A method to combine non-probability sample data with probability sample data in estimating spatial means of environmental variables.

In estimating spatial means of environmental variables of a region from data collected by convenience or purposive sampling, validity of the results can be ensured by collecting additional data through probability sampling. The precision of the pi estimator that uses the probability sample can be increased by interpolating the values at the nonprobability sample points to the probability sample points, and using these interpolated values as an auxiliary variable in the difference or regression estimator. These estimators are (approximately) unbiased, even when the nonprobability sample is severely biased such as in preferential samples. The gain in precision compared to the pi estimator in combination with Simple Random Sampling is controlled by the correlation between the target variable and interpolated variable. This correlation is determined by the size (density) and spatial coverage of the nonprobability sample, and the spatial continuity of the target variable. In a case study the average ratio of the variances of the simple regression estimator and pi estimator was 0.68 for preferential samples of size 150 with moderate spatial clustering, and 0.80 for preferential samples of similar size with strong spatial clustering. In the latter case the simple regression estimator was substantially more precise than the simple difference estimator.

Mapping the probability of exceeding critical thresholds for cadmium concentrations in soils in The Netherlands.

The probability of exceeding critical thresholds of Cd concentrations in the soil was mapped at a national scale. The critical thresholds in soil were based on food quality criteria for Cd in crops or in organs of cattle (Bos taurus), and were calculated by inverting a regression model for the Cd concentration in the crop, with the Cd concentration in soil, soil organic matter (SOM) content, clay content, and pH as predictors. The probability of exceeding the critical threshold for Cd in soil per node of a 500- x 500-m grid was approximated by Monte Carlo simulation, using the estimated cumulative distribution functions (cdf) of SOM, clay, pH, and Cd as input. The cdfs were estimated by simple indicator kriging with local prior means. For SOM, clay, and pH, detailed maps of soil type and land use were used to define subregions with assumed constant local means of the indicators (a priori distributions). The cdfs were sampled by Latin hypercube sampling. We accounted for correlation between the actual and critical Cd concentrations in soil by drawing Cd values from cdfs conditional on SOM and clay. The estimated probability for grassland is negligible, even in areas with high Cd concentrations in soil, and for maize (Zea mays L.) land the probability is almost everywhere smaller than 5%. For arable soils, however, these probabilities commonly are larger than 5% when sugar beet (Beta vulgaris L.) or wheat (Triticum aestivum L.) is taken as a reference crop, and locally exceed 50%.