On the Interpretation of Orthonormal Coordinates for Compositional Data

The simplex with the Aitchison geometry is a natural sample space for compositional data, that is, observations carrying only relative information (especially proportions, percentages, etc., often occurring in the geosciences). For this reason, standard statistical methods that rely on Euclidean structure of the real space cannot be used directly for statistical analysis. At first, compositional data need to be expressed in coordinates of an orthonormal basis on the simplex (with respect to the Aitchison geometry). The mathematical interpretation of the orthonormal coordinates is derived from the procedure by which they are constructed (called sequential binary partition), and they act as balances between groups of compositional parts. The goal of this paper is to describe the covariance structure of coordinates and, consequently, to provide a complementary interpretation based on log-ratios of parts of the original composition. It must be noted that, in a composition, the ratios themselves contain all the relevant information. The possibilities as well as the limitations of this approach are demonstrated through illustrative examples.     

Advances in Principal Balances for Compositional Data

Compositional data analysis requires selecting an orthonormal basis with which to work on coordinates. In most cases this selection is based on a data driven criterion. Principal component analysis provides bases that are, in general, functions of all the original parts, each with a different weight hindering their interpretation. For interpretative purposes, it would be better to have each basis component as a ratio or balance of the geometric means of two groups of parts, leaving irrelevant parts with a zero weight. This is the role of principal balances, defined as a sequence of orthonormal balances which successively maximize the explained variance in a data set. The new algorithm to compute principal balances requires an exhaustive search along all the possible sets of orthonormal balances. To reduce computational time, the sets of possible partitions for up to 15 parts are stored. Two other suboptimal, but feasible, algorithms are also introduced: (i) a new search for balances following a constrained principal component approach and (ii) the hierarchical cluster analysis of variables. The latter is a new approach based on the relation between the variation matrix and the Aitchison distance. The properties and performance of these three algorithms are illustrated using a typical data set of geochemical compositions and a simulation exercise.     

Correlation Between Compositional Parts Based on Symmetric Balances

Correlation coefficients are most popular in statistical practice for measuring pairwise variable associations. Compositional data, carrying only relative information, require a different treatment in correlation analysis. For identifying the association between two compositional parts in terms of their dominance with respect to the other parts in the composition, symmetric balances are constructed, which capture all relative information in the form of aggregated logratios of both compositional parts of interest. The resulting coordinates have the form of logratios of individual parts to a (weighted) “average representative” of the other parts, and thus, they clearly indicate how the respective parts dominate in the composition on average. The balances form orthonormal coordinates, and thus, the standard correlation measures relying on the Euclidean geometry can be used to measure the association. Simulation studies provide deeper insight into the proposed approach, and allow for comparisons with alternative measures. An application from geochemistry (Kola moss) indicates that correlations based on symmetric balances serve as a sensitive tool to reveal underlying geochemical processes.     

A Critical Approach to Probability Laws in Geochemistry

Probability laws in geochemistry have been a major issue of concern over the last decades. The lognormal on the positive real line or the additive logistic normal on the simplex are two classical laws of probability to model geochemical data sets due to their association with a relative measure of difference. This fact is not fully exploited in the classical approach when viewing both the positive real line and the simplex as subsets of real space with the induced geometry. But it can be taken into account considering them as real linear vector spaces with their own structure. This approach implies using a particular geometry and a measure different from the usual ones. Therefore, we can work with the coordinates with respect to an orthonormal basis. It could be shown that the two mentioned laws are associated with a normal distribution on the coordinates. In this contribution both approaches are compared, and a real data set is used to illustrate similarities and differences.     

