Towards a Theory of Predictability of Change Alberto Montanari (1) and Guenter Bloeschl (2)

1. Towards a Theory ofPredictability of Change Alberto Montanari(1) and GuenterBloeschl(2) (1) University of Bologna, alberto.montanari@unibo.it (2) Vienna University of Technology, bloeschl@hydro.tuwien.ac.at

2. What are the basic elements of a theory? • Why a theory? To establish a consistent, transferable and clear working framework. • In science, the term "theory" is reserved for explanations of phenomena which meet basic requirements about the kinds of empirical observations made, the methods of classification used, and the consistency of the theory in its application among members of the class to which it pertains. A theory should be the simplest possible tool that can be used to effectively address the given class of phenomena. • Basic elements of a theory: • Subject. • Domain (scales, domain of extrapolation, etc.). • Definitions. • Axioms or postulates (assumptions). • Basic principles. • Theorems. • Models. • ….. • Important: a theory of a given subject is not necessarily unique

3. The essential role of uncertainty • Hydrological predictions are inherently uncertain, because we cannot fully reproduce the chaotic behaviors of weather, the geometry of water paths, initial and boundary conditions, and many others. It is not only uncertainty related to lack of knowledge (epistemic uncertainty). It is natural uncertainty and variability. • Therefore determinism is not the right way to follow. We must be able to incorporate uncertainty estimation in the simulation process. • The classic tool to deal with uncertainty is statistics and probability. There are alternative tools (fuzzy logic, possibility theory, etc.). • A statistical representation of changing systems is needed. Important: statistics is not antithetic to physically based representation. Quite the opposite: knowledge of the process can be incorporated in the stochastic representation to reduce uncertainty and therefore increase predictability. • New concept:stochastic physically based model of changing systems. (AGU talk by Alberto, Thursday December 16, 1.40 pm). It is NOT much different with respect to what we are used to do. Understanding the physical system remains one of the driving concepts.

4. Towards a theory of hydrologic prediction under change • Main subject: estimating the future behaviours of hydrological systems under changing conditions. • Side subjects: classical hydrological theory, statistics,…. and more. • Axioms, definitions and basic principles: here is the core of the theory and the research challenge. We have to define concepts (what is change? How do we define it? What is stationarity? What is variability?) and driving principles, including statistical principles (central limit theorem, which is valid under change, total probability law etc.). • The key source of information is the past. We have to understand past to predict future. • What is stationarity? Its invariance in time of the statistics of the system but better to say what is non-stationarity: it is a DETERMINISTIC variation of the statistics. If we cannot write a deterministic relationship then the system is stationary. • Do we assume stationarity? Unless we can write a deterministic relationship to explain changes yes. A stationary system is NOT unchanging. In statistics a stationary system is defined through the invariance in time of its statistics, but it is subjected to significant variability and local changes that are very relevant. Past climate is assumed to be stationary but we had ice ages.

5. What is invariant? • Is future climate invariant?, Is the model invariant?, Are Newton laws still valid?, Can we identify additional optimality principles? • The research challenge is to identify invariant principles to drive the analysis of change. Merz, R. J. Parajka and G. Blöschl (2010) Time stability of catchment model parameters – implication for climate impact analysis. Water Resources Research, under review Fig. 1: Locations of the catchments and classification into drier catchments (red), wetter catchments (blue) and medium catchments (grey).

6. Fig. 2: 5 year mean annual values of climatic variables averaged for 273 Austrian catchments (black lines). The spatial of means of the wetter catchments are plotted as blue lines, the spatial of means of the drier catchments are plotted as red lines

7. Fig. 4: Model parameters (snow correction factor (SCF), Degree-day factor (DDF), maximum soil moisture storage (FC) and non-linearity parameter of runoff generation (B)) of 5 year calibration periods averaged for 273 Austrian catchments (black lines). The spatial of means of the wetter catchments are plotted as blue lines, the spatial of means of the drier catchments are plotted as red lines Fig. 5: Box-Whisker Plots of the Spearman Rank correlation coefficients of model parameters to climatic indicators. Temporal Correlation for the six 5-years calibration periods. (Box-Whisker Plots show the spatial minimum

8. Fig. 12: Cumulative distribution of the relative errors of observed and simulated low flows (Q95), mean flow (Q50) and high flows (Q5) for different 5 years period for a different time lag of calibration and verification period.

9. Towards a theory of hydrologic prediction under change • A first set of definitions • Hydrological model: • in a deterministic framework, the hydrological model is usually defined as a analytical transformation expressed by the general relationship: • where Qp is the model prediction, S expresses the model structure, I is the input data vector and e the parameter vector. • In the uncertainty framework, the hydrological model is expressed in stochastic terms, namely (Koutsoyiannis, 2009): • where f indicates the probability distribution, and K is a transfer operator that depends on model S and can be random. Note that passing from deterministic to stochastic form implies the introduction of the transfer operator.

10. Towards a theory of hydrologic prediction under change • A first set of definitions • Hydrological model: • if the random variables e and I are independent, the model can be written in the form: • Randomness of the model may occur because N different models are considered. In this case the model can be written in the form: • where wi is the weight assigned to each model, which corresponds to the probability of the model to provide the best predictive distribution. Basically we obtain a weighted average of the response of N different hydrological models depending on uncertain input and parameters.

11. Towards a theory of hydrologic prediction under change • Estimation of prediction uncertainty: • - Qo true (unknown) value of the hydrological variable to be predicted • - Qp(e,I,i) corresponding value predicted by the model, conditioned by model i, model parameter vector eand input data vector I • - Assumptions: • 1) a number N of models is considered to form the model space; • 2) input data uncertainty and parameter uncertainty are independent. - Th.: probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008): • where wi is the weight assigned to each model, which corresponds to the probability of the model to provide the best predictive distribution. It depends on the considered models and data, parameter and model structural uncertainty.

12. Towards a theory of uncertainty assessment in hydrology Setting up a model: Probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008) • Symbols: • - Qo true (unknown) value of the hydrological variable to be predicted • - Qp(e,I,i) corresponding value predicted by the model, conditioned by • - N Number of considered models • - e Prediction error • - e Model parameter vector • - I Input data vector • -wi weight attributed to model i

13. Conclusions and research challenges • Prediction of change needs to be framed in the context of a generalisedtheory. • Theory should make reference to statistical basis, although other solutions present interesting advantages (fuzzy set theory). • Research challenges: • a) Identify fundamental laws that are valid in a changing environment (optimality principles, scaling properties, invariant features. • b) Devise new techniques for assessing model structural uncertainty in a changing environment. • c) Propose a validation framework for hydrological models in a changing environment. • d) Devise efficient numerical schemes for solving the numerical integration problem.