Kriging with imprecise (fuzzy) variograms. I: Theory

Mathematical Geology, Vol. 22, No. 1, 1990

Kriging with Imprecise (Fuzzy) Variograms. h Theory 1

A. Bardossy, z I. Bogardi, 3 and W. E. Kelly 3

Imprecise voriogram parameters are modeled with fiLzzy set theory. The.fit of a variogram model to experimental variograms is oJqen subjective. The "accuracy" of the fit is modeled with bnprecise variogram parameters. Measurement data often are insufficient to create "good" experimental variograms, b7 this case, prior knowledge and experience can contribute to determination of the variogram model parameters. A methodology for kriging with imprecise variogram parameters is developed. Both kriged values and estimation variances are calculated as fuzzy numbers and characterized by their membership fimctions. Besides estimation variance, the membership functions are used to create another uncertainty measure. This measure depends on both homogeneity and configuration of the data.

KEY WORDS: kriging, variogram, fuzzy set theory, prior information, methodology.

INTRODUCTION

The purpose of this paper is to present a methodology for utilizing imprecise information on variogram parameters in geostatistical predictions. In Part I, the theory is developed, whereas Part II illustrates the methodology.

Commonly in applied geostatistics, only a small amount of spatial measurement data are available, and thus difficulties in estimation of the variogram arise. However, knowledge about possible values (or at least intervals) for the variogram parameters stemming from prior experience or the literature may be available. When traditional geostatistics cannot be used because of lack of a variogram model in areas such as geotechnical engineering (Baecher, 1984), petroleum engineering (Jones, 1984) and hazardous waste containment reliability analysis (Bogardi et a l . , 1987), the autocorrelation structure of key vari- ables often is neglecte& e.g., permeability (Gelhar, 1986).

~Manuscript received 5 May 1988; accepted 26 June 1989. 2Institute for Hydrology and Water Resources, University of Karlsruhe, Kaiserstr. 12, D-7500 Karlsmhe, F.R.G.

3Department of Civil Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska, 68588- 0531, U.S.A.

63 0882-8121/90/0100-0063$06.00/I © 1990 International Association for Mathematical Geology

64 Bardossy, Bogardi, and Kelly

The general goal for introducing fuzzy set methods is to handle different types of uncertain information in geostatistical modeling. Because available information can be of different type and quality, ranging from measurement data to geological experience, the treatment of such information should reflect its diversity. For the inexact (imprecise) information resulting from subjectNe as- sessments, a nonfrequentist approach could be used. The present approach uses fuzzy set theory as a nonfrequentist method of handling uncertainty. Diamond (1989) has used fuzzy valued random functions in kriging although in a different way than presented in this paper.

In a prior paper, use of imprecise data in kriging was introduced (Bardossy et al., 1988). Here, uncertainty in the variogram fit is addressed.

Probabilistic formulation of spatial processes is one approach to the problem of estimating the unique realization of a phenomenon. One cannot speak of a " t rue" variogram, but rather of an "exhaustive" variogram, provided by exhaustive sampling of the area considered (Joumel, 1985). Even in the case of large data sets, the experimental variogram cannot be fitted easily with a theoretical model. Experimental variograms from a simulated field (Fig. 1) shows a case where the fit is relatively simple and a more difficult (uncertain) fit (Fig. 2). Fits could be even more imprecise in the case of small data sets. Experimental variograms reflect the available data, however the fitted model must be representative of the entire population. This is another often neglected source of uncertainty. Several methods to robustify the calculation of experimental variograms exist (Cressie and Hawkins, 1980; Dowd, 1984; Joumel, 1988; Srivastava and Parker, 1989), but they still can lead to substantially dif-

(D

12

10

8

6

4

2

O O

emp i rical

fit

I I i i

1 O0 200 300 400

Lag Distance, h

Fig. 1. Experimental variogram with an "easy" fit.

Kriging with Imprecise Variograms. I 65

cD

12

10

8

8

4

2

0 O

fit

I _ I I J

100 200 3oo 400

Lag Distance, h

Fig. 2. Experimental variogram with a "difficult" fit.

ferent fits. As the variogram model plays a critical role in calculation of the reliability of an estimation, the uncertainty of its fit should not be neglected in determining that reliability. Loss of "uncertainty" in the variogram parameters may result in underestimated risks and a false confidence in the results.

