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Catena 33 Ž1998. 139–153
Spatial variability of soil properties at different
scales within three terraces of the Henares River
žSpain /
A. Saldana
˜ ) , A. Stein, J.A. Zinck
ITC International Institute for Aerospace SurÕey and Earth Sciences, P.O. Box 6, 7500 AA Enschede,
Netherlands
Received 26 September 1997; revised 27 August 1998; accepted 27 August 1998
Abstract
This paper applies statistical and geostatistical procedures to a soil chronosequence on the
terraces of the Henares River ŽNE Madrid. to analyse the spatial distribution of several soil
properties and use the contribution of geostatistics to establishing a landscape evolution model of
the area. Particle-size distribution, pH, calcium carbonate and organic carbon were analysed.
Statistical procedures focus on analysing differences between terraces. Geostatistical procedures
identify short- and medium-range variations within individual terraces at different scales. Standard
transitive variogram models describe the properties of the younger terrace, whereas the linear
intransitive model fits the majority of variograms of the older terrace. The analysis confirms and
quantifies the decrease in variability of soil properties from young to old deposits, showing thus an
increment of soil homogenisation with time. Ageing of the terraces causes the variables to show
nontransitive variogram models with unbounded variances within the observation range. q 1998
Elsevier Science B.V. All rights reserved.
Keywords: Chronosequence; Spatial variability; Variogram; Soil homogenisation; Spatial sampling
1. Introduction
Spatial variation in soil has been recognised for many years ŽBurrough, 1993.. A
useful distinction is that between random and systematic variation. Systematic variation
is a gradual or marked change in soil properties as a function of landforms, geomorphic
)
Corresponding author. Zaragoza 5, 28804 Alcala´ de Henares, Madrid, Spain. E-mail: [email protected]
0341-8162r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 3 4 1 - 8 1 6 2 Ž 9 8 . 0 0 0 9 0 - 3
140
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˜ et al.r Catena 33 (1998) 139–153
elements, soil-forming factors andror soil management ŽJenny, 1941.. Random variations entail either differences in soil properties which cannot be explained in terms of
known soil-forming factors, recognisable at a reasonable sampling density, or measurement errors at the scale of the study. Few attempts have been made so far to differentiate
between systematic and random soil variations in chronosequences ŽHarrison et al.,
1990..
The soils of the Henares River terraces are arranged in a topo-chronosequence. They
have been studied along transects to establish relationships between terrace surfaces and
soil properties, and to understand the evolution of the valley during the Pliocene and
Quaternary Žsee, e.g., Ibanez
´˜ et al., 1990, 1994.. However, few soil chronosequence
studies were based on sufficient data points within terraces and at different depths to
enable the degree and nature of soil variability within and between seemingly homogeneous land areas to be determined.
On a soil map, variation is displayed using geomorphic and soil knowledge, mainly in
terms of systematic variation. Map units often contain information on the degree and
nature of spatial variation, but the areal proportion occupied by each taxum is not always
precisely determined. Moreover, the patterns of soil distribution and the scale at which
the soil components are mapped may not be compatible.
In this paper, statistical methods were used to describe quantitatively the variation in
soil properties within and between map units. The coefficient of variation and the t-test,
for instance, help distinguish variation between units. Geostatistics, based on the theory
of regionalized variables, provides a basis for quantifying the spatial relation among
sample values within map units. It also allows to predict values at unvisited locations by
kriging and to design rational sampling schemes ŽWebster, 1985.. However, uncritical
use of geostatistics in soil survey has several drawbacks. The large number of data
required to estimate a variogram and the assumptions regarding stationarity of the
variation, necessary to measure spatial variation from a single set of observations
ŽJournel and Huijbregts, 1978., restrict the application of the variogram to small sections
of landscape ŽAgbu and Olsen, 1990.. Selection of the appropriate variogram model is
still largely done interactively, which may introduce some subjectivity in the process.
