Problems of sampling in geoscience

There are two kinds of problems concerning the elements of a population : the first is to estimate the proportion of some specific kind of element in the population, e.g. the amount of gold in a block of ground (equals population) or the proportion of mercury in a sample; the second is to estimate the length, breadth, volume, etc., of a specific kind of element, e.g. the grain size of quartz grains in a beach sand or the size, shape and arrangement of lenses of ore in an ore deposit. Given a specific objective, a bounded population, identified elements and the element characteristics of interest, it is then necessary to select a sample of the elements from the population and the problem of sampling requires a decision on how to select the sample. It is also necessary to establish, either by heuristic analysis or empirically, based on long experience, the model frequency distribution of the desired element characteristics: in general, constant-probability models are used as models, the binomial or Poisson for count data and the normal distribution for measurements (or their transformed equivalents). The objective of sampling then becomes to obtain an appropriate statistical estimate of the desired population parameter: for example, the estimators may be the mean (X) and variance (a2): these should be unbiased sufficient estimators of their corresponding parameters (u, a2, respectively). Statistical tests are performed in order to determine that the estimators are of the appropriate kind: all statistical tests are designed to test against bias of a specific kind, although they are stated as tests against randomness. Given a random sample of some population, the estimators are usually stable and adequate to solve the sampling problem. Random samples are simple to define but exceedingly difficult to achieve in practice: in selecting a sample, if every element in the population has an equal chance of occurring, then the sample is a random sample. Interaction between the arrangement of elements in a population, i.e. its structure, and the sample selection process frequently introduces bias—that is, some elements have more chance than others of appearing in the sample and, hence, the sample is not random. An algorithm for dealing pragmatically with this aspect of the sampling problem offers one way of achieving random samples and, therefore, adequate estimators. Examples of the use of this algorithm on various kinds of geological populations, in both field and laboratory, illustrate the requirements for solving sampling problems and the achievement of appropriate estimators.