Variability of igneous rocks and its significance

Variability of igneous rocks and its significance E. H. Timothy Whitten WHIITEN, E. H. T. 2000. Variability of igneous rocks and its significance. Proceedings of the Geologist' Association, 111, 1-15. The types of specimens and chemical data used and needed for establishing the nature, variability and genesis of igneous (particularly granitic) rocks are examined. Specialization (by professionals and amateurs) within subdisciplines and the explosion in acquisition and statistical analysis of petrological data have led to lack of the necessary integration between dissimilar approaches. Some significant topics stemming from mathematical geology that are relevant to petrology are reviewed briefly (e.g. genetic process-response models, relevant sample populations and sample-data variance, classification and partitioning, and various methods of avoiding inherent difficulties in use of percentage data). An example of using compositional data to test quantitatively a genetic hypothesis for a granitoid suite is given. Lower Bonehill Farm, Widecombe-in-the-Moor, Near Newton Abbot, Devon TQ13 TTD (e-mail: [email protected])

1. BACKGROUND AND HISTORY The geological sciences, being primarily derivative sciences, have had a long history of data gathering and natural history-type description - that is, inventory that encouraged naming rocks and recording distribution. Except for some volcanic phenomena, petrologists have few opportunities for reproducing experiments to test genetic hypotheses. By contrast, in chemistry and physics (and recently in experimental geochemistry) most experiments are reproducible. Also, most petrological phenomena are complex; interplay of dissimilar processes and variables is involved, and random or arbitrary events and interactions are sometimes deemed significant. Until development of rapid chemical analysis by X-ray fluorescence in the late 1940s, petrographic rock descriptions contained few chemical analyses. Early studies rarely contained more than one or two major-element analyses because 'wet chemistry' is very expensive. Modes (by weight or volume) were difficult to make until integrating-stage micrometers (later point counters) and the ability to make large thin sections evolved in the late 1940s and early 1950s. In the second half of the twentieth century, vast numbers of modes and major- and minorelement analyses of igneous rocks and their constituent minerals became available; assembly of huge datasets is now routine and common. Also, manual calculators used into the 1950s were replaced by computers that permit fast sophisticated statistical data manipulation on the desk top; the International Association for Mathematical Geology has blossomed since its foundation in 1968. Concomitantly, amateurs and professionals specialized so that gulfs between subdisciplines deepened and broadened; petrologists, for example, are generally not fully cognizant with statistical and computing techniques (developed independently by mathematical geologists), so their pragmatic value to petrology is slow to emerge, and vice Proceedings of the Geologists' Association, 111, 1-15.

versa (cf. Whitten, 1983, p. 241). Also, despite invaluable electronic bibliographic search tools, worldwide proliferation of journals, symposia, etc., has made acquiring a holistic view difficult. Because much of my work since 1945 involved granites, they are used for illustration, but the substance of this paper applies to most fields of geology. Science is increasingly under economic pressure and guided by cost effectiveness. Hence, it is crucial to define objectives explicitly before research starts, as opposed to collecting new information about some rocks, scrutinizing data that happen to be gathered, hoping interesting results emerge, and, if they do emerge, pronouncing them the objective of enquiry. Interesting results sometimes emerge from the latter approach; it would be nice to preserve opportunities for such, essentially random, pure research. However, for economic and engineering geology, and much academic and industrial work, identification of objectives at the outset is needed to define unequivocally the research programme and particular data needed. As late as 1952, in a volume on scientific objectives, Prof. H. H. Read asserted: 'The basis of all geological research is the geological map; ... what the geologist sees either in the field or in the laboratory depends almost entirely on his own personal character. Geology is individual and personal, no technical or instrumental assistance is possible .. .' Maps are inventories of information, but many more features are appraised in the field by map makers than are recorded. Earth scientists integrate mentally many properties that are rarely expressed objectively. Heretofore, aspiring geologists needed to see a similar sequence of features in the field or laboratory, and to emphasize mentally the same, subjectively evaluated, phenomena. Such experience is increasingly difficult to 0016-7878/00 $15·00 © 2000 Geologists' Association

2

E . H. T . WHITTEN

acquire; current research requires assessing the variability of many variables (e.g. oxygen isotope ratios) that cannot be seen (cf. Whitten, 1974, pp.183-4; Janousek, Rogers, Bowes & Vankova, 1997). Almost two decades after Read' s statement, Prof. Ager (1970, p. 421) claimed: '. . . . the expensive piece of ironmongery and the computer have taken the place of the hammer (and of the hand-auger) as the basic items of equipment for many geologists. Obviously it is laudable and desirable that we should try to make our subject more scientific in methodology as well as in approach. However, it must be said that this trend toward quantification and the exploration of the neighbouring disciplines has not always led to greater precision and validity.' Ager's contention remains true, as does Prof. Griffiths' (1962, p. 565) that 'Progress in scientific investigation in any specialized field is generally measured by the degree to which the subject is pervaded by mathematics'. Unfortunately, statistical techniques are frequently used after data collection that was essentially arbitrary; at best, this tends to be inefficient and, at Worst, ineffective (cf. Griffith s, 1962, p. 567; Whitten, 1984, p. 3). The following remarks apply to most fields of geology: 'Structural geology . .. is rapidly progressing to the stage in which observed elements can be viewed in the framework of a web of processes and responses (i.e., causes and effects) that will eventually be expressed mathematically (e.g. as differential equations) in terms of deterministic equations . . . . Geologists are still some distance from erecting deterministic models on the bases of observed data, but the rapid growth of quantitative experimental and observational data is making the study of models an urgent necessity . . .' (Whitten, 1966a, pp. vii and 545). In the earth sciences, observational data are used almost exclusively for two dissimilar, and rapidly diverging, tasks, namely: (a) development of inventories and/or databases, i.e. mapping and recording the nature and variability of rocks, and collation of results; and (b) development and evaluation of hypotheses, particularly for prediction, economic evaluation and petrogenesis (cr. Whitten, 1974, 1984). Without belittling the importance of basic surveys, mapping and data collection, understanding the genesis of geological features is commonly the primary challenge. The interrelationships of observed data and genetic models (particularly for granite s) are examined in this paper; experimental geochemistry, which affords an opportunity for repeated experiments, is not considered. How petrographic data can be used to evaluate and/or corroborate particular petrogenet ic models is illustrated. First, some establi shed concepts and definitions are reviewed in the next three sections. 2. INVENTORIES Maps are inventories of qualitative and quantitative informat ion. For petrography and petrology, tables,

databa ses, summary staustics, graphs and triangular diagrams are also frequently used inventories, in which spatial components are normally lost or ignored. Most mathematical geology publications dealing with igneous rocks comprised sophisticated inventory, standard statistical techniques having been used to describe, classify and/or partition data. Analyses-of-variance, factor, cluster and trend-surface analyses are common statistical tools for such purposes. In most cases, data that happen to have been available (or happen to have been collected) were used, without much regard to relevance of the (a) sampled to the target population (terms defined in section 4), or (b) available variables to the primary objective (if there was one, apart from use of the mathematical technique). Contouring methods (surfacefitting) provide inventories for samples actually studied, but have sometimes been used for (commonly misleading ) implicit or explicit interpolation and/or extrapolation from sampled to target populations. Weighted moving-average methods developed in the mining industry (Krige, 1964) have evolved into the widely used (but inappropriately named) specialist field of geostatistics (e.g. David, 1977; Journel & Huijbregts, 1978), which assumes data for spatially close samples tend to be more alike than those for more distant samples. Many common variables (Table 1) are described by closed data (e.g. percentage data) which introduce particular problems (see section 6). Successively more sophisticated mathematical method s for describing inventory data can lead to new techniques and interesting results, but not directly to new genetic models (cf. Whitten, 1984, p. 9). Geological databases permit speedy retrie val of information. They are especially valuable if the objectives being served are specified. For example, S. C. Smith (Smith, Moreno, Waldvogle & Huang, 1983) catalogued verbally the eight objectives of the Houston International Mineral Corporation's databank. Chayes' (1985) igneous rock database, IGBADAT, is well known and apparently used widely. However, many databa ses house all information generated by an agency, including similarly described, but otherwise unequivalent, data; this has inherent dangers. Many data have ephemeral interest and/or value. For example, chemical analyses for 114 granitoid samples which I collected for a special purpose in 1952 from the Older Granite of Donegal, Ireland, reflect my particular biases and objecti ves and are appropriate in my databank. Without full details of the sampled population, such chemical analyses are inappropriate to a database accessible to others, who may have dissimilar objectives (Whitten, 1984, p. 8). Morton, Baird & Baird's (1969) 162 chemical analyses for a precisely described, sampled population of Rattlesnake Mountain Pluton, California, exemplify material suitable for a general database. The sampled populations for other large datasets of excellent chemical analyses (e.g. 550 analyses for the Northern Peninsula and Transverse Range s batholiths, California - Baird, Baird & Welday,1979) are not so well documented and would be less useful.

