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Geography, Spatial Data Analysis, and
Geostatistics: An Overview
Robert P. Haining,1 Ruth Kerry,2 Margaret A. Oliver3
1Department of Geography, University of Cambridge, Cambridge, UK, 2Department of Geography, Brigham
Young University, Provo, UT, and CRSSA, Rutgers University, New Brunswick, NJ, 3Department of Soil
Science, University of Reading, Reading, UK
Geostatistics is a distinctive methodology within the field of spatial statistics. In the
past, it has been linked to particular problems (e.g., spatial interpolation by kriging)
and types of spatial data (attributes defined on continuous space). It has been used
more by physical than human geographers because of the nature of their types of data.
The approach taken by geostatisticians has several features that distinguish it from the
methods typically used by human geographers for analyzing spatial variation associ-
ated with regional data, and we discuss these. Geostatisticians attach much impor-
tance to estimating and modeling the variogram to explore and analyze spatial
variation because of the insight it provides. This article identifies the benefits of geo-
statistics, reviews its uses, and examines some of the recent developments that make it
valuable for the analysis of data on areal supports across a wide range of problems.
Introduction
As an introduction to this special issue, the purpose of this article is to provide an
overview of the core concepts and techniques of geostatistics, together with a short
literature review of its application in the environmental sciences and in geography.
Geostatistics has a long history of application in the environmental sciences where
data are on a point or small regular area support, but it is now being applied to re-
gional data where data are on an areal support that might be large and regular or
irregular. We describe the new tools associated with the latter type of data and contrast
them with techniques of spatial data analysis with which geographers, especially hu-
man geographers, are familiar. These techniques have descended more or less directly
from work that began in the 1960s by Dacey (1968) and Cliff and Ord (1969), also the
subject of a recent special issue of Geographical Analysis (2009, issue 4). Geostatistics,
by contrast, has a different lineage and uses a different set of tools and techniques.
Correspondence: Ruth Kerry, Department of Geography, 690 SWKT, Brigham YoungUniversity, Provo, UT 84602e-mail: [email protected]
Submitted: August 22, 2008. Revised version accepted: September 9, 2009.
Geographical Analysis 42 (2010) 7–31 r 2010 The Ohio State University 7
Geographical Analysis ISSN 0016-7363
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The use of the term spatial analysis in geography can be traced back to the
1950s (see, e.g., Berry and Marble 1968). It includes several distinctive elements
(Haining 2003, pp. 4–5), but the statistical analysis of spatial data is the focus here,
referred to by statisticians as spatial statistics (Ripley 1981) or statistics for spatial
data (Cressie 1993). Geographers often refer to these as methods for spatial data
analysis (Haining 1993), and many of these models and techniques figure prom-
inently in geographic information science (Goodchild and Haining 2004) and spa-
tial econometrics (Anselin 1988).
The roots of spatial statistics can be traced back to the early part of the twen-
tieth century to analyses of agricultural field trial data by statisticians. Geostatistics
is a component of spatial statistics, although its evolution has been led principally
by applied scientists and mathematicians rather than by classically trained statis-
ticians. This historical context may explain why little cross-fertilization occurred
with other branches of spatial statistics until quite recently (Cressie 1993; Diggle
and Ribeiro 2007) and why geostatistics is distinctive.
Any methodology for analyzing spatial data needs to recognize that such data
have the fundamental property of spatial dependence or spatial autocorrelation. For
many attributes, values recorded at locations close together in space are correlated
(autocorrelated); as the separating distance increases, autocorrelation weakens.1
The autocorrelation structure in a region may be complex, with several scales of
variation nested within, or superimposed on, one another, varying with direction
(anisotropic) and between subareas (spatially heterogeneous). Quantifying spatial
dependence matters, whether the purpose of an analysis is to interpolate, to fit a
regression model, or to test a hypothesis (Haining 2003, pp.33–36, 40–41). Differ-
ent branches of spatial statistics model spatial dependence in different ways.
Geographical data acquire other properties as a consequence of the chosen
representation of geographic space. The areal units into which a study region may
be partitioned for reporting attribute values often vary in size and shape (e.g., cen-
sus output areas). If the population denominator for rates (e.g., mortality rates) var-
ies, the standard errors of such statistics are not constant across a map. Therefore,
values obtained from irregularly sized areas may not be directly comparable, mak-
ing map interpretation potentially problematic. Data for areas with small popula-
tions suffer from the ‘‘small number problem’’ (Haining 2003, pp.196–99). Methods
must be able to deal not only with spatial dependence but also with the properties
acquired as a consequence of a chosen representation.
This article elucidates the distinctiveness of geostatistics and how it differs from
other branches of spatial statistics that are concerned with the same general prob-
lem of analyzing spatial variation and illustrates the relevance of geostatistics to
geographers. Physical geographers have been somewhat receptive to geostatistics,
partly because their problems and data are similar to those of other earth scientists,
who were among the early practitioners of geostatistics. Nevertheless, geostatistical
methods generally have been used only for basic interpolation. Human geogra-
phers, by contrast, generally have taken little interest in geostatistics for spatial data
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analysis because often attribute values are not defined everywhere in a region, and
data values are defined for irregular spatial units (e.g., census areas).
To verify some of these assertions, we searched the bibliographic databases
Geobase and Information Sciences Institute Web of Knowledge (ISI).2 These
searches produced 4377 and 1596 hits, respectively. Table 1 shows the journals
that publish most articles on geostatistics (also see Zhou et al. 2007). Physical geo-
graphers publish in these journals. However, of the institutions with authors who
have published more than three articles on geostatistics articles identified by the
Geobase search, only 9 out of 61 were geography departments, and only 50 of over
4000 articles found with this search included authors from geography departments.