Interpolation of data yields maps of single properties at one depth, whereas a real soil
body on the landscape integrates many soil properties at several depths. In spite of these
limitations, a combination of geostatistics with soil classification could improve the soil
survey method and, in particular, determine the observation density needed to properly
describe soil units, as suggested by several authors. Stein et al. Ž1988. applied
Žco.-kriging to existing soil map delineations to improve the accuracy of prediction of
land qualities at minimal effort and costs. Prior landscape stratification, based on the
correlation of soil types with major landforms and geological features, was used to
establish the soil map units. Water-table classes based on ground-water table measures
were also considered in the analysis. Goovaerts and Journel Ž1995. used indicator
kriging and the Markov–Bayes algorithm to establish the probability of copper and
cobalt deficiencies in soils. They showed that the use of soil map information improves
the delineation of deficiency areas, particularly where the sampling is sparse. On the
other hand, Voltz et al. Ž1997. proposed a method combining soil classification and
three interpolation methods Žkriging, inverse squared distance and nearest neighbour. to
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141
map soil properties at regional scale with acceptable precision. Sample information from
a reference area and soil observations distributed over the region were also used. They
found that estimates from soil classification combined with kriging were the most
precise.
Determining the number and location of the field observations is difficult in flat
alluvial systems, where the inherent spatial variation of soil properties is not easily
predicted from soil–landform relationships ŽDi et al., 1989.. Alluvial soils are often very
variable, both laterally and with depth, because changes in both dimensions can result
from differences in parent material and depositional processes.
This paper shows results obtained from the geostatistical analysis of soil properties
within the terraces generated by Quaternary evolution of the Henares River incision.
Stationarity is assumed provided the similar nature, origin of the parent material and
pedogenesis of the terraces. The study examines sampling at different spatial scales to
establish Ž1. differences among three selected terraces of lower, medium and upper
Pleistocene age and Ž2. short- and medium-range variations occurring within the
terraces.
2. Material and methods
2.1. Study area characteristics
The study area is in the provinces of Madrid and Guadalajara, between 40830X N and
40850X N and 3810X W and 3830X W ŽFig. 1., 40 km NE of Madrid, on the southern slope
of the Ayllon
´ mountain range. The altitude varies from 600 to 900 m above sea level.
The climate is continental Mediterranean, with hot dry summers and cold wet winters.
Fig. 1. Location of the three sample areas in the Henares River valley and structure of the 3-level sampling
scheme applied to each terrace.
A. Saldana
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142
The annual mean temperature is 148C and the annual mean rainfall is 400 mm ŽIMN,
1992.. The soil moisture regime is xeric and the soil temperature regime is mesic
ŽUSDA, 1994.. Past climate fluctuations, tectonic movements and lithologic-structural
controls have influenced the development of the Henares River valley, resulting in a
typical asymmetric valley of central Spain. As many as 20 terraces and a series of
incised glacis-terraces of Pleistocene age have been identified along the right and left
banks of the river, respectively. The granulometric and petrographic composition of the
terraces is very similar throughout, with quartzite, quartz and limestone pebbles within a
sandy matrix. Calcareous pebbles are absent from the higher terraces ŽITGE, 1990.. The
ages of the terraces probably range from late Pliocene to upper Pleistocene and
Holocene ŽGallardo et al., 1987.. Three terraces of lower, middle and upper Pleistocene
age were selected for description and sampling. The soil types developed on them
include Inceptisols and Alfisols ŽUSDA, 1994.. Calcixerollic, Fluventic and Typic
Xerochrepts are found on the lower and younger terrace. Haploxeralfs, Rhodoxeralfs and
Palexeralfs, with Calcic, Petrocalcic, Vertic and Typic subgroups, dominate the middle
and higher terraces. The land is mainly used for rainfed agriculture, in particular wheat,
barley and sunflowers. Irrigated sunflower and maize are produced on the floodplain and
lower terraces. Natural vegetation occurs only in marginal areas with poor agricultural
productivity; it is mainly the degradation stage of the original climax forest formation.
2.2. Statistics and geostatistics
2.2.1. Variogram estimation
Statistics, such as minimum, maximum, mean, median, standard deviation and
coefficient of variation summarise the data. Graphs of the cumulative relative variance
for increasing distances show the distances at which important increases in variance
occur. To analyse the spatial variability between observation points Žhorizontal. and
observations depth Žvertical., use was made of geostatistical methods ŽJournel and
Huijbregts, 1978; Cressie, 1991.. Each soil variable that is measured is associated with
its observation location x. For the ith variable, denoted as z i Ž x ., the variogram g i Ž h. is
the expected squared difference as a function of the distance h or lag between two
locations, defined by:
gi Ž h. s
1
2
E Zi Ž x . y Zi Ž x q h . ,
2
where x and x q h are two locations, separated by a distance h, at which the
regionalized variable is measured, and E denotes the mathematical expectation.
