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Concept
Variogram
Summary
In spatial statistics the theoretical variogram, denoted , is a function describing the degree of spatial dependence of a spatial random field or stochastic process . The semivariogram is half the variogram.
In the case of a concrete example from the field of 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. Samples taken far apart will vary more than samples taken close to each other.
The semivariogram was first defined by Matheron (1963) as half the average squared difference between the values at points ( and ) separated at distance . Formally
where is a point in the geometric field , and is the value at that point. The triple integral is over 3 dimensions. is the separation distance (e.g., in meters or km) of interest.
For example, the value could represent the iron content in soil, at some location (with geographic coordinates of latitude, longitude, and elevation) over some region with element of volume .
To obtain the semivariogram for a given , all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the empirical variogram is used instead.
The variogram is twice the semivariogram and can be defined, equivalently, as the variance of the difference between field values at two locations ( and , note change of notation from to and to ) across realizations of the field (Cressie 1993):
If the spatial random field has constant mean , this is equivalent to the expectation for the squared increment of the values between locations and (Wackernagel 2003) (where and are points in space and possibly time):
In the case of a stationary process, the variogram and semivariogram can be represented as a function of the difference 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 of the distance only (Cressie 1993):
The indexes or are typically not written.
Official source
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