Compositional Structures in 

Two Batholiths of

GEOLOGICAL SURVEY PROFESSIONAL PAPER 574- H
Compositional Structures in 

Two Batholiths of 

Circumpacific North America

By A. T. MIESCH and BRUCE L. REED

STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 5 74-H

A mathematical analysis of chemical data on 386 

samples of granitic rocks from the Sierra Nevada 

and Alaska-Aleutian Range batholiths, and discussion 

of the petrogenetic implications of the results

UNITED STATES GOVERNMENTPRINTING OFFICE, WASHINGTON: 1979
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Miesch, Alfred T.

Compositional structures in two batholiths of circumpacific North America.

(Statistical studies in field geochemistry)

(Geological Survey Professional Paper 574-H)

Bibliography: p. 30

Supt.ofDocs.no.: 119.16:574-H.

1. Batholiths Sierra Nevada. Mountains. 2. Batholiths Alaska Aleutian Mountain Range.

3. Granite Sierra Nevada Mountains. 4. Granite Alaska Aleutian Mountain Range. 5. Phase rule and

equilibrium. 6. Multivariate analysis.

I. Reed, Bruce L., 1934- joint author. II. Title. III. Series. IV. Series: United States Geological Survey Professional Paper 574-H. 

QE75.P9 no. 574-H [QE611.5.U6] [551.8'8] 557.3'08s 78-606014

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 

Stock Number 024-001-03184-1
CONTENTS

Page

Abstract........................................................................... HI

Introduction ....................................................................... 1

Compositional structure ............................................................ 2

Rock compositions represented as vectors ............................................ 3

Vector configurations............................................................... 4

Factor-variance diagrams ........................................................... 6

Compositional structures in the Sierra Nevada and Alaska-Aleutian Range batholiths .... 7

Speculation on end-member compositions............................................. 10

Formation of an individual pluton ................................................ 20

Compositions of partial melts from the crust...................................... 22

Conclusions ........................................................................ 27

Appendix .......................................................................... 29

References cited .................................................................... 30

ILLUSTRATIONS

Page

FIGURE 1. Vector diagrams representing three forms of the same data for five hypothetical rock samples .................. H4

2. Stereogram showing vectors representing compositions of hypothetical samples .............................. 5

3-5. Factor variance diagrams for the:

3. Data in tables 1 and 2 ......................................................................... 7

4. Pikes Peak batholith, Colorado ................................................................ 8

5. McCartys Basalt, New Mexico................................................................. 8

6. Index map of the Alaska-Aleutian Range batholith and the Sierra Nevada batholith........................... 9

7. Factor-variance diagram for the Sierra Nevada batholith.................................................... 9

8. Factor-variance diagram for the Alaska-Aleutian Range batholith ........................................... 10

9. Stereogram showing the three-dimensional vector system for the Sierra Nevada batholith..................... 13

10. Stereogram showing the three-dimensional vector system for the Alaska-Aleutian Range batholith............. 14

11. Fields of compositions for the Sierra Nevada batholith and Alaska-Aleutian Range batholith in the three-dimensional vector systems ................................................................................ 16

12. Vector diagram representing a process of formation of the magma for the Mount Givens pluton of the Sierra

Nevada batholith, and a process of differentiation of the magma ......................................... 21

13. Q-Or-Ab-An tetrahedron showing the surface containing possible melt compositions for samples from the AlaskaAleutian Range batholith............................................................................. 24

14. Q-Or-Ab-An tetrahedron showing the surface containing possible melt compositions for six groups of plutons

within the Sierra Nevada batholith.................................................................... 26

TABLES

Page

TABLE 1. Hypothetical data reflecting a complex compositional structure ............................................. H2

2. Hypothetical data reflecting a simple compositional structure ............................................... 3

3. Some end-member compositions and mixing proportions for the data in table 1................................ 3

4. Some end-member compositions and mixing proportions for the data in table 2 ................................ 3

5. Average compositions of the Sierra Nevada and Alaska-Aleutian Range batholiths and absolute variances accounted for and not accounted for by various mixing models ............................................. 10

in
IV CONTENTS

Page

TABLE 6. Average compositions of groups of plutons in the Sierra Nevada batholith.................................... Hll

7. Average compositions of groups of plutons in the Alaska-Aleutian Range batholith ........................... 11

8. Compositions (weight percent) represented by some hypothetical vectors in figure 9 ........................... 15

9. Compositions (weight percent) represented by some hypothetical vectors in figure 10 .......................... 15

10. Average compositions of hornblende from granitic rocks of the Sierra Nevada batholith ....................... 15

11. Chemical and normative compositions (weight percent) of end members for the three-end-member Sierra Nevada

batholith model...................................................................................... 16

12. Chemical and normative compositions (weight percent) of end members for the Alaska-Aleutian Range batholith

model............................................................................................... 16

13. Statistical summary of mixing proportions for the three-end-member Sierra Nevada batholith model............ 18

14. Statistical summary of mixing proportions for the Alaska-Aleutian Range batholith model.................... 18

15. Chemical and normative compositions (weight percent) of end members for the four-end-member Sierra Nevada

batholith model...................................................................................... 19

16. Statistical summary of mixing proportions for the four-end-member Sierra Nevada batholith model............ 20

17. Average compositions, in percent, of magmas for the Sierra Nevada batholith ................................ 20

18. Compositions of mafic mineral assemblages (end members ABH and ANH) required for the average samples from 

the two batholiths according to the derived models, and the average composition of mafic inclusions from the 

southern California batholith ......................................................................... 20

19. Mass balance computations for the average sample from the Alaska-Aleutian Range batholith................. 23

20. Data pertaining to points within tetrahedron of figure 13 .................................................... 25

21. Data pertaining to points within tetrahedrons of figure 14 ................................................... 28

22. Mixing proportions, original data, and corresponding data generated from the Alaska-Aleutian Range batholith

model............................................................................................... 30
STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF 

CIRCUMPACIFIC NORTH AMERICA

By A. T. MIESCH and BRUCE L. REED

ABSTRACT

The compositional structure within an igneous body or cogenetic 

series depends on the number of end members that are required to 

account for a major part of its compositional variation. If a large 

part of the compositional variation can be accounted for by mixing 

of only a few materials (magmas or magmas and country rocks), the 

compositional structure is simple. Or, the compositional structure is 

simple if the variation can be explained by the separation of only a 

few mineral phases from a magma.

Factor-variance diagrams constructed from data on major oxide 

constituents in samples from the Sierra Nevada and AlaskaAleutian Range batholiths show that these bodies have fairly simple compositional structures, and, therefore, probably resulted from 

a few processes that were dominant over others. There is no need to 

call on highly complex mechanisms of fractional crystallization, 

assimilation, or magma mixing to account for the oxide data. In 

fact, the relatively simple compositional structures of the 

batholiths make these processes seem unlikely.

Models containing three end members can account for most of the 

compositional variability in each batholith. One end member is a 

parent magma, or magma-source material, and the other two represent the extremes in a two-component range of mafic mineral 

assemblages that separated from the magma. The compositions of 

228 samples from the Sierra Nevada batholith and of 158 samples 

from the Alaska-Aleutian Range batholith can be closely approximated as linear combinations of these end members. These approximations are somewhat improved for the Sierra Nevada if a fourth 

end member is used; the end member representing the magma, or 

magma-source material, is replaced by two end members that are 

apparently required to account for variation in the composition of 

the magmas that formed across the present site of the batholith. 

The principal variation among these magmas is in K2 O con tent, 

which was slightly higher in magmas formed at a distance from the 

continental margin.

If it is assumed that the magmas were generated by partial 

melting of the crust or other material, the models may be used to 

estimate a range of compositions for the partial melts that formed 

each sample. The final composition of each sample was determined 

by the composition of this melt and by the composition of mafic 

mineral assemblages that were either retained in the melt after partial melting of the crust or precipitated from the melt during its 

ascension to the present site of each batholith. The melt compositions that are estimated on the assumption of minimum heat requirements correspond well with those that occur along cotectic 

lines on classical phase diagrams. Individual samples from the 

Alaska-Aleutian Range batholith required melting of 6 to 24 percent of the crust, with an average of about 9 percent melting. Most

individual samples from the Sierra Nevada batholith require 

melting of 11 to 27 percent of the crust; the average degree of 

crustal melting required for individual groups of plutons (sequences) ranges from 17 to 21 percent.

Although the models developed for the two batholiths are not unique, they appear to be in accord with major geologic observations 

that bear on the batholiths' origins. Other models could be 

developed that would account equally well for the compositional 

variabilities; however, these other models would also have to account for the fairly simple compositional structures present in each 

of the batholiths.

INTRODUCTION

Almost all of the observed compositional variation 

in igneous rock bodies has resulted from processes of 

mixing or unmixing. Mixing occurs where country 

rock is incorporated into the magma or where two or 

more magmas or melts come into contact. Unmixing 

results from magmatic differentiation. Because nearly 

all of the variation results from mixing and unmixing, 

it is helpful to examine the variation by some kind of 

vector analysis. Vector systems constructed from 

chemical and mineralogical data on samples from an 

igneous body reflect the compositional structure of the 

body. The compositional structure depends on the kind 

of mixing and unmixing that occurred that is, on the 

number of materials involved and the variations 

among the proportions added or separated.

The only mathematical difference between mixing 

and unmixing is in the signs of the mixing proportions. 

A rock may have formed by the mixing of, say, 0.6 

parts of material x and 0.4 parts of material y. It may 

also have formed by the unmixing of 0.4 parts of 

material z from 1.4 parts of material x; that is, mixing 

of 1.4 parts material x and -0.4 parts material z. The 

only requirement is that the mixing proportions sum 

to unity. In the discussion to follow, the term mixing 

refers to either type of process, unless mixing in 

positive proportions or unmixing (separation) is 

specified.

The purpose of this report is to illustrate the concept 

of compositional structure, to discuss its bearing on inHl
H2 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

terpretation of the origin of igneous rocks, and to 

describe the compositional structures in major parts of 

the Sierra Nevada batholith of California and the 

Alaska-Aleutian Range batholith. The method of vector analysis employed is based on a form of Q-mode 

factor analysis originally described by Klovan and Imbrie (1971) and extended for use in geochemistry and 

petrology (Miesch, 1976a, 1976b). Computer programs 

for the extended Q-mode have been given by Klovan 

and Miesch (1976) and Miesch (1976c). An attempt will 

be made here to employ the methods and concepts of 

Q-mode analysis, and to describe the results in terms 

familiar to petrologists who have had little or no experience with factor analysis. Other readers may wish 

to consult the references cited and the appendix for additional details or mathematical clarification. Excellent introductions to the concepts and terminology 

of factor analysis methods in geology were given by 

Imbrie (1963) and Klovan (1975).

We are grateful to Paul C. Bateman for selecting the 

228 analyses used in examination of the Sierra Nevada 

batholith and for helpful criticisms of early drafts of 

the manuscript. We also appreciate the helpful 

criticisms and suggestions provided by Z. E. Peterman, Priestley Toulmin 3d, and David R. Wones.

COMPOSITIONAL STRUCTURE

Multivariate statisticians commonly use the term 

structure in referring to the patterns of variation or 

order within a data matrix. If the variables 

represented in the matrix vary almost independently 

of one another, the matrix has little or no structure. If 

the variables are correlated at least some structure is 

present, and if the structure is pronounced it is possible that the information contained in the data matrix 

might be expressed approximately in a simpler matrix 

that could be more easily interpreted in terms of the 

subject matter problem. The oxide variables 

represented in petrologic data matrices always display 

some correlation with one another. Therefore, these 

variables can quite commonly be represented in alternative matrices with fewer columns, if the columns 

represent variables other than the original oxides, such 

as combinations of the oxides that may approximate or 

equal the compositions of rocks, magmas, and 

minerals. Thus, the term structure seems appropriate 

and useful for describing this property of petrologic 

data and the rock bodies that the data represent. 

Depending on the amounts of correlation present, the 

compositional structures may range from simple (pronounced) to complex (weak) and the structural complexity, in turn, determines the number of columns in 

the alternative data matrix. Of particular significance

to the petrologist is the fact that this complexity determines the number of end members required to account 

for the compositional variation in the petrologic 

system.

In spite of the mathematical justification for use of 

the term structure in the connotation explained above, 

we recognize that the term may have misleading implications to geologists. The choice of terms may be 

poor for this reason, but no better term has come to 

mind.

Two hypothetical data matrices are given in tables 1 

and 2. Corresponding constituent means and standard 

deviations are about the same from one matrix to the 

other. The main difference between the two is in the 

compositional structures they represent. The matrix in 

table 1 can be reproduced only by mixing seven end 

members; for example, by mixing the seven endmember compositions in table 3A in the proportions 

given in table 3B. In contrast, the data in table 2 may 

be reproduced exactly by mixing only three end 

members, such as those of table 4A in the proportions 

given in table 4B. The data in table 1, therefore, reflect 

a complex compositional structure, whereas the compositional structure represented by the data in table 2 

is relatively simple. Complex compositional structures 

result from processes that have involved a large 

number of end members being mixed in variable proportions. Simple compositional structures result from 

mixing a relatively small number of end members, or 

conceivably, from mixing a larger number of end 

members in rather constant proportions. However, the 

latter process effectively reduces the number of end 

members; if two end members are always mixed in proportions of, say, 3 to 1, then the two end members are 

effectively only one.

