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Element chemostratigraphy of the Devonian/Carboniferous boundary –

a compositional approach

K. FAČEVICOVÁ1*, O. BÁBEK2, K. HRON3, T. KUMPAN4

1Department of Mathematical Analysis and Applications of Mathematics – Palacký University

Olomouc, Czech Republic, kamila.facevicova@gmail.com

2Department of Geology – Palacký University Olomouc, Czech Republic

3Department of Mathematical Analysis and Applications of Mathematics – Palacký University

Olomouc, Czech Republic

4Department of Geological Sciences – Masaryk University Brno, Czech Republic

*corresponding author

The Devonian/Carboniferous (D/C) boundary is a critical interval in the Phanerozoic history, which is

associated with vigorous climatic perturbations, continental glaciation, global sea-level fall and rapidly

increased extinction rates in marine realms. In many sections world-wide, these global changes left a

marked lithological signature, in particular the Hangenberg black shale (products of deep-shelf anoxia)

and the overlying Hangenberg sandstone (sudden siliciclastic influx into predominantly carbonate

depositional environments). Both layers bear a distinct geochemical signature. Even though either or

both of these two lithologies are absent at many sections, their correlative counterparts can be

indicated by subtle geochemical markers. We studied elemental geochemistry of fourteen D/C

boundary sections in six key areas across Europe with the aim to select globally correlatable elemental

proxy for the D/C boundary. Analysis of raw/log-transformed geochemical data (EDXRF, c.p.s. units),

presenting the standard approach here, indicates that concentrations of terrigenous elements (Al, K,

Rb, Ti and Zr) are mainly controlled by diluted Ca (carried by marine calcium carbonate) in limestone

facies and, accordingly, their variations can be related to carbonate production in the sea rather than to

terrigenous input from continent. Nevertheless, due to the relative nature of geochemical observations,

reliance solely on statistical processing of raw data might lead to incomplete picture of multivariate

data structure and/or biased interpretations. For this reason, the aim of this contribution is to discuss

the logratio alternatives of the standard statistical methods, which may better reflect the relative nature

of the data. For this purpose, principal component analysis was employed to reveal main geochemical

patterns and while the geochemical signature of the D/C boundary was further analysed using Q-mode

clustering that leads to predicative orthonormal logratio coordinates – balances. The comprehensive

picture of the multivariate data structure provided by these statistical tools makes them a primary

choice for exploratory compositional data analysis. At the same time, it turns out that the standard and

compositional approaches have synergic effects. This fact can be extensively used in further

geochemical studies.

Keywords: element geochemistry; compositional biplot; dendrogram; logratio coordinates;

Devonian/Carboniferous boundary; variation matrix

1. Introduction

In geochemistry, most of data are compositional in nature (Aitchison, 1986; Pawlowsky-

Glahn and Buccianti, 2011; Pawlowsky-Glahn et al., 2015a). It is not just indicated by units,

in which observations are measured or expressed, like mg/kg, ppm, or percentages, but

inherently also in the fact that ratios between components form the primary source of

information. As a consequence, any sum of components (compositional parts) is irrelevant. In

particular, in geochemical practice it rarely happens that the input observations sum up to a

constant given by the mode of used representation. For example, to make mg/kg units sum up

to unity (one million) it would mean that all elements in the rock were identified and

analyzed. From this perspective, it is much more convenient to treat the data as

compositional, if their parts convey quantitatively expressed relative contributions on a

whole, formed by the given composition. The above properties also imply that compositional

data carry exclusively relative information (Pawlowsky-Glahn et al., 2015a).

Accordingly, when the relative structure of geochemical observations is of the sole interest,

applying standard statistical tools to the input concentrations may lead to misleading results,

because compositional data obey different geometrical rules. These inherent features can be

expressed by principles of compositional data analysis (Egozcue, 2009): scale invariance,

permutation invariance and subcompositional coherence. From the practical perspective, the

most important one is scale invariance stating that information in a composition does not

depend on the particular units in which the composition is represented. Specifically,

proportional positive vectors represent the same composition. The latter two principles

provide a solid theoretical basis of any reasonable (not exclusively) statistical processing, and

they all are needed to build a sound geometrical setting that would reflect inherent properties

of geochemical (compositional) data. It is provided by the Aitchison geometry (Billheimer et

al., 2001; Pawlowsky-Glahn and Egozcue, 2001; Egozcue et al., 2003), defining an algebraic-

geometrical structure of the sample space of compositions, formed by equivalence classes of

proportional positive vectors (or by simplex for a given constant-sum representation of

compositions). As the majority of statistical methods rely on Euclidean geometry in real space

(Eaton, 1983), it is necessary to transform the compositions prior to standard statistical

analysis with such as the principal component analysis (PCA) and/or visualization. The

definition of compositional data implies that any such transformation (from geometrical

reasons, referred to as coordinates) should be formed by ratios between parts that form the

elemental information in compositional data. Or, even better, by log-ratios (Aitchison, 1986)

that symmetrize positions of components in the ratio and are mathematically easier to handle.

