the catchment: mechanistic model andrea castelletti politecnico di milano nrml09
TRANSCRIPT
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Adriatic Sea
Fucino
VILLA VOMANO
PIAGANINI
PROVVIDENZA
CAMPOTOSTO
MONTORIO (M)
SAN GIACOMO (SG)
Irrigation district(CBN)
S. LUCIA (SL)
PROVVIDENZA (P)
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Identifying the Model
Definining the components and the system scheme
Identifying the models of the components Aggregated model
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Adriatic Sea
Fucino
VILLA VOMANO
PIAGANINI
PROVVIDENZA
CAMPOTOSTO
MONTORIO (M)
SAN GIACOMO (SG)
Irrigation district(CBN)
S. LUCIA (SL)
PROVVIDENZA (P)
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... and the input?
Precipitation Sunshine duration Temperature Air relative umidity Atmospheric pressure Wind velocity
Meteorological variables:
Describe and modulate energy and water exchanges between
atmosphere and the earth.
volume in the time interval [t, t+1)
average temperature in the interval [t, t+1)
How to proceed then?
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The catchment: block diagram
When the model is particularly complex even the simple identification of the causal network might be too difficult.
The system is first decomposed into sub-components, then a causal network is constructed for each component.
BLOCK DIAGRAM
Like causal networks, block diagrams describe cause-effect relationships between relevant variables. However, at a higher conceptual level, at which some complex processes and variables are not yet considered.
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Block diagram 1° step
Pt1
catchment
Tt1
dt1
air temperature
precipitation (solid and liquid)
outflow from the
catchment
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The water cycle
rainfallsnowfall
evaporation
total flow
infiltration
percolation
intercepted rainfall
evapotranspiration
capillary fluxhypodermic
flowdeep flow
surface flow
evaporation
snow
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Block diagram2° step: functional components
snow pack
Tt1
qt1n
ground
drainage net
qt1s
dt1
inflow to the ground
outflow from the ground
Pt1
outfllow from the
catchment
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Block diagram 3° step: orography
....band 1
Tt11
Pt11
qt11
band 2
Tt12
Pt12
qt12
band m
Tt1m
Pt1m
qt1m
+
Tt1 Pt1
snow pack
flow to the ground
qt1n
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Block diagram 4° step: sub-catchments
+
(c)
+
+
(c)
(a)
(b)(b)
(b) (b)
(a)
(a)(a)
The model of each sub-catchment is first identified, then combined with the other to form the aggregated model of the catchment.
The model of each sub-catchment is first identified, then combined with the other to form the aggregated model of the catchment.
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Didactic scheme
COMPONENT Reservoir Catchment Other
components
TYPES of MODELS
BBNs Mechanistic
DETAILS
Mechanistic Campotosto
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Didactic scheme
COMPONENT Reservoir Catchment Other
components
TYPES of MODELS
BBNs Mechanistic
DETAILS
Mechanistic Campotosto
Mechanistic intercept.1350
Typical structure of rainfall/runoff models
air temperature
precipitation (solid and liquid)
Usually rainfall/runoff model have the following structure.
snow pack
Tt1
qt1n
ground
drainge net
qt1s
dt1
inflow to the ground
outflow from the ground
Pt1
outfllow from the
catchment
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Snow pack: the state
• snow-pack depth
• density
• snow temperature
• water content of the snow
• color of the snow surface
• snow-pack depth
• density
• water content of the snow
Solid phase
(water equivalent)
Liquid phase
How is the state of the snow-pack made?
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snow pack
Snowpack: variables
= solid phase of precipitation
= liquid phase of precipitation1
RtP
1S
tP1 tP
Solid phase of the snow-pack (water equivalent) ts Liquid phase of the snow-pack th
flow to the ground1 n
tq
average air temperature1 tT
State variables
Outputs:
Inputs:
1tP1tT
tt
t
sx
h
1ntq
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Snowpacksolid phase dynamics
ts1 ts Pt1S -
melting
M
Tt+1
Tt+1min [ ]}max { 0,
meltingmelting
, ts
ts saturation to st
Net daily snow-meltNet daily snow-melt
M T
t1, h
t, s
t M T
t1, h
t, s
t
always non-negative
Assumption: snow-melt grows linearly with T.
mm of snow melt per °C and per day.
This approach is usually known as “degree-day”.
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M
Tt+1
the frozen volume is always non-
negative
Snowpacksolid phase dynamics
melting - freezing
Tt+1-max [ 0, ]} ( )
- freezing- freezing
ts
min { ,th
- th
For the sake of simplicity let’s assume the same for melting and re-freezing.
For the sake of simplicity let’s assume the same for melting and re-freezing.
