Geoderma 89 Ž1999. 1–45
Geostatistics in soil science: state-of-the-art and
perspectives
P. Goovaerts )
Department of CiÕil and EnÕironmental Engineering, The UniÕersity of Michigan, Ann Arbor,
MI 48109-2125, USA
Received 4 November 1997; accepted 6 July 1998
Abstract
This paper presents an overview of the most recent developments in the field of geostatistics
and describes their application to soil science. Geostatistics provides descriptive tools such as
semivariograms to characterize the spatial pattern of continuous and categorical soil attributes.
Various interpolation Žkriging. techniques capitalize on the spatial correlation between observations to predict attribute values at unsampled locations using information related to one or several
attributes. An important contribution of geostatistics is the assessment of the uncertainty about
unsampled values, which usually takes the form of a map of the probability of exceeding critical
values, such as regulatory thresholds in soil pollution or criteria for soil quality. This uncertainty
assessment can be combined with expert knowledge for decision making such as delineation of
contaminated areas where remedial measures should be taken or areas of good soil quality where
specific management plans can be developed. Last, stochastic simulation allows one to generate
several models Žimages. of the spatial distribution of soil attribute values, all of which are
consistent with the information available. A given scenario Žremediation process, land use policy.
can be applied to the set of realizations, allowing the uncertainty of the response Žremediation
efficiency, soil productivity. to be assessed. q 1999 Elsevier Science B.V. All rights reserved.
Keywords: geostatistics; spatial interpolation; risk assessment; decision making; stochastic
simulation
1. Introduction
During the last decade, the development of computational resources has
fostered the use of numerical methods to process the large bodies of soil data
)
Fax: q1-313-734-2275; E-mail: goovaert@engin.umich.edu
0016-7061r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 0 1 6 - 7 0 6 1 Ž 9 8 . 0 0 0 7 8 - 0
2
P. GooÕaertsr Geoderma 89 (1999) 1–45
that are collected across the world. A key feature of soil information is that each
observation relates to a particular location in space and time. Knowledge of an
attribute value, say a pollutant concentration, is thus of little interest unless
location or time of measurement or both are known and accounted for in the
analysis. Geostatistics provides a set of statistical tools for incorporating spatial
and temporal coordinates of observations in data processing.
Until the late 1980s, geostatistics was essentially viewed as a means to
describe spatial patterns by semivariograms and to predict the values of soil
attributes at unsampled locations by kriging, e.g., see review papers by Vieira et
al. Ž1983. , Trangmar et al. Ž 1985. , and Warrick et al. Ž 1986. . New tools have
recently been developed to tackle advanced problems, such as the assessment of
the uncertainty about soil quality or soil pollutant concentrations, the stochastic
simulation of the spatial distribution of attribute values, and the modeling of
space–time processes. Because of their publication in a wide variety of journals
and congress proceedings, these new developments are generally barely known
by soil scientists who must also struggle with different sets of notation to
establish links between all these techniques. This paper aims to provide a
coherent and understandable overview of the state-of-the-art in soil geostatistics,
refer to recent applications of geostatistical algorithms to soil data, and point out
challenges for the future.
I have extracted most of the material in this paper from my recent book
ŽGoovaerts, 1997a. on the application of geostatistics to natural resources
evaluation. The presentation follows the usual steps of a geostatistical analysis,
introducing tools for description of spatial patterns, quantitative modeling of
spatial continuity, spatial prediction and uncertainty assessment. Particular attention is paid to practical issues such as the modeling of sample semivariograms,
the choice of an interpolation algorithm that incorporates all the relevant
information available, or the incorporation of uncertainty assessment in decision
making. Some common misunderstandings regarding the modeling of cross
semivariograms, the use of the kriging variance or Gaussian-based algorithms
will also be reviewed. The different concepts will be illustrated using multivariate soil data related to heavy metal contamination of an area of the Swiss Jura
ŽAtteia et al., 1994; Webster et al., 1994. , kindly provided by Mr. J.-P. Dubois
of the Swiss Federal Institute of Technology.
2. Description of spatial patterns
Analysis of spatial data typically starts with a ‘posting’ of data values. For
example, Fig. 1 shows the spatial distribution of five stratigraphic classes and of
the concentrations of two heavy metals recorded, respectively, at 359 and 259
locations in a 14.5 km2 area in the Swiss Jura. For both continuous and
P. GooÕaertsr Geoderma 89 (1999) 1–45
3
Fig. 1. Locations of sampling sites superimposed on the geologic map, and concentrations in Cd
and Ni at 259 of these sites Žunitss mg kgy1 ..
categorical attributes, the spatial distribution of values is not random in that
observations close to each other on the ground tend to be more alike than those
further apart. The presence of such a spatial structure is a prerequisite to the
application of geostatistics, and its description is a preliminary step towards
spatial prediction or stochastic simulation.
2.1. Continuous attributes
Consider the problem of describing the spatial pattern of a continuous
attribute z, say a pollutant concentration such as cadmium or nickel. The
information available consists of values of the variable z at n locations u a ,
z Žu a ., a s 1,2, . . . ,n. Spatial patterns are usually described using the experimental semivariogram gˆ Žh. which measures the average dissimilarity between
4
P. GooÕaertsr Geoderma 89 (1999) 1–45
data separated by a vector h. It is computed as half the average squared
difference between the components of data pairs:
1 N Žh .
2
gˆ Ž h . s
z Žu a . y z Žu a q h.
Ž1.
Ý
2 N Ž h . as1
where N Žh. is the number of data pairs within a given class of distance and
direction. Webster and Oliver Ž 1993. showed that reliable estimation of semivariogram values requires at least 150 data, and larger samples are needed to
describe anisotropic Ž direction-dependent. variation. This does not mean, however, that geostatistics cannot be applied to smaller data sets. It is noteworthy
that geostatistics has become the reference approach for characterization of
petroleum reservoirs where the information available typically reduces to a few
wells. Sparse data are often supplemented by expert knowledge or ancillary
information originating from a better sampled area which is deemed similar to
the study area; see also latter discussion on semivariogram modeling.
Fig. 2 Žtop graphs. shows the omnidirectional semivariograms computed from
the heavy metal concentrations displayed in Fig. 1. These graphs point out
Fig. 2. Experimental omnidirectional semivariograms for Cd and Ni: original concentrations and
indicator transforms using thresholds corresponding to the second- Ž — ., fifth- Ž – – . and eighth-decile Ž- - -. of the sample histogram. To facilitate the comparison, indicator semivariogram values
were rescaled by the indicator variance.
P. GooÕaertsr Geoderma 89 (1999) 1–45
5
distinct spatial behaviors of the two metals: Ni concentrations appear to vary
more continuously than Cd concentrations, as illustrated by the smaller nugget
effect and larger range of its semivariogram. In combination with a good
knowledge about the phenomenon and the study area, such a spatial description
can improve our understanding of the physical underlying mechanisms controlling spatial patterns ŽOliver and Webster, 1986a; Goovaerts and Webster, 1994;
Webster et al., 1994. . In the present study, the long-range structure of the
semivariogram of Ni concentrations is probably related to the control asserted by
rock type, while the short-range structure for cadmium suggests the impact of
local man-made contamination ŽGoovaerts, 1997a, p. 38..
Description of spatial patterns is too often restricted to the spatial variation
over the full range of attribute values. Spatial patterns may, however, differ
depending on whether the attribute value is small, medium, or large. For
example, in many environmental applications, a few random ‘hot spots’ of large
concentrations coexist with a background of small values that vary more
continuously in space. Depending on whether large concentrations are clustered
or scattered in space, our interpretation of the physical processes controlling
contamination and our decision for remediation may change. The characterization of the spatial distribution of z-values above or below a given threshold
value z k requires a prior coding of each observation z Žu a . as an indicator
datum iŽu a ; z k ., defined as:
i Žu a ; z k . s
½
1
0
if z Ž u a . F z k
otherwise
Ž2.
Indicator semivariograms can then be computed by substituting indicator data
iŽu a ; z k . for z-data z Žu a . in Eq. Ž 1.:
gˆ I Ž h; z k . s
1
N Žh .
Ý i Žu a ; z k . y i Žu a q h; z k .
2 N Žh.
2
Ž3.
as1
The indicator variogram value 2gˆ I Žh; z k . measures how often two z-values
separated by a vector h are on opposite sides of the threshold value z k . In other
words, 2gˆ I Žh; z k . measures the transition frequency between two classes of
z-values as a function of h. The greater is gˆ I Žh; z k ., the less connected in space
are the small or large values.
Fig. 2 Žbottom graphs. shows the omnidirectional indicator semivariograms
computed for the second-, fifth- and eighth-decile of the distributions of
cadmium and nickel concentrations. To facilitate the comparison, all semivariograms values were standardized by dividing them by the indicator variance. For
both metals, indicator semivariograms for small concentrations have smaller
nugget effect than those for larger concentrations, which suggests that homogeneous areas of small concentrations coexist within larger zones where large and
P. GooÕaertsr Geoderma 89 (1999) 1–45
6
medium concentrations are intermingled. Two clusters of small concentrations
are indeed apparent on the location maps of Fig. 1 and correspond to the
Argovian rocks.
2.2. Categorical attributes
Many soil variables such as texture or water table classes take only a limited
number of states which might be ordered or not. Spatial patterns of such
categorical variables can also be described using geostatistics. Let S be a
categorical attribute with K possible states sk , k s 1,2, . . . , K. The K states are
exhaustive and mutually exclusive in the sense that one and only one state sk
occurs at each location u a . The pattern of variation of a category sk can be
characterized by semivariograms of type Ž 3. defined on an indicator coding of
the presence or absence of that category:
i Ž u a ; sk . s
½
1
0
if s Ž u a . s sk
otherwise
Ž4.
The indicator variogram value 2gˆ I Žh; sk . measures how often two locations a
vector h apart belong to different categories skX / sk . The smaller is 2gˆ I Žh; sk .,
the more connected is category sk . The ranges and shapes of the directional
indicator semivariograms reflect the geometric patterns of sk .
Fig. 3 shows the indicator semivariograms of two stratigraphic classes of Fig.
1 computed in four directions with an angular tolerance of 22.58. For both
classes the indicator semivariogram value equals zero at the first lag, which
means that any two data locations less than 100 m apart belong to the same
formation. The longer SW–NE range Žlarger dashed line. reflects the corresponding preferential orientation of these two lithologic formations.
Fig. 3. Experimental indicator semivariograms of Argovian and Sequanian rocks computed in four
directions Ž —: 22.58, – –: 67.58, – – –: 112.58, . . . : 157.58; angular tolerances 22.58..
P. GooÕaertsr Geoderma 89 (1999) 1–45
7
Fig. 4. Experimental omnidirectional cross semivariogram between Cd and Ni.
2.3. BiÕariate description
Soil information is generally multivariate, and geostatistics is increasingly
used to investigate how the correlation between two soil properties varies in
space. A measure of the joint variation of two continuous attributes z i and z j is
the experimental cross semivariogram:
gˆ i j Ž h . s
1
N Žh .
Ý z i Žu a . y z i Žu a q h. P z j Žu a . y z j Žu a q h.
2 N Ž h . as1
Ž5.
In other words, one looks at the joint variation of gradients of z i- and z j-values
from one location to another a vector h away. If both attributes are positively
related, an increase Ždecrease. in z i from u a to u a q h tends to be associated
with an increase Ždecrease. in z j , and so the cross semivariogram value is
positive as for the pair cadmium–nickel in Fig. 4.
