Science of the Total Environment

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1 Science of the Total Environment 46 (0) 96 0 Contents lists available at SciVerse ScienceDirect Science of the Total Environment journal homepage: The concept of compositional data analysis in practice Total major element concentrations in agricultural and grazing land soils of Europe Clemens Reimann a,, Peter Filzmoser b, Karl Fabian a, Karel Hron c, Manfred Birke d, Alecos Demetriades e, Enrico Dinelli f, Anna Ladenberger g and The GEMAS Project Team a Geological Survey of Norway, PO Box 635 Sluppen, N-749 Trondheim, Norway b Institute for Statistics and Probability Theory, Vienna University of Technology, Wiedner Hauptstrasse 8 0, A-040 Wien, Austria c Department of Mathematical Analysis and Applications of Mathematics, Palacký University, Faculty of Science, Listopadu, CZ-7746 Olomouc, Czech Republic d Federal Institute for Geosciences and Natural Resources (BGR), Branch office Berlin, Wilhelmstr. 5 30, D-3593 Berlin, Germany e Institute of Geology and Mineral Exploration, Entrance C, Olympic Village, Acharnae, Athens, GR-3677, Greece f University of Bologna, Department of Earth Science, Piazza di Porta San Donato, I-406 Bologna, Italy g Geological Survey of Sweden (SGU), Box 670, S-75 8 Uppsala, Sweden article info abstract Article history: Received 0 December 0 Received in revised form 4 February 0 Accepted 4 February 0 Available online April 0 Keywords: Agricultural soil XRF Major elements Europe Geochemistry Compositional data Applied geochemistry and environmental sciences invariably deal with compositional data. Classically, the original or log-transformed absolute element concentrations are studied. However, compositional data do not vary independently, and a concentration based approach to data analysis can lead to faulty conclusions. For this reason a better statistical approach was introduced in the 980s, exclusively based on relative information. Because the difference between the two methods should be most pronounced in large-scale, and therefore highly variable, datasets, here a new dataset of agricultural soils, covering all of Europe (5.6 million km ) at an average sampling density of site/500 km, is used to demonstrate and compare both approaches. Absolute element concentrations are certainly of interest in a variety of applications and can be provided in tabulations or concentration maps. Maps for the opened data (ratios to other elements) provide more specific additional information. For compositional data XY plots for raw or log-transformed data should only be used with care in an exploratory data analysis (EDA) sense, to detect unusual data behaviour, candidate subgroups of samples, or to compare pre-defined groups of samples. Correlation analysis and the Euclidean distance are not mathematically meaningful concepts for this data type. Element relationships have to be investigated via a stability measure of the (log-)ratios of elements. Logratios are also the key ingredient for an appropriate multivariate analysis of compositional data. 0 Elsevier B.V. All rights reserved.. Introduction Geochemistry aims to quantitatively determine the chemical composition of the Earth and its parts and to discover the factors that control the distribution of individual elements (Goldschmidt, 937, Corresponding author. Tel.: address: Clemens.Reimann@ngu.no (C. Reimann). S. Albanese, M. Andersson, A. Arnoldussen, R. Baritz, M.J. Batista, A. Bel lan, D. Cicchella, B. De Vivo, W. De Vos, M. Duris, A. Dusza-Dobek, O.A. Eggen, M. Eklund, V. Ernstsen, T.E. Finne, D. Flight, S. Forrester, M. Fuchs, U. Fugedi, A. Gilucis, M. Gosar, V. Gregorauskiene, A. Gulan, J. Halamić, E. Haslinger, P. Hayoz, G. Hobiger, R. Hoffmann, J. Hoogewerff, H. Hrvatovic, S. Husnjak, L. Janik, C.C. Johnson, G. Jordan, J. Kirby, J. Kivisilla, V. Klos, F. Krone, P. Kwecko, L. Kuti, A. Lima, J. Locutura, P. Lucivjansky, D. Mackovych, B.I. Malyuk, R. Maquil, M.J. McLaughlin, R.G. Meuli, N. Miosic, G. Mol, P. Négrel, P. O'Connor, K. Oorts, R. T. Ottesen, A. Pasieczna, V. Petersell, S. Pfleiderer, M. Poňavič, C. Prazeres, U. Rauch,. Salpeteur, A. Schedl, A. Scheib, I. Schoeters, P. Sefcik, E. Sellersjö, F. Skopljak, I. Slaninka, A. Šorša, R. Srvkota, T. Stafilov, T. Tarvainen, V. Trendavilov, P. Valera, V. Verougstraete, D. Vidojević, A.M. Zissimos, Z. Zomeni. 954). Geochemical studies need to be carried out from the atomic to the continental and finally global (some may argue cosmic) scale (for discussions of scale see: Darnley et al., 995; Reimann et al., 009, 00) to meet these aims. Geochemical data are usually reported as concentrations in units of mg/kg or weight percent (wt.%) and are thus a classical example of compositional (closed) data (CoDa Aitchison, 986). If all chemical elements in a sample are analysed, the analytical results sum up to a constant (,000,000 mg/kg or 00 wt.%). Thus no single variable is free to vary separately from the rest of the total composition. Even if not all chemical elements are analysed, the total element concentrations still depend on each other. The relevant information for each single variable in a geochemical dataset thus lies in the ratios between all variables and not in the measured element concentrations as such. An interpretation and statistical evaluation of the observed concentration values is only meaningful if the relationship to the values of the remaining variables is taken into account (Aitchison, 986; Filzmoser et al., 00). It could hence be argued that a multi-element geochemical /$ see front matter 0 Elsevier B.V. All rights reserved. doi:0.06/j.scitotenv

2 C. Reimann et al. / Science of the Total Environment 46 (0) dataset should only be analysed in multivariate space, without even considering the univariate case. However, a careful univariate data analysis has always been the starting point of statistical analyses of regional geochemical datasets (Reimann et al., 008). This is a reasonable approach because it helps to better understand the behaviour of the data before more sophisticated multivariate techniques are applied. For example, the very aim of a regional geochemical mapping project is to study and predict the distribution (concentration) of a chemical element in two-dimensional space. Such maps have been successfully used to aid geological mapping, for mineral exploration, for documenting contamination, and for detecting a multitude of additional processes that determine the distribution of chemical elements at the Earth's surface. It will be hard to convince a regional geochemist that all these maps are wrong and that dimensionless ratio maps (which ratio?) are the only correct maps. Thus, while many solutions to the closure problem exist for multivariate data analysis (e.g., Aitchison and Greenacre, 00; Buccianti and Pawlowsky-Glahn, 005; Buccianti et al., 006; Egozcue and Pawlowsky-Glahn, 0; Filzmoser and Hron, 008; Filzmoser et al., 009b; Hron et al., 00; Otero et al., 005; Pawlowsky-Glahn and Buccianti, 00, 0; Tolosana-Delgado and van den Boogaart, 0; von Eynatten et al., 003), a sensible approach to univariate and bivariate data analysis of