Evaluating Coinciding Anomalies Along a 

Fault Trace or Other Traverse~ 

Sim.ulations and Statistical Procedures 

U.S. GEOLOGICAL SURVEY BULLETIN 1802 

Evaluating Coinciding Anomalies Along a 

Fault Trace or Other TraverseSimulations and Statistical Procedures 

By RUSSELL L. WHEELER and KATHERINE B. KRYSTINIK .,, 

Methods for evaluating spatial associations of anomalies in several 

different types of data along a linear or curvilinear traverse to 

determine whether the anomalies coincide by chance 

U.S. GEOLOGICAL SURVEY BULLETIN 1802 
DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1987 

For sale by the 

Books and Open-File Reports Section 

U.S. Geological Survey 

Federal Center 

Box 25425 

Denver, CO 80225 

library of Congress Cataloging-in-Publication Data 

Wheeler, Russell L. 

Evaluating Coinciding Anomalies Along a Fault Trace or Other Traverse-Simulations 

and Statistical Procedures. 

(U.S. Geological Survey Bulletin ; 1802) 

"Methods for evaluating spatial associations of anomalies in several different types 

of data along a linear or curvilinear traverse to determine whether the anomalies 

coincide by chance." 

Bibliography: p. 

Supt. of Docs. no.: I 19.3:1802 

1. Geology-Statistical methods. 2. Spatial analysis (Statistics) I. Krystinik, Katherine 

B. II. Title. Ill. Series. 

QE75.B9 no. 1802 557.3 s 87-600288 

[QE33.2.M3] [550' .72] 
CONTENTS 

Abstract 1 

Introduction 1 

Coincident anomalies 2 

Associated variables and 1 accard coefficients 5 

Simulations and anomalous sections of the traverse 6 

Randomization tests 7 

Two-dimensional case 7 

Three-dimensional case 9 

Summary 12 

Acknowledgments 12 

References cited 12 

FIGURES 

1. Map of hypothetical study area 1 

2. Locations of several anomalies in one data type 3 

3. Identification of coincident anomalies 3 

4. Hypothetical anomaly pattern from four data types 4 

5. Use of Jaccard coefficient (J) 6 

6. Numbers of pairs, triplets, and quadruplets of coincident anomalies 7 

7. Numbers of coincident anomalies 8 

8. Orthogonal coordinate system for three-dimensional case 10 

9. Serial section 11 

10. Intercepts and slope of trace of S' in serial section 11 

TABLES 

1. Symbols used in representation of anomalies 3 

2. Descriptions of anomalies illustrated in figure 4 4 

3. Calculation of 1 accard coefficients 6 

4. Symbols used for three-dimensional case 10 

Contents iii 

Evaluating Coinciding Anomalies Along a Fault Trace or 

Other Traverse-Simulations and Statistical Procedures 

By Russell L. Wheeler and Katherine B. Krystinik 

Abstract 

If two or more types of data are mapped along the same 

traverse, such as a fault trace, anomalies in different data types 

can coincide. If the anomalies coincide because they have a 

common geological cause, the spatial coincidences are worth 

interpreting to understand that cause. Alternatively, the anomalies might coincide by chance, so that genetic interpretations 

of the observed patterns of anomalies would be misguided. 

Opinion is an unreliable guide for deciding which is the case. 

Numerical simulations and statistical tests aid in this decision. Each simulation produces a random scattering of the 

observed anomalies along the traverse and an anomaly pattern that resembles, to some degree, the one observed. Randomization tests are used to compare the observed pattern of 

anomalies to the collection of simulated patterns, in terms of 

numbers of triplets and larger groups of coincident anomalies. 

Test results show whether the anomalies of the observed pattern coincide more than should be attributed to chance. If so, 

the group of coincident anomalies is worth interpreting. 

Jaccard coefficients detect those data types that tend to 

have anomalies together along the traverse. Two such highly 

associated data types can be combined. Together they might 

detect the occurrence of whatever causes the coincident 

anomalies more effectively than can either data type alone. 

INTRODUCTION 

A problem that is commonly encountered in the 

earth sciences is the evaluation of spatial associations between features on maps of two or more kinds of data that 

are collected over the same study area. For features that 

are represented on the maps as points, geographers and 

statisticians have developed many procedures for detecting and evaluating spatial associations (for example, 

Lewis, 1977; Ripley, 1981). The problem becomes successively more complex if the mapped features are represented as lines, as patches of similar sizes and simple, 

similar shapes, or as irregular areas of dissimilar sizes and 

shapes. Figure 1 illustrates this last case, which is a common one in geology. Anomalies in all three data types 

appear to coincide in the upper left part of the study area. 

Pairs of anomalies appear to coincide at two places on 

traverse A -A I • Are these spatial coincidences worth interpreting, or should they be dismissed as chance overlaps 

of unrelated anomalies in different types of data? 

The general problem has four parts: (1) defining 

anomalies, (2) identifying coincident anomalies, (3) determining whether the coincident anomalies are likely to 

have coincided by chance instead of from some common 

cause, and (4) interpreting such coincidences. Solving 

each part requires solving all the preceding parts. The last 

part is usually the most fun, which might explain the common tendency to jump to it without explaining how (or 

if) the first three parts of the problem were solved. The 

resulting arguments are often more exciting than informative, perhaps because they might root in unrecognized 

disagreements about the first three parts of the problem. 

The evaluation of earthquake hazards encounters 

a simplified version of this problem. Instead of a map 

area, we have the trace of a fault on the Earth's surface, 

for example traverse A-A I in figure 1. Data were 

A 

Figure 1. Map made by superimposing three hypothetical maps 

of the same study area. Each superimposed map shows spatial 

distribution of a different type of geological, geophysical, or 

other earth-science data. Each data type has anomalies in one 

or more parts of ~udy area. Solid, dashed, and dashed-dotted 

lines outline anomalies in various data types, one line type per 

superimposed map. A-A' is traverse discussed in text. 

