Davis Directional Data

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I

Statistics

and

Data

Analysis

in Geology

hird dition

John

C avis

Kansas

Geological

Survey

The University of

Kansas



Statistics and Data Analysis in Geology Chapter 5

treated as though they occurred along a single, straight sampling line. This and

other methods for investigating the density of patterns of lines are reviewed by

Getis and Boots (1978). A computer program for computing nearest-neighbor dis-

tances, orientation, and other statistical measures of

patterns of

lines is given

by

Clark and Wilson (1994).

Analysis of Directional Data

Directional data are an impo rtant categoryof geologic information. Beddingplanes

fault surfaces, and joints are all characterized by their attitudes, expressed

strikes a nd dips. Glacial striations, sole marks, fossil shells, and water-laid peb-

bles may have prefe rred orientations. Aerial and satellite

photographs

may

show

·

oriented linear patterns. These features can be

measured and

treat ed quantitattvely

like m easurements of other geologic properties, but it is necessary to use special

statistics that reflect the circular (or spherical) nature of directional data.

Following the practice o f geographers, we can distin guish betwee n d i r e c t i o n ~ l

and oriented features. Suppose a car is traveling

north

along a highway; the car's ·

motion

has

direction, while the highway itself

has

only a

north-south

orientation;

Strikes

of

outcrops

and

the traces of faults are examples of geologic observations .

that

are oriented, while drumlins and certain fossils

such

as high-spired gastropods

have clear directional characteristics.

We

may also distinguish observations that are distributed

on

a circle, such

as paleocurrent measurements, and those

that

are distributed spherically, such'

as

measu reme nts of metamorphic fabric. The former data are conventionally shoWn

as rose diagrams a form of circular histogram,· while the latter are plotte ;l as

points on a projection of a hemisphere. Although geologists have plotted diretl·

tional measurements in these forms for many years, they have not

used

forinal.

statistical techniques extensively to test the veracity of the conclusions they have

drawn from their diagrams. This is doubly unfortunate; not only are these statis·

tic:al

tests useful, but the development of many of the procedures was origin.ally

inspired by problems

in

the Earth sciences. · ' '

Figure 5-13 is a map of glacial striations measured in a small area of south·

ern

Finland; the measurements are listed in Table

5-4

and contained

in

file

FIN·

LAND.TXT.

The directions indicated by the striations can be expressed by plottin&

them as unit vectors or on a circle of unit radius as in Figure 5-14

a.

If the circle

is.

subdivided into segments and the number

of

vectors within each segment counted,

the results can

be

expressed as the rose diagram, or circular histogram, shown

as

Figure

5-14

b.

Nemec (1988) pointed out that many of the rose diagrams published by ge-

ologists violate the basic principal

on

which histograms are

based

and, as a con·

sequence, the diagrams are visually misleading. Recall

that

areas of columns n a

histogram are proportional to the number (or percentage) of observations occurring

n

the corresponding intervals. For a rose diagram to correctly represent a circular

distribution, it

must

be constructed so that the areas of the wedges (or "petals") of ·

the

diagram are proportiona l to class frequencies. Unfortunately,

most

rose dia·

grams are drawn so that the radii of the wedges are proportiona l to frequency. The

resulting distortion may suggest the presence of a str ong directional trend where

none exists (Fig. 5-15). ·

.

~ p a t a l Ana ys s

~ : : t . . . l ~

~ : : t o . .

~

r

t

I

~

I

: : t o . . ~

. . .

r

Figure 5-13. Map showing location and direct ion of 51 measurements of glacial striations

in a 35-km

2

area

of

southern Finland.

Table 5-4. Vector directions

of

~ l c i l striations

measured

in

an

area of southern

~ m l a n d ;

measure-

ments

given

in degrees clockw1se

from

north.

23

105

127

144

171

27

113

127

145

172

53

113

128

145

179

58

114

128

146

181

64

117

129

153

186

83

121

132

155

190

85

123

132

155

212

88

125

132

155

93

126

134

157

99

126

135

163

100 126 137

165

If we define a radius for a sector of a rose diagram

that represents

either

one

observation, or 1 , we can easily calculate t ~ e appropriate radii that represent any

number

of

observations

or

relative frequencres,

lr 5.38)

r = uw

where

r

is

the unit radius

representing one observation or

1 :

f is the ~ r e q u e n c y

n c o ~ s or ercent) of observations within a class, and r IS the radius of the

. ~ l a s s sector. ~ o t h e r words, the radius s h o u l ~ be proportional to the square root

of the frequency rather than to the frequency Itself.

