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Ezio Todin symposium talk


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Still to be finished, this is the talk for the Ezio Todini 70th celebration.

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Ezio Todin symposium talk

  1. 1. About the compromiseamong conceptual, mathematical and numericaltractability in some hydrological modelsThe&NoAward2-CoverRiccardo RigonWednesday, June 5, 13
  2. 2. We discover ... that all our laws can bewritten in mathematical form; and thatthis has a certain simplicity and beautyabout it. So, ultimately, in order tounderstand nature it may be necessary tohave a deeper understanding ofmathematical relationships*R. Feynman*i.e. equations, and differential equationsWednesday, June 5, 13
  3. 3. 3Extract from the AbstractWe all either try to formulate laws at one scale byguessing them, using the available knowledge, or try todeduce them by a mix of algebraic treatment of thebasic laws of mass, energy and momentumconservation, and educated simplifications.It is important to have the true equations !mass, momentum and energy conservationarethe most “true” equations we knowI liked my abstract ... maybe because it was so obvious!R. RigonWednesday, June 5, 13
  4. 4. 4This is what we did with the GEOtop modelRigon et al. 2006Process based modelsR. RigonWednesday, June 5, 13
  5. 5. 5Picasso,DoraMaarBut since the perfect model does not exist•shallow water equations for surface flow•Richards’ equation for subsurface flowindeed we adopt:Freeze and Harlan, 1969 (but with much better numerics)Process based modelsR. RigonWednesday, June 5, 13
  6. 6. 6Actually our statement on Richards’ equationis that what it is true is thisMass conservation (no nuclear reactions) !but actually true if the continuum (a.k.a. Darcy) hypothesis is validProcess based modelsR. RigonWednesday, June 5, 13
  7. 7. Not necessarily this:7Se = [1 + ( ⇥)m)]nSe :=w r⇥s rC(⇥)⇤⇥⇤t= ⇥ · K( w) ⇥ (z + ⇥)⇥K( w) = Ks⇧Se⇤1 (1 Se)1/m⇥m⌅2SWRC +Darcy-BuckinghamParametricMualemParametricvan GenuchtenC(⇥) :=⇤ w()⇤⇥Process based modelsR. RigonWednesday, June 5, 13
  8. 8. 8The last representation of mass conservationis just matter of conveniencehabits, and ignorance of some phenomenaI bet that others talked about some other phenomena, but, in theprevious slide, we were also missing:•variable and changing temperature•soil freezing•transition to saturationProcess based modelsR. RigonWednesday, June 5, 13
  9. 9. Richards equation is “wrong” !9Sure. But then, what else I should use:•Green-Ampt ?•SCS ?•Topmodel ?I use all of them when I find convenient.However, all of them are even more “wrong” than Richards. So for the firstpart of this talk I stick with Richards’ assumptions.Take it as my null hypothesisBetter wrong than “not even wrong”R. RigonWednesday, June 5, 13
  10. 10. 10Richards’s eq. was said to be too much computationalexpensiveThis statement is also true but, the real truth is that up to 1990 (M.Celia et al.) we did not have an appropriate numerical method tosolve them, and the right numerics penetrated slowly in the community.Now we have many (including the recents Casulli and Zanolli, 2010, 2012),and we can, at least, explore with some confidence their behavior.Better wrong than “not even wrong”R. RigonWednesday, June 5, 13
  11. 11. 11To exaggerate•energy budget: turbulent flows, heat equation, soilfreezing, snow budgetwe addedstill Freeze and Harlan, 1969 ?EndrizziandMarsh,2010;Dall’Amicoetal.,2011,Endrizzietal.,2013Bertoldietal.,2010a,bBetter wrong than “not even wrong”R. RigonWednesday, June 5, 13
  12. 12. 12Is it feasible ?Is it usable ?Does it works ?We did it !It is useful ?e.g Beven, 2000, 2001 (for instance) criticized this approach but weneeded anyway a reference model to start withYes, it is!We “forecasted” decently well: water flows, soil moisture,landslides, terrain temperatures, evapotranspiration, snowcover ...Better wrong than “not even wrong”R. RigonWednesday, June 5, 13
