Application of multi-dimensional discrimination diagrams

Transcription

Versão online: http://www.lneg.pt/iedt/unidades/16/paginas/26/30/125
Comunicações Geológicas (2012) 99, 2, 79-93
ISSN: 0873-948X; e-ISSN: 1647-581X
Application of multi-dimensional discrimination diagrams
and probability calculations to acid rocks from Portugal and
Spain
Aplicação de diagramas discriminantes multi-dimensionais e
cálculos de probabilidades para rochas ácidas de Portugal e
Espanha
S. P. Verma1*
Artigo original
Original article
Recebido em 12/02/2012 / Aceite em 09/04/2012
Disponível online em Abril de 2012 / Publicado em Dezembro de 2012
© 2012 LNEG – Laboratório Nacional de Geologia e Energia IP
Abstract: Discrimination diagrams have been used to decipher tectonic
setting of old terrains. New discriminant-function based multi-dimensional
diagrams for acid magmas were recently proposed. I present eleven case
studies of Ediacaran-Early Cambrian to Permian acid magmas from Portugal
and Spain to highlight the application of these diagrams and probability
calculations. Two case studies on Rebordelo-Agrochão and Telões (Portugal)
show a continental arc to collision transitional tectonic setting during the Early
Carboniferous to Permian. The third case study on Oledo and Gouveia
(Portugal) also indicated a continental arc to collision setting during the Early
Ordovician. For the next three case studies on Gouveia, Castelo Branco, and
Guarda from Portugal, and one on Jalama from Spain, a collision setting was
clearly inferred during the Late Carboniferous to Permian. A collision setting
was also indicated for Ossa-Morena (Portugal) during the Ediacaran-Early
Cambrian. Finally, the last three case studies showed an island arc setting for
Albrenoa (Portugal) and island arc to collision transitional setting for Serra
Branca (Portugal) during the Devonian to Early Carboniferous, and an arc to
collision setting for Évora (Portugal) during the Early Carboniferous. Possible
reasons for generally consistent as well as some probably inconsistent
inferences were also briefly discussed.
Keywords: Tectonic setting; SINCLAS computer program; discriminantfunction based diagrams; log-ratio transformation; probability calculations.
Resumo: Diagramas de discriminação têm sido utilizados para decifrar o
enquadramento tectónico de terrenos antigos. Novos diagramas multidimensionais de função discriminante para magmas ácidos foram
recentemente propostos. Apresento onze estudos de caso de magmas ácidos do
Ediacariano-Câmbrico inferior até ao Pérmico de Portugal e Espanha para
destacar a aplicação destes diagramas e cálculos de probabilidade. Dois
estudos de caso em Rebordelo-Agrochão e Telões (Portugal) mostram um
enquadramento tectónico transitório entre arco continental e colisão durante o
início do Carbónico ao Pérmico. O terceiro estudo de caso em Oledo e
Gouveia (Portugal) também indicou arco continental a colisão como
enquadramento tectónico durante o início do Ordovícico. Para os seguintes
três estudos de caso em Gouveia, Castelo Branco e Guarda em Portugal e de
Jalama em Espanha, um enquadramento tectónico de colisão foi claramente
inferido durante o Carbónico tardio-Pérmico. Um enquadramento tectónico de
colisão também foi inferido para a Ossa-Morena (Portugal) durante o
Ediacariano-início do Câmbrico. Finalmente, os últimos três estudos de caso
mostraram um enquadramento de arco insular para Albrenoa (Portugal),
transitório entre arco insular e colisão para Serra Branca (Portugal) durante o
Devónico-início do Carbónico e um enquadramento de colisão para Évora
(Portugal) durante o início do Carbónico. Possíveis razões para as inferências
geralmente consistentes, bem como algumas provavelmente inconsistentes
foram também brevemente discutidas.
Palavras-chave: enquadramento tectónico, programa de computador
SINCLAS, diagramas baseados em função de discriminação, transformação
log-rácio; cálculos de probabilidades.
1
Departamento de Sistemas Energéticos, Centro de Investigación en Energía,
Universidad Nacional Autónoma de México, Privada Xochicalco s/no., Centro,
Apartado Postal 34, Temixco, Mor., 62580, México.
* Corresponding author / Autor correspondente: spv@cie.unam.mx
1. Introduction
Discrimination diagrams have provided a frequently used
geochemical tool for deciphering tectonic environment of old
terrains as well as of tectonically complex areas (Pearce & Cann,
1971, 1973; Wood, 1980; Shervais, 1982; Pearce et al., 1984;
Cabanis and Lecolle, 1989; Rollinson, 1993; Verma, 2010).
Recently, Verma (2010) extensively evaluated a large number of
available diagrams and inferred that those proposed recently
(Agrawal et al., 2004, 2008; Verma et al., 2006; Verma &
Agrawal, 2011) show the highest success rates (76%-97%).
Satisfactory functioning of these multi-dimensional diagrams was
also confirmed independently by Sheth (2008), Verma et al.
(2011), and Pandarinath & Verma (2013). Most of these recent
diagrams were based on linear discriminant analysis (LDA) of
natural logarithm of element ratios; this transformation constitutes
correct statistical treatment of compositional data (e.g., Aitchison,
1986; Aitchison et al., 2000; Thomas & Aitchison, 2005; Agrawal
& Verma, 2007; Verma, 2010; Verma et al., 2010). All these
multi-dimensional diagrams were proposed for the discrimination
of basic and ultrabasic magmas. A computer program TecD
(Verma & Rivera-Gómez, 2013) facilitates their efficient
application.
For acid magmas, only a few older bivariate-type diagrams
(Pearce et al., 1984) were available until the proposal of new multidimensional diagrams (Verma et al., 2012). These new diagrams
based on LDA of loge-transformed ratios were shown by Verma et
al., (2012) to work better than the older concentration-based
diagrams of Pearce et al. (1984).
Therefore, in the present work I have used the newest multidimensional diagrams (Verma et al., 2012) to illustrate their use for
the recognition of the most probable tectonic settings for acid rocks
of different ages and localities of Portugal and Spain. This set of
five new discriminant-function diagrams based on natural
logarithm-transformed ratios of major-elements has been proposed
for the discrimination of island arc (IA, group no. 1), continental
arc (CA, group no. 2), continental rift (CR, group no. 3), and
80
S. Verma / Comunicações Geológicas (2012) 99, 2, 79-93
collision (Col, group no. 4) tectonic settings. It is important to note
that this is the first set of multi-dimensional diagrams proposed to
discriminate the two very similar tectonic settings of island and
continental arcs in the presence of two other tectonic settings.
2. Database
Ten localities in Portugal and one in Spain were selected. A synthesis
of the relevant information (locality, approximate coordinates,
number of compiled samples, age, and literature references) is
provided in Table 1. A map of sample locations is substituted by
approximate geographical coordinates (Table 1). Detailed geology
and locations of samples can be consulted in the papers from which
the data were compiled. In summary, data for ten case studies from
northern (two cases), central (four) and southern (four) Portugal and
one from Spain were compiled in Statistica software.
All major-element chemical compositions were processed in
SINCLAS (Verma et al., 2002, 2003) to ascertain that the magma
type was acid and to obtain adjusted values of eleven oxides under
the Middlemost (1989) option for Fe-oxidation adjustment.
The first diagram discriminates the tectonic setting of IA and CA
together as arc (IA+CA), CR, and Col, for which the x and y
coordinates were calculated, respectively, as DF1(IA CA-CR -Col)m3
and
DF2 (IA  CA -CR -Col) m3
functions from equations 1 and 2, where
in the subscript m3, m stands for major-elements and 3 refers to
the third set of such multi-dimensional diagrams. The first two
sets of diagrams (Agrawal et al., 2004; Verma et al., 2006) were
proposed for the discrimination of basic and ultrabasic magmas,
for which the subscripts m1 and m2 were used by Verma and
Rivera-Gómez (2013) in their computer program (TecD).
