Decoupled isotopic record of ridge and subduction zone processes in oceanic basalts by independent component analysis

Abstract

[1] Isotopic variability in oceanic basalts indicates possible interactions among multiple mantle components or geochemical end-members. Beyond the standard principal component analysis, which has been used so far to identify mantle components, the relatively new independent component analysis is well suited for extracting independent features in multivariate compositional space. Radiogenic isotopic compositions of oceanic basalts from the Atlantic and South Indian oceans, including both mid-ocean ridge basalts (MORB) and ocean island basalts (OIB), show that two independent compositional vectors (referred to as independent components or ICs) account for most of the observed variations with three isotopic ratios of Pb (856 MORB and 781 OIB) or five isotopic ratios of Pb, Sr, and Nd (672 MORB and 597 OIB). In both cases, the first IC distinguishes OIB from MORB, while another maps the geographical distribution of a mantle component and in particular the DUPAL anomaly. This property shows that the two ICs indeed distinguish independent information and reflect two distinctive geodynamic processes, a feature which is not present in the conventional analysis of mantle isotopic variability. The first IC that distinguishes OIB from MORB is similar to the isotopic trend reproduced in the MORB-recycling model of Christensen and Hofmann (1994). The second IC that discriminates geographical distribution is characterized by simultaneous enrichment/depletion of Pb, Rb, and Nd relative to U-Th, Sr, and Sm, respectively, which can be explained by elemental fractionation associated with aqueous fluid-mineral reactions. These geochemical characteristics, together with the fact that most of the observed multidimensional isotopic space is spanned by the joint distribution of the two ICs, indicate independent but overlapping differentiation processes which mostly take place within the depleted mantle domain. They are likely to reflect ridge versus subduction zone processes, or melting versus interaction with aqueous fluid. We use the regional distribution of the second, “enriched” IC to redefine the DUPAL anomalous mantle and show that in addition to its Southern Ocean type locality, it also distributes itself broadly in the Northern Hemisphere.

1. Introduction

[2] One of the well-accepted concepts of mantle isotope geochemistry is that the isotope compositions of radiogenic elements (e.g., Sr, Nd and Pb) in terrestrial basalts can be broken down into individual mantle components. These components are inherited from geochemical reservoirs with a distinctive history, such as ancient residues of melting at ridge crests or recycled oceanic crust, and with characteristic isotopic properties resulting from long-term isolation. The geochemical nature and spatial distribution of these components are thought to provide key information on differentiation and convection of the Earth's mantle. For this reason, extensive efforts have been made to identify the geochemical mantle components present in mid-ocean ridge basalts (MORB) and ocean island basalts (OIB) [e.g., White, 1985; Zindler and Hart, 1986; Hofmann, 1997].

[3] Principal component analysis (PCA) has been regarded as the most efficient way to identify these mantle components [e.g., Zindler et al., 1982; Allègre et al., 1987; Blichert-Toft et al., 2005]. Principal components are those linear combinations of observables with the largest possible variance. As will be shown later, however, the core assumption of PCA, which holds that the data constitute a single multivariate Gaussian population, is clearly invalidated for the isotope compositions of oceanic basalts. In this case, the principal components (PCs) do not form a true base, i.e., a set of independent vectors describing uniquely the isotopic variability. A promising tool for the analysis of geochemical mantle components is independent component analysis (ICA), which has been established in Information Science over the past ∼15 years (e.g., see the textbooks by Hyvärinen et al. [2001] and Amari [2002]). As with PCA, the core assumption is that the data can be accounted for by a linear combination of mutually independent components, but without the condition that the population is unique with a multivariate Gaussian distribution. ICA deconvolves a data set into independent components (ICs) by finding the directions that maximize the non-Gaussianity through criteria such as a higher-order cumulant or negentropy (see below) of the projected data distribution [Hyvärinen et al., 2001]. A caveat is that the term “component” as used for both PCA and ICA refers to a vector or a direction, which unfortunately conflicts with the well-entrenched denomination of geochemical mantle components. In order to avoid such confusion, we hereafter use “component” for describing the statistical distribution in both PCA and ICA and “end-member” for the geochemical mantle components with specific compositions, such as Depleted MORB Mantle (DMM).

[4] The relevance of statistical distributions, typically normal versus lognormal, underlying elemental and isotopic data on oceanic basalts has been discussed by different authors under different perspectives. Allègre and Lewin [1995] investigated different geochemical properties in basalts and concluded that the underlying distributions could be normal, lognormal, fractal or multifractal. Meibom and Anderson [2004] conjured up the central limit theorem to argue that the complexity and multiplicity of melting and mixing events in the mantle lead to near-Gaussian histograms. Rudge et al. [2005] argued that isotopic ratios are not normally distributed and analytically elaborated statistical distributions from melting-recycling models. They derived expressions for the higher central moment and particularly the skewness of the distributions of isotopic ratios, which provides a background theory for non-Gaussian histograms. These authors only considered models with linearized radioactive decay, a constraint later released by Rudge [2006]. By examining databases, Albarède [2005] used statistical tests (notably quantile-quantile plots and comparison between corresponding means and modes) to show that normal distributions do not do justice to actual histograms of isotopic ratios in MORB. We will see in particular that the non-Gaussianity of isotope distributions appears nowhere more clearly than in two-dimensional diagrams of Pb-isotopic ratios in oceanic basalts.