Standardization of metal concentrations in sediments using regression residuals: an example from a large lake in Florida, USA

 A new method of standardizing metal concentrations in sediments was tested on samples from Lake Miccosukee, a large karstic lake in north Florida. Metal concentrations were analyzed in 222 sediment samples from 26 cores representing 9 sampling sites in the lake. Measured sedimentation rates in the lake are low. Percent organic matter strongly increases upward in all the cores. The C/N ratio remains constant throughout all the samples, with a mean value of about 13, regardless of depth or location. All of the geochemical variables are at least approximately log-normally distributed; thus, log-log or semi-log scattergrams were used and the data were log-transformed before statistical calculations were performed. Some elements (Mn, Zn, Hg, Cu, and Ca) are primarily associated with the organic fraction; others (La, Cr, Sr, and Ba) are clearly related to the terrigenous fraction; others show affinities for both fractions. Consequently, no bivariate scattergrams or plots of ratio versus depth – commonly used for standardization by plotting or ratioing a reference element (such as Al) to an element of interest – were found to be adequate for standardization of this dataset. The best method for standardization was found to be one based on multivariate (trivariate) linear regression, using log Al and log C as the independent variables (reference elements representing terrigenous and organic fractions, respectively), and the log of the element of interest as the dependent variable. Residuals (deviations) from the best-fit linear surface were then plotted versus depth in the cores to accomplish the standardization. The results indicate that, with the possible exception of Mn at two sites, there is little evidence of anthropogenic input of trace elements to the lake, and most trace-element concentrations in the lake can be considered as valuable baseline information. A significant finding is that different and erroneous conclusions might have been reached if other standardization methods, not based on trivariate regression, had been employed. Received: 28 August 1997 · Accepted: 24 November 1997     

Color display of well logs

Information displayed in image form is conventionally at least two-dimensional. One-dimensional data can also be displayed as imagery by amplitude slicing a function and representing these amplitude slices as grey levels or as intensities of a particular color. An image having length and width can be formed from one-dimensional data by arranging a sequence of such color levels along the length of the image, corresponding to the independent space or time variable, and by extending these color levels arbitrarily along the width of the image. Such an image is of arbitrary width and exhibits changes, or information, only along its length. If three such images are made, one in each of the three primary colors, each image representing a different one-dimensional function over the same range of a common independent space or time variable, standard primary color composition can be used to unite these three images into a single full-color image in either computer graphic or photographic media. When three different well logs from a single borehole are combined in this manner, the resulting full-color image is equivalent to a three-axis cross-plot of the logs involved. Because primary colors appear to the human eye to be orthogonal and miscible, the color image preserves all information present in the three original data sets.Further, because this color representation of data can be arranged conveniently in depth sequence, the color composite image presents both information best visualized in cross-plot presentation (e.g., lithologic clustering and differentiation) and information lacking in cross-plots that is best seen in conventional wiggle-trace displays of well logs (e.g., stratigraphic or depth sequence). Such color presentations of well log data reduce the number of displays competing for an interpreter's attention by uniting three wiggle traces into one display and relieve the burden of mentally combining data from several separate displays. Combined interpretation is particularly essential in well log interpretation where one must analyze individually ambiguous data sets; color log presentation automatically provides combination while leaving interpretation to the scientist. Color log images also present log data to the user in a form that differs from conventional wiggle trace and so appeals to a different and richer set of human perceptions and pattern recognition skills.     

Reduction of lithologic-log data to numbers for use in the digital computer

The development of a standardized system for conveniently coding lithologic-log data for use in the digital computer has long been needed. The technique suggested involves a reduction of the original written alphanumeric log to a numeric log by use of computer programs. This numeric log can then be retrieved as a written log, interrogated for pertinent information, or analyzed statistically.publication authorized by Director, U.S. Geological Survey.     

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.     

On an application of cluster analysis for classification of valley cross profiles using special orthonormal functions

Morphometric approaches deal with obtaining quantitative parameters of relief forms. In particular, such parameters can be used for the classification of diverse relief forms. There is a noticeable demand for such approaches in geosciences. For example, the differentiation of valley forms belongs to the research field of geomorphology. A valley form includes important information with respect to the valley genesis. It is assumed that the U-shape probably corresponds to a glacier-induced valley, but the V-shape reflects the fluvial genesis of a valley. A prior generalisation of valley cross-profiles is necessary in order to obtain their morphometric parameters. These parameters contribute to the valley differentiation. The conventional generalisation approaches are based on the approximation of such cross-profiles using a special function. This function can be a catenary curve, a special polynomial or a set of polynomials. An alternative model is the GPL model. In this paper we propose a new generalisation approach, which uses special orthonormal functions. Based on these functions, form coefficients can be obtained for use in cluster analysis, resulting in the shape classification of valley cross profiles. An application of our approach for a real data set is presented.     

