Gianluca Botter talked about the travel time distribution approach to catchment scale transport. A topic that intersects also the “old water paradox” querelle, but is, in general, pretty effective in getting the distribution of pollutants. This approach has a long story that put its roots, in Gedeon Dagan’s work, as well as in Rodriguez-Iturbe geomorphic unit hydrograph. Andrea own papers on Mass response function with Sandro Marani can also be considered at the foundations of this presentation.

1. TRAVEL TIME DISTRIBUTIONS AND
CATCHMENT-SCALE TRANSPORT MODELS:
RECENT ADVANCES AND NEW CHALLENGES
GIANLUCA BOTTER
(gianluca.botter@dicea.unipd.it)
UNIVERSITA’ DEGLI STUDI DI PADOVA
Department of Civil Architectural and Environmental Engineering (DICEA)

2. thethethethe chemical responsechemical responsechemical responsechemical response is much more “damped” compared to theis much more “damped” compared to theis much more “damped” compared to theis much more “damped” compared to the
hydrologic signalhydrologic signalhydrologic signalhydrologic signal –––– different processesdifferent processesdifferent processesdifferent processes
HYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALSHYDROLOGIC vs CHEMICAL SIGNALS
[Kirchner et al.., Nature 2000][Kirchner et al.., Nature 2000][Kirchner et al.., Nature 2000][Kirchner et al.., Nature 2000]

3. the hydrologic response to a rainfall event is chiefly made by waterthe hydrologic response to a rainfall event is chiefly made by waterthe hydrologic response to a rainfall event is chiefly made by waterthe hydrologic response to a rainfall event is chiefly made by water
particles alreadyparticles alreadyparticles alreadyparticles already in storage before the event (old water)in storage before the event (old water)in storage before the event (old water)in storage before the event (old water)
THE OLD WATERTHE OLD WATERTHE OLD WATERTHE OLD WATER PARADOXPARADOXPARADOXPARADOX
TRACKS OF PAST RAINFALL EVENTS INTRACKS OF PAST RAINFALL EVENTS INTRACKS OF PAST RAINFALL EVENTS INTRACKS OF PAST RAINFALL EVENTS IN STREAMS…STREAMS…STREAMS…STREAMS…
LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)LASTING FOR MONTHS/YEARS (LONG MEMORY)
EVENTEVENTEVENTEVENT WATERWATERWATERWATER

4. THE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATER & WATER& WATER& WATER& WATER QUALITY ISSUESQUALITY ISSUESQUALITY ISSUESQUALITY ISSUES
NUTRIENTSNUTRIENTSNUTRIENTSNUTRIENTS
PESTICIDESPESTICIDESPESTICIDESPESTICIDES ECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTS
SLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLS

5. THE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATER & WATER& WATER& WATER& WATER QUALITY ISSUESQUALITY ISSUESQUALITY ISSUESQUALITY ISSUES
NUTRIENTSNUTRIENTSNUTRIENTSNUTRIENTS
PESTICIDESPESTICIDESPESTICIDESPESTICIDES ECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTS
SLUDGE SPILLSLUDGE SPILLSLUDGE SPILLSLUDGE SPILL
[Rinaldo et al., WRR 1989][Rinaldo et al., WRR 1989][Rinaldo et al., WRR 1989][Rinaldo et al., WRR 1989]

6. THE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATERTHE AGE OF WATER & WATER& WATER& WATER& WATER QUALITY ISSUESQUALITY ISSUESQUALITY ISSUESQUALITY ISSUES
LAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCELAND MANAGEMENT AND CATCHMENT RESILIENCE

