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DIVISION S-6—SOIL & WATER MANAGEMENT & CONSERVATION

Comparing Ordinary Kriging and Cokriging to Estimate Infiltration Rate

Dep. of Soil Science, Faculty of Agriculture, Gaziosmanpasa Univ., 60250 Tokat, Turkey

^* Corresponding author (sersahin{at}gop.edu.tr).

	ABSTRACT

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Measuring infiltration within a landscape is important because it is one of the key processes controlling water budgets for the agricultural production and transport processes in the soil profile. Estimation of this process at an acceptable level of accuracy is important, especially in the case when it exhibits high variability, since its measurement is a time- and labor-consuming procedure. This study was conducted to evaluate and compare kriging and cokriging to estimate infiltration rate (IR) using limited available data on an 8.5-ha alluvial field (loamy mixed mesic Ustifluvent). Infiltration tests were conducted using double-ring infiltrometers until steady-state IRs were obtained at nodes of an irregular grid consisting of 24 columns and four rows. Values of field-measured IR ranged from 1.92 to 8.88 cm h^-1, with a mean of 5.11 cm h^-1. Bulk density of subsoil was significantly related to IR and, therefore, helped estimation of IR values at unobserved locations. Both kriging and cokriging adequately estimated IR when 50 observed IR values were used with both estimators (mean reduced error was 4.15 x 10^-3 for kriging and 4.10 x 10^-3 for cokriging). To determine the minimum number of observed IR values needed to estimate IR without losing significant spatial information, kriging and cokriging were repeated with 45, 40, 35, and 30 observed IR values. Forty-five measured IR values with kriging, and 40 values with cokriging were adequate to estimate IR. Therefore, cokriging was found superior to kriging in estimating IR in the case of limited available data.

Abbreviations: AWC, available water content • CV, coefficient of variation • D_b, bulk density • IR, infiltration rate

	INTRODUCTION

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INFILTRATION is one of the key processes controlling yields of crops and transport of water and chemicals in the soil profile. Infiltration rate on a landscape may vary from very low to very high because of variability in the soil physical characteristics (Jensen et al., 1987). Bosch and West (1998) found that infiltration characteristics varied on a transect across a 1-ha field with sand and loamy sand soils. A recent study (Rockström et al., 1999) showed that temporal and spatial variability in the percentage of infiltration, ratio between infiltration of rainfall (mm) and total rainfall (mm) x 100, greatly influenced the variability in the yields of pearl millet (Pennisetum glaucum L.) in a farmer's field in semi-arid Niger.

Finer detail and greater certainty are needed in simulation modeling of chemical transport and irrigation studies with precision farming programs. However, since measuring infiltration in a field is time- and labor-consuming work, it is difficult to estimate infiltration values at unobserved sites with an acceptable level of accuracy, especially when spatial variability of this property is high. This requires quantifying the spatial information for IR.

Spatial variability in IR can be assessed quantitatively and qualitatively using semivariograms calculated from the spatial data on IR. Moreover, a close relationship between infiltration and easily measured soil variables may be exploited to estimate infiltration with a reasonable accuracy at unobserved sites in the field. Ordinary kriging (or simply kriging) and cokriging can sometimes be used for this purpose. While kriging uses the spatial information on infiltration, cokriging uses spatial information on infiltration along with spatial correlation between infiltration and an auxiliary variable to make estimations on unobserved sites (Vieira et al., 1981; Vauclin et al., 1983; Isaaks and Srivastava, 1989).

