Models Mathematical for Environmental Systems
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MATHEMATICAL MODELS FOR ENVIRONMENTAL SYSTEMS
Chapter Goals
Role of mathematical models in the analysis of environmental systems Different approaches for building mathematical models
MATHEMATICAL MODELS AS SYSTEM ANALYSIS TOOLS
The environment is a complex structure with intricate networks and myriad interactions. Visualizing the environment of interest as a system provides a convenient method to focus on the problem at hand and eliminate unwanted complexity. The state variables succinctly exemplify the behavior of the system and provide useful indications of potential alterations to the system in time and space. These state variables also depend upon other factors that may be easier to measure and quantify. Sometimes, we may have estimates for these independent variables into the future and we want to use them to indicate the state of our system. For example, a weather report tells us what the temperatures will be over the next few days. We may want to use this information to forecast ground-level ozone values (which typically show strong dependency on the temperature). The ground-level ozone is an indicator (state variable) of air pollution in the region. A model establishes the relationship between the state variable(s) and the independent variables.
Physical and Mathematical Models
Environmental models come in various shapes and sizes. In some instances, the environmental system of interest (e.g., a river or a lake) is abstracted into
18 Applied Environmental Systems Modeling a beaker or even a test-tube to see how contaminant levels change over time. Physical models are scaled-down versions of actual environmental systems and are used to study the behavior of the pollutants in laboratory settings. These models are easy to visualize as can actually touch and feel them and they are the only recourse to modeling when we have no understanding of the system whatsoever. However, like the building models of architects, they cannot be easily ported from one place to another. Once these models are built they are rather difficult to change (imagine removing a couple of stories from an architects building model without actually dismantling it). Fortunately, the immutable theories of physical sciences, such as the law of mass conservation, enable us to conceptualize the behavior of systems using the language of mathematics. The advent of computers and associated computational advancements enable us to codify mathematical expressions and use them in a variety of different ways. With todays internet technologies, you may not even need specialized and sophisticated software on your computer to run these mathematical models. Mathematical models for environmental decision-making often utilize the principles of physics, chemistry and biology to understand and predict how pollutants move within a system of interest. Concepts from other physical and social sciences are also drawn as necessary. A big advantage of these mathematical models lies in their portability (i.e., they can be easily applied to different problems at different locations). In many instances, the size of the system is not an issue as well. We can use the same mathematical scheme to model a big lake or a small pond by using appropriate metrics to characterize the system. Generically, mathematical models utilize a set of inputs to provide us with one or more outputs (Figure 3-1).
Figure 3-1 system.
Conceptualization of inflows and outflows in choke canyon reservoir
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As an example, consider the movement of water and phosphorus into Choke Canyon Reservoir, TX. We have inflows of water and the pollutants into the reservoir, which is visualized to consist of reservoir (water), plant and bottom sediment compartments. There are inflows of water and pollutants in and out of the system and the model uses this information to tell us what the water level in the lake will be or how phosphorus will distribute itself among different phases. So in environmental modeling we typically have to provide various chemical properties and site characteristics to the model in order for it to predict how the contaminant will distribute itself in space and time. Of course, the first-step will be to develop a tool (usually a computer program of some sort) that relates these inputs to outputs. I call the logic connecting the inputs to the outputs as the inference system and the complete set of inputs, outputs and associated inference schemes and computer programs as the mathematical model (see Figure 3-2).
Figure 3-2
Basic elements of a model Inputs, inference engine and outputs.
HOW ARE ENVIRONMENTAL SYSTEMS MODELS BUILT?
There are two different approaches that can be used to build fate and transport models. Based on the scientific basis, mathematical models can be broadly classified into two groups 1) Theory driven models and 2) Data driven models.