Compositional Data in Geostatistics: A Log-Ratio Based Framework to Analyze Regionalized Compositions

Problems with compositional data, like spurious correlation and negative bias, are well known in the Geosciences. Not so well known is the fact that the same problems appear when dealing with regionalized compositions. Here, these problems are illustrated, and a solution, based on the principle of working in coordinates using orthonormal logratio representations, is presented. This approach offers a tool for standard geostatistical studies. One of the advantages the method has is that it allows the usual inconsistencies with indicator kriging to be overcome through simplicial indicator kriging. A general way of modelling crossvariograms of coordinates, based on the matrix valued variation variogram, is discussed. In summary, the main aspects related to the modelling and analysis of regionalized compositions have had satisfactory solutions found for them. The proposed methodology is illustrated with public data from a survey concerning arsenic contamination in underground water in Bangladesh.

     

Element chemostratigraphy of the Devonian/Carboniferous boundary – A compositional approach

The Devonian/Carboniferous (D/C) boundary is a critical interval in the Phanerozoic history, which is associated with vigorous climatic perturbations, continental glaciation, global sea-level fall and rapidly increased extinction rates in marine realms. In many sections world-wide, these global changes left a marked lithological signature, in particular the Hangenberg black shale (products of deep-shelf anoxia) and the overlying Hangenberg sandstone (sudden siliciclastic influx into predominantly carbonate depositional environments). Both layers bear a distinct geochemical signature. Even though either or both of these two lithologies are absent at many sections, their correlative counterparts can be indicated by subtle geochemical markers. We studied elemental geochemistry of fourteen D/C boundary sections in six key areas across Europe with the aim to select globally correlatable elemental proxy for the D/C boundary. Analysis of raw/log-transformed geochemical data (EDXRF, c.p.s. units), presenting the standard approach here, indicates that concentrations of terrigenous elements (Al, K, Rb, Ti and Zr) are mainly controlled by diluted Ca (carried by marine calcium carbonate) in limestone facies and, accordingly, their variations can be related to carbonate production in the sea rather than to terrigenous input from continent. Nevertheless, due to the relative nature of geochemical observations, reliance solely on statistical processing of raw data might lead to incomplete picture of multivariate data structure and/or biased interpretations. For this reason, the aim of this contribution is to discuss the logratio alternatives of the standard statistical methods, which may better reflect the relative nature of the data. For this purpose, principal component analysis was employed to reveal main geochemical patterns and while the geochemical signature of the D/C boundary was further analysed using Q-mode clustering that leads to predicative orthonormal logratio coordinates – balances. The comprehensive picture of the multivariate data structure provided by these statistical tools makes them a primary choice for exploratory compositional data analysis. At the same time, it turns out that the standard and compositional approaches have synergic effects. This fact can be extensively used in further geochemical studies.     

A normalization-based watermarking scheme for 2D vector map data

Copyright protection of 2-dimensional (2D) vector map data is of great importance in the community of geographic information sciences. However, no blind, imperceptible and robust algorithm has been proposed by far. To fill the gap, this paper puts forwards a normalization-based watermarking algorithm for 2D vector map data: firstly, each coordinate of the original vector map data is normalized, and the watermark is shuffled; then the shuffled watermark is multiply embedded in the normalized vector data. After this, the coordinates of the watermarked data is restored (i.e. anti-normalized) for normal use. Once a suspicious copyright piracy appears, the watermarked data is re-normalized and the watermark can be detected and extracted. Our mathematical proofs and experiments have shown that the scheme owns four features that a good watermarking algorithm should possesses, i.e. usability, invisibility, blindness and robustness: the precision of the watermarked data can be guaranteed (usability); the users cannot perceive the embedded watermark (invisibility); the watermark can be extracted without knowing the original data and the original watermark; and the algorithm can resist a number of attacks such as translation, scaling, vertices reordering, vertex deletion, vertex insertion, vertex modification, cropping, and data format transformation (robustness).     

