Tuesday, March 24, 2015

Probabilità in pillole

Per imparare bene la teoria delle probabilità, non c'e' dubbio lo studio approfondito di qualche buon libro è necessario. Per una piccola bibliografia personale, si veda il post "Learn Statistics and Probability!" Ma per una ripasso in pillole, ecco quanto detto e fatto a lezione:

1 - Introduzione (Audio 2015: 6Mb. YouTube 2017)
2 - Assiomi (Audio 2015: 12.6 Mb)
3 - Domini discreti e Spazi metrici (Audio 2015: 11.3 Mb)
4 - Distribuzioni uniforme e gaussiana univariate (Audio 2015: 13.9 Mb)
5 - Distribuzioni univariate di interesse idrologico (Audio 2015: 7.9 Mb)
6 - Random sampling
7 - Il teorema del limite centrale e la legge dei grandi numeri (Audio 2015: 4.2 Mb)

Altre risorse:

Le slides tutte assieme: Un ripasso di probabilità.

Audio 2014:

I Assiomi, Bayes etc. - 22Mb;
II - Distribuzioni di Probabilità - 5.6 Mb).

Qui un sintetico manualetto con alcuni temi "non standard" (dovuto a Falcioni Vulpiani).

Monday, March 23, 2015

Le precipitazioni

Ecco le slides divise sulle precipitazioni divise per argomenti.

1  - La circolazione generale e i gradienti barici
2  - Il gradiente adiabatico di temperatura
3  - La stabilità atmosferica
4  - L'evoluzione giornaliera dello strato limite
5  - Meccanismi di formazione delle precipitazioni

6  - Un po' di riassunto e di sintesi (Audio 2015. 18.3 Mb - YouTube 2017)
7  - Le caratteristiche delle precipitazioni al suolo (Audio 2015. 12.3 Mb - YouTube 2017)
8  - Le precipitazioni estreme e le curve di possibilità pluviometrica (Audio 2015. 14.9 Mb - YouTube 2017)
9  - La distribuzione di Gumbel (Audio 2015. 9.9 Mb-YouTube 2017)
10 - Gumbel - Metodo dei momenti (Audio 2015. 5 Mb - YouTube 2017)
11 - Gumbel - Massima Verosimiglianza (Audio 2015. 13.8 Mb -YouTube 2017)
12 - Gumbel - Minimi Quadrati (Audio 2015. 3.8 Mb - YouTube 2017)
13 - Test di Pearson (Audio 2015. 6.5 Mb - YouTube 2017)
14 - Chi quadro (Audio 2015. 5.3 Mb - YouTube 2017)
15 -  GEV (Audio 2015. 8.8 Mb - YouTube 2017)
16 -  Calcolo delle linee segnalatrici di possibilità pluviometrica con R^3
17 - - La misura delle precipitazioni (e gli errori della misura, secondo Lanza et al., 2005) - Audio 2014 (12.4 Mb);

Gli audio del 2014:

6/7 - Audio: Generalità (33.9 Mb);
8 - Precipitazioni estreme (19.7 Mb);
9 /11 - Gumbel Distribution, metodo dei momenti, metodo della massima verosimiglianza (13.2 Mb);
12 - Metodo dei minimi quadrati (5.3 Mb).
14 - Chi Quadro (20.7 Mb)

Note

^1 - Il mio post su R può servire per partire. Sul sito di R si trovano varie risorse per imparare ad usare R. Un gruppo italiano di utenti di R è Rante e li' vi si trova anche un manuale introduttivo ad R.  Tra gli altri strumenti, in inglese, ci sono quelli che potete trovare qui.   I contributi ad R si susseguono così velocemente che ogni giorno ce ne sono di migliori. Quindi tenete d'occhio il web. C'e' anche una versione del libro di Matloff, The Art of R programming.

^2 -  Trovate al link il file delle portate 1990-2005.txt  e il file della Pluviometria di Paperopoli (Unix/Mac o  MS-Windows) utilizzati a lezione. Qui, invece,  lo script di R con tutti i comandi eseguiti nella lezione del 2 Aprile 2012

^3 Gli script usati a lezione (ma molto più commentati) si trovano qui.
 

Esempi di relazione e materiale correlato

Una relazione sulle curve di possibilità pluviometrica
- Di Aaron Iemma (qui): in realtà bastava citare la teoria statistica, ma Aaron ha voluto essere qui un pò "scholar". Buona anche perchè Aaron ha fornito una serie di script per riprodurre tutto quanto ha fatto, mutata mutandis (qui gli script).

