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Uncertainty in Environmental Modeling

Ⓒ K. Fedra 2000

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

Treatment of uncertainty

A Monte Carlo approach to uncertainty analysis (Fedra, 1981, 1983):

- model structure identification by hypothesis testing
- parameter estimation
- error propagation (forecasting under uncertainty)

Ⓒ K. Fedra 2000

Modeling framework

- Hypothesis or universal statement (the model structure) F
- Set of initial conditions, including
- initial conditions: X(t0)
- parameter vector
- time variant forcings U
- Set of singular statements (output Y(t))
- Set of corresponding observations.

Ⓒ K. Fedra 2000

Analysis procedure:

- For a given model structure, a parameter vector is sampled randomly from an a priori defined parameter space;
- Each model run is classified or evaluated by a set of rules;
- The resulting subsets are analysed.

Ⓒ K. Fedra 2000

Analysis procedure:

Evolutionary approach:

- large number of random mutations (MC sample vectors)
- subjected to selection in a constrained environment: ecological niche, envelope of viability (Greppin 1978)
- resulting in a (genetic) pool of surviving instances.

Ⓒ K. Fedra 2000

Treatment of uncertainty

The model is a vector function f with Domain D(f) (set of all possible parameter vectors) and Range R(f) (set of all possible behavior vectors).

If RD is a subset of R, the invers image of RD under f is the subset of D(f):

Ⓒ K. Fedra 2000

Treatment of uncertainty

This subset is denoted PM, the set of all parameter vectors resulting in acceptable model results.

For identification, we define RD by a set of constraint conditions (rules) derived from the set of observations that capture the expected (allowable) system behavior.

Ⓒ K. Fedra 2000

Treatment of uncertainty

Ⓒ K. Fedra 2000

Model identification example

Marine pelagic foodweb example

15 year data set from the North Sea (Helgoland Reede) P-PO4, organic carbon representing plankton dynamics.

Models range from 2 to 5 compartment classical marine foodweb representations (extensive literature).

Ⓒ K. Fedra 2000

Model identification example

Data set shows the typical

annual variability, but also

a high degree of inter-

annual variability that can

mask annual patterns

when averaged.

Ⓒ K. Fedra 2000

Model identification example

Behavior definition examples:

- primary producers are below 4.0mgm-3 during month 1-3;
- between Julian day 120 and 270 biomass increases at least twofold
- there must be at least two peaks with the second at least 25% below the first
- the higher peak must not exceed 25mgm-3

continued …..

Ⓒ K. Fedra 2000

Model identification example

- yearly primary production must be between 300 and 700 gCm-2
- after day 270, biomass must again be below 4.0 mgm-3
- P-PO4 must be above 20mgm-3 between days 1 and 90
- all variables must be cyclically stable with a maximum 25% deviation between initial and final value for a given year

Ⓒ K. Fedra 2000

Model identification example

Model structure

alternatives:

P: phosphate

A: algae

Z: zooplankton

D: detritus

Z1: herbivores

Z2: carnivores

Ⓒ K. Fedra 2000

Model identification example

Model results H1

(simplest structure)

fails second peak

criterion, meets

average values

Ⓒ K. Fedra 2000

Model identification example

Data revisited:

when using different literature data (17.5 gN-2 or half the German value is quoted by Pichot and Runfola, 1975 for Belgian coast) for the primary production constraints, model H5 reproduces all required features.

“A nice adaptation of conditions will make almost any hypothesis agree with the phenomena.” (Black, 1803)

Ⓒ K. Fedra 2000

Treatment of uncertainty

“Our whole problem is to make the mistakes fast enough”(J.A.Wheeler, 1956)

Some MC statistics:

250,000 runs yield 219 acceptable solutions, 12 parameter model.

High number of runs yields valuable sensitivity data (parameter - output correlations and parameter cross-correlations).

Ⓒ K. Fedra 2000

Propagation of uncertainty

Monte Carlo simulation based on the ensemble of parameters identified, or based on a priori distributions around parameters, inputs, and initial conditions:

results in an ensemble of solutions that can be described in terms of its statistics (mean, median, S.D., 95% etc.)

Ⓒ K. Fedra 2000

Propagation of uncertainty

Application example:

chemical emergency, spill of toxic material, simulation of pool evaporation and subsequent atmospheric dispersion, fire, and soil contamination models with population exposure estimates.

Method:

cascading Monte Carlo simulation

Ⓒ K. Fedra 2000

Propagation of uncertainty

Primary model:

- loss term (spill) from a damaged container
- dual-phase release (gaseous, liquid)
- pool evaporation (dynamic)
- soil infiltration,
- monitoring of explosivity and flammability limits (near field)

Ⓒ K. Fedra 2000

Propagation of uncertainty

Scenario definition includes:

- type of probability density function (Gaussian, rectangular)
- standard deviation or range

for each parameter (physical, chemical, meteorological, geometrical)

Ⓒ K. Fedra 2000

Propagation of uncertainty

Each frequency class of the total mass evaporated corresponds to a set of trajectories of evaporation rates, with a corresponding frequency distribution of average evaporation rates.

