Towards a Theory of Predictability of Change Alberto Montanari (1) and Guenter Bloeschl (2)

This presentation can be downloaded at http://www.albertomontanari.it

Water Cycle Projections over Decades to Centuries at River Basin to Regional Scales: A Vibrant Research Agenda for

Systems in TransitionChapel Hill 21-22 October 2010


Towards a Theory ofPredictability of Change

Alberto Montanari(1) and Guenter Bloeschl(2)

(1) University of Bologna, [email protected](2) Vienna University of Technology, [email protected]

mailto:[email protected]

mailto:[email protected]


Water Cycle Projections over Decades to Centuries at River Basin to Regional Scales: A Vibrant Research Agenda for Systems in Transition

Chapel Hill 21-22 October 2010

What are the basic elements of a theory?• Why a theory? To establish a consistent, transferable and clear working framework.

• In science, the term "theory" is reserved for explanations of phenomena which meet basic requirements about the kinds of empirical observations made, the methods of classification used, and the consistency of the theory in its application among members of the class to which it pertains. A theory should be the simplest possible tool that can be used to effectively address the given class of phenomena.

• Basic elements of a theory:- Subject.- Domain (scales, domain of extrapolation, etc.).- Definitions.- Axioms or postulates (assumptions).- Basic principles.- Theorems.- Models.- …..

• Important: a theory of a given subject is not necessarily unique




The essential role of uncertainty• Hydrological predictions are inherently uncertain, because we cannot fully reproduce

the chaotic behaviors of weather, the geometry of water paths, initial and boundary conditions, and many others. It is not only uncertainty related to lack of knowledge (epistemic uncertainty). It is natural uncertainty and variability.

• Therefore determinism is not the right way to follow. We must be able to incorporate uncertainty estimation in the simulation process.

• The classic tool to deal with uncertainty is statistics and probability. There are alternative tools (fuzzy logic, possibility theory, etc.).

• A statistical representation of changing systems is needed. Important: statistics is not antithetic to physically based representation. Quite the opposite: knowledge of the process can be incorporated in the stochastic representation to reduce uncertainty and therefore increase predictability.

• New concept: stochastic physically based model of changing systems. (AGU talk by Alberto, Thursday December 16, 1.40 pm). It is NOT much different with respect to what we are used to do. Understanding the physical system remains one of the driving concepts.




Towards a theory of hydrologic prediction under change• Main subject: estimating the future behaviours of hydrological systems under changing

conditions.• Side subjects: classical hydrological theory, statistics,…. and more.• Axioms, definitions and basic principles: here is the core of the theory and the research

challenge. We have to define concepts (what is change? How do we define it? What is stationarity? What is variability?) and driving principles, including statistical principles (central limit theorem, which is valid under change, total probability law etc.).

1. The key source of information is the past. We have to understand past to predict future.

2. What is stationarity? Its invariance in time of the statistics of the system but better to say what is non-stationarity: it is a DETERMINISTIC variation of the statistics. If we cannot write a deterministic relationship then the system is stationary.

3. Do we assume stationarity? Unless we can write a deterministic relationship to explain changes yes. A stationary system is NOT unchanging. In statistics a stationary system is defined through the invariance in time of its statistics, but it is subjected to significant variability and local changes that are very relevant. Past climate is assumed to be stationary but we had ice ages.




What is invariant?Is future climate invariant?, Is the model invariant?, Are Newton laws still valid?, Can we identify additional optimality principles?

The research challenge is to identify invariant principles to drive the analysis of change.

Wetter catchments (PET/P<0.35)Drier catchments (PET/P>0.6)

0 1000 2000 3000 m a.s.l.

100 km

Merz, R. J. Parajka and G. Blöschl (2010) Time stability of catchment model parameters – implication for climate impact analysis. Water Resources Research, under review

Fig. 1: Locations of the catchments and classification into drier catchments (red), wetter catchments (blue) and medium catchments (grey).




Fig. 2: 5 year mean annual values of climatic variables averaged for 273 Austrian catchments (black lines). The spatial of means of the wetter catchments are plotted as blue lines, the spatial of means of the drier catchments are plotted as red lines

1980 1990 2000

200400600800

10001200140016001800

runo

ff (m

m/y

r)

1980 1990 20000.20.30.40.50.60.70.80.9

1

Q/P

1980 1990 2000

600

800

1000

1200

1400

1600

1800pr

ecip

itatio

n (m

m/y

r)

1980 1990 20000123456789

10

air t

emp.

