# An Automated Method to Generate and Evaluate Geochemical Tectonic Discrimination Diagrams Based on Topological Theory

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}O–Nb/Ba diagram, are recommended for use. This research supports the view that gabbroic rocks can also be used to discriminate between different tectonic settings. The method could also be applied to other rock types.

## 1. Introduction

_{2}O–Nb/Ba diagram, are discussed in detail.

## 2. Methodology

#### 2.1. Overall Framework

_{2}and SiO

_{2}can be combined as TiO

_{2}/SiO

_{2}and SiO

_{2}/TiO

_{2}. An element ratio should be regarded as a new element.

#### 2.2. Element (or Element Ratio) Pairs

#### 2.3. Delaunay Triangulation

#### 2.4. Coordinate Transformation of Samples

#### 2.4.1. Coordinate Transformation under Multiscale Coordinates

_{max}and y

_{min}are the maximum and minimum of the points along the y-axis and x

_{max}and x

_{min}are the maximum and minimum of the points along the x-axis. (x

_{i}, y

_{i}) is the original value of the ith point, and (x

_{i}’, y

_{i}’) is the zoomed valued of the ith point. By generating a Delaunay triangulation with the zoomed points, a set of connection rules for the points can be calculated. After this step, by connecting the original points according to these connection rules, a new triangulation can be made, as shown in Figure 5b. It is evident that Figure 5b is more consistent with the visual observation.

#### 2.4.2. Coordinate Transformation under Linear and Logarithmic Coordinates

#### 2.5. Confidence Region

#### 2.6. Evaluation of Discrimination Diagrams

_{i}is the region of the ith type of samples, A(R

_{i}) is the area of R

_{i}, γ

_{i}is the overlap rate of the region of the ith type of samples and other types of samples, and γ is the overlap rate of the whole discrimination diagram and is a value between [0,1]. Therefore, if the area of the overlap is large, the overlap rate is large.

## 3. Experiment and Analysis

#### 3.1. Data Collection and Preprocessing

_{2}> 56 wt% or SiO

_{2}< 35 wt% were filtered out. Moreover, because the tectonic setting information for the samples was ambiguous (some samples were labeled with geomorphologic terms, for instance, seamount, submarine ridge, and convergent margin), we carefully checked all the samples according to the locations, longitudes and latitudes, and the analytical results provided by the contributors, and then divided them into the correct tectonic settings. In addition, it should be noted that we did not remove the cumulate gabbroic rocks, altered samples, or contaminated samples. Further, considering that there might be some uncertain influencing factors in the data set (for example, erroneous records), the box-plot approach was used to remove the outliers and ensure that the remainder of the samples accounted for 85% of the whole data set. Figure 10 shows the distributions of the remaining samples on the plutonic Total alkali–silica (TAS) diagram.

_{2}, TiO

_{2}, Al

_{2}O

_{3}, FeO

^{T}, CaO, MgO, MnO, K

_{2}O, Na

_{2}O, P

_{2}O

_{5}, Sc, V, Cr, Ni, Cu, Zn, Rb, Sr, Y, Zr, Nb, Ba, La, Ce, Nd, and Sm were selected. The means and the standard deviations of the element contents for the three types of tectonic settings are presented in Table 1 and Table 2. In addition, the Mg# was calculated by Equation (3), and the result was 0.75.

#### 3.2. Designation and Evaluation of Discrimination Diagrams

#### 3.2.1. Discrimination Diagrams for IAG and Non-IAG

#### 3.2.2. Discrimination Diagrams for OIG and Non-OIG

_{2}O and Nb, Nb and Sm, Nb and Y are the most important elements (and element ratios) for logarithmic cases.

_{2}O are the most significant basic elements in logarithmic diagrams, as shown in Figure 16.

#### 3.2.3. Discrimination Diagrams for MORG and Non-MORG

_{2}O/Ba has the most significant effect on the discrimination process, as shown in Figure 18a; the word cloud of the logarithmic discrimination diagrams indicates that the element ratio pairs comprised by Na

_{2}O and Ba, Na

_{2}O and Rb are the most frequently occurring elements, as shown in Figure 18b.

_{2}O, Rb, Zn, and Sc contribute the most to linear discrimination, and that Ba, Na

_{2}O, and Rb are the most effective elements in logarithmic discrimination.

#### 3.2.4. Discrimination Diagrams for IAG–OIG–MORG

_{2}, K

_{2}O, Ni, Cu, La, and Ce also have certain effects on discrimination; (2) in logarithmic cases, the element ratios of Nb and Sc and of Ba and Sc contribute the most to the discrimination process and that Nb, Ba, and Sc are the most useful basic elements, while TiO

_{2}, Al

_{2}O

_{3}, FeO

^{T}, V, Cr, Ni, and Sr can slightly help to discriminate among different tectonic settings.