To control combinatorial explosion, a representative trajectory is sampled.

Ⓒ K. Fedra 2000

Propagation of Uncertainty

Ⓒ K. Fedra 2000

Propagation of uncertainty

2D Fire model:

Stochastic input:

flow rate

meteorology

pool geometry

Stochastic output:

temperature field

population exposure

Ⓒ K. Fedra 2000

Propagation of uncertainty

1D Soil and groundwater contamination:

Input and parameter uncertainty:

viscosity of the spilled substance

permeability of the soil

distance to the water table

Output uncertainty:

arrival time of contaminant

Ⓒ K. Fedra 2000

Propagation of uncertainty

Dynamic 2D atmospheric dispersion model (multi-puff, INPUFF 2.4) using a 3D diagnostic wind field.

Model uses the dynamic source term from the spill model.

Dynamic concentration overlay estimates population exposed above a threshold.

Ⓒ K. Fedra 2000

Propagation of uncertainty

Ⓒ K. Fedra 2000

Propagation of uncertainty

Default: most likely

median value from

the source model

output distribution

Impact:

9 hectare

96 people

Ⓒ K. Fedra 2000

Propagation of uncertainty

Ⓒ K. Fedra 2000

Propagation of uncertainty

Worst case:

maximum value

from the dynamic

source model.

Impact:

94 ha

819 people

Ⓒ K. Fedra 2000

Implications for decision making

Uncertainty analysis offers the possibility to take uncertainty explicitly into account: probability of population exposure, evacuation needs, timing of response measures etc:

- median, 95%, worst case
- explicit treatment of uncertainty in risk analysis: risk is the output variable

Ⓒ K. Fedra 2000

Risk Assessment

Risk contours around

a plant location

(a source of risk):

10-6 events/year

unacceptable

individual risk

10-8 events/year

negligible risk

Ⓒ K. Fedra 2000

Alternative approaches

Event tree: traces possible events from

loss of cooling to:

- safe shutdown
- discharge from

safety valve

- explosion

Ⓒ K. Fedra 2000

Alternative approaches

Symbolic logic:

rule-based expert systems

IF condition operator condition

AND ….

OR

THEN consequence

conditions can be formulated in terms of

symbols, ranges, distributions, fuzzy sets

Ⓒ K. Fedra 2000

Alternative approaches

IF wind_speed == very_low

AND radiation == strong

AND time == daytime

THEN stability = very_unstable

wind_speed:

very_low [0.1, 1.0, 2.0]

low [2.0, 2.5, 3.0]

medium [3.0, 4.0, 5.0]..

Ⓒ K. Fedra 2000

Alternative approaches

If we replace the first order logic by conditional probabilities, we arrive at probabilistic inference:

Ⓒ K. Fedra 2000

Alternative approaches

and Bayes ‘ rule (Bayes 1763)

which leads to Bayes networks, polytrees, belief networks and qualitative probabilistic networks (Drudzel 1996, Nilsson 1998) in the field of AI.

Ⓒ K. Fedra 2000

Implications for decision making

Error propagation can provide an estimate of reliability over time (forecasts):

- plotting the coefficient of variation against input change (decision variable) and time:
- uncertainty increases with the forecasting period (time horizon) and the input change (degree of extrapolation).

Ⓒ K. Fedra 2000

Implications for decision making

Ⓒ K. Fedra 2000

Implications for decision making

- Depend on the intended use of the information; other criteria include timeliness and cost
- Estimation of the robustness or reliability of decisions:
- robustness describes the distance of the efficient point from its nearest neighbor (alternative).

Ⓒ K. Fedra 2000

Multicriteria decision example

axes normalized as % of possible achievement

(the distance from nadir to utopia):

0%

utopia

reference point

(required safety,

acceptable cost)

cost

dominated

efficient

point

100%

safety

100%

nadir

Ⓒ K. Fedra 2000

Multicriteria decision example

axes normalized as % of possible achievement

(the distance from nadir to utopia):

0%

utopia

reference point

(required safety,

acceptable cost)

cost

dominated

100%

safety

100%

nadir

Ⓒ K. Fedra 2000

Uncertainty

- is an inherent element of environmental analysis, and modeling in particular
- the challenge is to incorporate and exploit it as additional useful information

Monte Carlo based uncertainty analysis is an appropriate tool for complex environmental systems represented by dynamic, non-linear, and spatiallydistributed models.

Ⓒ K. Fedra 2000

Uncertainty in environmental modeling

References available on request:

http://www.ess.co.at

Ⓒ K. Fedra 2000

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