[5] In this study, we examined the compositional space of a maximum of six isotopic ratios (²⁰⁴Pb/²⁰⁶Pb, ²⁰⁷Pb/²⁰⁶Pb, ²⁰⁸Pb/²⁰⁶Pb, ⁸⁷Sr/⁸⁶Sr, ¹⁴³Nd/¹⁴⁴Nd, ¹⁷⁷Hf/¹⁷⁶Hf) with an algorithm known as FastICA [Hyvärinen, 1999] for oceanic basalts from the Atlantic and South Indian oceans. The data set includes both MORB from the literature [Agranier et al., 2005; Meyzen et al., 2005, 2007, and references therein] (most of which can be found in the PetDB database http://www.petdb.org) and OIB from GEOROC database (http://georoc.mpch-mainz.gwdg.de/georoc/). On the basis of the geochemical characteristics of the two ICs in the five-dimensional space of Pb-Sr-Nd isotopic ratios, the origin of isotopic heterogeneity and the differentiation processes of the mantle are discussed. The detected ICs naturally leads to redefinition of the DUPAL anomaly, showing that the enriched signature distributes broadly into the Northern Hemisphere.

2. Principles of Independent Component Analysis

[6] Independent component analysis (ICA) is a statistical and computational technique designed for revealing the hidden sources and factors that underlie the distribution of multivariate observations [Hyvärinen et al., 2001]. In this model, the observed multivariate data are assumed to be linear mixtures of unknown latent variables. No assumption about the specific processes by which these variables mix is made. Contrary to principal components, the latent variables are required to be mutually independent but do not form a multivariate Gaussian distribution. They are referred to as independent components (ICs), or equivalently as the sources or factors of the observed data. The core concepts and principles of ICA will now be described. To avoid ambiguity, let us first review the definition of “independent” versus “correlated” variables. Two random variables X and Y are independent if their joint probability density function (pdf), f_XY(x, y), can be factored as the product of two pdfs of the single variables X and Y, i.e.,

[e.g., Hamilton, 1964]. Any distribution of n normal variables can always be decomposed in the linear combination of n independent variables. Noting E the expectation (mean value), two variables X and Y are correlated if their covariance

and the associated correlation coefficient are different from zero. From these equations, it follows that independent variables are not correlated, whereas uncorrelated variables are not necessarily independent. As a special case, uncorrelated Gaussian variables are independent, but this does not in general hold for non-Gaussian variables. Let us use a simple example to demonstrate that the PCs of non-Gaussian data are uncorrelated but may not be independent [Hyvärinen et al., 2001; Amari, 2002]. Figure 1a shows the joint distribution of two independent variables, s₁ and s₂, with uniform probability density in the range of s₁ ∈[−1,1] and s₂ ∈ [−1,1]. Clearly the original variables, s₁ and s₂, are independent and their correlation coefficient is zero. Let us mix s₁ and s₂ to produce two mixed variables x₁ = 2(s₁ + s₂) and x₂ = (s₂ − s₁) as in Figure 1b. In this case, the PCs lie along the diagonal axes, x₁ and x₂, which maximize the variance of the data distribution projected on each axis. Since the expectation E[x₁x₂] = E[(s₂² − s₁²)/2] = 0, x₁ and x₂ are uncorrelated. However, it is clear that x₁ and x₂ (hence the two PCs) are not independent: when x₁ departs from the mean value, we recognize that the modulus of x₂ becomes smaller, and therefore information on x₂ can be extracted from x₁. ICA achieves the goal of extracting the independent components, s₁ and s₂, from the multivariate data set as follows.

Figure 1

A simple example showing joint non-Gaussian distribution and the corresponding marginal probability density. (a) The joint distribution of two independent components, s₁ and s₂, with uniform probability density in the range of s₁ ∈ [−1, 1] and s₂ ∈ [−1, 1]. The diagonal axes, x₁′ and x₂′, correspond to linear transformation (i.e., whitening) of mixed variables x₁ and x₂ in Figure 1b. (b) The two variables x₁ = 2(s₁ + s₂) and x₂ = (s₂ − s₁) are produced by mixing of s₁ and s₂. The two nonorthogonal variables s₁′ and s₂′ correspond to the transformed (i.e., dewhitened) s₁ and s₂ in Figure 1a. (c) The marginal probability density, p, corresponding to the joint distribution shown in Figure 1a: p_s1(s₁) (probability density projected on s₁, red line) and p_x1′(x₁^′) (that projected on x₁′, green line) are plotted, together with a Gaussian distribution (black line) for reference.

[7] First, the observed multivariate variables and data (e.g., x₁ and x₂ in Figure 1b), which are assumed to be linear mixtures of unknown independent variables, are centered and scaled by the standard deviations along the PCs. This procedure is called “whitening” [e.g., see Hyvärinen et al., 2001, chapter 6], and transforms the variables and the data in Figure 1b to those in Figure 1a where the transformed variables x₁′ and x₂′ and the two PCs lie along the diagonal axes. In this whitened space of Figure 1a, any orthogonal pair of two variables, including x₁′ and x₂′, are uncorrelated but not necessarily independent. ICA searches for the independent pair from these orthogonal pairs by maximizing non-Gaussianity instead of maximizing variance as for PCA. Figure 1c shows the marginal probability density corresponding to the joint distribution shown in Figure 1a: p_s1 (s₁) (probability density projected on s₁, red line) and p_x1′ (x₁^′) (that projected on x₁′, green line) are plotted, together with a Gaussian distribution (black line) plotted for reference. It shows that p_s1(s₁) deviates significantly from the Gaussian distribution (large non-Gaussianity), whereas p_x1′ (x₁^′) deviates to a lesser extent (smaller non-Gaussianity). In the whitened space, the independent components have the maximum non-Gaussianity of all the possible sets of uncorrelated components. This is because, according to the central limit theorem, random mixing of non-Gaussian variables approaches Gaussian more than the original variables. Therefore, a linear combination of s₁ and s₂ (i.e., Σ_ia_is_i, where a_i are the mixing coefficients), such as x₁′ or x₂′ in Figure 1a, are closer to Gaussian than s_i.