## 4. Discussion

#### 4.1. The Use of Basalt in Discriminating among Different Tectonic Settings

#### 4.2. Using Gabbroic Rocks in Discrimination Tasks?

_{2}in MORB is 1.20–1.50%, the average content of TiO

_{2}in IAT is 0.80%, and the average content of TiO

_{2}is OIB more than 2.00% [31,47]. Generally, the content in TiO

_{2}of basalt is seldom less than 0.5%. Andesite is different, and its content of TiO

_{2}can be less than 0.5%. Moreover, the content of TiO

_{2}in boninite is less than 0.4% [55,56]. However, the content of TiO

_{2}in gabbroic rocks is generally low and can be as low as 0.2–0.3% [34]. This situation occurs because cumulate gabbroic rocks are mainly composed of cumulate minerals (olivine, pyroxene, and plagioclase) and interstitial melts that cool in the spaces between cumulate minerals, but these cumulate minerals do not contain Ti, and the Ti of gabbroic rocks mainly exists in the interstitial melts. However, the proportion of interstitial melt in gabbroic rock is uncertain, making the content of Ti decrease in varying degrees. If discriminating among different tectonic settings of gabbroic rocks with a basalt discrimination diagram based on Ti, all the gabbroic rock samples would be determined as island arc rocks, which is extremely unreasonable.

#### 4.3. New Gabbroic Rock Discrimination Diagrams

#### 4.3.1. La/Y–Nb/Ba Diagram

#### 4.3.2. Nb/Sc–Sc/Ba Diagram

^{−6}, Table 1), while in IAG and OIG, the contents of Sc are low (10–12 × 10

^{−6}, Table 1). This contrast might occur because gabbroic rocks are mainly composed of cumulates (mean Mg# = 0.75), so Sc enters the lattice of pyroxene and is removed by fractionation in island arc and ocean island systems. Ba is an incompatible element; the contents of Ba in IAG and OIG are high (Table 1), and that of MORG is extremely low (the contents of Ba in IAG, OIG, and MORG are 98 × 10

^{−6}, 140 × 10

^{−6}, and 7.7 × 10

^{−6}, respectively, Table 1). Therefore, in Figure 23b, the value of Sc/Ba in MORG is the highest (the mean is 4.0), while in IAG and OIG the values are 0.11 and 0.09, respectively. It follows that Sc/Ba can discriminate between MORG and non-MORG effectively, as shown in Figure 24b. Moreover, because of the depletion in Nb in IAG and the enrichment in Nb in OIG and the similar contents of Sc in IAG and OIG, Nb/Sc can easily distinguish IAG and OIG.

#### 4.3.3. Ba/Nb–Ba/Sc Diagram

#### 4.3.4. La/Na_{2}O–Nb/Ba Diagram

_{2}O in cumulate rocks are generally low. The contents of Na

_{2}O in IAG, OIG, and MORG are also found to be similar. Therefore, the differences between La/Na

_{2}O values of IAG, OIG, and MORG are mainly dependent on the values of La. In Figure 24d, La/Na

_{2}O is mainly used to distinguish between OIG and MORG. OIG is rich in La, while MORG is poor in La. Consequently, La/Na

_{2}O can discriminate between the two tectonic settings. IAG is extremely low in Nb and rich in Ba; thus, the Nb/Ba value of IAG is significantly higher than the Nb/Ba value of OIG and MORG. As a result, the La/Na

