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SOIL T E C H N O L O G Y vol. 1, p. 117-132 Cremlingen 1988 ]

KRIGING AND COKRIGING OF SOIL WATER PROPERTIES

M . H . Alemi , M . R . S h a h r i a r i & D . R . Nie l sen , D a v i s

Abstract

Hydraulic conductivity and the soil textural components were measured in-situ at 315 locations on a regular square grid within a 78 km ~- area in Azarbayjan, Iran. The theory of regionalized variables is used to estimate these soil properties at unobserved sites within the domain using Kriging and cokriging estimation techniques. The estimation variance was smaller when hydraulic conductivity was estimated employing hydraulic conductivity and clay content data compared to Kriging. The spatial variability of soil properties and the cost associated with the data collection make it necessary to collect the optimum number of data points with respect to both cost and data adequacy. The use of estimation variance in determining optimum number of samples is demonstrated.

Summary

Kriging is a linear interpolation technique that uses spatial autocorrelation among observations to estimate a variable at an unsampled location without bias and with minimum variance. Cokriging utilizes spatial autocorrelations of two variables and spatial cross-

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correlation between two variables to estimate the undersampled variable. Kriging was applied to estimate hydraulic conductivity and textural components measured in a 78 kilometer square area in Azarbayjan, Iran. Cokriging was used to estimate hydraulic conductivity using spatial cross-correlation of hydraulic conductivity and clay content. The estimation variance of cokriging was less than that of Kriging.

1 Introduction

Developing suitable schemes for de- sign and management of irrigation and drainage systems requires sufficient and reliable data of soil and water properties as well as other pertinent information. In practice, only limited data is available because in some cases the cost of obtain- ing data on a large scale is prohibitive. It is desirable to coIlect the minimum number of data points and estimate the data at unsampled location without sacrific- ing accuracy. Therefore, determining the optimum sampling scheme and the minimum number of required data points for estimating a variable with a predetermined level of reliability can be useful in a better characterization of spatially variable soil water systems.

MATHERON (1963) developed the method of Kriging, which estimates a variable for any coordinate position

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118 Alemi, Shahriari & Nielsen

within the domain measured, using the autocorrelation between adjacent points. Kriging has been used in estimation of ore reserves in mineral deposits (JOUR- N E L & HUIJBREGTS 1978). WEB- STER & BURGESS (1980) used Krig- ing to describe soil properties. HAJRA- SULIHA et al. (1980) used Kriging for salinity studies, VIEIRA et al. (1983) applied Kriging to agronomic properties, and ABOUFERASSI & MARINO (1983) applied Kriging to water table data.

In addition to Kriging JOURNEL & HIUJBREGTS (1978) developed the geostatistical cokriging technique of estimating a variable that is undersampled by considering its spatial correlation with other variables that are more densely sampled (DAVID 1977). Cokrig- ing was used by VIEIRA et al. (1983) to estimate agronomic properties while ABOUFERASSI et al. (1984) estimated the transmissivity of an aquifer utilizing the spatial correlation of transmissivity value with specific capacity. Kriging and cokriging estimates can be used to obtain the minimum number of samples if a knowledge of the spatial variability of the parameter in question is available.

We assume that the reader is relatively unfamiliar with the concept of Krig- ing and cokriging, and, hence, present the principles of these methods along with their application in estimating soil hydraulic conductivity and soil textural components. Kriging is used to estimate saturated hydraulic conductivity, sand, silt, and clay contents. Cokriging is used to estimate the hydraulic conductivity employing field measurements of hydraulic conductivity and clay contents as well as estimated (by Kriging) clay content. These techniques were examined in relation to their effectiveness of

reducing the cost of collecting data for ascertaining the value of soil hydraulic conductivity.

2 T h e o r y

In classical statistics the analysis of spatial data is based upon the assumption that the data are spatially independent. In this work we assume that the spatial variability is invariant with time and is composed of two components:

1. a random component which reveals the local irregularities due to experimental error and relative scale of observation and

2. a structural component which shows spatial features of the variable.

