statanalysis howarth 2010
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Statistical analysis and data display at the Geochemical ProspectingResearch Centre and Applied Geochemistry Research Group,
Imperial College, London
Richard J. Howarth1,* & Robert G. Garrett21Dept. of Earth Sciences, University College London, Gower Street, London WC1E 6BT, United Kingdom2Emeritus Scientist, Geological Survey of Canada, 601 Booth St., Ottawa, Ontario, K1A 0E8, Canada
*Corresponding author: (email: [email protected])
ABSTRACT: The Imperial College of Science and Technology, a constituent collegeof the University of London in the 1960s, had the good fortune to be one of the firstcolleges in the United Kingdom to have access to digital computing facilities. Thisreview traces the history of the application of computing in the GeochemicalProspecting Research Centre and its successor, the Applied Geochemistry ResearchGroup, as computing moved from being a frontier research area to becoming acommonplace tool. The three principal areas in which it was involved comprised: thequality control, and thereby assurance, of analytical data; the production ofpioneering atlases of regional geochemical variation in Northern Ireland ( 1973) andEngland and Wales (1978); and the application of methods introduced by workersin patternrecognition and statistics to the interpretation of landbased and marineregional geochemical data.
KEYWORDS: computers, computing, applied geochemistry, history of geochemistry, history of statistics, history of cartography, regional mapping, spatial filters, geochemical atlas, SC4020,LGP2703, multielement maps, data transformation, factor analysis, cluster analysis, discriminantanalysis, ridge regression, KleinerHartigan trees, robust statistics, quality assurance
The Geochemical Prospecting Research Centre (GPRC ) wasestablished in 1954, under the direction of Professor JohnStuart Webb (19202007), in the Mining Geology section ofthe Royal School of Mines (RSM), Imperial College of Scienceand Technology (ICST), London. Initial studies were concerned with mineral prospecting using soil and drainage sampling in Northern Rhodesia (Zambia), Uganda, Sierra Leone,Bechuanaland (Botswana), Tanganyika (Tanzania), BritishNorth Borneo (Sabah, East Malaysia), Burma (Myanmar) andthe Federation of Malaya (West Malaysia), and extended in the1960s to Southern Rhodesia (Zimbabwe), the PhilippineRepublic, Borneo (now divided between Malaysia and
Indonesia), Fiji, East Africa, Australia, and the UnitedKingdom. By 1960, its studies had broadened into regionalgeochemistry, based on the analysis of stream sediments. In1963, Webb initiated the first of a series of investigationsconcerning the relationship between regional geochemistry andagricultural problems in livestock in Eire (Webb 1964; Webb &
Atkinson 1965). The application of geochemistry to marinemineral exploration began in 1964 (Tooms 1967). Consequently, by 1963, the Centres name was changed to the
Applied Geochemistry Research Group (AGRG ) to reflect theincreasing breadth of its applications.
The work of the GPRC and AGRG was underpinned bydevelopments in two complementary spheres: methods andinstrumentation for chemical analysis (discussed in the paper by
Michael Thompson (2010) and computing (Fig. 1). The latterfacilitated: (i) statistical qualityassurance in the analytical lab
oratory; (ii) the display of large, multielement, data sets in mapform; and (iii) the interpretation of such multielement datasets.
First steps
Many of the early studies undertaken by research students inthe GPRC included simple, manuallybased, statistical analyses.
Fig. 1.Annual numbers of GPRC/AGRG publications (total n=76)and theses (n=24) with a substantial computing and/or statisticalcontent over the years 195488.
Geochemistry: Exploration, Environment, Analysis, Vol. 10 2010, pp. 289315 14677873/10/$15.00 2010 AAG/Geological Society of LondonDOI 10.1144/14677873/09238

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The situation in the early 1960s is summarized in Hawkes &Webb (1962 ). The use of histograms to display the frequencydistributions of element concentrations was commonplace,
while probability plots of cumulative frequency distributionswere less frequently prepared. In both cases, the data for therequisite plots were compiled by hand through the preparationof tally tables. At that time, analytical quality assurance wasbased on the use of statistical series samples. These were aseries of synthetic samples (each of which was composed ofknown proportions of two natural endmembers, one having alow concentration of the element of interest, the other a highconcentration) which were included in analytical batches following the procedure developed by the exRSM geologist andchemical engineer, Charles Alex Urton Craven (191893), withadvice from Professor George Alfred Bernard (19152002) ofthe Mathematics Department, ICST (Craven 1954), in order toestimate analytical accuracy and precision. For the largeamounts of photographicplate spectrographic data generatedat the GPRC, bins for data concentrationranges were selected(because of a tendency of operators to unconsciously interpo
late values which were biased towards those of the analyticalstandards used), using a logarithmic concentration scale, andbin boundaries were placed midway between the knownconcentrations of the geochemical standards. A tick (the tally)
was placed in the appropriate bin for each analysis falling in thatrange, every fifth count being drawn as a horizontal linethrough the previous four ticks. This facilitated counting thetotal numbers of analytical results falling into each bin. A book,
widely used by students at the time, was Moroneys (1960 )Facts from Figures, which gave formulae for the calculation ofmeans and standard deviations from such grouped data, asaccumulated in the tally tables.
For those more interested in statistical analysis, Dixon &Massey (1957) was the text of choice. However, in the early
and mid1960s, textbooks written by geologist and statisticiancoauthors started to be published on the topics of statisticaldata analysis and modelling, e.g. Miller & Kahn (1962) andKrumbein & Graybill (1965), and these, together with agrowing number of research papers, did much to exposestudents to the possibilities of the application of mathematicsand statistics to applied geochemical problems. In the early1960s such computations were carried out by means of tablesof logarithms and a sixinch (15 cm) sliderule, with whichstudents were as adept as todays are with pocket calculators.
To assist in the calculations (based on a linear regressionmodel) required by Cravens (1954) method of estimation ofanalytical accuracy and precision, preprinted worksheets wereused; one simply followed the steps and the results were arrived
at very much a black box approach. In order to meet therequirements of normality of residual errors in the regressionmodelling, and homogeneity of variance when the concentration levels in the statistical series samples spanned over anorderofmagnitude, it was desirable to carry out these calculations following a logarithmic transformation. This was thesubject of an MSc thesis by Stern (1959), but the routineapplication of his method was computationally complex, andessentially impractical for routine application, even using theMonroe electromechanical calculator available in the GPRC.Sterns supervisor in the Department of Mathematics, Dr. G.M. Jenkins (193382), who later became an expert in timeseries analysis and systems engineering, appears to have begun
work on improving the deficiencies he recognised in Sterns
approach, in an unpublished manuscript A statistical problemin geochemical prospecting (1959?, recently found in oldAGRG files). In 1970, an exmember of the GPRC staff,Clifford (Cliff) Henry James (19312003), published a version
of Cravens method still adapted to handcalculation, on thegrounds that one of the difficulties of the method as originallydescribed is that the calculations involved require a computeror an electronic calculator with a memory unit . . . manylaboratories do not possess these facilities (James 1970, B88).
