The question now is how to update those probability distributions using information from actual samples taken nearby. Classical statistical methods applied to data using random sampling do not account for spatial correlations in the data. In fact, as indicated above, most of those methods assume that the data are independent. The correlation between concentration values at nearby locations can be considered an additional source of information. This information is ignored by the classical model of independent measurements, but it is central to geostatistical models used in kriging.
Given a set of sample results at locations, zi , i = 1,...n, the kriging estimator of the result that would be expected at a nearby unsampled location is a weighted moving average of the sample results. The weights depend on the strength of the correlation of the concentrations at two locations as a function of the distance between them. A structural correlation model explicitly defines this correlation strength. Deriving this correlation model is an interim step in the kriging process. The parameters of this correlation model, along with the kriging results themselves, are also components in the eventual update of the prior distribution described in Updating the Soft Information using Hard Information. Currently, only an exponential correlation model may be used.
A full explanation of correlation modeling and geospatial methods, particularly ordinary kriging can be found in Overview of Geospatial Modeling, Spatial Correlation, Modeling Spatial Correlation, and Ordinary Kriging.
In the present case, we are interested in the probability distribution for p(z0), the probability that a measurement at a specific location, z0, in the survey unit will exceed the release criterion. Given the sample result at location zi, this probability is either zero or one, depending on the result of the measurement. That is, 
if the measurement at zi is less than the release criterion, and 
if the measurement at zi is greater than the release criterion. The measurement has been transformed into an indicator value. This transformation occurs for each sampled point ( I = 1 to n). A correlation model based on these transformed values is created and used with the ordinary kriging model to predict the probability distribution p(z0).
Typically, we are not just interested in a single specific location z0 but rather numerous unsampled locations usually on a regular grid. In particular, each unsampled location on the previous grid of prior probability and confidence values (:, F) described in the Bayesian Approach are important. Once the probability distribution p(z0) is estimated by kriging for each node on the grid, we are ready to update the corresponding : and F at each node.