Cross semivariograms can also be computed from indicator values of type Ž2.
in order to characterize the spatial connection of small or large values of two
soil properties ŽGoovaerts, 1997a, p. 52.. For example, a similarity in the spatial
distribution of large values for two heavy metals could indicate the existence of
common sources of contamination. Another application of indicator cross semivariograms is the study of the spatial architecture of categories such as soil types
in Goovaerts Ž 1994a..
3. Modeling spatial variation
Description of spatial patterns is rarely a goal per se. Rather, one generally
wants to capitalize on the existence of spatial dependence to predict soil
properties at unsampled locations. A key step between description and prediction is the modeling of the spatial distribution of attribute values. Most of
P. GooÕaertsr Geoderma 89 (1999) 1–45
8
geostatistics is based on the concept of random function, whereby the set of
unknown values is regarded as a set of spatially dependent random variables.
Each datum z Žu a . is then viewed as a particular realization of a random
variable ZŽu a .. An important characteristic of the random function is its
semivariogram which is modeled from the experimental values.
3.1. UniÕariate case
Consider the problem of modeling the semivariogram of a single attribute z
from the set of experimental values gˆ Žh k . computed for a finite number of lags,
h k , k s 1, . . . , K, in distance and direction. Most soil scientists are now aware
that semivariograms can be modeled using only functions that are conditionally
negative definite, such as the exponential or spherical models Ž McBratney and
Webster, 1986. . Fewer know that the Gaussian semivariogram model is generally unrealistic and leads to unstable kriging systems and artifacts in the
estimated maps Ž Wackernagel, 1995, pp. 109–111. . In many situations, two or
more permissible models must be combined to fit the shape of the experimental
semivariogram, as illustrated for cadmium in Fig. 5. Such nested model is
written as:
L
g Ž h . s Ý b l g l Ž h . with b l G 0
Ž6.
ls0
where b l is the positive sill or slope of the corresponding basic semivariogram
model g l Žh..
The way in which these permissible models are chosen and their parameters
Žrange, sill. are estimated is still controversial Ž Goovaerts, 1997a, pp. 97–107..
Several methods have been proposed, ranging from full black-box procedures
which involve an automatic choice and fitting of the model to visual approaches
in which the model is selected so that the fit looks satisfactory graphically. An
Fig. 5. Experimental omnidirectional semivariogram for Cd, and the nested model fitted which
includes a nugget effect and two spherical models with ranges of 200 m and 1.3 km.
P. GooÕaertsr Geoderma 89 (1999) 1–45
9
intermediate approach consists of an automatic Ž least-squares. estimation of
parameters of models chosen by the user Žsemi-automatic procedure..
Black-box procedures should be avoided because they cannot take into
account ancillary information that is critical when sparse or preferential sampling makes the experimental semivariogram unreliable. Differences between
semi-automatic and visual procedures reside in the criterion used to assess the
goodness of the fit. The user might feel more comfortable if the choice of a
particular model can be based on statistical criteria such as the sum of squares of
differences between experimental and model semivariogram values. Using such
criteria amounts to reducing semivariogram modeling to an exercise in fitting a
curve to experimental values, which I think is too restrictive. The objective of
the structural analysis is to build a permissible model that captures the major
spatial features of the attribute under study. Although experimental semivariogram values play an important role in this process, ancillary information such
as provided by physical knowledge of the area and phenomenon may be of great
interest. For example, strong prior qualitative information may lead one to adopt
an anisotropic model even if data sparsity prevents seeing anisotropy from the
experimental semivariograms.
In all situations one should avoid overfitting semivariogram models. The
more complicated model usually does not lead to more accurate estimates.
3.2. BiÕariate case
Modeling the coregionalization between two variables z i and z j requires the
modeling of the two direct semivariograms g ii Žh. and g j j Žh. plus the cross
semivariogram g i j Žh.. A difficulty lies in the fact that the three models must be
built together. More precisely, one must ensure that the matrix of semivariogram
models G Žh. s w g i j Žh.x is conditionally negative semi-definite for all lags h,
which requires the following inequality to be satisfied for all lags h:
< g i j Ž h . < F g ii Ž h . g j j Ž h . ;h
(
Ž7.
The apparent complexity of condition Ž 7. has discouraged some practitioners
from modeling cross semivariograms and so using multivariate interpolation. In
fact, condition Ž7. is easily checked as long as the three Ž cross. semivariograms
are modeled as linear combinations of the same set of basic models g l Žh.:
L
g i j Ž h . s Ý bil j g l Ž h . ; i , j
Ž8.
ls0
or, using matrix notation,
L
G Ž h . s Ý Bl g l Ž h .
Ž9.
ls0
where Bl s w bil j x is referred to as a coregionalization matrix. Sufficient conditions for the so-called linear model of coregionalization Ž LMC. to be permissible
P. GooÕaertsr Geoderma 89 (1999) 1–45
10
are: Ž1. the functions g l Žh. are permissible semivariogram models, and Ž 2. each
coregionalization matrix is positive semi-definite, which implies that the coefficients bil j satisfy the following constraints:
biil G 0, bjlj G 0 ; l
Ž 10.
(
< bil j < F biil bjlj ; l
Ž 11.
In practice, the modeling is done in two steps:
1. both direct semivariograms are first modeled as linear combinations of
selected basic structures g l Žh. ,
2. the same basic structures are then fitted to the cross semivariogram under the
constraint Ž11..
This approach was used to fit visually the following model to the Ž cross.
semivariograms of Cd and Ni displayed in Fig. 6:
g Cd Ž h .
s 0.3 g 0 Ž h . q 0.3 Sph Ž hr200 m . q 0.26 Sph Ž hr1.3 km .
g Ni Ž h .
s
11 g 0 Ž h . q 71 Sph Ž hr1.3 km .
g Cd – Ni Ž h .
s
0.6 g 0 Ž h . q 3.8 Sph Ž hr1.3 km .
Ž 12.
The requirement that all semivariograms must share the same set of basic
structures might seem a severe limitation of the linear model of coregionaliza-
Fig. 6. Experimental omnidirectional direct and cross semivariograms for Cd and Ni, and the
linear model of coregionalization fitted.
P. GooÕaertsr Geoderma 89 (1999) 1–45
11
tion. Variables that are well cross-correlated are however likely to show similar
patterns of spatial variability. In addition, there is no need for the direct and
cross semivariograms to include all the basic structures; for example, the Ni
semivariogram and the cross semivariogram Cd–Ni do not include the shortrange Ž200 m. spherical structure. The reader is warned against the use of
alternative models Že.g., Myers, 1982; Zhang et al., 1992. that are more flexible
but do not provide easy way to check their permissibility Ž Goovaerts, 1994b. . It
is unfortunate that such models have been mainly used in soil science.
3.3. MultiÕariate case
For more than two variables Ž NÕ ) 2., checking the permissibility of the
linear model of coregionalization becomes cumbersome because, for each
structure g l Žh., the NÕ = NÕ matrix of coregionalization Bl must be positive
semi-definite. Fortunately, Goulard Ž 1989. has developed an iterative procedure
that fits the linear model of coregionalization directly under the constraint of
positive semi-definiteness of all matrices Bl . Most applications of this innovative fitting technique have been in the field of soil science Ž Goulard and Voltz,
1992; Goovaerts, 1992; Voltz and Goulard, 1994; Webster et al., 1994. .
4. Spatial prediction
The main application of geostatistics to soil science has been the estimation
and mapping of soil attributes in unsampled areas. Kriging is a generic name
adopted by the geostatisticians for a family of generalized least-squares regression algorithms. The practitioner often gets confused in the face of the palette of
kriging methods available: simple, ordinary, universal or with a trend, cokriging,
kriging with an external drift . . . . This section presents a brief description of the
main methods and provides references to soil applications. A detailed presentation of the mathematics can be found in textbooks such as Isaaks and Srivastava
Ž1989, pp. 278–337. and Goovaerts Ž 1997a, pp. 125–258. .
4.1. UniÕariate kriging
Consider the problem of estimating the value of a continuous attribute z at
any unsampled location u using only z-data z Ž u a ., a s 1, . . . ,n4 . All kriging
estimators are but variants of the basic linear regression estimator Z ) Žu. defined
as:
n Žu .
Z ) Ž u . y m Ž u . s Ý la Ž u . Z Ž u a . y m Ž u a .
as1
Ž 13.
P. GooÕaertsr Geoderma 89 (1999) 1–45
12
where la Žu. is the weight assigned to datum z Žu a . interpreted as a realization
of the random variable ZŽu a . and located within a given neighborhood W Žu.
centered on u. The nŽ u. weights are chosen so as to minimize the estimation or
error variance s E2 Ž u. s Var Z ) Žu. y ZŽu.4 under the constraint of unbiasedness
of the estimator. These weights are obtained by solving a system of linear
equations which is known as ‘kriging system.’
Differences between kriging variants reside in the model considered for the
trend mŽ u. in expression Ž13..
Ž1. Simple kriging Ž SK. considers the mean mŽu. known and constant
throughout the study area A:
m Ž u . s m, known ; u g A
Ž2. Ordinary kriging ŽOK. accounts for local fluctuations of the mean by
limiting the domain of stationarity of the mean to the local neighborhood W Žu.:
m Ž uX . s constant but unknown ;uX g W Ž u .
Unlike simple kriging, the mean here is deemed unknown.
Ž3. Kriging with a trend model ŽKT., also known as ‘universal kriging,’
considers that the unknown local mean mŽuX . smoothly varies within each local
neighborhood W Žu., and the trend is modeled as a linear combination of
functions f k Žu. of the coordinates:
K
X
m Ž u . s Ý a k Ž uX . f k Ž uX . ,
ks0
with a k Ž uX . f a k constant but unknown ; uX g W Ž u .
The coefficients a k ŽuX . are unknown and deemed constant within each local
neighborhood W Ž u.. By convention, f 0 ŽuX . s 1, hence ordinary kriging is but a
particular case of KT with K s 0.
These differences are illustrated in the one-dimensional example of Fig. 7
where Cd concentration is estimated every 50 m using at each location the five
closest data. The middle graph shows the mean implicitly used by each kriging
variant: global mean for SK, constant mean within local search neighborhoods
for OK, local linear function of the x-coordinates for KT. For the latter two
Fig. 7. Impact of the kriging algorithm on the estimation of the trend Žmiddle graph. and of Cd
concentration Žbottom graph. along a transect. The vertical dashed lines delineate the segments
that are estimated using the same five Cd concentrations. For example, the first segment, 1–2.1
km, includes all estimates that are based on the data at locations u 1 to u 5.
P. GooÕaertsr Geoderma 89 (1999) 1–45
13
algorithms, the parameters of the trend model are constant within each segment
where the same five neighboring data are used, and these are delineated by
14
P. GooÕaertsr Geoderma 89 (1999) 1–45
vertical dashed lines. The corresponding estimates of Cd concentration are
displayed at the bottom of the figure. Simple kriging is inappropriate because it
does not account for the increase in Cd concentration along the transect.
Ordinary kriging with local search neighborhoods provides similar results to
kriging with a trend model while being easier to implement. Large differences
between the two estimators arise only beyond location u 10 and are due to the
arbitrary extrapolation of the trend model evaluated from the last five data. Note
that in this example the linear trend model yields negative estimates of concentration around 7 km. Such aberrant results show that the blind use of complex
trend models is risky and may yield worse estimates than the straightforward
ordinary kriging.
4.2. Accounting for exhaustiÕe secondary information
When measurements are sparse or poorly correlated in space, the estimation
of the primary attribute of interest is generally improved by accounting for
secondary information originating from other related categorical or continuous
attributes. This secondary information is said to be exhaustively sampled when it
is available at all primary data locations and at all nodes of the estimation grid,
e.g., a soil map or an elevation model. In this case, secondary data can be
incorporated using three variants of the aforementioned univariate methods:
kriging within strata, simple kriging with varying local means, and kriging with
an external drift.