REGIONAL MAPPING
Following extensive fieldwork over several thousand squaremiles of Africa in the mid1950s by Webb, Tooms and theirstudents, it became apparent that there was considerable scopefor regional geochemical surveys based on drainage reconnaissance surveys. By 1960, this hypothesis was confirmed throughfurther studies in what was then Northern Rhodesia, elsewherein Africa, and in S.E. Asia. In 1960, a suite of drainage samplescollected for a base metal drainage reconnaissance survey over3000 mi2 (7770 km2) of the LivingstoneNamwala Concessionarea, Zambia, were made available to the GPRC by NamwalaConcessions Ltd. These were analysed spectrographically andchemically for 17 elements in 196162. Following a study of theassociation between trace element concentrations in the drainagematerials and the geology (Harden 1962), it was apparent thatthe

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Fig.
2.Portionofapointsymbolmapoftheconce
ntrationofcoldextractablecopper(ppm)
inthe 0 and xi,i=1,k =100) to a new set of
variables y1, . . ., y(k1) where yj= loge[xi/x(k1)]; j=1, k1.In recent years, this transform has been widely promoted foruse in the earth sciences (most recently by Buccianti et al. 2006).Howarth tried on several occasions to apply the logratiotransform as a precursor to multivariate analysis of various
AGRG geochemical data sets but found that, in practice, the
results were often geochemically uninterpretable, and thatwhenever xi 0, the transform resulted in serious outlierproblems. More research, with a wide variety of data sets, isrequired on this subject.
In recent years, Dennis Helsel (Helsel & Hirsch 1992,357376; Helsel 2005), of the US Geological Survey WaterResources Division, has provided new approaches to theproblem of dealing with censored, i.e. below analytical detection limit (dl), data. In the days of GPRC/AGRG, any such
values in a data set were routinely set to the appropriate dl/2for the purposes of statistical analysis and, in practice, it isdoubtful whether it brought any geologically significant biasinto the geochemical interpretations arrived at.
Robust methods
The deleterious effect of outliers present in a data set, leading tobias in calculation of the mean, inflated variances, spurious
Fig. 15. Trial KolomogorovSmirnov
filtered map for zinc concentrations(ppm) in the

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correlation coefficients, and so on, has long been recognised.However, it was only in the mid1970s that methods whichcould automatically downweight the effects of outliers to
obtain robust estimates of both univariate statistics (such asthe mean and standard deviation) and the covariance matrixor correlation matrix (which underpin principal components,
Fig. 16.Qmode factor analysis maps for the geochemistry of the

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factor and classical linear discriminant analysis) began to bedeveloped (Andrews et al. 1972; Huber 1981), but it was a
while before their potential utility in applied geochemistry waspointed out (Campbell 1982; Garrett 1983; Howarth 1984).
Robust correlation matrices were calculated by Leech (1984),using software developed by the statistician Norman Campbell of the Commonwealth Scientific and Industrial ResearchOrganisation, Australia (Campbell 1980), who had taken hisdoctorate in the Statistics Department at Imperial College.
Turner (1986 ) implemented robust versions of both principalcomponents analysis and ridge regression software, whichproved immensely useful to AGRG research subsequently(e.g. Coward & Cronan 1987).
The extensive study of the application of robust principalcomponents and factor analysis by Turner (1986, 434548)concluded that factor analysis is preferable to principal components analysis because the use of a small number of factorsforces a grouping of the variables, reducing the dimensionality
of the problem and increasing interpretability. The greatestanomaly contrast is obtained using untransformed data; priorBoxCox transformation of the data is best if backgroundassociations and relationships are to be revealed.
Data displays
The arrival of the interactive statistical package MINITAB(Ryneret al. 1976) on the Colleges distributed terminal system
enabled routine data analysis to be used by both staff andstudents in AGRG and, because it embodied much of therecent thinking on graphicsbased Exploratory Data Analysis(Tukey 1977), boxplots, quantilequantile plots and othergraphical displays were soon taken up in AGRG work (Earle1982; Howarth 1984; Turner 1986). Earle (1982, 168183)developed a program (GIRAF) for the interactive dissection ofprobability plots into constituent subpopulations. Turner(1986, 166179) showed the utility ofmultivariate probabilityplots, based on the cuberoot of the Mahalanobis distance(Healy 1968; Campbell 1979) for detection of multivariateoutliers.
Use of two new multivariate graphics to portray multielement sample compositions for the purpose of comparison
were extensively investigated by Turner (1986), using theMorayBuchan data set: (i) Chernoff faces (Chernoff 1973),which assigns features of the human face ( e.g. position/style ofeyes, eyebrows, nose, mouth) to different variables to make
Fig. 17. Subtractive colourcombinedmap of the first three varimax rotatedfactors of the BoxCox transformedgeochemistry of the

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Fig. 18.Empirical discriminant classification map of the geochemistry of Pb, Ga, V, Mo, Cu, Zn, Ti, Ni, Co, Mn, Cr and Fe 2O3in the

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comparative displays, each face corresponding to a samplecomposition; and (ii) KleinerHartigan (KH ) trees (Fig. 23a;Kleiner & Hartigan 1981; Garrett 1983).
Turner found Chernoff faces to be unsatisfactory, in thatmuch work was required to find the best facial features to
which a particular element should correspond (which impliedthat the technique could be used to deliberately distort resultsby emphasis or suppression of any variable) and that, in orderto achieve the best visual emphasis for any anomalous samples,the analyst must have prior knowledge of which they are(Turner 1986, 346, 355). The KH trees were far moreeffective at portraying the multielement sample compositions,using a tree morphology based on the hierarchical clusteranalysis of a robust correlation matrix; branchlengths aredrawn proportional to the concentrations of the elements to
which each corresponds (Fig. 23a). It was likened to performing a visual factor analysis. Although the physical size of theplotted trees made it difficult to use them in a spatial context
with a large data set by plotting them at their correspondingsample location on a map, nevertheless, sidebyside comparison of the trees laid out as a graphic table, in numerical orderof sample numbers (Fig. 23b), proved quite satisfactory. KH
trees were also extensively used by Coward (1986).