Theory Driven Models
Theory driven models utilize the almost immutable laws of physics to describe the behavior of an environmental system. In all cases, these models utilize the law of conservation of mass in form or another. The law of conservation of mass states that matter cannot be created or destroyed
20 Applied Environmental Systems Modeling (except in very rare cases) and provides a fundamental framework to track the movement of pollutant mass into and out of the system. In addition to mass conservation, the law of conservation of energy and the law of conservation of momentum are also utilized in some applications. As the laws of physics are universal in nature, a theory-driven environmental model built for one system can be transferred to another similar one as long as the necessary inputs are available at the latter system. In other words, once the system is described using the fundamental laws and the mathematical equations are programmed into a computer, analyzing another similar system entails obtaining data specific to this new system and plugging it into the computer to see how it behaves. For example, elevated levels of phosphorus in a lake causes excessive algae growth and diminishes the water quality of the lake and is of concern to water quality managers and engineers. Lake Corpus Christi and Choke Canyon reservoir are two man-made lakes built on the Nueces River to provide water supply to the City of Corpus Christi, TX. A theory-driven model can be developed for simulating phosphorus transport into and out the Choke Canyon reservoir (see Figure 3-1 for a conceptual depiction of the system). Such a model would use phosphorus concentrations in the lake as the state variable. The conservation of mass principle would be used to quantify how much phosphorus accumulates in the Choke Canyon Reservoir in different compartments. To compute this accumulation we need to know how much phosphorus comes into the lake from various sides and how much phosphorus exists from all sides. In addition, losses or gains to phosphorus due to plants within the lake must also be accounted as well. Also, as phosphorous does not come into the lake as a pure substance but is transported as a dissolved constituent of the water or is sorbed on to suspended sediments in water that enters and exits the lake. Therefore, is also required to track the water entering and exiting the lake from all sides. If it is difficult to measure all the water entering and exiting the lake, one may have to use energy or momentum balance expressions to figure out water movement into and out of the lake. Also, the conservation of mass can be utilized to study how algae flow in and out of the lake. Such a model would include biokinetic principles to characterize the uptake phosphorus during algae grow and release of phosphorus back into the lake upon their death. A more complex model could include uptake and release due to zooplankton grazing and death as well. At the first sight, the process for modeling phosphorus in Choke Canyon reservoir, as described above, may seem very difficult and entail a lot of work. But, once all the mass balance and any required energy and/or momentum balance expressions are derived and coded into the computer the
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accumulation of phosphorus in the lake (or the changes in phosphorus concentration over time) can be used to predict phosphorus and test out different land management practices for phosphorus reductions without having to resort to costly and probably risky experimentation. Another big advantage in building the model is that the same model (i.e., conservation expressions and the associated computer programs) can be used to estimate the accumulation of phosphorus in another lake (say Lake Corpus Christi) as well. Again, one would have to know how much water and phosphorus enter and exit the Lake Corpus Christi as well as how much phosphorus is exchanged with the plants in the lake. However, there is no need to develop new programs to carry out the computations. While the theory-based modeling approach, does appears to be cumbersome, it actually provides the highest level of flexibility and as such is pursued at length in this text. A wide range of environmental systems can be studied by mastering a few concepts related to mass and energy conservation. However, mathematical expressions for mass, energy and momentum conservation are in the form of ordinary and partial differential equations and as such advanced mathematical skills are required to set up and solve these models. In some instances, the underlying mathematical equations are nonlinear and as such cannot be solved using exact mathematical schemes. In such instances, numerical approximations have to be resorted to. Therefore some training and background in numerical analysis is essential as well. Most engineers and physical scientists are exposed to many of these mathematical and numerical skills in the early part of their undergraduate education. The Part B of this book discussed various mathematical techniques that should be part of the fate and transport modelers toolkit to develop theory driven models.
Data Driven Models
Data driven models are developed by correlating empirical observations of the state variable with one or more independent variables. For example, to study how phosphorus affects algae growth in the Choke Canyon reservoir, one could simultaneously measure the levels (concentrations) of algae and phosphorus at different times of the year. The measured data can be plotted and a relationship between phosphorus and algae can be established from this paired dataset using statistical regression. Modern day spreadsheets have routines for regression and as such there is no need for solving complicated mathematical expressions. Clearly, a chart between the two variables is simple to interpret and easy to understand as well. However, the data driven modeling approach suffers from several limitations. Firstly, the amount of algae in the lake does not just depend upon
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Figure 3-3
Theory and Data-driven approaches to fate and transport modeling.
