Use of a covariance variogram to investigate influence of subsurface drainage on spatial variability of soil–water properties

Agricultural Water Management 37 (1998) 1±19

Use of a covariance variogram to investigate influence of subsurface drainage on spatial variability of soil±water properties Mahmoud M. Moustafa*, Atsushi Yomota Faculty of Environmental Science and Technology, Okayama University, 2-1-1, Tsushima-naka, 700 Okayama City, Japan Accepted 12 March 1998

Abstract Knowledge of the spatial variability of soil±water properties is of primary importance for management of agricultural lands. This study was conducted to examine which spatial structure measure, the semi-variogram or the covariance variogram, is appropriate for inference of the spatial structure and performing interpolation of soil±water properties from sample data sets. Using the appropriate spatial structure measure, the spatial variability of these properties (saturated hydraulic conductivity, water table depth, groundwater salinity, and soil salinity and sodicity) as affected by subsurface drainage is also evaluated. The soil±water properties were sampled before and after the installation of subsurface drainage on a regular square grid of 500 m at 61 locations within 1470 ha in the Nile Delta of Egypt. The results showed that the covariance variogram reveals the character of spatial structure and that it is more appropriate for interpolation than the semi-variogram. Subsurface drainage has highly affected the spatial variability of soil±water properties. On average, the spatial correlation range increased by approximately 29%, whereas the ratio of structural heterogeneity to the total variation (relative structured heterogeneity) was doubled 4 years after drainage installation. Moreover, the nugget effect increased and was present for all soil±water properties with noticeably high values. Uneven spatial distributions were also observed. Further study of long-term spatial variation of soil±water properties as affected by subsurface drainage is suggested. # 1998 Elsevier Science B.V. All rights reserved. Keywords: Covariance function; Covariance variogram; Semi-variogram; Spatial variability; Subsurface drainage

* Corresponding author. Tel.: +81 86 251 8366; fax: +81 86 251 8385; e-mail: [email protected] 0378-3774/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 7 7 4 ( 9 8 ) 0 0 0 4 6 - 8

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1. Introduction Soil properties are inherently variable in both space and time. Although the impact of subsurface drainage on soil properties has long been recognized in most parts of the world where irrigated agriculture is practiced (Wesseling and van Wijk, 1957; Reeve and Fausey, 1974; Abdel-Dayem et al., 1987), data on its effect on the spatial variability of soil±water properties is scarce. Knowledge of the spatial variability of soil±water properties is of great importance for determining a soil sampling strategy, for understanding and modeling of water and chemical movement, for designing field experiments and for many other investigations associated with the management of agricultural lands. Freeze (1980), Bouwer and Jackson (1974) and Milly and Eagleson (1987) showed that the spatial variability of soil properties may induce significant effects on soil±water management of agricultural lands. Classical statistical methods are inadequate to interpolate spatial variables, since they do not take into account spatial correlation and the relative locations of samples. Geostatistical methods are increasingly used to describe the spatial dependence structure of soil±water phenomena (Burgess and Webster, 1980a, b; Vieira et al., 1981; Trangmar et al., 1986; Gallichand and Marcotte, 1993; Agrawal et al., 1995) providing a greater precision as well as an error of estimation. Most geostatistical research has been based on determining a spatially correlated function (i.e., the semi-variogram) which provides the basis for interpolation by kriging. This is a technique for calculating optimal, unbiased linear estimation of properties at unsampled locations with minimum estimation error variance (Journel and Huijbregts, 1978). However, Isaaks and Srivastava (1988), Journel (1988) and Srivastava and Parker (1989) indicated that the semi-variogram should not be the spatial continuity measure. Until recently, little work has been done on the possibility that one of the other alternative functions to describe the spatial continuity of a property may be preferable. The objectives of this study are twofold: first, to investigate the performance of the semi-variogram and covariance functions for identification of spatial structure and precision of interpolation; second, to evaluate the effect of subsurface drainage on spatial variability of soil±water properties (saturated hydraulic conductivity, soil salinity and sodicity, water table depth, and groundwater salinity).