Table 1 also shows specifically which geography journals2 have published the most
articles on geostatistics; none is devoted solely to human geography. Figures 1a and
1b show that the number of articles has increased over time, both in general and in
geography; but for the latter, the increase seems to have leveled out since 2000.
Table 1 Number of geostatistics articles identified by Geobase and ISI searches in the top 10
journals that publish most geostatistics articles in general and in geography
Number of articles in top 10 journals publishing most geostatistics articles:
In general In geography
Journal name Geobase ISI Journal name Geobase ISI
Mathematical Geology 310 144 International Journal of
GIS
23 10
Geoderma 161 51 Acta Geoglogica Sinica 13 0
Water Resources
Research
143 46 Geographical Analysis 12 9
Computers and
Geosciences
138 47 Geographical and
Environmental
Modelling
9 0
Journal of Hydrology 105 30 Cartography and GIS 7 0
Soil Science Society of
American Journal
78 16 Journal of
Geographical Science
7 1
Environmentrics 66 22 Journal of
Biogeography
6 1
International Journal of
Remote Sensing
65 22 The Professional
Geographer
6 2
Stochastic
Environmental Research
and Risk Assessment
47 17 Progress in Physical
Geography
6 3
The Environmental
Monitoring and
Assessment
42 13 Physical Geography 4 2
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
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Between 1990 and 2008, only 3.4% of the Geobase results and 2.1% of the ISI
results identified geostatistics articles that were published in geography journals.
Geostatistics: Its core concepts, techniques, and relationship with other
approaches to spatial data analysis
An historical perspective on geostatistics
The use of the term geostatistics stems from Matheron’s development of a compre-
hensive theory for the prediction of properties in geographical space. Matheron’s
(1963) theoretical framework for geostatistics was developed from D. G. Krige’s
empirical ideas for improving predictions of the amount of gold in rock (Krige 1951)
by using neighboring samples. Matheron (1963) uses the term kriging for the
method of optimal prediction or estimation in geographical space—a spatial best
linear unbiased predictor (BLUP). Matheron’s fundamental contribution is to define
the covariance or variogram of a random field based on a probabilistic or stochastic
approach to the analysis of spatial variation, which recognizes its complexity and
treats as random that aspect of the variation that appears to be random. Matheron’s
approach led to the formulation of models of spatial variation that provide weights
for the BLUP (see Bilodeau, Meyer, and Schmitt 2005 for more detail on Matheron’s
contribution to geostatistics).
Matheron’s ideas had been anticipated in earlier work. The article by Mercer
and Hall (1911), as well as the appendix to it by Student (1911), anticipates some of
the fundamental features of modern geostatistics: support, spatial dependence,
correlation range, and the nugget effect. Kolmogorov (1941) devises the ‘‘structure
function’’ to represent both spatial correlation, which we now recognize as the
variogram, and his method of interpolation now known as kriging (Ripley 1981,
pp. 44–50). Matern (1960) theoretically derives some of the familiar ‘‘permissible’’
y = 23.649x – 47011R2 = 0.657
0
100
200
300
400
500
600
700
800 (a) (b)
1990 1995 2000 2005 2010Year
Tot
al n
umbe
r of
geo
stat
istic
alar
ticle
s in
all
jour
nals
(G
eoba
se)
y = 0.7812x – 1553R2 = 0.5735
0
5
10
15
20
25
1990 1995 2000 2005 2010Year
Num
ber
of g
eost
atis
tical
art
icle
s in
geog
raph
y jo
urna
ls (
Geo
base
)
Figure 1. Number of geostatistics articles identified by a Geobase search published (a) in
general and (b) in geography journals as a function of time.
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functions to describe spatial covariance from random point processes. These are
equivalent to Jowett’s (1955) ‘‘serial variation function,’’ which now is known as
the variogram.
The geostatistical approach to describing spatial dependence: the covariance
and variogram
Most spatial properties vary in such a complex way that variation cannot be defined
deterministically, and thus the basis of geostatistics is to treat the variable of interest
as a random variable. Traditionally, a variable is defined on a continuous surface
such that at each point, s, in space, a range of values exists for an attribute, Z(s), and
the one observed, z(s), is drawn at random from a probability distribution. The set of
random variables, fZ(s): sADg, where D is a subset of two-dimensional space (R2),
is a random field, and the actual values of Z observed are for just one of a poten-
tially infinite number of realizations of it (the ‘‘superpopulation’’ view). Geostatis-
tics is based on regionalized variable theory (RVT), which provides a sound model
of how properties vary in space. It recognizes gradual change across R2, locally
erratic and structured components of variation, and uncertainty.
To describe the variation of an underlying random field and to estimate the
mean and variance of an attribute, the spatial autocovariance (or covariance) is
estimated to describe quantitatively the relation between pairs of points a given
distance apart. This covariance is given by
Cðsi ; sjÞ ¼ E½ ZðsiÞ � mðsiÞf g ZðsjÞ � mðsjÞ� �
� ð1Þ
where m(si) and m(sj) are the means of Z at si and sj, and E[.] is the expected value.
Because only one realization of Z exists at each point, these means are unknown.
