Rills in Utah

Going from Zion National Park to Grand Arches National Park, I had the occasion to see several astonishing geological landscapes. Particularly exciting to me were the rillings that I could observe along the road, eroded in different types of lithology. Here below some images at low resolution (by clicking on the image you can get the higher resolution). Unfortunately I took them with a cellular phone and therefore they are not so clear as they could.

In the first image, taken close to the capitol Reef national park, on 24th National Road, where red sedimentary rocks are present, above rills are in the central part of the image. On top other processes than erosion dominate, i.e. rockfall, but in the bottom deposits rills and fluvial type of forms appear everywhere. 

Same as above, but in a different material (taken close to Hanksville). On the bottom, rock formations that repeat sequentially are also present. The geometry there is convex-divergent but still filled with rills.

Same material as above. Different geometries, more pronounced aggregative formations.

Same are as above a complete network eroded in the center of the image. Remarkably a sedimentary (?) layer across the formation but without very much effect on the rilling structure.

I do not know if a laser altimeter survey of the area is available, but in the case, it would be really interesting for  geomorphologists to analyse the literally thousands of rills and river networks that formed in this arid environments.

"Two Rivulets side by side,
Two blended, parallel, strolling tides,
Companions, travelers, gossiping as they journey."
W. Withman

River Networks as Ecological Corridors for Species Populations and Water borne Diseases

This was the talk given by Andrea Rinaldo for the opening day of the 2013-2014 Doctoral Academic year of our doctoral School. Andrea is possibly the most influential Italian hydrologist ever (if we do not count as proto-hydrologist Leonardo da Vinci).  

 It was a fascinating and passionate travel through river networks structure (their topology, their physical origin), a research of which I was a active witness for some period, and the spreading of diseases or populations (but without loosing appropriate generality in the description of the phenomena) along the networks. 

Please find clicking on the figure the presentation.

Cited References (The ones I could reconstruct)

Ammerman & Cavalli Sforza, The Neolithic transition and the Genetics of population in Europe, Princeton Univ. Press 1984^1

Bertuzzo, E., A. Maritan, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo (2007), River networks and ecological corridors:Reactive transport on fractals, migration fronts, hydrochory, Water Resour. Res., 43, W04419, doi:10.1029/2006WR005533. 

Bertuzzo, E., S. Azaele, A. Maritan, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo (2008), On the space-time evolution of a cholera epidemic, Water Resour. Res., 44, W01424, doi:10.1029/2007WR006211. 

E. Bertuzzo,Casagrandi  R. , Gatto M. ,Rodriguez-Iturbe  I., 

and Rinaldo A. , On spatially explicit models of cholera epidemics, J. R. Soc. Interface (2010) 7, 321–333 doi:10.1098/rsif.2009.0204

E Bertuzzo, L Mari, L Righetto, M Gatto, R Casagrandi, M Blokesch, I Rodriguez‐Iturbe, A Rinaldo, Prediction of the spatial evolution and effects of control measures for the unfolding Haiti cholera outbreak, Geophysical Research Letters, Vol 38, No 6, 2011

Campos, D., Fort, J., & Méndez, V. (2006). Transport on fractal river networks: Application to migration fronts. Theoretical Population Biology, 69(1), 88–93. doi:10.1016/j.tpb.2005.09.001

Carrara, F., Altermatt, F., Rodriguez-Iturbe, & Rinaldo, A. (2012). Dendritic connectivity controls biodiversity patterns in experimental metacommunities, 1–6. doi:10.1073/pnas.1119651109/-/DCSupplemental/pnas.201119651SI.pdf

F. Carrara, A. Rinaldo, A. Giometto and F. Altermatt. Complex Interaction of Dendritic Connectivity and Hierarchical Patch Size on Biodiversity in River-Like Landscapes, in The American Naturalist, vol. 183, num. 1, p. 13-25, 2014.

Chao, D., Halloran, M. E., & Longini, I. M. (2011). Vaccination strategies for epidemic cholera in Haiti with implications for the developing world. Pnas, 1–5. doi:10.1073/pnas.1102149108/-/DCSupplemental/sapp.pdf

 Codeço, C. T. (2001). Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infectious Disease, 1–14. 

Capasso V, Paveri-Fontana SL., A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev Epidemiol Sante Publique. 1979 Sep 18;27(2):121-32.