Use of the variogram for interpolation requires Ew Zi Ž x .x to be constant in the area
and that the g i Ž h. do not depend upon x, according to the so-called intrinsic hypothesis.
To estimate g i Ž h. using n observations of Zi Ž x . with values z i Ž x 1 ., z i Ž x 2 ., . . . , z i Ž x n .,
the expectation E is replaced by the average value and a sample variogram gˆ i Ž h. is
computed by:
gˆ i Ž h . s
1
Ni Ž h .
2 Ni Ž h .
js1
Ý Ž zi Ž x j . y zi Ž x j q h. .
2
,
A. Saldana
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143
where z i Ž x j . and z i Ž x j q h. form a pair of points separated by a distance h of which
there are Ni Ž h.. Commonly, pairs of observations are grouped into a limited number of
distance classes to ensure that Ni Ž h. is sufficiently large. Each class contains pairs with
approximately the same distance. The sample variogram was estimated using the
programme SPATANAL ŽStein, 1993..
2.2.2. Model fitting
The parameters of a variogram contain the spatial information required for prediction;
they are estimated for distances h and sample variogram values gˆ i Ž h.. A variogram is a
mathematical function that must be able to characterise three important parameters: the
nugget variance, the sill variance and the range. The nugget is the positive intercept of
the variogram with the ordinate and represents unexplained spatially dependent variation
or purely random variance. Transitive variograms reach a sill value at which they level
out, at a distance known as the range of spatial dependence. Common transitive models
are the spherical, exponential, Gaussian, hole effect Žor wave. and pure nugget ŽCressie,
1991.. Continuous, gradually varying attributes are often described by a Gaussian
variogram. Attributes with abrupt boundaries at discrete and regular spacings Žthe range.
are described by the linear model with a sill. The spherical model describes variables
similar to the previous ones when the distance between abrupt changes is not clearly
defined. Attributes characterised by abrupt changes at all distances are described by the
exponential model. The hole effect model reflects repetition in the data related to the
periodicity of parent material deposition and consequently to the repetition of landform
sequences in space. The pure nugget model indicates that there is no spatial dependence
at the scale of investigation. In contrast, a common nontransitive model is the linear one,
which is suitable to describe attributes varying at all scales ŽJournel and Huijbregts,
1978; Burrough, 1983, 1987; Oliver, 1987..
Model fitting is required for interpolation procedures and is a previous step to the
creation of soil property maps. Model selection was based on a combination of the R 2 ,
or unadjusted coefficient of determination, of a weighted nonlinear regression Žvalues
close to 1 indicate a good fit., and interactive interpretation of the sample variogram
values. For example, both the hole effect and the Gaussian models yielded a similar R 2
value for the pH at depth d 2 in sample area A 1 , but the experimental variogram did not
show any evidence of periodicity. Therefore, the Gaussian model was selected. All
parameters were estimated by a weighted nonlinear regression procedure using the
Statistical Analysis System ŽSAS, 1985..
2.3. Spatial sampling at different scales
A previous knowledge of soil properties and variation relationships with landscape
features and statistical sampling can be used to collect spatial information. The collected
data z i Ž x i1 ., . . . , z i Ž x i n ., including their sampling locations x i1 , . . . , x i n , can be
summarised by the sampling design for the ith variable Si s Ž z i , x i ., distinguishing
between the variable-specific part of the design and the location-specific part. For the
location-specific part, random sampling, grid sampling or any other sampling procedure
144
A. Saldana
˜ et al.r Catena 33 (1998) 139–153
can be applied. The final sampling design S s jSi is the collection of the individual
sampling designs. The density of observations depends on the variation: a very variable
property may need to be collected with a greater density than one that is less variable.
The sampling density for the random design and for the grid sampling is governed by
criteria such as the standard deviation of the observations z Ž x i ..
Sampling is complicated by the fact that the data are spatially dependent, usually with
an unknown degree of spatial dependence, hence the need to cover several scales of
spatial resolution for several variables in a single sample design. van Groenigen and
Stein Ž1998. distinguish between Ž1. designs for estimating spatial dependence, Ž2.
designs for even spacing of data throughout the area using previous observations and
ancillary information such as irregular boundaries of the area, and Ž3. designs for
optimising spatial interpolation. These designs might be totally different even for a
single variable. The design for estimating spatial dependence leads to a clustering of
observation locations in an area, whereas this is generally avoided when an even cover
of samples or a design for optimising spatial interpolation is applied. All these
considerations therefore produce a sampling scheme that has to serve several objectives,
several variables and an unknown relation between a variable and its observation
locations.