Rose diagrams, even if properly scaled, suffer

f r o ~

~ h e

same

p r o b ~ e m s

as or

din

histograms; their appearance is e x t ~ e m e l ~ s . e n s i t l ~ e the chmce. of class

w i d ~

and

starting

point

and they exhibit variations similar to

the histogram

317



Statistics and

Data

Analysis

in

Geology-

Chapter 5

270°

360 °

oo

360 °

Figure 5-14. Directions of glacial striations shown on Figure 5-13.

a)

Directions plotted

as unit vectors.

b)

Directions plotted as a rose diagram showing

numbers

of vectors

within successive .0° segments. ·

a

b

Figure 5-15. Rose diagram of glacial striations shown on Figure

5-13

plotted

in

lOo seg

ments .

a)

Length of petals proportional to frequency. b) Area of

petals

proportional

to frequency.

examples shown in Figure 2-11 on p. 30. Wells (1999) provides a computer program

that quickly constructs rose diagrams with different conventions

and

also includes

an assortment

of graphical alternatives that may be superior to conventional rose

diagrams for some uses

(Fig. 5-16).

To compute statistics that describe characteristics of an entire set of vectors,

we

must

work directly with the individual directional measurements

rather

than

with a graphical summary such as a rose diagram. (Note that the following dis• ·

cussion

uses

geological

and

geographic conventions

in

which angles are measured

clockwise from north, or f rom the positive end of the Y-axis. Many

papers

on di-

rectional statistics follow a mathematical conven tion in which angles are measured'

counterclockwise from east, or from the positive end of the X-axis.)

Spatial Analysis

a b

c

d

e

f

Figure 5-16. Effect of choice of segment size and origin on appearance of rose diagrams.

Data

are directions of glacial

striations

from file FINLAND.TXT:

a)

so

segments, oo

origin,

outer

ring 20 ;

b)

5o segments, 0° origin,

outer

ring

30 ; (c)

30o segments,

oo origin,

outer

ring 40 ; d) 15° segments, 10° origin-compare to b) . Alternative

graphical forms include (e) kite diagram, 15° segments, oo origin-sometimes used

in statistical presentations; (f) circular histogram, 15° segments, 0° origin-widely

used

to

plot wind directions.

The dominant direction

in

a

set

of vectors can

be

found by computing

the

vector resultant.

The

X-

andY-coordinate s of the end point of a

unit

vector whose

direction is given by the angle

e

are

xi= cos i

rt =

sinei

(5.39)

Three such vectors are shown plotted in Figure 5-17. Also shown is the vector

resultant,

R,

obtained by summing the sines

and

cosines of the individual vectors:

Xr =

L:r=l

cos ei

Yr

= L:f=

1

sin i

(5.40)

From the resultant, we can obtain the mean direction,

8

which is the angular av

erage of all of the vectors in a sample. t is directly analogous to the mean value of

a set of scalar measurements

1 1

('n

.

/ n )

=tan (Yr Xr)

=tan Li=l smei Li=l cos

ei

(5.41)

Obviously, the magrlitude or length of the resultant depends

in

part on the

amount

of dispersion in the sample of vectors,

but it

also

depends

upon

the number of

31 9



Statistics and

ata

Analysis

in Geology

Chapter 5

b

~

a b

Figure

5-17.

e t e ~ m i n a t i o n

of mean direction of

a set

of unit

vectors.

a)

Three

taken from

F1gure

5-16.

b)

Vector resultant,

R,

obtained by

combining

the t h r i ~ o t

unit

vectors. Order

of

combination is immaterial.

a

-

.)o.,

--

----------. .

c

a

b

Figure

5-18. Use of

length

of

resultant to

express

dispersion

in

a collection of unit

a)

Three vectors tightly clustered

around

a

common

direction . Resultant R

is

tively

long, approaching the

value

of n .

b)

Three widely

disperSed

vectors·

length is

less

than 1.0. ·

vectors. In order to compare resultants from samples of different sizes, they

be converted into a standard ized form. This is done simply by dividing

the

~ ~ ~ -

nates of the resultant by the numb er of observations, n ·

- 1

n

\

c = Xrln = nLi=l cos 8i

- 1

n

. .

s = Yrln = n Li=1 smei

Note

that these

coordinates also define the centroid of the

end points of the

vidual unit vectors.

The resultant provides information not only about the average direction

set of

vectors, but also

on

the spread of the vectors about this a:verage.

5-18a shows three vectors that deviate only slightly from the

mean

direction:

resultant is almost equal in length to the sum of the lengths of the three vec:rors

.:.;

In contrast, three vectors

in

Figure

5-18

b are Widely dispersed; their resultant

very short. The length of the resultant,

R,

is given by the Pythagorean theorem:

.R = + Yi = ~ ( I ~ = l co ei}

2

+ I:

1

sin ei)

2

;

The ength of the resultant can be standardized

by

dividing by the

number of

vations. The

standardized

resultant length can also be found from

the

end

J?Oints

:._

R

2

R . = C

+S

n

j

Spatial Analysis

fbe quantity R called the mean resultant.length will range from zero to one. It i

a measure

of

dispersion a n a l o ~ o u s to the variance, but expressed in the opposite

sense. That is, large values of R indicate that the

b s e r v a t i o n s

are tightly bunched

together with a small dispersion, while values of R near zero indicate that the vec

tors are widely dispersed. Figure 5-19 shows sets

of

vectors having different values

ofR.

