Several commercially available software tools can be used to perform Monte Carlo simulation. These tools can be stand-alone or used as add-ins to commonly used spreadsheet programs. Many software tools offer an option of different sampling methods, including random Monte Carlo simulation and variations of Latin Hypercube Sampling (LHS), which can produce 'smoother' looking model output distributions for sample sizes of only a few hundred samples. The disadvantage of using LHS is that one must decide ahead of time how many iterations to use. This is because two or more LHS simulations cannot be combined since they will use overlapping strata, leading to difficulties in interpreting results. In some cases, LHS can yield underestimates of the higher moments of PDFs, since the stratification method also can preclude clustering of very high or low values as can occur in random data sets. The overall suggestion is to use random Monte Carlo simulation as the default method, because it will give flexibility to continue a random simulation to larger and larger simulation sample sizes if necessary until the model output distribution converges14 .

14 Cullen and Frey (1999) provide more information on the comparison of LHS and Monte Carlo simulation (pp. 207-213).

The number of iterations can be determined either by setting the number of model runs, a priori, such as 10,000 and allowing the simulation to continue until reaching the set number, or by allowing the mean to reach a relatively stable point before terminating the simulation. For example, when the estimate for the 95 percent confidence range is determined to within ± 1%, then an adequately stable result has been found. This can be checked by plotting a frequency plot of the estimates of the emission. This plot should be reasonably smooth (see Figure 3.8).

Another alternative is to assess the precision of the current number of replicates based on the standard errors of the percentiles that were used to construct 95 percent confidence intervals. If the range of the confidence intervals for each percentile (2.5 and 97.5) is less than the reported precision, then the number of iterations should be adequate (e.g., emissions are reported values to a single digit after the decimal and the percentile confidence intervals are less than 0.1, such as 0.005). Therefore, the Monte Carlo percentile estimates are unlikely to change in the reported digits for other simulations with the same number of iterations.

Figure 3.8 Example frequency plots of the results of a Monte Carlo simulation

Figure 3.8 Example frequency plots of the results of a Monte Carlo simulation

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