In estimating a variable from experimental values by Kriging, the existence of spatial correlation of measured values is necessary. In case of cokriging, spatial cross-correlation of the two variables is also required. To determine the spatial correlation of one variable the semi-variance is used. The semi-variance is estimated by

N(h) y*(h) = 1/2N(h) E [Z(x,) - - g(x,+h)] 2

i=1

(1)

where N(h) is the number of pairs of observations, Z(x~) and Z(xi+h) are values of the variable Z at space locations xi and xi + h, a distance h apart. A plot of semi-variance versus h is called the semi-variogram. The semi-variogram is called isotropic if semi-variograms for various directions are the same. When h approaches zero, the experimental variogram extrapolated may approach zero. But in practice it will approach a value

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Kriging and Cokriging 119

which is called the nugget, Co, caused by measurement error or variability at distances less than the sampling distance. The semi-variance is expected to increase with increase in h. At a cer- tain value of h the semi-variance may remain constant which is called sill and is equal to the total variance. The range of a semi-variogram is the distance over which the observations are assumed correlated. An appropriate theoretical model should be fitted to define the experimental variogram using equation (1). D E L H O M M E (1976) reviews several models to describe the variogram. Four models are often used:

1. linear,

2. spherical,

3. exponential, and

4. Gaussian.

For example a spherical model is defined by the following expression:

y(h) = Co + Ct[1.5(h/a) - 0.5(h/a) 3] (2)

where Co is nugget, Co + C l is the sill and a is the range of the variogram.

Kriging is an interpolation procedure that uses spatial autocorrelation between adjacent points for estimation. With Kriging the best linear unbiased estimator for estimation of a property is selected. Kriging has some advantages over other interpolation techniques, because of the fact that the number of observations, spatial configuration of observations, and their distances to the estimation point are taken into consider- ation. In addition Kriging provides the estimation variance of the Kriged variable.

Suppose there are N points where soil property Z (hydraulic conductivity, clay content, etc) is measured. Kriging can be used to estimate Zxo from the known Z(x,)(i=l.2,3,...,N), where xo is the space location where an estimation is desired (DELHOMME 1976). The Kriging estimate is

Z * (.~o) E 2~Z(.~,} (3) i= l

where

Zt'~o)

~i I S

N

= the estimator of the unknown

true value of Zlx0} = weighted coefficients to

be determined = the number o f neighboring points

used in Kriging.

The problem is to choose 2i in a way where the estimation variance is minimal and estimation is unbiased. Unbiased here means that the expected value of the estimate equals that of the true value (E[Z * ( x o ) - Z ( : , u ) ] ----- 0).

The estimation variance at point xo is

N

°~o) = # + E 2,7 (x; - xo) (4) i=l

where ~L is the Lagrangian multiplier obtained from minimizing the kriging variance and y ( x i - xo) is semi-variance for Z at distance (xi - x0).

The experimental variogram is used to solve the Kriging system to obtain 2i and # which are used in equations (3) and (4) to calculate the Kriged value and its estimation variance, respectively.

Cokriging is an extension of Krig- ing to the case of more than one variable (DAVID 1977, J O U R N E L & HUI- JBREGTS 1978). Cokriging considers cross-correlation between two variables

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120 Alemi, Shahriad & Nielsen

with one of the variables estimated utilizing autocorrelations and the cross- correlat ion between the two variables.

Consider a set of experimental values of Z1 : [zli, i = 1 ..... NIl (i.e., clay content) and a set of experimental values of Z2 : [z2j,j = 1 ..... N2] (i.e., hydraulic conductivity) over a field where N2 is less than N 1. We are interested in estimating the value of Z2cv/), (the undersampled variable) at space location x0 from Z 1 and Z2. The minimum variance unbiased estimator is

NI N2

Z2~x0) = 2 j l iZI (x l3 + E/~2jZ2(x2j) i=l j=l

(5)

where x l i and x2j are space locations of experimental values of 21i and z2j and 21t and 2aj are coefficients to be determined.

The estimation variance at x0 is

Z2(x2j+h) are values of experimental variables Z1 and Z2 at space locations x2j and x2j + h, respectively. A plot of the cross-variance versus distance is called the cross-variogram. Theoretical models similar to the ones used for the semi-variogram are fitted to the experimental cross-variance. The range of a cross-variogram indicates the distance over which the two variables are cross- correlated.

