## Abstract

Mineral deposit models strategically guide exploration. The lithologies from which these models are built have genetic connotations. Thus, rock classification must be accurate to ensure that mineral exploration is effective and successful.

Rock classification is based on mineral proportions, and these are commonly determined by: (1) visual inspection, which is subject to large errors; (2) point counting, which is tedious and time-consuming; (3) image analysis of stained slabs or polished thin sections, which is expensive and constrained by the availability of appropriate stains; and (4) image analysis of spectrometric data, which is expensive. These features make rock classification difficult and undermine its quality, thereby negatively impacting geological conclusions and mineral exploration results.

A novel alternative procedure for igneous rock classification involves using whole rock lithogeochemical data for classification on Streckeisen ternary diagrams. This approach employs several calculations that transform: (1) mass-based element concentrations (the original lithogeochemical data produced by the laboratory) sequentially into (2) unstandardized (do not sum to unity) molar element numbers; (3) unstandardized molar mineral numbers; (4) unstandardized volume mineral numbers; and finally (5) standardized (closed; sum to unity) volume mineral concentrations that estimate the mineral modes in rocks. These mineral mode estimates can then be plotted on (projected onto) Streckeisen ternary diagrams, to classify the rocks in the normal manner.

This new approach has advantages over conventional classification strategies, in that it is relatively inexpensive, adaptable to all forms of igneous rocks, quantitative, accurate, and precise. Required petrographic information necessary to conduct such a classification includes only knowledge of chemical formulae of the ‘essential’ mineral assemblage. Essential minerals are, here, considered those minerals having concentrations exceeding 5% in 5% of the rocks under consideration. This criterion allows this lithogeochemical classification procedure to be applicable to a wide variety of igneous rocks.

This lithogeochemical classification procedure has additional applications beyond the classification of plutonic igneous rocks. For example, if an essential mineral assemblage can be identified or hypothesized, classification of felsic or mafic volcanic rocks can also be achieved. Additionally, an essential mineral assemblage does not have to consist exclusively of igneous minerals. As a result, conversion from molar element numbers to molar mineral numbers can be undertaken using many mineral assemblages. This allows analogous lithogeochemical classification to be undertaken for almost any rock type (e.g. clastic sedimentary rocks, using the calculated proportions of quartz, feldspar, and clay minerals). Consequently, lithogeochemical calculation of the essential mineral modes in rocks can be used to establish mineral zoning maps in space or time, allowing exploration geoscientists to create down-hole logs depicting hydrothermal alteration mineral abundances, or surface maps of hydrothermal alteration zones on a mineral property.

To demonstrate this new procedure, results from classifications of metaluminous, peraluminous, and alkaline felsic plutonic and volcanic rocks, and mafic and ultramafic plutonic and volcanic rocks are compared with mineral modes acquired by independent means (visual estimates, point counts, image analysis, spectrometry). These case studies demonstrate that the proposed lithogeochemical classification procedure is as or more accurate than conventional classification methods. Furthermore, because lithogeochemical samples are far larger, and thus more representative than the surfaces used to estimate mineral modes by conventional means, this lithogeochemical classification procedure is also far more precise. The resulting classification is thus especially effective when working with fine-grained rocks where mineral identification and volume estimation is difficult.

Effective mineral exploration requires the collection of reliable geological information from outcrop and drill core. Central to this objective is accurate rock classification and the clues it provides a geoscientist regarding the genesis of the target mineralization, including the environment it formed in.

Unfortunately, accurate rock classification represents a significant challenge to geoscientists because the diagnostic properties of rocks may not be easy to determine. Specifically, the mineral modes used in rock classification are commonly difficult to estimate, and thus susceptible to large errors. Fine-grained texture is also a significant impediment to accurate mineral mode estimates, and obscuring geological processes, such as hydrothermal alteration, metamorphism, and weathering, can alter the mineralogy of rocks. All of these factors complicate rock classification and can reduce its accuracy to inadequate levels.

Use of a novel igneous rock classification procedure that relies on lithogeochemical data, instead of conventional mineral mode estimates, can substantially improve igneous rock classification accuracy and precision and, with more reliable geological data, can significantly improve the effectiveness of mineral exploration.

## Conventional igneous rock classification

A very large number of igneous rock classification methods have been proposed to facilitate both mapping of plutonic and volcanic rocks, and investigating their environments and genesis (Meschede 1986; Sun & McDonough 1989; Pearce 1996; Hollings & Wyman 2005; Wyman & Hollings 2006). Some methods employ both major oxide and trace element geochemical concentrations to assist in: (1) assigning names to igneous rocks (i.e. classification; e.g. Winchester & Floyd 1977; Ewart 1982; Hallberg 1984; Le Bas *et al.* 1986; Pearce 1996); (2) establishing the geochemical affinities of igneous rocks (e.g. Irvine & Baragar 1971; Miyashiro 1974; Floyd & Winchester 1975; Pearce 1996; Frost & Frost 2008); and (3) determining the tectonic environment in which the igneous rocks formed (e.g. Pearce & Cann 1973; Wood 1980; Pearce *et al.* 1981, 1984; Shervais 1982; Pearce 1983, 1996). These methods have generally been used effectively to understand the nature of igneous rocks and provide insights into their genesis.

However, probably the most fundamental igneous rock classification method presently available is that of Streckeisen (1974; Streckeisen *et al.* 2002; Figs 1 and 2). This method traditionally requires quantitative volume mineral mode estimates to assign proper names to (classify) both felsic and mafic magmatic and volcanic rocks on a variety of ternary diagrams. As a result, classification using the Streckeisen technique is different from the above approaches in that it does not take advantage of the generally superior accuracy and precision that lithogeochemical data can provide. As a result, errors in estimates of mineral abundances can significantly undermine the accuracy and precision of Streckeisen classification, forcing its application to be undertaken using expensive and time-consuming point count, image analysis or spectrometric procedures.

One method that attempts to resolve this problem is that of Le Maitre (1976), which uses CIPW norms calculated from lithogeochemical data to classify both plutonic and volcanic rocks on Streckeisen ternary diagrams. This method first requires conversion of the mass-based oxide concentrations into mass mineral proportions using the conventional normative calculation, and then further conversion into the volume proportions necessary for Streckeisen classification. Although this effort represents a laudable and rigorous attempt at igneous rock classification using lithogeochemical data, and exploits the accuracy and precision advantages that lithogeochemistry data can provide, it suffers from several problems that have limited its application. First, some of the minerals used in the norm calculation (e.g. corundum, hypersthene) may not be present in the igneous rocks to be classified. Conversely, some relatively abundant igneous minerals commonly found in magmatic and volcanic rocks (e.g. biotite, hornblende, muscovite) are not normative minerals. Furthermore, none of the normative mafic minerals (olivine, hypersthene, diopside) used to accommodate the ferro-magnesian components of granitoid rocks contain Al, in contrast to the mafic minerals that are most common in granitoid rocks (biotite, hornblende). Unfortunately, these mineral assemblage and mineral composition differences undermine the appropriateness and relevance of Le Maitre's approach because they distort the subsequent lithogeochemical calculations, causing inaccuracies (biases) in subsequent classifications. As a result, the normative-Streckeisen ternary diagram classification method (Le Maitre 1976) has not gained widespread favour as a reliable classification method.

## A new igneous rock lithogeochemical classification method

To address the accuracy and precision problem of mineral mode-based igneous rock classification, a general approach has been developed to provide the volume-based mineral mode estimates of rocks necessary to employ the Streckeisen classification philosophy for igneous rocks on ternary diagrams (Figs 1 and 2) using lithogeochemical data. This procedure is inspired by Le Maitre's CIPW norm-based approach (1976), and it utilizes the matrix methods and strategies employed by Pearce element ratio analysis (Pearce 1968; Russell & Nicholls 1988; Stanley & Russell 1989; Russell *et al.* 1990; Russell & Stanley 1990*a*, *b*; Madeisky & Stanley 1993; Nicholls & Gordon 1994; Stanley & Madeisky 1994). It undertakes a novel strategy, using the set of minerals that are present in substantial proportions in the rocks under consideration, to classify igneous rocks, instead of using the fixed, artificial set of normative minerals employed by Le Maitre (1976). This allows conversion of the mass-based major oxide concentrations into realistic volume-based mineral modes necessary for mineralogical classification on ternary diagrams (Streckeisen 1974; Streckeisen *et al.* 2002).

The proposed procedure uses several intermediate steps (Fig. 3), the first of which involves the transformation of major oxide/element mass concentrations into molar element numbers through division by the corresponding molecular weights. The resulting molar element numbers have the advantage of sharing the same format as mineral compositions, as both are expressed in molar terms. As a result, simple linear combinations of molar *element* numbers can be used to produce molar *mineral* numbers via a change of basis calculation (Strang 1993).

The linear combinations used to convert molar element numbers into molar mineral numbers are constrained (as vectors) to be independent of (projections from) each other. The appropriate linear combinations to achieve these conversions are determined using matrix algebra and the principles of projective geometry, and are further constrained by the compositions of the ‘essential’ minerals in the rocks (those minerals comprising at least 5% of the rock – by volume – in at least 5% of the rocks). As a result, petrographic information derived by macroscopic, microscopic, image analysis, or spectrometric means, or alternatively a petrologic hypothesis confirmed by analysis of lithogeochemical data using molar element ratio analysis methods (Pearce 1968; Russell & Nicholls 1988; Stanley & Russell 1989; Russell *et al.* 1990; Madeisky & Stanley 1993; Stanley & Madeisky 1994), is typically necessary to allow identification of the essential mineral assemblage so that correct conversion from molar element numbers to molar mineral numbers can be achieved. A useful criterion to determine whether the full range of essential mineral assemblage in rocks under consideration has been correctly identified is that the sum of the concentrations of the elements included in the formulae of the essential mineral suites exceeds 95% in 95% of the rocks. Use of these essential minerals to undertake the classification effectively amounts to the calculation of a ‘custom norm’ tailored to the mineral assemblage present in the rocks under consideration. This approach thus provides a significant adaptive advantage over Le Maitre's norm-based classification (1976), because a static mineral assemblage does not have to be used in all cases, as will be demonstrated in the following examples.

Finally, by including in the essential mineral assemblage the minerals used in conventional Streckeisen classification procedures (e.g. quartz, alkali feldspar, and plagioclase (QAP) or feldspathoid, alkali feldspar, and plagioclase (FAP) for classification of felsic igneous rocks – Figure 1; and olivine, pyroxene, and plagioclase (OPP), orthopyroxene, clinopyroxene, and plagioclase (OCP), olivine, clinopyroxene, and orthopyroxene (OCO), olivine, pyroxene, hornblende (OPH), or pyroxene, plagioclase, and hornblende (PPH) for classification of mafic and ultramafic igneous rocks – Fig. 2; Streckeisen 1974; Streckeisen *et al.* 2002), estimates of the molar abundances of the minerals used in igneous rock classification can be obtained. These estimates can be achieved using simple and straightforward matrix algebra techniques that ‘change the basis’ of the rock compositions, from elements to minerals (Strang 1993). Then, after conversion of the resulting molar mineral numbers into volume mineral numbers through multiplication of the corresponding molar volumes (equal to a mineral's gram formula weight divided by its density), and finally standardizing these volume mineral numbers to sum to unity, the results can be plotted on Streckeisen ternary diagrams for classification (Figs 1 and 2; Streckeisen 1974; Streckeisen *et al.* 2002).