DF1(IACA-CR -Col)m3  (0.36077  ln(TiO 2 /SiO 2 ) adj )  (0.95693  ln(Al2 O 3 /SiO 2 ) adj ) 
(-2.09239  ln(Fe2 O 3 /SiO 2 ) adj )  (0.93391 ln(FeO/SiO2 ) adj ) 
(0.42703  ln(MnO/SiO2 ) adj )  (0.18732  ln(MgO/SiO2 ) adj ) 
(0.45615  ln(CaO/SiO2 ) adj )  (0.56098  ln(Na 2 O/SiO 2 ) adj ) 
(-1.65167  ln(K 2 O/SiO 2 ) adj )  (-0.15580  ln(P2 O 5 /SiO 2 ) adj )  1.58259
(1)
DF2(IA CA -CR -Col)m3  (0.472353  ln(TiO2 /SiO 2 ) adj )  (-0.954629  ln(Al2O3/SiO 2 ) adj ) 
(0.109516  ln(Fe2O3 /SiO 2 ) adj )  (0.699238  ln(FeO/SiO2 ) adj ) 
(0.739533 ln(MnO/SiO2 ) adj )  (-0.027717  ln(MgO/SiO2 )adj ) 
(-0.244687  ln(CaO/SiO2 ) adj )  (0.231677  ln(Na 2O/SiO 2 ) adj ) 
3. Multi-Dimensional Diagrams
(0.173552  ln(K 2O/SiO 2 ) adj )  (-0.353797  ln(P2O5 /SiO 2 )adj )  6.691035
The discriminant functions for five diagrams were calculated
from equations 1-10 summarised in this section. Five diagrams
are required to discriminate four tectonic settings of IA, CA, CR,
and Col (Verma et al., 2012). For each diagram, two functions
must be calculated for each compiled sample.
The second diagram discriminates the tectonic setting of IA, CA
and CR, for which equations 3 and 4 were used for calculating
the DF1(IA - CA - CR) m3 and DF2(IA - CA - CR) m3 functions. Note the
Col setting is absent from it.
Table 1. Synthesis of database on acid rocks from Portugal.
Tabela 1. Síntese da base de dados das rochas ácidas em Portugal.
For the zone of Portugal, the abbreviations are as follows: N−northern, C−central, S−southern. The
number of areas corresponds to samples grouped for the interpretation synthesised in Table 3.
(2)
Multi-dimensional discrimination diagrams in acid rocks from Portugal and Spain
The samples are plotted and counted for the different
tectonic fields of these diagrams. Alternatively, the plotting of
samples can be complemented and, in fact, replaced by the
probabilities calculations for individual samples.
DF1(IA-CA -CR) m3  (0.4786  ln(TiO2 /SiO 2 ) adj )  (-0.0871 ln(Al2O3/SiO 2 ) adj ) 
(2.7433  ln(Fe2O3/SiO 2 ) adj )  (1.0663  ln(FeO/SiO2 ) adj ) 
(-0.1389  ln(MnO/SiO2 ) adj )  (-0.1907  ln(MgO/SiO2 ) adj ) 
(-0.8516  ln(CaO/SiO2 ) adj )  (-0.7139  ln(Na 2O/SiO2 ) adj ) 
(1.7166  ln(K 2O/SiO 2 ) adj )  (0.3386  ln(P2O5 /SiO 2 ) adj )  6.2573
(3)
4. Probability Calculations
DF2(IA-CA -CR) m3  (0.3204  ln(TiO2 /SiO 2 ) adj )  (-1.7585  ln(Al2O3 /SiO 2 ) adj ) 
(-3.2046  ln(Fe2O3/SiO 2 )adj )  (1.1210  ln(FeO/SiO2 ) adj ) 
(0.2170  ln(MnO/SiO2 )adj )  (-0.0745  ln(MgO/SiO2 ) adj ) 
(1.2505  ln(CaO/SiO2 ) adj )  (1.3142  ln(Na 2O/SiO2 )adj ) 
(1.6616  ln(K 2O/SiO2 )adj )  (0.0186  ln(P2O5 /SiO 2 ) adj )  0.9984
(4)
The third diagram discriminates the tectonic setting of IA, CA
and Col (the absent setting is CR) and the DF1(IA - CA - Col) m3 and
DF2(IA - CA - Col) m3 functions were calculated from equations 5 and 6.
DF1(IA- CA - Col) m3  (0.3620  ln(TiO2 /SiO 2 ) adj )  (-0.0342 ln(Al2 O3 /SiO 2 ) adj ) 
(0.5198  ln(Fe2 O3 /SiO 2 ) adj )  (-0.4980  ln(FeO/SiO2 ) adj ) 
(-0.7223 ln(MnO/SiO2 ) adj )  (-0.1229  ln(MgO/SiO2 ) adj ) 
(-0.1388 ln(CaO/SiO2 ) adj )  (-0.8174  ln(Na 2O/SiO 2 ) adj ) 
(1.5074  ln(K 2 O/SiO 2 ) adj )  (0.2684  ln(P2 O5 /SiO 2 ) adj )  3.0829
(5)
DF2(IA- CA -Col) m3  (0.142  ln(TiO2 /SiO 2 ) adj )  (1.984  ln(Al2 O3 /SiO 2 ) adj ) 
(1.747  ln(Fe2 O3 /SiO 2 ) adj )  (-0.735  ln(FeO/SiO2 ) adj ) 
(-1.226 ln(MnO/SiO2 ) adj )  (0.062  ln(MgO/SiO2 ) adj ) 
(-1.152 ln(CaO/SiO2 ) adj )  (-3.189 ln(Na 2O/SiO2 ) adj ) 
(-2.339 ln(K 2O/SiO2 ) adj )  (0.495  ln(P2O5 /SiO 2 ) adj )  18.190
(6)
The fourth diagram discriminates the tectonic setting of IA, CR
and Col (the absent setting is CA) and the DF1(IA - CR - Col) m3 and
DF2(IA-CR-Col)m3 functions were calculated from equations 7 and 8.
(-2.6406  ln(Fe2 O 3 /SiO 2 ) adj )  (2.9494  ln(FeO/SiO2 ) adj ) 
(0.1970  ln(MnO/SiO2 ) adj )  (0.0673  ln(MgO/SiO2 ) adj ) 
(0.0620  ln(CaO/SiO2 ) adj )  (0.6219  ln(Na 2 O/SiO 2 ) adj ) 
(7)
mdf2g1 ); ( mdf1g2 , mdf2g2 ); and ( mdf1g3 , mdf2g3 ) ─ of the
(0.8267  ln(Fe2O3/SiO 2 ) adj )  (0.3032  ln(FeO/SiO2 ) adj ) 
tectonic groups g1, g2, and g3 (Table 2), respectively, in a given
diagram were calculated as follows:
(0.4084  ln(MnO/SiO2 ) adj )  (-0.0905  ln(MgO/SiO2 ) adj ) 
(-0.3260  ln(CaO/SiO2 ) adj )  (0.1518  ln(Na 2O/SiO2 ) adj ) 
(8)
Finally, the fifth diagram discriminates the tectonic setting of
CA, CR and Col (the absent setting is IA), for which the
DF1(CA - CR - Col) m3 and DF2(CA - CR - Col) m3 functions were calculated
from equations 9 and 10.