Where major scales of spatial variation are related to changes in land use, soil
type or lithology, secondary categorical information such as land use, soil or
geologic maps can be used to stratify the study area Ž e.g., see Stein et al., 1988;
Voltz and Webster, 1990; Van Meirvenne et al., 1994. . Kriging is then
performed within each stratum using a semivariogram model and data specific to
that stratum. When data sparsity prevents computing reliable semivariogram
estimates within each stratum, experimental values can be combined into a
single pooled within-stratum semivariogram. Fig. 8 shows a stratification of the
transect of Fig. 7 based on geology. Within-stratum semivariograms computed
from the entire study area indicate that Cd concentration varies more continuously within the stratum A1. Kriging within strata yields estimates Ž bottom
graph, solid line. that suddenly change at the strata boundaries, as opposed to
ordinary kriging without stratification Ž dashed line. .
A second approach is simple kriging where the global mean is replaced by
local means derived from a calibration of the secondary information. For
example, the calibration of the geologic map of Fig. 1 yields an average Cd
concentration for each rock type, see Fig. 9 Žleft top graph.. Residual data can
be computed by subtracting from each Cd measurement the local mean of the
prevailing rock type. Residual values are then estimated at unsampled locations
P. GooÕaertsr Geoderma 89 (1999) 1–45
15
Fig. 8. Kriging within strata. The transect is first split into two strata A1 and A2 , according to
geology Žtop graph.. Within each stratum, the Cd semivariogram is inferred and modeled Žmiddle
graph., and Cd concentrations are estimated using ordinary kriging and stratum-specific data
Žbottom graph, solid line.. Vertical arrows depict discontinuities at the strata boundaries. The
dashed line represents the OK estimate without stratification.
16
P. GooÕaertsr Geoderma 89 (1999) 1–45
P. GooÕaertsr Geoderma 89 (1999) 1–45
17
using simple kriging and the semivariogram of residuals, see Fig. 9 Ž third row. .
The final estimates of Cd concentration are obtained by adding the local means
to the residual estimates. Unlike kriging within strata, data across geologic
boundaries are used in the estimation, which attenuates discontinuities at
boundaries. A similar approach was used by Bierkens Ž 1997. to incorporate soil
map information in the mapping of heavy metal concentrations. To account for
the accuracy of the soil map information in mapping, Heuvelink and Bierkens
Ž1992. proposed a weighted average of soil map predictions and kriged estimates where more weight is given to the information with the smallest prediction error variance. A case study showed this heuristic method to produce a
more accurate map of mean water table than either kriging or soil map
prediction as long as the soil map accuracy is correctly estimated and point
observations are scarce.
If the secondary attribute is continuous, the local means in simple kriging can
be derived by regression procedures. For example, Odeh et al. Ž1997. used a
combination of multiple linear regression and ordinary Ž instead of simple.
kriging to account for landform attributes derived from a digital elevation model
in the prediction of percent topsoil organic carbon.
Kriging with an external drift is but a variant of kriging with a trend model
where the trend mŽu. is modeled as a linear function of a smoothly varying
secondary Žexternal. variable y Žu. instead of a function of the spatial coordinates:
m Ž u . s a 0 Ž u . q a1Ž u . y Ž u .
K
as opposed to: mŽu. s a0 Žu. q Ý ks1
a k Žu. f k Žu. for kriging with a trend model.
Besides the difficult inference of the residual semivariogram, this method
requires that the relation between primary trend and secondary variable is linear
and makes physical sense. One of the few applications of the method to soil
science is a paper by Bourennane et al. Ž1996. where the slope gradient is used
as external drift for predicting the thickness of a pedological horizon. Gotway
and Hartford Ž1996. used a similar approach to account for corn yield measurements and soil nitrate concentration data in the prediction of the amount of
nitrogen left in the soil after harvest. In Heuvelink Ž1996., the external drift
takes the form of a polygonal map of mean highest water table derived from a
calibration of a soil map.
Fig. 9. Simple kriging with varying local means. The trend component at each location is
estimated by the average Cd concentration for the rock type prevailing at that location. Residuals
are interpolated by simple kriging, and the results Žthird row. are added to the trend estimates to
yield the Cd estimates Žbottom graph..
P. GooÕaertsr Geoderma 89 (1999) 1–45
18
4.3. Cokriging
When the secondary information is not exhaustively sampled, the estimation
can be done using a multivariate extension of the kriging estimator ŽEq. Ž 13..
which is referred to as cokriging:
NÕ n i Žu .
Z1) Ž u . y m1Ž u . s Ý Ý la iŽ u . Zi Ž u a i . y m i Ž u a i .
Ž 14.
is1 a is1
where la iŽu. is the weight assigned to the primary datum z 1Žu a i . and la iŽu.,
i ) 1, is the weight assigned to the secondary datum z i Žu a i .. Like kriging, three
cokriging variants can be distinguished according to the models adopted for the
trends of primary and secondary variables: simple, ordinary, and universal or
with trend models. To be complete, one must also mention a variant of ordinary
cokriging, called standardized ordinary cokriging Ž Isaaks and Srivastava, 1989,
p. 416., which uses a single unbiasedness constraint that calls for all primary
and secondary data weights to sum to one. To be unbiased though, the method
requires a prior rescaling of all secondary variables to the primary mean.
Because it does not call for the secondary data weights to sum to zero, ordinary
cokriging with a single unbiasedness constraint gives a larger weight to the
secondary information while reducing the occurrence of negative weights
ŽGoovaerts, 1998.. Numerous examples of cokriging can be found in the soil
literature, e.g., McBratney and Webster Ž 1983. , Yates and Warrick Ž 1987. , Stein
et al. Ž1988., Leenaers et al. Ž1990.. Most of these studies, however, do not
check the permissibility of the coregionalization model which is used in
prediction; recall previous remark on the linear model of coregionalization.
Cokriging is much more demanding than kriging in that NÕŽ NÕ q 1.r2 direct
and cross semivariograms must be inferred and jointly modeled, and a large
cokriging system must be solved. As emphasized by Wackernagel Ž 1994. on the
occasion of Pedometrics’92, the additional modeling and computational effort
implied by cokriging is not worth doing when primary and secondary variables
are recorded at the same locations Žisotopic case. and the direct semivariogram
g 11Žh. of the primary variable is proportional to the cross semivariograms with
the secondary variables, g 1i Žh., i ) 1. More generally, practice has shown that
cokriging improves over kriging only when the primary variable is undersampled with regard to the secondary variables and those secondary data are well
correlated with the primary value to be estimated Ž Journel and Huijbregts, 1978,
p. 326; Goovaerts, 1998. .
Cokriging can also be used to incorporate secondary information that is
exhaustively sampled. Several studies Ž Asli and Marcotte, 1995; Goovaerts,
1998. have shown that secondary data that are close or even co-located with the
estimated location tend to screen the influence of further away secondary data.
Thus, in the presence of highly redundant secondary information, one gains little
by retaining more than one secondary datum in cokriging. Besides the screening
P. GooÕaertsr Geoderma 89 (1999) 1–45
19
effect, the use of redundant information can make the cokriging system unstable.
Cokriging using the single secondary datum collocated with the location being
estimated is referred to as collocated cokriging Ž Almeida and Journel, 1994. .
Whereas kriging with an external drift uses the secondary exhaustive information only to inform on the shape of the trend of the primary variable, cokriging
exploits more fully the secondary information by directly incorporating the
values of the secondary variable and measuring the degree of spatial association
with the primary variable through the cross semivariogram. Moreover, in some
situations, it is more realistic to use the secondary information as a full covariate
than just an indicator of the trend shape ŽGotway and Hartford, 1996. . For
example, it is important to account for the actual values and magnitude of local
fluctuations of hydrologic parameters, such as specific capacity, in the prediction of conductivity.
Another application of cokriging is the combination of different measurements of the same attribute. For example, a few precise laboratory measurements of clay content can be supplemented by more numerous field data
collected using cheaper measurement devices. Measurement errors are likely to
be larger for field data, and their semivariogram is likely to have a larger
relative nugget effect. To account for such a difference in the patterns of spatial
continuity, precise laboratory measurements and less precise field data are
weighted differently through cokriging. Note that the secondary information can
take the form of constraint intervals indicating that the primary attribute is
valued between specific bounds, e.g., reference colors on a colorimetric paper to
evaluate acidity levels. Secondary data are thus coded into indicators of type Ž2.
prior to their incorporation in the cokriging estimator Ž Goovaerts, 1997a, pp.
241–244..
4.4. Smoothing effect and kriging Õariance
Estimation by kriging is best in the least-squares sense because the local error
variance Var Z ) Žu. y ZŽu.4 is minimum. A shortcoming of the least-squares
criterion, however, is that the local variation of z-values is smoothed: estimated
values are typically much less variable than actual values, which is expressed by
an overestimation of small values while large values are underestimated. Another drawback of the estimation is that the smoothing depends on the local data
configuration; it is small close to the data locations and increases as the location
being estimated gets farther away from sampled locations. This uneven smoothing yields kriged maps that artificially appear more variable in densely sampled
areas than in sparsely sampled areas. For all these reasons, interpolated maps
should not be used for applications sensitive to presence of extreme values and
their patterns of continuity, typically soil pollution data and physical properties
Žpermeability, porosity. that control solute transport in soil. A better alternative
is to use simulated maps which reproduce the spatial variability modeled from
the data, see later section.
20
P. GooÕaertsr Geoderma 89 (1999) 1–45
Kriging provides not only a least-squares estimate of the attribute but also the
attached error variance. The so-called kriging variance is unfortunately often
misused as a measure of reliability of the kriging estimate, as reminded by
several authors Ž Journel, 1993; Armstrong, 1994. . By doing so, one assumes that
the variance of the errors is independent of the actual data values and depends
only on the data configuration, a situation referred to as ‘homoscedasticity.’ In
the example of Fig. 10, the kriging variance is similar at locations uX1 and uX2
with similar data configurations, although the potential for error is expected to
be greater at location uX2 , which is surrounded by a very large value and a small
one, compared with location uX1, which is surrounded by two consistently small
Cd values. Homoscedasticity is rarely met in practice because the local variance
of data usually changes across the study area Ž nonstationarity. . For example, the
Fig. 10. Ordinary kriging estimates of Cd concentration with the associated kriging variance.
P. GooÕaertsr Geoderma 89 (1999) 1–45
21
sills of the within-stratum semivariograms of Fig. 8 indicated that the variance
of Cd concentrations is larger within the stratum A1. In such a case, one might
consider the possibility of subdividing the study area into more homogenous
zones which are then treated as different statistical populations. Such subdivision, however, is conditioned by our ability to delineate the different zones in
the field, and the availability of enough data to infer statistics such as semivariograms for each zone. Stationarity is a property of the random function model
that is needed for statistical inference, and in most situations the ‘stationarity
decision’ is taken by the user simply because he cannot afford to split data into
smaller subsets! An interesting approach consists of rescaling locally a relative
semivariogram computed from all the data so that its sill equals the variance of
the data within the kriging search neighborhood Ž Isaaks and Srivastava, 1989,
pp. 516–524.. Whereas this local rescaling does not change the kriging weights
and so the estimated value, it might provide kriging variances that better inform
on the actual estimation errors.
Another way to correct for the lack of homoscedasticity is to transform the
data to stabilize the variance, typically when the sample histogram is asymmetric and the local variance of data is related to their local mean Ž proportional
effect. . A frequent transform consists of taking the logarithm of strictly positive
measurements, y Žu a . s ln z Ž u a .. The problem lies in the back-transform of the
estimated value y ) Žu. to retrieve the estimate of the original variable z at u. For
example, the unbiased back-transform of the simple lognormal kriging estimate
is:
2
z ) Ž u . s exp y ) Ž u . q s SK
Ž u . r2
Ž 15.