## Matrix algebra solutions

Critical to the above lithogeochemical classification procedure is the identification of the appropriate linear combinations that convert molar element numbers into molar mineral numbers. These linear combinations can easily be calculated using MATLAB® matrix algebra software. First, a matrix (*C*) is defined that contains the elements describing the compositions of all essential minerals in the rocks to be classified, plus those minerals used for classification on Streckeisen ternary diagrams. For example, if classification of a biotite-bearing granitoid intrusion is the objective, the rock will be classified using the QAP Streckeisen ternary diagram (Fig. 3). This requires consideration of the abundances of quartz, plagioclase, and alkali feldspar (the minerals comprising the vertices of this ternary diagram), and biotite (because it is also an essential mineral, and thus must be projected from to correctly classify the granitoid rock). The corresponding matrix *C* (equation 1) thus will contain a row for each of the end-member compositions of these four essential minerals (quartz, *QZ*; anorthite, *AN*; albite, *AB*; K-feldspar, *KS*; biotite, *BT*) and have columns for each analyzed element in the formula of these minerals:
(1)Note that in matrix *C*, Fe and Mg have been combined (as *FM*) and thus phlogopite and annite have been combined (as *BT*) to simplify the matrix calculations. This can be done because Fe and Mg undergo a simple exchange substitution in biotite. Note also that plagioclase and alkali feldspar compositions are not represented in matrix *C*; rather, the end-member feldspar compositions, *AN*, *AB*, and *KS*, are included. Reasons for this substitution will be made clear in the following section. Lastly, note that H has not been included in *C* because H_{2}O^{+} is typically not analyzed for in lithogeochemical analysis.

In order to achieve a valid rock classification, an additional constraint is imposed on the *C* matrix. This constraint is that the rank of the *C* matrix is equal to the number of rows in the *C* matrix. If this constraint is not met, and the number of rows exceeds the rank of the matrix, one or more chemical reactions can be written between the essential minerals, and the linear combinations necessary to classify the rocks in this manner cannot be calculated. As a result, if the rank of *C* does not equal the number of rows in matrix *C*, it is wise to determine the chemical reaction(s) that exist(s) between the essential minerals and to omit one essential mineral in each reaction from the *C* matrix to allow classification. To ensure that the best possible igneous rock classification is produced, the essential mineral(s) to be omitted should have the lowest concentrations in the rocks under consideration, and should not be one of the minerals used to classify the rocks on Streckeisen ternary diagrams.

In this lithogeochemical adaptation of the Streckeisen classification approach, the molar amounts of each element in the minerals under consideration (matrix *C*) will be multiplied by three sets of unknown coefficients (*V _{QZ}*,

*V*

_{PL}_{,}

*V*; vectors defining linear combinations of elements) to produce quantitative estimates of the molar amounts of each mineral on the vertices of the Streckeisen ternary diagram. To derive the desired linear combinations specific to these three vertex minerals, the dot products of the mineral composition (row) vectors in matrix

_{AF}*C*and the unknown coefficient vectors for each ternary diagram vertex (

*V*,

_{QZ}*V*,

_{PL}*V*) must equal ‘1’ for the corresponding mineral, and ‘0’ for all of the other essential minerals. This ensures that the resulting linear combinations produce molar mineral numbers that are independent of each other. These ‘0’ and ‘1’ dot product values are stored in an identity matrix (

_{AF}*P*), labelled as: (2)By configuring matrices

*C*and

*P*in this manner, one can form a characteristic matrix equation (

*C × A = P*), the solution of which (matrix

*A*) contains the linear combinations that convert the molar element numbers into molar mineral numbers. In cases where the number of columns in matrix

*C*equals the rank of matrix

*C*, a unique solution is defined by matrix

*A*. In other cases, where the number of columns in matrix

*C*exceeds the rank of matrix

*C*, an infinite number of solutions are possible. Regardless of the number of columns in matrix

*C*, all possible solutions to this matrix equation can be obtained using the following matrix procedure. First, append the

*C*and

*P*matrices side-by-side to produce an augmented matrix: (3)Then, derive the reduced-row-echelon form (

*RREF*) of this augmented matrix (a function for this is available in MATLAB®; Strang 1993) to produce: (4)Finally, drop the leading identity sub-matrix (

*I*), append a negative identity sub-matrix of appropriate size (in this case, 1 × 1) to the bottom of the null space sub-matrix (

*N*; between the two vertical dashed lines), and append a zero sub-matrix of appropriate size (1 × 5) to the bottom of the square

*M*sub-matrix (to the right of the right-most vertical dashed line). This produces a matrix

*A*containing column vectors from which all possible solutions to the characteristic matrix equation (

*C × A = P*) can be derived: (5)In the resulting

*A*matrix, two types of column vectors exist. Vector

*V*(left of the vertical dashed line; referred to as

_{N1}*A*) comprises a basis for the null space of matrix

_{N}*C*. This null space basis vector is by definition perpendicular to all of the row vectors in

*C*(their dot products with

*V*all equal to zero). As a result, linear combinations of this vector (and any other null space basis vectors that might exist in other examples) define all possible, non-trivial (non-zero) solutions to the corresponding homogeneous matrix equation:

_{N1}*C*×

*A*= 0: (6)The other five column vectors (

_{N}*V*,

_{QZ}*V*,

_{AN}*V*,

_{AB}*V*, and

_{KS}*V*); collectively referred to as

_{BT}*A*) describe linear combinations of elements that can be used to convert the molar element numbers into corresponding molar mineral numbers for quartz, anorthite, albite, K-feldspar, and biotite. Each of these vectors, when multiplied by the corresponding row vector(s) in matrix

_{M}*C*, will equal one, but will equal zero when multiplied by any of the other row vectors (as defined originally in matrix

*P*). As a result, these other column vectors (

*V*,

_{QZ}*V*,

_{AN}*V*,

_{AB}*V*, and

_{KS}*V*) collectively represent one set of independent solutions (

_{BT}*A*) to the characteristic matrix equation

_{M}*C × A*. Thus: (7)These five mineral column vectors (

_{M}= P*V*,

_{QZ}*V*,

_{AN}*V*,

_{AB}*V*, and

_{KS}*V*) each create projections from all other essential minerals in the granite, and thus produce independent molar mineral numbers for

_{BT}*QZ*,

*AN, AB*,

*KS*, and

*BT*from the molar element numbers. In mathematical terms, the conversion of molar element numbers to molar mineral numbers is called a ‘change of basis’, and such calculations merely change the way the rock compositions are described (in terms of mineral concentrations instead of in terms of element concentrations). As a result, this change of basis is really no different than changing the manner in which a point in space is described from Cartesian coordinates (

*X,Y*) to radial coordinates (

*θ,r*).

These molar mineral numbers provide estimates of the molar amounts of plagioclase (*PL = AN + AB*) and alkali feldspar (*AF = AB + KS*). In this hypothetical case, *PL* and *KS* are used to describe feldspar in these rocks, because two feldspars are present (see below). These quantities, after conversion into standardized volume mineral proportions, can thus be used in subsequent classification on a QAP Streckeisen ternary diagram. As a result, after multiplying the resulting molar mineral numbers (*QZ*, *PL* and *KS*) by their corresponding molar volumes, and standardizing the resulting volume mineral numbers to unity, these quantities can be plotted on the QAP Streckeisen ternary diagram for classification.

A fundamental feature of the matrix algebra employed in the lithogeochemical classification procedure is that, for this and many other essential mineral assemblages, there is commonly more than one set of linear combination vectors that solve the characteristic equation (*C × A = P*). In the above biotite granitoid example, because the null space matrix (*A _{N}*) exists (it has more than 0 columns), there are an infinite number of linear combinations (

*A*) that solve equation 7. As a result, each of these linear combinations can be used to convert the molar element numbers into molar mineral numbers for quartz, plagioclase, K-spar, and biotite, and thus provide alternative Streckeisen classifications. These infinite solutions consist of the set of vectors for each essential mineral (

*V*,

_{QZ}*V*,

_{PL}*V*,

_{KS}*V*) plus a linear combination of the null space vectors (

_{BT}*V*, where

_{QZ*}= j_{1}V_{N1}+ V_{QZ}*j*is an arbitrary coefficient in column vector

_{1}*J*).

This is because:

if

*C*×*A*= 0, then_{N}*C*×*JA*=_{N}*J*0 = 0; andif

*C*×*A*=_{M}*P*, then: (*C*×*JA*) + (_{N}*C*×*A*) = 0 +_{M}*P*=*P*; and thus(

*C*×*JA*) + (_{N}*C*×*A*) =_{M}*C*× (*JA*×_{N}*A*) =_{M}*P*;

making every linear combination (*J*) of the column vectors in *A _{N}* plus the vectors in

*A*also a solution.

_{M}For example, if *j*_{1} = 3, then an alternative solution for quartz (*V _{QZ*}*) can be determined:
(8)This linear combination (

*V*) produces exactly the same result as the intuitively derived linear combination for quartz (

_{QZ*}*V*; Fig. 3), and is also projected from plagioclase, alkali feldspar, and biotite.

_{QZ}Analogously, if we use different values for the linear combinations of the null space vectors plus the K-spar vector (*V _{KS*} = j_{1}V_{N1} + V_{KS}*), where now

*j*, we obtain an alternative

_{1}= −*1**V*solution vector: (9)Again, this linear combination (

_{KS*}*V*) produces exactly the same result as the intuitively derived linear combination for (

_{KS*}*V*; Fig. 3).

_{KS}Clearly, because there are an infinite number of linear combinations that can be used to convert molar element numbers into molar mineral numbers, it is useful to create a labelling system for at least some of the simplest of these matrix equation solutions. In the above hypothetical example, the linear combinations are:
(10)(from equation 7). Note that in this case, the vectors *V _{AN}* and

*V*have already been added together to produce

_{AB}*V*. The one characteristic common to all four mineral column vectors is that the coefficients for potassium are all zero. As a result, we can refer to this set of linear combinations as the ‘K-absent’ solution (

_{PL}*A*) of the matrix equation, borrowing this nomenclature from metamorphic petrology.

_{K-absent}The alternative solution, presented in Figure 3 and utilizing *V _{QZ*}*,

*V*, and

_{PL}*V*instead of

_{KS*}*V*,

_{QZ}*V*, and

_{PL}*V*, is: (11)This solution has zero coefficients for Al in all four mineral vectors, and thus can be called an ‘Al-absent’ solution (see Fig. 3). Analogous linear combinations can be derived for ‘Ca-absent’ (

_{KS}*A*; where

_{Ca-absent}*V*= 2

_{QZ**}*V*

_{N1}+

*V*

_{QZ}and

*V*= ½

_{PL*}*V*

_{N1}+

*V*; equation 12) and ‘Na-absent’ (

_{PL}*A*; where

_{Na-absent}*V*1

_{PL**}=*V*; equation 13) solutions: (12)and: (13)Note that a ‘Si-absent’ solution cannot be derived because there is only one oxide that occurs in quartz (SiO

_{N1}+ V_{PL}_{2}), so Si must be used to describe the amount of quartz present and thus cannot be ‘absent’. Similarly, an ‘Fe-Mg-absent’ solution cannot be derived because no linear combination of

*V*plus each of the four original mineral vectors can produce a result that has zero coefficients for Fe + Mg.

_{N1}Clearly, an advantage of having multiple ways of calculating the abundances of the ternary diagram vertex minerals is that if one or more elements have not been analyzed, or cross-cutting veins or hydrothermal alteration have partially or completely added or removed one or more elements from the rock, classification can still be undertaken using solutions that do not involve these problematic element(s).

Another advantage of this approach is that, although ideal compositions of the essential minerals have been used in this example, the actual compositions, determined by electron microprobe analysis, could be used to derive the appropriate linear combinations. Although this would likely result in non-integer coefficients in the linear combinations, it would precisely tailor the classification to the compositions of the minerals present in the rocks.