Ridge regression
Linear multivariate regression analysis has long been used inapplied geochemisty to correct for the effects of elementinteraction (e.g. enhancement of element concentration levelsas a result of iron and manganese scavenging), and to empirically explain the behaviour of an element in terms of others.Emphasis is often placed on the regression residuals (theobserved concentration minus that predicted by the fittedregression model) as a means of identifying anomalous behaviour. For example, Moorby et al. (1987) fitted quadratic trendsurfaces (see above) to the residuals of Pb and Zn as predicted
by separate regression models fitted to the suite of elements{Ca, Mg, Al, Fe, Mn} in order to delineate broad trends ofbackground variation in carbonaterich marine sediments (andhence the spatial setting of anomalous concentration values) in
two areas of the continental shelf of Greece. Stable anomalypatterns were shown to exist off the Sounion Peninsula, aknown area of mineralisation.
However, where it is crucial that the relative importance ofa number of elements in controlling the behaviourof another isdetermined, Hoerl & Kennard (1970) recognised that wheneverthe supposedly independent predictors in a linear regressionmodel are correlated (as is always the case where geochemicaldata are concerned) it will lead to the coefficients of somepredictors in the fitted regression equation which will be toolarge, and may even be of the wrong sign. Consequently, theyintroduced the ridge regression method to overcome suchundesirable features. The existence of their work was firstbrought to the attention of geologists by Jones (1972). It wasprogrammed for use in AGRG by Turner in 1979 (Turner1980), and the RIDGE11 program was subsequently extendedby Howarth in 198182, during work on the NURE contract
with the University of Georgia (see above; Howarth 1984;Howarth & Koch 1986) to include interactive selection of theridge parameter, choice of variables, progressive deletion of
outliers, and resubstitution of the entire data set, using the finalfitted equation, to obtain the residuals. The method wasextensively investigated in an exploration context by PhilipDavies (1983), and proved to be equally helpful in derivinginterpretational models in relation to the occurrence of bovinehypocupraemia (Leech et al. 1983; Leech 1984), and in theanalysis of a suite of marine mineral exploration data from thesouthwest Pacific (Coward 1986; Coward & Cronan 1987).
Turner (1986, 549593), using Ba, Pb and Zn as responsevariables for the 23element MorayBuchan data set, demonstrated the efficacy of robust ridge regression, and showed thatanomaly (regression residual) contrast was maximised ifuntransformed data were used.
Other work
Miscellaneous applications have included: analysis of variance(ANOVA) to quantify variability attributable to both fieldsampling and analysis (Garrett 1969; Howarth & Lowenstein1971, 1972) and in the doctoral thesis by Richard Duff (1975),and the application of robust ANOVA by Ramseyet al. (1992);development of statisticallybased criteria for the recognition ofuraniferous granitoids from NURE HSSR data (Koch et al.1981a,b; Howarthet al. 1981); and the application of numericalmodelling in vapour geochemistry by Ruan Tianjian (Ruan1981; Ruanet al. 1985a, b). In more recent years, GeographicalInformation Systems have been applied in studies ofenvironmental and urbangeochemistry by workers in the
Environmental Geochemistry Research Group, the successorto AGRG at Imperial College (TristanMontero 2000; Tristan etal. 2000; Thums & Farago 2001; Thums 2003; Li et al. 2004;
Appletonet al. 2008).
ANALYTICAL QUALITY ASSURANCE
The development of analytical methods and related qualityassurance and interpretation methods in the GPRC and AGRGare discussed by Thompson (2010) but, for the sake ofcompleteness, brief details are also included here. As wasmentioned in the Introduction, Cravens statistical seriesapproach continued to be used into the 1970s (Stanton 1966;
James 1970), but it came to be recognized that the low and
highconcentration endmembers of a statistical series mightnot be representative, so far as their nature and matrix wereconcerned, of the field samples being analysed, and that themethod could only provide either an estimate of analytical
Fig. 20.Nonlinear mapping onto 2dimensions of the geochemistryof manganese nodules from the Pacific Ocean on the basis ofnormalised Mn, Fe, Co, Ni, Cu, Pb and Ti. The compositionalcontinuum is divided into 6 classes for purposes of interpretation.(Redrawn from Glasbyet al. 1977, fig. 1).
Statistical analysis and data display at GPRC and AGRG 309

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precision ( repeatability) at a particular concentration, or anaverage precision value over the concentration range.
Thompson & Howarth (1973, 1976, 1978), Howarth &
Thompson (1976 ), and Thompson (1978, 1981, 1983), developed an alternative approach, based on duplicate analysis ofrandomised splits of routine field samples in which it was
Fig. 21. Spatial disposition in thePacific Ocean (Lambert equalareaprojection) of the 6 classes from thenonlinear mapping of Fig. 20. (Redrawnfrom Glasbyet al. 1977, fig. 3).
Fig. 22. Comparison of the BoxCoxtransform in reducing skewness (s) andkurtosis (k; shown ask) for a data set
with the same parameters for theuntransformed and logtransformed data
values (Howarth & Earle 1979, fig. 8).
R. J. Howarth & R. G. Garrett310

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assumed: (i) that analytical error could generally be wellmodelled by the normal distribution (Thompson & Howarth
1980); and (ii) that analytical precision varied as a linearfunction of concentration in the analytical system (Thompson1988) which, it turns out, was also assumed by Jenkins in hisunpublished (1959?) manuscript mentioned p. 290. The dupli
cate analysis method rapidly became established within AGRG,alongside the use of classical Shewhart ( 1931) control charts to
control analytical batch performance through the monitoring ofanalyte concentration levels in splits of longterm house reference materials (Thompson 1981, 1983). The ThompsonHowarth chart, as it became named, was subsequently adopted
Fig. 23. (a) KleinerHartigan ( KH)tree morphology for theMorayBuchan, Scotland, streamsediment data set based on Wards(1963) agglomerative clusteringalgorithm applied to a robustcorrelation matrix of BoxCoxtransformed data (redrawn from Turner1986, fig. 7.76); (b) Examples of KHtrees for actual samples from theMorayBuchan data set (portion of
Turner 1986, fig. 7.102).
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by the wider geochemical and chemical community (e.g. Analytical Methods Committee 2002) and their approach continuesto be extended in scope (e.g. Stanley 2006; Stanley & Lawie2008).