2 Ž .
where s SK
u is the simple lognormal kriging variance. Because of the exponentiation of both kriging estimate and variance, final results are very sensitive to
the semivariogram model which controls the kriging variance!
4.5. Factorial kriging
Semivariogram models used in earth science are frequently nested in that
several structures with different ranges or slopes are combined to fit the
experimental curves. Recall expression Ž6. for the nested model:
L
g Žh. s Ý b l g l Žh.
Ž 16.
ls0
The random function ZŽu. with a nested semivariogram g Žh. can be interpreted
as the sum of Ž L q 1. independent random functions Z l Žu., each with zero mean
and semivariogram b l g l Žh. :
L
Z Žu. s Ý Z l Žu. q m Žu.
ls0
Ž 17.
P. GooÕaertsr Geoderma 89 (1999) 1–45
22
where the trend component mŽu. is assumed locally constant as in the practice
of ordinary kriging. According to Burrough Ž 1983. , the variation of soil properties appears to be consistent with the hypothesis of the nested model, and
examples of sources of variation affecting soil at different spatial scales are
earthworms, geology, relief, to which one can add tree-throw and man’s
divisions into farms and fields ŽOliver and Webster, 1986b. . Once several
spatial scales have been identified on the semivariograms, the corresponding
spatial components Z l Žu. can be estimated and mapped using a variant of
kriging, known as factorial kriging or kriging analysis ŽMatheron, 1982;
Goovaerts, 1992..
Several authors ŽGoovaerts, 1994c; Webster et al., 1994. have used factorial
kriging to separate local variation in soil properties due to field-to-field differences or local sources of pollution from regional variation related to different
soil types or geological classes. In these studies, the maps of spatial components
served mainly as descriptive tools to improve our understanding of the sources
of spatial variation. More recently, Bourgault et al. Ž1995. proposed to use the
map of the regional component of soil electromagnetic response as secondary
information in the cokriging of electrical conductivity. They capitalized on the
fact that these two soil properties were better correlated at a regional scale
Ž r s 0.63. than at a local scale Ž r s 0.30.. Such scale-dependent relations are
frequent in soil science, e.g., see Goulard and Voltz Ž1992., Goovaerts and
Webster Ž1994., Dobermann et al. Ž1995, 1997.. Many soil properties of the soil
are controlled by the same physical processes which operate at different spatial
scales and influence these properties in different ways. Accounting for the
spatial scale in the study of correlations may enhance a relation between
variables that is otherwise blurred in an approach where all different sources of
variation are mixed, leading to a better understanding of the physical underlying
mechanisms controlling spatial patterns.
Multivariate factorial kriging, also called factorial kriging analysis Ž Matheron,
1982; Wackernagel, 1988, 1995, pp. 160–165., allows one to analyse relations
between variables at the spatial scales detected and modeled from experimental
semivariograms. Like factorial kriging, which is based on the linear model of
regionalization Ž Eq. Ž 16.., multivariate factorial kriging is based on the specific
linear model of coregionalization fitted to the experimental direct and cross
semivariograms:
L
g i j Ž h . s Ý bil j g l Ž h . ; i , j
ls0
Under that particular model, each random function Zi Žu. can be interpreted as
the sum of independent random functions Zil Žu.:
L
Zi Ž u . s Ý Zil Ž u . q m i Ž u .
ls0
Ž 18.
P. GooÕaertsr Geoderma 89 (1999) 1–45
23
Characterizing the linear relation between the two variables Zi and Z j at a
particular scale l amounts to computing the covariance between the two
corresponding spatial components Zil Žu. and Z jl Žu., which is but the contribution
Žsill, slope. of the structure g l Ž h. to the cross semivariogram g i j Žh.:
Cov Zil Ž u . ,Z jl Ž u . s bil j
½
Ž 19.
5
A standardized measure of the spatial relation is the structural correlation
coefficient defined as:
r ilj s Corr Zil Ž u . ,Z jl Ž u . s
½
5
bil j
(b P b
l
ii
l
jj
Ž 20.
Application of expression Ž20. to the linear model of coregionalization Ž12.
yields the following structural correlation coefficients for the pair Cd–Ni: 0.33,
0.0, and 0.88 at the micro- Ž nugget effect. , local Ž 200 m. and regional Ž 1.3 km.
scales, respectively. The linear correlation between the original concentrations
which embody all three different scales is 0.49. Taking into account the spatial
scale in the correlation analysis reveals a strong correlation of the two metals at
the regional scale that matches the scale of the stratigraphy, suggesting that the
source of these metals is geochemical.
Whereas multivariate analysis is usually done without regard to spatial
position of observations, multivariate factorial kriging accounts for the regionalized nature of variables by analyzing the Ž L q 1. matrices of structural correlation coefficients separately. Each correlation structure is thus distinguished by
filtering the structures belonging to other scales of spatial variation. Factorial
kriging has been increasingly used these last five years in soil science. One must
be aware that the decomposition into spatial components, and hence any
subsequent interpretation, depends on the somewhat arbitrary decision of whether
to include a particular component in the linear model of Ž co. regionalization.
Therefore, during the structural analysis, it is crucial to take into account any
physical information related to the phenomenon and the study area.
5. Modeling of local uncertainty
There is necessarily some uncertainty about the value of the attribute z at an
unsampled location u. In geostatistics, the usual approach for modeling local
uncertainty consists of computing a kriging estimate z ) Žu. and the associated
error variance, which are then combined to derive a Gaussian-type confidence
interval ŽIsaaks and Srivastava, 1989, pp. 517–519., see Fig. 11 Žright top
24
P. GooÕaertsr Geoderma 89 (1999) 1–45
Fig. 11. Different models of uncertainty about the Cd concentration at the unsampled location u:
95% confidence interval derived from the ordinary kriging estimate and the associated kriging
variance, and local distributions of probability Žccdf. established using either a multi-Gaussian or
an indicator approach Žbottom graphs..
graph. . A more rigorous approach is to model the uncertainty about the
unknown z Žu. before and independently of the choice of a particular estimate
for that unknown. The interpretation of the unknown as a realization of a
random variable ZŽu. leads one to model the uncertainty at u through the
conditional cumulative distribution function Ž ccdf. of that variable:
F Ž u; z < Ž n . . s Prob Z Ž u . F z < Ž n . 4
Ž 21 .
where the notation ‘ <Ž n.’ expresses conditioning to the local information, say,
nŽ u. neighboring data z Žu a .. The ccdf fully models the uncertainty at u since it
gives the probability that the unknown is no greater than any given threshold z,
see Fig. 11 Ž bottom graphs. . This section presents the main algorithms for
P. GooÕaertsr Geoderma 89 (1999) 1–45
25
modeling these ccdfs, with an emphasis on indicator methods. The use of ccdf
models in spatial prediction and decision making is discussed.
5.1. The parametric approach
Determination of ccdfs is straightforward if an analytical model defined by a
few parameters can be adopted for such distributions. Disjunctive kriging
ŽMatheron, 1976; Webster and Oliver, 1989. and multi-Gaussian kriging Ž Verly,
1983. are two parametric approaches capitalizing on the congenial properties of
the Gaussian model. The theory underlying disjunctive kriging is rather complex
and requires the use of orthogonal polynomial expansions with their related
convergence problems. The second approach is easier to implement in that,
under the multi-Gaussian model, the mean and variance of the Gaussian ccdf at
an unsampled location are but the simple kriging estimate and variance, see Fig.
11 Žleft bottom graph..
The congeniality of the parametric approach is balanced by strong assumptions about the normality of the two-point cdf for disjunctive kriging and of the
multi-point cdf for multi-Gaussian kriging. The normality of the one-point
distribution of data Žsample histogram. can be easily checked and, if required,
data can be normalized using a transform which amounts at replacing the
original values by the corresponding quantiles in the standard normal distribution ŽDeutsch and Journel, 1998, p. 141..
The normality of the one-point cdf is a necessary but not sufficient condition
to ensure that a random function is bivariate, and a fortiori multivariate,
Gaussian. A common misunderstanding in the soil literature about disjunctive
kriging is that the normal score transform ensures the normality of the two-point
distribution. In fact, such a univariate transform has no impact on the bivariate
properties of the random function Ž Deutsch and Journel, 1998, p. 143; Rivoirard,
1994, p. 46.. Deutsch and Journel Ž 1998, p. 142. provide a way to check the
normality of the two-point distribution from the shape of indicator semivariograms computed for a series of threshold values, see also Goovaerts Ž1997a, p.
271.. Unfortunately, such checks are rarely performed, and the adoption of a
Gaussian-based approach is essentially a subjective decision too often driven by
the simplicity of the corresponding algorithms.
Multi-Gaussian models have their own merits but they should not be adopted
if critical features of the data are not reproduced. In particular, the Gaussian
model does not allow for any significant spatial correlation of very large or very
small values, a property known as destructuration effect. Connected strings of
small or large values are very important features in particular for soil pollution,
and so the Gaussian model should never be considered when the structural
analysis Žindicator semivariograms. or qualitative information indicates that very
large or very small values are spatially correlated. Even in the absence of
information about the connectivity of extreme values, the Gaussian model is not
P. GooÕaertsr Geoderma 89 (1999) 1–45
26
conservative in the sense that it leads one to understate the potential for hazard,
Ž 1997. for mass transport in an aquifer section
as shown by Gomez-Hernandez
´
´
with heterogeneous hydraulic conductivity.
5.2. The nonparametric approach
Unlike the Gaussian-based techniques, the nonparametric algorithms do not
assume any particular shape or analytical expression for the conditional distributions. Instead, the value of the function F Ž u; z <Ž n.. is determined for a series of
K threshold values z k discretizing the range of variation of z:
F Ž u; z k < Ž n . . s Prob Z Ž u . F z k < Ž n . 4
k s 1, . . . , K
Ž 22.
The resolution of the discrete ccdf is then increased by interpolation within each
class Ž z k , z kq1 x and extrapolation beyond the two extreme threshold values z 1
and z K . For example, Fig. 11 Ž right bottom graph. shows the model fitted to the
nine ccdf values estimated at location u.
Nonparametric geostatistical estimation of ccdf values Ž Journel, 1983. is
based on the interpretation of the conditional probability Ž Eq. Ž 22.. as the
conditional expectation of an indicator random variable I Žu; z k . given the
information Ž n.:
F Ž u; z k < Ž n . . s E I Ž u; z k . < Ž n . 4
Ž 23.
with I Žu; z k . s 1 if ZŽu. F z k and zero otherwise. Ccdf values can thus be
estimated by least-squares Žkriging. interpolation of indicator transforms of data.
Practical implementation of the indicator approach involves the following steps
ŽGoovaerts, 1997a, pp. 284–328..
Ž1. Code each observation z Ž u a . into a vector of K indicator values:
1 if z Ž u a . F z k
k s 1, . . . , K
Ž 24.
0 otherwise
The set of K threshold values is typically chosen such that the range of z-values
is split into Ž K q 1. classes of approximately equal frequency, e.g., the nine
deciles of the sample cumulative distribution.
Ž2. For each threshold z k , compute the experimental indicator semivariogram
ŽEq. Ž3.. , and model it using a linear combination Ž Eq. Ž 6.. of permissible
semivariogram models.
Ž3. At each unsampled location u, the following should be done.