## Accommodating feldspar on Streckeisen ternary diagrams

Streckeisen classification of felsic igneous rocks uses modal volume estimates of plagioclase and alkali feldspar. Unfortunately, these two feldspar solid solutions share albite as an end-member composition (Fig. 4) and it is not immediately clear how to partition albite among plagioclase and alkali feldspar for the purposes of classification (Streckeisen *et al.* 2002).

The recommended approach used when classifying igneous rocks via Streckeisen's volume-based classification method involves assigning Na-Ca feldspar to ‘plagioclase’, unless the Na-Ca feldspar composition in the igneous rock to be classified is end-member ‘albite’ (*X _{AN}* < 0.05), in which case this albite is assigned to alkali feldspar (Streckeisen

*et al.*2002). This approach leads to a significant and sometimes problematic compositional discontinuity that can result in substantial classification disparities when Na-Ca feldspar compositions in granitoid rocks are close to

*X*= 0.05. Nevertheless, the above approach does have theoretical justification in that most plagioclase compositions (with

_{AN}*X*greater than

_{AN}*c.*0.05) have a higher liquidus temperature than any alkali feldspar composition (Fig. 5; Fuhrman & Lindsley 1988; Elkins & Grove 1990).

Thus, in most felsic melts, plagioclase generally crystallizes before alkali feldspar, and this ensures that albite compositional components are mostly removed from the melt via plagioclase crystallization before significant alkali feldspar crystallization commences (thus these albite compositional components are not generally available to go into alkali feldspar). This prevents significant albite component from occurring in alkali feldspar when sufficient anorthite component is present in the melt, and thus also ensures that the composition of alkali feldspar in equilibrium with calcic, and even intermediate, plagioclase compositions is relatively close to end-member potassium feldspar (Fig. 4; Fuhrman & Lindsley 1988; Elkins & Grove 1990). As a result, the compositional characteristics of feldspar petrogenesis closely match the way feldspar compositions are treated numerically in the silica saturated and silica under-saturated Streckeisen ternary diagram classification procedures (Streckeisen *et al.* 2002).

When using the lithogeochemical classification procedure, a numerically equivalent approach can be undertaken. This requires first establishing the bulk molar composition (*X _{AN}*,

*X*,

_{AB}*X*) of feldspar in the rocks. Using the

_{KS}*C*matrix appropriate for the rocks under consideration, the molar mineral numbers for anorthite, albite, and K-feldspar can be determined. Standardizing these to sum to unity allows one to plot the bulk molar feldspar composition on a ternary diagram (Fig. 4).

Then, depending on whether the granitoid rocks contain essential plagioclase or not, two different strategies can be employed. If the granitoid rocks contain essential plagioclase (with or without alkali feldspar, *V _{AB} + V_{AN}* (≈

*V*) and

_{PL}*V*(≈

_{KS}*V*) linear combinations are used to describe the feldspars in the lithogeochemical classification procedure. In this case, the composition of the plagioclase used in classification can be obtained by drawing a line from the

_{AF}*KS*vertex through the bulk feldspar composition (grey circle; Fig. 6) to the plagioclase join on left side of the feldspar ternary diagram. The intersection of this line with the plagioclase join estimates the plagioclase composition in the rock, and the molar proportions of plagioclase and K-feldspar are determined using the lever rule.

Alternatively, if essential calcic plagioclase is not present, then *V _{AN}* (≈

*V*) and

_{PL}*V*(≈

_{AB}+ V_{KS}*V*) linear combinations are used in the lithogeochemical classification procedure. In this alternative case, the bulk composition of the alkali feldspar used in classification can be obtained by drawing a line from the AN vertex through the bulk feldspar composition (grey square; Fig. 6) to the alkali feldspar join on the feldspar ternary diagram. The intersection of this line with the alkali feldspar join estimates the bulk alkali feldspar composition, and the proportion of anorthite (plagioclase) and alkali feldspar can again be determined using the lever rule. Using the above numerical strategy, this lithogeochemical classification approach mathematically mimics the petrologic criteria employed to address feldspar in the QAP and FAP Streckeisen ternary diagram classification procedures, and conforms to the experimental feldspar phase relations (Fuhrman & Lindsley 1988; Elkins & Grove 1990).

_{AF}To illustrate the equivalence of this approach with Streckeisen's classification procedure, two ‘synthetic’ bulk feldspar compositions are considered. The first synthetic rock contains two feldspars: plagioclase and alkali feldspar. It has a bulk (molar) feldspar composition of *X _{AN}*,

*X*,

_{AB}*X*= 0.50, 0.25, 0.25 (grey circle; Fig. 6). Based on experiments (Fuhrman & Lindsley 1988; Elkins & Grove 1990), the corresponding plagioclase and alkali feldspar compositions for this bulk feldspar composition would be approximately

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.66, 0.31, 0.03 and (

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.03, 0.11, 0.86 at 650°C, respectively (black circles; Fig. 6). Because the experimentally estimated plagioclase composition –

_{KS}*X*

_{AN}/**(**

*X*

_{AN}+X_{AB}**)**– is calcic (= 0.68), Streckeisen's classification procedure would plot calcic plagioclase and alkali feldspar with molar proportions of

*X*,

_{PL}*X*= 0.73, 0.27 on the silica-saturated ternary diagram, as determined via the lever rule (note that the corresponding volume proportions are what would actually be plotted).

_{AF}In contrast, the criteria used in the lithogeochemical classification procedure described herein (via projection from potassium feldspar on Fig. 6) would employ plagioclase and alkali feldspar compositions of *X _{AN}*,

*X*,

_{AB}*X*= 0.67, 0.33, 0.00 and

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.00, 0.00, 1.00, respectively, because all of the albite component in the bulk feldspar composition would be assigned to plagioclase, leaving potassium feldspar as the alkali feldspar composition. These ideal feldspar compositions have molar proportions of

_{KS}*X*,

_{PL}*X*= 0.75, 0.25 (white circles; Fig. 6), also determined by the lever rule. Thus, these pairs of plagioclase and alkali feldspar proportions, defined by the experimentally determined and lithogeochemically-calculated approaches, do not differ significantly (by only 2 mole % units).

_{AF}Alternatively, a second synthetic granitoid rock containing only essential alkali feldspar has a bulk feldspar composition of *X _{AN}*,

*X*,

_{AB}*X*= 0.03, 0.67, 0.30 (grey square; Fig. 6). The corresponding experimentally-determined plagioclase and alkali feldspar compositions would be

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.04, 0.68, 0.28 and

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.03, 0.37, 0.60 at 650°C, respectively (Fuhrman & Lindsley 1988; Elkins & Grove 1990). These two feldspar compositions are on opposite sides of the alkali feldspar solvus (black squares; Fig. 6) and are, in reality, both alkali feldspar in proportions of approximately [90

_{KS}*AB*: 10

*KS*]. Thus could be expected to exhibit an anti-perthitic texture. In this case, the sodic plagioclase would be considered alkali feldspar, and the experimentally determined molar proportions of plagioclase and alkali feldspar would be

*X*,

_{PL}*X*= 0.00, 1.00. Because, ideally, no plagioclase exists, this ‘synthetic’ rock would ultimately be classified as alkali feldspar syenite, alkali feldspar quartz syenite, or alkali feldspar granite, depending on the amount of quartz present (Fig. 1).

_{AF}In contrast, the plagioclase and alkali feldspar compositions that would be employed using the lithogeochemical classification procedure described herein via projection from anorthite are *X _{AN}*,

*X*,

_{AB}*X*= 1.00, 0.00, 0.00 and

_{KS}*X*,

_{AN}*X*,

_{AB}*X*= 0.00, 0.69, 0.31 because all of the albite component would be assigned to alkali feldspar. Although these feldspar compositions are significantly different from those based on experiments (because alkali feldspar exsolution is not accommodated; Fuhrman & Lindsley 1988; Elkins & Grove 1990), the molar proportions of plagioclase and alkali feldspar used in lithogeochemical classification are nevertheless virtually identical

_{KS}*X*,

_{PL}*X*= 0.00, 1.00 to those used with Streckeisen's approach

_{AF}*X*,

_{PL}*X*= 0.03, 0.97 (differing by only 3 mole % units).

_{AF}Because only the proportions of plagioclase and alkali feldspar are used in the lithogeochemical classification procedure, classification by Streckeisen's method and this new lithogeochemical procedure produce similar results. Consequently, the numerical calculations in this lithogeochemical classification approach effectively treat plagioclase and alkali feldspar in a manner similar to that employed in Streckeisen's method, leading to consistent and comparable classification.

## Discussion

To illustrate how classification can be undertaken using this lithogeochemical classification procedure, several examples are presented below involving a number of different igneous rock bodies. These examples illustrate a number of important features inherent to the new lithogeochemical classification procedure.

### Metaluminous Sloggett Pluton Granitoid, New South Wales

The first example involves a single rock composition (OZCHEM # 72464; Sedgmen *et al.* 2007) from the Sloggett Pluton, a 7 km^{2} intrusion within the Oberon Granite in the Lachlan fold belt of New South Wales, Australia (McCormack 1985; Pogson & Watkins 1998; Glen *et al.* 2006). This rock was originally classified as ‘biotite granite’ by Stuart-Smith (Glen *et al.* 2006) based on hand sample descriptions, and ‘adamellite’ by McCormack (1985) based on thin section point counts. It contains coarsely porphyritic pink alkali feldspar (*OR*_{81} to *OR*_{88}) in a coarse grained groundmass of quartz, plagioclase (*AN*_{26} to *AN*_{29}), alkali feldspar (*OR*_{73}), and biotite. Analysis of this granite sample was by fused disc-XRF for major oxides and a limited suite of trace elements (Glen *et al.* 2006), and it has a molar Al_{2}O_{3}/(CaO + Na_{2}O + K_{2}O) (Shand) ratio = 1.02, causing it to be described as having metaluminous, I-type, magnetite-series affinities (McCormack 1985).

The abundances of quartz, plagioclase, alkali feldspar, and biotite in this Sloggett Pluton sample are all greater than 5%, and so likely control the composition of the intrusion. Although minor to trace amounts of igneous magnetite, zircon, apatite and titanite are also present, these occur in concentrations far less than 5 volume % (McCormack 1985; Pogson & Watkins 1998; Glen *et al.* 2006). In addition, deuteric alteration has caused the incipient replacement of plagioclase cores and rims by sericite, and biotite by minor chlorite and epidote. Sparse, thin calcite veins also locally cut the granitoid (McCormack 1985). Fortunately, none of these alteration minerals exist in concentrations exceeding 5%, and thus are not likely to substantially influence the rock composition or resulting classification.

Obviously, lithogeochemical classification of this Sloggett Pluton sample can be undertaken using any set of solution vectors described in the above illustration of matrix algebra. As a result, classification has been undertaken in five ways using the four (Al-H-, Ca-H-, Na-H-, and K-H-absent) linear combinations solutions (equations 10 through 13; note that H_{2}O^{+} concentrations were not analyzed), and an alternative Al-H-absent solution where molar CO_{2} is subtracted in amounts equal to the Ca coefficient so that the resulting linear combination is also a projection from calcite.

Figure 7 presents the bulk molar feldspar compositional estimates used to address feldspar appropriately, and Figure 8 presents the resulting classification on a QAP Streckeisen ternary diagram. Results indicate that the Sloggett Pluton sample should be classified as a ‘monzogranite’ by the Streckeisen-type lithogeochemical classification method, as all five solutions result in essentially the same result. These classifications are consistent with hand sample descriptions originally made by Stuart-Smith (Glen *et al.* 2006) and McCormack (1985). As a result, the various lithogeochemical classifications applied to this example produce entirely consistent and likely highly accurate results, indicating that such lithogeochemical classification could be used in lieu of more traditional visual estimation, point counting, image analysis, or spectrometric mineral mode-based Streckeisen classification methods.