In

order to have a measure of dispersion

that

increases with increaslng scatter,

JUs sometimes expressed as its complement, the circular variance

=

R

=

n

R fn

(5.45)

J; j

other directional statistics can be computed , including circular analogs

of

the stan-

dard deviation, mode ,

and

median. Equations for these are given

in

convenient table

fof,rn

by Gaile and Burt (1980).

c

d

e

f

l

t>

l

Figure 5-19. Sets of

unit vectors

illustrating the

value

ofR produced y different dispersions

of vectors.

In

all

examples,

the mean direction

is

52°:

a)

R = 0.997, b) R = 0.90,

(c) R =

0.75,

d) R =

0.55,

(e) R =

0.40, f)

R =

0.10.

Orientation data

must

be modified before mean directions or measures of dis

p e r s can be calculated .Since the orie ntation

of

any feature·

may be

expressed

as . ither of two opposite directions,

some

convention

must

be

adopted

. to avoid

I J U I ~ t i n g

the dispersion of the measurements. Krumbein (1939) hit upqn a novel

soJution

to this problem

while studying

the

orientations

of stream

pebbles. If all

of

the measured angles are doubled, the same _angles

will

be recorded regardless of

which directional sense of the oriented features is used.

As

an example, con sider

a fault tra ce

that

strikes northeast southwest. Its orientation could equally well be

~ ? 1



Statistics and Data Analysis

in

Geology- Chapter 5

recorded as 45 • or as 225 · .

f

we double the angles, we obtain 45 • x 2

=

90•

225 • x 2 = 4 50•, which becomes (450. - 360•) =go·.

Mean direction, mean resultant length, and circular variance can be foun d n

usual manner after the orientation angles have been doubled. To recover the t r l l l

mean orientation, simply divide the calculated mean direction by two . This

be illustrated using the data in file

CAROLINA.

TXT, which contains

the major axes of

99

Carolina bays -ellipsoidal depressions

o.n

the southern

of the U.

S.

Atlantic Coastal Plain. The origin of these geomorph ic features

5-20)

was at one time the subject of intense controversy; the depressions

attributed to causes as diverse as meteorite impact, karstic solution, or a e i a ~ i 6 n , i i l t '

(Prouty, 1952). Subsequent studies (Rasmussen, 1959) have favored a cornpc:>Stt

1

ii

origin involving differential solution by groundwater and eolian removal

rial. Figure 5-21 shows rose diagrams of the axial orientations of the bays

(a)

incorrectly as vectors (resulting in a bimodal distribution) ,

(b)

as vectors

angles have

been

doubled, and

then

(c) as vectors after dividing the doubled

of (b) by two, and also plotting the complements of the vectors. Although.

measurement is plotted twice in this diagram, it yields the correct impression of

symmetrical distribution. ·

• j

; .

Testing hypotheses about circular

directional data

n order to test statistical hypotheses about circularly distributed data, we

have a probability model of known characteristic s against which we can test.

are circular analogs of the univariate distributions. discu ssed in Chapter

2.

The

useful of these is the von Mises distribution t is a circular equivalent of the

distribution and similarly posses ses only two parame ters, a mean direction, e

concentration parameter,

K.

The von Mises distri bution is unimo dal and

about the mean direction. As the concentration parameter increases, the I < « ~ I i b o o ~

of observing a directional e r s u r e m e n t ~ r y close to t h ~ mean direqion

inr · r"" 'c"

'

If

K

is equal to zero, all directions are equally probable, and a circular uuJL.lUl·J.U

distribution results.

Figu

re '

5-22

a shows t he form o f the von Mises

several values o

K.

The distribution can also be shown

in

conventional form

Figure

5-22

b;

note

that the horizontal scale is giv

en

1n degrees

and

o r r e s p o n ~ s

a comp lete circle. . ·

It is difficult to determine

K

directly, but the concentration

parameter

can

estimated

from R i f we assume that the

data

are a sample from a population

a von Mises distribution; AppendiX Table A.9 gives maximum likelihood <>ctirn,,t

.

of K for a calculated R We will use these estimated values of K in some ,

.

1 u ' " " Y l u c u

statistical tests. ,· ,

.

I·

·\ ' ' ; .

Test for

randomness.-

The simplest hypothesis that can be statistically tested

that the directional observations are randoin.

·"

Iri other words, there is no .

direction, or. the probability of occurrence is the same for all directions. If,

.

sume that the observations come from a von Mises distribution, the h•nnnlth<>-ct

is equivalent to 'statiilg that the concentration parameter, K; is eqtial ·to zero,

l

cause

then

the

distribution becomes a circular·unifofm.·

n

formal:terms,

the

hypothesis and alternative 'are · ; , .

- ·

· ·. · . · ''

··1H 1· J , • • · Ho

·

k=O

f'l

: 1 c

. .

' '

, .

. . . .