Introduction 
collected or measured along the trace, or within a narrow strip around it. The problem and its four parts are 

unchanged. 

For anomalies that might coincide along a study 

traverse like A-A' in figure 1, we present a general procedure for solving the second and third parts of the problem. Most such traverses will be along fault traces, but 

the procedure can be used along any kind of linear or 

curvilinear traverse. 

We will not treat the first part of the problem, that 

of defining the anomalies, because the methods for doing that depend on the types of data to be analyzed, on 

the ways in which the data are represented or summarized on maps, and on the goals of the analysis. For example, this report emerged from an attempt to identify 

places along the Wasatch fault zone in Utah where the 

fault zone might be segmented into lengths that tend to 

rupture independently of each other (Schwartz and Coppersmith, 1984; Wheeler, 1984; Machette and others, 

1986). The goal was to detect boundaries between 

segments. The approach was to examine kinds of data 

that ought to record the presence of a segment boundary 

as anomalies of predictable sorts. This goal imposed conditions on the kinds of data to be analyzed and on the 

kinds of anomalies to be sought in each data type. 

Wheeler and Krystinik (1987a, b, in press) used anomalies 

along the north-striking fault zone that include easttrending gravity and aeromagnetic gradients and large 

cross faults, and changes along the fault zone in abundance of earthquake epicenters, in the geometry of the 

fault zone, and in the height and width of the upthrown 

fault block. Because many anomalies were identified subjectively by inspection of maps of points, contour maps, 

and geologic maps, the anomalies had to be shown to be 

reliable, for example by being recognized by two or more 

independent and competent workers. Wheeler and 

Krystinik (in press) described how the data and anomalies 

from the Wasatch fault zone satisfied these conditions. 

Peculiarities of other investigations will impose their own 

conditions on data and anomalies. 

The second part of the problem, that of identifying coincident anomalies, requires specifying objective 

rules for identification. We did this by developing a procedure for identification of coincident anomalies, programming the procedure, and testing the program on 

artificial data. 

The third part of the problem, determining whether 

the observed coincidences of anomalies are or are not 

likely to have occurred by chance, is a statistical effort. 

If anomalies are randomly scattered, the exact form of 

the distribution that underlies the numbers of coincident 

anomalies is unknown, so we solve this part of the problem with simulations and statistical tests of the simulated 

anomaly patterns. (We describe the simulation and 

statistical procedures in detail because this report is 

2 Evaluating Coinciding Anomalies 

directed toward geophysicists and quantitatively oriented 

geologists, many of whom have little training in statistics.) 

The statistical nature of this part of the problem imposes 

two more conditions on the anomalies (Wheeler and 

Krystinik, in press). Some version of these two conditions 

will apply to any investigation of any types of data that 

can be represented as in figure 1. The anomalies in any 

one data type had to be identified without reference to 

any other data type used in the analysis, and also without 

reference to the segment boundaries that Schwartz and 

Coppersmith (1984) suggested in the report that prompted 

our investigation. 

The fourth part of the problem is that of interpreting the group of coincident anomalies that is unlikely to be an artifact of chance. For the Wasatch fault zone, 

Wheeler and Krystinik (1987a, b) applied the methods of 

this report to the data of Wheeler and Krystinik (in press) 

and interpreted the results. 

COl NCI DE NT ANOMALIES 

For each data type, the presence and absence of 

anomalies along the length of the fault trace or other 

study traverse are summarized graphically. In an example (fig. 2), three anomalies of different widths are located 

where variable v(i) has a value of 1. Along the other four 

sections of the traverse a value of 0 for v(i) records the 

absence of any anomaly in this data type. 

We need a rule to determine when anomalies coincide. We define two anomalies as coincident if either 

anomaly contains the center of the other (fig. 3). For example, the two anomalies labelled 1 in figure 3 coincide 

by this rule, because the wide anomaly in v(2) includes 

the center of the narrower anomaly in v(1). This example illustrates why the rule does not require each anomaly to contain the center of the other. The narrow anomaly 

in v(1) does not contain the center of the wide anomaly 

in v(2), but these anomalies coincide in any geologically 

reasonable sense of the word. The second two anomalies 

coincide because each contains the center of the other. 

The third two anomalies do not coincide because neither 

contains the center of the other. Finally, two anomalies 

in the same variable cannot coincide because they cannot merge or overlap without becoming a single anomaly. 

We use a hypothetical pattern of anomalies in four 

data types (fig. 4) to illustrate procedures throughout this 

report. Each data type contains from one to six anomalies, which vary in width and separation. The 17 anomalies have a median width of 3 km, and range from 1 to 

6 km in width except the single 30-km wide anomaly (K) 

in data type 3 (table 2). Depending on the data type, from 

70 to 83 percent of the traverse's length lacks anomalies. 

Pairs of coincident anomalies are found at four places 

along the traverse (fig. 4). At about d = 32 three types 
NORTH 

0 .... 

d 

D 0 

SOUTH 

I 

v(i) 

Figure 2. Sketch showing locations 

of several anomalies in one data type 

(such as along a fault trace or other 

traverse of a study area). Table 1 defines symbols. 

of data appear to be anomalous, and at about d = 55 all 

four types appear to be anomalous. 

A triplet of coincident anomalies can be identified 

by extending the rule for coincident pairs in this manner: if two anomalies in different variables coincide, a 

third coincides with both if it coincides with each. The 

extended rule has several consequences that will form the 

bases of later analyses. For the three anomalies at d = 

32 (fig. 4), there are three possible pairs of coincident 

anomalies, and all three coincident pairs exist. The 

anomaly (C) in v(l) spans km 30-35, (H) in v(2) spans 

Table 1. Symbols used in representation of anomalies 

Symbol Definition 

n 

d 

Number of data types to be examined 

for anomalies. 

Distance along traverse in km, from 

d = 0 km at north end of traverse 

to d = D km at south end. 

v(i) Binary variable that can take 

values of 0 or 1 along traverse. 

