Improving anomaly selection by statistical estimation of background variations in regional geochemical prospecting

Journal of Geochemical Exploration, 29 (1987) 295-316 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

295

Improving Anomaly Selection by Statistical Estimation of Background Variations in Regional Geochemical Prospecting C. ROQUIN 1and H. ZEEGERS 2

~C.N.R.S.- Centre de Sddimentologie et Gdochimie de la Surface, 1, rue Blessig, 67084 Strasbourg Cedex, France 2Bureau de Recherches Gdologiques et Mini~res, Ddpartement des G~tes Mindraux, B.P. 6009, 45060 Orldans Cedex 02, France (Received November 4, 1985; revised and accepted July 15, 1986)

ABSTRACT Roquin, C. and Zeegers, H., 1987. Improving anomaly selection by statistical estimation of background variations in regional geochemical prospecting. In: R.G. Garrett (Editor), Geochemical Exploration 1985. J. Geochem. Explor., 29: 295-316. The first part of this study, based on numerous data collected during the Mineral Resources Inventory Programme in France, investigates the influence of environmental factors on geochemical background in regional prospecting. Within the 30 lithological units selected in basement areas of the Massif Armoricain, Vosges and Massif Central, two main differentiation factors are commonly identified: (1) a dilution effect of trace elements by a barren siliceous phase related to various environmental parameters such as the nature of the substratum and overburden, or the type of material sample; (2) a coprecipitation effect of Zn, Ni, Co, Cu and P with Fe-Mn hydroxides, marked by a frequent association between these elements and, in Brittany, by their enrichment in stream sediments compared to the soils. Both factors appear to be closely related to the mineralogy of the samples, and are better characterized by multielement analytical data than by coded field observations. In the second part of the study, an example of Zn background estimation and correction is given for 1530 samples collected in the Nort-sur-Erdre district (Brittany). A simple technique of regression by local estimation, known as "neighbourhood regression", has been applied on Zn in the chemical space of a suite of "background characteristic" elements defined previously. A reappraisal of the Zn anomalies is carried out on the residual component of Zn and, in order to give importance to the most diluted samples, this index is also normalized, either by the raw Zn value or by its local standard deviation. Synthetic mapping of these various indices helps in the final stages of anomaly interpretation and selection.

INTRODUCTION

In geochemical prospecting, background values of the base metals Cu, Pb and Zn generally exhibit a wide range of concentrations which are related to 0375-6742/87/$03.50

© 1987 Elsevier Science Publishers B.V.

296

variations in primary and secondary environmental factors. This noisy background is the main problem in interpreting the results of a survey when attempting to discriminate anomalous samples as metal contents related to high background samples and low contrasted anomalies are not systematically different. In fact, there is often a wide range of overlap between the two populations, so that using a single cutoff value is not a very efficient way to separate them. This problem is particularly important during the first stage of exploration at the regional scale, when, for economic reasons, large areas are covered at a low sampling density. In such a case, an orebody that is not well exposed at the surface or is too distant from any sampling point, may contribute only a small part to the composition of the weathered and transported material collected in the field. An adjustment of the background value at each sampling site is therefore required in order to allow for the influence of environmental factors and to obtain a better assessment of the useful component of the geochemical signal. The risk of failure in the sorting of geochemical anomalies may then be reduced in two ways: (1) by eliminating false anomalies, corresponding to high background values; and (2) on the contrary, by selecting low level anomalies diluted by an important amount of barren material within the samples. Regression techniques have been introduced for this purpose in stream sediment geochemistry several years ago by Dahlberg (1969), Rose et al. (1970), Rose and Suhr (1971), Brundin and Nairis (1971) and Chatupa and Fletcher (1972), and since have been applied and discussed by many workers. The first problem in background modelling is to characterize the environment of a sample with a few relevant parameters. Regional characteristics are sometimes available on maps but local features must be recorded in the field at each sampling site. Systematic field data recording is generally limited to a few easily measured parameters or qualitative observations. Some examples of standard procedures for field data acquisition are given by Garrett et al. (1980). However, in practice, as Culbert (1976) noted, environmental control on metals background is better characterized by physical or chemical properties of the samples themselves. This fact was also recognized in France by Beguinot et al. (1979) who effected a real improvement in the geochemical criteria for anomaly selection with the use of multielement analysis. Then, a second problem is the definition of the relationship between environmental parameters and indicator elements. A linear least-square fit by multiple regression analysis is generally used, with some accomodations often needed to respect the underlying assumptions of both geochemical and statistical models. A logarithmic transformation of the variables is often used to reduce the influence of higher values responsible for the skewness of elements distribution and heteroscedasticity of the variance. Techniques of stepwise regression or ridge regression have been devised to limit the effect of correlation between the dependent variables representative of environmental factors.

