My Past Research on the Evolution of River Networks

Chronologically, one of my first interests was modeling the evolution of channel networks according to principles of minimal energy dissipation and self-organization by critical states. These two types of models proved to be capable of reproducing the two- and three-dimensional statistical characteristics of channel networks and natural basins, as well as the fractal and multifractal characteristics. This work, born with the intent of identifying a minimum set of characteristic dynamic elements in the evolution a hydrographic basin, has always been carried out in parallel to the refinement of measurement and analysis techniques of topographic data [J3]. 

The concept of optimality of a hydrographic basin was introduced in [J3, J4, J5]. In these works, the three postulates of optimal channel networks (OCNs) are stated and developed, proving how such principles can have quantitative effects on the morphology of river networks, particularly affecting the structure of slopes and of contributing areas, the geometry of the channels, and the characteristic velocity of the peak flow of a basin. All of these results explain numerous empirical laws and are still the basis of measurement campaigns.

In [J4], that which was postulated in [J3, J5] was verified by numeric simulation. It should be noted that the minimization of dissipated energy generates fractal forms that reproduce the quantitative characteristics of real basin. In [J6] the concept of optimality is further refined by introducing the hillslope contribution and presenting some case studies. In [J6], more tools are introduced for the qualitative comparison between the numeric models and the natural data. In [J8, J9, A6] a model of the evolution of river basins is presented that is based on the concepts of self-organization by critical states. This model proved to be equivalent to optimization model of [J3-J6]. In [J13] the impact of climatic variability on the morphology of the fluvial landscape is simulated, so offering an interpretative framework for some fluvial forms that can be found in nature.

Subsequently, the concept of optimality was refined observing that real basins do not have the configuration that would give an absolute minimum of dissipated energy, but rather that of states of local minimum that are dynamically accessible. From here the concept of feasible optimality was derived [A10, J18, J19]. It was also demonstrated that the states of absolute minimum, dynamically unreachable, have statistical properties that are not realistic, while accessible minimum states have the desired statistical characteristics. A relevant characteristic of the space-time dynamics of hydrographic networks is that they can be described by means of a parameter that can be linked to temperature [J16]. It is therefore possible to define the thermodynamics of the river networks. As with the thermodynamics of other physical systems, the relevant quantities are energy (dissipated in unit time), entropy, and the temperature.

It should be noted that the space-time evolution of river networks happens with an intermittent behavior similar to the concept of point equilibrium proposed for the evolu- tion of the biological species. It was also demonstrated that the temporal dynamics of river networks is coupled with the spatial activity at all scales and that natural networks, therefore, evolve according to conditions of minimum dissipation of energy but in the presence of a great variety of possible dynamic states.

In [J10] an accurate analysis of the fractal and multifractal properties of optimal river networks was carried out. The note [J18] is a review article, sent to the Annual Review of Earth and Planetary Sciences, that treats the aforementioned topics. In [J20] the results of a theorem on network topology that relates the sum of the contributing areas with the contributing areas themselves and hypothesizes that these quantities are analogous to the ratio of metabolic rhythm and mass of living beings. The two quantities are linked an exponential law with an exponent that was proved to be Hack’s exponent.

This work, born with the intent of identifying a minimum set of characteristic dynamic elements in the evolution of a hydrographic basin, has always been carried out in parallel to the refinement of measurement and analysis techniques of topographic data [s2,eb3]. Recently, this field of study has produced a work [J26] where the morphometric statistics of tributaries of natural rivers and OCNs are studied. These are related to the characteristics of peak flows and they have ecological implications such as, for example, the velocity of diffusion of waterborne diseases and the diffusion of species along the river network. In Rigon’s work, the morphological relations between the different parts of fluvial basins have been analyzed with ever more refined numeric instruments, to the point of creating a series of GIS methods known as the Horton Machine [eb3].

