Variogram on arvau

Wikipedia / Variogram

A variogram is the graphical representation of the spatial dependence between pairs of data points, commonly used in geostatistics and spatial statistics. The term is sometimes used synonymously with semivariogram, but the latter is also used by some authors to refer to half of a variogram, and should therefore be avoided. Likewise, the term semivariance can be misleading, since the values shown in a variogram are entire variances of observations at a given spatial separation (lag).

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. This explains why only a limited number of variogram models are used: most commonly, the linear, the spherical, the Gaussian, and the exponential models.

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. Samples taken far apart will vary more than samples taken close to each other.

Definition

. Formally

(with geographic coordinates, 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, differently, as the variance) across realizations of the field (Cressie 1993):

If the spatial random field are points in space and possibly time):

In the case of a stationary process between locations only, by the following relation (Cressie 1993):

If the process is furthermore isotropic only (Cressie 1993):

is sometimes used for the variogram, which brings some confusion.

Properties

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

, since it is the expectation of a square.
.
it holds:

is given by the negative of this double sum and must be nonnegative.

If the covariance function C of a stationary process exists, it is related to variogram by
If the variance V and correlation function c of a stationary process exist, they are related to semivariogram by
Conversely, the covariance function C of a stationary process can be obtained from the semivariogram and variance as
everywhere except at the origin, where it is zero.
The semivariogram is a symmetric function.
Consequently, the isotropic semivariogram is an even function.
If the random field is stationary and ergodic corresponds to the variance of the field. The limit of the semivariogram with increasing distance is also called its sill.
As a consequence the semivariogram might be non continuous only at the origin. The height of the jump at the origin is sometimes referred to as nugget or nugget effect.

Parameters

In summary, the following parameters are often used to describe variograms:

nugget: The height of the jump of the semivariogram at the discontinuity at the origin.
sill: Limit of the variogram tending to infinity lag distances.
range: The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.

Empirical variogram

for a 2D sample space can be applied to each to obtain a modified Euclidean distance.

total different locations (the sample size can be calculated for each bin:

) are found. These form the set of points

(the set size. By definition the result is divided by 2 for the semivariogram at this separation.

] do not provide any additional information.

Variogram models

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. Some important models are:

The exponential variogram model
The spherical variogram model
The Gaussian variogram model

). The indicator function and 0 otherwise.

Applications

The empirical variogram is used in geostatistics as a first estimate of the variogram model needed for spatial interpolation by kriging.

Empirical variograms for the spatiotemporal variability of column-averaged carbon dioxide was used to determine coincidence criteria for satellite and ground-based measurements.
Empirical variograms were calculated for the density of a heterogeneous material (Gilsocarbon).
Empirical variograms are calculated from observations of strong ground motion from earthquakes. These models are used for seismic risk and loss assessments of spatially-distributed infrastructure.

Related concepts

, can be replaced with different powers: A madogram is defined with the absolute difference, and a rodogram is defined with the square root. Estimators based on these lower powers are said to be more resistant to outliers. They can be generalized as a "variogram of order α",

in which a variogram is of order 2, a madogram is a variogram of order 1, and a rodogram is a variogram of order 0.5.

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 variograms are used in indicator kriging.