A-36 
the vector a will be assumed to have expectation 0 and variance ^ a an< J 
each vector ^ will be assumed to have expectation 0 and variance 
i is the ordinal index for days and 
s is a vector valued ordinal for spatial location. 
Under this model, defines the correlation over time at the segment-day level and 
2/Ji defines correlation over space that occurs cell to cell within a day. 
Let C j(Sj) be a collection of threshold limits that define the acceptable criterion for 
the measured attribute. If Y j(sj) exceeds C j($j) in a cell, that cell is called degraded. 
The criterion is allowed to vary in both time and space so that in theory each Y j(Sj) 
might be compared to a unique C j(Sj).. It may vary over time because different 
levels of Y may be acceptable in different seasons. It may vary over space because 
different levels of Y may be acceptable in different salinity regimes so that even 
within a segment, C may be a function of salinity. As a rule, it is anticipated that 
C j(s'j) will be constant for regions of space and time such as salinity zones and 
seasons. 
Now convert the measured attribute Y j(Sj) to a Boolean response as follows 
TY i(sj) = I(Y jOj) > C j(Sj)) = 1 if Y j(Sj) > C s (Sj) Eqn 5.1.1.2 
= 0 otherwise 
Thus TY takes the value 1 when Y exceeds the threshold defined by C. Using TY, 
we summarize the state of a segment on one day as the fraction of that segment that 
is out of compliance 
Pi = (l/N)2” i TY l (s i ) 
Eqn 5.1.1.3 
The CFD that we wish to estimate is one minus the cumulative distribution function 
of the Pj’s. If P {i) represents the ordered values of the P,’s for any assessment period, 
then let 
G(p)-(l/M)]T i=| I(P,i)3p) Eqn 5 l lA 
G defines the CFD that if it were known would be used for an exact assessment. The 
cumulative distribution function is determined by the mean and variance of the ideal 
population. This population is defined with a spatial variance component and a 
temporal variance component. The final CFD shows the cumulative percent of time 
that a certain percent of space is below the criterion threshold. If the CFD shows that 
water quality in a segment is beyond the threshold for too much space and too much 
time, then the segment is classified as impaired. 
For one assessment period, G can be considered exact as defined above, but recog¬ 
nize that even this is only one observation of the many possible observations of G 
that could result from sampling different assessment periods. 
Assume for simplicity that Y is normal. If were 0 so that Y had constant expec¬ 
tation over time and if were of the form a 2 1 then each cell on each day would 
have constant probability of exceeding a constant value of C given by 1 - <J>(C) 
appendix a 
The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
