Variogram

 ( Geostatistics ) 

In spatial statistics the theoretical variogram, denoted $2\backslash gamma(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)$, is a function describing the degree of spatial dependence of a spatial random field or stochastic process $Z(\backslash mathbf\{s\})$. The semivariogram $\backslash gamma(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)$ is half the variogram.

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.

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Definition The semivariogram $\backslash gamma(h)$ 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 $h$. Formally

$\backslash gamma(h)=\backslash frac\{1\}\{2\}\backslash iiint\_V\; \backslash leftf(M+h)^2dM,$

where $M$ is a point in the geometric field $V$, and $f(M)$ is the value at that point. The triple integral is over 3 dimensions. $h$ is the separation distance (e.g., in meters or km) of interest. For example, the value $f(M)$ could represent the iron content in soil, at some location $M$ (with geographic coordinates of latitude, longitude, and elevation) over some region $V$ with element of volume $dV$. To obtain the semivariogram for a given $\backslash gamma(h)$, 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 of the difference between field values at two locations ($\backslash mathbf\{s\}\_1$ and $\backslash mathbf\{s\}\_2$, note change of notation from $M$ to $\backslash mathbf\{s\}$ and $f$ to $Z$) across realizations of the field (Cressie 1993):

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

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

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

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

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

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

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

The indexes $i$ or $s$ 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 $\backslash gamma$ is sometimes used for the variogram, which brings some confusion.

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

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

:$\backslash sum\_\{i=1\}^N\backslash sum\_\{j=1\}^N\; w\_\{i\}\backslash gamma(\backslash mathbf\{s\}\_i,\backslash mathbf\{s\}\_j)w\_j\; \backslash leq\; 0$

which corresponds to the fact that the variance $\backslash operatorname\{var\}(X)$ of $X=\backslash sum\_\{i=1\}^N\; w\_i\; Z(x\_i)$ 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

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

• If the variance V and correlation function c of a stationary process exist, they are related to semivariogram by

$\backslash gamma(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)=V(1\; -\; c(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2))$

• Conversely, the covariance function C of a stationary process can be obtained from the semivariogram and variance as

$C(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)=V-\backslash gamma(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)$

• If a stationary random field has no spatial dependence (i.e. $C(h)=0$ if $h\backslash not=\; 0$), the semivariogram is the constant $\backslash operatorname\{var\}(Z(\backslash mathbf\{s\}))$ everywhere except at the origin, where it is zero.
• The semivariogram is a symmetric function, $\backslash gamma(\backslash mathbf\{s\}\_1,\backslash mathbf\{s\}\_2)=E\backslash left|Z(\backslash mathbf\{s\}\_1)-Z(\backslash mathbf\{s\}\_2)|^2\backslash right=\backslash gamma(\backslash mathbf\{s\}\_2,\backslash mathbf\{s\}\_1)$.
• Consequently, the isotropic semivariogram is an even function $\backslash gamma\_s(h)=\backslash gamma\_s(-h)$.
• If the random field is stationary and ergodic, the $\backslash lim\_\{h\backslash to\; \backslash infty\}\; \backslash gamma\_s(h)\; =\; \backslash operatorname\{var\}(Z(\backslash mathbf\{s\}))$ 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.

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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.

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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 $\backslash mathbf\{s\}=(x,y)$ 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 $\backslash mathbf\{s\}$ 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 $Z(\backslash mathbf\{s\}\_i)=z\_i$. Observations may be taken at $M$ total different locations (the sample size). This would provide as set of observations $z\_1,\backslash ldots,z\_M$ at locations $\backslash mathbf\{s\}\_1,\backslash ldots,\backslash mathbf\{s\}\_M$. Generally, plots show the semivariogram values as a function of separation distance $h\_k$ for multiple steps $k=1,\backslash ldots$. In the case of empirical semivariogram, separation distance interval $h\_k\; \backslash pm\; \backslash delta$ is used rather than exact distances, and usually isotropic conditions are assumed (i.e., that $\backslash gamma$ is only a function of $h$ and does not depend on other variables such as center position). Then, the empirical semivariogram $\backslash hat\{\backslash gamma\}(h\; \backslash pm\; \backslash delta)$ can be calculated for each Data binning:

$\backslash hat\{\backslash gamma\}(h\_k\; \backslash pm\; \backslash delta):=\backslash frac\{1\}\{2N\_k\}\backslash sum\_\{(i,j)\backslash in\; S\_k\}\; |z\_i-z\_j|^2$

Or in other words, each pair of points separated by $h\_k$ (plus or minus some bin width tolerance range $\backslash delta$) are found. These form the set of points

The number of these points in this bin is $N\_k=|S\_k|$ (the set size). 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 $(z\_a,z\_b),(z\_c,z\_d)$ taken from locations with separation $h\; \backslash pm\; \backslash delta$ only $(z\_a,z\_b),(z\_c,z\_d)$ need to be considered, as the pairs $(z\_b,z\_a),(z\_d,z\_c)$ do not provide any additional information.

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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 geostatistics 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 exponential variogram model
• : $\backslash gamma(h)=(s-n)(1-\backslash exp(-h/(ra)))+n\; 1\_\{(0,\backslash infty)\}(h).$
• The spherical variogram model
• : $\backslash gamma(h)=(s-n)\backslash left(\backslash left(\backslash frac\{3h\}\{2r\}-\backslash frac\{h^3\}\{2r^3\}\backslash right)1\_\{(0,r)\}(h)+1\_\{[r,\backslash infty)\}(h)\backslash right)+n1\_\{(0,\backslash infty)\}(h).$
• The Gaussian variogram model
• : $\backslash gamma(h)=(s-n)\backslash left(1-\backslash exp\backslash left(-\backslash frac\{h^2\}\{r^2a\}\backslash right)\backslash right)\; +\; n1\_\{(0,\backslash infty)\}(h).$

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 indicator function $1\_A(h)$ is 1 if $h\backslash in\; A$ and 0 otherwise.

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Discussion Three functions are used in geostatistics 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 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.

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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 . These models are used for seismic risk and loss assessments of spatially-distributed infrastructure.

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Related concepts The squared term in the variogram, for instance $(Z(\backslash mathbf\{s\}\_1)\; -\; Z(\backslash mathbf\{s\}\_2))^2$, can be replaced with different powers: A madogram is defined with the absolute difference, $|Z(\backslash mathbf\{s\}\_1)\; -\; Z(\backslash mathbf\{s\}\_2)|$, and a rodogram is defined with the square root of the absolute difference, $|Z(\backslash mathbf\{s\}\_1)\; -\; Z(\backslash mathbf\{s\}\_2)|^\{0.5\}$. based on these lower powers are said to be more resistant to . They can be generalized as a "variogram of order α",

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

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.

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.

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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.

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External links

• 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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