V. Pawlowsky-Glahn | Geology Reports

USGS2015

Cokriging of compositional balances including a dimension reduction and retrieval of original units

Compositional data constitutes a special class of quantitative measurements involving parts of a whole. The sample space has an algebraic-geometric structure different from that of real-valued data. A subcomposition is a subset of all possible parts. When compositional data values include geographical locations, they are also regionalized variables. In the Earth sciences, geochemical analyses are a common form of regionalized compositional data. Ordinarily, there are measurements only at data locations. Geostatistics has proven to be the standard for spatial estimation of regionalized variables but, in general, the compositional character of the geochemical data has been ignored. This paper presents in detail an application of cokriging for the modelling of compositional data using a method that is consistent with the compositional character of the data. The uncertainty is evaluated by a Monte Carlo procedure. The method is illustrated for the contents of arsenic and iron in groundwaters in Bangladesh, which have the peculiarity of being measured in milligrams per litre, units for which the sum of all parts does not add to a constant. Practical results include maps of estimates of the geochemical elements in the original concentration units, as well as measures of uncertainty, such as the probability that the concentration may exceed a given threshold. Results indicate that probabilities of exceedance in previous studies of the same data are too low.

Journal of the Southern African Institute of Minin↗

USGS2008

Kolmogorov-Smirnov test for spatially correlated data

The Kolmogorov-Smirnov test is a convenient method for investigating whether two underlying univariate probability distributions can be regarded as undistinguishable from each other or whether an underlying probability distribution differs from a hypothesized distribution. Application of the test requires that the sample be unbiased and the outcomes be independent and identically distributed, conditions that are violated in several degrees by spatially continuous attributes, such as topographical elevation. A generalized form of the bootstrap method is used here for the purpose of modeling the distribution of the statistic D of the Kolmogorov-Smirnov test. The innovation is in the resampling, which in the traditional formulation of bootstrap is done by drawing from the empirical sample with replacement presuming independence. The generalization consists of preparing resamplings with the same spatial correlation as the empirical sample. This is accomplished by reading the value of unconditional stochastic realizations at the sampling locations, realizations that are generated by simulated annealing. The new approach was tested by two empirical samples taken from an exhaustive sample closely following a lognormal distribution. One sample was a regular, unbiased sample while the other one was a clustered, preferential sample that had to be preprocessed. Our results show that the p-value for the spatially correlated case is always larger that the p-value of the statistic in the absence of spatial correlation, which is in agreement with the fact that the information content of an uncorrelated sample is larger than the one for a spatially correlated sample of the same size. ?? Springer-Verlag 2008.

Stochastic Environmental Research and Risk Assessm↗