Disjunctive Kriging
3. Cokriging
(U.S.) Robert S. Kerr Environmental
Research Lab., Ada, OK
1986
PB87-166260
L
J
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4. TITLE AND SUBTITLE
DISJUNCTIVE KRIGING 3.
(Journal Version)
TECHNICAL REPORT DATA
{Please read Instruetiont on the revene before complel1
, REPORT NO.
EPA/600/J-86/232
2.
Cokriging
PBB7-166260
5. REPORT DATE
1986
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
S. R. Yates
8. PERFORMING ORGANIZATION REPORT NO.
«. PERFORMING ORGANIZATION NAME AND ADDRESS
Robert S. Kerr Environmental Research Laboratory
U.S. Environmental Protection Agency
Post Office Box 1198
Ada, Oklahoma 74820
10. PROGRAM ELEMENT NO.
ABWD1A
11. CONTRACT/GRANT NO.
In-House
12. SPONSORING AGENCY NAME AND ADDRESS
Robert S. Kerr Environmental Research Lab. - Ada, OK
Post Office Box 1198
Ada, Oklahoma 74820
13. TYPE OF REPORT AND PERIOD COVERED
Journal Article
14. SPONSORING AGENCY CODE
EPA/600/15
15. SUPPLEMENTARY NOTES
Published in: WATER RESOURCES RESEARCH, 22(10):1371-1376, September 1986.
16. ABSTRACT
The disjunctive kriging (DK) method described in the first paper of this
series is extended to account for more than one random function. In the deriva-
tion contained herein, two random functions are considered, but this is easily
generalized to any number. An example is presented using disjunctive cokriging
(DCK) where the surface gravimetric moisture content is estimated using the bare
soil temperature as an auxiliary random function. The results indicate that the
DCK procedure produces a better estimator than ordinary cokriging in terms of
reduced variance of errors and exactness of estimation. Also, using DCK, an
estimate of the conditional probability that the level of a property is greater
than a known cutoff value can be obtained. In general, this conditional proba-
bility is better than the DK probability by virtue of the additional information
contained in the second, auxiliary random function.
17.
KEY WORDS AND DOCUMENT ANALYSIS
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RELEASE TO PUBLIC.
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UNCLASSIFIED
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UNCLASSIFIED
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EPA/600/J-36/232
JOURNAL ARTICLE
WATER RESOURCES RESEARCH, VOL. 22. NO. 10. PAGES 1371-1376. SEPTEMBER 1986
Disjunctive Kriging 3. Cokriging
S. R. YATES
R. S. Kerr Environmental Research Laboratory. Ada, Oklahoma
The disjunctive kriging (DK) method described in the first paper of this series is extended to account
for more than one random function. In the derivation contained herein, two random functions are
considered, but this is easily generalized to any number. An example is presented using disjunctive
cokriging (DCK) where the surface gravimetric moisture content is estimated using the bare soil temper-
ature as an auxiliary random function. The results indicate that the DCK procedure produces a better
estimator than ordinary cokriging in terms of reduced variance of errors and exactness of estimation.
Also, using DCK, an estimate of the conditional probability thai the level of a property is greater than a
known cutoff value can be obtained. In general, this conditional probability is better than the DK
probability by virtue of the additional information contained .in the second, auxiliary random function.
INTRODUCTION
In a recent paper [Yaies et a/., 19860] the theoretical basis
for the disjunctive kriging (DK) method was reviewed. In a
following paper [Vales et al., 19865] an example of the DK
method was presented using the electrical conductivity data
collected by Al-Sanabani [1982]. The example and review ma-
terial assumes the presence of only one random function. At
times, however, it may be desirable to utilize an additional
variables) in the estimation process. This might be necessary
to fill in missing information or to increase the accuracy of the
estimation. When two or more random functions are used for
kriging it is termed cokriging [Journel and Huijbregts, 1978;
Vauclin el a/., 1983; Carr el a/., 1985; Myers. 1982,1984].
