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Choosing between Abrupt and
Gradual Spatial Variation?
Gerard B.M. Heuvelinkl and Johan A. ~ u i s r n a n ~
Abstract.-Two basic models of spatial variation are widely used in
present-day soil survey practice. The discrete model of spatial variation
(DMSV) forms the basis of the traditional soil map, in which
homogeneous soil mapping units are separated by abrupt boundaries. The
continuous model of spatial variation (CMSV) originates from
geostatistics, where kriging is used to map gradual changes in soil
properties. Neither of the two models is capable of handling situations in
which abrupt and gradual spatial variation are both present in the same
area. Therefore, recently a straightforward combination of the two
models was introduced, known as the mixed model of spatial variation
(MMSV). The MMSV contains the two basic models, suggesting that it
may perform well on the whole range of spatial variation. In this paper
we investigate the anticipated flexibility of the MMSV using a simulation
study. As expected, the MMSV is superior to the DMSV and the CMSV
when both types of spatial variation are present. But the MMSV is also
as suitable as the DMSV in case of a 'discrete' reality, and as suitable as
the CMSV in case of a 'continuous' reality. From this we conclude that
the MMSV should be recommended for situations where an a priori
choice between abrupt and gradual spatial variation cannot easily be
made.
INTRODUCTION
Traditional soil survey is based upon the presumption that soil behaves
uniformly within soil mapping units and changes fairly abrupt at the boundaries
between them (Voltz and Webster 1990, Webster and Oliver 1990, Burrough
1993). However, the practical validity of this conventional representation of soil
variability has repeatedly been questioned (e .g. Webster and Cuanalo 1975,
Nortcliff 1978, Campbell et al. 1989, Nettleton et al. 1991). Major drawbacks
of the conventional model of soil spatial variation are that it cannot represent
gradual boundaries and that it ignores spatial autocorrelation within mapping
units.
As an alternative to the conventional model, some fifteen years ago the use
of geostatistical techniques for modelling spatial variation was introduced to soil
science. Burgess and Webster (1980) were among the first to apply kriging, an
'Geostatistician, University of Amsterdam, The Netherlands
'Student, University of Amsterdam, The Netherlands
exponent of this theory, to soil survey. Since then the geostatistical approach to
modelling soil spatial variation has flourished and kriging is now routinely being
adopted for the mapping of soil properties (Webster 1994).
Recently, however, it is more and more being realized that the outright
abandoning of the conventional approach to soil spatial variation is perhaps too
drastic. Kriging definitely has disadvantages as well, such as its inadequacy to
deal with sharp boundaries. In order to bridge the gap between the conventional
and geostatistical representation of soil spatial variation, several models have
been developed that can handle discrete (abrupt) as well as continuous (gradual)
spatial variation in the same area (Stein et al. 1988, Voltz and Webster 1990,
Heuvelink and Bierkens 1992, Rogowski and Wolf 1994, Goovaerts and Journel
1995, Heuvelink 1996).
In this paper we examine one such a model, known as the mixed model of
spatial variation (MMSV). In Heuvelink (1996) it was anticipated that the
MMSV should work well on the whole range of spatial variation, from purely
discrete to purely continuous. We analyze the anticipated flexibility of the
MMSV using nine simulated 'realities'. But before describing the exact
procedure of the simulation exercise, we will first briefly review the three
models of spatial variation used in this study.
THREE MODELS OF SPATIAL VARIATION
The Discrete Model of Spatial Variation (DMSV) first divides the
geographical domain D into K separate units D,. It then makes the following
assumptions on the behaviour of a spatially distributed attribute Z(-):
1
2
3
Z(x) =p, +E(x) for all x E D,
E ( . ) has zero mean and is spatially uncorrelated
Var(~(x))=C, for all x E D
Thus the DMSV assumes that Z(x) is the sum of a unit-dependent mean p, and a
residual noise ~ ( x ) .The DMSV will usually be adopted when the units D, are
available in the form of a polygon map, such as a soil map, a landuse map or a
geological map, and. when the within-unit variability is expected to be small in
comparison with the between-unit variability. In other words, the DMSV
represents the conventional model of soil spatial variation and is appropriate
when major jumps in the attribute Z(.) take place at the boundaries of the mapping units.