In lengths of traverse that cross 

an anomaly in data type i, where 

i=1, ... , n, v(i)=1. In lengths 

between anomalies in data type 

NORTH 

0 

d 

D ....._ 

SOUTH 

i, v(i)=O. Where v(i)=1, v(i) is 

said to have an anomaly or to be 

anomalous. 

v(1) v(2) 

0 0 0 I I 

2 

__ j2 ___. 

d 

3 

0~.-... 

Figure 3. Sketch illustrating identification of coincident anomalies. Table 1 defines symbols. Two 

types of data occur along traverse, and anomalies 

in them are located by values of v(1) and v(2). 

Numerals identify pairs of anomalies that fall at 

about the same places along traverse. 

Coincident Anomalies 3 
NORTH 

0 0 

20 

30 

40 

d 50 

60 

70 

80 

90 

D 100 

SOUTH 

v(1) 

0 1 

A 

8 

c 

0 

E 

v(2) 

0 1 

F 

G 

H 

J 

v(3) 

0 1 

K 

Variables that contain 

coincident anomalies 

v(4) 

0 1 

L 

1&2 

1&2 

M 1,2,&4 

N 3&4 

0 1, 2, 3, &4 

p 

Q 1&4 

Figure 4. Hypothetical anomaly pattern in four data types along 

a traverse. Table 1 defines symbols. Data of types 1-4 have 

17 anomalies along length of traverse. Anomalies are where 

variables v(1) to v(4) have values of 1, by the convention illustrated in figure 2. Letter to right of each anomaly keys it to 

table 2. Column at right shows where and which anomalies 

coincide. 

km 28-33, and (M) in v(4) sp~ns km 30-33; thus all three 

span and have centers within km 30-33 (table 2). The first 

two anomalies contain each others' centers, and so define 

a coincident pair. The anomaly in v(4) coincides with each 

of the other two anomalies. Thus, the three anomalies 

satisfy the extended rule that defines a triplet. However, 

suppose that the anomaly in v(l) were one km farther 

south, so that it spanned km 31-36. Its center would lie 

at d = 33.5. The anomaly in v(2) centers at km 30.5, so 

neither anomaly would contain the center of the other. 

The anomalies in v(l) and v(2) would no longer coincide. 

The triplet would then degenerate into two pairs, one linking v(2) and v(4), and another linking v(l) and v(4). 

Identifying a quadruplet or larger coincidence of 

anomalies requires a more general rule, which also applies to triplets. For m (the number of coincident 

anomalies) greater than 2, an m-tuplet of coincident 

anomalies exists if and only if all possible coincident pairs 

4 Evaluating Coinciding Anomalies 

Table 2. Descriptions of anomalies illustrated in figure 4 

Distance from 0 (km) 

Anomaly Ends Center 

A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

p 

Q 

2-5 

8-12 

30-35 

52-56 

80-83 

2-7 

8-11 

28-33 

54-57 

90-94 

v ( 1 ) 

3.5 

10 

32.5 

54 

81.5 

v(2) 

4.5 

9.5 

30.5 

55.5 

92 

v(3) 

40-70 55 

15-17 

30-33 

44-46 

51-57 

73-74 

80-83 

v(4) 

16 

31.5 

45 

54 

73.5 

81.5 

Width 

(km) 

3 

4 

5 

4 

3 

5 

3 

5 

3 

4 

30 

2 

3 

2 

6 

1 

3 

Coincident 

with 

F 

G 

H,M 

I,K,O 

Q 

A 

B 

C,M 

D,K,O 

N,D,I,O 

C,H 

K 

D,I,K 

E 

of its component anomalies also exist. The number of 

possible coincident pairs is the number of ways that m 

anomalies can be chosen two at a time, or m(m-1)/2. A 

triplet requires three coincident pairs, a quadruplet requires six, and a quintuplet requires 10. This more general 

rule shows that the four anomalies at km 55 of figure 

4 do define a quadruplet. Six coincident pairs create the 

quadruplet. However, if the anomaly in v(2) at d = 55 

were one km farther south, then the anomalies in v(l) 

and v(2) would no longer coincide. The quadruplet would 

degenerate into two triplets, one linking v(l), v(3), and 

v(4), and another linking v(2), v(3), and v(4). Each of 

these triplets would contain three coincident pairs of 

anomalies, and so would not degenerate. 

This extended rule for m-tuplets avoids a phenomenon that we call chaining. Chaining is illustrated by 
considering the hypothetical triplet that degenerated into two pairs. If a pair links v(2) and v(4), and if a second 

pair links v(1) and v(4), then a chain of two pairs links 

all three anomalies in these variables. However, the 

anomalies in v(l) and v(2) do not coincide. If the anomaly 

in v(4) were wide enough, the anomalies in v(l) and v(2) 

might even be several km apart. To accept such chained 

anomalies as a triplet would blur the idea of coincident 

anomalies beyond usefulness. 

On cursory inspection of figure 4, v(1) and v(2) appear to be associated variables in the sense that most of 

their anomalies coincide. Variable 3 appears least associated with most other variables. However, the single 

anomaly of v(3) is part of the only quadruplet of coincident anomalies. 

How much of the pattern described in the preceding 

paragraphs reflects some underlying structure that is 

worth trying to interpret, and how much should be 

dismissed as arising from chance? Two aspects of coincident anomalies are of interest: individual places on the 

traverse where unusually many anomalies coincide 

(anomalous sections of the traverse), and variables that 

are highly associated over the whole length of the traverse 

(associated variables). 

Anomalous sections of the traverse differ in some 

way from adjacent sections. What this difference might 

be depends on what phenomenon the data were chosen 

to reveal. For the Wasatch fault zone, data were chosen 

to reveal segment boundaries, so anomalous sections of 

the fault zone could be segment boundaries. Highly 

associated variables might be more effective together than 

are single variables at identifying locations of the 

phenomenon that is under study. For example, if cross 

faults are thought to have localized mafic intrusions and 

volcanic rocks in an area, then coincident gravity and 

magnetic highs could identify hidden cross faults better 

than could either gravity or magnetic data alone. 

