It is now over 25 years since the first papers started to appear on stochastic process representations in hydrology, longer since the first stochastic time series and storage models appeared. In general the principles followed in the use of such models have continued to be those of traditional statistics: that is, to assume that a correct model structure of the process can be identified and then to generate realizations that are consistent with this model and to evaluate the resulting variability in the model responses. This approach has been used in virtually all stochastic modelling, from random pulse rainfall models to Bayesian conditional simulation of groundwater systems.
There is, however, another view of the problem that is not nearly so widespread but might be worthy of greater consideration. That view suggests that the real system, whether it be characterized by a time series or by spatial data or both, is not the outcome of a mean process or model structure but is better considered as a single unique realization that may have resulted from a variety of different model structures. In general, we do not have adequate data samples with which to identify a 'correct' model structure. The traditional approach does recognize this to some extent of course in allowing that the parameters of any model of the process have to be considered uncertain, but this is only done within the formulation of that chosen model structure.
Consider the example of flood frequency analysis. Even where data are available from a discharge measurement site, the data series are generally shorter than the return periods for which we would like to predict flood frequencies. The series of floods over a particular period is just one realization of all possible realizations over periods of similar length, and there is no guarantee that the period we have observed is representative, it just happens to be the period that has been measured. In fact, it is well known from records of greater length that the frequency characteristics of many rivers have changed significantly over the last 120 years or so. Is it therefore adequate as an analysis to treat the (limited) measurements as if they were representative of the mean process, or should they be treated as a possible single realization from a wider range of processes?
Consider also the example of a spatial field of hydraulic conductivities. This was the context of my original comment about the single realization