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Giuh2020

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This treats from a historical point of view, first, the GIUH theories. Then It introduces the new theories of travel time, residence time distribution. Finally propose how to work out a modern statistical mechanical theory of the hydrological budgets

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Giuh2020

  1. 1. Geomorphological Aspect of Hydrological modelling in 2020 Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera, Giuseppe Formetta and Alban de Lavenne Perugia, Giornate dell’idrologia, 7 Ottobre 2015 ParcoQuerini,Ottobre2015,Vicenza
  2. 2. !2 What was Geomorphology in hydrological modelling before 1979 ? but this is a post-reinterpretation that there was geomorphology there A little of (biased) history R.B.A.F.L
  3. 3. !3 What was geomorphology ? Other Ancestors Zoch RT, 1937. On the relation between rainfall and stream flow. Monthly Review 55: 135–147. Clark C, 1945. Storage and the unit hydrograph. Transactions of the American Society of Civil Engineers 110(1): 1419–1446. Not to go back to the old papers on the: Mulvany, T. J. (1851) On the use of self-registering rain and flood gauges in making observations of the relations of rain fall and of flood discharges in a given catchment. Proceedings of the Institution of Civil Engineers of Ireland 4, 18–33. Ross, C. N. (1921) The calculation of flood discharge by the use of a time contour plan. Transactions of the Institution of Engineers, Australia 2, 85–92. See Also: “Rainfall-runoff modelling, IAHS Benchmark papers in Hydrology, 2010, J. McDonnell (Ed.) R.B.A.F.L
  4. 4. !4 A hybrid Surkan,A.J.,Wrr,1976What was geomorphology ? R.B.A.F.L
  5. 5. !5 1979 Clearly there is not a unique thread in this history, and possibly Rodriguez-Iturbe and Valdes did not know, when they wrote their paper, neither Dooge’s nor the other works, and they were more inspired by other contributions, as those in stochastic hydrology by V. Yevievich
 GIUH R.B.A.F.L
  6. 6. !6   Iturbe and Valdés, marked the beginning of a new era in rainfall- runoff models. The use of geomorphological information to assist in definining the unit hydrograph (or more general hydrological response functions such as travel time distributions), and the conceptualisation   of hydrologic response as the convolution of travel time distributions, was represented with a mathematically neat method. 
 It Was really new GIUH Q(t) = A Z t 0 p(t ⌧)Je(⌧)d⌧ p(t) = X 2 (p 1 ⇤ · ⇤ p ⌦ )(t) R.B.A.F.L
  7. 7. !7 Or, really syntethic Q(t) = A X 2 p (Je ⇤ p 1 ⇤ · ⇤ p ⌦ )(t) R.B.A.F.L GIUH
  8. 8. !8 Notably there was an effort to get information from the knowledge of terrain R.B.A.F.L GIUH Rigon et al., ESP&L, 2015
  9. 9. !9 BUT Geomorphological information at that time was limited NO SRTM R.B.A.F.L GIUH
  10. 10. !10 Horton Laws Geomorphology was daTarboton:www.cuahsi.org • The bifurcation law … just for illustrative purposes R.B.A.F.L
  11. 11. !11 Somewhat the interpretation given by researchers was rigid • Partition was taken according to Horton Laws • Pdfs were chosen exponential And obviously the problem of choosing a right effective rainfall was ubiquitous R.B.A.F.L A somewhat limited interpretation
  12. 12. WFIUH Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera, Giuseppe Formetta and Alban de Lavenne Perugia, Giornate dell’idrologia, 7 th October 2015