I

,,

. · , ,

~ ' ; , Hr:

>

· ·J.:· . ~ · # u t ' J

Spatial Analysis

1 km

Figure

5-20.

Aerial photo of "Carolina bays,"

subparallel

ellipsoidal depressions on Atlantic

Coastal Plain of southeastern U.S.

in

Bladen County, North Carolina (Prouty, 1952).

c

Figure

5-21.

Effect of doubling angular direction order to calculate mean orientation.

(a) Orientations of major axes of 99 "Carolina bays" plotted as vector directions. Re-

sultant mean direction is 20i• and s near

zero length R

= 0.008). (b) Orientation

measurements plotted as vector directions after angles are

doubled

. Distribution is

no

longer

bimodal.

Resultant reflects correct. trend of d c i u b l a·

ngles

and

is near

unity in

length

(mean direction

is 97

.4°; R = 0.98):. (c). OrientatiOilS replotted at

original angles, and their complemenr

True

r e s l l ~ a n t direction,

{48

.7•)

is

found by

halving resultant airecticin in (b). . ' ' ' ' ' ' '

The test is ~ x t r e m e l y simple and involv_s only the ~ a l c u l a t i o n of R accorcling to

Eqqation (5.44).. This statistic. is compared, o a. criticai value of

1

R

for

th

e . e s i r ~ d ·

level

of

significance.

if

he

observations

do

come

from

·a circular

uniform

distribu

tl ?n. we would exi>e_t R to $mall, as in Figure'

5-19

f. Ho,wever,

i f

the computed

statistic

is

so large

that

it exceeds the critical value, the null hypothesis

niust be

rejected and the observations may be presumed to come from a population having

323

.



Statistics and Data Analysis in Geology

Chapter 5

oo

/

180°

a

0.40

0.30

0.20

0.10

0.00

180° 225° 270° 315° 360° 45° . 90° 135° .180°

b

I

Figure 5-22. ·Von Mises distributions having different concentration parameters.

. tributiort plotted

in

. polar form.

(b)

Distribution plotted as conventional

uru ud unn

distribution. Note that horizontal axis

is

given in degrees. After Gumbel 1

and Durand (1953). . , .

Spatial Analysis

a preferred orientation. This test was originally developed by Lord Rayleigh at

the turn

of

the nineteenth century; a

modem ~ r i v t i o n

is given by Mardia (1972).

Appendix Table A.lO gives critical values of R for Rayleigh s te st of a preferred

trend for various levels of significance and numbers of observations.

Remember tha t Rayleigh s te st presumes that the observed vectors are sampled

from a von Mises distribution. That is, the population o f vectors is either unifo rm

(if K = 0) or has a single mode or preferred direction. If the vectors .are·actually

sampled from a bimodal distribution such as that shown

in

Figure

5-21

a, the test

will

give misleading results.

We will test the measurements of Finnish glacial striations at a 5 level of

significance to determine i they have a preferred direction. Since there are 51

observations, Appendix Table A.lO yields a critical value of R

5 6

= 0.244. The

test statistic is simply the normalized resultant,

R

The sum of the cosines of the

vectors is

Xr

=

-25.793

and the sum of the sines is

Yr

= 31.637. The resultant

length is

R =

- 2 5 . 7 9 3 )

+ (31.637)

2

=

40.819

which, when divided by the sample size, yields a mean resul tant length of

R = 40.819/51 = 0.800

Since

the computed value of R far exceeds the critical value, we reject the null

hypothesis that the concentration parameter is equal to zero. The striations

must

have a preferred trend.

Test for a specified trend.-On some occasions

we

may wish to test the hypothesis

that the observations correspond to a specified trend. For example, the area of

. Finland where the measurements

of

glacial striations were taken is located wi thin a

broad topographic depression aligned northwest-southeast at approximately

105•.

Does the mean direction of ice movement. as indicated

by

the striations, coincide

with the axial direction of this depression? ·

c. Exact tests of the hypothesis that a sample of vectors has been taken: from a

population having a specified mean direction require the use o f extensive chart s

in

order td set the critical value (Stephens, 1969). A simpler alternative is to

detel IlliD.e

a confidence angle around the mean direction of the sample and see

i

his angle is

sufficiently broad to.encompass the hypothetical mean direction. This confidenc e

angle is based

on

the standard error o f the·estimate of the mean direction, e and·

thus considers

both

the size

of

the sample and its dispersion. · ·

L; Before computing the confidence angle, the Rayleigh test should .be applied

to confirm that a statistically significant mean direction does exist. Then the

mean

resultant l ength R mustbe computed and the concentration parameter·

K

estimated

using Appendix Table A.9. The approximate standard er ror of the mean direction(

given in radiansi is

1,

r

t i

· ··1

(5.46)

_, . .