  13. 13. 13For instance a service for forecastingsnow heightsbased on GEOtop is currently operationalseehttp://www.mountain-eering.comBetter wrong than “not even wrong”R. RigonWednesday, June 5, 13
  14. 14. 14So that’s the end of the story ?certainly not !The criticism to this type of modelling have foundations.GEOtop NewAge Boussinesq PeakFlowSHALSTAB GEOtop-FS The Horton Machineand we have several models that we use at different scales and fordifferent purposesWe did not marry process based modelsR. RigonWednesday, June 5, 13
  15. 15. 15.. In practice, when in the hands of hydrologists both the approachescontaminate each other, and represent some compromise amongexperimental evidence, scientific knowledge, mathematicalconvenience, and computational tractability ... and the natural lazinessthat everybody has.Extract from the AbstractGEOtop NewAge Boussinesq PeakFlowSHALSTAB GEOtop-FS The Horton MachineWe did not marry process based modelsR. RigonWednesday, June 5, 13
  16. 16. 16In some models wetreat just one process,i n o t h e r , l i k e i nNewAGE*, we treatthem all again.The goal here was to simplify theequations as much as possible butmaintaining a spatially variabledescription of the modelsFormetta et al., 2011, 2013a,b,cBut I am not going to talk about it!Please look at the Poster session.We did not marry process based modelsR. RigonWednesday, June 5, 13
  17. 17. 17Using  modelsSimulaon:  imitate  one  process  by  another  processProcess:  temporal  sequence  of  states  of  a  system“In  computer  simulaons  of  physical  systems,  the  construcon  of  models  is  guided,  but  not  determined,  by  theory.  At  the  same  me  simulaons  models  are  o>en  constructed  precisely  because  data  are  sparse.  They  are  meant  to  replace  experiments  and  observaons  as  sources  of  data  about  the  world;  hence  they  cannot  be  evaluated  simply  by  being  compared  to  the  world.  So  what  can  be  the  source  of  credibility  for  simulaon  models?  I  argue  that  the  credibility  of  a  simulaon  model  comes  not  only  from  the  credenals  supplied  to  it  by  the  governing  theory,  but  also  from  the  antecedently  established  credenals  of  the  model  building  techniques  employed  by  the  simulaonists.  In  other  words,  there  are  certain  sorts  of  model  building  techniques  which  are  taken,  in  and  of  themselves,  to  be  reliable.  Some  of  these  model  building  techniques,  moreover,  incorporate  what  are  somemes  called  ‘‘falsificaons’’.  These  are  contrary-­‐to-­‐fact  principles  that  are  included  in  a  simulaon  model  and  whose  inclusion  is  taken  to  increase  the  reliability  of  the  results.  The  example  of  a  falsificaon  that  I  consider,  called  arficial  viscosity,  is  in  widespread  use  in  computaonal  fluid  dynamics.  Arficial  viscosity,  I  argue,  is  a  principle  that  is  successfully  and  reliably  used  across  a  wide  domain  of  fluid  dynamical  applicaons,  but  it  does  not  offer  even  an  approximately  ‘‘realisc’’  or  true  account  of  fluids.  Arficial  viscosity,  therefore,  is  a  counter-­‐example  to  the  principle  that  success  implies  truth  –  a  principle  at  the  foundaon  of  scienfic  realism.  It  is  an  example  of  reliability  without  truth.”  (Winsberg,  2006)Hartmann,  S.  (1996),  The  World  as  a  Process:  Simula=ons  in  Natural  and  Social  Sciences,  in:  Hegselmann,  R.,  U.  Mueller,  K.  Troitzsch  (eds.),  Simulaon  and  Modelling  in  the  Social  Sciences  from  the  Philosophy  of  Science  Point  of  View,  Kluwert,  77-­‐100.Winsberg,  E.  (2006),  Models  of  success  versus  the  success  of  models:  Reliability  without  truth,  Synthese,  152,  1–19.Computa=onal  eraI like philosophersrobbed from M.ToffolonWednesday, June 5, 13