To proceed, geostatistics invokes assumptions of stationarity, which means that
certain properties of a random field are assumed to be the same everywhere. We
assume that the mean, m5 E[Z(s)], is constant for all s, and hence m(si) and m(sj) can
be replaced by m, which can be estimated by repetitive sampling. When si and sj
coincide, equation (1) defines the variance or the a priori variance of a field,
s2 5 E[fZ(s)� mg2] which is assumed to be finite and, as for the mean, the same
everywhere. When si and sj do not coincide, their covariance depends on their
separation and not on their absolute positions, a property that applies to any pair of
points separated by lag h (a vector in both distance and direction). Therefore, given
two points si and sj separated by lag h,
Cðsi; sjÞ ¼E½ ZðsiÞ � mf g ZðsjÞ � m� �
�¼E½fZðsÞgfZðsþ hÞg � m2�¼CðhÞ
ð2Þ
which is also constant for any given h. This constancy of the first and second mo-
ments of a random field constitutes second-order or weak stationarity. Equation (2)
indicates that the covariance is a function of the spatial lag and describes quan-
titatively the dependence between values of Z with changing separation or lag
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
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distance. Often the autocovariance is converted to the dimensionless autocorrela-
tion by
rðhÞ ¼ CðhÞ=Cð0Þ
where C(0) 5s2 is the covariance at lag 0.
The mean often appears to change across a region, and the variance will then
appear to increase indefinitely as the extent of an area increases. The covariance
cannot be defined because no value for m exists to insert into equation (2). This
situation is a departure from weak stationarity. Matheron’s (1965) solution is the
weaker intrinsic hypothesis of geostatistics. Although the general mean might not
be constant, it would be constant for small distances, and so the expected differ-
ences would be zero:
E½ZðsÞ � Zðsþ hÞ� ¼ 0
and the expected squared differences for those lags define their variances:
E½ ZðsÞ � Zðsþ hÞf g2� ¼ var½ZðsÞ � Zðsþ hÞ� ¼ 2gðhÞ ð3Þ
The quantity g(h) is known as the semivariance at lag h, or the variance per
point when points are considered in pairs. As for the covariance, the semivariance
depends only on the lag and not on the absolute positions of the data points. As a
function of h, g(h) is the semivariogram or, more usually, the variogram. The va-
riogram can be applied when the assumptions of second-order stationarity do not
hold or when uncertainty exists about whether they do or do not. This result makes
the variogram a valuable tool, and accordingly it has become the cornerstone of
geostatistics. If the field fZ(s): sADg is second-order stationary, the semivariance
and covariance are equivalent.
The usual method of computing the empirical semivariances from data,
fz(s1), z(s2), . . .., z(sn)g, at sample points s1, s2, . . ., sn, is Matheron’s (1965) method
of moments (MoM) estimator:
gðhÞ ¼ 1
2mðhÞXmðhÞi¼1
fzðsiÞ � zðsi þ hÞg2 ð4Þ
where z(si) and z(si1h) are the actual values of Z at locations si and si1h, which
are separated by the lag h. The sum is over m(h), which is the number of paired
comparisons separated by h. By changing h, an ordered set of semivariances is
obtained; these semivariances constitute the experimental or sample variogram.
Other estimators of the variogram exist, most notably the residual maximum
likelihood (REML) estimator (Pardo-Iguzquiza 1998a, b). Diggle and Ribeiro
(2007) also suggest a model-based approach to geostatistics where expert knowl-
edge, in addition to properties of the data, plays a role in determining an appro-
priate variogram model.
Assuming sufficient data from which to compute a reliable empirical variogram
(see Webster and Oliver 1992, pp. 178, 190–91), a best-fitting model is selected
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from what are known as authorized (or permissible) functions (see Webster and
Oliver 2007, pp. 82–95). Several fitting procedures exist from which to choose
(Cressie 1993, pp. 90–104).
The variogram can take on a variety of forms (Fig. 2), and the reader is referred
to Webster and Oliver (2007, pp. 56–60) for a discussion of the most common
forms and their interpretations and for explanations of commonly used terms that
describe important features of the variogram, such as its ‘‘sill,’’ ‘‘range,’’ and ‘‘nug-
get’’ variance. Introductions to basic geostatistical methodology and terms are also
given by Armstrong (1998), Christensen (2001), Goovaerts (1997), and Isaaks and
Srivastava (1989). The variogram is a valuable exploratory data tool, regardless of
whether an analyst wishes to use other geostatistical tools. For example, if data
result in a variogram that appears as a horizontal set of points (i.e., pure nugget)
(a) (b)
Lag distance (h) Lag distance (h)
(c) (d)
Lag distance (h)
Range (a)
c1
Sill variance
c1
a1a2
c2
0 40 80 120Lag distance (h)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Var
ianc
e
c0
160
(c0+c1)
Nugget (c0)
Figure 2. Examples of (a) a bounded variogram fitted with a spherical model, (b) a bounded
variogram fitted with an exponential model (nugget highlighted), (c) an unbounded vario-
gram fitted with a power model, and (d) a bounded variogram with nested variation fitted
with a double spherical model.
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
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interpolation would merely give the local observed mean. In such cases, all that is
essentially being done is the application of a local smoothing function. The vario-
gram shape may also indicate whether the variation has a spatially random com-
ponent (nugget, c0; Fig. 2a–d) and whether more than one scale of variation is
present, which requires a nested model that can be used for factorial kriging (Fig.
2d). If a variogram has no upper bound or sill (Fig. 2c), disjunctive kriging and
empirical-BLUP are precluded. Anisotropy in variation can be explored by com-
puting a variogram in different directions. The variogram map (semivariances plot-
ted against separation distance in the x and y directions [Webster and Oliver 2007,
p. 75]) can indicate the directions of greatest and least variation.
