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.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Bancheri M. , A travel time model for the water budgets of complex catchments
Thank you
Thank you for your attention!