### Alkaline Emerald Lake Pluton Granitoid, Yukon

The Mid-Cretaceous composite Emerald Lake Pluton occurs in southeastern Yukon, Canada, and is part of the alkalic Tombstone Plutonic Suite. Mapping of this elongate body reveals that it consists of four phases: augite syenite (AS), hornblende quartz syenite (HQS), hornblende quartz monzonite (HQM), and biotite granite (BG), with gradational contacts only between the HQS and HQM phases and cross-cutting contacts between the others (Smit 1984; Smit *et al.* 1985; Duncan *et al.* 1998*a*, *b*; Duncan 1999; Coulson *et al.* 2001, 2002; Fig. 9).

Thirty-six 1 kg samples from these four phases were analyzed by fusion-XRF methods for major oxides and petrographically examined in thin section. These samples were also photographed before geochemical analysis, subjected to potassium cobaltinitrate staining, and image analysis to determine the rock's mineral modes and to classify them (Duncan 1999; Fig. 10).

Classification of these samples was also undertaken using the lithogeochemical strategy described above, except that the *C* and *P* matrices for each of the phases were different, as determined by the different ‘essential mineral assemblages’. For the augite syenite, quartz-alkali feldspar-plagioclase-clinopyroxene (*CP*)- Na-rich hornblende (defined by the pargasite (*PG*) end-member) were used in matrix *C* (note that H was not included in the compositional descriptions of this matrix because H_{2}O was not analyzed). The *C* matrix:
(17)yields the following linear combinations:
(18)Note that, because *C* is a square 6 × 6 matrix with rank of six, no null space vectors were derived from this *C* matrix, and thus only one solution to the characteristic matrix equation is possible.

In contrast, for the hornblende quartz syenite and hornblende quartz monzonite, an essential mineral assemblage of: quartz-alkali feldspar-plagioclase-hornblende (defined by both actinolite (*AC*) and pargasite end-members) was used in matrix *C*:
(19)and this resulted in the following linear combinations:
(20)Again, no null space vectors are derived from this *C* matrix, so only one solution is possible.

Lastly, an essential mineral assemblage of: quartz-alkali feldspar-plagioclase-biotite was used for biotite granite, and the set of linear combinations used to obtain the mineral modes was the K-absent solution (equation 10) described above. Results for all three of these Streckeisen-style lithogeochemical classifications are presented ensemble in Figures 11 and 12.

A comparison of Figure 10 (stained slab image analysis ≈ petrography) and Figure 12 (lithogeochemistry) reveals several important conclusions. First, the rocks in Figure 12 are more clustered within lithologically-consistent groups (exhibiting lower modal variance), and these clusters are more separated between lithologies than the image analysis data in Figure 10. This indicates that the lithogeochemical samples are likely substantially more representative than the image analysis estimates of the mineral modes because they have less ‘sampling error’. In fact, based on the masses of these two samples (thin sections: *c.* 0.065 g = 2 cm × 4 cm × 0.003 cm × 2.7 g/cm; lithogeochemical samples: *c.* 1000 g), the lithogeochemical samples are more than 15 000 times larger, and thus their sampling variance is 15 000 times smaller (Stanley 2007), resulting in a far more ‘consistent’ classification. Second, the HQS and HQM compositions overlap in Figure 12, and are distinct from the augite syenite and biotite granite compositions. These features cannot be recognized on Figure 10, as all of the data are overlapping and far more dispersed across the diagram. The overlapping and non-overlapping relationships between phases on Figure 12 are entirely consistent with the contact relationships between these phases in the Emerald Lake Pluton, as the HQS and HQM are the only two phases with a gradational contact, suggesting a transitional igneous relationship.

Lastly, samples of augite syenite on Figure 10 plot in a location suggesting that this phase has no quartz in its mode. This does not appear to be the case in Figure 12, as significant quartz is predicted based on the rock compositions, and this alters the classification for this lithology from augite syenite to ‘augite quartz syenite’. Similarly, the HQS and HQM units largely plot in the syenogranite field and overlap substantially, so these units could be re-classified as a single phase of ‘hornblende syenogranite’ because they of their overall compositional similarities. Finally, the biotite granite samples plot in the monzogranite field, so they should likewise be re-classified as ‘biotite monzogranite’. In conclusion, based on the above observations, the lithogeochemical classification procedure appears to produce a far more representative and at least as accurate a classification as stained slab image analysis results for rocks from the Emerald Lake Pluton.

Before considering another test of this classification procedure, one should note that linear combinations of molar element numbers sometimes produce negative molar mineral numbers. Such a situation should theoretically be ‘impossible’. Nevertheless, negative molar mineral numbers were derived in 14 of 22 samples from the above example for the actinolite component of hornblende in the calculation to classify the HQM and HQS units of the Emerald Lake Pluton. Experience in such lithogeochemical classification reveals that these negative results are not uncommon, and in this case likely derive from the fact that some of the linear combination coefficients for actinolite are negative (Al = −2/3, FM = −1/3; equation 20). As a result, analytical error that exists in these data can produce a slightly negative result if the concentrations of Al_{2}O_{3}, FeO and MgO are over-estimated by the laboratory and the concentrations of other elements in the linear combination (CaO, Na_{2}O, and K_{2}O; note that the SiO_{2} coefficient = 0) are under-estimated by the laboratory. In this example, negative molar mineral numbers occur in roughly half of the samples. However, all of these molar mineral numbers cluster very close to zero (ranging from −0.015 to 0.017; Fig. 13), and are very small relative to the molar mineral numbers for other minerals in the essential mineral assemblage, as the mean molar mineral numbers for these rocks are 0.357 for quartz, 0.070 for plagioclase, 0.192 for alkali feldspar, −0.001 for actinolite, and 0.024 for pargasite. This suggests that the mode for actinolite in these rocks is likely 0, but that random measurement error has caused some of the actinolite molar mineral numbers to be under-estimated (producing ‘impossible’ negative modes). Confirmation of this hypothesis can be obtained by examining the frequency distributions of the unstandardized molar mineral numbers for the HQM and HQS units of the Emerald Lake Pluton (Fig. 13). These clearly illustrate that actinolite molar mineral numbers cluster about 0 relative to the molar mineral numbers for the other essential minerals, indicating actinolite is likely absent in these rocks (i.e. the amphibole composition in these rocks is pargasitic).

To avoid having to deal with negative mineral modes for a mineral that is likely absent from the rock, the negative molar mineral numbers produced for those minerals can simply be recoded to zero, and the calculations carried out as before. This has little or no effect on the subsequent calculations, depending on whether the mineral is used in the classification, or not.

If the mineral with negative molar mineral numbers is not on the vertices of the ternary diagram used in classification, then that mineral is being projected from during classification. Consequently, such recoding has no impact on the classification, as three other minerals are used in classification, and their molar mineral numbers are summed to one after multiplication by their corresponding molar volumes. As a result, in this case, recoding negative molar mineral numbers to zero does not change the classification at all. In this example, both actinolite and pargasite are projected from to produce this classification on a QAP Streckeisen ternary diagram, so recoding the negative molar mineral numbers for actinolite to zero had no impact on the classification.

However, if the mineral with negative molar mineral numbers is on a vertex of the ternary diagram used in classification, a different behaviour occurs. This behaviour can be understood by considering all possible scenarios. When the three minerals used in classification of a rock all have positive molar mineral numbers, then they will have positive volume numbers, and the rock composition will plot in X-Y-Z space within the positive orthant of Figure 14 (yellow diamond). Subsequent ‘closing’ of these data to sum to unity projects the rock composition onto the X + Y + Z = 1 plane, which is the ternary classification diagram (Fig. 14, yellow circle). However, if one of the three minerals has slightly negative X, Y or Z molar mineral numbers, the corresponding molar volume numbers will also be slightly negative, and the rock compositions will plot just outside of the positive orthant (in either the [+ + −], [+ − +], or [− + +] orthants; Fig. 14, red diamond). After summing to unity, the rock compositions will lie outside the ternary diagram (Fig. 14, red circle), a consequence that is to be avoided. When these slightly negative molar mineral numbers are recoded to zero, the compositions are moved to rest exactly on the plane between the positive orthant and one of the [+ + −], [+ − +], or [− + +] orthants (Fig. 14, orange diamond). Subsequent summing of the data to unity moves the rock compositions onto one of the boundary lines of the ternary diagram (Fig. 14, orange circle), indicating that the rock composition does not contain one of the minerals used in classification.

Note that analogous behaviour exists when two of the minerals used in classification have slightly negative molar mineral numbers. In this case, however, the rock compositions lie in either the [+ − −], [− + −], or [− − +] orthants, and recoding of the two slightly negative molar mineral numbers to zero will cause the resulting rock compositions to lie precisely on either the X, Y or Z axes, and thus on one of the ternary diagram vertices, indicating that the rock contains only one of the minerals used in classification.

Lastly, if all three minerals used in classification have slightly negative molar mineral numbers, then the rock composition lies in the negative orthant [− − −]. Although projection to the ternary plane: X + Y + Z = 1 is possible using a negative standardizing factor, recoding of the slightly negative molar mineral numbers to zero will cause the rock to have none of any of the classifying minerals, rendering classification using those minerals impossible.

Consequently, when one or two molar mineral numbers are slightly negative, recoding of these slightly negative values to zero moves the rock composition only a small distance in X-Y-Z space, causing little impact on the resulting classification.

### Mafic-Ultramafic Stillwater Layered Intrusion, Montana

The Stillwater Complex, located in the Tobacco Root Mountains of southern Montana, is a late Archean layered mafic-ultramafic intrusion composed of three parts: (1) a Basal Series consisting of mafic dykes and noritic rocks *c.* 100 m thick; (2) an overlying 2200 m thick Ultramafic Series consisting of olivine−, and olivine + orthopyroxene-bearing cumulate rocks in its lower 1000 m, and orthopyroxene-bearing cumulate rocks, locally hosting chromitite, in the upper 1200 m; and (3) a 4300 m thick superjacent Banded Series consisting of largely plagioclase cumulates subdivided into three parts (Lower, Middle, and Upper; McCallum *et al.* 1980; Fig. 15). The Middle Banded Series is *c.* 800 m thick and comprises an anorthosite cumulate zone between two sequences of olivine-bearing cumulate rocks (OB-3 and OB-4). These rocks have been examined in detail (Meurier 1995; Meurier & Boudreau 1996, 1998), and contain anothositic, troctolitic, olivine gabbroic and gabbronoritic lithologies (Fig. 16).

Two surface traverses through the Middle Banded Series have provided 96 un-weathered (2 kg) rock samples from a continuous section for study (Meurier 1995). Petrographic classification (olivine, plagioclase, clinopyroxene and orthopyroxene) of these samples was undertaken using image analysis of enlarged photographs of 2 × 4 cm thin sections as input into a conventional Streckeisen classification (Streckeisen 1974; Streckeisen *et al.* 2002). Average grain sizes were 2 mm, so *c.* 200 grains were present on each thin section. These estimates indicate that the samples consist of anorthosite, troctolite, norite, olivine norite, clinopyroxene norite, olivine gabbronorite, gabbro, olivine gabbro, and orthopyroxene gabbro. These petrographic results are presented on the two Streckeisen ternary diagrams (OPP and OCP) of Figure 17.