2. Materials and methods 2.1. Subsurface drainage and sampling description The study was conducted in a 1470 ha area located in the Nile Delta within the Sharkyia Governorate of Egypt (Fig. 1). The topography of the area is plain with a gentle slope running S±N. The soils are alluvial and alluvio-marine deposits. The predominant textural class is clay. The cropping pattern in the area includes rice in rotation with dryfoot crops. A composite subsurface drainage system was installed in January 1985, to address waterlogging and salinity problems. The system consists of PVC laterals and concrete collectors. A steady state drainage rate of 1.0 mm/day and a dewatering depth of

M.M. Moustafa, A. Yomota / Agricultural Water Management 37 (1998) 1±19

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Fig. 1. Location of the study area and soil sampling sites.

1.0 m midway between drains were adopted as drainage design criteria in Egypt (AbdelDayem and Ritzema, 1990; Amer, 1990). Laterals of 8 cm diameter were installed with 40 m spacing at an average depth of 1.25 m on a 0.1% average grade to achieve the dewatering criterion at the design discharge rate. Their length varied from 200 to 300 m. No envelope materials were used. Collectors comprised of 0.75 m concrete pipes with diameters varying from 0.15 m at upstream end to 0.25 m at the outlet, were situated at an average depth of 1.75 m. The collector drains were designed for a capacity of 4.0 mm/day to maintain adequate drainage conditions of the dry-foot crops (Amer et al., 1990). Manholes are placed at a maximum interval of 180 m to allow inspection and maintenance operations. Soil sampling was carried out before drainage installation in October 1981 and after drainage installation in January 1987 and April 1989 at 61 locations on a regular grid with 500 m spacing (Fig. 1). Saturated hydraulic conductivity was measured in situ at a depth of 2 m by the auger hole method using a 8 cm auger. The water table in the hole was allowed to reach equilibrium and its depth was measured. Water samples were then taken for electrical conductivity analysis. Soil samples were taken from depths of 0±0.25 and 0.25±0.50 m. Three samples were drawn randomly within 1 m2 around each grid point, and a composite sample was prepared. Soil salinity and Sodium Adsorption Ratio (SAR) were determined on saturated paste extracts at the Drainage Research Institute, Cairo,

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Egypt. Due to lack of data for groundwater salinity and SAR values in October 1981 before the subsurface drainage installation, they were evaluated only after the installation of the system in January 1987 and April 1989. These three data sets (one before and two after drainage installation) were used to achieve the stated objectives of the study. 2.2. Semi-variogram and covariance functions Let ‰z…xi ‡ h†; z…xi †Š be a pair of soil property measurements at the sampling positions xi ‡ h and xi, and therefore separated by a vector h. Each z…xi † is, in turn, a realization of the random variable Z…xi †, where xi is a fixed position. The set of random variables [Z(x), when x varies within field under study] constitutes a random function (Journel and Huijbregts, 1978). A natural phenomena can often be characterized by the distribution in space of one or more variables, called `regionalized variables'. 2.2.1. Semi-variogram The semi-variogram (h) of a regionalized variable is by definition (Vieira et al., 1981), o 1 n (1)

…h† ˆ E ‰Z…xi ‡ h† ÿ Z…xi †Š2 2 and it is estimated by (Journel and Huijbregts, 1978), 1 X ‰z…xi ‡ h† ÿ z…xi †Š2 2N…h† iˆ1 N…h†

…h† ˆ

(2)

where E is the expected value and N(h) is the number of experimental pairs ‰z…xi †; z…xi ‡ h†Š of data separated by a vector h. 2.2.2. Covariance variogram The covariance function, C(h), can be calculated as (Vieira et al., 1983), 1 X z…xi ‡ h†  z…xi † ÿ mÿh  m‡h N…h† iˆ1 N…h†

C…h† ˆ

(3)

where mÿh and m‡h denote the means of the data values z…xi ‡ h† and z…xi †, respectively. The covariance function typically decreases with distance, in contrast to the semivariogram which typically increases with distance. The two functions are linearly related and interchangeable within the probabilistic framework under the assumption of secondorder stationarity. However, within the deterministic framework, this is not true. In practice, these two approaches can lead to significantly different results (Isaaks and Srivastava, 1988). For detailed descriptions of the theoretical development, use and limitations of the relationships between the two functions, the reader is referred to Vieira et al. (1983), Journel (1985) and Isaaks and Srivastava (1988). In this study we will use a covariance function in the form of a variogram, hence increasing with distance h, being called the `Covariance Variogram',