Vieira et al. (1981) characterized spatial variability of 1280 field measurements of infiltration rate on Yolo Loam (fine-silty, mixed, nonacid, thermic, Typic Xerortents) by applying geostatistical concepts. They found that the observations separated by 50 m or less were dependent on each other as indicated by the range of the semivariogram, and concluded that 128 field-measured values were sufficient to obtain essentially the same information as with 1280 values. Vauclin et al. (1983) used semivariograms for available water content (AWC), water content at -0.03 MPa (pF_2.5), and sand content; and cross-semivariograms for the spatial correlation between AWC and sand content and between pF_2.5 and sand content to estimate AWC and pF_2.5 at unsampled locations. They concluded that cokriging was superior to kriging in minimizing estimation variance. Greminger et al. (1985) found that small differences in the percentage of sand in the topsoil yielded significant differences between (h)-curves, and concluded that improved estimates of (h) across a field of the Yolo Loam could be achieved using geostatistical methods for lags <10 m. Yates and Warrick (1987) used bare soil temperature and the percentage of sand as auxiliary variables of gravimetric water content to estimate gravimetric water content on a 1-ha field. They found that cokriging gave better predictions than kriging when sample correlations exceeded 0.5 and when the auxiliary variable was oversampled. In a recent study (Triantafilis et al., 2001) ordinary kriging, regression kriging, three-dimensional kriging, and cokriging were compared on the basis of precision and bias in soil salinity estimates. Although regression kriging performed best overall, mean and standard deviation of ranks showed that cokriging ranked highest for these criteria. While many studies (Stein et al., 1988; Stein and Corsten, 1991; Zhang et al., 1992, 1997; Istok et al., 1993) showed superiority of cokriging to ordinary kriging, others (Shouse et al., 1990; Martinez, 1996) showed that cokriging was only minimally superior to ordinary kriging when auxiliary variables were not highly correlated to primary variables. This suggests that use of a correct auxiliary variable is important to obtain successful results from cokriging. In addition, to ensure the validity of the estimates made by kriging and cokriging, the semivariograms and cross-semivariograms of the variables used must accurately describe the spatial structures.

The objectives of this study were (i) to describe spatial variability in IR, (ii) to assess the spatial relationship between IR and some selected properties of topsoil (0–30 cm) and subsoil (30–60 cm), and (iii) to evaluate and compare the geostatistical procedures kriging and cokriging in estimating IR on unobserved sites in the case of limited available data on IR.

	REVIEW OF THE STATISTICAL METHODS

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Coregionalization
Detailed discussions on this topic can be found elsewhere (Journel and Huijbregts, 1978; Vauclin et al., 1983; Yates and Warrick, 1987; Isaaks and Srivastava, 1989); therefore, only a brief discussion will be given here.

The Semivariogram and Cross-Semivariogram Functions
The semivariogram and cross-semivariogram functions describe the spatial correlation. The estimator for the semivariogram and cross-semivariogram is

[1]

where, _ij is the semivariance (when i = j) with respect to random variable z_i, h is the separation distance, n(h) is the number of pairs of z_i(x_k) and z_j(x_k) in a given lagged distance interval of (h + dh). When i j, _ij is the cross-semivariogram which is a function of h (Yates and Warrick, 1987).

Semivariograms and cross-semivariograms were fit using the spherical (Eq. [2]) and Gaussian (Eq. [3]) models:

[2]

and

[3]

where, C₀ is the nugget variance, C₁ is the sill, a is the range, and h is the lagged distance.

The spatial distribution of IR along with soil properties was predicted by applying the best-fit mathematical functions of the semivariograms and cross-semivariograms. The version 1.3 of software package ToolBox (Froidevaux, 1990) was used to perform all the geostatistical computations.

	MATERIALS AND METHODS

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Materials
This study was conducted as a component of a project to investigate the spatial variability of nitrate leaching parameters on an 8.5-ha (850 by 100 m) level (1–2% slope) and well-drained alluvial field (loamy mesic Ustifluvent) located 25 km northeast of Downtown Tokat in Central Anatolia of Turkey. The topsoil (0–30 cm) is characterized by dark grayish brown color (10YR 4/2; dry); moderate fine to moderate medium granular structure; common fine and very fine roots; common fine pores; and a clear wavy boundary.

A plow layer is present between the 31- and 60-cm depths, which formed as a result of tillage. The plow layer is characterized by very dark grayish brown color (10YR 2/2; moist); moderate fine and weak fine platy structure; few root channels filled with dark material from the upper horizon; few, fine tubular pores; common fine Fe and Mn concretions; and many prominent dark brown (10YR 3/3; moist) argillans on ped faces.