TABLE 1. Hypothetical data reflecting a complex compositional 

structure

Sample Constituent (weight percent)

1

2

3

4

5

6

7

8

9

10

Mean

Standard

deviation

50.16

52.85

16.53

30.45

50.97

63.25

29.09

40.88

44.22

32.55

41.1

13.2

3.54

10.17

9.40

15.61

7.06

10.84

.25

5.03

14.85

14.90

9.2

4.9

4.78

8.75

4.31

.05

5.20

3.32

4.17

3.95

8.77

7.40

5.1

2.5

5.33

9.79

5.34

11.74

8.33

10.05

6.53

7.54

9.54

11.59

8.6

2.2

34.47

15.11

61.70

37.14

24.92

9.58

57.09

39.33

17.44

29.57

32.6

16.3

0.41

.75

1.14

1.32

.56

.10

.59

.73

1.68

0

.7

.5

1.41

2.58

1.58

3.69

2.96

2.86

2.28

2.54

3.50

4.00

2.7

.8
COMPOSITIONAL STRUCTURES IN TWO B ATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H3

TABLE 2. Hypothetical data reflecting a simple compositional 

structure

Sample Constituent (weight percent)

a b c d e f g

1 51.25 7.08 4.01 6.64 27.62 1.07 2.33 

2 29.15 8.40 5.75 10.20 42.10 1.45 2.95 

3 76.83 4.72 1.35 2.49 12.41 .46 1.74 

4 53.08 14.46 9.68 6.61 12.47 2.57 1.13 

5 49.60 15.50 10.60 7.20 13.20 2.80 1.10

6 31.41 2.44 1.05 9.63 51.47 .22 3.78 

7 36.11 6.32 3.91 9.02 40.64 .99 3.01 

8 47.77 8.12 4.93 7.23 28.35 1.30 2.30 

9 54.73 6.04 3.09 6.05 26.89 .84 2.36 

10 25.06 6.98 4.78 10.80 47.88 1.18 3.32

Mean 45.5 8.0 4.9 7.6 30.3 1.3 2.4

Standard 14.7 3.9 3.0 2.3 14.0 .8 .8 

deviation

TABLE 3. Some end-member compositions and mixing proportions 

for the data in table 1

A. End-member compositions (weight percent)

End member Constituent

a b c d e f g

A 68.1 4.8 4.3 8.9 10.8 0 3.1 

B 28.0 .1 4.5 6.0 57.6 .8 3.0 

C 29.0 12.3 1.2 13.1 36.2 1.6 6.6 

D 38.0 1.6 1.4 9.1 44.7 .1 5.1 

E 42.9 2.2 5.8 9.7 32.0 1.2 6.2 

F 30.6 30.7 5.9 .6 29.4 .4 2.4 

G 52.3 2.7 3.2 3.0 28.6 1.0 9.2

B. Mixing proportions

Sample End member

A B C D E F G

1 0.8 1.0 0.1 -0.5 -0.5 0 0.1 

2 .5 -.1 -.2 -.3 1.3 .3 -.5 

3 0 1.3 .4 -.4 -.4 .2 -.1 

4 .5 .6 1.3 -.4 -1.0 0 0 

5 .6 .3 .1 -.2 .2 .1 -.1 

6 1.0 -.1 .3 0 -.2 .1 -.1 

7 .1 1.0 0 .1 -.1 0 -.1 

8 .5 .8 .3 -.3 -.3 0 0 

9 .4 .2 .3 -1.0 1.1 .3 -.3 

10 -.5 -1.2 -.7 1.5 2.2 .7 -1.0

ROCK COMPOSITIONS REPRESENTED AS 

VECTORS

The method employed in Q-mode factor analysis for 

representing rock compositions as vectors can be illustrated best by considering a series of five entirely 

unrealistic rock samples composed of just SiO2 and 

A12O3 . However, the hypothetical samples will be considered to have SiO2 far in excess of A12O3 , as in common igneous rocks. The samples are represented as

TABLE 4. Some end-member compositions and mixing proportions 

for the data in table 2

A. End-member compositions (weight percent)

End member Constituent

a b c d e f 8

A 49.6 15.5 10.6 7.2 13.2 2.8 1.1 

B 84.4 5.1 1.4 1.3 5.9 .5 1.4 

C 8.7 1.3 .9 13.2 71.0 .1 4.8

B. Mixing proportions

Sample End member

ABC

1 0.3 0.4 0.3 

2 .5 0 .5 

3 0 .9 .1 

4 .9 .1 0 

5 1.0 0 0

6 0 .3 .7 

7 .3 .2 .5 

8 .4 .3 .3 

9 .2 .5 .3 

10 .4 0 .6

vectors in figure L4; a possible problem with this sort 

of straightforward vector representation is that the 

vectors cluster near the reference axis representing the 

variable (SiO2 in this case) with the highest mean. That 

is, the configuration of the vector cluster may be controlled dominantly by the variable means and standard 

deviations. The result is that the vector analysis of 

most igneous-rock data will be dominated by SiO2 and, 

to a lesser extent, by A12O3 . Relatively minor constituents such as iron, magnesium, calcium, and especially trace elements will exert little influence on the vector configuration, even though these constituents can 

be more diagnostic of igneous processes than SiO2 and 

A12O3 . Manson and Imbrie (1964) recognized this difficulty and scaled all variables to range from zero to 

unity, and Klovan and Imbrie (1971) equipped their 

factor analysis computer program (CABFAC) with the 

facility to scale the original data to either proportions 

of the variable range (that is, to range from zero to unity) or proportions of the maximum value for each 

variable. The mathematical effect of these transformations is to give each compositional variable approximately the same mean and standard deviation and to 

allow the sample vectors to occupy the full dimensions 

of the space in which they are represented 

mathematically (fig. LB). The important consequence is 

that minor constituents can be just as influential on 

the factor analysis outcome as major ones, and, 

therefore, just as diagnostic of processes of rock origin.
H4

100

STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

1

20 40 60 

PERCENT AI2Os

0.2 0.4 0.6 0.8 

PROPORTION Al20s

0.4 0.6 

AI2Os

1.0

A ORIGINAL DATA B SCALED DATA C NORMALIZED SCALED DATA 

FIGURE 1. Vector diagrams representing three forms of the same data for five hypothetical rock samples.

Scaling of the jth variable (oxide) to proportions of 

the total range is according to:

x =

where Xj is the original data value (generally the 

weight percent of an oxide), jcmax,- and jcmin,- are, 

respectively, the maximum and minimum values for 

thejth variable, and x'j is the scaled data value. Scaling to proportions of the maximum is performed simply by dividing each value of Xj by

In the treatment of major-element (oxide) data in 

petrology, transformation of the variables by scaling is 

generally helpful, although not in all cases. The decision to scale or not to scale this type of data should be 

based on experimentation. If the data are not scaled, 

the mixing proportions derived from the extended 

Q-mode method for a given set of end-member compositions are the same as those derived by the leastsquares method of Bryan, Finger, and Chayes (1969).

Whether the variables are scaled or not, a final data 

transformation is used in Q-mode factor analysis to adjust all vectors to unit length (that is, row-normalize). 

This is done by dividing each original or scaled 

analysis through by the square root of the sum of 

squares. This is for mathematical convenience and has 

no effect on the positions of the vectors relative to each 

other (fig. 1C).

VECTOR CONFIGURATIONS

Although the vector systems represented in figure 1 

are based on only two compositional variables (SiO2 

and A12O3), the same procedures can be used in 

mathematical, rather than graphical, form no matter

how many variables are treated. Other mathematical 

procedures can be used to determine the extent to 

which the vectors occupy various dimensions of the 

multidimensional space; the total number of dimensions equals the number of oxides being used in the 

analysis. The procedures are principally those of eigenvalue analysis. The eigenvalues 1 for a vector system 

are measures of the variability within the system that 

occurs within the major dimension of the vector 

cluster, the second greatest dimension, the third, and 

so forth. Thus, if all but two of the eigenvalues are 

zero, the vector system occupies two dimensions only 

(that is, the vectors all lie within the same plane), even 

though the analytical data were mathematically 

represented in a number of dimensions equal to the 

number of oxide variables. Similarly, if all but three of 

the eigenvalues are zero, the vectors occupy a threedimensional subset of the original multivariate space.

The eigenvalues for the vector systems representing 

the data in tables 1 and 2 are given below. They are 

listed in order of decreasing magnitude:

Eigenvalues

Eigenvalue No.

1

2

3

4

5

6

7

Table 1

7.68

1.17

.62

.24

.17

.10

.02

Table 2

7.52

1.57

.91

0

0

0

0

Total 10.00 10.00

In most applications of eigenvalue analysis, it happens that most of the eigenvalues are nonzero, but that 

some are small relative to others. Thus, if the first m 

eigenvalues are distinctly large relative to the others,

'All eigenvalues referred to in this report were computed for the minor product of the 

scaled (zero to one) and row-normalized data matrix and its transpose. Computer programs 

for the derivation of eigenvalues are available at most computer installations.
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA

the vectors are known to cluster about an 

m-dimensional subspace rather than lie precisely 

within it. Each of the vectors can then be projected 2 into the subspace. The projected vectors then become 

somewhat less than unity in length, and because they 

are in different positions, represent compositions 

somewhat different from the original compositions of 

the corresponding samples. However, if all but the first 

m eigenvalues are close to zero, and the amount of projection, therefore, is small, the projected vectors remain close to unity in length and the modified (recomputed) compositions are not greatly different from the 

original sample compositions. The square of the length 

of a projected vector is referred to as the vector communality (h2 ), and is the conventional measure of the 

degree to which a specific composition corresponds to 

the compositional series represented by a vector 

system.

If the first m eigenvalues are distinctly large relative 

to the others, the projection of the vectors into 

m-dimensional space is done on the presumption that 

m end members have been dominant in controlling the 

compositional variation within the rock body 

represented by the samples. Assuming that the 

laboratory errors are uncorrelated among the various 

constituents determined, there is no other circumstance that could cause the m eigenvalues to be 

large relative to the others. Those who may doubt the 

practical value of eigenvalues can explore their 

usefulness easily by means of simulation. Simply form 

some data by mathematical mixing of any given endmember compositions and then determine the eigenvalues of the minor product of the row-normalized data 

matrix and its transpose. It will be observed, for example, that when a number of end members equal to or 

greater than the number of oxides (M) are mixed in any 

proportions that are not perfectly correlated, the eigenvalues tend to be all nonzero. If only m end members 

(m<M) are mixed, only m eigenvalues will be nonzero.

The interpretation of vector systems representing 

rock compositions can be explained by reference to 

figure 2. Figure 2 is a stereogram constructed by projecting vectors from an upper hemisphere vertically 

onto the plane of the diagram. (The same projection 

was used for constructing other stereograms given in 

this report. See figures 9,10, and!2.)Nine hypothetical 

sample compositions, ranging from basalt (vector B) to 

rhyolite (vector R), are represented by vectors that all 

lie within the same plane (XR). The first two eigenvalues describing the configuration of the vector 

system are positive and the remainder are zero. The only likely natural process that would lead to a vector 

system such as this with all sample vectors in the 

same plane is one involving the mixing of two, and

H5

FIGURE 2. Stereogram showing a configuration of vectors 

representing the compositions of some hypothetical samples.

Compositions represented by vectors, in weight percent

Oxide

SiO2 . . 

A1 2O3 "FeO" 

MgO . 

CaO .. 

Na2O . 

K2O ..

Oxide

SiO2 . . 

A1 2O3 "FeO" 

MgO . 

CaO .. 

Na2O . 

K2O ..

B

51.25 

16.40 

11.71 

6.45 

9.35 

3.25 

1.59

6

69.72 

14.17 

4.21 

1.60 

2.87 

3.38 

4.06

i

61.64 

15.15 

7.49 

3.72 

5.70 

3.32 

2.98

7

72.02 

13.89 

3.27 

.99 

2.06 

3.40 

4.37

2

62.79 

15.01 

7.02 

3.42 

5.30 

3.33 

3.13

R

74.33 

13.62 

2.33 

..39 

1.25 

3.41 

4.68

3

65.56 

14.67 

5.89 

2.69 

4.33 

3.35 

3.50

X

41.39 

17.59 

15.72 

9.03 

12.81 

3.18 

.27

4

66.25 

14.59 

5.61 

2.51 

4.08 

3.35 

3.60

Y

62.72 

15.27 

6.87 

3.23 

5.12 

3.43 

3.36

5

67.41 

14.45 

5.14 

2.21 

3.68 

3.36 

3.75

z

68.47 

13.97 

4.97 

2.22 

3.59 

3.23 

3.56

2The sample vectors are projected by rotation of the first m principal component axes according to the varimax criterion of Kaiser (1958).

only two, end members. Moreover, the two end 

members must have had compositions that can be 

represented as vectors in the same plane (XR). For 

example, the composition of sample 2 (fig. 2) can be 

reproduced by mixing 0.5 parts of composition B and 

0.5 parts of composition R, and the composition of 

sample 6 can be reproduced by mixing 0.2 parts of 

composition B and 0.8 parts of composition R. Alternatively, the composition of sample 2 can be reproduced by separating 1.1702 parts of composition X from 

2.1702 parts of composition B, and the composition of 

sample 6 can be reproduced by separating 1.8723 parts 

of composition X from 2.8723 parts of composition B. 

In contrast, a sample formed by the mixing of composition B with some composition represented by a vector
H6 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

outside the plane XR, say vector Y, would have a composition represented by a vector in the plane BY.

It is mathematically possible to generate the compositional series represented by vectors in the plane 

XR by mixing more than two compositions 

represented by vectors outside of the plane XR, but 

the process, in effect, would be the same as mixing only 

two end members. For example, samples 1 and 2 in 

figure 2 could result from the mixing, in positive proportions, of compositions B, Y, and Z, and samples 3 

through 7 could result from the separation of composition B from a mixture of compositions Y and Z. 

However, in each case, the proportion of composition Y 

would have to be exactly 1.417 times greater than the 

proportion of composition Z no more and no less. The 

result would be a composition that could be 

represented by a vector in the plane XR, at the intersection of the planes XR and YZ. Thus, the two end 

members, Y and Z, effectively would be only one.

The same principles of vector addition and subtraction can be extended to situations where the sample 

vectors fall not on a plane, but within a factor space of 

three dimensions, four dimensions, or more.

FACTOR-VARIANCE DIAGRAMS

Although the eigenvalues serve to describe important aspects of the vector configurations, they give little information about the differences between the compositions represented by projected vectors and the actual compositions of the samples that the vectors 

represented before projection. If these differences are 

larger than can be attributed to analytical errors and 

to minor petrologic processes, the number of end 

members for the model, m, will have to increased. If 

the differences are acceptable, however, the projected 

vector configuration may provide a basis for developing a petrologic model, or for testing a previously 

developed model. The extended Q-mode procedures 

(Miesch, 1976a, 1976b) always lead to matrices of 

recomputed data (represented by the projected vectors) that have column (oxide) means essentially the 

same as those of the original data. The correlations 

between corresponding columns, from one matrix to 

the other, however, range from near zero to unity, 

depending on the degree of compositional modification 

that occurred with projection. The square of each correlation, r2/, is a measure of the maximum proportion 

of the variance in each oxide that can be accounted for 

by a model with m end members.

