the catchment: empirical model nrml10 lena - delta andrea castelletti politecnico di milano
TRANSCRIPT
The catchment: empirical model
NRMNRML10L10
Lena - Delta
Andrea CastellettiPolitecnico di Milano
2
The catchment
control section
3
Traditional models
• Rational method (Mulvany, 1850)
• Unit hydrograph Sherman (1932)
• Nash model (1957)
4
pioggia
d(t)t
P
impulsive unit rainfall
Rational method (Mulvany, 1850)
P
t
stepwise unit rainfall
t
A d
tc
ttc
dA
dt d
A(t)
t
dA
t+dt
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Unit hydrograph (Sherman, 1932)
Postulates
For a given catchment:
1. Runoff duration is equal for rainfall events of equal duration, regardless of their total volume.
2. At time t from the beginning of the event, rainfall events having the same temporal distribution generate runoff volumes in the same relation as the total volumes of rainfall.
3. Runoff temporal distribution does not depend on the previous history of the catchment
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Convolution integral
Unit hydrograph (Sherman, 1932)
d(t) h( )
0
t
P(t - )d
t
P
d
ttc
Cumulative hydrograph curveCumulative hydrograph curve unithydrograph
h( )
rainfall
P(t - )
The catchment is a linear system
The catchment is a linear system
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Nash model (Nash 1957)PtPt
dtdt
The catchment is modelled as a cascade of n (e.g. n=4) reservoirs
State transition
xt11 (1- k
1)x
t1 h
1P
t
xt12 (1- k
2)x
t2 k
1x
t1 h
2P
t
.....................
xt1n (1- k
n)x
tn k
n 1x
tn 1 h
nP
t
h1Pt
k1xt1
x1
h2Pt
k2xt2
x2
h3Pt
k3xt3
x3
x4
h4Pt
k4xt4
= dtdt
dtk
nx
tn
Output transformation
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Modello di Nash, 1957
The catchment is modelled as a cascade of n (e.g. n=4) reservoirs
State transition
xt11 (1- k
1)x
t1 h
1P
t
xt12 (1- k
2)x
t2 k
1x
t1 h
2P
t
.....................
xt1n (1- k
n)x
tn k
n 1x
tn 1 h
nP
t
dtk
nx
tn
Output transformation
dt1a1d
t a2d
t 1 ..... and
t n1 b1P
t b2 P
t 1 ..... bn P
t n1
Linear system
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ARX models (‘ 70)
dt+1 = dt persistent runoff
dt+1 = 2dt - dt-1 AR(2)
dt+1 = a1 dt +….+an
dt-n+1 AR(n)
ARX
dt+1 = Pt-n+1 n = time of concentration
dt+1 = b1Pt+ …+bnPt-n+1
3) Complete model:
1) Runoff prediction using rainfall data:
2) Runoff prediction using runoff data:
t
d
t-2 t-1 t t+1
measurecomput.
dt1 a1d
t a2d
t 1 K and
t n1 + b1P
t K bn P
t n1 dt1
a1dt a2d
t 1 K and
t n1 + b1P
t K bn P
t n1
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Some remarks
• ARX ≡ Sherman : a linear model described through the impulse response.
• ARX ≡ Mulvany : the catchment is a linear system (the impulse response is the derivative of the step response)
These 4 models are identical by a mathematical point of view.
They can be traced back to the same linear equation.
• ARX is the I/O relation of a linear discrete model
ARX ≡ Nash dt1
a1dt a 2 d
t 1 K a n d
t n1 + b1P
t K bn P
t n1
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Some remarks
The main difference in the 4 approaches is the way the parameters are estimated
Mulvany: performs a qualitative estimate based on the topographic characteristic of the cacthment.
Sherman: either assumes a given a priori shape for the unit hydrograph or estimates it starting from observed impulsive rainfall events.
Nash: uses a trial-and-error approach for estimating the parameters
ARX: uses the least square algorithm
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but then …
the functions of the models could be identified directly, without caring about the cause-effect relationships in the real physical process
more precisely, we could directly identify the relationship between inputs and outputs
For instance, we could describe the dynamics of the output with an expression of the following form
y
t1y(y
t,...,y
t ( p 1),u
t,...,u
t (r ' 1), w
t,..., w
t (r '' 1),
t1,...,
t (q 1))
usually called either input/output form or external representation
empirical models
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Mechanistic models
Reality
Empirical models
Mechanistic and empirical models
How to find it?
SPACE OF THE MODELS
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Empirical models
They simply reproduce the relationship between the system inputs and outputs
Rainfall time series
Empirical model
Flow rate time series
structural changes in the water system can not be modelled.
