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Iibrary of Congress Cataloging-in-PublicationData
ISBN 978-0-19-605013-4
QE33.2.M3183 1989 551’.72-d~20 89-34891 CIP
29 28 2 26 25
24 3 22
on acid-free paper
1 Introduction 3
2 Univariate Description 1
Cumulative Frequency Tables and Histograms 12
Normal and lognormal Probability Plots 13
Summary Statistics 16
h Sca t t e rp lo t s 52
Cor relat ion Fun ct ions. Covariance Fun ct ions. and Vari
og rams 55
Notes 64
T he Di s t ribu tion of V 67
T he Di s t ribu tion of U 70
T he Di s t ribu tion of T 73
75
78
Sp at ia l Descr ipt ion of U 80
Moving W indow Sta t i s t ics 90
Notes 106 
Spa t ia l Cont inu i ty 93
6 The Sample Data Set 1 7
D a t a Errors 109  
T h e Sampl ing Hi story 10
Un ivariate Descr ipt ion of V 12
Un ivariate Descr ipt ion of U 123
27
129
Fur the r Read ing 138
T h e E f f e ct of t h e T T y p e
T h e V U Rela t ionship 27
S p a t a1 D esc rip t on
7 The Sample Data Set: Spatial Continuity 14
Sam ple h Sca t te rp lo ts and The i r Sum mar ie s 1 4 1  
An Out l ine of Spa t ia l C ont inu i ty Analysi s
43
Choos ing the Di s t ance Pa ramete r s 146
49
Choo s ing the Direct iona l T olerance 154 
54
Relat ive Variograms 163
T h e Covariance Function a nd the Correlogram 70 
Directional Covariance Functions for 173 
Cross Variograms 175 
Notes 181 
Further R eading 182
E s t i m a t i o n 84 
Weighted Linear Combinations 85 
Point an d Block E st imates 90 
Notes 194 
Further R eading 194 
9 R a n d o m F u n c t i o n M o d e l s 196 
T h e Necessi ty of Modeling 196 
Deterministic Models 198 
Probabalistic Models 200
Random Variables 202 
Parameters of a Random Variable 06 
Joint Random Variables 210
W eighted Linear Com binations of Random Variables 215 
Random Functions 218 
T he Use
An Example o the Use of a Probabalistic Model 31 
Further R eading 236 
1 G l o b a l E s t i m a t i o n 237 
Polygonal Declustering 38 
Declustering Three Dimensional Data 247 
Further R eading 248 
Inverse Dis tance M ethod s 57 
Search Neighborhoods 59 
Es t ima t ion Cr i t e r ia 26
Case Studies 266
12 Ordinary Kriging 278
T h e L a g r an g e P a r a m e t e r 84
Ordinary Kriging Using y or p
An Example of Ord inary Kr ig ing 9
An In tu i t ive Look a t O rdinary K r ig ing 99
Variogram Model Parameters 1
Compar i son of Ordina ry Kr ig ing to Othe r Es t ima t ion
M e t h o d s 313
Notes 321
Fur the r R ead ing 322
T h e Ran dom Func t ion Mode l and Unb iasedness
279
T h e R a n d o m F u n c ti o n M o d e l an d Error Variance 81
Minimiza t ion of t h e Error Variance
86
89
Ordin ary Kr ig ing and th e Model of Spa t i a l Con t inu i ty 296
1 Block Kriging 323 
Block E st imates Versus th e Averaging of Poin t Es t ima tes3 2 7
Varying th e Gr id of Point Locat ions With in a Block
327
T h e Block Kr ig ing S ys tem 24
A C a s e S t u d y 33
1 4 Search S trategy 338
Search Neighborhoods 39
Quadrant Search 344
Relevance of Nearby Samples and Sta t ionary Models 349
Notes 349
Cross Validation 352
Cross Validation as a Qu ant i ta t ive Tool 52
Cross Validation as a Qual i ta t ive Tool 359
Cross Validation as a Goal Oriented Tool
64 
369 
Rest r ic t ions on th e Var iogram Model 7
Positive Definite Variogram Models 72
Models in O ne Direction 75
Models of Anisotropy 377
M a t r i x N o t a t io n 386
Coo rd ina te Trans fo rma t ion by Ro ta t ion 88
T h e Linear Model of Coregionalization 39
Models For th e T h e Walker Lake Samp le Variograms 391
Notes 397
F u r t h e r R e a d i n g 398
17 Cokriging 4
T h e Cokriging Sys tem 4 1
A C okr ig ing Examp le 5
A C a s e S t u d y 4 7
Notes 16
F u r t h e r R e a d in g 416
18 Estimating a Distribution
Cumula t ive Dis t r ibut ions 18
T h e I n a de q u a cy of a Naive Dis t r ibut ion 19
T h e I n a de q u a cy of Poin t Es t ima tes 2
Cu mu la t ive Di s t ribu t ions . Cou n t ing and Ind icato r s 21
Est im at ing a G lobal Cum ula t ive Dis t r ibut ion 24
Es t ima t ing Oth e r Pa ram ete r s of t h e Global D i s t r ibu t ion 428
Est imat ing Local Dis t r ibut ions 433
Choosing Ind ica tor Thresholds 435
Case S tud ies 438
442
Case S tud y Resu l t s 448
Notes 456
19 Change O f Support 458 
T h e P r a c ti c a l I m p o r t an c e of the Su pp or t Ef fect 58
T h e Effect of Sup por t on Sum mary S ta t is t i c s 62
Correc t ing For th e S up po r t Effect 68
Trans fo rming One D is tr ibu t ion to Ano the r 69
Affine Correction 471
Dispersion Variance 476
Est ima t ing Dispersion Variances F rom a Variogram Model480
C a s e S t u d y : G l o b al C h a n g e of S u p p o r t
83
2 Assessing Uncertainty 489
Repor t ing Uncer ta in ty 92
Rank ing Uncer ta in ty 497
Case Stud y: Ran king Sample Da ta Configurat ions 99
Assigning Confidence Intervals 504
th e Global Mean 506
A Dubious Use o Cross Validation 14
Local Confidence Intervals 17
Variograms 519
Notes 523
Description an d D a ta Analysis 25 
Es t ima t ion 528
Globa l Es t ima t ion 528
Local Es t imat io n 528
Accom mod at ing Dif ferent S amp le Su pp or t 30
Search St ra tegy 531 
Incorpora t ing a Trend 31 
Cross V alidation 533 
Using Other Variables 35
O the r Uses of ndica tors 36 
Bibliography 5 8
T h e Digita l Elevation M odel 42 
T h e E x h a u s t i v e D a t a S et 45 
Art i fac ts 545 
B Continuous Random Variables 5 48 
T h e Pro bab i l i ty Dis tr ibut ion Funct ion 48 
Paramete r s of a Cont inuous R andom Vari ab le 549
Join t R an do m Variab les 5
M argina l Dis t r ibut ions 5
Con di t iona l Dis t r ibut ions 52 
Paramete r s of Join t R and om Variab les 552 
Index 55
INTRODUCTION
This book presents an introduction to the set of tools that has become
known commonly as
a
nomena; many others can be used to develop quantitative answers
to specific questions. Unfortunately most classical statistical meth-
ods make no use of the spatial information in earth science data sets.
Geostatistics offers
essential feature of many natural phenomena and provides adaptat ions
of classical regression techniques to take advantage of this continuity.
The presentation of geostatistics in this book is not heavily mathe-
matical. Few theoretical derivations or formal proofs are given; instead
references are provided to more rigorous treatments of the material.
The reader should be able to recall basic calculus and be comfortable
with finding the minimum of a function by using the first derivative
and representing a spatial average as an integral. Matrix notation is
used in some of the later chapters since it offers a compact way of writ-
ing systems of simultaneous equations. The reader should also have
some familiarity with the statistical concepts presented in Chapters 2
and 3
Though we have avoided mathematical formalism the presentation
is not simplistic. The book is built around a series of case studies on
a distressingly real data set. As we soon shall see analysis of earth
science data can be both frustrating and fraught with difficulty. We
intend to trudge through the muddy spots stumble into the pitfalls
 
A n Introduction to Applied Geostatistics
tackled a geostat is t ical s tu dy will sy m pa thiz e with us in ou r man y
dilemmas.
O ur case s tudies d if fe rent f rom those t h a t prac t i t ioners encoun ter
in only one aspect ; throughout our s tudy we wil l have access to the
correc t answers. T h e d a ta se t w i th which we perform th e s tudies is in
fact a sub set of a much larger , completely known d a ta set . T hi s gives
us a yardstick by which we can me asure the success of several different
approaches.
A warn ing is appropr i a t e here. The solutions we propose in the
various case s tudies a re par t icu lar to th e d a ta se t we use. I t i s no t o ur
intent io n t o propose these as general recipes. T h e hal lmark of a good
geostat is t ical s tu dy is customizat ion of th e approa ch t o th e problem at
hand. All we in tend in these s tudies is t o cu l t iva te an und ers tand ing of
wh at various geos tat is tica l too ls can d o an d , more im por tan t ly , wh at
the i r l imi ta t ions a re .
The Walker Lake Data Set
T h e focus of this book is a d a ta se t tha t was der ived f rom a digi ta l
elevat ion model f rom th e western United Sta tes; the W alker Lak e are a
in Nevada.
We will not be using the original elevation values as variables in
ou r case s tudies . T h e variab les we d o use , however, a re re la ted t o th e
elevat ion and, as we shal l see, their maps exhibi t features which are
related to th e topog raphic fea tures in F igure 1.1. For this reason, we
will be referring t o specific su b areas within th e W alker Lake are a by
th e geographic names given in Fig ure 1.1.
T h e original digi ta l elevation m odel contained elevat ions for abo ut
2 million points on a regular gr id . Th ese elevat ions have been t rans-
formed to produce a data set consis t ing of three var iables measured
a t e a c h of 78,000 poin ts on a 260
x 300 rectang ular gr id . T h e f irst
t w o variables a re cont inuous an d their values rang e from zero to sev-
eral thou sand s. T h e third var iable is discrete an d i ts value is e i ther
one or two. Detai ls on how t o ob tain th e digita l e levat ion model a nd
reproduce th is d a ta se t a r e g iven in A ppend ix A.
W e have tried to avoid w riting a book th at is too specific
t o one fie ld of application.
For this reason the var iables in the
Walker Lake da ta se t a re re ferred t o anonymously as V U a n d T. Un-
 