297 For the same purpose, Closs and Nichol (1975) prefer to apply the regression to geologically meaningful orthogonal factor scores derived by R-mode principal component analysis. An iterative procedure is also presented by Malmquist (1978) to remove the influence of outliers from the background estimation. However, regional geochemical data are generally very heterogeneous and a linear model often seems inappropriate. Therefore, Besnus (1975) and Zeegers (1979) used regression curves computed by moving averages to describe the behaviour of some elements related to iron, aluminium or silica. A rather similar technique of local estimation in a multidimensional space and known as "r~gression par voisinage" (or "neighbourhood regression") was introduced by Bordet (1973) and Lebeaux (1974), and has been applied in geochemistry by Cazes and Reyre (1976) and Baraton and Roquin (1986). The present study, based on numerous data collected as part of the French Mineral Resources Inventory programme, is a contribution on the following two aspects of geochemical background modelling: (1) the description of the influence of environmental factors on geochemical data; and (2) background estimation and correction, presented here for zinc on the Nort-sur-Erdre 1/50,000 sheet in Brittany. DESCRIPTION OF THE INFLUENCE OF ENVIRONMENTALFACTORS Material and methods In 1975, systematic multielement geochemical surveys utilizing soils and stream sediments were commenced, as part of a major programme of mineral exploration, in the basement areas of France, i.e., Brittany, Vosges, Massif Central, Pyr~ndes and Corsica. Thirty lithological units corresponding mainly to schist and granite formations, were selected from several 1/50,000 sheets already surveyed in 1980 (Fig. 1 ), so that on each unit the influence of environmental factors could be studied independently from the main lithological differentiations. In total 4971 samples were selected, and each unit is represented by a few tens to a few hundreds of samples. Every one of them is characterized by two kinds of information: (1) The composition of its fine fraction ( < 125 ]~m) analyzed by plasma emission spectrometry for (Fe203) and 21 trace elements (Mn, Ba, P, Pb, Zn, Cu, Ag, As, Sb, Cd, B, V, Ni, Cr, Co, Be, Sn, Mo, W, Y, Nb) according to the procedure described by Boucetta and Fritsche (1979). (2) Its environment as characterized by four kinds of parameters recorded in the field: the type of sampled material, the lithology, the presence or absence of cultural features and marshes. Four kinds of materials have been sampled in order to achieve uniform sampling density of about 2.5-3 samples/km2: (1) active stream-sediments; (2)

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alluvium soils taken on the stream banks; (3) thalweg soils in streamless valleys; (4) soils developed on the slopes of a watershed. The influence of environmental factors on geochemical background was then assessed in two ways: (1) directly, by comparing element contents between groups of samples defined by coded field observations (lithology, sampling media and vegetation) ; and (2) indirectly, by interpreting the multielement data displaying geochemical associations between Cu, Pb, Zn and other elements normally unrelated to mineralization.

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Influence of lithology The geochemical background for each lithological unit was first estimated by the median values of the elements. A simple comparison of the backgrounds is given by principal components analysis of the medians. The projection of elements and lithological units in the first factorial plane, which explains about 58% of the total variance, displays two well-defined clusters for the granite and schist units with the migmatite and gneiss units projecting in an intermediate position (Fig. 2).

