T. E. Unny

Stochastic nature of outputs from conceptual reservoir model cascades

Abstract Assuming that the time distribution of outflows from catchments fed by rainfall is described by routing through cascades of reservoirs, this paper discusses the stochastic characteristics of the outputs from these reservoir cascades when the physical inputs of rainfall, catchment abstractions and evaporation from reservoir surfaces are considered random. In effect, the paper expands the Nash cascade into a stochastic reservoir cascade. It is found that variations in no more than two parameters representing the nature of the conceptual reservoir are sufficient to describe catchment outflows with widely varying moment properties and correlation characteristics.

Stochastic model of first-order bod kinetics

Abstract The mathematical theory of stochastic differential equations is used to obtain models for the expectation and variance of the first-order BOD equation. The methods used to develop the stochastic models are the moment equation, the Fokker-Planck equation, stochastic integration and the Riccati equation. The variance model calibration and verification steps are performed on BOD curves obtained from sampling the raw influent of the Waterloo Pollution Control Plant.

Stochastic synthesis of hydrologic data based on concepts of pattern recognition: I. General methodology of the approach

Abstract In stochastic models for streamflow synthesis no consideration is given to the presence of groups of data corresponding to wet and dry periods. The existence of such groups in geophysical records including hydrologic records was well emphasized by Hurst. This paper is the first in a series of three, which develops a feature synthesis model for hydrologic records utilizing properties assignable to groups and the concepts of pattern recognition. Part II discusses the application of the model to natural streamflow records, and Part III demonstrates the efficacy of the proposed model through testing the synthesized streamflows from the viewpoints of various statistical and hydrological tests.

Stochastic models for first-order kinetics of biochemical oxygen demand with random initial conditions, inputs, and coefficients

Abstract Biochemical oxygen demand (BOD) is a parameter of prime importance in surface water pollution studies and in the design and operation of waste-water treatment plants. A general, stochastic analytical model (denoted S1) is developed for the temporal expectation and (heteroscedastic) variance of first-order BOD kinetics. The model is obtained by integrating the moment equation, which is derived from the mathematical theory of stochastic differential equations. This model takes into account random initial conditions, random inputs, and random coefficients, which appear in the model formulation as initial condition (σO2), input (σl2), and coefficient (σc2) variance parameters, respectively. By constraining these three variance parameters to either vanish or to be nonnegative, model S1 is allowed (under appropriate combinations of the constraints) to split into six stochastic “submodels” (denoted S2 to S7), with each of these submodels being a particular case of the general model. Model S1 also degenerates to the deterministic model (denoted D) when each of the variance parameters vanish. The deterministic parameters (i.e., the rate coefficient and the ultimate BOD) and the stochastic variance parameters of the seven models are estimated on sets of replicated BOD data using the maximum likelihood principle. In this study, two (S5 and S7) of these seven stochastic models are found to be appropriate for BOD. The stochastic input (S5) model (i.e., null initial condition and coefficient variance parameters) shows the best prediction capabilities, while the next best is the stochastic initial condition (S7) model (i.e., null input and coefficient variance parameters).

Dissolved oxygen concentrations in Lake Erie (U.S.A.-Canada): 1. Study of spatial and temporal variability using cluster and regression analysis

Abstract A non-hierarchical nearest-centroid clustering method was used to separate data pairs consisting of the dissolved oxygen (DO) concentration and temperature into four groups corresponding to hypolimnetic and non-hypolimnetic water of the Central and Eastern Basins of Lake Erie. For the stations which were common to all cruises within a year and were classified as being in the hypolimnion, initial DO concentrations and depletion rates were calculated and tests about their constancy were performed using weighted regression analysis and regression models with the time structure of the data explicitly incorporated in the models. The yearly uncorrected depletion rates for 1967–1980 were similar to values previously reported by several authors, indicating that this semi-objective clustering procedure provides a practical alternative to subjective selection of data. The conclusions about constancy of initial concentrations and depletion rates based on an unweighted regression analysis were shown to differ from those of weighted regression. Using regression with empirical weights, it was found that neither the initial DO concentration nor the depletion rate remained constant between 1967 and 1980 in the Central Basin but that the depletion rate remained constant and the initial DO concentration varied in the Eastern Basin.