IlawUlane
Introduction
5
NEVADA
l
A location map of the Walker Lake area in Nevada The small rectangle
on the outline of Nevada shows the relative location of the area within the state
The larger rectangle shows the major topographic features within the area
in; this reflects both the historical roots of geostatistics as well as the
experience of the authors. The methods discussed here however are
quite generally applicable to any dat a set in which the values are spa-
tially continuous.
The continuous variables V and U ould be thicknesses of a geo-
logic horizon or the concentration of some pollutant; they could be soil
strength measurements or permeabilities; they could be rainfall mea-
surements or the diameters of trees. The discrete variable T can be
viewed as a number that assigns each point to one of two possible cate-
 
An Introduct ion to Appl ied Geostat is t ics
species; i t could s ep ar at e different rock typ es or different soil litholo-
gies; i t cou ld record som e chemical difference such as th e presence or
abs enc e of a part icula r e lement .
For th e sak e of convenience an d consistency we will refer t o V a n d
U as co ncent ra t ions of some mater ia l and will g ive bo th of th em uni t s
of pa rts per mill ion pp m ). We will t reat T as an indica tor of two
types that will be referred to as type a n d type 2 Finally, we will
assign u ni ts of meters t o our gr id even tho ug h i ts or iginal dimensions
a re much l a rger t han 260 x 300 m2
T h e Walker Lake da ta se t cons is ts of V U a n d T m e a s u r e m e n ts a t
eac h of 78 ,000 points on a x 1 m2 grid. From this extremely dense
d a t a s e t a subse t of 470 sam ple poin ts has been chosen t o represent a
t yp ica l s ample d a t a se t . To dis t inguish be tween these two da ta se t s ,
th e comple te se t of al l information for th e 78 ,000 poin ts is called th e
exhaustive d a ta se t , while th e smaller subse t of 470 poin ts is cal led th e
sample d a t a s e t.
Goals of the Case Studies
Using th e 470 samples in th e samp le d a t a se t we will address th e fol-
lowing problems:
1. T he desc rip tion of t h e imp or t an t f ea tu res of th e da ta .
2 T h e e s t i m a ti o n of an average value over a l a rge a rea .
3 T he es tima t ion of an unknow n value at a par t icu lar loca t ion .
4 T h e e s t i m a ti o n of a n av erage value over sm all areas .
5 T h e use of t he ava i lab le sampl ing t o check th e per formance of a n
est im at ion methodology.
6. T h e use of samp le values of one variable to im prove th e est ima-
t ion of another var iable .
7 T h e es t ima t ion of
a
distribution of values over a l a rge a rea .
8. T h e es t ima t ion o f a d i s t r ibu t ion of values over small area s.
9. T he es t ima t ion of a distribution of block averages.
 
The first question despite being largely qualitative is very impor-
tant. Organization and presentation is a vital step in communicating
the essential features of a large data set. In the first part of this book
we will look a t descriptive tools. Univariate and bivariate description
are covered in Chapters 2 and 3 In Chapter 4 w e will look at various
ways of describing the spatial features of a data set. We will then take
all of the descriptive tools from these first chapters and apply them
to the Walker Lake dat a sets. The exhaustive at a set is analyzed in
Chapter 5 and the sample data set is examined in Chapters 6 and 7.
The remaining questions all deal with estimation which is the topic
of the second part of the book. Using the information in the sample
data set we will estimate various unknown quantities and see how well
we have done by using the exhaustive dat a set to check our estimates.
O u r approach to estimation as discussed in Chapter 8 is first to con-
sider what i t is we are trying to estimate and then to adopt a method
that is suited t o tha t particular problem. Three important consider-
ations form the framework for our presentation of estimation in this
book. First do we want an estimate over a large area or estimates for
specific local areas? Second are we interested only in some average
value or in the complete distribution of values? Third do we want our
estimates to refer to a volume of the same size as our sample data or
do we prefer to have our estimates refer to a different volume?
In Chapter 9 we will discuss why models are necessary and intro-
duce the probabilistic models common to geostatistics. In Chapter
10 we will present two methods for estimating an average value over
a large area. We then turn to the problem of local estimation. In
Chapter 11 we will look at some nongeostatistical methods that are
commonly used for local estimation. This is followed in Chapter 12
by a presentation of the geostatistical method known as ordinary point
kriging The adaptation of point estimation methods to handle the
problem of local block estimates is discussed in Chapter 13.
Following the discussion in Chapter 14 of the important issue of
the search strategy we will look a t cross validation in Chapter 15
and show how this procedure may be used to improve an estimation
methodology. In Chapter 16 we will address the practical problem of
modeling variograms an issue that arises in geostatistical approaches
to estimation.
In Chapter 17 we will look at how to use related information to
 
A n Introduction to Applied Geostatistics
pract ice when one variable is undersampled . W hen we ana lyze th e
s a m p l e d a t a s e t in C h a p t e r 6, we will see tha t th e measurements of the
second variable,
U
a re miss ing a t m any sample locat ions. T h e me thod
of cokriging presented in Ch ap ter 17 allows us to inco rpo rate th e mo re
a b u n d a n t V sample values in the estimation of U t ak ing advan tage
of th e re la tionship be tween th e two t o improve our e s t imat ion of t h e
more sparsely samp led U variable.
T h e es t imat ion of a complete distribution is typically of more use
in prac t ice than i s the es t imat ion of a single average value. In m an y
applicat ions one is interested not in an overal l average value but in
th e average value abo ve som e specified th reshold. Th is th reshold is
often some ext reme va lue and th e es t imat ion of t he d i s t r ibu t ion above
ex tre m e values cal ls for different techniques th an th e est im atio n of t h e
overall mean. In Cha pter 18 we wil l explore the est imat ion of local
and global dis t r ibut ions. We wil l present the indicator approach o n e
of several adv anc ed techniques developed specifically for t h e est im atio n
of local distributions.
A further complication arises if we want our es t im ates t o refer to
a volume different from t h e volume of ou r samples. T hi s is commonly
referred to as t h e
support
problem and frequently occurs in practical
appl icat ions. For exam ple, in a model of a petroleum reservoir o ne does
no t need e stim ated permeabilit ies for core-sized volumes bu t ra th er for
much la rger blocks. In a mine, one will be mining and processing vol-
umes much la rger th an th e volume of t he samples tha t a r e typ ica lly
available for a feasibi l i ty s tudy. In Chapter 19 we wi l l show tha t the
d i s t r ibu t ion of poin t values is no t th e same as th e d is t r ibu t ion of av-
erage block values an d present tw o me tho ds for accou nt in g for this
discrepancy.
I n C h a p t e r 20 we will look a t t h e assessment of uncer ta in ty , a n i ssue
th at is typical ly muddied by a lack of a c lear objec t ive meaning for th e
various uncertainty measures that probabi l is t ic models can provide.
W e will look a t several com mo n problems, discuss how o ur p robabilist ic
model might provide a relevant answer , and use th e exhaus t ive d a ta
set t o check th e performan ce of various m ethod s.
The f ina l chapter provides a recap of the tools discussed in the
boo k, recal ling th eir s t reng ths and their l imitat ions. Since this book
a t t e m p t s a n i n tr o du c ti o n to bas ic methods , many advanced methods
have not been touched, however, the types of problems that require
 
Before we begin exploring some basic geostatisticd tools we would
like to emphasize that the case studies used throughout the book are
presented for their educational value and not necessarily to provide a
definitive case study of the Walker Lake data set. It is o u r hope that
this book will enable a reader to explore new and creative combinations
of the many available tools and to improve on the rather simple studies
we have presented here.
UNIVARIATE DESCRIPTION
D at a spea k most c learly when they a r e organized . M uch of s ta t i s t ics ,
therefore , dea ls w i th th e organiza t ion , p rese nta t ion , an d s um m ary of
da ta . I t is hoped th a t much of th e ma ter ia l in these ch ap ters will
a l r eady b e f ami li ar t o the r eade r . Thou gh some not ions pecu l i a r t o
geos ta t i s tics will b e in t roduced , th e presenta t ion in th e fo llowing cha p-
ters is in tende d pr im ari ly as review.
In th is ch apter we will dea l wi th univar ia te descr ip tion . In th e
following ch ap te r we will look a t ways of descr ibin g the re lat ionships
between pairs of variables. In Chapter 4 we inco rpora t e th e loca t ion
of the d a ta an d cons ider ways of descr ib ing th e spa t ia l fea tures of th e
d a t a s e t .
T o mak e i t easy to fo llow and check the various ca lcu la t ions in
the next th ree chapters we wi l l use a smal l 10 x 10 m 2 p a t c h o f t h e
exh aus t ive d a ta se t in a ll of our examples [l]. n the se ex amples, all of
t h e U a n d V values have been rounded off t o t he nea re s t i n tege r . Th e
V values for these 100 poin ts a r e shown in F igure 2.1. T h e g oa l of this
cha p te r w ill b e to desc r ibe the d i s t r ibu t ion of these 100 values.
Frequency Tables and Histograms
O n e of th e most c omm on an d usefu l p resenta t ions of d a ta se t s is t he f re -
quency t ab le and i ts co r re spond ing g raph , t he h i s tog ram. A frequency
 
@pm)
Figure 2.2 Histogram of the 1 0 0 selected V data.
classes. Table 2.1 shows a frequency table that summarizes the 100 V
values shown in Figure 2.1 .
The information presented in Table 2.1 can also be presented graph-
ically in a histogram, as in Figure 2.2. It is common to use a constant
class width for the histogram so that the height of each bar is propor-
 