Drawback
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Empirical models
Does the external representation always exist?In practice ....• empirically assume that it exists
• fix the order (p,r’,r’’,q) of the model a priori and perform the parameter estimation using the appropriate algorithm;
• if the model outputs fit the reality well, the external representation have been found; otherwise, go back to the previous step and increase the model order
• ... go on until the external representation has been found or the model order is “too high”.
hth(s
t)
st1 f (s
t,u
t, w
t,
t1)
internal representation of the reservoir
external representation h
t1h(h
t,...,h
t ( p 1),u
t,...,u
t (r ' 1),w
t,...,w
t (r '' 1),
t1,...,
t (q 1))
Examples
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Empirical modelsEmpirical models do not aim at increasing our knowledge of the physical process (scientific purpose), but only at predicting as accurately as possible the system ouputs given some inputs (engineering purpose).
black-box models
The external form is sought for into an a priori given class of functions
When all the variables are scalar the linear form is often used
yt1
t1y
t ...
tp y
t ( p 1)
t1'u
t ...
tr 'u
t (r ' 1)
+t1''w
t ...
tr ''w
t (r '' 1)
t1
t1
t ...
tq
t (q 1)
PARMAX
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Empirical models
The linear form is a very simple one and it is supported by powerful calibration algorithms, however not always is it the more suitable …
st1s
t u
t(w
t
t1) but...
s
t
N(g)
r
t1
u
t
s
not-linear!
NOT!
A non-linear class of functions would be by far a better choice, e.g.
ARTIFICIAL NEURAL NETWORKS
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Stochastic empirical models
Moreover, it is worth while considering the stochastic form
st1s
t u
t(w
t
t1) t1
… some time using a “coloured noise” (non-white noise)
s
t1s
t u
t(w
t
t1)
t1
t1
t ...
tq
t (q 1)
yt1
y(yt,...,y
t ( p 1),u
t,...,u
t (r ' 1), w
t,...,w
t (r '' 1),
,t1
,...,t (q ' 1)
,t,...,
t (q '' 1))
t1
In general
process noise
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Remarks
The external representation can be properly identified only when the available input/ouput historical time series are long enough
Empirical models can never be used when the alternatives considered do include structural changes to the system, as no data are available that describe their effects.
The prediction potential of a model highly depends on the class of functions adopted for it.
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In conclusion …
Mechanistic models tend to be too much complicated and often describe details that are useless if they are used for management purposes only, i.e. details with no effect on the I/O relationship.
Identifying empirical models does require the class of function they belong to be specified a priori. This migth strongly affect the quality of the model.
IDEA (relatively new 1994)
Use a mechanistic model, but infer the shape of its characteristic functions directly from data, ignoring any a priori information available.
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An exampleCunning River – Australia
?
The soil is too dry and the rainfall
completely absorbed
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measured runoff
estimated runoff
A tentative model of the Cunning river
yt1
ty
t
tw
t
t1Let’s try with a PARMAX
rainfall
NO
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Data Based Mechanistic (DBM) models
yt1 y
t( y
t)w
t
t1Let’s try with a
DBM model
The value of depends on the runoff rate, which in turn depends on the soil moisture.
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Data Based Mechanistic (DBM) models
yt1 y
t( y
t)w
t
t1Let’s try with a
DBM model
The value of depends on the runoff rate, which in turn depends on the soil moisture.
measured runoff
estimated runoff
the prediction is now accurate
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Disturbances
Ultimately, models are to be used to simulate the system behaviour under a given alternative
xt1
ft(x
t,u p ,u
t,w
t,
t1)
yt1
ht(x
t,u p ,u
t, w
t,
t1)
Input trajectories are required to run simulations.
provided by the policy
deterministically know at time t, but at the time of the project?
random: who will provide it?
N.B. The disturbances we are dealing with here are the
disturbances of the
GLOBAL MODEL
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Who does provide it?
Two options:
1. Using the historical trajectorybut it could be too short.
2. Identifying a model of the disturbance however without inputs otherwise... we fall into a vicious circle...... that sometime could be useful.
Sooner or later the disturbance must be described without introducing further models, thus using its previous values only and, at the most, some state or control variables of the systems.
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t1y
t(
t,...,
t ( p 1),u
t,...,u
t (r ' 1), x
t,..., x
t (r '' 1),
t,...,
t (q 1))
t1
y
t1y
t( y
t,..., y
t ( p 1),u
t,...,u
t (r ' 1),x
t,...,x
t (r '' 1),
t,...,
t (q 1))
t1
The model must be an empirical one
and suitably changing the notation …
NOT!
vicious circleunless ...
is a white noise
process noise
How to model the disturbances
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White noise
A time series for which a model can be formulated is said
algorithmically compressible
An algorithmically un-compressible time series is awhite noise
With stochastic disturbances this means a self-correlogramm
equal to zero .
In conclusion: distubances must be modeled as white noise
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Reading
IPWRM.Theory Ch. 4/Ap.6-7
IPWRM.Practice Sec. 6.5