compositional data, satisfying the statistician as well as the geochemist, is still under development (Filzmoser et al., 009a, 00). The administration of the new European REACH (Registration, Evaluation and Authorisation of Chemical EC, 006) regulation, which came into force on the st of June, 007, requires knowledge about soil quality at the European scale. The GEMAS (Geochemical mapping of agricultural and grazing land soils) project, a cooperation project between EuroGeoSurveys and Eurometaux, aims at providing such data for Europe. Samples of agricultural soil and of soil under permanent grass cover were taken during 008 at an average density of site/500 km, covering the member states of the European Union (exception Malta and Romania) and several neighbouring countries (e.g., Norway, Serbia, Ukraine). In total, over 4000 samples were collected, prepared and analysed (see also Reimann et al., 0a). The total concentrations of the major elements (Al O 3,CaO,Fe O 3,K O, MgO, MnO, Na O, P O 5,SiO, and TiO, plus Loss on Ignition (LOI)) in the soil samples, reported for the GEMAS project analysed by X-ray fluorescence spectrometry (XRF LOI gravimetric) are a classical example of a closed dataset. This dataset is used here: () To report the concentration of major elements in European agricultural soils. () To study the regional distribution of the major elements in order to better understand the processes governing the distribution of chemical elements in European agricultural soils and their relative importance at the continental scale. (3) To investigate effects of data closure and to understand which evaluation procedures may be applied to such data, and which should be avoided. (4) To compare alternative data analysis techniques to the classical way of treating geochemical data. (5) To further develop recommendations for the uni-, bi- and multivariate investigation of compositional datasets... The survey area Maps covering topography and land use for Europe can be found in almost any atlas. A number of maps covering different themes at about the scale of the GEMAS project (topography, geology, tectonics, fault and fracture zones, distribution of different rock types, distribution of the main sedimentary basins, precipitation and population density) are collected in Reimann and Birke (00). Fig. shows a simplified geological map including the main geological structures discussed in this paper. For Europe an excellent source of land use information is the CORINE land cover map of Europe (GLC000 database, 003). A detailed geological map of Europe is provided by Asch (003), and concise descriptions of the geology of Europe can be found in Ziegler (990), Blundell et al. (99) and McCann (008). The soil atlas of Europe provides a wealth of information on European soils, but also contains maps of average precipitation, temperature, land use, population density, extent of the last glaciation, and soil texture (Jones et al., 005).. Material and methods.. Project background and sampling GEMAS is a cooperation project between the Geochemistry Expert Group of EuroGeoSurveys (EGS) and Eurometaux. The GEMAS project aims to produce consistent soil geochemistry data at the continental scale in accordance to REACH (EC, 006) requirements. REACH specifies that industry must prove that it can produce and handle its substances safely. Risks due to the exposure to a substance during production and use at the local, regional and European scale all need to be assessed. Industries handling metals needed harmonised data on the natural distribution of chemical elements, and of soil properties governing metal availability in soils at the continental scale. REACH requires that risk assessment is performed according to land use. The GEMAS project focused on agricultural soils from arable and grazing land, both linked to the human food chain. According to REACH the sample depth should be 0 0 cm for agricultural soils (arable land, Ap-horizon) and 0 0 cm for grazing land soils (land under permanent grass cover) and the b mm grain size is the fraction to be analysed. With the exception of the sample density, the sampling requirements were thus rigidly fixed by external requirements. With regard to sample density it was decided to follow the example of an earlier project, covering Northern Europe (the Baltic Soil Survey: Reimann et al., 003) and to sample one site per 500 km (50 50 km grid). The grid cells were centrally provided, but the sample teams were free to decide where in a grid cell the two samples of agricultural and grazing land soil were taken. Sample materials and especially the bags used for storing the samples were centrally provided to all field teams. Samples were taken as composites from 5 sites spread over a ca. 00 m area in a large agricultural field (Ap-sample) and on land under permanent grass cover (Gr sample). The average weight of a sample was 3.5 kg. It was attempted to find sample sites for the Ap and Gr samples in as close proximity as possible. The average distance between the two sites is 500 m, but, depending on land use, single sample pairs, where the sites are more than 50 km apart do occur. All sites and the soil profile at any one site were documented in a series of photographs. Field procedures are detailed in a field handbook which is freely available on the internet (EGS, 008). For quality control purposes, a field duplicate was taken at every 0th sample site with an offset distance of ca. 0 0 m from the original sample site... Sample preparation All samples were prepared in a central laboratory (Geological Survey of the Slovak Republic). The samples were air dried and sieved to pass a mm nylon screen. All samples were then randomised and analytical duplicates and project standards were introduced at a rate of in 0. All samples were then split into ten aliquots using a Jones Rifflesplitter. Four splits of ~00 g each are stored for future reference, and 6 splits of g each were sent to the different contract laboratories for the immediate analytical work. For analysis by XRF the samples needed to be milled prior to further sample preparation. One of the small sample splits was milled to less than 63 μm in an agate disc mill at BGR's laboratory in Germany. Loss on Ignition (LOI) was then determined on all samples via slowly heating to 030 C, keeping them at this temperature for 5 min in a muffle furnace, letting them cool to room

