Theory Of Sampling (TOS) short review

Sometimes grab sampling may, purely by chance, could be have the true chemical lot composition, but it is never possible to identify when this happens, and in any event the situations where it fails fatally simply dominate. At the outset of the present study it was considered likely that the current sampling approach is problematic and not providing an outcome consistent with reality. It was therefore considered a critical success factor to invoke the Theory of Sampling (TOS) (Gy 1998) as providing the framework in which the specific HHXRF evaluation should be carried out.

Fundamental Sampling principle (FSP)

All units of the lot (minerals, increments (see below)) must be fully and practically accessible by the sampling procedure in use. There cannot be any “restricted zones” in any lot from where sampling is not possible. Thus, TOS stipulates that the entire lot volume must be available for sampling; this is called the Fundamental Sampling Principle (FSP). Occasionally, in practice, the entity responsible for the use of the final analytical results, may decide that samples from the surface of stationary lot will suffice for a “fit-for-purpose” characterization. Such a decision will always be at the express behest of the user – and most emphatically not of TOS! In mining and minerals processing sectors there may be an opinion, or a widely accepted practical assumption, that surficial samples over the entire surface of a stationary lot will be able to characterize the entire, full volume lot. While this is a tacit assumption often found within these application fields, there is no guarantee, from TOS nor from any other reliable source, that this holds true for all lots in question, or to all successive lots in time. Whether such is the case is of course completely dependent upon the effective heterogeneity of the lot(s) in question. Perhaps prior knowledge is at hand that possibly could support continuation of such operative assumptions? In any case – indeed in every case – this could constitute a breach of due diligence the Fundamental Sampling Principle. For smaller lots, e.g. sub-samples compliance with FSP becomes more and more easy – it is very much easier to get full access to a 55kg sub-lot, or a 10-kg sub-sub-lot etc. than the whole lot who could be represent over than 40+ ton.

Lot 

TOS uses the term “lot” to designate the sampling target; i.e. a stockpile, a barrel, a truck or a railroad load or a geological outcrop. The meaning of the term lot is dependent upon the situation. All lots are made up by a specific material, occupies a specific geometrical volume and has a specific lot mass, density etc. The crucial common characteristic of all lots is heterogeneity, which makes sampling of however disparate and different types of lots, a matter of a focused and unified approach, the Theory of Sampling (TOS).

Composite sampling – increments 

TOS stipulates the mandate always to use composite sampling using a heterogeneitydetermined number of increments (Q). An increment designates an individual extracted unit (often defined by the volume of the sampling tool), i.e. a partial lot unit that, when combined with other such units, makes up a composite sample. The only free parameter in composite sampling is the necessary number of increments needed, Q, needed to result in fit-for-purpose sampling.

Sampling errors

TOS specifies five errors occurring during sampling stationary lots; there are two more, specific errors when sampling moving (dynamic) lots, (Esbensen and Julius, 2013; Minkkinen and Esbensen, 2009; Pitard, 2009). The first five errors are listed below, not in order of importance, but in their logical order in the sampling process.
1) The Fundamental Sampling Error (FSE)
2) The Grouping and Segregation Error (GSE)
3) The Increment Delimitation Error (IDE)
4) The Increment Extraction Error (IEE)
5) The Increment Preparation Error (IPE)

The last three sampling errors (IDE, IEE, IPE) are collectively termed the “Incorrect Sampling Errors” (ISE), which are sampling bias-generating errors. These are fatal sampling errors because, if not eliminated from the sampling process, they give rise to a fatal sampling bias. Full details in the TOS literature cited above. All these errors contribute to the Total Sampling Error total (TSE). To this could be added the Increment Weighing Error (IWE) but this is not directly involved in the HHXRF sensor sampling and in conventional sampling increment weight variability less than +/-20% is usually considered acceptable. The Fundamental Sampling Error (FSE) is irreducible without physical modifications of the material. The only way to reduce this error is to reduce the particle size (crushing, comminution). This error is inversely proportional to the realized mass of the sample, but increasing this by itself will not help much representative – any grab sample is still but a miniscule part of the entire lot – another reason for using composite sampling. The Grouping and Segregation Error (GSE) will be higher as a direct consequence of high heterogeneity; most often this is in the form of local “hot spots” (with either high, or lower, average concentration(s), grouping, and/or stratification, segregation. This error is also partially caused by temporal heterogeneity whenever transportation plays a role (i.e. transport induced vibration can displace smaller fragments towards the bottom of the pile). This error can be reduced, or sometimes almost canceled, by effective composite sampling and/or for smaller lots, by mixing. The Increment Delimitation Error (IDE) is occurring when one is not able to reproduce exactly the same geometric delineation of the increments being sampled. The extraction error (IEE) is the result of extracting increments with a sampling tool which is selective, meaning that it does not allow equal opportunity for all particles to be extracted by the tool exactly and only from the delineated volume. This type of error can be reduced by carefully monitoring and maintaining the performance of the sampling tools. IPE covers all errors occurring in handling, mixing, comminution or similar in the laboratory, which are due to:
1) Loss or addition of matter (dust, moisture, poor cleaning etc.)
2) Chemical or physical alteration
3) Negligence or non-adherence to GLP by the laboratory operator

Referral is made to the international standard (DS 3077, 2013), in which the complete TOS framework is laid out for sampling of lots of all dimensionalities and with all levels of heterogeneity. The way TOS has been applied in the present work is described below in the practical context of the present project.

Replication experiment 

In order to quantify the effects of the total sampling, processing and analytical errors combined, there exists an approach called the Replication Experiment (RE) (Juran and Godfrey, 1998; DS 3077, 2013; Esbensen and Julius, 2009; Esbensen and Wagner, 2016). The variability of repeated sampling can be quantified by extracting and analyzing a number of replicate samples; this term needs to be carefully defined, see Esbensen and Wagner (2014). A replication experiment can be applied to any existing sampling procedure, as well as to any new sampling operation. It is critical to understand that the RE must start, must replicate, from the primary sampling stage, or else the primary sampling error will not be included in the resulting estimate of the total measurement uncertainty, (Juran and Godfrey, 1998; DS 3077, 2013; Esbensen and Julius, 2009; Esbensen and Wagner, 2016). The RE approach can either be applied in the above primary sampling setting, and it can additionally be applied at each individual sampling, processing or analytical stage in a hierarchical fashion.