A-43 
grids of data were simulated to represent 36 months in a three year assessment 
period. The temporal and spatial trends were added to the simulated data by adding 
in means computed for each month and river kilometer during the period Jan 1, 1991 
to Dec 31, 1993. Simulated data were created using the “grf’ function of the Geosta- 
tistical Package “geoR" of the R-package. 
After the full population of data was simulated for 3 year assessment period, a 
sampling experiment was conducted to assess the effect of sample size on the shape 
of the CFD. First, as a benchmark, a CFD was computed using all of the simulated 
data. To simulate the effect of sampling, a sample of fixed size was randomly 
selected from each the 36 5x60 grids of data. Using these samples, kriging 
(krige.conv function of geoR) was used to populate each monthly grid with esti¬ 
mates. These estimated chlorophyll surfaces were used to compute an estimate of the 
CFD which was graphically compared to the benchmark (Figure 5.5). For a fixed 
sample size, the process was repeated until it was clear whether the differences 
between the benchmark CFD and the estimated CFDs were due to variance or bias. 
To assess the effect of sample size, the process was repeated for several sample sizes. 
The effect of sample size on the shape of the CFD is consistent with expectations 
based on the relation of the CFD to the empirical distribution function (Figure 5.5). 
As sample size decreases, the variance of the estimated values of fraction of space 
increases. This increase in variance results in the estimated CFD being to the left of 
the true curve for low values of fraction of space and to the right of the true curve 
for high values of fraction of space. This assessment has been repeated many times, 
varying the threshold criterion, systematic vs. random sampling, the level of vari¬ 
ability in the simulated data, and so on. This sample size effect persists for every case 
where realistic estimation is employed. 
Sampling Scale and Shape 
As shown above (Figures 5.2-5.4) the shape of the CFD is a function of the ratio of 
temporal and spatial variance. To the extent that the ratio of these variance compo¬ 
nents in the data represent the true state of nature, this is acceptable. However, under 
a model with strong spatial and temporal dependence, the ratio of these variance 
components might be influenced by the scale of sampling in the spatial and temporal 
dimensions. For example, samples collected far apart in time might reflect higher 
variance than samples collected close in time. If the ratio of temporal and spatial 
variance is influenced by the density of sampling in each dimension, then experi¬ 
mental design will have an effect on the asymmetry of the CFD estimate. 
5.5 CONFIDENCE BOUNDS AND STATISTICAL INFERENCE 
An investigation into the use of conditional simulation to obtain confidence bounds 
for the CFD showed that not only is this a promising technique for statistical infer¬ 
ence, but also has potential in correcting bias associated with sample size effects that 
has been identified as a central problem in implementing the CFD approach. 
Correcting the bias of the CFD due to the sample size effect is important in obtaining 
confidence bounds on the CFD that cover the true CFD for a segment. Because bias 
correction is an important first step, this aspect of the conditional simulation exper- 
appendix a 
The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