For a standard geochemical analysis, it is preferred to deal with the original compositional

parts, not their log-ratios. However, with all consequences that imply from the nature of

compositions considered, this is frequently not satisfactory. Nevertheless, if interpretation in

terms of single original part is required, centred logratio (clr) coordinates provide a

compositional alternative to raw data. Each part of vector of these coordinates

(1)

where g(x) stands for geometric mean

, (2)

represents a dominance of with respect to the complete composition. Because the resulting

vector has components, a redundancy condition arises, , leading to

singular covariance matrix of . Although this affects applicability of some statistical

approaches, like the class of robust methods (Filzmoser et al., 2009a), it is still possible to

employ them for most of exploratory tools including PCA. For more discussion about the use

of clr coordinates in geochemical context, see McKinley et al. (2016).

Not all geochemical data must necessarily be treated as compositional, especially if total

abundances are relevant for the analysis. However, from the methodological perspective,

analyzing solely raw data still cannot be recommended. Instead, compositional data should be

treated as standard positive observations that induce again specific geometrical features, e.g.,

their relative scale (Mateu-Figueras and Pawlowsky-Glahn, 2008). Beside the (clr) coordinate

representation, the usual log-transformation seems to be meaningful

for these situations (Pawlowsky-Glahn et al., 2015b).

The D/C boundary geochemical dataset provides a suitable working material to test the

applicability of the compositional approach. The data comprise marine carbonate rocks, which

alternate with several siliciclastic layers each confined to a specific stratigraphic interval.

These layers are correlatable across the six studied areas and provide thus a common feature

of all the studied sections. They comprise the Hangenberg black shale, sandstone and grey

shale, collectively referred to as the Hangenberg event sensu lato, HBS s.l. (costatus-kockeli

Interregnum, uppermost Famennian) and the Lower Alum Shale, LAS (crenulata Zone,

middle Tournaisian) (Kaiser et al., 2011). Their geochemical signature reflects rapid

siliciclastic influx into shelf seas (sandstones and grey shales of the HBS s.l.), increased

organic productivity in pelagic settings and global development of water bottom anoxia (black

shales of the HBS s.l. and LAS) and effects of dilution by biomineralized calcium carbonate

(interlayered carbonates) (Kumpan et al., 2014a,b, 2015; Bábek et al., 2016). Developed

more-or-less in all of the studied sections, this signature is modulated by such local factors as

different source of the siliciclastics and variable depositional settings. All these factors

contribute to a complex structure of the geochemical data.

The aim of the manuscript is to demonstrate, that not just statistical analysis based on the

compositional approach, being definitively the most relevant from the theoretical perspective,

but also analysis of raw data or their log-transformation bring some benefits to the overall

analysis. The reason is that the effect of the concrete scale of data cannot be frequently simply

removed as it is the case of the compositional approach (scale invariance). Practical

experiences indicate that even considering absolute scale of the original (raw) data, being

methodologically rather incorrect, can reveal some further interesting features. The aim of the

practical part is not to provide an exhaustive comparison, but to demonstrate that just

combination of different approaches (though not necessarily theoretically sound) leads to

complex understanding of the multivariate data structure. Nevertheless, following the

compositional approach that reflects predominant geometrical features of data at hand, one is

on the “safe side”. As an adjacent goal, inspired by quasimetric structure of the variation

matrix, the paper extends the use of Q-mode clustering, introduced in van den Boogaart and

Tolosana-Delgado (2013) for alternative agglomerative clustering procedures. All statistical

calculations were conducted with software R (R Core Team, 2016) and its package

compositions (van den Boogaart et al., 2013; van den Boogaart et al., 2013).

2. Materials and Methods

2.1. Data description

The study material includes elemental geochemistry of carbonate and siliciclastic bed

successions across the Devonian/Carboniferous (D/C) boundary. The dataset includes 1884

samples from 14 sections studied in six major Variscan massifs of Europe (Rhenish Massif,

Germany; Namur-.Dinant Basin, Belgium and northern France; Moravo-Silesian Zone of the

Bohemian Massif, Czech Republic; Carnic Alps, Austria and Italy; Montagne Noire of the

Massif Central, southern France and Pyrenees, southern France) . Figure representing position

of the studied sections in the major outcrops of Variscan massifs in Europe is attached as a

Supplementary Material. The geological settings and stratigraphy of the sections were

recently summarized by Kumpan et al., (2014a,b, 2015) and Bábek et al. (2016). Samples for

element geochemistry were taken from the studied sections with a vertical step of 5 to 25 cm

(rarely with 1 cm or 50 to 60 cm) depending on overall section thickness and required detail.

The average sampling density was one sample per 11.5 cm.