In other applications, simulation and regression techniqueshave been applied to evaluation of matrix correction andinterference effects (Howarth 1973d; Thompson et al. 1979)and to the comparison of analytical accuracy between analyticalmethods (Thompson 1982). More recently, robust ANOVAhas been used to determine the magnitude of analytical variancein relation to other sources of variance in geochemical data(Ramseyet al. 1992).
When John Webb initiated the pioneering series of multielement multipurpose geochemical atlases in the mid1960s,there was inevitable tradeoff between the analytical methodused, expected analytical precision, and rapidity of turnround;this was not what traditional geochemists were used to, and thematter proved controversial. AGRG staff had to justify thisnew approach (Howarth & Lowenstein 1971, 1972; Webb &
Thompson 1977; Webb et al. 1978). Even today, despite
considerable advances in analytical methods, such a fitnessforpurpose approach to analysis requires explanation (Thompson& Fearn 1996; Fearn et al. 2002).
Looking back now, it is probably impossible for youngergeochemists to realise just how difficult it was, not only toimplement many of the statistical techniques, where we werebreaking new ground in applied geochemistry, but to convincepotential users of the utility of the results. In a broaderperspective, Garrett et al. (2008) reviewed the development ofinternational geochemical mapping to date; it is pleasing tothink that AGRG pioneered many of the methods that subsequently became adopted.
The development and implementation of the computerbasedmethods over the years described here was enabled by many bodies.
We principally have to thank the Department of Scientific andIndustrial Research and its successor, the Natural EnvironmentResearch Council in Britain for their support to AGRG over manyyears; other contributions have come from the Anglo AmericanCorporation (South Africa) Ltd.; the Institute of GeologicalSciences/British Geological Survey; Roan Selection Trust TechnicalServices; Sierra Leone Geological Survey; Ministerio de Economia,Industria y Comercio de Costa Rica; Wolfson Foundation; and theU.S. Department of Energy, National Aeronautics and Space
Administration, and Rome Air Development Center (New York).We are grateful to them all for their assistance, whether throughresearch contracts, support for studentships, or other help. Theauthors are most grateful to the Editor, Gwendy Hall, and the
Association of Applied Geochemists for their assistance with thefunding of the colour illustrations in this paper.
REFERENCES
A, L.H. 1954. The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary). Geochemica et CosmochimicaActa, 5 , 4973; 6 , 121131.
A,J. 1982. The statistical analysis of compositional data. Journal of theRoyal Statistical Society, B44, 139177.
A,J. 1986.The Statistical Analysis of Compositional Data. Chapman andHall. London and New York.
A, H. 1978a. A method of bivariate interpolation and smooth surfacefitting for irregularly distributed data points.ACM Transactions on MathematicalSoftware, 4 , 148159.
A,H. 1978b. Algorithm 526. Bivariate interpolation and smooth surfacefitting for irregularly distributed data points.ACM Transactions on MathematicalSoftware, 4 , 160164.
A,H. 1979. Remark on Algorithm 526. ACM Transactions on MathematicalSoftware, 5 , 242243.
A,P. & K,W.C. 1962. Secondary trend components in the TopAshdown Pebble Bed: A case history. Journal of Geology, 70 , 507538.
ANALYTICAL METHODS COMMITTEE 2002. A simple fitnessforpurpose control chart based on duplicate results obtained from routine testmaterials. Analytical Methods Committee Technical Brief no. 9, at the website:www.rsc.org/Membership/Networking/InterestGroups/Analytical/AMC/TechnicalBriefs.asp
A,D.F., B, P.J., H, F.R., H, P.J., R,W.H. &T, J.W. 1972. Robust estimates of location. Survey and advances. PrincetonUniversity Press, Princeton, NJ.
A, J.D., R, B.G. & T, I. 2008. National scaleestimation of potentially harmful elements background concentrations intopsoil using parent material classified soil:stream sediment relationships.Applied Geochemistry, 23, 25962611.
AB, A. 1971. Provincial and regional geochemical studies in Zambia.Unpublished PhD thesis, University of London, UK.
AB, A. & N, I. 1970. Regional geochemical reconnaissance and the location of metallogenic provinces. Economic Geology, 65,312330.
B,G.H. & H,D.J. 1965. ISODATA, a novel method of data analysisand pattern classification. Stanford Research Institute, Menlo Park, CA.Research Report, AD699616, April 1965.
B,P. & D,B.R. 1955. Numerical weather map analysis. Tellus,7, 1660.
B, A. 1943. On a measure of divergence between twostatistical populations defined by their probability distributions. Bulletin ofthe Calcutta Mathematical Society, 35, 99109.
B,G.E.P. & C,D.R. 1964. An analysis of transformations. Journal of theRoyal Statistical Society, B26, 211252.
BGS. 1991. Regional geochemistry of the East Grampiansarea. British Geological Survey, Keyworth, Nottingham.
B,A., MF,G. & PG,V. (eds) 2006.Compositional Data Analysis in the Geosciences. From Theory to Practice. Geological Society, London, Special Publication, 264.
C, N.A. 1979. Canonical variate analysis: Some practical aspects. Unpublished PhD thesis, University of London, UK.
C, N.A. 1980. Robust procedures in multivariate analysis. I. Robustcovariance estimation. Applied Statistics, 29 , 231237.
C, N.A. 1982. Statistical treatment of geochemical data. In: S,R.E. (ed.) Geochemical Exploration in Deeply Weathered Terrain. CSIROInstitute of Energy and Earth Resources, Floreat Park, WA, 141144.
CM,R. 1973.Application of discriminant and cluster analysis to regionalgeochemical surveys. Unpublished PhD thesis, University of London, UK.
CM, R. & H, R.J. 1976. Application of the empiricaldiscriminant function to regional geochemical data from the UnitedKingdom.Bulletin of the Geological Society of America, 87, 15671581.
C,P.E. 1998.A history of modern computing. MIT Press, Cambridge, MS.
C, F. 1971. Ratio Correlation. A Manual for Students of Petrology andGeochemistry. The University of Chicago Press, Chicago and London.
C,H. 1973. The use of faces to represent points in Kdimensionalspace graphically. Journal of the American Statistical Association. 68 , 361368.
C,R. & K,P.G. 1976. Digital image processing on a microfilmplotter. Unpublished report, University of London Computer Centre,London.
C, R.N. 1986. A statistical appraisal of regional geochemical data from thesouthwest Pacific for mineral exploration. Unpublished PhD thesis, University ofLondon, UK.