Using the experimental semi-variograms and cross-variogram the cokriging system is solved to obtain the values of ~t2, 2 , , and )],2j which in turn are used to obtain values of hydraulic conductivity and estimation variances from equations (5) and (6), respectively.

3 Experimental

N[

CY3xo) #2 -~ Z )CliVl2(Xli- XO) i=1

N2

q" E ~-2jY22 (X2j -- XO) (6) j=l

where #2 is the Lagrangian multiplier obtained from minimizing the variance and ~)22 is the semi-variance for Z2. Hence 712(= 721) is the cross-variance estimated by

yt2(h) =,

Values of hydraulic conductivity and the percentages of sand, silt, and clay were measured at 315 locations within an area of 78 km 2 in Azarbayjan Province, Iran I. The field was rectangular in shape and consisted of 15 columns and 21 rows of measurements. The observations were made on a regular grid with 0.5 km spacing, The hydraulic conductivity (K) was measured in situ at a depth of approx- imately 1.2 meters using the auger hole method introduced by DISERENS (see BOERSMA 1965). The depth of the water table at all locations at the time o f N (h)

. .measurements was less than 1 meter. Soil 1/2N (h) Z.~ [Z l(x2j+h)- L t(.%)lsamples collected from depths below the j=l

water table were analyzed for sand, silt, [Z2(x2j+h) -- Z2(~2j-)] (7)and clay content.

where N(h) is the number of pairs of observations separated by distance h and Zllx2j), Zl/,:2j+h), Z2(x20 and

JData Courtesy of Agricultural Engineering Service, Ministry of Agriculture - - Iran

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122 Alemi, ShahriaH & Nielsen

4 Results and Discussion

Values o f the hydraulic conductivity given in fig.la were found to be lognor- really distributed using the Kolmogorov- Smirnov test (BARR et al. 1979). The hydraulic conductivity, except for the central part of the field where the values are low, exhibits considerable variability. Measurements of sand, silt, and clay content given in steps of 5% in figs.lb, lc, and ld were found to be normally distributed. The soil texture is predom- inantly clay except near the boundaries of the region. The low values of hydraulic conductivity are mainly associated with high clay percentages. The mean and variance of the hydraulic conductivity and textural components are given in tab.1.

Soil Property Mean Variance

K* -0.74 3.40 Clay (%) 48.6 315 Silt (%) 22.0 98 Sand (%) 30.0 420

[I * Values of K (m/day) expressed as ln(K).

Tab. i : Statistical data.

Directional semi-variances of each property computed from equation (1) in each of the principal directions (0 °, 45 °, 90 ° and 135 °, with angular regulariza- tion in each direction of 45 °) indicated an isotropic behavior. The average semi- variogram model of ln(K) using entire data set represented by the solid line with symbol in fig,2a is:

~l(h) =

71(h) = 3 . 4

2.0 + 1.4[1.5h/3.0 - 0.5(h/3.0) 3]

fo rh_<3 km (8)

for h > 3 km

The upper and lower values repre- sent the range of values calculated for the four principal directions. The semi- variogram of In(K) indicates that there is a spherical structure in the semi-variance in all directions as well as the average semi-variance. Furthermore, the average semi-variances were computed for 90, 70, and 50% of the hydraulic conductivity data (fig.2b) and clay content data (not presented), This random re- moval of data did not change the shape of the variogram.

Figs.3a and 3b show experimental semi-variograms of sand and clay percentages, respectively. Cross-variograms for In(K) with sand and clay percentages using equation (7) are presented in figs. 4a and 4b, respectively. The semi- variograms of sand, silt, and clay as well as the cross-variograms between In(K) and sand and clay percentages indicated spherical structure with a rather large nugget, the large nugget is due to the soil variation, method and scale of sampling and measurement errors. Having examined all of the theoretical models, the spherical model was fitted to all experimental semi-variograms and cross- variograms (see tab.2). A range of about 3.0 km was observed for all the variograms which indicates that pairs of observations closer than 3.0 km are in- deed correlated. It is expected that at the scale of sampling in this particular data, the value of hydraulic conductivity is influenced by soil textural variation. Cross-variograms of ln(K) and sand content exhibited positive correlation while that of ln(K) and clay content was neg- atively correlated. The range of 3.0 km for cross-variograms also indicates that the degree of spatial cross-correlation between In(K) and clay and sand contents is consistent with the degree of spatial

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e,l

@

.p.d

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Upper • Lower

Average

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Distance, km

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[] 70~ OF DATA 0 50~. OF BTAT

b

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Fig. 2: a) Semi-variograms of In(K) in various directions, b) semi-rariograms of In(K) using 50, 70, and 90% of data.