VARIABILITY OF IGNEOUS ROCKS AND ITS SIGNIFICANCE

3

Table 1. Partial list of variables that can be measured for an igneous rock hand sample Modal volume and weight percent for each major and each minor mineral phase. Weight per unit volume and weight percent of each major oxide, each major cation and each trace element (including rare earths). Weight per unit volume and weight percent of each major oxide, each major cation and every trace element (including rare earths) in each major and minor primary and secondary mineral phase. Optical properties of every primary and secondary mineral component (e.g. optic-axial angle, triclinicity, refractive indices, birefringence). Specific gravity, electrical resistivity, alpha-radiation of hand sample. Grain size of whole rock sample and of individual component minerals (including mean, variance, skewness, kurtosis and statistical moments of size of each mineral phase, where 'size' is largest diameter, diameter of sphere of equivalent volume, lengthlbreadth ratio, etc.). Texture of whole rock (e.g. Chayes' Identity-Change Index, Vistelius' Chi-square-value based on markovity) including porosity, vesicularity and permeability. Amount and type of alteration of each mineral phase. Zonation patterns of component minerals. Content and partitioning of sulphur, oxygen, strontium, lead, etc. isotopes. Differences in chemistry, optics, physical properties, grain size, etc. between megacrystals and ground mass phases. Variance (and other statistical measures) of all physical and chemical variables within sample etc.

3. GENETIC HYPOTHESES AND MODELS After identifying the nature, composition and variability of some igneous rocks, commonly a primary objective is to understand how and why those rocks originated (i.e. the petrogenesis). This is valuable in itself, but is also pivotal for predicting correctly features of economic or geological importance that are currently unobserved or unmeasured. Because nature is complex, understanding is eased by breaking natural systems into component parts for which cause-and-effect relationships can be expected. From all the possible causes and all the possible attributes of the effects, it is useful to identify those necessary and sufficient to account for such components. Models relating such characteristics are useful. Potter & Pettijohn (1963, p. 226) wrote: '. . . . a "sedimentary model" . . . is an intellectual construct which, as in much of geology, is based on a prototype. The concept of a geosyncline is based on the Appalachian model; ... the model concept embodies the idea that the fill of sedimentary basins is an organized response to a relatively few major dispersal patterns which can be defined and identified by systematic study. With the proper choice of model one should be able to make more successful predictions about those portions of a basin which are concealed and unexplored.' Within a broader setting, Krumbein & Sloss (1963, p. 501) opined that in the search for' ... generalizing principles it is a useful philosophical device to recognize models actual or conceptual frameworks to which observations are referred as an aid in identification and as a basis for prediction' . Whitten (1964a) defined several types of geological model: (a) process model refers to geological processes that operate in the specified place studied, (b) response

model refers to the actual rocks and samples studied; and (c) process-response model expresses the linkages of those processes to the rocks. As Krumbein & Graybill (1965, pp. 15 et seq.) showed formally, the objects (rocks, etc.), forces, events, etc., must be replaced by variables, parameters and constants in equations if the goal of expressing systems mathematically is to be achieved. This involves sufficient understanding of the physical and chemical processes involved to permit writing initial equations. Building realistic numerical models is one of the greatest challenges for petrology and mathematical geology. The objectives of petrological investigations can remain vague until the petrogenetic processes involved, and responses to those processes, are expressed quantitatively in a unifying model. An initial hypothesis about a lava flow could be expressed as a conceptual processresponse model. Interrelationships of the simultaneously varying process factors (temperature at surface, speed of flow, lava thickness, gases emitted, etc.) and observed features of an actual flow can be included in such a model, that might be treated in a statistical manner later. Then, using the model, observations of ancient lavas permit predictions about their genetic processes, or observations of current eruptions could lead to predictions about the rocks that will result (cf. Krumbein, 1964). Igneous rocks and petrogenetic processes are (or were) real entities, but process models are concepts. For most petrological studies, processes can only be inferred. For surface phenomena, uniformitarian principles may be useful and suggest that current processes can be correlated with ancient responses. However, there are few opportunities for observing or measuring processes involved with plutonic rocks. Inferences can be made, but those of different scientists are likely to be dissimilar, due to different hypotheses about subsurface conditions and silicate chemistry (cf. Bateman, Clark,

4

E . H . T. WHITTEN

Huber, Moore & Rinehart, 1963, pp. 30-2). Logically, only one set of processes was involved for each granite pluton, but which set must be deduced from study of the rock s that remain, experimental and theoretical geochemistry, etc. Building and testing proces s-response models for a pluton involves defining: (a) possible petrogenetic hypothe ses (conceptual process models) incorporating relevant structural, physical, chemical and temporal factors (with quantitati ve limits for each); aureole rocks may need inclusion too; (b) a conceptual response model appropriate to each pro cess model; this defines which observable variables are relevant for discrimination between rival process models; and (c) the spatial variability of all relevant chemical, mineralogical, textural, structural and temporal variables to test the response models. New data can prove the incorrectness and/or incompleteness of a model, whereas compatible data indicate only that a model is not violated by current information. Eventually, it may be possible to refine the models as deterministic models incorporating all relevant characteristi cs in differential equations that can be tested and modified as new data demand . Such models could be used for accurate description of, and prediction about, parts of the rock formation not yet examined. Probabilistic, rather than deterministi c, response models incorporating randomness and stochastic processes are sometimes necessary (cf. Dacey, 1975; James, 1975); incorporating fuzzy-set concept s in models can also have considerable value (cf. Fang & Chen, 1990; Kacewicz, 1993). Prof. A. B. Vistelius' and his colleagues' significant contributions (e.g. Vistelius, 1945; Vistelius & Sarmano v, 1947) to the philosophy of models are not widely known in English; they were reviewed briefly by Whitten (1977, pp. 326-8). Bardossy (1997, p.l7), in exploring whether there is a bridge between geologist s and mathematicians, without reference to older literature, introduced a dissimilar classification and terminology for models. Some scientists have asserted there is inherent randomne ss in geological data and processes that cannot be avoided in deciphering reality; Mann (1993) reviewed this subject. There are two oppos ing views, namely: (a) the natural world incorporates irreducible elements of chance and indeterminism (cf. Leopold & Langbein, 1963; Krauskopf, 1968, p.17), and (b) physical-chemical laws govern cause-and-effect processes and control every happening and product, although microconditions vary. Use of process-response models involves reliance on the latter.

4. OBSERVED DEPENDENT DATA

Thousands of discrete variables for igneous rocks can be analysed (Table I); some can only be expressed qualitatively, but most are capable of quantitative (or at least

semi-quantitative) measurement. 'Continuingly, additional variables are measured, mapped and their inter- and intrarock unit variability evaluated; not all mea surable variables have identified significance for particular objectives. Recording all possible variables (because equipment is available and of excellent quality) is unrealistic in terms of time and expense; includ ing variables arbitrarily can involve significant redundanc y and/or irrelevance. Erecting a process-response model identifies which variables are relevant and may suggest eliminating other variables, including some measured frequently (Whitten, 1974, pp.186-7). On the premise that rock composition varies from place to place (spatial location), variables describing samples are called dependent variables (with respect to space), while coordinates of samples' locations are independent variables. Time can be an important fourth independent variable (e.g. Bateman et al., 1963) as, for example, in modelling autometasomatism or the repartitioning of cations during magma cooling (Whitten, 1964b). In most field s of endeavour, the natural human tendency is to notice, record and collect the unusual, rather than the ordinary and the average. For a mass described and mapped as granite, a ' typical' sample is likely to approximate the collector 's concept of granite, irrespective of compositional variation within the whole lithic unit; bias is introduced either involuntarily or voluntarily. If a granite mass were essentially homogeneou s, petrologists could readil y colle ct a single "typical" sample, and its chemical analysis would typify the mass adequately. Because comp osition varies significantly from place to place in most granites, the arithmetic mean of chemical analyses of single specimens from M plutons has little precise relationship to the average compo sition of the M pluton s. If, from a collection of hand-specimens from P different plutons, N similar samples were selected by trained geologists, the average of their individual full chemical analyses might represent the average composition of those N specimens, but not of the P plutons. However, this applies only to variables that happen to be closely correlated with characteristics visible in hand specimen; many features (e.g. accessory minerals, isotopes, trace elements, rare earths) have few visible features to guide collection of representative material (cf. Whitten, 1960, 1962, 1963; Janousek et al., 1997). In practice, adequate sampling of an intru sion does not involve collecting numerou s arbitrary specimens, but use of a sampling design adjusted to the particular unit. No matter how much detailed knowledge is obtained about the composition of individual specimens, it has relatively little value until viewed in relation to the target population of specimens comprising the whole rock-mass under study (Whitten, 1961a, p. 619). The rich and large data arrays now available have been used extensively by mathematical geologists, but their techniques have progressed more rapidly than an understanding of the (a) underlying geological phenomen a and (b) variance of, and between, the geological variables (dependent variables). Statistical techniques developed for