300

The granite backgrounds differ from those of the shales by lower contents in Cr, Ni, Co, V, Mn, Zn, and Ba and higher values for Be, and Pb. Such differences have already been reported by Barbier (1979) on several geochemical maps in the Massif Central. However, two exceptions can be noticed: the "granite des Crates", in the districts of Munster and Remiremont (Vosges), is characterized by high scores on both factors and the "granite de Pouzauges" in Brittany clusters with the schist group. For nine schist areas in Brittany and the Massif Central, the importance of geochemical differentiation between units has been compared to the withinunits variance components by analysis of variance. For each element, four components are distinguished (Fig. 3) : (a) an anomalous component (1) corresponding to samples with very high concentrations: greater than m (mean) + 3 s (standard deviation) ; (b) at the background level, variations within units (2), partly due to the mixture of four sample types, are compared to variations between units ( 3 ) ; (c) the percentage within pairs variance (4) of 377 duplicate samples represents the variation due to sampling and analytical errors as indicated by Garrett (1973). Figure 3 reveals a rather similar contribution by anomalous samples to the global variance of each element in Brittany and the Massif Central with the exception of Zn. The anomalous contribution is very high for Pb and As, medium for Cu, Mn, Ba, P, B, Ni, Co and low for Fe, V, Cr, Y, Nb. This is an indication of the somewhat erratic behaviour of these elements in relation to local environmental factors. On the contrary, the Zn anomalies appear to have a greater contrast in the Massif Central than in Brittany. In the geochemical background range of values, regional differentiation between schist units is higher for Fe, Ba, Ni, V, Y, Nb than for Mn, Co, Pb, As. The differentiation between units ibr Cr is more important in Brittany than in the Massif Central. Within units, the local variations of background are still significantly higher than measurement errors for every element. These variations can be due either to lithological facies variations and perhaps some weak influence of mineralization, or to secondary environmental factors such as differences between the four sample types.

Influence of the sampling media A comparative study of element distributions in the four kinds of materials sampled was carried out for each unit by examination of their cumulative frequency distribution (Roquin, 1984). The sampling bias corresponding to the diversity of material sampled is rather variable from one area to the other, with a particularly contrasted geographical distribution in the Massif Armoricain. In Normandy, for all the northeastern part of Massif Armoricain, element contents are generally lower in stream sediments than in soils, particularly for

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302 Cu, Pb, Mn, P, Fe, Cr, V and B. The same trend exists in the Vosges and Massif Central and can be interpreted as a dilution effect of the trace elements by an increased concentration of the more siliceous detritic minerals and a removal of fine clay particles by running water. In Brittany, on the contrary, there is a systematic increase of Zn, Fe, Mn, Co, Ni and P in stream sediments compared to the soils. It is well known that this group of mobile elements is frequently trapped by adsorption or coprecipitation with Fe and Mn oxides and their simultaneous enrichment probably reflects the strong influence of these scavenging factors in the stream sediments of Brittany. Such regional disparities between the element distributions in soils and stream sediments of Brittany and N o r m a n d y are really outstanding and not yet fully explained. However, the dilution effect observed in stream sediments in Normandy is suspected to be enhanced by contamination with siliceous eolian loess material.

Influence of cultivation In some areas of Brittany, soils of uncultivated areas are sufficiently well represented for a reliable comparison with cultivated soils. There is generally a considerable enrichment of phosphorus in cultivated soils. For instance, on the Brioverian Schist of Loudeac in Brittany, the median value of phosphorus content in uncultivated soils is 315 ppm against 770 ppm for cultivated soils. This difference certainly reflects the influence of fertilizers, but natural differentiation may explain a slight increase for some other elements such as Mn, V, Cr, Zn in cultivated soils. In fact, uncultivated soils come mainly from forest areas developed on poor land unfavourable to cultivation and related to a thick overburden of silt and sand. This illustrates the difficulty of measuring the specific influence of an environmental parameter on geochemical background without any interference from its surroundings.

