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Average variograms to guide soil sampling
International Journal of Applied Earth Observation and Geoinformation 5 (2004) 307–325 www.elsevier.com/locate/jag
Average variograms to guide soil sampling R. Kerry, M.A. Oliver* Department of Soil Science, University of Reading, PO Box 233, Whiteknights, Reading RG6 6DW, UK Received 28 March 2003; accepted 12 July 2004
Abstract To manage land in a site-specific way for agriculture requires detailed maps of the variation in the soil properties of interest. To predict accurately for mapping, the interval at which the soil is sampled should relate to the scale of spatial variation. A variogram can be used to guide sampling in two ways. A sampling interval of less than half the range of spatial dependence can be used, or the variogram can be used with the kriging equations to determine an optimal sampling interval to achieve a given tolerable error. A variogram might not be available for the site, but if the variograms of several soil properties were available on a similar parent material and or particular topographic positions an average variogram could be calculated from these. Averages of the variogram ranges and standardized average variograms from four different parent materials in southern England were used to suggest suitable sampling intervals for future surveys in similar pedological settings based on half the variogram range. The standardized average variograms were also used to determine optimal sampling intervals using the kriging equations. Similar sampling intervals were suggested by each method and the maps of predictions based on data at different grid spacings were evaluated for the different parent materials. Variograms of loss on ignition (LOI) taken from the literature for other sites in southern England with similar parent materials had ranges close to the average for a given parent material showing the possible wider application of such averages to guide sampling. # 2004 Elsevier B.V. All rights reserved. Keywords: Soil sampling; Average variograms; Parent material; Topographic units
1. Introduction To manage land in a site-specific way for agriculture and other uses requires detailed maps of the within-site variation of several soil properties. Geostatistical methods can provide reliable estimates at unsampled locations provided that the sampling * Corresponding author. Tel.: +44 1189 316557; fax: +44 1189 316666. E-mail address: [email protected] (M.A. Oliver).
interval resolves the variation at the level of interest. The scale of variation in soil properties within fields can be determined by the variogram, therefore, if one exists it can be used to guide sampling. A practical approach is to sample at an interval of just less than half of the variogram range. Alternatively an optimal interval can be determined using the kriging equations to provide predictions with a given tolerable error (McBratney et al., 1981). Kerry and Oliver (2003) showed that variograms from certain types of sensed data, particularly aerial
0303-2434/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jag.2004.07.005
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photographs of the bare soil, could be used to suggest an appropriate sampling interval for soil survey. McBratney and Pringle (1997, 1999) suggested an alternative where there are no variograms; examples of variograms of a property from the literature could be
averaged and the resulting average variogram used to guide sampling. They indicated that such global average variograms would be appropriate only if the conditions in the soil at a particular site reflected the global average conditions. It seems unlikely that the
Fig. 1. Sampling scheme and landscape units for each site: (a) Shuttleworth clay (Oxford Clay), (b) Shuttleworth sand (Lower Greensand), (c) Wallingford (Gravel) and (d) Yattendon (Chalk).
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Chalk, were sampled on 30 m grids. The Yattendon site comprised three neighbouring fields (Fields 214, 215 and 217). The sites on these different parent materials were divided further into landscape units (Fig. 1a–d). There were 80–100 sampling points for each landscape unit at all sites. At each node of the sampling grid six cores of soil (0–15 cm depth) were taken with an auger within a 1 m2 area, and bulked. Loss on ignition (LOI) was determined on the air-dry <2 mm fraction at 500 8C (Avery and Bascomb, 1982). Although LOI is the focus of this paper, several other standard field and laboratory soil properties are also considered briefly.
global average would be suitable for many sites, but McBratney and Pringle (1999) suggested that with the development of a world database of variograms for different types of soil, average variograms could be based on a narrower range of soil forming conditions. This paper assesses whether average variograms provide a sound basis to guide sampling. The effects of parent material and topography have been taken into account in this evaluation from the results of four intensive soil surveys on arable fields in southern England.
2. Methods 2.2. Statistical and geostatistical analysis 2.1. Site description and soil analysis The summary statistics of LOI for each landscape unit on each parent material are given in Table 1. The statistical distribution of LOI was close to normal apart from that for all topography at SS (SSall), and the south slope (Yssl) and Field 215 (Y215) at Yattendon. The LOI data for SS were not transformed because the variograms of the raw and transformed data were similar. For Yssl and Y215 one and four outliers, respectively, were removed which reduced the
The soil was sampled and observed on 20 m grids at two sites on the Shuttleworth Estate, Bedfordshire, England: Shuttleworth clay (SC) which is on the Oxford Clay and Shuttleworth sand (SS) on the Lower Greensand. The SS site comprised two fields: Football (slope) and Cricket Meadow (valley). The sites at Wallingford (Oxfordshire), on flinty and calcareous gravel, and the Yattendon Estate (Berkshire), on the
Table 1 Summary statistics for loss on ignition (LOI, %) for different landscape units Landscape unit Shuttleworth clay Slope Valley All topography Shuttleworth sand Slope Valley All topography Wallingford Plateau Slope Valley All topography Yattendon North slope Plateau South slope Valley All topography Field 214 Field 215
Number of values
Mean
Minimum
Maximum
Range
Variance
Coefficient of variation (%)
Skewness
Linear trend (%)
Quadratic trend (%)
105 100 205
5.118 5.868 5.484
3.11 4.73 3.11
6.83 6.83 6.83
3.72 2.10 3.72
0.735 0.202 0.614
16.75 7.66 14.29
0.39 0.23 0.90
42.9 21.3 35.8
46.2 30.4 54.1
159 109 268
3.580 4.100 3.794
1.80 2.52 1.80
6.22 6.87 6.87
4.42 4.35 5.07
0.300 1.040 0.667
15.39 24.87 21.52
0.49 0.63 1.11
62.1 34.8
62.2 41.7
91 120 108 296
3.947 4.609 4.497 4.352
2.76 3.06 3.24 2.76
5.51 6.26 6.16 6.26
2.75 3.20 2.92 3.50
0.289 0.574 0.286 0.456
13.61 16.44 11.89 15.52
0.22 0.08 0.21 0.30
0.0
26.6
55 96 86 87 307 100 120
11.12 8.705 5.808 6.258 7.620 11.11 5.700
6.24 3.75 3.49 2.25 2.25 5.73 3.49
14.27 14.24 13.97 10.45 14.27 14.27 13.97
8.03 10.49 10.48 8.20 12.02 8.54 10.48
2.019 9.597 1.552 0.989 7.588 2.595 1.347
12.78 35.59 21.45 15.90 36.15 14.50 20.36
0.47 0.04 3.11 0.01 0.77 0.52 2.87
65.0
66.2
35.8
59.9
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skewness (Table 2). The histograms for the plateau (Ypla) and the whole site (Yall) at Yattendon had bimodal distributions, therefore the data were stratified. The pooled within-class variogram was computed for these sites from the residuals of the class means. Omnidirectional experimental variograms were computed for each landscape unit and for each entire site (all topography) using the usual computing equation and appropriate data (see Webster and Oliver, 2001). There were not enough data for each landscape unit to explore any anisotropy in the variation reliably. Therefore, it was investigated with the more intensive digital data available from aerial photographs. The results from this suggested that the anisotropy is zonal and is associated with the landscape units, therefore it was not investigated further. Several experimental variograms suggested that regional trend might be present in the variation; the semivariances increased rapidly after reaching a sill or they had upwardly concave shapes. We examined these data further by fitting first- and second-order polynomials on the coordinates. If the percentage variance accounted for by these functions was greater than 20%, we computed the variogram
afresh on the residuals because geostatistics assumes that the underlying process is random. If the variogram of the residuals was different from that of the raw data we used the former for further analyses. The experimental variograms were modelled for each landscape unit using weighted least squares approximation (GenStat, Payne, 2000). An average of the variogram ranges of all properties for each landscape unit for a given parent material was calculated. In addition, average variograms were determined in two ways. Simple average variograms were calculated from the modelled rather than the experimental variances of variograms for each landscape unit (see Fig. 2). This was done because if variograms from the literature were to be used the information would be derived from the model parameters rather than the experimental semivariances. Fig. 2 shows that the simple average variograms calculated from the experimental (a) and the modelled variances (b) are similar. Simple average variograms can be calculated from variograms of the raw data only, which excludes those computed from transformed data or residuals from a trend surface. This means that for some sites or variables there might
Table 2 Variogram model parameters of LOI (%) for different landscape units Landscape unit Shuttleworth clay Slope Valley All topography Shuttleworth sand Slope Valley All topography Wallingford Plateau Slope Valley All topography Yattendon North slope Plateau South slope Valley All topography Field 214 Field 215
Maximum lag (m)
Data used
Model type
c0
c
a(3r) (m)
120 140 200
Lresid. Lresid. Qresid.