DF1(CA-CR -Col)m3  (0.0645  ln(TiO2 /SiO 2 ) adj )  (-1.7943 ln(Al2O3/SiO 2 ) adj ) 
(11)
d g2  (df1s - mdf1g2 ) 2  (df2s - mdf2g2 ) 2
(12)
d g3  (df1s - mdf1g3 ) 2  (df2s - mdf2g3 ) 2
(13)
under evaluation in a given diagram. The mean values mdf1g1 ,
(-0.3265  ln(CaO/SiO2 ) adj )  (0.1063  ln(Na 2O/SiO2 ) adj ) 
(9)
mdf2g1 , etc., for all five diagrams of Verma et al. (2012) are
given in Table 2.
New functions sg1, sg2, and sg3 based on these distances
( d g1 , d g2 , and d g3 ; equations 11-13) for that particular sample
DF2(CA -CR -Col) m3  (0.8760  ln(TiO2 /SiO 2 ) adj )  (0.8018  ln(Al2O3/SiO 2 ) adj ) 
(0.2472  ln(Fe2O3/SiO 2 ) adj )  (-0.8796  ln(FeO/SiO2 ) adj ) 
were computed from equations 14-16 as follows:
(-0.0624  ln(CaO/SiO2 ) adj )  (-0.2052  ln(Na 2O/SiO2 )adj ) 
(-3.3091 ln(K 2O/SiO 2 ) adj )  (-0.3526  ln(P2 O5 /SiO 2 ) adj )  3.8959
d g1  (df1s - mdf1g1) 2  (df2s - mdf2g1) 2
where df1s and df2s are the coordinates or scores of the sample
(0.5264  ln(Fe2O3 /SiO 2 ) adj )  (0.6385  ln(FeO/SiO2 ) adj ) 
(1.8098  ln(K 2O/SiO2 ) adj )  (-0.0338  ln(P2O5 /SiO 2 ) adj )  8.2616
from equations 11 to 19 to keep them relatively simple and useful
for all five diagrams; otherwise, 36 more equations, being nine for
each diagram, had to be listed.
The distances ( d g1 , d g2 , and d g3 ) of a sample under
evaluation from the three group mean values ─ ( mdf1g1 ,
DF2(IA-CR -Col) m3  (0.2786  ln(TiO2 /SiO 2 ) adj )  (-1.0544  ln(Al2O3/SiO 2 ) adj ) 
(0.6698  ln(K 2O/SiO2 ) adj )  (-0.2261 ln(P2O5 /SiO 2 ) adj )  6.5170
The probability calculations were recently highlighted by Verma
& Agrawal (2011) for their immobile element based diagrams,
although probability-based boundaries were used in all such
diagrams (Agrawal et al., 2004, 2008; Verma et al., 2006;
Verma et al., 2012). In this work, the concept of posterior
probability calculations for individual samples was used for
drawing statistical inferences.
A sample plotting at the boundary of two fields in a given
diagram will have approximately equal probability of 0.5000 for
the two fields. But a sample that plots at the triple point, being
the point of intersection of the three boundaries in a diagram,
will have an equal probability value of about 0.3333 for each of
the three fields. Thus, the probability of a sample plotting at the
boundary of two fields will change from 0.5000 to 0.3333 as the
sample moves along the boundary towards the triple point.
However, the probability increases very rapidly for a given
field as the sample plots somewhat away from the boundary in
the interior of that particular field (for more details, see Verma
& Agrawal, 2011). The procedure of probability calculations for
a sample in a discrimination diagram was not presented by
Verma et al. (2012), but is explained in this section.
To calculate the probabilities of a sample for the three
tectonic settings discriminated in a given diagram, the procedure
is from Verma & Agrawal (2011); a few nomenclature errors
were corrected. The subscripts, such as (IA CA -CR -Col) m3 ,
(IA -CA -CR) m3 and (IA - CA - Col) m3 , etc., are purposely eliminated
DF1(IA-CR -Col)m3  (0.0226  ln(TiO 2 /SiO 2 ) adj )  (1.2877  ln(Al2 O 3 /SiO 2 ) adj ) 
(-2.0579  ln(K 2 O/SiO 2 ) adj )  (-0.0751 ln(P2 O 5 /SiO 2 ) adj )  2.1790
81
(10)
sg1  e
-d  / 2
g1
2
(14)
82
-d  / 2
(15)
Table 2. Means of discriminant functions for different tectonic settings in five
diagrams.
-d  / 2
(16)
Tabela 2. Médias das funções discriminantes para os diferentes enquadramentos
tectónicos nos cinco diagramas.
sg2  e
sg3  e
g2
g3
2
2
Finally, the probabilities for belonging to each of three
groups ( P1s , P2s , and P3s ) were calculated from the above
parameters (sg1, sg2, and sg3) as follows:
P1s 
sg1
sg1  sg2  sg3
(17)
P2s 
sg2
sg1  sg2  sg3
(18)
P3s 
sg3
sg1  sg2  sg3
(19)
These probability estimates (P1s, P2s, and P3s) directly
provide the inferred tectonic setting for the sample under
consideration; the inferred setting is the one for which the
corresponding probability (P1s, P2s, or P3s) is the highest. A
sample will plot in the tectonic field for which it has the
highest probability. The actual value of the highest probability
also indicates how far away from the tectonic field boundary
the sample will actually plot in the field of the inferred
tectonic setting. Thus, a simple comparison of the three
probabilities will provide the inferred tectonic setting for a
given sample or a set of samples, without any special need to
plot the data in a discrimination diagram. Therefore, it is not
necessary to plot and count the samples in a diagram, the
number of samples is simply determined from the highest
probability counts for a given tectonic setting. It is more
important, however, to evaluate the probability values for a set
of samples.
Nevertheless, these calculations must be carried out five
times to obtain probabilities for all five discrimination
diagrams, for which the DF1-DF2 mean values (Table 2) were
used. These diagrams are as follows (Table 2): (a) IA+CA-CRCol; (b) IA-CA-CR; (c) IA-CA-Col; (d) IA-CR-Col; and (e)
CA-CR-Col. For actual applications, it is mandatory to use
precise mean values (i.e., with many significant digits; Table
2) in the probability calculations; otherwise, the probability
based decisions of sample assignment to a group or class may
not fully agree with the actual plotting of samples in the
diagrams, particularly for samples that plot very close to the
field boundaries.
The posterior probability estimates from equations 11-19
(and Table 2) for a given set of samples analyzed from the area
under study, their range, mean and standard deviation values,
as well as total probabilities for the different tectonic settings,
were calculated and used to replace the plots. A new concept
of total percent probability was also introduced for interpreting
the data more objectively than simply counting of samples.
This probability based evaluation procedure, not hitherto
reported in any paper on multi-dimensional diagrams (Agrawal
et al., 2004, 2008; Verma et al., 2006; Verma & Agrawal,
2011; Verma et al., 2012), nor in their use (Verma, 2010;
Verma et al., 2011; Pandarinath & Verma, 2012), has allowed
me to better discuss the case studies for which the diagrams
show a complex or transitional setting. Besides, the cases of a
more definitive result of tectonic setting are also better
visualised or understood, in which the inapplicable diagram
can be clearly identified from the other four useful diagrams as
explained in the next section.