C, R.N. & C, D.S. 1987. A statistical evaluation of geochemicaldata in regard to bedrock and placer mineral exploration in the S.W.Pacific. Marine Mining, 6 , 205221.
C, C.A.U. 1954. Statistical estimation of the accuracy of assaying.Transactions of the Institution of Mining & Metallurgy, London, 63, 551563.
C, D.A. 1974. Application of multivariate methods to regional geochemistry: theevaluation of a new technique. Unpublished MSc thesis, University of London,UK.
D,M. 1977. Geostatistical ore reserve estimation. Elsevier, Amsterdam.
D, P.R. 1983. Geochemical applications of ridge regression for tinmineralisedgranitoids. Unpublished MSc thesis, University of London, UK.
D, J.C. & S, R.J. 1973. Statistics and data analysis in geology. JohnWiley & Sons, New York.
D,W.J. & M,F.J. 1957.Introduction to Statistical Analysis. 2nd edition.McGrawHill Book Co, New York.
D, J.R.V. 1975. Variability in some stream sediment geochemical data from
Australia. Unpublished PhD thesis, University of London, UK.E,S.A.M. 1982. Geological interpretation of the geochemistry of stream sediments,
waters and soils in the Bristol district, with particular reference to the Mendip Hills,Somerset. Unpublished PhD thesis, University of London, UK.
R. J. Howarth & R. G. Garrett312

7/25/2019 StatAnalysis Howarth 2010
25/27
E, S.A.M. 1978. Spatial presentation of data from regional geochemicalstream surveys. Transactions of the Institution of Mining and Metallurgy, London,B87, 6165.
F, T., F, S.A., T, M. & E, S.L. 2002. A decisiontheory approach to fitness for purpose in analytical measurement. TheAnalyst, 127, 818824.
F, T.L. 1975. Feature selection in multiclass pattern recognition. UnpublishedMSc thesis, University of London, UK.
G, R.G. 1966. Regional Geochemical Reconnaissance of Eastern Sierra Leone.Unpublished PhD thesis, University of London, UK.
G, R.G. 1967. Two programs for the factor analysis of geologic and remotesensing data. National Aeronautics and Space Administration, NorthwesternUniversity Report, 12 .
G, R.G. 1969. The determination of sampling and analytical errors inexploration geochemistry. Economic Geology, 64 , 568569.
G, R.G. 1983. Opportunities for the 80s. Mathematical Geology, 15,385398.
G, R.G. & N, I. 1967. Regional geochemical reconnaissance ineastern Sierra Leone. Transactions of the Institution of Mining & Metallurgy,London, B76, 97B112.
G, R.G. & N, I. 1969. Factor analysis as an aid in theinterpretation of regional geochemical stream sediment data. In: C,F.C. (ed.) Proceedings of the International Geochemical Exploration Symposium
(April 1720, 1968, Colorado School of Mines, Golden, Colorado).Quarterly of the Colorado School of Mines, 64, 245264.
G, R.G., R,C., S,D.B. & X,X. 2008. From geochemical prospecting to international geochemical mapping: a historical overview. Geochemistry: Exploration, Environment, Analysis, 8 , 205217.
G,G.P., T,J.S. & H,R.J. 1974. Geochemistry of manganese concretions from the northwest Indian Ocean. New Zealand Journal ofScience, 17, 387407.
G, J.G.S. 2010. Early years in the Geochemical Prospecting ResearchCenter, Imperial College of Science and Technology, London: explorationgeochemistry in Zambia in the late 1950s; a personal recollection.Geochemistry: Exploration, Environment, Analysis, 10 , 237249.
H, G. 1962.Geochemical dispersion patterns and their relation to bedrock geologyin the Nyawa area, N. Rhodesia. Unpublished PhD thesis, University ofLondon, UK.
H,H.E. & W,J.S. 1962.Geochemistry in Mineral Exploration. Harper &
Row, New York.HR, B.H. 1968. Geochemical dispersion of tin in marine
sediments. Mounts Bay, Cornwall. Unpublished PhD thesis, University ofLondon, UK.
H, M.J.R. 1968. Multivariate normal plotting. Applied Statistics. 17,157161.
H,D.R. 2005.Nondetects And Data Analysis. Statistics for Censored Environmental Data. WileyInterscience, John Wiley, Hoboken, NJ.
H, D.R. & H, R.M. 1992. Statistical Methods in Water Resources.Studies in Environmental Science, 49 . Elsevier, Amsterdam, London andNew York.
H,A.E. & K,E.W. 1970. Ridge regression: biased estimation fornonorthogonal problems. Technometrics, 12 , 5567, 6982.
H, W.S. 1966. Automatic photointerpretation and target location.Proceedings of the IEEE, 54 , 16791686.
H, H. 1933. Analysis of a complex of statistical variables intoprincipal components. Journal of Educational Psychology, 24, 417441,498520.
H, R.J. 1971a. FORTRAN IV program for greylevel mapping ofspatial data. Mathematical Geology, 3 , 95121.
H, R.J. 1971b. An empirical discriminant method applied to sedimentary rock classification from majorelement geochemistry. MathematicalGeology, 3 , 5160.
H, R.J. 1971c. Empirical discriminant classification of regionalstreamsediment geochemistry in Devon and east Cornwall. Transactions of the Institution of Mining and Metallurgy, London, B80, 142149.
H, R.J. 1972. Empirical discriminant classification of regionalstreamsediment geochemistry in Devon and east Cornwall. Discussion.Transactions of the Institution of Mining and Metallurgy, London, B81,115119.
H, R.J. 1973a. FORTRAN IV programs for empirical discriminantclassification of spatial data. Geocom Bulletin, 6 , 131.
H, R.J. 1973b. Preliminary assessment of a nonlinear mapping
algorithm in a geological context. Mathematical Geology, 5 , 3957.H, R.J. 1973c. The pattern recognition problem in applied geochem
istry.In:J,M.J. (ed.)Geochemical Exploration 1972. Institution of Miningand Metallurgy, London, 259273.
H, R.J. 1973d. Monte Carlo simulation of matrix correlation effects.The Analyst, 98 , 777781.
H, R.J. 1974. The impact of pattern recognition methodology ingeochemistry [Abstract]. Proceedings of the Second Joint Conference on PatternRecognition. Copenhagen, August 1974, 411412.
H, R.J. 1977. Approximate levels of significance for the cos thetacoefficient. Computers & Geosciences, 3 , 2530.
