COMPOSITION AND DISCRIMINATION OF SANDSTONES: A STATISTICAL EVALUATION OF DIFFERENT ANALYTICAL METHODS

HILMAR VON EYNATTEN1, CARLES BARCELO-VIDAL 2, AND VERA PAWLOWSKY-GLAHN 2

2 1 Institut fur Geowissenschaften, FSU Jena, Burgweg 11, D-07749 Jena, Germany Departament d Informatica i Matematica Aplicada, Universitat de Girona, Lluis Santalo, s/n, E-17071 Girona, Spain ` ` e-mail: eynatten@geo.uni-jena.de

ABSTRACT: The discriminative power of four analytical approaches to sandstone composition is evaluated with respect to the separation of different formations and source areas. The case study is Cretaceous synorogenic sandstones (litharenites) from the Eastern Alps of Europe, which belong to four different formations and are derived from two source areas. Methods evaluated are light-mineral analysis (petrographic framework composition), heavy-mineral analysis, major-element XRF analysis, and trace-element XRF analysis. The statistical parameters calculated (percentages of well-classied samples, Mahalanobis distance) applying the logratio approach suggest that light-mineral analysis has a signicantly lower discriminative power than the other three methods. Taking into account the analytical expenditure for data acquisition, trace-element analysis appears to be the most efcient method for discrimination of at least the sandstone units examined. Although based on a single case study, these results are interpreted to have a more general meaning with respect to sandstone discrimination based on composition. Concerning sandstone provenance, trace-element analysis provides a quick tool to estimate the discriminative potential of a sample suite, i.e., the potential to discriminate between contrasting source areas. If a provenance model already exists and discriminate functions between contrasting source areas are calculated, trace-element analysis is considered to be most efcient in correctly assigning an unknown sample to its source area. These results cannot be extended to all kinds of sands and sandstones, but they cast serious doubt on the belief that petrographic point-count methods are the best approach to discriminate between sandstones.

the most widely applied analytical technique in the determination of the major-element and trace-element chemistry of rocks (Rollinson 1993). Major-element and trace-element analysis are here treated separately because they are performed using partly different analytical methods. Trace-element data constitute a small fraction of the whole-rock chemical composition. Apart from cases in which one component is absolutely missing for one sample group or the separation of groups is obvious because of very large differences, the determination of signicant differences between sandstone compositions requires statistical analysis. All data mentioned are compositional data, which means that they are proportions, subjected to the constant-sum constraint, and therefore should not be analyzed by standard statistical methods (Aitchison 1986). Special techniques are necessary to rigorously analyze compositional data, and several studies have already demonstrated their usefulness (e.g., Butler and Woronow 1986; Rollinson 1992; Heins 1993; Cardenas et al. 1996; Weltje et al. 1996; Barcelo-Vidal et al. 1997). The aim of this paper is to evaluate the discriminative power of four different analytical approaches (light-mineral and heavy-mineral analysis and major-element and trace-element analysis) to the composition of sandstones and relate the outcome to the analytical expenditure involved with each method. The case study is a well-documented example of synorogenic Cretaceous sandstones from the Northern Calcareous Alps in Austria (von Eynatten et al. 1996; von Eynatten and Gaupp 1999). The applied statistical methods take into account the specic nature of compositional data. Although based on a single case study, the results are interpreted to have a more general meaning with respect to the discrimination of sandstones based on composition, with special emphasis on provenance discrimination.
CASE STUDY

INTRODUCTION

The complex topic of sandstone composition is treated in the literature in several different ways concerning both analytical techniques and lines of interpretation. Interpreting the composition of sandstones in terms of, for example, their belonging to different lithologic units (e.g., Fuchtbauer 1964), their derivation from different sources on both global (e.g., Potter 1978; Dickinson and Suczek 1979) and regional scale (e.g., van de Kamp and Leake 1995), as well as contrasting climatic (e.g., Suttner et al. 1981) or diagenetic (e.g., Milliken 1988) conditions requires the conrmation that these sandstones display signicant differences in composition. The composition of sandstones can be ascertained in two fundamental ways: petrographically (mineralogy and texture) or chemically. The petrographic composition of sandstones generally is obtained by analyzing their framework components using point-count techniques on thin sections (e.g., Ingersoll et al. 1984). We refer to this method as light-mineral analysis, in contrast to heavy-mineral analysis. The latter constitutes a small fraction (mathematically: subcomposition) of the whole-rock mineralogical composition, but is usually treated separately because the data are obtained by different analytical methods (e.g., Morton 1985). Point-count methods on thin sections allow us to differentiate between primary detrital grains and secondary diagenetic processes, e.g., authigenic phases and compaction. The chemical composition of sandstones provides whole-rock data. Accordingly, they do not allow a differentiation between detrital or diagenetic origin of certain elements. XRF analysis of a powdered rock specimen is
JOURNAL OF SEDIMENTARY RESEARCH, VOL. 73, NO. 1, JANUARY, 2003, P. 4757
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Geological Setting The case study used in this paper is Cretaceous synorogenic sandstones from the Northern Calcareous Alps in Austria (Fig. 1A). The Cretaceous orogeny of the Alps documents the early stages of convergence between Africa-derived plates (Adriatic plate, Austroalpine microplate) and the European plate, which were separated since the Early to Middle Jurassic by the Penninic ocean (Fig. 1B). Ongoing convergence led to subduction of Penninic oceanic crust in the Late Cretaceous and a nal continentcontinent collision in the Eocene (e.g., Froitzheim et al. 1996). In the Northern Calcareous Alps (belonging to the Upper Austroalpine unit) the Cretaceous is characterized by the formation of various nappes which were thrusted onto each other top-to-the-northwest. Thrusting started in the southeast in the latest Jurassic and subsequently propagated towards the northwest (Fig. 2). The sandstones under investigation were deposited by turbidity currents in elongated basins situated on individual nappes, with depositional axes striking parallel to nappe fronts (piggy-back basins, cf. Ori and Friend 1984). Two source areas were distinguished in a previous study (von Eynatten and Gaupp 1999): the rst is situated to the southeast, the second to the northwest of the depositional sites (Fig. 2). Both source areas were found to be composed largely of Paleozoic metasediments, Mesozoic carbonates, and ultrabasic rocks, but the relative contribution of each of these source rocks varied with time and between the two source areas. A distinct