The paper [J41] is partially a review of old results, that were not collected before, and were overlooked by people because they did not appear in Rodriguez-Iturbe and Rinaldo 1997 book. It includes however some new set of simulation were injection of rainfall is assigned with certain distributions (with given correlation structure) producing differentiated power laws for discharge and contributing areas. Clearly a result to further explore.

References

In English:

[J3] - Rodriguez-Iturbe, I. , A. Rinaldo, R. Rigon, R. L. Bras, A. Marani and E.J. Ijjasz-Vasquez, Energy dissipation, runoff production, and the 3-dimensional structure of river basin, Water Resources Research, (28)4, 1095-1103, 1992.

[J4] - Rinaldo, A., I. Rodriguez-Iturbe, R. Rigon, R.L. Bras, E. J. Ijjasz-Vasquez e A. Marani, Minimum energy and fractal structures of drainage networks, Water Resources Research, (28), 2183, 1992.

[J5] - Rodriguez-Iturbe, I. , A. Rinaldo, R. Rigon, R. L. Bras, A. Marani and E.J. Ijjasz-Vasquez, Fractal structure as least energy patterns: The case of river networks, Geophysical Res. Letters, (19)9, 889-892, 1992.

[J6] - Rigon R., A. Rinaldo, I. Rodriguez-Iturbe, R. L. Bras and E. Ijjasz-Vasquez, Optimal channel networks: a framework for the study of river basin morphology, Water Resources Research, 29(6), 1635-1646, 1993.

[J7] - Ijjasz-Vasquez, E., R.L. Bras, I. Rodriguez-Iturbe, A. Rinaldo and R. Rigon, Are river networks OCN?, Advances in Water Resources, 16, 69-79, 1993.

[J8] - Rinaldo, A., I. Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez, and R.L. Bras, Self organized fractal river networks, Physical Review Letters,70(6), 822-26, 1993.

[J9] - Rigon, R., A. Rinaldo and I. Rodriguez-Iturbe, On landscape self-organization, Journal of Geophysical Research, 99(B6), 11971-11993,1994.

[J10] - Rodriguez-Iturbe, I., M. Marani, R. Rigon and A. Rinaldo, Self-organized river basin landscapes: fractal and multifractal characteristics,Water Resources Research, 30(12), 3531-3539,1994.

[J16] - Rinaldo, A. Maritan, A. Flammini, F. Colaiori, R. Rigon, I. Rodriguez-Iturbe and J. R. Banavar, Thermodynamics of fractal networks, Physical Review Letters, 76(18), 3364-3367, 1996.

[J18] Rinaldo, A., I. Rodriguez-Iturbe and R. Rigon, Channel Networks, Annual Review of Earth and Planetary Sciences, 26, 289-327, 1998

[J27] - Convertino, M, Rigon, R; Maritan, A; I. Rodriguez-Iturbe and Rinaldo, A, Probabilistic structure of the distance between tributaries of given size in river networks, Water Resour. Res., Vol. 43, No. 11, W11418, doi:10.1029/2007WR006176, 2007

[J41] -Rinaldo,  A., Rigon R., Banavar, J., Maritan, A. and Rodriguez-Iturbe, I., Evolution and selection of river networks: Statics, dynamics, and complexity, PNAS 2014


In Italian:

[A04]- Rigon, R., Il clima è scritto nella forma del reticolo idrografico?, Rapporti e studi della commissione di studio dei provvedimenti per la conservazione e la difesa della cittá di Venezia, Tomo CLI, Classe di Scienze ff. mm. e nn., 1-21, 1992.