Colaiori et al., PRE, 2003 ?

Gatto, M., Mari, L., Bertuzzo, E., Casagrandi, R., Righetto, L., Rodriguez-Iturbe, I., & Rinaldo, A. (2012). Generalized reproduction numbers and the prediction of patterns in waterborne disease. PNAS, 1–6. doi:10.1073/pnas.1217567109/-/DCSupplemental

M Gatto, L Mari, E Bertuzzo, R Casagrandi, L Righetto, I Rodriguez-Iturbe, and A. Rinaldo, Spatially explicit conditions for waterborne pathogen invasion, The American Naturalist 182 (3), 328-346

A Giometto, F Altermatt, F Carrara, A Maritan, A Rinaldo, Scaling body size fluctuations

Proceedings of the National Academy of Sciences 110 (12), 4646-4650

Hartley et al, PNAS, 2006

Hubbel, 2001

Greg Huber, Scheidegger's rivers, Takayasu's aggregates and continued fractions, Physica A: Statistical Mechanics and its Applications, Volume 170, Issue 3, 15 January 1991, Pages 463–470

A Kolmogorov, L Petrovsky, N Piskunov, An investigation of the diffusion equation combined with an increase in mass and its application to a biological problem - Bull. Uni. Moscow Ser. Int. A, 1937

Leopold, L. B.; Langbein, W. B., The concept of entropy in landscape evolution

1962,  USGS Professional Paper: 500-A 

Marani, A., R. Rigon and A. Rinaldo, A Note on fractal channel networks,Water Resources Research, (27)5, 3041-3049, 1991.

L Mari, E Bertuzzo, R Casagrandi, A Rinaldo, M Gatto,
A spatially-explicit model for the spread of the zebra mussel (Dreissena polymorpha) along river networks: some preliminary results

L Mari, E Bertuzzo, L Righetto, R Casagrandi, M Gatto, I Rodriguez-Iturbe, and A. Rinaldo,

Modelling cholera epidemics: the role of waterways, human mobility and sanitation

Journal of The Royal Society Interface 9 (67), 376-388

L Mari, E Bertuzzo, R Casagrandi, M Gatto, SA Levin, I Rodriguez‐Iturbe, and A. Rinaldo, Hydrologic controls and anthropogenic drivers of the zebra mussel invasion of the Mississippi‐Missouri river system

Water Resources Research 47 (3) 

L Mari, R Casagrandi, E Bertuzzo, A Rinaldo, M Gatto,
Metapopulation persistence and species spread in river networks

Ecology letters

R Muneepeerakul, JS Weitz, SA Levin, A Rinaldo, I Rodriguez-Iturbe,

A neutral metapopulation model of biodiversity in river networks

Journal of Theoretical Biology 245 (2), 351-363 2007

R Muneepeerakul, A Rinaldo, I Rodriguez‐Iturbe,
Effects of river flow scaling properties on riparian width and vegetation biomass

Water Resources Research 43 (12)

R Muneepeerakul, SA Levin, A Rinaldo, I Rodriguez‐Iturbe,
On biodiversity in river networks: A trade‐off metapopulation model and comparative analysis

Water Resources Research 43 (7)

R Muneepeerakul, E Bertuzzo, HJ Lynch, WF Fagan, A Rinaldo, ...
Neutral metacommunity models predict fish diversity patterns in Mississippi–Missouri basin,

Nature 453 (7192), 220-222, 2008

DW Murray, AJ Carr et al., Survival analysis of joint replacements,  Journal of Bone & Joint …, 1993 - bjj.boneandjoint.org.uk

    

Pascual M, Bouma MJ, Dobson AP (2002) Cholera and climate: Revisiting the quantitative evidence. Microbes Infect 4: 237–245

Pascual, M. and  Dobson A., Seasonal Patterns of Infectious Diseases, 2005, DOI: 10.1371/journal.pmed.0020005

Renaud Piarroux, Robert Barrais, Benoît Faucher, Rachel Haus, Martine Piarroux, Jean Gaudart, Roc Magloire, and Didier Raoult, Understanding the Cholera Epidemic, Haiti, Emerg Infect Dis. 2011 July; 17(7): 1161–1168. doi:  10.3201/eid1707.110059

PMCID: PMC3381400

Rinaldo et al., PNAS, in press

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.

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.

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.