 …h† ˆ C…0† ÿ C…h†

(4)

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The value of C(0) is the sample variance and *(h) is the covariance variogram which can be modeled and used in kriging in the same way as the semi-variogram. 2.3. Spatial modeling and cross-validation To ensure the mathematical consistency required for kriged estimations of soil±water properties, it is necessary to fit sample variograms to simple theoretical models. Model fitting to the sample variograms is preferably done `by eye' to avoid complications resulting from least square curve fitting (Journel and Huijbregts, 1978; Isaaks and Srivastava, 1989; Vieira et al., 1983). Two transitive spatial models (spherical and exponential) defined as, Spherical model:

1 …h† ˆ c0 ‡ c1

1 …h† ˆ c0 ‡ c1 Exponential model:

1 …h†



h ÿ
3 h 2 a

ÿ 12



ÿh
3 i a

ÿ3h ˆ c0 ‡ c1 1 ÿ exp a

0ha h>a

(5)

 0  h  3a

(6)

are fitted to the sample variograms. Here 1 …h† is the estimated variogram for lag h, c0 equals the nugget effect, c1 is the structured heterogeneity component, c0‡c1 is the sill, h is the distance between measurement points (lag), and a is the correlation range of the variogram. The selected models were validated using the cross-validation technique (Vieira et al., 1983). This was done by omitting the observation points one at a time and then kriging the removed point, using the variogram model and the remaining data. The model's validity is measured by the value of the reduced mean error (S1) and the reduced variance (S2) which must be close to 0 and 1, respectively. There are no simple and reliable rules to judge whether the computed values of S1 and S2 are close enough to 0 and 1 for the model to be accepted. Kitanidis (1993) mentioned that it is a common practice to reject a model if the computed values of S1 and S2 differ from their correct values by 0.15±0.20.

3. Results and discussion 3.1. Summary statistics Soil sampling was carried out in different seasons. Therefore, the effect of short or long-term seasonal variation of water and soil management, climatic conditions and other agricultural factors on soil±water properties cannot be assessed in a straightforward fashion unless a comparison is made simultaneously with a similar undrained field at the same times. Nevertheless, if large changes of soil±water properties are observed after installation of the subsurface drainage, it should be possible to attribute a major part of

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these changes to the drainage system. In this study, the influence of seasonal variation on soil properties and sampling measurements was assumed to be negligible. This agrees with the results of a nearby similar area in the eastern part of the Nile Delta as showed by Abdel-Dayem et al. (1987). Analysis of soil±water properties, based upon 61 grid samples, showed an increase in mean values of saturated hydraulic conductivity and water table depth and a decrease in soil salinity and sodicity (SAR), indicating soil improvement over the study period (Table 1). An increase of groundwater salinity was observed from January 1987 to April 1989, probably caused by salt leaching from the surface layers following the subsurface drainage. The low values of saturated hydraulic conductivity are mainly associated with high clay percentages in the study area. Its mean value in October 1981 was 0.11 m/day which increased to 0.21 and 0.30 m/day, respectively in January 1987 and April 1989, Table 1 Experimental statistics of the data batch of soil±water properties Variable

Unit

Saturated hydraulic conductivity K81 K87 K89

(m/d)

Water table depth WTD81 WTD87 WTD89

(cm)

Groundwater salinity GWS87 GWS89

(dS/m)

Soil salinity 0±0.25 m (SS)1 81 (SS)1 87 (SS)1 89 0.25±0.50 m (SS)2 81 (SS)2 87 (SS)2 89

(dS/m)

SAR-value 0±0.25 m (SAR)1 87 (SAR)1 89 0.25±0.50 m (SAR)2 87 (SAR)2 89

(Ð)

a

Minimum 0.01 0.01 0.02 10 25 46

0.65 1.00 0.79 131 190 198

Variance

CV (%)