All the samples were analysed by energy-dispersive X-ray fluorescence (EDXRF) using a

MiniPal 4.0 instrument (PANalytical, Netherlands) with an Rh lamp (30 kV) and Peltier

cooled Si PIN detector. The samples were ground to <63 m particle size and filled into

plastic cells 25 mm in diameter and with Mylar foil bottoms. Eighteen elements were

analysed (Al, Ca, Cr, Cu, Fe, K, Mn, Ni, P, Pb, Rb, S, Si, Sr, Ti, Y, Zn and Zr). Al and Si

signals were acquired for 300 s at 5 kV/400 μA with a Kapton filter under He flush (99.996%

purity); K, Ti, Fe, and Mn and Fe for 200 s at 12 kV/200 μA with a thin Al filter in air; and Zr

for 500 s at 30 kV/200 μA with an Ag filter in air. Total analysis time was set at 800 s per

sample. The EDXRF results are given in counts per second (cps). The EDXRF analytical

results for Ca, Si, Al, K, Ti, Mn, Fe, Sr, and Zr (in cps) were calibrated through inductively

coupled plasma mass spectrometry (ICP-MS) analysis of 11 samples (Lesní lom, Křtiny, and

Grüne Schneid sections) by an accredited analytical laboratory of the Technical University of

Ostrava, Czech Republic, using an X Series 2 ICP-MS instrument (Thermo Scientific). The

quality of the ICP-MS analytical data was checked by measuring 2709a standard reference

material (SRM) (San Joaquin Soil, NIST, USA). EDXRF results for Cr, Rb, Y, Cu, Zn, Ni,

and Pb were calibrated by ICP-MS (Element2, Thermo Scientific) at the Geological Institute,

Czech Academy of Sciences, Prague, using calibration equations from an external set of 17

samples from lower Devonian carbonates and shales of the Prague Basin, Czech Republic.

Data quality from the Geological Institute was checked by analysing 1d SRM (Argillaceous

limestone, National institute of Standards and Technology). Calibration curves of the EDXRF

vs. ICP-MS results suggest that all target elements were above EDXRF detection limits while

the high correlation coefficients (R2 = 0.93 to 0.99 generally; only slightly lower for Mn: R2 =

0.893, Cr: R2 = 0.887, and Ni: R2 = 0.722) indicate the good reproducibility of the EDXRF

signal. P and S were not calibrated. All element ratios and enrichment factors mentioned

throughout this paper are based on uncalibrated EDXRF (cps) data.

The elemental composition sensitively reflects basic lithological and paleoenvironmental

changes at the D/C boundary. Particularly effective are the following element groups with

similar geochemical behavior:

a) Ca, which is driven by marine CaCO3 production in benthic and pelagic settings (Sageman

and Lyons, 2005);

b) Al, K, Rb and Ti, which are bound to siliciclastic minerals derived from continent, in

particular clay minerals and silt-sized heavy minerals and phyllosillicates (Ross and Bustin,

2009; Sageman and Lyons, 2005; Vijver et al., 2008);

c) Si and Zr, which tend to concentrate in coarse-grained siliciclastic minerals (Schnetger et

al., 2000; Jones et al., 2012) also derived from continent (heavy minerals, quartz, feldspars)

but Si is also linked to organic pelagic production in the sea (radiolarians and sponge

spicules);

d) Zn, Ni, Cu, Pb, S and P - productivity-sensitive elements, which tend to concentrate in

organic matter-rich sediments such as the Hangenberg black shale and Lower Alum shale

(Bout-Roumazeilles et al., 2013; Fralick and Kroberg, 1997; Sageman and Lyons, 2005;

Śliwiński et al., 2010; Tribovillard et al., 2006).

e) redox-sensitive elements such as Fe and Mn, which are highly mobile across sub-bottom

redox gradients (Haese et al., 1998).

The advantage of the dataset is in the detailed knowledge about distinct sedimentary layers

with expected geochemical behavior, which include: layer 1) Upper Devonian carbonates;

layer 2) Uppermost Devonian black shales of the Hangenberg (HBE) event interval; layer 3)

Uppermost Devonian sandstones and non-black shales of the Hangenberg (HBE) interval;

layer 4) Lower Carboniferous carbonates and layer 5) Lower Carboniferous black shales and

cherts of the Lower Alum Shale interval. Figure representing chronostratigraphy,

biostratigraphy, lithostratigraphy and thickness of the studied sections is attached as a

Supplementary Material.

2.2. Exploration of compositional variation structure

Covariance structure of compositional data reflects the fact that the source information is

contained in pairwise log-ratios. Accordingly, in the compositional context the multivariate

variability is captured by the variation matrix (Aitchison, 1986), defined as

(3)

Its interpretation is intuitive. A non-diagonal element of the variation matrix is zero, or nearly

so, if and only if the respective compositional parts are proportional, or nearly so. In other

words, proportionality here replaces covariance (correlation) between variables from standard

multivariate statistics. Consequently, elements of variation matrix can also be used as a

measure of dissimilarity between compositional parts, for example for the purpose of

clustering of compositional parts (Montero-Serrano et al., 2010, Pawlowsky-Glahn et al.,

2011; van den Boogaart and Tolosana-Delgado, 2013; McKinley et al., 2016). From the basic

(metric) properties of distances the following are obviously fulfilled,

(4)

i.e., non-negativity, identity of indiscernibles and symmetry. On the other hand, triangular

inequality is not fulfilled in general, only its generalized form

(5)

for a constant that corresponds to quasimetrics (Xia, 2009) can be derived. According

to Fišerová and Hron (2011) it holds true that

, (6)

resulting in . It is shown (Xia, 2009) that many well-known results for the usual metrics

still hold true in quasimetric space that makes them a natural generalization of the basic

metric settings. These findings are inspirative for Q-mode clustering, introduced in the next

section. Finally, a matrix relationship exists between the variation matrix and covariance

matrix of clr coordinates (Aitchison, 1986), which is useful for practical computations.