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H. VON EYNATTEN ET AL. and Wagreich 1992). Thus, a quick tool is required to assign a given sandstone of ambiguous origin to its correct stratigraphic formation and source area. All of the sandstone formations are affected by postsedimentary nappe tectonics, including faulting, folding, and, sometimes, boudinage of competent sandstone layers. On microscopic scales this deformation is reected in considerable compaction and squeezing of soft lithoclasts. Illitization and chloritization of ne-grained lithoclasts is also a commonly observed feature (von Eynatten 1996). Calcite cements are derived from dissolution of carbonate clasts and reprecipitation. Data on vitrinite reectance and illite crystallinity indicate that temperatures during the burial stage of the sandstones did not exceed 150200 C (e.g., Gaupp and Batten 1985). The thermal overprint increases from northwest to southeast. Because of the considerable diagenetic overprint, identication of individual grain types cannot be as sophisticated as in sands and sandstones that were not affected or only little affected, by diagenetic processes. Sandstone Composition From the Cretaceous sandstones described above a set of 52 sandstones is available, which were analyzed by all of the four analytical methods. On the basis of light-mineral data, all of the samples are classied as litharenites following classications of Zuffa (1980) and McBride (1963) (von Eynatten and Gaupp 1999). The QmFLt diagram (Fig. 4) illustrates the predominance of lithoclasts and the relatively low amounts of feldspars, which range from 0.3 to 5.3%. Lithoclasts commonly include carbonate extrabasinal clasts (21.775.3%), subdivided into dolomite (D), micritic calcite (Cm) and sparry calcite (Cs), and silicate lithoclasts such as metasedimentary clasts (Lsm, 0.722.0%) and serpentinite clasts (Lu, 0.0 31.7%). The latter are an important characteristic of these successions in terms of both variability and source-rock information. The occurrence of serpentinite clasts is related to common chrome spinel in heavy-mineral composition (von Eynatten et al. 1997). Other quartzose framework grains are chert (Qc, 2.339.0%), polycrystalline quartz (Qp, 0.024.0%), and monocrystalline quartz (Qm, 1.024.7%). Minor constituents are volcanic lithoclasts, sedimentary intraclasts, mica, chlorite, glauconite, and heavy minerals. A better separation of formations based on light-mineral data compared to the QmFLt-diagram is obtained by using ratios of various lithoclasts in a logratio diagram (Fig. 5). Relative contributions of serpentinite (Lu), metasedimentary lithoclasts (Lsm), and dolomite (D) suggest that both source areas show considerable changes in the relative contribution of individual source rocks with time. For the southeastern source area the data reect on average decreasing Lu/Lsm ratios from the Rossfeld Formation to the Lech Formation, suggesting a lower relative contribution of ultrabasic source rocks to the younger Lech sandstones. For the northwestern source area the data reect on average increasing D/Qm and Lu/Lsm ratios from the Losenstein Formation to the Brandereck Formation, suggesting a higher relative contribution of dolomite and ultrabasic source rocks to the younger Brandereck sandstones (Fig. 5; see also von Eynatten and Gaupp 1999). Using the chemical classication scheme of Herron (1988) the samples are classied as litharenites, wackes, Fe-sandstones, and sublitharenites (Fig. 6). Sandstones from the Brandereck and Losenstein formations are mostly litharenites and wackes, whereas sandstones from the Rossfeld and Lech formations are mostly Fe-sandstones. This differentation is due to higher Fe2O3 /K2O ratios for sandstones from the latter two formations. These higher ratios are due largely to lower K2O contents of these sandstones compared to sandstones from the Brandereck and Losenstein formations. Sandstones from the Rossfeld Formation also are characterized by higher SiO2 /Al2O3 ratios on average.

