residence time theories of pollutants transport
DESCRIPTION
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.TRANSCRIPT
TRAVEL TIME DISTRIBUTIONS AND
CATCHMENT- SCALE TRANSPORT MODELS:
RECENT ADVANCES AND NEW CHALLENGES
GIANLUCA BOTTER([email protected])
UNIVERSITA’ DEGLI STUDI DI PADOVADepartment of Civil Architectural and Environmental Engineering (DICEA)
the the the the chemical response chemical response chemical response chemical response is much more “damped” compared to the is much more “damped” compared to the is much more “damped” compared to the is much more “damped” compared to the
hydrologic signal hydrologic signal hydrologic signal hydrologic 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]
the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water the hydrologic response to a rainfall event is chiefly made by water
particles already particles already particles already particles 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 WATER THE OLD WATER THE OLD WATER THE OLD WATER PARADOXPARADOXPARADOXPARADOX
TRACKS OF PAST RAINFALL EVENTS IN TRACKS OF PAST RAINFALL EVENTS IN TRACKS OF PAST RAINFALL EVENTS IN TRACKS 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)
EVENT EVENT EVENT EVENT WATERWATERWATERWATER
THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER & WATER & WATER & WATER & WATER QUALITY ISSUESQUALITY ISSUESQUALITY ISSUESQUALITY ISSUES
NUTRIENTSNUTRIENTSNUTRIENTSNUTRIENTS
PESTICIDESPESTICIDESPESTICIDESPESTICIDES ECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTSECOSYSTEM IMPACTS
SLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLSSLUDGE SPILLS
THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER THE 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]
THE AGE OF WATER THE AGE OF WATER THE AGE OF WATER THE 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
XXXX0
XXXXt(t;X0)
X1
X3
X2
INJECTION
AREA
CONTROL CONTROL CONTROL CONTROL
VOLUMEVOLUMEVOLUMEVOLUMELagrangianLagrangianLagrangianLagrangian transporttransporttransporttransport model: model: model: model:
water water water water particlesparticlesparticlesparticles travelingtravelingtravelingtraveling throughthroughthroughthrough
a control volume a control volume a control volume a 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 of FORMULATION of FORMULATION of FORMULATION of TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE TRANSPORT AT THE CATCHMENT SCALE
),();(
tXVdt
XtdXt
t =0
particle’s trajectory:
INPUT
OUTPUT
“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 heterogeneityof the relevant processes and enviroments)
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
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
RESIDENCE TIME PDF: TEMPORAL EVOLUTION
STORAGE S(t)INPUT OUTPUT
low
high
N(t
R,t)
N(tR,t) = S(t) pRT(tR,t)
number of particles with age tR at time t
RESIDENCE TIME tR
IN(t
)
OU
T(t)
t
t
PURE AGEING: 0=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
PLAYPAUSE
RESIDENCE TIME PDF: TEMPORAL EVOLUTION
STORAGE S(t)INPUT OUTPUT
low
high
N(t
R,t)
N(tR,t) = S(t) pRT(tR,t)
number of particles with age tR at time t
RESIDENCE TIME tR
IN(t
)
OU
T(t)
t
t
PURE AGEING: 0=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
PLAYPAUSE
RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(t
R,t)
RESIDENCE TIME tR
BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
PURE AGEING: 0=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
INPUTIN
(t) t
STORAGE S(t) OUTPUT
low
high
OU
T(t)
t
PAUSEPLAY
RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(t
R,t)
RESIDENCE TIME tR
BOUNDARY CONDITION: N(tR=0,t)=IN(t) + AGEING: 0=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
=∂
∂+
∂∂
R
RR
tttN
tttN ),(),(
INPUTIN
(t) t
STORAGE S(t) OUTPUT
low
high
OU
T(t)
t
AGEING + SAMPLING VIA OUTPUT )(),(' tOUTttp ROUT−
PAUSEPLAY
RESIDENCE TIME PDF: TEMPORAL EVOLUTION
N(t
R,t)
RESIDENCE TIME tR
INPUTIN
(t) t
STORAGE S(t) OUTPUT
low
high
OU
T(t)
t
)(),(')],()([)],()([ tOUTttpt
ttptSt
ttptSROUT
R
RRTRRT −=∂
∂+
∂∂
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 LOW LOW AVAILABILITY or LOW LOW AVAILABILITY or LOW LOW AVAILABILITY or LOW PREFERENCE PREFERENCE PREFERENCE PREFERENCE IMPLIES LOW IMPLIES LOW IMPLIES LOW IMPLIES LOW SAMPLINGSAMPLINGSAMPLINGSAMPLING
