4.3 Geostatistical interpolation methods
4.3.5 Authorized variogram models
The need for a variogram model comes from the fact that, for estimation purposes, a vari- ogram value is needed for distances for which the sample variogram does not have a value. The use of a model also guarantees the uniqueness of the covariance matrices for each dis- tance h used to calculate estimation weights. To attempt this condition of uniqueness, only positive definite covariance models must be used. The positive definite condition is as a guarantee that the variance of RV formed by a weighted linear combination of other RVs will be positive.
One way of satisfying the positive definiteness condition, is to use only a few covari- ance functions that are known to be positive definite. The basic variogram models can be conveniently divided into two types; those that reach a plateau or sill and those that do not. Variogram models of the first type are often referred to as transition models. The plateau they reach is called the sill and the distance at which they reach this plateau is called the range. Some of the transition models reach their sill asymptotically (like the Gaussian model). For such models, the range is arbitrarily defined and corresponds to a distance at which 95% of the sill is reached. In the group of transition models, the most common ones are the pure nugget, the spherical, the Gaussian and the exponential models.
Pure nugget model (nug)
As seen from sample variograms there is an obvious spatial discontinuity at the origin. While the variogram value for h = 0 is strictly 0, the variogram value at very small separation distances may be significantly larger than 0 giving rise to a discontinuity. We can model such discontinuity using a discontinuous positive definite transition model that is 0 when h is equal to 0 and 1 otherwise. This is the nugget effect model and its equation is given by:
γ(h) = (
0, h = 0 C0, h 6= 0
Spherical model (sph)
Perhaps the most commonly used variogram model because of its validity in 3D space, its well-marked range, and its ease of calculation (8). Its equation is:
γ(h) = (
C0+ C · (3h2a −12(ha)3), if h ≤ a
C0+ C, if h > a
Where a is the range. It has a linear behavior at small separation distances near the origin but flattens out at larger distances, and reaches the sill at a. In fitting this model to a sample variogram it is helpful to remember that the tangent at the origin reaches at the origin reaches the sill at about two thirds of the range. Up to the range distance correlation exists. The spherical variogram reaches its sill with zero derivative; this should be the statistical (a priori) variance.
Exponential model (exp)
Its equations are:
γ(h) = (
0, h = 0 C0+ (C − C0) · (1 − exp(−ha )), if h 6= 0
This model reaches its sill asymptotically, with the practical range a defined as the dis- tance at which the variogram value is 95 % of the sill. Like the spherical model, the expo- nential model is linear at very short distances near the origin, however it rises more steeply and then flattens out more gradually. It is helpful to remember that the tangent at the origin reaches the sill at about one fifth of the range.
Gaussian model (gauss)
The Gaussian model is a transition model that is often used to model extremely continuous phenomena. Its equation is
γ(h) = C0+ C · (1 − exp(
−h2
Like the exponential model, the Gaussian model reaches its sill asymptotically, and the parameter a is defined as the practical range or distance at which the variogram value is 95% of the sill. It has a parabolic behavior near the origin, and is the only model with an inflection point. This model is associated with a high regularity of variations. It is often used to model extremely continuous phenomena and is hardly encountered in the earth sciences (8).
Description of the variogram models
Throughout the present thesis a coding was adopted to describe semivariogram parameters. It follows, in some way, the sequence of parameters adopted in the GSLIB software (14). Nested variogram structures are joined with a + sign. Each variogram structure is initially described by an abbreviation of the model type; for example, nug stands for nugget or gauss for Gaussian. Beside to the model, comes the contribution of the corresponding structure. For example, nug0.5 indicates a nugget model with 0.5 variance.
If the model is anisotropic, the ranges of the semivariogram structure are specified for an azimuth angle with a maximum range and another direction for the minimum range. The direction with maximum continuity and range is called the principal range (R) and the or- thogonal direction is called the secondary range of continuity (r). With the use of a variogram rose graphic (37), the direction having the most continuity can be better identified.
The maximum range is indicated following an uppercase R and the minimum range after a lowercase r. For example the coding sph0.3(90)R1000r500 indicates that the structure in the 90 degrees azimuth has a range of 1000 meters, while the orthogonal direction has a range of 500 meters. When the structure is isotropic, only the maximum range is indicated with an R. For example, the coding: nug0.5+sph0.3R1000+sph0.2R5000 represent an isotropic semivariogram model with a nugget model and 2 nested structures.