A-23 
other statistical distributions and data transformations (Christenson et al. 2001). The 
spatial correlation in e(s) is represented by positive definite functions. These func¬ 
tions can be assumed isotropic where correlation decay depends just on distance, or 
anisotropic where correlation decay depends on distance and direction. Variograms 
are another special type of mathematical function closely related to spatial correla¬ 
tion functions that can and are more often used to represent spatial correlation. For 
purposes here and in many kriging applications, variograms and spatial correlation 
functions provide equivalent representations of spatial structure. For consistency in 
what follows only the term variogram will be used in discussions of spatial structure. 
While there is considerable flexibility in implementing the error structure of a 
kriging model, it is possible to generalize somewhat with respect to the error struc¬ 
ture of Chesapeake Bay water quality data. Of the three water quality parameters 
being assessed, chlorophyll and clarity measures tend to follow the log-normal 
distribution and dissolved oxygen is reasonably approximated by the normal distri¬ 
bution. The horizontal decay rate of spatial correlation does not tend to be 
directionally dependent. Thus if the bay is viewed as a composite of horizontal 
layers, isotropic variograms are appropriate for kriging each layer. In a vertical direc¬ 
tion, water quality can change rapidly and thus spatial correlation can decay over a 
short distance. A 3-D interpolation procedure would benefit from use of an 
anisotropic variogram in order to differentiate the vertical correlation decay from the 
horizontal correlation decay. 
Note, in the literature model (1) is referred to as a universal kriging model. When 
covariates (the X’s) are not considered to influence interpolation of Y the right hand 
side of model (1) contains just the constant term /J 0 and e(s). The resulting model is 
referred to as the ordinary kriging model. When the spatial structure (variogram) for 
model (1) is known, statistically optimal predictions for the variable Y at unsampled 
locations (outside of estimation of possible regression effects) can be derived using 
standard statistical principles. The optimality criteria results in spatial predictions 
that are linear in the data, statistically unbiased, and minimize mean squared predic¬ 
tion error, hence referred to as best linear unbiased predictions (BLUPs). The 
minimized mean squared prediction error is also taken as a measure of prediction 
uncertainty. In practice, however, spatial structure of the data is unknown, the esti¬ 
mation of which via the variogram function is cornerstone to kriging applications. 
To demonstrate let {>’(si), . . . , y(s n )} represent a set of spatial data, for example a 
water-quality parameter such as dissolved oxygen sampled at a set of n spatial loca¬ 
tions S|, . . . , s n . Assume this data to be a realization of the ordinary kriging version 
of model (1). The first step in kriging is variogram estimation. There are several 
methods available, method of moments and statistical likelihood based being two of 
the more common, all of which though are based on the sample data {_y(sj), . . . , 
v(s n )}. Without going into detail, this process ends with a chosen variogram function 
and its parameter estimation, describing the shape and strength (rate of decay) of 
spatial correlation. There is also a determination, again based on the sampled data, 
of whether the spatial structure is isotropic or anisotropic. The estimated variogram 
is then assumed known and kriged interpolations and their interpolated uncertainty 
are computationally straight forward to generate at numerous locations where data 
appendix a 
The Cumulative Frequency Diagram Method tor Determining Water Quality Attainment 