Fusion-ICP analyses of 250 g powders of these same rocks (Meurier 1995; Meurier & Boudreau 1996, 1998) were used in the lithogeochemical classification procedure described above, using the ideal formulae for olivine, plagioclase, orthopyroxene, and clinopyroxene as essential minerals to produce the *C* matrix:
(21)The resulting linear combinations used to convert the Stillwater Complex molar element numbers into molar mineral numbers consist of:
(22)These linear combinations were used to produce lithogeochemically-derived mineral modes that were used to make the lithogeochemical classifications presented in Figure 18.

Comparison of Stillwater Complex petrographic classifications (Fig. 17) and lithogeochemical classifications (Fig. 18) reveals very few differences. In general, the classifications made by image analysis and lithogeochemical calculations are consistent. As a result, accurate and precise mafic and ultramafic igneous rock classification is also possible using the lithogeochemical classification procedure.

### Peraluminous Halifax Pluton, South Mountain Batholith, Nova Scotia

The South Mountain Batholith (SMB) of southwestern Nova Scotia is the largest granitoid batholith in the Appalachian Orogen, has a Devonian age (280 Ma – 276 Ma; Kontak *et al.* 2003; Reynolds *et al.* 2004), and was intruded in two stages as 13 distinct plutons (MacDonald & Horne 1987). Rocks in the batholith exhibit a range of evolution, from granodiorite (with up to *c.* 25 volume % biotite) to muscovite-topaz leucogranite (with as little as 2 volume % combined mafic minerals, mostly biotite). The batholith is entirely peraluminous based on Shand indices that are greater than one, and rocks contain abundant quartz, plagioclase, and alkali feldspar with varying proportions of peraluminous minerals, including: aluminous biotite, muscovite, cordierite, andalusite, garnet, topaz and tourmaline (MacDonald & Horne 1987; Horne *et al.* 1989; MacDonald *et al.* 1989, 1992*a*, *b*; MacDonald 2001).

On the SE side of the South Mountain Batholith in Halifax County, Nova Scotia, is the Halifax Pluton, an approximately ovum-shaped intrusion of roughly 60 km by 30 km dimension oriented WNW-ESE (Fig. 19). Minerals contained in the six phases of this pluton include quartz, plagioclase, alkali feldspar, and biotite, but some of the more evolved phases contain significant concentrations of muscovite and cordierite as well; andalusite and garnet are accessory phases.

The least evolved phase in the Halifax Pluton is a marginal biotite granodiorite (MBG; average SiO_{2} concentration = 68 wt. %, average Shand index = 1.14, average [molar Al_{2}O_{3}/(Na_{2}O + K_{2}O] index = 1.48), and this grades inward into Sandy Lake and Peggy's Cove Biotite Monzogranite (SLBM and PCBM; average SiO_{2} concentrations = 70 and 70 wt. %, average Shand indices = 1.15 and 1.20, average alkali indices = 1.39 and 1.44, respectively) on the north/west and south sides of the pluton, respectively. The Halifax Two-Mica Cordierite Monzogranite (HMCM; average SiO_{2} concentration = 73 wt. %, average Shand index = 1.20, average alkali index = 1.34) occurs interior to these monzogranites, and the Harrietsfield Cordierite-Biotite Monzogranite (HCBM; average SiO_{2} concentration = 71 wt. %, average Shand index = 1.17, average alkali index = 1.38) occurs to the east of the Halifax phase. All of the above phases were intruded during Stage I of the South Mountain batholith emplacement history and have gradational contacts. In contrast, the Tantallon Fine-Grained Two-Mica Leucomonzogranite (TAML; average SiO_{2} concentration = 75 wt. %, average Shand index = 1.27, average alkali index = 1.36) demonstrably cross cuts these Stage 1 phases, and was intruded during Stage II as small dykes, plugs and irregular masses (Fig. 19).

The Stage 1 phases of the Halifax Pluton are megacrystic, and comprise a lithogeochemical and petrological gradationally zoned sequence of igneous rocks from outside to inside. Marginal biotite granodiorite contains <2% alkali feldspar megaphenocrysts, and these increase to 15 – 50% in the HMCM, but then decrease to 10 – 15% in the HCBM. Similarly, MBG contains 12 – 15% biotite, and this decreases subtly inward, ultimately to 8 – 10% in the HCBM. In contrast, muscovite concentrations increase from <1% in the granodiorite to 5% in the HMCM. Cordierite concentrations are generally less than 1% in these zoned phases, except for the north and south margin of the HCBM and parts of the HMCM, where anomalous cordierite concentrations reach 10% (MacDonald & Horne 1987).

The TAML is petrologically distinct from these Stage 1 phases, as it is commonly pink and texturally diverse, exhibiting fine-grained, medium-grained, aplitic, pegmatitic, equigranular, porphyritic, seriate, and/or graphic textures. This phase occurs as small intrusions up to several km in diameter that intrude all previous phases, as well as the wall rocks on the eastern margin of the intrusion (MacDonald & Horne 1987; Horne *et al.* 1989; MacDonald *et al.* 1989, 1992*a*, *b*; MacDonald 2001).

The six Halifax Pluton igneous phases were sampled by the Nova Scotia Department of Natural Resources during a regional fusion-XRF lithogeochemical survey (Ham *et al.* 1989, 1990; MacDonald *et al.* 1989, 1992*a*, *b*; MacDonald 2001) undertaken during the late 1980's and early 1990's; Fig. 19). The overall coarse grain size and presence of alkali feldspar megaphenocrysts in most rocks collected in this survey make it particularly difficult to obtain representative samples of this intrusion, especially for some of the coarser and sparser megaphenocryst-bearing rocks. As a result, samples collected as part of both of this reconnaissance lithogeochemical survey were on the order of 20 kg in mass in order to be representative (MacDonald *et al.* 1992*a*, *b*; MacDonald 2001).

In addition to whole rock lithogeochemical analysis, Nova Scotia Department of Natural Resources samples were petrographically classified using a macroscopic point counting procedure implemented using stained slabs of variable dimension (averaging 30 × 30 cm) and a ½ cm grid for point counting. This produced *c.* 900 point counts per sample, with errors (describing constitutional heterogeneity; Pitard 1989*a*, *b*) of ±3 volume % for quartz, plagioclase and alkali feldspar for the granodiorite and monzogranite phases, and ±2 volume % for biotite (two standard deviations, assuming binomial sampling errors). Because thin sections were available for only a small subset of these samples, the composition of plagioclase was not determined. Thus, in all cases, feldspar that did not stain was considered to be plagioclase, even though it may have been albitic and thus should have been considered alkali feldspar, as per Streckeisen's protocol (pers. comm., Mike MacDonald, Nova Scotia Department of Natural Resources). Consequently, some of these rocks may not have been accurately classified.

Given the very coarse overall grain size of these granitoid samples, slabs of these rocks may not be representative of the material sampled. This is because: (1) the slab surfaces represent only a small proportion of the slab volume; and (2) local modal constitutional heterogeneity (variations effectively attributable to a nugget effect; Pitard 1989*a*, *b*) impose significant variability. Ideally, the volume represented by each slab for surface evaluation purposes is estimated to equal the surface area of the slab times the dimension of the average mineral grain in these samples. Assuming rock densities of 2.7, a 20 kg sample, and average grain sizes of 1 cm for the GLBG, GLBM, and SLBM (a collective mean of the alkali feldspar megaphenocrysts and coarse-grained matrix of quartz, plagioclase, alkali feldspar and biotite) and 2 mm for the GRML, these slabs represent only about 1.66 and 0.34% of the original sample masses, respectively, and thus cannot be considered adequately representative of them. As a result, in addition to not being accurate, classification of these rocks by the macroscopic point counting procedure employed is probably not very precise.

Nevertheless, in all cases, the Nova Scotia Geological Survey defined a name for each lithology based on the average plutonic rock classification obtained for each phase (in these cases, ‘granodiorite’, ‘monzogranite’, and ‘leucomonzogranite’). Unfortunately, the measured mineral modes obtained from these macroscopic point counts were never published.

In an effort to recover these point count data, photographs of stained slabs from 55 samples retained at the Nova Scotia Department of Natural Resources’ Stellarton Core Storage Facility were subjected to image analysis (Fig. 20). Results from these classifications (Fig. 21) exhibit significant scatter across several classification fields, probably as a result of the coarse grain size of the rocks and relatively ‘small’ stained slab masses. Nevertheless, results do suggest that the average petrologic compositions of the six phases of the Halifax Pluton are either granodiorite or monzogranite.

Lithogeochemical classification of these rocks represents a possible means to obtain more reliable estimates of these rock compositions. In order to classify the rocks collected using lithogeochemistry, an essential mineral assemblage for each phase must be identified. All phases of the Halifax Pluton include quartz, plagioclase, alkali feldspar, and biotite, but the HMCM also contains significant muscovite and cordierite, the HCBM also contains significant cordierite, and the TAML also contains significant muscovite. The presence of muscovite and cordierite suggests that these rocks exhibit strong peraluminosity, and this likely causes biotite to exhibit aluminous compositions via the tschermak exchange [Al_{2}(Fe,Mg)_{−1}Si_{−1}]. As a result, the biotite in the Halifax Pluton should probably be described using ideal biotite and eastonite-siderophyllite: K(Fe,Mg)_{5/2}Al_{2}Si_{5/2}O_{10}(OH)_{2}] (aluminous biotite). Unfortunately, when muscovite, cordierite, and these two biotite end-members are considered for the HMCM, HCBM, and TAML phases, eight essential minerals exist, and the rank of the resulting *C* matrix (7) is less than the number of rows in the *C* matrix. Consequently, a reaction exists amongst these minerals:
As a result, in Halifax Pluton phases (HMCM, HCBM, and TAML) containing muscovite and/or cordierite, only seven minerals can be used in the *C* matrix (QZ, AN, AB, KS, MS, CD, BT). As a result, biotite can only be described using one composition. To identify this appropriate biotite composition, electron microprobe results were used (MacDonald & Horne 1988). Biotite compositions in these rocks have an *X _{SD}* = 0.60 and

*Mg#*= 0.25 [averaging approximately K

_{2}Mg

_{1.5}Fe

_{4}Al

_{3}Si

_{5.5}O

_{20}(OH)

_{4}; BT*]. This aluminous biotite composition is consistent with the higher peraluminosity of these later phases and was thus used as an essential mineral composition in the following matrix calculations.

For the other phases that do not contain essential muscovite or cordierite (MBG, SLBM, PCBM), the essential mineral assemblage could include both biotite (BT) and aluminous biotite (SD) along with QZ, AN, AB, and KS. This allows calculation of an average biotite composition for these more primitive intrusive rocks from the lithogeochemical data. As a result, two different *C* matrices containing two different essential mineral assemblages were used in the classification of these rocks.

The first *C* matrix was employed to classify the MBG, and SLBM, and PCBM rocks, and considered quartz, plagioclase, alkali feldspar, biotite and siderophyllite-eastonite as essential minerals:
(24)This *C* matrix and a *P* identity matrix were then used to obtain the *A* matrix:
(25)A second *C* matrix was employed to classify the HMCM, HCBM, and TAML rocks, and considered quartz, plagioclase, alkali feldspar, an approximation of the EMP analyses of biotite (BT*), muscovite, and cordierite as essential minerals:
(26)Again, this matrix and a *P* identify matrix was used to create the resulting *A* matrix:
(27)

Results from each of these classifications are plotted on two molar bulk feldspar composition ternary diagrams in Figure 22. These reveal that as the Halifax Pluton crystallized and evolved from the outside in, the plagioclase composition became subtly but progressively more albitic. Results are also plotted on a QAP Streckeisen ternary classification diagram (Fig. 23). These indicate that the MBG consistently classifies lithogeochemically as ‘granodiorite’, but all other phases straddle the ‘granodiorite-monzogranite’ boundary.

