Though the processes influencing the spatial variation of a measured variable in the atmosphere obey physical laws, the number of forces acting on the measured variable can make it appear random. There is still, however, an underlying structure relative to its location. Such as variable is known as a regionalized variable and is best described using a random function. A single realization of a regionalized variable Z for all locations in x (x1, x2…xn) constitutes the random function which is defined as

Z(x) = µ + ε(x)

where µ is a stationary mean and ε(x) consists of both spatially autocorrelated residuals and a random noise component. The expected difference between Z at location x and x + h is zero

E [ Z(x) - Z(x+h) ] = 0

where h is a spatial separation vector. This means that the variance of Z measured at two locations in x is a function of distance h

var [ Z(x) - Z(x+h) ] = 2γ(h)

where the quantity 2γ(h) is the variogram.

Plotting the variogram reveals characteristics of the spatial nature of the random variable. Typically, as the spatial separation distance h increases, as does the variance until a point to where the variance levels off. This is known as the sill. It is at this point that spatial autocorrelation between samples dissipates. The distance h at which the sill occurs is known as the range. The range component of the variogram is important as it speaks to the spatial dependence of the random variable. Such information can lead to better informed sampling of the random variable, or used to assign weights for spatial interpolation.

References:

Cressie, N., 1993. Statistics for spatial data, New York, Wiley.

Burrough, P.A., McDonnell, R., McDonnell, R.A., & Lloyd, C.D. 2015. Principles of geographical information systems, New York, Oxford.