Treatment of this uncertainty is possible using traditional probabilistic approaches; however, to do so, strong assumptions on the underlying distributions have to be made. Kitanidis (1986) in using Bayesian kriging, for example, assumes multivariate normal distribution. The adequateness of such assumptions for a particular data set is often doubtful. Jackknife cross validations can justify variogram parameters, but several theoretical models can perform similarly. Also variogram parameters validated in this way refer to the limited data available and not to the entire underlying population.

In the present paper, imprecision refers to vagueness in boundaries of a set. This definition corresponds to a generalized notion of imprecision which is characterized by so-called "sof t" tolerance interval boundaries in tolerance analysis (Dubois and Prade, 1980). This is in contrast with the usual notion of imprecision, where tolerance interval boundaries are well-defined (so-called crisp) numbers. Imprecision will be treated mathematically using the notion of fuzziness and techniques of fuzzy set theory (Hipel, 1982; Kaufmann and Gupta, 1985).

FUZZY VARIOGRAM PARAMETERS

The choice of the variogram type (spherical, exponential, or Gaussian) and parameters (nugget effect, sill, and range) can influence strongly the kriging


results (Diamond and Armstrong, 1984; Armstrong, 1984; Brooker, 1986; Bar- dossy, 1988). However, the limited amount of data and/or the nonsymmetric distribution of the parameter under study makes it difficult to define robust [with regard to the assumed parameter distributions, as defined by Huber (1977)] variogram estimators. Different techniques have been proposed to calculate robust variogram estimates (Cressie and Hawkins, 1980; Dowd, 1984; Omre, 1984).

On the other hand, the general problem of small data sets has received relatively little attention. No wonder that Armstrong (1984) questions the need for more sophisticated methods for robust variograms, claiming that "common sense and ordinary data cleaning techniques" would help, in most cases, to estimate a "sensible" variogram. The proposed use of fuzzy set theory is in- tended to model (i.e., quantify) "common sense" and "sensible" variograms.

Two basic tools of fuzzy set theory will be used: the definition of fuzzy sets in R" and the extension principle (Appendix A).

Imprecision in variogram estimation is related to two causes:

1. Uncertainty due to the choice of a particular analytical model.

2. Given an analytical model, the uncertainty about the corresponding variogram parameters.

The second reason is addressed first, then the model choice problem is considered.

Given an analytical model, the variogram parameters p~ . . . . . Pk cannot be assessed as exact (crisp) numbers. This can be due to the behavior of the experimental variogram (difficult fit) and the small amount of available data to calculate the experimental variogram.

The theoretical variogram selected depends on its parameters and the sep- aration vector h.

V(h) = V(P, . . . . . pk, h) (1)

The set of possible variogram parameters is supposed to be a fuzzy set (P in R k. Each parameter vector (p~ . . . . . Pk) is assigned a membership value in (P showing how acceptable the parameter vector is. For an experimental variogram with difficult fitted models (Fig. 3), different membership values correspond to the imprecise parameters. The membership function of (P can be constructed by assessing membership functions for each parameterpi, and then combining them. This approach ignores possible dependencies between individual parameters. Another approach consists in constructing the membership function using functional dependencies between parameters, and then combining the memberships corresponding to these functional expressions. This second approach is illustrated with two examples.


8

(_9

4-

1 2 -

t0

high fuzzy

100 2OO ,30O 40O

Lag Distance, h

Fig. 3. Experimental variogram and fits with three different membership val- ties,

An example of verbal definition of the set (P is 7 ( h ) , a spherical model with a nugget effect, with a sill approximately 1, a range around 100, and a relative nugget constant of about 0.33. Thus, the variogram expression is:

I 0 ( ) if h=0

h 3 7(p j ,P2 , p3, h) = Pl +P2 1.5--h _ 0 . 5 5 i f 0 < h < P 3

/)3 P3

PJ + P 2 i fh __p3

The membership function o f the fuzzy set corresponding to the above verbal definition can be constructed using the extension principle (Appendix A):

#6,(Pl, P2, P3)

+ p2 ' F p 3 - 1001 = rain 1 - I Pl + p 2 - I t 1 - , l

0.05 ' 0.03 10

[/*~(P~, P2, P3) = 0 if the above minimum is negative. ] The first term in the min quantity reflects uncertainty in the sill, with the tolerance in the sill of 0.05, [in other words, the sill will not be outside the interval (0.95, 1.05)]. The second term corresponds to the relative nugget constant, and the third one to the range.