Kriging
Kriging is a generic term for a range of BLUP least-squares methods of spatial in-
terpolation in geostatistics. It can be linear or nonlinear, although the former is
more common. Kriging provides not only predictions but also the kriging errors or
kriging variances at each prediction location. The original formulation of linear
kriging, now known as ordinary kriging (Journel and Huijbregts 1978), is the most
robust and most used method. Ordinary kriging assumes that the mean is unknown
but constant and that the random field is locally stationary. In linear kriging, the
estimate at any location s0, Z ðs0Þ, is a weighted linear combination of the data:
Zðs0Þ ¼Xn
i¼1
lizðsiÞ ð5Þ
The weights, li, are chosen to minimize E½fZðs0Þ � Zðs0Þg2�, or the kriging
variance, and to ensure that the estimates are unbiased, the weights are constrained
to sum to 1 (Webster and Oliver 2007, pp. 155–59). Kriging uses the spatial infor-
mation described by a variogram function together with the data to predict opti-
mally. The weights depend on the variogram and the spatial positions of the data,
fz(si)g, in relation to one another and to the target point or block (s0). Observations
that are nearest to a prediction point, s0, have the largest weights, but clusters of
adjacent observations that are highly correlated are individually down-weighted.
Kriging is essentially a local predictor that, depending upon the aims of the pre-
diction, can be applied to point (punctual kriging) or block supports of various sizes
(block kriging) and shapes (area-to-area [ATA] kriging) even if the sample informa-
tion is for points. This theory also provides the basis for designing optimal sampling
schemes for kriging using a variogram (Webster and Oliver 2007, pp. 186–88).
Since its original formulation, kriging has been elaborated to tackle increas-
ingly complex problems. Disjunctive (Matheron 1973) and indicator (Journel 1982)
kriging are nonlinear forms that give probabilities that attribute values are above or
below a given threshold. Both techniques have been widely used as part of the risk
assessment of contaminated sites based on thresholds that define serious contam-
ination (Brus et al. 2002; Gaus et al. 2003). Oliver, Webster, and McGrath (1996)
also discuss the merits of disjunctive kriging for environmental management. Kerry
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and Oliver (2007) use indicator kriging to map soil structure from categorical field
observations, and Lloyd and Atkinson (2001) use it to assess the uncertainty of
digital elevation model (DEM) estimates. Lark and Ferguson (2004) use both indi-
cator and disjunctive kriging in an agricultural context to map the probability of
serious nutrient deficiency. Matheron (1969) originally introduced universal kriging
to deal with data with a strong deterministic component (i.e., trend), but the state-
of-the-art is empirical BLUP (Stein 1999). The latter is based on a variogram esti-
mated by REML (Lark, Cullis, and Welham 2006). For situations in which some
prior knowledge about a drift or trend exists, Omre (1987) introduces Bayesian
kriging.
When two or more variables are spatially correlated or co-regionalized, and
one is more expensive to obtain than the other, predictions of the less densely
sampled variable can be improved by ordinary cokriging (CK) (Matheron 1965).
When the cross-variogram has been computed and modeled, the strength of
co-regionalization and the potential benefit of CK for the dependent variable can
be assessed. Several other methods in geostatistics incorporate secondary informa-
tion to improve the accuracy of predictions of a primary variable. Examples include
regression kriging, simple kriging with locally varying means (SKlm), kriging with
an external drift (KED), and multivariate factorial kriging. These methods often have
been compared with ordinary kriging (Odeh, McBratney, and Chittleborough 1994;
Odeh and McBratney 2000; Rawlins et al. 2009). We refer interested readers to
Chiles and Delfiner (1999), Goovaerts (1997), McBratney et al. (2000), Wacker-
nagel (1995), and articles in this special issue for more detail on these methods,
which take advantage explicitly of the co-regionalization between variables. Cross-
variogram analysis also can describe how the relationship between properties
varies with scale in the form of structural correlations (Webster and Oliver 2007,
p. 240) or of regionalized correlation coefficients (Wackernagel 1995). These
multivariate geostatistical methods have interpretive as well as predictive power,
especially if the primary variable is regarded as analogous to the dependent vari-
able and if the secondary variables are treated as the independent variables.
Factorial kriging (Matheron 1982) was developed for nested variation. Long-
and short-range components of variation are estimated in a single analysis and can
be filtered out. Oliver, Webster, and Slocum (2000) use factorial kriging to filter
different scales of variation from remotely sensed imagery and to determine the
sources of variation at each scale. Goovaerts and Webster (1994) use this approach
to isolate different scales of variation in topsoil copper and cobalt concentrations
and to examine the correlation between them at various scales.
The kriging equations can be used with an existing variogram to design a new
optimal sampling scheme for kriging (McBratney, Webster, and Burgess 1981). No
data are required for this task, and it is advantageous when variables are expensive
to obtain and where previous sampling has been either too intensive or insufficient.
Atkinson (1991) uses geostatistics to optimize ground sampling strategies for re-
motely sensed investigations. The structure of spatial autocorrelation characterized
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
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by the variogram also can be used to improve the spatial continuity of classifications
(Oliver and Webster 1989). Atkinson and Lewis (2000) use this approach to classify
remotely sensed imagery, and Frogbrook and Oliver (2007) use it to identify spatially
coherent management zones for precision agriculture. Geostatistical methodology
also has been implemented for compressing image data (Oliver, Shine, and Slocum
2005) and for identifying redundant bands in hyperspectral imagery.
Kriging tends to smooth variation in data (see Webster and Oliver 2007, p.
267–68, for further explanation). Often this outcome is required, but sometimes
the uncertainty or likely variability of observed patterns needs to be established.
Determining uncertainty involves geostatistical simulation by methods such as
turning bands, sequential Gaussian simulation, and LU decomposition. Given a
specific variogram function and histogram, data with the desired characteristics can
be simulated to produce multiple equi-probable realizations with the same vario-
gram and histogram. Simulation also can be conditioned on existing data where the
characteristics of the conditioning data are taken into account. Goovaerts (2001)
provides an overview of the relative merits of kriging and simulation. Often sim-
ulation is used for risk assessment; for example, Bierkens (2006) uses conditional
simulation to determine uncertainties associated with groundwater pollution and
the associated costs of installing a monitoring network. Webster and Oliver (1992)
use turning bands simulation to create fields with a known variogram to determine
how well the generating variogram is reproduced by samples of different sizes.