Few examples of cokriging using linear estimators exist in
the soil water literature. Exceptions include Vaudln et al.
[1983], who used ordinary cokriging (OCK) to estimate the
available water content where the sand content was included
in the estimation process, S. R. Yates and A. W. Warrick
(unpublished manuscript, 1986) used cokriging to estimate the
surface gravimetric moisture content (CMC) in a 1-ha field.
The auxiliary random functions used along with the CMC
were the bare soil surface temperature, which was collected
with an infrared nhcrmomeler, and percent sand content. Their
results indicate that the ordinary cokriging estimator is su-
perior to the ordinary kriging estimator in terms of reduced
kriging variance as well as reduced sum of squares error.
The purpose of this paper is to extend the disjunctive krig-
ing method to allow more than one random function to be
used in the estimation process. In the first part of the paper
the theory will be discussed; this is followed by an example.
The example will include a comparison between the disjunc-
tive cokriging (DCK) and OCK estimators in terms of vari-
ance of errors and exactness of estimation. The conditional
probability distributions will be calculated for DCK and com-
pared to the DK estimator of one random function.
THEORY
Consider two second-order stationary random functions
sampled on point support in two dimensions where a value of
the property of primary interest is obtained al p locations and
is denoted as Z(x,J. i - I. 2. .... p. Further, an auxiliary
Copyright 1986 by the American Gcophvocal Union.
Paper number 6W4396.
if>4J-IJ97/86/a>6W-O9MOiOO
random function, V(x), is sampled at q locations: ^(xjj),;' = 1,
2,.... q. The 1 and 2 in the subscripts for x are used to identify
with which random function the x location is associated. The
sampling positions of Z(x) need not be the same as V(x), but it
is assumed that at sufficiently many locations both Z(x) and
Y(x) are sampled so that the cross-correlation structure can be
determined.
The DK method utilizes transformed variables which are
assumed to be uni- and bivariate normally distributed. For
DCK, two such transform functions must be defined, one for
each random function (i.e., Z(.x) and V(x)):
t-o
t-o
(1)
(2)
where H4 is a Hermite polynomial of order Jc, and Y(xu) and
U(x2J) are random variables which designate the transformed
data. The random variables V(xH) and U(x2J) have standard
normal distributions which are obtained from the transforms,
and \l/, respectively. Although Y(xlt) and f(x,j) have the
same distribution, they are given different symbols in order to
delineate from which data set they are derived. The same is
true for the transform relationships.
Disjunctive Cokriying Estimator
The disjunctive cokriging estimator
is defined as
ZDC*'(*O)
i-i
+ I
(3)
where /j and ht are unknown functions to be determined ; m
and n are the number of data points (nearest neighbors) used
in the estimation; and, in general, m need not equal n.
Expanding the unknown functions in a series of Hermite
polynomials gives
t-O I-I
where the
pansion.
+ I I **«»[(/(*„)] (4)
t-o ;-i
and h are the coefficients of the Hermite ex-
1371
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1372
YATES: DISJUNCTIVE KRIGINO, 3
zi, i
Fig. 1. Schematic of the space £>„„., where
tion of Z(x0) onto D.».
X,, - x,yc»
o CO)
In (9) and (10), p»(x0 — x<) and p,j(x0 — x() are the correlation
and cross-correlation functions for the separation distance xa
Since both (9) and (10) must be satisfied for all k the disjunc-
is the projec- jjve cokriging equations can be written as
Disjunctive Cokriging Equations
The derivation contained in the next few paragraphs follows
the same approach that was used by Journel and Huijbregts
[1978] in deriving the disjunctive kriging equations for one
random variable. The symbolism and many of the definitions
that will be used are given in the work by Yates et al. [1986o].
