1. Introduction. 2. Modelling elements III. CONCEPTS OF MODELLING. - Models in environmental sciences have five components:

Transcription

1 III. CONCEPTS OF MODELLING 1. INTRODUCTION 2. MODELLING ELEMENTS 3. THE MODELLING PROCEDURE 4. CONCEPTUAL MODELS 5. THE MODELLING PROCEDURE 6. SELECTION OF MODEL COMPLEXITY AND STRUCTURE 1 1. Introduction 2 2. Modelling elements - Models in environmental sciences have five components: 1- forcing functions or external variables 2- state variables 3- mathematical equations 4- parameters 5- universal constants 3 1

2 2. Modelling elements: 1. Forcing functions or external variables - Functions or variables of an external nature that influence the ecosystem. => If forcing functions are varied (temperature, precipitations), how will this influence the state of the ecosystem?? - The model is used to predict the response of ecosystem on the changes of forcing functions. -The forcing functions under our control are called control functions. ==> examples: - ecotoxicology: inputs of toxic substances - eutrophication models: inputs of nutrients - other cases: climatic variables (temperature, radiation, precipitaiton) 4 2. Modelling elements: 2. State variables Describe the state of ecosystem. The selection of the state variables is crucial => defines the structure and complexity of the model Cases: - to model the bioaccumulation of toxic substances: organisms from food chain, concentrations of toxins in these organisms - to model eutrophication: concentration of nutrients, concentration of phytoplankton - models used in management context: the values of the state variables predicted by changes of forcing functions 5 2. Modelling elements: 3. Mathematical equations - Used to represent the biological, chemical and physical processes. - They describe relationships between forcing functions and state variables. - The same processes can be found in different environmental conditions => the same equations can be used in different models!! - It doesn t imply that the same process is always described by the same equation!! => description of the same process with different equation because of the influence of other factors => number of details included in the mode may be different from case to case due to complexity of the model or system or/and problem - Mathematical description of the processes can be referred as sub-models (library construction) 6 2

3 2. Modelling elements: 4. Parameters - coefficients in mathematical equations - they may be considered as constants (they don t vary) - in causal models parameters have a scientific definition (extraction rate of cadmium from fish) - many parameters are indicated as ranges Application of parameters as constants is unrealistic: => processes in ecosystem are dynamic with many feedbacks => new generation of the models attempts to use parameters varying according to some ecological principles of ecosystem 7 2. Modelling elements: 5. Universal constants - Gas constant, atomic weights, speed of light, 8 3. The modelling procedure: Definition of the problem Definition of the problem = the first modelling step Problem => decomposition of the problem on the elements => linking the elements with processes described with mathematical equations => MODEL=> Formal expressions of the essential elements of the problem in mathematical terms. First recognition/description of the problem is often verbal => essential preliminary step in the modelling procedure Verbal problem is hard to visualize => translation into conceptual diagram which contains: state variable, forcing functions and mathematical formulation of their interactions 9 3

4 3. The modelling procedure: Definition of the problem Problem must be bound by constrains of space, time and subsystems => defines the complexity of the model number of sub-models or modeling elements > complexity =/= > more accurately simulation of the real system Not necessarily : ==> more parameters ==> increases the level of uncertainty because of the errors in estimations of the parameters values ==> higher complexity == larger uncertainty To select right model complexity is importation goal!!! The modelling procedure: The adjacency matrix - The list of state variables - List has a form of matrix: variables are listed vertically and horizontally - Links between variables are indicated in matrix: direct link 1, no link 0 (Question: Is the link between two variables possible or not? Yes:1, No:0 - Adjacency matrix is set up before the conceptual diagram. - The idea behind is to get the very first overview of the state variables and their interactions - It defines the complexity of the model and requests for data. 12 4

5 3. The modelling procedure: The adjacency matrix Adjacency matrix for the nitrogen cycle model. Does exist link from (row) to (column)? - Yes:1 - No: The modelling procedure: The conceptual diagram - Links between state variables indicated by adjacency matrix are visualized into conceptual diagram - It gives information which state variables, forcing functions and processes are required in the model. 15 5

6 3. The modelling procedure: The conceptual diagram CONCEPTUAL MODELS To present different conceptual methods > their advantages and disadvantages. Selection of the conceptual method depends on: - the problem - the ecosystem - the application of the model - habits of the modeler Conceptual diagrams Conceptualization is one of the early steps in the modelling procedure. A conceptual model can not only be considered as a list of state variables and forcing functions of importance to the system and the problem in focus BUT it will also show how these components are connected by processes. Conceptual diagrams are tools to create abstraction of reality in ecosystem and to delineate the level of organization (complexity) that best meet the objectives of the model. 18 6