  18. 18. 18A warning to myselfWhen hydrologists play to do philosophers, eventhe best, they do not do their jobI like philosophers butR. RigonWednesday, June 5, 13
  19. 19. 19... The modeling of some processes, i.e. rainfall-runoff, soil stormflow, snowpack evolution, are presented here according to differentdegree of simplifications, and the simplifications brieflydiscussed. ...Extract from the AbstractAs in Ezio workWe were attracted however by determining the structure of modelssimplification by theory, more than “inventing” by analogy“processes lines” to simplifyLess is moreR. RigonWednesday, June 5, 13
  20. 20. 202D - de Saint Venant equationswith some smart subgrid parameterization(e.g. Casulli, 2009)1D - Kinematic equationSo many to cite here but ... Liu andTodini, 2002Various aggregation strategiesfor runoff, including residencetime theories (a.k.a GIUH)Rodriguez-Iturbe and Valdes, 1979;Rinaldo et al., 1991,D’Odorico and Rigon, 2003R. RigonLess is moreWednesday, June 5, 13
  21. 21. 213D-Richards’ equation(Richards, 1931; Celia et al. 1990)1D-Richards + BoussinesqTopkapiHsBTopog/TopmodelCordanoandRigon,2008(Citations from Cordano and Rigon, 2013)Liu and Todini, 2002Troch et al., 2003O’Loughlin, 1986; Beven and Kirkby, 1979Less is moreR. RigonWednesday, June 5, 13
  22. 22. 22Dalton’s Equatione.g. Brutsaert 1982PenmanPenman, 1948MonteithMonteith, 1965Priestley-TaylorPriestley and Taylor, 1972Less is moreR. RigonWednesday, June 5, 13
  23. 23. 23Energy BudgetJordan, 1991Radiation + TemperatureBrubaker et al., 1996Degree-day (Justtemperature)Martinec and Rango, 1975Less is moreR. RigonWednesday, June 5, 13
  24. 24. 24* But not anymore necessarilyR. RigonLess is moreWednesday, June 5, 13
  25. 25. 24Models “complexity” and computational time increasegoing from bottom up.More complexity, more processes physics.Scales of application usually* decrease from top tobottom* But not anymore necessarilyR. RigonLess is moreWednesday, June 5, 13
  26. 26. 25Less is moreR. RigonWednesday, June 5, 13
  27. 27. 25Parameters pretend to be estimated ex-ante(measured) in more complex models (with a lot ofdisclaimers ... obviously)Are certainly calibrated (ex-post) in the simplestmodels (but in some models preserve a physicalsignificance)From top to bottom heuristic and statisticssubstitute processes analysisLess is moreR. RigonWednesday, June 5, 13
  28. 28. Just an example of top down derivationThe case of Richards’ equationChimpanzeeCongopaintingWednesday, June 5, 13
  29. 29. Iverson,2000;CordanoeRigon,200827The Richards equation on a plane hillslopeRichardsonianaR. RigonWednesday, June 5, 13
  30. 30. Iverson,2000;CordanoeRigon,200828The Richards equation made dimensionlessRichardsonianaR. RigonWednesday, June 5, 13
  31. 31. Iverson,2000;CordanoeRigon,200829Richards eq. solution expressed in terms ofthe asymptotic hydrostatic solution and a transientterm:See also. D’Odorico et al., 2003RichardsonianaR. RigonWednesday, June 5, 13
  32. 32. and one equation forIverson,2000;CordanoeRigon,200830So Richards equation isdivided into one equation forRichardsonianaR. RigonWednesday, June 5, 13
  33. 33. 31In turn“Short termsolution” Taylor’sexpansionWater tableequation Taylor’sexpansionSlope normal flowtime scale Lateral flowtime scaleSee also. D’Odorico et al., 2003RichardsonianaR. RigonWednesday, June 5, 13
  34. 34. 32Neglecting some detailsthat can be found in Cordano and Rigon, 2008Zeroth perturbation orderFirst perturbation order+ analogous for d*RichardsonianaR. RigonWednesday, June 5, 13
  35. 35. 33Integrating zeroth order solution in the columnMaking a long story shortTopkapi modelLiu and Todini, 2002RichardsonianaR. RigonWednesday, June 5, 13
  36. 36. 34Integrating first order solution slope-parallelMaking a long story short - IIBoussinesq equation(e.g. Cordano and Rigon, 2013)RichardsonianaR. RigonWednesday, June 5, 13