Bibliografia

Albertson, J., and M. Parlange, Surface Length Scales and Shear Stress: Implications for Land-Atmosphere Interaction Over Complex Terrain, Water Resour. Res., vol. 35, n. 7, p. 2121-2132, 1999

Burlando, P. and R. Rosso, (1992) Extreme storm rainfall and climatic change, Atmospheric Res., 27 (1-3), 169-189.

Burlando, P. and R. Rosso, (1993) Stochastic Models of Temporal Rainfall: Reproducibility, Estimation and Prediction of Extreme Events, in: Salas, J.D., R. Harboe, e J. Marco-Segura (eds.), Stochastic Hydrology in its Use in Water Resources Systems Simulation and Optimization, Proc. of NATO-ASI Workshop, Peniscola, Spain, September 18-29, 1989, Kluwer, pp. 137-173.

Burlando, P. e R. Rosso, (1996) Scaling and multiscaling Depth-Duration-Frequency curves of storm precipitation, J. Hydrol., vol. 187/1-2, pp. 45-64.

Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin hydrology. 1. Precipitation scenarios for the Arno River, central Italy, Hydrol. Process., 16, 1151-1175.

Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin hydrology. 2. Impacts on runoff variability of the Arno River, central Italy, Hydrol. Process., 16, 1177-1199. 

 Coles S.,‘‘An Introduction to Statistical Modeling of Extreme Values, Springer, 2001

 Coles, S., and Davinson E., Statistical Modelling of Extreme Values, 2008

Foufula-Georgiou, Lectures at 2008 Summer School on Environmental Dynamics, 2008

Fréchet M., Sur la loi de probabilité de l'écart maximum, Annales de la Société Polonaise de Mathematique, Crocovie, vol. 6, p. 93-116, 1927

Gumbel,  On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, Phil. Mag. vol. 6, p. 157-175, 1900

 Houze, Clouds Dynamics, Academic Press, 1994

Kleissl J., V. Kumar, C. Meneveau, M. B. Parlange, Numerical study of dynamic Smagorinsky models in large-eddy simulation of the atmospheric boundary layer: Validation in stable and unstable conditions, Water Resour. Res., 42, W06D10, doi:10.1029/2005WR004685, 2006

Kottegoda and R. Rosso,  Applied statistics for civil and environmental engineers, Blackwell, 2008

Kumar V., J. Kleissl, C. Meneveau, M. B. Parlange, Large-eddy simulation of a diurnal cycle of the atmospheric boundary layer: Atmospheric stability and scaling issues, Water Resour. Res., 42, W06D09, doi:10.1029/2005WR004651, 2006

Lettenmaier D.,  Stochastic modeling of precipitation with applications to climate model downscaling, in von Storch  and, Navarra A.,  Analysis of Climate Variability: Applications and Statistical Techniques,1995


Salzman, William R. (2001-08-21). "Clapeyron and Clausius–Clapeyron Equations" (in English). Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. http://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html. Retrieved 2007-10-11.

von Storch H, and Zwiers F. W, Statistical Analysis in climate Research, Cambridge University Press, 2001

Whiteman, Mountain Meteorology, Oxford University Press, p. 355, 2000

Misura e rappresentazione dei dati idrologici

Ecco qui la presentazione sui dati idrologici, suddivisa in parti:




Tutto il blocco di argomentii assieme nella versione 2014. Audio (21.6 Mb)