Ø Estimate each of the K ccdf values as a linear combination of neighboring
indicator data by kriging as for continuous attributes; for example, the ordinary
indicator kriging estimator is:
i Žu a ; z k . s
½
)
F Ž u; z k < Ž n . . O IK
s
)
I Ž u; z k . OK
n Žu .
s
Ý la Žu; z k . I Žu a ; z k .
as1
Ž 25.
P. GooÕaertsr Geoderma 89 (1999) 1–45
27
Ø Correct the estimated probabilities w F Žu; z k <Ž n..x ) that do not meet the
following constraints:
F Ž u; z k < Ž n . .
)
F Ž u; z k < Ž n . .
)
g 0,1 ; z k
F F Ž u; z kX < Ž n . .
Ž 26.
)
; z kX ) z k
Ž 27 .
Ø Interpolate or extrapolate ccdf values to build a continuous model for the
conditional cdf, which allows one to retrieve the probability of being no greater
than any threshold z, in addition to the K original thresholds z k .
The indicator approach appears much more demanding than the multi-Gaussian approach both in terms of semivariogram modeling and computer requirements. This additional complexity is balanced by the possibility of modeling
spatial correlation patterns specific to different classes of attribute values
through indicator semivariograms. In particular, the connectivity of extreme
values can be accounted for. Note that in many applications the objective is not
to model the whole ccdf but rather to assess the probability of exceeding a
particular value, say a regulatory threshold in soil pollution Ž Leonte and
Schofield, 1996. or a critical value for soil quality Ž Smith et al., 1993. . In this
case, the indicator approach is as straightforward as the parametric approach
since a single indicator semivariogram and kriging system need to be considered.
Fig. 12 Žfirst two rows. shows the different steps of the indicator approach to
map the probability of contamination by cadmium in the region Žregulatory
threshold z c s 0.8 mg kgy1 .. A usual criticism of the indicator approach is that
the indicator coding amounts to discarding much of the information in the data.
In the example of Fig. 12, all concentrations ranging from 0.81 mg kgy1 to 5.2
mg kgy1 yield the same unit indicator value, while all concentrations no greater
than 0.8 kgy1 are translated into zero indicator values. In theory, this loss of
information can be compensated by accounting for indicator values defined at
different thresholds that is using indicator cokriging instead of kriging. Practice
has shown, however, that indicator cokriging improves little over indicator
kriging Ž Goovaerts, 1994d. because cumulative indicator data carry substantial
information from one threshold to the next one, and all indicator values are
available at each sampled location Žisotopic or equally-sampled case. . An
alternative to indicator cokriging is probability kriging Ž Journel, 1984; Goovaerts,
1997a, p. 301. where ccdf values are estimated as a linear combination of
neighboring indicator and uniform transforms of the data:
)
F Ž u; z k < Ž n . . PK
s
)
I Ž u; z k . PK
n Žu .
s
n Žu .
Ý la Žu; z k . I Žu a ; z k . q Ý na Žu; z k . X Žu a .
as1
a s1
Ž 28.
28
P. GooÕaertsr Geoderma 89 (1999) 1–45
Fig. 12. Estimation of the probability of contamination by cadmium using the indicator approach.
The original concentrations are transformed into indicators of exceedence of the regulatory
threshold 0.8 mg kgy1 that are then kriged using an indicator semivariogram. The bottom
probability map accounts for additional information in the form of soft probabilities derived from
a calibration of the geologic map of Fig. 1.
The standardized rank x Žu a . of observation z Žu a . is computed as r Žu a .rn,
where r Žu a . g w1,n x is the rank of the datum in the sample cumulative distribution. The data ranks valued in w0,1x allow one to discriminate Cd concentrations
P. GooÕaertsr Geoderma 89 (1999) 1–45
29
with similar indicator transforms 0 or 1, and so corrects for the loss of resolution
caused by the use of a single threshold in indicator kriging.
A limitation of the indicator approach is the a posteriori correction of
estimated probabilities that do not meet constraints Ž 26. and Ž 27. , although
practice has shown that order relation deviations are generally of small magnitude ŽGoovaerts, 1994d. . A more elegant solution would consist of implementing these constraints directly into the indicator kriging system, as proposed by
de Gruijter et al. Ž1997. for categorical variables. A potential pitfall is the
interpolation or extrapolation of the corrected probabilities to derive a continuous ccdf model. Characteristics of the ccdf such as the mean or variance may
overly depend on the modeling of the upper and lower tails of the distribution
ŽGoovaerts, 1997a, p. 338.. A linear model is usually adopted for interpolation
within each class Ž z k , z kq1 x, whereas power or hyperbolic models are used for
extrapolation beyond the two extreme threshold values z 1 and z K ŽDeutsch and
Journel, 1998, pp. 135–138.. The choice of these models is fully arbitrary, and I
prefer to capitalize on the higher level of discretization of the cdf Ž i.e., the
cumulative histogram. to improve the within-class resolution of the ccdf
ŽGoovaerts, 1997a, p. 327.. For example, the resolution of the discrete ccdf of
Fig. 11 Žright bottom graph. has been increased by performing a linear interpolation between tabulated bounds provided by the histogram of 259 Cd concentrations. An alternative to the piecewise interpolationrextrapolation of the ccdf
model consists of fitting a continuous parametric model to the set of estimated
probabilities. In all cases, the impact of extrapolation models can be reduced by
selecting more threshold values within the two tails of the distribution Ž Deutsch
and Lewis, 1992; Chu, 1996..
The major advantage of the indicator approach is its ability to incorporate soft
information of various types Ž e.g., soil map or qualitative field observations such
as the smell or color of contaminated soil. in addition to direct measurements on
the attribute of interest. The only requirement is that each soft datum must be
coded into a vector of K cumulative probabilities of the type:
Prob Z Ž u . F z k <specific local information at u 4 k s 1, . . . , K
Ž 29.
Goovaerts and Journel Ž1995. showed how probabilities ŽEq. Ž29.. can be
derived from the calibration of a soil map and combined with precise measurements of metal concentration to map the risk of deficiency of these metals in the
soil. Similar applications in soil science are presented in Colin et al. Ž 1996. ,
Goovaerts et al. Ž1997. , and Journel Ž 1997. . Fig. 12 Ž bottom graphs. illustrates
the use of the geologic map as soft information in mapping the risk of
contamination by cadmium. Probabilities of contamination derived from a
calibration of geology are used as local means in the simple kriging of indicator
data; recall similar approach for continuous attributes in Fig. 9. Taking the
geology into consideration produces a more detailed map and enhances the
30
P. GooÕaertsr Geoderma 89 (1999) 1–45
contrast between Argovian rocks, on which the soil contains little cadmium, and
other rocks with larger probabilities of exceeding the regulatory threshold.
Another interesting feature of the indicator approach is its ability to account
for a secondary continuous variable, say another metal content, that is nonlinearly related to the primary variable. The idea consists of discretizing the two
variables using two series of K thresholds Že.g., deciles of the sample cumulative distributions., then combining primary and secondary indicator data at each
threshold using a cokriging algorithm similar to the one introduced for continuous variables Ž Zhu and Journel, 1993; Goovaerts, 1997a, p. 308..
5.3. Using the model of local uncertainty
Once the uncertainty about an unsampled value has been modeled using
either parametric or nonparametric approaches, an estimate for that unknown
can be retrieved from the ccdf, say the mean or the median of the conditional
distribution. For example, the map in Fig. 13 Ž left top graph. depicts the mean of
the conditional cdfs of Cd concentration modeled using an indicator approach.
Fig. 13. Ordinary kriging estimates of cadmium concentration and the corresponding estimation
variances Žright column.. Left maps show the mean ŽE-type estimate. and variance of ccdfs
modeled using an indicator approach.
P. GooÕaertsr Geoderma 89 (1999) 1–45
31
This map of conditional means, also known as E-type estimates, looks similar to
the map of Cd concentrations estimated using ordinary kriging Ž Fig. 13, right
top graph.. The advantage of the indicator approach over ordinary kriging is that
it provides a measure of uncertainty that accounts for the local data, whereas the
kriging variance depends only on the data configuration and semivariogram
model; recall previous discussion and Fig. 10. These differences are clear on the
two maps at the bottom of Fig. 13 which depict the variance of the conditional
cdfs and the kriging variance, respectively. The conditional variance is larger in
the high-valued parts of the study area where the Cd measurements fluctuate the
most and so the largest uncertainty is intuitively expected. The uncertainty is
smaller on Argovian rocks where Cd concentrations are consistently small. In
contrast, the kriging variance map indicates greater uncertainty in the extreme
west corner of the study area where data are sparse, whereas the uncertainty is
smallest near data locations. Elsewhere the kriging variance is about the same
whatever the surrounding data values. As mentioned previously, several methods could be used to correct for the lack of stationarity of the variance, such as
data transformation, stratification of the study area, or local rescaling of a
relative semivariogram.
Mapping metal concentrations or other soil properties is often a preliminary
step towards decision making, such as the delineation of polluted areas or the
identification of zones that are suitable for crop growth. For soil pollution, a
straightforward approach consists of declaring contaminated all locations where
the pollutant concentration estimate exceeds the regulatory threshold. Similarly,
a farmer may decide to grow a given crop wherever the estimated value of the
limiting factor Že.g., depth to parent material, soil acidity. exceeds some
threshold. Such an approach has two drawbacks: Ž1. the estimation error is
ignored: a contaminated location can be declared safe on the basis of a wrong
estimate of pollutant concentration which is slightly less than the regulatory
threshold, Ž2. the decision rule requires crip thresholds such as provided by
environmental protection agencies for soil pollution. For land evaluation, however, the use of crisp thresholds is generally inappropriate in that the crop
response is continuous from a slight reduction in crop performance through to
crop failure ŽBurrough, 1989. .
For environmental applications where crisp thresholds exist the uncertainty
about the predicted soil attribute can be accounted for by estimating and
mapping the probability of exceeding those thresholds. A common question is
then above which level of risk should we decide to clean a polluted area or
develop specific land use policies ŽGoovaerts, 1997b. . When the probability is
very large or very small, the risk-based decision is quite straightforward.
Decision making is much more difficult for locations with intermediate probabilities, say in the interval w0.3,0.7x. Depending on the resources available, complementary investigation could be made to reduce the uncertainty at these locations,
which amounts to decreasing or increasing the intermediate probabilities. Never-
P. GooÕaertsr Geoderma 89 (1999) 1–45
32
theless, the choice of a probability threshold is mainly subjective: a given
probability of contamination may be unacceptable for residential areas, yet
tolerable for industrial yards. One must also keep in mind that probabilities are
but estimates and so are subject to errors as the estimates of attribute values.
The practical use of probability maps in decision making faces the difficult
choice of crisp thresholds for attribute values and probability thresholds. Instead
of the abstract notion of risk, decision makers usually prefer dealing with a
financial assessment of the different options. For soil pollution, they would like
to know the financial costs that might result from a wrong decision, i.e.,
declaring safe a contaminated location or cleaning a safe location. The idea is to
go beyond a mere assessment of the risk and provide decision makers with a set
of alternative solutions and the corresponding potential costs Ž Srivastava, 1987;
Journel, 1987; Goovaerts and Journel, 1995; Goovaerts et al., 1997. . This
requires, however, that decision makers provide scientists with the impact or
cost functions specific to the problem at hand. Such an approach supports the
assertion of Bouma Ž1997. that researchers must interact with stakeholders
Žfarmers, planners and politicians. to analyze problems and propose cost-effective solutions.
Fig. 14 Žleft top graph. shows an example of cost functions for soil contamination by cadmium Ž regulatory threshold z c s 0.8 mg kgy1 .. Consider first the
decision of taking no remedial measure at a location u. If this location is
actually safe, z Ž u. F z c , then the decision is correct and there is no cost.
