© k.fedra 2000 1 systems analysis and the systemic approach: analytical methods: environmental...

© K.Fedra 2000

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Systems Analysis and Systems Analysis and the Systemic Approach:the Systemic Approach:Systems Analysis and Systems Analysis and

the Systemic Approach:the Systemic Approach:

Analytical Methods: Environmental Modeling

Analytical Methods: Environmental Modeling

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Systems AnalysisSystems AnalysisSystems AnalysisSystems Analysis

… … many of these socio-technical many of these socio-technical problems can be addressed by … the problems can be addressed by … the logical, quantitative, and structurallogical, quantitative, and structural tools for modern science and tools for modern science and technology. The craft that does this technology. The craft that does this well we call well we call applied systems analysisapplied systems analysis..

Quade and Miser, Quade and Miser, Handbook of Systems AnalysisHandbook of Systems Analysis

… … many of these socio-technical many of these socio-technical problems can be addressed by … the problems can be addressed by … the logical, quantitative, and structurallogical, quantitative, and structural tools for modern science and tools for modern science and technology. The craft that does this technology. The craft that does this well we call well we call applied systems analysisapplied systems analysis..

Quade and Miser, Quade and Miser, Handbook of Systems AnalysisHandbook of Systems Analysis

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Systems AnalysisSystems AnalysisSystems AnalysisSystems Analysis

……it brings together:it brings together:– Methods of modern science and Methods of modern science and

technologytechnology

– Social goals and valuesSocial goals and values

– Elements of judgment and skillsElements of judgment and skills

– Consideration of the larger contextConsideration of the larger context

– UncertaintyUncertainty

……it brings together:it brings together:– Methods of modern science and Methods of modern science and

technologytechnology

– Social goals and valuesSocial goals and values

– Elements of judgment and skillsElements of judgment and skills

– Consideration of the larger contextConsideration of the larger context

– UncertaintyUncertainty

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Operations ResearchOperations ResearchOperations ResearchOperations Research

Applied Systems Analysis derived from Applied Systems Analysis derived from System Theory and Operations Research:System Theory and Operations Research:

Operational Research Operational Research Robert Watson-Watts, 1937:Robert Watson-Watts, 1937:

……investigation by scientific method on actual investigation by scientific method on actual operations – current, recent, or impending operations – current, recent, or impending – and explicitly directed to the better, more – and explicitly directed to the better, more effective and more economical conduct of effective and more economical conduct of similar operations in the future.similar operations in the future.

Applied Systems Analysis derived from Applied Systems Analysis derived from System Theory and Operations Research:System Theory and Operations Research:

Operational Research Operational Research Robert Watson-Watts, 1937:Robert Watson-Watts, 1937:

……investigation by scientific method on actual investigation by scientific method on actual operations – current, recent, or impending operations – current, recent, or impending – and explicitly directed to the better, more – and explicitly directed to the better, more effective and more economical conduct of effective and more economical conduct of similar operations in the future.similar operations in the future.

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Environmental modelingEnvironmental modelingEnvironmental modelingEnvironmental modeling

Central tool of Systems Analysis:Central tool of Systems Analysis:

ModelingModelingModels provide:Models provide:

• Explanatory powerExplanatory power

• Predictive capabilitiesPredictive capabilities

Central tool of Systems Analysis:Central tool of Systems Analysis:

ModelingModelingModels provide:Models provide:

• Explanatory powerExplanatory power

• Predictive capabilitiesPredictive capabilities

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Environmental Modeling:Environmental Modeling:Environmental Modeling:Environmental Modeling:

a mathematical representation a mathematical representation of environmental processes, of environmental processes, and relationships.and relationships.

Modeling Modeling is an extension of is an extension of scientific analysis by other scientific analysis by other means. (means. (Von Clausewitz ….)Von Clausewitz ….)

a mathematical representation a mathematical representation of environmental processes, of environmental processes, and relationships.and relationships.

Modeling Modeling is an extension of is an extension of scientific analysis by other scientific analysis by other means. (means. (Von Clausewitz ….)Von Clausewitz ….)

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what do we want to model ?what do we want to model ?

CHANGE:CHANGE:

• In parameter spaceIn parameter space

• In timeIn time

• In physical spaceIn physical space

what do we want to model ?what do we want to model ?