0.11 0.21 a 0.30 a,d

0.018 0.079 a 0.035 a,c

126 134 63

82 128 a 114 a,d

811 1954 a 1535 a,d

35 35 35

0.40 0.85

11.30 11.50

3.53 4.01 d

8.004 7.413 d

80 68

0.70 0.03 0.10

2.60 3.00 3.50

1.60 1.49 b 1.25 a,d

0.206 0.410 b 0.511 a,d

28 43 57

0.80 0.44 0.10

2.78 3.56 2.00

1.66 1.62 b 1.23 a,c

0.261 0.615 a 0.323 b,d

31 49 46

1.19 2.25

8.00 6.60

4.67 4.24 d

2.337 1.294 d

33 27

2.40 1.01

10.80 9.96

7.15 4.55 c

4.067 4.104 d

28 45

Significantly different with the year 1981 (at 1% level). Not significantly different with the year 1981 (at 5% level). c Significantly different with the year 1987 (at 1% level). d Not significantly different with the year 1987 (at 5% level). b

Maximum Mean

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indicating an increase of about 91% and 173% during the period of 2 and 4 years after installation of the subsurface drainage. The mean value of water table depth increased by 39%, whereas soil salinity and SAR value decreased by about 25% over the study period. The hypotheses of equality of means and equality of variances of soil±water properties between each year were tested using the Student's t and F statistical tests. Results indicated significant differences (at 1% level) for the means and the variances of soil± water properties after the installation of subsurface drainage (Table 1). However, depending upon soil properties and soil depth, no significant differences were found (at 5% level) for most of soil±water properties between 1987 and 1989 after the drainage installation. Such results show distinct variations among the soil±water properties as affected by the subsurface drainage. The coefficient of variation (CV) of saturated hydraulic conductivity was highest among the soil±water properties. It decreased from 126% in October 1981 to 63% in April 1989. Variation of soil salinity as indicated by the CV was observed to increase with depth for 1981 and 1987, whereas for 1989 it decreased. In contrast, variation of SAR decreased with depth in 1987 and increased with depth in 1989. On the other hand, the CV of water table depth was the same over the study period. These observations revealed that soil±water properties improved non-uniformly during the study period. Providing the area with subsurface drainage leads to control of the water table and effective leaching of salts present in the soil or introduced with the irrigation water. This leaching process leads to low soil salinity and sodicity (Table 1) which result in an increase of the saturated soil hydraulic conductivity. Moreover, after drainage installation, the clay soil of the study area is on the average drier than before, leading to an increase of biological activity and macropore areas in the soil which have great influence on increase of saturated soil hydraulic conductivity. Clay soils often show an increase of saturated hydraulic conductivity after subsurface drainage installation. El-Mowelhi and van Schilfgaarde (1982) showed that saturated hydraulic conductivity of an area in the Nile Delta of Egypt increased from 0.075 to 0.085 m/day in average after 2 years of subsurface drainage installation. 3.2. Performance of semi-variogram and covariance variogram To investigate the performance of the semi-variogram and the covariance variogram in inference of spatial structure, plots of sample semi-variogram and sample covariance variogram versus lag (h) of the 61 grid samples on saturated hydraulic conductivity and water table depth in 1981 (K81, WTD81) are presented in Fig. 2. As shown in this figure, the sample covariance variogram is much smoother than the semi-variogram. However, the two functions produce almost the same nugget value, which is noticeably high. Due to the relatively narrow width of the grid samples in the E±W direction, the omnidirectional semi-variogram becomes directional (N±S) at large lags. The significant decrease of the semi-variogram of saturated hydraulic conductivity data set at large lags (Fig. 2(a)) is probably due to the existence of zones along the N±S direction (as shown later by the kriged map) which may suggest the presence of a weak nonstationarity at large lags in this direction. The covariance variogram obviously eliminated this effect and produced better interpretable spatial structures than the semi-variogram since it

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Fig. 2. Sample semi-variogram and sample covariance variogram of saturated hydraulic conductivity (a) and water table depth (b) of October 1981.

incorporates the possible nonstationarity in the lag means and combines products of data values rather than squared differences, leading to more stability at large lags. The frequency distribution of saturated hydraulic conductivity is high positively skewed distribution. Based on the cumulative frequency distribution function for a normal population, observations of saturated hydraulic conductivity was found to be lognormally distributed when the plot of its logtransform values ranked on increasing order versus the normal standard deviates, is linear. Since the poor performance of the semi-variogram was found due to presence of a weak nonstationarity at large lags as

