Variogram
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Three functions are used in geostatistics for describing the spatial or the temporal correlation of observations: these are the correlogram, the covariance and the semivariogram. The last is also more simply called variogram. The sampling variogram, unlike the semivariogram and the variogram, shows where a significant degree of spatial dependence in the sample space or sampling unit dissipates into randomness when the variance terms of a temporally or in-situ ordered set are plotted against the variance of the set and the lower limits of its 99% and 95% confidence ranges.
The variogram is the key function in geostatistics as it will be used to fit a model of the spatial/temporal correlation of the observed phenomenon. One is thus making a distinction between the experimental variogram that is a visualisation of a possible spatial/temporal correlation and the variogram model that is further used to define the weights of the kriging function. Note that the experimental variogram is an empirical estimate of the covariance of a Gaussian process. As such, it may not be positive definite and hence not directly usable in kriging, without constraints or further processing. This explains why only a limited number of variogram models are used like the linear, the spherical, the gaussian and the exponential models, to name only those that are the most frequently used.
When a variogram is used to describe the correlation of different variables it is called cross-variogram. Cross-variograms are used in co-kriging. Should the variable be binary or represent classes of values, one is then talking about indicator variograms. Indicator variogram is used in indicator kriging.
The experimental variogram is computed by measuring the mean-squared difference of a value of interest z evaluated at two points x and x+h. This mean squared difference is the semivariance and is assigned to the value h, which is known as the lag. A plot of the semivariance versus h is the variogram.
[edit] Controversy
In mathematical statistics, a set of n measured values gives df=n-1 degrees of freedom whereas the in situ or temporally ordered set gives df(o)=2(n-1) degrees for the first variance term. The variogram and semivariogram are both invalid measures for variability, precision and risk because the sum of squared differences between x and x+h is divided by n, the number of data in the set rather than by df(o)=2(n-1), the degrees of freedom for the first variance term of the ordered set.
[edit] References
- Burrough, P A and McDonnell, R A, 1998, Principles of Geographical Information Systems
- David, M 1978 Geostatistical Ore Reserve Estimation, Elsevier Publishing
- Journel, A G and Huijbregts, Ch J 1978 Mining Geostatistics, Academic Press
- Clark, I 1979, Practical Geostatistics, Applied Science Publishers
