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比较传统的Kriging法和Cokriging来估计入渗率

比较传统的Kriging法和Cokriging来估计入渗率
比较传统的Kriging法和Cokriging来估计入渗率

Comparing Ordinary Kriging and Cokriging to Estimate Infiltration Rate

Sabit Ersahin*

ABSTRACT

Spatial variability in IR can be assessed quantitatively and qualitatively using semivariograms calculated from Measuring infiltration within a landscape is important because it is the spatial data on IR.Moreover,a close relationship one of the key processes controlling water budgets for the agricultural between infiltration and easily measured soil variables production and transport processes in the soil profile.Estimation of may be exploited to estimate infiltration with a reason-this process at an acceptable level of accuracy is important,especially in the case when it exhibits high variability,since its measurement is able accuracy at unobserved sites in the field.Ordinary a time-and labor-consuming procedure.This study was conducted to kriging (or simply kriging)and cokriging can sometimes evaluate and compare kriging and cokriging to estimate infiltration be used for this purpose.While kriging uses the spatial rate (IR)using limited available data on an 8.5-ha alluvial field (loamy information on infiltration,cokriging uses spatial infor-mixed mesic Ustifluvent).Infiltration tests were conducted using dou-mation on infiltration along with spatial correlation be-ble-ring infiltrometers until steady-state IRs were obtained at nodes tween infiltration and an auxiliary variable to make of an irregular grid consisting of 24columns and four rows.Values of estimations on unobserved sites (Vieira et al.,1981;field-measured IR ranged from 1.92to 8.88cm h ?1,with a mean of Vauclin et al.,1983;Isaaks and Srivastava,1989).

5.11cm h ?1.Bulk density of subsoil was significantly related to IR Vieira et al.(1981)characterized spatial variability of and,therefore,helped estimation of IR values at unobserved locations.1280field measurements of infiltration rate on Yolo Both kriging and cokriging adequately estimated IR when 50observed Loam (fine-silty,mixed,nonacid,thermic,Typic Xeror-IR values were used with both estimators (mean reduced error was 4.15?10?3for kriging and 4.10?10?3for cokriging).To determine tents)by applying geostatistical concepts.They found the minimum number of observed IR values needed to estimate IR that the observations separated by 50m or less were without losing significant spatial information,kriging and cokriging dependent on each other as indicated by the range of the were repeated with 45,40,35,and 30observed IR values.Forty-five semivariogram,and concluded that 128field-measured measured IR values with kriging,and 40values with cokriging were values were sufficient to obtain essentially the same adequate to estimate IR.Therefore,cokriging was found superior to information as with 1280values.Vauclin et al.(1983)kriging in estimating IR in the case of limited available data.

used semivariograms for available water content (AWC),water content at ?0.03MPa (pF 2.5),and sand content;and cross-semivariograms for the spatial correlation be-I

nfiltration is one of the key processes controlling tween AWC and sand content and between pF 2.5and yields of crops and transport of water and chemicals sand content to estimate AWC and pF 2.5at unsampled in the soil profile.Infiltration rate on a landscape may locations.They concluded that cokriging was superior vary from very low to very high because of variability to kriging in minimizing estimation variance.Greminger in the soil physical characteristics (Jensen et al.,1987).et al.(1985)found that small differences in the percent-Bosch and West (1998)found that infiltration character-age of sand in the topsoil yielded significant differences istics varied on a transect across a 1-ha field with sand between ?(h )-curves,and concluded that improved esti-and loamy sand soils.A recent study (Rockstro ¨m et al.,mates of ?(h )across a field of the Yolo Loam could be 1999)showed that temporal and spatial variability in achieved using geostatistical methods for lags ?10m.the percentage of infiltration,ratio between infiltration Yates and Warrick (1987)used bare soil temperature of rainfall (mm)and total rainfall (mm)?100,greatly and the percentage of sand as auxiliary variables of influenced the variability in the yields of pearl millet gravimetric water content to estimate gravimetric water (Pennisetum glaucum L.)in a farmer’s field in semi-content on a 1-ha field.They found that cokriging gave arid Niger.