In the lithogeochemical classification of the Halifax Pluton rocks, the MBG, PCBM, and SLBM samples were classified using an essential mineral assemblage of quartz, plagioclase, alkali feldspar, biotite and siderophyllite-eastonite (aluminous biotite), allowing one to estimate the composition of biotite in these rocks. The average biotite composition calculated by the classification of these rocks is very aluminous, with an *X _{SD}* = 0.93 and

*Mg#*of 0.48. As a result, the average biotite composition in these phases is approximately that of siderophyllite-eastonite with a balanced Fe/Mg ratio [KMg

_{5/4}Fe

_{5/4}Al

_{2}Si

_{5/2}O

_{10}(OH)

_{2}]. This very aluminous composition illustrates that even these relatively un-evolved rocks are substantially peraluminous.

In the above classification, the MBG, SLBM, and PCBM modes were calculated using an essential mineral assemblage of quartz, plagioclase, alkali feldspar, biotite and siderophyllite-eastonite. However, because the composition of biotite in these phases was approximately that of siderophyllite-eastonite, they could have been classified using an essential mineral assemblage of QZ, PL, AF, BT, MS, and CD because a reaction (equation 23) can be written between Al-biotite and QZ, BT, MS and CD. As a result, if MBG, SLBM, and PCBM samples had been classified using this alternative essential mineral assemblage, they should produce a classification similar to that obtained using the original essential element assemblage (QZ, PL, AF, BT, SD), albeit with a slightly different amount of quartz, because quartz is the only essential mineral in the reaction of equation 23 that is also used in classification. The very small amounts of biotite, muscovite and cordierite in the MBG, SLBM, and PCBM will cause only minor differences in quartz abundance, and thus will have minimal impact on classification.

To test this hypothesis, Figure 24 presents the classifications for MBG, SLBM, and PCBM rocks obtained using the essential mineral assemblage of QZ, PL, AF, SD, MS, and CD. These can be compared with the classifications undertaken on the same samples using an essential mineral assemblage of QZ, PL, AF, BT, and SD (Fig. 23; top). In both cases, the MBG classifies as ‘granodiorite’ and the SLBM and PCBM classify as transitional between ‘granodiorite’ and ‘monzogranite’. Furthermore, there is no apparent difference in the amounts of quartz in the rocks on these two diagrams. Thus, both essential mineral assemblages produce essentially the same classification. Clearly, the change of basis lithogeochemical classification procedure described above can be undertaken using any superset of essential minerals in a rock, as the extra minerals included in the assemblage, but not present in the rock, have no substantive impact on the resulting classification because they are projected from.

Finally, using the results calculated during lithogeochemical classification (Figs 23 and 24), total biotite modes in the Halifax Pluton range from 12% in the MBG to <5% in the TAML. As a result, the TAML is confirmed to be a leucogranite (with BT <5%). Muscovite modes range from 3 to 10%, but only the MBG phase has average muscovite modes less than 5%, so all other phases can be considered to be two-mica granites.

### Peraluminous Salmontail Lake Pluton, South Mountain Batholith, Nova Scotia

The Salmontail Lake Pluton is located on the north margin of the South Mountain Batholith, in Kings County, Nova Scotia. The irregularly shaped intrusion is roughly 50 km by 25 km in dimension with a long axis oriented NE-SW (Fig. 25). Several different igneous phases are present in the Salmontail Lake Pluton, including the Gaspereau Lake Biotite Granodiorite (GLBG; average SiO_{2} concentration = 68 wt. %, average Shand index = 1.14, average alkali index = 1.57), the Gaspereau Lake Biotite Monzogranite (GLBM; average SiO_{2} concentration = 69 wt. %, average Shand index = 1.18, average alkali index = 1.52), and Salmontail Lake Biotite Monzogranite (SLBM; average SiO_{2} concentration = 69 wt. %, average Shand index = 1.18, average alkali index = 1.50), all of which emplaced during an early intrusive episode (Stage 1). The Gold River Two-Mica Leucomonzogranite (GRML; average SiO_{2} concentration = 72 wt. %, average Shand index = 1.23, average alkali index = 1.32) was emplaced during a later, but geochronologically indistinguishable, intrusive episode (Stage 2). It nevertheless exhibits distinct crosscutting relationships with the SLBM (Kontak *et al.* 2003; Reynolds *et al.* 2004).

The GLBG, GLBM, and SLBM phases are all very similar, containing anti-perthitic alkali feldspar megaphenocrysts up to 2 × 5 cm in dimension set in a coarse-grained groundmass of quartz, plagioclase, alkali feldspar and biotite (Fig. 26; left). Collectively, these three units exhibit a crude trachytic texture with orientation parallel to the pluton margin, and ostensibly exhibit evidence of concentric zoning within the pluton, with as much as 27 volume % modal biotite on the outside of the intrusion and as little as 9 volume % modal biotite on the inside (MacDonald & Horne 1987).

In contrast, the GRML is petrologically different, as it is commonly fine-grained to aplitic textured, pink, and typically contains only up to 5 volume % biotite and 5 – 10 volume % muscovite (Fig. 26; right). It occurs as isolated plugs of limited dimension (<1 km across) that clearly intrude the SLBM along a NE-SW trend through the centre of the pluton (MacDonald & Horne 1987; Horne *et al.* 1989; MacDonald *et al.* 1989, 1992*a*, *b*; MacDonald 2001).

The four Salmontail Lake Pluton igneous phases were sampled during: (1) a regional fusion-XRF lithogeochemical survey (Ham *et al.* 1989, 1990; MacDonald *et al.* 1989, 1992*a*, *b*; MacDonald 2001) undertaken by the Nova Scotia Department of Natural Resources during the late 1980s and early 1990s, and (2) later in another more limited regional lithogeochemical survey of similar sample size and density (Gladwin 2000, *unpublished data*; Fig. 25). The overall coarse grain size and presence of alkali feldspar megaphenocrysts in most rocks collected in both surveys make it particularly difficult to obtain representative samples of this intrusion, especially for some of the coarser and sparser megaphenocryst-bearing rocks. As a result, samples collected as part of both of these reconnaissance lithogeochemical surveys were on the order of 20 kg in mass to be representative (MacDonald *et al.* 1992*a*, *b*; Gladwin 2000; MacDonald 2001).

In addition to whole rock lithogeochemical analysis, Nova Scotia Department of Natural Resources samples were petrographically classified using the point counting procedure described above for the Halifax Pluton. Results suffer from the same, significant problems as those described for the Halifax Pluton. Nevertheless, the Nova Scotia Geological Survey, as it did for the Halifax Pluton, defined a name for each lithology based on the average plutonic rock classification obtained for each phase, in these cases, ‘granodiorite’, ‘monzogranite’, and ‘leucomonzogranite’).

In the manner described above, an attempt was made to undertake image analysis of the photos of stained slabs from samples retained at the Nova Scotia Department of Natural Resources’ Stellarton Core Storage Facility. Unfortunately, because of the storage conditions of these slabs, the potassium cobaltinitrate stains on many of the slabs from this pluton had faded, limiting their use in classification. As a result, useful photos from only 20 slabs of the lithogeochemical samples could be obtained, and results are so erratic and sparse that average compositions of each phase cannot be obtained.

As a result, lithogeochemical classification of these rocks effectively represents the only remaining immediately available means to obtain more reliable estimates of these rock compositions. In order to classify the rocks collected in these two surveys using lithogeochemistry, an essential mineral assemblage for each phase was identified. The GLBG, GLBM, and SLBM phases all contain essential quartz, plagioclase, alkali feldspar, and biotite, whereas the GRML contains essential muscovite as well. The presence of muscovite supports the chemical evidence suggesting this pluton is peraluminous, and as a result, it is likely that biotite in all four of these phases is Al-rich. Thus, a single essential mineral assemblage for these phases includes quartz, plagioclase, alkali feldspar, biotite, aluminous biotite (siderophyllite-eastonite; SD), and muscovite (MS), and this assemblage can be used to simultaneously classify all four phases of the Salmontail Lake Pluton. Thus, for the Salmontail Lake Pluton, the essential mineral assemblage defines a *C* matrix of:
(28)This *C* matrix can be used to obtain a corresponding matrix (*A _{M}*) containing the coefficients required for classification using Streckeisen ternary diagrams that are projections from biotite, aluminous biotite (siderophyllite-eastonite) and muscovite. This

*A*matrix is: (29)Results derived using this

_{M}*A*matrix are presented in the molar feldspar ternary diagram in Figure 27. These illustrate that the GLBG, GLBM, and SLBM phases, which have distinct plagioclase and alkali feldspar, plot well above samples from the GRML. This compositional break is consistent with the intrusive contacts between the GRML and SLBM phases. Interestingly, this compositional break is not apparent on the QAP Streckeisen ternary diagram on which these samples have been plotted after classification using the lithogeochemical classification procedure (Fig. 28). On this diagram, samples of GLBG, GLBM, SLBM, and GRML phases plot together near the boundary between granodiorite and monzogranite, but should probably all be formally lithogeochemically classified as ‘granodiorite’.

_{M}The petrologic zoning hypothesis suggested to occur in the GLBG, GLBM, and SLBM phases of the Salmontail Lake Pluton (MacDonald & Horne 1987) can be addressed using results derived from the lithogeochemical classification procedure, as the *V _{BT} + V_{SD}* coefficients can be used to obtain parameters describing the amount and composition of biotite in these rocks, and the

*V*coefficients can be used to estimate the muscovite modes. Figure 29 (top) presents a bubbleplot illustrating where high volume concentrations of biotite (BT plus SD volume modes) exist in the Salmontail Lake Pluton (note that samples from the GRML have been omitted because of the cross-cutting nature of this phase). Large bubbles, corresponding to biotite concentrations of up to 27 volume %, occur predominantly in the SW lobe of the intrusion, whereas small bubbles, corresponding to biotite concentrations as low as 9 volume %, occur in the interior of the Salmontail Lake Pluton. Notably, these high biotite concentrations do not occur in granodiorite rocks to the north, east, and SE, suggesting that the mafic zoning observed during mapping is not perfectly concentric.

_{MS}Interestingly, when calculated modal amounts of muscovite are plotted in space, GLBG, GLBM, and SLBM samples on the margin of the pluton, particularly to the SW, report very low to negligible muscovite concentrations, whereas those in the pluton core exhibit higher muscovite concentrations (Fig. 29; bottom). Consequently, spatial petrological patterns for biotite and muscovite suggest that the SW portion of the Salmontail Lake Pluton exhibits a distinct compositional contrast with the central and NW portions of the intrusion, as it is more mafic and less peraluminous than the core. This pattern represents a refinement to the mineral zoning originally recognized during mapping (MacDonald & Horne 1987).

Lastly, the median biotite composition in the GLBG, GLBM, and SLBM phases, derived using the lithogeochemical classification procedure is not as aluminous as that determined for the Halifax Pluton, yet it still has an intermediate magnesium number (*X _{SD}* = 0.55 and

*Mg#*= 0.49). Consequently, the Salmontail Lake Pluton does not appear to be evolved as the Halifax Pluton, being less peraluminous. It also contains lower lithophile element concentrations than the Halifax Pluton, with median Ba, Rb, Li, F, and Zr abundances that are 10 times less than those of the Halifax Pluton.

### Melts of the Sudbury Igneous Complex, Ontario

The Sudbury Igneous Complex (SIC) is located in northern Ontario on the Canadian Shield and has been studied intensively for over 50 years. Two major portions of the complex comprise norite and granophyre. Norite has been recognized as having multiple compositions on the north and south margins of the complex (felsic norite, South Range norite, mafic norite, quartz-rich norite; Naldrett *et al.* 1970). Granophyre overlies and is separated from the norite by quartz gabbro and upper gabbro units, and intrudes overlying breccia of the Onaping Formation. It also is recognized as having at least two crosscutting compositions, improperly referred to as ‘granophyric micropegmatite’ and ‘plagioclase-rich rock’ (Peredery & Naldrett 1975).