  13. 13. !13 What was Geomorphology in hydrological modelling in eighties ? The Digital Elevation Models Era The old concept of isochrones Ross (1921), or, e.g. Beven (2011), could be derived from the definition of width function (Kirkby, 1976), under the hypothesis of constant celerities throughout the network , from digital data. Figure 2: A basin can be continuously subdivided into strips of terrain at the same distance from the outlet. These strips are not necessarily continuous.They are physically connected to the outlet by many channels, but of these channels the physically significant quantity is R.B.A.F.L WFIUH
  14. 14. !14 A finer view of some isochrones At steps of 2.5 km apart R.B.A.F.L WFIUH
  15. 15. !15 R.B.A.F.L WFIUH Width function after, Kirkby 1976
  16. 16. !16 f(t|x) = x p 4⇡Dt3 exp  (x uct)2 4Dt Q(t) = Z t 0 p(t ti)Je(ti)dti WFIUH pw(t ti) = Z L 0 w(x)f(t ti|x)dx R.B.A.F.L WFIUH Rinaldo et al., 1991; Mesa and Mifflin, 1986
  17. 17. !17 In case channel celerity and hillslope celerity were separated R.B.A.F.L WFIUH
  18. 18. !18 x0 = xc + r xh r := uh uc R.B.A.F.L Rescaled width function Rinaldo et al, 1995; D’Odorico and Rigon, 2003; Rinaldo and Botter, 2003
  19. 19. !19 Achievements • It was possible to assess the role of geometry/topology and water dynamics by showing that geometry counts more than dynamics (geomorphological dispersion, Rinaldo et al., 1991) • It was possible to assess the role of hillslope and channels, by showing that hillslopes count more than channels, e.g. D’Odorico and Rigon, 2003. • It was possible to give semi-analytical formula for the peak flows (Rigon et al., 2011) R.B.A.F.L The good
  20. 20. !20 R Rn Q Total flowpath travel time Baseflow travel time (a) 1F1U1P R Rn Q Network travel time Hillslope travel time Baseflow travel time (b) 1F2U2P R Rn Q Total flowpath slow travel time Total flowpath fast travel time Baseflow travel time (c) 2F2U3P R Rn Q Hillslope slow travel time Network travel time Hillslope fast travel time Baseflow travel time (d) 2F3U4P Figure 3: Graphical description of the di↵erent model’s structure. 2.2.2 Production function In order to be able to implement those transfer functions, a production function and a baseflow Various complications were tried R.B.A.F.L Various complications de Lavenne et al., in preparation, 2015
  21. 21. !21 and is only use to describe the discharge between the events which are, as for them, modelled by the transfer function. For Co¨et-Dan at Naizin basin, the baseflow function can even be totally removed from the global structure of the model (↵1 closed to 0) which underline that a simple transfer function which is considering dispersion could be enough to describe the basin’s response. In that case, model’s structure would be greatly simplified. 0.0 0.1 0.2 0.3 01 Nov 2007 05 Nov 2007 12 Nov 2007 19 Nov 2007 26 Nov 2007 03 Dec 2007 10 Dec 2007 17 Dec 2007 24 Dec 2007 31 Dec 2007 Runoff simulations on Fremeur at Plumeliau Discharge[m³/s] KGE 0.72 0.72 0.5 0.77 0.77 0.79 0.66 0.66 0.53 0.75 0.75 0.74 Models 1F1U1P 1Frv1U1P 1Fd1U2P 1F2U2P 1Frv2U2P 1Fd2U3P 2F2U3P 2Frv2U3P 2Fd2U4P 2F3U4P 2Frv3U4P 2Fd3U5P Observation Figure 11: Runo↵ simulation on Fremeur at Plumeliau from 2007-11-01 to 2007-01-01 using the di↵erent transfer function. Performances R.B.A.F.L Good Performances in events predictions de Lavenne et al., in preparation, 2015
  22. 22. !22 Performances 2006−10to2010−10 2006−11to2007−01 2007−05to2007−07 2007−11to2008−01 2008−05to2008−07 2008−11to2009−01 2009−05to2009−07 2009−11to2010−01 2010−05to2010−07 2010−11to2011−01 2011−05to2011−07 2011−11to2012−01 2012−05to2012−07 1F1U−d 2F2U−d 1F1U−rv (a) Calibration periods, colorized by columns 2006−10to2010−10 2006−11to2007−01 (b) Calibra 2007−01to2007−03 2007−07to2007−09 2008−01to2008−03 2008−07to2008−09 2009−01to2009−03 2009−07to2009−09 2010−01to2010−03 2010−07to2010−09 2010−10to2012−10 2011−01to2011−03 2011−07to2011−09 2012−01to2012−03 2012−07to2012−09 2F3U 2F3U−rv 