Variation structure of compositional data can be visualized using compositional biplots

(Aitchison and Greenacre, 2002). Similarly as for a standard PCA biplot (Gabriel, 1971),

which is used to reduce dimensionality of input data, compositional biplot displays as well

scores and loadings of the first two principal components in one planar graph. The scores are

usually marked as points in order to capture multivariate data structure; loadings are

represented by arrows and stand for the input variables. While in the standard biplot original

(log-transformed) variables are considered, in the compositional case clr coordinates are

usually represented. This also affects the interpretation of loading vectors. While in the

standard case the length of the arrow and the cosine of the angle between two arrows

approximate standard deviation of the corresponding variable and correlation between

variables, respectively, interpretation of clr variables needs to be taken into account in the

compositional case. Consequently, the length of the arrow cannot be interpreted as a single

original part but as a representation of its dominance to an “average part” in the composition

(Filzmoser et al., 2012). Instead of interpreting correlation between two clr coordinates, which

is affected by the zero sum constraint of variables, it is preferred to consider links between

vertices, approximating pairwise logratio variances (elements of the variation matrix). In

particular, link between vertices of and approximates

; if the vertices

coincide, or nearly so, then and are proportional, or nearly so. An enhanced

interpretation of compositional biplot in terms of covariance structure of pairwise log-ratios

can be found, e.g., in Pawlowsky-Glahn et al. (2015a).

2.3. Pattern identification

In order to assess the log-ratio patterns in the geochemical signature of the D/C boundary

rocks, it is necessary to search for such variables which are responsible for the general

geochemical patterns. In particular they include the detrital-input sensitive elements (Al, Ti,

Rb, Zr), grain-size sensitive ones (Si, Zr), organic productivity sensitive ones (P, S, Ca, Ni,

Zn, Cu) and redox-sensitive ones (Fe, Ni, Mn) (see section 2.1). For the case, when total

abundances are informative as well, the log-transformed original variables or their

combinations would be appropriate for this purpose. Nevertheless, it seems that for carbonate

rocks rather relative contributions of the EDXRF signal (in cps) are of interest. This is

reflected also by recent practice in the field, where ratios between elementary components

such as Zr/Rb, K/Al and Rb/K are considered to indicate the lithological changes across the

D/C boundary (Kumpan et al., 2015; Bábek et al., 2016). The previous section implies that,

referring to relative structure of compositional data, single compositional parts are not

appropriate to serve as feature variables, because they necessarily (directly or indirectly) rely

on the other components in the actual composition. All relative information about single

compositional parts is extracted using clr coordinates (1); on the other hand, this information

might be too complex due to different and possibly antagonistic patterns of log-ratios

aggregated there (Reimann et al., 2012; McKinley et al., 2016). Taking simply all possible

pairwise log-ratios into account would need an exhaustive search, therefore not very practical

for data with moderate or even larger number of components. Although expert knowledge can

be used to select such log-ratios, a data-based approach can facilitate finding possible further

interesting geochemical markers by mutual considering specific geochemical behavior of

elements. Consequently, such an unsupervised approach can help to extract log-ratios (not

necessarily belonging to any of previously mentioned extreme cases, pairwise log-ratios

versus clr coordinates) able to recognize the geochemical signature.

For this purpose, in van den Boogaart and Tolosana-Delgado (2013) a compositional Q-mode

clustering was proposed. This clustering method is based on idea to obtain easily interpretable

orthonormal coordinates with respect to the Aitchison geometry, referred to as principal

balances (Pawlowsky-Glahn et al., 2011). It is essentially hierarchical clustering, where the

variation matrix plays the role of a measure of association between compositional parts. The

resulting graphical output, dendrogram, can be applied to define a sequential binary partition

of compositional parts into groups of parts (Egozcue and Pawlowsky-Glahn, 2005); each

horizontal link is used to set a new variable that expresses balance between the corresponding

groups of compositional parts. By denoting those on the left side of the link by plus sign and

parts on the right side by minus sign, the balance (orthonormal coordinate) is defined as

, (7)

where g(x+) and g(x-) stand for the geometric mean of parts from the first and second group,

respectively; together coordinates are obtained by such a procedure. Their

interpretation can be enhanced by suppressing the normalization constant and changing the

base of logarithm. It is also worth to note that each balance aggregates all pairwise log-ratios

between both groups of parts (Fišerová and Hron, 2011), the fact that can be used to search

for simpler log-ratios responsible for the geochemical signature of the D/C boundary.

According to McKinley et al. (2016), the resulting clusters of compositional parts will contain

elements behaving proportionally throughout the dataset. Log-ratios between parts of two

different clusters should thus be similar to other log-ratios of the elements of the same

clusters. Therefore, as indicated above, one of these log-ratios or a balance of one cluster

against the other might be representative for many log-ratios, and consequently may represent

a process influencing many elements in the same way. Balances of elements within the cluster

will filter out these large-variability effects and focus on differences between elements

behaving similarly with respect to major processes.

