Uncertainty in prediction and interpretation of spatially variable data on soils

Abstract

Fuzzy set theory is a useful tool for dealing with uncertainty in the interpretation of quantitative information on soil properties, particularly when this is automated in an expert system or geographical information system (GIS). A fuzzy set of soil conditions suitable for some purpose may be defined on the universe of possible values of a soil property. A particular value of the property corresponds to a `membership' in the fuzzy set, determined by a membership function, which may be 1 (if such a soil is unambiguously suitable for the purpose), 0 (if it is unambiguously unsuitable) or some intermediate value when the complexity of the processes involved means that the interpretation is not clear cut. In practice when a user of a GIS asks a question about a site or regions of interest, the values of soil properties used to answer the query are likely to be estimates derived by methods of spatial prediction such as kriging from actual data held within the GIS. The estimate is obtained with error, so uncertainty of prediction (generally stated in statistical terms) and uncertainty of interpretation (as discussed above) both attend the output of the GIS. It is proposed here that the uncertainty of prediction and of interpretation for a soil property at a site may both be represented by a ‘weighted membership’ in a fuzzy set. This is calculated by dividing the range of values of the property into discrete intervals. The conditional probability that the true value of the property lies within each interval is estimated by disjunctive kriging. The weighted membership is obtained by summing over all intervals the product of the conditional probability and the value of the membership function corresponding to the value of the soil property central to the interval. If both the uncertainties of prediction and of interpretation are low at any location then the weighted membership will be close to 0 or 1. If the weighted membership takes a value nearer to 0.5 this indicates that the uncertainty of prediction and/or of interpretation prevent an unambiguous conclusion. The concept of weighted membership is illustrated using two data sets. It is concluded that it could be a useful way of combining the two sources of uncertainty in certain situations. However, in some situations it may be preferable to develop methods for using information on the two sources of uncertainty separately. Other possible methods for combining uncertainty of prediction and of interpretation may merit further research.