3 98 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Fig.. Simplified geological map of the survey area (modified from Reimann et al., 0b). AM: Armorican Massif, BF: Black Forest, BM: Bohemian Massif, Co: Cornwall, H: Harz, IV: Iberian Variscides, ISC: Irish-Scottish Caledonides, MC: Massif Central, RS: Rhenish Slate Mountains, SC: Scandinavian Caledonides, V: Vosges. TESZ: Trans European Suture Zone. temperature in a dessicator and reporting the weight loss. For soil samples with a LOI b5% g of sample was mixed with 5 g lithium metaborate and 5 mg lithium bromide in Pt95-Au5 crucibles and fused for 0 min at 00 C in an automatic fluxer (HAG 500). For soil samples with LOI >5% a mixture of g of sample with.5 g lithium metaborate and.5 g lithium tetraborate was used for fluxing..3. Analysis Total concentrations of the 0 major elements reported here (Al O 3, CaO, Fe O 3,K O, MgO, MnO, Na O, P O 5,SiO and TiO ) were determined by wavelength dispersive X-ray fluorescence spectrometry (WD-XRFS) using PAN400 and AXIOS WD-XRFs with Cr- and Rh-anode X-ray tubes, respectively. To correct for matrix effects and spectral interferences calibration curves were constructed using 30 certified reference materials..4. Quality control Quality control (QC) was based on (a) a field duplicate taken at a rate of in 0 samples, (b) an analytical replicate produced from each field duplicate and (c) the frequent ( in 0) insertion of a project standard. Results of QC are documented in a report which is freely available on the internet (Reimann et al., 0b). Precision for all elements/ parameters reported here is better than 3%..5. Data analysis Data analysis and map plotting were carried out in R, an open source software, which can be freely downloaded from the CRAN server at The R scripts used for producing the graphics in this article can be found in the supplementary material. All results from the GEMAS project will be published in form of a book in 03. All project data will accompany that book in the form of excel files on an attached CD-ROM. 3. Results and discussion 3.. Tabulation of statistical distribution measures Table summarises the analytical results. In an attempt to provide valid measures for compositional data the table was built around percentiles (quantiles of the distribution). Although percentiles of the distribution are not influenced by a log-transformation, they can change

4 C. Reimann et al. / Science of the Total Environment 46 (0) Table Summary statistics, major elements, agricultural soil (Ap-samples, 0 0 cm, b mm fraction, N=08) and grazing land soil (Gr samples, 0 0 cm, b mm, N=04). All analytical results in weight percent (wt.%). Analysis of oxides by WD-XRF, LOI: gravimetric. Mat.: material, DL: detection limit; Min: minimum; Q: quantiles (Q50=median); Max: maximum; MAD: median absolute deviation,.log : for the log-transformed values,.ilr : for the ilr tansformed results; powers: orders of magnitude variation. Oxide Mat. DL Min Q Q5 Q0 Q5 Q50 Q75 Q90 Q95 Q98 Max MAD.log MAD.ilr Powers Al O 3 Ap Gr CaO Ap Gr Fe O 3 Ap Gr K O Ap Gr MgO Ap Gr MnO Ap Gr Na O Ap Gr 0.0 b P O 5 Ap Gr SiO Ap Gr TiO Ap Gr Parameter MAT. DL Min Q Q5 Q0 Q5 Q50 Q75 Q90 Q95 Q98 Max MAD.log MAD.ilr Powers LOI Ap Gr under a logratio-transformation. Thus, already the validity of Table could be questioned, because it reports percentiles of compositional data. To improve this situation, it would be desirable to calculate the percentiles of the logratio-transformed data, and to afterwards transform the results back to the original data scale for ease of interpretation. Unfortunately, such a back-transformation would only be unique if the concentrations for all samples would sum up to, which is usually not the case. An alternative is to work with all possible pairwise logratios, and derive their percentiles. However, having to consider all pairs makes the analysis quite complex when many compositional parts are involved. The information collected in Table is needed if absolute values rather than relative ratios are of interest. Note that Table does not provide values for mean and standard deviation, which both are based on Euclidean distances. Compositional data, however, do not belong to the classical Euclidean space, but need to be considered in their own Euclidean geometry on the simplex (see Aitchison, 986; Filzmoser et al., 009a, 00; Egozcue and Pawlowsky-Glahn, 0), even for univariate data analysis. Therefore, all classical statistical tests for comparison of the mean (median) of the two datasets will deliver faulty results because they are based on Euclidean distances. In terms of logratio- (the logarithm of a ratio) transformation three different approaches to open the data are possible: (a) an additive logratio (alr)-transformation (Aitchison, 986), sacrificing one variable, e.g., TiO and presenting all other results asalogratiototio (but why to TiO, different results must be expected when another variable is sacrificed); (b) a centred logratio (clr)-transformation (Aitchison, 986) where, in order to construct the logratios, each variable is divided by the geometric mean of all elements measured, followed by a log-transformation; (c) an isometric logratio (ilr)-transformation (Egozcue et al., 003) which has preferable geometrical properties for multivariate data analysis but where the direct relation to the elements is lost. In the following, the clr-transformation is applied to perform the statistical analysis of compositional data. However, there exists a severe disadvantage of the clr-transformation. The resulting clr-variables have a certain information overlap because the geometric mean is used as a common divisor. A scatter plot of a pair of clr-variables could thus be interpreted in a misleading way. For this reason, in such plots only the single clr-variables will be interpreted later on, but not the relation between them. It has been demonstrated that the single clr-variables are proportional to ilr-variables using a special class of ilr transformations (see Filzmoser et al., 0). Each of these ilr-variables (and thus each clr-variable) contains all the relative information of the corresponding element to the remaining elements, and is, therefore, fully informative about the compositional information of the underlying element. The ilr-(clr-) variables are a statistically correct representation of a compositional dataset as long as each of these variables is considered separately. Table shows the CoDa analysis equivalent to Table, the statistical parameters for the clr-transformed variables. Each clr-variable in Table contains all relative information of the studied element to the remaining elements. Since each clr-variable should be considered separately, any use of correlation analysis with clr-variables would not be meaningful. These clr-variables are ratios and consequently dimensionless numbers without any obvious meaning to the geochemist. There exist no values to compare in the literature, and these ratios do not provide the desired information on how much of an element occurs at a given location in space. The variables do no longer provide absolute but relative information. Median values and variances of the single CoDa variables remain comparable. It is worth noting that, when accepting that one is looking at different and dimensionless numbers, Table provides information which is not that different from Table. Table shows how dominant an element is in the composition. Elements with high concentrations are also characterised by high clr(element)-values; even the sequence of elements remains the same. Just like in Table, the differences observed between the two datasets (Ap and Gr) are minimal. The median value of clr(al O 3 )of 0.74 explains that on average the concentration of Al O 3 is =5.5 times larger than the geometric mean of all concentrations (for ease of interpretation the log to the base 0 rather than the natural logarithm was used for the clr-transformation). The median value clr(mno) of.35, in contrast, signifies that MnO has an abundance of only /5 of the geometric mean. The maximum value observed for clr(mno) is still lower than the minimum value for clr(al O 3 ). The interquartile range (IQR) and median absolute deviation (MAD) provide estimates