As a default agglomerative clustering procedure the Ward method (Ward, 1963) is used in

Pawlowsky-Glahn et al. (2011) and van den Boogaart and Tolosana-Delgado (2013), where

those two clusters are fused which result in the least increase in the sum of the (squared)

distances from each observation to the centroid of cluster contained it. Consequently, this

methods leads to spherical, tightly bound clusters that might in general be intuitively

interpretable. Moreover, because the Ward criterion corresponds to minimizing the total

within-cluster variance, it seems be recommendable for clustering of variables (compositional

parts). On the other hand, as the variation matrix has all properties of quasimetics, it is

meaningful to consider also other clustering methods that might lead even to better

interpretable clusters. For example, smaller clusters might be geologically easily interpreted.

Large numbers of small, tightly bound clusters are obtained using complete linkage, therefore

being a candidate for such an alternative agglomerative clustering algorithm. By considering

all these aspects, the dendrogram output might help to reveal such coordinates (balances) that

contain log-ratios responsible for the geochemical signature of the D/C boundary, represented

by presence of HBE black shale and sandstone layer.

3. Results and discussion

Despite of strong limitations of univariate analysis of raw compositional data (Filzmoser et

al., 2009b; McKinley et al., 2016), summaries of elemental concentrations are popular starting

point of any geochemical study. According to Pawlowsky-Glahn and Egozcue (2002) and

Mateu-Figueras and Pawlowsky-Glahn (2008), geometric mean is used to compute mean

concentrations; for a raw impression about variability of elements, interquantile ranges (IQR)

are applied.

The mean concentrations of the elements in the composition (in the order of its geometric

mean) are as follow: Ca (geometric mean = 15.55 %; IQR = 21.57 %), Si (5.05 %; ),

Al (0.75 %; 2.69 %), Fe (0.70 %; ), K (0.29 %; 0.77 %), Ti (734.04 ppm; 2110.46

ppm), Mn (557.11 ppm; ppm), Sr (208.40 ppm; 181.30 ppm), Zr (39.19 ppm; 84.14

ppm), Cr (28.39 ppm; 65.26 ppm), Rb (25.76 ppm; 81.29 ppm), Ni (22.19 ppm; 33.53 ppm),

Y (21.02 ppm; 18.99 ppm), Zn (15.50 ppm; 27.16 ppm), Cu (11.65 ppm; 16.21 ppm), and Pb

(8.58 ppm; 10.21 ppm). Taking the relative scale into account, the element concentrations

show a strong variance across the studied areas and sections. This largely reflects the

principal lithology where pure carbonate, shale, sandstone, and siliceous sediments represent

the ideal end members.

The Upper Devonian carbonate-dominated successions (layer 1) are generally characterized

by high concentrations of Ca (geometric mean: 22.60 %) and low concentrations of Al, Si, K,

Ti, Fe, Rb, and other elements. The highest concentrations of Ca were detected in the pelagic

successions of the Rhenish Massif, Carnic Alps and Montagne Noire. In contrast, coeval

strata of the Namur–Dinant Basin (Gendron-Celles, Les Ardennes sections) have relatively

lower Ca concentrations.

In the following HBE black shale and sandstone interval (layers 2 and 3), the element

composition at the majority of sections changes rapidly, consistently with the lithology

change. Compared to the underlying strata, the mean concentrations of Ca in the HBE shales

and sandstones are extremely low (geometric mean 3.58 %) while the mean concentrations of

typically terrigenous elements are much higher (Al = 4.62 %, K = 1.31 %, Fe = 3.16 %, Ti =

4500 ppm, Rb = 190 ppm, S = 15 cps). In addition, the HBE black shales (layer 2) have very

high mean concentrations of Zr (221 ppm), S (59 cps), Zn (51 ppm), Ni (114 ppm) and Pb

(95 ppm).

The lower Tournaisian carbonate successions (layer 4) have relatively high concentrations of

Ca (25.13 %) and relatively low concentrations of elements such as Al (0.39 %), K (0.16 %),

Fe (0.43 %), Ti (470 ppm) and Rb (15 ppm), which is similar to those of the upper Famennian

succession (layer 1).

The youngest strata of the LAS interval (layer 5) were reached only at several sections of the

Namur-Dinant Basin (Rivage), Rhenish Massif (Oese, Drewer), Montagne Noire (Puech de la

Suque), Pyrenees (Saubette) and Carnic Alps (Kronhofgraben). These sediments are shaly,

often siliceous and phosphatic. They have high mean concentrations of Al (2.69 %), Si (25.05

%), K (0.65 %), Fe (2.46 %), Ti (0.22 %), Rb (104 ppm) and low concentrations of Ca (3.69

%). Compared to the all underlying layers, they are markedly enriched in elements such as P

(11 cps), Cu (47 ppm), Zn (44 ppm) and Ni (85 ppm).