Factor-variance diagrams are constructed by plotting r2,- for each oxide against the number of end 

members in the model, and show the maximum proportion of the total variance in each oxide that can be accounted for by mixing m end members. Factorvariance diagrams were given previously (Miesch,

1976b) for four suites of samples from (1) a rhyolitebasalt complex in Yellowstone National Park, Wyoming (Fenner, 1938), (2) a granitoid intrusive in eastern 

Nevada (Lee and Van Loenen, 1971), (3) lavas and 

pumices from the 1959 summit eruption at Kilauea, 

Hawaii (Murata and Richter, 1966), and (4) the layered 

series of the Skaergaard intrusion, Greenland (Wager 

and Brown, 1968). Another factor-variance diagram 

was given previously for a single pluton of the 

southern California batholith (Miesch and Morton, 

1977). The implications of these diagrams with respect 

to the origins of the rocks they represent are discussed 

in the cited references.

Factor-variance diagrams for the hypothetical-data 

matrices in tables 1 and 2 are given in figure 3. The differences between the diagrams clearly indicate the difference in compositional structure referred to previously. Figure 3 shows that seven end members are required to account for all of the variance in all the constituents for the data in table 1, whereas all of the 

variance in each of the constituents for the data in 

table 2 can be explained by mixing only three end 

members.

Factor-variance diagrams representing two real igneous bodies are given in figures 4 and 5. The diagram 

in figure 4 pertains to the Pikes Peak batholith in Colorado; the data used in constructing it are from Barker 

and others (1975, tables I and IV), and consist of 

measurements for 7 oxides in 41 samples, principally 

of quartz syenites, fayalite granites, and fayalite-free 

granites. The diagram shows that two- and three-endmember models would fail to account for much of the 

compositional variability in the batholith, and that 

even a four-end-member model would leave 20 percent 

of the variability in KiO and MgO unexplained. The 

compositoinal structure in the batholith, therefore, appears to be complex. This is in accord with the complex 

model of origin that Barker and others have developed 

on conventional petrologic and geochemical grounds. 

The model includes the incorporation of lower and intermediate crusts into, respectively, basaltic and 

syenitic magmas, with the crystallization and separation of phases such as olivine, clinopyroxene, and 

feldspars. Also, both of the crustal materials and each 

of the separated phases could have been of variable 

composition.

The factor-variance diagram in figure 5 pertains to 

the McCartys Basalt (Holocene) in northwestern New 

Mexico. The data used are from Garden and Laughlin 

(1974) and consist of measurements on 7 oxides in 37 

samples. The diagram indicates that two- to four-endmember models would be inadequate to account for 

any appreciable proportion of the compositional 

variance in the flow. The compositional structure in 

the flow is complex. This is in accord with the model of 

Garden and Laughlin, which involves crystal fractiona-
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H7

Data from table 2

6723 

NUMBER OF END MEMBERS

FIGURE 3. Factor-variance diagrams for the data for constituents a-g in tables 1 and 2.

tion of plagioclase, olivine, and pyroxene, all of 

variable composition, in addition to several accessory 

minerals.

COMPOSITIONAL STRUCTURES IN THE

SIERRA NEVADA AND ALASKA-ALEUTIAN

RANGE BATHOLITHS

The geology of the Sierra Nevada batholith in 

eastern California (fig. 6) has been summarized by 

Bateman and others (1963). The batholith consists 

primarily of quartz-bearing granitic rocks that range in 

composition from quartz diorite to alaskite. The rocks 

occur as discrete plutons, and groups of plutons form 

sequences. Plutons within a sequence tend to be 

spatially continuous with each other and of about the 

same age. Definitions of the sequences within the 

batholith were by Bateman and Dodge (1970), 

although they apply the less formal term "rocks" in 

place of the term "sequence" wherever the age relations are not entirely certain. The same authors retain 

a previously assigned name, "Toulumne intrusive 

series," for one group of plutons that has the 

characteristics of a sequence.

The chemical data on the Sierra Nevada batholith 

pertain to 228 samples collected from a large number

of plutons in the central part by Paul C. Bateman and 

his colleagues. About one-half of the samples are from 

six previously defined sequences: White Mountains 

rocks, Palisade Crest sequence, Scheelite sequence, 

Toulumne intrusive series, Yosemite rocks (called 

Yosemite intrusive series by Wright, 1975), and the 

Shaver sequence (Bateman and Dodge, 1970, fig. 2). 

The area represented is about the same as that 

reported on by Bateman and Dodge (1970) in their examination of chemical variations across the batholith. 

Seven major oxides (SiO2 , A12O3 , TeO' 3 , MgO, CaO, 

Na2O, and K2O), recalculated to sum to 100 percent, 

are represented in the data. These oxides were selected 

because they are essential in the principal rockforming minerals.

The eigenvalues describing the dimensionality of the 

vector system representing the data, before projection, 

are 198.3, 24.3, 2.8,1.4, 0.7, 0.4, and 0.2. Thus, the vectors cluster about a two-dimensional space, even 

though they were mathematically represented in 7 

dimensions. The factor-variance diagram in figure 7, 

however, indicates that the compositions represented

3As used in this report, 'FeO' equals FeO+0.9Fe2O3 , and thus represents total iron.
H8 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

34567 

NUMBER OF END MEMBERS

FIGURE 4. Factor-variance diagram for the Pikes Peak 

batholith, Colorado.

by the vectors after they have been projected into a 

plane are greatly different from the original compositions with respect to K2O, A12O3 , and especially 

Na2O. But, the diagram also shows that if the vectors 

are projected into a three-dimensional space, rather 

than into a plane, all of the recomputed compositions 

are at least moderately close to the observed compositions of the samples. Three-end-member models can account for 77 to 97 percent of the variance in each oxide. 

If the vectors are projected into four-dimensional 

space, the situation is improved still further. Four-endmember models can account for 90 to 99.9 percent of 

the variance in each oxide.

The eigenvalues and the factor-variance diagram indicate, therefore, that the compositional structure of 

the Sierra Nevada batholith is fairly simple. Three end 

members can account for most of the variation in the 

data and an additional end member improves this

NUMBER OF END MEMBERS

FIGURE 5. Factor^ariance diagram for the McCartys 

Basalt (Holocene), New Mexico.

substantially. The remaining variation can be attributed to minor geochemical processes and to 

analytical variance. Some of it may also be attributed 

to small compositional variations within each of four 

principal end members.

The Alaska-Aleutian Range batholith (fig. 6), like the 

Sierra Nevada batholith, is composed of discrete 

plutons that can be assigned to sequences (Reed and 

Lanphere, 1969,1973). The rocks range in composition 

mostly from diorite to granite. The analytical data used in the vector (Q-mode factor) analysis are from 158 

samples, and consist of measurements of the same 

seven oxides used in the Sierra Nevada analysis. Data 

for 79 of these samples are given in Reed and Lanphere 

(1974); the remaining 79 samples were collected more 

recently, chiefly from the northern part of the 

batholith, and were analyzed for this study. These 

rocks are classified into 9 groups according to their age
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H9

165° 150° 135° 120° 105° 90° 75° 60°

Alaska-Aleutian

Range batholith - _

15C

0°

FIGURE 6. Index map of the northeastern circumpacific region, the 

Alaska-Aleutian Range batholith and the Sierra Nevada batholith.

and chemical characteristics (Reed and Lanphere, 

1973). Of the 79 additional samples, 53 are from the 

same groups of rocks discussed by Reed and Lanphere 

(1974) and 26 are from the granodiorite of Mount 

Estelle, the Hartman sequence, and the Yentna sequence. (See Reed and Lanphere, 1973, fig. 2.) The 

oldest group consists of samples from plutonic rocks of 

Jurassic age. Seven groups are of samples from rocks 

of Late Cretaceous-early Tertiary age. The final group 

is from the Merrill Pass sequence of middle Tertiary 

age. The rocks of Jurassic age range mostly from 

diorite to granodiorite, whereas quartz monzonites and 

granites make up much of the younger groups.

The eigenvalues for the vector system constructed in 

seven dimensions on the basis of the seven oxides are 

129.2, 22.0, 5.4, 0.7, 0.4, 0.2, and 0.1. Like those for the 

Sierra Nevada rocks, these values indicate that the 

vector system occurs principally in two or three dimensions. The factor-variance diagram (fig. 8), however, 

shows that models with three end members are required to account for all of the seven oxides. Three-endmember models can account for 92 to 98 percent of the 

variance in each oxide. Unlike the factor-variance 

diagram for the Sierra Nevada batholith, figure 8 

shows that a fourth end member would do little to improve the model for the Alaska-Aleutian Range 

batholith.

The average compositions of both batholiths, 

estimated from the data referred to, are given in table 

5 along with the amount of absolute variance in each 

batholith that can be accounted for, and the amount

1.0

0.9 -

0.8

0.7

D 0.6 O

§

u

0.5

_ O 0.4

0.3

0.2

0.1

-7

234567 

NUMBER OF END MEMBERS

FIGURE 7. Factor-variance diagram for the Sierra 

Nevada batholith.

that cannot be accounted for, by mixing models with 

the indicated number of end members. The average 

compositions of groups of plutons within each 

batholith are given in tables 6 and 7.

As comparison of figures 7 and 8 shows, the compositional structure in the Alaska-Aleutian Range 

batholith is similar to that in the Sierra Nevada 

batholith, although probably even less complex. With 

the exception of Na2O, more than half of the variance 

in each oxide can be accounted for by only two end 

members for each batholith (figs. 7 and 8). Three end 

members can account for more than 90 percent of the 

variances in all oxides in the Alaska-Aleutian Range 

batholith and for more than three-fourths of the 

variances in all oxides in the Sierra Nevada batholith; 

a fourth end member increases the variances accounted for in the Sierra Nevada batholith to more 

than 90 percent.
H10 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

34567 

NUMBER OF END MEMBERS

FIGURE 8. Factor-variance diagram for the AlaskaAleutian Range batholith.

The simplicity of the compositional structures in 

both batholiths, compared to those determined for 

some other igneous bodies, suggests that the major 

features of their compositional variation in both space

and time can be attributed to only a few dominant 

geologic processes. For example, these variations 

could arise from a single, homogeneous magma or melt 

if only two mineral assemblages were separated from it 

by fractional crystallization and settling, or if one 

mineral assemblage that varied in composition within 

a two-end-member system were separated out. Alternatively, the processes may have consisted of the mixing of three magmas or of a magma and two other 

materials, possibly by assimilation of crustal or subcrustal materials. The important point is that a process involving three dominant end members can explain major parts of the observed compositional variation in both batholiths. The remaining variation can be 

attributed to processes or factors that were of relatively minor importance in causing compositional variation. These minor factors include small compositional 

variations in each of the end members, minor 

petrologic processes, and variation caused by 

analytical procedures.

A fourth end member is required in the model for the 

Sierra Nevada batholith in order to account for about 

the same proportions of the total variances that are accounted for by three-end-member models for the 

Alaska-Aleutian Range batholith. However, the fourth 

end-member should be one that serves only to refine a 

basic three-end-member model.

SPECULATION ON END-MEMBER 

COMPOSITIONS

The relatively simple compositional structures of the 

Sierra Nevada and Alaska-Aleutian Range batholiths 

suggest that the processes of their formation were not 

highly complex, at least not complex in terms of the 

number of independent phases involved, and this leads 

us to speculate what these end members might have 

been.

The only end-member compositions that will serve to 

reproduce the original data to the degrees indicated by

TABLE 5. Average compositions of the Sierra Nevada and Alaska-Aleutian Range batholiths and absolute variances

accounted for and not accounted for by various mixing models

Sierra Nevada batholith Alaska-Aleutian Range batholith

Three-end-member model Four-end-member model Three-end-member model

Constituent

SiO2 

A1 203 'FeO' 

MgO 

CaO 

Na2O 

K2O

Average 

concentration 

(percent)

69.49 

15.51 

3.33 

1.25 

3.16 

3.49 

3.76

Total 

variance

20.5363 

1.7182 

2.8348 

.7762 

2.4649 

.1936 

.8610

Variance 

accounted 

for

19.2936 

1.3871 

2.6871 

.7389 

2.3812 

.1551 

.6601

Variance 

not accounted 

for

1.2427

.3311 

.1477 

.0373 

.0837 

.0385 

.2009

Variance 

accounted 

for

20.5065 

1.5512 

2.7092 

.7438 

2.3964 

.1817 

.8545

Variance 

not accounted 

for

0.0298 

.1670 

.1256 

.0324 

.0685 

.0119 

.0065

Average 

concentration 

(percent)

68.60 

15.89 

3.62 

1.53 

3.44 

3.87 

3.05

Total 

variance

39.9159 

3.4611 

4.4610 

2.1159 

5.2624 

.4575 

1.7363

Variance 

accounted 

for

39.1547 

3.1885 

4.2504 

1.9757 

5.1472 

.4230 

1.6084

Variance 

not accounted 

for

0.7612 

.2726 

.2106 

.1402 

.1152 

.0345 

.1279
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA Hll

TABLE 6. Average compositions of groups of plutons in the Sierra Nevada batholith

[All analyses were recomputed to 100 percent before computation of averages]

Number 

of

Group of plutons

Scheelite sequence ...........