5 00 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Table Summary statistics, clr-transformed major elements, agricultural soil (Ap-samples, 0 0 cm, b mm fraction, N=08) and grazing land soil (Gr samples, 0 0 cm, b mm, N=04). All values are ratios, i.e. dimensionless. Mat.: material, Min: minimum; Q: quantiles (Q50=median); Max: maximum; MAD: median absolute deviation, IQR: interquartile range. Oxide Mat. Min Q Q5 Q0 Q5 Q50 Q75 Q90 Q95 Q98 Max RANGE MAD IQR clr(al O 3 ) Ap Gr clr(cao) Ap Gr clr(fe O 3 ) Ap Gr clr(k O) Ap Gr clr(mgo) Ap Gr clr(mno) Ap Gr clr(na O) Ap Gr clr(p O 5 ) Ap Gr clr(sio ) Ap Gr clr(tio ) Ap Gr Parameter Mat. Min Q Q5 Q0 Q5 Q50 Q75 Q90 Q95 Q98 Max RANGE MAD IQR clr(loi) Ap Gr of the variance of the clr-transformed variables. The higher the variance, the higher is the influence of this variable on the multivariate data ensemble. Interestingly, due to this effect, clr(cao) and clr(na O) have both more influence than the dominating element clr(sio ). The reason is that the relative concentrations of CaO and Na O vary more than that of SiO (see MAD or IQR in Table ). To better understand the effects of clr-transformation, the original concentration values are compared to the clr values of the same variable. If opening the data would be a minor correction only, one would expect a well-defined one-to-one relationship between concentration and clr values. By showing the plots for Al O 3,CaO,P O 5 and LOI (all plots are collected in the Supplementary material), Fig. demonstrates clearly that no such well-defined relation exists. It would be very difficult to predict the effect of opening the data on any one variable. In contrast to a widespread misconception, the non-uniqueness does not depend on the absolute concentration. In the four examples shown, the largest differences must be expected for Al O 3,P O 5 and Fe O 3 while for CaO the results probably do not change much. 3.. Plots of the cumulative data distribution One of the most powerful tools to study the data distribution is a cumulative robability (CP) diagram. For the above examples (Al O 3, CaO, Fe O 3 and P O 5 ) comparing the CP's for original data and clrtransformed variables in Fig. 3 results in qualitatively similar distributions. At the level of data interpretation, the statistical distributions of the Ap and Gr datasets are generally surprisingly well comparable for the major elements. Only LOI is significantly higher for the Gr samples (compare Table ). This has probably several reasons: () under permanent grass cover a thin organic layer develops at the top of the soil profile, () the dilution of organic by minerogenic material is less at the Gr sample depth of 0 0 cm than at the 0 0 cm used for Ap soils, and (3) in parts of Europe the very reason that agricultural land is used as grazing land is that it is too wet to plough, which implies that it is rich in organic material. One purpose of CP plots is to detect breaks in the data structure which may indicate subpopulations or certain geochemical processes (Reimann et al., 008). As already visible in Fig. the CP plots show that the statistical distributions displayed by concentration versus clr-transformed variables contain different information. Most prominently it is visible that the breaks shift to different positions in the plots Comparing the two sample materials In a next step, the results from the two sample materials are compared in more detail than provided by data tabulation or statistical distribution, where Tables and,andfig. 3 indicate that the statistical behaviour of Ap and Gr are almost identical. This can be done graphically by plotting for each variable the Ap result against the result of the neighbouring Gr sample, and adding a : line for ease of comparison. Fig. 4 compares the datasets sample-by-sample using XY plots: comparisons are done both in log scale and in clr scale. This reveals that substantial local differences between neighbouring Ap and Gr samples occur. However, the majority of sample pairs return well comparable results, and this fact dominates the overall statistical appearance in Fig. 3. The statistical similarity of the two datasets has an important implication for low density geochemical mapping at the continental (European) scale. It demonstrates that two independent sample sets, collected at the same density throughout Europe, essentially reflect the same information, even if sampling at the very low sample density of one site per 500 km (see also the discussion in Smith and Reimann, 008). It thereby confirms that robust geochemical maps can be expected by this procedure. In a way, it is surprising that the two different sample materials (Ap and Gr), collected from sites with dissimilar land use at diverse depths (0 0 cm vs. 0 0 cm), still deliver such comparable results. The differences between the sample materials, as outlined above, explain for the scatter in the XY plots (Fig. 4). While the supplement contains all graphics for both sample materials, the spatially more representative Ap-sample set will be exemplarily presented in the following sections Mapping One of the main aims of regional geochemistry is to visualise the spatial data structure on a map (Reimann, 005). Different approaches to geochemical mapping are outlined in Reimann (005), Reimann et al. (008). Reliable and informative geochemical maps are needed in

6 C. Reimann et al. / Science of the Total Environment 46 (0) Fig.. Scattergram of measured element concentration versus the clr-transformed values of the same variable. mineral exploration, in environmental studies, and for communicating relevant geochemical findings to environmental regulators and policy makers. The map of SiO versus all other elements on Fig. 5 highlights the problem of working with compositional (closed) data, because it is dominated by a belt of very high SiO concentrations in northern central Europe (N-Germany, Poland, Baltic States). These high SiO concentrations are related to the occurrence of sandy, coarse grained soils in these areas. These soils are young (about 8000 years) and developed on the sediments (end moraines) of the last glaciation. They predominantly consist of quartz and feldspar, such that SiO concentrations in the soils reach over 80 wt.% (Fig. 5). Accordingly, in a closed system summing up to 00 wt.