In its simplest form, the Continuous Model of Spatial Variation (CMSV)
makes the following assumptions:
1
2
E[Z(x)]=pforallxED
Cov(Z(x),Z(x h)) =C,( I h I ) for all x,x h E D
+
+
Thus the CMSV assumes that Z(*) is second-order stationary, meaning that it
has a constant mean and that its spatial autocovariance is a function only of the
distance between the locations. The CMSV embodies the geostatistical model of
spatial variation. In geostatistics it is customary to use the variogram y,(*) to
characterize the spatial autocorrelation of Z($ It is related to the autocovariance
by the identity yz( 1 h 1 ) =C,(O)-Cz( 1 h I).
The assumptions underlying the Mixed Model of Spatial Variation (MMSV)
are a combination of those underlying the DMSV and CMSV:
1
2
3
+
Z(x) =p, ~ ( x for
) all
E(-) has zero mean
x E D,
Cov(~(x),~(x+h))=C,(lhJ)
for all x,x+hED
Second-order stationarity is thus imposed on E(-)instead of on Z(.). The MMSV
is more general than the DMSV and CMSV, and in fact it contains both models.
However, the DMSV and CMSV are included here as separate models because
they are very often used in practice.
APPLICATION TO NINE SIMULATED REALITIES
In order to study the suitability of the three MSV's for mapping under
various circumstances, nine different 'realities' were created. This was done by
adding maps generated using unconditional Gaussian simulation (Deutsch and
Journel 1992) to an artificially constructed 'soil map'. The nine realities are
given in figure 1. The A-maps in figure 1 (top row) are strongly dominated by
the discrete soil map, the B-maps (middle row) to a much lesser extent, and the
C-maps (bottom row) bear no influence from the discrete soil map. The degree
of spatial autocorrelation in the added residual decreases from the 1-maps (left
column) to the 3-maps (right column).
Mapping the soil property from 200 observations
From each of the nine simulated maps data sets were created by collecting
observations at 200 randomly selected locations. From these observations the
soil property was mapped using the three MSV's. This resulted in 27 maps of
predictions and 27 maps of prediction error standard deviations.
Mapping using the DMSV is done simply by calculating the mean of the
observations for all mapping units separately, and using the unit mean as a
prediction for all points lying in the same unit. Mapping with the CMSV is done
by ordinary kriging, in which a variogram is used computed from the 200
observations. Mapping is somewhat more complicated in case of the MMSV.
First a variogram is computed from the 200 residuals obtained by subtracting the
unit means from the observations (Kitanidis 1994). Next the attribute is mapped
using universal kriging (Cressie 1991).
Figure 1.-Nine simulated realities. Letter indicates influence from the artificial soil
map: A =large, B =moderate, C =none. Number indicates degree of spatial
autocorrelation in the added residual: 1 =large, 2 =moderate, 3 =none.
In figure 2 the results of the mapping are given for a selection of three out
of the nine simulated realities. Note that these are the prediction maps, and that
the corresponding maps of prediction error standard deviations are not given
here. The DMSV maps necessarily follow the delineations of the soil map,
which is quite all right for map A3 but less so for map B2 and definitely
inappropriate in case of map C1. Conversely, the CMSV is suitable for mapping
C1, but it is not appropriate for mapping B2 and even less so for mapping A3.
The most important observation from figure 2 is that the MMSV is indeed
capable of an adequate mapping in all three cases. It is interesting to observe
that the MMSV mimics the DMSV in case of a 'discrete reality' and the CMSV
in case of a 'continuous' reality.
A3
-
DMSV
A3
-
MMSV
A3
-
CMSV
B2
-
DMSV
B2
-
MMSV
B2
-
CMSV
Cl
-
DMSV
Cl
-
MMSV
Cl
-
CMSV
I90
Figure 2.-Mapping three simulated realities (maps A3, B2 and C1) from 200
observations using the DMSV, MMSV and CMSV.