Bibliografia

  • Agnoli, P.,  Il senso della misura, la codifica della realtà tra filosofia, scienza ed  esistenza umana, Armando Editore, 2004            
  • Agnoli, P., Breve introduzione storica alle prime unità di misura, http://www.roma1.infn.it/~dagos/SSIS/PaoloAgnoli_appuntimisure.pdf , 2006, last retrieved 2011/03/18
  • AA.VV, Le misure nella scienza, nella tecnica, nella società, Manuale di metrologia,  a cura di S. Sartori,   Paravia, Torino,1979
  • AA. VV,  Le misure di grandezze fisiche, a  cura di E. Arri e S. Sartori, Paravia, 
  • Torino,1984 
  • Burroughs, W., J, Weather Cycles,  Cambridge U. P.,  2003
  • Grünewald, T., Schirmer, M., Mott, R., & Lehning, M. (2010). Spatial and temporal variability of snow depth and ablation rates in a small mountain catchment. The Cryosphere, 4(2), 215-225. doi:10.5194/tc-4-215-2010
  • Loreti, M., Teoria degli Errori e Fondamenti di Statistica: Introduzione alla Fisica Sperimentale, 2006, http://wwwcdf.pd.infn.it/labo/INDEX.html, last retrieved 2011/03/18
  • Roth, K. (2007). Soil Physics Lecture Notes (p. 1-340), 2007, http://www.iup.uni-heidelberg.de/institut/forschung/groups/ts/soil_physics/students/lecture_notes05/sp.pdf, last retrieved 2011/03/18
  •  Shuttleworth, W. James (January/February 2008). "Evapotranspiration Measurement Methods". Southwest Hydrology (Tucson, AZ) 7 (1): 22–23. Retrieved 2009-07-22.
  • Western, Andrew W. (2005). "Principles of Hydrological Measurements". In Anderson, Malcolm G.. Encyclopedia of Hydrological Sciences. 1. West Sussex, England: John Wiley & Sons Inc.. pp. 75–94
  • Tabony, R. C.  (1979), A spectral filter analysis of long period records in England and Wales, Meterol. Mag. 108, 97-119 
  • Zambrano-Bigiarini, M. (2010). On the effects of hydrological uncertainty in assessing the impacts of climate change on water resources, 1-293.

Links to web sites

Saturday, March 21, 2015

Four interesting papers on Hydrological modelling

I met the first time Martyn Clark in Fort Collins last summer. USGS scientist Stacey Archfield organised a meeting for modellers (to which I was not invited :-( ), and Martyn was part of the crew.
I did not his work up to recently. and after our meeting, he came out with three Water Resources Research Papers, that I annotating here for subsequent readings.


Clark, M. P., Kavetski, D., & Fenicia, F. (2011). Pursuing the method of multiple working hypotheses for hydrological modeling. Water Resources Research, 47(9), n/a–n/a. doi:10.1029/2010WR009827

Pablo A. Mendoza, Martyn P. Clark, Michael Barlage, Balaji Rajagopalan, Luis Samaniego, Gab Abramowitz and Hoshin Gupta, Are we unnecessarily constraining the agility of complex process-based models? , Water Resources Research, Volume 51, Issue 1, pages 716–728, January 2015

Clark, M. P., B. Nijssen, J. Lundquist, D. Kavetski, D. E. Rupp, R. A. Woods, J. E. Freer, E. D. Gutmann, A. E. Wood, L. D. Brekke, J. A. Arnold, D. Gochis, R. Rasmussen. 2015. A unified approach for process-based hydrologic modeling: Part 1. Modeling concept, Water Resources Research, doi:10.1002/2015WR017198.

Clark, M. P., B. Nijssen, J. Lundquist, D. Kavetski, D. E. Rupp, R. A. Woods, E. D. Gutmann, A. W. Wood, D. Gochis, R. Rasmussen, D. Tarboton, V. Mahat, G. Flerschinger, D. Marks. 2015. A unified approach for process-based hydrologic modeling: Part 2. Model implementation and case studies, Water Resources Research, doi:10.1002/2015WR017200.


A late addition, the technical note regarding this SUMMA stuff.

Thursday, March 19, 2015

JGrass-NewAGE essentials

Also JGrass-NewAGE posts have grown too much. So where to start, when you want to know something about this system ? 

A good reading is for sure, the history of JGrass-NewAGE.  The second step is giving a look to the papers at in the three references here following their order.

Finally browsing the rest of the posts, and the rest of references you find would complete the task.
For the documentation of software modules (which includes test file), please refer to the GEOframe blog.

Codes can be found at the GEOframe repository.
NewAGE users can send inquiries at newageusers@googlegroups.com
NewAGE developers discussions happen at jgrass-newage-dev@googlegroups.com

References

1 - Formetta G., Antonello A., Franceschi S., David O., and Rigon R., Hydrological modelling with components: A GIS-based open-source framework, Environmental Modelling & Software, 5 (2014), 190-200

2 - Formetta, G.; Mantilla, R.; Franceschi, S., Antonello A., Rigon R., The JGrass- NewAge system for forecasting and managing the hydrological budgets at the basin scale: models of flow generation and propagation/routing, Geoscientific Model Development Volume: 4 Issue: 4 Pages: 943-955, DOI: 10.5194/gmd-4- 943-201, 2011 

3 - Formetta, G., Kampf, S. K., David, O., and Rigon, R.: Snow water equivalent modeling components in NewAge-JGrass, Geosci. Model Dev., 7, 725-736, doi:10.5194/gmd-7-725-2014, 2014.