_{2}O–Nb/Ba diagram also has good performance in discriminating among IAG, OIG, and MORG.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, P.; Glover, L., III. A tectonics test of the most commonly used geochemical discriminant diagrams and patterns. Earth-Sci. Rev.
**1992**, 33, 111–131. [Google Scholar] [CrossRef] - Song, S.; Niu, Y.; Su, L.; Xia, X. Tectonics of the north Qilian orogen, NW China. Gondwana Res.
**2013**, 23, 1378–1401. [Google Scholar] [CrossRef] - Wilson, M. Igneous Petrogenesis; Springer: Dordrecht, The Netherlands, 1989; 466p. [Google Scholar]
- Xia, L.; Li, X.M. Basalt geochemistry as a diagnostic indicator of tectonic setting. Gondwana Res.
**2019**, 65, 43–67. [Google Scholar] [CrossRef] - Pearce, J.A.; Cann, J.R. Ophiolite origin investigated by discriminant analysis using Ti, Zr and Y. Earth Planet. Sci. Lett.
**1971**, 12, 339–349. [Google Scholar] [CrossRef] - Pearce, J.A.; Cann, J.R. Tectonic setting of basic volcanic rocks determined using trace element analyses. Earth Planet. Sci. Lett.
**1973**, 19, 290–300. [Google Scholar] [CrossRef] - Pearce, J.A.; Norry, M.J. Petrogenetic implications of Ti, Zr, Y, and Nb variations in volcanic rocks. Contrib. Mineral. Petrol.
**1979**, 69, 33–47. [Google Scholar] [CrossRef] - Roser, B.P.; Korsch, R.J. Determination of Tectonic Setting of Sandstone-Mudstone Suites Using SiO
_{2}Content and K_{2}O/Na_{2}O Ratio. J. Geol.**1986**, 94, 635–650. [Google Scholar] [CrossRef] - Pearce, J.A.; Peate, D.W. Tectonic implications of the composition of volcanic arc magmas. Annu. Rev. Earth Planet. Sci.
**1995**, 23, 251–286. [Google Scholar] [CrossRef] - Vermeesch, P. Tectonic discrimination diagrams revisited. Geochem. Geophys. Geosystems
**2013**, 7, 1–55. [Google Scholar] [CrossRef] - Jankovics, M.É.; Taracsák, Z.; Dobosi, G.; Embey-Isztin, A.; Batki, A.; Harangi, S.; Hauzenberger, C.A. Clinopyroxene with diverse origins in alkaline basalts from the western Pannonian Basin: Implications from trace element characteristics. Lithos
**2016**, 262, 120–134. [Google Scholar] [CrossRef] - Sánchez-Muñoz, L.; Müller, A.; Andrés, S.L.; Martin, R.F.; Modreski, P.J.; Moura, O.J.M.D. The P–Fe diagram for K-feldspars: A preliminary approach in the discrimination of pegmatites. Lithos
**2016**, 272, 116–127. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Zhao, G.; Eizenhöfer, P.R.; Sun, M.; Han, Y.; Hou, W.; Xu, B. Varying contents of sources affect tectonic-setting discrimination of sediments: A case study from permian sandstones in the eastern tianshan, Northwestern China. J. Geol.
**2017**, 125, 299–316. [Google Scholar] [CrossRef] [Green Version] - Verma, S.P.; Pandarinath, K.; Verma, S.K.; Agrawal, S. Fifteen new discriminant-function-based multi-dimensional robust diagrams for acid rocks and their application to Precambrian rocks. Lithos
**2013**, 168, 113–123. [Google Scholar] [CrossRef] - Verma, S.P.; Rivera-Gómez, M.A.; Díaz-González, L.; Pandarinath, K.; Amezcua-Valdez, A.; Rosales-Rivera, M.; Verma, S.K.; Quiroz-Ruiz, A.; Armstrong-Altrin, J.S. Multidimensional classification of magma types for altered igneous rocks and application to their tectonomagmatic discrimination and igneous provenance of siliciclastic sediments. Lithos
**2017**, 278, 321–330. [Google Scholar] [CrossRef] - Stepanova, A.V.; Stepanov, V.S.; Larionov, A.N.; Azimov, P.Y.; Egorova, S.V.; Larionova, Y.O. 2.5 Ga gabbro–anorthosites in the Belomorian Province, Fennoscandian Shield: Petrology and tectonic setting. Petrology
**2017**, 25, 566–591. [Google Scholar] [CrossRef] - Yamasaki, T.; Nanayama, F. Enriched mid–ocean ridge basalt–type geochemistry of basalts and gabbros from the Nikoro Group, Tokoro Belt, Hokkaido, Japan. J. Mineral. Petrol. Sci.
**2017**, 112, 311–323. [Google Scholar] [CrossRef] [Green Version] - Gavryushkina, O.A.; Kruk, N.N.; Semenov, I.V.; Vladimirov, A.G.; Kuibida, Y.V.; Serov, P.A. Petrogenesis of Permian-Triassic intraplate gabbro–granitic rocks in the Russian Altai. Lithos
**2018**, 326, 71–89. [Google Scholar] [CrossRef] - Liu, X.L.; Zhang, Q.; Li, W.C.; Yang, F.C.; Zhao, Y.; Li, Z.; Chen, W.F. Applicability of large-ion lithophile and high field strength element basalt discrimination diagrams. Int. J. Digit. Earth
**2018**, 11, 752–760. [Google Scholar] [CrossRef] - Jiao, S.; Zhang, Q.; Zhou, Y.; Chen, W.; Liu, X.; Gopalakrishnan, G. Progress and challenges of big data research on petrology and geochemistry. Solid Earth Sci.
**2018**, 3, 105–114. [Google Scholar] [CrossRef] - Di, P.F.; Chen, W.F.; Zhang, Q.; Wang, J.R.; Tang, Q.Y.; Jiao, S.T. Comparison of global N-MORB and E-MORB classification schemes. Acta Petrol. Sin.
