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# A travel time model for estimating the water budget of complex catchments

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This is the presentation given by Marialaura Bancheri for her admission to the final exam to achieve a Ph.D. in Environmental Engineering. It contains a synthesis of her studies about spatially integrated models of the water budget, and about travel time theory. A model structure is also presented preliminarily containing five reservoirs.

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### A travel time model for estimating the water budget of complex catchments

1. 1. A travel time model for water budget of complex catchments Candidate: Supervisors: Marialaura Bancheri Prof. Riccardo Rigon Matr.: 169091 Eng. Giuseppe Formetta Doctoral School in Civil, Environmental and Mechanical Engineering - 29°Cycle
2. 2. A travel time model for water budget of complex catchments Getting the right answers for the right reasons: toward many “embedded” reservoirs. Age-ranked hydrological budgets and a travel 6me descrip6on of catchment hydrology JGrass-NewAge: a replicable hydrological model Bancheri M., A travel time model for the water budgets of complex catchments Overview
3. 3. Travel time T Residence time Tr Life expectancy Le Injection time tin Exit time tex t Time Travel time: the time a water particle takes to travel across a catchment T = (t tin) | {z } Tr + (tex t) | {z } Le Bancheri M., A travel time model for the water budgets of complex catchments Travel times as random variables
4. 4. dS(t) dt = J(t) Q(t) AET (t) S(t) = Z min(t,tp) 0 s(t, tin)dtin AET (t) = Z min(t,tp) 0 aeT (t, tin)dtin J(t) = Z min(t,tp) 0 j(t, tin)dtin Q(t) = Z min(t,tp) 0 q(t, tin)dtin ds(t, tin) dt = j(t, tin) q(t, tin) aeT (t, tin) Bancheri M., A travel time model for the water budgets of complex catchments “Bulk” water budget VS “age-ranked” water budget
5. 5. Backward probability conditioned on the actual time t Travel time T Exit time tex t TimeInjection time tin Looks backward to tin Bancheri M., A travel time model for the water budgets of complex catchments Backward probabilities
6. 6. Time Time J(t) S(t) S(t) Residence time ttin s(t, tin) Residence time backward probabilities pS(t tin|t) := s(t, tin) S(t) [T 1 ] Bancheri M. , A travel time model for the water budgets of complex catchments Backward probabilities
7. 7. Time J(t) ttin TimeResidence time Q(t) Q(t) Travel time backward probabilities q(t, tin) pQ(t tin|t) := q(t, tin) Q(t) [T 1 ] Bancheri M. , A travel time model for the water budgets of complex catchments Backward probabilities
8. 8. On the shape of the backward pdfs Z min(t,tp) 0 pQ(t tin|t)dtin = 1 •  Time-variant •  •  not always true for other classical distributions , e.g., Z min(t,tp) 0 (t tin)↵+1 e (t tin) ↵ (↵) dtin 6= 1 8t⇤ 2 [0, min(t, tp)] Bancheri M. , A travel time model for the water budgets of complex catchments Backward probabilities
9. 9. After the previous deﬁnitions and some proper substitutions, the water budget equation for the control volume can be written as: d dt S(t)pS(Tr|t) = J(t) (t tin) Q(t)pQ(t tin|t) AEt(t)pET (t tin|t) obtaining a linear ordinary differential equation that can be solved exactly, once assigned the SAS values : d dt S(t)pS(Tr|t) = J(t) (t tin) Q(t) SAS z }| { !Q(t, tin) pS(Tr|t) | {z } pQ(t tin|t) AEt(t) SAS z }| { !ET (t, tin) pS(Tr|t) | {z } pET (t tin|t) Bancheri M. , A travel time model for the water budgets of complex catchments Backward probabilities
10. 10. The formalism developed is applicable, in principle to any substance, say indicated by a superscript i. If the substance is diluted in water, it is usually treated as concentration in water, which is known once the concentration of the solute in input is known together with the backward probability: d dt Si (t)p(t tin|t) = Ji (t)pi J (t tin|t) Qi (t)!Q(t, tin)p(t tin|t) Ci (t) = Z t 0 p(t tin|t)Ci J (tin)dtin Bancheri M. , A travel time model for the water budgets of complex catchments Passive solute transport
11. 11. Forward probability conditioned on the injection time tin Travel time T Exit time tex t TimeInjection time tin Looks forward to t Bancheri M. , A travel time model for the water budgets of complex catchments Forward probabilities