5. Application Results for Eleven Case Studies
The data for samples listed in Table 1 were plotted in Figures 13. Figure 1 is for samples from four localities (two - RebordeloAgrochão and Telões from northern Portugal and two - Oledo
and Gouveia from central Portugal). Figure 2 presents samples
from three localities (Castelo Branco and Guarda from central
Portugal and Jalama from Spain), whereas Figure 3 shows
samples from four localities (Ossa-Morena, Albernoa, Serra
Branca, and Évora) from southern Portugal. Table 3 includes
probability estimates and synthesis of the number of samples
plotting in each diagram (Figs. 1-3) from the highest probability
values. Thus, these plots (Figs. 1-3) are for reference purpose
only, because the probability estimates (Table 3) are fully
autonomous. Their interpretation does not require the
presentation or examination of the corresponding multidimensional x-y type plots. I must clarify that although both
probability estimates (Table 3) as well as diagrams (Figs. 1-3) are
presented here to convince the reader about this novel probability
based approach, the plots can be totally eliminated in future. The
probability values alone can be used for drawing inferences.
This is an extremely important aspect of the new probability
based multi-dimensional diagrams, because such probability
based discussion is likely to commence a new trend in geological
sciences. Traditionally, samples are plotted in conventional
discrimination diagrams and after a visual examination of the
plots, qualitative inferences are made (e.g., Pearce & Cann, 1971,
1973; Wood, 1980; Shervais, 1982; Pearce et al., 1984; Cabanis
and Lecolle, 1989; Rollinson, 1993). For the new multidimensional diagrams, Agrawal et al. (2004, 2008) and Verma et
al. (2006) encouraged the counting of the number of samples
plotting in different tectonic fields and the use of the success rate
parameter to quantify the results (see also Verma, 2010). Later,
Verma & Agrawal (2011) and Verma et al. (2012) suggested
probability values to complement the success rate parameter.
Therefore, from the probability values for a set of samples from a
given area, the mean x and standard deviation s values were
calculated for each tectonic field corresponding to each diagram.
Thus,
one
set
of
values
in
square
brackets
[ x  s of probability values ] are included in Table 3. However,
because these two statistical parameters ( x and s ) belong to the
category of outlier-based methods, the basic assumption of
discordant outlier-free data should be fulfilled before these
statistical calculations are performed (Barnett & Lewis, 1994;
Verma, 2012). Therefore, single-outlier type discordancy tests at
a very strict 99.5% confidence level were applied from a
modified version of DODESSYS software (Verma & DíazGonzález, 2012). Discordant outlying probability values were
observed and separated in some cases, for which the statistical
parameters are also reported in a second set of square brackets
"[]". This second set of probability values also includes (within
the square brackets) the number of discordant outlier-free
remaining samples after the application of DODESSYS. This
procedure was repeated for all five diagrams.
In this work, I propose to fully replace the multi-dimensional
diagrams from the probability estimates, introducing new
parameters of total number of samples {Σn}, total probability
{Σprob} and total percent probability [%prob]. This method
provides a quantitative estimate of how far from the tectonic field
boundaries the samples plot in a given tectonic field (Figs. 1-3).
The method consists of first calculating the total number of
samples plotted in all five diagrams {Σn}; this simply amounts to
five times the total number of samples available with complete
dataset and this number is reported in the column of number of
samples (Table 3), whereas the subsequent columns contain the
sum of the samples plotting in a given tectonic field in all
diagrams (e.g., Fig. 1a-e; see the tectonic diagram column in
Table 3).
It is advisable for the overall picture that the total probability
should be calculated for each tectonic field occupied in all five
diagrams. Thus, the total probability {Σprob} of all samples
plotted in a given tectonic setting in all five diagrams is the sum
of the individual probability values for that particular setting. The
smaller values of probability of these samples for the remaining
two tectonic settings were ignored. Nevertheless, the total
probability of samples that plotted in the combined arc field (e.g.,
Fig. 1a; Table 3) must be subdivided and assigned
proportionately to the two types of arc fields IA and CA. This
was done from the weighing factors of total probabilities for
these two arc fields in all the remaining diagrams (Fig. 1b-e;
Table 3). Thus, from these total probability estimates, the total
percent probability (%prob) values for the four tectonic settings
(IA, CA, CR, and Col) were calculated. The results are included
in the final row of each case study (Table 3). The probability
based evaluation procedure is explained for the first case study of
Rebordelo-Agrochão.
5.1. Northern Portugal
The data compiled for both localities (Rebordelo-Agrochão Early Carboniferous and Telões - Late Carboniferous-Permian;
Table 1) showed a complex, probably transitional tectonic setting
(Fig. 1; Table 3) as explained below. To familiarise the reader
with this innovated probability based procedure for the
geological sciences, the first case study is described in detail.
Gomes & Neiva (2005) reported mean concentration values
(their Table 1) for granitoids from Rebordelo-Agrochão;
unfortunately, compositional data for individual samples (total of
37 samples) were not available to me, which rendered this
application less appropriate. However, this first example serves
the purpose of motivating investigators and journal editors to
make the complete datasets available to other researchers to
allow them to seek alternative interpretations and promote
exchange of ideas.
The procedure would be to evaluate the data for individual
samples in discrimination diagrams. Nevertheless, this
application highlights the difficulty of evaluating a small number
of average analyses from unequal number of samples (N; note the
symbol N to avoid confusion with n used in this work). Out of
the seven sets of mean values, one set (tonalite B1; N=3) proved
to be of intermediate magma type, whose results are not included
in Table 3. Thus, the application relies on only six sets of mean
values. No sample size dependent weighting factors were
considered for these mean values. The samples plotted mainly in
continental arc, continental rift and collision fields (Table 3). The
83
tonalitic and granitic enclaves E1 and E2 (each of these analyses
being average of N=2; Gomes & Neiva, 2005) and one
granodiorite B2 (N=7) plotted mainly in the continental arc field
(all three in IA-CA-CR and IA-CA-Col diagrams and one in CACR-Col diagram; Table 3). Their total probability {Σprob} was
4.8119 for {Σn}=7 (Table 3). The granitic rocks consisting of one
overall average value B3 (for N=18), as well as the most and the
least deformed sample averages (a) and (b) (N=2 and N=3,
respectively; Gomes & Neiva, 2005) plotted mainly in the
collision field ({Σn}=11; relatively high total probability {Σprob}
of 6.0927; Table 3). One sample ({Σn}=1; probability of 0.8206)
plotted in the island arc field (IA-CR-Col diagram) and a total of
eight samples ({Σn}=8; {Σprob} of 4.7373; Table 3) plotted in
the continental rift field. The three samples that plotted in the
combined arc (IA+CA) field (IA+CA-CR-Col diagram) showed
the total probability of 2.0301. In order to calculate the total
percent probability (%prob) for the four fields (IA, CA, CR, and
Col), this total probability of 2.0301 for the combined arc field
was subdivided in the proportion of the total probabilities for the
IA and CA fields (0.8206 and 4.8119, respectively), i.e., 2.0301
was
multiplied
by
(0.8206/(0.8206+4.8119))
and
(4.8119/(0.8206+4.8119)), respectively, to obtain values of
0.2957 and 1.7344, which were added respectively to the IA and
CA total probability values of 0.8206 and 4.8119. The total
probability {Σprob} values for IA, CA, CR, and Col, were about
1.1163 (=0.2957+0.8206), 6.5463 (=1.7344+4.8119), 4.7373, and
6.0927, respectively, which when expressed in percentage give us
the following total percent probability [%prob] values: about
6.0%, 35.4%, 25.6%, and 32.9% (see Reboldelo-Agrochão in
Table 3).
Although a clear-cut result is not obtainable from the
multidimensional diagrams (Fig. 1) nor from the probability
calculations (Table 3), a transition from continental arc to
collision setting can be tentatively inferred for RebordeloAgrochão area during the Early Carboniferous (about 357 Ma).
