Modelling environmental systems
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Modelling environmental systems
Some words on modelling the hitchhiker’s guide to modelling
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
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
Systems thinking • for seeing wholes, counteract reductionism • relationships rather than things • patterns of change rather than static snapshots • seeing circles of causality • dealing with complexity and delays • acknowledging both hard and soft components
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
Modelling: the hardest
part
Needed Unnecessary
Sorting the essential from the nonessentials!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
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”)One question – one
model!
Never trust a Swiss Army knife
model!Model categories and classification
Breeds of models • Models are • conceptual • physical • mathematical
mental/
Models are physical mathematical
conceptual
system identification
definition of system boundary, components, interactions
encompasses..
a conceptual, verbal a scaled coupling of
The model is... description of reproduction of a functions, rules,
system behaviour real system equations
premises, mathematical
Elements of a model
conclusions, a physical object functions and (state)
are..
syllogisms variables
conclusions are an experiment in a
validation and
Plausibility check is.. tested on real-world controlled
sensitivity analyses
cases environment
a thinking a physical a numerical solution
A simulation is..
experiment experiment of the equation sets
(adapted from Seppelt, 2003)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
Characteristic Mathematical
Process Variables
time model
Biomass, nitrogen
Microbial growth 30 minutes ODE
content
Nitrogen
Nitrification,
compoundes, 1 day to 1 week Systems of ODE
denitrification
micrrobial activity
Density of eggs,
DAE, DDE, Systems
Population dynamics juveniles, larvae, Weeks
of ODE
adults
Biomass, nitrogen
Crop growth content, leaf area Month Systems of ODE
index
Water transport in
Water content 1 hour PDE
unsaturated soil
Concentration in
Solute transport in large up to several PDE coupled with
liquid and solute
aquifers years ODE system
phase
(Seppelt, 2003)Spatial scale • It is the spatial extent • how many dimensions? • what is the grid size?
Model use • Descriptive models • Decision models • Prescriptive models • Forecast models
Conceptual models
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
Types of conceptual
models
• Word models
• Picture models
• Box-models
• Feedback dynamics, Casual Loop Diagrams
• Energy Circuit Diagrams (Odum):7='>71%,- 6%%1+$5?' F4G3#H .%#)*'J%#- A4%)71%,- >71%,-
974-3/%)
!"#$#%&'($)*$+,%-
"#$#%'($)
"#$#%($) "#$#% "#7)$Causal loop diagrams
Feedback dynamicsWhat 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 behaviourReinforcing feedback
• Reinforcing behaviour R
• Something that causes an amplified
condition
• the larger the population the more
births
• the more money in the bank, the more
in interestBalancing feedback
• Balancing behaviour
B
• Something that causes a change which
dampens/opposes a condition,
• Limited amounts of nutrients
• Intensity of competitionReinforcing
An example of system in growth over time
100
75
• A self-reinforcing system is a
system in growth, e.g. bank
quantity
account, economic growth or 50
population growth, exponential
growth 25
0
2001 2002 2003 2004
timeBalancing
An example of system that balances over time
100
• In a balancing system
there is an agent which
retards the growth or is 75
a limiting factor to the
quantity
reinforcing growth, e.g. 50
limited resources in the
soil, limited light or space 25
for growth etc.
0
2001 2002 2003 2004
timeThe structure of
Causality
Variables change:
”in the same direction”
”in the opposite direction”A very simple example
+
Photosynthesis R Growth
+Another simple
example
-
Nutrient Nutrients
uptake B
available
+a bit more complex
Some practice with
CLDAtmospheric system
Natural system
Social system
Economic system
Combined system
The difficult transition from conceptual to mathematical models
Problem formulation • Conceptual model construction • System boundaries • CLD • Actors, Drivers and Conditions • Reference behaviour
Model construction
• From conceptual model to quantitative
model
• Parameterization
• Sensitivity and robustness testing
• Model validationThe modelling process
Scope/
Purpose
Conceptualisation Data collection
Calibration
Validation
UseProblems 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 limitationsConceptual 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 itConceptual 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
experiencesProblems 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 componentsProblems in the model validation phase • Finding data for validation • Robustness of model • Qualitative components • Appropriate time and space boundaries
Adding causes to
model
From: Sverdrup &
Haraldsson, 2002Model performance
From: Sverdrup &
Haraldsson, 2002Model cost and
performance
From: Sverdrup &
Haraldsson, 2002System Levels From: Sverdrup & Haraldsson, 2002
Mathematical models
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 transformationThe equations General model equation xt+!t (z) = M!t (xt (z), ut (z), "(z), z) yt (z) = ft (xt (z)) Initial condition x0 (z) and boundary conditions
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
Randomness
Hydrological
Process control Ecological Social models
processes models
Electrical Nuclear reactors Air pollution Economical
engineering
modelsModel 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
Mechanistic Models • Ordinary Differential Equations • Difference Equations • Partial Differential Equations • Stochastic models
Empirical Models
• Completely data-driven
• No insight on the model causal structure
• Input-output models
!
yt+1 = yt yt , . . . , yt−(p−1) , ut+1 , . . . , ut−(r" −1) , wt+1 , . . . . . . , wt−(r"" −1) , !t+1 , . . . , !
. . . , wt−(r"" −1) , !t+1 , . . . , !t−(q−1)
• Neural NetworksData 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, 1994An input-output model
fails
runoff PARMAX forecast
4
4
3 3.5
Deflusso
2
3
1
2.5
0
01.02.85 11.05.85 19.08.85 27.11.85 07.03.86 15.06.86 24.09.86 02.01.87
Deflusso
2
80
60 1.5
Precipitazione
40
1
20
0 0.5
01.02.85 11.05.85 19.08.85 27.11.85 07.03.86 15.06.86 24.09.86 02.01.87
rainfall 0
01.02.85 11.05.85 19.08.85 27.11.85 07.03.86
Giorno
15.06.86 24.09.86 02.01.87
yt+1 = !yt + "wt + #t+1The DBM approach
Parameters may depend on the state!
!"!(
!"!#
!"!'
!"!&
)*+,
!"!%
!"!$
!
!!"!$
! !"# $ $"# % %"# & &"# '
-*./0112
yt+1 = !yt + "(yt )wt + #t+1Using a DBM
• The structure is
discovered from data
4
3.5
•
3
The rainfall contribution
depends from the runoff!
2.5
Deflusso
2
•
1.5
When the soil is dry,
1
rainfall is absorbed, but
0.5
when saturation is
0
reached, runoff can 01.02.8511.05.8519.08.8527.11.8507.03.8615.06.8624.09.8602.01.87
Giorno
increaseNext 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 reasoningYou can also read