In this study, a multi-scale sampling grid was used to quantify and model the spatial
variability of soil properties. For such a grid, s grid meshes d j , j s 1, . . . , s, are decided
upon in advance, and sampling grids S Ž1., . . . , S Ž s. are defined such that for i - j the
average distance between points in S Ž i. is less than that in S Ž j.. A multi-level grid has
some advantages over a single grid as single observations may belong to more than one
level of the design. For example, each design may be concentrated around a single point,
that is the centre of all of the designs S Ž i.. This enables the spatial dependence of the
data to be analysed at different spatial scales and a comparison between the scales is
then easy to determine. This provides a compromise where there are variables with
different spatial behaviour and when different research objectives are pursued. In such a
scheme, there is some clustering and, at the same time, the area is fairly evenly covered
with observations.
Three areas of 540 m = 540 m were sampled ŽFig. 1.. Sample area A 1 was located on
a low Pleistocene terrace ŽT-29., A 2 on a middle Pleistocene terrace ŽT-25. and A 3 on a
high Pleistocene terrace ŽT-15., with terrace labelling according to ITGE Ž1990.. Three
sampling intervals were selected and the observations arranged in a nested scheme ŽFig.
1. with:
Ø 10 m intervals, to sample the short-range variation at the intra-polypedon scale,
giving 49 observations on a square grid of 7 by 7 points;
Ø 30 m intervals, to sample the medium-range variation, giving observations on a
square grid of 7 by 7 points, with the innermost nine locations coinciding with the
locations on the 10-m-interval grid. A distance of 30 m was considered appropriate to
describe the transverse structure of alluvial systems, e.g., variation from the levee to the
basin. It corresponds to the sampling distances used by Campbell Ž1978. and Weitz et al.
Ž1993.;
Ø 90 m intervals, to sample medium-range variation, giving 49 observations on a
square grid of 7 by 7 points, with the innermost nine locations coinciding with the
A. Saldana
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145
locations on the 30-m-interval grid. This interval is appropriate for terrace fragments of
limited flat areas, as higher terraces have been strongly dissected by erosion.
The total number of observations was 129, as there was some overlap in the central
locations. At each observation point, samples were taken at three standard depths:
0.1–0.2 m Ž d1 ., 0.4–0.5 m Ž d 2 . and 0.9–1.0 m Ž d 3 .. Variables measured in all areas
included sand, silt, clay, calcium carbonate and soil reaction ŽpH.. Organic carbon was
determined in all areas at d1 and in A 1 also at the other depths. Particle-size distribution
was determined by the Bouyoucos method, organic carbon by the Walkley–Black
method, calcium carbonate with the Bernard calcimeter, and pH with a pH meter in
1:2.5 soil–water mixtures.
A test was developed to investigate the significance of the differences in mean values
between strata when the observations of a regionalized variable are Žspatially. related.
Suppose p strata are investigated, from every stratum it is known that the spatial
dependency structure is given by the variograms g i Ž r . for r G 0, i s 1, . . . , p. As an
estimator for the mean and the variance within the ith stratum we have:
m
ˆ is
1XnGy1
y
i
1XnGy1
i 1n
,
where the matrix Gi contains values of the variogram in the ith stratum; Gi depends on
the variable under study. The variance of the mean is equal to:
y1
1
Var Ž m
ˆ i . s X y1 s .
gi
1 n Gi 1 n
The null hypothesis H0 that no differences exist between the different strata and the
alternative hypothesis H1 can be formulated as:
H0 : m 1 s m 2 s . . . s m p
H1 : at least one m j differs from the other mXi s, i / j.
When the spatial structure is known, this hypothesis is tested with the test statistic:
2
p
Ts Ý
is1
žÝ /
g i m̂ i
p
g i m2i y
ˆ
is1
p
,
Ý gi
is1
2
which has under H0 a x -distribution with p y 1 df. Of course, in practical studies, the
spatial structure has to be estimated from the data. As the test value will only slightly
change, the same x 2-distribution can be used ŽStein et al., 1988..