**2018**, 34, 264–274. [Google Scholar] - Snow, C.A. A reevaluation of tectonic discrimination diagrams and a new probabilistic approach using large geochemical databases: Moving beyond binary and ternary plots. J. Geophys. Res. Solid Earth
**2006**, 111. [Google Scholar] [CrossRef] [Green Version] - Delaunay, B. Sur la sphère vide. Izv. Akad. Nauk SSSR Otd. Mat. Estestv. Nauk
**1934**, 7, 1–2. [Google Scholar] - Lee, D.T.; Schachter, B.J. Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci.
**1980**, 9, 219–242. [Google Scholar] [CrossRef] - Edelsbrunner, H.; Tan, T.S.; Waupotitsch, R. An O (n
^{2}log n) time algorithm for the minmax angle triangulation. SIAM J. Sci. Stat. Comput.**1992**, 13, 994–1008. [Google Scholar] [CrossRef] [Green Version] - Cao, T.T.; Edelsbrunner, H.; Tan, T.S. Proof of correctness of the digital Delaunay triangulation algorithm. Comput. Geom. Theory Appl.
**2015**, 48, 507–519. [Google Scholar] [CrossRef] - Su, T.; Wang, W.; Lv, Z.; Wu, W.; Li, X. Rapid Delaunay triangulation for randomly distributed point cloud data using adaptive Hilbert curve. Comput. Graph.
**2016**, 54, 65–74. [Google Scholar] [CrossRef] - Buccianti, A.; Mateu-Figueras, G.; Pawlowsky-Glahn, V. Frequency distributions and natural laws in geochemistry. Geol. Soc. Lond. Spec. Publ.
**2006**, 264, 175–189. [Google Scholar] [CrossRef] [Green Version] - PetDB Search: Find & Select Samples & Data. Available online: https://search.earthchem.org/ (accessed on 8 January 2020).
- Geochemistry of Rocks of the Oceans and Continents. Available online: http://georoc.mpch-mainz.gwdg.de/georoc/ (accessed on 8 January 2020).
- Middlemost, E.A.K. Naming materials in the magma/igneous rock system. Earth Sci. Rev.
**1994**, 37, 215–224. [Google Scholar] [CrossRef] - Bowen, N.L. The Evolution of Igneous Rocks; Princeton University Press: New York, NY, USA, 1928; 334p. [Google Scholar]
- Best, M.G.; Christiansen, E.H. Igneous Petrology; Blackwell Science: Oxford, UK, 2001; 458p. [Google Scholar]
- Wager, L.R.; Brown, G.M. Layered Igneous Rocks; Oliver and Boyd: Edinburgh, UK; Oliver and Boyd: London, UK, 1968; 588p. [Google Scholar]
- Rollison, H.R. Using Geochemical Data: Evaluation, Presentation, Interpretation; Routledge: London, UK, 1993; 384p. [Google Scholar]
- Irvine, T.N.; Baragar, W.R.A. A guide to the chemical classification of the common volcanic rocks. Can. J. Earth Sci.
**1971**, 8, 523–548. [Google Scholar] [CrossRef] - Miyashiro, A. The Troodos ophiolitic complex was probably formed in an island arc. Earth Planet. Sci. Lett.
**1973**, 19, 218–224. [Google Scholar] [CrossRef] - Glassley, W. Geochemistry and tectonics of the Crescent volcanic rocks, Olympic Peninsula, Washington. Geol. Soc. Am. Bull.
**1974**, 85, 785–794. [Google Scholar] [CrossRef] - Pearce, J.A. Basalt geochemistry used to investigate past tectonic environments on Cyprus. Tectonophysics
**1975**, 25, 41–67. [Google Scholar] [CrossRef] - Pearce, J.A. Statistical analysis of major element patterns in basalts. J. Pet.
**1976**, 17, 15–43. [Google Scholar] [CrossRef] - Pearce, J.A. Trace element characteristics of lavas from destructive plate boundaries. Andesites
**1982**, 8, 525–548. [Google Scholar] - Pearce, J.A. Role of the Subcontinental Lithosphere in Magma Genesis at Active Continental Margins. In Continental Basalt and Mantle Xenoliths, Nantwich, England; Hawkesworth, C.J., Norry, M.J., Eds.; Shiva Publications: Chandigarh, India, 1983; pp. 230–249. [Google Scholar]
- Pearce, J.A. Supra-Subduction Zone Ophiolites: The Search for Modern Analogues. In Ophiolite Concept and the Evolution of Geological Thought: Geological Society of America Special Paper; Dilek, Y., Newcomb, S., Eds.; Geological Society of America: Boulder, CO, USA, 2003; Volume 373, pp. 269–293. [Google Scholar]
- Pearce, T.H.; Gorman, B.E.; Birkett, T.C. The relationship between major element chemistry and tectonic environment of basic and intermediate volcanic rocks. Earth Planet. Sci. Lett.
**1977**, 36, 121–132. [Google Scholar] [CrossRef] - Wood, D.A.; Joron, J.L.; Treuil, M. A re-appraisal of the use of trace elements to classify and discriminate between magma series erupted in different tectonic settings. Earth Planet. Sci. Lett.
**1979**, 45, 326–336. [Google Scholar] [CrossRef] - Wood, D.A. The application of a Th–Hf–Ta diagram to problems of tectonomagmatic classification and to establishing the nature of crustal contamination of basaltic lavas of the British Tertiary Volcanic Province. Earth Planet Sci. Lett.
**1980**, 50, 11–30. [Google Scholar] [CrossRef] - Capedri, S.; Venturelli, G.; Bocchi, G.; Dostal, J.; Garuti, G.; Rossi, A. The geochemistry and petrogenesis of an ophiolitic sequence from Pindos, Greece. Contrib. Mineral. Petrol.
**1980**, 74, 189–200. [Google Scholar] [CrossRef] - Mullen, E.D. MnO–TiO
_{2}–P_{2}O_{5}: A minor element discriminant for basaltic rocks of oceanic environments and its implications for petrogenesis. Earth Planet. Sci. Lett.**1983**, 62, 53–62. [Google Scholar] [CrossRef] - Pearce, J.A.; Lippard, S.J.; Roberts, S. Characteristics and tectonic significance of supra-subduction zone ophiolites. Geol. Soc. Lond. Spec. Publ.