Circular Circular Circular
0.0232 0.0628 0.0575
0.3890 0.1170 0.2230
66.46 82.30 71.07
200 120 200
Raw Lresid. Qresid.
Pentaspherical Circular Pentaspherical
0.0965 0.1500 0.0970
0.1870 0.2430 0.2710
124.6 72.80 120.2
180 200 270 390
Qresid. Raw Raw Raw
Circular Spherical Circular Pentaspherical
0.0702 0.1127 0.1199 0.0535
0.1420 0.4260 0.1860 0.4610
66.13 135.5 244.5 226.1
120 300 160 175 180 160 120
Raw Within-class Outliers removed Raw Within-class Raw Outliers removed
Spherical Circular Circular Circular Circular Spherical Spherical
0.5329 0.6357 0.0601 0.3292 0.9286 0.8171 0.2659
1.475 1.972 0.6940 0.6850 0.7070 1.6940 0.5069
51.80 105.0 65.20 66.50 64.17 80.10 77.57
c0 is the nugget variance, c the correlated component, a the range of spatial dependence and 3r the effective range for the exponential model; Lresid. the residuals from a linear trend function; Qresid. the residuals from a quadratic trend function.
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Fig. 2. Average variograms calculated from variograms computed for Yattendon based on: (a) experimental semivariances and (b) modelled semivariances.
be no simple average variogram. As an alternative to include all types of data (raw, transformed and residuals), variogram models for the individual landscape units were standardized by setting the sill variance to 1 and by calculating the nugget:sill ratio. These standardized variograms of LOI were then averaged to provide standardized average variograms. A suitable sampling intensity to map soil properties can be determined from just less than half the range of spatial dependence described by the variogram model.
This was based on the average range of several conventional variograms and the range of standardized average variograms. The simple average variograms were not used to guide sampling because they were less reliable than those above when based on few variograms. The model parameters of variograms of the raw data based on all topography and of the standardized average variograms were also used with the kriging equations to determine an optimal sampling interval for different parent materials
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Fig. 3. Experimental variograms and fitted models of LOI for the landscape units at each site: (a–c) Shuttleworth clay, (d–f) Shuttleworth sand, (g–j) Wallingford, and (k–o) Yattendon. The symbols are the experimental semivariances and the line is the fitted model.
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(McBratney et al., 1981). A tolerable kriging error of 25% of the sill variance, as suggested by Lark (2000), was used to determine an optimal sampling interval from graphs of block kriging variance plotted against grid spacing for estimates over blocks of different size. To assess the suitability of the sampling intervals suggested by the methods described above, the original LOI measurements were sub-sampled to provide a range of sampling intervals. The subsampled LOI data were then used with the variogram model parameters from the original data to predict LOI at unsampled locations by block kriging. The predictions from the different sampling intensities were then mapped so that the variation could be compared.
3. Results and discussion We describe in detail here only the results for loss on ignition (LOI), but the same approach was used for all soil properties in the survey and the results were similar. The results are summarized in Table 4. 3.1. Summary statistics Summary statistics for LOI are given for each site and landscape unit, Table 1. The mean values show that Yattendon had the largest LOI, followed by SC, and SS had the smallest values. This order generally reflects the clay and calcium carbonate content of the soil at the different sites. Clay content increases LOI values because of the loss of water from the interlayer spaces of clay minerals during ignition. Clay soil also tends to have a larger organic matter content because less is oxidized and translocated because of the chemical bonds between the clay particles and organic matter. The statistical range of LOI values and coefficients of variation (CVs) were the largest for the plateau and all topography at Yattendon (Ypla and Yall), which was due to marked differences in the calcium carbonate content of the soil on the plateau areas of Fields 214 and 215. Free calcium carbonate in the soil (dissolved and precipitated) protects organic matter from rapid microbial action. The differences between the means and variances of LOI for the four parent materials were shown to be significant at the 0.05 level by a Kruskal–Wallis H-test (Ebdon, 1994).
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Therefore, the difference in mean LOI values for different parent materials could be used to indicate whether the soil forming conditions at a particular site are similar to those at sites where standardized average variograms are available. Although differences between the means and variances for the landscape units on a particular parent material cannot be assessed statistically in the absence of replicates, some observations can be made. At SC and SS the mean LOI values are larger for the valleys than slopes. The valleys are in receiving positions and drainage is more likely to be impeded at times, which would reduce the breakdown of organic matter. At Wallingford, the mean LOI value for the valley is larger than that for the plateau, but less than that for the slope where there is some calcareous gravel. The relations between LOI and landscape were complicated at Yattendon because the calcium carbonate content of the soil varied from one landscape unit to another. The significant differences between the means and variances of LOI on different parent materials shows the importance of parent material as a soil forming factor. The differences in mean LOI also show the importance of topography in soil formation. These results suggest that some generalization about the characteristics of soil developed on similar parent materials and topographic positions might be possible. These effects should also be evident in the model parameters of variograms for different parent materials and topographic positions. Here, the range only will be considered as it can be used to guide future sampling intensity. 3.2. Variogram analysis The variograms of LOI for different landscape units are shown in Fig. 3 and their model parameters are given in Table 2. Table 1 gives the percentage variance that linear and quadratic functions accounted for in those landscape units where residuals from the trend were used to calculate variograms. For sites such as SC and SS where the variograms were computed on the residuals the sampling interval suggested by the range would be relevant only to the non-deterministic component of the variation. In such cases, it might be possible to reduce the number of sampling points required further, especially where the percentage
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Table 3 The average range of the models fitted to the variograms of LOI, and the range of the standardized average variogram Site
Average range of fitted models (m)
Range of standardized average variogram
Shuttleworth clay Shuttleworth sand Wallingford (Gravel) Yattendon (Chalk) All parent materials All plateaus All slopes All valleys
73.3 105.9 168.1 72.9 104.8 85.6 88.7 116.5
114.1 72.2 189.7 93.6 82.7 116.8 68.6 82.4
a1 (m)
a2 (m)
203.9 128.6 269.8
a1 is the short-range component and, a2 is the long-range component of the nested model.
variance accounted for by trend is large. To assess this would require further detailed research, however. The autocorrelation range of a variogram essentially describes the extent of patches of similar soil; therefore, one would expect less variable areas to have variograms with a larger range. The variogram ranges of the landscape units were similar for SC, but that for the valley was the longest. It also had the smallest CV of 7.66% (Table 1). For SS the slope had the longest variogram range and the smallest CVof 15.39% (Table 1). The ranges of variograms for Wallingford varied from 66 to 244 m although most were over 100 m, and that for the valley was the longest. The latter area also had the smallest CV of 11.89% (Table 1). For Yattendon the variogram ranges were between about 50 and 100 m; that for the plateau was longest. Unlike the other sites, the latter had a large CV of 36% (Table 1). This was probably due to differences in the calcium carbonate and organic matter contents of Fields 214
and 215. These differences were taken into account by computing the pooled within-class variogram for the plateau. The variances within these classes were small, showing that the longer variogram range was again associated with the less variable soil within the strata on the plateau, Table 2. A Kruskal–Wallis H-test showed that the differences in the variogram ranges of LOI with parent material were significant at the 0.1 level. The variogram ranges of LOI for each landscape unit were averaged for each parent material; they are given in Table 3. The smallest average range was for Yattendon (Chalk) and the longest was for Wallingford (gravel). These differences between parent materials are more evident in the average variogram ranges for several properties, Table 4. There is also a marked difference in the average range for the selected soil properties for each parent material (Table 4, last row). This difference has the order of gravel (150 m) >
Table 4 Average variogram range for all properties Property
Chalk (Y) average
Gravel (W) average
Oxford Clay (SC) average
Lower Greensand (SS) average
Overall average
Depth LOI Stoniness Munsell value VWC Clay Sand pH Average of above soil properties
73.01 72.91 85.58 102.7 77.34 103.4 88.19 – 86.06
123.1 168.1 204.9 160.5 141.2 136.0 118.6 145.8 149.8
– 73.28 133.4 170.6 108.4 114.8 94.99 128.5 117.7
– 105.9 90.34 74.42 87.23 98.62 140.4 82.26 97.01
98.03 104.9 128.6 127.1 103.6 113.2 110.5 118.9 –
LOI is the loss on ignition; VWC the volumetric water content.
