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The function described here is applicable to quality and quantity parameters with a spatial character, like rainfall, temperature, evaporation, etc., but sampled at a number of stations (point measurements).
The two correlation functions, the correlation and the semivariance, give an indication of similarities in mechanisms causing a phenomenon to exhibit. Both functions use the co-ordinates of the selected series in their calculation together with the measured values.

The correlation and semivariance functions both make use the covariance. The covariance of two random variables is defined as the expectation of the product between the respective deviations from their mean. The sample estimator C(x~1~ x₁ ,x~2~ x₂ ) for the population covariance is computed from:

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The values for r(0) and a must be entered, these values are input values for the correlation distance function. The maximum inter-station distance is the maximum distance between stations for which the correlation will be calculated. This distance is also the maximum distance in the graph.

Semivariance

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An estimation of the semivariance can be made from the available n measurements _Zi_ with locations _X_ _~i~_ and _X_ _~j~_ . Each pair of measurements can be associated with a lag, or distance vector _h=X_ _~i~_ _-X_ _~j~_ . The pairs are then grouped in a limited number of lag classes in order to have a significant number of pairs in each class. The mean value of the semivariance of all pairs in a class with a mean distance _h_ _~m~_ , may be estimated of half the spatial variance _var\[Z\ X _i and X _j . Each pair of measurements can be associated with a lag, or distance vector h=X _i -X _j . The pairs are then grouped in a limited number of lag classes in order to have a significant number of pairs in each class. The mean value of the semivariance of all pairs in a class with a mean distance h _m , may be estimated of half the spatial variance var[Z(x)-Z(x+h)\]_ divided by the possible pairs of measurements in that distance class. The traditional non-parametric estimator is: !image009.gif!
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where:
g(h) = the estimated semivariance for the distance class _h_ ,
Z(x~i~ x_i ),Z(x~i~ x_i +h) = the measured values within a distance class _h_ .
n(h) = the number of pairs in the distance class _h_ . \\ The resulting plot is called an experimental semivariogram. The semivariogram shows the expected difference in value of two points against its distance. \\ Any found experimental semivariogram is only a reflection of the true semivariogram. Kitanidis \[1993\] gave some useful guidelines to obtain a reasonable .
The resulting plot is called an experimental semivariogram. The semivariogram shows the expected difference in value of two points against its distance.
Any found experimental semivariogram is only a reflection of the true semivariogram. Kitanidis [1993] gave some useful guidelines to obtain a reasonable semivariogram:

• use three to six intervals,
• make sure you have at least ten pairs in each interval,
• include more points at distances where the differences between calculated semivariances is larger. Especially for large values of h there may be significant differences between semivariances calculated from different subsets of data.
  In practice, the experimental semivariogram must be fitted into an idealised model of the semivariogram, a theoretical semivariogram. The idealised curves are defined as simple mathematical functions which relate

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• ? to h .
  The distance at which the maximum semivariance, also called the sill', C , is reached is called the range', a , of the phenomenon. The range characterises the zone of spatial dependency of the data. Almost all experimental semivariograms show an apparent discontinuity at the origin. This intercept, C ₀ , is called the nugget variance.
  Appropriate semivariogram models can be based on a linear fit, a spherical fit, an exponential fit, a Gaussian fit, a cubic fit or even a mathematical formula taking anisotropy into account.
  Spherical model :
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  Gaussian model :
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  Exponential model :
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  Power model :
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  The next step is crucial, the fitting of a smooth curve through the calculated values in order to express the semivariance by a mathematical formula. By iterative changing of the lag distance, in order to optimise the compilation of the semivariogram expressed in C ₀ , C and a , an optimum fit will be achieved. It should be noticed that there must be sufficient lag classes to obtain a reasonable estimate of the semivariogram.