CHANGE:CHANGE:

• In parameter spaceIn parameter space

• In timeIn time

• In physical spaceIn physical space

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Environmental modelingEnvironmental modelingEnvironmental modelingEnvironmental modelingWhat do we want to model ?What do we want to model ?PROCESSES:PROCESSES:• Growth, decay Growth, decay • Transformation Transformation • Dispersion (transport, diffusion)Dispersion (transport, diffusion)• feedback, constraints, delaysfeedback, constraints, delaysMechanismsMechanisms: by design, evolution; : by design, evolution; • Physical, chemical (gradients), Physical, chemical (gradients), • Biological: reproduction, Biological: reproduction, • Cultural: learning, technologyCultural: learning, technologyBoundary conditionsBoundary conditions: conservation laws, thermodynamics: conservation laws, thermodynamics

What do we want to model ?What do we want to model ?PROCESSES:PROCESSES:• Growth, decay Growth, decay • Transformation Transformation • Dispersion (transport, diffusion)Dispersion (transport, diffusion)• feedback, constraints, delaysfeedback, constraints, delaysMechanismsMechanisms: by design, evolution; : by design, evolution; • Physical, chemical (gradients), Physical, chemical (gradients), • Biological: reproduction, Biological: reproduction, • Cultural: learning, technologyCultural: learning, technologyBoundary conditionsBoundary conditions: conservation laws, thermodynamics: conservation laws, thermodynamics

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Types of ModelsTypes of ModelsTypes of ModelsTypes of Models

• scale models scale models (architecture, (architecture, construction, mechanical engineering)construction, mechanical engineering)

• conceptual models conceptual models (qualitative, (qualitative, block and flow diagrams, show major block and flow diagrams, show major components and interrelationships)components and interrelationships)

• mathematical models:mathematical models:– analytical, analog, digitalanalytical, analog, digital

• scale models scale models (architecture, (architecture, construction, mechanical engineering)construction, mechanical engineering)

• conceptual models conceptual models (qualitative, (qualitative, block and flow diagrams, show major block and flow diagrams, show major components and interrelationships)components and interrelationships)

• mathematical models:mathematical models:– analytical, analog, digitalanalytical, analog, digital

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A model is any abstraction or A model is any abstraction or simplifications of a SYSTEM.simplifications of a SYSTEM.

Modeling is done to aid in the con- Modeling is done to aid in the con- ceptualization and measurement of ceptualization and measurement of complex systems, and … to predict complex systems, and … to predict the consequences of an action that the consequences of an action that would be expensive, difficult, or would be expensive, difficult, or destructive to do with the real system.destructive to do with the real system.

A model is any abstraction or A model is any abstraction or simplifications of a SYSTEM.simplifications of a SYSTEM.

Modeling is done to aid in the con- Modeling is done to aid in the con- ceptualization and measurement of ceptualization and measurement of complex systems, and … to predict complex systems, and … to predict the consequences of an action that the consequences of an action that would be expensive, difficult, or would be expensive, difficult, or destructive to do with the real system.destructive to do with the real system.

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Models are devices for predicting the Models are devices for predicting the behavior of a complicated, poorly behavior of a complicated, poorly understood entity from parts that are understood entity from parts that are well understood.well understood.

……models must be checked frequently models must be checked frequently against the real world to assure that against the real world to assure that their representation of the real world is their representation of the real world is accurate, or at least accurate, or at least inaccurate in inaccurate in ways which we are aware ofways which we are aware of..

Models are devices for predicting the Models are devices for predicting the behavior of a complicated, poorly behavior of a complicated, poorly understood entity from parts that are understood entity from parts that are well understood.well understood.

……models must be checked frequently models must be checked frequently against the real world to assure that against the real world to assure that their representation of the real world is their representation of the real world is accurate, or at least accurate, or at least inaccurate in inaccurate in ways which we are aware ofways which we are aware of..