[8] In turn, a linear combination of the observed mixture variables (e.g., Σ_ib_ix_i′, where b_i are the mixing coefficients) will be maximally non-Gaussian if it equals to one of the independent components [Hyvärinen et al., 2001, chapter 8]. In the example of Figure 1a, x₁′ and x₂′ are rotated around the center to find the ICs (i.e., s₁ and s₂) that give maximum non-Gaussianity. Then the ICs can be linearly backtracked to the original space (e.g., s₁′ and s₂′ in Figure 1b). The two ICs are therefore nonorthogonal in the original space (Figure 1b), which contrasts with the orthogonal relationship between PCs. Independent components may be orthogonal, but on the condition that the variables are uncorrelated Gaussian variables. In such a case, however, non-Gaussianity is zero for all the components and ICA cannot extract a unique set of ICs. Nonorthogonal ICs in the original space and oblique PCs with respect to ICs therefore reflect the non-Gaussian character of the observed data. We will show the data distribution and the extracted ICs in both original and whitened spaces in the following analyses of the oceanic basalts.

[9] Whitening removes correlation from the original data set and also determines the proportion of the total variance that the components account for. As is commonly assumed in both PCA and ICA, components which account for a small proportion of the variance are judged to be unimportant signals. In this study we follow this conventional procedure to determine the number of ICs, although ICA can potentially extract ICs as many as the number of the observed variables based on non-Gaussianity criteria.

[10] At this point, application of ICA to the isotopic space of oceanic basalts is straightforward: first, the observed isotopic compositions are centered and scaled by the standard deviations along the PCs (i.e., whitened), and second the original axes are rotated until the projection of the whitened data gives histograms with maximum non-Gaussianity. Such axes correspond to a set of independent compositional base vectors that create the observed compositional space. Entropy H is an information theory parameter which characterizes the randomness and lack of structure of a random variable Y with pdf f_Y(y) and defined as:

where is the domain of definition of the variable Y. Normal variables have the largest entropy of all the real-valued random variables with the same mean and variance [Hyvärinen, 1999]. In this study, the non-Gaussianity is measured by the negentropy, which is the difference between the entropy of the observed measurements and that of a normal variable with the same mean and variance. Negentropy J(y) is approximated by

where y is the whitened and projected data, c is an arbitrary constant, ν is the Gaussian variable of zero mean and unit variance, and the contrast function G(y) = exp(−ay²/2) with a constant a is selected among the choices suggested in the FastICA algorithm to hold robustness against outliers [Hyvärinen, 1999]. J(y) is zero if y is exactly Gaussian, and increases as y deviates from Gaussian. It should be noted that, unlike principal components, the independent components cannot be ranked according to the proportion of variance they account for. Because the ICs are independent, there is no common measure of their respective importance. The FastICA software package for MATLAB/Octave is available at http://www.cis.hut.fi/projects/ica/fastica/.

3. Independent Components of Isotopic Compositions of Oceanic Basalts

[11] In total, 1637 data sets are available with the three Pb isotopic ratios (856 MORB and 781 OIB), while 1269 data sets are available with the five ratios of Pb, Sr and Nd (672 MORB and 597 OIB), and 461 data sets exist for all six ratios (393 MORB and 68 OIB) (Figure 2). First, we examine the three Pb-isotopic ratios, for which mixing of different mantle geochemical end-members forms a truly linear trend suitable for ICA, as a large number of high-quality data recently have become available. Because counting statistics and thermal noise affect small signals much more so than strong signals, couples of isotopic ratios with a minor or otherwise noisy isotope as the denominator, such as ²⁰⁶Pb/²⁰⁴Pb, ²⁰⁷Pb/²⁰⁴Pb, and ²⁰⁸Pb/²⁰⁴Pb, are strongly correlated. Albarède et al. [2004] calculated the correlation coefficient between ²⁰⁶Pb/²⁰⁴Pb and ²⁰⁷Pb/²⁰⁴Pb introduced by Poisson counting statistics on ²⁰⁴Pb variables and found it equal to 0.96. Such a correlation of purely analytical origin interferes with geochemical correlations, e.g., because of the mixing of mantle geochemical end-members. The effect of analytical error correlations is particularly critical because the range of ²⁰⁷Pb/²⁰⁴Pb variations cannot be neglected with respect to analytical errors. In contrast, the correlation coefficient between ²⁰⁴Pb/²⁰⁶Pb and ²⁰⁷Pb/²⁰⁶Pb due to counting statistics is only 0.16, which makes the correlation between counting uncertainties essentially negligible with respect to those of more geochemical significance. The normalization to ²⁰⁶Pb is routinely used for early Solar System chronology and the parameters of the ²⁰⁷Pb/²⁰⁶Pb versus ²⁰⁴Pb/²⁰⁶Pb isochron are easily derived from those of the more conventional ²⁰⁷Pb/²⁰⁴Pb versus ²⁰⁶Pb/²⁰⁴Pb isochron [Tera and Wasserburg, 1972] (see Appendix A). For these reasons, we use ²⁰⁶Pb as the denominator for the Pb isotopic ratios, although the detected ICs are almost identical in both ²⁰⁶Pb- and ²⁰⁴Pb-normalized spaces (Appendix A and Figure A1).

