Variogram
Encyclopedia
In spatial statistics the theoretical variogram 
is a function describing the degree of spatial dependence of a spatial random field
or stochastic process
. It is defined as the variance of the difference between field values at two locations 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
itself is called the semivariogram. In 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 variogram and semivariogram can be represented by a function
of the distance 
only (Cressie 1993):
The indexes
or 
are typically not written. The terms are used for all three forms of the function. Moreover the term variogram is sometimes used for semivariogram and the symbol 
for the variogram, which brings some confusion.
at locations 
the empirical variogram 
is defined as (Cressie 1993):
where
denotes the set of pairs of observations 
such that 
, and 
is the number of pairs in the set. (Generally an "approximate distance" 
is used, implemented using a certain tolerance.)
The empirical variogram is used in geostatistics
as a first estimate of the (theoretical) variogram needed for spatial interpolation by kriging
.
According (Cressie 1993) for observations
from a stationary
random field
the empirical variogram with lag tolerance 0 is an unbiased estimator of the theoretical variogram, due to
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
has different values in different references, due to the ambiguity in the definition of the range. E.g. 
is the value used in (Chiles&Delfiner 1999). The 
function is 1 if 
and 0 otherwise.
for describing the spatial or the temporal correlation of observations: these are the correlogram
, the covariance
and the semivariogram. The last is also more simply called variogram. The sampling variogram, unlike the semivariogram and the variogram, shows where a significant degree of spatial dependence in the sample space or sampling unit dissipates into randomness when the variance terms of a temporally or in-situ ordered set are plotted against the variance of the set and the lower limits of its 99% and 95% confidence ranges.
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 visualisation 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.
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 variogram is used in indicator kriging.
is a function describing the degree of spatial dependence of a spatial random fieldRandom field
A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
or stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
. It is defined as the variance of the difference between field values at two locations 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
itself is called the semivariogram. In case of a stationary processStationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
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
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
, then variogram and semivariogram can be represented by a function
of the distance only (Cressie 1993):The indexes
or are typically not written. The terms are used for all three forms of the function. Moreover the term variogram is sometimes used for semivariogram and the symbol 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:- The semivariogram is nonnegative
, since it is the expectation of a square. - The semivariogram
at distance 0 is always 0, since 
. - A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights
subject to 
and locations 
it holds:
which corresponds to the fact that the variance
of 
is given by the negative of this double sum and must be nonnegative. - 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.
- If the covariance functionCovariance functionIn probability theory and statistics, covariance is a measure of how much two variables change together and the covariance function describes the variance of a random variable process or field...
of a stationary process exists it is related to variogram by
For a non-stationary process the square of the difference between expected values at both points must be added:
- If a stationary random field has no spatial dependence (i.e.
if 
) the semivariogram is the constant 
everywhere except at the origin, where it is zero. -
is a symmetric function. - Consequently
is an even function. - If the random field is stationaryStationary processIn the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
and ergodic, the
corresponds to the variance of the field. The limit of the semivariogram is also called its sill.
Empirical variogram
For observations at locations the empirical variogram is defined as (Cressie 1993):
where
denotes the set of pairs of observations such that , and is the number of pairs in the set. (Generally an "approximate distance" is used, implemented using a certain tolerance.)The empirical variogram is used in geostatistics
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
as a first estimate of the (theoretical) variogram needed for spatial interpolation by kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
.
According (Cressie 1993) for observations
from a stationaryStationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
random field
Random field
A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real or integer valued "time", but can instead take values that are multidimensional vectors, or points on some manifold....
the empirical variogram with lag tolerance 0 is an unbiased estimator of the theoretical variogram, due to
Variogram parameters
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..
Variogram models
The empirical variogram cannot be computed at every lag distance and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some GeostatisticalGeostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
methods such as kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
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 exponential variogram model
- The spherical variogram model
- The Gaussian variogram model
The parameter
has different values in different references, due to the ambiguity in the definition of the range. E.g. is the value used in (Chiles&Delfiner 1999). The function is 1 if and 0 otherwise.Discussion
Three functions are used in geostatisticsGeostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
for describing the spatial or the temporal correlation of observations: these are the correlogram
Correlogram
In the analysis of data, a correlogram is an image of correlation statistics. For example, in time series analysis, a correlogram, also known as an autocorrelation plot, is a plot of the sample autocorrelations r_h\, versus h\, ....
, the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
and the semivariogram. The last is also more simply called variogram. The sampling variogram, unlike the semivariogram and the variogram, shows where a significant degree of spatial dependence in the sample space or sampling unit dissipates into randomness when the variance terms of a temporally or in-situ ordered set are plotted against the variance of the set and the lower limits of its 99% and 95% confidence ranges.
The variogram is the key function in geostatistics
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology,...
as it will be used to fit a model of the temporal/spatial correlation
Spatial Correlation
Theoretically, the performance of wireless communication systems can be improved by having multiple antennas at the transmitter and the receiver. The idea is that if the propagation channels between each pair of transmit and receive antennas are statistically independent and identically...
of the observed phenomenon. One is thus making a distinction between the experimental variogram that is a visualisation of a possible spatial/temporal correlation and the variogram model that is further used to define the weights of the kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
function. Note that the experimental variogram is an empirical estimate of the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
of a Gaussian process
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
. As such, it may not be positive definite and hence not directly usable in kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
, 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.
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 variogram is used in indicator kriging.