In
spatial statistics
Spatial statistics is a field of applied statistics dealing with spatial data.
It involves stochastic processes (random fields, point processes), sampling, smoothing and interpolation, regional ( areal unit) and lattice ( gridded) data, poin ...
the theoretical variogram, denoted
, is a function describing the degree of
spatial dependence
Spatial analysis is any of the formal techniques which study entities using their topological, geometric, or geographic properties, primarily used in Urban Design. Spatial analysis includes a variety of techniques using different analytic appro ...
of a spatial
random field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...
or
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
. The semivariogram
is half the variogram.

For example, in
gold mining
Gold mining is the extraction of gold by mining.
Historically, mining gold from Alluvium, alluvial deposits used manual separation processes, such as gold panning. The expansion of gold mining to ores that are not on the surface has led to mor ...
, 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
The semivariogram
was first defined by Matheron (1963) as half the average squared difference between a function and a translated copy of the function 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
A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude. It is the simplest, oldest, and most widely used type of the various ...
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, differently, as the
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
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
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
, then the 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 to denote the semivariogram, and the symbol
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:
* 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.
* If the
covariance function
In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a dom ...
''C'' of a stationary process exists, it is related to variogram by
:
* If the
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
''V'' and
correlation function
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables ...
''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
:
* 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.
* The semivariogram is a symmetric function,
">math>(z_a,z_b),(z_c,z_d)taken from locations with separation
h \pm \delta only
(z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)need to be considered, as the pairs
(z_b,z_a),(z_d,z_c)">math>(z_b,z_a),(z_d,z_c)do not provide any additional information.
Variogram models
The empirical variogram cannot be computed at every lag distance
h and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some
geostatistical
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 petro ...
methods such as
kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
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
*:
\gamma(h)=(s-n)(1-\exp(-h/(ra)))+n 1_(h).
* The spherical variogram model
*:
\gamma(h)=(s-n)\left(\left(\frac-\frac\right)1_(h)+1_(h)\right)+n1_(h).
* The Gaussian variogram model
*:
\gamma(h)=(s-n)\left(1-\exp\left(-\frac\right)\right) + n1_(h).
The parameter
a has different values in different references, due to the ambiguity in the definition of the range. E.g.
a=1/3 is the value used in (Chiles&Delfiner 1999). The
1_A(h) function is 1 if
h\in A and 0 otherwise.
Discussion
Three functions are used in
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 pet ...
for describing the spatial or the temporal correlation of observations: these are the
correlogram
In the analysis of data, a correlogram is a chart of correlation statistics.
For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram.
If cross-correlation is plotted ...
, the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
, and the semivariogram. The last is also more simply called variogram.
The variogram is the key function in geostatistics as it will be used to fit a model of the temporal/
spatial correlation
In wireless communication, spatial correlation is the correlation between a signal's spatial direction and the average received signal gain.
Theoretically, the performance of wireless communication systems can be improved by having multiple anten ...
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
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
function. Note that the experimental variogram is an empirical estimate of the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
of a
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
. As such, it may not be
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of w ...
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.
Applications
The empirical variogram is used in
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 pet ...
as a first estimate of the variogram model needed for spatial interpolation by
kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
.
* Empirical variograms for the spatiotemporal variability of column-averaged
carbon dioxide
Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
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
In seismology, strong ground motion is the strong earthquake shaking that occurs close to (less than about 50 km from) a causative fault. The strength of the shaking involved in strong ground motion usually overwhelms a seismometer, forci ...
from
earthquake
An earthquakealso called a quake, tremor, or tembloris the shaking of the Earth's surface resulting from a sudden release of energy in the lithosphere that creates seismic waves. Earthquakes can range in intensity, from those so weak they ...
s. These models are used for
seismic risk
Seismic risk or earthquake risk is the potential impact on the built environment and on people's well-being due to future earthquakes. Seismic risk has been defined, for most management purposes, as the potential economic, social and environment ...
and loss assessments of spatially-distributed infrastructure.
Related concepts
The squared term in the variogram, for instance
(Z(\mathbf_1) - Z(\mathbf_2))^2, can be replaced with different powers: A ''madogram'' is defined with the
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y, and is a special case of the Lp distance fo ...
,
, Z(\mathbf_1) - Z(\mathbf_2), , and a ''rodogram'' is defined with the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the absolute difference,
, Z(\mathbf_1) - Z(\mathbf_2), ^.
