What is a Model?


Who develops these models?


Why are they such a powerful tool?


Why were they developed?


What do they do?


Why should you know how to use them?

Approach taken in this Environmental Chemistry class

These models were produced by the U.S. EPA
This creates a unique situation due to U.S. copyright laws.
(Exception - selected examples are commercial derivatives)

You are encouraged to retain copies of the models you use.

Objective and Approach -
You are training to be the next generation environmental scientist - manager and you must be trained with the state of the art tools and skills.

You will need these skills in your profession
Learn to use them.

Where possible you will be provided with

• a model (several various important types)
  
• scientific relationships
  (frequently included in the model
  
• unique approaches used
  
• scientific principles upon which the model is based
  
• limitations and scope of the models assumptions
  
• validation of the models
  

What is Environmental Modeling & What constitutes an environmental model

Elements of a (Mathematical) Model

• Identification of the problem and variables
  

• Constructing appropriate relationships between these variables (chemical relationships)
  

• Collect data evaluating the size, significance, and interrelationships of variables
  (to used in constructing the model)
  

• Collect data and decide the key parameters necessary to provide the models critical inputs
  

• Establish the values of parameters within the model that cannot be measured or calculated from data
  

• Construct relationships including relational (chemical equilibrium and their interactions) and mathematical (algorithmic)
  

• Validate assumptions and relationships
  (all models must be validated before use)
  (each new use not originally planned must be validated separately)
  

• Limit the use of the model to appropriate situations
  

This is science and philosophy, like the scientific method.
It has changed the way science is done and is the basis of a modern technologically based approach.

Computers have altered many areas of science.

Note:
With all of the complex equilibrium occurring simultaneously you have two options:

1. Measure what is there - the final result
   Know that you are not sure how it happened, but that it is the current state of the system under study.
   

2. Model the situations attempting to make the model as complete and comprehensive as possible to account for all the simultaneous equilibrium.
   

Then:
Validate the Model
Once you have a model you must measure (collect data) to verify the model(known as model validation).

All Models must be verified or validated.

Therefore:
The Modeling approach is a combination

So prediction through modeling is a combination of defining the important relationships.

Many are equilibrium that are shifted and effected by one another.


Relating data and measurements to the model to validate it and to obtain data to put into the model and to confirm the relationships used by the model.

These are the key requirements of modeling and data relationships.

Example of a model I once encountered

Goal:

System being modeled:

Why model first, why not just test?


Can I perform a test measurement testing the stability of the disposal method covering the 10,000 years necessary?

Data and assumptions used by the investigator for modeling deep ocean disposal of high level nuclear waste canisters.

• Canisters contain glass made with high level nuclear waste inside stainless steel.
  

• Nuclear waste will be dangerous if it enters the water and food chain for 10,000 years.
  

• Canisters will be approximately 90 °C for over 100 years due to radioactive decay energy levels.
  

• Depth approximately 2 to 3 miles below sea level.
  

• Temperature of the ocean at these locations
  approximately 4 °C.
  

• Ocean Floor
  Varies with rocky, sand, sediment, shells, muds
  

• Ocean currents
  Low to 20 miles per hour
  

• Accessibility
  Not accessible for retrieval through current technology once implanted in ocean floor
  

Model Proposed by Researcher

1. The canisters will penetrate the ocean floor to a minimum depth of 50 Meters.
   

2. No disturbance of the ocean water will be caused by the presence of the canisters.
   

3. Canisters will last for over 1,000 years.
   

4. After 1,000 years, leaching will occur.
   

5. A diffusion model was constructed to distribute the radioactive leached isotopes. Taking into consideration exclusively brownian movement and diffusion aspects of the dispersion in the water.
   

Questions:

How was the model constructed?


What are the important parameters?


What are the relationships?


What is the mathematical relationship?
(an example of a typical diffusion algorithm)


The convection-dispersion equation for diffusion


The process of solute movement are commonly described by the convection-dispersion equation.


c/ t = D ²c/delta z ² - v c/ z


Where:
c/ t is the change in concentration with the change in time,


and
c is solute concentration
t is time
D is the dispersion coefficient (D = D_o+ ev)


where: e is the coefficient of solute dispersivity in soil, and D_o is the molecular diffusion coefficient

v is poor water velocity (given by water flux (q) divided by the volumetric water constant (theta)


Ref. Modeling Chemical Transport in Soils: Natural and Applied Contaminants, Hossein Chadiri and Calvin Rose, Lewis Publishers, pg. 147, 1992.