FIG. 1.A) Structural sketch map of the European Alps with location of the study area in the Northern Calcareous Alps (NCA). B) Paleographic sketch showing positions of major structural units at around Valaginian to Hauterivian time, 140130 Ma (modied after Froitzheim et al. 1996, von Eynatten and Gaupp 1999). Stippled line indicates position of cross sections from Figure 2. The present-day Northern Calcareous Alps form the Upper Austroalpine (UAA) sedimentary cover, which was thrusted over Lower Austroalpine (LAA) and Penninic units in Late Cretaceous to Tertiary time.

difference between the two source areas is the exclusive occurrence of highpressure metamorphic rocks in the northwestern source area, as evidenced by the chemistry of detrital blue amphiboles (glaucophane) and white mica (phengite) in sandstones derived from this source (von Eynatten and Gaupp 1999). Rare blue amphiboles in sandstones derived from the southeastern source area are chemically different in composition (crossite to riebeckite) and white micas are exclusively muscovites (von Eynatten and Gaupp 1999). The sandstones belong to four formations ranging in age from Valanginian to Santonian (Fig. 3): the Rossfeld and Lech formations, which were derived from the southeastern source area, and the Losenstein and Brandereck formations, which were derived from the northwestern source area (Fig. 2). Macroscopically these sandstones often are quite similar, and their regional distribution is obscured because of complicated tectonics in the present-day nappe pile. On the other hand, these sandstones are important recorders of Cretaceous synsedimentary tectonics (e.g., Gaupp 1982; Faupl

COMPOSITION AND DISCRIMINATION OF SANDSTONES

49

FIG. 2.Tectonosedimentary model of Valanginian to Coniacian sedimentation in the Northern Calcareous Alps (modied from von Eynatten and Gaupp 1999) illustrated in three NWSE oriented cross sections. For approximate position of cross sections see stippled line in the paleogeographic sketch of Figure 1B. Sandstones from Rossfeld and Lech formations were derived from the southeastern source area, sandstones from Losenstein and Brandereck formations were derived from the northwestern source area.
METHODS

Analytical Methods In this section we give a brief description of the methods used for data acquisition. For further information on the mineralogical data (details of methodology and data tables), see von Eynatten and Gaupp (1999). Tables of the chemical data are available in JSRs digital archive (see Acknowledgments). The measured variables obtained by each of the four methods are listed in Table 1. The methods are ordered by decreasing analytical expenditure, starting with heavy-mineral analysis and ending up with traceelement XRF analysis.

Heavy-Mineral Data.Sandstones were disaggregated using acetic acid to remove carbonate cement. Heavy minerals were obtained from the 63 125 m sieve fraction of the disintegrated sand by gravity settling in tribromoethane. At least 200 non-opaque non-micaceous minerals were counted using the ribbon counting method (Mange and Maurer 1991; Morton and Hallsworth 1994), except for four samples from which only 100 grains were counted because of very high amounts of opaque heavy minerals. Light-Mineral Data.Light-mineral data were obtained by point counting of at least 300 framework grains. Thin sections were stained with Alizarin red S for distinguishing between calcite and dolomite. In contrast to 63 m within lithoclasts were the GazziDickinson method, minerals

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FIG. 4.Sandstone light-mineral compositions illustrated in QmFLt diagrams. All samples (n 52) are litharenites.

FIG. 3.Stratigraphic range of sandstones from the four sedimentary successions.

counted as the type of lithoclast in which they occur (e.g., Decker and Helmold 1985). Monomineralic grains and lithoclasts were distinguished using the 0% cutoff proposed by Ingersoll et al. (1984). 4 mm Major-Element Data.Sandstones were crushed to pieces using a jaw crusher and then powdered using a corundum ball mill. Loss on ignition (LOI) was determined by heating the dried samples up to 950 C for two hours. Sample powder was then mixed (1:10) with a 2:1 mixture of lithium tetraborate (Li2B4O7) and lithium metaborate (LiBO2) to prepare fusion discs. Major-element data were determined on fusion discs by Xray uorescence spectrometry with a Philips PW 2400 wavelength-dispersive spectrometer system using a Rh tube. Relative errors on major elements are usually less than 2%. For statistical calculations, major-element oxides were recalculated to 100% with the LOI included. Total iron is expressed as Fe2O3. Trace-Element Data.The data were obtained using the same analytical equipment as for major elements but trace elements were measured on pressed powder pellets. Relative errors on trace elements are usually less than 5%. Pressed powder pellets are much easier and faster to produce than fusion discs, and therefore we treat trace-element analysis as an independent method. This is because, if samples are already well discriminated by trace elements, there is no need to analyze major elements for discrimination purposes. Statistical Methods In order to analyze the data obtained by the four different methods with statistical rigor we follow the method suggested by Aitchison (1986). All data sets are compositional data, which means that they are restricted to values between 0 and 1 (or 100%) and are subjected to the constant-sum constraint. This constraint means that all variables sum to a constant (e.g., 100%) and, consequently, cannot vary independently from each other. This implies that compositional raw data cannot follow a multivariate normal distribution and therefore fail a major prerequisite of parametric statistical methods, such as standard discriminant analysis. The method of Aitchison (1986) is based fundamentally on the logratio transformation of the compositional data. This means, that a d-dimensional composition x (x1, x2, . . . , xd ) is transformed to y (ln(x1 /xd ), ln(x2 /xd ), . . . , ln(xd 1 /xd )). The choice of the denominator (here: xd ) of the logratio transformation is not critical to the results (Aitchison 1986). This operation transforms the data from their constrained sample space, the simplex S d, into the real space R d 1, where parametric statistical methods can be applied to the transformed data (Aitchison 1986). For applying logratio transformations to the

data sets, zero values must be replaced by small positive values. In the chemical data sets no zeros occur, but zero values are present in both mineralogical (light and heavy minerals) point-count data sets. We applied the method of replacement suggested by Martn-Fernandez et al. (2000) assuming that zero values are not essential zeros. As a reasonable input value for zero replacement we choose 0.1%, which corresponds to 2033% of the lowest measurable value according to 200300 point counts. Biplot Analysis.Biplots describe graphically the pattern of relative variation of a multivariate data set by projection onto a plane xed by principal components. It is traditionally dened (Gabriel 1971) using the rst two principal components, but there is no need to restrict the diagram to these two axes. For a detailed description of biplot techniques see Krzanowski (1988). Aitchison (1990, 1997) applied the biplot to compositional data using the centered logratio transformation, i.e., the denominator of the ratios is given by the geometric mean of each composition. This implies

FIG. 5.Logratios using specic lithoclasts and monocystalline quartz (Qm) grains allow a better seperation of formations than QmFLt diagrams. D, dolomite; Lu, serpentinite; Lsm, metasedimentary lithoclasts. Symbols are the same as used in Figure 4.