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[24 AlemL Shahriari & Nielsen

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0 r - (13

> ,&

E ID

¢J)

450

400

350

300

250

200

150

100

50

0 0

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"r-

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3O0

250

200

150

100

50

0 0

[] tn

[]

[] fil

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1 2 3

Distance, Km

Fig. 3: a) Average semi-variograrn of sand Content, b) average semi-variogram of clay content.

autocorrelat ion of K, clay, and sand contents. Cross-variograms of In(K) versus clay or sand indicate that for small distances local scale variations of clay and sand are sufficient to affect K. The spatial variat ion in K due to variation in clay content manifests a larger value of --11.1 for the sill o f the cross-variogram of ln(K) and clay content.

The Kriging system was used to estimate K, sand, silt, and clay at 157 locations where measurements were available. Every other measured value was removed (total of 157) and the aver-

age variogram model computed from the complete data set was used to estimate the missing variables. The mean differ- ence between the observed and estimated values was zero at statistical level of sig- nificance of less than 0.01 for the four variables. The estimated plus measured values are plotted in fig.5 for K, sand, silt and clay, respectively. Comparing figs.1 and 5, the estimated plus measured maps for K, sand, silt, and clay percentages closely resemble the measured maps. The maps of Kriging estimates plus measured values as expected are

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¢.j

e~

d~

12

10

8

6

4

2

0 0

El

i t

1 2

Distance, km

I~1 El

e~

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L

r,.)

0

-2

- 4 -

- G '

-10 '

-12 0

rn

I~1 N m N N m

2 3 4 5 6

Distance, km

Fig. 4: a) Cross-variogram of In(K) versus sand content, b) cross-variogram of ln(K) versus clay content.

smoother than the original data. The average estimation variance for Kriging K, sand, silt, and clay contents were 2.24 (ln(K)) 2, 310(%) 2, 72(%) 2, and 237(%) 2, respectively. The hydraulic conductivity, sand, silt, and clay percentages were also estimated at the centroid of 280- 0.25 km 2 cells where observations were not available (fig.6). Having tested the

adequacy of Kriging procedure/ 'or estimation (as mentioned above) fig.6 indicates that Kriging can be used for estimating these variables at unsampled locations.

Selecting a minimum radius of !..5 km for the Kriging neighborhood provided sufficient data (between 12 to 30 points depending on the distance of estimat-

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126 AlemL Sh~lhriari & Nielsen

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128 Alemh Shahri~u4 & Nielsen

Property Semi-variograms, (%)2 Lag, km

Clay 72(tl) = 225 + 90[1.5h/3.0 -0 .5 (h /3 .0 ) 3] h < 3.0 72(t0 = 315 h > 3.0

Sand ?3(h) = 290 + 130[1 .5h/3 .0- 0.5(/1/3.0) 3] h < 3.0 73(t0 = 420 h > 3.0

Silt 74(h) = 71 + 27 [1 .5h /3 .0 - 0.5(h/3.0) 3] h < 3,0 74(tl) = 98 h > 3.0

Cross-variograrns of In(K) vs clay and sand contents, In(K)(%) Clay 712(/0 = -5 .0 -- 6.1 [1.5h/3.0 - 0.5(/1/3.0) 3] h _< 3.0

"}'t2(h) = - l l . t h > 3.0 Sand 713(tl) = 5.0 + 5.5[1.5h/3.0 - 0.5(h/3.0) 3] h < 3.0

713(tl) = 10.5 h > 3.0

Tab. 2: Semi-variogram and cross-variogram models.

ing point to the boundary of region) for estimation. The Kriging neighborhood selected was sufficient for this particular data with a sampling interval of 0.5 km. Increasing the radius did not improve the estimates significantly since in a regularly spaced square grid, the four closest neighbors will constitute a large percentage of the weighting coefficients. In this case the four closest neighbors contributed about 80% to the weighting coefficients. Beyond 1.5 km the weighting coefficients 2i become negligible and due to increase in the number of neighbors the cost of computations become prohibitive.