Other two ERC Grants in Water related topics

They escape to my attention so far, but other two ERC grants were given to water scientists. First grant was given to Dani Or (GS, Blog)

“The Hidden Frontier: Quantitative Exploration of Physical and Ecological Origins of Microbial Diversity in Soil”



and, as the name let's to understand is related to the soil ecology of and microbiology.
The other ERC grant was given to Marco Dentz (GS) on Transport and mixing:

Sunday, March 8, 2015

The geomorphic unit hydrograph from a historical-critical perspective

I have just submitted to the review process a paper on the geomorphic unit hydrograph. (Now published here). It crosses close to four decades of progress the field of describing the treatment of the hydrologic response with the travel time concept, and hopefully was able to convey the core of the ideas behind this very nice (and effective) theory, and to open the way to new researches. The paper was accepted as a "State of  art paper" by Earth Surface Processes and Landforms.

Its abstract:
"In this paper we present a brief overview of Geomorphic Instanteneous Unit Hydro- graph (GIUH) theories and analyze their successful path without hiding their limitations. The history of the GIUH can be subdivided into three major chapters. The first chap- ter is based on the pioneering work by Rodríguez-Iturbe and Valdés (1979), and Gupta and Waymire (1983), which recognized that a treatment of water discharges with ”travel times” could give a rich interpretation of the theory of the Instantaneous Unit Hydrograph (IUH). We show how this was possible, what assumptions were made, which of these assumptions can be relaxed, and which have become obsolete and been discarded. The second chapter focuses on the Width Function Based IUH (WFIUH) approach and its achievements in assessing the interplay of the topology and geometry of the network with water dynamics. The limitations of the WFIUH approach are described, and a way to work around them is suggested. Finally, a new formal approach to estimating the water budget by ”travel times”, which derives from a suitable use of the water budget equation and some mixing hypotheses, has been disentangled and presented."


The little post on the width function can be considered a complementary reading too.  The pre-print of the paper is here.

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Tarboton DG, Bras RL, Rodriguez-Iturbe I, 1991. On the extraction of channel networks from digital elevation data. Hydrological Processes 5(1): 81–100. ISSN 1099-1085. doi: 10.1002/hyp.3360050107.

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van der Tak LD, Bras RL, 1990. Incorporating hillslope effects into the geomorphologic instantaneous unit hydrograph. Water Resources Research 26(10): 2393–2400. ISSN 1944-7973. doi:10.1029/WR026i010p02393.

Wang CT, Gupta VK, Waymire E, 1981. A geomorphologic synthesis of nonlinearity in sur- face runoff. Water Resour. Res. 17(3): 545–554. doi:198110.1029/WR017i003p00545.

Wilkinson L, Friendly M, 2009. The History of the Cluster Heat Map. The American Statistician 63(2): 179–184. doi:10.1198/tas.2009.0033.

Woods R, Sivapalan M, 1999. A synthesis of space-time variability in storm response: Rainfall, runoff generation, and routing. Water Resour. Res. 35(8): 2469–2485. doi: 199910.1029/1999WR900014.

Zoccatelli D, Borga M, Viglione A, Chirico GB, Blöschl G, 2011. Spatial moments of catch- ment rainfall: rainfall spatial organisation, basin morphology, and flood response. Hydrol- ogy and Earth System Sciences 15(12): 3767–3783. doi:10.5194/hess-15-3767-2011.


Zoch RT, 1937. On the relation between rainfall and stream flow. Monthly Weather Review 55: 135–147.

The width function

The width function is said to have been introduced in 1976 by Mike Kirkby (one of his many achievements). If you consider the map of the distances from any point in a basin to the outlet, following the drainage direction, you get the map of distances to outlet, a map easy to obtain from DEMs, as below:
The width function is actually the distribution of the above distances, which can be normalized by the total area to obtain a probability distribution function (of distances from the outlets along the drainage path).  Below, please find the width function of the above basin.