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Fig. 4. Standardized average variograms of LOI: (a) Shuttleworth clay, (b) Shuttleworth sand, (c) Wallingford, (d) Yattendon, (e) all parent materials, (f) all plateaus, (g) all slopes, and (h) all valleys.
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Oxford Clay (118 m) > Lower Green Sand (97 m) > Chalk (86 m). A Kruskal–Wallis H-test showed a significant difference between the average variogram range of the selected soil properties in Table 4 according to parent material at the 0.05 level. This suggests that it should be feasible to determine a suitable sampling interval for future soil surveys in similar pedological settings from the average variogram range. In contrast to the average range for each parent material mentioned above there is little variation in the overall average range for different soil properties (Table 4, final column). Table 3 (last four rows) also shows that for LOI there is little difference in average variogram range when all parent materials are considered, and when all plateau, slope and valley areas are grouped regardless of parent material. Therefore, a global average variogram for a given soil property, as described by McBratney and Pringle (1999), would not suggest appropriate sampling intervals for all possible parent materials; it would lead to over-sampling on some and under-sampling on others. 3.3. Standardized average variograms The standardized average variograms of LOI for all landscape units at each site are shown in Fig. 4, and their autocorrelation ranges are given in Table 3. That for Wallingford had the longest range, as did the average range for the fitted models (Table 3). These variograms for each parent material were fitted with single functions. The standardized average variograms for all parent materials, and the landscape units on all parent materials except for the plateaus, were fitted by
nested functions with short- and long-range components. For these latter functions the short-range component accounted for most of the variation, therefore, the range of this structure was used as a guide to sampling interval. Where variation is nested and both structures account for similar proportions of the variance it is recommended that an average of the two ranges be used to determine a suitable sampling interval. One might wonder if additional samples at an interval smaller than half the range should be obtained where the nugget:sill ratio is large because of the unresolved variation at distances less than the sampling interval. It was not possible to assess this here because the nugget:sill ratios of the standardized average variograms for each parent material were similar (20–40%) because of the averaging (Fig. 4). However, Webster and Oliver (1992) showed that the confidence intervals on a variogram with a nugget component were larger than those with no nugget. They showed that with more intensive sampling this difference in the confidence intervals decreased. Based on this we recommend that where standardized average variograms have a large nugget:sill ratio the sampling at half the variogram range should be supplemented by some additional samples at a shorter interval. In fact, we consider that this is generally advisable to improve the reliability of the variogram at short lag distances. A Kruskal–Wallis H-test showed a significant difference between the ranges of the standardized average variograms for all soil properties (not shown here) on the different parent materials at the 0.05 level. These results suggest that standardized average variograms could be used to guide sampling as described above.
Table 5 Sampling intervals for LOI suggested by less than half the variogram range Site
Shuttleworth clay Shuttleworth sand Wallingford (Gravel) Yattendon (Chalk) All parent materials All plateaus All slopes All valleys
Sampling interval (m) Half average range of fitted models
Half range of standardized average variogram
35 50 80 35 50 40 40 55
55 35 90 45 40 55 30 40
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Fig. 5. Graphs of kriging error plotted against grid spacing for LOI based on the conventional variogram for all topography at: (a) Shuttleworth clay, (c) Shuttleworth sand, (e) Wallingford, (g) Yattendon, and on the standardized average variogram for: (b) Shuttleworth clay, (d) Shuttleworth sand, (f) Wallingford, and (h) Yattendon.
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3.4. Sampling intervals Based on the notion that a sampling interval of just less than half the variogram range should be adequate to resolve the main features of the variation in the soil at the level of the investigation, sampling intervals were determined based on the average variogram range and on the range of standardized average variograms for different parent materials, Table 5. In addition, the kriging errors were computed for a number of grid spacings and block sizes with the conventional variograms for each site based on the data from all topographic units, and with the standardized average variograms. Graphs of kriging error plotted against grid spacing for LOI at each site are shown in Fig. 5. These graphs were used with a tolerable error of 25% of the sill variance to determine optimal sampling intervals for different block sizes, Table 6. Table 5 shows that sampling intervals of 35 m (average range of several variograms) and 55 m (range of standardized average) were suggested for LOI at SC. However, based on the average range of the selected soil properties the interval would be about 55 m, Table 4. The optimal sampling intervals for SC for a block support of 25 m were 45 m (conventional variogram based on all topography) and 50 m (standardized average variogram), Table 6. For SS a sampling interval for LOI of between 35 and 50 m was suggested by the methods given in Table 6 Optimal sampling intervals suggested for LOI using the variogram and the kriging equations with a tolerable error of 25% of the sill variance Variogram
Sampling interval suggested by threshold of 25% of sill height 25 m block
50 m block
75 m block
Shuttleworth clayall Shuttleworth claystdav Shuttleworth sandall Shuttleworth sandstdav Wallingfordall Wallingfordstdav Yattendonall Yattendonstdav
50 45 50 50 70 50 50 40
70 60 65 70 90 75 – 60
– 80 85 95 105 95 – 100
Shuttleworth clayall for example is the sampling interval based on the variogram for all topography and Shuttleworth claystdav the sampling interval based on the standardized average variogram.
Table 5, and an optimal interval of 50 m for blocks of 25 m, Table 6. The sampling interval would be about 45 m (Table 4) at this site if the other soil properties were taken into account. Sampling intervals of 80– 90 m were indicated for LOI at Wallingford in Table 5, whereas the optimal sampling intervals for 25 m blocks were 50–70 m, Table 6. Half the average range of the variograms for other soil properties suggested that an interval of about 70 m would be appropriate at Wallingford (Table 4). The methods in Table 5 suggested sampling intervals of between 35 and 45 m for LOI at Yattendon, and intervals of 40 and 45 m were suggested for 25 m blocks using the optimal approach, Table 6. Finally, half the average range of the variograms for the other soil properties (Table 4) suggested that an interval of 40 m would be appropriate at Yattendon. Tables 5 and 6 show that there is some consistency in the sampling intervals suggested for LOI using half the average range of several variograms, half the range of standardized average variograms, and the optimal approach. Table 4 also shows that the intervals suggested by the average variogram range of several soil properties on a given parent material are of a similar order of magnitude to those for LOI. Table 5 shows that the intervals suggested for LOI by the average variogram range and standardized average variograms for all parent materials or all slopes from different parent materials, for example, are between 34 and 58 m. This confirms that average variograms that are not parent material specific would not provide a suitable guide to sampling on all parent materials. 3.5. Kriging with the sub-sampled data Fig. 6a shows the maps of ordinary kriged predictions for SC with the original data on a 20 m grid, and Fig. 6b–d those with sub-sampled data on 40, 60 and 80 m grids, respectively. The main features of the variation in LOI are retained in the maps based on the 40 and 60 m grids (Fig. 6b and c, respectively), but there is a loss of detail in that for the 80 m grid (Fig. 6d). These results suggest that for the 60 m grid interval additional sampling at a 30 m interval at randomly selected grid nodes would avoid some loss of detail, which supports the intervals indicated in Tables 5 and 6. These maps also show the effect on the predictions of the configuration of sampling sites in
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Fig. 6. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth clay.
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Fig. 7. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth sand (Cricket Meadow).
the sub-samples, especially when the sampling interval is larger. Frogbrook and Oliver (2000) also observed the effect that sample position for different grid spacings had on the patterns of variation described in their fields. The kriged maps of LOI for SS (Fig. 7) suggest that a 60 m grid would be adequate for sampling Cricket Meadow (Fig. 7c) because of the large amount of trend in LOI in this field. Trend surface analysis can predict the deterministic component of variation, and Frogbrook and Oliver (2001) showed that as few as nine carefully chosen points (i.e. placed in areas of large, medium and small LOI values) can be used to describe this component of the variation accurately. It is notable that the 60 m grid for this field contained only 12 points. For Football field (Fig. 8), however, a 40 m grid interval (Fig. 8b) would be needed to resolve the variation adequately. These findings are also consistent with the intervals suggested for these sites in Tables 5 and 6. The kriged maps of LOI for Wallingford, Fig. 9, suggest that sampling intervals of 80–90 m would
resolve the variation adequately. The main features of the variation based on kriging the 30 m data (Fig. 9a) are reproduced well by kriging with the data on the 60, 90 and 120 m grids (Fig. 9b–d, respectively), but there is some loss of detail for the 150 m grid (Fig. 9e). The kriged maps of LOI for Yattendon based on the residuals from the class means (Fig. 10) show that a sampling interval of 35–40 m (Table 5), as indicated by the variograms, would be required. The variation in LOI based on a 60 m grid, Fig. 10b, is poorly resolved although some of the main features of the variation are still evident. The sampling intervals of 40–50 m suggested by the variograms for all parent materials (Table 5, row five) would be adequate to guide sampling at most sites, except at Wallingford, where they would lead to over-sampling. The sampling intervals suggested by variograms for all plateaus, slopes and valleys (Table 5, last three rows) would not be appropriate for individual landscape units (see ranges for landscape units in Table 2). This shows that if topography is to be included as a soil forming factor it should be done in
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Fig. 8. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth sand (Football field).
