Variogram

All the previous examples conclude the fact that Variogram analysis will clearly reveal all the length scales that characterize a surface under study. As it was stated before with RIMPAS technique, if several physical processes act on a surface producing different distinctive patterns, the Variogram method is able to identify the corresponding length scales.  

The same topographic patterns generated on metallic plates and were studied with RIMAPS technique (Figs. 6-9), are used here to give examples of the Variogram analysis. In all cases, the figures present (a) the log-log representation of the Variance 

Zinc-plate surfaces with different time of exposure to chemical etching, are shown in Figs. 15-17. Three distinctive patterns cover the whole surface under study in each case. In the example of a linear-type pattern of Fig. 15, Variogram analysis gives the width of the facets, the length of its different segments and the separation between them. 

The case of the pattern that results after chemically etching and resembles a combination of straight line-segments, is shown in Fig. 16. From a Bright Field 

Illumination image, Variogram detects the size of each segment that can be identify on the surface. 

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A sample is representative of a neighborhood measured by the range of correlation. For example, a soil sample could represent a circular area in the field centered at the sample site with a radius less than or equal to the zone of influence. This has always been intuitively obvious to the environmental scientist but now can be described statistically. The zone of influence is defined by the theoretical semi-variogram and is easily estimated from an empirical semi-variogram. 

Several different types of semi-variograms are useful. The spherical model with nugget, the random model, and the spherical model with no nugget are discussed below. 

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 ... 

Figure 3.23 The three basic variogram types increasing, flat and periodic. In the lower frame a weakly expressed period of approximately 5 lags can be observed superposed on an increasing variogram.

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... 

Figure 3.24 Principal variogram illustrating the three key parameters nugget effect [MPE, V(0)h range and sill.

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). 

The periodic variogram (fluctuations superposed on an increasing or a flat variogram). 

The principal variogram types are illustrated in Figure 3.23. When the basic variogram type has been identified, information on optimized 1-D sampling can be derived. The increasing variogram will be used as an example below. 

The variogram function is not defined for lag = 0 - which would correspond to extracting the exact same increment twice. Even though this is not physically possible, it is highly valuable to obtain information of the likely variance corresponding to this zero-point variability (i.e. what if it would have been possible to repeat the exact same physical sampling). The TOS identifies this variance as MPE (minimum practical error), which is also known as V(0). 

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.

This comparison revealed not only an inconvenient truth regarding the production process at site D, but also other surprising distinctions between the five sites/processes hitherto assumed to be operating with about equal reliability and efficiency. With reference to the basic decomposition features of any variogram (Figure 3.24), the following can be concluded from the results presented in Figure 3.26. 

Sites A and D experience TSE to such a magnitude that no process variation can be observed at all (MPE equals the sill - a flat variogram). This is caused either by a production process in complete control, site A, or by the least efficient process sampling within this particular corporation, site D, actually caused by a completely unacceptable sampling procedure. 

Figure 3.26 Variogram analysis of one identical product from six different production sites - from six different countries, each employing local raw materials. Six variograms allow for critical corporate comparison of process stability/quality versus sampling (see text for full details). Sill levels for all six cases are highlighted. Panels are labelled top to bottom as A, B, C (left column) and D, E, F (right column).

Comprehensive quality control could here be carried out at on unambiguous quantitative basis, with full site specificity, based on only six experimental variograms. 

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