Table 2 1
of
1 1
0 0
2 2
13 13
110
120 < V
140 5
Cumulative F’requency Tables and Histograms
Most s ta t i s t ica l tex ts use the convent ion th a t d a t a a re ranked in as -
cending order t o pro duc e cum ulat ive frequency tables an d descr ipt ions
of cum ulat ive frequency d is t r ibut ion s. For man y ea r th science appl i -
cat ion s, such as ore reserves an d pol lution s tudie s , th e cum ulat ive fre-
quency above a lower limit is of mo re in te res t . For such s tudies , cum u-
la t ive f requency tab les a nd h istograms may be prepared af te r ranking
th e d a ta in descending order.
In Tab le
2 2
we have taken the information from Table 2.1 a n d
presented it in cum ula t ive form. R ath er tha n record th e numbe r of
values within cer tain classes , we record the total number of values
below certain cutoffs [3]. The corresponding cumula t ive h is togram,
shown in Figure 2.3 , is a nondecreasing function between 0 a n d 100 .
T h e percent f requency and cum ula t ive percent f requency forms are
 
width of 10 ppm.
Cumulative frequency table of the 100 selected V values using a class
Class Number Percentage
Normal and Lognormal Probability Plots
Some of the estimation tools presented in part two of the book work
better if the distribution of data values is close to a Gaussian or nor
mal distribution. The Gaussian distribution is one of many distribu-
tions for which a concise mathematical description exists [4]; lso it
has properties that favor its use in theoretical approaches to estima-
tion. It is interesting therefore to know how close the distribution
of one’s data values comes to being Gaussian. A normal probabil-
ity plot is a type of cumulative frequency plot that helps decide this
question.
On a normal probability plot the y-axis is scaled in such a way that
the cumulative frequencies will plot as a straight line if the distribution
is Gaussian. Such graph paper is readily available at most engineering
supply outlets. Figure 2 4 shows a normal probability plot of the 100
 
A n Introduction to Applied G eostat is t ics
100
75
5
25
n
b p m )
Figure 2.3 Cumulative histogram of the 100 selected V data .
f I I
V b p m )
Figure 2.4 A normal probability plot of the 100 selected V data . The y-axis has
been scaled in such a way that the cumulative frequencies will plot as a straight
line if the distribution of V is G aussian.
th at a l thou gh most of th e cum ulat ive frequencies plot in a relatively
strai gh t l ine, t h e smaller values of V depar t f rom th is t rend .
M any variables in ea r th science d a ta sets have dis t r ibu t ions th at
ar e not even close to normal. I t is common t o have m any q ui te smal l
values and a few very large ones. In Chapter 5 we will see several ex-
amples of th is type f rom th e exhaus t ive Walker Lake d a ta se t . Thou gh
th e norm al d is tr ibut ion i s of ten inappro pr ia te as a model for this type
 
t
15
Figure 2.5 lognormal probability plot of the 100 selected V data. The y-axis is
scaled so that the cumulative frequencies will plot as a straight line if the distribution
of the logarithm of V is Gaussian.
ma1 distribution can sometimes be a good alternative. A variable
is distributed lognormally if the distribution of the logarithm of the
variable is normal.
By using a logarithmic scale on the x-axis of a normal probability
plot one can check for lognormality. As in the normal probability plot
the cumulative frequencies will plot as a straight line if the data values
are lognormally distributed. Figure 2 5 shows a lognormal probability
plot of the 100 V values using the same information that was used to
plot Figure 2.4. The concave shape of the plot clearly indicates that
the values are not distributed lognormally.
Assumptions about the distribution of data values often have their
greatest impact when one is estimating extreme values. If one intends
to use a methodology that depends on assumptions about the distri-
bution one should be wary of casually disregarding deviations of a
probability plot a t the extremes. for example it is tempting to take
the normal probability plot shown in Figure 2 .4 as evidence of normal-
ity disregarding the departure from a relatively straight line for the
smaller values of V . Departures of a probability plot from approximate
 
A n Introduction to Applied Geostat is t ics
overlook when t h e rest of th e plot looks relatively stra igh t. However,
the es t imates der ived us ing such a “close f i t ted” dis t r ibut ion model
m ay be v astly different from reality.
Probabi l i ty plots are very useful for checking for the presence of
mu lt iple pop ulat ions. Altho ugh kinks in th e plots d o no t necessari ly
indica te mul t ip le popula t ions , they represent changes in the charac-
ter is t ics of th e cu mu lat ive frequencies over different intervals and th e
reasons for this should be explored.
Choos ing a theoret ical mode l for t h e dis t r ib ut io n of d a ta values is
not always a necessary s te p pr ior t o es t imat ion , so one should not read
too much in to a probabi l i ty p lot . T h e s t ra ightness of a l ine on a proba -
bility plot is no gua ran tee of a good e s t im ate and th e crookedness of a
l ine should not condem n dis t r ibu t ion-based approaches t o es t imat ion .
Cer t a in m e thods l ean more heav ily o n a s sumpt ions ab ou t th e d i s tr i-
bu t ion tha n d o o the rs . Some es tima t ion too ls bu i lt on a n a s sum pt ion
of normal i ty ma y s t il l be useful even when th e da ta a re no t normally
dis t r ibuted .
Summary Statistics
T h e i m p o r t a n t f ea tu r es of most h is tograms can be captured by a few
sum m ary s ta ti s t ics . T h e sum m ary s ta ti s t ics we use here fa ll in to three
categories: measures of locatio n, measures of spread an d measures of
shape .
T h e s ta t i s t ics in th e f ir st g roup g ive us informat ion ab ou t where
various parts of t he d i s tr ibu t ion lie. T he mean , t h e med ian , and the
mo de can g ive us some idea where t he center of the distribution lies.
The loca t ion of other par t s of the d is t r ibu t ion are g iven by var ious
quant i les . T h e second group includes the var iance , th e s tan da rd de-
v ia tion , and th e in t e rqua r ti l e r ange. These a re used t o descr ibe th e
variabi li ty of th e d a ta values. T h e sh ap e of th e dis t r ibu t ion is de-
scribed by the coefficient of skewness and the coefficient of variation;
th e coefficient of skewness provides info rm ation on th e sy m m et ry while
the coefficient of variation provides information on the length of t h e
tai l for cer ta in types of dis t r ibut ions . Tak en toge th er , these s ta t i s -
t ics provide a valuable sum ma ry of the informat ion conta ined in th e
his tog ram .
a
Measures of Location
Mean. T h e m e a n , m, i s the a r i thm et ic average of t h e d a t a values [5]:
T h e number o f d a t a is n a n d 21 . ,x, a r e t h e d a t a v alu es. T h e m e a n
of o u r 100
V values is 97.55 ppm.
Median. T h e m e d ia n , M is the midpoint of the observed values if
they ar e arran ged in increasing order . Half of th e values are below th e
med ian and half of th e values a r e above the med ian . Once the d a t a
a re o rde red so t h a t x1 5 5 . . 5 x t h e med ian can be ca l cu lat ed
f rom one of th e fol lowing equat ions:
3-' if n is odd
+ 2 if n is even
= { xsi
T h e median c an easi ly b e read f rom a proba bi l i ty plot. Since th e
y-axis records th e cu mu la t ive f requency, the median i s th e va lue on t h e
x-axis th a t cor responds t o 50 on th e y-axis (F igure 2 .6 ) .
Bo th th e mean and th e med ian a re measures of th e loca t ion of the
 
18
A n In t rdu c t ion to Appl ied G eos tat i st i c s
values. If the 145 ppm va lue in ou r d a t a se t had been 1450 p p m , t h e
mea n would change t o 110.60 ppm . T h e med ian , however, would be
unaffected by this ch ang e bec ause i t dep end s only on how m an y values
a r e a b o v e or below i t ; how much above or below is no t considered.
For t he 100 V values tha t appear in F igure 2.1 the median i s
100.50 p p m .
Mode. T h e mo de is th e va lue th a t occurs m os t f requently . T h e c lass
wi th th e ta lles t ba r on th e h is togram gives a qu ick idea where th e m ode
is. From th e h is togram in F igure 2 .2 we see th a t th e 110-120 pp m class
ha s the m ost values. W ithin this c lass, the value 111 pp m occurs more
t im e s t h a n a n y o t h e r .
O ne of the drawbacks of th e mo de is th a t i t change s wi th t he pre-
cis ion of th e da ta values. In Figure
2.1
V
values
t o th e neares t in teger . Had we kept two dec imal p laces on a l l our mea-
su rem en t s , no two wou ld have been exac t ly the sam e and th e mode
could then be a ny one of 100 equally common values. For this reason,
t h e m o d e is no t par t icular ly useful for da ta sets in which t he measure-
m en ts hav e several significant digits. In such cases, when w e speak of
th e m ode we usua lly m ean some approx ima te va lue chosen by f inding
th e t a ll e s t ba r on a h i s tog ram. Some p rac ti ti one r s in t e rp re t t h e m ode
t o be th e ta lles t b ar i tself .
Minimum. T h e sma lle st va lue in t he d a t a se t is t he m in imum . In
m an y p rac t ica l s i tua t ions the smal les t va lues a re recorded s imply as
be ing be low som e de tec t ion limi t . In such s i tua t ions , i t m at te rs l i t t l e
for descr ip t ive purposes w hether th e minim um is given as 0
or as some
arb i t rar i ly small value. In som e est ima t ion m eth od s, as we will discuss
in la te r cha pte rs , i t is convenient to use a nonzero value (e.g. , half the
de tec t ion l imi t ) or t o assign s l ight ly different values to those d a ta th a t
were below th e detec t ion l imit . For ou r 100 V values , the minimum
value is 0 p p m .
Maximum. T h e l a r ge s t v alu e in t h e d a t a s e t is t h e m a x i m u m . T h e
ma x imum of ou r 100 V
values is 145 p p m .
Lower and Upper Quartile. In th e same way th a t t he med ian sp l it s
th e d a t a i n to halves, t he qua r t il e s sp l it t he d a t a in to qua r t e r s . If t h e
d a t a va lues a re a r ranged in increas ing order , then a q u a r t e r of t h e d a t a
falls below the lower
or
f i rs t quar t i le , Q1, and a q u a r t e r of t h e d a t a
 