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Environmental ModelingEnvironmental ModelingEnvironmental ModelingEnvironmental Modeling

considerable tradition: considerable tradition:

• 1856 Darcy’s Law, fundamental 1856 Darcy’s Law, fundamental equation describing groundwater flowequation describing groundwater flow

• 1871 St.Venant equations describing 1871 St.Venant equations describing unsteady open channel flowunsteady open channel flow

• 1924 Lotka’s 1924 Lotka’s Elements of Physical Elements of Physical BiologyBiology

considerable tradition: considerable tradition:

• 1856 Darcy’s Law, fundamental 1856 Darcy’s Law, fundamental equation describing groundwater flowequation describing groundwater flow

• 1871 St.Venant equations describing 1871 St.Venant equations describing unsteady open channel flowunsteady open channel flow

• 1924 Lotka’s 1924 Lotka’s Elements of Physical Elements of Physical BiologyBiology

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• 1960-70 first computer models1960-70 first computer models• 1971 B.Patten, 1971 B.Patten, Systems Analysis and Systems Analysis and

Simulation in Ecology Simulation in Ecology (linear systems)(linear systems)• 1972 J.Forrester, 1972 J.Forrester, Principles of Systems Principles of Systems

(Systems Dynamics)(Systems Dynamics)• 1972 H.T.Odum, Energy flow modeling1972 H.T.Odum, Energy flow modeling• 1974,79 CLEANER multi-compartment 1974,79 CLEANER multi-compartment

lake models, Park et.al.lake models, Park et.al.

• 1960-70 first computer models1960-70 first computer models• 1971 B.Patten, 1971 B.Patten, Systems Analysis and Systems Analysis and

Simulation in Ecology Simulation in Ecology (linear systems)(linear systems)• 1972 J.Forrester, 1972 J.Forrester, Principles of Systems Principles of Systems

(Systems Dynamics)(Systems Dynamics)• 1972 H.T.Odum, Energy flow modeling1972 H.T.Odum, Energy flow modeling• 1974,79 CLEANER multi-compartment 1974,79 CLEANER multi-compartment

lake models, Park et.al.lake models, Park et.al.

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Development through increasing Development through increasing complexity: number of interacting complexity: number of interacting compartments, types of interactions.compartments, types of interactions.

No explicit spatial distribution in early No explicit spatial distribution in early process models.process models.

First spatially explicit models in the First spatially explicit models in the physical domain (flow), linkage of physical domain (flow), linkage of transport and ecological processes by transport and ecological processes by the mid 70’s and 80’s.the mid 70’s and 80’s.

Development through increasing Development through increasing complexity: number of interacting complexity: number of interacting compartments, types of interactions.compartments, types of interactions.

No explicit spatial distribution in early No explicit spatial distribution in early process models.process models.

First spatially explicit models in the First spatially explicit models in the physical domain (flow), linkage of physical domain (flow), linkage of transport and ecological processes by transport and ecological processes by the mid 70’s and 80’s.the mid 70’s and 80’s.

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Simplified blockSimplified block

diagram of the diagram of the

aquatic ecosystemaquatic ecosystem

model CLEANERmodel CLEANER

(Park et al., 1975)(Park et al., 1975)

Simplified blockSimplified block

diagram of the diagram of the

aquatic ecosystemaquatic ecosystem

model CLEANERmodel CLEANER

(Park et al., 1975)(Park et al., 1975)

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Why Modeling:Why Modeling:Why Modeling:Why Modeling:

• conceptualising, organisingconceptualising, organising

• communicatingcommunicating

• understanding, assessing understanding, assessing

• testing field measurementstesting field measurements

• forecasting, early warningforecasting, early warning

• optimising decision makingoptimising decision making







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Why Modeling:Why Modeling:Why Modeling:Why Modeling:













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mathematical modelsmathematical models• conceptual or empiricalconceptual or empirical

• deterministic or stochasticdeterministic or stochastic

• steady-state or dynamicsteady-state or dynamic

• analytical or numericalanalytical or numerical

• spatially aggregated or distributedspatially aggregated or distributed

mathematical modelsmathematical models• conceptual or empiricalconceptual or empirical

• deterministic or stochasticdeterministic or stochastic



• spatially aggregated or distributedspatially aggregated or distributed

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• conceptual or empiricalconceptual or empirical• based on basic laws of nature or based on basic laws of nature or

theoretical conceptstheoretical concepts

• derived from observations derived from observations (input-output relations), providing (input-output relations), providing

phenomenological descriptionsphenomenological descriptions

• conceptual or empiricalconceptual or empirical• based on basic laws of nature or based on basic laws of nature or

theoretical conceptstheoretical concepts

• derived from observations derived from observations (input-output relations), providing (input-output relations), providing

phenomenological descriptionsphenomenological descriptions

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laws of nature, theoretical conceptslaws of nature, theoretical concepts• conservation laws: conservation laws:

