controls are a sustainable solution for stormwater management in a scenario of continuous urban area growth. Structures that manage storage volumes through infiltration include extra environmental benefits, such as groundwater recharge, evapotranspiration, and pollutant load removal. Permeable pavement systems are among these controls and can easily integrate into dense urban areas, resulting in paved surfaces contributing to stormwater management. The shift toward on-source strategies is encouraged through regulations, policies, incentives, and awareness campaigns, which are substantially increasing their dissemination. Optimizing the design of on-source storage controls with infiltration, such as permeable pavement systems, through robust methodologies can reduce reservoir depth, reducing environmental impact and costs without impact on reliability. The analytical-probabilistic (AP) method using derived probability distribution theory from rainfall event characteristics and the mathematical description of hydrologic processes within the permeable pavement systems provides an analytical equation that can be used as a design tool, proving robustness analogous with continuous simulations. Results obtained with the AP method were compared with traditional event-based methodologies and continuous simulation, assessing the reliability of the proposed method in optimizing permeable pavement systems' reservoir depth.

Analytical Equation for Estimating the Stormwater Capture Efficiency of Permeable Pavement Systems

Permeable pavement systems have been increasingly used for onsite storm water management. In this paper, an analytical equation is derived for calculating the long-term average storm water capture efficiency of permeable pavement systems. This analytical equation is obtained by using the derived probability distribution theory on the basis of the probabilistic models of rainfall event characteristics and the mathematical representations of the hydrologic processes occurring in permeable pavement systems. Simplifying assumptions are made in establishing these mathematical representations. The validity of these assumptions and the accuracy of the analytical equation are demonstrated by comparing the analytical equation results with those determined from a series of continuous simulations where no similar simplifying assumptions are made.

Optimal volume of non-point sources management detention considering spatio-temporal variability of land surface moisture condition

This study presents the method to estimate the optimal volume of a non-point sources (NPS) management detention considering spatio-temporal variability of urban land surface moisture condition. In order to consider statistical characteristics of rainfall events, 3-parameter mixed exponential probability density function (PDF) is applied, and rainfall events are transformed to runoff events through Natural Resources Conservation Service–curve number method. Urban drainage catchment is divided into square-type grid cells, where runoff curve number is estimated by cells. Using the analytical model developed previously with derived probability distribution theory, the PDF of rainfall event depth is transformed to the PDF of stormwater depth. By cells, in consideration of changes in antecedent soil moisture condition and spatial variations in CN of drainage catchment, stormwater capture curve that can represent the drainage area is drawn and from which the optimal volume of a NPS management detention is estimated.

Estimating the Payback Period of In-line Micro Turbines with Analytical Probabilistic Models

This paper presents a new analytical probabilistic model that can characterize the energy recovery and economic payback period of in-line micro turbines in water distribution applications.The analytical probabilistic model incorporates elements of derived probability distribution theory. A probability distribution function for the power produced by the turbine is first derived assuming that the power produced by the turbine is functionally dependent on the flow rate through the turbine. A probability distribution function for the payback period of the turbine is also derived assuming that the payback period is both functionally dependent and inversely proportional to the power produced.The analytical probabilistic model was applied to a water distribution network to evaluate the expected values and standard deviation of payback period. The case study was used to examine the impact of the coefficient of variation of turbine flow, initial capital cost of the turbine, and electricity price on the expected value and standard deviation of payback period for an optimized turbine location. The results suggested that the coefficient of variation of turbine flow had little impact on the expected value and standard deviation of payback period. The initial capital cost and electricity price had a significant impact on the expected value of payback period but only a small impact on the standard deviation of payback period. The results will help municipalities better understand how the uncertainties associated with demand, initial capital costs, and electricity pricing will affect the economic viability of in-line micro-turbine projects in their systems.