Figure 2.7 The quartiles of a normal probability plot.
As with the median, quartiles can easily be read from a probability
plot. The value on the x-axis, which corresponds to 25 on the y-
axis, is the lower quartile and the value tha t corresponds to 75 is the
upper quartile Figure 2 .7 ) . The lower quartile of our 100 V values is
81.25 ppm and the upper quartile is 116.25 ppm.
Deciles Percentiles and Quantiles. Th e idea of splitting the data
in to halves with the median or into quarters with the quartiles can be
extended t o any other fraction. Deciles split the da ta into tenths. One
tenth of the data fall below the first or lowest decile; two tenths fall
below the second decile. The fifth decile corresponds to the median. In
a similar way, percentiles split the data into hundredths. The twenty-
fifth percentile is the same as the first quartile, the fiftieth percentile
is the same as the median and the seventy-fifth percentile is the same
as the third quartile.
Quantiles are a generalization of this idea to any fraction. For ex-
ample, if we wanted to talk about the value below which one twentieth
of the da ta fall, we call it 4.05 rather than come up with a new -ile name
for twentieths. Just as certain deciles and percentiles are equivalent to
the median and the quartiles, so too can certain quantiles be written as
one of these statistics. For example 4 25 is the lower quartile, 4.5 is the
 
A n Introduction to Applied Geos a t istics
qua nt i les ra th er th an deciles an d percent iles , keeping only t h e mediar ,
and th e tw o quar t i les as spec ia l measures of location.
Measures of Spread
Variance. T h e va ri ance, u 2 s given by [6]:
n
m
kl
It is the average squared difference of the observed values from their
me an. Since i t involves squared differences, th e var ianc e is sensi t ive t o
erra t ic high values . T h e var iance of th e 100 V values is 688 p p m 2 .
Standard Deviation. T h e s t a n d a r d d e v i a t i o n , 0 s s imply the
squ are root of th e variance. I t is of ten used instead of th e var iance
s ince i t s un i ts a r e the sam e as th e uni t s of the var iab le be ing descr ibed .
For the 100 V values th e s tan da rd de via t ion is 26.23 p p m .
Interquartile Range. Another usefu l measure of the spread of the
observed va lues is th e in te rq uar t i le range . T h e in te rquar t i le rang e or
IQR, is th e difference between t h e upp er and lower quar t i les and is
given by
(2 .4 )
Unlike th e va ri ance an d th e s t and a rd dev ia tion , t he in t e rqua r t i l e r ange
does not use the mean as th e cen ter of th e d is t r ibu t ion , and is therefore
often preferred if a few erratically high values strongly influence the
mea n. T h e in te rqu ar t i le range of o u r 100 V values is 35.50 p p m .
Measures of Shape
Coefficient of Skewness. One fea tu re of t he h i s tog ram th a t t he
previous s ta t i s t ics do not ca p tu re is i ts symmet ry . T h e mos t commonly
used s ta t is t ic for summar iz ing the symm et ry is a qu an t i ty ca lled t he
coefficient of skewness which is defined as
; ;=l .i 77-43
coefficient of skewness =
u
T h e nu m era tor is th e average cubed d if fe rence be twee n th e da ta va l-
ues an d the ir m ean , and th e denomina to r is t h e cube of t he s t an da rd
devia t ion .
Univar ia te D escr ip t ion 21
T h e coefficient of skewness suf fers even more than the mean and
var iance f rom a sensi t ivi ty t o errat ic high values. A single large value
can heavily influence th e coefficient of skewness since the difference
be tween each da ta va lue and th e mean is cubed .
Quite of ten one does not use the magnitude of the coeff ic ient of
skewness b ut ra t he r only i t s s ign t o descr ibe th e symmetry . A pos-
i t ively skewed his togram has a long tail of high values to th e r igh t ,
making th e median less th an th e mean. In geochemica l d a t a se ts ,
posit ive skewness is typical when the variable being described is t h e
conce nt ra t ion of a minor e lement . If there is a long tail of sm all values
as
ma jor elem ent concen trat ions, th e his togram is negatively skewed. If
th e skewness is close t o zero, th e his togram is approxim ately sym m etr ic
and th e med ian i s close to th e mean .
For t h e 100 V values w e ar e describing in this c ha pt er t h e coefficient
of skewness is close to zero ( -0 .779) , indica t ing a dis t r ibut ion tha t i s
only s l ight ly asymmetr ic .
Coefficient of Variation. T h e coefficient of va riat ion , CV , s a s t a t i s -
t ic th at is of ten used as an a l t e rna t ive to skewness t o desc r ibe th e shap e
of th e dis trib uti on . I t is used prim arily for distrib utio ns whose values
a re al l posi t ive an d whose skewness is a lso posi tive; thou gh i t ca n be
ca lcu la ted for o ther types of distr ibut ions, i ts usefulness a s a n index of
sh ap e becomes questionable. I t is defined as t he r at io of t,he s ta nd ard
d e v ia ti o n t o t h e m e a n [7]:
c v =
m
If est imation is the f inal goal of a study, the coefficient of varia-
t ion c an provide som e warning of upcoming problems. A coefficient of
var ia t ion grea ter t ha n one indica tes the presence of some er ra t ic h igh
sample values th a t may have a s ignif icant imp act on th e f ina l es t imates .
T h e coefficient of variatio n for ou r 100 V values is 0.269, which
ref lects th e fact t h a t th e h is togra m does not have a long tail of high
values.
Notes
[ l ] T h e coordina tes of th e corners of th e 10 x 10 m 2 pa tch used to i l lus-
t ra te th e var ious descr ip t ive tools a re (11,241) , (20,241) , (20,250 ) ,
a n d ( 11,250) .
22
A n Introduction to Appl ied Geos ta t i s t i cs
[2] If the c lass widths a re var iab le i t i s impor tan t to remember tha t
on a histogram it is th e a re a (no t th e he ight ) of the b ar t h a t i s
propor t iona l t o the f requency.
[3] T h e exam ple in t h e tex t is des igned to mak e i t easy to follow how
Tab le 2.2 re la tes to Table 2.1. Thou gh th e choice of classes is nec-
essary for a frequency tab le and a histogram, it is not required for
cum ulat ive frequency tables or cumulat ive his tograms. Indeed, in
practice o ne typically chooses cutoffs for t h e cu m ula tive frequencies
th a t correspond to th e ac tu a l d a ta values .
[4 ] For a description of th e norm al dis t r ibu t ion an d i ts prop ert ies see:
Johnson , R. A. and Wichern , D . W .
Applied M ult ivariate Stat is
tical Analysis.
Englewood Cliffs, New Jersey: Prentice-Hall , 1982.
[5] Though the a r i thmet ic average i s appropr ia te for a wide variety
of appl icat ions, there are important cases in which the averaging
process is not ar i thmetic , For example, in fluid flow studies the
effective pe rm eab ility of a strat if ied sequence is th e ar i th m etic mean
of th e permeabil it ies within th e various s t ra ta if th e flow is parallel
t o th e s t r a ta . If th e flow is perpendicular to the s t r a t a , however,
th e harmonic mean , mH , is more appropr ia te :
1 1 1 l
k g mH --EK= l
where the k are the permeabi l i t ies of t h e n s t r a t a . For th e case
where th e flow is neither stric tly parallel nor str ictly perp en dic ula r
t o the s t ra t if i ca t ion , or where the different facies are not clearly
strat ified , some stud ies suggest th at th e effective permeabili ty is
close t o th e geometr ic me an, mG:
[6] Some readers will recall a formula for a rom classical statist ics
th a t uses instead of $. This classical formula is designed to
give an unbiased es tim ate of th e population variance i t h e d a t a
ar e uncorrelated. T h e formula given here is inten de d only t o give
th e sam ple variance. In la ter ch ap ter s we will look at th e problem
 
a percentage
New York:
Wiley, 1973.
Koch, G . and Link, R. Statistical Analysis of Geologicul Da ta . New
York: Wiley, 2 ed., 1986.
Mosteller, F and Tukey, J . W . Da ta Analysis and Regression. Read-
ing, Mass.: Addison-Wesley, 1977.
Ripley, B . D. S p t i a l S t at is ti cs . New York: Wiley, 1981.
Tukey, J . Exploratory Data Analysis. Reading, Mass.: Addison-
Wesley, 1977.
BIVARIATE DESCRIPTION
T h e univar ia te tools d iscussed in th e las t chap ter can b e used to d e-
scrib e th e distrib utio ns of ind ivid ual variables. W e get a very l imited
view, however, if we analyze a mul t iva r ia t e d a t a se t one va ri ab le a t a
t ime. Som e of the most imp or tan t and in teres ting fea tures of ear th
science da ta se ts are th e relat ionships an d dependencies betw een vari -
ables.
T h e Walker Lake d a ta se t conta ins two cont inuous variab les . F ig-
u re 3 .1 shows the 100 V values we saw in Fig ure 2.1 a long wi th th e U
values at t h e s a m e 100 locat ions. In this chap ter we look a t ways of
describing th e relat ionsh ip between these tw o variables .
Comparing Two Distributions
In th e analysis of ea r th sc ience d a ta se t s we will o f ten w ant to com pare
two d is t ribu t ions . A presenta t ion of the i r h i s tograms a long wi th som e
su m m ary s ta t is t ics will reveal gross differences.
Unfortunately, i f the
two d is t r ibu t ion s a re very s imilar , th i s meth od of com parison will not
b e helpful in uncovering th e inte restin g su btl e differences.
T h e h is tograms of the V a n d
U
values shown in Figure 3.1 a r e
given in Figure 3.2, and the i r s ta t i s t ics a re presented in Table 3 .1
Th ere a re som e ra the r m ajor d iffe rences be tween th e d is t r ibu t ions of
the two var iables .
T h e U distr ib ut io n is posit ively skewed; th e V
dis t r ibut ion , on the o ther hand, i s negat ive ly skewed. Also, t h e V
 