7. XXXX0
XXXXt(t;X0)
X1
X3
X2
INJECTION
AREA
CONTROLCONTROLCONTROLCONTROL
VOLUMEVOLUMEVOLUMEVOLUME
LagrangianLagrangianLagrangianLagrangian transporttransporttransporttransport model:model:model:model:
waterwaterwaterwater particlesparticlesparticlesparticles travelingtravelingtravelingtraveling throughthroughthroughthrough
a control volumea control volumea control volumea control volume
(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)(MAINLY HILLSLOPES PRCESSES)
[e.g., Taylor, 1921;[e.g., Taylor, 1921;[e.g., Taylor, 1921;[e.g., Taylor, 1921; DaganDaganDaganDagan, 1989], 1989], 1989], 1989]
FORMULATION ofFORMULATION ofFORMULATION ofFORMULATION of TRANSPORT AT THE CATCHMENT SCALETRANSPORT AT THE CATCHMENT SCALETRANSPORT AT THE CATCHMENT SCALETRANSPORT AT THE CATCHMENT SCALE
),(
);(
tXV
dt
XtdX
t
t
=0
particle’s trajectory:
INPUT
OUTPUT

8. “Time interval elapsed from the entry to the leave”
TRAVEL (or TRANSIT) TIME tT : the AGE when you die
“ Time spent by a water/solute particle inside a catchment since entry”
RESIDENCE TIME tR : the AGE
RESIDENCE TIMES vs TRAVEL TIMES
...RANDOM VARIABLES (space-time heterogeneity
of the relevant processes and enviroments)

9. RESIDENCE TIME PDF evaluated at time t: pRT (tR , t )
( ) ( ) ( )∫∞−
−=
t
iiRTiINs
dttttptCtC ,
average storage concentration
(STORAGE MEMORY of the INPUT)
RESIDENCE TIMES vs TRAVEL TIMES
AGE DISTRIBUTION of living people

10. RESIDENCE TIME PDF evaluated at time t: pRT (tR , t )
( ) ( ) ( )∫∞−
−=
t
iiRTiINs
dttttptCtC ,
average storage concentration
(STORAGE MEMORY of the INPUT)
RESIDENCE TIMES vs TRAVEL TIMES
TRAVEL TIME PDF conditional to the exit time: p’OUT (tT , t )
( ) ( ) ( )∫∞−
−=
t
iiOUTiINOUT
dttttptCtC ,'
output flux concentration
(OUTPUT MEMORY of the INPUT)
AGE DISTRIBUTION of living people AGE DISTRIBUTION of people dying

11. RESIDENCE TIME PDF: TEMPORAL EVOLUTION
STORAGE S(t)INPUT OUTPUT
low
high
N(tR,t)
N(tR,t) = S(t) pRT(tR,t)
number of particles with
age tR at time t
RESIDENCE TIME tR
IN(t)
OUT(t)
t
t
PURE AGEING: 0=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
PLAYPAUSE

12. RESIDENCE TIME PDF: TEMPORAL EVOLUTION
STORAGE S(t)INPUT OUTPUT
low
high
N(tR,t)
N(tR,t) = S(t) pRT(tR,t)
number of particles with
age tR at time t
RESIDENCE TIME tR
IN(t)
OUT(t)
t
t
PURE AGEING: 0=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
PLAYPAUSE

13. RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(tR,t)
RESIDENCE TIME tR
BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
PURE AGEING: 0=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
INPUTIN(t)
t
STORAGE S(t) OUTPUT
low
high
OUT(t)
t
PAUSEPLAY

14. RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(tR,t)
RESIDENCE TIME tR
BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
=
∂
∂
+
∂
∂
R
RR
t
ttN
t
ttN ),(),(
INPUTIN(t)
t
STORAGE S(t) OUTPUT
low
high
OUT(t)
t
AGEING + SAMPLING VIA OUTPUT )(),(' tOUTttp ROUT
−
PAUSEPLAY

15. RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(tR,t)
RESIDENCE TIME tR
INPUTIN(t)
t
STORAGE S(t) OUTPUT
low
high
OUT(t)
t
)(),('
)],()([)],()([
tOUTttp
t
ttptS
t
ttptS
ROUT
R
RRTRRT
−=
∂
∂
+
∂
∂