They also determine confidence limits for experimental variograms computed from
different sample sizes.
Finally, geostatistical methods have been used to investigate temporal as well
as spatial autocorrelation. Applications in this context include studies by Kyriakidis
and Journel (2001a, b) of sulfate deposition over Europe and by Heuvelink, Musters,
and Pebesma (1996) of soil water content. Kyriakidis and Journel (1999) give an
overview of these methods.
Geostatistics for regional data
More recently, the basic theory of geostatistics has been adapted to predict and map
regional data and to model spatial variation in an attribute (a dependent variable) in
terms of other variables (independent variables), as in linear regression. This rep-
resents a particularly interesting and important development in geostatistics for
human geographers.
Oliver et al. (1998) develop binomial CK to analyze the risk of childhood can-
cer in the West Midlands of England. To estimate the variogram of risk for binomial
CK, population size is taken into account to give more weight to pairs of areas with
larger populations and hence more reliable rates (Oliver et al. 1998, pp. 286–87).
In addition to the kriging weights li, summing to 1, the weighted sum of the risk
covariances between the target point s0 and the centroids of the data supports si are
constrained to equal the variance of the underlying risk (Oliver et al. 1998, pp.
284–85).
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When the population at risk is large and the probability is small, the binomial
distribution approaches the Poisson. Monestiez et al. (2005, 2006) introduce Pois-
son kriging for rates where the small number problem is an issue. This approxi-
mation has since been applied to health data (Goovaerts 2005; Ali et al. 2006).
Again, areas with larger populations are given more weight when estimating the
variogram and when solving the kriging equations. An ‘‘error variance’’ term, de-
rived from the Poisson distribution (Goovaerts 2005, 2006b) and associated with
the reliability of the rates based on population size, is introduced.
Articles by Gotway and Young (2002, 2005), Kyriakidis (2004), and Goovaerts
(2006b, 2008) address problems associated with change of support in the case of
irregularly sized and shaped areas. This situation involves deconvolution of the
variogram obtained from areal data to estimate a point-support variogram. Then
area-to-point (ATP) kriging is applied where the measurement support is an area,
but the prediction support is a point. Goovaerts (2008) describes an approach to
ATP kriging that involves discretizing each area, where the number of discretizing
points for any area will depend on its size. Each discretizing point has an associated
population count obtained from small area census data; therefore, population het-
erogeneity is taken into account in estimating the deconvoluted variogram (Goo-
vaerts 2008, p. 109). The distance between any two areas (required for calculating
the variogram) is measured as a population-weighted average of the straight-line
distances between all the points that discretize the two areas (Goovaerts 2008, p.
106). Setting aside in this review the ‘‘supplementary and unverifiable hypotheses’’
(Journel and Huijbregts 1978, p. 231) that ATP kriging (downscaling) involves, the
creation of such maps of disease rates helps to reduce the visual bias of choropleth
maps, where physically large areas dominate.
Goovaerts (2006b, 2008) combines Poisson kriging with ATP kriging (Ky-
riakidis 2004) for the analysis of cancer rate data. ATA Poisson kriging is applied
when both the measurement and prediction supports are areas (blocks). Taking
population into account filters out the influence of the small number problem.
Comparing geostatistics with other approaches to spatial data analysis
Regional data are encountered widely in human geography, and methods have
been developed that, in many respects, contrast with those of geostatistics.
The initial interest in spatial dependence among human geographers in the
1960s took the form of adapting existing tests for spatial autocorrelation developed
by Moran (1950), Geary (1954), and Krishna Iyer (1949) on regular lattices to the
irregular shapes of geographical units (Cliff and Ord 1973). Moran’s I and Geary’s c
statistics are
I ¼ nXn
i;j¼1
dði; jÞðzðiÞ � �zÞðzðjÞ � �zÞ
24
35, Xn
i;j¼1
dði; jÞ
0@
1AXn
i¼1
ðzðiÞ � �zÞ224
35 ð6Þ
and
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
17
Page 12
c ¼ ðn � 1ÞXn
i;j¼1
dði; jÞðzðiÞ � zðjÞÞ224
35, 2
Xn
i;j¼1
dði; jÞ
0@
1AXn
i;j¼1
ðzðiÞ � zðjÞÞ224
35 ð7Þ
where z(i) is the observation for area i (i 5 1, . . ., n) and d(i,j) is 1 if i6¼j and i and j
are contiguous, and 0 otherwise. The numerator of equation (6) shares similarities
with an estimator for autocovariance (compare it with equation (1)), whereas the
numerator of equation (7) is similar to the MoM estimator for computing the semi-
variances (compare it with equation (4)). Cliff and Ord’s (1973) adaptation of these
statistics is achieved by replacing fd(i, j)g by the more general fw(i, j)g, which
specifies a connectivity or weights matrix. This specification can be based on
topological and/or geometric properties of the shapes or inter-area flow (or similar)
data as a measure of the ‘‘strength’’ of the connections between areas (Haining
2003, pp. 74–87). The resulting n-by-n matrix also could be used to identify the
different orders of contiguity (similar to the vector h in geostatistics; see equation
(4)), but in practice many analyses consider only the first-order or closest nearest
neighbors. Testing takes the form of a null hypothesis of no spatial autocorrelation
against a nonspecific alternative hypothesis that spatial autocorrelation is present.