Lithogeochemical data from both norite and granophyre are abundant. As a result, for each lithology several major oxide datasets suitable for lithogeochemical classification were assembled (*norite*: Beswick, unpublished, used with permission, *n* = 31; Binney 1994, *n* = 113; Lightfoot *et al.* 1997, *n* = 67; Therriault *et al.* 2002, *n* = 25; Dessureau 2003, 332; and Zeig, unpublished, used with permission, *n* = 64 ; *granophyre*: Beswick, unpublished, used with permission, *n* = 58; Binney 1994, *n* = 43; Lightfoot *et al.* 1997, *n* = 27; Therriault *et al.* 2002; *n* = 25; and Zeig, unpublished, used with permission, *n* = 46). These lithogeochemical analyses were used to classify both lithologies after the corresponding sets of essential minerals were identified for norite and granophyre using petrographic descriptions of each magma (Naldrett *et al.* 1970; Peredery & Naldrett 1975).

The petrography of the two granophyre lithologies, ‘granophyric micropegmatite’ and ‘plagioclase-rich rock’, consist of: micrographic intergrowth of *QZ* and *KS* (60 – 85% and 20 – 50%), plagioclase (5 – 15% and 25 – 40%), augite (5 – 10% and 20 – 30%), ilmenite (1 – 3% and 1 – 5%), and apatite (<1% and <1%), respectively (Peredery & Naldrett 1975). As a result, an appropriate essential mineral assemblage for these two lithologies consists of *QZ*, *AN*, *AB*, *KS*, *CP*, and ilmenite (IL), as apatite occurs in insufficient abundances to affect classification. The *C* matrix for this essential mineral assemblage is:
(30)This *C* matrix was used to obtain the *A* matrix:
(31)which defines a Ti-absent solution to the matrix equation. However, because TiO_{2} was likely analyzed well in these rocks, and because isochemical uralization of augite (Oliver 1951) resulted in the formation of actinolite, chlorite, and epidote, both ferric and ferrous Fe are present in the rocks, even though only total Fe (as Fe_{2}O_{3}) was measured. As a result, a possibly more robust solution than that presented in equation 31 would be an Fe + Mg-absent solution. Consequently, *V _{N1}* was subtracted from

*V*to produce an alternative

_{IL}*A*matrix containing such an Fe + Mg-absent solution involving (

_{M}*V*): (32)Because SIC granophyre contains at most 30% mafic minerals, it should be classified using a QAP Streckeisen ternary diagram. Lithogeochemical classification results using this

_{IL}*A*matrix are presented in Figure 30 (a molar feldspar ternary diagram). On this figure, a distinct compositional break exists along a vertical line at 35% K-spar and separates non-plagioclase-rich rocks from plagioclase-rich rocks. These two groups of rocks are interpreted to correspond to the two granophyre units described above (‘granophyric micropegmatite’ and ‘plagioclase-rich rocks’).

_{FM-absent}On Figure 31, a QAP Streckeisen ternary diagram with results from the lithogeochemical classification procedure plotted, these two granitoid compositions cluster in two places, suggesting that the ‘granophyric micropegmatite’ should be classified as a transitional ‘granodiorite-monzogranite’, whereas the plagioclase-rich rocks should be classified as a ‘granodiorite’. These classifications are consistent with the petrographic data collected from these units (Peredery & Naldrett 1975).

In order to classify the Sudbury norites, the essential mineral assemblage of those units must also be determined. Lithogeochemical data are available from only the felsic norite, mafic norite, and South Range norite, and the essential minerals in these three units are identical, consisting of *QZ*, *PL*, *OP*, *CP*, ulvospinel (*UV*) and a micrographic intergrowth of *QZ* and *KS*, although different proportions of these minerals exist in each unit. Incipient to minor replacement of *OP* and *CP* by biotite, actinolite, and epidote during uralization has occurred, but this it not interpreted to have substantially modified the rock composition, other than causing it to be more hydrous (Oliver 1951).

The *C* matrix for the essential mineral assemblage for these norites is thus:
(33)This *C* matrix was used to obtain the *A* matrix:
(34)The norites contain between 70 and 90% mafic minerals, and so can be classified using the QAP Streckeisen ternary diagram (Fig. 32). The three different norites all plot in the same field (quartz monzodiorite/gabbro), and so depending on their plagioclase composition (>*AN*_{50} or <*AN*_{50}), they should be called ‘quartz monzogabbro’ or ‘quartz monzodiorite’, respectively. To determine which name is appropriate, the plagioclase compositions have to be estimated petrographically or measured instrumentally (e.g. using electron microprobe analysis). Alternatively, the rock compositions could be plotted on a molar feldspar ternary diagram, so that after projection from *KS*, the average plagioclase composition in each rock can be estimated. This latter approach has been undertaken (Fig. 33), and reveals that although estimated plagioclase compositions in these norites range from *AN*_{8} to *AN*_{75}, largely due to several outliers, the bulk of the estimated plagioclase compositions are between *AN*_{35} and *AN*_{60} (corresponding to the dotted lines on Fig. 33). As a result, plagioclase compositions in SIC norites overlap the *AN*_{50} threshold used to distinguish between diorite and gabbro (the dashed line on Fig. 33). Fortunately, the individual norite units plot in different locations in Figure 33. For example, South Range norite mostly has calculated plagioclase compositions greater than *AN*_{50} (consistent with results from Naldrett *et al.* 1970 and Mungall 2003), and so this rock should be called a ‘quartz monzogabbro’, whereas mafic norite mostly has calculated plagioclase compositions less than *AN*_{50}, and so it should (perhaps ironically) be called a ‘quartz monzodiorite’. Felsic norite has calculated plagioclase compositions across a wider range (from *AN*_{35} to *AN*_{60}), so it is not clear whether it should be called ‘quartz monzogabbro’ or ‘quartz monzodiorite’. Measured plagioclase compositions of felsic norite by Naldrett *et al.* (1970) from the Fecunis Traverse and Strathcona Section on the North Range, and the McLennan Traverse on the East Range are almost unanimously greater than *AN*_{50}, so it is possible that plagioclase compositions in felsic norite vary systematically across the Sudbury Igneous Complex.

Because SIC norites contain significant amounts of mafic minerals (from 70 to over 90%), it is probably worthwhile examining these rocks using other Streckeisen ternary diagrams designed to classify mafic and ultramafic rocks. An appropriate diagram is the OP-CP-PL Streckeisen ternary diagram of Figure 34. Samples plot on this diagram in places that classify felsic and South Range norite as transitional between ‘leuconorite’ and ‘clinopyroxene leuconorite’ and mafic norite as transitional between ‘norite’ and clinopyroxene norite’.

A last point to make in this examination of the Sudbury Igneous Complex norites involves the procedural consequences of incorrect identification of the essential mineral assemblage. Such an error can result in the calculation of negative molar mineral numbers, as described above for the Emerald Lake Pluton. In this case, though, this error has a different manifestation. If, for example, the SIC norites are errantly thought to be ‘olivine tholeiites’, instead of ‘quartz tholeiites’, as above (Yoder & Tilley 1962), then the appropriate essential mineral assemblage for the norites would be olivine (*OL*), *PL*, *KS*, *OP*, *CP*, and *UV* instead of *QZ*, *PL*, *KS*, *OP*, *CP*, and *UV*. Thus, these rocks would plot to the left of the plane of silica saturation, instead of to the right, on the basalt tetrahedron (Fig. 35, Yoder & Tilley 1962). For the SIC norites, this error in the selection of the essential mineral assemblage results in negative molar mineral numbers for olivine in virtually every sample during classification (Fig. 36). This is because the presence of quartz in these rocks causes the linear combinations for olivine (equation 35, column 1):
(35)to always be less than zero. Effectively, the only way these mathematics can describe the amount of quartz that is actually present in these rocks is using negative olivine and positive orthopyroxene molar mineral amounts. This is because:
(36)Consequently, this example illustrates that when negative molar mineral numbers that are significantly different from zero occur in virtually every sample, it is likely that the essential mineral assemblage for a suite of rocks has been misidentified. Consequently, when such a situation occurs, an alternative essential mineral assemblage should be sought that produces positive molar mineral numbers (Fig. 37; *QZ*, *PL*, *CP*, *OP*, *KS*, *UV*), to ensure that a correct classification is obtained.

### Uwekahuna Laccolith Rocks, Hawaii

Lavas emanating from the Kilauea volcano on Hawaii have been intensely studied, largely because they could easily be sampled as they flow by. As a consequence, unaltered rock compositions can be confidently obtained to investigate igneous processes. A small lithogeochemical dataset was collected by Murata & Richter (1961) from the subvolcanic Uwekahuna Laccolith, which outcrops for 300 m along the Kilauea caldera wall immediately below the USGS Hawaiian Volcano Observatory on the big island.

The five samples collected by Murata & Richter (1961) from a vertical traverse across the laccolith's 15 m thickness were analyzed for whole rock major and trace element concentrations by wet chemical means. Petrographically, the rocks contain abundant olivine phenocrysts (from 15 to 40 volume %) that appear to have settled to the bottom half of the magma chamber. The magma's groundmass consists of very fine-grained plagioclase, clinopyroxene, and minor orthopyroxene (Murata & Richter 1961).

Pearce & Stanley (1991) demonstrated using Pearce element ratio analysis that this picritic magma exhibits compositional variations consistent with olivine sorting, and that variations attributable to the fractionation of other minerals within the laccolith (plagioclase, clinopyroxene, orthopyroxene) are negligible. Consequently, classification of these magma compositions can be achieved using an essential mineral suite of *OL*, *PL*, *CP*, and *OP* (the same suite used to classify rocks within the Stillwater Complex; equation 22). Classification of these sub-volcanic rocks will essentially predict the proportions of *OL*, *PL*, *CP*, and *OP* that would have formed had the laccolith cooled slow enough to fully crystallize.

Figures 38, 39 and 40 present the lithogeochemical classification of these Uwekahuna Laccolith rocks plotted on OL-OP-CP, PL-PX-OL, and PL-OP-CP Streckeisen ternary diagrams. In Figures 38 and 39, the rock compositions plot on lines that pass through the olivine vertex, whereas on Figure 40 (a projection from olivine), they all plot at a single point. These patterns illustrate the olivine sorting is the only process responsible for compositional variability, confirming the olivine sorting hypothesis proposed by Murata & Richter (1961) and tested compositionally by Pearce & Stanley (1991).

Because the plagioclase mineral modes estimated by the lithogeochemical classification procedure described above range from 35 to 51% plagioclase, Figure 38 should not be used for classification of the Uwekahuna Laccolith samples. This is because the ternary diagram of Figure 38 is valid only for rocks containing <10% plagioclase. Figure 39 illustrates that all samples plot precisely on the *OP* gabbro-*CP* norite boundary, allowing use of the term ‘gabbronorite’ as an appropriate name for these rocks. On Figure 40, samples with relatively high concentrations of olivine plot in the ‘olivine melagabbronorite field’, whereas samples with relatively low concentrations of olivine plot in the ‘olivine gabbronorite’ field. Consequently, the bottom, *OL*-rich portion of the Uwekahuna Laccolith has compositions equivalent to OL melagabbronorite, whereas the top, *OL*-poor portion of the Uwekahuna Laccolith has compositions equivalent to *OL*-gabbronorite. Clearly, use of the above lithogeochemical classification procedure not only reveals the controls on compositional variability in these rocks, but also allows identification of appropriate lithology names for these sub-volcanic rocks.