[A06] - Rigon, R. - Principi di auto-organizzazione nella dinamica evolutiva delle reti idrografiche, Tesi di Dottorato, Università degli Studi di Genova, Firenze, Padova, Trento, 1994

[A12] - Rigon, R., Che cosa guida i processi morfologici nei bacini fluviali? Reti ottime di canali e la legge di Hack, Atti XXVI Convegno di Idraulica e Costruzioni Idrauliche, Vol II, 121, 1998

My Past Research on Rainfall-Runoff (Peak Flows) Modelling and related topics

These works of mine reagards event base prediction of discharges based on the Geomorphological Unit Hydrograph. They show that the detailed knowledge of a river basin's morphology allows one to frame the main features of the  hydrological response in terms of a minimal set of dynamical parameters.  This is relevant insomuch as the form of river networks can now be derived with automatic high resolution and objective remote-sensing techniques.  Typically, the required dynamical parameters are the mean flow velocity in the network and distribution of residence times of water in the hillslopes.

In this context, the variance of the GIUH is proven to depend mostly on the structure of the pathways followed by the single volumes of effective rainfall from their release points to the control cross-section (geomorphological dispersion) [J1] rather than on the hydrodynamic dispersion; the latter becoming  relevant only at the large scale.
Generally, it is possible to determine with precision the first moment, the variance, the  skewness, and the kurtosis of the hydrological response of a river basin  as a whole [A3, A9].  In [A7, J12] the production mechanisms of effective rainfall and the  characteristic contributions of the hillslopes are studied. As a result it was observed that rarely is the  response time of the hillslopes negligible when calculating the hydrological response of the  river basin as a whole.
In  [A18, A19, J21] the use of width functions in the construction of the GIUH and the concept
of including information about initial moisture conditions for the basis are further developed.
In this way it was observed that, with varying fractions of saturated river basin, the hillslopes
and channels contributed different fractions to the flood wave;  the hillslopes being particularly
important under conditions of extreme saturation of the basin [J21].
The formulation of the GIUH on the basis of width functions has also given semi-analytical
results regarding peak times and maximum discharges for a basin [J31].
All of these studies brought to the implementation of part of the Horton Machine [eb-3], and on the model Peakflow (e.g. http://www.jgrasstools.org).

The post on the lecture given at Montpellier contains the rational and an explicitation of the assumption made in such type of modelling.

More recently, the study of the hydrological response was directed mainly towards the investigation
of runoff production mechanisms on hillslopes (actually in researches related to the hillslope stability),  in relation to the soil depth [J33, J35, J37] and brought new insights to the concept of hydrological connectivity. These studies overcome the results in [A29] that, while interesting, assume simplistic hillslope setups. Parallel efforts, which are reported in Physico-Statistical Modelling of the Hydrological Cycle, were made in overcoming the limitations of event based modelling.

The paper [j47] is a review taken from a historical-critical point of view of the theory of the geomorphological unit hydrograph that also enlarge the view to the modern theories for describing water fluxes by travel time. It also serves as the starting point for future research in this directions.

References

In English:

[J1] - Rinaldo, A., A. Marani and R. Rigon, Geomorphological dispersion, Water Resources Research, 27(4), 513-525, 1991

[J12] - Rinaldo A., G. K. Vogel, R., Rigon and I. Rodriguez-Iturbe, Can one gauge the shape of a basin?, Water Resources Research, (31)4, 1119-1127, 1995.

[A18] - Rigon, R., Cozzini A., Pisoni S. Getting the Rescaled Width Function and the Derived WGIUH. The Geomatic Workbooks, (http://geomatica.ing.unico.it), 2001

[A19] - Rigon, R., Cozzini A., Pisoni S. Looking for a new method of estimating solid discharges in small alpine watersheds. The Geomatic Workbooks, vol. 2, (http://geomatica.ing.unico.it), 2001

[J21] - D’Odorico, P. e R. Rigon, Hillslope and channel contributions to the hydrologic response, submitted to Water Resour. Res., 2003

[A29] - Panciera, R., Chirico G.B., Rigon R., Grayson R. Contributing Area Dynamics produced by Saturation Excess Runoff. Atti del XXIX Convegno di Idraulica e Costruzioni Idrauliche, Settembre 2004