7 4.1 Conceptual diagrams Focal studied system must be placed into hierarchical structure of environment: - to determine a system one step higher: => determines the affects of environmental constrains on studied system/model - to determine one level lower: => submodules => relevant to a good description of the processes => easier understanding the relationships between elements of the model Conceptual diagrams Case: Photosynthesis: - it is determined by the processes going on in the individual plants chlorophyll (level -1), and processes related with CO 2 in atmosphere (level +1) Atmosphere Photosynthesis Chlorophyll Types of conceptual diagrams 1. Box models -Simple and commonly used conceptual diagrams for ecosystem models -Each block represents a component and arrows between boxes indicate processes. - If the model contain numbers (stocks and flows) then we got steady state model 21 7

8 4.2 Types of conceptual diagrams Types of conceptual diagrams Types of conceptual diagrams 24 8

9 4.2 Types of conceptual diagrams Types of conceptual diagrams - Black box models: internal process are not known (causality is not known), relationships between inputs and outputs are analyzed. - White box models: models are constructed based on causality for all processes => this type is not applicable on ecosystem because we are not able to capture all causalities in the system. - Gray models: most of our models, we know for some causality but no for all of them Types of conceptual diagrams 2. Adjacency models -Adjacency matrix (connections between elements): 1= direct link 0 = no link - Input/output matrix (matter and information flows). To X 1 X 2 X 3 From X 1 X 2 X 3 To X 1 X 2 X 3 From X 1 X 2 X

10 4.2 Types of conceptual diagrams 3. Signed diagram model - extend the adjacency matrix - positive and negative interactions are marked with + and A B C F E D A B C D E F A + B + C - D - + E F Types of conceptual diagrams 4. Energy circuit diagrams (H.T. Odum) - designed for ecological energetic analysis

11 5. The modelling procedure: Verification - The step that follows the selection of mathematical equations. - Answer to the following question is needed: Is the model stable in long term? State variables should are maintained at approximately the same levels despite variations of forcing functions ==> first running period the model is sensitive to initial conditions stability of the model must be achieved also under long term vales The modelling procedure: Verification Does the model react as expected? To compare model results with observations and changes of the model are made according to the modeler s intuition and knowledge of the reactions of the model. Its time to play with a model and evaluate the model outputs. ==> we use the knowledge about the system which we simulate ==> model should be useful description of real ecosystem The modelling procedure: Verification Are applied units in the model consistent? -Equations should use consistent unites 33 11

12 5. The modelling procedure: Sensitivity analysis Get an overview of the most sensitive components of the model. To provide a measure of sensitivity of either parameters, forcing functions or submodels to the state variables of greatest interest in the model. The relative change in parameter value is chosen based on knowledge of the certainty of the parameters. It enables to distinguish between high (high impact on the system behavior) and low (low impact) leverage variables. Low leverage variables may be excluded from the model => effect on complexity and structure of the model => lower complexity => simpler dataset => higher certainty of the model The modelling procedure: Calibration The scope is to improve the parameter estimation (exact value or range of values of the interval): the parameter set that gives the best agreement between model output and measured values is chosen. Selection of parameter s values: - from literature - from experiments - one value or set of values from an interval The modelling procedure: Calibration -To select the most important parameters and experiment with the variation of their values (software for automatic calibration) => time consumable or not applicable at all: (10 parameters, 15 possible values of each parameter = experiments = this is higher number then is the number of seconds since the big bang - The quality of calibration depends the most on the quality of data

13 5. The modelling procedure: Calibration Recommendations for the calibration procedure: - Find as much as possible parameters from the literature, otherwise determine them by experiments in situ -Use sensitivity analysis to find the most important parameters. - Use an intensive data collection program for the most important state variables to provide better estimation for the most important parameters - Perform calibration using data not yet applied in order to get acquainted with the model s reactions to changes in parameters. - Repeat sensitivity analysis and try to make fine tuning of parameter s values The modelling procedure: Validation - To test the model against an independent set of data to see how well the model simulations fit these data. - The validation is always required (to get a picture about the reliability od the model) - Data for validation must not be the same as was used for calibration. - Data need to be from wide range of forcing functions - Validation criteria (i.e. how good is the model) must be formulated on objectives of the model and the quality of the available data

14 6. SELECTION OF MODEL COMPLEXITY AND STRUCTURE The base model (Zeigler) => the model that will be capable of accounting for the complete input output behavior of real ecosystem and be valid for all frames => ideal model => ecosystem is very complex => the base models should be very complex as well => it is impossible to make a base model of ecosystem. Central question: How could we select the complexity and the structure of the model to ensure the optimum knowledge gained or the best answer to the question posed by the model? SELECTION OF MODEL COMPLEXITY AND STRUCTURE New knowledge Very good data Complexity of the model SELECTION OF MODEL COMPLEXITY AND STRUCTURE Balance between complexity and expectations: 1. Good definition of the problem 2. Good knowledge about the system 3. Right complexity of the model Model is a simplification of real system! => it s impossible to make a model of real ecosystem, => models of some aspects of ecosystem may be made

15 6. SELECTION OF MODEL COMPLEXITY AND STRUCTURE Selection of the complexity must not be random because: It affects the outputs from the model (main product of the modeler) Enough time need to be taken to select right level of complexity