COMPOSITION AND DISCRIMINATION OF SANDSTONES

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FIG. 6.Chemical classication scheme of sandstones based on major elements (Herron 1988). Symbols are the same as used in Figure 4.

that the origin of the biplot corresponds to the center (geometric mean) of the whole data set. The axes of the biplot correspond to principal components of the logcentered data (Fig. 7). A principal advantage of biplots is that they represent both the samples and the variables of compositional data. The former are termed cases, the latter vertices. For the interpretation of a biplot it is important to note that: (1) The squared distance between a vertex and the origin corresponds to the variance of the logcentered variable. If the angle between the line from a vertex to the origin and an axis is small, the variable has a strong inuence on the corresponding principal component. The larger the distance of the vertex to the origin and the smaller the angle, the stronger the inuence. For example, the logcentered variable Lu (serpentinite clasts) of the light-mineral dataset (Fig. 7B) shows the highest relative variability of all light-mineral variables and strongly determines the rst principal component. This is in contrast to most of the other variables, which have either close to no inuence at all or more inuence on the second component (Fig. 7B). (2) The squared distance between two vertices corresponds to the variance of the logratios of these vertices (variables), which implies that nearly coincident vertices means that the variance of the logratios of these variables is near zero and, thus, the ratio is almost constant. A good example
TABLE 1.Measured variables
Heavy Mineral Analysis 11 chrome spinel (csp) zircon (zr) tourmaline (to) rutile (rt) garnet (gt) chloritoid (cd) blue amphibole (gl) epidote minerals (ep) green amphibole (ac) apatite (ap) others* 11 monocrystalline quartz (Qm) polycrystalline quartz (Qp) microcrystalline quartz, chert (Qc) feldspar (F) metasedimentary lithoclasts (Lsm) ultrabasic clasts (serpentinite, Lu) volcanic lithoclasts (Lv) micritic calcite extraclasts (Cm) sparitic calcite extraclasts (Cs) dolmite extraclasts (D) others** Light Mineral Analysis Major Element Analysis 11 SiO2 TiO2 Al2O3 Fe2O3 (t) MnO MgO CaO Na2O K2O P2O5 LOI Trace Element Analysis 15 Ba Co Cr Cu Ga Nb Ni Pb Rb Sc Sr V Y Zn Zr

Method No. of variables Variables

* Traces of brown amphibole, tremolite, pumpellyite, brookite/anatase, barite, and anhydrite. ** Traces of sedimentary intraclasts, bioclasts, mica, chlorite, glauconite, and heavy minerals.

can be seen in both trace-element biplots (Fig. 7D) where the logcentered variables Cr and Ni lie close together, implying that the ratio Cr/Ni is relatively constant. (3) The distance between two samples (cases) is a measure of the similarity of the two samples and, thus, strong clustering of samples implies that these samples show strong similarities in composition. It must be stressed that biplots serve as a descriptive tool for a rst evaluation of the data. All statements based on biplots should be regarded as hints for further quantitative examinations but not as a nal result. This is because only a portion of the total variability is explained by the twodimensional projection. The proportion of the variability explained can be taken as a measure of the strength of an individual biplot in interpreting the data. Obviously, if the proportion is exactly or near to 100%, results obtained by graphical biplot interpretation are very robust. Discriminant Analysis.We performed the linear discriminant analysis using standard software routines (MINITAB, SPSS) applied to the additive logratio transformed data. For principles of multivariate discriminant function analysis the reader is referred to, e.g., Krzanowski (1988). The choice of the denominator of the logratio transformation is not critical to the results (see above). Choosing a denominator that is assumed to be valuable for discrimination may be helpful if a later reduction of the variables necessary for discrimination is intended. For linear discriminant analysis we have to assume that the transformed data of individual classes are samples of multivariate normal distributions with the same covariance matrix. In the general procedure, the best linear discriminant functions between the classes (e.g., the four formations) were calculated for each of the four methods using all of the available samples. Then each individual sample was classied according to the discriminant functions and a rate of wellclassied samples was calculated for each class (formation) and each method. In a second step, the procedure was repeated for each individual sample with linear discriminant functions calculated using all of the samples minus this individual one. The latter technique is known as cross-validation and satises the condition that the data to be classied should not be used for the formulation of the classication rules. To provide additional information on differences in composition between the four classes (formations), we use the Mahalanobis distance. Given that the Mahalanobis distance includes all of the variability of the sample classes, it can be used as a measure of the separation of classes (Krzanowski 1988). Because the method that discriminates best among all of the four formations is not necessarily the one that discriminates best between any two formations, we also

