The Theory of Geostatistics

• Geostatistics as we now know it has developed from Matheron’s (1963) coherent theoretical underpinning of Krige’s empirical observations.
• The spatial variation of most properties on, above or beneath the Earth’s surface is so complex that it led Matheron to find an alternative approach to the traditional deterministic one for their analysis.
• The approach he adopted was one that could deal with the inherent uncertainty of spatial data in a stochastic way.
• The basis of modern geostatistics is to treat the variable of interest as a random variable
• This implies that at each point, x, in space there is a series of values for a property, Z.(x), and the one observed, z(x), is drawn at random according to some law, from some probability distribution.
• At x, a property Z(x) is a random variable with a mean,μ, and variance, σ 2.
• The set of random variables, Z(x1), Z(x2), …, is a random process, and the actual value of Z observed is just one of potentially any number of realizations of that process.
• To describe the variation of the underlying random process, we can use the fact that the values of regionalized variables at places near to one another tend to be auto correlated.
• Therefore, we can estimate the spatial covariance to describe this relation between pairs of points; for a random variable this is given by

C(x₁, x₂) = E[{ Z(x₁)- μ(x₁)}{Z(x₂)- μ(x₂)}]

Where μ(x1) and μ (x2) are the means of Z at x1 and x2, and E denotes the expected value. As there is only ever one realization of Z at each point, this solution is unavailable because the means are unknown. To proceed we have to invoke assumptions of stationarity