A Parametric Approach for Dealing with Compositional Rounded Zeros

In this work, a parametric approach for replacing data below the detection limit, also known as rounded zeros, in compositional data sets is proposed. Compositional rounded zeros correspond to small proportions of some whole that cannot be reliably detected by the analytical instruments under given operating conditions. This kind of zeros appear frequently in the data collection process in geosciences. They must be treated in an adequate way before some multivariate analysis can be applied. Our procedure results from a modification of the Expectation-Maximization (EM) algorithm and is based on the additive log-ratio transformation. Its coherence with the nature of compositional data and with basic operations in the simplex sample space is checked. Using real data sets, we find that this approach improves other parametric and non-parametric techniques for compositional rounded zeros.     

Estimation of regionalized compositions: A comparison of three methods

A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable.     

南海珊瑚礁光学浅水海域多光谱遥感水深反演

海洋水深信息对研究珊瑚礁海域资源与环境具有重要作用。南海珊瑚礁海域测深数据受多种条件限制施测困难,在时间与空间方面数量非常有限。文章针对南海岛礁海域以I类水体为主导的海水光学特性,以南沙群岛库归沙洲海域为例,使用Sentinel-2多光谱卫星遥感影像和同期过境的MODIS卫星数据,构建底质光谱,采用半分析半经验模型计算海水表面遥感反射率与海水叶绿素浓度,通过对数比值模型进行该地区光学浅水海域遥感水深反演分析,并进一步通过多时相反演水深融合提升精度。经与多波束实测水深数据验证,研究区域反演水深总体均方根误差和平均相对误差分别为2.68 m 和9.99%。该方法通过叶绿素浓度推演部分海水光学特性,可以从多光谱卫星影像中快速获取南海岛礁光学浅水海域初步水深信息,供相关海洋领域分析与应用。     

Statistical evaluation of tectonomagmatic discrimination diagrams for granitic rocks and proposal of new discriminant-function-based multi-dimensional diagrams for acid rocks

The four tectonic discrimination diagrams of Pearce et al. [Journal of Petrology, v. 25, p. 956–983] for granitic rocks were first evaluated using the literature cited by these authors as well as from our new database. The first diagram (Y?Nb) cannot discriminate volcanic-arc and collision settings. Both Y?Nb and Yb?Ta diagrams have an overlapping field for within-plate and ocean-ridge granitoids. The remaining two diagrams (Y?+?Nb?Rb and Yb?+?Ta?Rb) use a mobile element (Rb) in their y-axis. Although these diagrams successfully discriminate volcanic-arc and within-plate granites, they perform less well for collision tectonics. Besides, felsic or acid rocks are scarce in ocean-ridge settings, which limits the usefulness of these diagrams for this geological environment. Therefore, using an extensive database, we proposed a set of five new discriminant-function-based multi-dimensional diagrams for acid magmas from four tectonic settings (island arc, continental arc, continental rift, and collision). The very similar tectonic settings of island and continental arcs are discriminated for the first time. These diagrams are based on correct statistical treatment of compositional data, because they use natural logarithm transformation of major-element ratios and linear discriminant analysis (LDA). The use of discordant outlier-free samples prior to LDA improved the success rates by about 3–5%. Success rates of these diagrams as inferred from a testing set were between 76% and 88% for island arc, 60% and 92% for continental arc, and 72% and 84% for both continental rift and collision settings. Finally, application of these new diagrams to case studies not compiled in our initial database used for constructing these diagrams provided the following results: a collision setting for the Himalayas at about 30 Ma; an island arc setting for Quaternary acid rocks from geothermal boreholes in El Salvador; an island- or continental-arc setting for northern Italy at 35–52 Ma; a continental-arc setting for the Italy–Austria border at about 30 Ma; either a rift or a collision setting for northern Nigeria at about 164 Ma; a collision setting for central Nigeria at about 144 Ma and for the Cretaceous Masirah ophiolites of Oman; and an island arc setting for the Cretaceous Semail ophiolites of Oman. In spite of the relative mobility of major elements, these applications suggest utility of the new discrimination diagrams for all four tectonic settings.     

Identification of Archaean plate tectonic processes from multidimensional discrimination diagrams and probability calculations

We applied our group's previously published multidimensional diagrams in 2006–2012 and corresponding probability estimates in 2011–2012 to geochemical data for Archaean rocks compiled from cratons in Australia, South Africa, Brazil, Canada, and India. Tectonic processes similar to present-day plate tectonics evidently were active at least since the Palaeoarchaean (?3570 Ma). This seems to be true in spite of a presumably hotter Earth at that time. For the eastern part of the Pilbara craton (Australia), a Palaeoarchaean (3570–3450 Ma) and Mesoarchaean (2900 Ma) continental arc setting apparently evolved to a collision (Col) setting during the Neoarchaean (2600 Ma). We infer an island arc (IA) environment for Kambalda (Australia) during the Neoarchaean (2700 Ma). For the Barberton belt (South Africa), a transition from a mid-ocean ridge regime during the older part of the Palaeoarchaean (3470 Ma) to an IA setting during the younger part (3300–3260 Ma) is likely. We inferred an arc environment for the São Francisco craton (Brazil) and the Rio Maria terrane (Brazil) during the Mesoarchaean (3085–2983 Ma and 2870 Ma, respectively), whereas a within-plate setting is clearly indicated for the Carajás metallogenic province (Brazil) during the Neoarchaean (2740–2700 Ma). We also recognize an IA regime for the Mesoarchaean (3000 Ma) North Caribou and Neoarchaean (2700 Ma) Abitibi greenstone belts (Canada), and for the Gadwal greenstone belt (India) during the Neoarchaean (2700–2500 Ma). A Col setting was inferred for the Archaean sanukitoid suite (Canada) and the Kolar suture zone (India) during the Neoarchaean (2700–2660 Ma and 2630–2520 Ma, respectively).     

