The experimental variogram

A dimensionless lag parameter, j, describes the normalized distance between any two increments  

The variogram function, V(j), is defined as Vi times the average squared difference in heterogeneity contributions between the sum of pairs of increments, as a function of j  

The proper setting for variographic analysis is a set of 60 representative increments (this is a minimnm requirement, 100 samples is always preferable if possible, the minimum number of increments ever snc-cessfnlly snpporting an interpretable variogram is 42), in order to cover well the specific process variations to be characterized. It is important to select a 0niin that is smaller than the most probable sampling frequency likely to be used in rontine process monitoring and QC. It will be the objective of the data analysis to 

Practical interpretation of the experimental variogram is the first item to address. The variogram level and form provide valuable information on the process variation captured, as well as the quality of the sampling procedure employed. There are only three principal variogram types encountered in process sampling, but many more combinations hereof  

The increasing variogram (the most often, normal variogram shape). 

---

Figure 3.25 Illustration of comprehensive variographic analysis. Top panels show original data series (left) and its rendition in heterogeneity contributions (right). Bottom panels, (left) the experimental variogram (with two auxilliary functions) (right) TSE estimations for the user-specified set of sampling rate, r, and Q.

The experimental variogram models constructed for each of the six cutpoints indicated that a simple omnidirectional variogram model was an appropriate descriptor of the continuity relationship for each of the six cutpoints. Figure 3 presents the variogram model appropriate for the 50 ppb cut point. 

Model Fitting When experimental variograms reveal the structure and distribution pattern in an ore body then for any further estimation, it is necessary to fit a mathematical model to experimental variograms, which are called theoretical variograms. These mathematical models will be used in Kriging estimation. Several predefined models (Linear, Spherical, Gaussian, etc.). 

An experimental variogram of estimated residuals Y (h) can then be calculated. However, this variogram differs from the underlying variogram of the true residuals, Y(h), and the bias is a function of the form of the estimator m ( x). In order to find Y(h) from Y (h), Y (h) is graphically compared with a set of YQ(h) defined from... 

The experimental isotropic variograms for each variable are shown in Fig. 4a (soil) and Fig. 4b (plants). To compare the semivariograms of each variable, the data are normalized (division of the semi variance by the sample variance). The main properties of each variogram are given in Table 2. Note that in contrast to Fig. 4 the model parameters in Table 2 originated from the original values. 

In the next step, a facies type has to be assigned to each finite-difference model cell within the simulation domain. For this task, the experimental histogram of the clusters (clusterl - clusterM) and the cluster variogram models are used to generate conditioned equiprobable three-dimensional realizations of the facies fields described by categorical variables. For the facies-based approach used here, the three-dimensional conditional sequential indicator simulation method (SIS) for categorical variables... 

A variogram model can be constructed as a sum of known positive-definite licit variogram functions called nested stmctures. Each nested stmcture explains a fraction of the variability. All nested stmctures together describe the total variability, a. Interactive software is typically used to fit a variogram model to experimental points in different directions. 

---