relation to parent material. For example, the LOI values at Yattendon and Wallingford were not always the largest in valley positions because of the variability in the parent material; the presence of calcium carbonate changed the usual pattern of variation associated with the landscape positions. 3.6. Validation sites To determine whether the variogram ranges from the four field sites in this study could be used to guide sampling at other arable sites in southern England with similar parent materials, we examined the model
parameters of LOI for six sites from other studies. The sites at CEDAR, Berkshire (Frogbrook, 2000) and Warren, Bedfordshire (Baxter, 2003) have soil developed on clay parent materials; the former is on London Clay and the latter on Oxford Clay. The sites of Cashmore (Baxter, 2003) and Northpark (Frogbrook, 2000), Bedfordshire, have soil developed on the Lower Greensand. Finally, the Clays, Oxfordshire (Oliver and Carroll, 2004) and Stubbs (Oliver et al., 2003), Berkshire, have soil developed on Chalk. First, we compared the mean values of LOI (Table 7) at the validation sites with those for all topography in the present study (Table 1). A Kruskal–Wallis H-test
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Fig. 9. Maps of kriged predictions of LOI from data on sampling grids of: (a) 30 m, (b) 60 m, (c) 90 m, (d) 120 m and (e) 150 m for Wallingford.
based on the means of LOI from both sets of sites showed a significant difference in relation to the three parent materials at the 0.05 level. The average ranges of the variograms for the validation sites have a similar order of magnitude (clay, 123.9 m > Chalk, 103 m > sand, 84.9 m) to those for the study sites (Table 4).
Frogbrook’s (2000) research also showed the same order and similar average ranges (clay, 140 m > Chalk, 100 m > sand, 68 m) for a group of soil properties. The average variogram ranges for LOI at the validation sites suggest sampling intervals of 50 or 60 m for sites with clay parent material, 40 m for
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Fig. 10. Maps of kriged predictions of LOI from data on sampling grids of: (a) 30 m, (b) 60 m, (c) 90 m and (d) 120 m for Yattendon.
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Table 7 Variogram model parameters for LOI at validation sites Site
Parent material
Mean (%)
Variogram
Model type
c0
c
a(3r) (m)
CEDAR97 CEDAR98 Warren Cashmore Northpark97 Northpark98 Northpark99 Clays Stubbs
Oxford Clay Oxford Clay Clay Lower Green Lower Green Lower Green Lower Green Chalk Chalk
4.49 4.39 5.71 3.78 4.00 3.93 3.85 4.90 3.96
Raw Raw Lresid. Lresid. Raw97 Raw98 Raw99 Lresid. Within-class
Pentaspherical Pentaspherical Spherical Exponential Pentaspherical Pentaspherical Pentaspherical Exponential Circular
0.0899 0.0717 0.6816 0.0284 0 0.0254 0.0507 0.0723 0.0467
0.0781 0.0818 0.7125 0.2871 0.4618 0.4229 0.3879 0.0954 0.1217
139.9 134.8 96.89 49.26 95.49 99.58 95.33 113.0 93.60
Sand Sand Sand Sand
Validation sites: Cashmore and Warren (Baxter, 2003), CEDAR and Northpark (Frogbrook, 2000), and the Clays (Oliver and Carroll, 2004) and Stubbs (Oliver et al., 2003).
sandy ones and 50 m for Chalk sites. These intervals would also be adequate to describe the variation in LOI for SC, SS and Yattendon, respectively (Figs. 6–8 and 10). The reverse is also true; the sampling intervals indicated by the average variogram range and the range of standardized average variograms for SC, SS and Yattendon would be appropriate for survey at the validation sites on the same type of parent material.
4. Conclusions There are significant differences in the means and variances, and variograms of LOI for soil developed on different parent materials. This suggests that it might be possible to make generalizations about the variation of LOI in relation to parent material to guide future sampling for mapping where there are no variograms. The optimal sampling intervals based on a tolerable error of 25% of the sill variance were similar to those based on less than half the variogram range. The latter method is recommended because it is more straightforward and the results are similar. The patterns of variation in LOI on the kriged maps based on subsampled data showed that the range of standardized average variograms can be used to determine suitable sampling intervals for future soil surveys based on less than half the range or on the kriging variances and a tolerable error of 25% of the sill variance. A comparison of the sampling intervals suggested by this study with those indicated by the data for other sites from the literature with comparable topography, parent materials and climate were similar for a
particular parent material. Use of the average variogram range or standardized average variograms from similar sites is suggested as a guide to sampling at a site only when there are no existing variograms of soil properties or appropriate ancillary data. They avoid the need for a reconnaissance soil survey when nothing is known about the scale of variation. We also recommend that the sampling intervals suggested by half the variogram range are supplemented by some additional samples at shorter intervals to resolve the spatial structure at short lag distances more reliably. This study has shown that the average variogram range and standardized average variograms for a given parent material could be used to guide sampling on the four parent materials examined in southern England. The approach is particularly useful where several variograms are available to calculate the average and where the range of soil forming conditions at sites is narrow. It could be applied more widely if there were databases of variogram parameters for different soil forming conditions. The mean value of a soil property could be used as a guide to the degree of similarity between sites. Suitable sampling intervals were suggested by variograms from the study sites at other sites in southern England where the parent material was similar. The results from this study could be applied to sites elsewhere with similar soil forming conditions, landscape features, climate and land use. If a reliable standardized average variogram exists it could be used to predict properties for mapping from a much smaller sample than that needed to compute a variogram. This should reduce the cost of producing accurate soil maps for land management even further. Nevertheless, it must be remembered the data still
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need to be spatially dependent for kriging or any other kind of interpolation. We will examine the viability of using standardized average variograms for kriging in another study.
Acknowledgements We thank the University of Reading and the Fertiliser Manufacturers’ Association for supporting this research.
References Avery, B.W., Bascomb, C.L. (Eds.), 1982. Soil Survey Laboratory Methods. Soil Survey Technical Monograph, No. 6. Harpenden, England. Baxter, S.J., 2003. Spatial variation of plant available nitrogen within arable fields. Unpublished Ph.D. Thesis. The University of Reading, Reading, England. Ebdon, D., 1994. Statistics for Geographers, Blackwell, Oxford. Frogbrook, Z.L., 2000. Geostatistics as an aid to soil management for precision agriculture. Unpublished Ph.D. Thesis. The University of Reading, Reading, England. Frogbrook, Z.L., Oliver, M.A., 2000. The effects of sampling on the accuracy of predictions of soils properties for precision agriculture. In: Heuvelink, G.B.M., Lemmens, M.J.P.M. (Eds.), Accuracy 2000. Delft University Press, Delft, Netherlands, pp. 225–232.
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Frogbrook, Z.L., Oliver, M.A., 2001. Comparing the spatial predictions of soil organic matter determined by two laboratory methods. Soil Use Manage. 1, 235–244. Kerry, R., Oliver, M.A., 2003. Variograms of ancillary data to aid sampling for soil surveys. Precis. Agric. 4, 253–270. Lark, R.M., 2000. Designing sampling grids from imprecise information on soil variability, an approach on based on the fuzzy kriging variance. Geoderma 98, 35–59. McBratney, A.B., Pringle, M.J., 1997. Spatial variability in soilimplications for precision agriculture. In: Stafford, J.V. (Ed.), Precision Agriculture’97, vol. 1. BIOS Scientific Publishers, Oxford, pp. 3–31. McBratney, A.B., Pringle, M.J., 1999. Estimating average and proportional variograms of soil properties and their potential use in precision agriculture. Precis. Agric. 1, 125–152. McBratney, A.B., Webster, R., Burgess, T.M., 1981. The design of optimal sampling schemes for local estimation and mapping of regionalized variables. I. Theory and method. Comput. Geosci. 7, 331–334. Oliver, M.A., Carroll, Z.L., 2004. Description of spatial variation in soil to optimize cereal management. Project Report No. 330. HGCA, London. Oliver, M.A., Heming, S.D., Gibson, G., Adams, N., 2003. Exploring the spatial variation of take-all (Gaeumannomyces graminis var. tritici) for site-specific management. In: Stafford, J.V., Werner, A. (Eds.), Precision Agriculture. Wageningen Academic Publishers, Wageningen, pp. 481–486. Payne, R.W., 2000. The Guide to GenStat. Part 2. Statistics. VSN International, Oxford. Webster, R., Oliver, M.A., 1992. Sample adequately to estimate variograms of soil properties. J. Soil Sci. 43, 177–192. Webster, R., Oliver, M.A., 2001. Geostatistics for Environmental Scientists, Wiley, Chichester.