Figure 2

Index map showing the distribution of oceanic basalts used in the analysis. The four colors correspond to the four geographical regions: north of 47°N (black), intermediate latitude between 47°N and 35°S in the Atlantic Ocean (red), south of 35°S in the Atlantic Ocean (green), and Indian Ocean (blue). Crosses correspond to mid-ocean ridge basalts, while solid dots correspond to ocean island basalts. These colors and symbols are used in Figures 3–5.

[12] The principal components are determined for reference (Figure 3). The first component, PC1 (97.3% of the population variance), corresponds to the longest axis of the overall data distribution as it is defined to give the maximum variance of the projected data. However, the second component, PC2 (2.7% of the population variance), is not useful in describing the data, either individually or in groups. These features are similar to those determined for North Atlantic MORB [Blichert-Toft et al., 2005]. Two independent components (ICs), i.e., independent compositional base vectors to represent the observed compositional space, also cover 99.9% of the population variance, but are clearly oblique with respect to the PCs (Figure 3a).

Figure 3

Independent components (compositional vectors/lines labeled IC1 and IC2) in the ²⁰⁴Pb/²⁰⁶Pb-²⁰⁷Pb/²⁰⁶Pb-²⁰⁸Pb/²⁰⁶Pb system, for 1637 data sets (856 MORB and 781 OIB data). The ICs have been calculated according to the FastICA algorithm based on the approximation of negentropy as a measure of non-Gaussianity [Hyvärinen, 1999]. Principal components (PC1 and PC2) are also shown. The isotopic data are compiled from literature (the PetDB database, Agranier et al. [2005], and Meyzen et al. [2005, 2007]) for MORB and the GEOROC database for OIB. From the GEOROC database, basaltic rocks with SiO₂ contents between 53 and 35 wt % were selected. (a) ²⁰⁴Pb/²⁰⁶Pb-²⁰⁸Pb/²⁰⁶Pb plot of the data, ICs and PCs. (b) IC1-IC2 plot where the data and the vectors are decomposed into the two ICs. In both diagrams, the data distribution is sharply cut off at the edges, with several internal subgroups or array-like structures. These characteristics clearly show a strong non-Gaussianity of the data distribution. In each figure, the labels “IC1” and “IC2” are placed along the positive axes of ICs. Widths of the ICs in Figure 3a represent the ranges obtained by perturbing the data points with their analytical uncertainties (1σ for ²⁰⁴Pb/²⁰⁶Pb, ²⁰⁷Pb/²⁰⁶Pb and ²⁰⁸Pb/²⁰⁶Pb has been estimated to be 1800, 800 and 800 ppm, respectively, on the basis of the maximum uncertainties reported in the references of the data sources). The distribution in slope of the perturbed ICs shows a normal distribution, and the ranges shown in Figure 3a approximately correspond to 3σ of the distribution. The colors and symbols are the same as in Figure 2. Approximate locations of conventional mantle geochemical end-members are also shown by light grey symbols: asterisk, HIMU [Zindler and Hart, 1986]; circle, FOZO [Hart et al., 1992; Stracke et al., 2005] or C [Hanan and Graham, 1996]; plus, depleted MORB mantle (D-DMM) [Workman and Hart, 2005]; square, EM-I [Zindler and Hart, 1986]; cross, EM-II [Zindler and Hart, 1986].

[13] This feature is similar to Figure 1b in the simple example for homogeneous joint distribution. Although the overall elongation of data distribution in the original space obscures the relationship between the PCs and ICs, the obliquity is obvious in the whitened space (Figure 3b), where the data points and vectors are broken down into the two ICs. The obliquity between PCs and ICs always occurs when the data constitute a multivariate non-Gaussian population. Figure 3b is similar to Figure 1a (homogeneous joint distribution of the two independent components in the whitened space) in terms of the overall data distribution and the two PCs close to the diagonal axes, indicating that the observed isotopic data is clearly non-Gaussian.

[14] In Figure 3b, most OIB lie in the field of positive IC1 values, except for Iceland, which is situated on a spreading ridge, whereas most MORB have negative IC1, except for the plume-influenced ridge basalts [e.g., Schilling, 1973; White et al., 1976; Hanan et al., 1986]. In contrast, IC2 discriminates the geographical distribution: most of the basalts from northern and central latitudes have negative IC2 values, while basalts from the South Atlantic and Indian Ocean mostly plot in the field of positive IC2 values. Such a clear separation indicates that, contrary to the PCs, the two ICs may be independent and reflect the effect of two separate geodynamic processes to span most of the observed compositional space. On the other hand, although a linear trend nearly parallel to IC2 through EM-I (enriched mantle 1 [Zindler and Hart, 1986]), FOZO (focal zone [Hart et al., 1992]) or C (common component [Hanan and Graham, 1996]), and HIMU (high-U/Pb mantle [Zindler and Hart, 1986]) roughly limits the data distribution on the positive-IC1 margin, some data lie outside the polyhedron defined by connecting the conventional mantle geochemical end-members, especially for negative IC1 values. This indicates that mixing of the end-members cannot fully explain the observed compositional space. An alternative interpretation of Figure 3b will be proposed later.

[15] As a next step, the five-dimensional isotopic space with Pb, Sr and Nd was explored using ICA, and again two dominant components were identified, which together account for 97.7% of the population variance. An additional 1.9% of the total variance is covered by inclusion of the third component. It is not clear whether such a small share actually indicates the presence of an additional geochemical component or reflects the nonlinear character of mass balance relationships in 5 dimensions including Sr and Nd. The inclusion of three components creates metastable solutions that locally maximize non-Gaussianity, indicating that the major robust features are adequately represented by the two ICs shown in Figure 4.