Otherwise, the cost of misclassifying a contaminated location Ž potential ill
health, legal liability. is modeled as proportional to the actual contamination
w z Žu. y z c x:
L1 Ž z Ž u . . s
½
0
if z Ž u . F z c
z Žu. y z c
otherwise
Ž 30.
The second cost function expresses the potential consequences of taking the
decision of cleaning a location. If this location is actually safe, there is undue
application of remedial measures and the corresponding cost is here modeled as
a constant value of 2.5:
L2 Ž z Žu. . s
½
0
2.5
if z Ž u . ) z c
otherwise
Ž 31.
The actual cost attached to either type of decision cannot be computed because
the actual concentration z Ž u. of pollutant is unknown. A ccdf model allows one
to account for the uncertainty about the unknown value and determine the
expected cost for the two alternatives:
q`
wi Žu. s E Li Ž Z Žu. . < Ž n . s
Hy` L Ž z Žu. . d F Žu; z < Ž n . . i s 1,2
i
Ž 32 .
P. GooÕaertsr Geoderma 89 (1999) 1–45
33
Fig. 14. Classification of locations as contaminated by cadmium on the basis that the resulting
expected cost Žunnecessary cleaning. is smaller than the cost associated with wrongly classifying a
location as safe Žpotential ill health.. Expected costs are computed using ccdf models and cost
functions specific to each type of decision.
This integral is, in practice, approximated by the discrete sum:
Kq 1
w i Ž u . , Ý L i Ž z k . F Ž u; z k < Ž n . . y F Ž u; z ky1 < Ž n . . i s 1,2
Ž 33.
ks1
where z k , k s 1, . . . , K, are K threshold values discretizing the range of
variation of z-values, and z k is the mean of the class Ž z ky1 , z k x, e.g.,
z k s Ž z ky1 q z k .r2. Fig. 14 Žbottom graphs. shows the two maps of expected
costs. At each location the health cost and the remediation cost are compared,
and the decision of cleaning the location is taken if one can expect the health
cost to be larger than the cost of unnecessary remediation. Rautman Ž 1997.
presented a more sophisticated risk-based economic-decision model to evaluate
the performance of different technologies for measuring uranium activity of
in-situ soils.
The concept of a cost function in soil pollution is analogous to that of a
membership function introduced by Burrough Ž 1989. for land evaluation problems. The membership function is here used to express the suitability of a soil
34
P. GooÕaertsr Geoderma 89 (1999) 1–45
for a given purpose, say a given crop A, as a function of the value of a soil
property. For example, the following function illustrates the impact of soil
acidity on the growth of crop A:
°1
if z Ž u . F 5
if z Ž u . ) 7
L Ž z Ž u . . s~0
¢Ž7 y z Žu. . r2 if 5 - z Žu. F 7
The membership ranges from 1 to 0 and reflects the possibility that soil is too
acid for crop A. This gradual response of the crop to soil pH is more realistic
than crop failure below a crisp pH threshold. The actual membership cannot be
computed since the actual pH value at u is unknown. However, the expected
membership can be computed from the ccdf model using the same expression as
for the expected cost ŽEq. Ž33.. . The concept of expected membership was
introduced by Lark and Bolam Ž 1997. under the name of ‘weighted membership.’ This approach allows one to combine two types of uncertainty: the
uncertainty about the value of a soil property at an unsampled location and the
imprecision Ž fuzziness. of the impact of that soil property on land suitability.
Note that one still faces the difficult problem of choosing a membership
threshold for decision making, which could be overcome by converting memberships into economic impacts.
5.4. Categorical attributes
Unlike continuous variables, categorical attributes such as texture or water
table classes cannot be estimated as a mere linear combination of neighboring
observations. In many situations, the unsampled location is simply allocated to
the same category as the nearest observation, i.e., in the same Thiessen polygon
or Dirichlet tile. Such an approach has two serious weaknesses: it ignores spatial
correlation and transition probabilities between categories, and it provides no
measure of the reliability of the prediction.
Qualitative information can be handled by indicator algorithms as long as it is
coded into indicator values, say 1 if the category is present and 0 otherwise;
recall expression Ž4.. Soft indicators, i.e., probabilities valued between zero and
one, can also be considered to account for the uncertainty in the classification of
sampled locations Žfuzzy classification.. Then, indicator kriging is used to
estimate the probability for each state sk of the attribute s to occur at the
unsampled location u:
p Ž u; sk < Ž n . . s Prob S Ž u . s sk < Ž n . 4 k s 1, . . . , K
Ž 34.
Secondary information such as provided by differences in lithology, landform or
drainage can also be incorporated in the estimation of conditional probabilities.
The prediction process amounts to allocating u to a single category on the
basis of the set of conditional probabilities Ž Eq. Ž 34.. . Bierkens and Burrough
Ž1993a,b. proposed to use as predictor the category with the largest probability
P. GooÕaertsr Geoderma 89 (1999) 1–45
35
of occurrence which defines the ‘map purity.’ Such a criterion typically leads
one to allocate most of the locations to the most frequent categories since the
probability of occurrence is likely to be larger if the corresponding global
proportion is large. Conversely, the less frequent categories tend to be underrepresented. This approach is thus inadequate if one aims at reproducing sample
proportions that are deemed representative of the entire area. One solution
consists of preferentially allocating locations to the category with the largest
probability of occurrence under the constraint of reproduction of global proportions Ž Soares, 1992. . An application to the mapping of land uses is given in
Goovaerts Ž 1997a, p. 357..
6. Stochastic simulation
As illustrated by the map of Fig. 15 Ž left column. , kriging tends to smooth out
local detail of the spatial variation of the soil attribute. The variance of ordinary
kriging estimates is much smaller than the sample variance sˆ 2 s 0.83, and the
experimental semivariogram has a much smaller relative nugget effect than the
semivariogram model, which indicates the underestimation of the short-range
variability of Cd values. Unlike kriging, stochastic simulation does not aim at
minimizing a local error variance but focuses on the reproduction of statistics
such as the sample histogram or the semivariogram model in addition to the
honoring of data values. In the example of Fig. 15, the same information Ž data,
semivariogram model. is used by the kriging and simulation approaches, but the
simulated map looks more ‘realistic’ than the map of statistically ‘best’ estimates because it reproduces the spatial variability modeled from the sample
information. Stochastic simulation is thus increasingly preferred to kriging for
applications where the spatial variation of the measured field must be preserved,
such as the delineation of contaminated areas Ž Desbarats, 1996; Goovaerts,
1997c. or the modeling of solute transport in the vadose zone Ž Vanderborght et
al., 1997. .
6.1. Modeling the spatial uncertainty
One may generate many realizations that all honor the same data and match
reasonably well the same statistics. For example, Fig. 16 shows three realizations of the spatial distribution of Cd values that all honor the 259 measurements
of Cd concentration displayed in Fig. 1 and reproduce approximately the
semivariogram model. The three images are consistent with the sample information, and their differences provide a measure of spatial uncertainty. Features
such as zones of large values are deemed certain if seen on most of the
realizations, and their probability of occurrence Ž i.e., probability that a given
threshold is jointly exceeded at a series of locations. can be computed as long as
36
P. GooÕaertsr Geoderma 89 (1999) 1–45
Fig. 15. Ordinary kriging estimates and simulated values of Cd concentration over the study area.
Bottom graphs show the corresponding histograms and standardized experimental and model
Žsolid line. semivariograms. The smoothing effect of kriging leads to underestimation of the
short-range variability of Cd values.
the realizations are equiprobable. Such joint probabilities cannot be derived from
the ccdfs introduced in the previous section: each ccdf is specific to a single
location and thus provides only a measure of local uncertainty.
In many situations, decision making concerns areas or blocks that are much
larger than the measurement support, such as the delineation of 1-ha remediation
units from soil core measurements. Provided the soil variable averages linearly,
P. GooÕaertsr Geoderma 89 (1999) 1–45
37
Fig. 16. Three realizations of the spatial distribution of Cd values over the study area and the
corresponding standardized semivariograms. Differences between realizations provide a model for
the uncertainty about the distribution in space of Cd values.
the average value over the large support can be estimated by block kriging or,
indirectly, by the arithmetical mean of kriging estimates at a series of locations
discretizing this support. This linear averaging, however, does not apply to
probability values: the probability that the average attribute value exceeds a
given threshold Ž block probability. is not equal to the linear average of local
probabilities of exceedence defined at a series of discretizing locations! Such an
upscaling can be done using disjunctive kriging under the stringent assumption
of bivariate normality ŽWebster, 1991. . A nonparametric alternative consists of
approximating numerically the block probability by generating many simulated
block values as linear averages of simulated point values, and counting the
proportion of block values that exceed the critical threshold Ž Isaaks, 1990;
Kyriakidis, 1997. . The major advantage of the simulation-based approach is that
it provides a nonparametric measure of the uncertainty attached to the prediction
of a single block or multiple spatially dependent blocks Ž Goovaerts, 1999. .
38
P. GooÕaertsr Geoderma 89 (1999) 1–45
Stochastic simulation can also be used to upscale properties like permeability
that do not average linearly in space.
Note that the use of block probability may lead to over-optimistic decisions in
that large local probabilities of contamination or large local pollutant concentrations are smoothed out through the averaging. One would then prefer dealing
with local probabilities of contamination and, for example, decide to clean the
block if the probability threshold is exceeded for a given proportion of locations
within the block. Such a proportion can be derived using block-indicator kriging
ŽGoovaerts, 1997a, p. 306.. Other decision making criteria for soil contamination are discussed in Goovaerts Ž 1997b. and Kyriakidis Ž 1997..
The impact of a given scenario, such as a particular remediation process or
land use policy, can be investigated from a simulated map that reproduces
aspects of the pattern of spatial dependence or other statistics deemed consequential for the problem at hand Že.g., connectivity of large values, spatial
correlation with secondary attributes.. Moreover, having many equiprobable
realizations allows one to assess the uncertainty about the consequences of this
particular scenario. Consider the problem of assessing the economic impact of
declaring the study area safe with respect to Cd. For each realization, this global
cost is obtained as the sum of local costs computed from each simulated value
using the cost function Ž Eq. Ž 30.. . The three realizations of Fig. 16 yield,
respectively, a cost of 1058, 1001, and 825. These differences reflect the
uncertainty about the economic impact of taking no remedial measure which
results from our imperfect knowledge of the spatial distribution of Cd values in
the area. The same operation can be repeated for 100 realizations, and the
resulting histogram of costs allows a probabilistic assessment of financial
consequences, e.g., there is here a 75% probability that the cost will be larger
than 930 Ž Fig. 17. .
Fig. 17. The distribution of costs resulting from a wrong decision to declare the study area safe
with respect to Cd. This distribution is obtained by applying the cost function of Fig. 14 Žsolid
line. to 100 realizations of the spatial distribution of Cd values.
P. GooÕaertsr Geoderma 89 (1999) 1–45
39
6.2. Simulation algorithms
Like estimation, simulation can be accomplished using a growing variety of
techniques which differ in the underlying random function model Ž multi-Gaussian or nonparametric. , the amount and type of information that can be
accounted for, and the computer requirements. Goovaerts Ž 1997a. Ž pp. 376–424.
provide a detailed description of the main simulation algorithms, and these are
available in the public domain geostatistical software library GSLIB ŽDeutsch
and Journel, 1998. . The key concept of each class of algorithms is briefly
outlined below.