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s based on these lower powers are said to be more
resistant to
outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s. They can be generalized as a "variogram of order ''α''",
:
2\gamma(\mathbf_1,\mathbf_2)=E\left Z(\mathbf_1)-Z(\mathbf_2)\^\alpha\right/math>,
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.
References
Further reading
* Cressie, N., 1993, Statistics for spatial data, Wiley Interscience.
* Chiles, J. P., P. Delfiner, 1999, Geostatistics, Modelling Spatial Uncertainty, Wiley-Interscience.
* Wackernagel, H., 2003, Multivariate Geostatistics, Springer.
* Burrough, P. A. and McDonnell, R. A., 1998, Principles of Geographical Information Systems.
Isobel Clark, 1979, Practical Geostatistics, Applied Science Publishers
* Clark, I., 1979, ''Practical Geostatistics'', Applied Science Publishers.
* David, M., 1978, ''Geostatistical Ore Reserve Estimation'', Elsevier Publishing.
* Hald, A., 1952, ''Statistical Theory with Engineering Applications'', John Wiley & Sons, New York.
* Journel, A. G. and Huijbregts, Ch. J., 1978 ''Mining Geostatistics'', Academic Press.
Glass, H.J., 2003, Method for assessing quality of the variogram, The Journal of The South African Institute of Mining and Metallurgy
External links
AI-GEOSTATS: an educational resource about geostatistics and spatial statistics
{{Commons category, Variogram
Geostatistics
Statistical deviation and dispersion
Spatial processes>Z(\mathbf_1)-Z(\mathbf_2), ^2\right\gamma(\mathbf_2,\mathbf_1).
* Consequently, the isotropic semivariogram is an
even function
In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain.
They are named for the parity of the powers of the ...
\gamma_s(h)=\gamma_s(-h).
* If the random field is
stationary and
ergodic
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies th ...
, the
\lim_ \gamma_s(h) = \operatorname(Z(\mathbf)) 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''
n: The height of the jump of the semivariogram at the discontinuity at the origin.
* ''sill''
s: Limit of the variogram tending to infinity lag distances.
* ''range''
r: 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
Generally, an empirical variogram is needed for measured data, because sample information
Z is not available for every location. The sample information for example could be concentration of iron in soil samples, or pixel intensity on a camera. Each piece of sample information has coordinates
\mathbf=(x,y) for a 2D sample space where
x and
y are geographical coordinates. In the case of the iron in soil, the sample space could be 3 dimensional. If there is temporal variability as well (e.g., phosphorus content in a lake) then
\mathbf could be a 4 dimensional vector
(x,y,z,t). For the case where dimensions have different units (e.g., distance and time) then a scaling factor
B can be applied to each to obtain a modified Euclidean distance.
Sample observations are denoted
Z(\mathbf_i)=z_i. Observations may be taken at
M total different locations (the
sample size
Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences abo ...
). This would provide as set of observations
z_1,\ldots,z_M at locations
\mathbf_1,\ldots,\mathbf_M. Generally, plots show the semivariogram values as a function of separation distance
h_k for multiple steps
k=1,\ldots. In the case of empirical semivariogram, separation distance interval
h_k \pm \delta is used rather than exact distances, and usually isotropic conditions are assumed (i.e., that
\gamma is only a function of
h and does not depend on other variables such as center position). Then, the empirical semivariogram
\hat(h \pm \delta) can be calculated for each
bin:
:
\hat(h_k \pm \delta):=\frac\sum_ , z_i-z_j, ^2
Or in other words, each pair of points separated by
h_k (plus or minus some bin width tolerance range
\delta) are found. These form the set of points
:
S_k=S(h_k \pm \delta) \equiv \
The number of these points in this bin is
N_k=, S_k, (the
set size
In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term ''cardinal ...
). Then for each pair of points
i,j, the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (
, z_i-z_j, ^2). These squared differences are added together and normalized by the natural number
N_k. By definition the result is divided by 2 for the semivariogram at this separation.
For computational speed, only the unique pairs of points are needed. For example, for 2 observations pairs
(z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)taken from locations with separation
h \pm \delta only
(z_a,z_b),(z_c,z_d)">math>(z_a,z_b),(z_c,z_d)need to be considered, as the pairs
(z_b,z_a),(z_d,z_c)">math>(z_b,z_a),(z_d,z_c)do not provide any additional information.
Variogram models
The empirical variogram cannot be computed at every lag distance
h and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some
geostatistical
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 petro ...
methods such as
kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
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
*:
\gamma(h)=(s-n)(1-\exp(-h/(ra)))+n 1_(h).