Validation
In order to evaluate the three prediction methods the mean error (ME), root
mean square error (RMSE) and standardized root mean square error (SRMSE)
were computed for each of the 9 realities. These statistics were computed from
all remaining points in the map. The results are given in figure 3.
The mean error is in all cases quite small. This is not surprising, because
unbiasedness conditions are included in all three mapping procedures.
Differences between the three mapping procedures are also negligible.
The SRMSE values are on average somewhat larger than one, particularly
when the CMSV is applied to situations in which the soil map influence is
dominant. This may be caused by forcing the CMSV upon a non-stationary
MMSV or DMSV reality, and perhaps also because the kriging variance does
not include the uncertainty in estimating the variogram (Christensen 1991).
Most interesting are the RMSE results. In this case we do see meaningful
differences between the three mapping procedures. The results demonstrate that
the CMSV is inappropriate for the A-maps, whereas the DMSV is inappropriate
for the C-maps. An exception is the pure nugget map C3, where all mapping
procedures are equally good (bad). The results also confirm that the MMSV is
superior for the B-maps. Note that comparison of RMSE values between maps is
difficult here because these are affected by differences in spatial autocorrelation.
H DMSV
1.5
1.o
MMSV
0.5
Q CMSV
0.0
H DMSV
15
I0
MMSV
5
CMSV
0
1.50
1.25
H DMSV
1.oo
0.75
H MMSV
0.50
0.25
0.00
Figure 3.-Validation results for the three prediction methods.
I=] CMSV
DISCUSSION AND CONCLUSIONS
The application to simulated 'realities' shows that the MMSV interpolates
well on the entire range of spatial variation. In all cases is the MMSV at least as
good as the DMSV and the CMSV, and it is superior in situations where there
is abrupt and gradual spatial variation.
The simulation exercise also shows that, depending on the situation, the
DMSV may perform much worse than the CMSV, and vice versa. This means
that if a choice between these two models is to be made (and these are the two
models most often used in practice), then it must be taken with care. This may
seem obvious, but in practice the choice of model is often dominated by
irrelevant factors, such as background and experience of the user. And even
when care is taken, it may not always be easy to decide beforehand whether
abrupt or gradual spatial variation prevails. Therefore the flexibility of the
MMSV demonstrated here is of clear importance, because it implies that by
adopting the MMSV one can protect oneself against using the wrong model. It is
as if one can leave the choice between abrupt and gradual spatial variation to the
MMSV.
The MMSV is especially advantageous in situations where abrupt and
gradual spatial variation are both present. Some indication of whether this is the
case can be obtained from the intra-class correlation (Webster and Oliver 1990),
but more informative is the comparison of the variograms of the original
attribute and its residual. A mixed form of spatial variation yields a residual
variogram that is substantially lower than the original variogram. Since both
variograms can easily be computed one can thus quickly decide whether the
MMSV is superior to the CMSV and DMSV for a given situation.
Another approach to handling abrupt and gradual spatial variation both
present in the same area is to adopt the CMSV separately per mapping unit
(Stein et al. 1988, Voltz and Webster 1990). The main difference with the
MMSV is that this approach excludes the presence of spatial autocorrelation
across mapping unit boundaries. Thus it is likely to create boundaries even when
they are not really there. As mentioned, the MMSV does not suffer from this
problem because it mimics the CMSV under such circumstances. We consider it
a major advantage of the MMSV that, although it is meant for situations in
which gradual and abrupt spatial variation are both present, it will also perform
well when spatial variation is exclusively gradual or exclusively abrupt.
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BIOGRAPHICAL SKETCH
Gerard B.M. Heuvelink is a geostatistician with the Landscape and
Environmental Research Group, Faculty of Environmental Sciences, University
of Amsterdam. He holds an M.S. in Applied Mathematics from Twente
Technical University and a Ph.D. in Geography from Utrecht University.
Johan A. Huisman is a graduate student in Physical Geography at the
Faculty of Environmental Sciences, University of Amsterdam.