### Kilauea Volcanic Rocks, Hawaii

Another lithogeochemical dataset from Kilauea volcano was assembled by Russell & Stanley (1990*b*). This purely volcanic rock dataset was derived from numerous authors (MacDonald & Eaton 1964; Murata & Richter 1966; Richter & Murata 1966; Richter *et al.* 1970; Wright & Fiske 1971; Wright 1973; Ho & Garcia 1988) who sampled the 1959 Kilauea Iki summit eruption lavas, and the 1955 and 1960 SE flank eruption lavas of Kilauea. These samples were collected while still hot, and whole rock analyses of major and trace elements were analyzed by wet chemical methods.

Petrographically, the 1959 summit lavas contain abundant olivine phenocrysts (up to 19%), whereas the 1955 and 1960 flank eruptions contain low concentrations of olivine, as well as plagioclase, and clinopyroxene. Only the 1955 SE rift flank eruption lavas contain orthopyroxene phenocrysts. The groundmass in all three lavas is glassy. Using the lithogeochemical data from these lavas, Russell & Stanley (1990*b*) were able to demonstrate that these three suites of rocks are cogenetic, and thus were likely derived from the same parent melt. Furthermore, Russell & Stanley (1990*b*) demonstrated using Pearce element ratio diagrams that compositional variation in the 1959 lavas was predominantly due to olivine sorting, whereas in the 1955 and 1960 lavas, a combination of olivine, plagioclase and clinopyroxene sorting explains the compositional variations. Considering the lavas collectively, Russell & Stanley (1990*b*) were able to convincingly identify the precise composition when the melt reached a cotectic using Pearce element ratio diagrams. They further demonstrated that the both plagioclase and clinopyroxene join the crystallizing mineral assemblage at approximately the same temperature in these lavas.

As above, using an essential mineral assemblage of *OL*, *PL*, *CP*, and *OP*, the linear combinations of equation 22 can be used to classify these Kilauea volcanic rocks by predicting, based on the lithogeochemistry, the proportions of *OL*, *PL*, *CP*, and *OP* that would have formed if these lavas had cooled slow enough to fully crystallize.

Figures 41, 42, and, 43 present the lithogeochemical classification of these Kilauea lavas plotted on OL-OP-CP, PL-OP-CP, and PL-PX-OL Streckeisen ternary diagrams. When results are plotted on Figure 41, four data trends (A, B, C, and D; defined by the dashed lines) are recognizable that suggest several petrologic processes (olivine fractionation through the clinopyroxene and plagioclasecotectic and the orthopyroxene peritectic, along with self mixing) are responsible for the observed compositional variations within these rocks:

Trend A (Fig. 41) passes through the olivine vertex and through all of the 1959 summit lavas, indicating that the sorting of olivine is the principle compositional control in these rocks;

Trend B (Fig. 41) passes through the olivine-poor end of the 1959 summit lava trend and the 1960 olivine-bearing SE rift lavas, indicating that the 1960 lavas may have evolved directly from the 1959 lavas after passing through the olivine-clinopyroxene-plagioclase cotectic;

Trend C (Fig. 41) is defined by olivine-rich 1960 SE rift lavas, and passes through 1959 olivine-rich summit lavas of Trend A, and 1960 olivine-rich SE rift lavas of Trend B, indicating that the olivine cumulate portion of the magma chamber (erupted in 1959) mixed with an evolved, fractionated portion of the magma chamber saturated in olivine, clinopyroxene, and plagioclase (erupted in 1960), and that these mixed lavas also erupted from the SE rift of Kilauea in 1960.

Trend D (Fig. 41) is represented by all of the samples from the 1955 and 1960 lavas that contain virtually no olivine (i.e. they lie on the CP-OP join and contain orthopyroxene phenocrysts), indicating that these lavas are the most evolved compositions in the subjacent magma chamber.

Again, because the plagioclase concentrations in these Kilauea lavas range from 39 to 60 volume %, Figure 41 should not be used to classify these rocks. On Figure 42, samples plot on a short horizontal trend ranging from OP gabbro to CP norite. On Figure 43, all samples plot on a line through the olivine vertex, but 1959 summit lavas plot within the OL gabbro field, the 1960 lavas plot within the OL gabbro and gabbro/norite fields, and the 1955 lavas plot within the gabbro/norite field. Consequently, 1959 lavas have compositions consistent with OL-OP gabbros, 1960 lavas have compositions equivalent to OL-OP gabbros and OP-gabbros, and 1955 lavas have compositions equivalent to OP gabbros and CP norites.

### Passamaquoddy Bay Rhyolite Volcanic Rocks, New Brunswick

Subaqueous quartz- and feldspar-porphyritic felsic volcanic flows, tuffs, and sub-volcanic intrusions of the Eastport Formation (Silurian age) outcrop of the shores of Passamaquoddy Bay, New Brunswick. These rocks were mapped and sampled by Van Wagoner *et al.* (2002), and analyzed for whole rock major and trace elements by XRF methods. Flows, tuffs, and non-peperitic sub-volcanic intrusive rocks in this suite exhibit little petrographic evidence of hydrothermal alteration with seawater, and so represent an ideal dataset to test the change of basis lithogeochemical classification procedure on felsic volcanic rocks.

Stanley (2002) examined these rocks using Pearce element ratio methods and demonstrated that these rhyolites derive from a common parent melt composition, display compositional variations consistent with quartz and alkali feldspar sorting, and exhibit only incipient hydrothermal alteration of feldspar to muscovite and quartz. Consequently, these rocks were classified by lithogeochemical means using an essential mineral assemblage of quartz, anorthite, albite, K-spar, and biotite and the linear combinations of equation 11 (the Al-absent solution in Fig. 3). Results are presented in Figure 44. These felsic volcanic rocks plot along a broad trend from the quartz monzodiorite field, through the granodiorite and monzogranite field, into the syenogranite field. This suggests that the magma chamber(s) that was (were) the source of these rocks likely evolved over time, and these intrusive compositions are consistent with descriptions of the felsic volcanic rock compositions as silicic dacite and rhyolite (Van Wagoner *et al.* 2002; Pickerill & Pajari 2011).

The molar mineral numbers of these rock compositions have been converted into mass mineral numbers via multiplication of their gram formula weights, and then ‘closed’ by dividing each by the sum of the mass mineral numbers, forcing the results to sum to 100%. This allows plotting of these data on Tuttle & Bowen's (1958) granite minimum melt, mass concentration ternary diagram (QZ-AB-KS at 500 bars water pressure; Fig. 45). Results exhibit a dense cluster of samples near the eutectic composition with scatter across the low temperature portions of the granite field (probably due to the crystal sorting during volcanic emplacement; observed using PER analysis of the lithogeochemistry). This pattern confirms that these rhyolites are products of partial melting, and that, even without macroscopic, microscopic, or spectroscopic ground truth, the magmatic classification derived using the change of basis lithogeochemical classification procedure described above is highly plausible.

## Conclusions

The lithogeochemical classification procedure for igneous rocks described herein was originally conceived in 2007, and has since undergone 10 years of comprehensive testing and evaluation. A large number of publically available lithogeochemical datasets with associated and comprehensive petrographic ground truth information were examined using this technique in order to assess this its effectiveness.

Results indicate that this new change of basis lithogeochemical classification procedure is highly effective. Positive (accurate, precise, consistent, robust, and conclusive) classification results have been obtained for a large number of felsic alkaline, peraluminous, and metaluminous magmatic and volcanic rocks, as well as a number of other mafic and ultramafic magmatic and volcanic rocks using this new lithogeochemical classification procedure. These results indicate that high quality igneous rock classification can be undertaken using lithogeochemical means, if constrained by basic petrographic information.

Features associated with the change of basis lithogeochemical classification procedure enhance its application. For example, in many cases, multiple linear combination solutions can be identified, and these allow one to avoid the use of elements that either are not available (were not analyzed) or underwent material transfer during post-emplacement alteration, and so would otherwise produce inaccurate results. Additionally, mineral compositions used to determine the linear combinations can be constrained by electron microprobe or molar element ratio analysis methods, but in the absence of these, successful classification has often resulted when the mineral compositions are assumed to be ideal. Furthermore, because this lithogeochemical classification procedure requires only lithogeochemical data and knowledge of the essential mineral assemblage in order to classify a suite of rocks, if the essential mineral assemblage is unknown but can be postulated (such as with volcanic rocks), classification still can be undertaken.

Moreover, application of the ‘change of basis’ lithogeochemical classification procedure to describe rock compositions in terms of the minerals contained in the rocks need not be used exclusively for classification. Such change of basis calculations can be undertaken to estimate the abundances of many mineral assemblages in rocks (whether they result from igneous, sedimentary, diagenetic, or hydrothermal processes). As a result, these ‘change of basis’ calculations can be used to create quantitative mineral mode estimates that can be plotted in space to help understand the mineral zoning that exists in a study area. Such quantitative mineral proportions can also be plotted against trace element concentrations to determine what minerals control various trace element concentrations.

Lastly, because the mineral abundance estimates can be translated from volume proportions to mass and molar proportions using mineral densities and molar volumes, results from the ‘change of basis’ calculation have substantial versatility. This is because a number of rock and mineral properties can be estimated for any suite of rocks using these different concentration formats. For example, by multiplying the volume proportions of each mineral times their corresponding densities, one can estimate the bulk density of a rock (except for any porosity present). Alternatively, using the molar proportions of minerals that produce and neutralize acid during weathering (e.g. pyrite, calcite), one to readily calculate the net neutralizing potential of a rock during acid-base accounting investigations.

The ‘change of basis’ calculation used in this lithogeochemical classification procedure does suffer from two challenges, both related to negative calculation results. When the lithogeochemical classification procedure produces slightly negative molar mineral numbers in a proportion of the samples, analytical error is likely the cause, and recoding the negative molar mineral numbers to zero (producing only a small change to the result) avoids any impossibly negative results. In contrast, when the change of basis lithogeochemical classification procedure produces significantly negative molar mineral numbers in virtually all samples, this is likely a consequence of the selection of the wrong essential mineral assemblage. When this occurs, re-examination of the petrology of the rock suite should allow identification of the correct essential mineral assemblage, so that properly positive mineral modes are obtained.

A final caveat that the reader should heed is that although this lithogeochemical classification procedure is based on the geochemistry of the rocks under consideration, its results will not necessarily match those resulting from a classification undertaken by the conventional Streckeisen method. Consequently, this change of basis lithogeochemical classification procedure should be considered as an alternative way to classify igneous rocks. It should not be considered a substitution of the conventional Streckeisen method unless, the rocks under consideration are fresh, and no sub-solidus isochemical reactions have taken place to modify the original igneous mineralogy. If sub-solidus, isochemical reactions have taken place to modify the mineralogy, differences in results from this and conventional Streckeisen classification methods will provide insight into the sub-solidus reactions that have occurred. However, if sub-solidus, material transfer reactions (e.g. hydrothermal alteration) have taken place, both the mineralogy and geochemistry of the rocks will have been modified. In this case, the lithogeochemical classification procedure cannot be considered reliable. Nevertheless, in such cases, accurate mineral modes can still be calculated if the appropriate essential mineral assemblage and the correct mineral compositions are employed in the change of basis calculation.

## Acknowledgements

The author would like to thank Rob Bowell and Kelly Russell for their helpful reviews of this manuscript, and the many Acadia University geology students, supervised by Dr Sandra Barr, who generously provided datasets from their theses used to perform additional tests of the lithogeochemical classification procedure that are not presented herein.

- © 2017 The Author(s)