The width function has a quite irregular form, and it has been suggested by Marani, S. et al, 1991 that he can be seen as the product of a multiplicative dissection of the area and having, in particular conditions, the structure of a multifractal (as in the case of the Peano curve) - see also Marani, M. et al., 1994.  Anyway, looking at its parts you have a picture like this ones below. In the first, "strips" of five kilometers are considered:
In the second, strips of 2.5 kilometers are considered:
Wow, in the third, strips of one kilometer are eventually considered:

What happens here is that point at the same (kilometric) distance to outlet do not actually constitute continuous strips, and are, in some cases far apart. This is clearly a manifestation of how the river network structure is non trivial. Figures are by Giuseppe Formetta. The basin is the Little Washita in Oklahoma. Reproducing the same pictures is easy with STAGE

References

Kirkby MJ, 1976. Tests of the random network model, and its application to basin hydrology. Earth Surface Processes 1(3): 197–212. ISSN 1931-8065. doi:10.1002/esp.3290010302.

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

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.  


Wednesday, March 4, 2015

And the Winner is ... QGIS (?)

Quest'anno anzich`è usare la tradizionale piattaforma uDig, che è bloccata nel suo spostamento da Refractions a Locationtech (una scelta che credo sia stata corretta, ma lo si vedrà su tempi un pò più lunghi), useremo Quantum GIS o, più amichevolmente, QGIS.
QGIS è uno dei GIS open source di maggior successo, è supportato da un "vibrant" user group che lo ha dotato di manuali e tutorial di vario grado e complessità, include un collegamento al buon vecchio GRASS, di cui può, quindi, usare vari moduli. E' dotato di un porting delle librerie Sextante, e di TauDEM, le librerie di analisi del terreno di David Tarboton. Come se non bastasse, sempre più amministrazioni lo adottano come alternativa open Source alla famiglia commerciale di ESRI.  Non resta che arrendersi ?
Noi naturalmente non ci arrendiamo del tutto ed useremo QGIS solo per visualizzare e stampare le mappe, mentre le produrremo con STAGE. Cliccando sul logo siete diretti ad una semplice introduzione a QGIS, ma le risorse sul web sono molte e varie

Installing STAGE and using JGrasstools

Questo post presenta l'applicazione STAGE (Spatial Toolbox And Geoscripting Environment) che e lo strumento che useremo quest'anno nel corso di Idrologia per delineare i bacini idrografici e fare alcune analisi geomorfologiche.  STAGE è la più recente versione dello Spatial Toolbox, nell'attesa che uDig riacquisti le funzionalità perdute e STAGE possa esservi integrato.

Cliccando sull'immagine sopra, si accede direttamente alla presentazione di introduzione ai JGrasstools. Altre informazioni sui JGrasstools, in Italiano, si possono trovare in questo vecchio post di cui però riassumo gli elementi importanti qui sotto.

Il materiale sulla Horton Machine, ovvero per il trattamento dei dati digitali del terreno, è in un post separato. Altre risorse sono certamente utili. Per esempio, la serie di screen-cast che si puo' trovare sul canale Youtube udiggis

benche' siano in Inglese. Quanti volessero aiutare, li invito a leggere il post di Andrea Antonello.

To early stage hydrologists (ESH)

Dear ESH,

you ask me to advise a young hydrologists. I just think that  he/she needs a master, first. I had the fortune to work at the beginning with Alessandro Marani, and subsequently Andrea Rinaldo and Ignacio Rodriguez-Iturbe. They had strong ideas on what is “the gold medal” that we have to gain, and had their own methods to pursue it (get ideas, get methods!). So, at least for me, the matter was to follow their own inclination, challenging myself to follow them, and see how much I liked it. 

So my first hint is: look for (one) some masters. You soon realise that some guys (or some places) are far far beyond the present state-of-art of the topic you are caring about.  Try to work with them. If you cannot, try to follow them and see what they do,  and understand what they think it is important. 


Copying from “smart people, can help you to save a lot of time, and avoid to spend huge time resources following unimportant topics.   In my academic life I saw very many “brilliant” guys (gals) spending their youth and hundreds of hours on irrelevant problems. However, if you do not give up, you will get the hot topic (and the people) that copes with you. The research company you get along, is going to stick with you for all of your life, and change you, especially  if you start an Academic path.

Not very much, indeed (meet the right people!), as an advise, but in a previous post you can find my decalogue for a hydrologist that can complete the topic.