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COMPOSITION AND DISCRIMINATION OF SANDSTONES have applied hierarchic discriminant analysis. This means that we separate rst one formation from all the others, then a second one from the rest, and nally the two remaining formations from each other. All variables of each analytical method (see Table 1) are used for the discriminant function analysis except for others (light-mineral and heavymineral data sets), LOI (major-element data set), and Cs (light-mineral data set). The former comprise the sum of mostly rare and poorly constrained variables. LOI is strongly related to CaO (see also Figure 7C: coincidence of CaO and LOI) for mineralogical reasons because high carbonate contents (calcite, dolomite) imply both high CaO and high CO2 (resulting in high LOI) contents. The latter (Cs, sparitic calcite) was not used because it may be biased by diagenetic processes: differentiation between sparitic calcite grains and calcite cements is sometimes ambiguous in diagenetically altered sandstones.
RESULTS

53

ples in the heavy-mineral biplot (Fig. 7A, principal components 1 and 2) or Brandereck Formation samples in the major-element biplot (Fig. 7C, principal components 1 and 3). At rst glance, separation of formations seems to work best in the heavy-mineral biplot (Fig. 7A, principal components 1 and 2), but only 56% of the total variability is explained within this diagram. The separation is not convincing on the heavy mineral biplot of principal components 1 and 3. In general, the data show distinct differences in compositions with respect to the four formations, but a discrimination of all of them seems to be questionable. Such a full discrimination can be tested by applying discriminant analysis. Statistical Inference: Discriminant Analysis The results of linear discriminant analysis are expressed as percentages of well classied samples both without (DA) and with cross-validation (DAX). Additional information comes from the squared Mahalanobis distance (MD 2), a measure of separation of different groups (Table 2). If MD 2 9 and DAX 80%, the specic discrimination is considered to satisfy the conditions of sufcient discrimination. These limits were chosen because (1) values of MD 2 9 correspond to a 3 sigma distance between the means of two groups in the univariate case and is thus considered to indicate a reasonable separation of groups, and (2) values of well classied samples which exceed 80% are generally considered to be acceptable (e.g., Herron 1988) and correspond to values obtained by classical methods of provenance determination (Molinaroli et al. 1991). None of the analytical methods allow a perfect discrimination of all of the four formations. Using standard linear discriminant analysis (DA) all methods fulll the 80% criterion, except for heavy-mineral analysis, which fails this criterion for Brandereck Formation samples (78%). Using cross validation (DAX), both of the mineralogical methods (light-mineral and heavy-mineral analysis) fail the 80% criterion for two out of four formations, whereas both of the chemical methods (major-element and traceelement analysis) fail this criterion for one formation. DAX values are lowest using light-mineral data (50% and 39%), and this method also displays some very low MD 2 values. On the basis of the latter, light-mineral data demonstrate a very poor separation between the Lech and Losenstein 2) as well as between the Lech and Brandereck formations (MD 2 5, Table 2). The other three analytical methods all formations (MD 2 display sufciently high MD 2 values to separate between the four formations. Summarizing the simultaneous discrimination of all formations, we can state that (1) the discriminative power of chemical methods is better than that of mineralogical methods, and (2) heavy-mineral analysis discriminates better than light-mineral analysis. Because hierarchic discriminant analysis may lead to a better discrimination of classes, this procedure is also applied. To avoid the confusion of demonstrating all possible steps, we only present all possible rst (discrimination of one formation from the other formations) and last steps (discrimination between two formations). The parameters of the resulting ten discriminant analyses for each of the methods are given in Table 2 and are illustrated in Figure 8. DA values mostly exceed 90% regardless of the method. Using the more critical parameters DAX and MD 2 the worst results are again given by light-mineral analysis, with 6 out of 10 analyses not satisfying the criteria of sufcient discrimination. Again both chemical methods are quite similar (3 out of 10 analyses fail), but using hierarchic discriminant analysis heavy-mineral analysis gives the best discrimination (1 out of 10 analyses fails).

Descriptive Statistics: Biplot Analysis Biplots of the data obtained by the four methods are shown in Figure 7. For each method we choose two biplots (dened by the rst and second and by the rst and third principal components). Together these three components explain between 68% and 81% of the total variability. The heavy mineral data show several mineral phases with comparably high relative variability spread in all directions of the biplot (Fig. 7A). A similar pattern is observed for the major-element data (Fig. 7C). Here, the small distance between the CaO and LOI vertices in both major-element analysis biplots indicate that the ratio CaO/LOI is relatively constant, which results from the high carbonate content (CE clasts) of the sandstones (von Eynatten and Gaupp 1999). The variability pattern of the light-mineral data (Fig. 7B) is quite different, because one variable (Lu) shows a much higher relative variability than all the other variables. This variable also has a strong inuence on the rst principal component, followed by Qp, whereas the other variables have either a small relative variability in general (e.g., Qs, Lsm, Cs, Qm, F) or a stronger inuence on the second principal component (e.g., Cm, D, Lv). A similar pattern is observed for the trace-element data (Fig. 7D), where two variables (Cr, Ni) have a much higher relative variability than do the others and these two variables also inuence strongly the rst principal component. The distance CrNi is very small in both trace-element biplots, indicating that their ratio is relatively constant. The similarity of light-mineral and trace-element biplot patterns is related to the nearly exclusive occurrence of Cr and Ni in serpentinite lithoclasts (Lu): a high relative variability of Lu in the light-mineral data set forces a high relative variability of Cr and Ni in the trace-element data set. This cannot be seen in the major-element data because the specic major-element component of serpentinite is Mg, which is obscured because of its concurrent occurrence in volcanic (Lv) and dolomite lithoclasts (D). It also is not seen in the heavy-mineral analysis biplot, although nearly all of the trace-element Cr should be derived from the heavy mineral chrome spinel (csp). The reason may be that the trace-element data were derived from the whole rock, whereas the heavy-mineral analysis data consider only a specic part of the whole-rock mineralogy, i.e., heavy minerals in sand size fraction (here: 63125 m). Thus, the greater part of Cr is probably derived from microcrystalline chrome spinel within serpentinite lithoclasts (Lu), which cannot be recognized by heavy-mineral analysis. With regard to discrimination, the biplots demonstrate that there are clusters of samples belonging to a given formation, e.g., Lech Formation sam