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mentioned before, we concluded that the skewed distribution did not contribute significantly to the poor performance of the semi-variogram. However, Armstrong (1984) presented the skewed distribution as one of the common problems seen in semivariogram estimation. Therefore, semi-variogram estimation from a natural logarithm transformation of saturated hydraulic conductivity data set was performed. The covariance variogram yielded a better interpretable spatial structure than the semivariogram of the transformed data (Fig. 2(a)), indicating that the covariance variogram can be used to squeeze the effect of high values of data set and to reduce the kriging analysis associated with the semi-variogram of the natural logarithm transform of the data set. The covariance variogram, as mentioned before, incorporates possible nonstationarity in the data set. However, when using it in the kriging estimation, stationarity must be assumed. Discounting the large lag distances, the semi-variogram and the covariance variogram presented in Fig. 2 show little differences indicating stationarity at the other lags. Olea (1974) indicated that treatment of nonstationarity in the data set for kriging estimation should be used if it represents a systematic appearance in the data set rather than at sporadic details. Burgess and Webster (1980a), Yost et al. (1982) and Trangmar et al. (1986) indicated that nonstationarity at large lags does not significantly affect kriging estimation, provided stationarity exists within the range of the variogram. Furthermore, Yost et al. (1982) and Trangmar et al. (1985) indicated that ordinary kriging is quite robust to the presence of even strong nonstationarity in the data set. Ordinary kriging was therefore used to assess the performance of the semi-variogram and the covariance variogram for estimation. A cross-validation technique was performed with K81 data set for both functions. The estimates obtained with the covariance variogram have a lower standard error than those obtained with the semi-variogram (Table 2). Thus, a spatial model built with the sample semi-variogram is expected to have a higher estimation error than one built with the sample covariance variogram. Moreover, there is one strong advantage in using the covariance, which is related to the numerical method used to obtain a more robust solution of the kriging system. This is mainly due to the fact that the diagonal of the kriging matrix, based on semi-variogram contains zeros which requires the so-called `pivoting' (Shoup, 1983) to be performed prior to the actual solution to avoid division by zero. On the other hand, when the covariance is used, the zeros of the main diagonal of the kriging matrix are replaced by the variance, and therefore has the largest numerical values in the matrix since the covariance function is a decreasing function. This does not require pivoting to solve the matrix, and thus, may yield savings in terms of computer time and lead to combat the computational difficulties involved with the use of semi-variogram.

Table 2 Standardized residuals of K81 data set with semi-variogram and covariance variogram Spatial function

Average error

Standard deviation of the error

Semi-variogram (h) Covariance variogram *(h)

0.008 ÿ0.009

1.131 1.012

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The foregoing results reveal that the covariance variogram provides a better method of identifying the spatial structure and performing interpolation of data sets than the semi-variogram. These results are supported by the results of Isaaks and Srivastava (1988) who found from both theoretical and practical considerations that the covariance function is better than the semi-variogram for most geostatistical problems. The covariance variogram was therefore used to evaluate the change in the spatial structures of soil±water properties as affected by subsurface drainage and to perform kriging estimation. 3.3. Spatial variability of soil±water properties Directional variograms of each property were computed from Eq. (4) (at the azimuths 08, 458, 908 and 1358, with angular regularization in each direction of 458) and indicated an isotropic behavior. Similar isotropic behavior was also observed by Gallichand et al. (1991) and Gallichand and Marcotte (1993) for saturated hydraulic conductivity and clay content in the Nile Delta of Egypt. Figs. 3 and 4 present the estimated variograms of saturated hydraulic conductivity and water table depth at different separation distance h with spherical and exponential models fitted to them, respectively, for the 3 years

Fig. 3. Values of covariance variogram and the fitted spherical model of saturated hydraulic conductivity of October 1981, January 1987 and April 1989.