[eb-3] - R.Rigon, E. Ghesla, C. Tiso and A. Cozzini, The Horton Machine, pg. viii, 136, ISBN 10:88-8443-147-6, University of Trento, 2006

[J31] - R. Rigon, P. D’Odorico, and G. Bertoldi, The geomorphic structure of the runoff peak, Hydrol. Earth Syst. Sci. Discuss., 8, 1031-1058, doi:10.5194/hessd-8- 1031-2011, 2011

[J33] - Lanni, C.; McDonnell, J. J.; Rigon, R., On the relative role of upslope and downslope topography for describing water flow path and storage dynamics: a theoretical analysis, Hydrological Processes Volume: 25 Issue: 25 Pages: 3909-3923, DEC 15 2011, DOI: 10.1002/hyp.8263

[J35] - Lanni C., J. McDonnell JJ, Hopp L., Rigon R., "Simulated effect of soil depth and bedrock topography on near-surface hydrologic response and slope stability" in Earth Surface Processes and  Landforms, v. 2012, (In press). - URL: http://onlinelibrary.wiley.com/doi/10.1002/esp.3267/abstract . - DOI: 10.1002/esp.3267

[J37] - Lanni C., Borga M., Rigon R., and Tarolli P., Modelling catchment-scale shallow landslide occurrence by means of a subsurface flow path connectivity index, Hydrol. Earth Syst. Sci. Discuss., 9, 4101-4134, www.hydrol-earth-syst-sci- discuss.net/9/4101/2012/ doi:10.5194/hessd-9-4101-2012,
HESS

[J47] - Rigon R.,  Bancheri M.,  Formetta G.,  deLavenne A. , The geomorphic unit hydrograph from a historical-critical perspective, accepted in Earth Sci. Proc. & Landforms, 2015


In Italian:

[A3] - Rigon, R., Influenza della morfologia di un bacino montano sui caratteri della risposta idrologica, Atti del XXXII Convegno di Idraulica e di Costruzione idrauliche, Firenze, 1992.

[A7] - Rigon, R., Formulazione del trasporto per tempi di residenza: un‘alternativa ai modelli di pioggia efficace nel calcolo della risposta idrologica, Atti del XXIV Convegno di Idraulica e di Costruzione idrauliche, Napoli, 1994

[A9] - Rigon, R., P. D’Odorico e L. Parra, Metodi geomorfologici di inferenza della risposta idrologica, Atti del XXV Convegno di Idraulica e di Costruzioni idrauliche, Torino, 1996.

[A13] - D’Odorico, P., M. Marani e R. Rigon, Questioni geomorfologiche e previsione delle piene nei bacini fluviali, Atti XXVI Convegno di Idraulica e Costruzioni Idrauliche, Vol II, 73, 1998

[A24] - Rinaldo A., M. Marani, A. Fornasiero, G. Botter, S. Silvestri, A. Bellin, Rigon R., M. Ferri, F. Baruffi, A. Rusconi. Modelli geomorfologici - Montecarlo per la valutazione del tempo di ritorno delle piene fluviali: fiume Brenta chiuso a Bassano. Atti del XXVIII Convegno di Idraulica e Costruzioni Idrauliche, vol. 1, pp.271-278, 2002

Soil Depth Estimation

Estimation of soil depth is crucial for the assessment of hillslope hydrological processes (e.g.
Tromp van Meerveld and McDonnell, 2006) and landslide stability (e.g Lanni et al., 2011, 2012).  Is a topic that had a lot of attention in the community of geomorphologists but indeed it remain still an open. In a recent paper (e.g. Lanni et al., 2012, under the final stage of review in HESS) we wrote a very short review:

"The spatial distribution of soil depth is controlled by complex interactions of many factors (topography, parent material, climate, biological, chemical and physical processes) (e.g., Summerfield, 1997, Pelletier and Rasmussen, 2009, Nicotina et al., 2011). As a result, soil depth is highly variable spatially and its prediction at a point is difficult. Moreover, soil depth survey is time consuming and soil depth is difficult to measure even for small basins (Dietrich et al., 1995). Various methods have been explored to allow the estimation of soil depth over landscapes. A process-based approach was suggested by Dietrich et al. (1995) for predicting the spatial distribution of colluvial soil depth. Based on this approach, topographic curvature may be considered a surrogate for soil production. Heimsath et al. (1997, 1999) validated the relationship between curvature and soil production based on observations of cosmogenic concentrations from bedrock in their Tennessee Valley site in California. This approach was incorporated into a landscape evolution model by Saco et al. (2006) to evaluate the dependence of soil production on simulated soil moisture. Roering et al. (1999) supported the idea that soil production follows a non linear equation and, therefore, modified the dependence of soil depth relationships. However, the various modeling approaches for predicting soil depth over landscapes, described above, showed only partial success (Tesfa et al., 2009).

In contrast to the process-based approaches, a number of studies have applied statistical methods to identify relationships between soil depth and landscape topographic variables (e.g., slope, wetness index, plan curvature, distance from hilltop, or total contributing area) (e.g. Gessler et al., 1993, Tesfa et al., 2009, Catani et al., 2010). Some of these works reported good predictive capabilities for these statistical relationships. For instance, Tesfa et al. (2009) report that their statistical models were able to explain about 50% of the measured soil depth variability in an out-of-sample test. This is an important result, given the complex local variation of soil depth."

Our short review possibly miss some interesting reference as the one by D'Odorico (2000) on a possible bi-stable evolution equation, and our little work in Bertoldi et al. (2006) that generalize Heimsath's to include random variaility in depth. 

However, reduction of soil depth formation simply to a geometrical factor (as implied by using equations with homogenous parameters) is clearly not enough. Pedologists know it. But so far I found only a few able to enter in the strict path of learning our equations. 

Anyway,  looking for R based hydrological resources I found this explanatory map by Roudier and Beaudette:

Looking at it gives clearly the idea that soil depth does not depend (only) on geometry. Where slope and curvature remain constant, anyway soil depth varies.

It is easy to think that it depend on variability in the geologic substrate,  soil cover (grass, plants), and soil use (for instance grazing or the presence of animals). But there is any pedologist out there which could help us to built a consistent quantitative theory ?

As usual, the references could be a starting point for a more in-depth literature search.

References

Catani, F., Segoni, S., and Falorni, G.: An empirical geomorphology-based approach to the spatial prediction of soil thickness at catchment scale, Water Resour. Res., 46, W05508, doi:10.1029/2008WR007450, 2010. 

Dietrich, W. E., Reiss, R., Hsu, M.-L., and Montgomery, D. R.: A process-based model for colluvial soil depth and shallow landsliding using digital elevation data, Hydrol. Processes, 9, 383 – 400, doi:10.1002/hyp.3360090311, 1995.

D'Odorico, P. (2000), A possible bistable evolution of soil thickness, J. Geophys. Res., 105(B11), 25,927–25,935, doi:10.1029/2000JB900253.

Gessler, P. E., Moore, I. D., McKenzie, N. J., and Ryan, P. J.: Soil landscape modeling and spatial prediction of soil attributes, Int. J. Geogr. Inf. Syst., 9, 421– 432, doi:10.1080/02693799508902047, 1995.

Heimsath, A. M., Dietrich, W. E., Nishiizumi, K., Finkel, R. C.: The soil production function and landscape equilibrium, Nature, 388, 358– 361, doi:10.1038/41056, 1997.

Heimsath, A. M., Dietrich, W. E., Nishiizumi, K., Finkel, R. C.: Cosmogenic nuclides, topography, and the spatial variation of soil depth, Geomorphology, 27, 151– 172, doi:10.1016/S0169-555X(98)00095-6, 1999.