better predictions than kriging when sample correla-Finer detail and greater certainty are needed in simu-tions exceeded 0.5and when the auxiliary variable was lation modeling of chemical transport and irrigation oversampled.In a recent study (Triantafilis et al.,2001)studies with precision farming programs.However,ordinary kriging,regression kriging,three-dimensional since measuring infiltration in a field is time-and labor-kriging,and cokriging were compared on the basis of consuming work,it is difficult to estimate infiltration precision and bias in soil salinity estimates.Although values at unobserved sites with an acceptable level of regression kriging performed best overall,mean and accuracy,especially when spatial variability of this prop-standard deviation of ranks showed that cokriging erty is high.This requires quantifying the spatial infor-ranked highest for these criteria.While many studies (Stein et al.,1988;Stein and Corsten,1991;Zhang et mation for IR.

al.,1992,1997;Istok et al.,1993)showed superiority of cokriging to ordinary kriging,others (Shouse et al.,1990;Dep.of Soil Science,Faculty of Agriculture,Gaziosmanpasa Univ.,Martinez,1996)showed that cokriging was only mini-60250Tokat,Turkey.Received 19Nov.2001.*Corresponding author mally superior to ordinary kriging when auxiliary vari-(sersahin@https://www.wendangku.net/doc/3f12647441.html,.tr).

ables were not highly correlated to primary variables.

Published in Soil Sci.Soc.Am.J.67:1848–1855(2003). Soil Science Society of America

Abbreviations:AWC,available water content;CV,coefficient of vari-ation;D ,bulk density;IR,infiltration rate.

677S.Segoe Rd.,Madison,WI 53711USA

ERSAHIN:ESTIMATING INFILTRATION RATE WITH KRIGING AND COKRIGING

1849

This suggests that use of a correct auxiliary variable is MATERIALS AND METHODS

important to obtain successful results from cokriging.Materials

In addition,to ensure the validity of the estimates made This study was conducted as a component of a project to by kriging and cokriging,the semivariograms and cross-investigate the spatial variability of nitrate leaching parame-semivariograms of the variables used must accurately ters on an 8.5-ha (850by 100m)level (1–2%slope)and well-describe the spatial structures.

drained alluvial field (loamy mesic Ustifluvent)located 25km The objectives of this study were (i)to describe spatial northeast of Downtown Tokat in Central Anatolia of Turkey.variability in IR,(ii)to assess the spatial relationship The topsoil (0–30cm)is characterized by dark grayish brown color (10YR 4/2;dry);moderate fine to moderate medium between IR and some selected properties of topsoil granular structure;common fine and very fine roots;common (0–30cm)and subsoil (30–60cm),and (iii)to evaluate fine pores;and a clear wavy boundary.

and compare the geostatistical procedures kriging and A plow layer is present between the 31-and 60-cm depths,cokriging in estimating IR on unobserved sites in the which formed as a result of tillage.The plow layer is character-case of limited available data on IR.

ized 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 REVIEW OF THE STATISTICAL METHODS

tubular pores;common fine Fe and Mn concretions;and many prominent dark brown (10YR 3/3;moist)argillans on ped Coregionalization

faces.

The annual precipitation is 420mm most of which falls Detailed discussions on this topic can be found else-between October and May,and the annual average tempera-where (Journel and Huijbregts,1978;Vauclin et al.,1983;ture is 12?C.Winter wheat (Triticum aestivum L.)was grown Yates and Warrick,1987;Isaaks and Srivastava,1989);in the study area in 1998at the discretion of the farmer.

therefore,only a brief discussion will be given here.

Methods

The Semivariogram and Soil Sampling and Infiltration Tests

Cross-Semivariogram Functions

The field was intensively sampled on a regular grid spacing The semivariogram and cross-semivariogram func-of 25m in March 1998.At each site,soil samples were obtained tions describe the spatial correlation.The estimator for from topsoil (0–30cm)and subsoil (31–60cm).The soil sampling the semivariogram and cross-semivariogram is

sites were marked for future use.All 280samples were analyzed for clay,silt,and sand contents by the hydrometer method (Gee and Bauder,1986),for organic matter content by the method ?ij (h )?

?

n (h )k ?1

{[z i (x k ?h )?z i (x )][z j (x k ?h )

of Nelson and Sommers (1982),for CaCO 3content and pH with a Scheibler Calcimeter (McLean,1982),and water con-?z j (x )]}/2n (h )

[1]

tents at ?0.033and ?1.5MPa soil water pressure were deter-mined with a pressure plate apparatus (Klute,1986).In addi-where,?ij is the semivariance (when i ?j )with respect tion,at each test site,undisturbed soil samples were obtained to random variable z i ,h is the separation distance,n (h )from the topsoil and the subsoil using 100-cm 3steel cores to is the number of pairs of z i (x k )and z j (x k )in a given determine soil bulk density (Blake and Hartge,1986).

