17.10: Modeling

Essential to Know

• To understand environmental processes, we generally must create a conceptual or mathematical model of the system (or parts of the system) that describes the components of the system and their interactions. The parts are often called “cells,” “boxes,” or “compartments.”
• There are many types of interaction. Examples include heat transfer between the ocean and the atmosphere, and the effect of changes in food supply on the population of an animal species.
• Conceptualized models must be parameterized and tested. Parameterization includes applying data to each model cell and developing equations to describe the flow of the modeled parameter between each adjacent pair of cells. These equations must be based on the physical laws that apply to each interaction. Testing consists of running the model to see how it fits the measured data. Parameters are adjusted between tests until the model fits the data set.
• Models must be validated by being applied to a data set that was not used in testing the model. This is often difficult to do for models for which the environmental data sets are sparse or limited to short time scales.
• Sensitivity analysis is performed by changing model inputs or assumptions and observing the changes in output. This type of analysis can help identify the parameters or processes that most affect the system’s behavior and, thus, are important for field studies or monitoring.
• To facilitate model creation, interactions are often assumed to be linear. Interactions are linear if one component of a system varies in direct proportion to a change in the other. However, most, if not all, natural systems include many nonlinear interactions.
• The nonlinear nature of natural systems limits the degree to which models can be used to make accurate predictions of future changes in the modeled system. Models of systems that are nonlinear in nature can be used only to predict a range of likely future conditions, and these predictions become more uncertain with increasing time into the future.

Understanding the Concept

The Earth, oceans, atmosphere, and the biota that inhabit these environments are all intimately linked in a system governed by physical laws. Each atom interacts with many others according to these laws, and it is the sum of these interactions and the atoms themselves that constitute the system. No matter how much information we obtain or how many computers we construct and program, describing this system accurately in all its details is impossible. Therefore, we must create a highly simplified system, or model, that replicates the real system in as much detail and as accurately as possible. We can create this model system either conceptually in our minds or mathematically by listing the components of the system and the mathematical relationships between each connected pair of these components.

Models that represent the entire Earth can be constructed at only a very gross level of detail. Hence, we generally construct models of parts of the system (subsystems) that we believe to be somewhat distinct from the larger system. We then represent the rest of the system by inputs and outputs to the subsystem. The descriptions of plate tectonics, atmospheric and ocean circulation, and marine biology, as well as other discussions in this textbook, are conceptual models of the real world.

Many different types of interactions must be described to construct models of the environment. Each model must take into account different interactions depending on which subsystem is being modeled. For example, models of atmospheric circulation must include a representation of the processes of heat transfer between ocean and atmosphere, land surface and atmosphere, and sea ice and atmosphere, and a representation of the internal convective processes (CC3) due to differential heating of air masses at different locations. Similarly, models of the changes of fish populations over time must represent the effects of varying food supply and varying abundance of predatory species on the growth, survival, and reproduction rates of the species involved. Models, including the two just mentioned, can be simple or extremely complicated, especially if they represent many different processes within a subsystem.

One example of a highly simplified model, and of how it can be useful, is the model describing the possible effects of changing the rate of input of a contaminant to an estuary or other body of water that is illustrated in Figures CC8-1 and CC8-2. At the other extreme, in coupled ocean-atmosphere global climate models, which are used to investigate, among other things, the likely consequences of the greenhouse effect, the ocean is typically segmented into “cells” that are formed by division of the ocean depths into 20 to 60 horizontal layers, or slices, and subdivision of those slices into segments of 2° to 4° of latitude and longitude. The atmosphere is typically represented by up to 90 horizontal levels subdivided at 1° to 5° intervals of latitude and longitude. The result is models that can have several million to tens of millions of adjoining cells. For each interface between two cubic cells (6 per cell), the model must have equations, based either on observed data or on physical laws, that reflect the rate of transport of water, heat, water vapor, and other parameters between the two cells as the properties of the water and air in each cell change.

These massive global climate models can be run on only the fastest computers, which operate at more than a exaflop (a exaflop is a measure of computer speed and is equivalent to  a billion billion or 1018 floating point operations per second). Even with these fast computers, running a model through as few as 100 years of climate can take days to months of computer time, and the output data from such a model run can consist of tens to hundreds of terabytes. A terabyte is 1024 gigabytes, or 1,099,511,627,776 (more than a million million) bytes.

The development and application of models involves three fundamental steps, each of which must be revisited and modified as needed when new information is available. The first step is to conceptualize the system under study as a series of separate but interacting cells (often called “compartments” or “boxes”). Most often, these cells are spatial areas, but they can be, for example, individual species within an ecosystem. The diagrams of the phosphorus and nitrogen cycles in Figures 12-6 and 12-7 are models in which the cells are both spatial (positioned within the atmosphere, mixed layer, or deep layer) and representative of the different chemical forms (e.g., dissolved ammonia, nitrate, nitrite, dissolved organic nitrogen, nitrogen in animal tissues, nitrogen in bacteria and detritus). Figures 12-10, 12-11, 12-12 are models in which the cells are individual species in a food chain or food web.