12. 12. Thanks to Niemi’s relationship (Niemi, 1977) we can connect the backward and forward pdfs: Where: We can also deﬁne the forward travel time pdfs as: pQ(t tin|tin) := q(t, tin) ⇥(tin)J(tin) ⇥(tin) := lim t!1 ⇥(t, tin) = lim t!1 VQ(t, tin) VQ(t, tin) + VET (t, tin) Q(t)pQ(t tin|t) = ⇥(tin)pQ(t tin|tin)J(tin) Bancheri M. , A travel time model for the water budgets of complex catchments Forward probabilities
13. 13. Bancheri M. , A travel time model for the water budgets of complex catchments Getting the right answers for the right reasons: toward many “embedded” reservoirs. R S Ssnow M SCanopy E Tr SRootzone TRZ SRunoﬀ TR Re SGroundwater QR QG U The entire model is based on the assumption that the water budget has been solved and the ﬂuxes are known. Flux Expression Tr(t) H(Scanopy(t) Imax)ac Scanopy(t) E(t) Scanopy SCanopymax (1 SCF) ETp U(t) p SRootzone TRZ(t) SRootzone SRootzonemax ETp Re(t) Pmax SRootzone SRootzonemax QR(t) A R t 0 uW(ut ⌧)↵(⌧)Tr(⌧)d⌧ TR(t) SRunoff SRunoffmax ETp QG(t) a SGroundwater
14. 14. Process Component Geomorphological model setup Jgrastools Meteorological interpolation tools Kriging IDW, JAMI Energy balance Shortwave radiation balance Clearness Index Longwave radiation balance Evapotranspiration Penmam-Monteith Priestley-Taylor Fao-Etp-model Snow melting Rain-snow separation Snowmelt and SWE model Runoff production Adige "Embedded" reservoirs Travel times description Backward travel times pdfs Forward travel times pdfs Solute trasport Automatic calibration LUCA Particle swarm Dream JGrass-NewAge Bancheri M. , A travel time model for the water budgets of complex catchmentsBancheri M. , A travel time model for the water budgets of complex catchments JGrass-Newge: hydrological modelling with components •  Rewrote according to the Java object orienting programming; •  Increased their ﬂexibility using design patterns; •  Gradle integrated; •  Travis CI integrated; •  Documentation wrote to obtain a variety of modelling solutions; •  OMS project example published for reproducing the results.
15. 15. Source code Project examples Community blog Documentation Bancheri M. , A travel time model for the water budgets of complex catchments Replicability of JGrass-NewAge
16. 16. http://geoframe.blogspot.com Bancheri M. , A travel time model for the water budgets of complex catchments GEOframe: a system for doing hydrology by computer
17. 17. Bancheri M. , A travel time model for the water budgets of complex catchments Application to real cases: River Net3 for the Posina river case 14 HRUs A= 36 km2 42 HRUs A= 112 km2
18. 18. Bancheri M. , A travel time model for the water budgets of complex catchments Applications: Posina River 0 100 200 300 1995 1996 1997 1998 1999 Precipitation[mm] 0 10 20 30 40 1995 1996 1997 1998 1999 Time [h] Discharge[m3/s] Measured Simulated
19. 19. Bancheri M. , A travel time model for the water budgets of complex catchments Applications: Posina River 0 5 10 15 20 1995 1996 1997 1998 1999 Rainfall[mm] Upper layer 0 50 100 150 1995 1996 1997 1998 1999 MeanTT[d] ω Preference for new water Uniform preference Preference for old water Beta(↵, ) : prob(x|↵, ) = x↵ 1 (1 x) 1 B(↵, ) B(↵, ) = Z 1 0 t↵ 1 (1 t) 1 dt T ω Uniform preference: α=1,β=1 1 T ω 1 Preference for new water α=0.5,β=1 T ω 1 Preference for old water α=3,β=1
20. 20. Bancheri M. , A travel time model for the water budgets of complex catchments Applications: Posina River 0 10 20 1995 1996 1997 1998 1999 Precipitation[mm] Precipitation [mm] 0 10 20 30 40 1995 1996 1997 1998 1999 MeanTT[d] Canopy 0 25 50 75 100 1995 1996 1997 1998 1999 MeanTT[d] Rootzone
21. 21. Bancheri M. , A travel time model for the water budgets of complex catchments Applications: Posina River 0.25 0.50 0.75 1.00 Gen 1994 Apr 1994 Lug 1994 Ott 1994 Gen 1995 Time PartitioningcoefficientΘ January February March April May June July August September October November Jan 94 Apr 94 Jan 95Oct 94Jul 94
22. 22. Bancheri M. , A travel time model for the water budgets of complex catchments Applications: Posina River Further valida6ons of the travel 6mes theory are required, especially to test the solute transport. However, since the lack of data, it was not possible 6ll now. Therefore I asked to a deferral of 6 months of the submission of the thesis. Hopefully the isotope data are arriving in the weeks… (maybe with Santa!)