lagged distance interval of (h ?dh).When i ?j,?ij is Infiltration tests were conducted during 1Apr.through 15Apr.1998at 50sampling sites where soil samples were ob-the cross-semivariogram which is a function of h (Yates tained on irregular grids of 24columns and four rows.These and Warrick,1987).

were done with double-ring variable-water level infiltrometers Semivariograms and cross-semivariograms were fit (the internal diameter was 30cm for inner and 60cm for outer using the spherical (Eq.[2])and Gaussian (Eq.[3])models:

ring)until final (steady state)IR was reached.The infiltration test sites along with sampling points are presented in Fig.1.Descriptive Statistics

Descriptive statistics including minimum,maximum,mean,?(h )

?

C 0?C 1?

1.5h

a ?0.5

?h a

?3?

if h ?a C 0?C 1

if h ?a

[2]

and coefficient of variation were calculated for IR and soil properties.The Shapiro-Wilks normality test was conducted to test the hypothesis that assumes each property has a normal distribution.In addition,simple correlation coefficients between and

IR and each of topsoil and subsoil properties were calculated.?(h )?C 0?C 1?

1?exp ??h 2

a

2

??

[3]

Geostatistical Analysis

Soil properties highly correlated (P ?0.05)with IR at h (0)where,C 0is the nugget variance,C 1is the sill,a is the were selected as potential auxiliary variables for IR for use range,and h is the lagged distance.

in the cokriging procedure.

Sample semivariogram and cross-semivariogram functions The spatial distribution of IR along with soil proper-for the IR and potential auxiliary variables were calculated ties was predicted by applying the best-fit mathematical (Isaaks and Srivastava,1989).Hypothetical semivariogram functions of the semivariograms and cross-semivario-models were fit to experimental semivariograms and cross-grams.The version 1.3of software package ToolBox semivariograms.The best models were determined by the

1850

SOIL SCI.SOC.AM.J.,VOL.67,NOVEMBER–DECEMBER 2003

https://www.wendangku.net/doc/3f12647441.html,yout of the experiment:x represents locations of soil sampling and represents locations of soil sampling plus infiltration tests.

one known value at a time from the data set and estimate in some areas of the field,loamy clay,and silty clay this value from a neighborhood around it but not itself.This textures were present in topsoil,and loamy clay in the procedure allows comparisons of the different models and subsoil.The average organic matter content was higher search strategies on the results of interpolation (Goovaerts,in the topsoil,and the average bulk density was higher 1997).Directional variograms for 0,45,90,and 135?were in the subsoil probably due to the plow layer (Tables 1calculated to check geometric anisotropy (David,1977;Isaaks and 2).

and Srivastava,1989).Plant AWC was the most and pH was the least vari-able in the topsoil,while organic matter content was Spatial Estimations

the most variable and silt content was the least variable Kriging and cokiriging procedures were applied to estimate in the subsoils.Water contents at ?1.5MPa were more IR at unobserved locations of the field,using the version 1.3variable than that at ?0.03MPa.Similar results were of Geostatistical Toolbox (Froidevaux,1990).The IR values reported by Greminger et al.(1985),and Mallants et al.were estimated within a 25by 25m grid using three ap-(1996).

proaches.(i)kriging using 50,45,40,35,and 30measured values of IR,(ii)cokriging using 50measured IR values along The field-measured IRs were moderately variable with 60,70,80,90,100,120,and 140measured bulk density with a mean of 5.11cm h ?1and a coefficient of variation (D b )values,and (iii)cokriging using 50,45,40,and 35mea-(CV)of 36.5%(Table 3).Vieira et al.(1981)found a sured IR values along with 120D b values.To determine the mean of 6.98mm h ?1and a CV of 39.9%for 1280field-data points to omit,the study area was divided into five equal measured IRs.Sisson and Wierenga (1981)measured subareas,and each time a datum was removed randomly from IR in a field plot with infiltrometers having diameters each subarea.The estimates in both kriging and cokriging of 5,25,and 127cm.In their study,IR exhibited a with original and reduced data sets were subjected to the lognormal distribution for all ring sizes.In this study,procedure cross validation that resulted in the smallest neigh-the Shapiro-Wilks normality test (results not shown)borhood (Vieira et al.,1981).Each time,mean reduced error and value of kurtosis indicated that IR values were and mean reduced variance calculated by cross validation test were considered,and the normality of residuals from cross normally distributed with a relatively small variance validation was tested to check the kriging and cokriging esti-(Table 3).

mations.The maps of original data set,estimates from kriging and cokriging with original and reduced data,and residuals Spatial Relationships Between Soil Properties

calculated at the corresponding cross validation tests were and Infiltration Rate

compared to qualitatively evaluate the estimations.