Stochastic synthesis of hydrologic data based on concepts of pattern recognition: II. Application of natural watersheds

Abstract The feature synthesis model is utilized to synthesize streamflow realizations from three natural rivers. The practical usefulness of the various techniques employed in consonance with the application of the model is explained with illustrative examples dealing with streamflow records. The salient peculiarities associated with the construction and operation of the model for streamflow synthesis are presented.

Conjunctive use of deterministic and stochastic models for predicting sediment storage in large reservoirs: 2. Deterministic model for the sediment deposition process

Abstract The papers presented discuss the development of a model to predict the deposition of sediment in reservoirs as a function of time (in years). The papers consist of three parts which are published together: Part 1 dealt with the development of the stochastic model for the total deposition of sediment in reservoirs. This cumulative deposition of sediment in time was shown to be an additive process defined on a finite Markov chain. The model was analysed to obtain the mean and the range of the cumulative sediment deposition as a function of discrete time in years. The present Part 2 discusses a deterministic model for the deposition process along the length of the reservoir. The pattern of deposition is a function of the inflow and the concentration of sediment in the inflow as well as the initial level in the reservoir and the initial composition of the sediment. The inflow in the reservoir is treated as unsteady and non-uniform. As much of the physical characteristics of the reservoir as possible is included in the analysis. Part 3 discusses the application of the two models in conjunction to the John Martin Reservoir, Bent County, Colorado, for which measurements are available (though limited in extent). The comparison between results obtained from the model and the actual measured data is found to be in good agreement with each other. The symbols used and references are provided separately in the Notation and References Section of each part.

Stochastic synthesis of hydrologic data based on concepts of pattern recognition: III. Performance evaluation of the methodology

Abstract Synthetic realizations of monthly streamflows obtained by utilizing a feature synthesis model are tested from statistical and hydrological viewpoints. In terms of statistical considerations the synthetic realizations are shown to possess relevant properties of the historical streamflows at the series level as well as at the monthly level. From hydrological considerations, the synthetic realizations are found to be embedded in properties that are associated with clusters of low flows and clusters of peak flows that are comparable to those in the historical sample. The model results suggest that the synthesis of streamflows based on concepts of pattern recognition is a potentially viable approach and warrants further investigation.

On the Outputs of the Stochasticized Nash-Dooge Linear Reservoir Cascade

By randomizing the inputs to the deterministic Nash-Dooge linear reservoir cascade, linear stochastic conceptual response models suitable for small catchments are formulated as simple linear stochastic dynamical systems within the formalism of stochastic differential equations [SDE’s]. The system driving process, rainfall and evapotranspiration losses (negative input) are modelled respectively as a compound Poisson process and a mean zero white Gaussian noise superposed on a deterministic mean. Moments and autocovariance functions for the steady-state system outputs are obtained via application of the Ito differential rule. Results for cascades of one to five reservoirs reveal the additional reservoirs progressively attenuate system response. Generalizations to an n reservoir cascade are given for the variance and autocovariance function.

Conjunctive use of deterministic and stochastic models for predicting sediment storage in large reservoirs: 1. A stochastic sediment storage model

Abstract The papers presented discuss the development of a model to predict the deposition of sediment in reservoirs as a function of time in years. The papers consist of three parts which are published together: The present paper, Part 1, deals with the development of a stochastic model for the total deposition of sediment in reservoirs. This cumulative deposition of sediment in time is shown to be an additive process defined on a finite Markov chain. The model is analysed to obtain the mean and the range of the cumulative deposition as a function of discrete time in years. Part 2 discusses a deterministic model for the deposition process along the length of the reservoir. The pattern of deposition is shown to be a function of the inflow and the concentration of sediment in the inflow as well as the initial level in the reservoir and the initial composition of sediment. The inflow in the reservoir is treated as unsteady and non-uniform. As much of the physical characteristics of the reservoir as possible is included in the analysis. Part 3 discusses the application of the two models in conjunction to the John Martin Reservoir, Bent County, Colorado, for which measurements are available (though limited in extent). The comparison between results obtained from the model and the actual measured data is found to be in good agreement with each other. The symbols used and references are provided separately in the Notation and References Section of each part.