The second example from northern Portugal is concerned
with only four analyses (three microgranular enclaves and one
host granite) of somewhat younger age (Late CarboniferousPermian, about 299 Ma) reported from Telões (Gomes, 2008). In
this example also, the number of samples is very small (only
four) and only one host granitic rock sample is included. The first
two analyses (tonalitic and granodioritic enclaves; see Table 1 of
Gomes, 2008) consistently showed an arc setting (relatively high
minimum total probability of 3.4546 for continental arc; Table 3),
whereas the other two (monzogranitic enclave and the host
granite) indicated a collision setting (relatively high total
probability of 5.1627; Table 3). The total percent probability
[%prob] for the collision setting (41.0%; Table 3) was somewhat
higher than that for the continental arc setting (34.5%). From this
example also, a transitional setting from continental arc to
collision may be considered for Telões during the CarboniferousPermian boundary.
5.2. Central Portugal and Spain
The data were plotted in Figure 1 for sixteen samples of the Early
Ordovician Oledo pluton (Antunes et al., 2009) and one sample
of Early Ordovician granodiorite and seven samples of Variscan
two-mica granite from the Gouveia area, central Portugal (Neiva
et al., 2009). Similarly, probability values were independently
calculated from equations 11-19 and listed in Table 3.
Thus, sixteen samples from Oledo and one from Gouveia
(Early Ordovician age) constituted the third example or case
study. This is a better example of the importance of probability
calculations and transition of tectonic setting than the first
example. The total number of samples plotting in island arc
84
({Σn}=27; Table 3) is slightly greater than for continental arc
({Σn}=22) and collision ({Σn}=25). No sample was observed
in the continental rift field (Fig. 1; Table 3), which makes the
discussion and inference for this case study simpler than for
the first example of Rebordelo-Agrochão. However, the total
probability for the Col setting ({Σprob} of 22.9329 for 25
samples; Table 3) is somewhat greater than that for the IA
({Σprob} of 19.7827 for 27 samples), which implies that the
samples plotted more inside the Col than the IA field. This is
also clear from the generally greater mean probability values
for collision than for IA (Table 3). Nevertheless, for the two
arc fields (IA and CA), the total number of samples ({Σn}=11)
plotting in the combined arc fields (IA+CA in IA+CA-CR-Col
diagram) and their total probability ({Σprob} of 9.3787)
should be proportionately added to IA and CA. Then, the total
percent probability ([%prob]; Table 3) for the IA field was
about 35.5%, somewhat greater than the collision field
(32.8%) or the continental arc field (31.7%). These
probabilities are only slightly different from each other;
therefore, a transition from an arc to collision setting can be
tentatively inferred for the Oledo area during the Early
Ordovician age.
On the other hand, for the Gouveia area of central
Portugal (fourth case study) the diagrams (Fig. 1a-e; Table 3)
clearly indicated a collision setting during the Late
Carboniferous-Permian (about 305-290 Ma), with an
extremely high total percent probability [%prob] for this
setting of 82.0% (Table 3; for the remaining tectonic settings
the total percent probability is significantly lower than for the
collision setting, being 0% for island arc, 2.3% for continental
arc, and 15.7% for continental rift). Another interesting
observation is that the diagram (Fig. 1b; IA-CA-CR; Table 3),
from which the inferred tectonic setting of collision is absent,
can be clearly identified as inapplicable, because all seven
samples plot in the collision tectonic field in all the remaining
four diagrams (Fig. 1a, c-e; IA+CA-CR-Col, IA-CA-Col, IACR-Col, and CA-CR-Col; Table 3). Note that for all three
earlier case studies, it was not possible to identify any diagram
as inapplicable.
Three additional case studies (fifth to seventh; Tables 1
and 3; Fig. 2) include two from central Portugal (Castelo
Branco, 10 sample of granites, Late Carboniferous, 310 Ma;
Antunes et al., 2008; and Guarda, a total of 22 samples–14
from Guarda-Sabugal and 8 from Guarda-Belmonte, Late
Carboniferous-Permian, 309-<299 Ma; Neiva & Ramos, 2010;
Neiva et al., 2011) and one from Spain (Jalama, a total of 79
samples of granites, Late Carboniferous to Permian, 319-279
Ma; Ramírez & Grundvig, 2000; Ruiz et al., 2008).
Surprisingly, all samples from all these localities,
independent of rock type or age from the Late Carboniferous
to Permian (319-279 Ma) plot in the collision setting (Fig. 2a,
c-e; IA+CA-CR-Col, IA-CA-Col, IA-CR-Col, and CA-CRCol; Table 3). Not even a single sample plots in the arc field
(island arc, continental arc, or combined IA+CA setting). The
diagram, from which the inferred collision setting is absent
(Fig. 2b; IA-CA-CR; Table 3), can be clearly declared as
inapplicable. In this diagram (IA-CA-CR) all samples plot in
the continental rift field. The total probability estimates
({Σprob}) also show that the combined total probability is
much higher for the collision setting than for the rift. The total
percent probability [%prob] values for the collision setting
were also very high (about 81.3%, 77.9%, and 79.5% for
Castelo Branco, Guarda, and Jalama, respectively). In these
examples, therefore, a clear identification of the tectonic
setting has been possible.
5.3. Southern Portugal
The final set of four case studies (eighth to eleventh) is for southern
Portugal. The data were compiled from Ossa-Morena (1 trachydacite
and 5 rhyolite samples of Ediacaran-Early Cambrian; Pereira et al.,
2006), Albernoa (32 acid volcanic rock samples of Devonian-Early
Carboniferous; Rosa et al., 2004), Serra Branca (48 acid volcanic
rock samples of Devonian-Early Carboniferous; Rosa et al., 2006),
and Évora (19 granitoid samples of Early Carboniferous, Moita et al.,
2009).
The volcanic rocks from Ossa-Morena were defined by Pereira
et al. (2006) as reworked andesitic tuff (one sample), rhyolite (three
samples), rhyodacite (one sample), and dacite (one sample), but the
andesitic tuff resulted as trachydacite and the other five proved to be
rhyolite from the SINCLAS computer program (Verma et al., 2002).
In spite of the small number of samples (six only), the multidimensional diagrams (Fig. 3; Table 3) showed that most samples
plotted in the collision field. The total percent probability [%prob] for
this field was about 52.4% and was followed by only 24.1% for the
continental rift setting. A collision setting can be clearly inferred for
Ossa-Morena during the Ediacaran-Early Cambrian.
For all samples from the Albernoa area, an island arc setting
was indicated during the Devonian-Early Carboniferous, because
most samples (20 to 24 out of 32) plotted in this field, and the total
percent probability was also the highest for this setting (about
51.2%), followed by 28.4% for the continental arc and 18.0% for the
collision setting. When these 32 samples were divided in two subsets
(17 drilled and 15 surface samples), and plotted in the multidimensional diagrams (see different symbols in Fig. 3 and
independent probability-based counting in Table 3;), an island arc
setting was also discernible from both subsets although the total
percent probability [%prob] for this setting was considerably less for
the drilled samples (41.3%, followed by 32.3% for the continental arc
field) than for the surface ones (61.4%, followed by 24.3% for the
continental arc field).
For the next case study of Serra Branca, the multi-dimensional
diagrams probably showed a transition from island arc to collision
setting during the Devonian-Early Carboniferous, because total
number of samples for these two settings ({Σn}=65 and 98 samples,
respectively; Table 3) and the respective total probabilities
({Σprob}= 58.4346 and 83.7003) were the highest. The total percent
probability [%prob] for collision was about 39.7%, somewhat higher
than the island arc setting (32.9%), but much higher than the
continental arc (18.3%) and rift (9.1%) fields. When only quartzfeldspar-phyric porphyry (21) samples were separately considered,
the island arc field showed the total percent probability [%prob] of
about 42.6% followed by 31.4% for the collision field. However, the
remaining 27 samples of microporphyry and pumice samples were
more consistent with a collision setting because the total percent
probability [%prob] of 45.9% for this setting was higher than 25.6%
for the island arc setting.