16. The particles leaving the system are sampled among those in storage,The particles leaving the system are sampled among those in storage,The particles leaving the system are sampled among those in storage,The particles leaving the system are sampled among those in storage,
and so their age:and so their age:and so their age:and so their age:
w(tR, t)p’OUT(tR,t) = pRT(tR, t)
AGES SAMPLED AGES AVAILABLE PREFERENCE
AGE FUNCTIONAGE FUNCTIONAGE FUNCTIONAGE FUNCTION
(MIXING)(MIXING)(MIXING)(MIXING)
LOW AVAILABILITY or LOWLOW AVAILABILITY or LOWLOW AVAILABILITY or LOWLOW AVAILABILITY or LOW PREFERENCEPREFERENCEPREFERENCEPREFERENCE IMPLIES LOWIMPLIES LOWIMPLIES LOWIMPLIES LOW SAMPLINGSAMPLINGSAMPLINGSAMPLING
–––– AGESAGESAGESAGES NOT REPRESENTED IN THE OUTPUTNOT REPRESENTED IN THE OUTPUTNOT REPRESENTED IN THE OUTPUTNOT REPRESENTED IN THE OUTPUT
MIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’sMIXING: LINKING RESIDENCE and TRAVEL TIME PDF ’s
1
SAMPLING through AGE functions
11
uniform preference
over all ages
ωωωω decreases for
older ages
,,, =const
ωωωω increases for
older ages
randomrandomrandomrandom samplingsamplingsamplingsampling preference for old waterpreference for old waterpreference for old waterpreference for old water preference for new waterpreference for new waterpreference for new waterpreference for new water

17. RAINFALL
DISCHARGE
EVAPO-
TRANSPIRATION
THE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDF
)()()(
)(
tQtETtJ
dt
tdS
−−=
),(),(
),(),(
ttFttp
t
ttp
t
ttp
RRRT
R
RRTRRT
=
∂
∂
+
∂
∂
MASTER EQUATION FOR pRT:






−−+−=
)(
)(
)],([
)(
)(
)],([
)(
)(
),(
tS
tJ
tt
tS
tQ
tt
tS
tQ
ttF RETRQR
ωωωωωωωω 11
(WATER FLUXES, TYPE OF MIXING)
where:

18. RAINFALL
DISCHARGE
EVAPO-
TRANSPIRATION
THE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDFTHE MASTER EQUATION FOR THE RESIDENCE TIME PDF
)()()(
)(
tQtETtJ
dt
tdS
−−=
),(),(
),(),(
ttFttp
t
ttp
t
ttp
RRRT
R
RRTRRT
=
∂
∂
+
∂
∂
MASTER EQUATION FOR pRT:






−−+−=
)(
)(
)],([
)(
)(
)],([
)(
)(
),(
tS
tJ
tt
tS
tQ
tt
tS
tQ
ttF RETRQR
ωωωωωωωω 11
where:
................ ANALYTICAL SOLUTIONS forANALYTICAL SOLUTIONS forANALYTICAL SOLUTIONS forANALYTICAL SOLUTIONS for ωωωω=1=1=1=1
TRAVEL & RESIDENCE TIME PDF’s MUST BETRAVEL & RESIDENCE TIME PDF’s MUST BETRAVEL & RESIDENCE TIME PDF’s MUST BETRAVEL & RESIDENCE TIME PDF’s MUST BE CONSISTENTCONSISTENTCONSISTENTCONSISTENT WITH THEWITH THEWITH THEWITH THE
SET OF AGES AVAILABLESET OF AGES AVAILABLESET OF AGES AVAILABLESET OF AGES AVAILABLE ANDANDANDAND THE INPUT/OUTPUTTHE INPUT/OUTPUTTHE INPUT/OUTPUTTHE INPUT/OUTPUT FLUXESFLUXESFLUXESFLUXES
(WATER FLUXES, TYPE OF MIXING)

19. THE CASE OFTHE CASE OFTHE CASE OFTHE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))






−
−
−
= ∫−
t
tt
T
T
TOUT
T
dx
xS
xJ
ttS
ttJ
ttp
)(
)(
exp
)(
)(
),('
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF
[Botter et al., WRR 2010]

