Transcript Document

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
Alberto Montanari(1) and Guenter Bloeschl(2)
(1) University of Bologna, [email protected]
(2) Vienna University of Technology, [email protected]
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 Transition
Chapel Hill 21-22 October 2010
• 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
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 Transition
Chapel Hill 21-22 October 2010
• 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.
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 Transition
Chapel Hill 21-22 October 2010
• 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.
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 Transition
Chapel Hill 21-22 October 2010
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
Wetter catchments (PET/P<0.35)
Drier catchments (PET/P>0.6)
0
1000
2000
3000
Fig. 1: Locations of the
catchments
and classification into drier catchments (red), wetter catchments (blue) and
100 km
medium catchments (grey).
m a.s.l.
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 Transition
1400
1200
1000
800
600
1990
2000
1800
1600
1400
1200
1000
800
600
400
200
1980
1990
2000
700
600
500
400
1980
Q/P
runoff (mm/yr)
1980
10
9
8
7
6
5
4
3
2
1
0
1990
2000
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
mean catchment area
covered by snow
1600
PET (mm/yr)
1800
air temp. (°C)
precipitation (mm/yr)
Chapel Hill 21-22 October 2010
1980
1990
2000
1980
1990
2000
1980
1990
2000
0.7
0.6
0.5
0.4
0.3
0.2
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
This presentation can be downloaded at http://www.albertomontanari.it
DDF
SCF
1.15
Water Cycle Projections
1.75 over Decades to Centuries at River Basin to Regional Scales:
A Vibrant Research Agenda for Systems in Transition
1.1
1.05
1.5
1980
1990
2000
Chapel Hill 21-22 October 2010
1980
1990
2000
350
Temporal co
10
Fig. 4: Model parameters (snow correction factor
1 (SCF), Degree-day factor
8 (DDF), maximum soil moisture storage (FC) and non-linearity parameter of
0.5
averaged for 273 Austrian
6 runoff generation (B)) of 5 year calibration periods
Correlation SCF
250
B
catchments (black lines). The spatial of means of the wetter catchments are
200
0
4 plotted as blue lines, the spatial of means of the
drier catchments are plotted
100
1990
2000
1980
1990
Correlation B
Correlation FC
0.5
0
-0.5
-0.5
0.5
0
-0.5
-1
Q
Q/Prec
1
1
1
This presentation can be downloaded at http://www.albertomontanari.it
PET
Temp
Prec
-1
0
0
1
1
Fig. 5: Box-Whisker Plots of the Spearman Rank correlation coefficients of
model parameters to climatic indicators. Temporal Correlation for the six 5years calibration periods. (Box-Whisker Plots show the spatial minimum
0.5
-1
-1
2000
Prec
1980
-0.5
1
Q/Prec
as red lines
Q
2
PET
150
Temp
FC
300
Correlation DDF
1
CDF
0.75 Cycle Projections over Decades to Centuries at River Basin to Regional Scales:
Water
A Vibrant Research Agenda for Systems in Transition
0.5
Chapel Hill 21-22 October 2010
0.25
0
-0.4
-0.2
0
Q95
0.2
0.4
Timelag
-0.4
0 yr
-0.2
5 yrs
0
Q50
10 yrs
0.2
-0.4
0.4
15 yrs
-0.2
0
0.2
Q5
0.4
25 yrs
20 yrs
1
CDF
0.75
0.5
0.25
0
0
0.1
0.2
0.3
abs(Q95)
0.4
0.5 0
0.1
0.2
0.3
abs(Q50)
0.4
0.5 0
0.1
0.2
0.3
abs(Q5)
0.4
0.5
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.
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 Transition
Chapel Hill 21-22 October 2010
• 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:
Q p  S ( ε, I )
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):
f Qp   K (ε, I ) f (ε, I )
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.
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 Transition
Chapel Hill 21-22 October 2010
• A first set of definitions
 Hydrological model:
if the random variables e and I are independent, the model can be written in the form:
f Qp   K (ε, I ) f (ε) f ( I )
Randomness of the model may occur because N different models are considered. In
this case the model can be written in the form:
f Q p    K i (ε, I ) f (ε ) f ( I ) wi
N
i 1
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.
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 Transition
Chapel Hill 21-22 October 2010
• 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.
- Th.: probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008):


f (Q0 )     [ f (Q p  e | ε, I , i )] f (ε ) f ( I ) d (ε ) d ( I ) wi
iN  ε I

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.
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 Transition
Chapel Hill 21-22 October 2010
Setting up a model: Probability distribution of Qo (Zellner, 1971; Stedinger et al., 2008)
• Symbols:
- Qo
- Qp(e,I,i)
-N
-e
-e
-I
-wi
true (unknown) value of the hydrological variable to be predicted
corresponding value predicted by the model, conditioned by
Number of considered models
Prediction error
Model parameter vector
Input data vector
weight attributed to model i


f (Q0 )     [ f (Qpp  e | ε, I , i))]
)] f (ε ) f ( I ) d (ε
(ε) d ( I ) wi
iN  I ε

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 Transition
Chapel Hill 21-22 October 2010
 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.
This presentation can be downloaded at http://www.albertomontanari.it