The annual precipitation is 420 mm most of which falls between October and May, and the annual average temperature is 12°C. Winter wheat (Triticum aestivum L.) was grown in the study area in 1998 at the discretion of the farmer.

Methods
Soil Sampling and Infiltration Tests
The field was intensively sampled on a regular grid spacing of 25 m in March 1998. At each site, soil samples were obtained from topsoil (0–30 cm) and subsoil (31–60 cm). The soil sampling sites were marked for future use. All 280 samples were analyzed for clay, silt, and sand contents by the hydrometer method (Gee and Bauder, 1986), for organic matter content by the method of Nelson and Sommers (1982), for CaCO₃ content and pH with a Scheibler Calcimeter (McLean, 1982), and water contents at -0.033 and -1.5 MPa soil water pressure were determined with a pressure plate apparatus (Klute, 1986). In addition, at each test site, undisturbed soil samples were obtained from the topsoil and the subsoil using 100-cm³ steel cores to determine soil bulk density (Blake and Hartge, 1986).

Infiltration tests were conducted during 1 Apr. through 15 Apr. 1998 at 50 sampling sites where soil samples were obtained on irregular grids of 24 columns and four rows. These were done with double-ring variable-water level infiltrometers (the internal diameter was 30 cm for inner and 60 cm for outer ring) until final (steady state) IR was reached. The infiltration test sites along with sampling points are presented in Fig. 1 .

View larger version (10K): [in this window] [in a new window]   Fig. 1. Layout of the experiment: x represents locations of soil sampling and represents locations of soil sampling plus infiltration tests.

Geostatistical Analysis
Soil properties highly correlated (P < 0.05) with IR at h(0) were selected as potential auxiliary variables for IR for use in the cokriging procedure.

Sample semivariogram and cross-semivariogram functions for the IR and potential auxiliary variables were calculated (Isaaks and Srivastava, 1989). Hypothetical semivariogram models were fit to experimental semivariograms and cross-semivariograms. The best models were determined by the leaving-one-out method of cross validation (Yates and Warrick, 1987; Shouse et al., 1990). To cross validate we remove one known value at a time from the data set and estimate this value from a neighborhood around it but not itself. This procedure allows comparisons of the different models and search strategies on the results of interpolation (Goovaerts, 1997). Directional variograms for 0, 45, 90, and 135° were calculated to check geometric anisotropy (David, 1977; Isaaks and Srivastava, 1989).

Spatial Estimations
Kriging and cokiriging procedures were applied to estimate IR at unobserved locations of the field, using the version 1.3 of Geostatistical Toolbox (Froidevaux, 1990). The IR values were estimated within a 25 by 25 m grid using three approaches. (i) kriging using 50, 45, 40, 35, and 30 measured values of IR, (ii) cokriging using 50 measured IR values along with 60, 70, 80, 90, 100, 120, and 140 measured bulk density (D_b) values, and (iii) cokriging using 50, 45, 40, and 35 measured IR values along with 120 D_b values. To determine the data points to omit, the study area was divided into five equal subareas, and each time a datum was removed randomly from each subarea. The estimates in both kriging and cokriging with original and reduced data sets were subjected to the procedure cross validation that resulted in the smallest neighborhood (Vieira et al., 1981). Each time, mean reduced error and mean reduced variance calculated by cross validation test were considered, and the normality of residuals from cross validation was tested to check the kriging and cokriging estimations. The maps of original data set, estimates from kriging and cokriging with original and reduced data, and residuals calculated at the corresponding cross validation tests were compared to qualitatively evaluate the estimations.

A maximum of eight and a minimum of four neighboring data points were used along with semivariograms and cross-semivariograms within a search ellipse with a radius of 150 m in both kriging and cokriging estimations.

	RESULTS AND DISCUSSION

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Characterization of Soil Properties and Infiltration Rate Values
The average textural classification of both topsoils (0–30 cm) and subsoils (31–60 cm) was loam. However, in some areas of the field, loamy clay, and silty clay textures were present in topsoil, and loamy clay in the subsoil. The average organic matter content was higher in the topsoil, and the average bulk density was higher in the subsoil probably due to the plow layer (Tables 1 and 2).