基于单个钻孔孔壁电视图像确定地下岩体结构面产状的普适数学方法

岩体结构面产状是影响岩体力学性质及工程稳定性的重要地质因素.从钻孔孔壁电视平展图上可较精确地提取结构面迹线上不共线三点的相对二维坐标(d, h),再利用空间解析几何理论确定其三维空间相对坐标(x, y, z),并求得结构面法向矢量.根据钻孔孔壁与坡面及结构面产状的空间几何关系,在保持原点不变的情况下,采用空间坐标变换的方法,将三维坐标系z轴和y轴分别转化为竖直方向z′和正北地理方向y′,并确定转化后坐标系空间的结构面法向矢量.再从结构面产状的定义出发,根据结构面法向矢量与结构面倾向、倾角的相关性,建立了一个空间几何模型,通过该模型可求得任意平直坡面上任一钻孔内局部岩体结构面的产状.      

Orthonormal residuals in geostatistics: Model criticism and parameter estimation

In linear geostatistics, models for the mean function (drift) and the variogram or generalized covariance function are selected on the basis of the modeler's understanding of the phenomenon studied as well as data. One can seldom be assured that the most appropriate model has been selected; however, analysis of residuals is helpful in diagnosing whether some important characteristic of the data has been neglected and, ultimately, in providing a reasonable degree of assurance that the selected model is consistent with the available information. The orthonormal residuals presented in this work are kriging errors constructed so that, when the correct model is used, they are uncorrelated and have zero mean and unit variance. It is suggested that testing of orthonormal residuals is a practical way for evaluating the agreement of the model with the data and for diagnosing model deficiencies. Their advantages over the usually employed standardized residuals are discussed. A set of tests are presented. Orthonormal residuals can also be useful in the estimation of the covariance (or variogram) parameters for a model that is considered correct.     

Porphyroblast rotation: eppur si muove*?

In a number of recent papers, the theory has been postulated that porphyroblasts as a rule do not rotate with respect to geographical coordinates, and can be used to determine the original orientation of older foliations. Complex inclusion patterns in spiral garnets have even been used to advocate a new model of orogenesis, involving several alternating phases of horizontal shortening and extension. Critical assessment of the assumptions and data used to support the theory of irrotational porphyroblasts reveals numerous flaws. Millipede structures, used as proof for flow partitioning, can also form by other flow geometries. Evidence quoted to support irrotational behaviour of porphyroblasts is unsound. Porphyroblasts do occur in sets with a preferred orientation of the internal foliation trace, but these cannot be shown to represent original orientations. Microstructures which resemble truncation planes in spiral garnets are used as evidence that these structures developed by several phases of deformation and as proof for periodic extension and horizontal shortening in orogenesis. They can, however, also be explained by intermittent growth of a rotating porphyroblast during a single phase of deformation. Finally, porphyroblast sets in which orientation is a function of aspect ratio indicate that porphyroblast rotation with respect to kinematic axes does occur in at least some situations.     

测井曲线剩余变化的分维描述火成岩裂缝发育情况

大庆深层火成岩段岩性复杂，偿仅具有从基性到酸性的各种熔岩，还包括不同喷发期次的凝灰岩。岩性变化是影响各种测井曲线响应特征变化的最主要因素。裂缝的响应往往被岩性的影响所淹灭。为此从测井曲线中提取分维来描述裂缝发育程度，必须考虑岩性的影响。为了消除岩性的影响，对分辨率较高且对裂缝反映灵敏的微球形聚焦测井曲线进行滤波处理，获得原始曲线与滤波曲线间的剩余变化，再从中提取分维，从而突出裂缝的影响。实际处理表明，从剩余曲线中提取的分数维，比从原始测井曲线中提取的分维能更好地指示出复杂火成岩地层裂缝发育情况。     