ASSOCIATED VARIABLES AND 

JACCARD COEFFICIENTS 

Associated variables are those that tend to have 

anomalies and lack anomalies together along the traverse, 

so that most of their anomalies coincide. Examples are 

v(l) and v(2) in figure 4. Association is observed, not inferred, and for now we make no genetic interpretations 

of any observed association. In contrast, unassociated 

variables are those that tend to have and lack anomalies 

without regard to each other. For example, in figure 4, 

v(3) appears to be relatively unassociated with v(1) and 

v(2). 

The degree of association between pairs of variables 

that can take only values of 0 or 1 is measured by J, the 

Jaccard coefficient (Cheetham and Hazel, 1969). J 

measures the fraction of the anomalies that coincide. In 

a notation similar to that of Cheetham and Hazel (1969, 

p. 1131) J is defined as C/N(t). Cis the number of coincident anomalies in each variable, that is, the number of 

sections of the traverse in which both variables have 

anomalies together. N(t) is the total number of anomalies 

present in both variables together, with each pair of coincident anomalies counted only once. That is, N(t) is the 

number of sections of the traverse where one or more 

anomalies are present. N(t) is calculated as N(i) + N(j)-

C, where N(i) and N(j) are the numbers of anomalies in 

variables i and j. For example, variables 1 and 2 of figure 

4 have five anomalies each, and four of them coincide. 

Therefore, C = 4, N(t) = 6, and J = 0.67. For most applications J = 0 for variables that have no coincident 

anomalies, and J = 1 for variables whose anomalies coincide completely (Cheetham and Hazel, 1969). Values of 

J that are near 0 characterize relatively less associated 

variables, and values near 1 characterize relatively more 

associated variables. 

In most applications of the Jaccard coefficient, J 

cannot exceed 1. Then, by the definition of J, C cannot 

exceed the average of N(i) and N(j). In this application, 

if all anomalies in v(i) and v(j) are about equally wide, 

then probably no anomaly will be wide enough to coincide with more than one other anomaly. Then C is unlikely to exceed the average of N(i) and N(j), and J is unlikely 

to exceed 1. However, if some anomalies are much 

wider than others, then a wide anomaly might coincide 

with more than one narrow anomaly, so that C increases 

and J might exceed 1. For an example, consider v(3) and 

v(4) and only kilometers 40-70 of figure 4. The wide 

anomaly K in v(3) coincides with both narrow anomalies 

Nand 0 in v(4). In this case N(3)= 1, N(4)= 2, and C= 

2. C exceeds the average of N(3) and N(4), and J = 

2/(1 +2-2)= 2. However, the effect of the wide anomaly would be diluted if the entire traverse were considered. 

Then N(4) would increase to 6, so that C = 2 would 

no longer exceed the average of N(3) and N(4), and J 

would decrease to 0.40. Thus, J can exceed 1 if some 

anomalies are much wider than others, but such large 

values of J will be few if wide anomalies are few. In any 

case, large values of J will still identify highly associated 

variables. 

We use J to examine the structure of the anomaly 

pattern of figure 4 (fig. 5, table 3). Variables 1 and 2 appear to be the most strongly associated (fig. 4), and the 

point (1 ,2) that represents this pair plots in the right part 

of figure 5 as it should. Variable 3 is comparatively 

unassociated with variables 1 and 2 (fig. 4). Accordingly, 

variable pairs involving v(3) should plot in the left part 

of figure 5 as they do. Plotting J in figure 5 appears to 

separate pairs of variables in ways that identify relatively more associated and relatively less associated variables. 

Associated Variables and Jaccard Coefficients 5 
I I I 

e(2,4) 

•(2, 3) •(3,4) 

e(1, 3) • (1' 4) •(1,2) 

I I I 

0.00 0.25 0.50 0.75 1.00 

j 

Figure 5. Illustration of use of jaccard coefficient (J). 

Data are from figure 4. Calculations are summarized 

in table 3. Numerals to right of plotted points identify 

pairs of variables that produce each point. Vertical axis 

has no meaning and serves only to separate points for 

legibility. 

Table 3. Calculation of jaccard coefficients to produce 

figure 5 from figure 4 

[C, number of coincident anomalies in a pair of variables; N(t), 

number of anomalies in both variables together, with coincident 

anomalies counted once; J, Jaccard coefficient, C/N(t)] 

Variables c N(t) J 

paired 

and 2 4 6 0.67 

and 3 1 5 .20 

and 4 3 8 .38 

2 and 3 1 5 .20 

2 and 4 2 9 .22 

3 and 4 2 5 .40 

SIMULATIONS AND ANOMALOUS SECTIONS 

OF THE TRAVERSE 

Simulations can help to identify sections of the 

traverse where anomalies in unusually many variables 

coincide. As explained earlier, the observed anomalies are 

identified and their widths determined by methods that 

depend on the data and goals of the investigation. 

However, once this is done, the number of anomalies in 

each data type and their widths are fixed. A simulation 

is performed by taking the observed number and widths 

of anomalies in each data type and placing the anomalies 

randomly along the traverse. The observed anomaly pattern is unlikely to have arisen by chance if some specified 

aspect of the pattern, such as a large number of coincident pairs, is found in the observed pattern but only in 

a small percentage of the simulated patterns. Any aspect 

of the patterns can be specified as the basis for comparing observed and simulated patterns. In this section we 

6 Evaluating Coinciding Anomalies 

examine separately the number of pairs, the number of 

triplets, and the number of quadruplets of coincident 

anomalies. In a later section we shall compare observed 

to simulated patterns on the basis of all three numbers 

considered together. The basis for comparison must be 

specified before the observed and simulated patterns are 

examined to avoid distortion of results by prior inspection (Freedman and others, 1978, p. 494; Moore, 1979, 

p. 294-295; Wheeler, 1985). 