The field-measured IRs were moderately variable with a mean of 5.11 cm h^-1 and a coefficient of variation (CV) of 36.5% (Table 3). Vieira et al. (1981) found a mean of 6.98 mm h^-1 and a CV of 39.9% for 1280 field-measured IRs. Sisson and Wierenga (1981) measured IR in a field plot with infiltrometers having diameters of 5, 25, and 127 cm. In their study, IR exhibited a lognormal distribution for all ring sizes. In this study, the Shapiro-Wilks normality test (results not shown) and value of kurtosis indicated that IR values were normally distributed with a relatively small variance (Table 3).

The IR values were spatially dependent over a distance of approximately 165 m (Table 5 and Fig. 2) . Vieira et al. (1981) reported a range of 50 m for 1280 measured IR values within a 160 by 55 m field. The geostatistical range of values calculated for IR in the present study was far greater than that calculated by Vieira et al. (1981). The geostatistical range of values obtained for IR and subsoil D_b were greater than the distance between any two nearby test sites and thus could provide a useful information about the spatial structure of IR and subsoil D_b.

View larger version (13K): [in this window] [in a new window]   Fig. 2. Experimental semivariograms for infiltration rate (IR) and subsoil bulk density (Db), and cross-semivariogram for IR/Db. Solid lines represent the spherical model fitted to experimental values.

Reynolds and Zebchuk (1996) characterized spatial relationship between soil texture, organic C, and surface topography, and values for field measured hydraulic conductivity (K_f) on a 1.2-ha texturally uniform silty clay soil that had stable soil structure. They concluded that K_f was primarily affected by a well-developed and stable soil structure, and not by the texture, organic C, or surface topography. They further found a close spatial relationship between initial water content and infiltration capacity. In this study, although a close relationship was detected between subsoil D_b and IR, no clear spatial relationships were apparent between soil texture and IR.

Estimation
Kriging and cokriging procedures were used along with isotropic semivarograms and cross-semivariogram to estimate IR values at 90 unobserved points. Based on the results from the cross validation procedure, a search ellipse with a radius of 150 m was used in both kriging and cokriging estimations, and a minimum of four and a maximum of eight data values within the search ellipse were included in the estimations of which results shown in Table 6.

[4]

should be close to zero, and the reduced variance defined by

[5]

should be close to unity (Vauclin et al., 1983). In the Eq. [4] and [5], _e and S²_Re are mean reduced error and reduced variance, respectively; z(x_i) and z*(x_i) are the observed and estimated values of the variable z at the location (x_i); and _k(x_i) is the estimation variance. Both parameters are important measures of estimation accuracy (Yates and Warrick, 1987).

View larger version (52K): [in this window] [in a new window]   Fig. 3. (a) Spatial patterns in the observed values of infiltration rate (IR), (b) and (c) kriging-estimated values with 50 and 45 field–measured IR values, respectively; and (d) cokriging-estimated values with 40 field-measured FIR values.

The cokriging procedure was applied to determine whether any advantage could be gained over kriging. The best auxiliary variable was determined using the cross validation. Cokriging was then repeated with 60, 70, 80, 90, 100, 120, and 140 measured D_b values along with 50 observed IR values to estimate IR at 90 unobserved locations. Each time, the mean reduced error and reduced variance from cross validation were calculated and normality of the residuals was tested to ensure the validity of the estimations. Results of cross validation showed that as the number of D_b supplementing the 50 IR values in the cokriging procedure gradually increased from 60 to 120, the mean reduced error generally decreased slightly at each step. However, the mean reduced error and reduced variance calculated with 140 D_b values was identical to that calculated with 120 D_b values. The mean reduced error calculated from the cokriging cross validation results were slightly less than that calculated from those of kriging (Table 6). Furthermore, the spatial pattern in the cokriging estimated values (data not shown) were quite similar to the Fig. 3b where the spatial pattern in the kriging estimated values are seen. These suggested that cokriging had no advantage over kriging when full data set used in the calculations.