Finally, the last case study for Évora also indicated a transition
from an island arc to collision setting for this area (localities of
Almansor, Valverde, and Alto de São Bento) during the Early
Carboniferous, because the highest number of samples plotted in
these two tectonic fields (Fig. 3) and showed the highest probabilities
for them (Table 3). The total percent probability [%prob] values for
island arc and collision were about 34.8% and 33.0%, respectively,
although the probability for continental arc (30.1%) was not much
less than these values. If the samples from the three areas were
separately considered, the inferred tectonic setting for Almansor (6
samples) would be collision, for Valverde (4 samples) island arc, and
for Alto de São Bento transitional from island arc to collision (see
Fig. 3; results are not separately presented in Table 3, because this is
not a recommended procedure for routine work).
Table 3. Application of the set of five discriminant-function-based multi-dimensional discrimination diagrams to granitic or acid rocks from Portugal and Spain.
Tabela 3. Aplicação dos diagramas discriminantes baseados em função discriminante às rochas graníticas ou ácidas de Portugal e Espanha.
85
86
87
88
IA – island arc; CA – continental arc; CR – continental rift; Col – collision; x  s ─ mean±one standard deviation of the probability estimates for all samples discriminated in
a given tectonic setting, these are reported in [], the second set of values are also included in [] when the single-outlier type discordancy tests identified one or more probability
values as discordant; * ─ inapplicable diagram identified whenever it is clearly established; the final row gives a synthesis of results as {Σn} {Σprob} [%prob] where {Σn} ─
total number of samples or data points plotting in all five diagrams are reported in the column of total number of samples whereas the sum of samples plotting in a given
tectonic field are reported in the respective tectonic field column, {Σprob} – sum of probability values for all samples plotting in a given tectonic field are reported in the
respective tectonic field column, and [%prob] – total probability of a given tectonic setting expressed in percent after assigning the probability of IA+CA to IA and CA (using
weighing factors explained in the text); Θ ─ one of these samples from Telões was the host granite.
Fig. 1. The set of five new discriminant-function diagrams based on natural logarithm-transformed ratios of major-elements for the
discrimination of island arc (IA), continental arc (CA), continental rift (CR), and collision (Col) tectonic settings, showing samples
from northern and central Portugal. In the first diagram, four groups are represented as three groups by combining IA and CA
together. The other four diagrams are three groups at a time. The symbols are explained as inset in Figure 1a (nP─northern Portugal;
cP─central Portugal). The subscript m3 is used here to distinguish these diagrams from previous two sets of major-element based
diagrams proposed by Agrawal et al. (2004; subscript m1) and Verma et al. (2006; subscript m2). The coordinates of the field
boundaries are: (a) IA+CA-CR-Col diagram, (3.0914, 8.00) and (-0.52237, 0.105108) for IA+CA-CR, (-8.00, -1.6511) and (-0.52237,
0.105108) for CR-Col, and (6.5177, -8.00) and (-0.52237, 0.105108) for IA+CA-Col; (b) IA-CA-CR diagram, (4.1608, 8.00) and
(0.41929, -0.66705) for CA-CR, (-8.00, 4.7147) and (0.41929, -0.66705) for IA-CA, (1.0939, -8.00) and (0.41929, -0.66705) for IACR; (c) IA-CA-Col diagram, (-6.8768, -8.00) and (0.13893, 1.18829) for IA-CA, (0.39469, 8.00) and (0.13893, 1.18829) for IA-Col,
and (4.1472, -8.00) and (0.13893, 1.18829) for CA-Col; (d) IA-CR-Col (1-3-4) diagram, (4.7956, 8.00) and (0.20518, -0.01689) for
IA-CR, (-8.00, 1.61186) (2.1584, -8.00) and (0.20518, -0.01689) for IA-Col, and (0.20518, -0.01689) for CR-Col; and (e) CA-CRCol diagram, (4.6620, 8.00) and (0.22442, 0.015552) for CA-CR, (-8.00, 0.53675) and (0.22442, 0.015552) for CA-Col, and (3.3907,
-8.00) and (0.22442, 0.015552) for CR-Col.
89
Fig. 1. O conjunto de cinco novos diagramas de função discriminante baseados em rácios naturais de logaritmos transformados dos elementos maiores para a discriminação de
ambientes tectónicos de arcos insulares (IA), arcos continentais (CA), rifte continental (CR) e colisão (Col), mostrando exemplos do norte e centro de Portugal. No primeiro diagrama,
quatro grupos estão representados como sendo três grupos ao combinar IA e CA juntos. Os outros quatro diagramas são três grupos de cada vez. Os símbolos estão explicados no
quadro da Figura 1a (nP─norte de Portugal; cP─centro de Portugal). O subscrito m3 é usado para distinguir estes diagramas dos previamente propostos por Agrawal et al. (2004;
subscrito m1) e Verma et al. (2006; subscrito m2). As coordenadas dos limites de cada campo são: (a) diagrama IA+CA-CR-Col, (3.0914, 8.00) e (-0.52237, 0.105108) para IA+CACR, (-8.00, -1.6511) e (-0.52237, 0.105108) para CR-Col, e (6.5177, -8.00) e (-0.52237, 0.105108) para IA+CA-Col; (b) diagrama IA-CA-CR, (4.1608, 8.00) e (0.41929, -0.66705)
para CA-CR, (-8.00, 4.7147) e (0.41929, -0.66705) para IA-CA, (1.0939, -8.00) e (0.41929, -0.66705) para IA-CR; (c) diagrama IA-CA-Col, (-6.8768, -8.00) e (0.13893, 1.18829)
para IA-CA, (0.39469, 8.00) e (0.13893, 1.18829) para IA-Col, e (4.1472, -8.00) e (0.13893, 1.18829) para CA-Col; (d) diagrama IA-CR-Col (1-3-4), (4.7956, 8.00) e (0.20518, 0.01689) para IA-CR, (-8.00, 1.61186) (2.1584, -8.00) e (0.20518, -0.01689) para IA-Col, e (0.20518, -0.01689) para CR-Col; e (e) diagrama CA-CR-Col, (4.6620, 8.00) e (0.22442,
0.015552) para CA-CR, (-8.00, 0.53675) e (0.22442, 0.015552) para CA-Col, e (3.3907, -8.00) e (0.22442, 0.015552) para CR-Col.
from central Portugal (cP) and Spain (S). For more details, see explanation of Figure 1. (a) IA+CA-CR-Col diagram; (b) IA-CA-CR
diagram; (c) IA-CA-Col diagram; (d) IA-CR-Col (1-3-4) diagram; and (e) CA-CR-Col diagram.
Fig. 2. Conjunto de cinco novos diagramas de função discriminante baseados em rácios naturais de logaritmos transformados dos
elementos maiores para a discriminação de ambientes tectónicos de arcos insulares (IA), arcos continentais (CA), rifte continental
(CR) e colisão (Col), mostrando exemplos do centro de Portugal (cP) e Espanha (S). Para mais detalhes, ver explicação na Fig. 1. (a)
diagrama IA+CA-CR-Col; (b) diagrama IA-CA-CR; (c) diagrama IA-CA-Col; (d) diagrama IA-CR-Col (1-3-4); e (e) diagrama CACR-Col.
from southern Portugal (sP). The symbols are explained in Figure 2a, where the age abbreviations are: Ed-Cam─EdiacaranCambrian; D-EC─Devonian-Early Carboniferous; other abbreviations are mp-pum─microporphyry and pumice. For more details, see
explanation of Figure 1. (a) IA+CA-CR-Col diagram; (b) IA-CA-CR diagram; (c) IA-CA-Col diagram; (d) IA-CR-Col (1-3-4)
diagram; and (e) CA-CR-Col diagram.