VARIABILITY OF IGNEOUS ROCKS AND ITS SIGNIFICANCE

5

other scientific domains have been applied extensively to geological data that happen to be available. Target populations and sampled populations Target population has technical meaning in statistical usage (Cochran, Mosteller, Tukey & Jenkins, 1954) that was introduced into geology by Rosenfeld (1954) and Krumbein (1960). It is the whole population of specimens of interest about which a geologist wishes to make inferences or draw conclusions (Whitten, 1961a, p. 1333). The target population might comprise all N potential specimens of the uppermost layer parallel to the eroded surface of a pluton (if the population is defined to exclude the subsurface and material lost by erosion). Again, a target population might comprise, say, all 1 kg specimens contained in the 10 em thick layer at 250 m above sealevel (i.e. at the ground surface, underground and already eroded away) of the Newer Granites of Scotland (which are mapped as outcrops unconnected at the present level of exposure). Such finite, but vast, numbers of potential samples are illustrated in Fig. 1. Lack of exposure (due to overburden, lakes, bog, etc.) restricts sample collection to actual exposures, drill cores, etc.; in this case, all n samples composing the entire surface of natural and artificial exposures, the drill cores, etc. comprise the sampled population; inaccessible and unvisited exposures, like unexposed areas, are not part of that population. If a grid were laid across most plutons, actual samples could be collected from only a limited number of intersections (cf. Fig. I), so there are severe restrictions on direct probability sampling of target populations. Theoretically, a probability sample could be taken from each area of continuous outcrop, i.e. from the sampled population. The sampled population commonly comprises the potential samples forming the surface of all actual outcrops that the geologist succeeds in visiting and sampling. Although a grid can be laid out for the sampled part of a pluton, and sample sites identified with the aid of, say, random-number tables to yield probability samples, commonly there is not complete freedom to collect at all prescribed grid points. In most cases, only upper semiprobability sampling is accomplished because, although grid intersections can be located at random, the actual specimens cannot be chosen at random; factors like ease of collection, lack of weathering and other practical problems influence the actual samples collected; portable drilling equipment, etc., can sometimes minimize such problems. On the scale of Fig. lA, the area in Fig. lB would be continuous exposure, but sample collection at grid intersections is impossible; at three intersections, there is no exposure and at the fourth there is a younger dyke; in practice, such sites might be abandoned, or sites near the grid intersections selected subjectively. Thus, in favourable circumstances, rigorous statistical inferences can be drawn about a sampled population on the basis of samples actually examined. Later, a geologist can use these to make subject-matter inferences about the target population on the basis of previous geological

Fig. 1. Target and sampled populations (after Whitten, 1961a, fig. 1). (A) Map of a hypothetical granite intrusion showing areas of actual exposure and inferred contact (dashed line) between granite and metasediments; a 200 m grid is represented symbolically at the bottom left comer which should continue across the whole map. (B) Detailed map of a bronzititeharzburgite-dunite complex, Blakely Creek area, Sweetgrass County, Montana (after Howland, Garrels & Jones, 1949, pI. 38); blank area is unexposed; solid black is doleritic dyke; other patterns indicate various ultramafic igneous-rock outcrops; grid lines are 500 feet apart. (C) Detailed map showing rock exposures in part of T. 63 N., R. 11 W., Minnesota (after Clements, 1903, sheet 22); all observed rock exposures - solid black; Ely greenstone - blank; Soudan Formation - grey shading; Giants Range Granite - dots; water - ruled lines; grid lines are 0.25 miles apart.

experience. As statisticians Cochran et at. (1954, p.19) noted: 'The step from sampled population to target population is based on subject-matter knowledge and skill, general information, and intuition - but not on statistical methodology'. This is an extremely important limitation in the earth sciences, except in rare cases where target and sampled populations are identical; such a situation might occur with the surface of desert sand dunes, where every sample desired is readily available. The target population has sometimes been regarded erroneously as of 'academic' interest only, and represented vaguely by 'typical specimens' or the average of miscellaneous samples whose precise three-dimensional spatial relationships are unimportant. Griffiths (1962, p. 606) used these concepts (with different terminology) and added hypothetical population for the total rock that existed prior to erosion, etc. He

6

E. H . T. WHITTEN

noted a further unfortunate population , commonly introduced inadvertently by inadequate sampling, which the collected samples happen to represent. Variance of individual variables Krumbein & Slack (1956), in one of the few detailed studies of levels of variance of a geological variable, showed the maximum variance of low-level radioactivity of a Pennsylvanian shale across Illinois occurs at the smallest (thin section) level of sampling. Without knowledge of a variable's variance, manual contouring of field data involves wholly subjective interpolation, as illustrated by Fig. 2. In 1964, Whitten (1972, p. 32) questioned whether major- and trace-element analyses for duplicate granite samples weighing 50 g would yield different results. Would 25 g samples give reproducible results, or should 1 kg samples be used? Would variability be greater between adjacent hand samples from the same outcrop, or between specimens collected 500 m apart? Little is known about levels of variance of igneous-rock variables, and most variables actually measured cannot be appraised visually in the field. Analysis-of-variance techniques are well suited to determining the proportion of variance attributable to each level of sampling (e.g. between and within sampling localities). In the only detailed study for a granite, Baird, McIntyre & Weiday (1967) showed that, if variance of attributes is large at the smallest level of

sampling (hand-specimen level), adjacent samples yield dissimilar values and thus dissimilar contoured maps. In general, each variable has dissimilar variance in samples of a specified size. Variance tends to be large between very small samples, especially when grain size is large. As sample size increa ses, variance between samples decreases to a minimum, before increasing again with extremely large samples. For a given igneous rock, the variance of each variable is probably a minimum in samples of dissimilar size; it is unknown whether variance and sample-size relationships differ between types of igneous rock. There are no sound bases at present for deciding what sample sizes are the most desirable; if a specified level of reproducibility for each variable is required, samples of dissimilar size are probably needed for each variable. Sample size used and information about variance have rarely been recorded by petrographers. Additional complications stem from the impossibility of every geologist measuring with equal accuracy and precision. Two investigators studying the same samples may provide equally accurate, but wrong, measures because, for example, they observed very accurately with measuring tapes that stretched differently with age, or with spectrometers calibrated wrongly. Again, precision may be poor because measurement error with the tools available is large, relative to the variability and amount of a component. For contouring (surface fitting), interpolation, etc., many mining, petroleum and other geologists use the

Fig. 2. Variation of ~O wt% in part of Older Granite, Donegal, Ireland, based on 110 analysed samples; without any knowledge of the variables' levels of variance, both maps show equally valid manual contours (after Whitten, 1966b, fig. 4).

VARIABILITY OF IGNEOUS ROCKS AND ITS SIGNIFICANCE

weighted-moving-average methods of Kriging (geostatistics), for which construction of variograms is an initial step. Variograms portray the variance of a dependent variable across the study area, commonly representing data for samples of one convenient size. It is unusual for levels of variance in samples of dissimilar size to be examined.

5. PARTITIONING AND CLASSIFICATION Naming schemes are provided in most petrology and petrography textbooks. It is necessary to differentiate clearly between naming based on classification and on partitioning (Whitten, 1987; Whitten, Bornhorst, Li, Hicks & Beckwith, 1987), processes that are distinct mathematically (see summary in Jernigan & Srihari, 1983, p. 1112). By definition, classification differentiates genetically distinct rocks. Sedimentary rocks are defined either genetically in terms of conditions during formation and/or distinctive parental materials (classification), or descriptively in terms of observed physical and chemical composition and spatial variability (partitioning). Most traditional and modem igneous-rock nomenclatures (e.g. Johannsen, 1931; Cox, Bell & Parkhurst, 1979) used descriptive variables to erect arbitrary classes and terminology (partitioning) that may fortuitously have genetic significance. When attempting to classify igneous rocks, genetically significant variables are unlikely to be obvious intuitively, without reference to process-response models, experimental data, etc. Variables measured most commonly may, or may not, be useful. Although Si02 wt% is of prime importance for most published nomenclature systems, rocks with large Si02 range can evolve from a common parent magma (e.g., by fractional crystallization, partial melting). If separation into classes is based on arbitrary variables (independent of genetic differences), the resulting partitioning can be misleading, especially if deemed to be classification. Classification of plutonic rocks involves distinguishing, a priori, on the basis of previous knowledge of, or theory about, petrogenesis. Defining variables, variable values, and representative examples are then specified for each class to allow new samples to be classified in the scheme. Examples of classification include use of saturation with respect to Si0 2 , normative Q-Ab-Orfeldspathoid diagrams, or the basalt tetrahedron. Shand (1947) advocated strongly the classification of eruptive rocks on the basis of phase petrology. Vistelius' (e.g. 1972) use of mineral-grain transition probabilities to differentiate 'ideal granites' from other granitoid rocks was classification, because critical mineral-grain transition probabilities were based on Tuttle & Bowen's (1958) experimentally established phase relationships. The importance of Vistelius' thinking has not been recognized widely; his use of simple, but somewhat outdated, experimental data may have distracted attention from his useful approach. Correlation of observed tectonic setting with certain analysed variables (trace- and majorelements and petrography) of basaltic and andesitic lavas