  37. 37. 35Integrating BoussinesqMaking a long story short - IIIHsBTroch et al. 2003RichardsonianaR. RigonWednesday, June 5, 13
  38. 38. 36Simplifying HsB assuming stationarity of fluxesand neglecting diffusive termsMaking a long story short - IV and VTopogO’Loughlin, 1986assuming an exponential decay of vertical hydraulicconductivityTopmodelBeven and Kirkby, 1979RichardsonianaR. RigonWednesday, June 5, 13
  39. 39. 37That is how we obtained:RichardsonianaR. RigonWednesday, June 5, 13
  40. 40. Just kidding!Wednesday, June 5, 13
  41. 41. 39Did you care about hypotheses ?Is it for any occasion realistic ? Look at the following sandy-loam:Hypotheses countsR. RigonWednesday, June 5, 13
  42. 42. 39Did you care about hypotheses ?Is it for any occasion realistic ? Look at the following sandy-loam:Hypotheses countsR. RigonWednesday, June 5, 13
  43. 43. constant diffusivity40The Decomposition of the Richards equationis possible under the assumption that:Time scale of infiltrationsoil depthtime scale of lateral flowhillslope lengthreference conductivityreference hydraulic capacityIverson,2000;CordanoandRigon,2008Hypotheses countsR. RigonWednesday, June 5, 13
  44. 44. Assuming hydrostatic conditions41Initial condition is then:Consequently, at surfaceHypotheses countsR. RigonWednesday, June 5, 13
  45. 45. 42For the sandy-loam soilassuming the water table at one meter depthwe have a vertical variation of hydraulic conductivity of one order of magnitude !Hypotheses countsR. RigonWednesday, June 5, 13
  46. 46. 43D0 which characterizes the time scales of flow is varyingwith depthHypotheses countsR. RigonWednesday, June 5, 13
  47. 47. 44Thereforeat surfaceso, lateral flow at the water table levelhas the same time scale vertical flow atthe surface (at least if we believe toRichards’ equation)Hypotheses countsR. RigonWednesday, June 5, 13
  48. 48. 45igure 2: Experimental set-up. (a) The infinite hillslope schematization. (b) The initial suction head pril-pixel hillslope numeration system (the case of parallel shape is shown here). Moving from 0 to 900sponds to moving from the crest to the toe of the hillslopeThe OpenBook hillslope in a 3DsimulationComparing with 3DR. RigonWednesday, June 5, 13
  49. 49. 46- 54 LANNI ET AL.: HYDROLOGICAL ASPECTS IN THE TRIGGERING OF SHALLOW LANDSLIDES(a) DRY-Low (b) DRY-MedSimulations resultComparing with 3DR. RigonWednesday, June 5, 13
  50. 50. 47At the beginning the pressure is constantalong the whole transect (except forphenomena at the divide’s edgeComparing with 3DR. RigonWednesday, June 5, 13
  51. 51. 48After a certain amount of time (25h in thissimulation) pressures along the slopedifferentiate. With a little of analysis wec a n d i s t i n g u i s h t w o r e g i o n s o fdifferentiation. One controlled by theboundary conditions at the bottom.The second generated by lateral waterflow accumulation.Comparing with 3DR. RigonWednesday, June 5, 13
  52. 52. 49(a) (b)Figure 6: Temporal evolution of the vertical profile of hydraulic conductivity (a) and hydraulic conductivity at the soil-bedrock interfaceHidraulic conductivity is varying by three order of magnitudeat the bedrock interface.The key to understand this phenomenologyLannietal.,2012Comparing with 3DR. RigonWednesday, June 5, 13
  53. 53. 50When simulating is understandingcourtesyofE.CordanoT’L can be very small indeed .....InterpretationsR. RigonWednesday, June 5, 13
  54. 54. 51Understanding from simulationsAt the beginning of the infiltration process the situation in surface ismarked by the blue line, the situation at the bedrock is marked by thered linecourtesyofE.CordanoR. RigonInterpretationsWednesday, June 5, 13
  55. 55. 52When lateral flow start we are in the following situationcourtesyofE.CordanoUnderstanding from simulationsR. RigonInterpretationsWednesday, June 5, 13
