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Uncertainty in Environmental Modeling

Uncertainty in Environmental Modeling. 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. Treatment of uncertainty.

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Uncertainty in Environmental Modeling

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  1. Uncertainty in Environmental Modeling Ⓒ K. Fedra 2000

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. Treatment of uncertainty Ⓒ K. Fedra 2000

  10. Treatment of uncertainty selection/classification: RS’: accepted RS’’: rejected Ⓒ K. Fedra 2000

  11. Demonstration example Model: y(t)=at Parameter range: 0.5<a<2.5 Behavior definition: 2.5<y(2)<5.0 7.0<y(8)<9.0 Ⓒ K. Fedra 2000

  12. Demonstration example fails for: y(t)=ta y(t)=eat Model 2: y(t)=at+b Constraint: 0.0<b<2.0 Ⓒ K. Fedra 2000

  13. 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

  14. 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

  15. 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

  16. 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

  17. Model identification example Constraint conditions formalised: Ⓒ K. Fedra 2000

  18. Model identification example Model structure alternatives: P: phosphate A: algae Z: zooplankton D: detritus Z1: herbivores Z2: carnivores Ⓒ K. Fedra 2000

  19. Model identification example Model results H1 (simplest structure) fails second peak criterion, meets average values Ⓒ K. Fedra 2000

  20. Model identification example H5 with multiple peaks, fails on cyclic stability Ⓒ K. Fedra 2000

  21. 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

  22. 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

  23. 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

  24. Ⓒ K. Fedra 2000

  25. Propagation of uncertainty in terms of min-max envelopes Ⓒ K. Fedra 2000

  26. 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

  27. 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

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  29. 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

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  33. 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

  34. Propagation of Uncertainty Ⓒ K. Fedra 2000

  35. Propagation of uncertainty 2D Fire model: Stochastic input: flow rate meteorology pool geometry Stochastic output: temperature field population exposure Ⓒ K. Fedra 2000

  36. Ⓒ K. Fedra 2000

  37. 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

  38. Ⓒ K. Fedra 2000

  39. 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

  40. Propagation of uncertainty Ⓒ K. Fedra 2000

  41. Propagation of uncertainty Default: most likely median value from the source model output distribution Impact: 9 hectare 96 people Ⓒ K. Fedra 2000

  42. Propagation of uncertainty Ⓒ K. Fedra 2000

  43. Propagation of uncertainty Worst case: maximum value from the dynamic source model. Impact: 94 ha 819 people Ⓒ K. Fedra 2000

  44. 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

  45. Risk Assessment Spatial risk analysis: location of plants, zoning Ⓒ K. Fedra 2000

  46. 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

  47. Alternative approaches Event tree: traces possible events from loss of cooling to: • safe shutdown • discharge from safety valve • explosion Ⓒ K. Fedra 2000

  48. 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

  49. 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

  50. Alternative approaches If we replace the first order logic by conditional probabilities, we arrive at probabilistic inference: Ⓒ K. Fedra 2000

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