Modelling environmental systems

Transcription

1 Modelling environmental systems

2 Some words on modelling the hitchhiker s guide to modelling

3 Problem perception Definition of the scope of the model Clearly define your objectives Allow for incremental model definition (don t start with a model which is too complex) Work in strict co-operation with the Decision Makers

4 Limits to modelling We tend to think linear System structure influences behaviour Structure in human system is subtle Leverage often comes from new ways of thinking Reductionist thinking is often hampering

5 Systems thinking for seeing wholes, counteract reductionism relationships rather than things seeing circles of causality dealing with complexity and delays patterns of change rather than static snapshots acknowledging both hard and soft components

6 What is a model? A model is any understanding which is used to reach a conclusion or a solution Only mental models exist; all models rest in the human mind There are no computer models, these are mere mechanical and mathematical pictures of mental models If a model is wrong, then the underlying understanding is to blame

7 Modelling: the hardest part Needed Unnecessary Sorting the essential from the nonessentials!

8 The quality of a model is determined by how useful it is for it s purpose how well users understand the model and have trust in it NOT the number of details

9 Simplicity and participation The major result is understanding (not the models themselves) Simple models ensure understanding Modelling is not a one man work! The process is everything! ( The road is the goal )

10 One question one model! Never trust a Swiss Army knife model!

11 Model categories and classification

12 Breeds of models Models are conceptual physical mathematical

13 Models are mental/ conceptual physical mathematical system identification encompasses.. definition of system boundary, components, interactions The model is... a conceptual, verbal description of system behaviour a scaled reproduction of a real system coupling of functions, rules, equations Elements of a model are.. premises, conclusions, syllogisms a physical object mathematical functions and (state) variables Plausibility check is.. conclusions are tested on real-world cases an experiment in a controlled environment validation and sensitivity analyses A simulation is.. a thinking experiment a physical experiment a numerical solution of the equation sets (adapted from Seppelt, 2003)

14 Temporal scale Defined by the time constant τ of the system In relation with the integration step t τ=1/ t Choice of the temporal scale and stiff systems

15 Process Variables Characteristic time Mathematical model Microbial growth Nitrification, denitrification Population dynamics Crop growth Water transport in unsaturated soil Solute transport in aquifers Biomass, nitrogen content Nitrogen compoundes, micrrobial activity Density of eggs, juveniles, larvae, adults Biomass, nitrogen content, leaf area index 30 minutes ODE 1 day to 1 week Systems of ODE Weeks Month DAE, DDE, Systems of ODE Systems of ODE Water content 1 hour PDE Concentration in liquid and solute phase large up to several years PDE coupled with ODE system (Seppelt, 2003)

16 Spatial scale It is the spatial extent how many dimensions? what is the grid size?

17 Model use Descriptive models Decision models Forecast models Prescriptive models

18 Conceptual models

19 A conceptual model is presented graphically as a compartment system compartments are defined w.r.t morphology, and physical, chemical and biological states connections denote exchange of matter, energy, information compartments may contain sub-models

20 Types of conceptual models Word models Picture models Box-models Feedback dynamics, Casual Loop Diagrams Energy Circuit Diagrams (Odum)

21 !"#$#%&'($)*$+,%-.$)$/%#%) #'6345#*74-.)75%--%- 6,78-974#)7,'($)*$+,%- :7341$);'9741*#*74-67)5*4<'6345#*74- :7='>71%,- A4%)<;' 6,78' >71%,- "#$#%'($) B$#% 974-#$4#C 6345#*74 "*4?D"73)5% "#$#%($) 974-#$4# 6345#*74 6,78 AE%4# AE%4# F4G3#H I3#G3# >71%,- B$#% "#$#%.%#)*'J%#-.,$5% 6,78 K)$4-*#*74 F4#%)$5#*74 A=5L$4<%'7M' F4M7)/$#*74 "3+H";-#%/- "73)5% "*4? K)$4-$5#* /%) "#7)$<% 0/G,*M;%) 9;5,*4<'B%5%G#7) Paradigms for Conceptual Modelling (Seppelt, 2003)

22 Causal loop diagrams Feedback dynamics

23 What is a Causal Loop Diagram? A simplified understanding of a complex problem A common language to convey the understanding A way of explaining cause and effect relationships Explanation of underlying feedback systems Helps us understanding the overall system behaviour

24 Reinforcing feedback Reinforcing behaviour Something that causes an amplified condition the larger the population the more births the more money in the bank, the more in interest R

25 Balancing feedback Balancing behaviour Something that causes a change which dampens/opposes a condition, B Limited amounts of nutrients Intensity of competition

