Uncertainty modelling and analysis of environmental systems: a river sediment yield example

Throughout the last decades uncertainty analysis has become an essential part of environmental model building (e.g. Beck 1987; Refsgaard et al., 2007). The objective of the paper is to introduce stochastic and setmembership uncertainty modelling concepts, which basically differ in the assumptions that are made with respect to the uncertainty characterization. Stochastic uncertainty modelling is most frequently applied and is characterized by probability density functions (pdf's) or simply by means and (co)variances. Typical approaches are the Bayesian and the Monte Carlo Markov Chain methods. Alternatively, a set-membership or bounded-error characterization, as opposed to a stochastic characterization, is favoured when assumptions about distribution or estimates of mean and covariance cannot be satisfactorily tested, as with small data sets or heavily structured (modelling) errors. The bounded-error characterization is in essence deterministic. Both approaches, using tools as DREAM, GLUE, exact and approximate bounding, MCSM and a pavement-based technique, were tested on a real-world example. The example, based on Wasson's (1994) sediment yield - area data and after a log-log transformation of the data, is a linear static problem with two parameters. There is a continuing debate on the use of formal and informal uncertainty methods in hydrology (see e.g. Beven et al. 2008). GLUE has been criticized especially for the use of an informal likelihood function and a subjective choice of threshold to separate behavioural (feasible) and rejected (infeasible) parameter vectors. It has also been pointed out (Qian et al., 2003) that the inefficient sampling inherent in naive Bayesian estimation techniques of this type is very likely to yield uninformative results for the posterior distribution of the parameters. On the other hand, it has been known for a long time that assumptions about the residual errors in formal approaches are often violated, making inference unreliable (Kuczera, 1983). Studies have compared GLUE using an informal likelihood measure with a method using a formally correct likelihood function (see e.g. Freni et al., 2009). Vrugt et al. (2009a) conclude that although GLUE and a statistically formal (DREAM) Bayesian approach can give very similar results, the main advantage of a formal approach is that it allows to disentangle different contributions to total uncertainty. The set-membership approach to uncertainty modelling and analysis, as an alternative to the stochastic uncertainty modelling approaches, has a firm theoretical basis, see Walter (1990), Norton (1994, 1995), Milanese et al (1996) and Keesman (2011). In real applications, the critical point, however, is the choice of the bounds on the error vector. The stochastic and set-membership methods were evaluated with respect to (i) Accuracy or precision of methods, (ii) Time and computational requirements, (iii) Skill requirements and (iv) Range of applicability. From an application point of view, we conclude that uncertainty analysis is of paramount importance in scenario studies of (complex) environmental systems if one wants to implement robust measures. It is also worthy to be eclectic in methods for uncertainty analysis, because any method makes assumptions and it is therefore valuable to consider uncertainty quantification under different conditions