Toulumne intrusive series

All plutons ..................

samples

21

1Q

23

7

7

36

115

228

SiO2

66.84

66.62

72.30

69.72

74.01

70.71

69.22

69.49

A1 2O,

16.40

15.85

14.32

15.58

14.61

15.20

15.68

15.51

'PeO'

3.84

4.31

2.50

2.97

1.57

3.21

3.41

3.33

MgO

1.55

1.70

.81

.82

49

1.05

1.36

1.25

CaO

3.22

3.96

2.47

3.39

9 07

2.87

3.30

3.16

Na2O

3.61

3.48

3.10

3.94

3.67

3.27

3.58

3.49

K2O

4.53

4.09

4.50

3.57

3.65

3.68

3.46

3.76

TABLE 7. Average compositions of groups of plutons in the Alaska-Aleutian Range batholith

[All analyses were recomputed to 100 percent before computation of averages]

Group of plutons

Number

of 

samples SiO2 A1 2O3 'FeO' MgO CaO Na2O K2O

Quartz monzonite of Tired Pup

Granodiorite of Mount Estelle . 

Yentna sequence .............

Plutonic rocks of Jurassic age. . 

All plutons ...................

29

14 

22

5

13

8

35

3

29 

158

75.10

73.41 

74.08

66.79

66.77 

64.31

64.73

70.38

62.41 

68.60

13.78

14.24 

14.24

15.76

16.28 

16.50

17.66

15.19

17.66 

15.89

1.76

2.55 

1.99

4.32

4.32 

4.91

4.36

2.93

5.63 

3.62

0.38

.39 

.31

2.64

2.01 

2.89

2.01

1.76

2.76 

1.53

1.39

1.44 

1.24

4.32

3.61 

4.25

5.05

2.99

5.81 

3.44

3.88

3.31 

3.94

3.01

3.21 

3.21

4.44

3.22

4.09 

3.87

3.72

4.67 

4.19

3.25

3.81 

3.93

1.75

3.52

1.63 

3.05

the factor-variance diagrams (figs. 7 and 8) are those 

that can be represented by vectors in the corresponding vector spaces. These vectors then can be used as 

reference vectors for describing the compositional 

systems. Mathematically satisfactory reference vectors, representing end-member compositions, can be 

found for the three- and four-end-member models by 

determining the compositions represented by 

hypothetical vectors throughout the three- and fourdimensional vector spaces. This must be done at increments inasmuch as the number of hypothetical vectors is infinite. Reference vectors may also be found by 

representing selected compositions as vectors in the 

three- and four-dimensional systems of sample vectors 

and determining the vector communalities. If the compositions can be represented in this manner (that is, if 

the vector communalities are close to unity), they can 

be accepted as mathematically suitable end members 

for the model. After plausible end-member compositions have been identified in either manner, it remains 

only to determine the required mixing proportions. If 

these proportions are geologically unreasonable or 

unlikely, another set of end-member compositions can 

be tried until the end-member compositions and the 

mixing proportions are not incompatible with 

whatever is known about the rock body and petrologic

processes in general. It will be apparent from this that 

the procedures lead to a petrologic model only, and 

that a unique solution to the origin of the rock body, by 

this means, is impossible. The computer programs used in the modeling procedure are referred to in the appendix.

Representation of selected compositions as vectors 

in the three- and four-dimensional vector systems for 

the two batholiths indicate that none of the end 

members could have had the compositions of common 

hornblende, biotite, quartz, or any feldspar, even 

though it seems obvious from field observations that 

these minerals were involved in fractional crystallization of the magmas. Moreover, none of the end 

members could have had the composition of any pyroxene or olivine. Consequently, it is likely that the endmember compositions were those of mineral mixtures, 

either clusters or assemblages of minerals or compositionally equivalent liquid phases.

If a major part of the compositional variability in 

each batholith is the result of the separation or 

redistribution of mineral assemblages, the observed 

compositional structures place constraints on the 

nature of the compositional variations within these 

assemblages. At least one of the end members for each 

batholith must be reserved to represent the parent
H12 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

magma or the magma-source material. The compositional variation in the mineral assemblages, therefore, 

must be of a kind that can be described largely in 

terms of a two-component system, inasmuch as only 

three end members controlled a major part of the variation in each batholith. There is no doubt that the 

mineral assemblages varied in composition in both 

space and time.

A model consisting of a parent magma or a magmasource material from which mineral assemblages 

separated or were redistributed is not the only one that 

might be accepted as geologically plausible. However, 

this type of model is relatively simple compared with 

other three-end-member models that might be proposed, and it seems reasonable to consider the most simple 

model first. Also, it is apparent from field observations 

that important amounts of compositional variation 

within and among plutons were caused by the 

redistribution of crystalline phases.

If the possibility that the determined compositional 

structures of the batholiths arose by chance is discounted, and if the end members contributng to the 

batholiths' formation are accepted as a magma, or 

magma-source material, and mineral assemblages of 

varying composition, other processes must be excluded as quantitatively important causes of variation 

within the batholiths. These other processes include 

magma mixing and assimilation of crustal and subcrustal materials. This may be the most important consequence of the determination that both batholiths 

have simple compositional structures.

Furthermore, if we accept that the compositional 

structures of the batholiths resulted from the separation of mineral assemblages from a magma or magmasource material, then there is little room in the model 

for compositional variations among the magmas or 

within the magma-source material. Bateman and 

Dodge (1970) and Reed and Lanphere (1974) showed 

that the K2O contents of the granitic rocks vary across 

the batholiths, and they interpreted the variation to 

reflect compositional variations in the magma-source 

materials. This interpretation still seems valid 

although it should be recognized that only a small part 

of the variance in K2O, and even less of the variance in 

most of the other oxides, is associated with the 

variance across the batholith. Reed and Lanphere 

(1974, p. 351) gave the correlation coefficient for K2O 

content and distance across the Alaska-Aleutian 

Range batholith as 0.53. The square of this value 

(0.28) gives the proportion of the total variance in K2O 

associated with distance. Clearly, other factors were 

far more important in controlling the compositions of 

the granitic rocks. Reed and Lanphere (1974) made the 

additional observation that trends in K2O and other

oxides within groups of plutons are commonly opposite to those observed across the entire AlaskaAleutian Range batholith. It will be shown that compositional variations among the magmas or magmasource materials can be represented in a four-endmember model for the Sierra Nevada batholith, but 

this model is only a refinement of a three-end-member 

model that accounts for most of the compositional 

variation.

Stereograms of the three-dimensional vector 

systems for the two batholiths are given in figures 9 

and 10. All vectors have been restored to unit length 

after projection into three-dimensional space, and are 

shown as points that have been projected vertically 

from an upper hemisphere. The natures of the compositional systems represented by the diagrams are shown 

by the compositions of selected hypothetical vectors in 

tables 8 and 9. Any hypothetical vectors that might occur outside the areas enclosed by dotted lines on 

figures 9 and 10 would represent compositions that are 

partly negative; only vectors within or near these areas 

need be considered.

A search for the compositions of mineral 

assemblages that might have separated from or been 

redistributed in the magmas was made by examination 

of the compositional system formed by hornblende, 

albite, anorthite, and magnetite. It is expected that the 

earliest formed assemblages, at least, would have had 

total compositions within the range of compositions of 

these phases. The composition of hornblende was 

taken as the average composition of 22 hornblendes 

from the central part of the Sierra Nevada batholith 

(table 10). The hornblende data are from Dodge, 

Papike, and Mays (1968), who pointed out that the 

data may not be representative of hornblendes that 

precipitated early. However, experimentation with 

other hornblende compositions, including those given 

for "early" hornblendes by Nockolds and Mitchell 

(1946, table XIV, analyses 2-6), has led to results 

substantially the same as those described here. Ideal 

compositions were assumed for albite and anorthite, 

and the composition of magnetite in terms of the oxides being considered is 100 percent 'FeO'. The fourcomponent compositional system was represented as a 

tetrahedron and examined throughout at increments 

of 2 weight percent; each of the 23,426 compositions 

was represented as a vector in the seven-dimensional 

vector space (based on the seven oxides) and then projected into the same three-dimensional space as the 

sample vectors (figs. 9 and 10). This procedure required 

less than one minute of computer processor time for 

each batholith.

Of the 23,426 compositions examined for the Sierra 

Nevada batholith, 385 could be represented by vectors
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA

III

H13

FIGURE 9. Stereogram showing the three-dimensional vector system for the Sierra Nevada batholith. Dots represent 228 sample vectors. 

Open circles labeled I, II, and III represent the varimax reference axes. Lettered triangles represent hypothetical vectors referred to 

in text and in table 8. Dashed line represents boundary of area in which all vectors represent compositions that are entirely nonnegative.

with communalities greater than the arbitrary value of 

0.98 in the three-dimensional vector system. The field 

containing these 385 compositions is indicated in 

figure IIA. The compositions represented by vectors 

with the highest communalities (as high as 0.9964) are

represented by points that occur along the line X-Y. 

Thus, line X-Y is taken as the most likely trend of compositional variation within the mineral assemblages 

that separated from the Sierran magmas. Compositions along this trend are those within the system
H14 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

III

FIGURE 10. Stereogram showing the three-dimensional vector system for the Alaska-Aleutian Range batholith. Dots represent 158 sample 

vectors. Open circles labeled I, II, and III represent the varimax reference axes. Lettered triangles represent hypothetical vectors 

referred to in text and in table 9. Dashed line represents boundary of area in which all vectors represent compositions that are entirely 

nonnegative.

albite-anorthite-hornblende-magnetite that are the 

most compatible with the compositional variations in 

the granites of the Sierra Nevada batholith.

Of the 23,426 compositions examined for the AlaskaAleutian Range batholith, only 156 could be

represented by vectors with communalities greater 

than 0.98 in the three-dimensional vector system. The 

trend of compositions represented by vectors with the 

highest communalities (as high as 0.9943) is shown by 

the line X-Y in figure 11B.
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H15 

TABLE 8. Compositions (weight percent) represented by some hypothetical vectors in figure 9

Vector

Oxide

D E G H

TABLE 9. Compositions (weight percent) represented by some hypothetical vectors in figure 10

Vector

Oxide

H

M

SiO2

A1 203

'FeO'

MgO

CaO

Na2O

K2O

46.43

21.48

11.85

5.65

11.24

2.95

.41

57.41

19.16

7.49

3.29

7.34

3.93

1.24

60.84

18.47

6.19

2.59

6.19

4.22

1.49

63.47

17.92

5.16

2.03

5.27

4.46

1.69

67.69

17.04

3.51

1.14

3.79

4.83

2.01

72.77

15.97

1.52

.06

2.01

5.27

2.40

58.81

17.87

7.47

3.47

6.92

2.68

2.79

62.12

16.90

6.30

2.88

5.76

2.61

3.43

64.67

16.15

5.40

2.43

4.87

2.55

3.92

68.54

15.02

4.03

1.75

3.52

2.47

4.67

71.70

14.10

2.92

1.20

2.41

2.40

5.28

77.41

12.43

.89

.19

.42

2.27

6.38

M

SiO2

A1 2O3

'FeO'

MgO

CaO

Na2O

K2O

46.56

21.58

10.88

6.42

11.22

3.33

.01

54.20

20.31

7.91

4.41

8.76

4.23

.19

56.87

19.86

6.87

3.71

7.90

4.54

.25

59.37

19.45

5.89

3.05

7.10

4.83

.31

63.88

18.70

4.14

1.87

5.65

5.36

.41

70.51

17.60

1.56

.12

3.52

6.13

.56

55.08

18.63

8.57

4.87

7.95

2.76

2.14

58.17

17.56

7.74

4.30

6.77

2.56

2.92

61.09

16.55

6.94

3.77

5.64

2.36

3.65

63.87

15.59

6.19

3.26

4.58

2.17

4.34

69.02

13.80

4.79

2.32

2.61

1.83

5.63

74.80

11.81

3.23

1.26

.39

1.44

7.07

TABLE 10. Average composition of hornblende from granitic rocks

of the Sierra Nevada batholith

[Data from Dodge, Papike, and Mays (1968) for 22 samples. All analyses were recalculated 

to sum to 100 before computation of average]

Constituent Average (in percent)

SiO2 ..................

Al 203 .................

'FeO' .................

MgO ..................

CaO ...................

Na2O ..................

K2O ...................

............... 48.89

............... 8.14

............... 16.42

............... 12.55

............... 12.26

............... 1.06

............... .68

Although the compositions along the trends X-Y in 

figure 11 are the compositions of best fit within the 

range of compositions examined, it is not expected 

that this range is all inclusive, because it is defined by 

ideal compositions for albite, anorthite, and magnetite 

and because the compositions of hornblendes can vary 

to some extent. Rather, the X-Y trends in figure 11 are 

taken as approximations of the most likely compositions of mineral assemblages that may have existed at 

an early stage within each magma. The compositions 

of points X and Y are represented on the stereograms 

of figures 9 and 10. It will be seen that the point X falls 

outside the area enclosed by the dotted line on each 

stereogram, indicating that even though the vector 

communalities for these compositions are high (and so 

the projections were minor), the projected vectors 

represent compositions that are partly negative. The 

points labeled Z in figures 9 and 10 define vectors most 

like vectors X that are entirely nonnegative.

The compositions represented by points Z and Y (or 

ABH and ANH, respectively) in figures 9 and 10 are 

given in tables 11 and 12 for the Sierra Nevada and 

Alaska-Aleutian Range batholiths, respectively. It will 

be noted that the compositions given for vectors ANH 

in these tables differ from those given for points Y in 

figure 11. The differences are the result of projection of 

vectors representing the compositions given for points 

Y into the three-dimensional vector systems.

The compositions for vectors ANH are interpreted 

as those of assemblages that consist largely of calcic 

(anorthitic) plagioclase, hornblende, and possibly other 

mafic minerals. The compositions given for vectors 

ABH in tables 11 and 12 are interpreted as those of 

mineral assemblages that consist largely of intermediate (more albitic) plagioclase and hornblende. 

These compositions are reflected in the vector names.

Although the minimum acceptable communality 

(0.98) used in searching the hornblende-albiteanorthite-magnetite tetrahedron was selected 

somewhat arbitrarily, the selection does not have any 

important consequences on the final models or 

geologic interpretation. If the selected critical value is 

much less than 0.98, the silica contents of the ANH 

end members will be improbably low; if much greater 

than 0.98, the range of compositions for the mafic 

assemblages becomes improbably small.