Analytical Probabilistic Model for Evaluating the Hydrologic Performance of Green Roofs

An easy-to-use and physically based analytical probabilistic model is developed to evaluate the long-term average hydrologic performance of green roofs. The probabilistic models of local rainfall characteristics are introduced first, the hydrologic and hydraulic proc- esses occurring on and inside a green roof system are then described mathematically, and the closed-form mathematical expressions depicting the storm-water management performance of a green roof system are finally obtained using the derived probability distribution theory. Simplifying assumptions are made to mathematically describe the hydrologic and hydraulic processes. The validity of these assumptions and the overall probabilistic approach is demonstrated by comparing its outcomes with results from a series of continuous simulations using long-term rainfall data from Detroit, Michigan and observations from a real case study in Portland, Oregon. DOI: 10.1061/(ASCE)HE.1943- 5584.0000593. © 2013 American Society of Civil Engineers. CE Database subject headings: Green buildings; Stormwater management; Runoff; Probability; Sustainable development. Author Keywords: Green roof; Storm-water management; Runoff reduction rate; Probabilistic methods.

Hydrological model for urban catchments – analytical development using copulas and numerical solution

A model based on analytical development and numerical solution is presented for estimating the cumulative distribution function (cdf) of the runoff volume and peak discharge rate of urban floods using the joint probability density function (pdf) of rainfall volume and duration together with information about the catchment's physical characteristics. The joint pdf of rainfall event volume and duration is derived using the theory of copulas. Four families of Archimedean copulas are tested in order to select the most appropriate to reproduce the dependence structure of those variables. Frequency distributions of runoff event volume and peak discharge rate are obtained following the derived probability distribution theory, using the functional relationship given by the rainfall–runoff process. The model is tested in two urban catchments located in the cities of Chillán and Santiago, Chile. The results are compared with the outcomes of continuous simulation in the Storm Water Management Model (SWMM) and with those from another analytical model that assumes storm event duration and volume to be statistically independent exponentially distributed variables. Citation Zegpi, M. & Fernández, B. (2010) Hydrological model for urban catchments – analytical development using copulas and numerical solution. Hydrol. Sci. J. 55(7), 1123–1136.

Development of analytical models for estimation of urban stormwater runoff

The estimation of urban stormwater runoff volume is usually of fundamental importance in the determination of the relative magnitudes of storage capacity and treatment rate for various storage/treatment facilities and in the evaluation of system runoff quantity and quality control performances. Recognizing the increasingly complex nature of urbanized watersheds, stormwater runoff volumes estimated without adequate consideration of the rainfall–runoff transformation process may be not reliable, and desired levels of system performance control may not be achieved. In contrast to the continuous simulation approach, derived probability distribution theory is employed in the development of event-based analytical probabilistic models for runoff volume. As a key element in the model development, different types of the rainfall–runoff transformations are formulated in light of various hydrological considerations associated with conceptualizing runoff generation mechanisms. The emphasis of this paper is to illustrate methodologies in the development of analytical probabilistic models for runoff volumes and to demonstrate that analytical models can be derived with various degrees of complexity based on different types of rainfall–runoff transformations. The verification of the developed analytical models is performed in case studies. By comparing the results from analytical models to those from continuous simulation, it is concluded that the closed-form analytical models not only demonstrate the capability of providing comparable results to continuous simulation, but also can be highly efficient and flexible in the estimation of urban stormwater runoff volumes.

Analytical Urban Storm Water Quality Models Based on Pollutant Buildup and Washoff Processes

This paper presents methodology and major procedures for the development of analytical urban storm water quality models following derived probability distribution theory, which involves conceptualization of the three major components, i.e., the rainfall–runoff transformation, pollutant buildup, and washoff processes. In this study, two different types of the rainfall–runoff transformations are employed in an attempt to improve model performance by considering spatial variations of parameters associated with runoff generation mechanisms. By integrating different types of the rainfall–runoff transformations and pollutant buildup function with washoff function, two different types of pollutant washoff load models are formulated. Thereby, the probability distributions of the rainfall characteristics are mathematically transformed to create system storm water quality control measures, such as the average pollutant event mean concentration and long-term pollutant loads to receiving waters. These storm water qua...

