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, email@example.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
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
Keywords: element geochemistry; compositional biplot; dendrogram; logratio coordinates;
Devonian/Carboniferous boundary; variation matrix
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
where g(x) stands for geometric mean
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
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
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
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,
i.e., non-negativity, identity of indiscernibles and symmetry. On the other hand, triangular
inequality is not fulfilled in general, only its generalized form
for a constant that corresponds to quasimetrics (Xia, 2009) can be derived. According
to Fišerová and Hron (2011) it holds true that
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
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)
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
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
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
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
Figure 2 Standard biplots of log-transformed data from locations Oberrödinghausen (left) and Oese (right), Rhenish Massif,
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
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
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
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.
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.
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,
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
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
(5) Triangular inequality of elements of the variation matrix.
(6) Relation between elements of the variation matrix and variance of a logratio.