23. 23. Bancheri M. , A travel time model for the water budgets of complex catchments Research outcomes Journals paper Rigon, R., Bancheri, M., Formetta, G., and de Lavenne, A. (2016) The geomorphological unit hydrograph from a historical-critical perspective. Earth Surf. Process. Landforms, 41: 27–37. doi: 10.1002/esp.3855. Rigon R., Bancheri M., Green T., Age-ranked hydrological budgets and a travel time description of catchment hydrology, in discussion, HESSD, 2016 Formetta, G., Bancheri, M., David, O., and Rigon, R.: Performance of site-speciﬁc parameterizations of longwave radiation, Hydrol. Earth Syst. Sci., 20, 4641-4654, doi:10.5194/hess-20-4641-2016, 2016. Bancheri, M., Seraﬁn, F., Abera, W., Formetta, G., Rigon R., A well engineered implementation of Kriging tools in the Object Modelling Sisytem v.3., in preparation, 2016
24. 24. Bancheri M. , A travel time model for the water budgets of complex catchments Research outcomes Conference abstract M. Bancheri, G.Formetta, W.Abera, R. Rigon, Componenti della radiazione solare ad onda lunga: NewAge-LWRB, XXXIV Convegno nazionale di Idraulica e Costruzioni Idrauliche, 2014 W. Abera, G. Formetta, M.Bancheri, R.Rigon, 2014, The effect of spatial discretization on hydrological response, the case of Semi-distributed Hydrological modelling, AGU chapman conference, 2014. M.Bancheri, W. Abera, G. Formetta, R.Rigon & F. Seraﬁn , Implementing a Travel Time Model for the Entire River Adige: the Case on JGrass-NewAGE, American Geophysical Union, Fall Meeting 2015, abstract #H11K-03. M.Bancheri, Rigon, R., Formetta, G. & Green T.R., Implementing a travel time model for water and energy budgets of complex catchments: Theory, software, and preliminary application to the Posina River, Hydrology Days 2016. Seraﬁn, F., Bancheri M., Rigon R. and David O. "A Java binary tree data structure for environmental modelling." (2016), International Congress on Environmental Modelling and Software, 2016 Bancheri, M., et al. "Replicability of a modelling solution using NewAGE-JGrass.", International Congress on Environmental Modelling and Software, 2016 Bancheri M. and Rigon R. “Implementing a travel time model for the water budget of complex catchment: theory and preliminary results.” , XXXV Convegno nazionale di Idraulica e Costruzioni Idrauliche, 2016 Bancheri M., Formetta G., Seraﬁn, F., and Rigon R. “Rasearch reproduciblity and replicability: the case of JGrass- NewAge”, XXXV Convegno nazionale di Idraulica e Costruzioni Idrauliche, 2016
25. 25. Bancheri M. , A travel time model for the water budgets of complex catchments Research outcomes Organized meetings - PhD Days di Ingegneria delle Acque 2015, University of Trento, Italy - Hydrological Modeling with the Object Modelling System (OMS) International Summer Class Short Course, University of Trento, Italy, July 18-21, 2016 Teaching activity - Supervision of undergraduates at Hydrology course A.A 2013-2014 - Supervision of undergraduates at Hydraulic Construction course A.A 2013-2014 - Supervision of undergraduates at Hydrology course A.A 2014-2015 - Supervision of undergraduates at Hydraulic Construction course A.A 2014-2015
26. 26. Bancheri M. , A travel time model for the water budgets of complex catchments Thank you Thank you for your attention!