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Uncertainty in Environmental Modeling. Kurt Fedra. Uncertainty . Websters: the quality or state of being uncertain Handbook of Mathematics and Computational Science (Harris & Stocker, 1998) : ultrafuzzy unbiased underdeterminate uniqueness unknowns. Uncertainty .

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Uncertainty in environmental modeling l.jpg
Uncertainty in Environmental Modeling

Kurt Fedra

© K. Fedra 2000


Uncertainty l.jpg
Uncertainty ...

Websters:

the quality or state of being uncertain

Handbook of Mathematics and Computational Science (Harris & Stocker, 1998) :

  • ultrafuzzy

  • unbiased

  • underdeterminate

  • uniqueness

  • unknowns

© K. Fedra 2000


Uncertainty3 l.jpg
Uncertainty ...

Elements of Mathematical Biology (A.J.Lotka, 1956): ---

A History of Western Philosophy (B.RusselI, 1945) ---

Objective Knowledge (K.R.Popper, 1972):

Of Clouds and Clocks :

irregular, disorderly, unpredictable

© K. Fedra 2000


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Uncertainty ...

The Logic of Scientific Discovery (1959):

Uncertainty: see Hypothesis

The problem of induction:

  • from singular statements (observations)

  • to universal statements (hypothesis, theories, models)

    … difficulties of inductive logic … are insurmountable.

© K. Fedra 2000


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Uncertainty ...

Reichenbach (Erkenntnis, 1930)

… for it is not given to science to reach either truth or falsity (quoted in Popper, op.cit)

Wittgenstein (Tractatus, 1918)

5.634 Alles was wir überhaupt beschreiben können, könnte auch anders sein.

Xenophanes (6th cent BC)

… But as for certain truth, no man has known it. For all is but a woven web of guesses.

© K. Fedra 2000


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Uncertainty ...

Bernoulli, Daniel:

Mathematical exercises (1724) includes discussion of uncertainty in the context of Faro; probability and political economy: moral value of income, probabilistic income: applications to insurance.

Euler, Paul:

Calculus of variations (1730), advisor on state lottery and insurance (Berlin, 1740)

Poisson, Simeon:

Recherches sur la probabilite des jugements (1837)

Poisson distribution: probability of a random event in a time or space interval, when the probability is low but the number of cases (trials) is large.

© K. Fedra 2000


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Uncertainty ...

Theory of Science (G.Gale, 1979)

Correspondence theory of truth:

A statement is true if and only if it corresponds to what it refers to

Reference to objects, real or ideal(Plato)

introduces perception (subjective), or

measurement: Uncertainty Principle (Heisenberg, 1925)

© K. Fedra 2000


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Uncertainty ...

uncertainty relation:

where x and p refer to position and momentum and, h is Planck’s quantum of action, a constant (Heisenberg, 1925).

© K. Fedra 2000


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Gödel’s first incompleteness theorem

For any formal system S which is

consistent (free from contradictions) and

rich enough (to contain number theory)

there are statements which

can be formulated in S yet

cannot be proved or refuted in S but

are true and

can be proved to be true by richer means.

© K. Fedra 2000


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Formal systems

are triplets <L,A,R> consisting of

a formal language L

a set A of axioms (formulated in L)

a set R of rules of derivation (formation, deducation, possible moves) such that

there is at least one axiom or one rule (i.e., AR 0)

© K. Fedra 2000


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Uncertainty ...

In summary:

large amount of literature, no satisfactory operational definition, but obvious intuitive understanding of the concept.

Environmental models, like any other formal method describing reality and relying on observations, contain uncertainty.

© K. Fedra 2000


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Uncertainty ...

One can write formally:

for a dynamic system with state vector X(t) and external forcings U(t)

© K. Fedra 2000


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Uncertainty ...

and expand to:

where is an error term, usually assumed to be white or Gaussian noise,

© K. Fedra 2000


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Uncertainty ...

or alternatively expand to:

where E is a vector representing initial state uncertainty, and G is a constant diagonal diffusion matrix corresponding to the vector Wiener process W(t) with independent components (from Filar and Haurie, 1998).