20. THE CASE OFTHE CASE OFTHE CASE OFTHE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))






−
−
−
= ∫−
t
tt
T
T
TOUT
T
dx
xS
xJ
ttS
ttJ
ttp
)(
)(
exp
)(
)(
),('
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF
[Botter et al., WRR 2010]
Relationship with
advection/dispersion models:

21. THE CASE OFTHE CASE OFTHE CASE OFTHE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))
ANALYTICAL (WELL MIXED)
NUMERICAL 3D SIMULATION
C=Cmax
C=0






−
−
−
= ∫−
t
tt
T
T
TOUT
T
dx
xS
xJ
ttS
ttJ
ttp
)(
)(
exp
)(
)(
),('
EMERGING FEATURES:
1. TIME VARIANCE
2. DEPENDENCE ON ECO-
HYDROLOGICAL PROCESSES
SERIES/PARALLEL OF MIXEDSERIES/PARALLEL OF MIXEDSERIES/PARALLEL OF MIXEDSERIES/PARALLEL OF MIXED STORAGES,STORAGES,STORAGES,STORAGES, ifififif necessarynecessarynecessarynecessary
[Rinaldo et al., WRR 2011]
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF

22. GENERAL MODELGENERAL MODELGENERAL MODELGENERAL MODEL STRUCTURE: TWOSTRUCTURE: TWOSTRUCTURE: TWOSTRUCTURE: TWO RS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTS
(Selective) ET(Selective) ET(Selective) ET(Selective) ET
may increase c(t)
)()()(
)(
tLtETtJ
dt
tds
nZR
−−=
)()()(
)(
tMtMtF
dt
tdM
LET
s
−−=
ROOT ZONE
DEEPER STORAGELEACHINGLEACHINGLEACHINGLEACHING

23. APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)
CHLORIDES introduced
through FERTILIZERS...
...inducing high Cl
concentrations in
STREAMS

24. APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)
NEGATIVE
CORRELATION
between Q and C

25. APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)APPLICATION 1. CHLORIDE TRANSPORT IN THE HUPSEL BROOK (NL)
NO FERTILIZATIONS
NEGATIVE
CORRELATION
between Q and C
PERSISTENCY of
the chemical
signal, despite the
input seasonality
(long memory)

26. OBSERVEDOBSERVEDOBSERVEDOBSERVED AND MODELEDAND MODELEDAND MODELEDAND MODELED DISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOK

27. OBSERVEDOBSERVEDOBSERVEDOBSERVED AND MODELEDAND MODELEDAND MODELEDAND MODELED Cl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOKCl CONCENTRATIONS @ HUPSEL BROOK
SHORT TERM FLUCTUATIONS RELATED TO
THE ROOT ZONE (short travel times)
in WINTER the Clin WINTER the Clin WINTER the Clin WINTER the Cl concentrationconcentrationconcentrationconcentration isisisis sustainedsustainedsustainedsustained by GW (longby GW (longby GW (longby GW (long traveltraveltraveltravel timestimestimestimes))))

28. TIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDFTIME VARIANCE OF THE RESIDENCE TIME PDF
ttttRRRR [[[[ ]]]]ttttRRRR [[[[ ]]]]
ppppRTRTRTRT((((ttttRRRR,t,t,t,t))))[[[[]]]]ppppRTRTRTRT((((ttttRRRR,t,t,t,t))))[[[[]]]]
ttttRRRR [[[[ ]]]]
ppppRTRTRTRT((((ttttRRRR,t,t,t,t))))[[[[]]]]
WINTERWINTERWINTERWINTER
(WET)(WET)(WET)(WET)
SUMMERSUMMERSUMMERSUMMER
(DRY)(DRY)(DRY)(DRY)

29. APPLICATIONAPPLICATIONAPPLICATIONAPPLICATION 2.2.2.2. CHLORIDE TRANSPORTCHLORIDE TRANSPORTCHLORIDE TRANSPORTCHLORIDE TRANSPORT @ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)
INCREASE or DECREASE of
chloride concentrations during
floods (depending on the season)