To determine whether cokriging has any advantage over kriging when data are limited, the cokriging procedure was repeated using 120 D_b values along with 45, 40, 35, and 30 IR values to estimate IR at 95, 100, 105, and 110 unobserved points, respectively, assuming that 5, 10, 15, and 20 measured IR values were missing. Similarly, the kriging procedure was repeated with same reduced data sets used with cokriging. Each time, the mean reduced error and reduced variance were calculated and the residuals from cross validation by both procedures were subjected to a normality test to ensure the validity of the assumptions made with kriging and cokriging estimations. The results are shown in the Table 6 and Fig. 3c and 3d.

Figure 4 shows that the pattern of the residuals calculated by kriging with original data set was quite similar to those calculated by kriging with 45, and by cokriging with 40 measured IR values. When the data used with kriging and cokriging were further reduced, they yielded residual maps highly different from that in Fig. 4a. This suggests that at least 40 measured IR values were needed with cokriging and 45 with kriging to maintain acceptable accuracy in estimating IR for the study area. A significant amount of spatial information was lost when IR was estimated by kriging with the reduced data set of 40 values. However, spatial information was maintained when cokriging was used instead of kriging. Therefore, the moderate spatial relationship between IR and D_b helped maintain the spatial information in IR at this point. However, with further reducing in data, cokriging could not maintain the spatial information in IR from that using the original data set.

View larger version (50K): [in this window] [in a new window]   Fig. 4. Spatial patterns in the residuals (measured-estimated) (a) calculated by kriging with 50 field measured infiltration rate (IR) values, (b) calculated by kriging with 45 field measured IR values, and (c) calculated by cokriging with 40 field measured IR values and 120 bulk density values.

To assess the effect of cross correlation between D_b and IR on the accuracy of the cokriging estimation in the reduced data sets, the cross correlation test was repeated with 50, 45, 40, 35, and 30 D_b and IR values. The resulting linear correlation coefficients were -0.60, -0.60, -0.58, -0.52, and -0.47 for the 50, 45, 40, 35, and 30 data points, respectively. All of the corresponding P values were <1.0 x 10^-3. The cross correlation results showed that the relationship between primary (IR) and auxiliary (D_b) parameters was maintained in the reduced data sets, and that it was more obscure with less data.

Kriging estimation variance (Isaaks and Srivastava, 1989, p. 278–322) is another good indicator of estimation accuracy. Results showed that as the number of observed data points decreased, the kriging and cokriging estimation variances gradually increased (Table 6). The estimation variance increased more rapidly in the kriging estimation than in the cokriging estimation due to the contribution from auxiliary variable (D_b).

	CONCLUSIONS

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Spatial variability in field-measured IR and soil properties having significant spatial correlation to IR were studied using kriging and cokriging procedures. Percentage of lime, silt, bulk density, water content at -1.50 MPa soil water potential in topsoil, and soil water contents at -0.03 and -1.50 MPa soil water potentials in subsoil were significantly correlated to IR within distances ranging from 165 to 215 m. Results from cross validation revealed that subsoil bulk density was the most representative auxiliary variable of IR.

The results showed that cokriging provided no advantage over kriging when data were sufficient. With kriging, 45 observed IR values were sufficient to obtain the same information as 50 observations. However, using cokiging with 120 bulk density values, 40 observed values of IR were sufficient to obtain the same information from that obtained with 50 field measurement of IR. This indicates that cokriging was more successful than kriging when IR is undersampled.

Infiltration is widely used in modeling of water and chemical transport in soils and irrigation practices. The spatial variability of this process on a landscape is important, affecting the accuracy of the modeling work and efficiency of the irrigation practices. However, measuring infiltration in the field is time- and labor consuming. Therefore, estimation of this process with a reasonable accuracy given a minimal observed values is very important. The result of this experiment illustrated the possibility of using kriging and cokriging.

	ACKNOWLEDGMENTS

 
The author thanks the Turkish Scientific and Technical Research Institute (TUBITAK) for the financial support to do this study (Project No: TOAG-TARP 1871).

Received for publication November 19, 2001.

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