Figure 4

Independent components (IC1 and IC2) in the ²⁰⁴Pb/²⁰⁶Pb-²⁰⁷Pb/²⁰⁶Pb-²⁰⁸Pb/²⁰⁶Pb-⁸⁷Sr/⁸⁶Sr-¹⁴³Nd/¹⁴⁴Nd system, for 1269 data sets (672 MORB and 597 OIB data). For the data sources and ICA procedure, see Figure 3. (a–d) The plot of data and ICs in the original space. (e) The IC1-IC2 plot where the data and PCs are decomposed into the two ICs. In each figure, the labels “IC1” and “IC2” are placed along the positive axes of ICs. Widths of the ICs in Figures 4a–4d represent the ranges obtained by perturbing the data points with their analytical uncertainties (1σ for ²⁰⁴Pb/²⁰⁶Pb, ²⁰⁷Pb/²⁰⁶Pb, ²⁰⁸Pb/²⁰⁶Pb, ⁸⁷Sr/⁸⁶Sr and ¹⁴³Nd/¹⁴⁴Nd has been estimated to be 450, 200, 200, 40 and 50 ppm, respectively, on the basis of those reported in the references of the data sources). The distribution in slope of the perturbed ICs shows a normal distribution, and the ranges shown in Figures 4a–4d approximately correspond to 3σ of the distribution. The colors and symbols are the same as in Figure 3, with an additional symbol: light grey triangle, average depleted MORB mantle (average DMM) [Workman and Hart, 2005]. The light grey dashed line in Figure 4c represents the trend reproduced in the MORB-recycling model of Christensen and Hofmann [1994].

[16] This result, i.e., the presence of only two major components covering more than 90% of the population variance, has already been shown by PCA for oceanic basalts [Zindler et al., 1982; Allègre et al., 1987; Hart et al., 1992], although the ICs are different from the PCs (Figure 4e). The two ICs in the space with five isotopic ratios are essentially the same as in Figure 3: IC1 separates OIB from MORB, and IC2 discriminates geographical distribution in both the OIB and MORB fields. We checked that essentially the same result is obtained for the subspace with only the three variables ²⁰⁴Pb/²⁰⁶Pb, ⁸⁷Sr/⁸⁶Sr and ¹⁴³Nd/¹⁴⁴Nd and with the four variables of the three Pb plus Nd isotopic ratios, which shows the robustness of the two ICs.

[17] The conventional mantle geochemical end-members do not span the entire space of observed compositions (Figure 4e). In addition, in the original space (e.g., Figures 4a–4d), some of the conventional end-members significantly deviate from the two-dimensional plane spanned by the two ICs. Note that, in Figures 3–5, the labels “IC1” and “IC2” are placed along the positive axes of ICs. EM-I and EM-II, which refer to those suggested by Zindler and Hart [1986] on the basis of extensive extrapolation of the data trends, plot in the field with positive IC1 and positive IC2, but do not plot in the same field in Figure 4d (and Figure 4b for EM-I). Assuming a ¹⁴³Nd/¹⁴⁴Nd for EM-I, its ⁸⁷Sr/⁸⁶Sr should be more radiogenic (by ∼0.002) if it is to be consistent with the actual data (Indian MORB or OIB) and the IC axes, i.e., to plot in the IC plane. Similarly, EM-II should correspond to lower ⁸⁷Sr/⁸⁶Sr values (by ∼0.002) so as to plot in the IC plane. In contrast, HIMU, FOZO/C and DMM plot consistently in all the diagrams: HIMU and FOZO/C plot in the field of positive IC1 and negative IC2, while DMM (D-DMM and A-DMM) falls along the IC1 axis. These end-members plot within the actual data or on a slight extension of the actual data trends and therefore plot in the two-dimensional IC plane, which accounts for 97.7% of the population variance. A slight deviation from the plane, however, distorts the data plot in the view nearly parallel to the IC plane: in Figure 4d, some of the Indian OIB (blue dots) plot slightly above the plane toward higher ⁸⁷Sr/⁸⁶Sr and ¹⁴³Nd/¹⁴⁴Nd values, which makes them appear closer to the IC2 axis.

Figure 5

Independent components (IC1 and IC2) in the ²⁰⁴Pb/²⁰⁶Pb-²⁰⁷Pb/²⁰⁶Pb-²⁰⁸Pb/²⁰⁶Pb-⁸⁷Sr/⁸⁶Sr-¹⁴³Nd/¹⁴⁴Nd-¹⁷⁷Hf/¹⁷⁶Hf system, for 461 data with the complete six ratios (393 MORB and 68 OIB data sets). For the data sources and ICA procedure, see Figure 3. (a–f) The plot of data and ICs in the original space. (g) The IC1-IC2 plot where the data and PCs are decomposed into the two ICs. In each figure, the labels “IC1” and “IC2” are placed along the positive axes of ICs. Widths of the ICs in Figures 5a–5f represent the ranges obtained by perturbing the data points with their analytical uncertainties (1σ for ²⁰⁴Pb/²⁰⁶Pb, ²⁰⁷Pb/²⁰⁶Pb, ²⁰⁸Pb/²⁰⁶Pb, ⁸⁷Sr/⁸⁶Sr, ¹⁴³Nd/¹⁴⁴Nd and ¹⁷⁷Hf/¹⁷⁶Hf has been estimated to be 450, 200, 200, 40, 50 and 35 ppm, respectively, on the basis of those reported in the references of the data sources). The distribution in slope of the perturbed ICs shows a normal distribution, and the ranges shown in Figures 5a–5f approximately correspond to 3σ of the distribution. The colors and symbols are the same as in Figure 4.

