# OSNACA Mathematical Techniques

The OSNACA mathematical technique quantifies differences in multi-element signatures and makes it possible to create a map of “magmato-hydrothermal space”. The co-ordinate system of magmato-hydrothermal space is defined by the number of metallic elements included in a particular investigation. For example, 24 ore and path-finder elements define a 24-dimensional mathematical space where each ore sample’s signature is defined by its “enrichment vector”. Every sample must be analysed for every element used in the calculations.

Raw data are normalised to average crustal abundance (ACA) after replacing all values below ACA with ACA. The normalised data are then logged and scaled to a fixed distance from the origin. This process is illustrated in two dimensions below (Fig. 1).

Data below ACA are replaced with ACA so that it is only metal enrichments that are measured. In other words, the analysis is restricted to the positive quadrant of Figure AA with average crustal abundance at the origin. Metal abundances below average crustal abundance may represent metal leaching but they may also represent the absence of this metal in the host rock to start with. In neither case is it desirable for this low metal abundance to affect the direction of the enrichment vector.

Log normalised data are scaled to a fixed distance from the origin (10 units) by dividing the vector by its length and multiplying by 10. With all data a fixed distance from the origin in 24-dimensional space, the Euclidian distance between any two sample points is a proxy for the angle between their enrichment vectors. Samples with similar metal signatures lie in similar directions from the origin, and have commensurately small Euclidian distances between scaled sample points. Dissimilar samples have large angles and commensurately large Euclidian distances between scaled data points.

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Fig. 1 Schematic illustration of scaling modified log normalised data

For elements with relatively high average crustal abundance (e.g., Cu (29 ppm) and especially Fe (56,000 ppm)) log10 measurements unduly restrict the log normalised score that these elements can reach (Fig. 2). Even for an abundance of 100% Fe, the log10 score is only 1.3. Therefore, for elements with ACA greater than 1 ppm, the base used to calculate the log normalised score has been adjusted so that 100% metal abundance gives a score of 6 (Fig. 3, Table 1). For elements with ACA below 1 ppm, scores higher than 6 have been “cut” to a value of six.

The resultant axes spread ore and path-finder geochemical values over a log score range of zero to six, in such a way that enrichment in one element can reasonably be compared to that of any other element. In Table 1, scores of 1 are universally recognised as “anomalous” in most ore environments, scores of 3 are around “ore grade”, and scores of 5 are universally regarded as “bonanza” grades. It is recognised that “ore grade” is an economic parameter that varies according to many factors but the attraction of this framework is its universality. It provides a fixed reference frame with which to quantify the difference between any two ore samples.

Fig 2 Log (base 10) scores for ACA-normalised data

Fig. 3 Modified log scores for ACA-normalised data

Table 1 Ore element abundances corresponding to scaled log-normalised scores of 0-6.

Worked Example

A worked example of the mathematical technique described above is provided in two downloadable files. Orogenic_INPUT.xlsx contains data for 79 Orogenic and Intrusion Related gold samples from the OSNACA database. The worksheet “Data” contains the raw data and all of the calculations through to scaled-log-normalised values. The worksheet “Matrix” calculates the Euclidean distance between every sample pair based on their scaled-log-normalised values. This matrix is the input for Hierarchical Clustering.

Hierarchical Clustering has been performed using the MultiDendrograms software package which can be downloaded free from the internet. The Joint Within Between algorithm was chosen to classify the samples.

Orogenic_OUTPUT.xlsx contains the same data for the 79 Orogenic and Intrusion-Related Gold samples but the samples have been ordered according to the results of Hierarchical Clustering. The dendrogram output is shown on worksheet “Dendrogram” where seven main groups (A-G) have been identified. The matrix on the worksheet “Matrix” shows the same groups and they are also plotted as spidergrams (worksheet Spidergrams) where the element enrichments that have led to these samples being grouped can be assessed visually.

The first division on the dendrogram is between Groups A-C (low As-Sb) and Groups D-G (high As-Sb). All of the intrusion related gold samples are in Groups A-C. Group A is distinct on the basis of elevated Cu and Bi ± Mo. Group B contains elevated Ag and Hg ± Mo, and Groups C contains elevated Bi ± Mo

Group D comprises samples that are not particularly closely related. This can be seen on the matrix and the spidergram. In contrast, Groups E and G form coherent spidergram patterns with commensurately small Euclidean distances between samples within each group. Groups E-G are marked by a fairly similar Au-As-Sb-Te-W signature.

Global Matrix

A matrix for 283 ore grade samples, that has been organised according to hierarchical clustering is available as a downloadable file; Matrix SEPT_2012.pdf, but requires an A0 plotter to print. It clearly demarcates distinct ore types such as Ni-Cu-PGE, Fe, Porphyry Mo and Sn mineralisation. It also separates different types of base metal deposit, different types of intrusion related deposits and two broad classes of orogenic gold deposit. Within each of the broad classes there are nested groups of more closely related samples (e.g., Zn-rich & Cu-rich VHMS deposits) and small groups of very closely related samples (e.g.,Crixas, Wiluna, Paddington and Lindsays samples within the Orogenic Au (As-Sb) Class. Naturally, there is overlap between different ore classes and there are also some samples which have been widely separated by hierarchical clustering that have relatively small Euclidean distances between them. These samples can be identified on the matric by coloured cells a long way from the matrix diagonal.