Further development of the models for the two 

batholiths will rest on the assumption that the mineral 

assemblages that separated from the magmas that 

formed each batholith ranged in composition from
H16 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

Ab

A COMPOSITIONS INCLUDE 2-6 PERCENT MAGNETITE An

Ab

B COMPOSITIONS INCLUDE CM PERCENT MAGNETITE An

TABLE 11. Chemical and normative compositions (weight percent) 

of end members for the three-end-member Sierra Nevada 

batholith model

A. Chemical compositions

End member

MGM .... 

ABH.....

ANH ....

SiO2

55.21 

53.95 

41.05

A12O3

19.60 

20.60 

21.94

'FeO'

8.43 

8.57 

14.27

MgO

3.81 

3.75 

7.07

CaO

8.16 

8.59 

13.13

Na2O

3.66 

4.53 

1.60

K20

1.14 

0 

.93

B. Normative compositions'

End member

MGM .. 

ABH . . . 

ANH . . .

Q Or

0.8 6.7 

0 0 

0 0

Ab An

31.0 33.7 

38.3 35.9 

0 49.9

Lc Ne

0 0 

0 0 

4.3 7.3

Wo

2.8 

2.8 

2.2

En Fs

9.5 15.5 

6.3 10.5 

.9 1.3

Fo Fa

0 0 

2.2 4.0 

11.7 19.2

Cs

0 

0 

3.1

FIGURE 11. Fields of composition (shaded) for the Sierra Nevada 

(A) and Alaska-Aleutian Range (B) batholiths in the hornblendealbite-anorthite-magnetite (H-Ab-An-M) system that have communalities higher than 0.98 when represented as vectors in the 

three-dimensional vector systems of figures 9 and 10, respectively. Fields have been projected from the magnetite (M) apex, in the 

foreground, onto the base of the tetrahedron. See table 10 for 

composition of the hornblende. The lines X-Y show the trends of 

compositions represented by vectors with the highest communalities.

'Normative mineral molecule abbreviations used throughout this report are as follows: Q, 

quartz; Or, orthoclase; Ab, albite; An, anorthite; Lc, leucite; Ne, nepheKte; Wo, 

wollastonite; En, enstatite; Fs, ferrosilite; Fo, forsterite; Fa, fayalite; Cs, Ca2SiO4.

TABLE 12. Chemical and normative compositions (weight percent) 

of end members for the Alaska-Aleutian Range batholith model

A. Chemical compositions

End member

MGM .... 

ABH..... 

ANH ....

Si02

53.58 

54.58 

45.09

A1 2O3 'FeO'

20.27 8.37 

20.40 7.66 

21.25 11.83

MgO

4.73 

4.24 

7.06

CaO

8.84 

8.69 

11.49

Na2O

3.81 

4.43 

2.58

K2O

0.60 

0 

.70

B. Normative compositions

End member

MGM .... 

ABH .....

ANH .....

ABH to

Or

3.6 

0 

4.2

ANH

Ab An Ne

32.2 35.9 0 

37.5 35.8 0 

8.6 44.3 7.2

as given in

Wo

3.3 

3.1 

5.3

tables

En

8.2 

9.0 

2.4

11

Fs Fo

10.7 2.5 

11.9 1.1 

2.9 10.7

and 12.

Fa

3.6 

1.6 

14.5

The

separated minerals, with total compositions in this 

range, will be referred to hereafter as mafic 

assemblages, or mafic-mineral assemblages, even 

though plagioclase of variable composition is interpreted to be a major component.

With this assumption, it is possible to define a range 

of compositions possible for the parent magmas or

It

Compositions represented 

by points X and Y

Constituent

SiO2 ......... AT n

'FeO' ........

MgO ........

CaO .........

Na2O ........

K2O .........

Sierra 

Nevada

X

59.87

21.14

3.97

1.51

5.50

7.93

.08

Y

46.36 

19.55 

12.21 

6.27 

13.79 

1.48 

.34

Alaska- Aleutian 

Range

X

57.26

20.92 

5.28 

2.51 

7.29 

6.60 

.14

Y

48.06 

19.89 

11.22 

5.52 

12.65 

2.36 

.30
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H17

magma compositions be represented by vectors to the 

right-hand side of planes passing through vectors 

ABH and ANH in figures 9 and 10. Otherwise, the 

magma would have to be subtracted from the mafic 

assemblages in order to arrive at the compositions 

represented by the sample vectors. Moreover, it is 

necessary that vectors representing the magma compositions occur to the left of planes passing through 

points ABH and MGM and points ANH and MGM. 

Otherwise, (1) the composition of the mafic 

assemblages required to form some samples will not be 

in the range ABH to ANH (tables 11 and 12), or (2) the 

mafic assemblages would have to be added to the 

magma rather than separated from it. The vector 

represented by point MGM is the vector of intersection of a plane through vectors ABH and a vector 

representing a sample of extreme composition with 

another plane through vector ANH and another extreme sample vector (figs. 9 and 10). Thus, the only 

possible magma compositions are those that can be 

represented by vectors intermediate to vectors MGM, 

ABH, and ANH on figures 9 and 10. Accordingly, the 

only possible magma compositions are linear combinations of the compositions given for MGM, ABH, and 

ANH in tables 11 and 12. Vector MGM is chosen to 

represent the parent magma composition for each 

batholith because these compositions are most like the 

sample compositions and require the least magmatic 

differentiation to form each of the samples.

The compositions represented by the MGM vectors 

in figures 9 and 10, as given in tables 11 and 12, closely 

resemble the average composition of andesite 

tabulated by Nockolds (1954, p. 1019) except for the 

high values for A12O3 , the low value for MgO in the 

case of the Sierra Nevada batholith (table 11), and the 

low value for K2O in the case of the Alaska-Aleutian 

Range batholith (table 12). Nockold's average 

andesite, with total iron as 'FeO' and recalculated to 

100 percent, is SiO2 =55.85, A12O3 =17.69, 'FeO'=8.88, 

MgO=4.49, CaO=8.16, Na2O=3.78, and K2O=1.14. 

Thus, the starting materials for the generation of the 

batholiths will be taken as alumina-rich andesitic 

magmas or magma-source materials.

The compositions of the three end members for each 

batholith, as given in tables 11 and 12, can be used to 

approximate the compositions of all 228 samples from 

the Sierra Nevada and all 158 samples from the 

Alaska-Aleutian Range. The required mixing proportions are positive for end member MGM, representing 

the parent magma or magma-source material, and 

negative for end members ABH and ANH. There is 

only one exception; sample 72 from the AlaskaAleutian Range (fig. 10) requires the separation of

mafic assemblages that have compositions outside the 

range from ABH to ANH. That is, all but one of the 

sample compositions can be approximated by starting 

with 1+p parts magma, of composition MGM, and 

subtracting p parts of mafic assemblages that are intermediate in composition between end members ABH 

and ANH. The one exception, sample 72 (See Reed and 

Lanphere, 1974, table 1), is of a poorly exposed amphibolite and consists of 98 percent hornblende and 

calcic plagioclase. It might represent a metamorphosed gabbro of Jurassic age or, possibly, a material of the 

same origin as the smaller mafic inclusions present 

throughout many parts of the batholith.

Statistical summaries of the mixing proportions required for the individual samples are given in tables 13 

and 14. Just as the individual sample compositions can 

be approximated by combining the end-member compositions in proportions determined for each sample, 

the average compositions for each group of plutons, as 

given in tables 6 and 7, can be approximated by combining the end-member compositions in the average 

proportions given for each group in tables 13 and 14. If 

either of these approximations are derived, it will be 

seen that they are only moderately good for A12O3 , 

Na2O, and K2O in the Sierran rocks. This is in accord 

with the factor-variance diagram in figure 7, which indicates that three-end-member models for the Sierra 

Nevada batholith account for these oxides to a lesser 

degree than they account for others. Hence, a four-endmember model will be sought.

As shown by the factor-variance diagrams in figures 

7 and 8, a fourth end member is required for the Sierra 

Nevada model in order to account for the compositional variability to about the same degree that a 

three-end-member model accounts for the variability in 

the Alaska-Aleutian Range batholith. The reason for 

this is not clear, and the selection of a fourth end 

member for the Sierran model is difficult. However, 

studies of the chemical and isotopic variations in the 

granitic rocks led Kistler and Peterman (1973) to suggest that the parent magmas were derived from deepseated materials that varied laterally in composition. 

If their interpretation is correct, the fourth end 

member might appropriately allow for compositional 

variations among the parent magmas or within the 

magma-source material. The average composition of 

all the magmas, or of the magma-source material, is 

taken as that of MGM in table 11. It remains only to 

find the manner in which the compositions varied. The 

major variation is expected to have been in a direction 

generally perpendicular to the continental margin.

An extreme parent-magma composition was sought 

by using the average composition of samples from the
HIS STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

TABLE 13. Statistical summary of mixing proportions for the three-end-member Sierra Nevada

Batholith model

[See table 1 1 for compositions of end members MGM, ABH and A NH]

Group of plutons

Palisade Crest sequence ......

Scheelite sequence ...........

Shaver sequence .............

Number 

of 

samples

21 

19 

23 

7 

7 

36

115 

228

MGM

3.95 

3.79 

4.81 

3.73 

4.39 

4.17 

3.76

3.97

Averages

ABH

-2.19 

-2.09 

-2.86 

-1.84 

-2.31 

-2.32 

-1.97

-2.16

Standard deviations

ANH

-0.76 

-.69 

-.94 

-.89 

-1.07 

-.85 

-.80

-.82

MGM

0.54 

.65 

.69 

1.15 

.93 

.91 

.89

.89

ABH

0.39 

.53 

.48 

.99 

.81 

.65 

.64

.67

ANH

0.22 

.19 

.26 

.17 

.15 

.31 

.28

.27

TABLE 14. Statistical summary of mixing proportions for the Alaska-Aleutian Range Batholith

model

[See table 12 for compositions of end members MGM, ABH and A NH]

Group of plutons

Quartz monzonite of Tired Pup

Hartman sequence ............

Granodiorite of Mount Estelle .

Undivided plutonic rocks ......

Plutonic rocks of Jurassic age. . 

All plutons ...................

Number 

of 

samples

29 

14 

22 

5 

13

8 

35 

3 '28

'157

MGM

10.48 

11.22 

10.44 

8.30 

8.47

8.02 

4.65 

9.31 

4.46

7.73

Averages

ABH

-6.23 

-7.04 

-6.25 

-5.21 

-5.26

-5.06 

-2.09 

-5.74 

-2.13

-4.44

Standard deviations

ANH

-3.25 

-3.17 

-3.18 

-2.09 

-2.20

-1.96 

-1.57 

-2.57 

-1.33

-2.29

MGM

1.47 

.92 

.95 

.93 

.84

1.61 

1.64 

1.43

1.85

3.15

ABH

1.07

.68 

.67 

.85 

.56

1.12 

1.15 

.74 

1.27 

2.25

ANH

0.41 

.27 

.30 

.09 

.31

.53 

.60 

.73 

.79

.98

'Sample 72 was excluded. See text.

White Mountains, (the easternmost group of plutons 

in the Sierra Nevada), as given in table 6, in order to 

estimate the magma composition required for this 

easternmost group of plutons. This average composition from table 6 is used to form a row vector, W. According to the three-end-member Sierran model, composition W resulted largely from the separation of 

mafic assemblages from a parent magma, or magmasource material. The average composition of the mafic 

assemblages was equal to a mixture of 2.19 parts ABH 

and 0.76 parts ANH (tables 11 and 13), and 3.95 parts 

of initial magma were required (1.00+2.19+0.76). The 

following approximate matrix equation can be written:

-z, or

The row vector U is the required magma composition 

for the White Mountains rocks, A is a row vector containing the mixing proportions for ABH and ANH 

(2.19 and 0.76) from table 13, and F is a matrix (with 2 

rows and 7 columns) containing the compositions of

end members ABH and ANH from table 11. Solution 

of the approximation for U gives:

____________Weight percent____________ 

SiO2 A12O3 'FeO' MgO CaO Na2O K,O

54.73 19.79 8.47 3.83 8.11 3.73 1.33

This composition was represented as a vector in the 

seven-dimensional vector space for the Sierra Nevada 

samples and projected into the four-dimensional vector 

system. The communality of the vector after projection was 0.99999, indicating that the composition, £7, 

is highly compatible with the compositional variability 

among the Sierran samples. The composition of the 

vector representing composition £7, after projection of 

the vector into the four-dimensional vector system, differs from composition U only in the second decimal 

places.

The composition of end member MGM (table 11) was 

also represented as a vector in the four-dimensional 

vector system, and the assumption was made that the 

two vectors to represent the extreme compositions 

among the parent magmas, or magma-source 

materials, occur in the plane of the U and MGM vec-
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H19

tors. The two vectors representing the extreme magma 

compositions, MGMl and MGM2, were taken as two 

extreme vectors in this plane. Each extreme vector 

was such that the mixing proportions for at least one 

sample indicated zero parts of the respective end 

member, and the mixing proportions for all other 

samples were positive for both of these end members. 

The compositions represented by these extreme vectors, MGMl and MGM2 are given in table 15 along 

with the compositions of end members ABH and ANH 

from table 11. It will be seen that the compositional 

variation required among the magmas, or magmasource materials, is small; SiO2 ranges only from 53.51 

to 56.12 percent. The greatest proportional variation, 

by far, is for K2O (0.77 to 1.81 percent), which is in accord with compositional variations among the granitic 

rocks as described by Bateman and Dodge (1970).

TABLE 15. Chemical and normative compositions (weight percent)

of end members for the four-end-member Sierra Nevada

batholith model

A. Chemical compositions 

End member SiO2 A12O3 'FeO' MgO CaO Na2O K2O

MGMl . . . MGM2 . . .

ABH.....

ANH ....