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Fig. 4. Values of covariance variogram and the fitted exponential model of water table depth of October 1981, January 1987 and April 1989.

of the study. Key parameters of the model fitted for each soil property are given in Table 3. The nugget effect (c0), a measure of variability within the sampling distance and other experimental uncertainties, showed noticeably high values for all the soil±water properties and soil depths. Higher values were observed after the drainage installation with relatively nonuniform variation among the soil±water properties throughout the study period. However, this variation could be reduced and be smoother with the years as in the case of K, WTD and (SS)2 (Table 3). The high nugget effect after the drainage installation may suggest the increase of short-scale soil variation in the study area with subsurface drainage. Large nugget values with a high sill value are common for the variograms as indicated by Chang et al. (1988). The sill (cs) and the structured heterogeneity component (c1) increased after installation of the subsurface drainage. Relative structured heterogeneity (c1/cs) in general had higher values after the installation of subsurface drainage. Depending upon the soil property and depth, it varied from 16% to 32% before the drainage installation with an average of 22.5%, and from 28% to 78% after the installation, with an average of 44.1%. However, higher values in 1987 than in 1989 were observed for some cases indicating non- uniform distribution over the study period.

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Table 3 Spatial dependence of soil±water properties of the study area Variable

Model

c0

c1

Saturated hydraulic conductivity K81 Spherical 0.014 K87 Spherical 0.041 K89 Spherical 0.022 Water table depth WTD81 Exponential WTD87 Exponential WTD89 Exponential Groundwater salinity GWS87 Spherical GWS89 Spherical Soil salinity 0±0.25 m (SS)1 81 (SS)1 87 (SS)1 89 0.25±0.50 m (SS)2 81 (SS)2 87 (SS)2 89

549 1285 339

cs

0.004 0.039 0.016 255 665 1196

0.018 0.080 0.038 804 1950 1535

a (m)

c1/cs (%)

Search S1 radius (m)

S2

2700 2900 3000

22.2 48.8 42.1

1500 1500 1500

ÿ0.009 0.012 0.001

1.005 1.002 0.998

3000 3800 3800

31.7 34.1 77.9

1500 1500 1500

ÿ0.006 ÿ0.009 ÿ0.001

1.003 1.001 1.001

4.700 4.565

3.658 3.094

8.358 7.659

3000 3500

43.8 40.4

2000 1500

ÿ0.015 0.006

1.001 1.007

Spherical Spherical Spherical

0.161 0.324 0.357

0.040 0.167 0.253

0.201 0.491 0.610

2500 2700 3400

19.9 34.0 41.5

1500 1500 2000

0.006 ÿ0.018 0.019

1.001 1.009 1.009

Spherical Spherical Spherical

0.229 0.363 0.215

0.044 0.325 0.138

0.273 0.688 0.353

2500 3000 3000

16.1 47.2 39.1

1500 1500 1000

ÿ0.009 ÿ0.003 0.019

1.007 1.005 1.003

38.3

1750

ÿ0.008

1.003

57.6 28.3

1500 1500

0.003 0.008

1.009 1.005

SAR-value 0±0.25 m (SAR)1 87 Spherical (SAR)1 89 0.25±0.50 m (SAR)2 87 Spherical (SAR)2 89 Spherical

1.443 0.896 2.339 3000 (pure nugget effect) at a value of 1.30 1.729 3.329

2.348 1.311

4.077 4.640

3000 4000

The correlation range (a) is the distance within which a soil property of two samples are related and beyond which they are independent. It provides therefore an estimation of areas of similarity. As shown in Table 3, this range shows continuous increase over the years. It varied from 2500 to 3000 m before installation of the subsurface drainage and from 2700 to 4000 m after drainage installation. These ranges are large relative to the distance over which soil±water properties are sampled (500 m). Therefore, sampling for soil±water properties can be relatively widely spaced. This argument may be supported by the results of Gallichand et al. (1992) who found that the optimum sampling density of saturated hydraulic conductivity in the Nile Delta can be determined from a preliminary survey with a grid spacing of 900 m rather than 500 m. Agrawal et al. (1995) have found that the correlation range of soil salinity decreased over the years after installation of subsurface drainage in a sandy loam soil. These observations indicate that the maximum sampling distance for which observations remain spatially correlated and can be used in designing soil sampling schemes is increasing after the subsurface drainage of clay soils as compared to sandy soils.

