Inspatial statistics the theoreticalvariogram, denoted${\displaystyle 2\gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)}$, is a function describing the degree ofspatial dependence of a spatialrandom field orstochastic process${\displaystyle Z(\mathbf{s})}$. Thesemivariogram${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)}$ is half the variogram.

For example, ingold 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.

Thesemivariogram${\displaystyle \gamma (h)}$ 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${\displaystyle h}$.^[1][2] Formally

${\displaystyle \gamma (h)=\frac{1}{2}{\iiint }_V{\left[f(M+h)-f(M)\right]}^{2}dM,}$

where${\displaystyle M}$ is a point in the geometric field${\displaystyle V}$, and${\displaystyle f(M)}$ is the value at that point. The triple integral is over 3 dimensions.${\displaystyle h}$ is the separation distance (e.g., in meters or km) of interest. For example, the value${\displaystyle f(M)}$ could represent the iron content in soil, at some location${\displaystyle M}$ (withgeographic coordinates of latitude, longitude, and elevation) over some region${\displaystyle V}$ with element of volume${\displaystyle dV}$.To obtain the semivariogram for a given${\displaystyle \gamma (h)}$, all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so theempirical variogram is used instead.

The variogram is twice the semivariogram and can be defined, differently, as thevariance of the difference between field values at two locations (${\displaystyle {\mathbf{s}}_1}$ and${\displaystyle {\mathbf{s}}_2}$, note change of notation from${\displaystyle M}$ to${\displaystyle \mathbf{s}}$ and${\displaystyle f}$ to${\displaystyle Z}$) across realizations of the field (Cressie 1993):

${\displaystyle 2\gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)=\text{var}\left(Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2)\right)=E\left[((Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2))-E[Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2)]{)}^2\right].}$

If the spatial random field has constant mean${\displaystyle \mu }$, this is equivalent to the expectation for the squared increment of the values between locations${\displaystyle {\mathbf{s}}_1}$ and${\displaystyle s_2}$ (Wackernagel 2003) (where${\displaystyle {\mathbf{s}}_1}$ and${\displaystyle {\mathbf{s}}_2}$ are points in space and possibly time):

${\displaystyle 2\gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)=E\left[{\left(Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2)\right)}^2\right].}$

In the case of astationary process, the variogram and semivariogram can be represented as a function${\displaystyle {\gamma }_s(h)=\gamma (0,0+h)}$ of the difference${\displaystyle h={\mathbf{s}}_2-{\mathbf{s}}_1}$ between locations only, by the following relation (Cressie 1993):

${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)={\gamma }_s({\mathbf{s}}_2-{\mathbf{s}}_1).}$

If the process is furthermoreisotropic, then the variogram and semivariogram can be represented by a function${\displaystyle {\gamma }_i(h):={\gamma }_s(h{e}_1)}$ of the distance${\displaystyle h=\Vert {\mathbf{s}}_2-{\mathbf{s}}_1\Vert }$ only (Cressie 1993):

${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)={\gamma }_i(h).}$

The indexes${\displaystyle i}$ or${\displaystyle s}$ 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${\displaystyle \gamma }$ is sometimes used for the variogram, which brings some confusion.^[3]

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

• The semivariogram is nonnegative${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)\ge 0}$, since it is the expectation of a square.
• The semivariogram${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_1)={\gamma }_i(0)=E\left((Z({\mathbf{s}}_1)-Z({\mathbf{s}}_1){)}^2\right)=0}$ at distance 0 is always 0, since${\displaystyle Z({\mathbf{s}}_1)-Z({\mathbf{s}}_1)=0}$.
• A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights${\displaystyle w_1,\dots ,w_N}$ subject to${\displaystyle \sum _{i=1}^N{w}_i=0}$ and locations${\displaystyle s_1,\dots ,s_N}$ it holds:

${\displaystyle \sum _{i=1}^N\sum _{j=1}^N{w}_i\gamma ({\mathbf{s}}_i,{\mathbf{s}}_j)w_j\le 0}$

which corresponds to the fact that the variance${\displaystyle var(X)}$ of${\displaystyle X=\sum _{i=1}^N{w}_{i}Z(x_i)}$ is given by the negative of this double sum and must be nonnegative.^{[disputed –discuss]}

• If thecovariance function of a stationary process exists, it is related to variogram by

  ${\displaystyle 2\gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)=C({\mathbf{s}}_1,{\mathbf{s}}_1)+C({\mathbf{s}}_2,{\mathbf{s}}_2)-2C({\mathbf{s}}_1,{\mathbf{s}}_2)}$

• If a stationary random field has no spatial dependence (i.e.${\displaystyle C(h)=0}$ if${\displaystyle h\ne 0}$), the semivariogram is the constant${\displaystyle var(Z(\mathbf{s}))}$ everywhere except at the origin, where it is zero.
• ${\displaystyle \gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)=E\left[|Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2){|}^2\right]=\gamma ({\mathbf{s}}_2,{\mathbf{s}}_1)}$ is a symmetric function.
• Consequently,${\displaystyle {\gamma }_s(h)={\gamma }_s(-h)}$ is aneven function.
• If the random field isstationary andergodic, the${\displaystyle \underset{h\to \mathrm{\infty }}{lim}{\gamma }_s(h)=var(Z(\mathbf{s}))}$ corresponds to the variance of the field. The limit of the semivariogram is also called itssill.
• 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 asnugget or nugget effect.

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

• nugget${\displaystyle n}$: The height of the jump of the semivariogram at the discontinuity at the origin.
• sill${\displaystyle s}$: Limit of the variogram tending to infinity lag distances.
• range${\displaystyle r}$: 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.

Generally, anempirical variogram is needed for measured data, because sample information${\displaystyle Z}$ 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${\displaystyle \mathbf{s}=(x,y)}$ for a 2D sample space where${\displaystyle x}$ and${\displaystyle y}$ 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${\displaystyle \mathbf{s}}$ could be a 4 dimensional vector${\displaystyle (x,y,z,t)}$. For the case where dimensions have different units (e.g., distance and time) then a scaling factor${\displaystyle B}$ can be applied to each to obtain a modified Euclidean distance.^[4]

Sample observations are denoted${\displaystyle Z({\mathbf{s}}_i)=z_i}$. Samples may be taken at${\displaystyle k}$ total different locations. This would provide as set of samples${\displaystyle z_1,\dots ,z_k}$ at locations${\displaystyle {\mathbf{s}}_1,\dots ,{\mathbf{s}}_k}$. Generally, plots show the semivariogram values as a function of sample point separation${\displaystyle h}$. In the case of empirical semivariogram, separation distance bins${\displaystyle h\pm \delta }$ are used rather than exact distances, and usually isotropic conditions are assumed (i.e., that${\displaystyle \gamma }$ is only a function of${\displaystyle h}$ and does not depend on other variables such as center position). Then, the empirical semivariogram${\displaystyle \hat{\gamma }(h\pm \delta )}$ can be calculated for each bin:

${\displaystyle \hat{\gamma }(h\pm \delta ):=\frac{1}{2|N(h\pm \delta )|}\sum _{(i,j)\in N(h\pm \delta )}|z_i-z_j{|}^2}$