53.51

56.12

53.95

41.05

20.26

19.24

20.60

21.94

8.64

8.31

8.57

14.27

3.86

3.78

3.75

7.07

8.01

8.24

8.59

13.13

3.90

3.53

4.53

1.60

1.81

.77

0

.93

B. Normative compositions

End member Q Or Ab An Lc Ne Wo En Fs Fo Fa Cs

MGMl . 0 10.7 33.0 32.4 0 0 3.1 2.7 4.4 4.9 8.8 0

MGM2 . 3.8 4.6 29.9 34.4 0 0 2.7 9.4 15.3 000

ABH ... 0 0 38.3 35.9 0 0 2.8 6.3 10.5 2.2 4.0 0

ANH... 000 49.9 4.3 7.3 2.2 .9 1.311.719.2 3.1

The compositions of each of the 228 samples from 

the Sierra Nevada can be closely approximated (to the 

degree indicated for a four-end-member model by 

figure 7) by mixing end members MGMl and MGM2 

in positive proportions and subtracting mixtures of 

ABH and ANH (table 15). A statistical summary of 

the mixing proportions is given in table 16. In addition, the average compositions for groups of plutons, 

given in table 6, can be approximated with this model 

(as was done with the three-end-member model in table 

13) by combining the end-member compositions in 

table 15 according to the average proportions given intable 16.

In summary, the major part (more than 92 percent) 

of the compositional variability in the Alaska-Aleutian 

Range batholith can be accounted for by starting with

an andesitic magma, or magma-source material, with 

the composition of MGMm table 12, and subtracting 

varying amounts of mafic mineral assemblages that 

range in composition from that of ABH and ANH, also 

given in table 12. A basically similar model will account for the major part (more than 90 percent) of the 

compositional variability in the Sierra Nevada 

batholith, but it is necessary to allow some variation in 

the composition of the andesitic parent magma or 

magma-source material. Table 17 was derived from the 

mixing proportions for end members MGMl and 

MGM2 (table 16) and the compositions of these end 

members (table 15) and shows the average composition 

of the parent magmas required by the model for the 

samples from each group of plutons. The greatest proportional variation among these compositions is with 

respect to K2O. The average composition required for 

the batholith as a whole (listed as "all plutons" in table 

17) is precisely that proposed in the thee-end-member 

model (table 11).

The variation in K2O among the magmas specified 

by the model for the various groups of plutons is small 

(table 17) but corresponds in a general way with the 

distance of the plutons from the continental margin. 

(See map of Bateman and Dodge, 1970, fig. 2.) This is 

in accord with the proposal of Moore (1959) that the 

source materials for granitic rocks of the western U.S. 

had K2O contents that increased progressively away 

from the continental margin.

The average compositions of the mafic assemblages 

separated from the magma, or magma-source material, 

for each batholith are given in table 18. These average 

compositions were computed from the average mixing 

proportions for end members ABH and ANH (tables 

14 and 16) and the compositions of these end members 

(tables 12 and 15). If the mafic assemblages are 

represented by the mafic inclusions that are 

widespread throughout many parts of both batholiths, 

the average compositions of the assemblages and the 

inclusions should be similar. Unfortunately, few data 

are available on the compositions of the inclusions in 

either batholith. However, Morton, Baird, and Baird 

(1969, p. 1558) gave the average composition of 13 

similar inclusions from a pluton in the southern 

California batholith. This average composition (recomputed to percents of oxides) is given in table 18 and is 

generally similar to the average compositions of the 

mafic assemblages derived from the factor models for 

both batholiths.

The average mixing proportions for the MGM end 

member as given in table 14 indicate that, according to 

the model, the Alaska-Aleutian Range batholith form-
H20 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

TABLE 16. Statistical summary of mixing proportions for the four-end-member Sierra Nevada

batholith model

[See table 15 for composition of end members MGMl, MGM2, ABH, and ANH\

Averages Standard deviations

Group of plutons

Number

of ________________________________________ 

samples MGMl MGM2 ABH ANH MGMl MGM2 ABH ANH

White Mountains rocks ......

Palisade Crest sequence ......

Scheelite sequence ...........

Toulumne intrusive series ....

Yosemite rocks ..............

Shaver sequence .............

All other plutons ............

All plutons ..................

21

19

23

7

7

36

115

228

2,

1,

1.

1,

1,

1.

1,

1,

.12

.65

.56

.35

.19

.26

.25

.40

1,

2,

3.

2,

3.

2,

2,

2,

.83

.14

.23

.38

.19

.92

.52

.57

-2.18

-2.10

-2.85

-1.84

-2.32

-2.32

-1.97

-2.15

-0.76

-.69

-.94

-.89

-1.06

-.85

-.80

-.82

0.50

.48

.53

.44

.50

.47

.45

.53

0.66

.53

.72

.77

.52

.54

.56

.69

0.43

.57

.46

.99

.78

.64

.63

.66

0.23

.20

.25

.17

.15

.30

.27

.27

TABLE 17. Average compositions, in percent, of magmas for the Sierra Nevada batholith model

Group of plutons A1 2O3

'FeO' MgO CaO Na2O K2O

White Mountains rocks .......

Toulumne intrusive series

All plutons ...................

54.72

54.99

55.28

55.18

55.42

55.34

55.26

55.21

19.79

19.69

19.57

19.61

19.52

19.55

19.58

19.60

8.49

8.45

8.42

8.43

8.40

8.41

8.42

8.43

3.82

3.82

3.81

3.81

3.80

3.80

3.81

3.81

8.12

8.14

8.17

8.16

8.18

8.17

8.16

8.16

3.73

3.69

3.65

3.66

3.63

3.64

3.65

3.66

1.33

1.22

1.11

1.15

1.05

1.08

1.11

1.14

TABLE 18. Compositions of mafic-mineral assemblages (end 

members ABH and ANH required for the average samples from the 

two batholiths according to the derived models, and the average 

composition of mafic inclusions from the southern California

batholith

Average composition (in percent)

Constituent

SiO2 ...........

A12O3 .........

'FeO' ..........

MgO ..........

CaO ...........

Na2O ..........

K2O ...........

Mafic mineral assemblages

Alaska- Aleutian 

Range batholith

.......... 51.35

.......... 20.69

.......... 9.08

.......... 5.20

Q fi4

.......... 3.80

.......... .24

Sierra Nevada 

batholith

50.39 

20.97 

10.14 

4.67 

9.84 

3.72 

.26

Mafic 

inclusions'

51.67 

19.31 

8.62 

5.34 

9.19 

3.01 

1.30

'Average of 13 analyses of samples from the Lakeview Mountains pluton (Morton and 

others, 1969, table 4).

ed from a starting material (magma or magma-source 

material) with a mass of about 7.73 times the mass of 

the part of the batholith represented by the 158 

samples. Thus, the batholith represents about 12.94 

(100/7.73) percent of the starting material. (Here and 

throughout the remainder of this report, many of the 

reported computational results contain more significant figures than are justified by the data. The additional figures are retained in order to allow the reader 

to follow the nature of the computations more easily.) 

A similar calculation for the Sierra Nevada batholith, 

based on the mixing proportions for all plutons given 

in tables 13 and 16, indicates that the batholith formed 

from about 25.19 percent of the starting material.

FORMATION OF AN INDIVIDUAL PLUTON

The models presented for the Sierra Nevada and 

Alaska-Aleutian Range batholiths call for a magma or 

source material of andesitic composition, and for 

subsequent separation of mafic mineral assemblages. 

However, field studies have revealed very clearly that 

mafic minerals not only separated from the magma, 

but accumulated in the marginal parts of many 

plutons, at least partly by accretion on the walls of the 

magma conduit (Bateman and others, 1963, p. 

D30-D31; Presnall and Bateman, 1973, p. 3196-3197). 

In fact, some samples from the marginal parts consist 

of collections of plagioclase, hornblende, and biotite 

with little quartz and essentially no potassium 

feldspar, even through potassium feldspar is abundant 

in the pluton's central part. The question here is how 

this process of crystal accumulation can be reconciled 

with a model that calls for a net loss of mafic mineral 

assemblages. (See mixing proportions in tables 13, 14, 

and 16.)

A possible explanation is given in figure 12. Figures 

12A and 12B are stereograms of the same type given in 

figures 2, 9, and 10. Figure 12A shows a representation 

of the earliest stages of the process. Vector MGM 

represents the composition of the parent magma, or 

magma-source material. Separation of mineral 

assemblages ranging in composition from a to 6 could 

change the magma composition to range from c to d 

Composition c would result from the separation of
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H21

2.746 parts of composition b from 3.746 parts of composition MGM. Composition d would result from the 

separation of 3.360 parts of composition a from 4.360 

parts of composition MGM. The modified magma, 

therefore, would represent about one-fourth of the 

original magma or magma-source material. The compositional variation in the mixture is small (ranging 

from c to d in fig. 12), but could have resulted from 

minor compositional variations in the separated mafic 

assemblages as represented in figure 12A. This variation, in turn, might have been caused by temperature 

and density gradients within the ascending magma.

The solid points in figure 12B represent vectors that 

correspond to the compositions of samples from the 

Mt. Givens pluton in the central part of the Sierra 

Nevada (Bateman and others, 1963). Some of these

samples are more felsic than the melt (points to the 

right of plane c-d), and some are more mafic (points to 

the left of plane c-d). Those that are more felsic may 

have formed by the separation of mafic minerals from 

the magma, and those more mafic formed by addition 

of the mafic minerals. For example, sample e might 

have formed by the subtraction of 0.115 parts of composition h from 1.115 parts of composition d. Sample / 

might have formed from the addition of 0.118 parts of 

composition g to 0.882 parts of composition c. Other 

samples, farther to the left of plane c-d and closer to 

the plane ABH-ANH, could result from the accumulation of mafic minerals containing only the amounts of 

melt that would occupy the mineral interstices.

The composition loadings, or mixing proportions, 

summarized in tables 13,14, and 16 may reflect the net

FIGURE 12. Vector diagrams representing a process of formation of the magma for the Mount Givens pluton of the Sierra Nevada 

batholith (A), and a process of differentiation of the magma (B). The magma may have ranged in composition from c to d, 

resulting from the separation of mafic assemblages from the magma, MGM. The mafic assemblages ranged in composition from 

a to 6. At a later stage, B, the magma differentiated by separation of mafic minerals (composition h) from some part of the magma 

(composition d) to form sample e, and by addition of mafic minerals (composition g) to another part (composition c) to form sampie/.

Constituent Compositions represented by vectors, in percent

ABH ANH MGM

SiO,

A1203

'FeO'

MgO

CaO

Na,O

K,0

53.95

20.60

8.57

3.75

8.59

4.53

0

41.05

21.94

14.27

7.07

13.13

1.60

.93

55.21

19.60

8.43

3.81

8.16

3.66

1.14

50.96

20.91

9.89

4.52

9.64

3.85

.22

49.86

21.03

10.38

4.81

10.03

3.60

.30

69.88

15.69

3.06

1.06

3.03

3.83

3.44

69.49

15.18

3.49

1.40

3.18

3.03

4.24

71.69

14.52

2.72

1.02

2.41

2.95

4.69

67.17

16.35

4.08

1.59

3.98

3.73

3.10

46.97

21.33

11.66

5.55

11.05

2.95

.51

50.28

20.98

10.20

4.70

9.88

3.70

.27
H22 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

effects of a two-stage process such as that represented 

in figure 12. This explanation requires that all 

separated mineral assemblages, both those separated 

early and those separated later, have total compositions in the range from ABH to ANH. However, if this 

were not the situation, the compositional structures in 

the batholiths would be more complex than indicated 

by the factor-variance diagrams in figures 7 and 8.

COMPOSITIONS OF PARTIAL MELTS 

FROM THE CRUST

Presnall and Bateman (1973) gave evidence that a 

major portion of the Sierra Nevada batholith 

originated by equilibrium fusion of the lower crust. 

The fusion processes caused the development of a 

crystal-liquid mush, the crystal portion consisting 

mostly of the more refractory minerals. They proposed 

that as the degree of crustal fusion reached some point, 

the crystal-liquid mush began to rise, and at least some 

of the crystal portion of the mush was retained in the 

liquid all the way to the observed sites of magma 

emplacement. Other parts of the crystalline material 

were either left behind at the sites of fusion or settled 

out of the mush, perhaps to form the mafic inclusions 

widespread throughout much of the batholith (Presnall 

and Bateman, 1973, p. 3197).

If the model of Presnall and Bateman and the factor 

model for the Sierra Nevada are both correct, certain 

consequences follow from the factor model regarding 

the compositions of the melts formed in the crust, the 

degree of melting, and the proportions of refractory 

material retained in the melt or precipitated from it. 

These consequences are described here so that they 

may be used in evaluating the models for both 

batholiths at some future time when more is known 

about the physicochemical aspects of the melting process and the conditions of melting that may have produced these batholiths. Experimental studies of 

melting have been directed at the petrologic system 

quartz-K-feldspar-albite-H2O, and at the artificial 

melting of mafic and ultramafic rocks. See James and 

Hamilton (1969) and Presnall and Bateman (1973, p. 

3187-3190) for reviews of this work and extensive 

bibliographies.

Previously in this report, we have referred to the 

MGM (also MGMl and MGM2) end members as 

representing either magma or magma-source material. 

If they represent magmas, the magmas must have 

been modified to the compositions of the samples by 

precipitation of mafic mineral assemblages within the 

compositional ranges from ABH to ANH. Alternatively, the models can be reconciled with the model of 

Presnall and Bateman by regarding the MGM end 

members as magma-source materials, rather than as

magmas, and the ABH and ANH end members as 

representing the extremes in the range of refractory 

mineral assemblages. Thus, end members MGMl and 

MGM2 (table 15) represent the range of compositions 

in the original crust beneath the Sierras, and the 

average compositions in table 17 indicate the average 

composition of the crustal materials that led to the formation of each group of plutons. Similarly, end 

member MGM for the Alaska-Aleutian Range model 

(table 12) represents a more uniform crustal material 

that originally may have been present beneath that 

batholith.

The factor models for the two batholiths specify the 

compositions of both the initial crustal materials and 

the mafic assemblages that comprise the more refractory phases. We do not know the abundance of the 

refractory phases in the partially melted crust or, from 

mathematical evidence alone, whether they were even 

present. If the melting of the crust had been nearly 

complete, for example, the refractory phases would 

have nearly disappeared. In this situation, the composition of the melt would have been close to the composition of the MGM end member and would have had 

to have been modified to the observed compositions of 

the samples by precipitation of mafic mineral 

assemblages. If, on the other hand, the degree of 

melting had been low, the composition of the melt 

would have been highly salic, and some refractory 

material (mafic assemblages) would have had to have 

been retained in the melt in order to produce the 

observed samples. Thus, it is only possible to estimate 

a range of melt compositions that could have led to 

each sample. The method of estimation will be illustrated for the theoretical average sample from the 

Alaska-Aleutian Range batholith.