Incorporation of uncertainty due to the choice of the model also is possible. A more complex definition of the fuzzy set 6) which reflects the model choice uncertainty is as follows. On the basis of prior knowledge, and given the limited number of samples, a variogram model with an approximate sill of 1, a relative nugget constant of 0.25, and an effective range of 100 is assumed. The variogram model is more likely to be spherical, but it could also be gaussian (or a combination of the two models). The membership value of a pure spherical is taken as 1, the pure gaussian has the membership value of #c. The complex variogram model is:

"Y(Pl, P2, P3, P4, Ps, h ) = Pl + P27s(P3, h) + P4"fG(Ps, h)

Here 7s(P3, h) is a spherical model with range P3 and 7c(P5, h) is a gaussian model with range Ps.

Following the above verbal description; the membership function of the corresponding fuzzy set 6 ) is constructed by assigning several membership grades Ug to the degree of fullfillment of the statements. These values are combined to construct the membership function of 6). Thus,

Pl Pl 0.25 / + P2 + P4

0, 1 - 0.05 Pl ~ max

reflecting uncertainty on the relative nugget constant,

p2 = m a x ( 0 , 1 - IP, +P20.10+P4--1 ] )

reflecting uncertainty on the sill,

u3 = m a x ( 0 , 1 I P 3 - 1001)i-0

reflecting uncertainty on the range of the spherical model, and

/'4 ~-- m a x ( 0 , 1 Ip5 - 57.7[)5.77

reflecting uncertainty of the range of the gaussian model (having the same effective range as the spherical model). Finally, the model choice uncertainty is described with:

P2 v5 - - - ( 1 - fie) + ~c

P2 + P4

Kriging with Imprecise Variograms. 1 69

Using these values, the membership function can be defined as:

#~ = min (vj, v> ~'3, v4, vs)

The membership function of the fuzzy set (P is constructed with the help of statements made on the variogram parameters. These statements can describe individual parameters (range in the above cases), or more parameters in dependence (relative nugget constant). These statements are combined, and no further dependence is assumed between them. Additional assumptions like a dependence between nugget constant and choice of variogram model also could be incorporated. (Note that in fuzzy formulation, dependence and independence have no probabilistic meaning. Dependence is expressed with the help of statements, which are used to construct the membership function of (P.) Uncertain- ties corresponding to individual statements can be assessed by answering the following questions: Which is the most likely value? What are the bounds between which the value is known with certainty? The answer to the first question gives the value with membership 1 ; the second answer gives end points of the interval with nonnegative membership values. Usually a linear interpolation between these points gives a useful membership value for intermediate points.

DEFINITION OF KRIGING EQUATIONS

Consider the estimation of an average value over a domain V from measurement data, using ordinary kriging. The kriged estimator, Z*(V) is

Z*(V) = Z X,Z(x,) (2) i = l

where

V:

X i

Z(xi): Xi:

the volume which average value is to be estimated

observation points (i = 1 . . . . . n)

measurement values at observation points (xi)

weights calculated from solution of the kriging equations.

/ t

x j = ] j = l

II

x i v ( x , - x j ) + = v ) i = 1 . . . . . j = l

here, ~ is the average variogram value:

l f v ) = - x ) dx

The kriging variance c~2x can be calculated as:

I I I I I t

a~ = - ~ ( V , V) - Z Z XsXiT(x , - . r / ) + 2 ~3 X,~(x,, V) j = l i = 1 i = [

(3)

(4)

(5)


here

~(V, V) - [vIZ v v T ( x - y) d x d y (6)

A kriging operator K is now defined to represent Eq. (2) (assuming that the variogram is predetermined):

Z * ( V ) : K [ V , p , . . . . . pk, x, . . . . . x , , , z ( x , ) . . . . . . z(x,)] (7)