The number and widths of anomalies in the observed pattern for each data type are held constant in the 

simulations. Only the locations of the observed anomalies 

are varied. The number of anomalies is held constant 

because each data type is measured reliably enough and 

its coverage is complete enough, that probably no 

anomalies have been missed. In terms of the variables v(i), 

this means that a value of 0 is as reliable as a value of 

1. In fact, for any variable the locations of the O's along 

the fault give exactly the same information as do the locations of the 1 's, because either set of locations can be used 

to generate the other set. Anomaly widths are held constant because the edges of most anomalies are located 

with an uncertainty that is small with respect to the width 

of a typical anomaly, and small with respect to the variation in anomaly widths. Wheeler and Krystinik (in press) 

described how the data from the Wasatch fault zone meet 

these conditions of data quality. 

As an example of this approach, 20 simulations 

were done using the data of figure 4. Randomly located 

anomalies were allowed to overlap the ends of the 

traverse, but could not overlap each other. The result was 

20 anomaly patterns, each resembling figure 4 to some 

degree. Whether the observed anomaly pattern has arisen 

by chance can be determined by comparison to the 20 

simulated patterns. Twenty simulations are too few to 

give conclusive results, but are enough to illustrate the 

procedure. In practice, many more simulations than 20 

would be used. For example, for the Wasatch fault zone 

we used 300 simulations. The number of simulations that 

is needed can be chosen by an experienced statistician. 

Alternatively, the number of simulations used can be increased gradually, say in increments of 50 simulations, 

until results stabilize and cease changing much with each 

added increment. 

The pattern that is observed in figure 4 has no more 

pairs or triplets of coincident anomalies than do many 

of the simulations (fig. 6). Despite the striking appearance 

of the triplet of figure 4, it would be unwise to spend 

much time trying to interpret it. The observed triplet is 

about as likely to reflect random processes as it is to 

betray the existence of any common cause of the three 

coincident anomalies. However, the observed pattern appears to be unusual in having a quadruplet. Only one of 

the 20 simulations has such a quadruplet. The observed 

quadruplet might merit investigation. 
~ w 

...J 

a.. 

=> 

a: 

0 

<! 

=> 

0 

A 

~ B 

0 2 3 4 5 6 7 

NUMBER OF PAIRS OF COINCIDENT ANOMALIES 

u5 

1-

w 

...J 

a.. 

a: 

1-

uj 

a: 

~ 

'J~L (!) 

z 

~ :I: 

(/) 

z 

0 

...J 

~ 

=> 

~ 

U5 

LL. 

0 

a: 

w 

CD 

~ 

=> 

z 

c 

20 

15 

10 

5 

0 

0 1 2 3 4 5 

NUMBER OF TRIPLETS OF 

COINCIDENT ANOMALIES 

* 

0 2 3 4 5 

NUMBER OF QUADRUPLETS OF 

COINCIDENT ANOMALIES 

10 

Figure 6. Histograms of numbers of pairs (A), triplets (8), and 

quadruplets (C) of coincident anomalies that occur in anomaly 

pattern of figure 4 (shown by asterisks) and in 20 randomized 

simulations of that pattern. Anomaly pattern of figure 4 has four 

pairs, one triplet, and one quadruplet. Thus, four simulated patterns contain same number of pairs (4) as does hypothetical observed pattern offigure 4. Four simulated patterns (not necessarily the 

same four) contain same number of triplets (1) as does figure 4. 

Only one simulation produced same number of quadruplets (1) 

as is observed. 

RANDOMIZATION TESTS quadruplets and triplets of coincident anomalies than one 

would expect to occur by chance. Note that whole patterns are to be compared, not individual coincidences of 

anomalies. Interpretation of an individual triplet or 

quadruplet would require information that is not included 

in this randomization test. 

Two-Dimensional Case 

Triplets and quadruplets are of more interest than 

are pairs, because we seek places along the fault where 

several anomalies coincide. We do this because two randomly located anomalies are more likely to coincide than 

are three or four. A two-dimensional graph like that of 

figure 7 allows triplets and quadruplets to be examined 

together, instead of separately as with figure 6. 

We examine the graph of figure 7 with a randomization test (Siegel, 1956; Conover, 1971; permutation test 

of Mosteller and Rourke, 1973) to determine whether the 

observed pattern of anomalies (fig. 4) contains more 

To avoid distortion of the test result by prior inspection, the test and testable hypothesis must be formulated before examining the data. The population 

consists of--the 20 randomized simulations of the observed 

anomaly pattern. The sample is the observed pattern. The 

goal of this randomization test is to tell whether the coincident anomalies in the sample are likely to have arisen 

from the same random processes that produced the 

population. 

Randomization Tests 7 
3 

~ ...J 

~2 

a: 

ILL. 

0 

a: 

w 

~ 1 

:::::> 

z 

0 

1 

8 

4 

6 

0 

* 

' I 

\ 

2 3 

NUMBER OF QUADRUPLETS 

Figure 7. Numbers of coincident 

anomalies counted in anomaly pattern of 

figure 4 (shown by asterisk) and in 20 randomized simulations of that pattern. Pattern of figure 4 has one place along 

traverse where anomalies in three variables 

coincide and one place where anomalies 

in four variables coincide. Dashed line has 

a slope of -t/q5

, where t

5 is total number 

of triplets in all 20 simulations together and 

q5 is total number of quadruplets. Dashed 

line is drawn to pass through the center of 

cell that contains asterisk and is used to 

determine whether asterisk plots significantly farther to upper right than do 

simulated patterns. 

The alternative hypothesis, H(a), is one-sided, and 

states that the observed pattern was drawn from a population of patterns that typically have more triplets and 

quadruplets than occur in the simulated patterns. The null 

hypothesis, H(o), states that typical patterns in the 

population that yielded the observed pattern do not contain more triplets and quadruplets than do the simulated 

patterns. 