Tuesday, March 3, 2015

Other types of numerics (for ecology)

When I think to numerics, I always think to the "Concrete Mathematics" that Donald Knuth taught to us, or to way to integrate partial differential equations. Recently, however, I came across the book by Legendre and Legendre called "Numerical Ecology", which is something different.
 The work is something in between statistics and classification or clustering stuff, interpretation of structures. Interpretation of spatial data. A hybrid of things, in my view, interesting to know, anyway.

Reference

Legendre, P and Legendre L, Numerical Ecology, Amsterdam, Elsevier, 2003

Monday, March 2, 2015

Introduzione alla geomorfologia

La geomorfometria è la scienza che studia in modo quantitativo la geomorfologia, specialmente attraverso l'uso dei nuovi strumenti digitali. Da anni ho sviluppato con i miei collaboratori degli strumenti (dei software) di analisi raccolti nella  Horton Machine.

La lezione è divisa in 5 parti:

1- Gli elementi di base. Audio  2014: (18.4 Mb). Audio 2015 (22.9 Mb)
2 - Le grandezze derivateAudio 2014 (24 Mb). Audio 2015 (10.4 Mb)
3 - Dove iniziano i canali. Audio 2015 (12 Mb)
4 - La sintesi digitale delle conoscenze sui bacini idrografici. Audio 2015 (4.6 Mb)
5 -Leggi geomorfologiche, processi idrologici e geomorfologiaAudio 2014 (22.8 Mb). Audio 2015 (15.4 Mb)

Tutto il materiale di supporto sulla Horton Machine, già nominato sopra,  si trova in quest'altro post.

Bibliografia (citata nelle slides)

Broscoe, A.J., 1995, Quantitative analysis of longitudinal stream profiles of small watersheds, Office of Naval Research, Project NR 389-042, Technical Report No. 18, Department of Geology, Columbia University, New York.

Howard A.D., A detachment-limited model of drainage basin evolution, Water Resources Research, vol. 30, n. 7, p. 2261-2285, 1994.

Leopold, L.B., and Maddock, T., Jr, The hydraulic geometry of stream channels and some physiographic implications: U.S. Geological Survey Professional Paper 252. 57p, 1953 

Maidment D.R., ed., Arc Hydro: GIS for Water Resources, ESRI Press, Redlands, Ca, 2002

Montgomery D.R. & Dietrich W.E., Channel initiation and the problem of landscape scale, Science, vol. 255, p. 826-830, 1992.

Moretti and Orlandini. Automatic delineation of drainage basins from elevation contour data using skeleton construction techniques.  (2007) pp. 1-39

Orlandini et al. On the prediction of channel heads in a complex alpine terrain using gridded elevation data. Water Resour. Res. (2011) vol. 47 (2) pp. W02538

Peckham S., New results for self-similar trees with applications to river networks, Water Resources Research, vol. 31, n. 4, p. 1023–1029, 1995

Peckham and Jordan. Digital Terrain Modelling. Lecture Notes In Geoinformation and cartography (2007) pp. 1-326

Rigon R., I. Rodriguez-Iturbe, A. Rinaldo, A. Maritan, A. Giacometti and D. Tarboton, On Hack's law, Water Resources Research, vol. 32, n. 11, p. 3367, 1996

Rigon R., Ghesla E., Tiso C. & Cozzini A., Cozzini The HORTON machine: a system for DEM analysis : the reference manual . Trento: Università di Trento. Dipartimento di ingegneria civile e ambientale, May 2006. - p. viii, 136, ISBN 10:88-8443-147-6, 

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

Rodriguez-Iturbe, I. and Rinaldo, A.: Fractal River Basins. Chance and Self-Organization, Cambridge University Press, New York, 1997.

Tarboton, D.G., A new method for the determination of flow directions and contributing areas in Grid Digital Elevation Models, Water Resources Research, vol. 33, n. 2, 309-319, http://www.engineering.usu.edu/cee/faculty/dtarb/dinf.pdf 

Tarboton, D.G., R.L. Bras and Rodriguez-Iturbe, 1992, A Physical Basis for Drainage Density, Geomorphology, vol. 5, n. 1/2

Venuleo, S., Analisi teoriche e di campo per la caratterizzazione di bacini idrografici montani,  Tesi di Laurea, Relatori: Rinaldo A., Passadorr, G., Padova, 2014

Wilson, J. P. and J. C. Gallant, (2000), Terrain Analysis: Principles and Applications, John Wiley and Sons, New York, 479 p.

Wood, J.D. (1996) The geomorphological characterisation of digital elevation models PhD Thesis, University of Leicester, UK