A derived probability distribution approach to stormwater quality modeling

The closed-form analytical stormwater quality models are developed for simulating urban catchment pollutant buildup and washoff processes. By integrating the rainfall–runoff transformation with pollutant buildup and washoff functions, stormwater quality measures, such as the cumulative distribution functions (CDFs) of pollutant loads, the expected value of pollutant event mean concentrations (EMCs) and the average annual pollutant load can be derived. This paper presents methodologies and major procedures for the development of urban stormwater quality models based on derived probability distribution theory. In order to investigate the spatial variation in model parameters and its impact on stormwater pollutant buildup and washoff processes as well as pollutant loads to receiving waters, an extended form of the original rainfall–runoff transformation which is based on lumped runoff coefficient approach is proposed to differentiate runoff generation mechanisms between the impervious and pervious areas of the catchment. In addition, as a contrast to the aggregated pollutant buildup models formulated with a single lumped buildup parameter, the disaggregated form of the pollutant buildup model is proposed by introducing a number of physically-based parameters associated with pollutant buildup and washoff processes into the pollutant load models. The results from the case study indicate that analytical urban stormwater management model are capable of providing results in good agreement with the field measurements, and can be employed as alternatives to continuous simulation models in the evaluation of long-term stormwater quality measures.

Probabilistic assessment of urban runoff erosion potential

At the planning or screening level of urban development, analytical modeling using derived probability distribution theory is a viable alternative to continuous simulation, offering considerably less computational effort. A new set of analytical probabilistic models is developed for predicting the erosion potential of urban stormwater runoff. The marginal probability distributions for the duration of a hydrograph in which the critical channel velocity is exceeded (termed exceedance duration) are computed using derived probability distribution theory. Exceedance duration and peak channel velocity are two random variables upon which erosion potential is functionally dependent. Reasonable agreement exists between the derived marginal probability distributions for exceedance duration and continuous EPA Stormwater Management Model (SWMM) simulations at more common return periods. It is these events of lower magnitude and higher frequency that are the most significant to erosion-potential prediction. Key words: erosion, stormwater management, derived probability distribution, exceedance duration.

Runoff Quality Analysis of Urban Catchments with Analytical Probabilistic Models

The assessment of stormwater runoff pollution from urban catchments is addressed within a probabilistic framework. The methodology utilizes probability density functions of rainfall event characteristics, a runoff coefficient based rainfall-runoff transformation, and commonly used pollutant buildup and washoff processes representations. Employing derived probability distribution theory, analytical probabilistic runoff quality models are derived using exponential buildup and washoff functions. The derived analytical models yield expressions for expected event pollutant washoff load, annual average washoff load, the probability distribution of event washoff load, and the long-term average pollutant event mean concentration (EMC). These derived models are verified with available field data from an urban catchment located in the Greater Toronto Area. The verification results suggest that these analytical probabilistic models can be used for the assessment of runoff pollutant loads from urban catchments during planning-level analyses. These are computationally efficient closed-form mathematical expressions that can be easily incorporated into the spreadsheets and are thus a useful tool as an alternative or complement to more resource-intensive simulation models.

Urban Storm Water Control Evaluation with Analytical Probabilistic Models

In urban storm water management planning, a common problem associated with the widespread implementation of storage/treatment facilities, is the determination of the relative magnitudes of the storage capacity and the controlled outflow capacity in conjunction with the desired levels of system performance. For this purpose, cost-effective designs in the configuration of storage/treatment facilities can benefit from analytical tools that are able to provide explicit analytical solutions to system control measures in terms of the catchment meteorological characteristics and the drainage system control elements (e.g., storage capacity and controlled release rate). This paper presents methodologies and necessary procedures for the development of system performance measures (such as average annual volume of spills and number of spills from storage facilities) for storm water runoff quantity control based on derived probability distribution theory. Essentially, these runoff control measures with closed-form mat...

Probabilistic Models for Analysis of Urban Runoff Control Systems

Given the significant urban runoff impacts on many receiving waters and the massive costs of future investments in drainage infrastructure, the design of urban runoff control systems must be cost-effective. Cost-effective design requires that various runoff control system alternatives be investigated at the planning stage so that cost-effective runoff control systems can be identified for design level analysis. To analyze the runoff control performance of various combinations of runoff control systems at the planning stage, efficient screening models are acutely needed. For this purpose, analytical probabilistic models were applied to analyze the runoff quantity/quality control performance of various combinations of storage and treatment systems. These analytical probabilistic models are developed with derived probability distribution theory whereby the input meteorology to the catchment is described by probability density functions (PDFs) of the meteorological characteristics that are transformed by hydrologic/hydraulic functions to PDFs of the system performance variables. The resulting PDFs are then used to determine the average performance conditions. These models provide closed-formed solutions of the performance equations that are highly efficient in both a conceptual and computational sense. As a result, they are particularly useful for the screening analysis of urban runoff control alternatives.