Average variograms to guide soil sampling R. Kerry, M.A. Oliver* Department of Soil Science, University of Reading, PO Box 233, Whiteknights, Reading RG6 6DW, UK Received 28 March 2003; accepted 12 July 2004
Abstract To manage land in a site-specific way for agriculture requires detailed maps of the variation in the soil properties of interest. To predict accurately for mapping, the interval at which the soil is sampled should relate to the scale of spatial variation. A variogram can be used to guide sampling in two ways. A sampling interval of less than half the range of spatial dependence can be used, or the variogram can be used with the kriging equations to determine an optimal sampling interval to achieve a given tolerable error. A variogram might not be available for the site, but if the variograms of several soil properties were available on a similar parent material and or particular topographic positions an average variogram could be calculated from these. Averages of the variogram ranges and standardized average variograms from four different parent materials in southern England were used to suggest suitable sampling intervals for future surveys in similar pedological settings based on half the variogram range. The standardized average variograms were also used to determine optimal sampling intervals using the kriging equations. Similar sampling intervals were suggested by each method and the maps of predictions based on data at different grid spacings were evaluated for the different parent materials. Variograms of loss on ignition (LOI) taken from the literature for other sites in southern England with similar parent materials had ranges close to the average for a given parent material showing the possible wider application of such averages to guide sampling. # 2004 Elsevier B.V. All rights reserved. Keywords: Soil sampling; Average variograms; Parent material; Topographic units
1. Introduction To manage land in a site-specific way for agriculture and other uses requires detailed maps of the within-site variation of several soil properties. Geostatistical methods can provide reliable estimates at unsampled locations provided that the sampling * Corresponding author. Tel.: +44 1189 316557; fax: +44 1189 316666. E-mail address: [email protected] (M.A. Oliver).
interval resolves the variation at the level of interest. The scale of variation in soil properties within fields can be determined by the variogram, therefore, if one exists it can be used to guide sampling. A practical approach is to sample at an interval of just less than half of the variogram range. Alternatively an optimal interval can be determined using the kriging equations to provide predictions with a given tolerable error (McBratney et al., 1981). Kerry and Oliver (2003) showed that variograms from certain types of sensed data, particularly aerial
0303-2434/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jag.2004.07.005
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photographs of the bare soil, could be used to suggest an appropriate sampling interval for soil survey. McBratney and Pringle (1997, 1999) suggested an alternative where there are no variograms; examples of variograms of a property from the literature could be
averaged and the resulting average variogram used to guide sampling. They indicated that such global average variograms would be appropriate only if the conditions in the soil at a particular site reflected the global average conditions. It seems unlikely that the
Fig. 1. Sampling scheme and landscape units for each site: (a) Shuttleworth clay (Oxford Clay), (b) Shuttleworth sand (Lower Greensand), (c) Wallingford (Gravel) and (d) Yattendon (Chalk).
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Chalk, were sampled on 30 m grids. The Yattendon site comprised three neighbouring fields (Fields 214, 215 and 217). The sites on these different parent materials were divided further into landscape units (Fig. 1a–d). There were 80–100 sampling points for each landscape unit at all sites. At each node of the sampling grid six cores of soil (0–15 cm depth) were taken with an auger within a 1 m2 area, and bulked. Loss on ignition (LOI) was determined on the air-dry <2 mm fraction at 500 8C (Avery and Bascomb, 1982). Although LOI is the focus of this paper, several other standard field and laboratory soil properties are also considered briefly.
global average would be suitable for many sites, but McBratney and Pringle (1999) suggested that with the development of a world database of variograms for different types of soil, average variograms could be based on a narrower range of soil forming conditions. This paper assesses whether average variograms provide a sound basis to guide sampling. The effects of parent material and topography have been taken into account in this evaluation from the results of four intensive soil surveys on arable fields in southern England.
2. Methods 2.2. Statistical and geostatistical analysis 2.1. Site description and soil analysis The summary statistics of LOI for each landscape unit on each parent material are given in Table 1. The statistical distribution of LOI was close to normal apart from that for all topography at SS (SSall), and the south slope (Yssl) and Field 215 (Y215) at Yattendon. The LOI data for SS were not transformed because the variograms of the raw and transformed data were similar. For Yssl and Y215 one and four outliers, respectively, were removed which reduced the
The soil was sampled and observed on 20 m grids at two sites on the Shuttleworth Estate, Bedfordshire, England: Shuttleworth clay (SC) which is on the Oxford Clay and Shuttleworth sand (SS) on the Lower Greensand. The SS site comprised two fields: Football (slope) and Cricket Meadow (valley). The sites at Wallingford (Oxfordshire), on flinty and calcareous gravel, and the Yattendon Estate (Berkshire), on the
Table 1 Summary statistics for loss on ignition (LOI, %) for different landscape units Landscape unit Shuttleworth clay Slope Valley All topography Shuttleworth sand Slope Valley All topography Wallingford Plateau Slope Valley All topography Yattendon North slope Plateau South slope Valley All topography Field 214 Field 215
Number of values
Mean
Minimum
Maximum
Range
Variance
Coefficient of variation (%)
Skewness
Linear trend (%)
Quadratic trend (%)
105 100 205
5.118 5.868 5.484
3.11 4.73 3.11
6.83 6.83 6.83
3.72 2.10 3.72
0.735 0.202 0.614
16.75 7.66 14.29
0.39 0.23 0.90
42.9 21.3 35.8
46.2 30.4 54.1
159 109 268
3.580 4.100 3.794
1.80 2.52 1.80
6.22 6.87 6.87
4.42 4.35 5.07
0.300 1.040 0.667
15.39 24.87 21.52
0.49 0.63 1.11
62.1 34.8
62.2 41.7
91 120 108 296
3.947 4.609 4.497 4.352
2.76 3.06 3.24 2.76
5.51 6.26 6.16 6.26
2.75 3.20 2.92 3.50
0.289 0.574 0.286 0.456
13.61 16.44 11.89 15.52
0.22 0.08 0.21 0.30
0.0
26.6
55 96 86 87 307 100 120
11.12 8.705 5.808 6.258 7.620 11.11 5.700
6.24 3.75 3.49 2.25 2.25 5.73 3.49
14.27 14.24 13.97 10.45 14.27 14.27 13.97
8.03 10.49 10.48 8.20 12.02 8.54 10.48
2.019 9.597 1.552 0.989 7.588 2.595 1.347
12.78 35.59 21.45 15.90 36.15 14.50 20.36
0.47 0.04 3.11 0.01 0.77 0.52 2.87
65.0
66.2
35.8
59.9
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skewness (Table 2). The histograms for the plateau (Ypla) and the whole site (Yall) at Yattendon had bimodal distributions, therefore the data were stratified. The pooled within-class variogram was computed for these sites from the residuals of the class means. Omnidirectional experimental variograms were computed for each landscape unit and for each entire site (all topography) using the usual computing equation and appropriate data (see Webster and Oliver, 2001). There were not enough data for each landscape unit to explore any anisotropy in the variation reliably. Therefore, it was investigated with the more intensive digital data available from aerial photographs. The results from this suggested that the anisotropy is zonal and is associated with the landscape units, therefore it was not investigated further. Several experimental variograms suggested that regional trend might be present in the variation; the semivariances increased rapidly after reaching a sill or they had upwardly concave shapes. We examined these data further by fitting first- and second-order polynomials on the coordinates. If the percentage variance accounted for by these functions was greater than 20%, we computed the variogram
afresh on the residuals because geostatistics assumes that the underlying process is random. If the variogram of the residuals was different from that of the raw data we used the former for further analyses. The experimental variograms were modelled for each landscape unit using weighted least squares approximation (GenStat, Payne, 2000). An average of the variogram ranges of all properties for each landscape unit for a given parent material was calculated. In addition, average variograms were determined in two ways. Simple average variograms were calculated from the modelled rather than the experimental variances of variograms for each landscape unit (see Fig. 2). This was done because if variograms from the literature were to be used the information would be derived from the model parameters rather than the experimental semivariances. Fig. 2 shows that the simple average variograms calculated from the experimental (a) and the modelled variances (b) are similar. Simple average variograms can be calculated from variograms of the raw data only, which excludes those computed from transformed data or residuals from a trend surface. This means that for some sites or variables there might
Table 2 Variogram model parameters of LOI (%) for different landscape units Landscape unit Shuttleworth clay Slope Valley All topography Shuttleworth sand Slope Valley All topography Wallingford Plateau Slope Valley All topography Yattendon North slope Plateau South slope Valley All topography Field 214 Field 215
Maximum lag (m)
Data used
Model type
c0
c
a(3r) (m)
120 140 200
Lresid. Lresid. Qresid.
