Many data used inprospect appraisal are distributed in an area. A common problem is to estimate , e.g. porosity, at a proposed drilling location, on the basis of data from nearby wells. Normally it is a problem of contouring, if at all possible. In our context we would like to know, not only the best estimate of porosity, but also the uncertainty, or variance. Geostatistical methods allow to find a best estimate and its variance. In this way we obtain an expectation curve of porosity for a given location. A fairly simple statistical principle is at the root of these methods.

First we consider that a random variable **Z**, measured at a location **(X,Y)** can have a probability distribution of some shape. This distribution is usually described by a mean and a variance and some analytical expression of the density curve ("pdf"). Assume that there are **n** locations (X_{i},Y_{i}) where a value **Z _{i}** is measured.

Statistical theory tells us that the sum of two random variables has a variance equal to the sum of the variances of the individual random variables. Equally, but less obvious, the variance of the difference of two random variables is also the sum of the individual variances! This latter fact is used in geostatistics for a particular "local" variance estimation.
In our theoretical example of data **Z** on a map at various locations, we can measure a "local" estimate of the variance of Z, by calculating the squared difference of **Z _{i} - Z_{j}**.:

Here we assume that the two

By taking all the possible pairs of points on the map (i not equal to j) and making these calculations, estimates of the variance become available (

Taking the square root of the variance turns this graph into a standard deviation versus

It is often observed that the sigmagram does not start at zero variance at zero distance, but somewhere higher on the vertical axis. This is called the

where

A complication arises when there is

For a more complete explanation of the geostatistical techniques we recommend the book by Harbough et al. 1995, Chapter 7 and the electronic book by Clark and Harper (2000). A very good introduction is found at the Spatial analyst website, which shows the formulas required and discusses the various types of kriging. Also worth reading is the introductory article by Olea

The variogram looks like this:

Here the zone of influence, if any, is at least smaller than the smallest interpoint distance. Contouring is always possible with geo-fantasy, but does not help!

A similar problem of estimation occurs for a one-dimensional series of data, e.g. log measurements. Can we interpolate intermediate values or would we be fooling ourselves?
A variogram of one-dimensional data can give an insight. Here we have a variogram of 21 TOC (Total Organic Carbon) values measured in a well. The mean TOC value is about 7 %. The variance is 7.56. Interpoint distances vary from 3 to 166 meter depth. In this case the variogram suggests a range of 42 m and a negligable nugget of 0.1, or 1.4% of the variance. A clear case of autocorrelation that can be used to interpolate TOC values between the actual observations.

Hoewever, it may be that part of this autocorrelation is due to a trend in the data, the so-called

More elaborate use of this method is in simulating reservoir architecture.

The full theory of geostatistics is immensely complex. Here we have touched only on a few of the principles:

- The possibility to use data to construct a "sigmagram" with the purpose of checking on autocorrelation in the data. Note that this is just as valid and practical for data along a borehole (one-dimensional) as for data on maps (two-dimensional) and can be extended to three dimensions as well (e.g. mining or reservoir engineering).
- Use the range from the sigmagram, c.q. variogram, to see where data become independent from each other and use this information in sampling schemes.
- Use geostatistics programs ("kriging") to make estimates of the mean value and the standard deviation of a variable for a point on a map.
- Use geostatistics to simulate values on a map and study the uncertainty of those estimates by conditional simulation.