**1984**, 16, 77–94. [Google Scholar] [CrossRef] - Harris, N.B.W.; Pearce, J.A.; Tindle, A.G. Geochemical characteristics of collision-zone magmatism. Geol. Soc. Lond. Spec. Publ.
**1986**, 19, 67–81. [Google Scholar] [CrossRef] - Meschede, M. A method of discriminating between different types of mid-ocean ridge basalts and continental tholeiites with the Nb, Zr, Y diagram. Chem. Geol.
**1986**, 56, 207–218. [Google Scholar] [CrossRef] - Workman, R.K.; Hart, S.R. Major and trace element composition of the depleted MORB mantle (DMM). Earth Planet. Sci. Lett.
**2005**, 231, 53–72. [Google Scholar] [CrossRef] - Galoyan, G.; Rolland, Y.; Sosson, M.; Corsini, M.; Melkonyan, R. Evidence for superposed MORB, oceanic plateau and volcanic arc series in the Lesser Caucasus (Stepanavan, Armenia). C. R. Geosci.
**2007**, 339, 482–492. [Google Scholar] [CrossRef] - Zhao, Z.H. How to use the trace element diagrams to discriminate tectonic settings. Geotecton. Metallog.
**2007**, 31, 92–103. [Google Scholar] - Hickey, R.L.; Frey, F.A. Geochemical characteristics of boninite series volcanics: Implications for their source. Geochim. Cosmochim. Acta
**1982**, 46, 2099–2115. [Google Scholar] [CrossRef] - Crawford, A.J.; Falloon, T.J.; Green, D.H. Classification, Petrogenesis and Tectonic Settings of Boninites. In Boninite; Crawford, A.J., Ed.; Unwin Hyman: London, UK, 1989; pp. 1–49. [Google Scholar]
- Kocaka, K.; Isıka, F.; Arslanb, M.; Zedef, V. Petrological and source region characteristics of ophiolitic hornblende gabbros from the Aksaray and Kayseri regions, central Anatolian crystalline complex, Turkey. J. Asian Earth Sci.
**2005**, 25, 883–891. [Google Scholar] [CrossRef] - Pollock, J.C.; Hibbard, J.P. Geochemistry and tectonic significance of the Stony Mountain gabbro, North Carolina: Implications for the Early Paleozoic evolution of Carolinia. Gondwana Res.
**2010**, 17, 500–515. [Google Scholar] [CrossRef] - Verma, S.P.; Guevara, M.; Agrawal, S. Discriminating four tectonic settings: Five new geochemical diagrams for basic and ultrabasic volcanic rocks based on log-ratio transformation of major-element data. J. Earth Syst. Sci.
**2006**, 115, 485–528. [Google Scholar] [CrossRef] [Green Version] - Agrawal, S.; Guevara, M.; Verma, S.P. Tectonic discrimination of basic and ultrabasic rocks through log-transformed ratios of immobile trace elements. Int. Geol. Rev.
**2008**, 50, 1057–1079. [Google Scholar] [CrossRef] - Verma, S.P.; Agrawal, S. New tectonic discrimination diagrams for basic and ultrabasic volcanic rocks through log-transformed ratios of high field strength elements and implications for petrogenetic processes. Rev. Mex. Cienc. Geológicas
**2011**, 28, 24–44. [Google Scholar] - Verma, S.P.; Torres-Alvarado, I.S.; Sotelo-Rodríguez, Z.T. SINCLAS: Standard igneous norm and volcanic rock classification system. Comput. Geosci.
**2002**, 28, 711–715. [Google Scholar] [CrossRef] - Egozcue, J.J.; Pawlowsky-Glahn, V.; Mateu-Figueras, G.; Barcelo-Vidal, C. Isometric logratio transformations for compositional data analysis. Math. Geol.
**2003**, 35, 279–300. [Google Scholar] [CrossRef] - Pawlowsky-Glahn, V.; Egozcue, J.J. Compositional data and their analysis: An introduction. Geol. Soc. Lond. Spec. Publ.
**2006**, 264, 1–10. [Google Scholar] [CrossRef] [Green Version] - Verma, S.P. Statistical evaluation of bivariate, ternary and discriminant function tectonomagmatic discrimination diagrams. Turk. J. Earth Sci.
**2010**, 19, 185–238. [Google Scholar] - Buccianti, A. Is compositional data analysis a way to see beyond the illusion? Comput. Geosci.
**2013**, 50, 165–173. [Google Scholar] [CrossRef] [Green Version] - Parent, S.É.; Parent, L.E.; Egozcue, J.J.; Rozane, D.E.; Hernandes, A.; Lapointe, L.; Hebert-Gentile, V.; Naess, K.; Marchand, S.; Lafond, J.; et al. The plant ionome revisited by the nutrient balance concept. Front. Plant Sci.
**2013**, 4, 39. [Google Scholar] [CrossRef] [Green Version] - Aitchison, J. The statistical analysis of compositional data. J. R. Stat. Soc. Ser. B
**1982**, 44, 139–160. [Google Scholar] [CrossRef] - Hart, S.R.; Dunn, T. Experimental cpx/melt partitioning of 24 trace elements. Contrib. Mineral. Petrol.
**1993**, 113, 1–8. [Google Scholar] [CrossRef]