* The spherical variogram model
*:
\gamma(h)=(s-n)\left(\left(\frac-\frac\right)1_(h)+1_(h)\right)+n1_(h).
* The Gaussian variogram model
*:
\gamma(h)=(s-n)\left(1-\exp\left(-\frac\right)\right) + n1_(h).
The parameter
a has different values in different references, due to the ambiguity in the definition of the range. E.g.
a=1/3 is the value used in (Chiles&Delfiner 1999). The
1_A(h) function is 1 if
h\in A and 0 otherwise.
Discussion
Three functions are used in
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 pet ...
for describing the spatial or the temporal correlation of observations: these are the
correlogram
In the analysis of data, a correlogram is a chart of correlation statistics.
For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram.
If cross-correlation is plotted ...
, the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
, and the semivariogram. The last is also more simply called variogram.
The variogram is the key function in geostatistics as it will be used to fit a model of the temporal/
spatial correlation
In wireless communication, spatial correlation is the correlation between a signal's spatial direction and the average received signal gain.
Theoretically, the performance of wireless communication systems can be improved by having multiple anten ...
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
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
function. Note that the experimental variogram is an empirical estimate of the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
of a
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
. As such, it may not be
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of w ...
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.
Applications
The empirical variogram is used in
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 pet ...
as a first estimate of the variogram model needed for spatial interpolation by
kriging
In statistics, originally in geostatistics, kriging or Kriging (), also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging g ...
.
* Empirical variograms for the spatiotemporal variability of column-averaged
carbon dioxide
Carbon dioxide is a chemical compound with the chemical formula . It is made up of molecules that each have one carbon atom covalent bond, covalently double bonded to two oxygen atoms. It is found in a gas state at room temperature and at norma ...
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
In seismology, strong ground motion is the strong earthquake shaking that occurs close to (less than about 50 km from) a causative fault. The strength of the shaking involved in strong ground motion usually overwhelms a seismometer, forci ...
from
earthquake
An earthquakealso called a quake, tremor, or tembloris the shaking of the Earth's surface resulting from a sudden release of energy in the lithosphere that creates seismic waves. Earthquakes can range in intensity, from those so weak they ...
s. These models are used for
seismic risk
Seismic risk or earthquake risk is the potential impact on the built environment and on people's well-being due to future earthquakes. Seismic risk has been defined, for most management purposes, as the potential economic, social and environment ...
and loss assessments of spatially-distributed infrastructure.
Related concepts
The squared term in the variogram, for instance
(Z(\mathbf_1) - Z(\mathbf_2))^2, can be replaced with different powers: A ''madogram'' is defined with the
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y, and is a special case of the Lp distance fo ...
,
, Z(\mathbf_1) - Z(\mathbf_2), , and a ''rodogram'' is defined with the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the absolute difference,
, Z(\mathbf_1) - Z(\mathbf_2), ^.
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on Sample (statistics), observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguish ...
s based on these lower powers are said to be more
resistant to
outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s. They can be generalized as a "variogram of order ''α''",
:
2\gamma(\mathbf_1,\mathbf_2)=E\left Z(\mathbf_1)-Z(\mathbf_2)\^\alpha\right/math>,
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.
References
Further reading
* Cressie, N., 1993, Statistics for spatial data, Wiley Interscience.
* Chiles, J. P., P. Delfiner, 1999, Geostatistics, Modelling Spatial Uncertainty, Wiley-Interscience.
* Wackernagel, H., 2003, Multivariate Geostatistics, Springer.
* Burrough, P. A. and McDonnell, R. A., 1998, Principles of Geographical Information Systems.
Isobel Clark, 1979, Practical Geostatistics, Applied Science Publishers
* Clark, I., 1979, ''Practical Geostatistics'', Applied Science Publishers.
* David, M., 1978, ''Geostatistical Ore Reserve Estimation'', Elsevier Publishing.
* Hald, A., 1952, ''Statistical Theory with Engineering Applications'', John Wiley & Sons, New York.
* Journel, A. G. and Huijbregts, Ch. J., 1978 ''Mining Geostatistics'', Academic Press.
Glass, H.J., 2003, Method for assessing quality of the variogram, The Journal of The South African Institute of Mining and Metallurgy
External links
AI-GEOSTATS: an educational resource about geostatistics and spatial statistics
{{Commons category, Variogram
Geostatistics
Statistical deviation and dispersion
Spatial processes