7

yielded classifications (e.g. Pearce & Cann, 1973; Rogers, Suayah & Edwards, 1984; Shervais & Kimborough, 1985). Partitioning involves identifying arbitrary classes in terms of variables specified by class definitions that are not influenced by a priori considerations. New samples can be included in the defined empirical classes. Commonly, many petrologists and petrographers have used easily measured variables for primary partitioning of igneous rocks, a process wrongly called classification. Because of its abundance, Si0 2 has been used frequently, together with Na20, K20, A1 203, CaO, etc. The space of binary, ternary, and quaternary diagrams based on such oxides and also on modes (or their sums, ratios, or transformations) has often been subdivided arbitrarily, with the resulting fields being used for igneous-rock partitioning (called 'classification') (e.g. Chayes, 1957; Streckeisen, 1976; Cox et al., 1979, fig. 2.2). Additional, or different, elements or modal components have been included frequently to subdivide particular rock groups. Cluster analysis 'classification procedures' of LeMaitre's (1982, p. 170) numerical-petrology textbook is an example of partitioning. In general, if different subsets of elements and/or modal variables (or functions or weighting of them) are used, wholly dissimilar 'classification' (i.e. partitioning) results; many such partitionings separate genetically related rocks. If lines within distribution diagrams (e.g. Harker-type or analogous diagrams) have explicit genetic significance, the separated domains represent classification. Chappell & White (1974) and Pitcher (1982) recognized 1- and S-type granites as derived by partial melting from, respectively, primary unweathered oceanic igneous crust and cratonic crustal material reflecting at least one weathering cycle. Barker (1984) advocated recognition of genetically different F-, FA-, G-, 1-,R-, and S-types without suggesting which variables permit the classification. If, indeed, such granitoid types are genetically dissimilar, a rational classification scheme would separate rocks consanguineous with each. Because processes resulting in plutonic rocks are commonly not observable, developing a significant classification for such rocks is almost always an ill-posed statistical problem (ct. Jernigan & Srihari, 1983, p. 1112; Whitten et al., 1987, p. 338).

6. COMPOSITIONAL DATA Compositional data for variables that have a constant sum are closed, but are open if not summing to a constant. Whole-rock chemical analyses expressed as weights percent are closed because the oxides (or elements, etc.) sum to 100%. A necessary result of closure is that at least one negative correlation between the variables must exist (Chayes, 1960). The consequence is obvious in, say, a quartz-feldspar aggregate where, if quartz increases, feldspar percentage decreases; the two components are negatively correlated. Inherent correlation relationships persist when more than two variables occur, but are much more complex (Chayes, 1971). Although relationships

8

E. H. T. WHITTEN

between trace elements have not been studied exhaustively, Chayes (1983) showed that some positive correlations commonly observed between them are probably products of closure because, if pre-closure variances of two trace elements are sufficiently small, their closed equivalents are correlated positively (cf. Rock, 1988, p. 204). Inventory is reasonably simple when observed variables are independent among themselves; if they are not independent, relationships are significantly more complex. Unfortunately, most compositional data used in petrography and petrology, being proportions expressed as percentages (e.g. MgO wt%, feldspar volume percent) or parts per million, are dependent on each other. Although, mathematical geologists and statisticians have cautioned against basing conclusions on percentage data for half a century, most petrologists (apart from a few involved with economic geology, metasomatism, and basalt petrology) rely on such data. Most standard statistical tests used in petrology assume the variables measured for each sample are normally distributed and independent of each other; results using dependent sample data are unsafe. The common practice of re-percentaging chemical data (sometimes several times) compounds the closure problem. Inherent correlations between closed data affect their relationships on standard distribution diagrams severely. Numerous statistical studies showed what should not be done with closed geological data (e.g. Chayes, 1971; Aitchison, 1982, 1986), or what has been done wrongly (Aitchison, 1990, p. 487). However, the precise impact of closure on petrological conclusions was made clear only recently (e.g. Pearce, 1968; Russell & Nicholls, 1988; Russell & Stanley, 1990; Aitchison, 1992). Closure can be avoided in at least three ways, namely: (a) transform the standard (closed) percentage data, (b) use different variables that are free of, or avoid, closure, or (c) use molar ratios. These possibilities and their advantages and problems are now examined.

Transformation of closed data for variables It is relatively easy to create open data by transforming closed data mathematically, as demonstrated by Aitchison (e.g. 1986; 1990). He asserted petrography is concerned only with relative magnitudes, or ratios, of constituents (variables) and that it is not meaningful to consider individual components in isolation. Because of this assertion, Aitchison and many following him unfortunately used compositional data in an extensive literature as a technical term (that means closed data by definition) for all data for igneous rocks. Although petrographers may reasonably not agree with the assertion, if accepted, it is appropriate to use ratios of the raw closed data; as Aitchison pointed out, this is simpler after transforming the ratios to log-ratios (following standard statistical practice). To this end, each variable in a closed array can be divided by a particular variable (e.g. divide all other major-oxide percentages by MgO percentage); logarithms

of such ratios comprise open data - in fact, a set of wholly new variables for the sample (cf. N. 1. Fisher in Aitchison, 1982, p.165). Other transformations yield open data (for additional new variables) and new dissimilar inter-sample relationships (e.g. different, mathematically correct, partitioning created by, say, cluster analysis, cf. Tauber, 1999). Whether the new variables produced by transformation (e.g. log-ratios of measured variables) are useful geologically depends on the objectives of the investigation. Process-response models prescribe the particular variables needed, so a log-ratio variable is irrelevant unless expressly dictated by the genetic model, which is unlikely. For inventory (e.g. maps, summary graphs and statistics, various types of partitioning), there is no reason why any convenient, interesting, compositional data (variable/s) should not be used. In general, each different set of variables (and particularly each different type of variable) yields dissimilar results and patterns when a particular inventory method is used. Statistical manipulation of transformed data commonly leads to results that are difficult (or impossible) to interpret in terms of the original samples, while extrapolation to the original sampled population can be fraught, and drawing geological inferences about the target population is unrealistic. Despite the numerous mathematical-geology publications and symposia in which log-ratio data (so-called compositional data) have been used for geological analyses, the practice is not recommended.

Different variables

When whole-rock specific gravity is different for each sample, chemical constituents expressed as weights per unit volume are open, because their sum is different for each sample (i.e. the total sample weight per 100 cm3 is different for each sample). Such open variables are appropriate for use in relevant process-response models and inventories. Commonly, the variance of whole-rock specific gravity is not small for granitoid samples of a target population. Complex partitioning of elements in different minerals during crystallization and neocrystallization of igneous rocks causes specific gravity to vary significantly from sample to sample, and not to have simple dependent relationships with the oxide-weight percentages. The variable content of (OH) within various less-dense minerals is also significant (Whitten, 1993, 1995). Grams 100 cm- 3 (i.e. wt% x whole-rock specific gravity) is not equivalent to an Aitchison-type transformation of wt%, unless specific gravity is a constant or correlated directly with weight percentages of the chemical constituents. Some mathematical geologists have contended g 100 crrr-' data are closed on the assumption that specific gravity is essentially constant within rock suites. However, the quantitative importance of correcting geochemical data for volume and density changes

9

VARIABILITY O F I GNEOUS ROCKS AND ITS SI GNIFICANCE

accompanying metasomatism and metamorphism was demon strated by Gresens ( 1967). It is widely recognized that amo unts in similar volumes of rock (e.g. g 100 crrr' ), rather than amount s in samples of equal weight (wt%) and dissimilar size, are needed in metamorphic, metasomatic , and economic geology (e.g. Attoh , 1973; Appleyard,1980; Gibson, Watkinson & Comba, 1983; Morton & Nebel, 1984; Elliott-Meadows & Appleyard, 1991). For similar reasons, Walker (1940) and Tyrrell (1948, 1952) used g 100 cm- 3 to portray correctly the differentiated Palisades and Lugar Sills, respectively. Unfortunatel y, few specific-gravity data for igneous-rock suites have been published; more are needed urgently. Modal data as used in petrography are closed. Cross, Iddings, Pirsson & Washington (1903 , p. 147) introduced mode for actual mineral composition and implied mineral wt% should be used. Currently, most modal data are recorded in volume percent. (cf. Chayes, 1956; Jarai, Kozak & Rozsa, 1997). Modal data would be open if expre ssed as weights per unit volume (modal wt% x whole-rock specific gravity) , but data for such variables are unavailable.