**Figure 2.**Example of Delaunay triangulation. The dashed line network is the Delaunay triangulation, and the colored polygon network is the Voronoi polygon.

**Figure 4.**Distributions of sample points when the length of the x-axis and the length of the y-axis are not of the same order of magnitude. (

**a**) Manually projecting a set of points onto a Cartesian coordinate system; (

**b**) Status of the points in the operation process of a computer; (

**c**) Delaunay triangulation without coordinate transformation.

**Figure 5.**Delaunay triangulations generated before and after scale zooming. (

**a**) Delaunay triangulation without coordinate transformation; (

**b**) Delaunay triangulation with coordinate transformation.

**Figure 6.**Generating boundaries of sample points with a linear coordinate and a logarithmic coordinate (the confidence coefficient is 0.85). (

**a**) Sample points and boundary using linear coordinate; (

**b**) Sample points and boundary using logarithmic coordinate.

**Figure 8.**Example of using the iteration algorithm to calculate the confidence field. (

**a**) Confidence field of a set of points; (

**b**) Iteration process of determining the confidence field.

**Figure 10.**Plutonic TAS diagram (the boundaries are from Middlemost [31]).

**Figure 11.**Discrimination diagrams for IAG and non-IAG. All four diagrams have a 0.85 confidence region. (

**a**,

**b**) are in linear coordinates; (

**c**,

**d**) are in logarithmic coordinates.

**Figure 12.**Word clouds of the top 20 important elements (or element ratios) for IAG and non-IAG discrimination diagrams. (

**a**) is for linear coordinates; (

**b**) is for logarithmic coordinates.

**Figure 13.**Statistics of the frequency of occurrence of the 26 basic elements in useful IAG and non-IAG discrimination diagrams.

**Figure 14.**Discrimination diagrams for OIG and non-OIG. All four diagrams have a 0.85 confidence region. (

**a**,

**b**) are in linear coordinates; (

**c**,

**d**) are in logarithmic coordinates.

**Figure 15.**Word clouds of the top 20 important elements (or element ratios) for OIG and non-OIG discrimination diagrams. (

**a**) is for linear coordinates; (

**b**) is for logarithmic coordinates.

**Figure 16.**Statistics of the frequency of occurrence of the 26 basic elements in useful OIG and non-OIG discrimination diagrams.