The first step is to define the vector space Dm+,, shown
diagrammatically in Figure 1, as the space spanned by the
(m + n) single-variable measurable functions /,[y(x,,)] and h}
[U(x2V)].Thatis,D,.+. = {/1[y(xu)] + ••• + /m[y(x1J] +
/i,[U(x21)] + ••• + h,[U(xln)']. Using the perpendicular pro-
jection of Z(x0) onto Dm+. will produce the estimate with the
minimum error. Two requirements which assure that the best
estimator will be produced are that ZKti* lies in D-+. and
that Z(x0) — ZJX-K^XO) must be orthogonal to any vector in
Dm+. (i.e., W in Figure 1). The latter condition can be written
as the scalar product
= 0
or [Journel and Huijbregts, 1978. p. 569]
£[Z(xo) | Y(xJ] = ECZocK'txo) | Y(xJ]
£[Z(.x0)
(5)
(6)
(7)
where x. = x,,;a, i = 1, 2, ...,m, and x, = x2J;P,j = 1. 2, ....
n. It is evident from (6) and (7) that disjunctive cokriging, like
linear cokriging, requires the solution of two simultaneous
equations.
The remainder of the solution is straightforward. Incorpor-
ating (1) and (3) into both (6) and (7) gives
£O[y
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YATES: DISJUNCTIVE KRIGINO, 3
1373
Solving for £[Z(.x0)ZDCK*(x0)] gives [see Journel and Huij-
bregts, 1978, especially p. 576]
£[Z(*o)Zo«*(*o)]
t-o
c I bikPll *(x0 - x,()
(17)
Incorporating (16) and (17) into (15) and noting that for k = 0,
EWxJZoc^Xof] is equal to E[Z(x0)2] (i.e., bi0 = ajo = \/(n
+ m)) gives the punctual disjunctive cokriging variance
*-i
1-1
(18)
which has a similar form to the linear cokriging variance.
Conditional Probability
The conditional probability (CP) that the unknown value at
a randomly located point is above a prescribed cutoff level can
be estimated using DCK. The derivation of the CP for cokrig-
ing is similar to the disjunctive kriging case, but due to the
added information from the second random function, it will, in
general, be an improved estimator. For purposes described
herein, the point CP will be discussed. However, if a block-
averaged value is required, it is only necessary to replace ajt
and bik with o,t and bik in the following discussion. The con-
ditional probability that the value of Z(.x0) is greater than a
cutoff level zc (or yc for the transformed variable) conditioned
on the available data is
zc I Z(xJ,
= P[y(x0) ;> yc |
(19)
In order to determine the CP using the disjunctive kriging
method it is necessary to transform the problem into a suit-
able form. Since disjunctive kriging is an estimator for the
conditional expectation, a means whereby (19) can be written
as the conditional expectation must be found. To do this an
indicator variable 0,,[K(x)] is defined such that
(20)
Using this indicator variable in (19) gives
,, = 1 1 y(xj. i/
E[0jy(x.),
(21)
The CP and the conditional expectation in (21) are the same
because the indicator variable has a value of zero for
- oo < y(x) < y, and unity for ye <. Y(x) < oo.
Expanding the unknown 0,([y(x)] in a series of Hermite
polynomials gives
(22)
4-0
where the 04's are the coefficients of the expansion which are
determined by using the orthogonality properties [see Journal
and Huijbregis, 1978; Yates et al., 1986a], and PO^* is the
estimator of the CP. For le = 0, Ok equals 1 — G(ye), and for
k> 0, Ok equals g(ye)Hk. ,(>>«)//£!, where C(u) and g(u) ate the
gaussian cumulative frequency and density functions, respec-
tively.
Incorporating the coefficients into (22) gives the estimator
for the CP at the point x0:
f DCK*(*O) = 1 - C(yc) + g(y,) £ Hk_ t(y,)H^Y(Xo)yk\ (23)
t-i
The only unknown in (23) is H4[y(x0)], which is estimated
using the disjunctive cokriging estimator (see equations (12)
and (13)):
(24)
Comparing (23) with (42) of Yates et al. [1986a] demonstrates
the similarity between the results for one and two variables.
The only difference is that for cokriging there is a contribution
to the estimate for W4[y(x0)] from the auxiliary random func-
tion.