114 130
8 1
110 131
149 7
3 2
83  4 947 195
88
 0
21 8 27 27
21
15 16 16
4
72 8p 3
55 1 1 3
15 12 24 27 30 a 2 18 18 18
Figure 3.1 Relative location map of the 100 selected V and U data. V values are
plotted above the u+ symbol and U are below.
than f ive t imes tha t of U . T h e V median and s t and a rd dev ia t ion a re
a lso g rea t e r tha n the i r U counterpar t s .
T h e s t a t is t i ca l sum ma ry p rov ided in Tab le 3.1 al lows us to com-
pare , am on g o the r th ings , th e medians and the quar t i les of the two
dis t r ibut ions . A mo re com ple te compar ison of the var ious quant i les
is given in Ta bl e 3.2, which shows th e V a n d U quantiles for several
cum ulat ive frequencies . T h e For example, the f i rs t entry te l ls us that
t h a t 5 of t h e V values a re below 48.1 pp m w hile 5 of the U values
fall below 3.1 ppm . T h e medians and quar t i les we saw earl ie r in Ta-
ble 3.1 a re also included in Ta ble 3.2. T h e fi rs t quart i le , 81.3 p p m for
V a n d 14.0 p p m for U corresponds to t he 0 .25 quan t i le ; th e m edian ,
100.5 ppm for V a n d 18.0 ppm for U , corresponds to q . 5 ; a n d t h e u p p e r
quart i le , 116.8 pp m for V and 25.0 ppm for U , corresponds to 4.75.
For a good visual comparison of two dis t r ibut ion s we can use a
graph cal led a q-q plot. T h i s is commonly used when there is s o m e
reason to expect tha t the d is t r ibu t ions are s imi la r . A q-q plot is a
 
@) 20
v bpm)
Y
a
Figure 3 2 T h e h i s to g r am of t h e 100 V values in a) a n d of t h e c o r r es p o n d i n g 100
U values in
(b).
one ano the r . T h e in fo rma t ion con ta ined in Tab le 3 . 2 is presented as
a q-q plot in Figure 3.3. The quan t i l e s of t h e V dis t r ibut ion serve
as th e x-coordina tes while those of t h e U dis t r ibut ion serve as t h e y -
coordina tes . If th e two d is t r ibu t ions be ing compared have the sam e
num ber of da ta , then th e calcu la tion of the q uant i les of each d is t r ibu-
t ion is no t a necessary s te p in m aking a q-q plo t . Ins tead , one can sor t
th e d a ta va lues f rom each d is t ribu t ion in ascending order and p lo t the
correspo nding pairs of values.
A q-q plot of tw o ident ical dis t r ibut ions will plot as th e s t ra ig ht line
x = y. For
dis t r ibut ions th a t a r e very s imi la r, th e smal l depar tures of
t h e q-q plot from t h e l ine z = y will reveal where they differ. As we
have a l ready noted , th e d is tr ibu t ions of the V a n d U values within our
selected a re a are very different; therefore, th eir q-q plot d oes n ot come
 
27
Table 3.1 Statistical summary of the V and values shown in Figure 3 . 1 .
V U
Frequency V U Frequency V U
0.05 48.1 3.1 0.55 104.1 19.0
0.10 70.2 7.0 0.60 108.6 20.0
0.15 74.0 8.1 0.65 111.0 21.0
0.20 77.0 11.2 0.70 112.7 22.7
0.25 81.3 14.0 0.75 116.8 25.0
0.30 84.0 15.0 0.80 120.0 27.0
0.35 87.4 15.4 0.85 122.9 29.0
0.40 91.0 16.0 0.90 127.9 33.8
0.45 96.5 17.0 0.95 138.9 37.0
0.50 100.5 18.0
If a q-q plot of two distributions is some straight l ine other t h an 2
= y , then th e two distribut ions have the same sha pe bu t their location
and spread m ay differ. We have already taken adv antag e
of
this prop-
er ty when we constructed th e normal probabili ty plots in Figure
2.4. In
fact, this is a q-q plot on which we compare th e quantiles
of
 
3
B
I +++
100 V values. Note the different scales on the axes.
A q-q plot of the distribution of the 100 special U values versus the
th e lognormal probabi l i ty p lo t we drew in F igure 2.5 is a compar ison
of the V quan ti les t o tho se of a s t and a rd lognormal d i s t ribu t ion . T he
similar i ty of a n observed d i s t r ibu t ion t o an y theore t ica l d i s t ribu t ion
model can be checked by th e s t ra ightness of their q-q plot.
Scatterplots
T he mos t common d i sp lay of b iva ri a te d a t a is t h e scatterplot which
is an x-y graph of the d a t a on which th e x-coordina te cor responds to
th e value of on e variable a n d the y -coord ina te to the value of t h e o t h e r
variable.
T h e 100 pairs of V - U values in Fig ure 3.1 are shown o n a sca t te rp lo t
in F igure 3 .4a. Tho ugh there is some scat ter in the cloud of points ,
the larger values of V tend to be associated with the larger values of
U and th e smal le r values of V t end to be assoc ia ted wi th th e smaller
values of U .
In add i t ion t o p rov id ing a good qu alit ativ e feel for how tw o vari-
ables a re rela ted, a sca t terplo t is a lso useful for draw ing ou r a t te nt i on
t o ab er ran t da ta . In th e early s tages of the s tud y of a spa t ia l ly cont in-
uous d a ta se t i t is necessary t o check and clean t h e da ta ; th e success of
 
29
P P )
Figure 3.4 Scatterplot of 100 U versus V values. The actual 100 data pairs are
plotted in (a) . In (b) the V value indicated by the arrow has been “accidentally”
plotted as 14 ppm rather than 1 4 3 ppm to illustrate the usefulness of the scatterplot
in detecting errors in the data.
have been cleaned, a few errat ic values may have a m a j o r i m p a c t o n
es t imat ion . T h e sca t te rp lo t can be used t o help bo th in th e val ida tion
of th e in i tia l da ta a nd in th e un ders tanding of la te r resu l ts .
T h e sca t te rp lo t shown in Figu re 3 .4a does not reveal a ny obvious
e r ro r s in t he V a n d U values . Th ere is one poin t th a t p lots in th e up per
r ight corner of Figure 3.4a with a U value of 55 p p m a n d a V vaIue
of 143 ppm . Had th e V value accidentally been recorded as 1 4 p p m ,
th i s pa i r of values would plot in the upper left corner all by i tself ,
as in Fig ure 3.4b, an d one’s suspicion would be aroused by such an
un us ua l pair . O ften , fur t he r invest igat ions of such unusu al pairs will
reveal errors th a t were most l ikely ma de w hen t he d a ta were collected
or recorded.
a
sca t -
terplot for erro r checking. We are relying on th e general re lationship
between th e two variables t o tell us if a particular pair of values is un-
usual . In the ex amp le g iven in the las t paragra ph, we expected th e V
value associated with a U va lue of 55 ppm to b e qui te h igh , somewhere
 
30
looking at the rest of the points on the scatterplot in Figure 3.4b and
extrapolating their behavior. In part two of this book we will present
an approach to estimation that relies on this same idea.
A n Introduction to Applied Geo statistics
Correlation
In the very broadest sense there are three patterns one can observe on
a scatterplot: the variables are either positively correlated, negatively
correlated, or uncorrelated.
Two variables are positively correlated if the larger values
of one variable tend to be associated with larger values of the other
variable and similarly with the smaller values of each variable. In
porous rocks porosity and permeability are typically positively cor-
related. If we drew a scatterplot of porosity versus permeability we
would expect to see the larger porosity values associated with the larger
permeability values.
Two variables are negatively correlated if the larger values of one
variable tend to be associated with the smaller values of the other.
In geological data sets the concentrations of two major elements are
often negatively correlated; in a dolomitic limestone for example an
increase in the amount of calcium usually results in a decrease in the
amount magnesium.
The final possibility is that the two variables are not related. An
increase in one variable has no apparent effect on the other. In this
case the variables are said t o be uncorrelated.
Correlat ion Coefficient. The correlation coefficient p is the statis-
tic that is most commonly used to summarize the relationship between
two variables. It can be calculated from:
The number of data is n; 21,.
. ,z, are the da ta values for the first vari-
able rn is their mean and a, is their standard deviation; y 1 , . . . , n
are the data values for the second variable my s their mean and ay
is their standard deviation.
 