[18] We also explored the six-dimensional space by adding Hf to the previous isotopic systems (Figure 5). Again two dominant components are identified, which together account for 95.2% of the population variance. However, in this case, the two calculated ICs are different from those of Figures 3 and 4, and are closer to the PCs (Figure 5g). The clear MORB-OIB separation and geographical discrimination are lost in the process. The geometries of data distribution and two ICs in the diagrams of Pb-Nd isotopic ratios (Figure 5c) and Pb-Hf isotopic ratios (Figure 5d) are similar, which is consistent with a high correlation between Nd- and Hf-isotopic ratios with almost identical slopes of the two ICs (Figure 5f). Also, most of the population variance (95.2%) is covered by the two components, which is similar to the result in the five-dimensional space with Pb-Sr-Nd isotopic ratios in Figure 4. A remarkable difference between Figures 4 and 5 lies with the number of data: because Hf isotope analysis has only recently become routine, the number of MORB data (393) far exceeds that of OIB data (68) (Figure 5), which contrast with the more balanced situation when Hf is omitted with 672 MORB and 597 OIB data (Figure 4). This imbalance significantly modifies the overall as well as internal structures of the data distribution, and results in the different ICs in Figure 5 relative to those in Figure 4. A larger number of data sets with the complete six isotopic ratios is nevertheless required to judge whether, as argued by Salters and Hart [1991] and Blichert-Toft et al. [2005], the Hf-isotopic ratio contains statistically unique information distinct from the information conveyed by the Pb-Sr-Nd systems.

4. Discussion

[19] Since ICA uses non-Gaussianity criteria throughout, a preliminary question is whether the data can safely be considered as nonnormal. In spite of convective stirring, the mantle is not homogeneous because new heterogeneities are continuously created by melting at ridges and subduction of the oceanic plates. As shown by Rudge et al. [2005], such a regime tends to a steady state, and the resultant distribution of geochemical variables are skewed and non-Gaussian.

4.1. Origin of Geochemical Independent Components

[20] The conventional view of mantle geochemical end-members is that their compositions are unique, although such compositions are only loosely constrained, namely by using either extreme data or an extension of observed trends where no actual datum exists. There is room, therefore, for shifting the composition of some mantle geochemical end-members to make them suitable as components of the ICs, as was discussed for EM-I and EM-II in the previous section. By contrast, the present study suggests that the literature mantle geochemical end-members may not necessarily represent unique compositions. The essential feature of Figure 3b and Figure 4e is that the variations along the two IC directions are created by two independent processes. When these two processes overlap, they will create the observed compositional variability. The ICA shown in Figures 3 and 4 suggest that the two components that have been identified are independent, which we take as an indication that they were created by distinct geodynamic processes. Since MORB and OIB roughly are symmetrically distributed around a depleted mantle composition in the IC space (although it is slightly more enriched compared to the average depleted MORB mantle (A-DMM) [Workman and Hart, 2005] as in Figure 4e), we argue that these processes occur as two differentiation processes mostly within the depleted mantle domain. The dual structure of the geochemical data demonstrated by ICA is robust and suggests that it reflects the two dominant geodynamic processes with distinct elemental fractionation processes in the Earth, the ridge and subduction zone processes. In addition, the overlap of two independent components reflects mutual processing: products from ridge activity are processed at subduction zones and ridges potentially remelt materials recycled through the subduction zones.

[21] Segregation and long isolation of MORB/eclogite from its harzburgitic residue and subsequent recycling can reproduce a broad trend in the Pb and Nd isotope space observed in oceanic basalts [Christensen and Hofmann, 1994], whose slope is similar to the IC1 direction. For example, in Figure 4c (²⁰⁴Pb/²⁰⁶Pb versus ¹⁴³Nd/¹⁴⁴Nd), the isotopic variation produced in the MORB-recycling model, which is slightly curved because of the faster rate of ²³⁸U decay relative to ¹⁴⁷Sm [Christensen and Hofmann, 1994], is nearly parallel to IC1. The slopes of IC1 in Figures 4a–4d suggest that IC1 originates from elemental fractionation associated with simultaneous increases (or decreases) in U/Pb, Th/Pb, Rb/Sr and Nd/Sm, which is consistent with that associated with melting [e.g., Beattie, 1993; Green, 1994; Salters and Longhi, 1999]. Although the range and slope of the trend depend on several physical parameters such as density contrast between MORB/eclogite and peridotite, the MORB/eclogite-rich portion can form a source region of OIB, whereas the residual harzburgitic portion represents DMM [Christensen and Hofmann, 1994]. These features coincide with those of IC1, separating OIB from MORB.

[22] What IC2 reflects in contrast is likely to be subduction zone processes. In Figures 4b and 4c, the slope of IC2 has an opposite sign to that of IC1 and MORB-recycling trend that reflects elemental fractionation associated with melting, while both IC1 and IC2 exhibit the same sign in Figures 3a, 4a, and 4d. These features indicate that IC2 originates from elemental fractionation with simultaneous increases (or decreases) in Pb/U, Pb/Th, Rb/Sr and Nd/Sm. Such fractionation can occur associated with aqueous fluid-mineral reactions [Brenan et al., 1995; Keppler, 1996; Kogiso et al., 1997], and therefore suggests that aqueous fluid processes in subduction zones create the IC2 variation superimposed on the IC1 variability.