-

since the standard error is a measure of the chance variation eXpected from sample

td sample

iri.

estimates

of.

the mean direction,·we can use it to define probabilistic

limits ori the location of the true·or population mean direction. Assuinirig

that

e s t i m t ~ o n errors are normally distributed; the interval · -

., f

0 ± ZaSe

. i (5.47)

325




in

Geology Chapter 5

should capture (or include) the true population mean direction

oc

of the i n ' J . ~ .

example, if

we

collected 100 random samples of the same size from a

...

.... ...u•u"'

of

vectors and computed the mean directions

and 95%

confidence intervals

each,

we

would expect that all

but

about

five

of th ose inte rvals would contciin

true mean direction. Of course, we would not know which five

of

the intervals

to capture the true direction, so we must assign a probabilistic caveat to an·

of

We

might,• or example, make the statement that "the interval, plus

and

many degrees around the mean direction

of

this particular sample,

cm1

t

atr.ts

' th

true population mean direction. The probability that this statement is

1nr·nn,,,. . .

5%X

' ··

t. ' I • • . ·: ; L{

We

have already applied Rayleigh's test an d rejected the hy pothesis

of nn •no,,.,.,.. ,

in

the observations

of

the striations. The approximate standard error

of

the ·

direct ion can now be found: ·

. ' r ; :;f

s

=

1

= --

- -

=

0.0924 radians

=

5.29° ·

1

e v'51 · 0 .8004 : 2.87129 10.826,

Therefore, tbe probability is

95%

that Jhe interval .,

. . . . .

±

1.9Q >.< 5·

2

.

9°::

contairls the population mean direction:. In

other

words, •

·

· ' ·'

- ; l j ; · ~ ,:jJ

I

•,

I .

118.8°

::5 e :5

139.6°

Since thil ·interval doe.s not include the direction of hligiunent of the t O J ~ O l f l " a l ~ h l i

depressionl

m u s t

conclude that the. axis

of

the. depression does

nd

v cotnotl[f8

With-the

m e a n : d i r e c ~ o n of

the striations:

\···, A

lr,

,. · ~ - ~ , · ·; . •· · •.

: : ( · f . · t J , ~ : \

..

. : . - . · ~ " : : J = .

r·· _}n_ j 1ti J,q-:. J

Test

of, gpodness

.

of

fit."-A simple nonparametric alternative to the Rayleigh

of

uiuformity involves dividing the

unit

circle into a convenient numbe r

of

.

s e g m e n t s If .

hese segments

are

equal in. size

and

tfie observed·vectors are

tributed at random, we should observe approxi.J:hately

e q u a l

n u m b e r s

o f . v e < : : t o r ~ J I I

each

e g m ~ n t

The.number actually observed ·

can.

be compared to· hose'

x ] ) e ~

t e a

by a test. The expected frequency

in ea<:h

·segment

must be

at

least

should:

be

·between

tt./

1S and n 15 .segment ? . The·xl statistic:is ... , : u ~ , J • u • • t : u . ..., ......

usuafin,anner (see 2.4

5)

and has

k -1

degrees offteedont, wl ere

k t h l e " ~ l l l J I 1 i b e 1

of

segments. ·, ·

,.

1·· .• :\.

·

' :.t.>'

.

··::: ::(: c .c :····: · · :' ·.t,..,., .,..:bi

• 'l',,The same procedure. can.

be ~ e d to test

the gobctlless

of fit of

·

he< n h ~ P r v e r J

vector's to other theoretical models, such as a.von Mises distri bution with a

.,.,,,,.,,.,n

concentration>parameter'

K

great.er; than

zerO> anci

>a

~ p e c i f i e d ;

niearr.

Cllr

•

ctJtO:Q.

Comp_uting'

the expected frequencies; howevei', can

be

complicated:t

. . J \ . O w . J . . I J • ~ ; . ,

given by Gumber; Greenwood, and Durand (1953)

and

Batschelet (1965)u;1 '4- ·

.J , ;; I: ' . ·-;., ' ~ \ \' :' :

' • l ~

} : . , _ ~ - )/.:_:,,.. ' ~ ~ . ·

Testing

the

equality

of

two sets of'dihktional vectors.'7'We may An1lt>tinu•<>•t.UUll,

to. E , , S t . . ~ Y P , o t p . ~ s e about the e q W . y a l ~ , . c ~ qf. o s ~ p ~ ~ ~ or

1

o l l ' c i o q ~

tiOJ.lal

m e ; ~ s u r ' i n e n t ~ . ,Fon e x a m ~ l e , we ,Jlli I.Y hayt:.

a l ~ p < l ; W e : q t ; s q . r e r Q . ~ d

~ < ; ) d i , f f e r ~ t

stratigraphic units ,

and

want,,tq;pete.rnWte

_f

the.lt

r u ~ t e l " t I D

are the same,

or

w,e may wish;

~ e e i f u:e

.

9 r i e . n t a t i o n ~

. . .

satellite image cdiricide with the

o r i e n t a t i o n ~

of faults known to exist

to raphed area. At a much smaller scale, we may want to comi?are the

a l i : J U . l ~ e n

Spatial Analysis

of elongated pores

in

thin sections from two cored samples

of

sandstone from a

petroleum

~ s e r v o i r .