  56. 56. 53At the beginningThe condition of the perturbative derivation are verifiedcourtesyofE.CordanoR. RigonInterpretationsWednesday, June 5, 13
  57. 57. 54At the endcourtesyofE.CordanoConditions for lateral flow are dominating. Actually the samephenomenology deducted by the perturbation theory! But obtained for adifferent reason.R. RigonInterpretationsWednesday, June 5, 13
  58. 58. 55Lateral Flow•Can be fast, ... very fast, much faster than what happens in vadoseconditions•In fact, to have the effects just described, we have to believe to the formthat Soil Water retention Curves have.•Other soils behave differently•If macropores or cracks are present, vertical infiltration can still remainfasterR. RigonInterpretationsWednesday, June 5, 13
  59. 59. 56Inappropriate numerics (or gridding)Can hide it!R. RigonInterpretationsWednesday, June 5, 13
  60. 60. Further investigationsMachaelLeong-Cuttingthetimewithaknife,2012Wednesday, June 5, 13
  61. 61. 58CAPITOLO 5. IL BACINO DI PANOLAFigura 5.2: Rappresentazione della profondit`a del suolo del pendio di Panola.costante su un campione prelevato a 10 cm di profondit`a, risulta pari a 64 [cm/h]; per ci`o che concerneil valore della conducibilit`a idraulica a saturazione del bedrock, non esistono misure dirette e↵ettuatesu campioni prelevati in sito; tuttavia si stima che il suo valore sia 2-3 ordini di grandezza inferiorerispetto a quella del terreno soprastante. Entrambi i valori di conducibilit`a idraulica satura (del bedrocke del terreno) saranno comunque oggetto di calibrazione numerica all’atto delle simulazioni svolte conGEOtop, utilizzando come valori di partenza quelli qui citati.Panola’s hillslopeR. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  62. 62. 59Terrain surface Bedrock surface Soil depth variesDepressionSoil (sandy loam) BedrockKsat = 10-4 m/s Ksat = 10-7 m/sPanola’s hillslopeR. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  63. 63. 60Q(m3/h)t=9ht=18ht=22hWith a rainfall of 6.5 mm/h and a duration of 9 hoursLannietal.,2011R. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  64. 64. 61t=6h t=9ht=7h t=14hLannietal.,2011With a rainfall of 6.5 mm/h and a duration of 9 hoursTromp Van Meerveld et al., 2006 call it filling and spillingR. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  65. 65. 62Q(m3/h)t=9ht=18ht=22hLannietal.,2011With a rainfall of 6.5 mm/h and a duration of 9 hoursR. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  66. 66. 631D3D No role played by hillslopegradientFirst Slope Normal infiltration worksThen Lateral flow startInfiltration front propagateDrainage is controlled by the bedrock formAs in the open book caseLannietal.,2011R. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  67. 67. 64Now we want a model that can run 100 times fasterIn which, we obviously use all the machinery of theRichards’ equation, i.e. hydraulic conductivity and soilwater retention curvesR. RigonRichards equation is still valid here ?Wednesday, June 5, 13
  68. 68. 65Ii.e. time to water tabledevelopmentTwt(x,y):= [Vwt(x,y)-V0(x,y)]/IInitial conditions(hydrostatic slope normal)boundary conditions(including rainfall, I)t> Twt(x,y)YESNOLannietal.,2012Slope Normalunsaturated flowA heuristic modelfor eachtimestepFaster is betterR. RigonWednesday, June 5, 13
  69. 69. 66YESt> Tmaxwt(x,y)hydrologicallyconnectedA(x,y) >0YESNOhydrologicallydisconnectedA(x,y) =0A heuristic modelLannietal.,2012R. RigonFaster is betterWednesday, June 5, 13
  70. 70. 67YESupdate soilpressurestart lateral flow update soilpressurenexttimestepA heuristic modelLannietal.,2012R. RigonFaster is betterWednesday, June 5, 13
  71. 71. 68* Is not completely true.I question also of personal attitude:I understand (fluid) mechanics throughequations and I try to interpret observationsthrough equations.Someone else (i.e. many of my students)simply did not have the training for that andprefer to rebuilt the physics of the problem bysmall pieces.This has a certain appealing to many (especiallyto natural scientists and geologists), and canindeed be useful to see thing from differentperspectives.Doodley,Muttley,andtheirflyingmachinesR. RigonAttitudesWednesday, June 5, 13