In developing a conceptual model, it is important to know exactly what question is being asked, so that the model can be as simple as possible. For example, if we were investigating the fate of anthropogenic nitrogen compounds released to the ocean, the model in Figure 12-7 would be adequate if we were interested only in how much of the nitrogen was transported to the deep layer and how fast. However, a much more complicated (and computer-intensive) model would be needed if we were interested in how the nitrogen released from North America was distributed to each ocean basin.

The second step in the modeling process is parameterization. In this process, values of the modeled parameter (e.g., nitrogen concentration, biomass, or heat content) must be assigned to each cell, and equations must be written to express the flows of the parameter between each pair of adjacent cells. These flow equations can be complex if they are based on physical principles. For example, heat transfer between two ocean water cells is a combination of heat conduction, diffusion, and mixing across the cell boundary, in addition to water transport between the cells. Usually, a single equation is written for such flows on the basis of observed data, as the processes involved are too complex to express as physical laws. Often, the observed data are very limited or available only for a small number of cells, particularly in complex models such as global climate models. After parameterization, the model is tested by being run with the parameters identified to see if it reproduces the data. For example, the global climate model would be examined for how closely it reproduces changes in temperature data at locations for which real-world data are available. Next, the parameters of the model are adjusted to improve the match to the real-world data. The model may be tested and parameters adjusted in this way many times.

The third step in model development is validation to determine whether the model actually does represent the way the natural system functions. In this step, the model must be tested with a data set that is completely different from the data used for the parameterization step. This is important because the process of parameterization forces the model to fit the input data. Thus, the model parameters may represent only the input set of conditions rather than the underlying processes involved.

Validating some models is relatively easy. For example, global climate models can be parameterized with decades of climate data and then validated by the use of data from a completely different time period. This is possible because relatively detailed historical atmospheric and ocean climate data are available for a number of decades in the past. Validating other models, especially complex ecological models, is much more difficult because the amount and historical coverage of the existing data are often very limited, and parameterization requires using most or all of the available data. Models that cannot be properly validated may still be useful for many research studies, especially sensitivity analysis, which is described in the next paragraph, but they are inherently unreliable if used to extrapolate beyond the existing data set, and they cannot be relied upon for prediction.

The most familiar use of models is to predict future conditions, particularly weather and climate. However, the most important use of models is to determine the parameters or processes that most affect a system’s behavior and, thus, are the most important for field studies or monitoring. To do this, we subject the models to sensitivity analysis. Sensitivity analysis is a simple, albeit sometimes tedious, process of repeatedly running the model, each time making small changes in a different input parameter or combination of input parameters (the initial value in a cell, or the coefficients of the equations relating flow between cells). This analysis can reveal which input parameters produce substantial changes in the model outputs and which produce only small changes. The former are sensitive parameters that must be accurately determined (or formulated for a flow equation) because small errors will greatly affect the accuracy of the model’s representation of the real world. They are also probably the most important parameters in controlling change in the real-world system that is being modeled. As a result, the most sensitive parameters of a model are often the most important parameters to focus on in field research and monitoring programs.

Models that have been adequately tested and validated can be used as predictive tools. The best-known examples of such uses of models are the weather forecasts seen nightly on television, the predictions of the future paths of hurricanes, long-term (months) climate predictions that are of particular importance to farmers, and predictions of future climate change associated with the past and future releases of greenhouse gases. These predictions are all based on weather and climate models that have been parameterized, tested, and validated with extensive historical data. However, as we all know, the predictions are always somewhat general in nature and are often inaccurate.

For example, as hurricanes travel across the Caribbean, several days away from the U.S., the model predictions of the storm’s path are shown as a wedge, with the future position more and more uncertain the further out in time the prediction is. The predicted range of possible landfall times and locations of the storm is narrowed as the storm approaches, but the predicted impact point often shifts and is no longer even within the range of the impact areas predicted several days earlier. In addition, predictions of strengthening or weakening of the storm as it approaches over the ocean are often incorrect, and sometimes the storm intensity suddenly and unexpectedly increases or decreases. Similar observations could be made about other model predictions.

Why are models not better predictive tools? There are several reasons. First, the models, although very sophisticated, are not the real world, and they do not account for variability within the cells of the model. Model cells often represent a large segment of the oceans or atmosphere, within which there is heterogeneity and variability that is not taken into account in the model. Second, the equations describing flow between cells are “fitted” to the testing and validation data and do not precisely describe all the factors that can affect these flows. Third, and this is the most important reason, natural systems include many nonlinear processes, and thus, these systems are chaotic, as discussed in more detail in CC11.

Even the most sophisticated models cannot make detailed and accurate long-term predictions for chaotic systems. In these systems, models will at best be able to accurately predict a range of likely future conditions. This is indeed exactly what hurricane path predictions do. These predictions are generally quite accurate for approximately a 24-h period ahead of the prediction, but they become progressively less accurate as the predictions extend out in time. Similarly, weather predictions are often quite accurate up to about 24 h ahead, but they become much less accurate farther out in time. In fact, there is a theoretical limit to future predictions in chaotic systems. Because this limit is only several days for weather predictions, these predictions will never be accurate for more than a few days into the future.