The variogram parameters were used in the program OPTIM ŽStein, 1996. that
determines the best sampling interval to obtain estimates at a given level of precision for
each soil property, i.e., the necessary sampling spacing to arrive at a preset kriging
variance s 02 . For square grids, the highest kriging variance occurs at the centre of four
grid points. Moreover, the kriging variance is independent from actual observations.
OPTIM uses an iterative optimisation procedure. It needs the size of the area, d a , as the
upper limit for grid spacing Ž d M ., as well as a minimum grid spacing, d m , initially set
A. Saldana
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146
equal to 0. It then starts with a grid mesh d1 s 1r2 d a , yielding a maximum kriging
variance s 12 . If s 12 ) s 02 then the second grid mesh d 2 equals 1r2 d1 , d m is fixed at
zero, and d M changes to d1. If, on the other hand, s 12 F s 02 , mesh d 2 is set equal to
1r2 Ž d1 q d a ., d m to d1 , and d M remains unchanged. Next, a grid mesh d 3 is
determined, yielding new values for d m and d M in a similar way as d 2 . Iteration stops
when s 02 is determined to a sufficient level of precision Že.g., 10y4 ., yielding an
optimal grid spacing d opt . Values of s 02 below the size of the nugget effect can never
be reached, even with a very small grid mesh. Inversely, values of s 02 above the sill
value are always reached, even if a single measurement point is used.
3. Results and discussion
3.1. Summary statistics and tests for significance
Table 1 shows summary statistics of sand, silt, clay, calcium carbonate, organic
carbon and pH for the three sample areas at the different sampling depths. The variables
sand, CaCO 3 and pH decrease with the age of the terrace presumably as a consequence
of weathering and leaching. On the contrary, the clay content increases both in depth
and from the lower to the higher terraces as a result of clay illuviation and weathering,
Table 1
Summary statistics of soil properties for the sample areas
Variable
Sand%
Silt%
Clay%
pH
CaCO 3%
O.C.%
Depth
d1
d2
d3
d1
d2
d3
d1
d2
d3
d1
d2
d3
d1
d2
d3
d1
d2
d3
A 1 Ž N s129.
A 2 Ž N s129.
A 3 Ž N s89.
m
s
CV
m
s
CV
m
S
CV
35
31
31
42
43
45
22
27
24
8
8.2
8.3
7
14
24
0.7
0.4
0.21
6
8
5
5
6
10
3
4
7
0.3
0.2
0.2
6
11
6
0.1
0.2
0.15
16
24
47
11
15
22
14
15
28
4
2
2
80
73
25
14
50
70
31
27
–
41
37
–
28
36
–
6.9
7.4
–
0
1
–
0.5
–
–
3
4
–
4
6
–
4
6
–
0.3
0.4
–
0
1
–
0.1
–
–
11
16
–
9
16
–
13
17
–
4
5
–
0
171
–
20
–
–
25
22
26
41
33
26
34
45
48
6.7
7.2
8.2
1
0.1
7
0.6
–
–
5
6
6
4
5
5
6
6
5
0.4
0.3
0.3
19
25
21
10
16
19
17
14
11
6
4
4
17
550
78
17
–
–
1
5
0.1
–
–
N s Number of data for each depth; ms mean; s sstandard deviation; CVs coefficient of variation.
A 1 , A 2 , A 3 are the sample areas in terraces T-29, T-25 and T-15, respectively.
d1 s10–20 cm depth; d 2 s 40–50 cm depth; d 3 s90–100 cm depth.
A. Saldana
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147
Table 2
Statistics of the x 2-test for soil properties at different depths Žsignificance at - 0.05 level in bold face.
Variable
Sand%
Silt%
Clay%
pH
CaCO 3%a
O.C.%
a
b
Depth
d1
d2
d 3b
607
11.3
38
0.01
–
0.0002
609
1850
625
0.02
485
–
142000
1750
3320
4.81
8530
–
Calculation considering A 1 and A 3 .
Calculation considering A 1 and A 3 .
whereas the silt content increases with depth in the younger terrace and decreases in the
older one. The clay distribution in the three terraces correlates with the presence of Bw
horizons in A 1 and Bt horizons in A 2 and A 3 at d 3 and sometimes d 2 depth. The
organic carbon of d1 shows little variation because of similar soil management practices.
The variation of the properties within terraces is generally small: the CV values are less
than 50% for texture, pH and organic carbon. There are large CV values Žup to 550%.
for CaCO 3 at d 2 either because of uneven decalcification or local recalcification in the
upper parts of the cambic and argillic horizons. The presencerabsence and concentration
of CaCO 3 are very variable at short-distances, even within individual pedons.
