Kriging and cokriging of soil water properties
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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
Hydraulic conductivity and the soil tex- tural 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 vari- ables is used to estimate these soil prop- erties at unobserved sites within the do- main using Kriging and cokriging esti- mation techniques. The estimation vari- ance was smaller when hydraulic conduc- tivity was estimated employing hydraulic conductivity and clay content data com- pared to Kriging. The spatial variability of soil properties and the cost associated with the data collection make it neces- sary to collect the optimum number of data points with respect to both cost and data adequacy. The use of estimation variance in determining optimum num- ber of samples is demonstrated.
Kriging is a linear interpolation tech- nique that uses spatial autocorrelation among observations to estimate a vari- able at an unsampled location with- out bias and with minimum variance. Cokriging utilizes spatial autocorrela- tions of two variables and spatial cross-
ISSN 0933-3630 @1988 by CATENA VERLAG, D-3302 Cremlingen-Destedt, W. Germany 0933-3630/88/5011851/US$ 2.00 + 0.25
correlation between two variables to esti- mate the undersampled variable. Kriging was applied to estimate hydraulic con- ductivity and textural components mea- sured in a 78 kilometer square area in Azarbayjan, Iran. Cokriging was used to estimate hydraulic conductivity us- ing spatial cross-correlation of hydraulic conductivity and clay content. The es- timation variance of cokriging was less than that of Kriging.
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 num- ber of data points and estimate the data at unsampled location without sacrific- ing accuracy. Therefore, determining the optimum sampling scheme and the min- imum number of required data points for estimating a variable with a prede- termined 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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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 proper- ties, and ABOUFERASSI & MARINO (1983) applied Kriging to water table data.
In addition to Kriging JOURNEL & HIUJBREGTS (1978) developed the geostatistical cokriging technique of es- timating a variable that is undersam- pled by considering its spatial correla- tion 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 ob- tain 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 conduc- tivity employing field measurements of hydraulic conductivity and clay contents as well as estimated (by Kriging) clay content. These techniques were exam- ined 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 spa- tial 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 exper- imental error and relative scale of observation and
2. a structural component which shows spatial features of the variable.
In estimating a variable from experi- mental values by Kriging, the existence of spatial correlation of measured val- ues is necessary. In case of cokriging, spatial cross-correlation of the two vari- ables 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
where N(h) is the number of pairs of observations, Z(x~) and Z(xi+h) are val- ues 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 vari- ogram extrapolated may approach zero. But in practice it will approach a value
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which is called the nugget, Co, caused by measurement error or variability at distances less than the sampling dis- tance. 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 dis- tance over which the observations are assumed correlated. An appropriate the- oretical model should be fitted to define the experimental variogram using equa- tion (1). D E L H O M M E (1976) reviews several models to describe the variogram. Four models are often used:
3. exponential, and
For example a spherical model is de- fined 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 esti- mator for estimation of a property is selected. Kriging has some advantages over other interpolation techniques, be- cause of the fact that the number of observations, spatial configuration of ob- servations, and their distances to the es- timation point are taken into consider- ation. In addition Kriging provides the estimation variance of the Kriged vari- able.
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 lo- cation where an estimation is desired (DELHOMME 1976). The Kriging esti- mate is
Z * (.~o) E 2~Z(.~,} (3) i= l
~i I S
= 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
~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 vari- able (DAVID 1977, J O U R N E L & HUI- JBREGTS 1978). Cokriging considers cross-correlation between two variables
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with one of the variables estimated uti- lizing 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 con- tent) 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 unbi- ased estimator is
Z2~x0) = 2 j l iZI (x l3 + E/~2jZ2(x2j) i=l j=l
where x l i and x2j are space locations of experimental values of 21i and z2j and 21t and 2aj are coefficients to be de- termined.
The estimation variance at x0 is
Z2(x2j+h) are values of experimental vari- ables 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 exper- imental cross-variance. The range of a cross-variogram indicates the distance over which the two variables are cross- correlated.
Using the experimental semi-vario- grams 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.
CY3xo) #2 -~ Z )CliVl2(Xli- XO) i=1
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 es- timated by
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 spac- ing, 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 wa- ter 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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\ ~ . -~ .~_~ ~'-I
o T"" c )
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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 variabil- ity. 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 hy- draulic conductivity are mainly associ- ated with high clay percentages. The mean and variance of the hydraulic con- ductivity 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:
71(h) = 3 . 4
2.0 + 1.4[1.5h/3.0 - 0.5(h/3.0) 3]
fo rh_ 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 aver- age semi-variances were computed for 90, 70, and 50% of the hydraulic con- ductivity 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 per- centages, respectively. Cross-variograms for In(K) with sand and clay percent- ages 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 sam- pling and measurement errors. Having examined all of the theoretical models, the spherical model was fitted to all ex- perimental semi-variograms and cross- variograms (see tab.2). A range of about 3.0 km was observed for all the var- iograms 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 conductiv- ity is influenced by soil textural variation. Cross-variograms of ln(K) and sand con- tent 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 be- tween In(K) and clay and sand contents is consistent with the degree of spatial
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