The CP in (23) can be written in terms of a conditional
probability density function, Pdf *(u)
r
Jr,
0)= Pdf(u)du
Jr.
(25)
yc
where the probability density function, which is evaluated at
x0 is written as
i.S
I.B
B.5
3 a.e
t—
GNC
BET
B.B I0.B 20.B 30.B
DISTANCE I.)
48.B
0.6
§
a •••
y, '•'
7 .6
B.B
GMC/BST .
BST/GMC
B.B 10.B 20.B 30.0 40.0
DISTANCE (.)
Fig. 2. Global correlation and cross correlation functions. The exes and asterisks indicate that less than 10 and more
than 65 pairs, were used to generate a value, respectively.
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1374
YATES: DISJUNCTIVE KRIGINO, 3
TABLE 1. Covariance Functions
Covariance*
Function
c,,
C22
c,,
C:,
Sill
42.5
33.5
-18.0
-22.0
Range
19.0
14.0
25.0
25.0
*,
-0.010
-0.052
-0.013
«...
0.95
1.01
0.84
Models are of the spherical type. The nugget for each model was
found to be zero.
•Definition of subscripts: 1 is for CMC, and 2 is for BST.
Pdf •(!!) =
(26)
*-»
EXAMPLE
The remainder of this paper will illustrate the DCK method
with an example. In order to use the cokriging method
(whether ordinary or disjunctive), a pair of correlated random
variables must be available. The following example uses a data
set which contains 71 values of the gravimetric moisture con-
tent (GMC) and 148 values of the bare soil surface temper-
ature (BST).
Sample statistics were obtained for the data sets. For the
GMC, the mean, variance, skew, and kurtosis are 10.76, 37.06,
0.64, and 7.21, respectively, and 2.14 < GMC (%) < 26.35. For
the BST data, the mean, variance, skew, and kurtosis are
38.36, 23.39, 0.39. and 1.70, respectively, and 31.71< BST
(°C) < 47.74. The correlation [Sokal and Rohlf, 1981], r, be-
tween the GMC and the BST is -0.744.
The hypothesis that the GMC and BST data are either
normally or lognormally distributed was tested using the
Kolmogorov-Smirnov (KS) test [So/ca/ and Rohlf, 1981; Rohlf
and Sokal, 1981; Rao et at., 1979]. A probability level of 0.1
was chosen in order to minimize the probability of the type II
error {Rao et at., 1979]. The KS critical values for the GMC
and BST data are 0.096 and 0.066, respectively. Comparing
the KS test values of 0.146 and 0.173, for the GMC and BST
data, respectively, to the KS critical values indicates that nei-
ther the GMC nor the BST was obtained from a normally
distributed random function. Taking the log,0 transform of
the data and calculating KS test values (0.107 and 0.167, for
the GMC and BST, respectively) indicate that neither the
GMC nor BST is lognormally distributed.
The spatial correlation functions for the GMC, BST, and
their crosscorrelation are plotted in Figure 2, where the solid
circles indicate the sample correlation functions and the solid
curve the corresponding spherical model. The exes and
asterisks indicate that less than 10 and more than 65 pairs of
samples fell into the lagged interval, respectively.
The method used to generate the correlation and cross-
correlation functions was to calculate the sample covariance
and cross covariance [Journel and Huijbregts, 1978, especially
pp. 40 and 194] from the original data and to fit a spherical
model to the sample function. The covariance models were
then "tested" for validity using the jackknifing procedure
[Rusxo, 1984a, b; Vaudin et al., 1983] where the reduced mean
and variance are calculated using an estimated and actual
value at each sample location. Based on the jackknifing
method, the spherical models listed in Table 1 were deemed
satisfactory representations of the true spatial correlation
functions. The reduced mean and variance for the model Co-
variance functions are also given in Table 1. A difficulty in
using this method for validating a spatial correlation function
is that it lacks an independent method for determining what is
an appropriate value for the reduced mean and variance.