%, there is little space for all other elements to vary.thisisdirectlyreflected on the maps of the other elements in Fig. 5 (maps of all elements in both sample materials are found in the Supplementary material): low concentrations necessarily appear in the belt where high SiO prevails. Mapping the other elements does not deliver intrinsic information for these elements. According to Aitchison (986), the information value of concentration data lies not in the measured values themselves but rather in the ratios between the variables. Although the geochemist intuitively is interested in the absolute concentrations of elements at any one location of a survey area, one has to question the intrinsic validity of single-element maps. This, of course, questions practically all classical geochemical maps. The statistical problem from the point of view of compositional data analysis is: single-element maps predominantly deliver results predictable from other elements. On the other hand, the geochemist argues that single-element maps convey a direct quantitative prediction of the expected element concentration at a specific location, and that they can be directly interpreted in terms of geology (occurrence of certain lithologies) and soil forming processes (occurrence of certain soil types). These two aspects are illustrated by the following examples (see supplementary material for the maps not shown in Fig. 5): Al O 3 : Scandinavia is with some exceptions covered by soils with moderate to high Al O 3 concentrations. The Caledonides and a belt of clay-rich soils running from southern Finland into southern Sweden are marked by high Al O 3. The soils developed on glacial sediments in central northern Europe show uniformly low Al O 3 concentrations, while variation is high in central/southern Europe. Some granitic intrusions as well as the alkaline volcanic provinces (Italy) are marked by high Al O 3 values. CaO: Even given the fact that one might expect severe disturbances of the natural geochemical patterns due to liming of agricultural fields, the map for CaO is certainly one of the most informative maps in terms of directly depicting geology. A clear break occurs between Scandinavia (moderate to high concentrations) and the rest of Europe, and in this case the break does not follow the glacial sediments but truly bedrock geology. In Scandinavia the Caledonides are marked by the highest CaO concentrations in the soils (note the continuation to Scotland and northern Ireland), and the

7 0 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Fig. 3. Cumulative probability (CP) diagrams of selected variables for the two sample materials: left hand: original data, right hand side clr-transformed data.

8 Gr LOI [wt%] Ap CaO [wt%] Ap PO5 [wt%] Ap LOI [wt%] 0 Ap clr(alo3) Gr clr(loi) Gr clr(po5) 0 4 Gr clr(cao) 0 Gr clr(alo3) 3 3 Ap AlO3 [wt%] 0.0 Gr PO5 [wt%] Gr CaO [wt%] Gr AlO3 [wt%] C. Reimann et al. / Science of the Total Environment 46 (0) Ap clr(cao) 4 3 Ap clr(po5) Ap clr(loi) Fig. 4. Scattergrams for element concentrations and clr(element) as determined in the two sample materials (Ap and Gr) from neighbouring sample locations. The line indicates a : relation and allows to better judge relative enrichment/depletion in one of the sample materials. Fig. 5. Maps of element concentrations in agricultural soils (Ap-horizon, 0 0 cm, b mm) for four selected elements based on the original data. Classes for the symbols are based on percentiles (0, 5, 5, 75, 95, and 00).

9 04 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Fennoscandian Shield mostly by moderate concentrations. The glacial sediments return a very uniform signal of low CaO values. In central/southern Europe variability is higher, exceptionally high CaO concentrations mark many of the areas underlain by limestones, dolomites and marble, while areas underlain by granites return uniformly low CaO values. Fe O 3 : The map in general appears rather noisy, with high and low concentrations occurring in close proximity. Clear patterns are shown by low Fe O 3 concentrations in soils on top of glacial sediments in central/northern Europe, and uniformly high Fe O 3 values over much of south-eastern Europe. K O: Again a disturbance of natural distribution patterns by input from fertilisers must be expected. The map for K O is in fact rather noisy. However, many areas underlain by granitic rocks are marked by unusually high K O concentrations (e.g., northern Spain/Portugal, massif Central, Bohemian massif). The alkaline volcanic rocks in southern-central Italy are also marked by high K O values. The soils developed on top of the glacial sediments in northern central Europe (but note the Baltic States!) give uniformly low K Oconcentrations. Soils in Sweden, southern Finland and the Baltic States show in contrast rather high K O concentrations. In southern Sweden and Finland these high concentrations are often due to the occurrence of clay-rich soils. The effects of agricultural practice on these agricultural soils are thus not immediately deducible from the map. LOI: A high value for Loss on ignition (LOI) can either be related to a high content of organic matter or to the occurrence of calcareous soils. It is thus not surprising that many high values are observed in parts of Scandinavia (organic matter) and in southern Europe (calcareous soil parent material), while soils developed on top of glacial sediments in northern central Europe are marked by uniformly low LOI values. MgO: The map is separated into three large zones: () predominantly high values in Scandinavia (with local exceptions e.g., on top of the sediments in Skåne in southernmost Sweden), () a belt of low values on top of the glacial sediments in northern/central Europe continuing towards northern France, Spain and into Portugal and (3) high values over south-eastern Spain southern France, Italy and most of south-eastern Europe. Local highs can be related to the occurrence of ophiolites (Greece) or dolomites (Spain, Italy). MnO: The map of MnO is rather noisy, one of the few clear features is the low MnO concentration in soils developed on top of the glacial sediments in northern/central Europe. Low values do also prevail over western France and all of Spain while much of south-eastern Europe is marked by high MnO concentrations. Na O: The map of Na O shows a clear and striking pattern: high values over all of Scandinavia (Baltic Shied and Caledonides), high values also on top of the Caledonides in Scotland and low values over much of south-western and central northern Europe, including the glacial sediments. Values in the median range (with local exceptions, e.g. much of Hellas with high values) mark large parts of south-eastern Europe. The exceptionally high Na O concentrations in northern Scandinavia indicate an important crustal boundary. P O 5 : This is another element where one might expect a major influence from the input of fertilisers to the agricultural soils. The map is in parts certainly noisy, however, the mostly high P O 5 concentrations throughout Scandinavia and Scotland are an outstanding feature and clearly linked to natural phenomena. These high values are partly due to the organic rich soils developed in the northern wet and cold climate and due to the occurrence of rocks rich in apatite. Soils developed on top of the glacial sediments in northern central Europe show uniformly low P O 5 concentrations. Local highs occur throughout the map in connection with some granitic intrusions and the volcanic rocks in Italy. SiO : The coarse grained glacial sediments in central/northern Europe bring about exceptionally high SiO concentrations. Values