**Figure 17.**Discrimination diagrams for MORG and non-MORG. All four diagrams have a 0.85 confidence region. (

**a**,

**b**) are in linear coordinates; (

**c**,

**d**) are in logarithmic coordinates.

**Figure 18.**Word clouds of the top 20 important elements (or element ratios) for MORG and non-MORG discrimination diagrams. (

**a**) is for linear coordinates; (

**b**) is for logarithmic coordinates.

**Figure 19.**Statistics of the frequency of occurrence of the 26 basic elements in useful MORG and non-MORG discrimination diagrams.

**Figure 20.**IAG–OIG–MORG discrimination diagrams. All four diagrams have a 0.85 confidence region. (

**a**,

**b**) are in linear coordinates; (

**c**,

**d**) are in logarithmic coordinates.

**Figure 21.**Word clouds of the top 20 important elements (or element ratios) for IAG–OIG–MORG discrimination diagrams. (

**a**) is for linear coordinates; (

**b**) is for logarithmic coordinates.

**Figure 22.**Statistics of the frequency of occurrence of the 26 basic elements in useful IAG–OIG–MORG discrimination diagrams.

**Figure 23.**Basic and ultrabasic discrimination diagrams [61]. In (

**a**), the DF1 = −0.6611 × ln (Nb/(TiO

_{2})

_{adj}) + 2.2926 × ln(V/(TiO

_{2})

_{adj}) + 1.6774 × ln (Y/(TiO

_{2})

_{adj}) +1.0916 × ln (Zr/(TiO

_{2})

_{adj}) + 21.3603, and the DF2 = 0.4702 × ln (Nb/(TiO

_{2})

_{adj}) + 3.7649 × ln (V/(TiO

_{2})

_{adj}) − 3.911 × ln (Y/(TiO

_{2})

_{adj}) + 2.2697 × ln (Zr/(TiO

_{2})

_{adj}) + 4.8487; in (

**b**), DF1 = −0.2646 × ln(Nb/(TiO

_{2})

_{adj}) + 2.0491 × ln(V/TiO

_{2})

_{adj}) + 3.4565 × ln(Y/TiO

_{2})

_{adj}) + 0.8573 × ln(Zr/(TiO

_{2})

_{adj}) + 32.9472, DF2 = 0.01874 × ln(Nb/(TiO

_{2})

_{adj}) + 4.0937 × ln(V/(TiO

_{2})

_{adj}) – 4.8550 × ln(Y/(TiO

_{2})

_{adj}) + 2.9900 × ln(Zr/(TiO

_{2})

_{adj}) + 0.1995. The subscript adj means adjust. The adjustment of the oxides refers to [68].

**Figure 24.**Recommended discrimination diagrams for gabbroic rocks. All four diagrams have 0.85 confidence regions. (

**a**) Nb/Ba–La/Y diagram; (

**b**) Sc/Ba–Nb/Sc diagram; (

**c**) Ba/Sc–Ba/Nb diagram; (

**d**) Nb/Ba–La/Na

_{2}O diagram.

**Table 1.**Mean element contents of island-arc gabbroic rocks (IAG), ocean-island gabbroic rocks (OIG), and mid-oceanic ridge gabbroic rocks (MORG) samples.

Basic Elements | IAG | OIG | MORG | Basic Elements | IAG | OIG | MORG |
---|---|---|---|---|---|---|---|