The randomization test will produce a P-value 

(Moore, 1979; descriptive level of significance of 

Mosteller and Rourke, 1973; associated probability of 

Siegel, 1956, and Gibbons, 1976). The P-value is the probability of getting a simulated pattern at least as extreme 

as the observed pattern if H(o) is true. The extremeness 

of a pattern is defined as how far the pattern plots toward 

the upper right of figure 7. We will need a quantitative 

definition of "toward the upper right": in a direction 

perpendicular to the straight, dashed line in figure 7. To 

allow comparison to the observed pattern, the dashed line 

passes through the center of the cell that contains the 

asterisk. The dashed line has an orientation that reflects 

the approximate shape of the cloud of points from the 

simulated patterns. 

8 Evaluating Coinciding Anomalies 

Use of a straight line to approximate the shape of 

the point cloud in figure 7 requires justification, because 

this approximation influences the P-value of the randomization test. The choice of the slope of the dashed 

line also needs justification, because the slope influences 

the P-value. The purpose of the randomization test is to 

compare the observed anomaly pattern to the collection 

of simulated patterns, by comparing numbers of coincident anomalies (here, triplets and quadruplets). The 

number of triplets and quadruplets in the collection of 

simulated patterns must be summarized in some way in 

figure 7. 

For the following reasons, the simplest though not 

necessarily optimal such summary is a straight line that 

slopes steeply down to the right or is vertical. First, random processes are more likely to cause anomalies in three 

variables to coincide than anomalies in four variables, 

so many simulations will have more triplets than 

quadruplets. Second, random processes are unlikely to 

produce a simulation with no triplets and no quadruplets, 

so comparatively few simulations will plot in the lower 

left corner of figure 7. Third, random processes are also 

unlikely to produce a simulation with many triplets and 

many quadruplets, so comparatively few simulations will 

plot in the central and upper right parts of figure 7. 

Fourth, the number of anomalies is fixed, so an increase 

in the number of triplets in a simulated pattern will usually mean a decrease in the number of quadruplets in that 

pattern. The result of these four reasons will usually be 

an elongated cloud of numerical entries in figure 7, sloping steeply down to the right and approaching or reaching 

the vertical in some cases. This elongated cloud of entries expresses the numbers of triplets and quadruplets 

in the various individual simulations. Because this cloud 

of entries is elongated, the simplest summary of its center 

is a straight line. The dashed line is drawn parallel to this 

center line to smooth out vagaries in the proportions of 

triplets and quadruplets in the individual simulations. The 

dashed line is offset to the right from the center line to 

pass through the center of the cell that contains the 

asterisk. 

The optimal way to determine the slope of the 

dashed line is unknown, because the distribution that 

underlies the numbers of triplets and quadruplets in the 

simulations is unknown. However, a simple and straightforward expression for the slope is the ratio of the expected values of the numbers of triplets and quadruplets 

in the simulations, or E(t) and E(q), respectively. There 

are 20 simulations, which together have ts triplets and qs 

quadruplets. Therefore, the slope of the dashed line is 

-E(t)/E(q), which is estimated by -(t/20)/(q/20) = -t/ 

q = -23. s 

The P-value is calculated as N(e)/N(p), where N(e) 

is the number of simulated patterns at least as extreme 

as the one observed and N(p) is the total number of 
simulated patterns. For the example of figure 7, 

N(p) = 20. No simulation plots on or to the upper right 

of the dashed line (the steeply sloping dashed line passes 

slightly to the right of the cell [1 ,0]). The P-value is 

0/20 = 0. P-values from as few as 20 simulations can be 

unstable. If this P-value had arisen from many more than 

20 simulations, the observed pattern would be concluded 

to have significantly more triplets and quadruplets than 

do the simulations. Then the triplet and quadruplet 

together could be interpreted as deserving further 

investigation. 

Three-Dimensional Case 

The example of figure 4 has four variables, so figure 

7 ignores pairs and plots triplets against quadruplets. If 

there were five variables, we would need to replace figure 

7 with a three-dimensional graph of triplets, quadruplets, 

and quintuplets against each other, again ignoring pairs. 

If there were six variables, we could ignore triplets as well 

as pairs and graph quadruplets, quintuplets, and sextuplets against each other. Alternatively, if the simulations for six variables contained few triplets, we might 

wish to consider them. If triplets are not abundant in the 

simulations, then sextuplets probably will be rare or absent. Then we could plot triplets, quadruplets, and quintuplets against each other, and could consider the few 

simulated patterns with sextuplets by adding them to the 

count for N(e). In any case, the replacement for figure 

7 is unlikely to need more than three dimensions. The 

following discussion is in terms of triplets, quadruplets, 

and quintuplets; it could as easily be in terms of 

quadruplets, quintuplets, and sextuplets, or septuplets, 

octuplets, and nonuplets. 

The problem is to recast figure 7 and its associated 

randomization test from two dimensions to three. Recall 

that the crux of the randomization test of figure 7 is to 

identify and count the simulated patterns that are at least 

as extreme as is the observed pattern, that is, the simulations that plot on or to the right of the dashed line that 

passes through the asterisk of figure 7. A more exact 

definition of extremeness could involve contouring the 

values of figure 7. Then, the simulated patterns that are 

at least as extreme as the observed pattern would be those 

that plot on or to the right of whichever contour passes 

through the center of the cell that contains the asterisk. 

For the example of figure 7, that contour would have a 

value of one, and two entries would plot on the contour, 

so N(e) = 2 and the P-value would be 2/20 or 0.1. Use 

of the dashed line gave a P-value of 0. Because the Pvalues are based on only 20 simulations the values might 

be unstable, but part of the difference between 0 and 0.1 

reflects the difference between the two definitions of extremeness, that using the dashed line and that using 

contours. 

At first glance it might seem that a three-dimensional version of figure 7, its dashed line, and its randomization test could be constructed straightforwardly 

from three two-dimensional figures and tests: triplets vs. 

quadruplets, triplets vs. quintuplets, and quadruplets vs. 

quintuplets. However, with much algebraic and geometric 

scribbling it can be shown that such an approach produces a result that is internally inconsistent. An alternative 

would be to contour in three dimensions. It is not clear 

to us how this could be done, either graphically or 

analytically. A remaining alternative is to construct directly a three-dimensional version of figure 7, its dashed line, 

and the associated randomization test (fig. 8, table 4). 

