Towards a Theory of - PowerPoint Presentation

Slide1

Towards a Theory of

Predictability of Change

Alberto

Montanari

(1)

and

Guenter

Bloeschl

(2)

(1)

University of Bologna,

alberto.montanari@unibo.it

(2)

Vienna University of Technology,

bloeschl@hydro.tuwien.ac.at

Slide2

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 uniqueSlide3

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.Slide4

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

.).

The

key source of information

is the past. We have to understand past to predict future.

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.

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.Slide5

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.

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).Slide6

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 Slide7

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

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

Slide8

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. Slide9

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

Q

p

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.

Towards a theory of hydrologic prediction under changeSlide10

A first set

of definitionsHydrological 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.

Towards a theory of hydrologic prediction under changeSlide11

Estimation of prediction uncertainty

: - Qo true (unknown) value of the hydrological variable to be predicted

-

Q

p

(

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.

where

w

i

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

Q

o

(

Zellner

, 1971;

Stedinger

et al., 2008):

Towards a theory of hydrologic prediction under changeSlide12

Towards a theory of uncertainty assessment in hydrology

Setting up a model: Probability distribution of

Q

o

(

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

-

w

i

weight attributed to model

iSlide13

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.