Molar ratios Pearce (I 968) and Nicholls (1988) showed convincingly that molar ratios with a common constant denominator can display actual chemical variability of igneous rocks unequi vocally (unlike chemical wt% data). Molar ratios can be calculated from oxide wt% and/or trace-element ppm data . For each of the i oxides in a rock, tlIe element fraction, e., is: (oxide wt%)(number of cations in oxide) / (molecular weight of oxide). One of the i element s, k, is identified as constant; its element fraction, ek, serves as denominator for the (e - I)

(A)

molar ratios, e j lek' of the other elements. (When element wt% or ppm is used, the arithmeti c for calculating e;lek is simplified.) The meth od requires , inter alia (a) evidence of geochemic al variability having developed within initially homo gene ous parent material compri sin g the target populati on (e.g. zonation developed within magma of a granitic pluton), (b) that at least one element did not participate in the material tran sfer (i.e. is cons tant throughout the target population), (c) high-precision chemical data, and (d) sampling adequate to establish spatial homogeneity of constant element/s in the target populati on. Molar ratios are no panacea for avoiding closed data, because at least one constant component is needed and very few data are available for granite components that might be candidates . Depending on the objective s of a study, the target population might be defined as the whole lithic unit, or the sampled population, etc. (Although molar ratios have mainly been used, the amount of an element, of its oxide or sulphide , the volume or mass of the system, or any other extens ive variable could be used as common denominator, Pearce, 1968, p. 148.) Russell & Nicholls (1988 ) and Stanle y & Russell ( 1989) used K, Ti and/or P (separately or in combination s) as constant denominator/s to evaluate models for basalt suites. Russell & Nicholls ( 1988), Nicholl s (1988), and Stanle y & Russell (1989) demon strated the effectiveness of molar ratios (with K, Ti, or P as con stant denominator/s) for portraying accurately the successive crystallization stages of various basic igneous rocks. Madei sky & Stanley (1993, p. 1137) also used molar ratios to show actual Ca, Na, K, and AI variations resulting from postulated fractionation, crystal sorting, etc., of feldspar and biotite in rhyolites. For samples from postulated granitoid suites, Lachlan Fold Belt, Australia, Whitten (I 996) compared actual chemical trends with those shown by traditional wt% diagrams (Fig. 3);

(B)

~1~~· ~':1(. ~ I

"""""-I. ""0~ .. ~

25

1 "">I: '1L c~~" 1 I

•

'"

~

.

;Y'

Q

~

2'

Fe 15

.>

.

.:

_ .-.fu-

ce :_ ...,....-

':" " " Mg

r:

e

N~ ~~i! .: :~ :.:: 1 dl: ~ ::- . -WL ·l: K

'f>.-"-

2

. .... .>: 1

I II

~5 ' - .- .: .S.!._..... ,

Si Wt%

II

I < >: ·tMg Na ~a. ,_: ..... K3 ' Y"" " .~.

- " '.-. 1,- "" -

1 .

21

Si /Ti

•• .

fsr

§L. . .•

.1 15

Fig. 3. (A) Jindabyne Suite (excluding Pendergast Pluton); lines indicate approximate slope for each variable. (8) Moruya Suite represented by Moruya and Tuross Head Pluton samples only. (a) Traditional element wt.% Harker diagram; (b) Molar-ratio Harkertype diagram with Ti (conserved element) as denominator for abscissa and ordinates (e.g. Alffi) (after Whitten, 1996, figs 4 and 5).

10

E. H. T. WHITTEN

Chappell, White & Wyborn (1987) contended the more felsic rocks of these suites evolved from the initial magmas, with quartz, potassic feldspar, and albitic plagioclase dominating later moieties, which implies that, as Si increases, so does K, Na, AI, and Sr, as is reflected by Fig. 3A(b) and B(b).

Some available granite datasets It is unrealistic to review all chemical and mineralogical data available for igneous rocks, but a few examples are appropriate. When 'rapid' X-ray analytical methods were developed, Mercy (e.g. 1960) demonstrated broad geographical variations of chemical components in some Donegal granites, Ireland, on the basis of aggregate samples collected from traverses extending a few kilometres. For Lakeview Mountains granitoid pluton, southern California, Morton et al. (1969) recorded 162 chemical analyses, whole-rock specific gravities and detailed information about their sampling plan. Baird et at. (1979) tabulated many hundred chemical analyses and specific gravities for the Peninsula and Transverse ranges, southern California. Major- and trace-element analyses and specific gravities for 305 Archaean submarine volcanic rocks, Ben Nevis Township, Ontario, are available on diskette from Dr Eric Grunsky (cf. Grunsky, Easton, Thurston & Jensen, 1992). Although not accompanied by specific-gravity data, several thousand excellent major- and trace-element analyses for granitoid and associated rocks have been made at the Australian National University; numerous subsets were published by Prof. B. W. Chappell and co-workers (e.g. White, Williams & Chappell, 1977; Chappell, 1978; Hine, Williams, Chappell & White, 1978; Beams, 1980). Chayes (e.g. 1985) made available enormous worldwide databases. In the majority of cases, recorded analyses are statements about the samples actually collected only, rather than about sampled or target populations. The target populations may, in fact, have been a set of exposures, or the whole originally crystallized pluton, lava flow, etc. In other cases, the object of enquiry (target population) comprised much more vast entities; for example, many Australian granitoid analyses were used to draw conclusions about suites thought to comprise several plutons (e.g. Chappell, 1984).

Granitic suites The term suite has been entrenched in petrological and petrographical literature for almost a century (e.g. Harker, 1909) having been used descriptively and/or genetically; it has rarely been well defined. For over two decades, suite has been used specifically for granitoid assemblages. For example, White, Chappell and co-workers (e.g. White & Chappell, 1983; Chappell, 1984) identified numerous granitoid suites that are differentiated clearly on various standard Harker-type diagrams. Their definition of suite

varied slightly, but Chappell, White & Hine (1988, pp. 506-7) asserted: 'Some recognizably separate plutons [in southeast Australia] share distinct textural, modal and chemical features, and these are grouped into suites .... Chemical criteria are a rigorous test of a suite; members of a suite show complete chemical coherence for every element, although the range in composition may vary from one unit to another .... Rocks within a suite must also have similar isotopic compositions .... Members of a suite are consanguinous [sic] and each suite is considered to have been derived from a specific narrow range of source-rock compositions.' Thus, partitioning was used to identify their suites, which are used as surrogates for groups resulting from classification. Where suites have not yet been postulated, similar objective or subjective partitioning would permit real suites to be identified for any large set of analyses for a batholith. Whitten et at. (1987, p. 336) concluded that: , .. Suite, as a descriptive entity, has no meaning apart from the specified variable, or set of variables, by which it is defined. Different sets of (descriptive) suites (defined by different variable/s) coexist in the same assemblage of igneous rocks. It is unrealistic to enunciate a genetic scenario to account for one set of descriptive suites, without concomitantly embracing the other coexisting sets of descriptive suites (defined by different variables or sets of variables)' . For example, cluster analysis might seem convenient for recognizing granitoid suites on the basis of chemical analyses of a sampled population. As a result of intensive mapping of Bega, Gabo and Moruya batholiths, SE Australia, by Prof. B. W. Chappell and co-workers, analyses for 32 elements and complete modes were available for 304 samples collected from arbitrary outcrops within 102 individual named plutons (with areas ranging from 970 to 0.1 krrr'); some maps and petrographic and chemical data were recorded by Beams (1980). Use of BMDP Cluster Analysis program P2M (Dixon,1981), which has an embedded normalizing algorithm, necessarily produced clusters of samples, some of which correspond to suites previously identified by Chappell and co-workers, and others which do not (Whitten, 1985). Commonly, in cluster-analysis-type computer programs, normalization gives each variable equal weight, although for igneous petrology there is no a priori reason why elements should be equally important and receive equal weight. Hence, particular suites identified descriptively, while real and of possible value for inventory and mapping, may have little petrogenetic or classificational significance. Different real suites result from use of different observed variables or weighting of the same variables (Whitten, 1991). Also, the results are questionable when closed data are used for cluster analyses and similar partitioning (cf. Aitchison, 1986, p. 300).

11

VARIABILITY OF IGNEOUS ROCKS AND ITS SIGNIFICANCE

7. AN EXAMPLE OF TESTING A MODEL Numerous hypotheses for granite genesis could be tested. For illustration, use is made of the restite (unmixing) model introduced by White & Chappell (1977). Although not accepted by all (cf. Wall, Clemens & Clarke, 1987), this model is chosen because it was used to interpret plutons of the Lachlan Fold Belt, Australia, and suitable original data are available.