This type of testing is of particular interest when examining the residuals from a
least-squares regression (for which some further modifications to equations (6) and
(7) and their distribution theory are needed), because model errors are required to
be independent. Investigations of the power of Moran’s I and Geary’s c by simu-
lation and evidence that the asymptotic relative efficiency of c to I is o1 have
resulted in Cliff and Ord’s (1973) modified I statistic becoming the spatial auto-
correlation test statistic of choice among geographers.
If spatial autocorrelation is detected in least-squares regression residuals, a
common response is to respecify the regression model using a simultaneous spatial
autoregressive model for the errors, where the value of the error term at location i is
modeled as a function of the values of the errors at adjacent locations plus an in-
dependent (white noise) term. Geographers analyzing regional data tend to model
spatial dependence using the simultaneous spatial autoregressive model (specified
either with respect to the dependent variable or the errors) rather than calculating
the empirical covariances and selecting a function that best fits these auto covari-
ances (for an overview, see Haining 2003, pp. 297–304, 350–58). This approach
taken by geographers is similar to that of econometricians when modeling time-
series data, which has led to the adoption, by some, of the term spatial economet-
rics (Anselin 1988).
Regression analysis is used to identify those independent variables that best
account for the variation in a dependent variable. The linear function element of a
regression model, which includes the independent variables, specifies the mean of
a dependent variable. It is variation in the set of significant independent variables
that model (or ‘‘statistically explains’’) variation in the mean of a dependent vari-
able. It is this variation usually that is of most interest to geographers. Spatial de-
Geographical Analysis
18
Page 13
pendence in the second-order sense, which is important in geostatistics, refers only
to that part of the variation in a dependent variable that remains ‘‘unexplained’’ by
regression. This remark applies not only to fixed-effects modeling but also to ran-
dom-effects modeling of spatial data because the random effects term is often
specified using some form of (conditional) spatial autoregressive mode (Besag,
York, and Mollie 1991).
Two further comparative remarks merit attention here. First, frequently the ef-
fects of population size are handled in regression through weighted least squares—
down-weighting most those data values with the largest error variances in model
fitting (e.g., those based on the smallest population denominators). In geostatistics,
this down-weighting is introduced both into modeling the variogram and into the
kriging weights. Second, geographers have paid particular attention to a form of
spatial heterogeneity where properties vary, or appear to vary, with location, es-
pecially when observed over large regions. For example, the mean varies in dif-
ferent map segments (though not in the form of a trend). In addition, or alternately,
the variance and covariance, or even the relationship between a dependent and a
set of independent variables, might vary in different parts of a map. To quantify this
heterogeneity, statistical tests for global- or map-wide spatial dependence, such as
Moran’s I, have been complemented with local statistics, such as local indicators of
spatial association that analyze spatially defined subsets of data using moving win-
dows (Anselin 1995). Sometimes geographically weighted regression is used with
local subsets of data (see Fotheringham, Brunsdon, and Charlton 2000). Some of the
geostatistical methods previously discussed can be seen as equivalent develop-
ments to deal with spatial heterogeneity. First, SKlm uses known strata from a ge-
ology map, for example, to inform about a locally varying mean, and the class
residuals are kriged and added to the strata means (Goovaerts 1997). Similarly,
SKlm can use regression between primary and secondary variables to inform about
a local mean. McBratney, Hart, and McGarry (1991) also acknowledge that the
structure of the variogram can change spatially and thus have partitioned regions
based on ancillary data and computed separate variograms for subregions within
a study area. Walter et al. (2001) and Haas (1990) use local or moving window
variograms and find improvements in estimates over area-wide variograms in sit-
uations where data display heterogeneity. Finally, when KED is employed, allow-
ance is made for spatial variation in the linear relationships between variables or
the regression coefficients (Goovaerts 1997).
The equivalent problem to spatial interpolation for regional data occurs when
there are no data in some areas. Although interest may exist in trying to estimate these
missing values, for either prediction or mapping purposes (see, e.g., Haining 2003,
pp.154–74 for an overview of methods, including those based on the
expectation-maximization algorithm, which has similarities to kriging), the focus
usually is again on fitting a regression model. The concern is less with estimating the
missing values per se (although estimates are generated) and more with estimating
the parameters of a regression model when there are missing values in a database.
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
19
Page 14
An overview of applications of geostatistics in geography
Table 2 classifies the results from the previously described Geobase search,
providing one relevant reference per category.3 The table shows that 40% of geo-
statistical references found in ‘‘mainstream’’ geography journals refer to the use
of kriging in relation to temperature, rainfall erosivity, various atmospheric
and groundwater pollutants, soil properties, vegetation and species distribu-
tion patterns, and DEM creation. By contrast, kriging has been little used
in human geography, an exception being De Cola’s (2002) mapping of Lyme
disease.
Few articles in geography journals consider the importance of the variogram as
an interpretive tool, for example, for quantifying ‘‘spheres of influence’’ or the
proportion of variation that is spatially uncorrelated and correlated at the scale of
sampling, or for identifying variation at different scales and directions as a first step
in understanding processes. This oversight might indicate that few researchers in
geography are as familiar as they could be with computing and modeling the va-
riogram and with choosing an appropriate method of kriging. A possible cause of
this problem is that certain software packages, such as Surfer, offer kriging as a
method of interpolation with default settings and do not provide users with access
to experimental variogram and modeling options. In ArcGIS’s geostatistical analyst,
a variogram model is fitted to the variogram cloud, but determining visually
whether a model is a sensible fit is difficult because of the inherent skewness in
the distribution of squared differences (Cressie 1993, p. 41).