–––– AGES AGES AGES AGES NOT REPRESENTED IN THE OUTPUT NOT REPRESENTED IN THE OUTPUT NOT REPRESENTED IN THE OUTPUT NOT 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 agesω ω ω ω decreases for
older ages
� ��, �� ��, �� ��, � =const
ωωωω increases for
older ages
random random random random 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
�� �� ��
� � �
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
)()()()(
tQtETtJdt
tdS −−=
),(),(),(),(
ttFttpt
ttpt
ttpRRRT
R
RRTRRT =∂
∂+∂
∂
MASTER EQUATION FOR pRT:
−−+−=)()(
)],([)()(
)],([)()(
),(tStJ
tttStQ
tttStQ
ttFRETRQR
ωωωωωωωω 11
(WATER FLUXES, TYPE OF MIXING)
where:
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
)()()()(
tQtETtJdt
tdS −−=
),(),(),(),(
ttFttpt
ttpt
ttpRRRT
R
RRTRRT =∂
∂+∂
∂
MASTER EQUATION FOR pRT:
−−+−=)()(
)],([)()(
)],([)()(
),(tStJ
tttStQ
tttStQ
ttFRETRQR
ωωωωωωωω 11
where:
.... .... .... .... ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ANALYTICAL SOLUTIONS for ωωωω=1=1=1=1
TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE TRAVEL & RESIDENCE TIME PDF’s MUST BE CONSISTENT CONSISTENT CONSISTENT CONSISTENT WITH THE WITH THE WITH THE WITH THE
SET OF AGES AVAILABLE SET OF AGES AVAILABLE SET OF AGES AVAILABLE SET OF AGES AVAILABLE AND AND AND AND THE INPUT/OUTPUT THE INPUT/OUTPUT THE INPUT/OUTPUT THE INPUT/OUTPUT FLUXESFLUXESFLUXESFLUXES
(WATER FLUXES, TYPE OF MIXING)
THE CASE OF THE CASE OF THE CASE OF THE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))
−
−−= ∫
−
t
ttT
T
TOUTT
dxxSxJ
ttSttJ
ttp)()(
exp)()(
),('
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF
[Botter et al., WRR 2010]
THE CASE OF THE CASE OF THE CASE OF THE CASE OF RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (RANDOM SAMPLING (ωωωω=1=1=1=1))))
−
−−= ∫
−
t
ttT
T
TOUTT
dxxSxJ
ttSttJ
ttp)()(
exp)()(
),('
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF
[Botter et al., WRR 2010]
��
���
��
�� �
� �
�
Relationship with
advection/dispersion models:
THE CASE OF THE CASE OF THE CASE OF THE 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
ttT
T
TOUTT
dxxSxJ
ttSttJ
ttp)()(
exp)()(
),('
EMERGING FEATURES:
1. TIME VARIANCE
2. DEPENDENCE ON ECO-
HYDROLOGICAL PROCESSES
� SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED SERIES/PARALLEL OF MIXED STORAGES, STORAGES, STORAGES, STORAGES, ifififif necessarynecessarynecessarynecessary
[Rinaldo et al., WRR 2011]
TRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDFTRAVEL TIME PDF
GENERAL MODEL GENERAL MODEL GENERAL MODEL GENERAL MODEL STRUCTURE: TWO STRUCTURE: TWO STRUCTURE: TWO STRUCTURE: TWO RS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTSRS COMPARTMENTS
(Selective) ET(Selective) ET(Selective) ET(Selective) ETmay increase c(t)
)()()()(
tLtETtJdt
tdsnZ
R−−=
)()()()(
tMtMtFdt
tdMLET
s −−=
ROOT ZONE
DEEPER STORAGELEACHINGLEACHINGLEACHINGLEACHING
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
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
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)
OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED AND MODELED AND MODELED AND MODELED DISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOKDISCHARGES @ HUPSEL BROOK
OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED AND MODELED AND MODELED AND 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 Cl in WINTER the Cl in WINTER the Cl in WINTER the Cl concentrationconcentrationconcentrationconcentration isisisis sustainedsustainedsustainedsustained by GW (long by GW (long by GW (long by GW (long traveltraveltraveltravel timestimestimestimes))))
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 [[[[�]]]]
pp ppR
TR
TR
TR
T(( (( tt tt
RR RR,t,t ,t,t
)) ))[[ [[���]] ]]
pp ppR
TR
TR
TR
T(( (( tt tt
RR RR,t,t ,t,t
)) ))[[ [[���]] ]]
ttttRRRR [[[[�]]]]
pp ppR
TR
TR
TR
T(( (( tt tt
RR RR,t,t ,t,t
)) ))[[ [[���]] ]]
WINTER WINTER WINTER WINTER
(WET)(WET)(WET)(WET)
SUMMER SUMMER SUMMER SUMMER
(DRY)(DRY)(DRY)(DRY)
APPLICATION APPLICATION APPLICATION APPLICATION 2. 2. 2. 2. CHLORIDE TRANSPORT CHLORIDE TRANSPORT CHLORIDE TRANSPORT CHLORIDE TRANSPORT @ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)@ PLYNIMON (UK)
INCREASE or DECREASE of
chloride concentrations during
floods (depending on the season)
APPLICATION APPLICATION APPLICATION APPLICATION 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 LOADS PESTICIDE LOADS PESTICIDE LOADS PESTICIDE LOADS introduced through farming...