Study of geochemical associations As shown for Pb, Zn and Cu, (Figs. 4, 5, 6, respectively), correlation coefficients between elements, calculated within each area are generally positive. This trend reflects a generally similar behaviour for all studied elements. This is also expressed by principal components analysis as a clustering of the elements around the first factor obtained before rotation. This factor explains 40-60% of the data variance, and can be interpreted as the expression of a dilution effect resulting from the mixing, in variable proportions, of the minerals bearing most of the trace elements with a more siliceous barren phase. Besides this important general factor, some more specific geochemical asso-

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ciations can be identified. They correspond mainly to the group of transition elements ( Fe, Cr, Ni, V ) and sometimes Cu or Zn, which seems to characterize, within the granites, the abundance of ferromagnesian minerals. In schist environment this association frequently includes Ba, B and Nb, and may be related to the abundance of clay minerals• Another type of association is characteristic of a group of elements affected

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by secondary processes and scavenging by hydroxides. It consists mainly of Mn, and Co, frequently linked with Fe, Zn, Ni, Cu, P and/or B. Thus two kinds of behaviour can be observed for the metals: (1) Cu and Zn generally enter into the two main element associations with a more pronounced "lithological" control for Cu and a more "supergene" affinity for Zn; and ( 2 Pb is more independent and often appears to be isolated on a single factor. Conclusion

The more general features of environmental control on geochemical background levels in basement areas of France appear to be directly related to the mineralogical composition of the samples. They are better characterized by multielement analysis than by coded field observations. The most important of the mineralogical factors is the dilution effect by one or more siliceous minerals, (probably mainly quartz ). It is responsible for: ( a ) chemical heterogeneity between soils and stream sediments; (b) differences in background level between schist units; and (c) a general correlation between all elements within the units. Another strong mineralogical effect is the scavenging of trace elements by Fe and Mn hydroxides. It is clearly observed in geochemical associations in many areas, and is particularly important in the stream sediments of Brittany. Multielement analysis may thus be very helpful in correcting for background variations of base metals, and more particularly for Zn and Cu.

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Data presentation T h e 1 / 5 0 , 0 0 0 N o r t - s u r - E r d r e s h e e t is l o c a t e d in t h e s o u t h o f t h e M a s s i f A r m o r i c a i n n e a r t h e m o u t h o f t h e L o i r e R i v e r (Fig. 1 ). G e o l o g i c a l f e a t u r e s o f t h i s a r e a a r e o u t l i n e d o n Fig. 7 a c c o r d i n g t o t h e 1 / 5 0 , 0 0 0 m a p a n d d e s c r i p t i v e notes from Barbaroux (1983). T h e P r e c a m b r i a n a n d P a l e o z o i c b a s e m e n t is d i v i d e d i n t o t w o d o m a i n s b y a

306

TABLE

1

Sampling statistics

Sampling media:

- stream sediments - alluvium soils - thalweg soils - soils

Lithology:

144 164 1081 141

- overburden

554

- sandstone

272

- schist

600

- limestone

11

- gneiss

85

- basic rocks

Total:

7 1530

1529

branch of the South Armoricain lineament: (a) in the north, volcanosedimen tary schist and sandstone formations of the St-Georges-sur-Loire synclinorium; (b) in the south, acid and metamorphic rocks (gneisses and leptynites) extend from east to west between Brioverian micaschist formations. Several sills of peridotite stretch along the major faults between these units. Tertiary sediments accumulated in the Nort-sur-Erdre and Saffr~ graben in the northwest part of the survey area. Pliocene marine sand and gravel deposits occur over all the northern part of the map sheet. Additionally, other residual laterite formations, inherited from lateritic profiles, and Quaternary deposits contribute to mask the nature of underlying bedrock. No economic mineralization is known in this area but the important Sn district of Abbaretz is located immediately to the north. The geochemical survey of this area of about 540 km 2 follows the general procedures of the French Inventory, which have been described above. The sampling statistics according to coded field observations are given in Table 1. The influence of sampling medium is mainly characterized by an increase of Mn, Co, Zn, Ni, B and Fe contents in stream sediments compared to the soils. As stated before, this kind of sampling bias is quite typical of Brittany. Spatial patterns of element geographical distributions revealed by regional geochemical mapping often transgress major geological boundaries in the basement units. In fact, the lithological units mapped during the geological survey do not have very different chemical compositions. This initial small contrast is probably further decreased in the geochemical samples by the presence of recent overburden. The weak heterogeneity of geochemical background observed between categories of samples as defined by coded field observations, is confirmed for Cu, Pb, and Zn by the analysis of variance results (Table 2). However, the base metals are highly correlated with the other elements, and their geochemical

307 TABLE 2 Percentage of variance explained by coded field parameters for Pb, Zn, Cu for the Nort-sur-Erdre map sheet

Sample type Lithology

Pb

Zn

Cu

4 3

8 4

2 8

associations are observable for the more characteristic background elements: Fe, Mn, Ba, P, V, B, Ni, Cr, Co, and Y. Those samples with concentrations greater than m + 3s for any of the background elements have been removed from the computation of principal components, and are later plotted on the factor plane as open circles. The elements Pb, Zn, Cu, As have been considered as supplementary variables in order to assess their geochemical association with background factors; they do not contribute to the definition of factorial axes but are projected on them afterwards. Factor loadings for the first two principal components for the logarithmically transformed background element suite are presented on Fig. 8. The first factor represents a high percentage of variance, about 50%, and is positively correlated with every element. This dominant factor reflects the general tendency of similar variations for all elements, and may thus be interpreted as a dilution effect. The host minerals for trace elements may include ferromagnesian, phyllites or oxides, but as a whole, they are similarly affected by the proportions of siliceous barren material in the samples. Here, this barren fraction F2

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308

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may have been introduced mainly by contamination with overburden silt and sand. It is noteworthy t h a t Zn and Cu contents are closely related to this first component of variation, indicating t h a t it can be chosen as an explanatory variable in a simple linear regression model for background corrections. However, a graphical display of the covariation between Zn and the sample scores on the first factor of the P.C.A. (Fig. 9) indicates t h a t this relation is not linear. In this case, a logarithmic transform of Zn seems to be better suited for linear background estimation, and 75% of Zn variations or 43% of Cu variations may thus be explained. However, we prefer to use a more general estimation technique, such as neighbourhood regression, which is based on the idea that one can expect to observe the same concentration level in some element, i.e., Zn, for all background samples of similar composition in other elements.

309

Estimation of zinc background variations Neighbourhood regression of Zn was applied directly in the space of elements considered as the most representative of background differentiation factors: Fe, Mn, Ba, P, V, B, Ni, Cr, Co, and Y. The estimation procedure permits a comparison of Zn content in any sample with values observed for its neighbours in this chemical space; it is like a moving average or local filtering technique applied in the multivariate space generated after standardization of the 10 background elements. The regression program, written by A. Baraton, yields an estimate of background and residual components of Zn; the precision of this estimation is characterized by the local standard deviation of Zn (Appendix 1). Here, we chose rather arbitrarily to limit the size of the neighbourhood to the fifteen nearest, i.e., most geochemically similar, samples. Several trials, with various numbers of neighbours (between 10 and 80) indicate that estimated and residual components are only slightly influenced by a change in this parameter (Baraton and Roquin, 1986). In comparison, the local standard deviation is much more affected by, and is very sensitive to, the presence of local "anomalies" in the vicinity of the estimated sample.