The equality

of

two mean directions may

be

tested

by

comparing the vector

resultants of the two groups to the vector resultant produce d when ,

the

two sets

of

rneasurements are combined

or

pooled.

If

the two samples actually are drawn from

die same population, the resultant

of

the pooled samples should

be

approximately

equal to the sum

of

their two resultants.

If

the

mean

directions

of

the:

Wo

1;amples

are significantly different, the pooled resulta nt will

be

shorter than the

suin of

their

resultants': ·

<

'

If

K

is a large value (greater

than

10) •.

an

F-test statistic can

be

computed by

_ ( ~ - 2 ) (R1 +Rz ~ R p )

- .· (n _ R

1

_

Rz)

J•:

'..,.

(5.48)

where ri

i ~ >

the total number of observations,

R1 and Rz

are the resultants

of the

. sampies

of

vectors,

and.Rp

is the resultant

of

the

set

of

e c t o ~ s

.after the two

grgups haye

been

pooled. , . . . . . · . . .

.

. ·

• .' Using Appendix Table

A.9, we

can estimate the value

of K

from

:R;,,

the

length

of

the mean resultant 0(

·

he

t}VO

pooled samples.

If K

is smallE_ r

than

1

b bu

,t greater

·} • . , ' . - . , I < •

than

2, then

a more accurate F-test is ; :

1· 1

··

· ..

' i ' / . ' _ -

t

,· ' . \ .

.,

,

/

_

:

. 3

-

)

n

2)

(R1

+

. . . .

Rp)

.

hn z

·=

·1

-

. .

(5 49)

' ; · i , K . :; (

n

- R1

-

Rz)

1 •

If K

is less\

han 2, s ~ e c i a t

tables ( s u c h ~ ; thosk mven

in

Mardia,1972)

eri-e

necessary.

It

is also possible t<;> test the equalitY

of

the concentration paramete:fs

of

two

sets

of\rectors ;

but

the. o m p u t ' a ~ o n s are invoived. Referito Mardia (197,2)

1

ror a detailed

discussion, and to Gaile and Burt

(I

980) for a o r ~ e d example from geomorphology.

· A fold be)t, expressed topographically as 'the Naga Hills and• herr extensions,

oecurs at the jun¢ture between the

India ?- s u b c o r l t i n e n ~ the

hidochiiiese

'pe.nin-

. sula. Apparently

related

to .compressive movements that created tile Himalayas,

the' fold

belt

bidudE_ s a

e r i ~ s

of subparallel anticlines along the eastern border

of Bangiadeslh

'OU

,arid gas

hci

e b e ~ n found

in

structural traps

in

t]Jis region; so

d ~ l i n e a t i ' o n of l d ~ -is

pf

economic weil' as scientific inte rest.

P r ~ s u m a b l y

t h e ~

f o ~ d s

occur,

perhaps

With:reduced magrutude·, to

the

west of

the

Naga

1

Hills,

but

are

concealed by

o ~ e m

~ e d i m e n t s deposited by the ·GcritgeS'.River and its tributaries .

Unfortunately, reflection seismic data

that

could reveal the bur ied

structures

are

. .

.t

. . }

<

·

t •

.. •• I • · ~ i

y \J '

v e r y s : p a r ~ e :

· .

...,I. · · · · '' · _;: .. ~ · . · < ·

·

..

Interpretations

of

Landsat satellit.e i m a g e

of

this ·region iridicate numerous

lineations

1

ofunkno'wn.

r i ~

It is possible that the

i l n e a t i o n ~ rt:fl'e

¢t subsurface

folds,

andj if

so, they may provide valuabl clues

tb

structural geology.

and

possible

petroleum deposits. : . ·

(.:,.

:,:/· ' .

:

· .

i, ·'

·

,.'

F i g u

5

-;

23 is a map

of easte.rJ1

Bangladesh showing

the

1

traces of_axial planes

of major e){posed anticlines·

and

the Hrrger lineaments'

measured on

Landsat

iffi,

a g ~ s . T h , ~

qrientations

of'

hese, two sets

are shown on

·

Figure 5.-24.:

Because

the

lines have

no

'seiise' of diTection, the plots are bilnodal;

a n d ~ ~

must" double '

the

~ b s e r v e d

an

gl

es

to

'obtain

the

cbrred distribution·

of

vectors. Table' 5-5

l i s t ~

the

o ~ e h t a t i 0 r i s 1 0 f

b'

oth

the axiru planes

and

the lineiUrients, which also

afe

-

contciirted

in file

BANGIJ\,

ix F

:.

h e f e <hi

.obvious

d i f f e r e n c e ~ t e ~ two t 1 ut

is

this difference statisticall y significant

or

could

it

have arisen

through the

vagaries.

of sampling? · . ' > . . · ·.. . ,'. :' ,,, •

327



Statistics and

ata

nalysis

in Geology- Chapter 5

.,

·

.