Latent Compositional Factors in The Llobregat River Basin (Spain) Hydrogeochemistry

The study of hydrogeochemical data sets frequently calls for statistical dimension reducing techniques. It is well known that hydrochemical parameters are compositions and, for this type of data, the direct application of classical statistical methods based on the correlation matrix yield spurious results. But new results on compositional data analysis have identified the sampling space, the simplex, with an Euclidean space, a fact that allows us to define a simplicial factor analysis strategy, thus overcoming the problem. For illustration, we use samples from the Llobregat River and its tributaries (NE Spain). Three unobservable or latent factorial components are extracted, which are identified with pristine waters, potash-mining influence and urban sewage influence. These three factorial components or compositional factors are plotted in a factorial ternary diagram, which reflects the relative influence of each one of these factors on each observation.     

Groups of Parts and Their Balances in Compositional Data Analysis

Amalgamation of parts of a composition has been extensively used as a technique of analysis to achieve reduced dimension, as was discussed during the CoDaWork'03 meeting (Girona, Spain, 2003). It was shown to be a non-linear operation in the simplex that does not preserve distances under perturbation. The discussion motivated the introduction in the present paper of concepts such as group of parts, balance between groups, and sequential binary partition, which are intended to provide tools of compositional data analysis for dimension reduction. Key concepts underlying this development are the established tools of subcomposition, coordinates in an orthogonal basis of the simplex, balancing element and, in general, the Aitchison geometry in the simplex. Main new results are: a method to analyze grouped parts of a compositional vector through the adequate coordinates in an ad hoc orthonormal basis; and the study of balances of groups of parts (inter-group analysis) as an orthogonal projection similar to that used in standard subcompositional analysis (intra-group analysis). A simulated example compares results when testing equal centers of two populations using amalgamated parts and balances; it shows that, in certain circumstances, results from both analysis can disagree.     

Application of 55 multi-dimensional tectonomagmatic discrimination diagrams to Precambrian belts

Fifty-five new multi-dimensional diagrams, mostly based on log-ratio transformations, were used to decipher tectonic settings for 17 cases of Precambrian belts in Canada, the USA, Poland, Finland, Jordan, Democratic Republic of Congo and Zambia, China, and India. The results of different sets of diagrams for basic-ultrabasic, intermediate, and acid magmas were generally internally consistent. Possible reasons for some inconsistencies may be related to the use of a sample group of mixed ages because of their uncertainties, extreme element mobility caused by metamorphism especially of high-grade type, analytical data quality, different petrogenetic processes for basic to acid magmas such as mantle versus crustal origin, and some deficiencies that still exist in the multi-dimensional diagrams. To partly overcome the last problem, one new multi-dimensional diagram is proposed and used to discriminate mid-ocean ridge and oceanic plateau settings.     

Exploring the joint compositional variability of major components and trace elements in the Tellus soil geochemistry survey (Northern Ireland)

The complexity of modern geochemical data sets is increasing in several aspects (number of available samples, number of elements measured, number of matrices analysed, geological-environmental variability covered, etc), hence it is becoming increasingly necessary to apply statistical methods to elucidate their structure. This paper presents an exploratory analysis of one such complex data set, the Tellus geochemical soil survey of Northern Ireland (NI). This exploratory analysis is based on one of the most fundamental exploratory tools, principal component analysis (PCA) and its graphical representation as a biplot, albeit in several variations: the set of elements included (only major oxides vs. all observed elements), the prior transformation applied to the data (none, a standardization or a logratio transformation) and the way the covariance matrix between components is estimated (classical estimation vs. robust estimation). Results show that a log-ratio PCA (robust or classical) of all available elements is the most powerful exploratory setting, providing the following insights: the first two processes controlling the whole geochemical variation in NI soils are peat coverage and a contrast between “mafic” and “felsic” background lithologies; peat covered areas are detected as outliers by a robust analysis, and can be then filtered out if required for further modelling; and peat coverage intensity can be quantified with the %Br in the subcomposition (Br, Rb, Ni).     

对数比变换和偏最小二乘法在地球化学组合异常提取中的应用——以湘西北铅锌矿为例

以湘西北水系沉积物测量常量元素含量数据为研究对象,针对湘西北铅锌矿床的层控特征,以研究区含矿层地球化学常量元素含量的统计规律为基础,利用对数比变换(alr、clr、ilr变换)对原始数据进行预处理,并在此基础上进行偏最小二乘(PLS)降维分析,提取了地球化学组合异常,并与常用的主成分分析、因子分析的结果进行对比。分析结果表明,等距对数比变换(ilr变换)相对于其他变换方法有明显的优势,变换后的数据对称特征明显,更趋于正态分布。相对于其他两种降维方法,偏最小二乘法提取的元素组合与研究区铅锌矿的地球化学基本特征更为接近,组合异常与已知矿床分布和构造特征耦合良好,与研究区地质特征吻合,比主成分分析与因子分析得到的结果更易于地质解释。