(°C

)

1980 1990 2000400

500

600

700

PE

T (m

m/y

r)

1980 1990 20000.2

0.3

0.4

0.5

0.6

0.7

mea

n ca

tchm

ent a

rea

cove

red

by s

now




Fig. 4: Model parameters (snow correction factor (SCF), Degree-day factor (DDF), maximum soil moisture storage (FC) and non-linearity parameter of runoff generation (B)) of 5 year calibration periods averaged for 273 Austrian catchments (black lines). The spatial of means of the wetter catchments are plotted as blue lines, the spatial of means of the drier catchments are plotted as red lines

1980 1990 20001

1.05

1.1

1.15

1.2

SC

F

1980 1990 20001.5

1.75

2

DD

F

1980 1990 2000100

150

200

250

300

350

FC

1980 1990 2000

2

4

6

8

10

B

-1

-0.5

0

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orre

latio

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p

Pre

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rec

-1

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K2

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/Pre

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eael

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rest

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-0.5

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rest

Tem poral corre la tion Spatia l corre la tion

Fig. 5: Box-Whisker Plots of the Spearman Rank correlation coefficients of model parameters to climatic indicators. Temporal Correlation for the six 5-years calibration periods. (Box-Whisker Plots show the spatial minimum




Fig. 12: Cumulative distribution of the relative errors of observed and simulated low flows (Q95), mean flow (Q50) and high flows (Q5) for different 5 years period for a different time lag of calibration and verification period.

-0.4 -0.2 0 0.2 0.4Q 5

-0.4 -0.2 0 0.2 0.4Q 50

-0.4 -0.2 0 0.2 0.4Q 95

0

0.25

0.5

0.75

1

CD

F

0 yr 5 yrs 10 yrs 15 yrs 20 yrsTim elag 25 yrs

0 0.1 0.2 0.3 0.4 0.5abs(Q 5)

0 0.1 0.2 0.3 0.4 0.5abs(Q 50)

0 0.1 0.2 0.3 0.4 0.5abs(Q 95)

0

0.25

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CD

F




• A first set of definitions

Hydrological model:in a deterministic framework, the hydrological model is usually defined as a analytical transformation expressed by the general relationship:

where Qp is the model prediction, S expresses the model structure, I is the input data vector and e the parameter vector.

In the uncertainty framework, the hydrological model is expressed in stochastic terms, namely (Koutsoyiannis, 2009):

where f indicates the probability distribution, and K is a transfer operator that depends on model S and can be random. Note that passing from deterministic to stochastic form implies the introduction of the transfer operator.

),( IεSQp

),(),( IεIε fKQf p

Towards a theory of hydrologic prediction under change




• A first set of definitions

Hydrological model:if the random variables e and I are independent, the model can be written in the form:

Randomness of the model may occur because N different models are considered. In this case the model can be written in the form:

where wi is the weight assigned to each model, which corresponds to the probability of the model to provide the best predictive distribution. Basically we obtain a weighted average of the response of N different hydrological models depending on uncertain input and parameters.

)()(),( IεIε ffKQf p

i

N

iip wffKQf )()(),(

1

IεIε





• Estimation of prediction uncertainty:- Qo true (unknown) value of the hydrological variable to be predicted- Qp(e,I,i)corresponding value predicted by the model, conditioned by

model i, model parameter vector e and input data vector I- Assumptions:

1) a number N of models is considered to form the model space;2) input data uncertainty and parameter uncertainty are independent.

iNi

p wddffieQf[Qf

ε I

IεIεIε )()()()()],,|()( 0

where wi is the weight assigned to each model, which corresponds to the probability of the model to provide the best predictive distribution. It depends on the considered models and data, parameter and model structural uncertainty.

- Th.: probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008):





Towards a theory of uncertainty assessment in hydrologySetting up a model: Probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008)

• Symbols:- Qo true (unknown) value of the hydrological variable to be predicted- Qp(e,I,i)corresponding value predicted by the model, conditioned by - N Number of considered models- e Prediction error - e Model parameter vector- I Input data vector-wi weight attributed to model i

iNi

p wddffieQfQf

I ε

IεIεIε )()()()()],,|([)( 0 )( 0Qf ),,|( ieQf p Iε)( 0Qf ε

εIε )()],,|([ fieQf p )(εd)( 0Qf I ε

IεIεIε )()()()()],,|([ ddffieQf p




Conclusions and research challenges Prediction of change needs to be framed in the context of a generalised

theory.

Theory should make reference to statistical basis, although other solutions present interesting advantages (fuzzy set theory).

Research challenges:

a) Identify fundamental laws that are valid in a changing environment (optimality principles, scaling properties, invariant features.

b) Devise new techniques for assessing model structural uncertainty in a changing environment.

c) Propose a validation framework for hydrological models in a changing environment.

d) Devise efficient numerical schemes for solving the numerical integration problem.

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Towards a Theory of Predictability of Change Alberto Montanari (1) and Guenter Bloeschl (2)

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