Selection of residual anomalies The influence of Zn background corrections on anomaly selection can be seen on the correlation diagram between residual and raw Zn values (Fig. 10). Two threshold values computed before and after regression and corresponding to rather high cutoff valuer of about three times standard deviation above the mean are also represented. They define four domains of interest corresponding to:

(a) confirmation of the initial recognition of well contrasted anomalies and low background samples; (b) rejection of false anomalies corresponding to high background zinc values with low residuals; (c) recognition of new anomalies with residuals more contrasted than raw values. On the same diagram, we have also considered an anomaly index given by the percentage of residual Zn that remains after background removal. A cutoff value of 50% residual ratio is represented that may be used to enhance low anomalies within the most diluted samples. Justification for such a criterion is that both background and anomalous components of the geochemical signal may be diluted in a similar way by a barren and possibly allochthonous phase like quartz. A similar effect occurs in the case of secondary hydromorphic anomalies where the amount of trapped metals depends on the availability of adsorbing materials.

310

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To deal with such a proportional relationship between background and anomalous components we can use a logarithmic transform before applying the neighbourhood regression. This is equivalent to estimating the background level of Zn as the local geometric mean, and considering the relative deviation of observed Zn about the mean. Figure 11 illustrates that there is a good approximate hyperbolic relationship between this geometric residual and the arithmetic residual ratio. As it is explained in Appendix 2, this relation results from the proximity of both geometric and arithmetic means, which could be improved by taking account of the logarithmic deviation as discussed, for example, by Agterberg (1974). Therefore, the arithmetic residual ratio, which is easier to compute and more suggestive than the logarithmic or geometric residual is chosen here as an anomaly index. Alternatively, the estimated local standard deviation can be used as a weighting factor to normalize the residual values. However, as this estimated

311 RZn

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Fig. 11. Correlation diagram between arithmetic Zn residual ratio (ZnRe/Zn) and an hyperbolic function of Zn geometric residual (1-1/ZnGR).

parameter is not very robust, the procedure is not reliable for samples in the vicinity of local anomalies. Finally, a synthetic map is presented (Fig. 12) to discuss the anomalous character of samples for Zn, in relation with their geographical distribution. Statistical anomalies reported on this map have been sorted by using two threshold levels and four indices corresponding to: raw Zn values; residual Zn values; residual ratio; and residual values normalized by the local standard deviation. Obviously, many kinds of geological and geochemical information have to be considered to interpret this map in detail. However, as expected, one can readily notice that several original Zn anomalies have disappeared or are reduced in contrast after regression; on the other hand some new wellclustered anomalies appear in the northwest corner of the map sheet which deserve further examination.

312

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6 KM -

I

Fig. 12. Map of Zn anomalous samples according to four statistical criteria and two threshold levels. Squares=Zn> 120 ppm (normal) or> 155 ppm (heavy); circles=residual Zn> 35 ppm (normal) or > 55 ppm (heavy) ; upright triangles = Zn residual ratio > 40% (normal) ; > 50% ( heavy ) ; inverted triangles = normalized residual: > 2.5 ( normal ) ; > 3.5 ( heavy ). In t h e s o u t h w e s t e r n p a r t of t h e m a p sheet, Zn g e o c h e m i c a l b a c k g r o u n d is v e r y low a n d a n o m a l i e s a p p e a r o n l y for t h e residual r a t i o a n d n o r m a l i z e d residual c o m p o n e n t , so t h a t in spite of a dilution effect b y c o n t a m i n a t i o n with a l l o c h t h o n o u s m a t e r i a l , s o m e w e a k influence f r o m m i n e r a l i z a t i o n m a y still be recognized in t h i s area. CONCLUSIONS S t a t i s t i c a l studies of m u l t i e l e m e n t d a t a for a b o u t 5000 soil a n d s t r e a m - s e d i m e n t s a m p l e s collected in v a r i o u s geological s e t t i n g s s h o w the influence of e n v i r o n m e n t a l f a c t o r s on g e o c h e m i c a l b a c k g r o u n d levels in regional p r o s p e c t ing. T w o m a i n d i f f e r e n t i a t i o n f a c t o r s are f o u n d in m o s t o f t h e studied areas: t h e dilution effect of t r a c e e l e m e n t s b y a b a r r e n siliceous p h a s e ( p r o b a b l y m a i n l y q u a r t z ) , a n d t h e s c a v e n g i n g of Zn, Ni, Co, Cu a n d P by F e - M n h y d r o x ides. B o t h f a c t o r s are r e l a t e d to t h e m i n e r a l o g i c a l c o m p o s i t i o n of t h e s a m p l e s ,