In the two-dimensional case (fig. 7) the orientation 

of the dashed line was determined by the expected values 

of the numbers of triplets and quadruplets in the simulated patterns. The position of the dashed line was determined by requiring the line to pass through the point with 

the coordinates of the observed pattern. This point is 

shown in the figure by an asterisk. The randomization 

test involved identifying and counting all simulations that 

plotted on or outside the dashed line. 

By analogy, the plane S' and the point P of figure 

8 correspond to the dashed line and the asterisk of figure 

7, respectively. The orientation of S' can be calculated 

from the values of t/n, q/n, and r/n, which estimate 

the expected values of ti, qi, and ri. The position of the 

plane S' can be calculated from the known coordinates 

of P. The randomization test will involve identifying and 

counting all simulations that plot on or outside S' (on 

the side of S' away from 0). 

Use of a planar significance surface S' requires 

justification. The ith simulated pattern has coincident 

anomalies that determine values of ti, qi' and ri. These 

three values define a point in three-dimensional space 

(fig. 8). There are n such points and they form a cloud 

of points. The significance of the observed pattern of 

anomalies depends on whether the point P lies farther 

from the origin 0 than does most of this cloud of points. 

Accordingly, we must construct a surface that passes 

through P and approximates the shape and orientation 

of this cloud of points. The plane S' is such a surface, 

for the following reasons. 

First, few if any simulated patterns will have so few 

coincident anomalies that they plot far inside S' , near 

0. Also, few if any simulations will have so many coincidences that they plot far outside S' . Most simulated patterns likely will have moderate numbers of triplets, 

quadruplets, and quintuplets, so that the cloud of points 

will tend to be thin in the direction perpendicular to S ' . 

Second, the number of anomalies that are available 

to coincide is fixed, so that any increase in one of ti, qi, 

or ri requires a decrease in one or both of the others. 

This relationship will be strictly true if all anomalies are 

involved in triplets, quadruplets, and quintuplets. The 

Randomization Tests 9 
+r 

...... ....... ...... ....... ....... ....... ..................... 

........... ........ ........ ........ ......... 

t' ....................... ts +t 

Figure 8. Sketch of oblique view of orthogonal coordinate 

system and points, lines, and planes used for three-dimensional 

case of anomalies of dissimilar widths. Table 4 defines symbols. Axes rand t lie in plane of page, axis q rises perpendicularly out of page. Taper of line from origin (0) to point P indicates 

that the line rises obliquely out of page toward reader. In 

general, line OP is not perpendicular to planes S' or 55

• Point 

Pis three-dimensional analogue of asterisk of figure 7, and is 

shown here as an asterisk to emphasize the analogy. For more 

than a few simulations t > t', q > q', and r > r', so that 5 

lies outside S', that is, ~n oppdsite side of S' from 0. Ran: 

domly located anomalies are more likely to coincide to form 

a triplet than a quadruplet, and more likely a quadruplet than 

a quintuplet. Therefore, for more than a few simulations, the 

cloud of points from simulations will be distributed asymmetrically with respect to axes: t

5 > q5 > r

5 and t' > q' > r'. 

However, single points need not share this asymmetry: it is not 

necessarily true that t0 > q > r 0 , or that ti > qi > ri for any 

single simulation. Values of t

0 , q0 , r

9

, t

5

, q5, r

5

, n, and (ti, qi, 

ri; i = 1, ... , n) are known. Values ott', q', and r' are to be 

calculated. 

relationship will also tend to be true if most anomalies 

are involved in such m-tuplets. The cloud will tend to 

spread out along the t, q, and r axes, so that the dimensions of the cloud in these directions will tend to exceed 

its thickness in the direction perpendicular to S 1 

• The 

cloud will tend to be wide and thin. 

Third, the wide, thin cloud of points will tend to 

undulate little, approximating a plane more than a highly 

curved surface. To see why, consider figure 8. For a constant number of quintuplets in a simulation, a quadruplet 

that loses one of its six component pairs will degenerate 

into three triplets. This conversion of quadruplets into 

triplets is roughly linear. Similarly, three triplets that are 

formed from five overlapping pairs can form one 

quadruplet if the missing sixth pair is added. This con10 Evaluating Coinciding Anomalies 

Table 4. Symbols used for three-dimensional case, which has 

anomalies of dissimilar widths 

Symbol Definition 

t Number of triplets of coincident 

anomalies . 

q Number of quadruplets of coincident 

anomalies . 

r Number of quintuplets of coincident 

anomalies. 

t 0 ,q0 ,r0 Values of t, q, and r, 

respectively, for observed 

pattern of anomalies. 

ts,qs,rs Values of t, q, and r, 

respectively, summed over all 

simulated patterns of anomalies. 

n Number of simulated patterns. 

ti,qi,ri Values oft, q, and r, 

respectively, in the ith 

simulated pattern, i = 1, ... ,n. 

0 Origin of orthogonal coordinate 

system with axes t, q, r. 

P End point of a vector from 0 to 

<to,qo,ro). 

Ss A plane with t-intercept ts, qintercept qs, and r-intercept rs. 

S' A plane through P, parallel to Ss, 

and called the significance 

plane. 

t' ,q' ,r' t-, q-, and r-intercept, 

respectively, of S'. 

version of triplets into quadruplets is also roughly linear. 

