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# Sampling of discrete materials quantitative approach sampling of zero dimensional objects

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isc pli rre de No ised f featu to be ade also showing how this relates directly to sampling practise (materials, equipment and procedures). A highlight of the latter issue concerns well as a theoretical standpoint, however, it may be dimensional model. From a physical and math- ematical standpoint, every element of the object can be well represented by a one-dimensional tolerance of, say, 20%): Here, conventional statistics can be applied. n No hypothesis of uniformity of the unit mass is cannot be applied. atory S is represented by its projection on a plane (often horizontal). We often have occasion to work on useful to represent a physical object by a model of a smaller number of dimensions. o A three-dimensional model alone can represent bulky lots L, e.g., an ore body and similar. o Flat objects, such as a sheet of paper, a steel sheet, the thickness of which is: n small in comparison with the two dimensions of its surface, n practically uniform (with a tolerance of, say, 20%) can often be well represented by a two- model. From a physical and mathematical stand- point, the lot is here represented by its projection on the axis of elongation. o Discrete objects such as lots made up of a large number of unspecified units, assumed to be inde- pendent from one another; i.e., populations of non- ordered units can, by extension and by convention, be defined as zero-dimensional objects. There are two cases: n Unit masses are more or less uniform (with a experimental estimation of the Fundamental Sampling Error (FSE). Part II is also fundamental for further developments in Part III, as it presents a complete overview discussion of the basic sampling operation of the one-dimensional object as well. D 2004 Elsevier B.V. All rights reserved. Keywords: Discrete; Quantitative approach; Zero-dimensional objects; Sampling of Particulate matter; Theory of Sampling 1. Joint introduction of parts II and III: three-, two-, one-, zero-dimensional models o Strictly speaking, all material objects, lots L, occupy a three-dimensional Cartesian space. From a practical as o Elongated objects such as a rail, a cable or a flux of matter whose length is: n very large in comparison with the two dimensions of its cross-section, n practically uniform (with a tolerance of, say, 20%) Parts II and III of this series are initiated by a joint discussion of the Theory of Sampling for zero-dimensional objects. It is necessary essential model rigour has been maintained. An attempt has been m Sampling of d II. Quantitative approach\u2014sam Pie Res. de Luynes, 14 Avenue Jean Received 28 August 2003; received in rev Abstract Chemometrics and Intelligent Labor lots L, which can be considered as practically two-dimensional. 0169-7439/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2004.05.015 rete materials ng of zero-dimensional objects Gy ailles, F-06400 Cannes, France orm 6 April 2004; accepted 28 May 2004 res related to the lot. Part II then delineates the central elements of brief within the limited format of the present tutorial series, but all to focus on the central mathematical theoretical core of TOS while ystems 74 (2004) 25\u201338 www.elsevier.com/locate/chemolab made. Conventional statistics We shall here deal exclusively with this most realistic case. errors and quantified heterogeneities. nt La respected, and taught everywhere. In this series, we will specifically study only the sampling of zero- and one- dimensional objects. Whereas the qualitative approach presented in Part I was common to all multi-elemental objects susceptible of being sampled (which excludes only compact solids), the present quantitative approach involves two different mathematical models corresponding to sampling of populations (zero- dimensional objects) or to time series (one-dimensional objects) respectively. It will be necessary to remind the reader of a few salient definitions (extracted from Section 4 of Part I). 1.1. Definitions and notations o LOT L: From a theoretical standpoint, a lot L of discrete material can be regarded as a set of units. We must take into consideration two kinds of sets. o SET: The set can be: ! Either a population of non-ordered units (e.g., batch of stationary material), ! Or a series of ordered units (e.g., a sequence of elementary cross-sections of a flowing stream. The order is then chronological; ordered spatial series can very often be treated with exactly the same mathematical apparatus). Different mathematical laws apply to these two kinds of sets. It is a grave mistake, which is often very costly, to use inappropriate laws or formulas! A population of non-ordered units is, by convention, described as a zero-dimensional object. The absence of dimension expresses the absence of order (chronological, spatial or otherwise). There may exist a concealed order, a o Bulky, flat or elongated objects have a common property: Their constituent elements are physically tied or correlated to one another. They are therefore liable, often likely, to show an essential spatial correlation within their specific dimensional framework. The prototype of a bulky three-dimensional compact object is a mineral deposit. Following the works of Krige, Sichel, de Wijs and more generally the South African school of mathematicians who were particularly interested in the first half of the 20th century in the gold deposits found in their country, the French mining engineer Georges Matheron (see Ref. [12]) founded a new science known as bgeostatisticsQ: the science of evaluation of mineral deposits. Michel David, another French mining engineer living in Canada (see Ref. [13]), helped significantly to diffuse the knowledge of geo- statistics in English-speaking countries. In the present work, we shall, however, not deal much with this science, which today is widely known, acknowledged and P. Gy / Chemometrics and Intellige26 correlation between units, but the model does not take this into account and is not meant to detect it either. The zero- 2. Zero-dimensional model\u2014contribution made to the heterogeneity of lot L by an unspecified unit Um In the zero-dimensional model, the lot L can be regarded in general terms as a population of unspecified units Um which will be defined as the case requires. A is the component of interest. The grade, or concentration, of component A in a given object (lot L, unit U, sample S, etc.) refers to its actual but unknown grade, not to some estimate of it. The following definitions are made: NU number of units Um in lot L, Mm mass of unit Um (with wet materials, Mm is generally the mass of dry solids) Am mass of component A in unit Um am mass proportion of component A in unit Um (i.e., grade of Um), defined by the identity Am ¼ amMm ð1Þ ML mass of lot L. ML is the sum Am of the masses Mm of the NU units Um. Am is a sum extended from m=1 to m=NU ML ¼ X m Mm ð2Þ AL mass of component A in lot L X X dimensional model presented here in Part II deals with this problem. A series of ordered units can be described as a one- dimensional object: With a flowing stream, the dimension is time: the order between consecutive units is chronological. There usually exists a correlation between the rank of a unit in the series and its composition. If such a correlation does not exist, the mathematical model detects it. The one- dimensional model presented in Part III of this article deals with this problem. Hence, the subdivision of the quantitative approach: Part II. Sampling of zero-dimensional objects Part III. Sampling of one-dimensional objects In Section 5 of Part I, we defined the concepts of homogeneity and heterogeneity. According to these defi- nitions, sampling would be an exact process if the material being sampled indeed was homogeneous. Unfortunately (but fortunately for the theoretician), homogeneity is a concept that can be defined mathematically