up to>95% SiO indicate that these soils are in part almost pure quartz sands. Most of the northern European soils, and large parts of southern European soils returned comparatively low SiO values (minimum values well under 5% SiO ). The reason for low SiO in the soil samples is not directly discernible on the map: the widespread occurrence of carbonates in the European south and of soils with high organic carbon content in the north, can result in both cases in exceptionally low SiO concentrations. TiO : In Scandinavia an east west gradient is observed in terms of TiO concentrations with the highest values occurring over the Caledonides in Norway. The soils on glacial sediments in northern central Europe are marked by uniformly low TiO.Thelargearea underlain by calcareous sediments in eastern Spain also corresponds to a large area with low TiO concentrations in soil. High TiO concentrations prevail in many of the soils from south-eastern Europe. Maps for the Ap and the Gr samples are with some local exceptions very well comparable (see Supplementary material). The geochemist will thus argue that all these maps are interpretable and deliver important and useful information. From the viewpoint of compositional data analysis, the maps of most elements add very little new, independent information after studying the spatial distributions of SiO, LOI and CaO. From this angle, it is more effective to plot maps for the clrtransformed variables, i.e. a map of the logratio of the investigated element to the geometric mean of all elements measured. As in theabovecaseofthetabulationofstatistical distribution measures, a map of a clr-transformed variable does not express direct information about the concentration of the investigated element at any one point within the map. Instead, it represents a relative abundance of the element with respect to the geometric mean of all elements measured. The comparison of such clr(element) maps, as shown in Fig. 6, with the corresponding maps in Fig. 5 demonstrates that for several elements (e.g., SiO, CaO, MgO) the mapped patterns (high versus low) remain almost the same, while for some other elements (e.g., K O, P O 5 ) the patterns on the map change significantly from element to clr(element). For example, it suddenly becomes visible that relative to the geometric mean these two major elements are actually highly abundant in the glacial sediments of northern central Europe. Their low absolute concentration in soil, which is emphasized by Fig. 5, is primarily due to the high concentrations of SiO in these soils (also known and discussed in the geosciences as the quartz dilution effect see, e.g., Bern, 009). Using the geometric mean as normalizer, effectively suppresses the predominant SiO signal, and therefore in these cases the two clr(element) maps deliver completely different information. Arguing from the compositional data side it is possible to state that the low K OandP O 5 information in the concentration maps is not the real information, because it is forced by the high SiO concentrations, and that one thus should not even bother to present singleelement maps, but rather directly move on to multivariate data analysis. In contrast, the geochemist will argue that the really measured K O concentrations are low in these areas, which is important information in itself, e.g., for assessing soil quality or fertility. Moreover, the patterns of some elements, like CaO and MgO, on both maps did not even change. This information would definitely be lost without mapping the concentration of the single elements first. In addition, practically all research in regional geochemistry during the last

10 C. Reimann et al. / Science of the Total Environment 46 (0) Fig. 6. Maps of the regional distribution of the clr-transformed variables for which concentration maps are presented in Fig. 5. Classes chosen for mapping are based on percentiles (0, 5, 5, 75, 95, and 00). 00 years is based on, and discussed in terms of absolute element concentrations, and not in terms of clr ratios. For example, all action levels for soils are set based on element concentrations. Following a clr-transformation one is studying abstract ratios instead. In the end, it appears that both maps are needed to understand the processes governing the distribution of the chemical elements in space. The classical concentration map is indeed needed for many practical applications the CoDa logratio map will obviously deliver truly new additional information in some, though not all, cases. It is not possible to extract this additional information from single-element distribution maps Correlation versus stability, scatterplot matrix, XY diagrams When analysing soil samples for their total element concentration, increasing the dominant element SiO must decrease the sum of the other oxides (and actually all elements in the sample) by virtue of the smaller space remaining for them, within the 00% total. It was correlation analysis and correlation based methods where the closure problem with compositional data was first detected and discussed (Pearson, 897; Chayes, 960), often under the name spurious correlation. Thinking in terms of correlation the requirement is that the correlation of two variables must not be affected by the influence of other variables. This requirement is never met when dealing with geochemical data. Even if only two elements are analysed, all other elements in the composition influence the measured concentrations of these two elements. Therefore, any method based on the absolute concentrations of the two parameters studied (like correlation) is bound to fail. When dealing with geochemical data, the measured concentration of any one element in a sample always depends on the concentrations of all other variables, and multivariate information rather than the bivariate information needs to be considered. When studying an XY diagram of absolute concentrations, it must always be kept in mind that there exist a multitude of further variables which influence the pattern visible in any one bivariate plot. Though they are a commonly used tool in geochemical data analysis of compositional data, XY plots should never be interpreted in terms of correlations. Fig. 7 (upper right) shows a scatterplot matrix for the log-transformed data of the Ap dataset. The boxplots of the lower triangle correspond to logratios of the variable in the column divided by the variable in the row. A scatterplot matrix, instead of a single XY diagram, shows the interactions of all the elements in the composition. The correlation between two elements neglects the effect of all the other elements in the composition on the concentration of these two elements. For example an increase of SiO will automatically lead to a decrease in the concentration of most other elements (see Fig. 7). In classical statistics this would be interpreted as a negative correlation, while in fact it is an artefact of the closed data structure and the dominance of the SiO variation. Thus, the whole concept of correlation in XY plots does not make sense when dealing with compositional data (Aitchison, 997; Filzmoser et al., 00).