Experimental semi-variograms of clay and cross-variogram of ln(K) versus clay (tab.2) were used in cokriging to estimate K. Using equation (5) estimated values of K were calculated for two cases. For the first case 157 values of measured K were removed and the remaining values of K along with measured clay content were used to estimate the missing K. The maps of estimated plus measured K are shown in fig.7a. Comparing figs. la and 7a, the two maps are nearly iden-

tical. For the second case clay content estimated by Kriging at the centroid of 280 cells along with measured K and clay content were used to cokrige values of K. Fig.7b shows the cokriged estimates of K at the centroid of the cells where observation of K was not available. The average estimation variance for cokriging was 2.05 (In(K)) 2. With the estimation variance for cokriging being less than that for Kriging, cokriging is more precise than Kriging. Based on the values of estimation variance for cokriging, soil water properties can be estimated more reliably with cokriging provided that a second property can be measured at locations where the estimation is desired. For this reason the cokriging estimates of K at the centroid of 280 cells are more reliable than correspond- ing Kriging estimates given in fig.6a. It is also demonstrated that a second variable (clay content) obtained by Kriging in conjunction with measured values of both variables may be used in cokriging to generate more information on an undersampled variable (in this case K).

Kriging and cokriging techniques are

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Z

30

lO

6 q~~.-.-_~->.~._@~

a

10

.,..j-"V"--.. _...1 | "~-_..

. ~ . , ~ . ~ ' . . ~-~.~

B 0

..v ~z4. ,,~:,-..~.'---~/%, " ~<t~ 2 "<' "~

b

Fig. 7: a) Cokriging estimates plus measured hydraulic conductivity, b) cokriging estimates of hydraulic conductivity using kriging estimates of clay content.

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¢ q <

4

_=

a

.=.

2

• i t i

0 2 3 4

Spacing, km

c q

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~y

3 .~- t _

e .

2

Fig. 8: Estimation variance and sampling cost for Kriging and cokriging.

attractive options to reduce the cost of collecting data for ascertaining K. How- ever, the effect of sampling density on estimation variance and cost of sampling need to be examined in advance. In or- der to assess the effect of sampling density on the reliability of estimating K, Kriging and cokriging were performed to estimate K at a given point located at the center of a block 4 km by 4 km, using four different grid spacings (0.5, 1, 1.5, and 2.0 km) and equal number of neighbors by removing the observed values. The average semi-variogram models of ln(K) (equation 8) and clay content (tab.2) and cross-variogram model o f In(K) versus clay content (tab.2) were used in the computations. Since semi- var iogram depends on the configuration of observed points and the distances among them, the estimation variances are related to the observation scheme and number of neighbors. The value of the semi-variance increases with distance, thus, the estimation variance is expected

to decrease as the distances between observed points and the estimating point is decreased. For a rectangular shaped domain (FL x FL km 2) if grid spacing (L) is decreased to U the number of samples at the grid points would increase from ( F L / L + 1) 2 to ( F L / L ' + 1) 2, where FL is an integer multiple of L and L'. The sampling cost is directly related to the percentage of increase in sampling defined by

R S = [ ( F L / L ' + 1) 2 - - ( F L / L + 1) 2 ( F L / L + 1) 2 ] 100(9)

where RS is the increase in sampling when grid spacing is reduced to L'.

Assuming a $20 cost per measurement for hydraulic conductivity and $2 per measurement for clay content, fig.8 shows the estimation variance and cost of sampling versus spacing for the rectangular shaped field of 4 km by 4 kin. The estimation variance increases as the distance between the observed points and the estimating point increases. The slope

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of the estimation variance versus spacing shows the increase in estimation variance per unit increase in spacing. The cost of sampling increases rapidly as spacing decreases. The cost of sampling is at its minimum when spacing is 4 km and estimation variance is equal to the total variance.