FIG. 7.Biplots of compositions obtained by the four methods. A) heavy-mineral data, B) light-mineral data, C) major-element data, D) trace-element data. Axes are rst and second (left side) and rst and third (right side) principal components; rst principal component as horizontal axis in both cases. Percentages indicate proportions of total variability explained by an individual biplot. Bold percentages indicate proportions of total variability explained by both of the biplots. Symbols are the same as used in Table 1 (variables) and Figure 4 (formations).

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TABLE 2.Summary of discriminant analysis

METHOD No. of Variables (logratios) linear discriminant analysis of all formations Brandereck Formation (BF) Losenstein Formation (LoF) Lech Formation (LeF) Rossfeld Formation (RF) Mahalanobis distance (MD 2): BF-LoF BF-LeF LoF-LeF BF-RF LoF-RF LeF-RF hierarchic linear discriminant analysis one from the others: BFothers LoFothers LeFothers RFothers pairwise: BFLoF BFLeF BFRF LoFLeF LoFRF* LeFRF linear discriminant analysis NW-source from SE-source n 52 (27 52 (6 52 (13 52 (6 33 (27 40 (27 33 (27 19 (6 12 (6 19 (13 25) 46) 39) 46) 6) 13) 6) 13) 6) 6) n 27 6 13 6 DA 78 100 100 100 13 10 14 DA 83 92 94 94 97 90 97 100 100 100 92

Heavy Minerals 9 DAX 63 100 85 50 DA 89 83 92 100 9 5 24 DA 96 85 81 100 97 95 100 90 100 100 83

Light Minerals 8 DAX 89 50 39 83 DA 85 83 92 100 14 25 12 DA 94 94 96 88 97 100 94 100 100 100 96

Major Elements 9 DAX 81 83 85 67 DA 92 83 92 100 16 16 41 DA 96 96 88 96 100 100 100 100 100 100 98

Trace Elements 14 DAX 89 83 69 83

26 37 MD 2 4 14 9 11 14 10 11 70 455 40 9

13 DAX 75 92 83 85 88 83 82 90 100 90 89

2 25 MD 2 6 4 2 21 11 6 20 7 37 42 3

19 DAX 90 75 71 94 91 83 97 68 67 95 75

25 21 MD 2 10 13 16 7 34 29 11 32 82 13 14

13 DAX 88 94 94 88 97 92 81 84 75 69 92

32 71 MD 2 11 20 7 17 33 21 57 43 272 42 18

16 DAX 84 94 75 92 88 82 91 69 100 63 88

55 (33 19)

DA percentage of well classied samples using standard linear discriminant analysis. DAX percentage of well classied samples using linear discriminant analysis with cross validation. MD 2 squared Mahalanobis distance. RF Rossfeld formation; LeF Lech formation, LoF Losenstein formation, BF Brandereck formation. Bold numbers indicate either MD 9 or DA 80% or DAX 80%. * Numbers of variables were decreased to eight for computational reasons (HMA without ln(ap/zr), MEA without ln(Na2O/Si2O), TEA without ln(Co/Zr), ln(Ni/Zr), ln(Nb/Zr), ln(Pb/Zr), ln(Sc/Zr), ln(Zn/Zr)).

The last calculations consider the discrimination between the two different source areas, which, in fact, means the discrimination of Brandereck and Losenstein formations from Lech and Rossfeld formations (Table 2, Fig. 8). Using light-mineral data both parameters (MD 2 3, DAX 75%) fail the criteria for sufcient discrimination, whereas using heavy-mineral, major-element, and trace-element data discrimination between source areas passes these criteria. The weak separation (MD 2 3) depends strongly on the very poor separation between Losenstein and Lech formations, because this separation is crucial for the separation of the two source areas (von Eynatten and Gaupp 1999). In summary, the discrimination of formations and source areas on the basis of heavy-mineral, major-element, and trace-element analysis mostly passes the criteria of sufcient discrimination as dened above, whereas discrimination based on light-mineral analysis mostly fails these criteria. The former three methods display no systematic differences with respect to both discrimination parameters. These results suggest that trace-element analysis exhibits the best relationship of low analytical expenditure and high discriminative power with respect to the analyzed case study.
DISCUSSION

