The Long-Term Area Source Model
The ISC Long Term Area Source Model is based on the numerical integration algorithm for modeling area sources used by the ISC Short Term model, which is described in detail in Section 1.2.3. For each combination of wind speed class, stability category and wind direction sector in the STAR meteorological frequency summary, the ISC Long Term model calculates a sector average concentration by integrating the results from the ISC Short Term area source algorithm across the sector. A trapezoidal integration is used, as follows:
Where:
Pi = the sector average concentration value for the ith sector
S = the sector width
fij = the frequency of occurrence for the jth wind direction in the ith sector
,(2) = the error term - a criterion of ,(2) < 2 percent is used to check for convergence of the sector average calculation
P(2ij) = the concentration value, based on the numerical integration algorithm using Equation (1-58) for the jth wind direction in the ith sector
2ij = the jth wind direction in the ith sector, j = 1 and N correspond to the two boundaries of the sector.
The application of Equation (2-6a) to calculate the sector average concentration from area sources is an iterative process. Calculations using the ISC Short Term algorithm (Equation (1-58)) are initially made for three wind directions, corresponding to the two boundaries of the sector and the centerline direction. The algorithm then calculates the concentration for wind directions midway between the three directions, for a total of five directions, and calculates the error term. If the error is less than 2 percent, then the concentration based on five directions is used to represent the sector average, otherwise, additional wind directions are selected midway between each of the five directions and the process continued. This process continues until the convergence criteria, described below, are satisfied.
In order to avoid abrupt changes in the concentrations at the sector boundaries with the numerical integration algorithm, a linear interpolation is used to determine the frequency of occurrence of each wind direction used for the individual simulations within a sector, based on the frequencies of occurrence in the adjacent sectors. This "smoothing" of the frequency distribution has a similar effect as the smoothing function used for the ISC Long Term point source algorithm, described in Section 2.1.8. The frequency of occurrence of the jth wind direction between sectors i and i+1 can be calculated as:
Where:
Fi = the frequency of occurrence for the ith sector
Fi+1 = the frequency of occurrence for the i+1th sector
1i = the central wind direction for the ith sector
1i+1 = the central wind direction for the i+1th sector
2ij = the specific wind direction between 1i and 1i+1
fij = the interpolated (smoothed) frequency of occurrence for the specific wind direction 2ij
The ISCLT model uses a set of three criteria to determine whether the process of calculating the sector average concentration has "converged." The calculation process will be considered to have converged, and the most recent estimate of the trapezoidal integral used, if any of the following conditions is true:
1) if the number of "halving intervals" (N) in the trapezoidal approximation of the sector average has reached 10, where the number of individual elements in the approximation is given by 1 + 2N-1 = 513 for N of 10;
2) if the estimate of the sector average has converged to within a tolerance of 0.02 (i.e., 2 percent), for two successive iterations, and at least 2 halving intervals have been completed (a minimum of 5 wind direction simulations); or
3) if the estimate of the sector average concentration is less than 1.0E-10, and at least 2 halving intervals have been completed.
The first condition essentially puts a time limit on the integration process, the second condition checks for the accuracy of the estimate of the sector average, and the third condition places a lower threshold limit that avoids convergence problems associated with very small concentrations where truncation error may be significant.
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