meltingmelting
saturation to
th
ts1 ts 1 StP -
Net daily snow meltNet daily snow melt
M T
t1, h
t, s
t M T
t1, h
t, s
t
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M
Tt+1
Snowpacksolid phase dynamics
snow melt - freezing- freezing- freezing
ts
- th
snow meltsnow meltsnow melt - freezing
Net daily snow meltNet daily snow melt
1, , t t tM T h s 1, , t t tM T h sts1 ts 1 StP -
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Snowpackliquid phase dynamics
45°
1 th th 1 RtP ts min{ , }
ts 1ntq
1 , , t t tM T h s
1th
th 1 RtP 1 , , t t tM T h s
flow to the ground
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1ntq
th 1 RtP 1 , , t t tM T h s
Snowpackflow to the ground
45°
1 ntq - ts max{ 0 , }
- ts
th 1 RtP 1 , , t t tM T h s
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min{ , }1 th th 1 RtP ts 1 , , t t tM T h s
Consistency check:it’s raining without snow-pack
System equations:
1 ntq - ts max{ 0 , } th 1 R
tP 1 , , t t tM T h s
ts1 ts 1 StP - 1, , t t tM T h s
It’s raining1 0S
tP
... without snow-pack 0ts
0th
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min{ , }1 th th 1 RtP ts 1 , , t t tM T h s
Consistency check:it’s raining without snow-pack
System equations:
1 ntq - ts max{ 0 , } th 1 R
tP 1 , , t t tM T h s
ts1 ts 1 StP - 1, , t t tM T h s
It’s raining1 0S
tP
... without snow-pack 0ts
0th
1
1
1 1
0
0
t
t
n Rt t
s
h
q P
The flow to the ground is the very rainfall.
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Ground
Evaporation
Surface flow
Flow to the ground
Hypodermic flow
Deep flow
Total runoffRoot zone
Water table
Soil
Percolation
Infiltration
1ntq
ground
1ctq
1tP
snow pack
1tT
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, min MS
Ground the soil
Evaporation
Surface flow
Infiltration
MS
tS 1 ntq
1
M
tS
S
1 1max 0, ( ) t t te P T SK
1tS
1 1max 0, Mt tS S S
tS
1 tS
1ntq
Flow to the ground
Surface flow
1max 0, t MS S
t
M
SS
Degree of saturation
% of inflow retained by the ground
1 t
M
SS
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Inflow retained by the ground
|1
100%-
γ = 1
γ >1
γ < 1
% of inflow retained by the soil
Degree of saturation t MS S
1 t
M
SS
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Ground root zone
Infiltration
Percolation
trK rHypodermic
flow
1tr
St
SM
qt1n tr ( 1- K
r)
min K
prt, R
M
tr
min K
prt, R
M
Percolation
rt
RM
KP
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Ground- drainage network
1 1max 0,
st t
r t f t
Mq S
K r K f
S
1 11 st t td dd qKd K
The total flow from the ground qst+1 is subject to a storing process in the drainage network.
r tK rHypodermic
flowtr
Deep flowf tK ftf
Total flow
Storing coeff.
Surface flow
1max 0, t MS S
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1. model structure
1a. snow pack
1b. ground
Mechanistic model
2. Analysis of the model properties
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Outflow from the catchment
Total flow
Water table
Roots
Soil
1 1(1 ) st d t d td K d K q
1 1 surf. flowst r t g t tq K r K f
1tr 1ntt
M
Sq
S
tr( 1- )rK min ,p t MK r R
tS1 n
tq
1 tS
MS
γ
tevap1 tS 1max 0, t MS S
1tS 1surf. flow t
Raining without snow pack
1tS
1surf. flow t
1stq
1td
1 (1 ) min ,t f t p t Mf K f K r R
1surf. flow t
1 1 n Rt tq P the inflow to the ground is the rainfall1 1 n Rt tq P the inflow to the ground is the rainfall
1R
tP affects dt+1
1ntq
1tS
1stq
The model is a improper one.
It can not be used for managing or forecasting.
It is uselles!
The model is a improper one.
It can not be used for managing or forecasting.
It is uselles!
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The model is an improper one
Total flow+
Surface flow
EvaporationFlow to
the ground
Root zone
Water table
Soil
Percolation
Infiltration
41Outlfow from the
catchment
Total flow
Water table
Roots
Soil
Proper model
f
t1(1 K
f) f
t min K
prt, R
M
q
t1s K
rrt K
gf
t
1tr
St
SM
qt1n tr ( 1- ) Kr
min Kprt, R
M 1 surf. flow t
tS1 n
tq
1 tS
MS
γ
tevap1tS 1max 0, t MS S
1tS 1surf. flow t
qt1n P
t1R flow to the ground is rainfall qt1
n Pt1R flow to the ground is rainfall
qt1n
1tS
1tS
1tr 1surf. flow t
rt+1 does not
affect qst+1
1R
tP does not affect dt+1
1surf. flow t
dt1K
dd
t (1 K
d)q
t1s
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1 1(1 ) st d t d td K d K q Outflow from the
catchment
Total flow
Water Table
Roots
Soil
tS1 n
tq
1 tS
MS
γ
tevap1 tS 1max 0, t MS S
1tS 1surf. flow t
1tr 1t
tM
Sy
S
tr( 1- )rK min ,p t MK r R1 surf. flow t
Rainining without snowpack (ground – proper model)
1 (1 ) min ,t f t p t Mf K f K r R
1 st r t g tq K r K f
1 1 n Rt tq P 1 1 n Rt tq P
1tS
1tr
2 1 1
2 2
1tr 2stq
2stq 2td
Rainfall is affecting only the outflow dt+2
1
The new model can be used in managemen and forecasting,
however ....
The new model can be used in managemen and forecasting,
however ....
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Typical model performance
1 Aug 10 Aug 20 Aug 30 Aug 10 Sep 20 Sep 30 Sep
0
250
500
750
1000
1250
Infl
ow
(
m³/
s )
simulatedobserved
A one-day delay due to the model properties.
A one-day delay due to the model properties.
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Solution to reduce the delay
There are two possible solutions:
1) Reducing the time step to a value smaller than the concentration time in the sub-catchment considered
(right solution)
2) Manipulating and transformin the model into a proper model. (wrong solution)
This latter solution is quite common in hydrology, but it precludes the use of the model in prediction.