An application of fuzzy sets and Dempster Shafter Theory (DST) in modeling the interpretational process of organic geochemistry data for predicting the level of maturities of oil and source rock samples is presented. This was accomplished by (1) representing linguistic imprecision and imprecision associated with experience by a fuzzy set theory, (2) capturing the probabilistic nature of imperfect evidences by a DST, and (3) combining multiple evidences by utilizing John Yen's generalized Dempster-Shafter Theory (GDST), which allows DST to deal with fuzzy information. The current prototype provides collective beliefs on the predicted levels of maturity by combining multiple evidences through GDST's rule of combination

Isaksen, G. H.

Kim, C. S.

English

NASA Technical Reports Server

APPLICATION OF
FUZZY SET AND DEMPSTER-SHAFER THEORY
TO ORGANIC GEOCHEMISTRY INTERPRETATION
C. SKIM and G. H. lsaksen
Exxon Production Research Co.
Houston, Tx. 77001
(713) 965-7645
ABSTRACT
This paper presents an application of fuzzy sets and Dempster
Shafer theory (DST) in modeling the interpretational process of
organic geochemistry data for predicting the level of maturities of oil
and source rock samples. This has been accomplished by (i)
representing linguistic imprecision and imprecision associated with
experience by a fuzzy set theory, (ii) capturing the probabilistic
nature of imperfect evidences by a DST, and (iii) combining multiple
evidences by utilizing John Yen's[1] generalized Dempster-Shafer
Theory(GDST), which allows DST to deal with fuzzy information. The
current prototype provides collective beliefs on the predicted levels
of maturity by combining multiple evidences through GDST's rule of
combination.
I. INTRODUCTION
Modeling the interpretation process of an expert requires
representation and management of uncertain knowledge. This is
because nearly every interesting domain contains knowledge that is
inherently inexact, incomplete, or unmeasurable.
In this paper we explicitly treat two forms of uncertainties. One form
of uncertainty is fuzziness related to linguistic imprecision. Based on
fuzzy set theory, Zadeh[2] developed possibility theory to express
this type of imprecision. The other form of uncertainty is the
probability with which a certain evidence correctly predicts a subset
of hypotheses. Dempster-Shafer Theory[3,4] (DST) deals with this
type of uncertainty and provides a mechanism for combining
multiple evidences for an overall belief in a subset of hypotheses.
Unlike classical probability theory, DST enables the degree of
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ignorance to be expressed explicitly and does not fix hypothesis
negation probability once occurrence probability is known.
In the past, several attempts[5,6] have been made to generalize DST
to deal with fuzzy information. While these attempts fall short of
fully justifying their approaches, John Yen[l] proposed a generalized
Dempster-Shafer Theory (GDST), in which the important principle of
DST is preserved: That the belief and the plausibility functions are
treated as lower and upper probability bounds.
In this paper, we demonstrate representation and management of
two types of uncertainties by GDST as applied to the interpretation of
organic geochemistry data. In the following sections, we review the
basics of GDST, and the development of a knowledge-based system
for geochemistry interpretation
II. BASICS OF A GENERALIZED DEMPSTER.SHAFER
THEORY
This review is not intended to describe detailed theory and
developments of DST and GDST. Rather, we plan to describe their
representation of imprecise information and the rule of combination
in a qualitative way. More interested readers should refer to the
references [1,3,4] cited.
In the DST, hypotheses in a frame of discernment must be mutually
exclusive and exhaustive, meaning that they must cover all the
possibilities and the individual hypothesis cannot overlap with
others. An important advantage of DST over classical probability
theory is its ability to express degree of ignorance associated with an
evidence. Also, unlike classical probability theory, a commitment of
belief to a hypothesis does not force the remaining belief to be
assigned to its compliment. Therefore, the amount of belief not
committed to any of the subsets of hypotheses represents the degree
of ignorance. In DST, a basic probability assignment(bpa) m(A), as a
generalization of a probability, indicates belief in a subset of
hypotheses A. This quantity m(A) serves as a measure of belief
committed to the subset A.
DST also provides a formal process for combining bpa's induced by
independent evidential sources, which is called the rule of
combination. This process is a tool for accumulating evidences to
274
narrow the hypothesis set. If ml, and m2 are two bpa's from two
evidential sources, a combined bpa is computed according to the rule
of combination:
ml(_m2(C) = _ ml(Ai)m2(Bj)/k
Aic_Bj= C
(1)
where k is a normalization factor,
k = 1 - _-'.ml(Ai)m2(Bj), (la)
Aic_Bj=¢