Circular Circular Circular
0.0232 0.0628 0.0575
0.3890 0.1170 0.2230
66.46 82.30 71.07
200 120 200
Raw Lresid. Qresid.
Pentaspherical Circular Pentaspherical
0.0965 0.1500 0.0970
0.1870 0.2430 0.2710
124.6 72.80 120.2
180 200 270 390
Qresid. Raw Raw Raw
Circular Spherical Circular Pentaspherical
0.0702 0.1127 0.1199 0.0535
0.1420 0.4260 0.1860 0.4610
66.13 135.5 244.5 226.1
120 300 160 175 180 160 120
Raw Within-class Outliers removed Raw Within-class Raw Outliers removed
Spherical Circular Circular Circular Circular Spherical Spherical
0.5329 0.6357 0.0601 0.3292 0.9286 0.8171 0.2659
1.475 1.972 0.6940 0.6850 0.7070 1.6940 0.5069
51.80 105.0 65.20 66.50 64.17 80.10 77.57
c0 is the nugget variance, c the correlated component, a the range of spatial dependence and 3r the effective range for the exponential model; Lresid. the residuals from a linear trend function; Qresid. the residuals from a quadratic trend function.
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Fig. 2. Average variograms calculated from variograms computed for Yattendon based on: (a) experimental semivariances and (b) modelled semivariances.
be no simple average variogram. As an alternative to include all types of data (raw, transformed and residuals), variogram models for the individual landscape units were standardized by setting the sill variance to 1 and by calculating the nugget:sill ratio. These standardized variograms of LOI were then averaged to provide standardized average variograms. A suitable sampling intensity to map soil properties can be determined from just less than half the range of spatial dependence described by the variogram model.
This was based on the average range of several conventional variograms and the range of standardized average variograms. The simple average variograms were not used to guide sampling because they were less reliable than those above when based on few variograms. The model parameters of variograms of the raw data based on all topography and of the standardized average variograms were also used with the kriging equations to determine an optimal sampling interval for different parent materials
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Fig. 3. Experimental variograms and fitted models of LOI for the landscape units at each site: (a–c) Shuttleworth clay, (d–f) Shuttleworth sand, (g–j) Wallingford, and (k–o) Yattendon. The symbols are the experimental semivariances and the line is the fitted model.
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(McBratney et al., 1981). A tolerable kriging error of 25% of the sill variance, as suggested by Lark (2000), was used to determine an optimal sampling interval from graphs of block kriging variance plotted against grid spacing for estimates over blocks of different size. To assess the suitability of the sampling intervals suggested by the methods described above, the original LOI measurements were sub-sampled to provide a range of sampling intervals. The subsampled LOI data were then used with the variogram model parameters from the original data to predict LOI at unsampled locations by block kriging. The predictions from the different sampling intensities were then mapped so that the variation could be compared.
3. Results and discussion We describe in detail here only the results for loss on ignition (LOI), but the same approach was used for all soil properties in the survey and the results were similar. The results are summarized in Table 4. 3.1. Summary statistics Summary statistics for LOI are given for each site and landscape unit, Table 1. The mean values show that Yattendon had the largest LOI, followed by SC, and SS had the smallest values. This order generally reflects the clay and calcium carbonate content of the soil at the different sites. Clay content increases LOI values because of the loss of water from the interlayer spaces of clay minerals during ignition. Clay soil also tends to have a larger organic matter content because less is oxidized and translocated because of the chemical bonds between the clay particles and organic matter. The statistical range of LOI values and coefficients of variation (CVs) were the largest for the plateau and all topography at Yattendon (Ypla and Yall), which was due to marked differences in the calcium carbonate content of the soil on the plateau areas of Fields 214 and 215. Free calcium carbonate in the soil (dissolved and precipitated) protects organic matter from rapid microbial action. The differences between the means and variances of LOI for the four parent materials were shown to be significant at the 0.05 level by a Kruskal–Wallis H-test (Ebdon, 1994).
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Therefore, the difference in mean LOI values for different parent materials could be used to indicate whether the soil forming conditions at a particular site are similar to those at sites where standardized average variograms are available. Although differences between the means and variances for the landscape units on a particular parent material cannot be assessed statistically in the absence of replicates, some observations can be made. At SC and SS the mean LOI values are larger for the valleys than slopes. The valleys are in receiving positions and drainage is more likely to be impeded at times, which would reduce the breakdown of organic matter. At Wallingford, the mean LOI value for the valley is larger than that for the plateau, but less than that for the slope where there is some calcareous gravel. The relations between LOI and landscape were complicated at Yattendon because the calcium carbonate content of the soil varied from one landscape unit to another. The significant differences between the means and variances of LOI on different parent materials shows the importance of parent material as a soil forming factor. The differences in mean LOI also show the importance of topography in soil formation. These results suggest that some generalization about the characteristics of soil developed on similar parent materials and topographic positions might be possible. These effects should also be evident in the model parameters of variograms for different parent materials and topographic positions. Here, the range only will be considered as it can be used to guide future sampling intensity. 3.2. Variogram analysis The variograms of LOI for different landscape units are shown in Fig. 3 and their model parameters are given in Table 2. Table 1 gives the percentage variance that linear and quadratic functions accounted for in those landscape units where residuals from the trend were used to calculate variograms. For sites such as SC and SS where the variograms were computed on the residuals the sampling interval suggested by the range would be relevant only to the non-deterministic component of the variation. In such cases, it might be possible to reduce the number of sampling points required further, especially where the percentage
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Table 3 The average range of the models fitted to the variograms of LOI, and the range of the standardized average variogram Site
Average range of fitted models (m)
Range of standardized average variogram
Shuttleworth clay Shuttleworth sand Wallingford (Gravel) Yattendon (Chalk) All parent materials All plateaus All slopes All valleys
73.3 105.9 168.1 72.9 104.8 85.6 88.7 116.5
114.1 72.2 189.7 93.6 82.7 116.8 68.6 82.4
a1 (m)
a2 (m)
203.9 128.6 269.8
a1 is the short-range component and, a2 is the long-range component of the nested model.
variance accounted for by trend is large. To assess this would require further detailed research, however. The autocorrelation range of a variogram essentially describes the extent of patches of similar soil; therefore, one would expect less variable areas to have variograms with a larger range. The variogram ranges of the landscape units were similar for SC, but that for the valley was the longest. It also had the smallest CV of 7.66% (Table 1). For SS the slope had the longest variogram range and the smallest CVof 15.39% (Table 1). The ranges of variograms for Wallingford varied from 66 to 244 m although most were over 100 m, and that for the valley was the longest. The latter area also had the smallest CV of 11.89% (Table 1). For Yattendon the variogram ranges were between about 50 and 100 m; that for the plateau was longest. Unlike the other sites, the latter had a large CV of 36% (Table 1). This was probably due to differences in the calcium carbonate and organic matter contents of Fields 214
and 215. These differences were taken into account by computing the pooled within-class variogram for the plateau. The variances within these classes were small, showing that the longer variogram range was again associated with the less variable soil within the strata on the plateau, Table 2. A Kruskal–Wallis H-test showed that the differences in the variogram ranges of LOI with parent material were significant at the 0.1 level. The variogram ranges of LOI for each landscape unit were averaged for each parent material; they are given in Table 3. The smallest average range was for Yattendon (Chalk) and the longest was for Wallingford (gravel). These differences between parent materials are more evident in the average variogram ranges for several properties, Table 4. There is also a marked difference in the average range for the selected soil properties for each parent material (Table 4, last row). This difference has the order of gravel (150 m) >
Table 4 Average variogram range for all properties Property
Chalk (Y) average
Gravel (W) average
Oxford Clay (SC) average
Lower Greensand (SS) average
Overall average
Depth LOI Stoniness Munsell value VWC Clay Sand pH Average of above soil properties
73.01 72.91 85.58 102.7 77.34 103.4 88.19 – 86.06
123.1 168.1 204.9 160.5 141.2 136.0 118.6 145.8 149.8
– 73.28 133.4 170.6 108.4 114.8 94.99 128.5 117.7
– 105.9 90.34 74.42 87.23 98.62 140.4 82.26 97.01
98.03 104.9 128.6 127.1 103.6 113.2 110.5 118.9 –
LOI is the loss on ignition; VWC the volumetric water content.