© K. Fedra 2000


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Uncertainty ...

the diffusion matrix

serves to excite the system state analogous to the random disturbance of particles in a Lagrangian model, i.e., a random component is added to a deterministically computed state.

© K. Fedra 2000


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Uncertainty ...

alternatively, in the context of calibration:

where x represents the dynamic system, and y its discrete observations at tk, with input disturbances and observation errors , respectively (Beck and van Straten, 1983)

© K. Fedra 2000


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Uncertainty ...

alternatively, in the context of decision making:

where denote controllable and uncontrollable forcings,and denotes all stochastic disturbances again (Young,1983)

© K. Fedra 2000


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Uncertainty ...

Questions:

  • sources and effects of uncertainty

  • how to estimate levels of uncertainty

  • how to reduce uncertainty

  • how to incorporate uncertainty into decision making processes.

© K. Fedra 2000


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Sources of Uncertainty ...

Observations of nature can not be made without error - in most cases.

  • Microscale(elementary uncertainty)Heisenberg (1925), Monod (1970)

  • Macroscale (Complex dynamic systems)

    Eigen and Winkler (1975), Gleick (1987)

© K. Fedra 2000


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Uncertainty ...

Macroscale: based on samples

  • sampling errors:

    • conceptual

    • temporal

    • spatial

      at different scales and levels of aggregation.

© K. Fedra 2000


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Uncertainty ...

If observations (empirical data) contain errors, hypothesis testing in a strictly Popperian sense (Popper, 1959, 1979) is no longer a binary process but must be a statistical one:

  • leads to model uncertainty

© K. Fedra 2000


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Uncertainty ...

If a hypothesis (models are compound hypotheses) can not unambiguously be falsified, we leave (in Poppers strict view) the realm of science.

Following pragmatic instrumentalism (Feyerabend 1975), the question becomes:

  • how useful is the hypothesis (model)

© K. Fedra 2000


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Types of Uncertainty in Modeling...

  • Uncertainty about the relationship among the variables (model structure)

  • Uncertainty about the parameters (coefficients) in the model (calibration)

  • Uncertainty associated with the prediction of future state/behaviour

© K. Fedra 2000


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Sources of Uncertainty ...

Uncertainty in Environmental Models: Dynamic Systems Perspective

(Filar and Haurie, 1998)

list sources of uncertainty in the context of computer implemented models of the environment (IMAGE climate impact model):

© K. Fedra 2000


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Sources of Uncertainty ...

  • errors in observations (which affect parameter estimation)

  • errors in parameter estimation

  • errors in the solution algorithms

  • errors in the computer implementation

  • errors in the modeling (model structure).

    (Filar and Haurie, 1998)

© K. Fedra 2000


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Uncertainty ...

Error (Webster’s):

  • an act or condition of ignorant or imprudent deviation from a code of behavior

  • an act involving unintentional deviation from truth or accuracy.

    Implies that a correct, error-free alternative exists !

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Data Uncertainty ...

  • elementary uncertainty

  • sampling errors

  • scale and

  • conceptual mismatch

© K. Fedra 2000


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Data Uncertainty ...

Elementary uncertainty:

macroscopic representation as variability:

  • genetic variability in a population,

  • behavior of turbulent systems

    Non-linear Systems (Lorenz, 1963)

    Fractal Geometry (Mandelbrot, 1977)

    Chaos Theory (Gleick 1987)

© K. Fedra 2000


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Data Uncertainty ...

Sampling errors:

systems with an inherent variability or diversity (parametric, spatial, temporal) must be sampled (or homogenised), which leads to sampling errors:

the true population statistics can only be estimated with some uncertainty.

© K. Fedra 2000


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Data Uncertainty ...

