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

5

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