SiO_{2} (wt%) | 47.9 | 46.5 | 49.6 | Ni (ppm) | 79.3 | 191 | 202 |

TiO_{2} (wt%) | 0.933 | 2.17 | 0.812 | Cu (ppm) | 47.5 | 77.5 | 65.4 |

Al_{2}O_{3} (wt%) | 17.6 | 15.6 | 16.4 | Zn (ppm) | 77.0 | 75.0 | 52.7 |

FeO^{T} (wt%) | 9.10 | 9.78 | 6.81 | Rb (ppm) | 12.3 | 13.9 | 0.512 |

CaO (wt%) | 11.4 | 12.1 | 12.0 | Sr (ppm) | 381 | 465 | 130 |

MgO (wt%) | 8.11 | 8.70 | 9.64 | Y (ppm) | 15.3 | 19.2 | 15.7 |

MnO (wt%) | 0.165 | 0.150 | 0.14 | Zr (ppm) | 44.4 | 111 | 31.3 |

K_{2}O (wt%) | 0.549 | 0.777 | 0.0762 | Nb (ppm) | 2.44 | 18.3 | 1.14 |

Na_{2}O (wt%) | 2.17 | 2.35 | 2.64 | Ba (ppm) | 157 | 181 | 9.59 |

P_{2}O_{5} (wt%) | 0.154 | 0.336 | 0.0890 | La (ppm) | 5.17 | 19.2 | 1.97 |

Sc (ppm) | 39.1 | 28.1 | 33.9 | Ce (ppm) | 14.3 | 40.6 | 6.22 |

V (ppm) | 261 | 269 | 177 | Nd (ppm) | 7.87 | 23.0 | 6.57 |

Cr (ppm) | 272 | 468 | 300 | Sm (ppm) | 2.04 | 5.06 | 2.18 |

Basic Elements | IAG | OIG | MORG | Basic Elements | IAG | OIG | MORG |
---|---|---|---|---|---|---|---|

SiO_{2} (wt%) | 3.67 | 3.25 | 2.63 | Ni (ppm) | 91.5 | 211 | 277 |

TiO_{2} (wt%) | 0.695 | 1.50 | 1.14 | Cu (ppm) | 35.1 | 71.2 | 39.0 |

Al_{2}O_{3} (wt%) | 3.83 | 4.60 | 3.09 | Zn (ppm) | 39.2 | 39.0 | 31.7 |

FeO^{T} (wt%) | 2.83 | 3.55 | 2.42 | Rb (ppm) | 17.9 | 19.5 | 0.570 |

CaO (wt%) | 2.69 | 2.97 | 1.87 | Sr (ppm) | 243 | 345 | 44.8 |

MgO (wt%) | 4.17 | 5.20 | 3.44 | Y (ppm) | 9.06 | 13.5 | 16.0 |

MnO (wt%) | 0.0581 | 0.05 | 0.06 | Zr (ppm) | 38.7 | 110 | 38.3 |

K_{2}O (wt%) | 0.666 | 0.90 | 0.13 | Nb (ppm) | 2.83 | 22.7 | 2.33 |

Na_{2}O (wt%) | 1.05 | 1.20 | 0.85 | Ba (ppm) | 144 | 191 | 7.78 |

P_{2}O_{5} (wt%) | 0.163 | 0.43 | 0.25 | La (ppm) | 4.93 | 21.6 | 3.58 |

Sc (ppm) | 21.5 | 12.5 | 13.2 | Ce (ppm) | 15.0 | 46.0 | 12.1 |

V (ppm) | 124 | 156 | 125 | Nd (ppm) | 5.99 | 25.2 | 12.8 |

Cr (ppm) | 418 | 600 | 363 | Sm (ppm) | 1.48 | 4.85 | 3.57 |

Type of Coordinates | Type of Important Elements | IAG vs. Non-IAG | OIG vs. Non-OIG | MORG vs. Non-MORG | IAG–OIG–MORG |
---|---|---|---|---|---|

Linear | Elements or element ratios | Ba/Nb, Rb/Nb | Nb/Sc | Na_{2}O/Ba | Sc/Nb, Nb/Sc |

Basic elements | Nb, Ba, Rb | Nb, Sc, Cu | Ba, Na_{2}O, Rb | Nb, Ba, Sc | |

Logarithmic | Elements or elements | Nb/Ba, Ba/Nb | Na_{2}O/Nb, Nb/Na_{2}O, Nb/Sm, Sm/Nb | Na_{2}O/Ba, Ba/Na_{2}O, Na_{2}O/Rb, Rb/Na_{2}O | Sc/Nb, Nb/Sc, Sc/Ba, Ba/Sc |

Basic elements | Ba, Nb, Sc | Nb, Sc, Sm, Na_{2}O | Ba, Na_{2}O, Rb | Nb, Ba, Sc |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Han, S.; Li, M.; Zhang, Q.; Song, L.
An Automated Method to Generate and Evaluate Geochemical Tectonic Discrimination Diagrams Based on Topological Theory. *Minerals* **2020**, *10*, 62.
https://doi.org/10.3390/min10010062

**AMA Style**

Han S, Li M, Zhang Q, Song L.
An Automated Method to Generate and Evaluate Geochemical Tectonic Discrimination Diagrams Based on Topological Theory. *Minerals*. 2020; 10(1):62.
https://doi.org/10.3390/min10010062

**Chicago/Turabian Style**

Han, Shuai, Mingchao Li, Qi Zhang, and Lingguang Song.
2020. "An Automated Method to Generate and Evaluate Geochemical Tectonic Discrimination Diagrams Based on Topological Theory" *Minerals* 10, no. 1: 62.
https://doi.org/10.3390/min10010062