Variogram

The (theoretical) variogram $$2\gamma(x,y)$$ is a function describing the degree of spatial dependence of a spatial random field or stochastic process $$Z(x)$$. It is defined as the expected squared increment of the values between locations $$x$$ and $$y$$ (Wackernagel 2003):

$$2\gamma(x,y)=E\left(|Z(x)-Z(y)|^2\right)$$

$$\gamma(x,y)$$ itself is called semivariogram. In case of a stationary process the variogram and semivariogram can be represented as a function $$\gamma_s(h)=\gamma(0,0+h)$$ of the difference $$h=y-x$$ between locations only, by the following relation (Cressie 1993):

$$\gamma(x,y)=\gamma_s(y-x)$$

If the process is furthermore isotropic, then variogram and semivariogram can be represented by a function $$\gamma_i(h):=\gamma_s(h e_1)$$ of the distance $$h=\|y-x\|$$ only (Cressie 1993):

$$\gamma(x,y)=\gamma_i(h)$$

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 for semivariogram and the symbol $$\gamma$$ 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 $$\gamma(x,y)\geq 0$$, since it is the expectation of a square.
 * The semivariogram $$\gamma(x,x)=\gamma_i(0)=E\left((Z(x)-Z(x))^2\right)=0$$ at distance 0  is always 0, since $$Z(x)-Z(x)=0$$.


 * A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights $$w_1,\ldots,w_N$$ subject to $$\sum_{i=1}^N w_i=0$$ and locations $$x_1,\ldots,x_N$$ it holds:

$$\sum_{i=1}^N\sum_{j=1}^N w_{i}\gamma(x_i,x_j)w_j \leq 0$$

which corresponds to the fact that the variance $$var(X)$$ of $$X=\sum_{i=1}^N w_i Z(x_i)$$ 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 function of a process exists it is related to variogram by

$$2\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y)$$


 * If a stationary random field has no spatial dependence (i.e. $$C(h)=0$$ if $$h\not= 0$$) the semivariogram is the constant $$var(Z(x))$$ everywhere except at the origin, where it is zero.


 * $$\gamma(x,y)=E(|Z(x)-Z(y)|^2)=\gamma(y,x)$$ is a symmetric function.


 * Consequently $$\gamma_s(h)=\gamma_s(-h)$$ is an even function.


 * If the random field is stationary and ergodic, the $$\lim_{h\to \infty} \gamma_s(h) = var(Z(x))$$ corresponds to the variance of the field. The limit of the semivariogram is also called its sill.

Empirical Variogram
For observations $$z_i,\;i=1,\ldots,N$$ at locations $$x_1,\ldots,x_N$$ the empirical variogram $$\hat{\gamma}(h)$$ is defined as (Cressie 1993):

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

where $$N(h)$$ denotes the set of pairs of observation $$i,\;j$$ placed at an approximate distance of $$h$$. Here "approximate distance $$h$$" is not exactly defined and typically implemented by 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 $$z_i=Z(x_i)$$ from a stationary random field $$Z(x)$$ the empirical variogram with lag tolerance 0 is an unbiased estimator of the theoretical variogram, due to

$$E[\hat{\gamma}(h)]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}E[|Z(x_i)-Z(x_j)|^2]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}2\gamma(x_j-x_i)=\frac{2|N(h)|}{2|N(h)|}\gamma(h)$$

See the controversy discussion on the correctness of the scaling factor below.

Variogram Parameters
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 gets neglectable. The exact position, where the difference gets "neglectable" is quite imprecise.

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 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

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


 * The spherical variogram model

$$\gamma(h)=(s-n)\left(\left(\frac{3h}{2r}-\frac{h^3}{2r^3}\right)1_{(0,r)}(h)+1_{[r,\infty)}(h)\right)+n1_{(0,\infty)}(h)$$


 * The Gaussian variogram model

$$\gamma(h)=(s-n)\left(1-\exp\left(-\frac{h^2}{r^2a}\right)\right)+n1_{(0,\infty)}(h)$$

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

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 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 spatial/temporal 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 like the linear, the spherical, the gaussian and the exponential models, to name only those that are the most frequently used.

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.

The experimental variogram is computed by measuring the mean-squared difference of a value of interest z evaluated at two points x and x+'h'. This mean squared difference is the semivariance and is assigned to the value h, which is known as the lag. A plot of the semivariance versus h is the variogram.

Controversy
In mathematical statistics, a set of n measured values gives df=n-1 degrees of freedom whereas the in situ or temporally ordered set gives df(o)=2(n-1) degrees for the first variance term. The variogram and semivariogram are both invalid measures for variability, precision and risk because the sum of squared differences between x and x+h is divided by n, the number of data in the set rather than by df(o)=2(n-1), the degrees of freedom for the first variance term of the ordered set.