313

and are better characterized by multielement data than by coded field observations. Therefore, a large portion of the Zn and Cu background variation may be estimated and corrected by regression analysis in the geochemical space of "background characteristic" elements. In regional prospecting, the data are rather heterogeneous and statistical dependency between trace metals and the other elements is generally complex and poorly defined. In this situation a local estimation technique such as "neighbourhood regression" is more appropriate than a "global" regression technique using a linear or curvilinear model. Furthermore, several kinds of criteria can be examined in order to obtain a better appraisal of the anomalous component when it is related to the background component: ( a ) If we assume that both components are independent and additive, we can use the raw residuals as an anomaly index. This kind of model may correspond to a simple mechanical dispersion of ore material in the environment. (b) If the components are related to each other by a proportional effect, the relative residual ( residual/raw variable ratio or residual/estimated component ratio) is a better indicator. This is the case when a secondary dilution process occurs because of the introduction of some siliceous allochthonous phase which reduces the proportion of both anomalous and background material within the samples. (c) If the relationship is more complex and responsible for heteroscedasticity of the variance, the residual component can be normalized by the local standard deviation. However, as the latter parameter is strongly affected by local anomalies, a more robust estimator would be preferable. Finally, it is shown that several anomaly criteria are available to the geochemist. One is generally not aware of which processes of ore metal dispersion are dominant at any location. Therefore all these indices should be considered together and compared with other relevant information on a synthetic map. ACKNOWLEDGMENTS

This work was supported by the Commission of the European Communities under the "Primary Raw Materials" research programme (contract nr. 08779-7-MPP/F). APPENDIX 1

Definition of the parameters used in the arithmetic neighbourhood regression For each sample, i, the following parameters are computed by the regression program:

314 Ni

-

neighbourhood of sample i, corresponds to its k nearest neighbours in the multivariate euclidean space, R p, generated after standardization of the p background elements:

Ni={j=l toknearestneighboursofsampleiinR p } Ri

-

Znesi

-

radius of the neighbourhood Ni, is the distance between sample i and its farthest neighbour, k, in RV; estimate of Zn background value for sample i, is given as the arithmetic mean of Zn values observed for its k neighbours: 1 k

Znesi : k j_~l Zn~, J Znrei

-

e Ni

residual value of Zn, is the difference between observed and estimated value of Zn:

Znre,, = Zni - Znesi Znsd~ -

local standard deviation of Zn, characterizes the dispersion of Zn in the neighbourhood N~, and the precision of the estimation: 1

h

~ (Znj-Znesi)2 Znsd ~ - k - 1 i= APPENDIX2

Comparison of the logarithmic and arithmetic approach in neighbourhood regression The neighbourhood regression of Zn logarithmic values yields the following parameters: 1 h logarithmic mean: Znle/= ~ j ~ In (Zn i), j e Ni logarithmic residual: Znlr,. = in (Zni) - Znlei They can be expressed as: geometric mean: Zngei = exp (Znlei) geometric residual: Zngri -- exp (Znlr,.) = Zni/Zngei The arithmetic residual ratio, rZn, is defined as the ratio between arithmetic residual and observed Zn values:

315 rZn = Znre/Zn

= 1 -- ( Z n e s / Z n )

and, as the arithmetic mean, Znes, and geometric mean, Znge, are not very d i f f e r e n t , w e h a v e t h e c l o s e r e l a t i o n s h i p , i l l u s t r a t e d o n F i g . 11: rZn-

1 - (1/Zngr)

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