The catchment: empirical model NRML10 Lena - Delta Andrea Castelletti Politecnico 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)d(t) t P impulsive unit rainfall Rational method (Mulvany, 1850) P t stepwise unit rainfall t A d tctc t tctc A(t)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

6 Convolution integral Unit hydrograph (Sherman, 1932) t P d t tctc Cumulative hydrograph curve unit hydrograph rainfall The catchment is a linear system

7 Nash model (Nash 1957) PtPt dtdt The catchment is modelled as a cascade of n (e.g. n=4) reservoirs State transition h1Pth1Pt k1xt1k1xt1 x1x1 h2Pth2Pt k2xt2k2xt2 x2x2 h3Pth3Pt k3xt3k3xt3 x3x3 x4x4 h4Pth4Pt k4xt4 =k4xt4 = dtdt Output transformation

8 Modello di Nash, 1957 The catchment is modelled as a cascade of n (e.g. n=4) reservoirs State transition Output transformation Linear system

9 ARX models (‘ 70) d t+1 = d t persistent runoff d t+1 = 2d t - d t-1 AR(2) d t+1 = a 1 d t +….+a n d t-n+1 AR(n) ARX d t+1 = P t-n+1 n = time of concentration d t+1 = b 1 P t + …+b n P t-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 measure comput.

10 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

11 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

12 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 usually called either input/output form or external representation empirical models

13 Mechanistic models Reality Empirical models Mechanistic and empirical models How to find it? SPACE OF THE MODELS

14 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

15 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”. internal representation of the reservoir external representation Examples

16 Empirical models Empirical 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 PARMAX

17 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 … but... not-linear! NOT! A non-linear class of functions would be by far a better choice, e.g. ARTIFICIAL NEURAL NETWORKS

18 Stochastic empirical models Moreover, it is worth while considering the stochastic form … some time using a “coloured noise” (non-white noise) In general process noise

19 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.

20 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.

21 An example Cunning River – Australia ? The soil is too dry and the rainfall completely absorbed

22 measured runoff estimated runoff A tentative model of the Cunning river Let’s try with a PARMAX rainfall NO

23 Data Based Mechanistic (DBM) models Let’s try with a DBM model The value of  depends on the runoff rate, which in turn depends on the soil moisture.

24 Data Based Mechanistic (DBM) models Let’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

25 Disturbances Ultimately, models are to be used to simulate the system behaviour under a given alternative 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

26 Who does provide it? Two options: 1.Using the historical trajectory but 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.

27 The model must be an empirical one and suitably changing the notation … NOT! vicious circle unless...  is a white noise process noise How to model the disturbances

28 White noise A time series for which a model can be formulated is said algorithmically compressible An algorithmically un-compressible time series is a white noise With stochastic disturbances this means a self-correlogramm equal to zero. In conclusion: distubances must be modeled as white noise

29 Reading IPWRM.Theory Ch. 4/Ap.6-7 IPWRM.Practice Sec. 6.5