Table 2 shows that multivariate geostatistical methods, which incorporate sec-
ondary information into the interpolation process (e.g., SKlm, KED, and CK), have
been used for estimating air temperature and bird diversity using the normalized
difference vegetation index. Factorial kriging has been used to filter different scales
of variation in remotely sensed imagery and plant and soil data and to explore
scale-dependent correlations in health geography. Indicator kriging has rarely been
used in geography, but examples include incorporating it into the classification of
hyperspectral imagery and estimating pre-settlement vegetation patterns. Although
indicator and disjunctive kriging are important in risk assessment, which is partic-
ularly important to geographers studying natural hazards, no applications of these
two specific techniques were found. Furthermore, disjunctive kriging appears not to
have been referred to at all in the geographical literature. Also, although adapta-
tions of kriging, such as binomial CK and Poisson kriging, are ideal for examining
animal count data, which are often highly skewed and involve small numbers, no
examples were found in the biogeography literature.
Geostatistical change of support has received some attention in the geograph-
ical literature; it links with scale issues and the modifiable areal unit problem. In
remote sensing and biogeography, geostatistics has been used to investigate the
effects of change of support in sampling, but most investigations are in human
geography and are relatively recent, applying techniques such as ATP kriging.
Geographical Analysis
20
Page 15
Tab
le2
Sum
mar
yof
aG
eobas
ese
arch
for
geost
atis
tics
1ar
ticl
esap
pea
ring
injo
urn
als
wit
hG
eogr
a�in
the
titl
e
Phys
ical
geogr
aphy
(subfi
elds)
Geo
stat
isti
cal
area
of
inve
stig
atio
n
Bio
geogr
aphy
Cli
mat
olo
gyG
eom
orp
holo
gyH
ydro
logy
Rem
ote
sensi
ng
Hum
an
geogr
aphy
Tota
lre
fs.
Kri
ging
for
inte
rpola
tion/
(com
par
ativ
e
inte
rpola
tion
studie
s)
9re
fs.
(1re
f.)—
Stra
nd
(1998)
Hes
slet
al.
(2007)
7re
fs.
(5re
fs.)
—
Wri
ght
etal
.
(2002)
Tat
alovi
ch,
Wil
son,
and
Cock
burn
(2006)
5re
fs.
(13
refs
.)—
Hock
and
Jense
n
(1999)
Lay
and
Wan
g(1
996)
3re
fs.—
Li,
Song,
and
Xia
o(2
005)
4re
fs.—
De
Cola
(2002)
28
refs
.(1
9
refs
.)
Inte
rpre
tive
valu
e
of
the
vari
ogr
am
8re
fs.—
Ken
t
etal
.(2
006)
4re
fs.—
Gom
ersa
ll
and
Hin
kel
(2001)
1re
f.—
Bia
n
and
Xie
(2004)
3re
fs.—
Ma,
Ma,
and
Xu
(2004)
16
refs
.
Mult
ivar
iate
geost
atis
tics
(KED
,
CK
,SK
lm)
7re
fs.—
Lin
etal
.(2
008)
2re
fs.—
Her
nan
dez
Rober
to(2
001)
9re
fs.
Fact
ori
alkr
igin
g2
refs
.—R
odge
rs
and
Oli
ver
(2007)
1re
f.—
War
r,
Oli
ver,
and
Whit
e(2
002)
1re
f.—
Goova
erts
,
Jacq
uez
,an
d
Gre
ilin
g
(2005)
4re
fs.
Indic
ator
and
dis
junct
ive
krig
ing
2re
fs.—
Wan
g
(2007)
1re
f.—
Goova
erts
(2002a)
3re
fs.
Bin
om
ial/
Pois
son
krig
ing
7re
fs.—
Goova
erts
(2005)
7re
fs.
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
21
Page 16
Tab
le2
Conti
nued
Phys
ical
geogr
aphy
(subfi
elds)
Geo
stat
isti
cal
area
of
inve
stig
atio
n
Bio
geogr
aphy
Cli
mat
olo
gyG
eom
orp
holo
gyH
ydro
logy
Rem
ote
sensi
ng
Hum
an
geogr
aphy
Tota
lre
fs.
Chan
geof
support
issu
es/A
toP,
Ato
A
krig
ing
1re
f.—
Bel
lehum
eur
and
Lege
ndre
(1997)
1re
f.—
Mas
on,
O’C
onai
ll,
and
McK
endri
ck
(1994)
3re
fs.—
Goova
erts
(2006b)
5re
fs.
Spac
e-ti
me
geost
atis
tics
1re
f.—
Janis
and
Robes
on
(2004)
1re
f.—
Su,
Song,
and
Zhan
g(2
003)
2re
fs.
Sam
pli
ng
schem
e
des
ign
1re
f.—
Lin
etal
.(2
008)
1re
f.—
Finle
y
etal
.(2
007)
2re
fs.
Dat
aco
mpre
ssio
n1
ref.
—A
tkin
son,
Curr
an,
and
Web
ster
(1990)
1re
f.
Spat
ial
wei
ghti
ng
of
clas
sifi
cati
on
1re
f.—
Goova
erts
(2002a)
1re
f.
Geo
stat
isti
cal
sim
ula
tion
4re
fs.—
Bis
hop,
Min
asny,
and
McB
ratn
ey(2
006)
1re
f.—
Goova
erts
(2002b)
2re
fs.—
Goova
erts
(2006a)
7re
fs.
Geo
stat
isti
cal
theo
ry/r
evie
w
arti
cles
1re
f.—
Lark
(2000)
1re
f.—
Dig
gle
and
Rib
eiro
(2002)
8re
fs.—
Corn
ford
,
Csa
to,
and
Opper
(2005)
3re
fs.—
Gri
ffit
h
(2002)
13
refs
.