Differences between terraces are significant for most variables as shown by the tests
at 0.95 confidence level ŽTable 2.. The only nonsignificant differences are for sand and
Table 3
Distance Žin metres. to largest cumulative relative variance for soil properties at different depths
Depth
Variable
A1
A2
A3
d1
Sand%
Silt%
Clay%
pH
CaCO 3%
O.C.%
Sand%
Silt%
Clay%
pH
CaCO 3%
O.C.%
Sand%
Silt%
Clay%
pH
CaCO 3%
O.C.%
30
10
10
10
10
30
30
30
90
30
30
10
30
30
30
90
90
10
10
90
30
90
–
90
30
90
30
30
90
–
–
–
–
–
–
–
90
90
90
90
90
90
90
90
90
90
90
–
90
90
90
90
90
–
d2
d3
A. Saldana
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148
Table 4
Best fitting variogram models and ranges Žin brackets. for selected soil properties
Area
Depth
Sand%
Silt%
Clay%
pH
CaCO 3%
O. C.%
A1
d1
d2
d3
d1
d2
d1
d2
d3
Hole Ž27.
Sph Ž131.
Nugget
Nugget
Nugget
Nugget
Linear
Linear
Gauss Ž66.
Sph Ž84.
Nugget
Hole Ž26.
Linear
Linear
Linear
Linear
Sph Ž76.
Exp Ž71.
Nugget
Nugget
Nugget
Linear
Linear
Linear
Hole Ž27.
Gauss Ž55.
Nugget
Sph Ž161.
Exp Ž50.
Sph Ž100.
Linear
Power
Sph Ž65.
Hole Ž23.
Nugget
–
–
–
Power
Linear
Sph Ž93.
Nugget
Nugget
Linear
–
Linear
–
–
A2
A3
Sph: spherical; Exp: exponential; Gauss: Gaussian.
organic carbon at d1 , and for pH at d1 in all areas as it is strongly influenced by the
homogenisation effect of land management.
The distance at which the highest cumulative relative variance occurs is an inherent
feature of each soil property, but is also controlled to a certain extent by the sampling
interval. For example, the distance at which this occurs might be 15 or 20 m, but the
latter were not used as sampling distances in this study ŽTable 3.. The effect of depth is
best illustrated within terrace A 1. At d1 , four variables show the greatest variation at a
distance of 10 m and the other two at 30 m. At d 2 , four variables show the most
variation at 30 m, whereas for organic carbon and clay this occurs at 10 and 90 m,
respectively. At d 3 , the particle size components show the largest variance at 30 m, pH
and CaCO 3 at 90 m and organic carbon at 10 m. There is little change in the distance of
maximum variance within terrace A 2 , with the largest variance mainly at 30 and 90 m at
both d1 and d 2 . For terrace A 3 , the maximum variance occurs at 90-m distance for all
soil properties and at all depths. Thus, in this respect, terrace A 2 is intermediate between
terraces A 1 and A 3 . The analysis of the sampling interval indicates that the degree of
variation in the soil decreases from the lower to the higher terraces.
Fig. 2. Depth to the gravel layer in terrace A 1 , showing large irregularities.
A. Saldana
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149
3.2. Spatial Õariation
Table 4 displays the best fitting models and ranges of the variograms within the
different sample areas. In A 1 , the youngest terrace, the spatial behaviour of the selected
soil properties is rather diverse and almost all common transitive models could be fitted
at depths d1 and d 2 . At d 3 , however, all the variograms were pure nugget effect, which
reflects the absence of spatial correlation at the sampling scale arising from large
point-to-point variation at short distances. This is probably related to the irregularity of
the underlying gravel layer ŽFig. 2.. Within A 3 , the oldest terrace, the most common
Fig. 3. Variograms and interpolated maps for CaCO 3% in area A 1 : Ža. depth d1 ; Žb. depth d 2 ; Žc. depth d 3 . As
the model fitting the latter is nugget Žhence the structure of the variation is not revealed at the scale of
sampling., it is not possible to create a map.