[23] Dehydration of the subducted oceanic crust concentrates more Pb, Rb and Nd in the aqueous fluid than U-Th, Sr and Sm, respectively, leading to a negative IC2 value of the dehydrated rocks, while hydration of the rocks can produce positive IC2. Clear differences between IC1 and IC2 also exist in relative magnitude of fractionation between the different parent/daughter pairs. Figures 3a, 4a, and 4d show that a differentiation process responsible for the IC2 variation fractionates U/Pb more than Th/Pb, and Rb/Sr more than Nd/Sm when compared to IC1. These differences can be explained by some of the experimental results on elemental partitioning among aqueous fluids, silicate melts and minerals [e.g., Beattie, 1993; Green, 1994; Brenan et al., 1995; Keppler, 1996; Kogiso et al., 1997; Salters and Longhi, 1999], although they are not readily inferred consistently from all the experimental data, since the partition coefficients significantly vary depending on a number of parameters, such as pressure, temperature, bulk composition, oxygen fugacity, alkali-chloride contents in fluids, mineralogy of the coexisting solid, and experimental configurations (e.g., static equilibrium or dehydration mobility experiments). However, considering the robust feature with simultaneous increases (or decreases) in Pb/U, Pb/Th, Rb/Sr and Nd/Sm, and that it must reflect a first-order differentiation process within the depleted mantle domain as dominant as ridge melting, we propose that IC2 is related to aqueous fluid processes in subduction zones as follows.

[24] During subduction of oceanic plates, altered MORB and oceanic mantle generate aqueous fluids and leave residues of dehydration. Fluids migrate upward and hydrate the overlying mantle wedge that may also contain recycled MORB and residual rocks circulated by corner flow, and cause flux melting and arc magmatism [Iwamori, 2007]. Hydration and dehydration at subduction zones of both the basaltic and refractory parts of the oceanic lithosphere and in the mantle wedge accounts for the overprinting of the IC1 geochemical variability by a distinct IC2 signature. The ubiquitous presence of this IC2 component in the source of both OIB and MORB (Figures 3b and 4e) simply reflects that most of the subducted oceanic lithosphere and the overlying mantle wedge go through the hydration-dehydration processes at subduction zones [Iwamori, 2007], in agreement with Li isotope evidence [Elliot et al., 2006].

[25] Melting in subduction zones also contributes to mantle differentiation, producing the continental crust and a residue [Tatsumi, 2000]. In terms of the nature of elemental differentiation, which reflects the effect of mineral/melt partitioning, the effect of melting in subduction zones may be similar to that at mid-ocean ridges. In fact, EM-II, which is analogous to continental crustal material recycled in the mantle [White and Duncan, 1996], has a positive IC1 and IC2 value (Figures 3b and 4e), reflecting both melting and aqueous fluid processes in subduction zones. Therefore, IC1 probably reflects melting processes, whereas IC2 reflects interaction of the mantle with aqueous fluid, regardless of the geodynamic sites where these processes take place. It is unclear at this stage how exactly the two elementary processes are separated in the IC space, partly because the aqueous fluid and melting processes are strongly coupled in subduction zones, as is suggested by numerical modeling of H₂O transportation and melting in subduction zones [Iwamori, 1998], trace element modeling of aqueous fluid, peridotite and island arc basalts [Ayers, 1998], and (²³¹Pa/²³⁵U)-(²³⁰Th/²³⁸U) variations of arc lavas [Thomas et al., 2002].

4.2. Geodynamic Implications

[26] The nature and possible origin of the ICs have several geodynamic implications. The existence of FOZO, or C, and the local trends toward it [Hart et al., 1992; Hanan and Graham, 1996] have been interpreted as representing mixing between the ambient mantle and a plume, or a diapir, rising from the core-mantle boundary [Hart et al., 1992] or the transition zone [Hanan and Graham, 1996]. In the IC space, FOZO, or C, is characterized by a positive IC1 and negative IC2 value (Figures 3b and 4e), corresponding to the melt component-rich (e.g., MORB/eclogite-rich) portion that has experienced dehydration in subduction zones, e.g., subducted oceanic crust. Because of its high density and viscosity, the subducted oceanic crust can be segregated from the subducted oceanic mantle and accumulated near the base of a convecting system for subsequent prolonged isolation [Christensen and Hofmann, 1994; Karato, 1997] to develop the isotopic characteristics suitable for FOZO or C. Its prevailing nature as a common source of oceanic basalts can be explained by the significant volume of subducted oceanic crust present in the mantle. However, the local trends toward FOZO or C, which are in general oblique to the ICs (Figures 3b and 4e), are minor structures within the IC space. Likewise, ICA shows no evidence for incorporation of what could represent a primordial mantle component or an early enriched reservoir into the observed depleted domain, which is consistent with ¹⁴²Nd evidence [Boyet and Carlson, 2005]. The lack of a clear correlation between Pb-isotopic compositions and ³He/⁴He in oceanic basalts [Hanan and Graham, 1996] also suggests that the incorporation of primordial components is largely decoupled from the major mantle processes identified by ICA.

