This paper reviews the role of uncertainty in the identification of mathematical models of water quality and in the application of these models to problems of prediction. More specifically, four problem areas are examined in detail: uncertainty about model structure, uncertainty in the estimated model parameter values, the propagation of prediction errors, and the design of experiments in order to reduce the critical uncertainties associated with a model. The review is rather lengthy, and it has therefore been prepared in effect as two papers. There is a shorter, largely nontechnical version, which gives a quick impression of the current and future issues in the analysis of uncertainty in water quality modeling. Enclosed by this shorter discussion is the main body of the review dealing in turn with (1) identifiability and experimental design, (2) the generation of preliminary model hypotheses under conditions of sparse, grossly uncertain field data, (3) the selection and evaluation of model structure, (4) parameter estimation (model calibration), (5) checks and balances on the identified model, i.e., model “verification” and model discrimination, and (6) prediction error propagation. Much time is spent in discussing the algorithms of system identification, in particular, the methods of recursive estimation, and in relating these algorithms and the subject of identification to the problems of prediction uncertainty and first-order error analysis. There are two obvious omissions from the review. It is not concerned primarily with either the development and solution of stochastic differential equations or the issue of decision making under uncertainty, although clearly some reference must be made to these topics. In brief, the review concludes (not surprisingly) that much work has been done on the analysis of uncertainty in the development of mathematical models of water quality, and much remains to be done. A lack of model identifiability has been an outstanding difficulty in the interpretation and explanation of past observed system behavior, and there is ample evidence to show that the “larger,” more “comprehensive” models are easily capable of generating highly uncertain predictions of future behavior. For the future of the subject, it is speculated that there is the possibility of progress in the development of novel algorithms for model structure identification, a need for new questions to be posed in the problem of prediction, and a distinct challenge to the conventional views of this review in the new forms of knowledge representation and manipulation now emerging from the field of artificial intelligence.

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Water quality modeling: A review of the analysis of uncertainty

[1] Hydrologic models use relatively simple mathematical equations to conceptualize and aggregate the complex, spatially distributed, and highly interrelated water, energy, and vegetation processes in a watershed. A consequence of process aggregation is that the model parameters often do not represent directly measurable entities and must therefore be estimated using measurements of the system inputs and outputs. During this process, known as model calibration, the parameters are adjusted so that the behavior of the model approximates, as closely and consistently as possible, the observed response of the hydrologic system over some historical period of time. In practice, however, because of errors in the model structure and the input (forcing) and output data, this has proven to be difficult, leading to considerable uncertainty in the model predictions. This paper surveys the limitations of current model calibration methodologies, which treat the uncertainty in the input-output relationship as being primarily attributable to uncertainty in the parameters and presents a simultaneous optimization and data assimilation (SODA) method, which improves the treatment of uncertainty in hydrologic modeling. The usefulness and applicability of SODA is demonstrated by means of a pilot study using data from the Leaf River watershed in Mississippi and a simple hydrologic model with typical conceptual components.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2004WR003059

Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation

Start-Up, Operation, Monitoring and Control of High-Rate Anaerobic Treatment Systems

Paper: An instrumental variable method for model order identification

When time‐series methods of parameter estimation are used to model the concentration of a pollutant in a natural waterway, the results may be grouped into a restricted class of models which are termed (j,n,m) models. It is shown that the dispersion as represented in conservative forms of these models has a temporal variance that increases linearly with time (thus mimicking the variance of a Fickian process) and is consistent with dead zone mixing within rivers and streams. Data from a variety of rivers are used to demonstrate that the (j,n,m) model provides an explanation of actual data that is superior to those obtained by manipulation of a Fickian equation. In addition, the model greatly simplifies computation of dead zone dispersive effects, is able to simply deal with non‐conservative pollutants and the results suggest that residence times are more meaningful physical parameters than dispersion coefficients in explaining dispersive phenomena.

Longitudinal Dispersion in Natural Streams

The Extended Kalman Filter (EKF) provides a logical statistically based extension to those existing approaches to model fitting based on deterministic model response error (surface) minimization. Its crucial feature as a basis for identification, however, is the recursive nature of the algorithm which permits the estimation of possible variations in the model parameters. Depending upon whether such estimated variations are realistic or not, bearing in mind the physical nature of the dynamic system, it is possible to formulate criteria for model adequacy. This approach to model identification was applied to the problem of modeling DO-BOD interaction in a freshwater river system based on daily field data. It was found that the basic model structure, as defined by a dynamic version of the Streeter-Phelps equations, was inadequate and, in the case of the river considered, it was necessary to introduce additional sustained sunlight terms to account for the effects of floating algal populations.

Bruce Beck

Papers

Systematic Identification of DO-BOD Model Structure

The modelling and control of water quality in a river system

The modelling and control of pollution in a river system