The derived average composition of the crust 

originally beneath the Alaska-Aleutian Range is given 

in column (a) of table 19, and the average composition 

of the separated mafic assemblages is given in column 

(b). If we assume minimum melting, and therefore 

minimum heat requirements, we will wish to retain as 

much of the crust as possible in the form of crystalline 

mafic assemblages. However, the proportion of 

crystalline material is limited by the relative MgO contents of the crust and the assemblages (columns a and 

b); if more than 90.96 percent of the crust consisted of 

the crystalline assemblages, the derived MgO content 

of the liquid portion would be negative. Columns (c) 

and (d) give the composition of the crystal-liquid 

mush with the assumption of minimum 

melting separated according to crystal and liquid 

portions. Column (c) was computed as 90.96 percent of 

column (b) and column (d) was derived as the difference 

between columns (a) and (c). Therefore, if the model is
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H23

interpreted in terms of partial melting of the crust, it 

must be concluded that at least 9.04 percent (table 

19.4, column d) of the crust had to melt to produce 

refractory-mineral assemblages of the composition 

given in column (b). The composition of the liquid portion of the mush is given in column (e), which simply 

shows the data from column (d) recalculated to 100 percent.

The average mixing proportions given in table 14 indicate that the Alaska-Aleutian Range batholith as a 

whole formed from 7.73 parts crust (in the context of 

this discussion), from which 6.73 parts of mafic 

assemblages separated. Thus, the batholith formed 

from 12.94 (100/7.73) percent of the crust. The computations that led to column (d) of table 19A, however, 

show that the melt, under minimum melting conditions, formed from only 9.04 percent of the crust. The 

difference, 3.90 percent, must have consisted of 

refractory-mineral assemblages that were retained and 

carried upward in the rising liquid. The refractory 

material, then, would have comprised 30 percent 

(100X3.90/12.94) of the liquid-crystal mixture. The 

total composition of the liquid plus retained crystals is 

derived in columns (/) and (g) and closely approximates 

the average composition of the total batholith given in 

column (h). Columns (/), (g), and (h) merely serve as a

check on the computations. The values in column (e) 

are of the most interest; they show the composition of 

the melt under conditions of minimum melting as 

derived from the Alaska-Aleutian Range model. The 

normative composition of the melt is 96 percent 

Q3 6Or26Ab34An4.

The computations shown in table 19A are based on 

the assumption that the degree of melting in the crust 

was minimal (9.04 percent) so that the melt contained 

no MgO. Similar computations can be made with 

assumptions that the degree of melting was greater. 

An example, based on the assumption of 20 percent 

crustal melting is given in table 19.B. The derived melt 

composition (column e) is less siliceous than that derived in table 19./4, and it is necessary to precipitate 7.06 

percent mafic assemblages in order to form the 

batholith from 12.94 percent of the crust 

(20-12.94=7.06). See table 19B, column (/). This constitutes precipitation of 35.3 percent of the melt fraction (100X7.06/20=35.3).

Computations similar to those outlined in table 19 

were carried out for each of the 158 individual samples 

from the Alaska-Aleutian Range batholith and for the 

average composition of each group of plutons listed in 

table 7. The composition of the crust used for each 

computation was the same as that given in column (a)

TABLE 19. Mass balance computations for the average sample from the Alaska-Aleutian Range batholith

Constituent

(a)

Composition of

crust (MGM from

table 12)

(b)

Average

composition

of mafic

assemblages'

(c) (d) 

Crystal-liquid mixture

Crystals

Liquid

(column a

minus column c

(e) 

Composition of

partial melt

(column d

recalculated

to 100

(/)

Liquid plus

(or minus)

crystals

(g)

Column /

recalculated

to 100

(h)

Average

composition

of batholith

(from table 5)

A. Assuming the lowest possible degree (9.04 percent) of crustal melting;

SiO2

A1 2O3

'FeO'

MgO

CaO

Na2O

K2O

Total

53.58

20.07

8.37

4.73

8.84

3.81

.60

100.00

B. Assuming 20 percent crustal melting;

SiO2

A12 03

'FeO'

MgO

CaO

Na2O

K2O

53.58

20.07

8.37

4.73

8.84

3.81

.60

51.35

20.69

9.08

5.20

9.64

3.80

.24

100.00

Column c =

0.9096 X

column b

46.71

18.82

8.26

4.73

8.77

3.45

.22

90.96

6.87

1.25

.11

0

.07

.36

.38

9.04

76.0

13.8

1.2

0

.8

4.0

4.2

100.00

Column c =

51.35

20.69

9.08

5.20

9.64

3.80

.24

0.80 X

column b

41.08

16.55

7.27

4.16

7.71

3.04

.19

12.50

3.52

1.10

.57

1.13

.77

.41

62.5

17.6

5.5

2.9

5.6

3.9

2.0

Column / =

column d

+0.0390

X column b

8.87

2.06

.46

.20

.45

.51

.39

12.94

Column / =

column d

-0.0706

X column b

8.87

2.06

.46

.20

.45

.51

.39

68.5

15.9

3.6

1.5

3.5

3.9

3.0

99.9

68.5

15.9

3.6

1.5

3.5

3.9

3.0

68.60

15.89

3.62

1.53

3.44

3.87

3.05

100.00

68.60

15.89

3.62

1.53

3.44

3.87

3.05

Total 100.00 100.00 80.00 20.00 100.00 12.94 99.9 100.00

'Equals a mixture of 4.44 parts of end member ABH and 2.29 parts of end member ANH (tables 12 and 14), recalculated to 100 percent.
H24 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

of table 19 and the compositions of the mafic 

assemblages (column 6) were derived from the mixing 

proportions for end members ABH and ANH as computed for the individual samples and for the average 

sample from each group of plutons (table 14). The 

CIPW norms were computed for each derived melt 

composition, and the normative values for Q, Or, Ab, 

and An were adjusted to sum to 100 percent. These adjusted values were then represented in a Q-Or-Ab-An 

tetrahedron and were found to define a surface as 

shown in figure 13. (Only 5 of the 158 samples gave 

results that deviated from this surface.) The surface

Lines for averages of groups of plutons are labeled with 

map symbols (T, Tertiary; K, Cretaceous; J, Jurassic) 

from Reed and Lanphere (1973, fig. 2)

contains all possible melt compositions that can be 

derived from the model assuming degrees of crustal 

melting ranging from the minimum for each sample to 

100 percent. The possible melt compositions for each 

individual sample occur on a line within the surface 

that extends from the line x-y-z to the point MGM. 

These lines are shown in figure 13 for the average of all 

samples from the batholith (line a-MGM) and for the 

average sample from each group of plutons. The two 

extreme samples are represented by lines x-MGM and 

z-MGM. The extreme sample represented by line 

z-MGM could have formed with as little as 5.6 percent

Tmp

TKt

TKc

TKh

TKme

TKy

TKsl

TKu

Jp

Merrill Pass sequence 

Quartz monzonite of Tired Pup 

Crystal Creek sequence 

Hartman sequence 

Granodiorite of Mount Estelle 

Yentna sequence 

Summit Lake rocks 

Undivided plutonic rocks 

Jurassic rocks

Ab 10 20 30 Or 40 50 60 70 80 90

FIGURE 13. Q-Or-Ab-An tetrahedron showing the surface containing possible melt compositions for samples from the Alaska-Aleutian 

Range batholith. The possible melt compositions were determined from the model by assuming various degrees of melting in the 

crust. The tetrahedron is viewed from directly above with the An apex in the foreground. The lines connecting the point MGM with 

the points for individual groups of plutons define the ranges of possible melt compositions for the average sample from each group. 

See table 20 for positions of points within the tetrahedron and other data.
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H25

melting in the crust, whereas the sample represented 

by line x-MGM requires crustal melting of at least 23.7 

percent. The surface shown in figure 13 is essentially 

planar in the range from 5.6 percent to more than 40 

percent melting and is curved convexly upward near 

the Ab-An-Or face of the tetrahedron. Data pertaining 

to each of the labeled points in figure 13 are given in 

table 20. The data for the average of all samples (line 

a-MGM) indicate that under minimum melting conditions the crustal melt retained 3.9 percent mafic 

assemblages and that if the degree of melting exceeded 

10 to 20 percent (about 13 percent by interpolation), 

the mafic assemblages would have had to have been 

precipitated from the melt in order to modify the melt 

composition to that of the average composition of the 

batholith.

The lines in figure 13 containing the possible melt 

compositions for the average sample from each group

of plutons (TKsl-MGM, Jp-MGM, and so forth) have 

positions that correspond in a general way to the 

geographic positions of the respective groups. The 

groups of plutons on the eastern margin of the 

batholith, nearer to the continental margin (Reed and 

Lanphere, 1973, fig. 2), tend to be represented by lines 

closer to the line x-MGM in figure 13. This indicates 

that, according to the model, groups of plutons that 

formed at a distance from the continental margin could 

have resulted from lower degrees of crustal melting 

than did those nearer the margin. The minimum 

degrees of melting for the average sample from each 

group are given in table 20; they range from 6.7 to 13.2 

percent.

Computations similar to those outlined in table 19 

and those used to construct figure 13 were also carried 

out for each individual sample from the 6 groups of 

Sierra Nevada plutons listed in table 6. The composiTABLE 20. Data pertaining to points within tetrahedron of figure 13

Group of

plutons

All plutons .......

Do ...........

Do ...........

Do ...........

Do ...........

Do ...........

Do ...........

Do ...........

Do ...........

Yentna sequence . .

Hartman sequence

Granodiorite of

Mount Estelle.

Undivided

plutonic rocks.

Quartz monzonite

of Tired Pup.

Crystal Creek

sequence.

Merrill Pass

sequence.

Jurassic rocks .... 

Summit Lake rocks

Point Percent Position of point

in Q+Or+Ab+An

fig. 13

z

y

X

a

b

c

d

e

MGM

TKy

TKh

TKme

TKu

Tkt

TKc

Tmp

TKsl

in melt

88.7

95.9

96.9

96.3

94.1

81.4

74.0

71.5

71.7

93.4

94.1

94.9

96.1

96.1

96.3

96.3

96.5 

96.6

Q

39

39

26

36

32

15

4

0

0

39

39

39

39

39

36

36

33 

31

in tetrahedron

Or

53

33

6

26

24

15

9

6

5

40

38

36

33

33

27

26

20 

16

Ab

8

28

52

34

35

40

44

45

45

21

23

25

28

29

34

35

40 

43

An

0

0

16

4

9

30

43

49

50

0

0

0

0

0

4

4

8 

10

Percent

melting

5.6

7.4

23.7

9.0

10.0

20.0

40.0

80.0

100.0

6.7

6.9

7.1

7.5

7.5

8.9

9.2

11.2 

13.2

Percent mafic 

assemblages 

retained in (+)

or precipitated

from ( ) melt

+ 19.4

+0.8

+ 3.3

+ 3.9

+ 2.9 -7.1

-27.1

-67.1

-87.1

+ 5.8

+ 5.2

+ 4.7

+ 3.3

+ 1.4

+ 0.7

+0.4

+ 11.2 

+ 8.3
Ab 10 20 30 40 50 60 70 80 90 OrAb 

A WHITE MOUNTAIN ROCKS

Or Ab

B PALISADE CREST SEQUENCE 

Q

C SCH

£> TOULUMNE INTRUSIVE SERIES E YOSEMITE ROCKS F SH

FIGURE 14. Q-Or-Ab-An tetrahedrons showing the surfaces containing possible melt compositions for six groups of pluton

Nevada batholith. The possible melt compositions were determined from the four-end-member model by assuming 

melting in the crust. The tetrahedrons are viewed from directly above with the An apex in the foreground. The lin

tetrahedron defines the range of possible melt compositions for the average sample from the corresponding group of pluto

positions of points within each tetrahedron and other data.
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H27

tions assumed for the crust, corresponding to column 

(a) in table 19, were different for each group of plutons 

and are given in table 17. The compositions of the 

mafic assemblages, corresponding to column (b) in 

table 19, were estimated from the mixing proportions 

for end members ABH and ANH for each sample and 

for the average sample from each group of plutons 

(table 16). The derived partial melt compositions occur 

within 6 different surfaces within the Q-Or-Ab-An 

tetrahedron, as shown in figure 14. Each of the surfaces is planar between the line representing minimum 

melting (x-a-z) and that representing melting of 80 percent. The possible melt compositions for the average 

sample from each group of plutons occur in the line a-m 

on each diagram. Each of these lines is close to the approximate trend of compositions of granites of the 

Sierra Nevada batholith shown by Presnall and 

Bateman (1973, line m-n in figure 4 of that report). The 

computations for individual samples indicate that the 

degree of melting ranged from 11 to 46 percent (11 to 

27 percent for all but one sample) and that, in general, 

melting of more than about 25 percent necessitates the 

precipitation of mafic mineral assemblages (table 21).

The minimum degree of melting for the average sample from each group of Sierra Nevada plutons is given 

in table 21 (see point a for each group); these minimum 

values vary only from 16.5 to 21.2 percent. The positions of the lines x-a-z for the different groups within 

the Q-Or-Ab-An tetrahedron (fig. 14), however, vary in 

a way that might be significant. Each of the lines is approximately parallel to the cotectic lines separating 

the quartz and feldspar fields on experimentally derived phase diagrams. The cotectic lines for higher partial 

pressures of H2O are farther from the quartz corner of 

the tetrahedron than are those for lower pressures. 

(See Luth and others, 1964, fig. 4; and Carmichael and 

others, 1974, p. 230.) If the melting process was one of 

equilibrium fusion, as concluded by Presnall and 

Bateman (1973), the initial melts may be expected to 

have had compositions near the cotectic lines. This 

would imply that the White Mountains rocks and the 

Palisade Crest sequence (figs. 14A and 14.B) formed at 

higher pressures than did some of the other groups of 

plutons located closer to the continental margin where 

the crust is probably thinner.