Consider the simulation of the continuous attribute z at N grid nodes uXj
conditional to the data z Žu a . , a s 1, . . . ,n4 . Sequential simulation amounts to
modeling the ccdf F ŽuXj ; z <Ž n.. s Prob ZŽuXj . F z <Ž n.4 , then sampling it at each
of the N nodes visited along a random sequence. To ensure reproduction of the
semivariogram model, each ccdf is made conditional not only to the original n
data but also to all values simulated at previously visited locations. Two major
classes of sequential simulation algorithms can be distinguished, depending on
whether the series of conditional cdfs are modeled using the multi-Gaussian or
the indicator formalism. The probability field approach also requires the sampling of N successive ccdfs. However, unlike in the sequential approach, all
ccdfs are conditioned only to the original n data. Reproduction of the semivariogram model is here approximated by imposing an autocorrelation pattern on the
probability values used for sampling these ccdfs. Unlike previous simulation
algorithms, in simulated annealing the creation of a stochastic image is formulated as an optimization problem without any reference to a random function
model. The basic idea is to perturb an initial Žseed. image gradually so as to
match target constraints, such as reproduction of the semivariogram model.
Stochastic simulation has been seldom used in soil science so far unlike in
mining or petroleum. Most applications relate to soil pollution and involve
diverse algorithms such as sequential Gaussian simulation Ž Desbarats, 1996. ,
sequential indicator simulation Ž Goovaerts, 1997c; Garcia and Froidevaux,
1997., probability field Ž Bourgault et al., 1995; Kyriakidis, 1997. . Simulated
annealing was used by Sterk and Stein Ž 1997. to generate unsmoothed maps of
wind-blown sediment transport. Bierkens and Burrough Ž 1993a,b. used the
categorical formalism of sequential indicator simulation to generate maps of
water table classes which were then used to assess land suitability for pastures.
7. Conclusions and perspectives
The growing interest of soil scientists in geostatistics arises because they
increasingly realise that quantitative spatial prediction must incorporate the
spatial correlation among observations. Also, geostatistics offers an increasingly
40
P. GooÕaertsr Geoderma 89 (1999) 1–45
wide palette of techniques well suited to the diversity of problems and information soil scientists have to deal with. The recent developments of data acquisition and computational resources have provided the geostatistician with large
amounts of information of different types Žcontinuous, categorical. , which can
be stored and managed in Geographic Information Systems, and which they can
process rapidly. Multivariate approaches, such as factorial kriging analysis, can
be used to investigate how the correlation between variables changes as a
function of the spatial scale, and to improve our understanding of scale-dependent physical processes. Multivariate geostatistical interpolation also allows one
to supplement a few expensive measurements of the attribute of interest Že.g.,
metal concentrations. by more abundant data on correlated attributes that are
cheaper to determine Že.g., pH or elevation. . In particular, indicator geostatistics
enables secondary categorical information such as provided by land use or soil
map to be accounted for in the prediction of continuous variables. There is still
research to be done on the incorporation of variables measured on different
supports, in particular the combination of field data with remote sensing
information. Another avenue of research is the three-dimensional spatial modeling of soil processes since the development of measurement techniques allows
the collection of a larger amount of data in both the horizontal and vertical
directions.
There is necessarily uncertainty about the attribute value at an unsampled
location, and its assessment is critical for some applications. The uncertainty can
be assessed from the kriging variance using a Gaussian-type confidence interval
centered on the kriging estimate. A more promising approach is to assess first
the uncertainty about the unknown, then deduce an estimate that is optimal in
some appropriate sense. This can be achieved using an indicator approach that
provides not only an estimate but also the probability of exceeding critical
values, such as regulatory thresholds in soil pollution or criteria for soil quality.
The last five years have witnessed a growing use of geostatistics to create
colorful probability maps, yet the practical use of these maps for decision
making, such as delineation of contaminated areas or areas of good soil quality,
has received little attention. We should now combine probabilistic models
provided by geostatisticians with expert knowledge in order to assess the
financial impacts of the different options available as is being done in mining. In
the future, more research should be devoted to this important topic, in particular
the incorporation of the uncertainty related to several correlated attributes such
as multiple criteria for land suitability.
Another way to model uncertainty is to generate numerous images Ž realizations. that all honor the data and reproduce aspects of the patterns of spatial
dependence or other statistics deemed consequential for the problem at hand. A
given scenario Ž remediation process, land use policy. can be applied to the set of
realizations, allowing the uncertainty of the response Ž remediation efficiency,
soil productivity. to be assessed. Stochastic simulation is one of the most active
P. GooÕaertsr Geoderma 89 (1999) 1–45
41
areas of research in geostatistics, and one can expect an increasing use of these
techniques to model soil spatial uncertainty.
The modeling of the space–time variability is also becoming one of the major
avenues of research is environmental geostatistics. The interested reader should
refer to the following papers for a brief overview of the techniques currently
available and references to the few space–time studies that have been conducted
Ž 1994. ,
so far in soil science: Goovaerts and Sonnet Ž 1993. , Papritz and Fluhler
¨
Heuvelink et al. Ž1997..
Finally, geostatistics was originally developed to solve practical problems,
namely, evaluating recoverable reserves in mining and its growth bears witness
to its utility and success. We should do well to keep this trademark of
practicality in soil science and to foster the increasing and successful application
of geostatistics to soil-related issues.
Acknowledgements
This work was partly done while the author was with the Unite´ Biometrie,
´
Universite´ Catholique de Louvain, Belgium. The author thanks Mr. J.-P. Dubois
of the Swiss Federal Institute of Technology at Lausanne for the data and the
National Fund for Scientific Research Ž Belgium. for its financial support. Data
can be downloaded from http:rrwww-personal.engin.umich.edur;goovaertr.
References
Almeida, A., Journel, A.G., 1994. Joint simulation of multiple variables with a Markov-type
coregionalization model. Math. Geol. 26, 565–588.
Armstrong, M., 1994. Is research in mining geostats as dead as a dodo? In: Dimitrakopoulos, R.
ŽEd.., Geostatistics for The Next Century. Kluwer Academic Publishers, Dordrecht, pp.
303–312.
Asli, M., Marcotte, D., 1995. Comparison of approaches to spatial estimation in a bivariate
context. Math. Geol. 27, 641–658.
Atteia, O., Dubois, J.-P., Webster, R., 1994. Geostatistical analysis of soil contamination in the
Swiss Jura. Environ. Pollut. 86, 315–327.
Bierkens, M.F.P., Burrough, P.A., 1993a. The indicator approach to categorical soil data: I.
Theory. J. Soil Sci. 44, 361–368.
Bierkens, M.F.P., Burrough, P.A., 1993b. The indicator approach to categorical soil data: II.
Application to mapping and land use suitability analysis. J. Soil Sci. 44, 369–381.
Bierkens, M.F.P., 1997. Using stratification and residual kriging to map soil pollution in urban
areas. In: Baafi, E.Y., Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer Academic
Publishers, Dordrecht, pp. 996–1007.
Bouma, J., 1997. The role of quantitative approaches in soil science when interacting with
stakeholders. Geoderma 78, 1–12.
Bourennane, H., King, D., Chery,
´ P., Bruand, A., 1996. Improving the kriging of a soil variable
using slope gradient as external drift. Eur. J. Soil Sci. 47, 473–483.
42
P. GooÕaertsr Geoderma 89 (1999) 1–45
Bourgault, G., Journel, A.G., Lesh, S.M., Rhoades, J.D., Corwin, D.L., 1995. Geostatistical
analysis of a soil salinity data set. In: Corwin, D.L., Loague, K. ŽEds.., Proc. of the 1995
Bouyoucos Conference on the Applications of GIS to the Modeling of Non-Point Source
Pollutants in the Vadose Zone, Riverside, CA. 1–3 May 1995. U.S. Salinity Lab., Riverside,
CA, pp. 53–114.
Burrough, P.A., 1983. Multiscale sources of spatial variation in soil: II. A non-Brownian fractal
model and its application in soil survey. J. Soil Sci. 34, 599–620.
Burrough, P.A., 1989. Fuzzy mathematical methods for soil survey and land evaluation. J. Soil
Sci. 40, 477–492.
Chu, J., 1996. Fast sequential indicator simulation: beyond reproduction of indicator variograms.
Math. Geol. 28, 923–936.
Colin, P., Froidevaux, R., Garcia, M., Nicoletis, S., 1996. Integrating geophysical data for
mapping the contamination of industrial sites by polycyclic aromatic hydrocarbons: a geostatistical approach. In: Rouhani, S., Srivastava, R.M., Desbarats, A.J., Cromer, M.V., Johnson, A.I.
ŽEds.., Geostatistics for Environmental and Geotechnical Applications. American Society for
Testing and Materials STP 1283, Philadelphia, pp. 69–87.
Desbarats, A.J., 1996. Modeling spatial variability using geostatistical simulation. In: Rouhani, S.,
Srivastava, R.M., Desbarats, A.J., Cromer, M.V., Johnson, A.I. ŽEds.., Geostatistics for
Environmental and Geotechnical Applications. American Society for Testing and Materials
STP 1283, Philadelphia, pp. 32–48.
Deutsch, C.V., Journel, A.G., 1998. GSLIB: Geostatistical Software Library and User’s Guide:
2nd edn. Oxford Univ. Press, New York, 369 pp.
Deutsch, C.V., Lewis, R., 1992. Advances in the practical implementation of indicator geostatistics. In: Proceedings of the 23rd International APCOM Symposium, Tucson, AZ, Society of
Mining Engineers, pp. 169–179.
de Gruijter, J.J., Walvoort, D.J.J., van Gaans, P.F.M., 1997. Continuous soil maps—a fuzzy set
approach to bridge the gap between aggregation levels of process and distribution models.
Geoderma 77, 169–195.
Dobermann, A., Goovaerts, P., George, T., 1995. Sources of soil variation in an acid Ultisol of the
Philippines. Geoderma 68, 173–191.
Dobermann, A., Goovaerts, P., Neue, H.U., 1997. Scale-dependent correlations among soil
properties in two tropical lowland rice fields. Soil Sci. Soc. Am. J. 61, 1483–1496.
Garcia, M., Froidevaux, R., 1997. Application of geostatistics to 3D modelling of contaminated
sites: a case study. In: Soares, A., Gomez-Hernandez,
J., Froidevaux, R. ŽEds.., geoENV
´
´
I—Geostatistics for Environmental Applications. Kluwer Academic Publishers, Dordrecht, pp.
309–325.
Gomez-Hernandez,
J.J., 1997. Issues on environmental risk assessment. In: Baafi, E.Y., Schofield,
´
´
N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer Academic Publishers, Dordrecht, pp.
15–26.
Goovaerts, P., 1992. Factorial kriging analysis: a useful tool for exploring the structure of
multivariate spatial soil information. J. Soil Sci. 43, 597–619.
Goovaerts, P., 1994a. Comparison of coIK, IK, and mIK performances for modeling conditional
probabilities of categorical variables. In: Dimitrakopoulos, R. ŽEd.., Geostatistics for The Next
Century. Kluwer Academic Publishers, Dordrecht, pp. 18–29.
Goovaerts, P., 1994b. On a controversial method for modeling a coregionalization. Math. Geol.
26, 197–204.
Goovaerts, P., 1994c. Study of spatial relationships between two sets of variables using multivariate geostatistics. Geoderma 62, 93–107.
Goovaerts, P., 1994d. Comparative performance of indicator algorithms for modeling conditional
probability distribution functions. Math. Geol. 26, 389–411.
P. GooÕaertsr Geoderma 89 (1999) 1–45
43
Goovaerts, P., 1997a. Geostatistics for Natural Resources Evaluation. Oxford Univ. Press, New
York, 512 pp.
Goovaerts, P., 1997b. Accounting for local uncertainty in environmental decision-making processes. In: Baafi, E.Y., Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer Academic Publishers, Dordrecht, pp. 929-940.