Similarly, another operator represents the calculation of the kriging variance

°2k = O ( V , Pl, . . . . Pk, x l , . . . , x,,) (8)

CALCULATION OF FUZZY KRIGED ESTIMATES AND THE ESTIMATION VARIANCE

Suppose that imprecise variogram parameters are defined. This means that instead of crisp parameters Pl, • • - , Pk, a fuzzy set (P of parameters is given. Using the extension principle for the kriging operator K, the membership value for any real number y resulting from kriging is:

~ sup { / ~ ( p , . . . . . Pk); Y = K[ V, p, . . . .

tx~(y) = ) pk, x, . . . . . x , , z ( & ) . . . . . z(x,,)]/ (9) / t, 0 if the above set is empty

Here, /x~ is the membership function for the fuzzy subset of variogram parameters.

This equation defines a membership function on the set of real numbers. The fuz zy number corresponding to this membersh ip func t ion will be called the

result o f f u z zy kriging. If the fuzzy subset (P fulfills the convexity conditions [Eq. (2), Appendix

A], the result of fuzzy kriging is a fuzzy number as demonstrated (Appendix B).

Use of the extension principle with operator U [Eq. (8)] results in an estimation variance expressed as a fuzzy number. The membership value of any real number s 2 for the estimation variance is defined as:

~sup [ ~ ( p , Pk); s2 = V ( V , p , . . . . . pk, x, . . . . . x,,)]

~v(s2) = (13 if the above set is empty

(10)

The above fuzzy set is a fuzzy number (Appendix B). In this way, the uncertainty (imprecision) of the variogram model is trans-

fered to the kriging estimate and estimation variance.

Kriging with Imprecise Variograms. 1 71

THE P R AC TIC E OF FUZZY K R I G I N G

At each location~ fuzzy kriging results in fuzzy numbers for the estimate and estimation variance. However, the definition of the membership functions [Eqs. (9) and (10)] as suprema is not convenient for computational purposes because it would require for each possible real number y, an optimization algorithm to retrieve the membership value #. To simplify calculations, instead of assigning a membership value to every specific value y, a value y could be assigned to each specific membership value t~. If fuzzy set 6 ~ is connected (Ap- pendix B), this can be done by using the fact that level sets of fuzzy numbers are intervals. For fuzzy kriging, the endpoints of these intervals can be deter- mined with the help of two optimization problems; namely, for any selected membership level 0 < t _< 1,

find: m i n K [ V , p l . . . . . pk , x , . . . . . x,,, z ( x , ) . . . . . z(x.)] (11)

subject to the constraint:

~ ® ( p , . . . . . pk) "--- t

and find: max K [ V. p t . . . . . Pk. x l . . . . . x,,. z ( x , ) . . . . . z(x. ) ]

(12)

subject to the same constraint. Results of these optimizations provide end points of intervals R~ These

optimizations for a selected finite set of t are sufficient because, by virtue of convexity, they provide greater and lesser bounds for membership of intermediate values. The same is done to calculate fuzzy estimation variances.

The problems defined by Eqs. (11) and (12) are constrained nonlinear optimization problems. Their solution requires fast numerical procedures. Also, the objective functions are the same for each possible t; thus, the feasible sets are included in one another. This allows a rapid optimization algorithm because solutions of t _> t' can be used as initial guesses for t.

P R O P E R T I E S O F K R I G I N G W I T H F U Z Z Y V A R I O G R A M S

The defined operator (9) for fuzzy set kriging has a number of interesting properties. Some are:

1. If the set of variogram parameter vectors (P is a crisp set, the result of fuzzy set kriging is a crisp number that equals the result of ordinary kriging. This property is a consequence of the extension principle.

2. If z(xl) = z(x2) . . . . . z(x,,), the result of fuzzy set kriging is a crisp number. The more the z ( x i ) values differ from each other the wider are the t-level sets of the results. This will be shown in the following numerical example.


3. I f the data configuration is totally symmetric (e,g., the average of a square is to be estimated from measurement values taken at the four corners), the result is a crisp number. The more symmetric the configuration, the wider are the t-level intervals of the results.

