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#### Kim H. Esbensen^{a} and Claas Wagner^{b}

^{a}KHE Consulting, www.kheconsult.com^{b}Sampling Consultant—Specialist in Feed, Food and Fuel QA/QC. E-mail: cw@wagnerconsultants.com

After the previous column’s introduction to the *why*, the *how* and the *technicalities* involved in process sampling and variographic analysis, it is time for a bonanza of applications and case histories covering as broad a practical scope as possible. In this column, we introduce the critical *prerequisites* for the *variographic experiment*, by focusing on the importance of TOS-correct increment extraction for proper variographics. This issue cannot be overemphasised.

## Moving, or static, 1-dimensional lots: increment cutting must be TOS-correct

Figures 1 and 2 illustrate how focus is on the extension dimension in process sampling (aka one-dimensional sampling), as long as each increment complies with TOS’ stringent demand for a *complete slice* of the two width–height dimensions. By securing increments of this geometric configuration, there is only the extension dimension heterogeneity left, i.e. the longitudinal in-between increment spatial heterogeneity (DH = distributional heterogeneity). All variographic characterisation is aimed at describing, and managing DH_{process}.

If this demand is not observed, see Figures 3 and 4, it is clear how there will be a fundamental compositional imbalance (see incorrect sampling errors in earlier sampling columns) from one increment to another, which therefore should not be used for the purpose of characterising the 1-dimensional DH.

Correct planar–parallel or curvy–planar cross-sections of a moving stream is the only correct delineation of process sampling increments, Figure 5 (cases “A” and “C”) and Figure 6 top panel, eliminating a potential incorrect delineation error [one of the three potential incorrect sampling errors (ISE)].

Incorrect increment delineation, and extraction, will give rise to an inflated nugget effect in variographic process sampling characterisation (see below and the previous sampling column). Non-compliance with these basics will give rise to incorrect sampling errors (IDE; IEE: Incorrect Delineation Error; Incorrect Extraction Error) which are unnecessary and which can in fact be eliminated from the sampling process.

Manual increment extraction is nearly always a bad idea, and can never be TOS-correct in practice, see horrific examples in Figure 7. Whenever all Incorrect Sampling Errors (IDE, IEE, IPE, IWE) have not been eliminated, the sampling process will invariably be fraught with a fatal, inconstant sampling bias, which can be never be corrected for, see, for example, References 1 and 2.

There is actually no excuse for not getting the fundamental increment sampling right—and from the first time. Figure 8 shows three examples that are all completely TOS-correct, which is the first condition for sampling representativity.

Only representative increment sampling processes are of interest in science, technology and industry. All examples shown in Figure 8 allow proper variographic process characterisation. Any violation of these simple requirements affects a given sampling process and will lead to an inflated nugget effect, see further below.

Observe above how 1-dimensional lots of both types, static and dynamic (moving), must live up to the same demands concerning the fundamental increment cutting requirements.

## “Sooner or later” …

Sooner or later, however, we are ready to perform proper process sampling—enter *variographics*. The variogram was introduced in the previous column in some detail, so most of what is lacking is simply a basic understanding of what the variogram portrays with respect to the process, and how this comes about. A very small matter of a mathematical equation is all it takes: the professional sampler has to understand the meanings and implication of the variographic master equation, ...., all will be revealed in the next column.

For readers who have been inspired to know more about variographics, and who can’t wait, there is salvation in the two standard references^{1,2} as well as the new professional introduction published recently by Minnitt & Esbensen.^{3} This latter also takes you through the matheatical intricacies; well suited as a follow-up to the present columns.

This column has presented the critical issue of correct increment delineation and extraction in great detail (eliminating the otherwise fatal Incorrect Sampling Errors contributing to a sampling bias)—for a good reason. Full attention to these issues is absolutely necessary before embarking on the powerful variographic process characterisation. The next two columns are filled with practical case histories in which both benefits and throwbacks will be revealed.

## References

- K.H. Esbensen (Chairman Taskforce F-205 2010–2013),
*DS 3077. Representative Sampling—Horizontal Standard*. Danish Standards (2013). http://www.ds.dk - K.H. Esbensen and C. Wagner, “Theory of Sampling (TOS) versus Measurement Uncertainty (MU)—a call for integration”,
*Trends Anal. Chem. (TrAC)*57, 93–106 (2014). https://doi.org/10.1016/j.trac.2014.02.007 - R. Minnitt and K. Esbensen, “Pierre Gy’s development of the Theory of Sampling: a retrospective summary with a didactic tutorial on quantitative sampling of one-dimensional lots”,
*TOS forum*7, 7–19 (2017). https://doi.org/10.1255/tosf.96