In order to reveal both overall and specific geochemical patterns, standard biplots of raw or

log-transformed data and compositional biplots were used at each stage of analysis and this

section presents those options which led to the most interesting results. Figure 1 shows a

standard biplot of raw data (left) and its compositional counterpart (right). In both plots, a

cluster of terrigenous elements (Al, Ti, Rb, K, Fe) stands in opposition to Ca along the first

principal component, which is interpreted as the effect of dilution of terrigenous input by

marine CaCO3 production. The first principal component (PC) in the biplot depicting raw data

explains 56.9 % of total variability indicating that the effect of CaCO3 dilution is the

dominant geochemical pattern in the whole dataset. The second PC in the standard biplot

shows the contents of such productivity- and redox- sensitive elements as S, P, and Mn, while

other productivity-sensitive elements (Zn, Cu) largely overlap with the terrigenous ones. The

effect of Ca dilution seems to override the subtle variations in element behaviors in the

standard biplot. On the other hand, the compositional biplot better differentiates between

samples from different locations (different provenance) particularly along the second

principal component, which represents the effects of provenance and marine productivity. In

addition, the typically detrital proxy elements (Al, Ti, Rb, K and Si) are more clearly

separated from productivity-sensitive elements (Zn, Pb and Cr) in the compositional biplot

than in the standard one. At the same time, the effect of Ca dilution is partly removed in the

compositional biplot as indicated by the fact the Ca and terrigenous elements are no more in

opposition (Figure 1).

Figure 1 Standard biplot of raw data (left) and clr biplot (right), with samples distributed according to its location.

The biplots showing the whole data set bring important information about the general data

structure. However, the clustered distribution of samples from different geographical settings

and, hence, different provenance (Figure 1) suggests that it is appropriate to analyze such data

clusters separately.

Log-transformation of the raw geochemical data reflects the relative scale of observations;

consequently, it should help to reveal further patterns, possibly hidden behind the raw data.

The Figure 2 shows the application of standard biplot on log-transformed data with the

distinct stratigraphic layers, from Upper Devonian carbonates to Lower Alum Shale,

indicated. As it will be shown later, such geochemical markers as element ratios are

instrumental in distinguishing these layers. The Figure 2, left, shows a standard biplot of log-

transformed data from the Oberrödinghausen section, Rhenish Massif, Germany with four of

the five layers (Uppermost Devonian /Famennian/ carbonates, the HBE black shale, HBE

sandstone, and Lower Carboniferous /Tournaisian/ carbonates) indicated. The layers group

together as distinct clusters; there is a general pattern of clustering of samples from the HBE

shales/sandstones and carbonates along the PC1 axis, which is interpreted as the dilution

effect of calcium carbonate (see above). In addition, samples representing the HBE black

shale event occur as distinct outlier observations due to the high concentrations of Pb, Zn, Ni

and Cu, typical productivity-sensitive elements. The overlaps between the HBE black shales

and HBE shales/sandstones in the area with negative PC1 scores and positive PC2 scores

suggests that the black shales and “normal” grey shales can be represented by very similar

geochemistry.

Similar patterns are shown for the Oese section, Rhenish Massif (Figure 2, right). The

carbonates are again well separated from the shales and sandstones and there are distinct

outliers representing the HBE black shales and Lower Alum Shales (negative scores on PC1

and PC2) suggesting high concentrations of productivity-sensitive elements such as Pb, Zn,

Cu, Ni and Y. This suggests that the black shales of the HBE and LAS interval share a similar

geochemical composition. Moreover, both the biplots show a distinct clustering of variables

(elements), in particular the terrigenous Al, Rb, Ti, K and Si, which is consistent with the

summary biplot of all observations (Figure 1) and their expected geochemical behavior (see

section 2.1).

Figure 2 Standard biplots of log-transformed data from locations Oberrödinghausen (left) and Oese (right), Rhenish Massif,

Germany.

An even better representation of sample clustering is shown in Figure 3 providing the D/C

boundary data from the Kronhofgraben section, Carnic Alps, Austria. This biplot of log-

transformed data clearly separates between the cherts and black cherty shales of the Lower

Alum Shale interval, enriched in Cu, Ni, Pb and Si (negative scores on PC1 and positive

scores on PC2) and black shales of the HBE black shale interval (negative scores on both PC1

and PC2). This is consistent with the dissimilar lithology patterns of the HBE and LAS layers

at the Kronhograben section, the former represented by typical black shales and the latter by

black cherts and cherty shales (Schönlaub et al., 1992).

Figure 3 Biplot of log-transformed data from location Kronhofgraben, Carnic Alps, Austria.

The applicability of raw data, which are somewhat inappropriate from purely methodological

perspective, and clr coordinates can be demonstrated on samples from the Rhenish Massif

(Oese, Oberrödinghausen and Drewer sections) (Figure 4). The raw-data biplot (Figure 4 left)

clearly separates the black shales enriched in Ni, Zn, Pb and Cu from the remainder of

samples. The latter samples mostly tend to align to a carbonate (Ca) – fine-grained siliciclastic

(Al, Rb, K, Ti) component line, which again reflects the degree of carbonate dilution of the

siliciclastic detrital input. In the compositional biplot, this pattern is generally lost, while the

differences between individual localities are highlighted.

Figure 4 Biplot of raw data (left) and compositional biplot (right) for locations from Rhenish Massif (Oberrödinghausen,

Oese, Drewer sections).