2F3U−d 1F2U−d 1F1U−d 2F2U−d 2F2U−rv 2F2U 1F2U−rv 1F2U 1F1U−rv 1F1U (c) Validation periods, colorized by columns 2007−01to2007−03 2007−07to2007−09 (d) Validat Figure 10: Heatmap of mean models’ simulations e ciency est tion and validation periods over the six studied basin models and sub-figure 10b is comparing periods of s relatively poorer performance, and darker color mean Heatmaps of figure 11 summarize model e ciency accordin average for all studied periods. Even if the description of each (because models are not compared according to simulation perio 2006−10to2010−10 2006−11to2007−01 2007−05to2007−07 2007−11to2008−01 2008−05to2008−07 2008−11to2009−01 2009−05to2009−07 2009−11to2010−01 2010−05to2010−07 2010−11to2011−01 2011−05to2011−07 2011−11to2012−01 2012−05to2012−07 1F1U−d 2F2U−d 1F1U−rv 1F1U 2F2U−rv 2F2U 2F3U 2F3U−rv 2F3U−d 1F2U−d 1F2U 1F2U−rv (a) Calibration periods, colorized by columns 2006−10to2010−10 2006−11to2007−01 2007−05to2007−07 2007−11to2008−01 2008−05to2008−07 2008−11to2009−01 2009−05to2009−07 2009−11to2010−01 2010−05to2010−07 2010−11to2011−01 2011−05to2011−07 2011−11to2012−01 2012−05to2012−07 1F1U−d 2F2U−d 1F1U−rv 1F1U 2F2U−rv 2F2U 2F3U 2F3U−rv 2F3U−d 1F2U−d 1F2U 1F2U−rv (b) Calibration periods, colorized by rows 1F2U−d 1F1U−d 2F2U−d 2F2U−rv 2F2U 1F2U−rv 1F2U 1F1U−rv 1F1U 1F2U−d 1F1U−d 2F2U−d 2F2U−rv 2F2U 1F2U−rv 1F2U 1F1U−rv 1F1U The darker, the better R.B.A.F.L Well … Custer heat maps de Lavenne et al., in preparation, 2015
  23. 23. !23 Issues • It is still an event-based model • Does not include evapotranspiration (the World has gone elsewhere) • Tracers tell another story (“The old water paradox”) R.B.A.F.L The idea that something is missing
  24. 24. The business as usual Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera and Giuseppe Formetta Already used in Padova, 23-24 September 2015
  25. 25. !25 Each HRU is a control volume • No lateral fluxes • No deep losses and recharge terms supplying deep groundwater S(t) : Water storage in the control volume V M(t) : Solute storage in the control volume V Figure From Catchment travel times distributions and water flow in soils, Rinaldo et al. (2011) HRUs level example R.B.A.F.L
  26. 26. !26 Water Budget dS(t) dt = J(t) Q(t) AET (t) HRUs level example R.B.A.F.L
  27. 27. !27 Water Budget Volume of water in the control volume Total precipitation = rainfall + snow melting Discharge Actual Evapotranspiration dS(t) dt = J(t) Q(t) AET (t) HRUs level example R.B.A.F.L
  28. 28. !28 The business as usual dS(t) dt = J(t) Q(t) AET (t) AET(t) = S(t) Smax ET (t) where ET(t) is potential evapotranspiration (maybe space-averaged) and a,b,Smax are parameters (in principle different for any HRU) Q(t) = k S(t)b HRUs level example R.B.A.F.L
  29. 29. !29 In this case: Let for a moment b=1, then the equation is linear and has a solution dS(t) dt = J(t) kS(t)b S(t) Smax ET (t) S(t) = e ( t k + 1 Smax R t 0 ET (t0 )dt0 ) Z t 0 e(s k + 1 Smax R s 0 ET (t0 )dt0 )J(s)ds if S(0) = 0 which is known, as soon as, ET(t) and J(t) are known HRUs level example R.B.A.F.L
  30. 30. !30 If we define S(t) := Z t 0 S(t, ⌧)d⌧ Storage at time t generated by precipitation at time Z t 0 S(t, s)ds = Z t 0 e (t s k + 1 Smax R t s 0 ET (t0 )dt0 )J(s)ds we have S(t, s) = e (t s k + 1 Smax R t s 0 ET (t0 )dt0 )J(s) HRUs level example R.B.A.F.L
  31. 31. !31 Q(t) := Z t 0 Q(t, ⌧)d⌧ AET (t) := Z t 0 AET (t, ⌧)d⌧ Discharge at time t generated by precipitation at time Actual evapotranspiration generated by precipitation at time We can also define HRUs level example R.B.A.F.L
  32. 32. !32 Is also Q(t, s) = ke (t s k + 1 Smax R t s 0 ET (t0 )dt0 )J(s) AET (t, s) = S 1 max h e (t s k + 1 Smax R t s 0 ET (t0 )dt0 )J(s) i ET (t) Given S(t, s) = e (t s k + 1 Smax R t s 0 ET (t0 )dt0 )J(s) HRUs level example R.B.A.F.L