A maximum of eight and a minimum of four neighboring No significant relationships were found between IR data points were used along with semivariograms and cross-and the percentage of sand,organic matter,and the semivariograms within a search ellipse with a radius of 150m pH values of topsoil and subsoil (data not shown).Per-in both kriging and cokriging estimations.

centage of CaCO 3,percentage of silt,and volumetric water content at ?1.50MPa soil water potential of top-RESULTS AND DISCUSSION soil;and bulk density,and volumetric water content at ?0.03and ?1.50MPa soil water potentials of subsoil Characterization of Soil Properties and

were significantly related to IR (P ?0.05)(Table 4).Infiltration Rate Values

Infiltration rate was highly influenced by the soil layer The average textural classification of both topsoils with lowest hydraulic conductivity in soil profile (Hillel,(0–30cm)and subsoils (31–60cm)was loam.However,

1980),which may explain the high negative correlation between IR and D b of subsoil indicated that the plow Table 1.Summary statistics for selected properties of topsoil.

layer reduced IR considerably in the study area.As

Soil property

Max.Min.Mean CV%?Sand,%46.9821.9433.4618.23Table 2.Summary statistics for selected properties of subsoil.

Silt,%52.2330.8342.0012.33Clay,%

31.0117.7924.6010.44Soil property

Max.Min.Mean CV%?Bulk density,Mg m ?3 1.43 1.03 1.258.16Water Content,m 3m ?3

Sand,%37.797.9320.9431.91Silt,%56.2535.2443.859.07?0.03MPa 47.3423.4736.7914.61?1.50MPa

34.567.5424.2720.06Clay,%

49.6422.0535.5014.57Water Content,m 3m ?3

PAWC?,m 3m ?321.39 4.1812.5225.43CaCO 3,%

20.957.2814.4217.15?0.03MPa 46.9315.6737.2417.58?1.5MPa

35.118.6525.8619.43Organic matter,% 5.61 1.91 3.8521.70PH

9.50

7.03

8.14

3.21

Bulk density,Mg m ?3 1.70 1.14 1.339.67

ERSAHIN:ESTIMATING INFILTRATION RATE WITH KRIGING AND COKRIGING

1851

Table 4.Simple correlation coefficients between field-measured Table 3.Summary statistics for field measured infiltration rate (cm h ?1).

infiltration rate (IR)values and properties of topsoil and subsoil.

Max.Min.Mean CV%?Skewness Kurtosis Soil variable

IR

Soil variable

IR 8.88

1.92

5.11

36.49

0.10

2.03

cm h ?1

cm h ?1

?CV,coefficient of variation.

Topsoil (0–30cm)Subsoil (31–60cm)

CaCO 3,%0.382**Bulk density,Mg m ?3?0.60**

indicated by Henderson and Haise (1987),this may re-?(?1.5MPa)?0.409**?(?0.03MPa)?0.452**sult in temporary excessively wet conditions during

Silt,%0.290*

?(?1.50MPa)?0.505**

heavy irrigation or rainfall that may cause injury to *Significant at the 0.05probability level.susceptible plants.The high positive correlation be-**Significant at the 0.01probability level.

?Volumetric water content (m 3m ?3)at ?1.5and ?0.03soil water pressure.

tween IR and water content at ?1.50MPa in subsoil is noteworthy.

clay soil that had stable soil structure.They concluded Table 5lists the geostatistical parameters for the IR that K f was primarily affected by a well-developed and and subsoil D b that were the most significantly related stable soil structure,and not by the texture,organic C,or to IR.Attempts to fit directional or anisotropic semiva-surface topography.They further found a close spatial riograms to the infiltration data resulted in no better relationship between initial water content and infiltra-cross validating regeneration of the spatial data than tion capacity.In this study,although a close relationship when simpler isotropic models were used.

was detected between subsoil D b and IR,no clear spatial A medium nugget effect (nugget variance/sill)was relationships were apparent between soil texture detected for IR (Table 5).However,the nugget effect and IR.

calculated for D b was very low,which suggested that this variable showed a considerable spatial dependence Estimation

within short distances.Goodness of fit for semivario-grams and cross-semivariogram (IR/D b )was obtained Kriging and cokriging procedures were used along with the cross validation procedure.