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Fig. 4. Standardized average variograms of LOI: (a) Shuttleworth clay, (b) Shuttleworth sand, (c) Wallingford, (d) Yattendon, (e) all parent materials, (f) all plateaus, (g) all slopes, and (h) all valleys.
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Oxford Clay (118 m) > Lower Green Sand (97 m) > Chalk (86 m). A Kruskal–Wallis H-test showed a significant difference between the average variogram range of the selected soil properties in Table 4 according to parent material at the 0.05 level. This suggests that it should be feasible to determine a suitable sampling interval for future soil surveys in similar pedological settings from the average variogram range. In contrast to the average range for each parent material mentioned above there is little variation in the overall average range for different soil properties (Table 4, final column). Table 3 (last four rows) also shows that for LOI there is little difference in average variogram range when all parent materials are considered, and when all plateau, slope and valley areas are grouped regardless of parent material. Therefore, a global average variogram for a given soil property, as described by McBratney and Pringle (1999), would not suggest appropriate sampling intervals for all possible parent materials; it would lead to over-sampling on some and under-sampling on others. 3.3. Standardized average variograms The standardized average variograms of LOI for all landscape units at each site are shown in Fig. 4, and their autocorrelation ranges are given in Table 3. That for Wallingford had the longest range, as did the average range for the fitted models (Table 3). These variograms for each parent material were fitted with single functions. The standardized average variograms for all parent materials, and the landscape units on all parent materials except for the plateaus, were fitted by
nested functions with short- and long-range components. For these latter functions the short-range component accounted for most of the variation, therefore, the range of this structure was used as a guide to sampling interval. Where variation is nested and both structures account for similar proportions of the variance it is recommended that an average of the two ranges be used to determine a suitable sampling interval. One might wonder if additional samples at an interval smaller than half the range should be obtained where the nugget:sill ratio is large because of the unresolved variation at distances less than the sampling interval. It was not possible to assess this here because the nugget:sill ratios of the standardized average variograms for each parent material were similar (20–40%) because of the averaging (Fig. 4). However, Webster and Oliver (1992) showed that the confidence intervals on a variogram with a nugget component were larger than those with no nugget. They showed that with more intensive sampling this difference in the confidence intervals decreased. Based on this we recommend that where standardized average variograms have a large nugget:sill ratio the sampling at half the variogram range should be supplemented by some additional samples at a shorter interval. In fact, we consider that this is generally advisable to improve the reliability of the variogram at short lag distances. A Kruskal–Wallis H-test showed a significant difference between the ranges of the standardized average variograms for all soil properties (not shown here) on the different parent materials at the 0.05 level. These results suggest that standardized average variograms could be used to guide sampling as described above.
Table 5 Sampling intervals for LOI suggested by less than half the variogram range Site
Shuttleworth clay Shuttleworth sand Wallingford (Gravel) Yattendon (Chalk) All parent materials All plateaus All slopes All valleys
Sampling interval (m) Half average range of fitted models
Half range of standardized average variogram
35 50 80 35 50 40 40 55
55 35 90 45 40 55 30 40
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Fig. 5. Graphs of kriging error plotted against grid spacing for LOI based on the conventional variogram for all topography at: (a) Shuttleworth clay, (c) Shuttleworth sand, (e) Wallingford, (g) Yattendon, and on the standardized average variogram for: (b) Shuttleworth clay, (d) Shuttleworth sand, (f) Wallingford, and (h) Yattendon.
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3.4. Sampling intervals Based on the notion that a sampling interval of just less than half the variogram range should be adequate to resolve the main features of the variation in the soil at the level of the investigation, sampling intervals were determined based on the average variogram range and on the range of standardized average variograms for different parent materials, Table 5. In addition, the kriging errors were computed for a number of grid spacings and block sizes with the conventional variograms for each site based on the data from all topographic units, and with the standardized average variograms. Graphs of kriging error plotted against grid spacing for LOI at each site are shown in Fig. 5. These graphs were used with a tolerable error of 25% of the sill variance to determine optimal sampling intervals for different block sizes, Table 6. Table 5 shows that sampling intervals of 35 m (average range of several variograms) and 55 m (range of standardized average) were suggested for LOI at SC. However, based on the average range of the selected soil properties the interval would be about 55 m, Table 4. The optimal sampling intervals for SC for a block support of 25 m were 45 m (conventional variogram based on all topography) and 50 m (standardized average variogram), Table 6. For SS a sampling interval for LOI of between 35 and 50 m was suggested by the methods given in Table 6 Optimal sampling intervals suggested for LOI using the variogram and the kriging equations with a tolerable error of 25% of the sill variance Variogram
Sampling interval suggested by threshold of 25% of sill height 25 m block
50 m block
75 m block
Shuttleworth clayall Shuttleworth claystdav Shuttleworth sandall Shuttleworth sandstdav Wallingfordall Wallingfordstdav Yattendonall Yattendonstdav
50 45 50 50 70 50 50 40
70 60 65 70 90 75 – 60
– 80 85 95 105 95 – 100
Shuttleworth clayall for example is the sampling interval based on the variogram for all topography and Shuttleworth claystdav the sampling interval based on the standardized average variogram.
Table 5, and an optimal interval of 50 m for blocks of 25 m, Table 6. The sampling interval would be about 45 m (Table 4) at this site if the other soil properties were taken into account. Sampling intervals of 80– 90 m were indicated for LOI at Wallingford in Table 5, whereas the optimal sampling intervals for 25 m blocks were 50–70 m, Table 6. Half the average range of the variograms for other soil properties suggested that an interval of about 70 m would be appropriate at Wallingford (Table 4). The methods in Table 5 suggested sampling intervals of between 35 and 45 m for LOI at Yattendon, and intervals of 40 and 45 m were suggested for 25 m blocks using the optimal approach, Table 6. Finally, half the average range of the variograms for the other soil properties (Table 4) suggested that an interval of 40 m would be appropriate at Yattendon. Tables 5 and 6 show that there is some consistency in the sampling intervals suggested for LOI using half the average range of several variograms, half the range of standardized average variograms, and the optimal approach. Table 4 also shows that the intervals suggested by the average variogram range of several soil properties on a given parent material are of a similar order of magnitude to those for LOI. Table 5 shows that the intervals suggested for LOI by the average variogram range and standardized average variograms for all parent materials or all slopes from different parent materials, for example, are between 34 and 58 m. This confirms that average variograms that are not parent material specific would not provide a suitable guide to sampling on all parent materials. 3.5. Kriging with the sub-sampled data Fig. 6a shows the maps of ordinary kriged predictions for SC with the original data on a 20 m grid, and Fig. 6b–d those with sub-sampled data on 40, 60 and 80 m grids, respectively. The main features of the variation in LOI are retained in the maps based on the 40 and 60 m grids (Fig. 6b and c, respectively), but there is a loss of detail in that for the 80 m grid (Fig. 6d). These results suggest that for the 60 m grid interval additional sampling at a 30 m interval at randomly selected grid nodes would avoid some loss of detail, which supports the intervals indicated in Tables 5 and 6. These maps also show the effect on the predictions of the configuration of sampling sites in
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Fig. 6. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth clay.