The restite model White & Chappell (1977) used the restite model to explain observed systematic variations of certain granites which they believed to be products of partial melting in the crust that yielded a granite-melt phase in equilibrium with residual phases (restite). Chappell et at. (1987, p. 1113) asserted: 'progressive separation of melt and restite ... during movement of the magma accounts for the very systematic linear chemical variations found in granites from both single plutons and groups of plutons, or suites. The process of differentiation amongst members of a suite is, according to this model, an unmixing process in which the melt phase of the magma progressively clears itself of restite but does not itself change its composition appreciably'. In this remagmatization process, '. . melt forms only a fraction of the magma which, before any fractionation of melt from restite occurs, has the same composition as the source rocks' (Chappell & Stephens, 1988, p. 77). Thus, essentially homogeneous material was thought to yield melts rich in components of quartz, potassic feldspar, sodic plagioclase, etc. The theoretical stoichiometry for such progressive enhancement can be depicted accurately on molar-ratio diagrams, and the restite model tested if suitable sample data are available.

Jindabyne granitoid suite Chappell and co-workers' various publications for the Jindabyne Suite, Lachlan Fold Belt, SE Australia, detailed numerous petrographic features interpreted by them as supporting the restite model. On the basis of analyses of 24 samples, mainly published by White et at. (1977), Hine et al. (1978) and Chappell (1984), it was shown (Whitten, 1996) that, using Ti, Zn, Ga, and/or Zr as constant (conserved) components, the available chemical data are compatible with a single petrogenetic hypothesis, and AI, Na, K, and Sr increase as Si increases, while Ca and Fe barely change with Si increase (Fig. 3). Following Nicholls' (1988) approach, and using Ti as conserved element, a plot of 3KfTi vs Si/Ti yields a linear trend with slope 1.0 if potassic feldspar (KAlSipg) is the only phase abstracted from, or added to, a homogeneous granitoid suite (assuming temporarily that K is not a component of another mineral in the system). Similarly, if plagioclase and K-feldspar are both added or subtracted, a

plot of [3K + (2Ca + 3Na)]ffi vs Si/Ti

(i)

has slope 1.0 (Table 2). For the restite hypothesis, quartz is also mobile, implying [3K + (2Ca + 3Na) + Sij/Ti vs Si/Ti

(ii)

is linear with slope 1.0 and provides an appropriate test of stoichiometry (provided elements in the numerator are not components of other minerals in the system). Jindabyne Suite data on a graph of equation (i) are linear with slope less than 1.0 (Fig. 4A(a)), while for equation (ii) the slope is greater than 1.0 (Fig. 4A(b)). If the latter had slope 1.0, the data would be compatible with the hypothesis that chemical variations in these samples are due to addition or subtraction of K-feldspar, plagioclase and quartz from a closed system. Use of Zn as conserved element (instead of Ti) gives analogous results (Fig. 4B(a) and (b)). Biotite (but not muscovite) also occurs in the Jindabyne Suite. The K:Si ratio in biotite and K-feldspar is the same, so trends in Figs 4A(b) and B(b) accommodate biotite involvement. However, petrographic evidence suggested hornblende and (biotite + quartz) are antipathetic in these rocks, so Ca and Na of hornblende must be involved in the stoichiometry. That is, [3K + (2Ca + 3Na) + Si - 4(Ca + Na)]ffi vs Si/Ti (iii) i.e. [3K + Si - (2Ca + Na)]ffi vs Si/Ti would yield slope 1.0 if biotite and hornblende, in addition to quartz and feldspars, are involved in the restite-model process. Available Jindabyne data are not incompatible with the restite model, as expressed by equation (iii), because they have a slope close to 1.0 (Fig. 4A(c)); using Zn as conserved element yields a similar slope (Fig. 4B(c)). If information were available about (a) the precision and accuracy of individual chemical analyses (particularly of Ti and Zn) and (b) the relationship of analysed samples to the sampled and/or target populations, it would have been realistic to use statistical tests of linearity, approach to slope 1.0 and approach to graph origin, etc. Such information being unavailable, formal tests are unrealistic. Because only arbitrary samples are available from the

Table 2. Mineral indices that produce linear slope 1.0 if the indicated minerals control chemical variation of a co-genetic granitoid assemblage Mineralls

Ordinate

Abscissa

K-feldspar or biotite Plagioclase Quartz Hornblende

3 KIn (2Ca + 3Na)/n

Sun Sun Sun Sun

n

Sun 4(Ca+ Na)/n

= conserved element (such as, e.g. Ti or Ga, if established)

12

E. H. T. WHITTEN

19

.'

/

/b

A

B

J

23 15

.. .'

I. :to

'

j 11

, ,7t

.j""

. £'\0

•/ ':-..0"'<

11

.!

,C /.

/ CO

8. CONCLUSION

;.. ....

7

possible models, can be tested formally, The molar-ratio approach is powerful. However, it can be used only when precisely analysed, conserved elements occur throughout the pluton, suite, or samples used, and the target population is reflected by the samples. Lack of suitable chemical analyses is a problem in applying the method more widely. Conserved elements may not exist in some (possibly many) granitoid and other rocks.

S·I/Ti "

12

' S I/Z n

Fig. 4. Testof conformity of Jindabyne Suite chemical data with the restite model, assuming different minerals are involved. (A) Conserved element n is Ti (units on axes x 10); (B) conserved element n is Zn (units on axes x 100). Abscissae are: (a) [3K + (2Ca + 3Na)]/n, representing K-feldspar and plagioclase; (b) [3K + (2Ca + 3Na) + Si]/n, for K-feldspar, plagioclase, quartz and biotite, and (c) [3K + Si - (2Ca + Na)]/n, as (b) but with hornblende undergoing replacement.

sampled population (itself dictated arbitrarily by outcrop availability, rather than representing accurately the whole target population), it is not meaningful to use statistical methods to identify trends on Fig. 4; the gradients are estimated. Although the molar ratio data are compatible with the model, this neither proves it represents the petrogenetic process that operated, nor that it would be compatible with a larger sample representative of the target population (suite).

Usefulness of molar ratios The preceding section illustrates one way in which a conceptual petrogenetic model, as one example of many

A pragmatic approach to petrography and petrogenesis suggests that the dissimilar demands of hypothesis testing and inventory require clear recognition when defining initial objectives for economic and research endeavours. Sampling and closure cannot be ignored when generating data to study igneous rocks and their genesis. 'Results' can always be obtained without proper attention to these issues, but it is necessary to distinguish between A result and THE correct result. These subjects have not proved popular heretofore because (a) incorporating them involves departing from entrenched procedures, and (b) two new, widely available, technologies have eclipsed attention. These technologies are powerful computing facilities that permit results to be obtained by manipulation of existing data and fast analytical facilities that generate data for the customary variables from easily available samples. The manner in which differentiation between partitioning and classification have been overlooked in igneous petrology, but not in metamorphic and sedimentary petrology, illustrates how entrenched procedures and thinking have been masked by modem technologies; it is interesting that, although its significance was not recognized widely at the time, Vistelius' (e.g. 1972) work on 'ideal granite' was an early attempt at addressing this issue.

ACKNOWLEDGEMENTS John C. Tipper made useful suggestions on an early draft of this paper. Extremely constructive comments by Donald B. McIntyre in the review process were greatly appreciated.

REFERENCES AGER,D. V. 1970. On seeingthe most rocks. Proceedings ofthe Geologists' Association, 81, 421-427. APPLEYARD, E. C. 1980. Mass balance computations in metasomatism: metagabbro/nepheline syenite pegmatite interactions in northern Norway. Contributions to Mineralogy and Petrology, 73, 131-144. AITCHISON, J. 1982. The statistical analysis of compositional data. Journal ofthe Royal Statistical Society, 44B, 139-177. - - 1986. The statistical analysis of compositional data. Chapman & Hall, London. - - 1990.Relative variation diagrams for describing patternsof compositional variability. Mathematical Geology, 22, 487-511. - - 1992.On criteriafor measures of compositional difference. Mathematical Geology, 24, 365-379.

AITOH, K. 1973. 'Metamorphic reactions in the Michigamme Formation, Iron County, Michigan'. PhD thesis, Northwestern University, Evanston, Illinois. BAIRD, A. K., BAIRD, K. W. & WELDAY, E. E. 1979. Batholithic rocks of the Northern Peninsula and Transverse Ranges, southern California: Chemical composition and variation. In (Abbott, P.L. & Todd, V.R.; eds) Mesozoic Crystalline Rocks, Peninsula Ranges Batholith and Pegmatites Point Sal Ophiolite. San Diego State Univ., Department of

Geological Sciences, San Diego,California, 111-132. - - , MCINTYRE, D. B. & WELDAY, E. E. 1967.Geochemical and structural studies in batholithic rocks in southern California, Part II: Sampling of the Rattlesnake Mountain Pluton for chemical composition, variability, and trend analysis. Geological Society ofAmerica Bulletin, 78, 191-222.