with isotropic semivarograms and cross-semivariogram The IR values were spatially dependent over a dis-to estimate IR values at 90unobserved points.Based tance of approximately 165m (Table 5and Fig.2).on the results from the cross validation procedure,a Vieira et al.(1981)reported a range of 50m for 1280search ellipse with a radius of 150m was used in both measured IR values within a 160by 55m field.The kriging and cokriging estimations,and a minimum of geostatistical range of values calculated for IR in the four and a maximum of eight data values within the present study was far greater than that calculated by search ellipse were included in the estimations of which Vieira et al.(1981).The geostatistical range of values results shown in Table 6.

obtained for IR and subsoil D b were greater than the Figure 3a presents the measured IR values,and distance between any two nearby test sites and thus Fig.3b estimated IR values with kriging.Figure 3b pro-could provide a useful information about the spatial duces a pattern similar to that shown in the Fig.3a,structure of IR and subsoil D b .

indicating that kriging approximated IR at unsampled The cross-semivariogram for IR/D b is given in Fig.2,locations of the study area when the full data set was and associated coefficients are presented in the Table https://www.wendangku.net/doc/3f12647441.html,ed in the estimation.Mean reduced error defined by,

The cross-semivariogram reflects the close spatial rela-tionship between IR and subsoil D b .For example,the R e ?1n ?n

i ?1

[z (x i )?z *(x i )]/?k (x i )

[4]

correlation coefficient between IR and D b values,which ignores spatial dependence,was ?0.60(P ?2.0?10?4)(Table 4).This relationship was represented in more should be close to zero,and the reduced variance de-detail by a rapidly decreasing spherical cross-semivario-fined by

gram calculated between IR and D b values (Fig.2),indicating that high values of IR were matched with low S 2R e

?1n ?n

i ?1

{[z (x i )?z *(x i )]/?k (x i )}2

[5]

values of subsoil bulk density,and vice versa.

Reynolds and Zebchuk (1996)characterized spatial should be close to unity (Vauclin et al.,1983).In the relationship between soil texture,organic C,and surface Eq.[4]and [5],R e and S 2R e are mean reduced error and topography,and values for field measured hydraulic conductivity (K f )on a 1.2-ha texturally uniform silty

reduced variance,respectively;z (x i )and z *(x i )are the

Table 5.Coefficients of the theoretical semivariogram and cross-semivariogram models of infiltration rate (IR)and subsoil bulk den-sity (D b ).

Variables Model

C 0?C s ?C 0/(C 0?C s )Range,m Sample variance Semivariogram models

IR,cm h ?1Spherical 1.60 2.00.44165 3.51D b ,Mg m ?3Spherical

1.0?10?4 1.7?10?2 6.0?10?31650.017

Cross-semivariogram model

Sample covariance

IR/D b

Spherical

?1.0?10?3

?0.12

8.0?10?3

165

?0.14

1852SOIL SCI.SOC.AM.J.,VOL.67,NOVEMBER–DECEMBER2003

The cross validation indicated that mean reduced er-

ror and reduced variance for the50measured IR values

were4.15?10?3and0.99,respectively(Table6).The

results of normality test(Shapiro-Wilks)indicated that

the residuals from the cross validation were normally

distributed.All these suggest that the conditions for the

assumptions made under kriging estimation were met.

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 with60,

70,80,90,100,120,and140measured D b values along

with50observed IR values to estimate IR at90unob-

served locations.Each time,the mean reduced error

and reduced variance from cross validation were calcu-

lated and normality of the residuals was tested to ensure

the validity of the estimations.Results of cross valida-

tion showed that as the number of D b supplementing

the50IR values in the cokriging procedure gradually

increased from60to120,the mean reduced error gener-

ally decreased slightly at each step.However,the mean

reduced error and reduced variance calculated with

140D b values was identical to that calculated with120D b

values.The mean reduced error calculated from the

cokriging cross validation results were slightly less than

that calculated from those of kriging(Table6).Further-

more,the spatial pattern in the cokriging estimated val-

ues(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 advan-

tage over kriging when full data set used in the calcu-

lations.

To determine whether cokriging has any advantage

over kriging when data are limited,the cokriging proce-

dure was repeated using120D b values along with45,

40,35,and30IR values to estimate IR at95,100,105,

and110unobserved points,respectively,assuming that

5,10,15,and20measured 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 calcu-

lated and the residuals from cross validation by both

procedures were subjected to a normality test to ensure Fig.2.Experimental semivariograms for infiltration rate(IR)and the validity of the assumptions made with kriging and subsoil bulk density(D b),and cross-semivariogram for IR/D b.Solid cokriging estimations.The results are shown in the lines represent the spherical model fitted to experimental values.Table6and Fig.3c and3d.