Environmental variables do not commonly follow normal distributions:

  • Patchiness

  • Diversity:

    Shannon index (derived from an information measure, Shannon and Weaver 1963)

© K. Fedra 2000


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Data Uncertainty ...

Scale (spatial and temporal):

because of spatial and temporal non-normal variability, scale affects sampling estimates.

Synoptic sampling of multiple parameters:

  • temperature (continuously)

  • temperature gradients (twice a day)

© K. Fedra 2000


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Data Uncertainty ...

Comparison of sampling sizes to:

  • real-world objects:

    • lake: sample size 0.001 m3

      lake volume 1,000,000 m3

  • model concepts:

    • air quality sample: 1 m3

      model cell: 10,000,000 m3

© K. Fedra 2000


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Data Uncertainty ...

Conceptual mismatch, calibration:

  • laboratory experiments (limited variability of forcings, stress)

  • measurement of proxies (remote sensing, ground truth)

  • valuation methods in environmental economics (contingent valuation, travel cost method)

© K. Fedra 2000


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Data Uncertainty ...

An experimental program allowing US national parks … to raise their fees has significantly boosted revenues without affecting the number of visitors …

Four agencies reported ... recreational fee revenues nearly doubled from 93 M$ in 1996 to 179 M$ in 1998. At the same time,….number of visitors to sites with higher fees increased by 5%.

(IHT, December 5, 1998)

© K. Fedra 2000


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Data Uncertainty ...

In summary:

(environmental) observation data can contain large errors;

but they more reliably provide

  • ranges, distributions

  • semiquantitative relationships

  • inequalities, constraints

  • patterns and Gestalt

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Model Structure Uncertainty ...

Models are simplifications

Simplification of complex systems:

  • bounding (excluding interactions)

  • aggregation (spatial, temporal, functional)

  • lower-order representation

  • linearisation, discretisation

© K. Fedra 2000


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Model Structure Uncertainty ...

Model Structure identification:

  • inductive or statistical methods, derivation from observation data with a minimum set of assumptions:

    • regression analysis, curve fitting,

    • Kalman filter,

    • neural nets, automatic learning.

  • limited explanatory value

  • problems of extrapolation

  • the problems of inductive logic ...

© K. Fedra 2000


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Model Structure Uncertainty ...

Model Structure identification:

  • hypothetico-deductive for mechanistic, causal models:

    • structure is assumed a priori

    • model must be subjected to bona fide test to destruction (falsification)

      In both cases, identification and calibration are inseparably linked.

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Parameter Uncertainty ...

If the system is well know (classical mechanics), initial and input conditions can be manipulated, and output can be observed with no error, calibration is trivial. Else, calibration is either

  • ignored

  • based on trial and error, subjective evaluation

  • based on some optimisation priciple (minimisation of an objective function)

© K. Fedra 2000


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Parameter Uncertainty ...

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Algorithmic Uncertainty ...

Numerical solution of (partial) differential equation systems is based on a discretisation method and possibly iterative solution.

This can introduce

an explicit error

that allows for efficient

convergence of the

iterative solution.

Relaxation accelerates convergence.

© K. Fedra 2000


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Algorithmic uncertainty

Example: Laplace equation, temperature distribution of a homogeneous plate:

on a grid (xi,yj) this becomes

© K. Fedra 2000


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Algorithermic uncertainty

difference approximation:

on the grid:

© K. Fedra 2000


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Algorithmic uncertainty

Difference equation on the grid

leads to N2 linear equations for the N2 unknowns Ti,j that can be solved iteratively with a defined error level.

© K. Fedra 2000


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Sources of Uncertainty ...

  • Data and observation uncertainty

  • Model structure uncertainty

  • Parameter uncertainty

  • Algorithmic uncertainty/error

  • Implementation errors

© K. Fedra 2000


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Implementation errors

Wind field calculation for an emergency model, introducing an order of magnitude error in the interpretation of the DEM input data (m for dm).

The results where favorably commented upon and explained by a group of experts including the model authors.

© K. Fedra 2000



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