VARI ABILITY O F I GNEOUS ROCKS AND ITS SIGNIFICANCE

BARDOSSY. G.1997 . Some fields of geoma thematics as seen by a geologi st (is there a bridge bet we en geo logists and mathematicians?). In (Pawlows ky-Gl ah n.V; ed.) Proceedings of IAMG '97. International Association for Mathematical Geology, Barcelona . 36-56. BARKER. F. 1984. Letter design ation of granites ; process vs proto lith. EOS. 65. 1151. BATEM AN . P. c.. CLARK, L. D.• HUBER . N. K.. MOORE. J. G . & RINEHART. C. D. 1963. The Sierra Nevada Batholith; a synthesis of recent work across the central part. US Geol ogical Su rvey Profes sional Paper 414D. BEAMS. S. D. 1980. ' Magmatic evo lution of the southeast Lachlan Fold Belt . Australia'. PhD thesis. La Trobe Universi ty. Australi a. CHAPPELL . B. W. 1978. Granitoids from the Moonbi district. New Eng land batholith. eastern Australia. Journal of the Geological Society ofAustralia. 25. 267- 283. - - 1984 . Source rocks of 1- and S-type granites in the Lachlan Fold Belt. southeastern Australia. Philosophical Transactions. Royal Society. series A. 310. 693-707. - - & STEPHENS, W. E. 1988. Origin of infracrustal (l-type) granit e ma gmas. Transactions, Royal Society of Edinburgh, Earth Sciences, 79 . 71-86. - - & WHITE, A. J. R. 1974. Two contr asting granite types . Pacific Geology. 8.173-174. - -. - - & HINE , R. 1988. Gran ite pro vinces and basement terranes in the Lachlan Fold Belt . southeas tern Australia. Australian Journal of Earth Sciences. 35 , 505-521. - -, - - & WYBORN, D. 1987 . The importance of residual sou rce mate rial (restite) in granite petr ogene sis. Journal of Petrology. 28. 1111-1138. CH AYES, F. 1956. Petrographic modal analysis. John Wile y & Sons. New York . - - 1957. A pro visional reclassification of granit e. Geological Magazine. 94 , 58--68. - - 1960 . On correlation between variables of constant sum. Journal of Geophysical Research. 65, 4185-41 93. - - 1971. Ratio correlation: A manual for students ofpetrology and geochemistry. Univers ity of Chi cago Press. Chicago. - - 1983. ' On the possible significance of strong positive co rrelations between trace elem ent s' . Unpubli shed type script , Geoph ysical Laboratory, Washington. DC. - - 1985. IGBADAT: A world dat a base for igneous petrology. Episodes. 8, 245-51. CL EMENTS. J. M. 1903. Atlas to accompany Monograph XLV on the Vermilion iron-bearing district of Minnesota. US Geolo gical Survey, Washington, DC. COCHRAN . W. G., MOSTELLER. E. TUKEY, J. W. & JENKINS. W. O. 1954. Statistical problems of the Kinsey Report on sexual behavior in the human male. American Statistical Associ ation . Washington . DC. COX. K. G.. BELL. J. D . & PARKHURST, R. J. 1979. The interpretation of igneous rocks. George Allen & Unwin. Londo n. CRO SS. C . W.. IDDINGS . J. P., PIRSSON. L. v. & WASHINGTON, W. S. 1903. Quantitative classification of igneous rocks. University of Chi cago Press, Ch icago. DACEY. M . F. 1975. Model of recu rring random walks for sediment tran sport . In (Wh itten. E. H. T.; ed .) Quantitative Studies in the Geological Sciences. Geological Society of America Memoir. 142 . 105-119. DAVID . M. L. R. 1977. Geostatistical ore reserve estimation. El sevi er Scientific Publishing Co .• New York . DIXON, W. J. 1981. BMDP Statistical Soft Ware. University of Cal iforn ia Press, Los Angel es. ELLIOTr-MEADOWS. S. R. & APPLEYARD, E. C. 1991. The a lteratio n geochemistry and petrology of the Lar Cu-Zn

13

dep osit. Lynn Lake Ar ea . Manitoba. Economic Geology, 86, 486-505. FANG, J. H. & CHEN, H . C. 1990. Uncert ainties are bett er handled by fuz zy arithmetic. Bulletin of the American Association of Petroleum Geologists, 74. 1228-1 233. GIBSON. H. L.. WATKINSON. D. H. & COMBA. C . D. A. 1983. Silicification: Hydrothermal alteration in an Archean geo therma l system within the Amulet Rhy olit e Formati on. Norand a, Quebec. Economic Geology. 78. 954-971 . GRES ENS. R. L. 1967. Composition-volume relati onship s of meta somatism. Chemical Geology, 2. 47--65. GRIFFITHS. J. C. 1962. Statistical method s in sedimentary petro graph y. In (Milner H. B.; ed .) Sedimentary petrography. I , Macmillan Co .• New York, 565-617. GRUNSKY. E. C .• EASTON. R. M .• THURSTON, P. C. & JENS EN . L. S. 1992 . Characterization and statis tic al clas sification of Archean volcanic rock s of the Superior Province using major element geochemistry. In (Thurston. P. C .• William s. H. R., Sutcliffe, R. H. & Stott, G. M .; eds) Geology of Ontario. Ontario Geological Survey, Special Volume , 4, part 2, 1397-1438. HARKER. A. 1909 . The natural history of igneous rocks. Methuen Co .• London. HINE, R.• WILLIAMS . 1. S., CHAPPELL. B. W. & WHITE. A . J. R. 1978. Contrasts between 1- and S-type gra nitoids of the Kosciusko Bath olith . Journal of the Geological Society of Australia, 25. 219-234. HOWLAND. A. L.. GARRELS , R. M. & JONES. W. R. 1949 . Ch romite dep osit s of the Boulder River are a, Swe etgrass County, Mont. US Geological Survey Bulletin, 948C, 63-82. JAMES . W. R. 1975. Multilayer Marko v mixi ng models for studies of coa stal contamination. In (Whitte n. E. H. T.; ed. ) Quantitative Studies in the Geological Sciences. Geol ogical Society of America Me moir. 142 , 12 1-136. JANOUSEK. v.. ROG ERS , G.. BOWES. D. R. & VANKOVA, V. 1997. Crypti c trace-element variation as an indicator of reverse zoning in a granitic pluton: the Rican y granite, Cze ch Republic. Journal of the Geological Society, London. 154, 807-8 15. JAR AI. A.• KOZAK, M . & ROZSA, P. 1997. Comp arison of methods of roc k-microscopic grain-size determination and qu ant itative analy sis. Mathematical Geology, 29, 977-99 1. JERNIGAN, M. E. & SRIHARI. S. N. 1983. Pattern recogn ition . In (Ralston, A. & Reilly. E. D .• Jr.; eds) Encyclopedia of computer science and engineering (2nd edn) . Von Nostrand Reinhold. New York. JOHANNSEN. A. 1931. A descriptive petrography ofthe igneous rocks (four volumes). University of Chicago Press. Chicago. JO URNEL, A. & HUIJBREGTS, C. J. 197 8. Min ing geostatistics. Academic Press , New York. KACEWICZ. M . 1993. Mathematics between source and trap: uncertainty in hydro carbon mi grat ion modeling. In (Davis. J. C. & Herzfeld. U. C.; eds) Computers in geology - 25 years of progress. Oxford University Press, New York. 69-84. KRAUSKOPF. K. B. 1968. A tale of ten plu tons. Geological Society of America Bulletin, 79, 1-18. KRIGE. D. G. 1964. Recent de velopments in South Africa in the appli cation of trend surface and multiple regre ssion techniques to gold ore valuation. Colorado School ofMines Quarterly, 59. 795- 809. KR UMB EIN , W. C. 1960 . The 'geological population ' as a framework for analyzing numerical data in geology. Liverpool & Manchester Geological Journal. 2. 34 1-368. - - J964 . A geo logical process-respon se model for analysis of beach phenomena. Bulletin of the Beach Erosion Board. 17 (for 1963). 1-15.