mass, energy, momentummass, energy, momentum

• Newtonian mechanics, diffusionNewtonian mechanics, diffusion• stoichiometric relationships, reaction stoichiometric relationships, reaction

kineticskinetics• trophic relations, growth, behaviortrophic relations, growth, behavior

laws of nature, theoretical conceptslaws of nature, theoretical concepts• conservation laws: conservation laws:

mass, energy, momentummass, energy, momentum

• Newtonian mechanics, diffusionNewtonian mechanics, diffusion• stoichiometric relationships, reaction stoichiometric relationships, reaction

kineticskinetics• trophic relations, growth, behaviortrophic relations, growth, behavior

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• deterministic or stochasticdeterministic or stochastic• all model inputs and parameters all model inputs and parameters

are assumed to be exactly knownare assumed to be exactly known• inputs and parameters can be inputs and parameters can be

represented by probability represented by probability distributions, resulting in distributions, resulting in probabilistic state and outputprobabilistic state and output

• deterministic or stochasticdeterministic or stochastic• all model inputs and parameters all model inputs and parameters

are assumed to be exactly knownare assumed to be exactly known• inputs and parameters can be inputs and parameters can be

represented by probability represented by probability distributions, resulting in distributions, resulting in probabilistic state and outputprobabilistic state and output

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• input and parameters are time-input and parameters are time-invariant, a solution independent of invariant, a solution independent of time can be derivedtime can be derived

• some model elements are some model elements are described as functions of timedescribed as functions of time


• input and parameters are time-input and parameters are time-invariant, a solution independent of invariant, a solution independent of time can be derivedtime can be derived

• some model elements are some model elements are described as functions of timedescribed as functions of time

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• dynamic simulation modelsdynamic simulation models

Address Address WHAT - IFWHAT - IF questions questions

Descriptive models analyzeDescriptive models analyze or forecast the evolution of or forecast the evolution of systems over time.systems over time.

• dynamic simulation modelsdynamic simulation models

Address Address WHAT - IFWHAT - IF questions questions

Descriptive models analyzeDescriptive models analyze or forecast the evolution of or forecast the evolution of systems over time.systems over time.

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• the model equations can be solved the model equations can be solved analytically and exactlyanalytically and exactly

• equations require a numerical equations require a numerical approximation for solution, based approximation for solution, based on some form of discretizationon some form of discretization


• the model equations can be solved the model equations can be solved analytically and exactlyanalytically and exactly

• equations require a numerical equations require a numerical approximation for solution, based approximation for solution, based on some form of discretizationon some form of discretization

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• spatially aggregated or distributedspatially aggregated or distributed• model is assumed to be model is assumed to be

independent of spatial locationindependent of spatial location• models uses average (lumped) models uses average (lumped)

values to describe a larger areavalues to describe a larger area• inputs, parameters or the transfer inputs, parameters or the transfer

function vary with location, state is function vary with location, state is defined for more than one location, defined for more than one location, spatial elements interactspatial elements interact

• spatially aggregated or distributedspatially aggregated or distributed• model is assumed to be model is assumed to be

independent of spatial locationindependent of spatial location• models uses average (lumped) models uses average (lumped)

values to describe a larger areavalues to describe a larger area• inputs, parameters or the transfer inputs, parameters or the transfer

function vary with location, state is function vary with location, state is defined for more than one location, defined for more than one location, spatial elements interactspatial elements interact

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Modeling processModeling processModeling processModeling process

• conceptualisation, identificationconceptualisation, identification• mathematical representationmathematical representation• numerical implementationnumerical implementation• parameter estimation, calibrationparameter estimation, calibration• hypothesis testinghypothesis testing• validationvalidation Q.E.D.Q.E.D.

• conceptualisation, identificationconceptualisation, identification• mathematical representationmathematical representation• numerical implementationnumerical implementation• parameter estimation, calibrationparameter estimation, calibration• hypothesis testinghypothesis testing• validationvalidation Q.E.D.Q.E.D.

