Variogram

For example, in gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples.

(with geographic coordinates of latitude, longitude, and elevation) over some region

, this is equivalent to the expectation for the squared increment of the values between locations

are points in space and possibly time): In the case of a stationary process, the variogram and semivariogram can be represented as a function

between locations only, by the following relation (Cressie 1993): If the process is furthermore isotropic, then the variogram and semivariogram can be represented by a function

[3] According to (Cressie 1993, Chiles and Delfiner 1999, Wackernagel 2003) the theoretical variogram has the following properties:

In the case of the iron in soil, the sample space could be 3 dimensional.

If there is temporal variability as well (e.g., phosphorus content in a lake) then

For the case where dimensions have different units (e.g., distance and time) then a scaling factor

Generally, plots show the semivariogram values as a function of sample point separation

In the case of empirical semivariogram, separation distance bins

are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that

can be calculated for each bin: Or in other words, each pair of points separated by

(plus or minus some bin width tolerance range

, the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (

These squared differences are added together and normalized by the natural number

For computational speed, only the unique pairs of points are needed.

and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above.

However some geostatistical methods such as kriging need valid semivariograms.

In applied geostatistics the empirical variograms are thus often approximated by model function ensuring validity (Chiles&Delfiner 1999).

Some important models are (Chiles&Delfiner 1999, Cressie 1993): The parameter

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 variogram is the key function in geostatistics as it will be used to fit a model of the temporal/spatial correlation of the observed phenomenon.

One is thus making a distinction between the experimental variogram that is a visualization 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.

, and a rodogram is defined with the square root of the absolute difference,

Estimators based on these lower powers are said to be more resistant to outliers.

[8] When a variogram is used to describe the correlation of different variables it is called cross-variogram.

Should the variable be binary or represent classes of values, one is then talking about indicator variograms.