In this paper we will demonstrate the use and efficiency of the bootstrap on a geologic problem. The tools of classical statistics are often not applicable because they strongly depend on certain conditions that are not fulfilled. Explicit mathematical formulas for standard errors and confidence intervals with respect to a parameter either require some specific (generally normal) distribution, or they do not exist at all. Hypothesis tests may also only be carried out if some conditions are satisfied. Using the bootstrap method one can simulate the unknown distribution of an arbitrary statistic by its bootstrap replicates; hence any characteristics (standard error, confidence intervals, and test significance levels) can be obtained through direct empirical calculations. We applied the bootstrap to the chemical composition data of rock samples from the Boda Claystone Formation, Hungary. First we investigated the distribution of 8 chemical components in a rock sample group of few elements, computing standard errors and confidence intervals for the mean, the standard deviation and the skewness of these distributions. Then two groups of rock samples from different sampling regions were compared using hypothesis tests.
The uncertainty in the semivariogram has hardly been investigated in previous geostatistical studies. This paper presents an efficient methodology of uncertainty assessment based on the bootstrap. By applying this computer-intensive statistical method one can easily simulate the distribution of the empirical semivariogram estimate for each lag. The lag-wise standard errors and confidence intervals of a given level can then be easily calculated from the bootstrap replicates. These estimations are valid in any situation when classical statistics fail. The bootstrap also provides a mathematical-statistical tool to decide whether the semivariogram reaches its maximum at a given lag or not. It leads directly to a simple determination of the range of influence. Effects beyond the range, such as the hole effect, can be explored with the same approach. The empirical semivariogram, supplied by measures of uncertainty, adequately describes the true spatial behavior of the studied variable. This universal method renders the customary theoretical semivariogram models obsolete.