26 Reinforcing An example of system in growth over time 100 A self-reinforcing system is a system in growth, e.g. bank account, economic growth or population growth, exponential growth quantity time

27 Balancing An example of system that balances over time In a balancing system there is an agent which retards the growth or is a limiting factor to the reinforcing growth, e.g. limited resources in the soil, limited light or space for growth etc. quantity time

28 The structure of Causality Variables change: in the same direction in the opposite direction

29 A very simple example + Photosynthesis + R Growth

30 Another simple example - Nutrient uptake + B Nutrients available

31 a bit more complex

32 Some practice with CLD

33 Atmospheric system

34 Natural system

35 Social system

36 Economic system

37 Combined system

38 The difficult transition from conceptual to mathematical models

39 Problem formulation Conceptual model construction System boundaries CLD Actors, Drivers and Conditions Reference behaviour

40 Model construction From conceptual model to quantitative model Parameterization Sensitivity and robustness testing Model validation

41 The modelling process Scope/ Purpose Conceptualisation Data collection Calibration Validation Use

42 Problems in conceptual modelling What is relevant? Sorting out essentials At what level? Micro- or Macro-level Static and dynamic factors? System boundaries? Time horizon Qualitative and/or quantitative factors? Problems to kill your darlings Perception limitations

43 Conceptual model building factors Deletion Select and filter according to preferences, mode, mood, interest, preoccupation and congruency Construction See something that is not there, filling in gaps Distortion Amplifying some parts and diminishing others, reading different meanings into it

44 Conceptual model building factors Generalisation One experience comes to represent a whole class of experiences One-sided experiences We tend to only remember one side of experiences

45 Problems in the CLD to model phase Including how many components? How to distinguish accumulations from processes? Units? Scales? Introduction of mass and energy balance principles? Non-linear relationships Qualitative components

46 Problems in the model validation phase Finding data for validation Robustness of model Qualitative components Appropriate time and space boundaries

47 Adding causes to model From: Sverdrup & Haraldsson, 2002

48 Model performance From: Sverdrup & Haraldsson, 2002

49 Model cost and performance From: Sverdrup & Haraldsson, 2002

50 System Levels From: Sverdrup & Haraldsson, 2002

51 Mathematical models

52 Systems theory approach A model, whatever mathematical formulation we choose, can be described by: state, input and output variables inputs can be controls and disturbances the dynamics of these variables is described by the state transition function the output transformation

53 The equations General model equation x t+!t (z) = M!t (x t (z),u t (z),"(z),z) y t (z) = f t (x t (z)) Initial condition x 0 (z) and boundary conditions

54 Dynamic vs static A dynamic system needs to store information in the state to evolve If the state at time t-1 is sufficient to compute the state at time t, then the system is Markovian If a system can be described only by its output transformation is static

55 Randomness Process control Hydrological processes Ecological models Social models Electrical engineering Nuclear reactors Air pollution Economical models

56 Model paradigms Scarce theoretical modelling knowledge, many data: Bayesian Belief Networks Good theoretical knowledge: mechanistic models Very little knowledge: empirical models Mixed knowledge: Data Based Mechanistic models

57 Mechanistic Models Ordinary Differential Equations Difference Equations Partial Differential Equations Stochastic models

58 Empirical Models Completely data-driven Input-output models No insight on the model causal structure y t+1 = y t ( yt,...,y t (p 1),u t+1,...,u t (r 1),w t+1,... Neural Networks...,w t (r 1),! t+1,...,! t (q 1)

59 Data Based Mechanistic models Mechanistic models are too complex and require too many details Empirical model use a-priori classes A new approach to model identification Input/Output relationships are extracted from data Proposed by Young and Beven, 1994

60 An input-output model 4 runoff fails 4 PARMAX forecast Deflusso Deflusso 2 Precipitazione rainfall Giorno y t+1 =!y t + "w t + # t+1

61 The DBM approach Parameters may depend on the state!!"!(!"!#!"!'!"!& )*+,!"!%!"!$!!!"!$!!"# $ $"# % %"# & &"# ' -*./0112 y t+1 =!y t + "(y t )w t + # t+1

62 Using a DBM The structure is discovered from data The rainfall contribution depends from the runoff! When the soil is dry, rainfall is absorbed, but when saturation is reached, runoff can increase Deflusso Giorno

63 Next steps Using models to perform scenario analysis and optimisation Learn models, policies, plans from data machine learning (bayesian networks, artificial neural networks) Learn models, policies, plans from human experience expert systems and case based reasoning