Lanni, C., McDonnell, J.J., and Rigon, R.: On the relative role of upslope and downslope topography for describing water flowpath and storage dynamics: a theoretical analysis, Hydrological Processes, 25(25), 2011.

Lanni, C., McDonnell, J., Hopp, L. and Rigon, R.: Simulated effect of soil depth and bedrock topography on near-surface hydrologic response and slope stability, Earth Surf. Process. Landforms, 2012, doi: 10.1002/esp.3267.

Lanni, C., Borga M., Tarolli P., Rigon R., Modelling shallow landslide susceptibility by means of a subsurface flow path connectivity index and estimates of soil depth spatial distribution , HESSD, 2012

Nicotina, L., Tarboton, D. G., Tesfa, T. K., Rinaldo, A.: Hydrologic controls on equilibrium soil depths, Water Resour. Res., 47, W04517, doi:10.1029/2010WR009538, 2011.

Liu, J., Chen, X., Lin, H., Liu, H., & Song, H. (2013). A simple geomorphic-based analytical model for predicting the spatial distribution of soil thickness in headwater hillslopes and catchments. Water Resources Research, n/a–n/a. doi:10.1002/2013WR013834

Pelletier, J. D. and Rasmussen, C.: Geomorphically based predictive mapping of soil thickness in upland watersheds, Water Resour. Res., 45, W09417, doi:10.1029/2008WR007319, 2009.

Roering, J.E., Kirchner, J.W., and Dietrich, W.E.: Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for landscape morphology, Water Resorces Research, vol. 35(3), 853–870, 1999.

Saco, P. M., Willgoose, G. R., and Hancock, G. R.: Spatial organization of soil depths using a landform evolution model, J. Geophys. Res., 111, F02016, doi:10.1029/2005JF000351, 2006.

Summerfield, M. A.: Global Geomorphology, 537 pp. Longman, New York, 1997

Tesfa, T. K., Tarboton, D. G., Chandler. D. G., and McNamara, J. P.: Modeling soil depth from topographic and land cover attributes, Water Resour. Res., 45, W10438, doi:10.1029/2008WR007474, 2009.

Tromp-van Meerveld, H.J., and McDonnell, J.J.: Threshold relations in subsurface stormflow: 2. The fill and spill hypothesis, Water Resources Research, 42, W02411. 2006.

Frost Heave

Frost heave is a phenomenon of accumulation of ice due  to peculiar thermodynamics circumstances. It is quite an important phenomenon that I encounter during the thesis of Matteo Dall'Amico, and we did not treat (but I  have some ideas about). Causally I found this youtube lecture that illustrate it, and can be used to understand a little more about.

References

1. Taber, S., Frost heaving, J. Geology 37 (1929) 429-461
2. Taber, S. The mechanics of frost heaving, J. Geology 38 (1930) 303-317
3. Dash, J.G., Fu, H. and Wettlaufer, J.S., The premelting of ice and its environmental consequences, Rep.Prog.Phys. 58 (1995) 115-167
4. Rempel, A.W., Wettlaufer, J.S. and Worster, M.G. Premelting dynamics in a continuum model of frost heave, J. Fluid Mech. 298 (2004) 227-244
5. Wettlaufer, J.S. and Worster, M.G. Premelting dynamics, Annu.Rev.FluidMech. 38 (2006) 427-452
6. Wettlaufer, J.S. and Worster, M.G. , Dynamics of premelted films: Frost heave in a capillary, Phys. Rev E., 51(5), 4679-4689, 1995

Henry Darcy

A nice video on Henry Darcy on youtube. You can easily take five minutes to watch it.

Thanks to Emanuele Cordano to have indicated it to me.

Summer School on bio-geo-dynamics and Earth Sytem Sciences: The biophysical processes that shape the Earth

This School, the BESS,  has a long tradition that goes back to 1990 and was recently renewed.
I usually suggests to my students that participating is an important experience for their formation. The lecturers are very distinguished colleagues. The location (Venice and the Istituto Veneto) are fantastic. The topics are of general interest and, if not strictly covering hydrology, they are certainly useful to open the view of hydrologists who think that hydrology is a little more that fitting some data set but want to understand "how nature works".