11 06 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Fig. 7. Scatterplot matrix for the log-transformed element concentrations (upper right). The dashed line indicates a constant ratio between the two elements (corresponds to the median of the logratio), and the parallel solid lines indicate a ratio that is twice/half as high. In the lower left part boxplots of the logratios of the element pairs are shown together with a coefficient of stability, which varies between 0 and. High stability is indicated by values >0.9. Which of the two elements in the pair dominates the composition is shown by the location of the boxplot in relation to the dashed line (above or below). The black symbols indicate the samples from the sandy soils (end moraines) in northern central Europe. A more surprising, and probably even more important, artefact is visible in Fig. 7. The dominant concentrations of SiO clearly outweigh the concentrations of most other elements (see Table, Fig. 7). This not only forces a negative relationship between the other elements and SiO, but it can also induce artificial positive correlations among the less dominant elements, when interpreted in terms of classical statistics. In the context of compositional data, only the ratios between the elements contain the relevant information. A simultaneous decrease or increase of a pair of variables is not of interest, but rather the stability of their ratio. Fig. 7 (upper right) contains this additional information: The dashed line indicates the constant ratio (corresponds to the median of the logratio), and the parallel solid lines indicate a ratio that is twice/ half as high. The closer the data points follow this band, the more stable is the ratio. In the lower left part of Fig. 7, boxplots of the logratios of the element pairs are shown together with a coefficient of stability, which varies between 0 and (Filzmoser et al., 00). This coefficient is computed as exp( var(z)), with z=/sqrt()*log(x i /x j ). Here, x i and x j are the parts i and j of the composition. Thus the coefficient is a measure of the variance of the logratio of two compositional parts. If the variance is low, the logratio is stable, and the resulting coefficient is close to. High instability leads to high variance and thus to a coefficient close to 0. As an example, for Al O 3 /TiO (lower left) a high (>0.9) coefficient of stability of 0.96 is shown. This indicates that this logratio is stable, i.e. all samples have a comparable logratio. This is also visible in the corresponding plot in the upper right corner of the diagram, where the points follow the dashed line of the median ratio. The boxplots indicate which of the two variables is more dominant in the composition. The median of a boxplot above the stipulated line (0) indicates that the first element in the ratio is more dominant (in the example Al O 3 is much more dominant than TiO ). Fig. 8 shows a plot of one selected pair of log-transformed variables from Fig. 7 (SiO and K O). In terms of correlation, there exists no obvious and strong relationship between SiO and K O (Pearson correlation coefficient 0.34). In terms of CoDa, only the ratio between the two variables is of interest. The median logratio SiO /K O of 33.6 corresponds to the dashed line shown in the XY plot. The band indicated by the solid lines contains all logratios between 6.8 and 67. in total 84% of the data, resulting in a coefficient of stability of 0.9 (shown on top of the boxplots in Fig. 7 see Filzmoser et al., 00). Although the Pearson

12 C. Reimann et al. / Science of the Total Environment 46 (0) Fig. 8. Scatterplots for SiO versus K O and MgO versus Na O (see Fig. 7) in the first case the logratio is highly stable (coefficient of stability: 0.9) in the right diagram it is unstable (coefficient of stability: 0.65). Note that the two variables are plotted using a log scale. correlation is rather low, the stability is very high. For comparison the pair MgO versus Na O delivers a comparable Pearson correlation of 0.37 while the band indicated in the plot (Fig. 8) contains only 55% of the data and consequently the coefficient of stability is only 0.65 (Fig. 7). The results for SiO and K O indicate that, although not visible when studying the plot in terms of a correlation, there exists in fact a close relation between these two variables, while the relationship between MgO and Na O is and remains weak. XY plots are usually shown to study the relation between the two variables plotted. Correlation, however, is definitely the wrong concept for quantifying relationships between two variables from compositional data. XY plots can still be used with care in the sense of truly exploratory data analysis (EDA - Tukey, 977) to detect unusual behaviour between two variables, or groups of samples with a different behaviour that may be indicative of different processes, as long as it is kept in mind that there exist many further variables that influence the visible diagram. The left diagram in Fig. 8 suggests a splitting of the samples in different groups. There exists at least one group of samples where both, SiO and K O, have unusually low concentrations. In fact, behind this trend (with a constant ratio!) are even two groups of soil samples hidden: (i) soils developed on the calcareous sediments, dominated by CaO (see Table, some soils can contain over 50% CaO) in combination with a high LOI (CO 3 ), that are prominent in southern Europe, and (ii) soils with an unusually high amount of organic material (indicated by high LOI but low CaO), typical for parts of northern Europe. In both cases, all other elements must decrease in view of CoDa. In contrast, the samples in Fig. 8 with very high SiO concentrations and decreasing K O concentrations are the sandy soils of the end moraines of the last glaciation in northern central Europe, where SiO concentrations of over 90% leave no room for other elements. It is possible to deduce that there exist samples with very low SiO and K OandadecreaseofSiO is accompanied by a decrease in K O. However, because the decrease is forced by other elements (CaO and LOI) it would be incorrect to suggest that there exists a direct relation between SiO and K O. A correct evaluation of the relation between the elements is only possible when looking at the ratios in case of the chosen example samples with low concentrations have about the same ratio as the majority of samples with higher concentrations and the relation is in fact strong Exploring new patterns in the clr(element) maps Several of the clr(element) maps (Fig. 6) showed distribution patterns for the ratios that were comparable to the element distribution maps (Fig. 5), while some maps displayed very different patterns. The spatial distribution displayed on the maps of P O 5 (Fig. 5) versus clr(p O 5 )(Fig. 6) is one example showing a large difference. The concentration map showed low P O 5 concentrations on the band of sandy soils dominating northern central Europe (the end moraines of the last glaciation), characterised by exceptionally high SiO concentrations (leaving no space for other elements to vary). Without intricate knowledge and understanding of the whole dataset it is close to impossible to determine the reason for the different spatial distribution observed on the clr(p O 5 ) map. These sandy soils from northern central Europe are highlighted in Fig. 7, the scatterplot matrix, in black. The problem is that the clr(p O 5 ) map is influenced by the ratios to all other elements what is then the reason for the high ratios (relative dominance of P O 5 in the composition) in this area? It is possible that the effect is caused by an increase of P O 5 or a decrease of all or any of the other elements. Studying the location of these soils in the scatterplot matrix in Fig. 7 it is clear that soils with low to moderate P O 5 concentrations are affected. The plot can now be used to investigate where these soils deviate from a constant ratio, indicated by the grey band in the graphic. The strongest deviation is visible in the plot P O 5 versus MgO, where MgO is depleted relative to P O 5. Smaller deviations in the same direction are also visible for CaO and Fe O 3. Fig. 9 shows two maps, the regional distribution of the logratios P O 5 /MgO and P O 5 /CaO. Here it becomes obvious that the effect is caused by a relative enrichment of P O 5 especially with respect to MgO. The Ca Mg P (Fe) ratios are important for plant fertility and production and these patterns may thus be the first clear indications of the effects of agricultural practice (use of fertilisers) on the studied soils. Note that simple logratio maps as shown in Fig. 9 are true representations of the data even in the CoDa sense Multivariate data analysis Many of the above discussed problems with data closure can be overcome in multivariate data analysis, when the data are automatically clr- or ilr-transformed. However, note that variables with poor data quality must be recognised and removed before any such transformation is carried out, otherwise they can dominate the result. A classical approach to PCA in geochemistry has been to log-transform all data, and to carry out the PCA with the log-transformed values. Fig. 0 shows the robust biplots for the log-transformed data (left), and the results for the conceptually correct version (right), based on ilr-transformed data, and back-transformed to the clr-space for a biplot representation (Filzmoser et al., 009b). Fig. 0 demonstrates that the log-transformation is not sufficient, because the first principal component (PC) is driven by the dominating (in terms of absolute concentration) variable SiO. In contrast, for the clr-transformed values, where relative information is considered, the biplots open up (see Filzmoser et al., 009b). The biplots for PC versus PC and PC versus PC3 for the log-transformed data clearly demonstrate the closure problem: with the exception of SiO all other elements draw into one direction.

