A-46 
2. A sample of size 40 is selected from each of the 36 simulations at random loca¬ 
tions on the grid. Ordinary kriging is used to estimate the spatial surface for 
each simulation and from these 36 estimates of the monthly spatial surfaces, a 
CFD is computed. This is called the ‘estimated’ CFD. 
3. For each of the kriged monthly surfaces, 1000 conditional surfaces are simu¬ 
lated based upon the mean and variogram estimated from the sample data. The 
Cholesky decomposition is used to reconstruct the covariance structure indi¬ 
cated by the estimated variogram. The conditionally simulated surfaces were 
processed to yield 1000 estimates of the proportion of noncompliance. The 
1000x36 noncompliance values are used to compute 1000 CFDs, which are 
called the population of “conditionally simulated” CFDs. 
4. Each “rank position” of the monthly ordered proportions of noncompliance 
has 1000 values in this simulated population. To assess variability in the simu¬ 
lated population, graphs of the miniumum, the 2.5th percentile, the 50th 
percentile, the 97.5th percentile, and the maximum at each rank position are 
plotted to illustrate a 95% confidence envelop for the CFD (Figure 5.6). 
To test this procedure under various conditions, this basic simulation exercise was 
repeated varying the sample size and adding temporal and spatial trend to the simu¬ 
lation of the “true” population to reflect conditions more similar to real populations. 
Conditional Simulation Results 
The results of this simulation exercise are presented graphically. In Figure 5.6 the 
line 1 represents the CFD computed for the true population computed from the orig¬ 
inal data. The line 2 is the estimated CFD computed from kriging estimates based on 
samples from the true population. The line 3 lines represent the min and max of the 
1000 conditionally simulated CFDs. The two line 4s represent the 2.5 and 97.5 
percentiles of the 1000 conditionally simulated CFDs, which is the proposed 95 
percent confidence interval. The line 5 is the median of the 1000 CFD curves. 
Bias Assessment 
The results in Figure 5.6 are unusual in several respects. First note that the line 2 
shows the typical sample size bias for the CFD as described above (n=40). Relative 
to the true CFD (line 1) the estimated CFD is below line 1 for half the curve and 
above line 1 for the remainder. The first unusual feature is that the distribution of the 
conditionally simulated CFD curves is not centered on estimated CFD. In fact the 
estimated CFD is not completely within the bounds (min, max) of the conditionally 
simulated population. A surprising feature is that the median of the simulated popu¬ 
lation tracks fairly well with the true CFD (line 1). It is clear that the simulated CFD 
population is estimating something other than what is estimated by the estimated 
CFD (line 2). At the same time, it appears that the median of the simulated popula¬ 
tion is a good estimator of the true CFD and the proposed confidence bands (line 3) 
is reasonable confidence envelop about the true CFD. 
What follows is a heuristic explanation for why CFD computed from conditional 
simulations might be a better estimator of the true CFD than a CFD computed from 
appendix a 
The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