Fig. 3. Conjunto de cinco novos diagramas de função discriminante baseados em rácios naturais de logaritmos transformados dos
elementos maiores para a discriminação de ambientes tectónicos de arcos insulares (IA), arcos continentais (CA), rifte continental
(CR) e colisão (Col), mostrando exemplos do sul de Portugal (sP). Os símbolos estão explicados na Fig. 2a onde as abreviaturas de
idade são: Ed-Cam─Ediacariano-Câmbrico; D-EC─Devónico-Carbónico inferior; outras abreviaturas são mp-pum─microporfirítica e
pedra-pomes. Para mais detalhes, ver explicação na Fig. 1. (a) diagrama IA+CA-CR-Col; (b) diagrama IA-CA-CR; (c) diagrama IACA-Col; (d) diagrama IA-CR-Col (1-3-4); e (e) diagrama CA-CR-Col.
90
6. Discussion
6.1 Case studies
The results of multi-dimensional diagrams will now be briefly
discussed in the light of evidence presented in the papers from
which the data were compiled and evaluated.
For the first case study, Gomes & Neiva (2005) postulated
that the origin of the Reboldelo-Agrochão granitoids may be
related to the Variscan collision during the late Paleozoic. It is
likely that the individual analyses could have provided a more
definitive conclusion from the multi-dimensional diagrams,
because the total number of data points would then have been
37, instead of only 6 acid and 1 intermediate averages.
Alternative interpretations might be that these acid rocks in fact
represent a transitional tectonic setting from continental arc to
collision, or a purely collision setting was operative but the
deformation of rocks documented by the Gomes & Neiva (2005)
modified the chemical compositions to change the results of
discrimination diagrams. It is pertinent to mention that the three
new sets of multi-dimensional diagrams for intermediate magma
based on major elements as well as immobile major or trace
elements (Verma & Verma, 2013) consistently showed a
collision setting for the only one average value (N=3) reported
by these authors (tonalite B1 in Table 1 of Gomes & Neiva,
2005).
The second case study is based on only one post-tectonic
host granite sample and its three enclaves (Gomes, 2008). The
host granite sample consistently showed a collision setting with
probabilities ranging from 0.5637-0.7691 (results are not
individually indicated in Table 3), whereas two of three enclaves
plotted in the continental arc setting and the remaining one in
the collision setting. It is likely that the host granite represents a
collision setting and the probably older enclaves a transition
from arc to collision or the results of the multi-dimensional
diagrams were affected by post-emplacement compositional
changes. Nevertheless, this second study also was concerned
with a very small number of samples (in fact, only one host
granite sample), which is not actually recommended for such
applications.
In the third case study of the Oledo pluton of Early
Ordovician age, the authors (Antunes et al., 2009) presented
analyses of host granite as well as numerous enclaves and
interpreted their data to decipher petrogenetic processes. They
did not comment on the probable tectonic setting. The results of
multi-dimensional diagrams to all samples (4 host granites and
13 enclaves) are not conclusive, or at best represent a transition
from an arc to collision setting. However, when the data for host
granite samples were separately considered (results are not
individually indicated in Table 3), three of the four samples (G2,
G3, and G4) consistently showed a collision setting (with high
probability values of 0.8461-0.9986) whereas granite G1
indicated an island arc setting (probability values of 0.64840.9463). According to Antunes et al. (2009), G1 is the most
deformed rock, which may have affected this granite sample to
behave differently from the other three granites on the
discrimination diagrams. Thus, from the study of only host
granites (4 samples) a collision setting could be inferred for this
area during the Early Ordovician.
The next four case studies were for Late Carboniferous to
Permian granitic rocks from Gouveia (Neiva et al., 2009),
Castelo Branco (Antunes et al., 2008), Guarda (Neiva & Ramos,
2010; Neiva et al., 2011), and Jalama (Ramírez & Grundvig,
2000; Ruiz et al., 2008). The multi-dimensional diagrams and
probability calculations consistently showed a collision setting
for all these areas of central Portugal and Spain. Neiva et al.
(2009) stated that large volumes of granitic rocks were emplaced
in Gouveia, mainly during the third Variscan deformation (D3)
from 320 to 300 Ma. The granitic rocks in Castelo Branco
(Antunes et al., 2008) and Guarda (Neiva et al., 2011) were also
emplaced during or after this main deformation event. The
Jalama batholith has one of the numerous granites of the Central
Iberian Zone with Sn- and W-associated mineralisation and
formed from a multi-phase intrusion of granites (Ramírez &
Grundvig, 2000; Ruiz et al., 2008). For all these four case
studies, the inferred collision setting is fully consistent with the
description in the original papers (Ramírez & Grundvig, 2000;
Antunes et al., 2008; Ruiz et al., 2008; Neiva & Ramos, 2010;
Neiva et al., 2011). However, no discrimination diagrams were
used by any of them to support their statements.
The final set of four case studies from southern Portugal
from Ossa-Morena (Pereira et al., 2006), Albernoa (Rosa et al.,
2004), Serra Branca (Rosa et al., 2006), and Évora (Moita et al.,
2009) are now briefly discussed.
For Ossa-Morena, the discrepancy in volcanic rock
nomenclature and magma types from Pereira et al. (2006) may
be due to the fact that the SINCLAS program follows strictly
the IUGS recommendations of using the adjusted SiO2 and
alkalis (Na2O and K2O) on an anhydrous 100% adjusted basis
after a proper assignment of Fe-oxidation ratio (Middlemost,
1989), whereas Pereira et al. (2006) might have used the
actually measured unadjusted data, although they did not
mention their procedure to assign volcanic rock names to their
samples. Pereira et al. (2006) presented a synthesis of plate
tectonic evolution as follows: (1) During the Ediacaran (570540 Ma), an active continental margin evolved through oblique
collision with accretion of oceanic crust, a continental
magmatic arc and the development of related marginal basins;
(2) the Ediacaran-Early Cambrian transition (540-520 Ma)
coeval with important orogenic magmatism and the formation
of transtensional basins with detritus derived from remnants of
the magmatic arc; and (3) Gondwana fragmentation with the
formation of Early Cambrian (520–510 Ma) shallow-water
platforms in transtensional grabens accompanied by rift-related
magmatism. Pereira et al. (2006) used ternary diagrams of
Bhatia & Crook (1986) for sedimentary rocks to infer an
inherited continental arc signature and the Y+Nb-Rb diagram
of Pearce et al. (1984) to show volcanic arc signature for their
volcanic rocks. Some diagrams for sedimentary rocks have
been criticised by Armstrong-Altrin & Verma (2005), whereas
the use of ternary diagrams has been recently discouraged by
Verma (2012). Similarly, Pearce et al. (1984) diagrams for
acid magmas generally show low success rates (Verma et al.,
2012). In fact, these diagrams were evaluated to work less well
for the collision setting (Verma et al., 2012), which may
explain the discrepancy of inferred tectonic settings for
volcanic rocks between Pereira et al. (2006) and the present
work. Finally, the precise age of the volcanic rocks analysed
by Pereira et al. (2006) is not known, which makes it difficult
to discuss any further the validity of a collision setting inferred
from the multi-dimensional diagrams for acid rocks.