Bivariate Description 31
and is often used itself as a summary statistic of a scatterplot. The
covariance between two variables depends on the magnitude of the data
values. If we took all of our V - U pairs from Figure 3.1 and multiplied
their values by 10, our scatterplot would still look the same, with
the axes relabeled accordingly. The covariance, however, would be 100
times larger. Dividing the covariance by the standard deviations of the
t w o variables guarantees that the correlation coefficient will always be
between -1 and 1, and provides an index that is independent of the
magnitude of the data values.
The covariance of our 100 V-U pairs is 216.1 ppm2, the standard
deviation of V is 26.2 ppm and of U is 9.81 ppm. The correlation
coefficient between V and U therefore, is 0.84.
The correlation coefficient and the covariance may be affected by
a few aberrant pairs. A good alignment of a few extreme pairs can
dramatically improve an otherwise poor correlation coefficient. Con-
versely, an otherwise good correlation could be ruined by the poor
alignment of a few extreme pairs. Earlier, in Figure 3.4, we showed
t w o scatterplots that were identical except for one pair whose V value
had been erroneously recorded as 14 ppm rather than 143 ppm. The
correlation coefficient of the scatterplot shown in Figure 3.4a is the
value we calculated in the previous paragraph, 0.84. With the change
of only one pair, the scatterplot shown in Figure 3.4b has a correlation
coefficient of only 0.64.
The correlation coefficient is actually a measure of how close the
observed values come to falling on a straight line. If p = 1, then the
scatterplot will be a straight line with a positive slope; if p = -1, then
the scatterplot will be a straight line with a negative slope. For lpl
< 1
the scatterplot appears as a cloud of points that becomes fatter and
more diffuse as IpI decreases from 1 to 0.
It is important to note that p provides a measure of the linear re-
lationship between two variables. If the relationship between two vari-
ables is not linear, the correlation coefficient may be a very poor sum-
mary statistic. I t is often useful to supplement the linear correlation
coefficient with another measure of the strength of the relationship,
the rank correlation coefficient [l] To calculate the rank correlation
coefficient, one applies Equation 3.1 to the ranks of the data values
rather than to the original sample values:
1
(3.3)
O ORy
A n Introduction to Applied Geostat is t ics
Rzi is the rank of x a m o n g a l l t h e o t h e r z values and is usually
ca lcu la ted by so r t ing the z values in ascend ing ord er an d seeing where
eac h value falls. T h e lowest of th e z values would appear first on a
sorted list and would therefore receive a ran k of 1; th e h ighest z value
would ap pe ar las t on th e l is t and would receive a r ank of n. By; s
t h e r a n k of
y; a m o n g all t h e o t h e r y values. M R ~ :s t h e m e a n of al l
of t h e r a n k s Rz1 . . ,R z a n d o h s the i r s tandard devia t ion . mRy
i s th e mean of al l of t h e r anks R y l , . ,Ry, a n d U R ~s the i r s t and a rd
devia t ion [2].
Large differences between prank a n d p are of ten qui te reveal ing
ab ou t t h e loca t ion of ex t rem e pa irs on th e sca t te rp lo t .
Unlike the
tra d iti on al correlation coefficient, th e ra nk correlation coefficient is no t
s trongly influenced by extre me pairs . Large differences betw een t h e tw o
A
value of prank a n d a low value of p m a y b e d u e t o t h e fa ct t h a t a few
errat ic pairs have adversely affected a n oth erwise good correlat ion. If,
on t he o th e r hand , i t is p th at is qu i te high while prank is qu ite low,
then i t is l ikely that the high value of p is d u e largely t o t h e influence
of a few e xtrem e pairs .
For th e scat terp lot shown in Figure 3.4b, t h e th e coeff ic ient of lin-
ear correlation is 0.64, while the rank correlation coefficient is 0.80.
The s ingle aberrant pair in the upper lef t corner has less of an inf lu-
ence on th e rank correlat ion th an i t does on th e t rad i t iona l correlat ion
coefficient.
Differences between p a n d prank may a lso reveal impor tan t fea-
tures of th e re lat ionsh ip between two var iables . If th e ran k correlat ion
coefficient is +1, the n th e ranks of th e two variables ar e ident ical: th e
larg est v alue of z correspo nds to th e largest value of y , an d th e smalles t
value of z correspo nds to th e smallest value of y. If the rank correla-
tion coefficient is +1, then th e relat ionship between 2 a n d y need not
be l inear. I t is , however, monotonic; if the value of z increases, then
the va lue of y also increases. T w o variables w hose rank correlation
coeff icient is not iceably higher th an their t ra di t io nal l inear correlat ion
coefficient ma y ex hib it a nonlinear relationship. For exam ple, tw o vari-
ables , X a n d Y which are related by the equat ion Y = X2 ill have
a value of p near 0 b u t a valu e of prank of 1.
T h e value of p is often a good ind icato r of how successful we mig ht
be in t ry ing t o predic t the va lue of one variab le from th e o th er wi th a
 
Bivariate Description 33
other variable is restricted to only a small range of possible values. On
the other hand, if lpl is small, then knowing the value of one variable
does not help us very much in predicting the value of the other.
Linear Regression
As we noted earlier, a strong relationship between two variables can
help us predict one variable if the other is known. The simplest recipe
for this type of prediction is linear regression, in which we assume that
the dependence of one variable on the other can be described by the
equation of a straight line:
y = a x + b (3 .4)
The slope, a , and the constant, b are given by:
b = my- a m
(3 .5 )
The slope, a is the correlation coefficient multiplied by the ratio of
the standard deviations, with oY being the standard deviation of the
variable we are trying to predict and Q, the standard deviation of the
variable we know. Once the slope is known, the constant, b can be
calculated using the means of the two variables, m and my.
If we use our 100
V - U pairs to calculate a linear regression equation
for predicting V from U , we get
26.2
9.81
(3 .6)
Our equation to predict V from a known U value is then
= 2.24 u + 54.7
(3 .7 )
In Figure 3.5b this line is superimposed on the scatterplot. Al-
though i t looks reasonable through the middle of the cloud, this regres-
sion line does not look very good a t the extremes. It would definitely
overestimate very low values of V . The problem is our assumption
that the dependence of V on U is linear. No other straight line would
do better than the one we calculated earlier[3].
Equation 3.7 gives us
a prediction for V if U is known. We might
 
-
+
@)
Figure 3.5 Linear regression lines superimposed on the scatterplot. The regression
line of U given V is shown in a) , and of V given in (b) .
In Equa t ion 3.5, y is the unknown variable and x is known, so t h e
calculation of a l inear regression equation that predicts U from V is:
9.81
(3.8)
T h e linear regression eq ua tion for predicting U from a known V value
is then
3.9)
T h is regression l ine is shown in Figu re 3.5a. In th is figure we have
plo t ted U on the y-axis an d V on th e x-axis t o emphasize the fac t th a t
i t i s U th a t is the unknow n in this case. We will cont inue with this
convent ion thro ugh out th e boo k; for scat terplo ts on which the re is a
known variable and a n unknown variable, we will plot th e unknown
A close look at Figure 3.5a and Figure 3.5b reveals that the two
regression l ines are not th e sam e; indeed Eq uat ion 3.9 is not s imply a
rear rangemen t of Eq uat io n 3 .7 .
T h e regression l ine shown in Figu re 3.5a ra ises an issue that we
 
within classes defined on the U value.
Number Mean
0 L U < 5 8 40.3
5 I U < 10 8 72.4
10 < U < 15 10 85 .5
20 < U < 25 15 106.9
25 < U < 30 12 113.5
30 < U < 35 7 125.7
35 <u< 7 133.9
15 s U < 20 33 97.5
wonder what the predicted value of U is for a V value of about 5 ppm.
Of course, the regression line continues into negative values for U and
if we substitute a value of 5 ppm for V into Equation 3.9 we get a
predicted value of -6.2 ppm for U . This is clearly a silly prediction; U
values are never negative. Simple linear regression does not guarantee
positive estimates, so where common sense dictates that the da ta values
are always positive, it is appropriate to set negative predictions t o 0, or
to consider other forms of regression which that respect this constraint.
Conditional Expectation
The formulas for calculating a linear regression equation are very sim-
ple but the assumption of a straight line relationship may not be good.
For example, in Figure 3.5a the regression line seems inadequate be-
cause the cloud of points has a clear bend in it.
An alternative to linear regression is to calculate the mean value of
y for different ranges of 2. In Table 3 .3 we have calculated the mean
value of V for different ranges of U . Each of our 100 U - V pairs has
been assigned to a certain class based on its U value, and the mean
value of V has been calculated separately for each class.
If we wanted to predict an unknown V value from its corresponding
 
60
40
3
20
0
Figure
3.6 A graph ‘of he mean values of V within classes defined o n values.
same class as o u r predicted value. This results in a predic t ion curve
th a t looks l ike the o ne shown in F igure 3.6 . T h e curve is d i scont inuous
becau se th e predicted value of V j u m p s t o a new value whenever we
cross a U class boundary.
This is a t ype of condit ional exp ecta t ion curve. W ithin cer ta in
classes of U values we have calculated a n expected value for V . T h o u g h
“expected value” has a precise probabilist ic meaning, i t is adequate
for our purposes here t o a llow i t t o keep i t s co lloquia l meaning , “ th e
value one expects t o get.” O ur exp ected values are cal led conditional
because they a re good only for a cer ta in range of
U values; if we move
t o a different class, we expect a dif fe rent va lue. T h e s ta i r s tep curve
shown in F igure 3.G is obtained by moving through al l the possible
classes of U and ca lcula ting an expected va lue of V for each class.
Ideally, with a huge number of data , one would l ike to make a
cond it ional exp ecta t ion curve w ith as m any classes a s possible . As t h e
n u m b e r of classes increases, th e wid th of each pa rtic ula r class would g et
narrower and the discontinuities in o u r condi t iona l expe cta t ion curve
would g et smaller . In th e l imit , when we have a huge number of very
narrow classes, ou r condit ional expe ctat ion curve would b e a s m o o t h
cur ve th a t would give us an expec ted value of V condi t iona l to known
 
Figure 3.7 Conditional expectation curves superimposed on the scatterplot . The
expected value of given V is given in (a) and the expected value of V given U is
shown in (b).
usually referring to this ideal limit. This ideal limit would serve very
well as a prediction curve, being preferable to the linear regression line
since it is not constrained to any assumed shape.
Regrettably, there are many practical problems with calculating
such an ideal limit. From Table 3.3 we can see that
if
the class width
was made any narrower, we would start to run out of pairs in the
highest and lowest classes. As the number of pairs within each class
decreases, the mean value of V from one class to the next becomes
more erratic. This erraticness also increases as the correlation between
the two variables gets poorer.
There are many methods for dealing with these practical compli-
cations. We have adopted one particular method for use throughout
this book [4].Whenever we present a conditional expectation curve,
it will have been calculated using the method that, for the curious, is
referenced in the notes at the end of this chapter.
We will not be relying on these conditional expectation curves for
prediction but will be using them only as graphical summaries of the
scatterplot. It will often be more informative to look at the conditional
expectation
curve
 