Or in other words, each pair of points separated by${\displaystyle h}$ (plus or minus some bin width tolerance range${\displaystyle \delta }$) are found. These form the set of points${\displaystyle N(h\pm \delta )\equiv \{({\mathbf{s}}_i,{\mathbf{s}}_j):|{\mathbf{s}}_i,{\mathbf{s}}_j|=h\pm \delta ;i,j=1,\dots ,N\}}$. The number of these points in this bin is${\displaystyle |N(h\pm \delta )|}$. Then for each pair of points${\displaystyle i,j}$, the square of the difference in the observation (e.g., soil sample content or pixel intensity) is found (${\displaystyle |z_i-z_j{|}^2}$). These squared differences are added together and normalized by the natural number${\displaystyle |N(h\pm \delta )|}$. 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 [${\displaystyle (z_a,z_b),(z_c,z_d)}$] taken from locations with separation${\displaystyle h\pm \delta }$ only [${\displaystyle (z_a,z_b),(z_c,z_d)}$] need to be considered, as the pairs [${\displaystyle (z_b,z_a),(z_d,z_c)}$] do not provide any additional information.

The empirical variogram cannot be computed at every lag distance${\displaystyle h}$ and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However somegeostatistical methods such askriging 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

  ${\displaystyle \gamma (h)=(s-n)(1-\exp (-h/(ra)))+n{1}_{(0,\mathrm{\infty })}(h).}$

• The spherical variogram model

  ${\displaystyle \gamma (h)=(s-n)\left(\left(\frac{3h}{2r}-\frac{h^3}{2r^3}\right)1_{(0,r)}(h)+1_{[r,\mathrm{\infty })}(h)\right)+n{1}_{(0,\mathrm{\infty })}(h).}$

• The Gaussian variogram model

  ${\displaystyle \gamma (h)=(s-n)\left(1-\exp \left(-\frac{h^2}{r^{2}a}\right)\right)+n{1}_{(0,\mathrm{\infty })}(h).}$

The parameter${\displaystyle a}$ has different values in different references, due to the ambiguity in the definition of the range. E.g.${\displaystyle a=1/3}$ is the value used in (Chiles&Delfiner 1999). The${\displaystyle 1_A(h)}$ function is 1 if${\displaystyle h\in A}$ and 0 otherwise.

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Three functions are used ingeostatistics for describing the spatial or the temporal correlation of observations: these are thecorrelogram, thecovariance, and thesemivariogram. The last is also more simply calledvariogram.

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 theexperimental variogram that is a visualization of a possible spatial/temporal correlation and thevariogram model that is further used to define the weights of thekriging function. Note that the experimental variogram is an empirical estimate of thecovariance of aGaussian process. As such, it may not bepositive 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.

Theempirical variogram is used ingeostatistics as a first estimate of the variogram model needed for spatial interpolation bykriging.

• Empirical variograms for the spatiotemporal variability of column-averagedcarbon dioxide was used to determine coincidence criteria for satellite and ground-based measurements.^[4]
• Empirical variograms were calculated for the density of a heterogeneous material (Gilsocarbon).^[5]
• Empirical variograms are calculated from observations ofstrong ground motion fromearthquakes.^[6] These models are used forseismic risk and loss assessments of spatially-distributed infrastructure.^[7]

The squared term in the variogram, for instance${\displaystyle (Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2){)}^2}$, can be replaced with different powers: Amadogram is defined with theabsolute difference,${\displaystyle |Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2)|}$, and arodogram is defined with thesquare root of the absolute difference,${\displaystyle |Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2){|}^{0.5}}$.Estimators based on these lower powers are said to be moreresistant tooutliers. They can be generalized as a "variogram of orderα",

${\displaystyle 2\gamma ({\mathbf{s}}_1,{\mathbf{s}}_2)=E\left[{\left|Z({\mathbf{s}}_1)-Z({\mathbf{s}}_2)\right|}^{\alpha }\right]}$,

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.^[8]

When a variogram is used to describe the correlation of different variables it is calledcross-variogram. Cross-variograms are used inco-kriging.Should the variable be binary or represent classes of values, one is then talking aboutindicator variograms. Indicator variograms are used inindicator kriging.

• 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., 1978Mining 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.

• AI-GEOSTATS: an educational resource about geostatistics and spatial statistics
• Geostatistics: Lecture by Rudolf Dutter at the Technical University of Vienna

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