Figure4shows that the pattern of the residuals calcu-observed and estimated values of the variable z at the

lated by kriging with original data set was quite similar location(x i);and?k(x i)is the estimation variance.Both

to those calculated by kriging with45,and by cokriging parameters are important measures of estimation accu-

racy(Yates and Warrick,1987).with40measured IR values.When the data used with Table6.Results of ordinary kriging and cokriging estimations,and values of mean reduced error and reduced variance calculated from the results of cross validation tests conducted with full and reduced data sets.

Estimated mean?Estimation variance Mean reduced error?Reduced variance?Number of Experimental

observed data mean Kriging Cokriging Kriging Cokriging Kriging Cokriging Kriging Cokriging 50 5.11 5.09 5.10 1.58 1.47 4.15?10?3 4.10?10?30.990.99 45 5.10 5.06 5.05 1.72 1.58?4.41?10?3?3.17?10?30.930.96 40 4.91 4.85 4.83 1.86 1.69?6.14?10?3?5.62?10?30.870.90 35 4.84 4.72 4.67 2.02 1.83?4.52?10?3?4.47?10?3 1.050.88 30 4.74 4.68 4.55 2.15 1.958.64?10?37.48?10?30.800.83

ERSAHIN:ESTIMATING INFILTRATION RATE WITH KRIGING AND COKRIGING1853

Fig.3.(a)Spatial patterns in the observed values of infiltration rate(IR),(b)and(c)kriging-estimated values with50and45field–measured IR values,respectively;and(d)cokriging-estimated values with40field-measured FIR values.

Fig.4.Spatial patterns in the residuals(measured-estimated)(a)calculated by kriging with50field measured infiltration rate(IR)values,(b)

1854SOIL SCI.SOC.AM.J.,VOL.67,NOVEMBER–DECEMBER2003

kriging and cokriging were further reduced,they yielded subsoil were significantly correlated to IR within dis-

residual maps highly different from that in Fig.4a.This

tances ranging from165to215m.Results from cross

suggests that at least40measured IR values were validation revealed that subsoil bulk density was the

needed with cokriging and45with kriging to maintain

most representative auxiliary variable of IR.

acceptable accuracy in estimating IR for the study area.The results showed that cokriging provided no advan-

A significant amount of spatial information was lost

tage over kriging when data were sufficient.With krig-

when IR was estimated by kriging with the reduced ing,45observed IR values were sufficient to obtain the

data set of40values.However,spatial information was

same information as50observations.However,using

maintained when cokriging was used instead of kriging.cokiging with120bulk density values,40observed val-

Therefore,the moderate spatial relationship between

ues of IR were sufficient to obtain the same information

IR and D b helped maintain the spatial information in from that obtained with50field measurement of IR.

IR at this point.However,with further reducing in data,

This indicates that cokriging was more successful than

cokriging could not maintain the spatial information in kriging when IR is undersampled.

IR from that using the original data set.

Infiltration is widely used in modeling of water and

Others have reported similar results to those above chemical transport in soils and irrigation practices.The

(Chien et al.,1997).Stein et al.(1988)found that the

spatial variability of this process on a landscape is impor-

mean variance of prediction error and mean squared tant,affecting the accuracy of the modeling work and

error of prediction for estimation of moisture deficit

efficiency of the irrigation practices.However,measur-

decreased only slightly when results from kriging were ing infiltration in the field is time-and labor consuming.

compared with cokriging with mean highest water table

Therefore,estimation of this process with a reasonable

as the auxiliary variable.They further concluded that accuracy given a minimal observed values is very impor-

the number of moisture deficit values used in cokriging

tant.The result of this experiment illustrated the possi-

could be reduced from400to160with only a small loss bility of using kriging and cokriging.

of accuracy in using the water table variable.In this

study,cross validation conducted for each reduced data ACKNOWLEDGMENTS

set resulted in a lower mean reduced error compared

The author thanks the Turkish Scientific and Technical with kriging(Table6).This indicates that estimates of Research Institute(TUBITAK)for the financial support to

cokriging were more precise compared with those from do this study(Project No:TOAG-TARP1871).

kriging.Zhang et al.(1992)showed that with limited

data,cokriging,as compared with kriging,significantly REFERENCES

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