A N U M E R I C A L E X A M P L E

Consider kriging of the average value of a 10 x 10 square block Vcentered at the origin. The variogram is linear with a nugget effect.

7(h) =p, +p21hl f o r h > 0 (13)

Sample locations are at three corners of V: xj = ( - 5 , - 5 ) , x2 = ( - 5 , 5), and x3 = (5, 5) (Fig. 4). The variogram is imprecise with p2 = 0.05 and Pl = 0.1. The corresponding membership functions are:

I 1 ]p, - 0.1] for 0.09 < p, < 0.11 ~, ( p , ) = 0.01 (14)

0 else

I i ' P 2 - 0"05' forO.O4 < p 2 < 0.06 /x2 (p2) = 0.01 (15)

else

The membership function of the fuzzy parameter set (9 is

, ~ ( P l , P2) = min [P,t(Pl) , / x2 (P2) ]

(Construction of this fuzzy set assumes that no influence of membership of one parameter exists on the other. I f any dependence exists, it could be incorporated as in the case of the two previously shown fuzzy sets (P,)

Kriging equations for this particular configuration can be written using the parameters Pl, P2:

)x2(10p2 -b Pl) q- X3(10"f2 P2 + Pl

Xl(lOp2 + p , ) + ~k3(10p2 + p,

)kl(lO@2 q-Pl) q- X2(10p2 + p,

) + ~ : ~(x, , v )

) q-/~ = ~(x2, v )

) + . : ~(x~. v )

k~ + X: + X 3 = 1

By symmetry Xl = k3 and ~(xi, V) = ~(xj, V). After some calculations

Xl =_ X3 = 10p2 + PE (16) 10(4 - x/2)P2 + 3p,

X 2 = 1 - 2)', 1 (17)


x2= (-5,5} xa=(5, 5) + - +

q-

! I 1 I I I

I L I l I

V

x I =(-5:5} ]

Fig. 4. Data configuration for the numerical example.

Therefore, the kriging estimator for block V using the notation ¢ = p~ is: P2

= ¢ + 1o 2 o - l O ~ + ¢ Z * ( V ) 40 - 10v/2 + 3¢ (z, + za) + 40 - 10v/-2 + 3¢ z2 (18)

Calculating the derivative of the above function with respect to ¢ shows that the function is strictly increasing or strictly decreasing depending on data values z~, z2, z3. Thus, extreme values of Z * ( V ) correspond to end points of the ¢ interval. For any selected membership level t,

rain [u; I~p~(U) >- t] ¢I 1> = (19)

max [v; .:,,(,:) > :]

and

ma~ [.; .,,,(.) >__ t] ¢I 2, = (20)

min [v; .p~(v) >- t]

The membership function of 2 * ( V ) can be constructed now. End points of the t level intervals for Z* (V) are the values obtained from Eq. (16) substi- tuting, respectively, ¢I 1) and ¢~2) for ¢.

For example, ifzj = z2 = 3, z3 = 4, then ~2.(3.3767) = 1 and [#2, (u) > 0] = [3.3738, 3.3788]. This shows that the imprecision of the variogram plays no important role in this case.

74 B a r d o s s y , Bogardi, and Kel ly

.D.. x=

I.O

0.8

0.6

0.4 ~-

0.2 l

0 L 3.320 3.340 3.360 3.380 3.400 3.420 3 .440

Kriging Esfimofor Z*(V)

Fig. 5. Membership functions A and B of 2 * ( V ) . B corresponds to dataz~ = z2 = 1, z3 = 10.63, A corresponds to data z~ = z2 = 3, Z3 = 4 .

If zl = z2 = 1, Z 3 = 10.63, then iz2.(3.3752 ) = 1 and [/z2.(u) > 0] = [3.3340, 3.4303], showing that the role of the imprecision becomes more important. The width as uncertainty measure depends on data values, and not only on the configuration, as is the case of the estimation variance (Fig. 5).

DISCUSSION

Fuzzy set theory can be used to account for imprecise knowledge of variogram parameters in kriging. This problem also can be addressed in a Bayesian framework, as was done by Kitanidis (1986) for the multivariate normal case. Both approaches have advantages and limitations. The problem with a Bayesian approach includes that prior distributions have to be selected, preferably from the conjugate family, and without further information, assumptions have to be made on the dependence-independence of the variogram parameters; also, the Bayesian approach requires extensive calculation, and its application is limited to the multivariate normal case. The fuzzy set approach has a similar difficulty in selection of the membership functions, but only simple dependence-independence assumptions are necessary, and computations are relatively simple.