The cross-covariance functions were also tested for validity
using the jackknifing procedure. The model covariance func-
tions, which were determined and validated prior to the cross-
covariance functions, were used in the validation procedure
and considered constants.
The correlation and cross-correlation functions were calcu-
lated from the covariance by dividing the covariance and
cross-covariance functions by C^O) and [C,,(0)C22(0)]0'5, re-
spectively, where C,, is used to represent the covariance func-
tion for the GMC and C22 for the BST. Sample variograms
were also calculated (not shown) and verify that the nugget is
approximately zero.
The correlation and cross-correlation functions were deter-
mined in this manner to facilitate comparison between the
DCK and OCK results. An alternate method for calculating
the spatial correlation functions directly would be to use the
transformed data.
After the spatial correlation functions have been deter-
mined, the next step in the disjunctive kriging method is to
determine the transform relationships, 4> and and ij/ transform relationships by
the solid and dashed curves, respectively. Ten and 30 coef-
1.0
m 0.5 -
0.0
10
20
MOISTURE CONTENT (X)
co 0.5 •
0.0
30
40
TEMPERATURE (°C>
50
Fig. 3. Comparison between empirical distribution (solid circles) and transform relationship (solid and dashed curves).
Ten and 30 coefficients, were used in the calculation for the solid and dashed curves (see equations (1) and (2)), respectively.
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YATES: DISJUNCTIVE KRIGINO, 3
1375
ficients (i.e., Ct or Dk) were used in generating the solid and
dashed curves in Figure 3. Although 30 coefficients produce a
good fit between the data and the transform relationships, the
improvement in the disjunctive kriging estimates diminishes
rapidly as k increases [Rendu, 1980; Yates et al., 1986b].
Therefore only 10 coefficients were used in subsequent calcula-
tions.
Figures 4 and 5 contain a comparison between OCK
(Figure 4) and DCK (Figure 5). A total of 931 estimates on a
3- by 3-m grid system superimposed over the field were used
to generate the contour maps. The estimates were calculated
using 10 Hermite polynomials and 5 nearest neighbors within
a maximum radius of 30 m. It is evident from these figures
that zones of high CMC tend to lie near the north and south
boundaries of the field and the lower levels of CMC near the
middle part of the field. Also, the contour diagram produced
using linear cokriging is more continuous when compared to
DCK (see, for example, the contour CMC = 7). Visual obser-
vation in the field during sampling tends to support the results
of DCK over those of ordinary cokriging as giving a more
accurate moisture profile, since it was noted that the wetter
zones appeared in small patches throughout the field and es-
pecially near the north and south borders.
Using the 931 estimates of the GMC, the mean value (of the
estimates) was calculated. The DCK method produced a value
of 10.38, whereas for OCK the mean was 9.64. For this data
set there appears to be more bias in the estimates for OCK
compared to DCK. This may be due to using all the available
data to determine the covariance functions and then making
comparisons to the same data. It would be better if a fraction
of the data were used to calculate the covariance function and
then the comparisons made to the remaining data. However,
adequate data were not available.
The kriging variance was calculated at each of the 931
points using DCK and OCK. From these values the average
kriging variance was calculated for each method. The DCK
method had an average kriging variance which was 7% lower
than for OCK (e.g., 23.53 versus 25.24 for OCK).
9O
60
30"
30
60
90
EAST (ml
Fig. 4. Contour diagram for the GMC based on ordinary cokrig-
ing. Estimates were calculated using five nearest neighbors within a
maximum radius of 30 m. The contours 7, 13, and 19% are solid
curves and 10 and 16% are dashed curves.
EAST (m)
Fig. 5. Contour diagram for the CMC based on disjunctive cok-
riging. Estimates were calculated using five nearest neighbors within a
maximum radius of 30 m. The contours 7, 13, and 19% are solid
curves and 10 and 16% are dashed curves.
Another comparison between the two methods is the sum of
squares between the actual sample values and the estimated
values (based on the jackknifing technique). This method has
an advantage over the average kriging variance in that the
value of the random variable as well as the spatial correlation
enter the calculation. The DCK method produced a value
which was 19% lower than for OCK (e.g., 16.3 versus 20.1 for
OCK).