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conceptualisation, identificationconceptualisation, identification

• assembling the constituent assembling the constituent hypotheses believed to govern hypotheses believed to govern system behaviorsystem behavior

• evaluation of hypotheses evaluation of hypotheses

• model structure identificationmodel structure identification

conceptualisation, identificationconceptualisation, identification

• assembling the constituent assembling the constituent hypotheses believed to govern hypotheses believed to govern system behaviorsystem behavior

• evaluation of hypotheses evaluation of hypotheses

• model structure identificationmodel structure identification

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mathematical representationmathematical representation• usually in the form of differential or usually in the form of differential or

algebraic equationsalgebraic equations

• possibly also as linguistic rules possibly also as linguistic rules (expert system)(expert system)

• rules for cellular automatarules for cellular automata

mathematical representationmathematical representation• usually in the form of differential or usually in the form of differential or

algebraic equationsalgebraic equations

• possibly also as linguistic rules possibly also as linguistic rules (expert system)(expert system)

• rules for cellular automatarules for cellular automata

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numerical implementationnumerical implementation• numerical solution algorithm for the numerical solution algorithm for the

system of coupled equations system of coupled equations (differential or difference equations)(differential or difference equations)

• inference engine and rule base for inference engine and rule base for and expert system implementationand expert system implementation

numerical implementationnumerical implementation• numerical solution algorithm for the numerical solution algorithm for the

system of coupled equations system of coupled equations (differential or difference equations)(differential or difference equations)

• inference engine and rule base for inference engine and rule base for and expert system implementationand expert system implementation

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parameter estimation, calibrationparameter estimation, calibration

• tuning of model parameters based tuning of model parameters based on data sets:on data sets:

this requires the availability of this requires the availability of measurements measurements commensuratecommensurate with with the model state variables.the model state variables.

parameter estimation, calibrationparameter estimation, calibration

• tuning of model parameters based tuning of model parameters based on data sets:on data sets:

this requires the availability of this requires the availability of measurements measurements commensuratecommensurate with with the model state variables.the model state variables.

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hypothesis testinghypothesis testing• testing the model output against testing the model output against

test conditions formulated for the test conditions formulated for the underlying hypothesesunderlying hypotheses

Strict testing (Strict testing (falsificationfalsification) in a ) in a Popperian sense is usually Popperian sense is usually impractical for a complex model: impractical for a complex model: it almost always can be falsified.it almost always can be falsified.

hypothesis testinghypothesis testing• testing the model output against testing the model output against

test conditions formulated for the test conditions formulated for the underlying hypothesesunderlying hypotheses

Strict testing (Strict testing (falsificationfalsification) in a ) in a Popperian sense is usually Popperian sense is usually impractical for a complex model: impractical for a complex model: it almost always can be falsified.it almost always can be falsified.

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validationvalidation• assuring the quality of the assuring the quality of the

instrument for the intended taskinstrument for the intended task• usually restricted to statistical usually restricted to statistical

goodness of fit goodness of fit measurement measurement against an independent data set against an independent data set not used in calibration.not used in calibration.

validationvalidation• assuring the quality of the assuring the quality of the

instrument for the intended taskinstrument for the intended task• usually restricted to statistical usually restricted to statistical

goodness of fit goodness of fit measurement measurement against an independent data set against an independent data set not used in calibration.not used in calibration.

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• dynamic simulation modelsdynamic simulation models state variablesstate variables, related by, related by operators operators ((SS), ), subject to subject to inputsinputs ( (X X (t)(t)),), producing producing outputsoutputs ( (Y Y (t)(t)):):

YYtt = = SXSXtt

• dynamic simulation modelsdynamic simulation models state variablesstate variables, related by, related by operators operators ((SS), ), subject to subject to inputsinputs ( (X X (t)(t)),), producing producing outputsoutputs ( (Y Y (t)(t)):):

YYtt = = SXSXtt

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• systems statesystems state LetLet

XX(t) = (t) = {x{x11(t), x(t), x22(t), …,x(t), …,xnn(t)}(t)} represent a system with represent a system with n n state state

variables variables x xii(t), i=1,2, …, n(t), i=1,2, …, n

Each state variable is Each state variable is a a function function of time of time t. t.

• systems statesystems state LetLet

XX(t) = (t) = {x{x11(t), x(t), x22(t), …,x(t), …,xnn(t)}(t)} represent a system with represent a system with n n state state

variables variables x xii(t), i=1,2, …, n(t), i=1,2, …, n

Each state variable is Each state variable is a a function function of time of time t. t.

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systems equations:systems equations:

systems equations:systems equations:

),...,,(

......................................