A n Introduction to Applied Geostatistics
J u s t as we had two regression l ines, one for predicting V from
U and another one for predic t ing U from V , so t o o a r e t h e r e t w o
cond it ional expec tat ion curves, on e th at gives th e expected value of V
given a partic ular value of U and ano the r tha t g ives the expected va lue
of
U
given a part ic ular value of V .
In F igure 3.7 we show th e condi tiona l expe c ta t ion curves t ha t our
part ic ular method produces. It is in te rest ing t o note th a t for predic t ing
V f rom U he condi t iona l expe c ta t ion curve is qu i te d if fe rent f rom t h e
regression l ine shown in Fig ure 3.5b, b u t for th e prediction of from V
th e regression l ine is qui te c lose t o th e condi t iona l expe c ta t ion curve .
Even thou gh th e condi tiona l expec ta t ion cu rve i s, in som e sense, th e
ideal prediction cu rv e, linear regression offers a very s im ple al tern at iv e
th a t i s of ten adequ ate .
Notes
[ l ] The linear coefficient of correlation given in Equation 3 . 1 is often
referred t o in th e s tat is t ica l l i teratu re as t h e Pea r son co r re l at ion
coefficient while the correlation coefficient of the ranks given in
Equa t ion 3.3 is often referred to as t h e Spea rman rank co r re la t ion
coefficient.
[ 2 ] All of th e num bers from 1 t o n app ear som ewhere in th e se t of x
ranks , R z l , .
. ,Rx,, and also in th e set of y ranks , R y l , . , . ,Ry,.
For this reason, th e univariate s ta t is t ics of the tw o sets a re identical .
In partic ular, for large values of n th e values of m z a n d m y a r e
bo th c lose to n/2 an d th e values of a;, a n d a i y ar e bo th c lose to
n/12.
[3] Th ere a re man y as sumpt ions bu il t i n to the theo ry th a t v iews th is
part ic ular line as th e best . Since a t this po int we are proposing this
only as a tool for summarizing a sca tter plo t, we defer th e discussion
of these impor tan t assumpt ions unt i l the second par t of t h e book
where we deal specif ical ly with methods that a im at minimizing
the variance of the estimation errors.
[4]Summariz ing a sca t te rp lo t wi th a condi t iona l expec ta t ion curve
is often a useful way of defining a nonlinear relat ionship between
two variables. O ften th e overall sh ap e of th e poi nt cloud clearly
reveals a re lat ionship between two var iables that can be more ac-
 
Bivariate Description 39
than it can by a straight line. For example, a scatterplot of y h)
and h (commonly called a variogram cloud), most often reveals a
nonlinear relationship between y h ) and h that is best described
by a smooth curve. There are a number of methods one can use
for estimating the conditional expectation curves of a scatterplot;
the algorithms are known generally as smoothers. Th e particular
smoother we have chosen
linear regression within a local sliding neighborhood. T he algorithm
provides an “optimal”
neighborhood size as well as an option for
curve estimation using methods resistant t o extreme values. A com-
plete description of the smoother with Fortran code is provided in:
Friedman, J. H. and Stuetzle, W. , “Smoothing of Scatterplots,”
Tech. Rep. Project Orion 003, Department of Statistics, Stanford
Further Reading
Chatterjee, S. and Price, B. , Regression Analysis b y Example. New
York: Wiley, 1977.
pp. 828-836, 1979.
Mosteller, F. and Tukey, J. W. ,Data Analysis and Regression. Read-
ing, Mass.: Addison-Wesley, 1977.
Royal Statistical Society Series B vol. 47, pp. 1-52, 1985.
Tukey, J. , Exploratory Data Analysis. Reading, Mass.: Addison-
Wesley, 1977.
SPATIAL DESCRIPTION
O n e of th e th ings th a t d is tinguishes e ar th sc ience da ta se t s f rom most
o thers is th a t th e da ta be long to som e loca tion in space . Sp at ia l fea-
tures of t h e d a t a s e t , su ch as th e location of ex trem e values, th e overal l
t r e n d , or the degree of cont inui ty, are of ten of considerable interest .
None of th e univ ar ia te and b ivar ia te descr ip t ive tools presented in t he
las t two chapters cap ture these spa t ia l fea tures. In th is ch apte r we
will look
at
t h e sp a t i a l a spec t s of ou r 100 se lec ted da ta and incorpo-
ra t e the i r loca t ion in to o u r description.
Data Postings
As with th e h istogram from Ch apte r 2 and the sca t te rp lo t from Cha p-
t e r 3, our most effective tools for spatial description are visual ones.
T h e s imples t d isp lay of spa t i a l da ta is a d a t a p o s ti n g, a ma p on which
each d a t a locat ion is p lo t ted a lon g wi th i t s cor responding da ta value .
F igure 2.1 was a d a t a p o s ti n g of t h e V values; Figure 3.1 a d d ed t h e U
values.
Pos t ings of t h e d a t a a r e a n i m p o r t a n t i n it ia l s t e p in a n al y zi n g s p a -
t ia l da ta se ts . Not only d o they revea l obvious e rrors in th e d a t a loca-
t ions , but they of ten a l so draw a t ten t ion to data va lues tha t ma.y be
erroneous. Lone high values surrounded by low values and vice versa
ar e wo r th rechecking . W i th i r regularly gr idded d a t a , d a t a p o st in g
 
81 77 103 112 123 19 40 111 114 120
82
61
+ + + + + + + + + +
88 70 103 1 1 1 122 64 84 105
113
123
118
127
+ + + + + + + + + +
+ + + + + + + + + +
+ + + + + + +
87 100 47 1 1 1 124 109 0 98 134 144
+ + + + +mI.J+ +
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + +
+ + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + m +
Figure
4.1 Location of t h e lowest V values in a) , and the highest values in b) .
map may have been inaccessible; heavily sampled areas indicate some
initial interest.
Locating the highest and lowest values on a posting of the dat a may
reveal some trends in the data. On Figure 4.la we have highlighted the
10 lowest values from Figure 2.1. Seven values, 19, 40, 52, 64, 48, 52,
and 0 are located in a north-south trough that runs through the area.
On Figure 4.lb, a similar display for the 10 highest values, no obvious
trend is apparent, with the highest values appearing in the southeast
corner.
Contour Maps
The overall trends in the data values can be revealed by a contour
map. Contouring by hand is an excellent way to become familiar with
a da ta set. Unfortunately, the size of many dat a sets makes automatic
contouring with a computer an attractive alternative. The contour
map of the V values shown in Figure 4.2 was generated by computer.
At this preliminary descriptive stage the details of the contouring
algorithm need not concern us as long as the contour map provides
a
helpful visual display. There are many algorithms that provide an ad-
 
42 A n Introduction to Applied Geostatistics
Figure 4.2 Computer generated contour map of 100 selected V data. T h e contour
lines are at intervals of 10 ppm and range from 0 140 ppm.
program will al so p ay a t ten t ion to aes the t ic de ta i l s such as th e use of
dow nhill t ick m arks t o show depressions.
S o m e of the fea tures we noticed earl ie r on the d a ta pos t ing become
clearer when co ntoured . T h e nor th -sou th t rough i s readily ap pa ren t ,
as are th e local maximum s. Also, som e features t h at were no t obvious
f rom th e da ta po s t ing a lone are now more prom inent . T h e c loseness
of the contour l ines in the southeastern corner indicates a s t eep g ra -
d ien t and d raws our a t t en t ion t o the f ac t t h a t t he h ighest d a t a va lue
(145 pp m ) is very close t o th e lowest da ta value 0 p p m ) .
A uto m atic conto uring of i rregular ly gr idded d a t a usually requires
the d a ta values t o be in t e rpo lat ed to a reg ula r grid. Inte rpo lated Val-
ues are usual ly less var iable than the or iginal data values and make
th e contoured sur face app ear smoo ther . Th is is an aes the t ic asse t , bu t
a sm ooth er sur face unders ta tes th e variab i li ty and may b e m isleading
f rom a qu an t i ta t iv e poin t of view. In this book we t re at o ur contou r
m aps a s we t rea t o ur condit ional ex pectat io n curves: as helpful quali-
 
5 5 6 7 8 1 2 7 7 8
5 4 7 8 7 5 3 7 7 8
5 4 6 7 7 6 4 7 7 8
5 4 6 7 8 4 5 7 7 8
5 5 6 7 7 7 4 7 7 8
5 5 5 6 7 7 5 6 8 8
4 5 5 6 6 6 6 4 8 8
5 5 5 5 6 6 6 3 9 9
5 5 4 7 8 9 6 3 9 9
5 6 3 7 8 7 0 6 8 9
0 -
0-14ppm
1 = 15-29
2= 33-44
3=45-59
4 60-74
5 =75-89
6= 90-104
7= 105-119
8= 120-134
9= 135-149
Figure 4.3 Symbol map of 100 selected V data. Each symbol represents a class
of data values as indicated by the legend on the right-hand side of the figure.
Symbol Maps
For many very large regularly gridded data sets, a posting of al l the
d a ta values may n ot be feasible, and a contour map may mask m any of
th e interesting local details. An alternative th at is often used in such
situations is a symbol map. Th is is similar t o a d a ta post ing with each
location replaced by a symbol th at denotes th e class to which th e da ta
value belongs. T he se symbols are usually chosen so that they convey
th e relative orderin g of th e classes by their visual density. This typ e of
display is especially convenient if one has access to a line printer but
not to a plo tting device. Un fortunately, th e scale on symbo l map s is
usually distorted since most line printers do not print th e same num ber
of cha racters p er inch ho rizontally as they d o vertically.
For a d at a se t as smal l as our 10 x 10 m2 grid, a symbol map is
probably not necessary since the actu al d a ta values ar e easy to post.
In order t o show a simple example, however, we present a symbol m ap
in Figure 4.3 that corresponds to the posting from Figure 2.1. In this
display we have used the digits 0 through 9 to denote which of t h e
ten classes the V value at each location belongs. An alternative to t h e
symbol map is a grayscale map. In such a m ap th e symbols have been
replaced with a suitable shade of grey as shown in Figure 4.4. These
m aps are much m ore pleasing t o th e eye and provide a n excellent visual
 