high ATRAZINE ATRAZINE ATRAZINE ATRAZINE CONCENTRATIONS CONCENTRATIONS CONCENTRATIONS CONCENTRATIONS in STREAMS during the SUMMER
(ATRAZINE is CANCEROUS)
ANNUAL
DYNAMICS of
DISCHARGE DISCHARGE DISCHARGE DISCHARGE and
ATHRAZINE ATHRAZINE ATHRAZINE ATHRAZINE
CONCENTRATIONCONCENTRATIONCONCENTRATIONCONCENTRATION
APPLICATION APPLICATION APPLICATION APPLICATION 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)
OBSERVED OBSERVED OBSERVED OBSERVED AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF AND MODELED At CONCENTRATIONS @ MONCHALTORF
OBSERVED
MODEL
CHALLENGESCHALLENGESCHALLENGESCHALLENGES
Derive analytical or numerical solutions for Derive analytical or numerical solutions for Derive analytical or numerical solutions for Derive analytical or numerical solutions for arbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functionsarbitrary mixing functions, , , ,
suitably parametrized suitably parametrized suitably parametrized suitably parametrized –––– relationship with relationship with relationship with relationship with spatially distributed spatially distributed spatially distributed spatially distributed modelsmodelsmodelsmodels
BetterBetterBetterBetter understandunderstandunderstandunderstand thethethethe rolerolerolerole ofofofof
evapotranspirationevapotranspirationevapotranspirationevapotranspiration andandandand plantplantplantplant
physiologyphysiologyphysiologyphysiology forforforfor thethethethe hydrohydrohydrohydro----
chemistrychemistrychemistrychemistry ofofofof riversriversriversrivers
Let the age functions be Let the age functions be Let the age functions be Let the age functions be dependent on the catchment’ dependent on the catchment’ dependent on the catchment’ dependent on the catchment’ state state state state (e.g. wet (e.g. wet (e.g. wet (e.g. wet
VS dry)VS dry)VS dry)VS dry)
FLOODSDROUGHTS
� AGE functions
CONCLUSIONSCONCLUSIONSCONCLUSIONSCONCLUSIONS
time time time time variance of travel variance of travel variance of travel variance of travel & & & &
residence residence residence residence time pdf’s time pdf’s time pdf’s time pdf’s @ @ @ @
multiple timescales multiple timescales multiple timescales multiple timescales
(variability (variability (variability (variability of of of of fluxes fluxes fluxes fluxes
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 comparison suggested by the comparison suggested by the comparison suggested by the comparison
with field datawith field datawith field datawith field data
travel and residence time pdf’s are travel and residence time pdf’s are travel and residence time pdf’s are travel and residence time pdf’s are
related objects related objects related objects related objects dependent on mixing dependent on mixing dependent on mixing dependent on mixing
processes and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxesprocesses and hydrologic fluxes
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
TRAVEL TIME DISTRIBUTIONS AND
CATCHMENT- SCALE TRANSPORT MODELS:
RECENT ADVANCES AND NEW CHALLENGES
GIANLUCA BOTTER([email protected])
UNIVERSITA’ DEGLI STUDI DI PADOVADepartment of Civil Architectural and Environmental Engineering (DICEA)