An attractive feature of fuzzy set formulation is that the probabilistic and fuzzy uncertainties are separated. Fuzzy uncertainty measures the uncertainty associated with actual measurement values and variogram parameter estimation. The estimation variance indicates the probabilistic uncertainty of the interpolation, assuming the variogram parameters exact. The interval width of kriged values can be used to measure the effect of variogram uncertainty. This width reflects both configuration and data values. The more a configuration differs


from full symmetry and the greater the variance of the data, the larger the fuzzy width will be.

These features of fuzzy formulation have important implications for the related observational network design problem. Classical network design gen- erally considers only estimation variance and endeavors to minimize it in con- junction with other objectives or under constraints related to measurement effort (Bogardi et al., 1985). As a result for traditional kriging, the preferred measurement scheme always would be a uniformly spaced grid. Experience has shown, however, that irregular measurement networks can be more effective whenever the measurements exhibit inhomogeneity (Rogowski and Simmons, 1988). A network design using a fuzzy set formulation can address this problem and consequently a dual objective can be stated, to minimize the width of the estimator and to minimize the estimation variance (or its fuzzy mean). Then, multicriterion decision-making can be used to trade off, first, the width with the estimation variance, then a composition of these two with the measurement effort (Bogardi et al., 1985), to find the most cost-effective network which may not be a regular grid.

CONCLUSIONS

l. Variogram parameters can be considered as fuzzy subsets with membership functions assessed from the experimental variogram, the data set, and prior knowledge.

2. The procedure herein called kriging with fuzzy variograms does not use specific weights as does traditional kriging but applies the extension principle of fuzzy numbers.

3. The extension principle is applied to an operator depending on actual measurement values. Thus, the results depend on measurement values, whereas in the case of ordinary kriging these weights are independent of measurement values.

4. Kriged values are expressed as fuzzy numbers which may be characterized by the fuzzy mean and width.

5. The width of kriged values can be interpreted as a new uncertainty measure originating from variogram uncertainty and depending on actual measurement values.

6. Estimation variance can be calculated also as a fuzzy number.

APPENDIX A: FUZZY SETS, FUZZY NUMBERS, THE EXTENSION P R I N C I P L E

Basic definitions of fuzzy sets, fuzzy numbers, and fuzzy operations (Du- bois and Prade, 1980) are summarized. Some definitions are not given in their

7 6 B a r d o s s y , B o g a r d i , a n d K e l l y

most general forms but, for convenience, are restricted to the form used in this paper.

Definit ion: Let X b e a set (universe). A is a fuzzy subset o f X ifA is a set of ordered pairs:

A = {Ix ,

where I*A(X) is the grade of membership o f x in A. Values of/*A (x) are in the closed interval [0, 1]. The closer/*A(x) is to 1, the more x belongs to A; the closer it is to 0, the less it belongs to A. If [0, 1] is replaced by the two element set { 0, 1 }, A can be regarded as a subset of X.

Definition: The q level set of the fuzzy subset A is the set of those ele- ments which have at least q membership:

Aq : {x; /~ (x ) _> q}

Definition: A fuzzy subset M is called normal if at least one x exists such that ~tM(x) = 1.

Definition: A fuzzy subset M of the set of real numbers is called convex if for e a c h 0 < ce < 1 and eachx, y ~ M

/~a4[oex + (1 - oe)y] _> min [~tM(x), /~M(Y)] (21)

Definition: A fuzzy subset M is called a fuzzy number, if M is a normal convex fuzzy subset of the set of real numbers.

The membership value of a real number reflects the "l ikeliness" of the occurrence of that number, the level sets (intervals in this case) reflect different sets of numbers with a given minimum likeliness (Kaufmann and Gupta, 1985). Any real number can be regarded as a fuzzy number, and often is called a crisp number in fuzzy mathematics.