44 n Introduction to pplied Geostatistics
0 15 30 45 6 75 90 105 120 135 150
Figure 4.4 Grayscale map of 100 selected V data. The value of each V datum is
indicated its shade of grey as shown by the scale at the top of the figure.
Indicator Maps
An indica tor m ap i s a symbo l map on which there a re only two symbo ls ;
in our examples here we use a black box a n d a white box. W ith only
two symbol s one can as sign each da ta po in t t o one of only tw o classes,
so an indica tor ma p s imply records where the d a t a va lues a re above
a cer ta in threshold and where they are below. Tho ugh th is may seem
at f irst t o b e ra the r restr ic t ive, a series of indicator m aps is of ten very
informat ive. Th ey sha re the advan tage wi th al l symbo l maps th a t t hey
show more de ta i l t ha n a conto ur m ap and th a t they avoid t h e d if ficu lty
of dis t inguish ing be tween symbols th a t ex is t wi th convent iona l symbo l
maps .
In Figures 4.5a-i we show a series of nine indica tor maps cor re-
s p o n d i n g to the n ine c lass boundar ies f rom our symbol map in F ig-
u r e 4.3 . Each m ap shows in wh i t e the da ta loca tions a t wh ich t he
V value is less than the given threshold and in black the locat ions at
which V is g rea ter than or equal to t h e threshold. Th is series of indica-
tor m aps records the t ransi t ion from low values th at tend t o be aligned
in a nor th-so uth d i rec tion to h igh va lues th a t tend t o be grouped in
 
Figure
4.5 Indicator maps of the 100 selected V data. Each figure is a map of
indicators defined using the indicated cutoff values. he pat tern of indicators in the
sequence of maps provides a detailed spatial description of the data. For example,
the indicator map defined at 75 ppm reveals the trough of low values seen earlier
 
I
, + + + + I + + + +
7  2  6 l ol 109 113 7+9 172
4 8+0 8+5 92 9+7 191 9+6 2
1 <7 ~ ~ @ ; ~ ~ ~ ~ - ~0 121
119 77 52 141
+ I +
2 5 4 ; 97 105 112 91 I 7 3 1
  f 701103 111 122 62 182 105
89 88 94 110 116 108 73 107
+ +
+ + + + + +
57 2 56 l+ol 10+9 1l+3 7,9 1 y Z
4 80 8+5 9+0 57 lfl 9 6 52
+ I-+-
overlapping moving windows for purposes of calculating
ferent spat ia l features . Figure 4.5f, for example , gives t he bes t image
of t he no r th - sou th t rough we noticed earlier, while Figure 4.5g gives
the best image of the local maximums. The usefulness of t hese map s
wi l l become more apparent in Chapter 5 where a series of indicator
map s a re used to explore several la rge d a t a se ts .
Moving Window Statistics
In th e analysis of ear th science d a ta se ts one is of ten m ost in te res ted in
th e anom alies i.e ., th e high g rade veins in a gold dep osi t ) or t h e i m p e r -
meab le layers t h at cond i t ion flow in a petroleum reservoir. A contour
map wil l help locate areas in which the average value is anomalous,
bu t anom alies in th e average value are not t h e only interest ing ones.
I t i s qu i te comm on t o find th a t th e da ta va lues in som e reg ions a re
more var iab le th an in o thers . T h e s ta t i s t ica l ja rgon for such anom a-
lies in the variability is heteroscedasticity. Such anomal i e s may have
serious practical implications. In a mine, very e rrat ic o re grades of ten
cause problems at th e mill bec aus e m ost m etallurgical processes bene-
f it f rom low variabil ity in t h e ore grade. In a petroleum reservoir , large
fluctuat ions in th e perm eabil i ty can ham per th e effect iveness of m a n y
seco nd ary recovery processes.
T h e ca lcu la t ion of a few su m m ary s ta t is t ics within moving windows
is f requen t ly used t o invest iga te anomalies bo th in t he av erage value
 
92.3
+
+ +
21.5 32.9 41.3
Figure 4.7 Posting of statist ics obtained from moving windows on the 100 V
d a t a . T h e m e an of each moving window is plotted above the “+”, and the standard
deviation below.
hoods of equal size and within each local neighborhood, or window,
summary statistics are calculated.
Rectangular windows are commonly used, largely for reasons of
computational efficiency. The size of the window depends on the av-
erage spacing between data locations and on the overall dimensions of
the area being studied [2]. We want to have enough data within each
neighborhood to calculate reliable summary statistics. If we make our
windows too large, however, we will not have enough of them to iden-
tify anomalous localities.
Needing large windows for reliable statistics and wanting small win-
dows for local detail may leave little middle ground. A good compro-
mise is often found in overlapping the windows, with two adjacent
neighborhoods having some data in common.
In Figure
we show an example of an overlapping moving window
calculation. We have chosen to use a 4 x 4 m2 window so that we will
have 16 data in each local neighborhood. By moving the window only
2 m each time so that it overlaps half of the previous window, we can
fit 16 such windows into our 10 x 10 m2 area.
Had we not allowed
the windows to overlap, we would have had only four separate local
neighborhoods.
 
A n Introduct ion to Applied Geostat is t ics
al ly n ot necessary. For smaller da ta sets or for ones in which the da ta
ar e i r regular ly spaced , over lapping becomes a useful tr ick. With irreg-
u larly spaced d a t a i t i s a l so im po r tan t to dec ide how ma ny d a t a will
be required within each window for a reliable calculation of t h e s u m -
m ary s ta t is t ics. If t he re a re to o few d a t a w i th in a par t icu lar window,
i t is of ten be t te r t o ignore th a t window in subsequent ana lys is th an t o
incorp ora te a n unre liab le s ta t is t ic .
Wi th enough da ta in any window, one can ca l cu la t e any of t h e
sum ma ry s ta t i s t ics we have previously d iscussed . T h e mean a nd th e
s ta nd ard devia t ion are commonly used , wi th on e provid ing a measure
of t he ave rage va lue and th e o the r a measure of the variability. If t h e
local means are heavily influenced by
a
few errat ic high values, one
could use th e med ian and in terqua r t i le range ins tead .
T h e means and s t anda rd dev iat ions wi th in 4 x 4 m2 windows are
shown in F igure 4 .7 . As shown in Figure 4 .6 and described earlier,
the windows overlap each other by 2 m , giving us a total of 16 local
ne ighborhood means and s t and ard devia tions . We have pos ted these
values in F igu re 4 .7 , where th e center of each window is marked wi th a
plus sign. T h e mean of each window is plot ted abo ve the + sign while
th e s ta nd ard deviat ion is plot ted below. If we had a l a rge r a rea an d
a
two con tour maps , w i th one showing th e means and the o th e r showing
th e s t anda rd dev ia t ions .
From th is pos t ing of moving window means and s tandard devia-
t ions we can see th a t bo th th e average value and th e variab i li ty change
locally across th e area. T h e windows with a high a vera ge value corre-
spond t o the highs we can see on the con tour map F igure 4.2). T h e
local changes in variabil ity, however, ha ve no t been cap tu red by a n y of
ou r previous tools . In th e south easte rn corner we see th e highest local
s t an da r d dev ia tions ,
a
resul t of th e very low values in th e t rou gh being
a d j a ce n t t o s o m e of th e highest values in the ent i re area. T h e very
low standard deviat ion on the western edge ref lects the very uniform
V values in that region.
I t i s in t e res ting to no te in th i s example th a t t he s t and a rd dev iat ions
vary much more across the area th an th e means . I t i s o f ten tempt ing
t o conclude t h at uniformity in t he local means indicates general ly well
behaved d a ta values. Here we see th a t even thou gh th e mean values ar e
83 .9 106 .7
 
Spatial Description 49
Figure 4.8 Hypothet ica l prof i les of d a t a va lues i l l u s tr a t i ng comm on re l a t i onsh ips
betw een th e loca l m ean an d loca l var iab il ity . In a ) the loca l mea n, represen ted by
t h e s t r a i g h t li ne , a n d t h e v a ri ab il it y a r e b o t h c o n s t a n t . C a s e b ) s h o w s a t r e n d i n
th e loca l mean wh il e t he va r i ab i li ty r em a ins cons t an t . Case c ) exh ib i t s a c o n s t a n t
loca l mean wh i le t he va r i ab i l i ty con ta ins a t r e n d a n d ca se d ) il l u s t r a te s a t r e n d i n
bo th t he l oca l mean an d t h e var iab il it y .
Proportional Effect
W h e n w e look at est im at io n in la ter cha pters , anomalies in t h e local
var iab i l i ty wi l l have an impact on the accuracy of our es t imates . If
we ar e in an a rea wi th very uni form va lues, the p rospec ts for acc ura te
es t ima tes a re qu i t e good . O n t he o th e r ha nd , if t he d a t a values f luc-
tu a t e wild ly ou r chances for acc ura te loca l es t imates a r e poor . Th is
has no th ing t o d o wi th the e s t ima t ion me thod we choose to use ; the
es t imates f rom any reasonable method will benefit from low variability
and suffer from high variability.
In a bro ad sense th ere are four rela t ionships one can ob serve be-
tween th e local average and th e local var iabil ity . Th ese ar e shown in
Figures 4.8a-d, which represent hypothetical profiles of the data val-
ues . O n each profile the l ine th at conn ects th e plus s igns represents
 
A n Introduct ion to Applied Geostat is t ics
In F igure 4 .8a t he average and the variab i li ty a r e both cons tan t .
The da ta va lues f luc tua te about the loca l average , bu t there i s no
obvious change in t he var iabil ity. In Fig ure 4.8b, th ere is a t r e n d t o
th e local average; i t r ises grad ually th en falls. T h e variabili ty, however,
is sti l l roughly co ns tan t. In F igu re 4.8c, we see th e reverse case, where
th e local average is con stan t while the variabi li ty changes. T h e most
common case for ear th science data is shown in Figure 4.8d, where
th e local averag e and variabi li ty b ot h chan ge together .
A s the local
ave rage increases, so, too, does the local variability.
For es t imation, the f i rs t two cases are the most favorable . If t h e
local var iabil ity is roughly con stan t , th en e st im ates in a ny par t ic ular
area wil l be as good as estimates elsewhere; no area will suffer more
than others f rom highly var iable data values .
It is more likely, how-
ever, th at th e variabil ity does chan ge not iceably. In such a case, it is
preferable t o be in a s i tua t ion l ike the on e shown in F igure 4A d, where
th e local variabil ity is related to th e local averag e an d is, therefore,
som ew hat predictable . I t is useful, therefore, t o es tabl ish in t he ini t ia l
d a t a analysis if such a predictable relationship does exist .
A sca t te rp lo t of th e loca l means an d th e loca l s ta nd ard devia t ions
from our moving window calculat