H, R.J. 1983. Mapping. In: H, R.J. (ed.), Statistics and dataanalysis in geochemical prospecting. Elsevier, Amsterdam, 111205.
H, R.J. 1984. Statistical applications in geochemical prospecting: Asurvey of recent developments. Journal of Geochemical Exploration,21, 4161.
H, R.J. 2004. Not just a petrographer: The life and work of FelixChayes (19161993). Earth Sciences History, 23, 343364.
H, R.J., C,D.S. & G,G.P. 1977. Nonlinear mapping ofregional geochemical variability of manganese nodules in the PacificOcean.Transactions of the Institution of Mining and Metallurgy, London,B86, 48.
H, R.J. & E, S.A.M. 1979. Application of a generalised powertransform to geochemical data. Mathematical Geology, 11, 4558.
H, R.J. & J,R.W. 1977. Multielement trends of variation ofthe South Bismark Sea rocks as shown by the nonlinear mappingalgorithm. In: J,R.W. (ed.) Distribution and majorelement chemistry of late Cainozoic volcanoes at the southern margin of the Bismark Sea, Papua NewGuinea. Australian Bureau of Mineral Resources, Canberra. 162170.
H, R.J. & K,G.S. jr 1986. Problems of using rockvolume data inpredictive resource studies. Economic Geology, 81, 617626.
H, R.J., K, G.S., C, C.Y., C, R.H. &S,J.H. 1980. Statistical map analysis techniques applied toregional distribution of uranium in stream sediment samples from thesoutheastern United States for the National Uranium Resource Evaluationprogram. Mathematical Geology, 12, 339366.
H, R.J., K, G.S. jr, P, J.A. & L, R.K. 1981. Identification of uraniferous granitoids in the USA using stream sedimentgeochemical data. Mineralogical Magazine, 44, 455470.
H, R.J. & L, P.L. 1971. Sampling variability of streamsediments in broadscale geochemical reconnaissance. Transactions of theInstitution of Mining and Metallurgy, London, B80, 363372.
H, R.J. & L, P.L. 1972. Sampling variability of streamsediments in broadscale geochemical reconnaissance. Discussion. Transactions of the Institution of Mining and Metallurgy, London, B81, 122124.
H, R.J. & L,P.L. 1974. Data Processing for the ProvisionalGeochemical Atlas of Northern Ireland. Applied Geochemistry Research Group,Imperial College of Science and Technology, London. Technical Communication,61.
H, R.J. & L, P.L. 1976. Threecomponent colour mapsfrom lineprinter output. Transactions of the Institution of Mining and Metallurgy,London, B85, 234237.
H, R.J. & T, M. 1976. Duplicate analysis in geochemicalpractice. II. Examination of the proposed method and examples of its use.The Analyst, 101, 699709.
H,P.J. 1981. Robust statistics. John Wiley, New York.I,J. 1963. Factor and vector analysis programs for analysing geologic data. Office
of Naval Research, Geography Branch. Northwestern University, Evanston,Illinois. Technical Report no. 6. ONR Task no. 389135.
I,J. & P,E.G. 1962. Classification of modern Bahamian carbonatesediments. In: H, W.E. (ed.) Classification of carbonate rocks: a symposium.Memoir 1, American Association of Petroleum Geologists, Tulsa, OK.253272.
INSTITUTE OF GEOLOGICAL SCIENCES 1978. Geochemical atlas of GreatBritain: Shetland Islands. Institute of Geological Sciences [British GeologicalSurvey], London.
J,C.H. 1970. A rapid method for calculating the statistical precision ofgeochemical prospecting analyses. Transactions of the Institution of Mining andMetallurgy, London, B79, 8889.
J,H. 1946. An invariant form for the prior probability in estimationsproblems. Proceedings of the Royal Society, London, A186, 453461.
J, T.A. 1972. Multiple regression with correlated independent variables.Mathematical Geology, 4 , 203218.
K,T. 1967. The Divergence and Bhattacharyya distance measures insignal selection. IEEE Transactions on Communication Technology, 15, 5260.
K, H.F. 1958. The varimax criterion for analytic rotation in factoranalysis. Psychometrika, 23, 187200.
K, J. 1969. The application of some data processing techniques to the
interpretation of geochemical data. Unpublished PhD thesis, University ofLondon, UK.
K,B. & H, J.A. 1981. Representing points in many dimensionsby trees and castles. Journal of the American Statistical Association, 76, 260269.
Statistical analysis and data display at GPRC and AGRG 313

7/25/2019 StatAnalysis Howarth 2010
26/27
K, G.S. Jr, H, R.J., C, R.H. & S, J.H.1979. Development of data enhancement and display techniques for streamsedimentdata collected in the National Uranium Resource Evaluation Program of the UnitedStates Department of Energy. U.S. Department of Energy, Grand Junction,Colorado. Openfile Report, GJBX28(80).
K,G.S. Jr, H, R.J. & S,J.H. 1981a.Uranium resourceassessment through statistical analysis of exploration geochemical and other data. FinalReport. U.S. Department of Energy, Grand Junction, Colorado. OpenfileReport, GJBX140(81).
K,G.S. Jr, H, R.J., S,J.H. & L, R.K. 1981b.Uranium resource assessment through statistical analysis of explorationgeochemical and other data. Economic Geology, 76 , 10561066.
K, A.N. 1933. Sulla determinazione empirico di una legge didistribuzione. Giornale dellIstituto Italiano degli Attuari, Rome, 4 , 8391.
K, W.C. 1959. Trend Surface analysis of contourtype maps withirregular controlpoint spacing. Journal of Geophysical Research, 64, 823834.
K,W.C. & G,F.A. 1965. An introduction to statistical models ingeology. McGrawHill Book Co, New York.
K,W.C. & I, J. 1963. Stratigraphic factor maps. Bulletin of theAmerican Association of Petroleum Geologists, 47, 698701.
L,A.F. 1984. The application of regional geochemistry to the causes and predictedincidence of bovine hypocupraemia. Unpublished PhD thesis, University ofLondon, UK.
L,A., T,I., H,R.J. & L,G. 1983. The incidence ofbovine hyprocupraemia in England and Wales and its relationship withgeochemistry. In: S, N.F., G, R.G., A, W.M., L,K.A. & W, G. (eds) Trace elements in animal production and veterinarypractice. British Society of Animal Production. Occasional paper, 7, 130131.
L,X.D., L,S.L., W,S.C., S,W.Z. & T,I. 2004. The studyof metal contamination in urban soils of Hong Kong using a GISbasedapproach. Environmental Pollution, 129, 113124.