  33. 33. Introducing residence/travel times Riccardo Rigon, Marialaura Bancheri October, 2015 ParcoQuerini,Ottobre2015,Vicenza
  34. 34. !34 Definitions Introduction Let’s define ⌧ ⌦ the injection time the exit time R.B.A.F.L
  35. 35. !35 Storage is usually defined as S. It is varying with time, so: S = S(t) of this storage, we can think to distinguish, the part that was injected at time tau and, further, the part that is expected to exit at time Omega. This part is: that integrated gives: S(⌦, t, ⌧) S(⌦, t) = Z 1 0 S(⌦, t, ⌧)d⌧ S(t, ⌧) = Z 1 0 S(⌦, t, ⌧)d⌦ R.B.A.F.L The Kinematics of travel time distributions
  36. 36. !36 Obviously: S(t) = Z 1 0 Z 1 0 S(⌦, t, ⌧)d⌦d⌧ Observe also that we can define such probabilities that *: p (⌧|t) := S(t, ⌧) S(t) !p (⌦|t) := S(⌦, t) S(t) * One obtains normalisation by integrating over R.B.A.F.L The Kinematics of travel time distributions ⌧
  37. 37. !37 Then: Define, analogously J(t, ⌧) Q(t, ⌧) AET (t, ⌧) S(t, ⌧) = S(t) p (⌧|t) S(⌦, t) = S(t)!p (t|⌦) R.B.A.F.L The Kinematics of travel time distributions
  38. 38. !38 When we are dealing with hydrology, we are doing water budgets. This can be presented in integrated form: dS(t) dt = J(t) Q(t) AEt(t) However, we can do it by any of the sub volumes of S: dS(t, ⌧) dt = J(t) Q(t, ⌧) AEt(t, ⌧) dS(⌦, t) dt = J(⌦, t) Q(t) AEt(t) R.B.A.F.L The Kinematics of travel time distributions
  39. 39. !39 Therefore, making the appropriate substitutions backward equation forward equation d dt S(t) p (⌧|t) = J(⌧) Q(t) p Q(⌧|t) AEt(t) p E(t, ⌧) d dt S(t)!p (⌦|t) = J(t, ⌦)!p J (⌦|t) Q(t) AEt(t) Note: both or them are probability conditional to the actual time, t. However, backward and forward refers to the fact that the free variable is before or after the actual time. R.B.A.F.LR.B.A.F.L The Kinematics of travel time distributions
  40. 40. !40 The above equations, once known S(t) are differential equations in p, but, to be solved, some trick must be made to express pQ and pAET as function of p. This can be done, introducing some StorAge Selection functions (SAS), each, for each output. p Q(⌧|t) := !(⌧, t) p (⌧|t) p AET (⌧|t) := !AET (⌧, t) p (⌧|t) !p AET (⌦|t) := !AET (⌦, t)!p (⌦|t) For the backward equation: For the forward equation: !p Q(⌦|t) := !Q(⌦, t)!p (⌦|t) R.B.A.F.L The Kinematics of travel time distributions
  41. 41. !41 Hence, the backward equation reads: d dt S(t) p (⌧|t) = J(⌧) !Q(t, ⌧)Q(t) p (⌧|t) !AET (t, ⌧)AET (⌧|t) p E(t, ⌧) R.B.A.F.L And can be solved as: The Kinematics of travel time distributions Notably from this distribution we can derive the average age of water and connect, for instance, tracers experiments with the model that gives Q and ET
  42. 42. !42 The Kinematics of travel time distributions R.B.A.F.L Botter, G. (2012). Catchment mixing processes and travel time distributions. Water Resources Research, 48(5), http://doi.org/10.1029/2011WR011160 Botter, G., Bertuzzo, E., & Rinaldo, A. (2010). Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox, 46(3). http://doi.org/10.1029/2009WR008371 Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time distributions: The master equation. Geophysical Research Letters, 38(11). http:// doi.org/10.1029/2011GL047666 Benettin, P., Catchment transport and travel time distributions: theoretical developments and applications, Ph.D. dissertation (A. Rinaldo & G. Botter, Supervisors.). Details on
  43. 43. Where is geomorphology here ? Riccardo Rigon, Marialaura Bancheri, Wuletawu Abera and Giuseppe Formetta Already used Padova, 23-24 September 2015 A.Bonomi Sketches of a theory
  44. 44. !44 Even in the new formalism we can think the single sub- catchments as part of system R.B.A.F.L The new theory