150
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model is the linear one, indicating that the sill variance has not been reached within the
maximum sampling distance of 540 m. This suggests that spatial correlation extends
beyond the size of the current sampling scheme. The oldest terrace, therefore, has a
long-range spatial dependence, which results from advanced homogenisation of the soil
cover during the Quaternary and correlates with the observed large distance at which the
highest variance of the selected soil properties Ž90 m. occurs.
Of particular interest is the spatial structure of CaCO 3 , because of its somewhat
deviant behaviour but also because of the important role it plays for soil evolution in the
valley. Figs. 3 and 4 show the variograms and interpolated maps obtained by ordinary
kriging in sample areas A 1 and A 3 . Different variogram models provided the best fit to
the same property at different depths within the same sample area. In area A 1 , a
spherical model with a range of 65 m is obtained at d1 ŽFig. 3a., whereas periodicity
related to the structure of the river depositional system is evident at d 2 ŽFig. 3b..
Homogenisation of calcium carbonate in the surface layer is due to farming practices.
The irregular distribution of the CaCO 3 in the gravel layer generates a pure nugget
effect in d 3 of A 1 ŽFig. 3c.. Within A 3 , power model is observed at d 2 ŽFig. 4a.
whereas the linear model fits the variable at d 3 ŽFig. 4b.. The quadratic model could
indicate a structural change of CaCO 3 within A 3 , resulting from the leaching of CaCO 3
from the upper terrace.
Fig. 4. Variograms and interpolated maps for CaCO 3% in area A 3 : Ža. depth d 2 ; Žb. depth d 3 . CaCO 3 is
absent in the upper part of the soils of this sample area.
A. Saldana
˜ et al.r Catena 33 (1998) 139–153
151
Table 5
Required sampling distances Žm. to predict CaCO 3% with various precisions
Precision Ž%.
A 1 , d1
A 1, d2
A 3 , d3
5
6
8
10
48
66
)1000
)1000
-1
-1
29
82
84
419
)1000
)1000
3.3. Effects of sampling at different scales
As a final analysis, the sampling interval required to estimate properties with a
prescribed precision was investigated. Optimal grid spacings, depending on the estimated variograms, were determined to obtain an estimated map of CaCO 3 with
precisions of 5, 6, 8 and 10% ŽTable 5.. The 5% precision can only be obtained at
terrace A 1 , depth d1 , by using a 48 m grid spacing. The 6% precision can be obtained at
terrace A 1 , depth d1 with a 66 m grid spacing and at terrace A 3 , depth d 3 with a 419-m
grid spacing, but it cannot be obtained at terrace A 1 , depth d 2 , because of the large
nugget effect. In the case of the terrace A 1 , depth d 3 , the sampling density should be
increased to reveal the spatial structure and shorter range of the soil variables. The 8 and
10% precisions require a grid interval of 29 and 82 m at terrace A 1 , depth d 2 ,
respectively, and are always obtained at terrace A 1 , depth d1 and at terrace A 3 , depth
d 3 . Small differences in percentage, even smaller than the determination errors in the
laboratory, have a large influence on the sampling distances: at A 1 , depth d1 , a
difference in precision from 6 to 8% leads to a difference in sampling distance from 66
to more than 1000 m.
4. Conclusions
As a consequence of soil evolution, increasing clay translocation and calcium
carbonate leaching are evident from younger to older terraces of the Henares River. Clay
contents increase with depth. A large coefficient of variation illustrates the irregular
distribution of calcium carbonate at depth mainly coinciding with Bwk or Btk horizons.
The analysis of spatial variation using variograms shows that many standard models
could be fitted to soil properties in the area. Several types of model describe the
properties of the younger terrace ŽT-29., while the linear model fitted most variograms
for the older terrace ŽT-15.. The older terrace has the largest range of spatial dependence, resulting from the homogenisation of soil properties with increasing time. This
results in unbounded models within the range of observation. Thus, the variability of the
soil properties decreases from younger to older deposits, as soil bodies converge to
increasing homogenisation as a function of age.
Development and application of a multi-scale sampling strategy have the advantage
that, with a shorter data set Žsome observations belong to more than one level, which
means cost reduction and time saving., a compromise can be achieved between shortand long-range variation, and that various targets of spatial analysis are met.
152
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Acknowledgements
This paper is funded by the project NAT89-0996 supported by the CICyT ŽSpain..
We are grateful to the Centro de Ciencias Medioambientales ŽCSIC, Spain., the
Regional Government of Madrid ŽSpain. and the ITC ŽThe Netherlands. for their
economic support.
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