A-24 
were not observed. Accounting for uncertainty in variogram parameter estimation 
has commonly been explored using Bayesian methods (Diggle and Ribeiro 2006). 
3.2 IDW OVERVIEW 
The inverse distance weighting method that is currently used in the CFD approach 
has already been described. Hence, this section provides a short review of IDW’s 
technical details and a comparison of IDW to alternative interpolation methods. 
The IDW method is essentially a deterministic, non-statistical approach to interpo¬ 
lating a two or three dimensional space. As a result it lacks statistical rigor so that 
estimates of the prediction errors are not calculable without additional assumptions. 
Similar to kriging, IDW predicts a value () at an unobserved site, say at location .v 0 , 
using a weighted average of the N nearest observed neighbors (N specified by the 
modeler): 
N 
Y(s 0 ) = ^w,Y(s,) 
i=l 
where the weights, w„ are inversely related to the distance between locations s 0 and s, 
d(s 0 ,s ,)' 2 
W , =“N- f 
J=1 
d(s 0 ,Sj) is the Euclidean distance between locations s 0 and s,, and the denominator of 
the weight is to ensure that the weights sum to 1. The IDW is an exact interpolator 
in that the predicted values for observed locations are the observed values and the 
maximum and minimum values of the interpolated surface can occur only at 
observed sites. 
Recent research has compared IDW to other interpolation techniques, most notably 
variations in kriging (Table 3.1). The authors found that in some cases kriging was 
at least as good an interpolator as IDW and in some instances better. The non-para- 
metric techniques (splines and similar methods) were not as precise as kriging and 
IDW. The method used for comparison in virtually all of the research was some 
variant of cross-validation, a method where some data are kept aside and not used in 
the model estimation phase and then using the resulting model to predict values for 
the data kept aside. The predicted and observed values are then compared and a 
statistic is calculated that summarizes the differences between the two sets of values 
(observed and predicted). 
None of these studies used datasets with highly irregular edges such as are found in 
the Chesapeake Bay nor did they use any distance metric other than Euclidean 
distance. Whether one method is preferable to another in these more difficult situa¬ 
tions remains unexplored. 
appendix a 
The Cumulative Frequency Diagram Method for Determining Water Quality Attainment 