  45. 45. !45 And we can apply the convolution formulas Q(t) = A X 2 p (Je ⇤ p 1 ⇤ · ⇤ p ⌦ )(t) but this is kind of a trivial application of the theory R.B.A.F.L The new theory
  46. 46. !46 Even if we applied it to a system complex like Adige River R.B.A.F.L Real applications
  47. 47. !47 To really introduce geomorphology we have to work out the storage problem for each hillslope, and make it dependent on some space variable. So that, for instance, we spatially distribute the input, the storage becomes where x is some space variable, for instance the distance to outlet. One possibility is taking: and, then: Q(t, ⌧) = Z L 0 J(x, ⌧)w(x)f(t ⌧|x)dx S(x, t, ⌧) for some appropriate probability distribution function f( ) S(x, t, ⌧) := J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧) R.B.A.F.L Storage is the key
  48. 48. !48 Integrating over the space variable S(t, ⌧) = Z L 0 J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧)dx and integrating over the injection time S(t) = Z t 0 Z L 0 J(x, ⌧) J(x, ⌧)w(x)f(t ⌧|x) AET (x, t, ⌧)dxd⌧ We can reconstruct all the relevant probabilities R.B.A.F.L Storage is the key
  49. 49. !49 We can do more … This scheme, in fact, correspond to a unique control volume R.B.A.F.L Too simple, too bad
  50. 50. !50 We would like to split it in more parts Figure 4: A HRU is “vertically” split into a surface and a subsurface domain. The overall response (to channel flow) is obtained just by the sum of the two contributions. In the same study, notwithstanding the generally optimal response of the WFIUH the- ory, an accurate analysis of the results over several events by means of ”Cluster Heat Maps” (Wilkinson and Friendly, 2009) clearly showed that there were basins and events say, two, for simplicity. R.B.A.F.L Too simple, too bad
  51. 51. !51 The Hymod-like way This is giving a distribution function for storages, for instance: F(C) = 1 ✓ 1 C Cmax ◆b C is the storage R.B.A.F.L Partitioning the control volume After Moore, 1985
  52. 52. !52 We can assume That (by saturation excess) those points where the capacity C is exceeded, goes into surface runoff, generated, for instance at any distance x. Surprise! You thought, by analogy, that the previous scheme, with one storage, was a kind of surface flow. Instead now it become subsurface storm flow. Runoff can be also routed through a geomorphologic scheme but also in a different way. R.B.A.F.L Partitioning the control volume
  53. 53. !53 I got you tired so I do not proceed further but clearly one can add layers at will, until necessary 0 100 200 300 2012−01−01 2012−02−01 2012−03−01 2012−04−01 2012−05−01 2012−06−01 2012−07−01 2012−08−01 2012−09−01 2012−10−01 2012−11−01 2012−12−01 monthly J(mm) −100 0 100 200 300 10-2011 11-2011 12-2011 01-2012 02-2012 03-2012 04-2012 05-2012 06-2012 07-2012 08-2012 09-2012 Months Watercomponent:Q,ET,S(mm) Q ET S components R.B.A.F.L It is too late. I close it here
  54. 54. !54 A few new things here R.B.A.F.L There is no need for finding the effective rainfall Evapotranspiration is included Snow modelling can be easily includes (it is just an added storage) Tracers and temperature dynamics is around the corner 1 2 3 4 Top storages can be set to describe the root zone5 6 Top storages can be set to be coupled with satellite information about soil moisture 7 …..
  55. 55. !55 Find this presentation at Ulrici,2000? Other material at Questions ? R. Rigon Mostly from Rigon et al., ESP&L, 2015 http://abouthydrology.blogspot.it/2015/10/geomorphological-modelling-in-2020.html http://abouthydrology.blogspot.it/search/label/Residence%20time

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