The most frequently used method to determine sandstone composition is thin-section petrography (light-mineral analysis). This is partly because several decades ago, when sedimentary petrology became a growing subdiscipline of geology (e.g., Pettijohn 1957), thin-section preparation and microscopy were accessible techniques for most geologists. However, lightmineral analysis still has strong advantages compared to chemical methods, because (1) analyzing the complete framework of a sandstone allows for the distinction between detrital and diagenetic phases, and (2) light-mineral analysis allows for the differentiation of textures of individual grains even if they are chemically and/or mineralogically similar. In fact, light-mineral analysis is capable of contributing to several specic problems in sedi-

mentary petrology and provenance analysis that cannot be solved by other methods, e.g., petrologic studies of sandstone diagenesis (e.g., Gaupp 1996) or the use of very detailed classications of unaltered lithic grains or quartz fabrics to evaluate unroong histories in the source areas (e.g., Dorsey 1988). On the other hand, light-mineral analysis also has several disadvantages. Concerning analytical expenditure, point-counting methods are quite timeconsuming compared to modern XRF techniques, which allow rapid acquisition of large numbers of precise chemical analyses (Rollinson 1993). There are relatively high methodical errors (counting statistics) in the quantication of individual variables (van der Plas and Tobi 1965) superimposed on possible operator bias due, in part, to subjective criteria for the separation of individual grain types (Dickinson 1970; Wolf 1971; Ingersoll et al. 1985; Suttner and Basu 1985). Despite these errors in quantication, light-mineral analysis and the models of classication or provenance determination relying on it (e.g., QFL and QmFLt diagrams of Dickinson 1985) often are used uncritically (Ingersoll 1990). A further pitfall of light-mineral analysis is the physical or chemical decomposition of relatively unstable grains in the course of diagenesis (e.g., Milliken 1988). Those grains carrying most information on provenance (feldspars and lithoclasts) are most prone to be degraded. This process generates ne-grained material mostly composed of clay minerals (pseudomatrix), which can no longer be identied by optical means. For a review of problems involved with such pseudomatrix we refer to Cox and Lowe (1996) and we use the term in their sense. Modifying the method by a combined microscopic-EDX approach (Bangs Rooney and Basu 1994) allows the determination of certain precursor minerals (or mineral aggregates) of the pseudomatrix but enhances the analytical expenditure and cannot be considered a purely petrographic method. Some disadvantages involved with light-mineral analysis commonly are thought to be circumvented by chemical analyses, especially when studying

COMPOSITION AND DISCRIMINATION OF SANDSTONES

55

FIG. 8.Squared Mahalanobis distance (MD 2) vs. percentage of well classied samples with cross validation (DAX) for the ten hierarchic discriminations (see Table 2) of each method. Gray area indicates insufcient discriminations with MD 2 9 and/or DAX 80%. Large symbols represent data from the discrimination between the two source areas (NW source vs. SE source).

muddy sandstones or altered arkoses and litharenites with high amounts of pseudomatrix (graywackes, e.g., Bhatia and Crook 1986; McLennan et al. 1993). But quantitative approaches comparing the results of both chemical and mineralogical methods applied to the same suite of samples are very rare. In a multi-method study of modern sediments of the Calabrian arc, Ibbeken and Schleyer (1991) statistically tested the discriminative power of different methods and grain sizes with respect to the separation of source areas. Considering the sand size fraction, heavy-mineral analysis discriminated best, major-element analysis intermediate, and light-mineral analysis worst. Molinaroli et al. (1991) applied discriminant analysis to mineralogical and chemical methods of provenance determination using case studies from the literature. They nd similar results for the individual data sets ranging from 74% to 85% (on average) of well classied samples, but this is not truly a comparative study in that different methods were applied to the same samples. Molinaroli et al. (1991) observed low percentages of well classied samples (5573%) for the magmatic-arc subprovinces of the light-mineral QFL and QmFLt provenance diagrams of Dickinson (1985). This observation is in agreement with the statement of Butler and Woronow (1986) that the subdivision of the magmatic-arc province may well be an artifact of the constant-sum constraint of compositional data. Our study suggests that the discriminative power of light-mineral analysis is signicantly lower than the other three applied methods, particularly when evaluating samples of a single sandstone class (e.g., litharenites). Taking into account the analytical expenditure for data acquisition, traceelement analysis appears to be the most efcient method for the discrimination of the analyzed sandstones. Although the results are based on only a single case study, we propose a more general applicability of this conclusion for the following reasons: (1) The analyzed sandstones are immature litharenites with a high proportion of lithoclasts. Although affected by a considerable tectonic and thermal overprint, lithoclasts still show a wide range of distinguishable types (e.g., serpentinite, quartzchloritemica aggregates, volcanics, chert, dolomite, micritic calcite). This high diversity of lithoclasts is due to a complex hinterland (von Eynatten and Gaupp 1999). Therefore, these sandstones should have a high potential for a good discrimination by light-