-

.

.

__ ____ _ ; · · .

igure

5 23.

Map

of

eastern Bangladesh

s h o w i ~ g

ax

ial

planes

of

major

~ t i c l i n e s

(solid

lines) and large lineaments interpreted from Landsat images (dashed lmes) . .

Spatial Analysis

a

e

g

igure

5 24. Rose diagrams of orientation data from eastern Bangladesh. Mean ori

entations

indicated by arrows. Top row shows plots

of

vector directions from file

BANGLA.TXT : a) Anticlinal axes (mean direction is 86.2° ; R = 0:05). b) Linea

ments (mean direction is 334 .6°; R ·0.15). (c) Pooled vectors mean direction is

352.5 ; R = 0.70) . Middle row shows plots of doubled vector directions:

d)

Anti

clinal axes (mean direction is 341.5 ; R = 0.85). e) Lineaments (mean direction is

30.1°; R = 0.77) .

f)

Pooled vectors (mean d j r ~ t i o n is 5.3°; R = 0.74). Bottom row

shows orientations replotted at original angles and their complements. True

resultant

directions found

by

halving resultant directions shown in middle row:

g)

Anticlinal

axes (mean direction is 350.8°;

R

= 0.85- .

h)

Lineaments

mean

direction is 15.0°;

R

=

0.77) .

i)

Pooled vectors (mean direction is 2.6°; R

=

0.74).

To test the hypothesis that the meari directions of the anticlinal axes and the

Landsat lineaments are the same,

we must

first compute the resUltants of each

of

th_ twogroups and the resultant of the two groups combined. The resultant

of

the

32




n Geology

Chapter 5

Table 5-5. Orientation

of

axial planes

of

anticlines

and Landsat lineations

in

eastern Bangladesh; measurements

given in degrees clockwise from north .

Anticlinal Axes,

n

= 34

Landsat Lineaments, n = 36

12

16

14

5 350

32

15 8

192

202 169 163 214

192

16

26

186

186

24

344 356 218 198

221

343 346 161 341 350 18 221 342

339

150 169

336 160 205

35

337

351

156 159

352

2 171 196 14

152 162 341 181 184

246

175

25

348

158 156 354 213 26 212

330

162

20

42 354

13

202

34

doubled measlll'ements of the fold axial planes is R1 =

38.97

and the

of the 36 doubled measlU'ements of the Landsat lineament is

R2

= 27.79 .

two groups can be combined into a pooled collection of

70

obser-Vations that has

resultant of

Rp

=

51.73.

The mean resultant of the pooled group is · \

Rp = = 0.74

and by use of Appendix Table

A.9,

we can estimate the concentration

K = 2.2893. . . .

Since

K

is greater than 2

but

less than

10,

the appropriate test staqstic is

by Equation

(5.49).

Substituting values we have calculated into that equatio11

F=

(1 .+ 3 ) ((70-2)(38.97.+27..79-51.73))

= ~ ~ .

' 8(2.2893)

(70-38

.97-27.79) ' ' .

The test has v

1

= 1 and v

2

= (70- 2) degrees of freedom. From the values ofF

Appendix Table A.3, we can interpolate to find the critical value for

F

at the 5

of

significance

(£X=

0.05) with

1

and 68 degrees of freedom; the value is

F

=

Since the test value far exceeds the critical value,

we

must regretfully conclude

the Landsat lineaments and the fold axes are not drawri from a con:llnon

Although Landsat lineaments may be useful guides for exploration, :In this

they apparently do not reflect the trends of structlll'al folds.

Spherical Distributions .

Statistical tests of directional data distributed in three dimensions have been

oped ot:lly in recent years, in part because the m a t h e ~ a t i c s of the distributions

very complicated. However, geologic problems that involve

·

·

. . .

· ·

.

'

tors are exceedingly common, and

we

should not shy away from the use of the

able statistical techniques for theit interpretation. Some of these.methods

matrix algebra, although· he matrices are no't large; and the' extraction df

·

values and eigenvectors. Tlie geometric' interpretation of eigenvectors' n ,

>C Olnn•t

in

Chapter 3

will

be of direct application. The mathematics ate closely

Spatial Analysis

multivariate procedlll'es described in Chapter 6. Here we deal with three physical

djnlensions; later we

will

apply the same steps to the analysis of multidimensional

data in which each dimension is a different geologic variable. ·

Examples of three-dimensional directional data in the Earth sciences include

measlll'ements of strike and dip taken for structlll'al analyses, vectorial measure

inents

of

the geomagnetic field, directional permeabilities measured

on

cores from

· petroleum reservoirs, measlll'ements of orientation a nd dip

of

crossbeds, and de-

terminations of crystallographic axes for petrofabric studies. ·

As

with two-dimensional data,

we

must

first establish a standard method of

notation.

can regard three-dimensional directional observations as consisting

ofvectors;

~ m c e

we are concerned primarily with their angular relationships, these

can be cons1dered to be of unit length. If all of the directional measurements from

' an area are collected together at a common origin, the tips of the un it vectors will

lie on the Slll'face of a sphere; hence the term spherical distribution

, Some oriented featlll'es do not have a sense of direction

md

can be referred

to as axes

Examples include the lines of intersections between sets .of dipping

planes, axes of revolution, md perpendiculars to plmes. In addition, it is some

times advmtageous to disregard the directional aspect

of

vectors

md

to treat

them

as.axes.

