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Details about Allocation Model Development
The model used in the fecal coliform case study is built around a pre-existing excel spreadsheet model called the "Pathogen Loading Estimation Tool". This tool estimates environmental loads of fecal coliform bacteria based on land use coverage and land use management characteristics. Impact coefficients are derived from input and output data from calibrated BASINS simulations (a full description of BASINS modeling simulations for the case study area can be downloaded). Optimization and Monte Carlo simulations are executed using excel addins: OptQuest and Crystal Ball 2000.
| Framework Application | Software |
|---|---|
| Framework | Excel Spreadsheets |
| Load Models | Pathogen Loading Estimation Tool |
| Impact Coefficients | BASINS |
| Optimizer | OptQuest (Excel Addin) |
| Monte Carlo Risk Simulations | Crystal Ball (Excel Addin) |
Impact Coefficients
Figure 1: How impact coefficients are calculated using daily concentration and load output from an environmental fate and transport model. Environmental model simulations are completed with and without a particular source load. Impact coefficients are equal to the difference in concentrations divided by the difference in the source load for each day. Probability density functions are fit to the population of daily impact coefficients.
In the case study, daily concentrations, with and without a particular source load, are obtained from BASINS simulations over a period of seven years. Impact coefficients for that source are calculated for each day by dividing the difference in concentration by the difference in load. Attempts are made to fit pre-defined probability density functions (pfd) (e.g., lognormal) to the sample of impact coefficients for each source for each month. If coefficients do not fit pre-defined pdfs, then customized pdfs are specified using impact coefficient histogram data. Figure 1 demonstrates how impact coefficients are derived for a POTW source.
The use of impact coefficients relies on the assumptions that
- the marginal impacts of any source load are not affected by other source loads and
- marginal impacts are constant with respect to instream concentrations.
These assumptions are likely to be acceptable for the range of pollutants likely to be encountered in TMDLs (exceptions might be concentrations near solubility, nutrient concentrations affected by biological growth, or multiple pollutant interactions such as dissolved solids and metals). The use of impact coefficients also assumes that coefficients are not correlated across sources. This third assumption may be violated in the case of nonpoint source pollutants, such as fecal coliform bacteria where impact coefficients from croplands and septic systems are likely to be correlated (i.e., both coefficients will be high during wet weather and low in dry weather). However, correlations between impact coefficients can be easily incorporated into the optimization process using regression equations or correlation matrices, and regression equations may even reduce simulation times.
In the consolidated optimization process (see figure 2), there is no need for links to external hydrologic models. A sample of impact coefficients are randomly drawn from the impact coefficient pdfs, using Monte Carlo simulations or other risk analysis applications available through addins such as Crystal Ball. Source loads are multiplied by the sample of impact coefficients to generate populations of concentrations, and mean and upperbound concentrations from this population are then be compared to criteria using optimizers such as OptQuest. The optimizer employs search allgorithms (e.g., evolutionary solver) to select a new set of values for decision variables if optimal conditions are not satisfied.
Figure 2: A flow chart explaining the relationship between the optquest optimizer, the Crystal Ball monte carlo simulator, loading models, and cost models in the consolidated modeling framework.