30. APPLICATIONAPPLICATIONAPPLICATIONAPPLICATION 3.3.3.3. PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)
Huge PESTICIDE LOADSPESTICIDE LOADSPESTICIDE LOADSPESTICIDE LOADS introduced through farming...
high ATRAZINEATRAZINEATRAZINEATRAZINE CONCENTRATIONSCONCENTRATIONSCONCENTRATIONSCONCENTRATIONS in STREAMS during the SUMMER
(ATRAZINE is CANCEROUS)

31. ANNUAL
DYNAMICS of
DISCHARGEDISCHARGEDISCHARGEDISCHARGE and
ATHRAZINEATHRAZINEATHRAZINEATHRAZINE
CONCENTRATIONCONCENTRATIONCONCENTRATIONCONCENTRATION
APPLICATIONAPPLICATIONAPPLICATIONAPPLICATION 3.3.3.3. PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)PESTICIDE TRANSPORT IN THE MONCHALTORF (CH)

32. OBSERVEDOBSERVEDOBSERVEDOBSERVED AND MODELED At CONCENTRATIONS @ MONCHALTORFAND MODELED At CONCENTRATIONS @ MONCHALTORFAND MODELED At CONCENTRATIONS @ MONCHALTORFAND MODELED At CONCENTRATIONS @ MONCHALTORF
OBSERVED
MODEL

33. CHALLENGESCHALLENGESCHALLENGESCHALLENGES
Derive analytical or numerical solutions forDerive analytical or numerical solutions forDerive analytical or numerical solutions forDerive analytical or numerical solutions for arbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functions,,,,
suitably parametrizedsuitably parametrizedsuitably parametrizedsuitably parametrized –––– relationship withrelationship withrelationship withrelationship with spatially distributedspatially distributedspatially distributedspatially distributed modelsmodelsmodelsmodels
BetterBetterBetterBetter understandunderstandunderstandunderstand thethethethe rolerolerolerole ofofofof
evapotranspirationevapotranspirationevapotranspirationevapotranspiration andandandand plantplantplantplant
physiologyphysiologyphysiologyphysiology forforforfor thethethethe hydrohydrohydrohydro----
chemistrychemistrychemistrychemistry ofofofof riversriversriversrivers
Let the age functions beLet the age functions beLet the age functions beLet the age functions be dependent on the catchment’dependent on the catchment’dependent on the catchment’dependent on the catchment’ statestatestatestate (e.g. wet(e.g. wet(e.g. wet(e.g. wet
VS dry)VS dry)VS dry)VS dry)
FLOODS
DROUGHTS
AGE functions

34. CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS
timetimetimetime variance of travelvariance of travelvariance of travelvariance of travel &&&&
residenceresidenceresidenceresidence time pdf’stime pdf’stime pdf’stime pdf’s @@@@
multiple timescalesmultiple timescalesmultiple timescalesmultiple timescales
(variability(variability(variability(variability ofofofof fluxesfluxesfluxesfluxes
and storages)and storages)and storages)and storages)
Robustness of the approach,Robustness of the approach,Robustness of the approach,Robustness of the approach,
suggested by the comparisonsuggested by the comparisonsuggested by the comparisonsuggested by the comparison
with field datawith field datawith field datawith field data
travel and residence time pdf’s aretravel and residence time pdf’s aretravel and residence time pdf’s aretravel and residence time pdf’s are
related objectsrelated objectsrelated objectsrelated objects dependent on mixingdependent on mixingdependent on mixingdependent on mixing
processes and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxes

35. THE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVATHE RESEARCH TEAM (PADOVA----LAUSANNE)LAUSANNE)LAUSANNE)LAUSANNE)
University of Padova, Italy
Gianluca Botter
Paolo Benettin
EPFL, Lausanne, CH
Andrea Rinaldo
Enrico Bertuzzo

36. TRAVEL TIME DISTRIBUTIONS AND
CATCHMENT-SCALE TRANSPORT MODELS:
RECENT ADVANCES AND NEW CHALLENGES
GIANLUCA BOTTER
(gianluca.botter@dicea.unipd.it)
UNIVERSITA’ DEGLI STUDI DI PADOVA
Department of Civil Architectural and Environmental Engineering (DICEA)