mineral analysis compared to more mature sandstones like sublitharenites or quartzarenites. Nevertheless light-mineral analysis mostly fails statistical criteria for sufcient discrimination. (2) Our results are supported by a previous study of modern sediments of the Calabrian arc by Ibekken and Schleyer (1991) that was based on a much larger number of samples. The clasts of these immature sediments are not modied by diagenetic processes, implying that all the information obtainable by light-mineral analysis is available for discrimination. Nevertheless, light-mineral analysis exhibits again the weakest discriminative power of all applied methods, including heavy-mineral analysis and majorelement analysis. (3) The larger the number of variables and the better these variables are dened, the better a multivariate discriminant analysis can be. Light-mineral analysis of diagenetically altered litharenites is quite limited in the number of precisely distinguishable variables and generally displays relatively high errors on individual variables. In contrast to counting methods, chemical analysis (e.g., XRF analysis of major and trace elements) usually displays much lower errors on individual variables, and in case of traceelement analysis the number of variables may be markedly larger. The latter point is considered to be valid for sandstones affected by a noticeable diagenetic overprint, but in case of unconsolidated sands or sandstones with low diagenetic imprint the number of variables obtainable by light-mineral analysis may be markedly larger than the number of variables obtainable by trace-element analysis. The results from this study cannot be generally extended to all kinds of sands and sandstones, but the type of sandstones considered here are very common (e.g., foreland basins) and form about 50% of all sands and sandstones (litharenites and graywackes; Pettijohn et al. 1987). As mentioned above, chemical whole-rock analysis does not provide information on texture and authigenesis of sandstones. For example, quartzarenites composed exclusively of varying contents of quartz and pure chert grains will probably not be discriminated succesfully by chemical methods. In some cases, diagenetic mineral phases may be the only cause for contrasting chemical compositions and, hence, their occurrence may cause discrimination of sandstones based on whole-rock geochemistry without any

56

H. VON EYNATTEN ET AL. based on data of this kind. The logratio approach of Aitchison (1986) provides a powerful tool for analyzing compositional data, and we therefore encourage its use whenever sandstone compositions are analyzed statistically.
ACKNOWLEDGMENTS

relation to sandstone provenance. However, these effects can be easily detected by qualitative petrography without using point-count techniques. The method with the greatest discriminative power is thought to discriminate best between groups of samples with different compositions. Because sandstone composition is controlled by geologic conditions such as source rocks, climate, and diagenesis, changes in composition reect changes in one or more of these conditions. Consequently, a signicant change in geologic conditions should be recorded in the sandstones, and the most prominent way to evaluate such changes is to look for statistically significant changes in sandstone composition. The higher the discriminative power of the chosen method to analyze sandstone composition, the higher is the chance to dectect such statistically signicant changes. Obviously, the method with the higher discriminative power does not necessarily contribute most to the interpretation of geologic conditions like, e.g., provenance. But (1) evaluating the potential for signicant differences in sandstone composition should be among the rst steps when trying to understand the geologic history of a suite of sandstones, and (2) use of statistical tools such as biplots of logratio-transformed compositional data gives useful hints to the further interpretation of the data (e.g., the high variability of Cr and Ni and their strong inuence on the rst principal component, Fig. 7D). The provenance model for the sandstones of this case study is based mainly on heavy-mineral analysis and single grain geochemistry of several detrital mineral phases (von Eynatten and Gaupp 1999). It would denitely not have been possible to establish the model solely on the basis of wholerock chemical analyses. This is in agreement with recent developments favoring multi-method approaches to sandstone provenance (e.g., Haughton et al. 1991; Morton and Hallsworth 1999; von Eynatten et al. 1999). Once a provenance model is established and appropriate discriminant functions between contrasting sources are calculated, chemical analysis of an unknown sample should allow us to assign this sample to its source. From our results, a high probability of correct classication combined with lowest analytical expenditure is achieved by using XRF trace-element analysis on pressed powder pellets.
CONCLUSIONS

This study was performed during a sabbatical stay of HvE at Universitat Politecnica de Catalunya in Barcelona and Universitat de Girona. HvE acknowledges funding by the Deutsche Forschungsgemeinschaft (grant EY 23/2). We appreciate discussions and presubmission reviews by Reinhard Gaupp and Thomas Voigt. The manuscript beneted from thorough and stimulating comments by JSR reviewers John Aitchison, Jose Arribas, Mark J. Johnsson, Ken Ridgway, and Alex Woronow. Careful editorial handling by David A. Budd and Mark J. Johnsson is greatfully acknowledged. The chemical data described in this paper have been archived, and are available in digital form, at the World Data Center-A for Marine Geology and Geophysics, NOAA/NGDC, 325 Broadway, Boulder, CO 80303; (phone: 303-4976339; fax: 303-497-6513; E-mail: wdcamgg@ngdc.noaa.gov; URL: http:// www.ngdc.noaa.gov/mgg/sepm/archive/index.html).
REFERENCES

Four different analytical methods were applied to a suite of diagenetically altered Cretaceous litharenites to evaluate the discriminative power of each method with respect to different formations and source areas. The results imply that light-mineral analysis has the lowest discriminative power of the applied methods. Trace-element analysis on pressed powder pellets provides a high discriminative power combined with the lowest analytical expenditure. On the basis of comparison with other studies, these results are interpreted to have a more general meaning with respect to the discrimination of sandstones based on composition. We are aware, however, that this is not proven to be valid for all kinds of sandstones. In the case of less mature and diagenetically altered sandstones (e.g., litharenites and graywackes) chemical analysis appears to be more precise and efcient with respect to discrimination purposes. Sandstone composition is strongly related to the provenance of sediment, which is largely controlled by source rocks, climate, and relief (e.g., Johnsson 1993). Because of a complex interaction of these factors, data based on a single analytical method mostly do not allow us to develop a precise provenance model. But if a provenance model already exists and discriminant functions are calculated for the sample suite the model relies on, traceelement analysis provides a fast and promising tool to assign an unknown sample to its appropriate source. If no such model exists, chemical analysis provides a quick tool for a rst estimate of the discriminative potential of a sample suite. Finally, we wish to emphasize that applying rigorous statistical methods to compositional data may enhance the strength of conclusions that are

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