- The Perceptual model: deciding on the processes
- The Conceptual model: deciding on the equations
- The Procedural model: developing the model code
- Model calibration: getting values of parameters
- Model validation: confirming applicability and accuracy”

6. Characterizing the model uncertainty

which is a requirement that certainly contemporary modelling demands.

Model structure includes a whole range of choices and assumptions made by the modeller either explicitly or implicitly in applying a hydrological model. Examples of different model structures include:

- different process choices and descriptions
- coupling of the processes
- numerical discretisation
- representations of the spatial variability-zones, grids, sub-catchments, etc.
- element scale and sub-grid process representations including distribution functions, different degrees of lumping, effective parameterisation, etc.
- interpretations and classifications of soil type, geology land use cover, vegetation, etc.

- the number of equation (let’s called them places according to our classification of such systems)
- the interactions between places (represented by the relative adjacency matrix)
- the form of fluxes laws
- the values parameters assume

Once these hydrologic heuristics are applied, we eventually find ourselves with the nude set of equations, and it could be interesting to see if there exist methods that can discover and classify the main properties of the dynamical systems which depends on their structure. This problem, indeed, has received a lot of attention in system and control theory (see for e.g. Ljung, 1999 and references therein), mostly to autonomous linear systems.

Some of the aspects, in this case is the discover of T-invariants and P-invariants, or, in a less obscure language, of loops and set of correlated quantities that remains globally (i.e. their sum) stationary (i.e. Gilbert and Heiner, 2006). Other aspects regards reachability, i.e. the prior understanding if a certain distribution of the state variables can be obtained. All these aspect are well dealt within traditional books in system and control theory. Unfortunately the resulting structure of hydrological models is usually non-autonomous (the system are open) and non-linear. All aspects that make investigations more complicate, but probably not unfeasible. A lot of digging in literature and research is necessary though.

References

- Beven, K. (2012). Rainfall-Runoff Modelling: a primer (pp. 1–472). Wiley-Blackwell.
- Butts, M. B., Payne, J. T., Kristensen, M., & Madsen, H. (2004). An evaluation of the impact of model structure on hydrological modelling uncertainty for streamflow simulation. Journal of Hydrology, 298, 242–266. http://doi.org/10.1016/j.jhydrol.2004.03.042
- Clark, M. P., & Kavetski, D. A. F. F. (2011). Pursuing the method of multiple working hypotheses for hydrological modeling. Water Resources Research, 47(W09301).
- Fenicia, F., Kavetski, D., Savenije, H. H. G., & Pfister, L. (2016). From spatially variable streamflow to distributed hydrological models: Analysis of key modeling decisions. Water Resources Research, 52(2), 954–989. http://doi.org/10.1002/2015WR017398
- Fenicia, F., Kavetski, D., Savenije, H. H. G., Clark, M. P., Schoups, G., Pfister, L., & Freer, J. (2014). Catchment properties, function, and conceptual model representation: is there a correspondence? Hydrological Processes, 28(4), 2451–2467. http://doi.org/10.1002/hyp.9726
- Fenicia, F., & Kavetski, D. A. S. H. H. G. (n.d.). Elements of a flexible approach for conceptual hydrological modeling: 1. Motivation and theoretical development. Water Resources Research, 47(11), W11510–1–13.
- Gilbert, D., & Heiner, M. (2006). From Petri Nets to Differential Equations - an Integrative Approach for Biochemical Network Analysis, 1–20.
- Ljung, L. (1999). System identification: Theory for the user (pp. 1-587). Prentice Hall.
- Kampf, S. K., & Burges, S. J. (2007). A framework for classifying and comparing distributed hillslope and catchment hydrologic models. Water Resources Research, 43(5), 1–24. http://doi.org/10.1029/2006WR005370
- Reggiani, P., & Schellekens, J. (2003). Modelling of hydrological responses: the representative elementary watershed approach as an alternative blueprint for watershed modelling. Hydrological Processes, 17(18), 3785–3789. http://doi.org/10.1002/hyp.5167

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