Application of multi-dimensional discrimination diagrams ... · transitório entre arco insular e colisão para Serra Branca (Portugal) durante o Devónico-início do Carbónico e

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* 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; discriminant-function based diagrams; log-ratio transformation; probability calculations. Resumo: Diagramas de discriminação têm sido utilizados para decifrar o enquadramento tectónico de terrenos antigos. Novos diagramas multi-dimensionais 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: [email protected]

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 multi-dimensional 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 multi-dimensional 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

Artigo original Original article

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.

3. Multi-Dimensional Diagrams

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 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

m3Col)-CR-CA(IADF1

and m3Col)-CR-CA(IADF2 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).

(1) 1.58259))/SiOOln(P (-0.15580))O/SiOln(K (-1.65167

))O/SiOln(Na (0.56098))ln(CaO/SiO (0.45615

))ln(MgO/SiO (0.18732))ln(MnO/SiO (0.42703

))ln(FeO/SiO (0.93391))/SiOOln(Fe (-2.09239

))/SiOOln(Al (0.95693 ))/SiOln(TiO(0.36077DF1

adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-CA(IA m3

(2) 691035.6))/SiOOln(P (-0.353797))O/SiOln(K (0.173552

))O/SiOln(Na (0.231677))ln(CaO/SiO (-0.244687

))ln(MgO/SiO (-0.027717))ln(MnO/SiO (0.739533

))ln(FeO/SiO 0.699238())/SiOOln(Fe (0.109516

))/SiOOln(Al (-0.954629 ))/SiOln(TiO0.472353(DF2

adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-CA(IA m3

The second diagram discriminates the tectonic setting of IA, CA and CR, for which equations 3 and 4 were used for calculating the

m3CR)-CA-(IADF1 and m3CR)-CA-(IADF2 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.

Multi-dimensional discrimination diagrams in acid rocks from Portugal and Spain 81

(3) 2573.6))/SiOOln(P (0.3386))O/SiOln(K (1.7166

))O/SiOln(Na (-0.7139))ln(CaO/SiO (-0.8516

))ln(MgO/SiO (-0.1907))ln(MnO/SiO (-0.1389

))ln(FeO/SiO .06631())/SiOOln(Fe (2.7433


adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22CR)-CA-(IA m3

(4) 9984.0))/SiOOln(P (0.0186))O/SiOln(K (1.6616



))ln(FeO/SiO .12101())/SiOOln(Fe (-3.2046


adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22CR)-CA-(IA m3

The third diagram discriminates the tectonic setting of IA, CA and Col (the absent setting is CR) and the

m3Col)-CA-(IADF1 and

m3Col)-CA-(IADF2 functions were calculated from equations 5 and 6.

(5) 0829.3))/SiOOln(P (0.2684))O/SiOln(K (1.5074


))ln(MgO/SiO (-0.1229))ln(MnO/SiO (-0.7223

))ln(FeO/SiO .49800-())/SiOOln(Fe (0.5198


adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CA-(IA m3

(6) 190.18))/SiOOln(P (0.495))O/SiOln(K (-2.339


))ln(MgO/SiO (0.062))ln(MnO/SiO (-1.226


))/SiOOln(Al (1.984 ))/SiOln(TiO0.142(DF2

adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CA-(IA m3

The fourth diagram discriminates the tectonic setting of IA, CR and Col (the absent setting is CA) and the

m3Col)-CR-(IADF1 and

m3Col)-CR-(IADF2 functions were calculated from equations 7 and 8.

(7) 1790.2))/SiOOln(P (-0.0751))O/SiOln(K (-2.0579


))ln(MgO/SiO (0.0673))ln(MnO/SiO (0.1970

))ln(FeO/SiO .94942())/SiOOln(Fe (-2.6406


adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-(IA m3

(8) 5170.6))/SiOOln(P (-0.2261))O/SiOln(K (0.6698





adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-(IA m3

Finally, the fifth diagram discriminates the tectonic setting of CA, CR and Col (the absent setting is IA), for which the

m3Col)-CR-(CADF1 and m3Col)-CR-(CADF2 functions were calculated

from equations 9 and 10.

(9) 2616.8))/SiOOln(P (-0.0338))O/SiOln(K (1.8098





adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-(CA m3

(10) 8959.3))/SiOOln(P (-0.3526))O/SiOln(K (-3.3091





adj252adj22

adj22adj2

adj2adj2

adj2adj232

adj232adj22Col)-CR-(CA m3

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.

4. Probability Calculations

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 m3Col)-CR-CA(IA ,

m3CR)-CA-(IA and m3Col)-CA-(IA , etc., are purposely eliminated

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 ( g1d , g2d , and g3d ) of a sample under

evaluation from the three group mean values ─ ( g1mdf1 ,

g1mdf2 ); ( g2mdf1 , g2mdf2 ); and ( g3mdf1 , g3mdf2 ) ─ of the

tectonic groups g1, g2, and g3 (Table 2), respectively, in a given diagram were calculated as follows:

(11) )mdf2-df2()mdf1-df1(d 2g1s

2g1sg1


2g2sg2


2g3sg3

where sdf1 and sdf2 are the coordinates or scores of the sample

under evaluation in a given diagram. The mean values g1mdf1 ,

g1mdf2 , 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

( g1d , g2d , and g3d ; equations 11-13) for that particular sample

were computed from equations 14-16 as follows:

(14) sg1

2/d- 2g1e


(15) sg2

2/d- 2g2e

(16) sg3

2/d- 2g3e

Finally, the probabilities for belonging to each of three

groups ( sP1 , sP2 , and sP3 ) were calculated from the above

parameters (sg1, sg2, and sg3) as follows:

(17) sg3sg2sg1

sg1P1s

(18) sg3sg2sg1

sg2P2s

(19) sg3sg2sg1

sg3P3s

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-CR-Col; (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.

Table 2. Means of discriminant functions for different tectonic settings in five diagrams.

Tabela 2. Médias das funções discriminantes para os diferentes enquadramentos

tectónicos nos cinco diagramas.

5. Application Results for Eleven Case Studies

The data for samples listed in Table 1 were plotted in Figures 1-3. Figure 1 is for samples from four localities (two - Rebordelo-Agrochão 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 multi-dimensional 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 multi-dimensional 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 [ y valuesprobabilit of sx ] 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 & Díaz-González, 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

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 CA-CR-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 Rebordelo-Agrochão 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 Carboniferous-Permian, 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 Carboniferous-Permian 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


({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, IA-CR-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-CR-Col; 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 multi-dimensional 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 multi-dimensional 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 quartz-feldspar-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.




IA – island arc; CA – continental arc; CR – continental rift; Col – collision; sx ─ 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 IA-CR; (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-CR-Col 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.


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+CA-CR, (-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.

Fig. 2. 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 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 CA-CR-Col.

Fig. 3. 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 southern Portugal (sP). The symbols are explained in Figure 2a, where the age abbreviations are: Ed-Cam─Ediacaran-Cambrian; 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 IA-CA-Col; (d) diagrama IA-CR-Col (1-3-4); e (e) diagrama CA-CR-Col.


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 0.6484-0.9463). 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 (570-540 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 Ossa-Morena 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 multi-dimensional 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

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 multi-dimensional 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 multi-dimensional 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 Fe-oxidation 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.


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.

References

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