),...,,(

),...,,(

21

2122

2111

nnn

n

n

QQQfdt

dQ

QQQfdt

dQ

QQQfdt

dQ

),...,,(

......................................

),...,,(

),...,,(

21

2122

2111

nnn

n

n

QQQfdt

dQ

QQQfdt

dQ

QQQfdt

dQ

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• state transitionstate transition

consider a system with n=16 possible consider a system with n=16 possible states, denotedstates, denoted

A B C D E F G H I J K L M N O PA B C D E F G H I J K L M N O P

• state transitionstate transition

consider a system with n=16 possible consider a system with n=16 possible states, denotedstates, denoted

A B C D E F G H I J K L M N O PA B C D E F G H I J K L M N O P

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• state transitionstate transitionThe behavior of the system can be The behavior of the system can be

described as the transition from described as the transition from one state to another:one state to another:

A --> DA --> D



A --> DA --> D

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A B C D E F G H I J K L M N O PA B C D E F G H I J K L M N O P D H D I P G P H A E E N B A N ED H D I P G P H A E E N B A N E



A B C D E F G H I J K L M N O PA B C D E F G H I J K L M N O P D H D I P G P H A E E N B A N ED H D I P G P H A E E N B A N E

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• systems state: systems state: vegetation examplevegetation example (high-mountain plateau, Utha; Ellison, 1954)

XX11 talus vegetation talus vegetation XX6 6 grasslandgrassland

XX2 2 ephemerals ephemerals X X7 7 low shrubslow shrubs

XX3 3 spruce-fir spruce-fir XX8 8 forbsforbs

XX4 4 tall shrubs tall shrubs XX99 erosion com.erosion com.

XX5 5 mixed herbs mixed herbs XX1010 rock crevice pl.rock crevice pl.

• systems state: systems state: vegetation examplevegetation example (high-mountain plateau, Utha; Ellison, 1954)

XX11 talus vegetation talus vegetation XX6 6 grasslandgrassland

XX2 2 ephemerals ephemerals X X7 7 low shrubslow shrubs

XX3 3 spruce-fir spruce-fir XX8 8 forbsforbs

XX4 4 tall shrubs tall shrubs XX99 erosion com.erosion com.

XX5 5 mixed herbs mixed herbs XX1010 rock crevice pl.rock crevice pl.

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• state transition state transition depends on depends on grazing as an external control:grazing as an external control:

1 no grazing1 no grazing 2 sheep only2 sheep only 3 cattle only3 cattle only 4 sheep with cattle4 sheep with cattle 5 sheep and cattle alternately5 sheep and cattle alternately

• state transition state transition depends on depends on grazing as an external control:grazing as an external control:

1 no grazing1 no grazing 2 sheep only2 sheep only 3 cattle only3 cattle only 4 sheep with cattle4 sheep with cattle 5 sheep and cattle alternately5 sheep and cattle alternately

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• state transition transition matrixstate transition transition matrix

XX11 X X2 2 XX3 3 XX44 X X55 X X6 6 X X77 X X88 X X99 X X1010

1 1 XX2 2 X X33 X X4 4 XX55 X X55 X X55 X X55 X X55 X X11 X X11

22 - X- X99 - - X - - X66 X X22 - - X - - X1010 - - 3 3 -- X X99 - - X - - X88 - - X - - X22 X X1010 - - 4 4 - - - - X- - - - X77 - X - X22 - - - - - - 5 5 - X- X99 - - - - - - X - - - - - - X1010 - -

• state transition transition matrixstate transition transition matrix

XX11 X X2 2 XX3 3 XX44 X X55 X X6 6 X X77 X X88 X X99 X X1010

1 1 XX2 2 X X33 X X4 4 XX55 X X55 X X55 X X55 X X55 X X11 X X11

22 - X- X99 - - X - - X66 X X22 - - X - - X1010 - - 3 3 -- X X99 - - X - - X88 - - X - - X22 X X1010 - - 4 4 - - - - X- - - - X77 - X - X22 - - - - - - 5 5 - X- X99 - - - - - - X - - - - - - X1010 - -

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state transitionstate transition

- discrete: - discrete: subsequent generationssubsequent generations

- continuous:- continuous: advection and diffusionadvection and diffusion

choice depends on the space and choice depends on the space and time resolution of the process -time resolution of the process -and the model.and the model.

state transitionstate transition

- discrete: - discrete: subsequent generationssubsequent generations

- continuous:- continuous: advection and diffusionadvection and diffusion

choice depends on the space and choice depends on the space and time resolution of the process -time resolution of the process -and the model.and the model.