[27] Finally, the detected ICs call for a redefinition of the criteria used to identify the DUPAL anomaly. Hart [1984] defined three criteria on the basis of Sr- and Pb-isotopic ratios and their deviations from a Northern Hemisphere reference line. Although the overall distribution of the contours clearly shows the anomaly in the Southern Hemisphere, the three types of contours are not consistent with each other, especially in the Atlantic Ocean [see Hart, 1984, Figure 2]. In the IC space, the five isotopic ratios of Pb, Sr and Nd have been considered simultaneously, in which case a simpler and more consistent criterion can be obtained. As a result, as shown by Figure 6, the realm of the enriched region defined by a positive IC2 value is different from the DUPAL domain of Hart [1984]: the enriched signature distributes itself in the Northern Hemisphere, which is consistent with occurrence of a DUPAL signature along the Nansen-Gakkel Ridge [Mühe et al., 1993]. This could have an important implication on mantle dynamics: if the enriched region was initially confined to a distinct domain (e.g., because it represents a mantle domain contaminated by extensive Pangeatic subduction [Anderson, 1982; Zindler and Hart, 1986]), the disposition of the IC2 domain and the amplitude of the anomaly may provide unique information on the long-term mantle flow and material transport. Accounting for mantle isotopic variability in terms of the independent differentiation processes identified by ICA rather than by interactions among mantle geochemical end-members with unique compositions is therefore bridging the gap between geochemical observations and mantle dynamics.

Figure 6

Distribution of IC2 values in oceanic basalts from the Atlantic and Indian oceans based on Figure 4e with the five isotopic ratios of Pb, Sr, and Nd. Circles correspond to MORB, while stars correspond to OIB. In each location, the IC2 variability is shown by the size of the color-coded symbols (smaller for the higher IC2 values). Contours for positive IC2 values are shown by solid lines, while broken lines have been drawn arbitrarily or with reference to the Pb-isotopic ratios of Figure 3b.

5. Conclusions

[28] Independent component analysis has been applied to the isotopic compositional space of oceanic basalts. Two statistically independent components have been found, which characterize two independent features: one opposes mid-ocean ridge basalts to ocean island basalts, while the other maps geographically regional isotopic properties. The distribution of the data in the IC space demonstrates that the observed compositional space is created by the joint distribution of the two ICs, indicating that two distinct differentiation processes mutually overlap, rather than by interaction among the conventional mantle geochemical end-members with unique compositions. The geochemical characteristics of the two ICs oppose the variations due to melting from those caused by interaction of aqueous fluid with the mantle, both at mid-ocean ridges and at subduction zones. The ICs provide a new quantitative measure for describing the isotopic heterogeneity of the mantle. As a result, a new criterion is proposed for the definition of the DUPAL anomaly. It is shown that the DUPAL signature is present both in the Southern Hemisphere and the Northern Hemisphere, which possibly reflects a large-scale feature of the mantle flow.

Acknowledgments

[31] We thank Matt Miller and Yoshi Tatsumi for discussion and help and Dan McKenzie, John Rudge, Vincent Salters, and Bill White for constructive reviews. Janne Blichert-Toft obliged with a careful editing of the manuscript. F.A. would like to thank the Earthquake Research Institute for their generous invitation to visit the University of Tokyo and the Program SEDIT of the Institut National des Sciences de l'Univers for financial support.

Appendix A:: ²⁰⁴Pb Normalization Versus ²⁰⁶Pb Normalization

[29] Let α = ²⁰⁶Pb/²⁰⁴Pb, β = ²⁰⁷Pb/²⁰⁴Pb and γ = ²⁰⁸Pb/²⁰⁴Pb. For a system evolving in time from 0 to t and with identical Pb isotope compositions at t = 0, the conventional ²⁰⁷Pb/²⁰⁴Pb versus ²⁰⁶Pb/²⁰⁴Pb isochron can be written as:

where

and λ are the decay constants of the different U isotopes. Dividing the first equation by α gives:

which is the equation of a straight line in a ²⁰⁷Pb/²⁰⁶Pb versus ²⁰⁴Pb/²⁰⁶Pb plot. In contrast with the conventional ²⁰⁷Pb/²⁰⁴Pb versus ²⁰⁶Pb/²⁰⁴Pb isochron, in which the age is obtained from the slope, the age in this so-called “inverse” isochron plot is read from the intercept.

[30] Let α = + Δα, β = + Δβ and γ = + Δγ, where , and represent the mean values of α, β and γ, respectively. Since Δα, ≫ Δβ and ≫ Δγ, the transformation from ²⁰⁴Pb-normalized variables (α, β, γ) to ²⁰⁶Pb-normalized variables (x, y, z) can be linearized as follows:

Since ICA searches for the components as linear combinations of original variables with maximum non-Gaussianity, any linear transformation, such as those expressed in (A4) to (A6), should lead to identical results. Figure A1 shows the result of ICA with ²⁰⁴Pb-normalized variables for the same data set as in Figure 3. In spite of the slight nonlinearity of transformation, the ICs obtained in the ²⁰⁴Pb (Figure A1) and ²⁰⁶Pb−normalized space (Figure 3) are essentially identical. However, because the transformation from ²⁰⁴Pb-normalized variables to ²⁰⁶Pb-normalized variables, while it is approximately linear, is not orthogonal, the PCs are different (Figure 3). In both cases described by Figures 3 and A1, the PCs are oblique with respect to the ICs, which reflects the non-Gaussian character of the data populations.

Figure A1

Independent components (compositional vectors/lines labeled IC1 and IC2) in the ²⁰⁶Pb/²⁰⁴Pb-²⁰⁷Pb/²⁰⁴Pb-²⁰⁸Pb/²⁰⁴Pb system. The data and symbols are the same as in Figure 3. (a) ²⁰⁶Pb/²⁰⁴Pb-²⁰⁸Pb/²⁰⁴Pb plot of the data, ICs and PCs. (b) IC1-IC2 plot where the data and the vectors are decomposed into the two ICs. (c) ²⁰⁶Pb/²⁰⁴Pb-²⁰⁸Pb/²⁰⁴Pb plot of the data, together with ICs and PCs which are obtained in the ²⁰⁴Pb-normalized space and are then transformed into the ²⁰⁶Pb-normalized space (compare with Figure 3a).