The conditional probability that the estimate is greater than
a cutoff level can be determined using DCK. Figure 6 contains
a comparison of the CP and conditional probability density
functions resulting from the DCK and DK methods for two
points in the field. The curves marked 1 and 2 indicate the
conditional probability distributions for points in the field lo-
cated at (66 and 39 m) and (81 and 45 m), respectively. The
solid and dashed curves represent the results from DCK and
DK, respectively. At the point marked 1, the probability dis-
tribution resulting from DCK is almost identical to DK. The
reason, to the first approximation, is that the GMC is domi-
nating the estimation process, since the GMC data is located
nearer to the estimation site than the BST data. The second
site (2) is a position in the field where the reverse is true (i.e.,
the BST is closer to the estimation site). The results for Figure
6 show the advantage of using an auxiliary variable in the
estimation process. At site 1 the CMC data supply most of the
information and thus the two methods are similar. At site 2
the BST is as important as the GMC data and therefore af-
fects the results more strongly.
CONCLUSIONS
The disjunctive cokriging method described in this paper
produces a nonlinear estimator which is better than the linear
(ordinary) cokriging estimator in terms of reduced variance of
errors and accuracy of estimation. This is expected, since a
nonlinear estimator should in general be superior to a linear
estimator except when the random variables are bivariate
normal. Estimates of the GMC based on 71 samples of GMC
and 148 samples of BST were produced using both the DCK
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1376
YATES: DISJUNCTIVE KRIGING, 3
5 0.5
1.0
CD 0.5
-3
-3
Fig. 6. (a) Conditional probability density and (b) cumulative distribution functions for two locations in the field.
Curves marked 1 and 2 correspond to the locations 66 and 39 m and 81 and 45 m, respectively. Solid and dashed curves
indicate the results from disjunctive cokriging and disjunctive kriging, respectively.
and OCK. Visual observation in the field during sampling
supports the results produced by the DCK method over those
by cokriging.
If in addition to the random function of primary interest
there is an additional highly correlated random function, it
can be included in the estimation process using the method
outlined in this paper. The advantage is that the DCK esti-
mator is a better estimator than the disjunctive kriging esti-
mator (of one variable) by virtue of the information added to
the problem by the auxiliary random function provided there
is a significant correlation between the two random functions.
A situation which lacks such a correlation will produce the
same results as the disjunctive kriging method (of one vari-
able) since the cross-correlation terms of (13) will be zero
which decouples the equations.
The CP that the value of a property is above a given cutoff
level can be based on cokriging. Since a better estimate of the
Wt[y(x0)] is obtained using DCK, in general, the CP based
on DCK will be better than the CP based on disjunctive
kriging. Using the DCK conditional probability can improve
the sampling efficiency by adding additional highly correlated
random functions which are easy to sample, if they exist, into
the estimation process.
The CP was calculated at two points in the field. The first
point is a location where the auxiliary variable adds little
additional information. For this situation, both the DCK and
DK methods are virtually identical. The second point is lo-
cated in an area of the field where there are few GMC data
nearby. For this location the DCK method produces different
conditional probability distributions compared to the DK.
In general, the disjunctive kriging method requires more
computational time compared to ordinary kriging. The same
is true for cokriging. Disjunctive kriging (both DK and DCK)
becomes an attractive alternative over ordinary kriging in
situations where the probability distributions are required at
the estimation site. For this situation the extra information
which is not readily obtainable using linear kriging methods
helps to offset the extra computational costs. However, as the
cost of computer time becomes less expensive (i.e., as with
persona] computers) the additional computational require-
ments necessary to implement DK will be a less important
consideration.
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S. R. Yates, R. S. Kerr Environmental Research Laboratory, P. O.
Box 1198, Ada, OK 74820.
(Received January 7,1986;
revised April 21,1986;
accepted April 28,1986.)
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