L, P.L. & H, R.J. 1973. Automated colour mapping ofthreecomponent systems and its application to regional geochemicalreconnaissance. In: J, M.J. (ed.) Geochemical Exploration 1972. Institution of Mining and Metallurgy, London, 297304.
M, S.J. 1980. Computerbased interpretation of large regional geochemical datasets. Unpublished PhD thesis, University of London, UK.
M, S.J. & H, R.J. 1978.Factor score maps of regional geochemical data
from England and Wales. Applied Geochemistry Research Group, ImperialCollege of Science, Technology and Medicine, London., 2 sheets.
M, S.J. & H, R.J. 1980. Power transform removal of skewnessfrom large data sets. Transactions of the Institute of Mining and Metallurgy,London, B89, 9297.
M,V. & I,J. 1964. FORTRAN program for factor and vector analysisof geologic data using an IBM 7090 or 7094/1401 computer system. KansasGeological Survey, Lawrence, KS. Special Distribution Publication, 13.
M, G. 1963. Principles of geostatistics. Economic Geology, 58 , 12461266.
MC, G.A. 1963. Optimization by function contouring techniques. Space andInformation Systems Division, North American Aviation Inc., Downey,CA. Report SID 63171.
MC,G.A. & DP,H.J. 1965.Improved FORTRAN IV function contouringprogram. Space and Information Systems Division, North American Aviation Inc., Downey, CA. Report SID 65672.
M,R.L. 1956.Trend surfaces: their application to analysis and descriptionof environments of sedimenation. I. The relation of sedimentsize parametersto currentwave systems and physiography. Journal of Geology, 64, 425466.
M,R.L. & K,J.S. 1962.Statistical Analysis in the Geological Sciences. JohnWiley & Sons, New York, USA.
M, S.A., H, R.J., S, P.A. & C, D.S. 1987. Aninvestigation of the applicability of trend surface analysis to marineexploration geochemistry.In: T,P.G., D,M.R., M,J.R. &S,U. (eds)Marine Minerals: Advances in Research and ResourceAssessment (NATO ASI series). Series C: Mathematical and Physical Sciences, 194. D. Reidel, Dordrecht, 559576.
M,M.J. 1960. Facts from Figures. Penguin Books, Harmondsworth.M,J.L. & A,D.C. 1968. Experimental evaluation of techniques for
automatic segmentation of objects in a complex scene. In: C, G.C.,L, R.S., P, D.K. & R, A. (eds) Pictorial patternrecognition. Thompson Book Co, New York, 313.
N,I., G, R.G. & W,J.S. 1966a. Studies in regional geochem
istry. Transactions of the Institution of Mining & Metallurgy, London, B75,106107.
N,I., G, R.G. & W,J.S. 1966b. Automatic data plotting andmathematical and statistical interpretation of geochemical data. In:
C, E.M. (ed.) Proceedings of the Symposium on Geochemical Prospecting,Ottawa, April, 1964. Geological Survey of Canada Paper 6654, 195210.
N, I., G,R.G. & W, J.S. 1969. The role of some statisticaland mathematical methods in the interpretation of regional geochemicaldata. Economic Geology, 64 , 204220.
N, I., T, I., W, J.S., F, W.K., H, R.J.,K,J. & T,D. 1970a. Regional geochemical reconaissanceof the Derbyshire area. Report 70/2. Institute of Geological Sciences[British Geological Survey], London.
N, I., T, I., W, J.S., F, W.K., H, R.J.,
K,J. & T,D. 1970b. Regional geochemical reconaissanceof the Denbighshire area. Report 70/8. Institute of Geological Sciences[British Geological Survey], London.
N, I., T, I., W, J.S., F, W.K., H,R.J. &
K, J. 1971. Regional geochemical reconaissance of the Devonand North Cornwall area. Report 71/2. Institute of Geological Sciences[British Geological Survey], London.
N,I. & W,J.S. 1967. The application of computerised mathematicaland statistical procedures to the interpretation of geochemical data.Proceedings of the Geological Society of London, 1642, 186199.
P, J.M. 1970. FORTRAN IV program for Qmode cluster analysis ondistance function with printed dendrogram. Computer Contribution no. 46.Kansas Geological Survey, Lawrence, KS.
R, M.H., T, M. & H, M. 1992. Objective evaluation ofprecision requirements for geochemical analysis using robust analysis ofvariance. Journal of Geochemical Exploration, 44 , 2336.
R,D.W., S,M.A. & H, R.J. 1973. Experimental geochemical maps a case study in cartographic techniques for scientific research.The Cartographic Journal, 10, 112118.
R, T. 1981. Some new approaches in vapour geochemistry. Unpublished PhDthesis, University of London, UK.
R, T., H,R.J. & H, M. 1985a. Numerical modelling experiments in vapour geochemistry. I: Method and FORTRAN program.Computers & Geosciences, 11, 5567.
R, T., H, M. & H, R.J. 1985b. Numerical modelling experiments in vapour geochemistry. II: Vapour dispersion patterns and exploration implications. Journal of Geochemical Exploration, 23, 265280.
R,T.A. jr, J,B.L. & R,B.F. 1976.MINITAB student handbook.Duxbury Press, North Scituate, MS.
S,J.W. jr 1969. A nonlinear mapping for data structure analysis. IEEETransactions on Computers, C18, 410409.
S,D. 1968. A twodimensional interpolation function for irregularlyspaced data. In: Proceedings of the 23rd National Conference of the Association forComputing Machinery. Brandon/Systems Press, Princeton, NJ, 517523.
S, W.A. 1931. The Economic control of manufactured product. D. VanNostrand Company, New York and London.
S,V.I. 1939. On the estimation of the discrepancy between empirical
curves of distribution for two independent samples. Bulletin Mathmatique delUniversit de Moscou, 2 , fasc. 2.
S, C. 1904. General intelligence, objectively determined andmeasured. American Journal of Psychology, 15, 201293.
S, D.F. 1967. Generation of polynomial discriminant functions forpattern recognition. IEEE Transactions on Electronic Computers, EC16,308319.
S, R.E. 1966. Rapid methods of trace analysis for geochemical applications.
Edward Arnold, London.S,C.R. 2006. On the special application of ThompsonHowarth error
analysis to geochemical variables exhibiting a nugget effect. Geochemistry:Exploration, Environment, Analysis, 6 , 357368.