T h e estimation variance and cost of sampling when cokriging is employed is also presented in fig.8. In this example we asume that clay content is measured at all locations where hydraulic conductivity measurements are available as well as the location where the estimation of hydraulic conductivity is desired. The estimation variance is smaller for cokriging while the costs are slightly higher in this example. However, when cokriging is employed the number of hydraulic conductivity measurements may be reduced which will reduce the total cost while the estimation variance would be im- proved compared with Kriging. Having the semi-variogram and cross-variogram models constructed the opt imum spacing between samples and the associated costs can be obtained for a predetermined level of reliability. On the other hand if the cost of sampling is the limiting factor, the precision of estimation for the sampling scheme can be determined, Similar approach can be used to test the estimation variances for a equilateral triangle grid or irregularly spaced observations.

In conclusion geostatistical analysis of the data can be useful in understand- ing the spatial distribution of soil water properties. The spatial structure may be used for estimation and for testing adequacy of quantitative information. Esti- mation with minimum variance can provide further data with known reliability and it can be used to identify the locations where more data are needed

to improve estimation. In the regularly square grid observation scheme of data used in this work the values of estimation variances at the center o f interior cells are the same because the configuration of the observation points are similar and the number of neighbors used in estimation are identical. The values of the estimation variances for both Krig- ing and cokriging were high due to the large spacing between observations indicating that further sampling at distances shorter than 0.5 km may provide more information about the structure of the variance and cross-variance. Cokriging K resulted in lower estimation variances than Kriging, indicating that cokriging is more reliable for estimating soil hydraulic conductivity. Kriging and cokriging can be used in ascertaining K and selecting the opt imum sampling scheme for reducing cost of sampling provided that the semi-variogram and cross-variogram are available.

References

ABOUFERASSI, M. & MARINO, M.A. (1983): Kriging of water levels in the Sous Aquifer, Morocco. Mathematical Geology, Vol. 15, No. 4, 537-551.

ABOUFERASSI, M. & MARINO, M.A. (1984): A geostatistical based approach to the identifi- cation of aquifer transmissivities in Yolo Basin, California. Mathematical Geology, Vol. 16. No. 2, 125-137.

BARR, A.J., GOODNIGHT, J.H., SALL, J.P., BLAIR, W.H. & CHILKO, D.M. (1979): SAS user's guide. SAS Institute, Inc., Raleigh, NC.

BOERSMA, L. (1965): Field measurements of hydraulic conductivity below a water table, In: C.A. Black et al. (ed.), Methods of Soil Analysis, Part 1. Agronomy 9, 222-233.

DAVID, M. (1977): Geostatistical ore reserve estimation. Elsevier Scientific Pubtication Co. 364 PR

DELHOMME, J.P. (1976): Kriging in hydro- sciences. Centre D'Informatique Geologique, Fountainbleau, France.

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HAJRASULIHA, S., BANIABBASSI, N., MET- THEY, J. & NIELSEN, D.R. (1980): Spatial variability &soil sampling for salinity studies in southwest [ran. Irrigation Science. 1, 197-208.

JOURNEL, A,G. & HUIJBREGTS, Ch.d. (1987): Mining Geostatistics. Academic Press.

MATItERON, G. (1963): Principles of Geo- statistics. Econ Geology 58, 1246-1266.

VIEIRA, S.R., HATFIELD, J.L., NIELSEN, D.R. & BIGGAR, 3.W. (1993): Geo~tatistical theory and application to variability of some agronomic properties. Hilgardia, Vol. 51, No. 3.

WEBSTER, R. & BURGESS, T.M, (1980): Opti- mal interpolation and isarithmic mapping of soil properties. I. The semi-variogram and punctual Kriging. J. Soil Science, Vol. 31, 315-331.

Addresses of authors: M.H. Aiemi LAWR Department, UC Davis Davis, Ca 95616, USA M.R. Shahriari Visiting Soil and Water Scientist Irrigation Specialist Ministry of Agriculture /ran D.R. Nielsen Professor of Water Science University of California Davis

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