Equivalences and differences among various Hydrological Dynamical Systems

In this paper we want to show that once the topology of a Hydrological Dynamical System is determined, the structure of the equations of the Water Budget Dynamical System is determined but with it also the travel time and residence time distributions are. This is obvious since our paper on age-ranked functions but here it is rigorously stated and worked out.

Ideally this paper is also a continuation of the paper on the representation of Hydrological Dynamical systems with Petri Nets of which in some sense, it represents an extension. In a ideal menù, the reader should read first the paper on the historical-critical approach to the GIUH, I would say excluded its last section, then the Age-Ranked paper, then the Petri Net paper and finally this one. Clicking on the Figure you can have the preprint, or on the bibliography below you can have access to all the manuscripts. This is part of my research program on trying a statistical-mechanical approach to hydrological modeling which across all my research activity since the last (almost) thirty years. 

References

• Rigon, Riccardo, Marialaura Bancheri, Giuseppe Formetta, and Alban de Lavenne. 2016. “The Geomorphological Unit Hydrograph from a Historical-Critical Perspective.” Earth Surface Processes and Landforms, EGU Reprint Series, 41 (1): 27–37.
• Rigon, Riccardo, Marialaura Bancheri, and Timothy R. Green. 2016. “Age-Ranked Hydrological Budgets and a Travel Time Description of Catchment Hydrology.” Hydrology and Earth System Sciences 20 (12): 4929–47.
• Bancheri, Marialaura, Francesco Serafin, and Riccardo Rigon. 2019. “The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN).” Water Resources Research 55 (11): 8895–8921.
• Marialaura, Bancheri, Francesco Serafin, and Riccardo Rigon. 2019. “Supporting Material for: The Representation of Hydrological Dynamical Systems Using Extended Petri Nets (EPN).” Water Resources Research.
• Rigon, Riccardo, and M. Bancheri, 2020, “Equivalences and Differences between the Hydrological Dynamical Systems of Water Budget, Travel Time and Tracers Concentration and the Legacy of Models’ Topology.”, submitted to Hydrol. Processes.
• Rigon, Riccardo, and Marialaura Bancheri, 2020. “Supplementary Material, Equivalences and Differences between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time, and Tracer Concentrations and the Legacy of Models’ Topology.” submitted to Hydrol. Processes.

Category theory

In feeling that Hydrology needs new mathematical-physics, I explored several directions (Information Theory, Statistics, Causality theory, Algebraic topology, Thermodynamics -!-, and not very much the usual paraphernalia that modern physics uses whose taste I kind think to know, Petri Nets) and I arrived to category theory. Category theory is the mathematics of connections and retro-actions represented in graphical form. I arrived here because I think that expanding  hydrology to Earth System Science, it opens to the world of connections, and understanding connections means understanding complexity. 

The posts by Tai-Danae Bradley are really and enjoyable starting. Especially if we do not really care to fully understand what the examples means (since they cover whole branches of mathematics).

• What is Category theory anyway

For other definitions, please give a look to this new book by 

• Brendan Fong (GS) and David I. Spivak (GS), Seven Sketches in Compositionality, an invitation to Applied Category Theory

Just at the first pages, it also address to other books, and I also found

• Emily Riehl, Category theory in context

I collected many other books and papers about CT. However, if you want to start, start from these two. Eventually you can subscribe to the Azimuth forum and enroll John Baez course.