It was stated previously that the factor models 

derived for the two batholiths are not unique, and it 

should be emphasized here that the observations 

described regarding the processes of melting and the 

resultant melt compositions are only as valid as the 

models themselves. These observations by no means 

confirm the validity of the models, but, on the other 

hand, do appear compatible with data from both field 

and experimental studies.

CONCLUSIONS

Igneous rock bodies that have formed by complex 

processes, involving the mixing and unmixing of a 

large number of end members, have complex compositional structures. Those that form, for example, by the 

mixing of a magma with only a few types of country 

rock or by the separation of only a few mineral phases 

from a magma, have simple compositional structures. 

The type of compositional structure in a rock body is 

revealed by a factor-variance diagram constructed 

from chemical or mineralogical data on representative 

samples. The diagram shows the proportion of the 

variance in each constituent that can be accounted for 

by mixing models with any given number of end 

members. It is unlikely that the relatively simple compositional structures revealed for some rock bodies 

arose by chance, even though statistical tests for this 

likelihood are not available. Factor-variance diagrams 

constructed for a wide range of igneous masses reveal 

an equally wide range of compositional structures.

A rock body that is highly variable in composition 

can have either a simple or complex compositional 

structure, and one that is nearly homogeneous, similarly, can be of either simple or complex compositional 

structure. The compositional structure is determined 

by the simplicity or complexity of the combination of 

processes that caused the compositional variation not by the amount of variation present. In fact, 

if an igneous body were perfectly homogeneous 

throughout, the only observed compositional variation 

would be that caused by analytical errors, and the compositional structure would probably appear to be quite 

complex. This would be particularly true if the 

analytical errors for the various oxides tended to be independent of each other.

The possibility should not be overlooked that the 

compositional structure within a magma might have 

been totally eradicated by thorough homogenization 

before the magma solidified. This could result, 

perhaps, from diffusion and convection operating over 

long periods of time. Such homogenization may have 

occurred in some magmas, but the fact that it does not 

occur in all of them is substantiated by the simple compositional structures that some of them display, and 

by the large-scale compositional variations that are 

present. Homogenization could not have been an important factor in the Sierra Nevada and AlaskaAleutian Range batholiths, which are composed of 

plutonic sequences that differ in age by as much as 100 

million years (Bateman and Dodge, 1970, p. 419; Reed 

and Lanphere, 1969, p. 39-41; 1973, p. 2605).

Factor-variance diagrams representing the 

batholiths of the Sierra Nevada and the Alaska-
H28 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

TABLE 21. Data pertaining to points within tetrahedrons of figure 14

Group of

plutons

White Mountains

rocks (fig. 144).. . . . .

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Palisade Crest

sequence (fig. 145). .

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Scheelite sequence

(fig. 14Q. ..........

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Toulumne intrusive

series (fig. 14D).. ....

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Yosemite rocks

(figure 14E. ........

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Shaver sequence

(figure 14F). ........

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Do ...............

Point Percent Position of point

in Q+Or+Ab+An

fig. 14

. . z

. . X

. . a

.. b

. . c

.. d

. . m

. . z

. . x

. . a

.. b

. . c

.. d

. . m

. . z

. . x

. . a

.. b

. . c

.. d

. . m

. . z

. . x

. . a

. . c

.. d

. . m

. . z

. . x

. . a

. . c

.. d

. . m

. . z

. . x

. . a

.. b

. . c

.. d

. . m

in melt

92.6

96.5

97.3

93.8

80.5

73.4

72.8

94.5

96.9

97.5

93.3

80.4

74.1

72.1

97.3

97.1

97.6

93.7

80.7

73.6

72.2

97.4

95.9

97.0

83.2

77.8

72.2

97.2

96.6

97.1

83.0

74.9

72.3

97.0

97.0

97.2

95.0

81.5

73.8

72.3

Q

29

25

28

24

10

1

0

37

30

34

28

12

3

0

44

35

39

33

15

4

1

37

26

34

16

7

1

39

33

38

18

6

2

46

37

39

35

16

4

2

in tetrahedron

Or

70

23

40

36

22

13

11

58

26

38

33

20

12

10

43

26

33

29

18

11

9

33

9

26

17

9

9

26

17

24

16

10

9

44

26

29

27

17

10

9

Ab

1

45

31

32

39

43

43

5

38

27

29

37

43

43

13

34

25

27

36

42

43

27

54

35

39

45

43

30

42

32

37

42

42

9

32

28

29

36

41

43

An

0

7

1

8

30

43

46

0

5

1

10

31

43

47

0

5

2

10

31

44

47

3

12

5

28

38

47

5

9

6

29

42

47

0

5

4

8

31

44

47

Percent

melting

11.2

27.0

17.1

20.0

40.0

80.0

100.0

11.8

22.5

16.5

20.0

40.0

80.0

100.0

13.5

20.8

16.7

20.0

40.0

80.0

100.0

17.3

46.3

21.2

40.0

80.0

100.0

19.3

27.6

20.7

40.0

80.0

100.0

12.9

23.6

18.1

20.0

40.0

80.0

100.0

Percent mafic 

assemblages 

retained in ( + )

or precipitated

from ( ) melt

+22.8

+8.5

+8.3

+5.4

-14.6

-54.6

-74.6

+6.6

+7.2

+9.9

+6.4

-13.6

-53.6

-73.6

+ 15.2 -0.5

+4.2

+0.9

-19.1

-59.1

-79.1

+ 1.2

+ 16.3

+5.6

-13.2

-53.2

-73.2

-0.4

+4.0

+2.1

-17.2

-57.2

-77.2

+35.5

+7.5

+5.8

+4.0

-16.1

-56.1

-76.1

Aleutian Range reveal compositional structures that 

are relatively simple. More than 90 percent of the 

variance in each major oxide can be accounted for by 

mixing three end members in the case of the AlaskaAleutian Range, and four in the case of the Sierra 

Nevada. There is no purely mathematical means for 

determining what the end members were, although 

proposed end-member compositions can be tested and
COMPOSITIONAL STRUCTURES IN TWO BATHOLITHS OF CIRCUMPACIFIC NORTH AMERICA H29

possibly rejected. Testing consists of determining the 

compatibility of the proposed composition with the 

compositional series formed by the representative 

samples, through estimation of a vector communality, 

and through determination and consideration of the 

mixing proportions required when that end-member 

composition is used in combination with others to form 

the mixing model.

The major implication of the models for the 

batholiths is that both could have originated from 

magmas or magma-source materials with andesitic 

compositions. Moreover, the models indicate that 

modification of the magmas, or of the magma-source 

material, could have been accomplished almost entirely by the separation of mafic mineral assemblages 

similar in composition to the mafic inclusions that are 

scattered throughout most of the granitic rocks or to 

mafic phases within these rocks. The mafic mineral 

assemblages could have consisted of minerals 

precipitated from the magmas or of refractory 

minerals in the magma-source materials.

If the Alaska-Aleutian Range batholith originated 

from partial melting, the minimum degree of melting 

required by the model ranges from about 5.6 to 23.7 

percent for the 158 samples. The average sample could 

have formed with as little as 9.0 percent melting if 3.9 

percent of the refractory material were retained in the 

melt. The model for the Sierra Nevada batholith indicates that all but one of the individual 228 samples 

could have formed with melting in the range from 11 to 

27 percent. The average sample from each group of 

plutons could have formed with as little as 16.5 to 21.2 

percent melting if 2.1 to 9.9 percent of the refractory 

material were retained. In a general way, groups of 

plutons within the Alaska-Aleutian Range batholith 

distant from the continental margin could have 

resulted from lower degrees of melting than those 

closer to the margin. If data from experimental 

petrology are applicable, the melts that formed the 

eastermost plutons in the Sierra Nevada batholith 

may have undergone higher partial pressures of H2O 

than did those closer to the continental margin.

The compositional structures of the batholiths indicate that three end members can account for most of 

the compositional variation in each batholith, and that 

the addition of a fourth end member to the Sierra 

Nevada model can improve this model substantially. If 

the batholiths formed from starting materials from 

which materials of varying composition were 

separated, at least three end members are required. 

Processes of magma mixing and assimilation would require additional end members in the models, which 

would contradict the determined compositional structures. It is concluded that these processes and others

could not have been quantitatively important in causing compositional variation in either batholith.

The two models leave less than 10 percent of the 

variance in each oxide constituent unexplained. The remaining variance could be attributed to rock-forming 

processes not represented in the models, to compositional variations in the proposed end members, and to 

analytical imprecision. Those who may have little feeling for what this means in terms of original data versus data computed from the models may refer to table 

22. Table 22 shows the determined mixing proportions 

for every tenth sample from the Alaska-Aleutian 

Range, the compositions reported by Reed and Lanphere (1974) (recalculated to sum to 100 percent), and 

the compositions computed for the indicated mixture 

of end members. A similar comparison of original and 

model data for the Sierra Nevada batholith shows even 

better agreement than indicated in table 22 because, as 

shown in table 5, all of the variances not accounted for 

by the Sierra Nevada four-end-member model are considerably smaller than those for the Alaska-Aleutian 

Range three-end-member model.

It has been assumed that both batholiths are of 

magmatic origin and that the compositional variations 

in each were brought about by differential separation 

of mafic mineral assemblages. This has led to 

hypothetical end-member compositions which might 

have been involved in the formation of the two 

batholiths. Although speculative, each model is one example that shows how knowledge of the compositional 

structures can serve to form and test the quantitative 

aspects of petrologic models. The primary intent of 

this report, however, has been to present the concept of 

compositional structure and to show the relatively 

simple compositional structures of the two batholiths 

of circumpacific North America. If the models can 

withstand the critical scrutiny of theoretical 

petrologists and other geologists who have studied the 

Sierra Nevada and the Alaska-Aleutian Range, they 

show that each batholith could have formed by a 

rather simple combination of processes. Even though 

the models may be rejected on valid grounds by other 

investigators, alternative models will have to be reconciled with the fact that both batholiths have simple 

compositional structures that call for three principal 

end members in any model that might be proposed.

APPENDIX

The preliminary Q-mode factor analyses of the 

chemical data for both batholiths were made with the 

CABFAC computer program of Klovan and Imbrie 

(1971) as extended for treatment of compositional data 

by Klovan and Miesch (1976). The extended CABFAC 

program provided the data for the construction of the
H30 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

TABLE 22. Mixing proportions, original data, and corresponding data generated from the Alaska-Aleutian Range batholith model for

selected samples 

[Original data are from Reed and Lanphere (1974, table 1) although total iron has been recomputed as 'FeO' and the seven major oxides have been recalculated to sum to 100]

Sample 

No.

10

20 

30 

40 

50 

60 

70

Mixing proportions'

MGM

9.9150 

11.0061 

5.4280 

4.9029 

5.1199 

4.9021 

3.4817

ABH

-5.7953 

-6.6850 

-2.5472 

-2.4060 

-2.3995 

-2.2687 

-1.6053

ANH

-3.1198 

-3.3211 

-1.8808 

-1.4969 

-1.7204 

-1.6334 

-0.8764

Model ...........

Model ...........

Original .........

Model ...........

Original .........

Model ...........

Model ...........

SiO2

74.46

74.25

75.11

75.06

67.13

66.99

64.15

63.87

66.13

65.78

63.67

65.17

59.10

59.41

Chemical compositions (weight percent)

A1 2O3

13.80

14.52

13.36 

14.00

17.19 

17.04

17.63 

17.53

17.55 

17.27

18.34 

17.42

17.79 

18.52

'FeO'

2.35 

1.72

2.01 

1.66

3.70 

3.68

5.00 

4.91

4.17 

4.13

5.32 

4.34

7.16 

6.48

MgO

0.38 

.24

.40 

.20

1.54 

1.56

2.36 

2.39

1.43 

1.87

1.34 

2.01

3.48 

3.45

CaO

1.63 

1.42

1.42 

1.02

3.99 

4.23

4.92 

5.23

4.59 

4.63

4.22 

4.84

6.95 

6.75

Na2O

4.02 

3.99

3.75 

3.68

4.50 

4.51

4.10 

4.13

4.29 

4.41

4.74 

4.38

4.09 

3.87

K2 0

3.35 

3.71

3.95 

4.22

1.94 

1.92

1.84 

1.87

1.84 

1.84

2.37 

1.78

1.43 

1.46

'See table 12 for compositions of end members MGM, ABH, and ANH.

factor-variance diagrams in figures 3-5 and 7-8 and 

produced output files containing other selected results 

from each Q-mode analysis. All further computations 

were performed using the interactive programs 

described in Miesch (1976c). The output file from the 

extended CABFAC program was transformed to 

another output file with program EQTRAN. The 

transformed output file was then used as input to the 

other interactive programs.

Program EQSPIN was used to derive the 

stereograms in figures 9 and 10, and program 

EQSTER was used to determine the compositions 

represented by vectors at various positions on the 

stereograms. (See, for example, tables 8 and 9.) Program EQEXAM was used to examine individual compositions, such as the compositions of selected 

minerals, for compatibility with the compositional 

series formed by the sample compositions when 

represented in three- or four-dimensional vector 

systems; compatibility is indicated by the projected 

vector communality. The compositional system formed by hornblende-albite-anorthite-magnetite (fig. 11) 

was examined for each batholith with program 

EQSCAN. After the points Y and Z were located on 

the stereograms of figures 9 and 10, point MGM on 

each stereogram was determined with program INSECT. The compositions of the end members and the 

corresponding CIPW norms (tables 11, 12, and 15) 

were determined with program EQMAIN. EQMAIN 

also provided the mixing proportions (composition 

loadings) summarized in tables 13, 14, and 16. The

final checks on the computations, as represented in 

table 22, were made with program EQCHEK.

Examination of the ranges of possible melt compositions called for by the models (figures 13 and 14; tables 

20 and 21) was made with other computer programs 

that have not been published. The methods of computation, however, are fully outlined in table 19.

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