A-29 
A third approach to interpolation in three dimensions is to implement 2-D interpola¬ 
tion in layers. Note that the IDW interpolator currently implemented by CBP 
(Section 2.2) employs a layered strategy by severely restricting (+/- 2m) the vertical 
distance that may be searched for nearest neighbors. A similar strategy could be 
implemented using 2-D kriging to interpolate the layers. Which of these approaches 
is best suited to 3-D interpolation for the bay will depend on the data available and 
assumptions related to vertical structure. Full 3-D kriging interpolation treats the 3rd 
dimension as a spatial dimension in the error term y ( s ). The covariate approach 
requires that the change over depth be a predictable trend. Interpolation in layers 
assumes that covariance decays so rapidly over depth that it is adequate to treat the 
layers as independent entities. Data sufficiency requirements increase for all 
approaches when considering three dimensional interpolations. When data are 
sparse, again a statistical based approach like kriging allows this to be reflected in 
prediction uncertainty. 
In many applications, attainment or lack of attainment will be so extreme that the 
assessment end point is clear even without optimizing the error estimation of the 
CFD. In these extreme cases, IDW or kriging simplified for automation could be 
sufficient to support the attainment ruling without precise quantification of estima¬ 
tion uncertainty. For these cases, the customized IDW algorithm that is currently 
implemented by CBP provides a tool with which to begin testing the CFD assess¬ 
ment procedure, but kriging simplified for automation may offer some advantages. 
Kriging can be simplified for automation by fixing the variogram model to one math¬ 
ematical form, say exponential, for all applications. With the variogram model fixed, 
kriging becomes like IDW in assuming the same mathematical form for the spatial 
dependence for all cases, but it is more flexible than IDW in that the rate of spatial 
correlation decay could be allowed to vary among applications. In addition, the 
simplified kriging opens the door for conditional simulation, with potential benefits 
that are discussed in Section 5. While a simplified kriging algorithm offers some 
advantages, there are also some potential drawbacks. Because variogram estimation 
typically entails use of an iterative procedure such as maximum likelihood or non¬ 
linear least squares, there is the potential that lack of convergence of these 
algorithms would be problematic for an automated implementation of kriging. 
In terms of computing, IDW is available in commercial GIS software, requiring GIS 
skills for application. Kriging is available in commercial statistical software and also 
in the free open source R Statistical Computing Environment (R Development Core 
Team 2005, Ribeiro and Diggle 2001) and requires programming skills for those 
software packages. 
In summary, kriging is more sophisticated than IDW, but requires greater expertise 
during implementation to fully exploit its full benefit. Table 3.2 provides a com¬ 
parison of the capabilities of assessments based simply on: 1) percent of samples, 
2) spatial interpolation based on IDW and 3) spatial interpolation based on kriging. 
appendix a 
The Cumulative Frequency Diagram Method tor Determining Water Quality Attainment 