How was the model constructed?


Evaluation, Validation, and Limitation:


Were the assumptions valid?


Is this an appropriate model?


If it was not a good model, why and how would you improve it?

Some Thoughts on Modeling the Environment & Models used in Environmental Evaluation

Essential skills for successful model development.

1. Identifying the problem variables accurately.
   

2. Constructing appropriate relations between these variables with as little bias as possible.
   

3. Taking measurements judging the size of quantities that are significant and the limitations for these relationships. Set limits.
   Example below: At certain pH there will be essentially no complexation, free salt, or etc.
   

4. Collecting data, decide how to use them and test your model before your finalize it. Modify the model with data and with additional theoretical relationships that you find through this process.
   

5. Know and document the limits of the model by estimating the parameters within the model that cannot be measured or calculated from data.
   

6. Limit the use of the model to its functional and appropriate capabilities excluding the inappropriate use of the model.
   

Some quotes and observations about modeling from an expert.

"Any model must have a definite purpose which is clearly stated at the start."

"A mathematical model is a model created using mathematical concepts such as functions and equations."

"It is also important to realize at the outset that mathematical modeling is carried out in order to solve problems."

"When we create mathematical models, we move from the real world into the abstract world of mathematical concepts, which is where the model is built. ... Finally we reenter the real world, taking with us the solution to the mathematical problem. Note that the start and end are in the real world."

"It must not therefore be thought that for a particular problem there is one right and proper model."

"Some models may be 'better' than others in the sense that they are more useful or more accurate, but this is not always the case." "It must not therefore be thought that for a particular problem there is one right and proper model."

"Generally the success of a model depends on how easily it can be used and how accurate are its predictions. Note also that any model will have a limited range of validity and should not be applied outside this range."

Reference:
Guide to Mathematical Modeling, Dilwyn Edwards and Mike Hamson, CRC Mathematical Guides, CRC Press. 1989.

Summary

1. Mathematical modeling consists of applying your mathematical skills to obtaining useful answers to real problems.
   
   They incorporate fundamental scientific relationships in appropriate, but limited ways.
   

   They remove the hand calculation of many simultaneous variables.
   

2. Learning to apply mathematical skills is very different from learning mathematics itself.
   

3. Models are used in a very wide range of applications, some of which do not appear initially to be mathematical in nature.
   

4. Models often allow quick and cheap evaluation of alternatives, leading to optimal solutions which are not otherwise obvious.
   

5. There are no precise rules in mathematical modeling and no 'correct' answers.
   

6. Modeling can be learned only by doing.
   

Ref.
Guide to Mathematical Modeling, Dilwyn Edwards and Mike Hamson, CRC Mathematical Guides, CRC Press. 1989.

Types of Environmental Models

• natural waters
  
• flowing water
  
• ground water
  
• stagnant water
  
• surface water
  
• stratified water
  
• soil
  
• soil and water mixtures
  
• soil with bed rock boundaries
  
• fate and transport, for soil, water, air; combinations
  
• atmospheric
  
• atmospheric layers
  
• stratified atmosphere
  
• atmospheric with impact on topography
  
• complex spheres and interactions between spheres
  
• anthropogenic vs. natural source
  
• flow tracking
  
• point and non-point sources
  
• other complexities...
  
• many others...
  

Examples

1. Chesapeake Bay Trace Elements Study
   

   Example of Environmental Complex Water System:
   

   • Sampling
     
   • Analysis
     
   • Modeling
     

2. Air Model Examples
   

   Examples of Environmental Air Systems:
   

   • Complex Structure
     
   • Various Components of Various Models
     
   • Use of Air Modeling
     

References:

EPA Exposure Models Library and IMES on CD-ROM
EPA Exposure Models Library and Integrated Model Evaluation System, Office of Research and Development, Office of Health and Environmental Assessment, Computer CD, EPA/600/C-92/002, Revised September 1993.

Guide to Mathematical Modeling, Dilwyn Edwards and Mike Hamson, CRC Mathematical Guides, CRC Press. 1989.

Air Pollution Modeling (Theories, Computational Methods and Available Software), Paolo Zannetti, Van Nostrand-Reinhold, 1990.

Modeling Chemical Transport in Soils: Natural and Applied Contaminants, Hossein Chadiri and Calvin Rose, Lewis Publishers, 1992.

Notes Table of Contents

Environmental Chemistry Home Page

Duquesne University Home Page

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Revised 6/14/98.