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Although we would expect spatial correlation scale to increase after drainage installation within the drain spacing, a larger scale was observed. The natural spatial variability of soil±water properties is a complex process consisting of many scales of variation superimposed upon one another and their interpretation is complicated by the fact that the variograms and spatial correlation range of a single soil property may well be a convolution sum of variograms of the various formative factors (Yost et al., 1982). Burrough (1983) mentioned that large and less gradual variations are more common in regions subject to changes by the human influence, e.g., subsurface drainage installation. The increase of biological activity and saturated hydraulic conductivity after the installation of subsurface drainage leads to improve part of the soil structure and to an increase of the areas of similarity over the study area, resulting in an increase of the correlation scale of soil±water properties. However, this variation could change over the soil depth. This is the case for surface and sub-soil layers of soil salinity (Table 3). The variation of the correlation range was larger after subsurface drainage in the sub-soil (0.25±0.50 m) than in the surface soil (0±0.25 m) in 1987, indicating greater continuity of soil salinity in sub-soil than in surface soil, while the reverse is true in 1989. This is probably due to insufficient time (4 years) after the drainage installation to smooth these variations by weathering or biological activity associated with installation of subsurface drainage. Furthermore, lowering of the water table depth as a result of the subsurface drainage installation can cause soil subsidence. This subsidence often varies in clay soils over relatively short distances, depending upon variation in the thickness and softness of the subsiding layer. The presence of these variation zones increases variability of soil samples and spatial dependence of macropore areas which may increase in turn the spatial correlation range of soil±water properties, particularly that of saturated hydraulic conductivity. This result reaffirmed the high nugget effect in the study area as affected by such short-scale variations as mentioned earlier. On the other hand, increase of values of the spatial properties after drainage installation was associated with an increase of the variance, indicating that this increase may be partially affected by the sampling schemes. This larger structural variation after drainage installation can be partially attributed to the variation in relative location of the grid samples to the drainage system. The variation was relatively higher in the water table depth than in the other soil properties (Table 3) as it may be affected by the elliptical shape of the water table variability between the lateral pipes. Comparing the short-scale variation with the increase of structural variation, we suggest that the soil sampling schemes should be designed, after installation of subsurface drainage, according to short-scale variation in the soil conditions. In summary, installation of subsurface drainage highly affected the spatial variability of soil±water properties after 4 years of installation. Further study of long-term variation after installation of subsurface drainage is suggested as such variation is expected to be more smooth and the true scale of spatial properties may be identified. The variogram of SAR for the surface soil (0±0.25 m) of the year 1989 showed a lack of regionalized relationship. Covariance was independent of lag distance and indicates a pure nugget effect at a value of 1.30. In this case, further samples are required at closer

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spacing to identify adequate structure. The nugget effect, structured heterogeneity component, and the sill of SAR were highest in the sub-soil (0.25±0.50 m). Similar results were also obtained by Samra et al. (1990). 3.4. Neighborhood search The search radius used to define the neighborhood for estimation determines the number of points used in the estimation and was obtained by a procedure similar to that of Vieira et al. (1981). Four locations were arbitrarily selected in the study area and were kriged based on all the neighbors within an increasing search radius until variations of the kriged estimates and estimation standard deviations were negligible. The resulting search radii are presented in Table 3 and correspond to approximately 12±35 points. Increasing the radius beyond these radii did not change the estimates or standard deviations significantly since in a regularly spaced square grid, the four closest neighbors will constitute a large percentage of the weighting coefficients (Alemi et al., 1988). In this case the four closest neighbors contributed about 65% to the weighting coefficients. 3.5. Cross-validation and kriging estimation Using the cross-validation procedure, each observation was estimated using the search radius and the best fitted model. The reduced mean error, S1, and the reduced variance, S2, were calculated (Table 3). The mean difference between the observed and estimated values and the reduced variance were approximately equal to zero and 1.0, respectively. This indicates that the covariance variogram is appropriate and confirms the results of other works (Journel and Huijbregts, 1978; Yost et al., 1982; Trangmar et al., 1986), e.g., that nonstationarity at large lags does not significantly affect kriging estimation. Kriging, which accounts for spatial patterns of variation, can be used to obtain a clear picture of the spatial distribution of soil±water properties before and after installation of the subsurface drainage. Since the nugget effect of soil±water properties is large, the use of punctual kriging may produce undesirably large estimation variances. These shortcomings of punctual kriging can be avoided by interpolation over areas, using block kriging which provides more reliable and smoother maps with smaller estimation variances (Samra et al., 1988; Burgess and Webster, 1980b). Thus, soil±water properties were interpolated at 100 meters intervals by block kriging using the search radii and variogram functions. For the study area with 61 observed points, 2720 data values were interpolated. The accuracy of block kriging depends on how to discretize the block being estimated. Since the spatial continuity of the soil±water properties is isotropic, the grid of discretizing points was regular. 20 block centers were arbitrarily selected for which values were kriged using ordinary block kriging based upon increasing the number of points used to discretize 100100 m2 block. Grid sizes of 22, 33 and 44 containing 4, 9 and 16 points were used for block estimates. The results indicate that the block estimates and the standard error of estimation did not show significant differences (0.01%) within the three grid sizes. A grid size of 22 (5050 m2) containing four points was therefore used for block kriging computations.