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analytical analytical solution solution vsvs difference difference equationsequations 10 and 2510 and 25 year time step year time step

analytical analytical solution solution vsvs difference difference equationsequations 10 and 2510 and 25 year time step year time step

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• change of statechange of state

XX(t+1) = (t+1) = XX(t) + (t) + XX

xxii = x = xii(t+1) - (t+1) - xxii(t)(t)

• change of statechange of state

XX(t+1) = (t+1) = XX(t) + (t) + XX


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• rate of changerate of change

xxi i = = xxii(t + (t + t) - t) - xxii(t)(t)tttt

and forand fort = 1t = 1


• rate of changerate of change

xxi i = = xxii(t + (t + t) - t) - xxii(t)(t)tttt

and forand fort = 1t = 1


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population growthpopulation growth

Let Let NN(t) be the size or density of a (t) be the size or density of a population with population with

birth rate birth rate bb and death rate and death rate dd:: ddNN//ddt = t = bN - dN bN - dN rrN N = (= (bb--dd))NN

where where r r is the specific growth rate. is the specific growth rate.


Let Let NN(t) be the size or density of a (t) be the size or density of a population with population with

birth rate birth rate bb and death rate and death rate dd:: ddNN//ddt = t = bN - dN bN - dN rrN N = (= (bb--dd))NN

where where r r is the specific growth rate. is the specific growth rate.

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the analytical solution ofthe analytical solution of

ddNN(t) = (t) = rrNN(t) is(t) is

NN(t) = (t) = NN(0) (0) eerrtt

given the initial density given the initial density NN(0).(0).


the analytical solution ofthe analytical solution of

ddNN(t) = (t) = rrNN(t) is(t) is

NN(t) = (t) = NN(0) (0) eerrtt

given the initial density given the initial density NN(0).(0).

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logistic growth: logistic growth: y y = 1/(= 1/(aa++bbeekxkx

))

(exponential growth(exponential growth with density with density dependence)dependence)

logistic growth: logistic growth: y y = 1/(= 1/(aa++bbeekxkx

))

(exponential growth(exponential growth with density with density dependence)dependence)

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Michaelis-Menten: Michaelis-Menten: yy = = kk((xx/(/(xx++KsKs))))

substrate substrate

saturation,saturation,

k k = max. rate,= max. rate,

KsKs half saturation half saturation

Michaelis-Menten: Michaelis-Menten: yy = = kk((xx/(/(xx++KsKs))))

substrate substrate

saturation,saturation,

k k = max. rate,= max. rate,

KsKs half saturation half saturation

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Analytical solutions:Analytical solutions:

Linear EquationsLinear Equations one several manyone several many

algebraic trivial easy algebraic trivial easy essentiallyessentially impossibleimpossibleordinary easy difficult ordinary easy difficult essentiallyessentiallydifferential differential impossibleimpossible

partial difficult partial difficult essentiallyessentially impossibleimpossibledifferential differential impossibleimpossible


Linear EquationsLinear Equations one several manyone several many

algebraic trivial easy algebraic trivial easy essentiallyessentially impossibleimpossibleordinary easy difficult ordinary easy difficult essentiallyessentiallydifferential differential impossibleimpossible

partial difficult partial difficult essentiallyessentially impossibleimpossibledifferential differential impossibleimpossible

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Nonlinear EquationsNonlinear Equations one several manyone several many

algebraic very very algebraic very very impossibleimpossible difficult difficultdifficult difficultordinary veryordinary very impossible impossible impossible impossible differential difficult differential difficult

partialpartial impossible impossible impossible impossible impossible impossibledifferential differential


Nonlinear EquationsNonlinear Equations one several manyone several many

algebraic very very algebraic very very impossibleimpossible difficult difficultdifficult difficultordinary veryordinary very impossible impossible impossible impossible differential difficult differential difficult

partialpartial impossible impossible impossible impossible impossible impossibledifferential differential

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Modeling DomainsModeling DomainsModeling DomainsModeling Domains

All environmental model domains All environmental model domains have an obvious have an obvious spatial dimensionspatial dimension..