13 08 C. Reimann et al. / Science of the Total Environment 46 (0) 96 0 Fig. 9. Maps of the P O 5 /MgO and the P O 5 /CaO ratio in the agricultural soils of Europe. The true data structure and the relations between the variables are only visible when the data are clr-transformed. Then suddenly the results are no longer dominated by SiO. The biplots for the opened data in Fig. 0 (right) show a strong relation between SiO,K O, and P O 5 (sandy and feldspar rich soils). A weaker relation exists between Fe O 3, MgO, and MnO, TiO (mafic minerals, mica rich soils), while CaO takes a separate position (calcisols, e.g., in Spain) which is also visible in Fig. 7, the scatterplot matrix in terms of low coefficients of stability in relation to all other elements. LOI is related to the occurrence of organic soils but also relatively high in areas with calcareous soils. 4. Conclusions Geochemical data are compositional and thereby closed data. Mathematically they define points in the Aitchison geometry on the simplex, and not in the usual Euclidean space for which all classical statistical methods are designed. For this reason, all calculations which explicitly or implicitly are based on Euclidean distances give misleading results. This already includes simple calculations, like computing mean or standard deviation. Even more, it can also affect simple visual assessments, like detecting groups in the data or suggesting patterns of linear relation, both are intimately linked to the Euclidean distance. All such graphics should thus only be used and interpreted with care. For the multivariate case appropriate transformations from the simplex to the usual Euclidean space overcome the problems of data closure. The problem with closed data is that they are multivariate by definition. A uni- or bivariate analysis of such data must always keep that multivariate structure in mind. From a practical point of view, however, geochemists are interested in the statistical and spatial distribution of single variables, e.g., elements or compounds. They also wish to use the correlation between two variables to draw conclusions about geochemical processes. Although the naïve application of uni- or bivariate analysis neglecting the closure problem is probably wrong, experience shows that in many cases this approach leads to interpretable and meaningful results, probably because the interest is often driven by high absolute concentrations of the measured variables and not by the relative contributions of the elements to the whole composition. Relative contributions are valuable because they will provide a deeper insight into the multivariate structure of the data. Opened variables can differ widely from their closed counterpart. Whether there will be a difference or not, does not depend on the concentration range. It is a misconception that closure is only a problem for elements with very high concentrations like SiO and Al O 3. Elements with very low concentration can and will be seriously affected by the relative information of other variables which are invisible in uni- or bivariate plots. In case of the GEMAS dataset, simple classical univariate element distribution maps of the major elements in European continental soils can be interpreted in terms of the main geological structures. Most prominent are a band of high SiO concentrations marking the end moraines of the last glaciations in northern central Europe, and high concentrations of CaO in the areas underlain by limestones, marls and dolomites (mostly in southern Europe); while uniform moderate CaO concentrations are characteristic for the Fennoscandian Shield. Many of the large granitic intrusions in Europe are related to high K O values. Soils developed on greenstone belts show high MgO concentrations. When mapping the opened, clr-transformed variables, the regional distribution does not change substantially for some elements, like, e.g., CaO and MgO; for other elements, like K OandP O 5, the patterns change significantly. The sandy soils in northern central Europe show suddenly high ratios of K OandP O 5 in relation to the geometric mean of all elements the low absolute values observed on the concentration maps are solely forced by the overall high SiO -concentrations. The new maps, however, are no longer concentration maps but rather show the regional distribution of a dimensionless ratio. The two maps () original element concentration versus () clr-transformed element deliver qualitatively different information: () the concentration of the studied elements at any location on the map and () the information whether the measured value is high or low in relation to the geometric mean of all elements. It is worth-while to study both maps. Classical correlation analysis should not be carried out on compositional data in XY plots. These may only be used with great care in an EDA sense to detect unusual data behaviour or subgroups of samples. The relation between all elements can be evaluated when adding information about the stability of a ratio in a scatterplot matrix. Used in an exploratory sense with indicated subgroups it is a powerful tool to understand complex multivariate relations in two dimensions. For multivariate data analysis the effects of closure must be overcome by applying a suitable logratio-transformation (ilr-transformation). When carrying out principal component or factor analysis, the effect of opening the data is immediately visible on a biplot. Only opened data provide information about the true relationships between the variables, relationships that are independent of the total concentrations of the elements. Acknowledgements The GEMAS project is a cooperation project of the EuroGeoSurveys Geochemistry Expert Group with a number of outside organisations (e.g., Alterra in The Netherlands, the Norwegian Forest and Landscape Institute, several Ministries of the Environment and University Departments of Geosciences in a number of European countries, CSIRO Land