Goovaerts, P., 1997c. Kriging vs. stochastic simulation for risk analysis in soil contamination. In:
Soares, A., Gomez-Hernandez,
J., Froidevaux, R. ŽEds.., geoENV I—Geostatistics for Envi´
´
ronmental Applications. Kluwer Academic Publishers, Dordrecht, pp. 247-258.
Goovaerts, P., 1998. Ordinary cokriging revisited. Math. Geol. 30, 21–42.
Goovaerts, P., 1999. Geostatistical tools for deriving block-averaged values of soil properties. J.
Environ. Quality, submitted.
Goovaerts, P., Sonnet, Ph., 1993. Study of spatial and temporal variations of hydrogeochemical
variables using factorial kriging analysis. In: Soares, A. ŽEd.., Geostatistics Troia
´ ’92, Kluwer
Academic Publishers, Dordrecht, pp. 745-756.
Goovaerts, P., Webster, R., 1994. Scale-dependent correlation between topsoil copper and cobalt
concentrations in Scotland. Eur. J. Soil Sci. 45, 79–95.
Goovaerts, P., Journel, A.G., 1995. Integrating soil map information in modelling the spatial
variation of continuous soil properties. Eur. J. Soil Sci. 46, 397–414.
Goovaerts, P., Webster, R., Dubois, J.-P., 1997. Assessing the risk of soil contamination in the
Swiss Jura using indicator geostatistics. Environ. Ecol. Stat. 4, 31–48.
Gotway, C.A., Hartford, A.H., 1996. Geostatistical methods for incorporating auxiliary information in the prediction of spatial variables. J. Agric. Biol. Environ. Stat. 1, 17–39.
Goulard, M., 1989. Inference in a coregionalization model. In: Armstrong, M. ŽEd.., Geostatistics,
Kluwer Academic Publishers, Dordrecht, pp. 397-408.
Goulard, M., Voltz, M., 1992. Linear coregionalization model: tools for estimation and choice of
cross-variogram matrix. Math. Geol. 24, 269–286.
Heuvelink, G.B.M., 1996. Identification of field attribute error under different models of spatial
variation. Int. J. GIS 10, 921–935.
Heuvelink, G.B.M., Bierkens, M.F.P., 1992. Combining soil maps with interpolations from point
observations to predict quantitative soil properties. Geoderma 55, 1–15.
Heuvelink, G.B.M., Musters, P., Pebesma, E.J., 1997. Spatio-temporal kriging of soil water
content. In: Baafi, E.Y., Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer
Academic Publishers, Dordrecht, pp. 1020-1030.
Isaaks, E.H., 1990. The Application of Monte Carlo Methods to the Analysis of Spatially
Correlated Data. PhD thesis, Stanford University, Stanford, CA.
Isaaks, E.H., Srivastava, R.M., 1989. An Introduction to Applied Geostatistics. Oxford Univ.
Press, New York, 561 p.
Journel, A.G., 1983. Non-parametric estimation of spatial distributions. Math. Geol. 15, 445–468.
Journel, A.G., 1984. The place of non-parametric geostatistics. In: Verly, G., David, M., Journel,
A.G., Marechal,
A. ŽEds.., Geostatistics for Natural Resources Characterization, Reidel,
´
Dordrecht, pp. 307-355.
Journel, A.G., 1987. Geostatistics for the Environmental Sciences. EPA project no CR 811893.
Technical report, US Environmental Protection Agency, EMS Laboratory, Las Vegas, NV.
Journel, A.G., 1993. Geostatistics: roadblocks and challenges. In: Soares, A. ŽEd.., Geostatistics
Troia
´ ’92, Kluwer Academic Publishers, Dordrecht, pp. 213-224.
Journel, A.G., 1997. Geostatistics: tools for advanced spatial modeling in GIS. In: Corwin, D.L.,
Loague, K. ŽEds.., Application of GIS to the Modeling of Non-point Source Pollutants in the
Vadose Zone, Special SSSA Publications, pp. 39–55.
Journel, A.G., Huijbregts, C.J., 1978. Mining Geostatistics. Academic Press, New York, 600 p.
Kyriakidis, P.C., 1997. Selecting panels for remediation in contaminated soils via stochastic
44
P. GooÕaertsr Geoderma 89 (1999) 1–45
imaging. In: Baafi, E.Y., Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer
Academic Publishers, Dordrecht, pp. 973-983.
Lark, R.M., Bolam, H.C., 1997. Uncertainty in prediction and interpretation of spatially variable
data on soils. Geoderma 78, 263–282.
Leenaers, H., Okx, J.P., Burrough, P.A., 1990. Employing elevation data for efficient mapping of
soil pollution on floodplains. Soil Use and Management 6, 105–114.
Leonte, D., Schofield, N., 1996. Evaluation of a soil contaminated site and clean-up criteria: a
geostatistical approach. In: Rouhani, S., Srivastava, R.M., Desbarats, A.J., Cromer, M.V.,
Johnson, A.I. ŽEds.., Geostatistics for Environmental and Geotechnical Applications. American
Society for Testing and Materials STP 1283, Philadelphia, pp. 133-145.
Matheron, G., 1976. A simple substitute for conditional expectation: the disjunctive kriging. In:
Guarascio, M., David, M., Huijbregts, C.J. ŽEds.., Advanced Geostatistics in the Mining
Industry, Reidel, Dordrecht, pp. 221-236.
Matheron, G., 1982. Pour une analyse krigeante de donnees
´ regionalisees.
´
´ Centre de Geostatis´
tique, Ecole des Mines de Paris, Report N-732, Fontainebleau.
McBratney, A.B., Webster, R., 1983. Optimal interpolation and isarithmic mapping of soil
properties: V. Co-regionalization and multiple sampling strategy. J. Soil Sci. 34, 137–162.
McBratney, A.B., Webster, R., 1986. Choosing functions for semi-variograms of soil properties
and fitting them to sampling estimates. J. Soil Sci. 37, 617–639.
Myers, D.E., 1982. Matrix formulation of co-kriging. Math. Geol. 14, 249–257.
Odeh, I.O.A., McBratney, A.B., Slater, B.K., 1997. Predicting soil properties from ancillary
information: non-spatial models compared with geostatistical and combined methods. In: Baafi,
E.Y., Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer Academic Publishers,
Dordrecht, pp. 1008-1019.
Oliver, M.A., Webster, R., 1986a. Semi-variograms for modelling the spatial pattern of landform
and soil properties. Earth Surface Processes and Landforms 11, 491–504.
Oliver, M.A., Webster, R., 1986b. Combining nested and linear sampling for determining the
scale and form of spatial variation of regionalized variables. Geogr. Anal. 18, 227–242.
Papritz, A., Fluhler,
H., 1994. Temporal change of spatially autocorrelated soil properties: optimal
¨
estimation by cokriging. Geoderma 62, 29–43.
Rautman, C.A., 1997. Geostatistics and cost-effective environmental remediation. In: Baafi, E.Y.,
Schofield, N.A. ŽEds.., Geostatistics Wollongong ’96. Kluwer Academic Publishers, Dordrecht, pp. 941-950.
Rivoirard, J., 1994. Introduction to Disjunctive Kriging and Non-linear Geostatistics. Oxford
Univ. Press, New York, 180 pp.
Smith, J.L., Halvorson, J.J., Papendick, R.I., 1993. Using multiple-variable indicator kriging for
evaluating soil quality. Soil Sci. Soc. Am. J. 57, 743–749.
Soares, A., 1992. Geostatistical estimation of multi-phase structures. Math. Geol. 24, 149–160.
Srivastava, R.M., 1987. Minimum variance or maximum profitability?. Canadian Industrial
Mining Bulletin 80, 63–68.
Stein, A., Hoogerwerf, M., Bouma, J., 1988. Use of soil-map delineations to improve Žco.kriging
of point data on moisture deficits. Geoderma 43, 163–177.
Sterk, G., Stein, A., 1997. Mapping wind-blown mass transport by modeling variability in space
and time. Soil Sci. Soc. Am. J. 61, 232–239.
Trangmar, B.B., Yost, R.S., Uehara, G., 1985. Application of geostatistics to spatial studies of soil
properties. Advances in Agronomy 38, 45–94.
Vanderborght, J., Jacques, D., Mallants, D., Tseng, P.H., Feyen, J., 1997. Analysis of solute
redistribution in heterogeneous soil: II. Numerical simulation of solute transport. In: Soares,
A., Gomez-Hernandez,
J., Froidevaux, R. ŽEds.., geoENV I—Geostatistics for Environmental
´
´
Applications. Kluwer Academic Publishers, Dordrecht, pp. 283–295.
P. GooÕaertsr Geoderma 89 (1999) 1–45
45
Van Meirvenne, M., Scheldeman, K., Baert, G., Hofman, G., 1994. Quantification of soil textural
fractions of Bas-Zaire using soil map polygons andror point observations. Geoderma 62,
69–82.
Verly, G., 1983. The multi-Gaussian approach and its applications to the estimation of local
reserves. Math. Geol. 15, 259–286.
Vieira, S.R., Hatfield, J.L., Nielsen, D.R., Biggar, J.W., 1983. Geostatistical theory and application to variability of some agronomical properties. Hilgardia 51, 1–75.
Voltz, M., Goulard, M., 1994. Spatial interpolation of soil moisture retention curves. Geoderma
62, 109–123.
Voltz, M., Webster, R., 1990. A comparison of kriging, cubic splines and classification for
predicting soil properties from sample information. J. Soil Sci. 41, 473–490.
Wackernagel, H., 1988. Geostatistical techniques for interpreting multivariate spatial information.
In: Chung, C.F., Fabbri, A.G., Sinding-Larsen, R. ŽEds.., Quantitative Analysis of Mineral and
Energy Resources. Reidel, Dordrecht, pp. 393–409.
Wackernagel, H., 1994. Cokriging versus kriging in regionalized multivariate data analysis.
Geoderma 62, 83–92.
Wackernagel, H., 1995. Multivariate Geostatistics: An Introduction with Applications. SpringerVerlag, Berlin, 256 p.
Warrick, A.W., Myers, D.E., Nielsen, D.R., 1986. Geostatistical methods applied to soil science.
In: Methods of Soil Analysis, Part 1. Physical and Mineralogical Methods. Agronomy
Monograph no. 9, 2nd edn., pp. 53–82.
Webster, R., 1991. Local disjunctive kriging of soil properties with change of support. J. Soil Sci.
42, 301–318.
Webster, R., Oliver, M.A., 1989. Optimal interpolation and isarithmic mapping of soil properties:
VI. Disjunctive kriging and mapping the conditional probability. J. Soil Sci. 40, 497–512.
Webster, R., Oliver, M.A., 1993. How large a sample is needed to estimate the regional variogram
adequately? In: Soares, A. ŽEd.., Geostatistics Troia
´ ’92, Kluwer Academic Publishers,
Dordrecht, pp. 155–166.
Webster, R., Atteia, O., Dubois, J.-P., 1994. Coregionalization of trace metals in the soil in the
Swiss Jura. Eur. J. Soil Sci. 45, 205–218.
Yates, S.R., Warrick, A.W., 1987. Estimating soil water content using cokriging. Soil Sci. Soc.
Am. J. 51, 23–30.
Zhang, R., Warrick, A.W., Myers, D.E., 1992. Improvement of the prediction of soil particle size
fractions using spectral properties. Geoderma 52, 223–234.
Zhu, H., Journel, A.G., 1993. Formatting and integrating soft data: Stochastic imaging via the
Markov-Bayes algorithm. In: Soares, A. ŽEd.., Geostatistics Troia
´ ’92, Kluwer Academic
Publishers, Dordrecht, pp. 1–12.