Z-axis

z --

-

-----

-

--

-

--

-

---

-

p

y

, Y:axis

0 ' -,, ,, _.

I

. __ .. .... .

', , ·I ••·

X

k.

X-axis M

Figure

~ - 2 5 . Notat

ional system for three-dimensional vector

OP in space

defined by

Caite-

lT. s1an . a x e ~ X

Y and Z. Angles between

OP

and the axes are

a,

b, and

c.. ,

Stmdard

mathematic<U notation utilizes three Cartesim ~ ~ o r c l l i J . a t e s to de

~ S } i a t o r in space

~ i g ;

~ 5 ) .

The

direction

.

of the vect.or OP is specified

DY

;}he·

o s i n

of ,the mgles Qetween the ve,ctor

md

each of the coordinate axes.

T i l ~ coordinates df the pointP

eire

equal .to ' ·'

.I_,

t

x

=cos

a

y

=cos

b

I

z =cos c

·

Since

the vector is considered to have unit e n ~ h ,

l· ..

x2 y2

z2

= 1

.

t'J

'I 1_

_

(5.51)



J

Statistics and Data nalysis

in

Geology

1

Appendi

Table

A.9.

Maximum likelihood estimates of the concentration parameter

Table A.lO.

Critical values of for Rayleigh s test for the resence

for calculated values of

R

(adapted from Batschelet, 1965;

of a preferred trend. From Mardia (

1972

). P

and Gumbel, Greenwood, and Durand, 1953).

Level

of Significance,

o

R

R

R

.10

.05

.025

.01

0.00

0.00000 0.35

0.74783 0.70

2.01363

Sample size,

, · , .

.01 .02000

.36 .77241 .71

2.07685

n

4 0.768 0.847

0.905

0.960

.02

.04001 .37 .79730

.72 2.14359

5 .677

.754

.816

.879

.03 .06003 .38 .82253 .73 2.21425 6 .618

.690

.753 .825

.04 .08006

.39 .84812

.74 2.28930

7

.572

.642

.702

.771

.OS

.10013 .40

.87408 .75 2.36930

8

.535 .602

.660

.725

.06

.12022 .41 .90043

.76 2.45490

9 .504

.569

.624

.687

.07

.14034 .42 .92720

.77 2.54686

1 .478

.540

.594

.655

.08

.16051 .43 .95440

.78 2.64613

11 .456

.516

.567

.627

.09

.18073

.44 .98207

.79 2.75382

12 .437

.494

.544

.602

.10

.20101 .45

1.01022 .80

2.87129

13

.420

.475

.524

.580

.11

.22134 .46 1.03889

.81 3.00020

14 .405

.458

.505

.560

.12

.24175 .47 1.06810

.82

3.14262

15

.391 .443

.489 .542

.13

.26223 .48

1.09788

.83 3.30114

16 .379

.429

.474

.525

.14

.28279 .49 1.12828

.84 3.47901

17 .367 .417 .460

.510

.15 .30344

.so 1.15932

.85

3.68041

18

.357 .405

.447

.496

.16 .32419 .51 1.19105 .86 3.91072 19 .348 .394 .436 .484

.17

.34503 .52 1.22350

;s?

4.17703

20 .339 .385 .425

.472

.18

.36599 .53

1.25672 .88 4.48876

21 .331

.375

.415

.461

.19 .38707

.54 1.29077

.89 4.85871

22 .323 .367

.405 .451

.20

.40828 .55

1.32570 .90 5.3047

23 .316 .359

.397 .441

.21

.42962 .56 1.36156

.91 5.8522

24 .309 .351

.389 .432

.22

.45110

.57 1.39842 .92 6.5394

25 .303

.344

.381

.423

.23

.47273

.58 1.43635

.93 7.4257

30 .277

. 315

.348 . .387

.24

.49453 .59

1.47543 .94 8.6104

35 .256

.292

.323

.359

.25 .51649

.60

1.51574 .95 10.2716

4 .240

.273

.302

.336

.26

.53863

.61 1.55738

.96 12.7661

45 .226

.257

.285

.318

.27

.56097

.62 1.60044

.97 16.9266

so

.214 .244

.270

.301

.28

.58350

.63 1.64506

.98 25.2522

.29 .60625

.64 1.69134

.Q9

50.2421

.30

.62922

.65 1.73945

1.00

.31

.65242

.66 1.78953

.32

.67587

.67 1.84177

.33

.69958

.68 1.89637

.34 .72356

.69 1.95357

Documents

Davis Directional Data