S,C.R. & L,D. 2008. ThompsonHowarth error analysis: unbiased alternatives to the largesample method for assessing nonnormally
distributed measurement error in geochemical samples. Geochemistry: Exploration, Environment, Analysis, 8 , 173182.
S,J.E. 1959.A statistical problem in geochemical prospecting. Unpublished MScthesis, Imperial College, University of London, UK.
S[W. S . G] 1908. Probable error of a correlation coefficient.Biometrika, 6 , 302310.
S, C.P. 1971. Phosphate deposits on the northwest African continentalshelf and slope. Unpublished PhD thesis, University of London, UK.
S, C.P. 1972. Geochemistry of continental margin sedimentsfrom northwest Africa. Chemical Geology, 10 , 137156.
T, L.L. 1947. Multiple factor analysis. University of Chicago Press,Chicago.
T, M. 1978. DUPAN3, a subroutine for the interpretation ofduplicated data in geochemical analysis. Computers & Geosciences,4, 333340.
R. J. Howarth & R. G. Garrett314

7/25/2019 StatAnalysis Howarth 2010
27/27
T,M. 1981. Quality control in the laboratory. In: F,W.K.(ed.) Analytical methods in geochemical prospecting. Handbook of ExplorationGeochemistry, 1 , 2546.
T,M. 1982. Regression methods in the comparison of accuracy. TheAnalyst, 107, 11691180.
T, M. 1983. Control procedures in geochemical analysis. In:H, R.J. (ed.) Statistics and data analysis in geochemical prospecting.Handbook of Exploration Geochemistry, 2 , 3958.
T, M. 1988. Variation of precision with concentration in ananalytical system. The Analyst, 113, 15791587.
T, M. 2010. Analytical methodology in the Applied GeochemistryResearch Group (19501988) at the Imperial College of Science andTechnology, London. Geochemistry: Exploration, Environment, Analysis, 10,251259.
T, M. & F, T. 1996. What exactly is fitness for purpose inanalytical measurement. The Analyst, 121, 275278.
T,M. & H,R.J. 1973. The rapid estimation and control ofprecision by duplicate determinations. The Analyst, 98 , 153160.
T, M. & H, R.J. 1976. Duplicate analysis in geochemicalpractice. I. Theoretical approach and estimation of analytical reproducibility. The Analyst, 101, 690698.
T,M. & H,R.J. 1978. A new approach to the estimation ofanalytical precision. Journal of Geochemical Exploration, 9 , 2330.
T, M . & H, R.J. 1980. The frequency distribution ofanalytical error. The Analyst, 105, 11881195.
T, M., W, S.J. & W, S.J. 1979. Statistical appraisal ofinterference effects in the determination of trace elements by atomicabsorption spectrophotometry in applied geochemistry. The Analyst, 104,299312.
T,I. 2010. Research in Applied Environmental Geochemistry, withparticular reference to Geochemistry and Health. Geochemistry: Exploration,Environment, Analysis, 103, 000000.
T, C.R. 2003. Geochemical associations and the spatial distribution of metals inurban soils. Unpublished PhD thesis, University of London, UK.
T, C. & F, M.E. 2001. Investigating urban geochemistry usinggeocgaphical information systems. Science Progress, 84 , 183204.
T, J.S. 1967. The inorganic mineral potential of the seafloor andproblems in its exploration. In: Proceedings of the British National Conference onthe Technology of the Sea and Seabed held at the Atomic Energy ResearchEstablishment, Harwell, April 5th, 6th and 7th, 1967; sponsored by the Ministry of
Technology. United Kingdom Atomic Energy Authority (Research Group),Harwell, Didcot, Berks. Report AERER 5500. Her Majestys StationeryOffice, London. Paper SB16, 120.
T, E., D, A., R , M.H., R, M.S.,S, P., T, I., V, E . & V, K. 2000.
Spatially resolved hazard and exposure assessments: an example of lead insoil at Laviron, Greece. Environmental Research, A82, 3345.
TM,E.E. 2000.Human health risk assessment for contaminated landin historical mining areas. Unpublished PhD thesis, University of London,UK.
T, J.W. 1977. Exploratory data analysis. AddisonWesley, Reading, MS[preliminary edition printed for private circulation, 1970].
T, M.St.J. 1980. A comparative study of multiple regression techniques ingeochemistry. Unpublished MSc thesis, University of London, UK.
T, M.St.J. 1986. Statistical analysis of geochemical data illustrated by reference tothe Dalradian of N.E. Scotland. Unpublished PhD. Thesis, University ofLondon, UK.
W, J.H. 1963. Hierarchical grouping to optimize an objective function.Journal of the American Statistical Association. 58, 236244.
W, J.S. 1964. Geochemistry and life. New Scientist. 23, 504507.
W, J.S. & A, W.J. 1965. Regional geochemical reconnaissanceapplied to some agricultural problems in Co. Limerick, Eire. Nature, 208,10561059.
W, J.S. & T, M. 1977. Analytical requirements in explorationgeochemistry. Pure and Applied Chemistry, 49, 15071518.
W,J.S., F,J., N,I. & T,J.S. 1964a.Regional geochemicalreconnaissance in the Namwala Concession area Zambia. To accompany theGeochemical Maps of the Namwala Concession Area published by theGeological Survey of Zambia in. 1964. Applied Geochemistry ResearchGroup, Imperial College of Science and Technology, London. TechnicalCommunication no. 47.
W, J.S., F,J. et al. 1964b. Regional geochemical maps of the NamwalaConcession area, Zambia based on a Reconnaissance Stream Sediment Survey.Geological Survey of Zambia, Zambia.
W,J.S., N,I., F,R., L,P.L. & H, R.J. 1973.Provisional Geochemical Atlas of Northern Ireland. Applied GeochemistryResearch Group, Imperial College of Science and Technology, London.Technical Communication,60.
W, J.S., T,I., T, M., H, R.J. & L,P.L. 1978. The Wolfson Geochemical Atlas of England and Wales. ClarendonPress, Oxford and London.
W,E.H.T. 1959. Composition trends in granite: Modal variations and
ghost stratigraphy in part of the Donegal Granite, Eire. Journal of GeophysicalResearch, 64 , 835848.
W,E.H.T. 1963.A surfacefitting program suitable for testing geological modelswhich involve areallydistributed data. Office of Naval Research, GeographyBranch. Northwestern University, Evanston, Illinois. Technical Report No.2, ONR Task No. 389135, Contract No. 1228(26).
Received 29 July 2008; revised typescript accepted 8 April 2009.
Statistical analysis and data display at GPRC and AGRG 315