phosphorus levels alone but also on other factors such as flows into and out of the lake, zooplankton in the lake, oxygen in the lake etc. Hence, a simple relationship between phosphorus and algae may not explain all the variations in the observed data. Secondly, even if the correlation between the two variables is high, it is not indicative of cause and relationships. effect Spurious correlations are common place in many statistical analyses when all the variables are not considered. In this case, both phosphorus and algae could increase in the lake due to increasing flowrate (i.e., an inflow into the lake brings in high levels of both phosphorus and algae). Finally, even if a statistically and conceptually valid relationship is established, the results cannot be transferred to another similar system. For example, a relationship between phosphorus and algae developed in Choke Canyon reservoir cannot (and should not) be used to predict algae levels in Lake Corpus Christi because these two lakes may differ in other characteristics that are not included in the statistical analysis. Despite these limitations, data driven models are easy to build and understand. They are best suited for situations where the analysis is restricted to one system. Data driven models are also useful when generalized theories are not available to explain a particular phenomenon. And obviously, a data driven model is useful when the required data are available or the costs required to collect the data are significantly lower that required to implement a theory driven model. Repeated application of data driven models in many similar systems may lead to the development of general theories about certain phenomenon which in turn can be
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incorporated into theory driven models. Data driven models can also be used to obtain preliminary insights before any mathematical models based on conservation principles are developed and can also be used to guide the development of more complex models. Sometimes, the output from the theory driven models may be so complex that it might be easier to extract the data and develop a simpler statistical model to explain the results to the general public. Thus, data driven and theory driven models can complement each other in the analysis of environmental systems. Hence, it is not advisable to be biased towards one form or another. From a practical standpoint, the choice of the model selected should be based on a variety of factors including, the most important being application needs, data availability, time and other project constraints. Data driven models are typically built using statistical techniques such as linear and non-linear regression. However, other advanced techniques such as neural networks and fuzzy set theory are being actively explored to develop data driven environmental models. Fuzzy logic models use a set of IF-THEN rules to map input-output correlations. Artificial neural networks emulate the structure and functioning of the brain to infer relationships between inputs and outputs as set of connected weights, while Genetic algorithms use the Darwinian concept of evolution (the survival of the fittest) to identify best possible relationship between a set of inputs and outputs. As data driven models cannot be easily generalized, we shall only pursue them cursorily in this text. The reader is referred to the textbooks by McBean and Rovers (1998) and Berthouex and Brown (2002) on using statistical methods for environmental modeling.
Further Classification of Mathematical Models
In addition to the scientific basis discussed above, mathematical models can be classified in many different ways as summarized in Table 3-1. For example, the transparency of the inference scheme has also been used to classify models. Mathematical models are described as black-boxes if the inference scheme is not at all transparent. Artificial neural networks and some statistical models fall into this category. On the other hand, white-box models are completely based on fundamental laws of physical sciences. Gray-box models are based on some simplified laws of physical sciences but also include some degree of empiricism in their formulation and are also sometimes referred to as conceptual models. As per these definitions, most environmental models would be of the gray-box type as certain fate processes can only be described empirically. Mathematical models can also be classified based on their intended use. Screening level models are used for preliminary environmental assessments.
24 Applied Environmental Systems Modeling Table 3-1: Model Categories and their Classification
Classification Basis
Scientific basis Transparency Solution Scheme Intended Usage Domain Characterization
Model Categories
Theory driven and data driven Black-box, gray-box and white-box Analytical, semi-analytical and numerical Linear, non-linear and combinatorial Screening level and refined Simulation and management Lumped-parameter (box) or distributed parameter Media-specific or multimedia Single phase or multiphase Steady-state or transient
Time Characterization
These models employ simple mathematical descriptions to quantify environmental processes and assume the underlying systems are fairly ideal (i.e., possess no variability). Despite their simplicity these models are quite useful, as they only require few inputs and can be easily set up and solved on spreadsheets or available mathematical software. In addition, these models provide valuable insights on how different environmental processes and phenomena affect the chemical concentration. While the estimates of screening level models may not be precise for detailed environmental analysis, they are quite useful to assess the relative behavior of different chemicals (hence the term screening). For example, a globe may not provide accurate estimates for distance between any two locations on the earth, but is sufficient to identify if San Diego, CA or Las Vegas, NV is closer to Corpus Christi, TX. Detailed (site-specific) environmental models utilize more complex (often non-linear) descriptions to characterize various environmental processes and phenomena and attempt to obtain the distribution of pollutants at a very high spatio-temporal resolution. This process evidently increases the complexity of the underlying mathematics and warrants the need for more sophisticated computer programs. Based on the intended use, mathematical models can also be classified as simulation models or management models. Simulation models are used to understand the behavior of pollutants within a given system while management models are used to identify appropriate environmental policies, remedial strategies or to carry out engineering design. Simulation and management models can vary significantly in their mathematical complexity. However, simpler mathematical conceptualizations (screening type models) are preferred as management models because they are easy to understand and implement with limited data.
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In most instances, the mathematical model is in the form of either one or more algebraic equations, ordinary differential equations or partial differential equations. I...