The topics of this year were:

• Spatial and temporal controls of soil function by Bridget Emmet
• The thermodynamics of the Earth System by Axel Kleidon
• Linking pattern formation and spatial ecology by Ehud Meron
• Complex Dynamics of Forest Ecosystems at Different Scales: From 100 square meters to the Global Scale by Herman Shugart

Fortunately all the lectures are now available as streaming videos (not the slides of the lectures unfortunately) and can be viewed here.

Work of students is available instead here. Thanks to Marco Marani and Andrea Rinaldo for organising it, and thanks to the Istituto Veneto di Lettere Scienze ed Arti for making all the material available, and for supporting all of it.

Sandro Marani 1936-2012

Sandro passed away last sunday for an heart attack. He was 76. As many knows, I due to him my being here in this field.

 After my master in Physics I had a grant for working with him to "Statistical Models of the Quality of the Atmosphere of Venice Lagoon". Apparently far from Hydrology. But not in the mind of Sandro who looked at the diffusion processes from a geometrical point of view. He believed,  after reading Mandelbrot works, that diffusion patterns could be understood better with fractal geometry. However, to understand these patterns we had to visualise them: and what better than rivers networks ? In fact,  river networks could have been thought, in his view, as a "reverse diffusion" pattern from which we could learn a lot. From diffusion patterns therefore we moved to study hydro-geomorphology, and published together with Andrea Rinaldo the paper: A note on fractal channel networks. Others arrived a little before us on the subject (e.g. Tarboton et al., 1988, and La Barbera and Rosso, 1989)  but Sandro path was quite independent, and the "discover" that the flow-path distances (i.e. the width function) have a multifractal statistical structure was ours.
He was always intrigued with geometry being the inner explanation of many phenomena and therefore his interested in the geomorphic unit hydrograph (Rodriguez-Iturbe and Valdes, 1979, Gupta et al., 1980) was natural. He saw in it both the mathematical way to fit geometry into equations, and as a theory suitable to generalisations for coupling water flow and nutrients transport (and diffusion). His work on the Mass Response Function with Andrea Rinaldo (Rinaldo et al,1988) was decades in advance with respect to the interest that eventually was raised on the topic.  I believe that also Geomorphological dispersion was quite an achievement that can be listed in the "gemoetrical" effort. That work reflected the idea that, at catchment scale, geometry is as much or more important than flow dynamics to produce the form of the hydrograph.

He was a man of innumerables ideas and initiatives. Always in advance of times (maybe too in advance). The School of Environmental Dynamics at IVSLA founded with Andrea Rinaldo was the field where many of us met with science not simply with hydrology. Any edition had an eye to new insights and paradigms (the recent editions were organized by his son Marco, and maintain the same standards and vision). How much I miss those times!
In his effort to promote modelling, he  around the end of the eighties organized a "modelling connection" among environmental scientists, hydrologist , geophysicists, economists, urban planners. The most exciting guys of the Universities of Venice who met for talking about quantitative modeling.
Some colleagues choose the particulars or the details of a discipline: he chose to look at the whole ! Ideas, ideas, ideas. That he was.

As everybody can realize, my recent work has been strongly influenced by working with him. What else was building JGrass, if not giving body to Sandro's vision about processes representation and  the idea that spatial explicit modeling  was necessary for any environmental problem? What else is my commitment with modeling frameworks like OMS3?  What else is my recent involvement with thermodynamics?

He used to say: "Models are wrong ? (Sbaiemo coi modei ?) We do mistake by doing models. (Sbaiemo.) But let's imagine without! (Ma figuresemose sensa!). We should then rely just on qualitative arguments, of ignorant people, based on unformalised belief ?"

Sandro I'll miss you.