Definition: IfX~ . . . . . X,, are sets and Aj . . . . . A,, are fuzzy subsets of X1 . . . . , X,,, respectively, the Cartesian product ofA~, . . . , A,,, that is Al x • . . X A,, is the fuzzy subset of X~ x . . . × X,, with the membership function

/*(xl . . . . . x,,) = min [tXAi(Xi); i = 1 . . . . . n]

The extension pr inciple (Zadeh, 1965) is a method of extending point-to- point operations to fuzzy sets. It is the basic tool for development of fuzzy arithmetic.

Definition: If X and Y are two sets, and f is a mapping from X to Y

f : X --+ Y for all x E X f ( x ) = y e Y

t h e n f c a n be extended to operate on fuzzy subsets of X in the following way: Let A be a fuzzy subset of X with membership function /*A, the image of

A in Y is the fuzzy subset B with the membership function

~ sup [/~A(X); x : y = f ( x ) , X ~ X]

/~B(Y) = (,0 if no x e X, such t h a t f ( x ) = y (22)


Applying this principle to functions l i k e f ( x , y) = x + y o f f ( x , y) = x • y, different operations on fuzzy sets can be defined.

Definition: The mean of a fuzzy number A is

i+~_ xlxA(x) dx M(A) = (23)

Definition: The width of a fuzzy number z{ is

W(A) = sup[x; tXA(X) > 01 - inf Ix; #A(X) > 0] (24)

The width of a fuzzy number is not necessarily finite, and the mean does not always exist. However, in this paper only fuzzy numbers with finite width are considered.

APPENDIX B: PROPERTIES OF FUZZY KRIGING

First, the kriged estimate will be shown to be a fuzzy number.

Proposition 1. I f @ is a fuzzy subset in R c such that:

1. one and only one (Pl . . . . . Pc) exists such that ~ ( P l . . . . . Pc) = 1,

2. for each 0 < t _< 1, the t level set (P,

@, = [ (P , . . . . . Pc); ,~,,(t'~ . . . . . Pc) >- t] of the fuzzy set of variogram parameters (P is connected in R *, then the result of fuzzy kriging is a fuzzy number.

Proof Because a fuzzy number is a normal convex fuzzy subset of real numbers, the result of fuzzy kriging has to proved to be

1. Normalized: I f p l . . . . . Pc is chosen such that tx~(p~ . . . . . Pk) = 1, then for

y = ~C[V, p, . . . . . p~, x, . . . . . x,,, Z ( x , ) . . . . . Z(x,,)]

# ( y ) = 1 from Eq. (9).

2. Convex: To show this, the level sets of the results have to be used.

R, = {s; .(>') t} Operator K(.) is continuous with respect to variogram parameters p~, . . . . Pc. This can be shown by using (12) or (16) in Diamond (1986).

Because of the continuity of K(.) with respect to p~ . . . . . Pk and the fact that the f level set (P, is connected, its image in R is also con-

78 Bardossy, Bngardi, and Kelly

nected, hence an interval. As the relation R t D Rt~, if t ~ > t also holds, the convexity is also proved.

Next, the fuzzy set defined for estimation variance [Eq. (10)] is proved also to be a fuzzy number.

Proposit ion 1. If (P is a fuzzy subset in R ~ with properties as in Proposition 1, the corresponding estimation variance is a fuzzy number.

P r o o f Similarly, the result of estimation variance has to be proved to be:

1. Normalized: If pl . . . . . px is chosen such that #~(p~ . . . . . Pk) = 1, for

y = U ( V , Pl . . . . . Pk, xi . . . . . X,,)

Ix(y) = 1 from Eq. (10).

2. Convex: Using Eq. (3), U(,) can be seen to be a continuous function of

variogram parameters p~ . . . . . Pk and because (Pt is connected in R k its image on R is also connected; hence it is an interval.

As the relation R, C Rt,, if t > t t also holds, the convexity is proved.

A C K N O W L E D G M E N T

Research on which this paper is based has been partially supported by the National Science Foundation (Fuzzy regression and interpolation with application to fuzzy reliability) and by the U.S. Geological Survey, Department of

Interior under award number 14-08-001-G1503. We thank reviewers, J. Haslett and A. G. Joumel.

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Kriging with imprecise (fuzzy) variograms. I: Theory

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