Geographical Analysis
22
Page 17
Geographers study both spatial and temporal variation, and spatio-temporal
variograms and space-time kriging can be used for predicting phenomena in these
domains. However, only two applications of space-time geostatistics were found:
one for determining the representativeness of air temperature records and one for
assessing the spatio-temporal variation of groundwater salt content.
Only two studies were identified where geostatistics has been applied to sam-
pling issues: one used geostatistics to determine the location of pollutants following
a bioterrorist attack, and the other to establish sampling schemes for mapping bird
diversity. Although the storage and classification of remotely sensed image data are
important in geography, only one article was found that uses geostatistics for data
compression and only one that uses spatial weighting by the variogram to improve
classification contiguity.
Most applications of geostatistical simulation are in geology or geomorphology
for risk assessment and uncertainty, but they are also used in remote sensing and
health geography. For the latter, geostatistical simulation is used to assess the un-
certainty associated with rare disease clusters.
Several theoretical articles were found that introduce new geostatistical con-
cepts or techniques or that provide reviews. These are the second most abundant
type of geostatistics article; however, the authors are not geographers but rather
prolific publishers of geostatistical applications in the applied sciences.
Concluding remarks
Geostatistical approaches to the analysis of spatial data have been underexploited
in geographical research, particularly in human geography. Two main reasons ap-
pear to account for this. First, traditional geostatistical techniques are applicable to
attributes with a continuous spatial index (Ripley 1981; Cressie 1993, pp. 8–9),
making these techniques less relevant to many problems in human geography,
where data frequently relate to areas. Second, geostatistics is often perceived as
being of use only for spatial interpolation or kriging, including mapping and sample
design when obtaining primary data. Other reasons sometimes cited for the un-
deruse of geostatistical approaches include lack of time allotted to teaching geo-
statistics in spatial data analysis courses in geography departments compared with
other forms of spatial analysis and lack of instruction about the appropriate use of
available software.
Geostatistics embraces a broad range of tools and modeling techniques that
can be applied to many spatial problems, including prediction, determination of
the scale of spatial variation, design of sampling for primary data collection,
smoothing of noisy maps, region identification, multivariate analysis, probability
mapping, and change of support. It has a research literature that includes many
disciplines. Most important, however, geostatistics provides the spatial analyst with
a statistically rigorous model of how properties vary in space (RVT) that recognizes
the different components of variation (from locally erratic to spatially structured
Geography, Spatial Data Analysis, and GeostatisticsRobert P. Haining et al.
23
Page 18
components) and furnishes permissible models for those components of variation,
as well as procedures for fitting them to real data.
Today, geostatistical methods can be applied to regional data, although their
use with this type of data does present challenges. Proximity measures are limited
to the distance between centroids, which cannot be defined uniquely when spatial
units have nonzero areas. Data based on rates and proportions are aggregated, and
the denominators often vary among spatial units reflecting differences in population
size. Some variation in data can be a consequence of these underlying differences
in sample support (Krivoruchko, Gotway, and Zhigimont 2003), and these issues
need to be taken into account when dealing with regional data.
More user-friendly software is now available. A list of free and commercial
software for geostatistical analysis is given at http://www.ai-geostats.org/index.
php?id=107, although it is by no means exhaustive, failing to include packages
such as GenStat (developed at Rothamsted Research), S-Plus, and Terraseer’s Space
Time Information System, which currently has a beta package that can accommo-
date different types of geographic data and has capabilities for ATA, ATP, and
Poisson kriging. In addition, many computer programs are available from articles
published in Computers & Geosciences, as well as authors making their code
freely available on their personal Web sites. Deutsch and Journel (1998) provide a
comprehensive set of Fortran programs for many geostatistical techniques in Geo-
statistical Software Library and User’s Guide, and most of these programs have
been adapted for a Windows environment in the freely available Stanford Geosta-
tistical Modeling Software (Remy, Boucher, and Wu 2009). However, the former
has no way of modeling a variogram once it has been computed without using
other software, and the latter employs visual fitting. Gstat is a freely available com-
mand line software package that is favored by many and includes the capability for
variogram model fitting both numerically and visually (Pebesma and Wesseling
1998). With new developments in methodology and software, exciting times lie
ahead for geographers exploring the benefits of geostatistics for solving geographic
problems.
Notes
1 This maps into what Tobler (1970) describes as ‘‘The First Law of Geography’’: ‘‘everything
is related to everything else, but near things are more related than distant things.’’ Bane-
rjee, Carlin, and Gelfand (2004, p. 39) refer to ‘‘The First Law of Geostatistics,’’ which
describes spatial structure in similar but more formal terms. This observation has statistical
roots that go back to the early twentieth century (see Student 1911).
2 The strategy used to search for geostatistics articles was as follows: Year range: 1990–June
2008; Document type: all; (Geobase) Subject/Title/Abstract includes: krig� OR variogram
OR geostat�; (ISI) Title includes: krig� OR variogram OR geostat�. Geostatistics articles
within geography were identified as those that contained the three preceding keywords in
the subject/title/abstract and also were found in a journal with geogra� in the title. We
recognize that the search terms are not exhaustive and will not identify all geostatistics
Geographical Analysis
24
Page 19
articles in the academic literature in general or within geography, but some basic patterns
may be observed that we think are informative. The terms used limit the number of articles
identified, but few, if any, nongeostatistics ones would be identified with these terms.
Simulation was not included because it would cause potential confusion. For more
information arising from the bibliometric search and for more references obtained from the
literature review in this article, interested readers should visit http://www.geog.cam.ac.uk/
people/haining/ or contact the corresponding author.
3 See note 2. We do not claim to have identified all areas of application. However, for more
details about references, go to http://www.geog.cam.ac.uk/people/haining/ or contact the
corresponding author.
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