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Fig. 5. Block kriged contour map of saturated hydraulic conductivity (m/day) of October 1981 (a), January 1987 (b), and April 1989 (c).

3.6. Mapping of kriged soil±water properties Kriged contour maps of the 3 years' data sets were used to diagnose the changes in soil±water properties after installation of the subsurface drainage. The block kriged contour maps of Fig. 5 indicate a distinct non-uniform increase of saturated hydraulic conductivity over the study period. Its distribution over the area was non-uniform and characterized by presence of zones along the northern direction. This reflects the poor performance of the semi-variogram at large lags as discussed earlier. The existence of these zones in January 1987 was higher than April 1989 which may explain the differences of relative increase of spatial correlation range after the drainage installation in January 1987 and April 1989 comparing to October 1981 before the drainage installation. The contour maps of January 1987 of saturated hydraulic conductivity (Fig. 5(b)), soil salinity (Fig. 6(a)), SAR value at depth of 0±0.25 m (Fig. 6(b)), and groundwater salinity (Fig. 6(c)) showed good relations between these soil±water properties. The highest values of saturated hydraulic conductivity approximately coincide with low soil and groundwater salinity and SAR value. Water table depth increased after installation of subsurface drainage as shown in Fig. 7 indicating good performance of subsurface drainage. This improvement was relatively uniform (Fig. 7(b) and (c)). However, there is a small area that had a water table depth less than 1.0 m in April 1989. This is probably due to some irrigation management and degree of soil stratification in the study area.

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Fig. 6. Block kriged contour map of January 1987 for soil salinity (dS/m) at depth 0±0.25 m (a), SAR value at depth 0±0.25 m (b), and groundwater salinity (dS/m) (c).

Fig. 7. Block kriged contour map of water table depth (cm) of October 1981 (a), January 1987 (b) and April 1989 (c).

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Kriged contour maps of soil salinity, SAR value and groundwater salinity (not shown) showed a decrease of soil salinity and SAR value and an increase of groundwater salinity over the study period. The soil salinity decreased with a non-uniform distribution after drainage installation. The soil was found as non-saline soil (0±2 dS/m) with no effect on crop yields (Van Hoorn and Van Alphen, 1994). SAR decreased with a relatively uniform distribution from 1987 to 1989 indicating a continuous improvement of soil stability after the installation of subsurface drainage. In contrast, groundwater salinity increased with a non-uniform distribution over those 2 years. This is mainly due to the continuous salt leaching processes from the surface layers caused by the subsurface drainage. 4. Conclusion This study revealed that the covariance variogram is superior to the semi-variogram in describing the spatial variability of soil±water properties of clay soils. The covariance variogram was found to be a useful tool to identify spatial structure and for interpolation of soil±water properties data sets. The skewed distribution of a soil property does not contribute to a poor performance of the semi-variogram as recognized by other researchers before. However, covariance variogram shows a stronger robustness to high values of the data set, resulting in smoother and clearer spatial distributions, as compared to the semi-variogram. Spatial structures of soil±water properties were characterized by a high nugget effect most likely due to random variability of heterogeneity and measurement errors. Spatial variability of soil±water properties was highly affected by the subsurface drainage. The spatial correlation range and relative structured heterogeneity increased after drainage installation, indicating an increase of the areal extent of the field over which two samples of a soil property are related (an increase of areas of similarity) and an increase of inherent heterogeneity of soil samples. The nugget effect also increased after drainage installation, most likely due to the increase of short-scale variations caused by subsurface drainage. Varied spatial distributions were also observed among the soil±water properties and the study years. Further study of long-term spatial variations of soil±water properties as affected by subsurface drainage is required to identify the true scale of the spatial properties.

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