Rodriguez-Iturbe & Rinaldo, Fractal River Basins: Chance and Self-Organization, Cambridge Univ, Press, 2007

M Takayasu, A Provata, G Huber,  Statistical models of river networks, J. Stat. Phys 65, 725-745, 1991

_________________________________________________________________________________

^1 - See also: M. Gkiasta, T. Russell, S.Shennan, and James Steele, Neolithic transiti on in Europe: the radiocarbon record revisited - UCL

Rivers in transitions

An editorial of Nature Geoscience to read.

"Rivers affect landscape structure and function to a much greater extent than might be expected from the fraction of the Earth's surface they cover. Rivers redistribute material as they flow, carving out canyons and building new land offshore. These morphological consequences of river flow are evident in any topographic map of the Earth's surface. ..."

Other links are also available at the page.

Browsing the same number also a commentary is available on the role of rivers, and a paper on river drainage patterns in New Zealand.

My Past Research on Hydro-geomorphology

The work on evolution of river networks, certainly enters also in this category. Here however, it is reported about those papers that deals directly with quantitative geomorphological analysis.
In most of the papers, the relationships between the various parts of a river basin are analyzed with fractal geometry techniques.  Such knowledge is useful not only for the evaluation of river basin evolution models, but also in  identifying the nature of their hydrological response to given events and their paleoclimate.  In [J2], using the Peano Basin, a mathematical reference structure, it is suggested that the amplitude function of natural networks can be reproduced with a multifractal multiplicative
process.   In [J11] this concept was rigorously formalized in the framework of random multifractal cascades theory.

The fractal properties, that is  power laws relating to  contributing areas and the length of stream reaches, were rigorously analyzed in [J15]. In [J17] more relationships between these characteristic quantities were found and an explanation of the nature of Hack's law is suggested.
In [J27] the structure of the river networks is further investigated by analysing the tributaries statistics. The relationships were verified experimentally by means of Digital Elevation Models (DEM).

Given programs for the extraction of digital terrain models (DEM) and their treatment, the natural development was the implementation of an open-source geographic information system: JGrass [s2].  JGrass, now part of uDig, contains within the package a large amount of GIS methods, jointly known as the Horton Machine [eb-3] to support the most common and some less common tools of analysis for river networks topology,  channel extraction, hillslope delineation. A review of these tools is in [a57].

Papers and work on landslide triggering are also concerned with geomorphology but referred in a different post.

References

[j2] - Marani, A., R. Rigon and A. Rinaldo, A Note on fractal channel networks,Water Resources Research, (27)5, 3041-3049, 1991.

[j11] - Marani, M., A. Rinaldo, R. Rigon and I. Rodriguez-Iturbe, Geomorphological width function and the random cascade, Geophysical Research Letters, 21(19), 2123-2126, 1994.

[j15] - Maritan, A., A. Rinaldo, R. Rigon, I. Rodriguez-iturbe and A. Giacometti, Scaling laws for river networks, Physical Review E , 53, 1510, 1996.

[j17] - Rigon, R., I. Rodriguez-Iturbe, A. Rinaldo, A. Maritan, A. Giacometti and D. Tarboton, On Hack’s law, Water Resources Research, 32(11), 3367, 1996

[s2] Jgrass (since 2002): it is a full featured GIS system built to support the hydro- geomorphological research of the group since the early 2000s. Recently it has been integrated in uDig. Current developments (which go far beyond my contribution) are available at: http://udig.refractions.net/

[eb3] - 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

[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

[a57]- W. Abera, A. Antonello, S. Franceschi, G. Formetta, R Rigon , "The uDig Spatial Toolbox for hydro-geomorphic analysis" in Geomorphological Techniques, v. 4, n. 1 (2014), p. 1-19

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

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.

Ignacio Rodriguez-Iturbe talk at Berkley in 2007 about Hydrology in 21st century

A talk of one of my masters, Ignacio Rodriguez-Iturbe, where he shows some of the work we did together more than ten years ago.

Obviously he, and Andrea Rinaldo were illuminating my way: but it was also my own way. It is always a pleasure to hear "El Jefe" talk.

Ignacio's talk starts around 6:10 m. The reference for the work to which I participated it is "The Book": Fractal River Networks, chance and self organization, Cambridge University Press, 1997.  Ecohydrology is partially covered in Ecohydrology of water controlled ecosystems: soil moisture and plants dynamics.