Similar roughly linear relationships link triplets with quintuplets, and quadruplets with quintuplets. These considerations suggest that several simulations that plot in 

figure 8 with the same value of r (or oft, or of q) might 

lie approximately along a straight line that is perpendicular to the r axis (or the taxis, or the q axis) especially if most or all anomalies are involved in triplets, 

quadruplets, or quintuplets. The degree to which these 

lines perpendicular to the axes are straight is the degree 

to which the cloud of points approximates the plane S 1 

• 

Use of t

5

, q5, and r5 to calculate the orientation of 

S 1 also requires justification. The expected value of the 

number of triplets in a typical simulated pattern is E(ti), 

which is estimated by t/n, and similarly for the 

numbers of quadruplets and quintuplets. Because 

multiplying the t-, q-, and r-intercepts of a plane by a 

constant, say 1/n, changes the position but not the orientation of the plane, a plane with intercepts E(t), E(qi), 

and E(r) is parallel to Ss and therefore to S 1 

• Therefore, 

determining the orientation of S 1 from the sums t5 , q5 , 

and rs is equivalent to determining the orientation from 
the estimated expected values. This orientation will be 

slightly susceptible to the distorting effects of a few 

unusual simulated patterns, such as one with no coincident anomalies or one with a large number of quintuplets. 

However, if the number of simulations is large this distortion will be small and negligible. 

The next step is to derive an equation for S 1 • S 1 

must be expressed in terms of the known values that are 

listed in the caption of figure 8. The intercept forms of 

the equations for Ss and S 1 are, respectively, 

(tit)+ (q/q5

) + (r/r5) = 1 

(tlt 1)+(qlq 1)+(rlr 1)= 1 (1) 

Because Ss and S I are parallel, the coefficients of these 

two equations are proportional. If A is a constant, 

(2) 

Now P lies in S 1

, so by substituting the coordinates of 

P and equation (2) into equation (1), A is found to be 

(3) 

Then, substituting equation (2) into equation (1) and 

multiplying through to clear fractions gives 

(4) 

Simulations will tend to have few quintuplets, so 

each simulation will plot in figure 8 as a point with an 

integral, usually small, value of ri. (For example, each 

simulation for the data along the Wasatch fault zone has 

ri = 0, 1, or 2.) Then all points that constitute the 

population of the randomization test can be plotted in 

a few serial sections through figure 8 perpendicular to 

the r axis, each serial section lying at a different integral 

value of r (fig. 9). If the trace of S 1 

can be graphed in 

each serial section then the number of simulations that 

plot on or outside S I can be counted easily by examining 

the serial sections. Setting r =kin equation (4) gives the 

equation of the linear trace of S 1 in the plane r = k, k = 0, 

1, ... , as 

(5) 

The equations for the intercepts and slope of the trace 

are given in figure 10. 

In terms of serial sections like those of figures 9-10, 

the P-value is N(e)/N(p). N(e) and N(p) are determined 

by counting the numbers of simulations that plot in 

various parts of the serial sections, analogously to the 

manner in which N(e) and N(p) were determined for the 

two-dimensional case by counting points in various parts 

of figure 7. N(e) is the number of simulations that plot 

+r 

r' 

+t 

Figure 9. Part of figure 8, showing a serial section. Heavy lines 

and associated labels are from figure 8. Dotted lines outline 

a serial section that intercepts r axis perpendicularly at r= k, 

where k= 0, 1, 2, .... Only serial section through point P is 

shown. Other parallel sections lie at larger and smaller values 

of r. Serial section intersects planeS' along short-dashed line 

which, in this section, passes through P analogously to dashed 

line of figure 7. Each simulated pattern that has k quintuplets 

plots in this serial section, either insideS' (for example, point 

B), outside S' (point C), or on dashed line (point D). 

+t 

At 5q5 r5 - kt 5q5 At 5 r5 -kt5 

t\ qsrs rs 

(Q; :?,j. k) +q 

Figure 10. Intercepts (t' k and q' k) and slope of trace 

(dashed line) of S' in serial section r= k. Table 4 defines 

symbols; geometry is shown in figures 8-9. Value of A 

is given by equation (3). Origin of this serial section is 

not 0 of figure 8, but instead is point (0, 01 k) on r axis 

of figure 9. 

Randomization Tests 11 
on or outside the dashed lines, summed over all serial sections. In the example of figure 9, points C and D contribute to N(e). N(p) is the number of simulated patterns. 

In the example of figure 9, all of points B-D contribute 

to N(p). 

SUMMARY 

Associated variables can be identified by plotting 

J for the various observed pairs of variables (fig. 5). For 

the example of figure 4, only variables 1 and 2 are likely 

to be strongly associated along the traverse. 

A pattern of coincident anomalies that is unlikely 

to have arisen by chance can be identified and evaluated 

visually by plotting counts of pairs and m-tuplets of coincident anomalies for observed and simulated anomaly 

patterns (figs. 6, 7). Numbers of triplets and quadruplets 

in the observed and simulated anomaly patterns are compared using a randomization test. If variables and 

simulated m-tuplets are too numerous to represent in two 

dimensions (fig. 7), a three-dimensional representation 

(figs. 8-10) should suffice. For the example studied here 

(fig. 4), results of the randomization test confirm suspicions derived from inspection of figure 7 and perhaps 

even from initial inspection of figure 4. The randomization test and inferences based on its result indicate that 

the observed pattern of anomalies contains more triplets 

and quadruplets than are likely to occur by chance. In 

practice, many more than 20 simulations would be needed 

to support statistical conclusions. 

These conclusions are guides to aspects of figure 

4 that are worth trying to interpret geologically. What 

interpretations are made depends on the natures of the 

variables and their anomalies. Further consideration of 

variables 1 and 2 together, and use of other geological 

or geophysical information, might suggest or confirm the 

existence of subtly expressed features at some or all of 

the places where variables 1 and 2 are anomalous 

together. The association of variables 1 and 2 along the 

length of the traverse indicates that these two variables 

together might be a more powerful investigative tool than 

is any of the four variables alone. 

ACKNOWLEDGMENTS 

W. F. Dummler wrote most of the simulation program and improved its design in numerous discussions. 

Comments by M.N. Machette and K.M. Shedlock improved the manuscript. 

:.t U.S. GOVERNMENT PRINTING OFFICE: 1988-573-047/66,025 REGION NO.8 

12 Evaluating Coinciding Anomalies 

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