  72. 72. 693968 C. Lanni et al.: Modelling shallow landslide susceptibility123Figure 7. Patterns of Return period TR (years) of the critical rainfalls for shallow landslide4triggering  (i.e.,  FS≤1)  and  associated  levels  of  landslide  susceptibility  obtained  by  means  5of QDSLaM.67Fig. 7. Patterns of return period TR (years) of the critical rainfalls for shallow landslide triggering (i.e. FS  1) and associated levels oflandslide susceptibility obtained by means of QDSLaM.Table 3. Percentages of catchment area (C) and observed landslide area (L) in each range of critical rainfall frequency (i.e. return period TR)for QDSLaM.SusceptibilityPizzano Fraviano CortinaTR level Ca Lb Ca Lb Ca LbYears Category % % % % % %Uncond Unstable 9.9 60.2 7.7 77.7 8.5 56.80–10 Very high 20.3 26.9 16.1 18.5 13.5 39.210–30 High 7.8 0.0 5.6 1.5 5.8 4.0Lannietal.,2012However, it worksR. RigonFaster is better if it works (Klemes fogive me!)Wednesday, June 5, 13
  73. 73. 70CAPITOLO 5. IL BACINO DI PANOLAFigura 5.4: Immagine tratta da Tromp-van Meerveld e McDonnell, (2006a) [24]; (a) deflusso sub-superficiale totale per i segmenti in cui `e stata suddivisa la trincea e (b) numero di eventi meteorici cheproducono deflussi misurabili.5.2.1 Il ruolo dei macroporiTrompVanMeerveldetal.,2006And finally macroporesR. RigonMacroporesWednesday, June 5, 13
  74. 74. 71Macropore FlowInitiationWater supply to themacroporesInteractionWater transfer betweenmacropores and thesurrounding soil matrixM.Weiler,fromMochaprojectMacropores!R. RigonMacroporesWednesday, June 5, 13
  75. 75. 720.00date (dd/mm) 200201/01 11/01 21/01 31/01 10/02 20/02 02/03 12/03 22/03 01/04 11/04 21/04 01/05 11/05 21/05Figura 5.16: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trinceaalla base del pendio. 0 - evento 6 febbraiodate (dd/mm) 2002portate[l/s]05/02 06/02 07/02 08/02 09/02 10/02 11/02 12/02Flussi misuratiSimulazione 0 - evento 30 marzodate (dd/mm) 2002portate[l/s]29/03 30/03 31/03 01/04 02/04 03/04 04/04 05/04 06/04 07/04Flussi misuratiSimulazione 0Figura 5.17: Confronto tra flussi misurati e computati attraverso la Simulazione 0 presso la trinceaalla base del pendio: a sinistra si riporta l’evento del 6 febbraio 2002, a destra quello del 31 marzo.pu`o essere causata da diversi fattori, quali un’errata assegnazione delle caratteristiche del suolo o delbedrock, oppure un errore nello stabilire la condizione iniziale circa la quota della falda.Un aspetto decisamente importante da considerare, tanto in questi risultati quanto in quelli presentatisuccessivamente, `e che nella creazione della geometria di calcolo 3D utilizzata da GEOtop non `eDaPrà,2013Certainly the volumes of water cannot besimulated with the only Richards equationNo way!R. RigonMacroporesWednesday, June 5, 13
  76. 76. ConclusionsTowrdsthecompleteworksofShakespeare(EssayofMonkeywriting)Wednesday, June 5, 13
  77. 77. 74.. It is concluded that all models, at any scale, are trulyinherently statistical, in the statisticians sense, and also inthe statistical-mechanical sense, since they derive from aninductive-deductive process compared to some evidences,and, at the same time, represent the emergent behavior ofsome smaller physical world.Extract from the AbstractI do not think I really illustrated this: but I believe it istrue, anyway.R. RigonEpilogueWednesday, June 5, 13
  78. 78. 75big thanks to EzioEventuallyfor his life-long coherent effort to work withequations and scientific rigor in a way that was anexample for me and for manyR. RigonEzio!Wednesday, June 5, 13
  79. 79. Thank you for your attentionG.Ulrici,2000?76These slides are available at http://abouthydrology.blogspot.comThank youR. RigonWednesday, June 5, 13