The island arc setting inferred from the multi-dimensional
diagrams for dacites and rhyolites from Albernoa would
support one of the several models proposed for the Iberian
Pyrite Belt as summarised by Rosa et al. (2004). According to
these authors, the proposed geotectonic settings include: an
island or continental arc; a forearc basin within an accretionary
prism; an intercontinental backarc basin; and a more complex
scenario of a basin formed by local extensional tectonics
caused by oblique continental collision following NE-dipping
subduction of the South Portuguese plate under the OssaMorena plate. Bivariate diagrams of Pearce et al. (1984) used
by Rosa et al. (2004) for their acid rocks showed an
overlapping volcanic arc and syn-collision (in Y-Nb diagram)
or a volcanic arc setting (in Y+Nb-Rb diagram). Nevertheless,
Rosa et al. (2004) argued in favour of a purely extensional
setting, without subduction, for their mafic rock samples, most
of which, however, were classified from SINCLAS as of
intermediate magma type. It may also be pertinent to mention
that immobile element based multi-dimensional diagrams for
these intermediate magmas (Verma & Verma, 2013) also
indicated an island arc setting for Albernoa.
For the next case study of Serra Branca, Rosa et al. (2006)
reported data for 52 samples, six of which were of
intermediate composition. In fact, the SINCLAS computer
program (Verma et al., 2002) identified 48 samples as acid
rocks (Tables 1 and 2), and only four samples as intermediate
magma type. No definitive inference of tectonic setting beyond
an arc to collision transitional setting was obtained from acid
rocks of Serra Branca from the multi-dimensional diagrams.
Although in bivariate diagrams (Y-Nb and Y+Nb-Rb) of
Pearce et al. (1984) a combined volcanic arc-collision or a
volcanic arc setting was, respectively, observed, Rosa et al.
(2006) discarded this indication. They argued that the rocks
from the Iberian Pyrite Belt showed a bimodal nature, which
will not be consistent with an arc setting and previous studies
(e.g., Mitjavila et al., 1997; Rosa et al., 2004) had indicated an
extensional setting for this belt.
For Évora massif, from the calc-alkaline signature of
magmatism Moita et al. (2009) suggested a continental arc
setting; no other evidence was, however, presented. Such a
tectonic conclusion based exclusively on the calc-alkaline
character of rocks has already been criticised by Sheth et al.
(2002). From the multi-dimensional diagrams the tectonic
regime of this area seems to be much more complex (probably
an arc to collision transition) than the simple subduction
process suggested by Moita et al. (2009).
6.2 Reasons for better functioning of multi-dimensional
diagrams
From the above discussion, a generally good functioning of the
multi-dimensional diagrams can be inferred (see also Verma et
al., 2012). Similar conclusions of good functioning of multidimensional diagrams were reached from studies for basic and
ultrabasic (Verma et al., 2006; Agrawal et al., 2008; Verma,
2010; Verma & Agrawal, 2011) as well as intermediate
magmas (Verma & Verma, 2013). The reasons for better
functioning of the newer multi-dimensional diagrams for all
kinds of magmas proposed during 2006-2012, compared to the
conventional bivariate and ternary diagrams are many fold. For
the functioning of older diagrams, it is mandatory (Rollinson,
1993; Pandarinath & Verma, 2013) that two basic assumptions
be fulfilled, which are: (1) the concentrations of the chemical
elements used in the discrimination diagrams show large
differences in the rocks from different tectonic settings; and
(2) these chemical elements are immobile in the rocks from the
time of rock formation up to the present. None of these two
assumptions need to be strictly valid in the case of the newer
multi-dimensional diagrams based on LDA of log-ratios. It
would suffice that there exist some (not necessarily large,
statistically significant) differences in the concentrations of the
chemical elements in rocks from different tectonic settings.
These small differences are, in fact, enhanced by loge-ratio
transformations (being the correct statistical methodology to
handle compositional data; Aitchison, 1986) and the
91
multivariate technique of LDA (Morrison, 1990). Thus, these
differences are rendered statistically significant at a high
confidence level (99% or more; see Verma & Agrawal, 2011).
On the other hand, because element ratios, and not actual
concentrations, are involved in the axis-variables, it is not
necessary that the element concentrations remain immobile,
but only the ratios of the elements used in the log-ratio
transformation should remain the same. Some petrogenetic
processes, such as fractional crystallisation and partial melting,
are known to maintain incompatible element ratios
approximately constant, which will therefore not seriously
affect the functioning of multi-dimensional discrimination
diagrams. Thus, element concentrations may change as long as
the ratios used in LDA are approximately constant. Finally,
because in the complex x- and y-axis equations (DF1-DF2; see
also Verma, 2010; Verma et al., 2012), the loge-ratio variables
have positive as well as negative multiplication constants, the
chemical variations in the ratios may cancel out or at least be
minimized in such multi-dimensional diagrams, which is
simply not possible in the conventional bivariate or ternary
diagrams (Verma, 2012).
Yet other reasons for the better functioning of newer
diagrams may be related to probability-based tectonic field
boundaries instead of eye-drawn subjective boundaries in older
diagrams (see Agrawal, 1999; Agrawal & Verma, 2007), as
well as the calculations of probabilities for individual samples
(Verma & Agrawal, 2011; see also the present work).
To promote a more efficient use of these new multidimensional diagrams and probability calculations (Verma et
al., 2012), a computer program will be written, which will
allow data input from an Excel spreadsheet. In the mean time,
potential users can send the author their data (in Excel) for
processing in Statistica software, or they can calculate the
discriminant functions (DF1 and DF2) from the equations 1-10
and probabilities from equations 11-19. For the earlier multidimensional diagrams proposed during 2004-2011, a computer
program TecD (Verma & Rivera-Gómez, 2013) is available on
request to the author of this work.
In the present work, only major element based diagrams
for acid magmas were used although, in some cases, immobile
element based diagrams for intermediate magmas were also
mentioned. The multi-dimensional diagrams are robust against
some post-emplacement compositional changes such as Feoxidation or moderate weathering effects, because the major
element data are always readjusted from SINCLAS computer
program (Verma et al., 2002) to 100% after Fe-oxidation
adjustment of Middlemost (1989). Nevertheless, new diagrams
for acid magmas based on immobile elements (currently under
preparation by Verma and colleagues) can be used to confirm
or rectify the results of the present set of major element based
diagrams proposed by Verma et al. (2012). This would
certainly reinforce such applications.
6. Conclusions
Generally good functioning of the new multi-dimensional
diagrams based on natural logarithm of ratios of major
elements is documented for acid rocks from Portugal and
Spain. New diagrams for acid magmas based on immobile
elements currently under preparation should reinforce this kind
of applications. Computer programs to facilitate their use and
calculation of probabilities for different tectonic fields would
eventually make this methodology widely applicable to rocks
of all ages and localities around the world.
92
Acknowledgements
I am grateful to the editor Telmo M. Bento dos Santos for the
invitation to contribute with a paper to the journal. The reviewers
Maria Manuela da Vinha G. da Silva and Maria Elisa Preto
Gomes and the editor Ana M. R. Neiva are especially thanked for
their comments and suggestions on an earlier version of this
paper, which helped me to improve my presentation. I am also
grateful to Sanjeet K. Verma and Pandarinath Kailasa for reading
a partially revised version of this paper and pointing out some
shortcomings. On my request Casilda Ruiz provided me their
electronic depository data on the Jalama batholith and Diogo R.
N. Rosa sent me an electronic copy of their 2004 IGR paper on
Albernoa; their immediate response is highly appreciated.
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Application of multi-dimensional discrimination diagrams

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