Most recent environmental models Most recent environmental models are spatially explicit (inputs and are spatially explicit (inputs and state are functions of space)state are functions of space)

XX (x,y,z,t) (x,y,z,t)

All environmental model domains All environmental model domains have an obvious have an obvious spatial dimensionspatial dimension..

Most recent environmental models Most recent environmental models are spatially explicit (inputs and are spatially explicit (inputs and state are functions of space)state are functions of space)

XX (x,y,z,t) (x,y,z,t)

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Distributed ModelsDistributed ModelsDistributed ModelsDistributed Models

are based on are based on

partial differential equationspartial differential equations; ; dependent variables are functions dependent variables are functions of two or more other variables:of two or more other variables:

dQ dQ

dx dy(continuity equation of 2D groundwater flow)

are based on are based on

partial differential equationspartial differential equations; ; dependent variables are functions dependent variables are functions of two or more other variables:of two or more other variables:

dQ dQ

dx dy(continuity equation of 2D groundwater flow)

+ = 0= 0 = 0= 0

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The partial differential equations are The partial differential equations are solved with a numerical scheme solved with a numerical scheme like finite elements like finite elements

or finite differences. or finite differences. This requires the This requires the solution domain to solution domain to be discretized.be discretized.

The partial differential equations are The partial differential equations are solved with a numerical scheme solved with a numerical scheme like finite elements like finite elements

or finite differences. or finite differences. This requires the This requires the solution domain to solution domain to be discretized.be discretized.

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DomainDomain discretisation:discretisation:

representingrepresenting continuouscontinuous processes andprocesses and distributions withdistributions with with discrete with discrete elementselements

DomainDomain discretisation:discretisation:

representingrepresenting continuouscontinuous processes andprocesses and distributions withdistributions with with discrete with discrete elementselements

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Examples of domain discretisation:Examples of domain discretisation:multiple layers in lake modelsmultiple layers in lake models

Examples of domain discretisation:Examples of domain discretisation:multiple layers in lake modelsmultiple layers in lake models

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Distributed ModelsDistributed ModelsDistributed ModelsDistributed Models Examples Examples of domainof domain discretisation:discretisation:

mulitplemulitple layers andlayers and volumesvolumes

Examples Examples of domainof domain discretisation:discretisation:

mulitplemulitple layers andlayers and volumesvolumes

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Examples of domain discretisation:Examples of domain discretisation: multiplemultiple

ssegmentegment

oror volumevolume

Examples of domain discretisation:Examples of domain discretisation: multiplemultiple

ssegmentegment

oror volumevolume

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Examples of domain discretisation:Examples of domain discretisation:block-centered regular gridblock-centered regular grid

Examples of domain discretisation:Examples of domain discretisation:block-centered regular gridblock-centered regular grid

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Examples of domain discretisation:Examples of domain discretisation:point-centered regular gridpoint-centered regular grid

Examples of domain discretisation:Examples of domain discretisation:point-centered regular gridpoint-centered regular grid

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Distributed ModelsDistributed ModelsDistributed ModelsDistributed Models complex curvi-linear FD gridcomplex curvi-linear FD grid

complex curvi-linear FD gridcomplex curvi-linear FD grid

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Examples of domain discretisation:Examples of domain discretisation:2D coupling of elements2D coupling of elements

Examples of domain discretisation:Examples of domain discretisation:2D coupling of elements2D coupling of elements

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Examples of domain discretisation:Examples of domain discretisation: 3D coupling3D coupling of elementsof elements

Examples of domain discretisation:Examples of domain discretisation: 3D coupling3D coupling of elementsof elements

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the spatial distribution further the spatial distribution further complicates model equations.complicates model equations.

Analytical solutions are therefor Analytical solutions are therefor restricted to steady-state solutions, restricted to steady-state solutions, e.g., Gaussian plume equation:e.g., Gaussian plume equation:

CC(x,y) (x,y) = Q = Q / / u*u*yyx x exp(-yexp(-y22/2/222yy))

the spatial distribution further the spatial distribution further complicates model equations.complicates model equations.

Analytical solutions are therefor Analytical solutions are therefor restricted to steady-state solutions, restricted to steady-state solutions, e.g., Gaussian plume equation:e.g., Gaussian plume equation:

CC(x,y) (x,y) = Q = Q / / u*u*yyx x exp(-yexp(-y22/2/222yy))

© k.fedra 2000 1 systems analysis and the systemic approach: analytical methods: environmental...

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