Loadings in the raw-data biplot again show a distinct clustering of elements, depending

largely on the expected geochemical behavior. In the Rhenish Massif, this clustering is very

well visible in areas where the particular genetic element groups (see above) are represented

by typical lithologies (black shales) with relatively high concentrations of their nominal

elements (Zn, Pb, Ni and Cu, Figure 4, left). There is again the overwhelming negative

correlation between Ca on one hand and Al, Rb, K and Ti on the other hand. In the

compositional biplot, however, the effects of a strong association with Ca (dilution effect) are

strongly suppressed and elements show clustering into several, genetically related groups: Ca

and Sr associated with marine calcium carbonate; terrigenous elements associated with

detrital rock-forming minerals, mainly phyllosilicates (Al, Si, Rb, Ti, Rb), redox-sensitive

elements associated with sulphidic phases (Cu, Ni, S, P) and other redox-sensitive elements

(Mn, Fe).

We also tested these patterns in the Ardennes (Gendron Celles section), where the HBE s.l.

layer is not represented by the distinct shale/sandstone lithology and the section is carbonate-

dominated (Figure 5). In the raw-data biplot (Figure 5 left) the effect of carbonate dilution is

very strong as indicated by the alignment of samples along the PC1 axis as well as the

negative correlation between Ca and the remainder of elements (with the exception of Sr, Mn

a Zn). In the compositional biplot, the effect of Ca dilution is again suppressed but the

variables (elements) tend to group together according to their expected geochemical behavior

(Al+Ti+Rb vs. Si+Zr vs. Pb+Zn+Cr+Mn) despite their very low concentrations. As indicated

in the previous sections, this seems to be the effect of relative scale of compositions, captured

by their clr coordinates.

Figure 5 Biplot of raw data (left) and compostional biplot (right) for location Gendron Celles, Ardennes, Belgium.

Another source of information about the data structure is represented by dendrograms

resulting from Q-mode clustering, whose construction was described in Section 2.3. Instead

of the originally proposed Ward clustering method, the complete linkage shown here and

based on the variation matrix provides more reliable results. Groups of elements identified by

this method can facilitate finding log-ratios, which optimally describe the presence of the

Hangenberg event layers and thus provide a compositional alternative to the standard proxies

as Zr/Al, K/Al and Rb/K ratios (Kumpan et al., 2014b, 2015).

Figure 6 shows a dendrogram for the Kronhofgraben section as the output of the Q-mode

clustering. According to this method, several balance coordinates were defined; three of them,

which lead to best separation of groups, are shown in Figure 7. The last one (Figure 7, right),

which excludes Ca (based on the assumption that Ca is strongly related to the dilution effect),

was selected, to see whether it can affect the values of the resulting balance. For comparison,

Figure 8 shows values of standard Hangenberg event proxies Zr/Al, K/Al and Rb/K. The

proposed log-ratios are useful to differentiate individual groups of observations; e.g., the low

values of ln(g(Ca,Sr)/g(rest)), where g(rest) stands for geometric mean of all components

except Ca and Sr, and log-ratio between Ca and Sr mark the presence of HBE black shale and

the LAS layer. Element ratios such as Zr/Al and K/Al are capable of distinguishing between

the HBE black shales and LAS black cherts/ cherty shales, the former having much higher

Zr/Al and lower K/Al values as compared to the latter (Figure 8). Nevertheless, when

comparing Figures 7 and 8 we can see that the compositional variables (balances) better

distinguish four groups of observations; the changes are really dramatic a provide a clear

structure. Note that similar patterns could be also seen for some other balances, indicated by

the dendrogram of Q-mode clustering (Figure 6), but the presented ones seem to provide the

best performance.

Figure 6 Dendrogram of Q-mode clustering for location Kronhofgraben.

Figure 7 Proposed log-ratios, based on dendrogram of Q-mode clustering from location Kronhofgraben, which optimally

discriminate layers.

Figure 8 Standard ratios used for discrimination between layers applied on location Kronhofgraben.

Q-mode clustering can also help to find suitable proxies only for subsets of observations, for

example for similar lithologies (shales and sandstones). Figure 9 shows the resulting

dendrogram for Oese section, in which only the HBE black shale and sandstone and Lower

Alum Shales were taken into account.

Figure 9 Dendrogram of Q-mode clustering for location Oese and layers Hangenberg black shale, Hangenberg sandstone and

Lover Alum Shale.

According to this clustering, combined with geochemical knowledge, we selected first the

log-ratio between subcompositions formed by parts Ti, Al, K and Cr, Rb, respectively (Figure

10, upper left) and one additional balance that links the previous elements with Cu and Zn

from the same branch of the dendrogram (Figure 10, upper right). Both balances clearly

separate the Lower Alum Shale layer from the HBE black shales. This represents the

advantage of log-ratio approach, compared to standard proxies, whose values are displayed on

Figure 10 (lower row) and which do not distinguish between layers at all. This is a direct

consequence of the fact that Q-mode clustering supplements the preliminary geochemical

knowledge with further possible candidates (balances) to reveal better the D/C boundary.

Figure 10 Proposed log-ratios, based on Q mode clustering from location Oese, which optimally discriminate layers (upper

row) and standard ratios used for discrimination between layers (lower row ).