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Fig. 7. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth sand (Cricket Meadow).
the sub-samples, especially when the sampling interval is larger. Frogbrook and Oliver (2000) also observed the effect that sample position for different grid spacings had on the patterns of variation described in their fields. The kriged maps of LOI for SS (Fig. 7) suggest that a 60 m grid would be adequate for sampling Cricket Meadow (Fig. 7c) because of the large amount of trend in LOI in this field. Trend surface analysis can predict the deterministic component of variation, and Frogbrook and Oliver (2001) showed that as few as nine carefully chosen points (i.e. placed in areas of large, medium and small LOI values) can be used to describe this component of the variation accurately. It is notable that the 60 m grid for this field contained only 12 points. For Football field (Fig. 8), however, a 40 m grid interval (Fig. 8b) would be needed to resolve the variation adequately. These findings are also consistent with the intervals suggested for these sites in Tables 5 and 6. The kriged maps of LOI for Wallingford, Fig. 9, suggest that sampling intervals of 80–90 m would
resolve the variation adequately. The main features of the variation based on kriging the 30 m data (Fig. 9a) are reproduced well by kriging with the data on the 60, 90 and 120 m grids (Fig. 9b–d, respectively), but there is some loss of detail for the 150 m grid (Fig. 9e). The kriged maps of LOI for Yattendon based on the residuals from the class means (Fig. 10) show that a sampling interval of 35–40 m (Table 5), as indicated by the variograms, would be required. The variation in LOI based on a 60 m grid, Fig. 10b, is poorly resolved although some of the main features of the variation are still evident. The sampling intervals of 40–50 m suggested by the variograms for all parent materials (Table 5, row five) would be adequate to guide sampling at most sites, except at Wallingford, where they would lead to over-sampling. The sampling intervals suggested by variograms for all plateaus, slopes and valleys (Table 5, last three rows) would not be appropriate for individual landscape units (see ranges for landscape units in Table 2). This shows that if topography is to be included as a soil forming factor it should be done in
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Fig. 8. Maps of kriged predictions of LOI from data on sampling grids of: (a) 20 m, (b) 40 m, (c) 60 m and (d) 80 m for Shuttleworth sand (Football field).
relation to parent material. For example, the LOI values at Yattendon and Wallingford were not always the largest in valley positions because of the variability in the parent material; the presence of calcium carbonate changed the usual pattern of variation associated with the landscape positions. 3.6. Validation sites To determine whether the variogram ranges from the four field sites in this study could be used to guide sampling at other arable sites in southern England with similar parent materials, we examined the model
parameters of LOI for six sites from other studies. The sites at CEDAR, Berkshire (Frogbrook, 2000) and Warren, Bedfordshire (Baxter, 2003) have soil developed on clay parent materials; the former is on London Clay and the latter on Oxford Clay. The sites of Cashmore (Baxter, 2003) and Northpark (Frogbrook, 2000), Bedfordshire, have soil developed on the Lower Greensand. Finally, the Clays, Oxfordshire (Oliver and Carroll, 2004) and Stubbs (Oliver et al., 2003), Berkshire, have soil developed on Chalk. First, we compared the mean values of LOI (Table 7) at the validation sites with those for all topography in the present study (Table 1). A Kruskal–Wallis H-test
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Fig. 9. Maps of kriged predictions of LOI from data on sampling grids of: (a) 30 m, (b) 60 m, (c) 90 m, (d) 120 m and (e) 150 m for Wallingford.
based on the means of LOI from both sets of sites showed a significant difference in relation to the three parent materials at the 0.05 level. The average ranges of the variograms for the validation sites have a similar order of magnitude (clay, 123.9 m > Chalk, 103 m > sand, 84.9 m) to those for the study sites (Table 4).
Frogbrook’s (2000) research also showed the same order and similar average ranges (clay, 140 m > Chalk, 100 m > sand, 68 m) for a group of soil properties. The average variogram ranges for LOI at the validation sites suggest sampling intervals of 50 or 60 m for sites with clay parent material, 40 m for
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Fig. 10. Maps of kriged predictions of LOI from data on sampling grids of: (a) 30 m, (b) 60 m, (c) 90 m and (d) 120 m for Yattendon.
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Table 7 Variogram model parameters for LOI at validation sites Site
Parent material
Mean (%)
Variogram
Model type
c0
c
a(3r) (m)
CEDAR97 CEDAR98 Warren Cashmore Northpark97 Northpark98 Northpark99 Clays Stubbs
Oxford Clay Oxford Clay Clay Lower Green Lower Green Lower Green Lower Green Chalk Chalk
4.49 4.39 5.71 3.78 4.00 3.93 3.85 4.90 3.96
Raw Raw Lresid. Lresid. Raw97 Raw98 Raw99 Lresid. Within-class
Pentaspherical Pentaspherical Spherical Exponential Pentaspherical Pentaspherical Pentaspherical Exponential Circular
0.0899 0.0717 0.6816 0.0284 0 0.0254 0.0507 0.0723 0.0467
0.0781 0.0818 0.7125 0.2871 0.4618 0.4229 0.3879 0.0954 0.1217
139.9 134.8 96.89 49.26 95.49 99.58 95.33 113.0 93.60
Sand Sand Sand Sand
Validation sites: Cashmore and Warren (Baxter, 2003), CEDAR and Northpark (Frogbrook, 2000), and the Clays (Oliver and Carroll, 2004) and Stubbs (Oliver et al., 2003).
sandy ones and 50 m for Chalk sites. These intervals would also be adequate to describe the variation in LOI for SC, SS and Yattendon, respectively (Figs. 6–8 and 10). The reverse is also true; the sampling intervals indicated by the average variogram range and the range of standardized average variograms for SC, SS and Yattendon would be appropriate for survey at the validation sites on the same type of parent material.
4. Conclusions There are significant differences in the means and variances, and variograms of LOI for soil developed on different parent materials. This suggests that it might be possible to make generalizations about the variation of LOI in relation to parent material to guide future sampling for mapping where there are no variograms. The optimal sampling intervals based on a tolerable error of 25% of the sill variance were similar to those based on less than half the variogram range. The latter method is recommended because it is more straightforward and the results are similar. The patterns of variation in LOI on the kriged maps based on subsampled data showed that the range of standardized average variograms can be used to determine suitable sampling intervals for future soil surveys based on less than half the range or on the kriging variances and a tolerable error of 25% of the sill variance. A comparison of the sampling intervals suggested by this study with those indicated by the data for other sites from the literature with comparable topography, parent materials and climate were similar for a
particular parent material. Use of the average variogram range or standardized average variograms from similar sites is suggested as a guide to sampling at a site only when there are no existing variograms of soil properties or appropriate ancillary data. They avoid the need for a reconnaissance soil survey when nothing is known about the scale of variation. We also recommend that the sampling intervals suggested by half the variogram range are supplemented by some additional samples at shorter intervals to resolve the spatial structure at short lag distances more reliably. This study has shown that the average variogram range and standardized average variograms for a given parent material could be used to guide sampling on the four parent materials examined in southern England. The approach is particularly useful where several variograms are available to calculate the average and where the range of soil forming conditions at sites is narrow. It could be applied more widely if there were databases of variogram parameters for different soil forming conditions. The mean value of a soil property could be used as a guide to the degree of similarity between sites. Suitable sampling intervals were suggested by variograms from the study sites at other sites in southern England where the parent material was similar. The results from this study could be applied to sites elsewhere with similar soil forming conditions, landscape features, climate and land use. If a reliable standardized average variogram exists it could be used to predict properties for mapping from a much smaller sample than that needed to compute a variogram. This should reduce the cost of producing accurate soil maps for land management even further. Nevertheless, it must be remembered the data still
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need to be spatially dependent for kriging or any other kind of interpolation. We will examine the viability of using standardized average variograms for kriging in another study.
Acknowledgements We thank the University of Reading and the Fertiliser Manufacturers’ Association for supporting this research.
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Frogbrook, Z.L., Oliver, M.A., 2001. Comparing the spatial predictions of soil organic matter determined by two laboratory methods. Soil Use Manage. 1, 235–244. Kerry, R., Oliver, M.A., 2003. Variograms of ancillary data to aid sampling for soil surveys. Precis. Agric. 4, 253–270. Lark, R.M., 2000. Designing sampling grids from imprecise information on soil variability, an approach on based on the fuzzy kriging variance. Geoderma 98, 35–59. McBratney, A.B., Pringle, M.J., 1997. Spatial variability in soilimplications for precision agriculture. In: Stafford, J.V. (Ed.), Precision Agriculture’97, vol. 1. BIOS Scientific Publishers, Oxford, pp. 3–31. McBratney, A.B., Pringle, M.J., 1999. Estimating average and proportional variograms of soil properties and their potential use in precision agriculture. Precis. Agric. 1, 125–152. McBratney, A.B., Webster, R., Burgess, T.M., 1981. The design of optimal sampling schemes for local estimation and mapping of regionalized variables. I. Theory and method. Comput. Geosci. 7, 331–334. Oliver, M.A., Carroll, Z.L., 2004. Description of spatial variation in soil to optimize cereal management. Project Report No. 330. HGCA, London. Oliver, M.A., Heming, S.D., Gibson, G., Adams, N., 2003. Exploring the spatial variation of take-all (Gaeumannomyces graminis var. tritici) for site-specific management. In: Stafford, J.V., Werner, A. (Eds.), Precision Agriculture. Wageningen Academic Publishers, Wageningen, pp. 481–486. Payne, R.W., 2000. The Guide to GenStat. Part 2. Statistics. VSN International, Oxford. Webster, R., Oliver, M.A., 1992. Sample adequately to estimate variograms of soil properties. J. Soil Sci. 43, 177–192. Webster, R., Oliver, M.A., 2001. Geostatistics for Environmental Scientists, Wiley, Chichester.