A derived probability distribution approach to stormwater quality modeling

Abstract

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.

Introduction

Different types of urban stormwater quality models are developed in the literature. These models with various degrees of complexity are based on different modeling approaches. Huber [21], [22] provided a review of these stormwater quality models and noted that there were six fully operational runoff quality models including the US EPA Stormwater Management Model (SWMM) [23], the most widely used continuous simulation model in North America. More generally, urban stormwater quality models fall into two broad categories, namely, deterministic models (including statistical and physically-based) and stochastic models.

Statistical models are normally developed with well-established multivariate pattern recognition techniques such as factor analysis, cluster analysis, correlation analysis, etc. Other frequently used techniques for developing stormwater quality models include linear, non-linear and stepwise regressions. Physically-based stormwater quality models usually involve a number of simplified and interconnected elements with some physical interpretation in attempting to describe the complex processes of pollutant generation and transport in urban catchments. Although considered as an alternative to physically-based approach, one apparent drawback of statistical modeling approach is its inability to take into account the antecedent dry period which can be considered as one of the most important parameters in conventional physically-based models [20]. In addition, the application of statistical models is somewhat restricted because of the onerous requirement of supporting data and the limited validation of results that cannot be extrapolated or transferred [30]. The determination of statistical relationships can be unduly influenced by high magnitude events with quite low frequencies of occurrence [38]. In contrast to deterministic models that describe independent variables and predict dependent variables with specific values of certainty, probabilistic models instead consider the possibility of occurrence of particular events and determine the likelihood of their occurrence. For the probabilistic models formulated with derived probability distribution theory, major inputs to an urban drainage system similar to many other physical phenomena, can be typically considered as independent random variables with specified known probability distributions. The probability distributions of these independent random variables are then mathematically transformed into the probability distributions of the dependent variables describing the system outputs.

The derived probability distribution approach has been widely employed in flood frequency analysis [18], [11], [44]. Gottschalk et al. [17] showed that the probability distribution function of low flows could be derived by combining a regional rainfall model developed from long rainfall record and a local model for runoff response. In addition to the modeling of stormwater quantity, the derived probability distribution approach has also been extended to include stormwater quality modeling as well [15], [27], [28], [37], [29]. Among which Segarra-Garcia and Loganathan [37] derived a mathematical expression for the expected pollutant washoff load by assuming that hydrologic variables such as runoff event depth, duration and interevent time, were exponentially distributed random variables. Loganathan and Delleur [27] applied the derived distribution approach in evaluating the effect of urbanization on the frequency of pollutant loads from storm sewer overflows. Akan [3] derived a mathematical frequency distribution for estimation of stormwater runoff suspended solids load washed off from the impervious surface of urban catchment. Li and Adams [26] developed analytical probabilistic models with derived probability distribution theory to analyze the runoff quantity/quality control performance of various combinations of storage/treatment systems. Xu et al. [45] proposed a probabilistic stormwater quality model in which stormwater runoff quantity was estimated by STORM, and statistical estimation methods such as regression and maximum likelihood methods were employed for estimating the mean of censored concentrations and pollutant loads.

This paper presents methodologies and major procedures for the development of urban stormwater quality model which consists of a number of functional components such as rainfall–runoff transformation, pollutant buildup and washoff, etc. By integrating rainfall–runoff transformations and pollutant buildup function with washoff model, 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 to receiving waters are developed with derived probability distribution theory. In order to evaluate the impacts of rainfall–runoff transformation on pollutant loads, two different types of the rainfall–runoff transformations are proposed and integrated with the disaggregated form of the pollutant buildup function in this study.