In general, statistical analysis of observations including elemental geochemistry of carbonate

and siliciclastic bed successions across the D/C boundary using both compositional and non-

compositional approaches has revealed several interesting features. Dimension reduction of

multivariate geochemical data through principal component analysis is nowadays a must for

any reasonable case study. In stratigraphy, statistical pre-treatment of geochemical data is also

a common approach prior to depicting of stratigraphic patterns of element concentrations

(Sedláček et al., 2013; Bábek et al., 2015). In the present paper, several approaches how to

process the input observations prior to PCA were presented. The work flow proceeds from the

raw compositions (which are often inappropriate due to the relative nature of compositional

data) to either log-transformation or clr coordinates that consider, or not, total abundances of

elements. Interestingly, raw-data biplots were found useful to depict the basic geochemical

patterns such as the dilution effect of Ca and enrichment of black shales with productivity-

sensitive elements (Cu, Zn, Ni, etc.). The possible reason is that these effects are

predominantly driven just by absolute concentrations (or cps signal) of components rather

than being inherently contained in their ratios. On the other hand, compositional biplots are

capable of filtering out these predominant geochemical trends (such as the Ca dilution effect)

and depicting subtle geochemical variations, which are obscured by the major trends.

Following the previous argumentation, depicting of compositional log-ratios in

vertical/horizontal logs, which is a common approach in stratigraphy, can be more appropriate

than depicting simple element ratios, because subtle geochemical trends can be obscured by

the predominant trends such as the Ca dilution. This applies, for example, for nodular

limestones (such as the Upper Devonian carbonates of the Rhenish Massif), in which

diagenetic Ca redistribution strongly affects the primary geochemical signal. Nevertheless,

analysis in the previous section has shown that even the dilution effect of Ca is desirably

suppressed, when sufficiently robust log-ratios are considered (Figure 7). Moreover, with Q-

mode clustering it is possible to cover both dimension reduction using PCA, performed in clr

coordinates, and D/C layer discrimination under one methodological framework.

5. Conclusions

Our recent experiments clearly indicate that the combination of raw/log-transformed data and

compositional analyses are capable of distinguishing D/C boundary sediment layers with

specific geochemical signature, in particular the HBE black shales and LAS shales. This is in

line with recent contributions from geochemistry (Montero-Serrano et al., 2010, Reimann et

al., 2012, Bábek et al., 2015, McKinley et al., 2016) showing that not just the standard

processing of geochemical data, represented mostly by log-transformation, but even the

compositional data analysis has some limitations that favour the use of the complementary

approach. It is unexceptionable that only the logratio methodology would be acceptable, if

exclusively relative information out of geochemical data were informative, but it rather seems

not to be the case in most practical situations. Consequently, neither the standard nor the

compositional approaches have prevalence, but it is advisable to use both of them to discern

predominant and subtle geochemical trends in large datasets. Using the fact that the standard

and compositional approaches have synergic effects is thus recommendable for future

developments, in sedimentology, and also in geochemistry in general. Nevertheless, one must

be aware of interpretational dangers by using the standard approaches (log-transformed or

even raw data) that do not occur with the logratio methodology; a deep understanding of the

underlying geological phenomena (being the case here) is a necessary presumption of their

possible use in geochemical practice.

Acknowledgements

Karel Hron gratefully acknowledges the support of the grant COST Action CRoNoS IC1408

and the grant IGA_PrF_2016_025 Mathematical Models of the Internal Grant Agency of the

Palacky University in Olomouc. This work was partly supported by the Czech Science

Foundation (GAČR) research project 14-18183S (O. Bábek). We thank to Tomáš Matys

Grygar (Institute of Inorganic Chemistry ASCR, v.v.i.,) for providing of EDXRF data

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List of Figures

Figure 1 Standard biplot of raw data (left) and clr biplot (right), with samples distributed

according to its location.

Figure 2 Standard biplots of log-transformed data from locations Oberrödinghausen (left) and

Oese (right), Rhenish Massif, Germany.

Figure 3 Biplot of log-transformed data from location Kronhofgraben, Carnic Alps, Austria.

Figure 4 Biplot of raw data (left) and compositional biplot (right) for locations from Rhenish

Massif (Oberrödinghausen, Oese, Drewer.)

Figure 5 Biplot of raw data (left) and compostional biplot (right) for location Gendron Celles,

Ardennes, Belgium.

Figure 6 Dendrogram of Q-mode clustering for location Kronhofgraben.

Figure 7 Proposed log-ratios, based on dendrogram of Q-mode clustering from location

Kronhofgraben, which optimally discriminate layers.

Figure 8 Standard ratios used for discrimination between layers applied on location

Kranhofgraben.

Figure 9 Dendrogram of Q-mode clustering for location Oese and layers Hangenberg black

shale, Hangenberg sandstone and Lover alum shale.

Figure 10 Proposed log-ratios, based on Q mode clustering from location Oese, which

optimally discriminate layers (upper row) and standard ratios used for discrimination between

layers (lower row).

.

List of Equations

(1) Centred log-ratio coordinates.

(2) Geometric mean.

(3) Variation matrix.

(4) Non-negativity, identity of indiscernibles and symmetry of elements of the variation

matrix.

(5) Triangular inequality of elements of the variation matrix.

(6) Relation between elements of the variation matrix and variance of a logratio.

(7) Balance.