14

E. H. T. WHITTEN

- - & GRAYBILL, F. A. 1965. An introduction to statistical models in geology. McGraw-Hili Book Co., New York. - - & SLACK, H. A. 1956. Statistical analysis of low-level radioactivity of Pennsylvanian black fissile shale in Illinois . Geological Society ofAmerica Bulletin, 67,739-762. - - & SLOSS, L. L. 1963. Stratigraphy and sedimentation (2nd edn). W. H. Freeman Co., San Francisco. LEMAITRE, R. E. 1982. Numerical petrology: statistical interpretation of geochemical data . Elsevier Scientific Publishing Co., Amsterdam. LEOPOLD , L. B. & LANGBEIN, W. B. 1963. Association and indeterminacy in geomorphology. In (Albritton , C. C., Jr.; ed.) The fabric of geology. Addison-Wesley Publishing Co ., Reading, Mass., 184-192. MADEISKY, H. & STANLEY, C. R. 1993. Lithogeochemical exploration of metasomatic zones assoc iated with volcanichosted massive sulfide deposits using Pearce element ratio analysis. International Geology Review, 35, 1121-1148. MANN, C. J. 1993. Uncertainty in geology. In (Davis, J. C. & Herzfeld, U. C,; eds) Computers in geology - 25 years of progress. Oxford University Press, New York, 241-254. MERCY, E. L. P. 1960. The geochemistry of the Older Granodiorite, Co. Donegal, Ireland. Transactions of the Royal Society of Edinburgh, 64, 101-127. MORTON, D. M., BAIRD, A. K. & BAIRD, K. W. 1969. The Lakeview Mountains Pluton, southern California batholith, Part II : Chemical composition and variation. Geological Society ofAmerica Bulletin, 80, 1553-1564. MORTON , R. L. & NEBEL , M. L. 1984. Hydrothermal alteration of felsic volcanic rocks at the Helen siderite deposit , Wawa, Ontario . Economic Geology, 79,1319-1333. NICHOLLS, J. 1988. The statistic s of Pearce element diagrams and the Chayes closure problem. Contributions to Mineralogy and Petrology, 99, 11-24. PEARCE , J. A. & CANN, J. R. 1973. Tectonic setting of basic volcan ic rocks determined using trace element analyses. Earth & Planetary Science Letters , 19, 290--300. PEARCE , T. H. 1968. A contribution to the theory of variation diagram s. Contributions to Mineralogy and Petrology, 19, 142-157. PITCHER , W. S. 1982. Granite type and tectonic environment. In (Hsu, K. J.; ed.) Mountain building processes. Academic Press, New York, 19--40. POTIER, P. E. & PETIUOHN, F .J. 1963. Palaeocurrents and basin analysis. Academic Press, New York. READ, H. H. 1952. The geologist as historian . In Scientific objectives. A selection from a series of lectures given at Imperial College, London, 1949-51. Butterworths Scientific Publicat ions, London, 52-67. ROCK , N. M. S. 1988. Numerical geology: A source guide, glossary and selective bibliography to geological uses of computers and statistics. Springer-Verlag , Berlin . ROGERS , J. J. W., SUAYAH, I. B. & EDWARDS, J. M. 1984. Tra ce elements in continental-margin magmatism: Pt. 4, Geochemical criteria for ' recognition of two volcanic assemblages near Auburn , western Sierra Nevada , California. Geological Society ofAmerica Bulletin, 95, 1437-1445. ROSENFELD, M. A. 1954. Petrographic variat ion in the Oriskany Sandstone. Geological Society of America Bulletin, 65,95-96. RUSSELL, J. K. & NICHOLLS , J. 1988. Analysis of petrologic hypotheses with Pearce element ratio s. Contributions to Mineralogy and Petrology, 99, 25-35. - - & STANLEY, C. R. 1990. A theoretical basis for the development and use of chemical variation diagrams. Geochimica et Cosmochimica Acta, 54, 24 19-2431.

SHAND, S. 1. 1947. Eruptive- rocks. Thos. Murby & Co., London. SHERVAIS, J. W. & KIMBROUGH, D. L. 1985. Geochemical evidence for the tectonic setting of the Coast Range ophiolite : A compo site island arc-oceanic crust terrane in western California . Geology, 13, 35-38. SMITH, S. c., MORENO, S. A., WALDVOGLE , G. & HUANG , C. I. 1983. Computer reporting of analytical geochemistry (CRAG) and its impact on the strategic mineral s industry. Mathematical Geology, 15, 400 . STANLEY, C. R. & RUSSELL, J. K. 1989. Petrologi c hypothesis testing with Pearce element ratio diagrams : derivation of diagram axes. Contributions to Mineralogy and Petrology, 103,78-89. STRECKEISEN, A. 1976. To each plutonic rock its proper name. Earth Science Reviews, 12, 1-33. TAUBER, F. 1999. Spurious clusters in granulometric data caused by logratio transformation. Mathematical Geology, 31, 491-504. TUTILE, O. F. & BOWEN, N. L. 1958. Origin of granite in the light of experimental studies in the system NaAISiP8-KAISiP8-SiOrH02' Geological Society of America Memoir , 74. TYRRELL, G. W. 1948. A boring through the Lugar Sill. Geological Society of Glasgow Transactions, 21, 157-202. - - 1952. A second boring through the Lugar Sill. Geological Society of Edinburgh Transactions, 15,374-392. VISTELIUS , A. B. 1945. Frequency distribution of porosity coefficients and epigenetic processes in Spiriferrous layers in the oil-bearing region of Buguruslan. CompoRendu (Doklady) Acad. Sci. SSSR., 44, 43--46. - - 1972. Ideal granite and its properties. Mathematical Geology, 4, 80--102. - - & Sarmanov, O.V. 1947. Stochastic basis of one geologically important probability distribution. Comp 'Rendu (Doklady) Acad. Sci. SSSR., 58, 631-634. WALKER, F. 1940. Differentiation of the Palisades Diabase , New Jersey. Geological Society of America Bulletin, 51, 1059-1106. WALL, V. J., CLEMENS, J. D. & CLARKE , D. B. 1987. Models for granito id evolution and source compos itions . Journal of Geology, 95, 731-749. WHITE, A. J. R. & CHAPPELL, B. W. 1977. Ultrarnetamorphism and granitoid genesis. Tectonophysics, 43 , 7-22. - - & - - 1983. Granitoid types and their distribution in the Lachlan Fold Belt, Southeastern Australia. In (Roddick, J. A.; ed.) Circum-Pacific Terranes. Geological Society of America Memoir, 159, 21-34. - - , WILLIAMS, I. S. & CHAPPELL, B. W. 1977. Geology of the Berridale 1:100000 Sheet 8625. Geological Survey of New South Wales, Department of Mines. WHITIEN, E. H. T. 1960. Average composition of granites, the genesis of tektites, and petrogenesis. Nature, 187, 867-868. - - 1961a. Quantitative areal modal analy sis of granitic complexes. Geological Society of Am erica Bulletin , 72 , 1331-1360. - - 1961b . Quantitative distribution of major and trace components in rock masses . American Institute of Mining & Metallurgical Engineers Transactions (Mining), 223, 239-246. - - 1962. The quantitative mineralogical compo sition and variation of the Lacorne, La Motte, and Preissac granitic complex , Quebec, Canada. Journal of Petrology, 3, 1-37. - - 1963 . Application of quantitative methods in the geochemic al study of granite massifs. Royal Society ofCanada Special Publication, 6, 76-123.

VARIABILITY OF IGNEOUS ROCKS AND ITS SIGNIFICANCE

- - 1964a. Process-response models in geology. Geological Society ofAmerica Bulletin, 75, 455--64. - - I964b. Models in the geochemical study of rock units. Colorado School of Mines Quarterly, 59, no. 4, 149-168. - - 1966a. Structural geology offolded rocks. 1. Wiley & Co., New York. - - - 1966b. Quantitative models in the economic evaluation of rock units: illustrated with the Donegal granite and the goldbearing Witwatersrand conglomerates. Transactions of Institution of Mining and Metallurgy, Section B, 75, 181-198. - - 1972. Conceptual models for three-dimensional variability of rock units. Proceedings XXII International Geological Congress, India, 1964, Section 16. International Geological Congress, New Delhi, India, 25-38. - - 1974. Scalar and directional field and analytical data for spatial variability studies. Mathematical Geology, 6, 183-198. - - 1977. Stochastic models in geology. Journal ofGeology, 85, 321-330. - - 1983. Twenty-five years of mathematical geology: a new threshold. Mathematical Geology, IS, 237-243. - - 1984. Objectives of mathematical geology. Proceedings of 27th International Geological Congress, USSR, 20, 1-15.

15

- - 1985. Suites within a granitoid batholith: a quantitative justification based on the Lachlan Fold Belt, S.E. Australia. Geologicky Sbornik (Bratislava), 36, 191-199. - - 1987. Classification and partitioning of igneous rocks. In (Prohorov, Yu. V. & Sazonov, V. V. eds), Proceedings 1st World Congress Bernoulli Society (International Statistical Institute) 1986 (Tashkent, USSR). International Science Publishers.Utrecht, 2, 573-577. - - 1991. Granitoid suites. GeologicalJournal, 26,117-122. - - 1993. A solution to the percentage-data problem in petrology. In (Davis, 1. C. & Herzfeld, U. c.; eds) Computers in geology - 25 years of progress. Oxford University Press, Oxford, 195-206. - - 1995. Open and closed compositional data in petrology. Mathematical Geology, 27, 789-806. - - 1996. Molar-ratio and Harker diagrams in portraying the actual chemical variability of granitoid suites. Journal of the Geological Society of London, 153, 121-125. BORNHORST, T. J., LI, G., HICKS, D. J. & BECKWITH, J. P. 1987. Suites, subdivision of batholiths, and igneous-rock classification: geological and mathematical conceptualization. American Journal of Science, 287, 332-352.

Manuscript received 28 May 1998; revised typescript accepted 17 February 1999