A robust model for pore-water chemistry of clayrock

The chemistry of pore water is an important property of clayrocks being considered as host rocks for long-term storage of radioactive waste. It may be difficult, if not impossible, to obtain water samples for chemical analysis from such rocks because of their low hydraulic conductivity. This paper presents an approach for calculating the pore-water compositions of clayrocks from laboratory-measured properties of core samples, including their leachable Cl and SO₄ concentrations and analysed exchangeable cations, and from mineral and cation exchange equilibria based on the formation mineralogy. New core sampling and analysis procedures are presented that reduce or quantify side reactions such as sample oxidation (e.g. pyrite) and soluble mineral dissolution (celestite, SrSO₄) that affect measured SO₄ concentrations and exchangeable cation distributions. The model considers phase equilibria only with minerals that are observed in the formation including the principal clay phases. The model has been used to calculate the composition of mobile pore water in the Callovo-Oxfordian clayrock and validated against measurements of water chemistry made in an underground research laboratory in that formation. The model reproduces the measured, in situ pore-water composition without any estimated parameters. All required parameters can be obtained from core sample analysis. We highlight the need to consider only those mineral phases which can be shown to be in equilibrium with contacting pore water. The consequence of this is that some conceptual models available in the literature appear not to be appropriate for modelling clayrocks, particularly those considering high temperature and/or high pressure detrital phases as chemical buffers of pore water. The robustness of our model with respect to uncertainties in the log K values of clay phases is also demonstrated. Large uncertainties in log K values for clay minerals have relatively small effects on modelled pore-water compositions.     

地质三维信息模型研究及其应用

地质三维模型研究与应用在地质构造研究、资源勘探与开发等领域具有十分重要的意义。针对地质勘探等原始离散数据采用UML建立数据静态结构模型,利于GIS空间数据库集中存储管理;根据原始数据的特征,采用矢量层面构模技术建立了三维层面模型,生成地层表面和地层层间实体的连续数据;三维可视化模型采用流行的OpenGL图形技术,实现了原始数据、特征层表面和地质体三维显示与交互式剖切操作等。基于组件开发技术建立了地质三维信息模型系统,成功应用于城市地质勘察三维建模与可视化分析,并指出可在三个层次对此模型进行重用与扩展,具有较高的推广应用价值。     

油田开发阶段储层参数精细评价

以胜索油田为例,详尽论述了油田开发阶段储层参数精细评价的思路和方向,包括测井资料预处理与标准化、不同开发时期储层四性参数变化分析、测井解释模型建立及参数求取、储层参数模糊评价。开发阶段应通过储层四性分析,利用生产测试资料和神经网络方法建立测井解释模型及处理,能有效提高解释度。精细处理结果及模糊数学方法的应用,准确地定量表征了储层的优劣,在油田剩余油挖潜中产生了较好的应用效果。     

Tracing the evolution and distribution of F and Cl in plutonic systems from volatile-bearing minerals: a case study from the Liujiawa pluton (Dabie orogen, China)

Fluorine and chlorine play an important role in magmatic differentiation, hydrothermal alteration, and related mineralization processes, but tracing their evolution in magmatic and especially plutonic systems is not an easy task. The F and Cl in melts can be estimated from F and Cl concentrations in minerals, provided that partitioning between minerals and melts are constrained. Based on available partitioning models between mineral/melt, mineral/fluid, and melt/fluid, a set of equations has been derived to determine F and Cl concentrations in melts from the compositions of amphibole, biotite, and apatite. The new calculation procedure has been applied to a plutonic system, the Liujiawa pluton, eastern Dabie orogen (China). Cl and F concentrations in amphiboles, biotites, and apatites from different rock types (gabbronorite, two-pyroxene diorite, clinopyroxene diorite, and hornblende gabbro) have been determined by electron microprobe. Most amphiboles show a negative correlation between log(Cl/OH) and Mg-number and a positive correlation between log(F/OH) and A-site occupation. Biotites from the gabbronorite and two-pyroxene diorite show a slight positive correlation between log(Cl/OH) and Mg, which is however not the case for the clinopyroxene diorite. Apatites from all the samples are rich in F and show negative correlations between Cl and F concentrations. In our case study, we demonstrate that the Cl concentration in melt remains approximately constant at 1,000–2,000?ppm over the major crystallization interval, but decreases strongly at near solidus temperatures as a result of fluid exsolution. The F concentration in melt remains nearly constant at ca. 2,000–3,000?ppm at high temperatures as well as near solidus conditions, indicating that it is not largely affected by fluid exsolution because of its strongly preferred incorporation into melt. Interestingly, the evolution of Cl and F concentrations in melt with magmatic differentiation is similar to that determined in volcanic systems, suggesting that the evolution of Cl and F in melts during crystallization and late magmatic stages at depth (plutonic systems) is similar to that observed in volcanic systems during decompression and degassing.