Disjunctive Kriging
      3. Cokriging
       (U.S.) Robert S. Kerr Environmental
      Research Lab., Ada, OK
      1986
                                                                  PB87-166260
L
J

-------
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
                  DESCRIPTORS
                                               b.lDENTlFIERS/OPEN ENDED TERMS
                                             c.  COSATi Field/Croup
It. DISTRIBUTION STATEMENT

   RELEASE TO PUBLIC.
                   19. SECURITY CLASS (This Rt port I
                        UNCLASSIFIED
              21. NO. OF PACES
                     7
                                               2O. SECURITY CLASS (ThJJ ptft>

                                                    UNCLASSIFIED
                                                                          22. PRICE
(PA
       2220-1 (••«. 4-77)   previous COITION if O»»OL(1C

-------
                                                                                     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

-------
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
-------
                                               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.

-------
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.

-------
                                             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

-------
 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.

                         REFERENCES
Abramowilz. M., and A. Stegun,  Handbook of Mathematical  Func-
  tions, Dover, New York, 1965.
Al-Sanabani, M., Spatial variability of salinity and sodium adsorption
  ratio in a Typic Haplargid Soil,  M.S. thesis, Univ. of Ariz., Tucson,
  1982.
Carr.  J.  R., D. E.  Myers, and C. E. Glass, Co-kriging—A computer
  program, Comput. Ceosci., 11, 111-128. 1985.
Journel, A. G., and Ch. J. Huijbregts, Mining Geostatistics, Academic,
  Orlando, Fla., 1978.
Myers, D. E., Matrix formulation of co-kriging. Math. Ceo/., 14, 249-
  257, 1982.
Myers, D. E., Co-kriging—New developments, in Geostatislics for Na-
  tural Resources Characterization, edited by G. Verly el al., pp. 205-
  305, D. Reidel, Hingham, Mass., 1984.
Rao, P. V., P. S. C. Rao, J. M. Davidson, and L. C. Hammond, Use of
  goodness-of-fil lesls for characterizing the spatial variability of soil
  properties, Soil Sci. Soc. Am. J., 43, 274-278, 1979.
Rendu, J. M., Disjunctive kriging: A  simplified theory. Math. Geol.,
  12,  306-321, 1980.
Rohlf, F. J., and R. R. Sokal, SlfllislicaJ Tables, 2nd ed., 219 pp., W.
  H. Freeman. New York, 1981.
Russo, D., Statistical analysis of crop yield-soil water relationships in
  heterogeneous soil under trickle irrigation, Soil Sci. Soc. Am. J., 48,
  1402-1410, 1984a.
Russo, D.,  A geostalistical approach to solute transport in hetero-
  geneous fields and its applications to salinity management,  Water
  Resour. Res., 20, 1260-1270, 1984fc.
Sokal, R. R., and F.  J.  Rohlf. Biometry,  2nd  ed..  859 pp., W. H.
  Freeman, New York, 1981.
Vauclin. M., S. R. Vieira, G. Vachaud, and D. R. Nielsen, The use of
  cokriging with limited field soil observations. Soil Sci. Soc. Am. J.,
  47,  175-184, 1983.
Yales, S. R., A. W. Warrick, and D. E. Myers, Disjunctive kriging, 1,
  Overview of  estimation  and conditional probability. Water Re-
  sourc. Res., 22,615-622,1986a.
Yates, S. R., A. W. Warrick, and D. E. Myers, Disjunctive kriging, 2,
  Examples, Water Resourc. Res., 22,623-630, 19866.
  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.)
                          NOTICE

This document  has been reviewed in  accordance  with
U.S.  Environmental Protection Agency policy and
approved  for publication.  Mention  of trade names
or  commercial  products does  not constitute endorse-
ment  or recommendation for use.
             This article documents research thtt was
           conducted under the auspices a) the US
           Environm«ntil Protection Agency (USEPA). At an
           arm of the government, tha USEPA ha* a paid up.
           non-exclusive, irravocabla. worldwide license lo
           raproduca. prapara derivative works, and distrib-
           uia lo tha public all of lha malarial contained
           herein, and to gram others acting on its b*half tha
           right to do so. The USEPA grant* to tha National
           Technical Information Service. Department of
           Commerce, the right to eel a* its agent in the
           distribution o( thie materiel.

-------