C-8 
variance. Spatial data, similar to the type sampled in Chesapeake Bay water quality 
criteria assessments, often exhibit a property known as (positive) spatial dependence; 
observations closer together are more similar than those further away. This property 
is accounted for in the model by allowing £'(s) to contain a spatial correlation 
structure. 
Common distributional assumptions on £(s) include normality and log normality, 
although kriging can be based on other statistical distributions and data transforma¬ 
tions. Functions of a specific mathematical type (positive definite) represent the 
spatial correlation in £(s) and are assumed isotropic (correlation depends only on 
distance) or anisotropic (correlation depends on both distance and direction). Vari- 
ograms constitute another special type of mathematical function—closely related to 
spatial correlation functions—that are more often used to represent spatial correla¬ 
tion. In this case, and in many kriging applications, variograms and spatial 
correlation functions provide equivalent representations of spatial structure. For con¬ 
sistency, only the term “variogram” is used here in discussions of spatial structure. 
In the literature, Equation C-l is referred to as a universal kriging model. When 
covariates (the X’s) don’t influence interpolation of Y, the right hand side of model 
(Equation C-l) contains only the constant term/3 0 . The resulting model is called the 
ordinary kriging model. When the spatial structure (variogram) for the model (Equa¬ 
tion C-l) 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 result in spatial predictions that 
are linear in the data, statistically unbiased, and minimize mean squared prediction 
error—known as best linear unbiased predictions. The minimized mean squared 
prediction error is also a measure of prediction uncertainty. In practice, however, the 
spatial structure of the data remains unknown. The estimation of the spatial structure 
using the variogram function, therefore, is critical to kriging applications. 
To demonstrate let (y(sj), . . . v(s n )} represent a sample set of spatial data such as 
dissolved oxygen collected at a set of n spatial locations Sj, . . .s n . Assume this data 
set to be a realization of the ordinary kriging version of model. The primary step in 
kriging is variogram estimation with several methods available; the method of 
moments and statistical likelihood based are two of the more common. All of these 
methods are based on the sample data (y(Si), . . . y(s n )}. This process ends with a 
chosen variogram function and its parameter estimation, describing the shape and 
strength (rate of decay) of spatial correlation. A determination, also based on the 
sampled data, is made of whether the spatial structure is isotropic or anisotropic. The 
estimated variogram is then assumed known. Kriged interpolations and their inter¬ 
polated uncertainty at numerous locations are computationally straightforward to 
generate. 
The following describes some of the benefits and potential limitations of kriging for 
the Chesapeake Bay Program to use in criteria attainment assessment application 
(with some comparisons to the IDW approach of spatial interpolation outlined in the 
previous section). A primary benefit of kriging compared to IDW is that it is a statis¬ 
tical technique. Statistics (including kriging) can make inferences from sampled data 
even in the presence of uncertainty; the quantity and quality of the sample data are 
appendix c 
Evaluation of Options for Spatial Interpolation 