mlOm2(C) is a combined bpa for a hypothesis C,
0 is a null set, and
Ai, Bj are hypotheses sets induced by the two
evidential sources.
In the GDST proposed by Yen[6], a basic probability m(A) is assigned
to a fuzzy subset of hypotheses. In this framework, each fuzzy subset
of hypotheses has bpa m(A), and fuzzy membership function _A(Xi),
where xi's are elemental hypotheses in the frame of discernment.
The rule of combination in GDST consists of two operations: a cross-
product operation and a normalization process. Basic probabilities are
first combined by performing a generalized cross-product including
fuzzy set operations:
ml2(C) = ml ® m2(C) = X ml(Ai) m2(Bj) (2)
Aic'_Bj=C
where ml2(C ) is an unnormalized bpa induced by two
evidences, and n denotes a fuzzy intersection operator.
Then, a normalization is performed on fuzzy subsets of hypotheses
whose maximum membership values are less than one. A detailed
procedure and justification of this normalization process can be
found in the reference [1]. Yen[l] also showed that this normalization
can be postponed until the last evidence without affecting the
computational results and the commutativity of the rule of
combination.
In case of combining only two fuzzy bpa's, a combined bpa using
GDST's rules of combination is:
275
mt• m2(C) = y. Max _AnB(Xi) ml(A)m2(B)/k
(_¢C xi
(3)
where
k = 1 - '_-'(1 - Maxp, AnB(xi)) ml(A)m2(B), and
xi
A,B
is a normalized AnB.
(3a)
As can be noticed in the equations above, GDST allows partially
conflicting evidences, while DST only allows either conflicting or
confirming evidences.
IlL BIOMARKER INTERPRETATION SYSTEM
In exploration for oil and gas, it is important to be able to assess the
maximum temperatures to which sediments or oils have been
exposed in the subsurface. This is referred to as the level of thermal
maturity. Organic chemical compounds known as biomarkers enable
the geochemist to assess the level of maturity (LOM) of oils and
sedimentary organic matter. In this paper, we focus our attention on
modeling the process of interpreting biomarker data to predict LOM.
The LOM scale ranges from 1 to 20, with LOM=I being least mature
and LOM=20 most mature. There exist more than 10 biomarkers
whose intensities have definite links to the maturity with varying
degrees of resolution and prediction power.
In our approach, these varying degrees of resolution among
biomarker evidences are represented by fuzzy subsets of maturity
intervals, and the probability with which an evidence correctly
predicts a fuzzy maturity interval is represented by a basic
probability in GDST. Therefore, evidential knowledge is represented
in fuzzy rules, and the confidence for a specific rule is represented
by a bpa. Moreover, GDST's rule of combination provide collective
belief in the predicted level of maturity. In the following, detailed
representation methods are presented along with actual application
results.
(A) Representing Two Types of Imprecision
Interpretation of geochemical data is based on experience as well as
theory. This interpretational knowledge is descriptive in nature, and
276
best represented by fuzzy logic and possibility theory. For example,
one may have an experience based correlation study between level
of maturity (LOM) and %C2920S, which is a ratio of the intensities of
several organic compounds. Then, the correlation curve in Figure 1
may be used by an interpreter as follow:
IF %C2920S is 40 %,
THEN expected LOM is about 8.
In the rule above, the concluding part is descriptive in that LOM = 8
is most possible, but LOM values of 6,7,9, and 10 are also possible
with lesser degree as shown in Figure 2. Another example is the case
where both premise and conclusion are best represented by fuzzy
membership functions. Based on theory and experience, Heptane
value can only predict maturity levels in four qualitative categories,
such as immature, early mature, mature, and over mature. Examples
of Heptane rules are:
IF Heptane value is medium,
THEN maturity is early mature
IF Heptane value is high,
THEN maturity is mature
IF Heptane value is very high,
THEN maturity is over mature
In the rules above, both the premise and the conclusions are
descriptive and best represented by membership functions for
Heptane value and maturity as depicted in Figure 3a and Figure 3b.
From the fuzzy rules above and the membership functions in Figures
3a and 3b, observation of a Heptane value of 19 will result in the
possibility values of 0.5, 1.0, 1.0, and 0.5 for LOM = 6, 7, 8, and 9
respectively:
1-I LOM = {0.5/6, 1/7, 1/8, .5/9} (4)
In the current system, LOM is predicted from 10 evidences each of
which predicts LOM with different degree of resolution as shown by
the two examples above.
In addition to the imprecision in the knowledge represented by
possibility theory above, there exists another type of uncertainty
associated with evidences. For example, rules associated with
%C2920S have higher probability of being true than the Heptane
277
rules. In our approach, the probability with which a proposition " If
A is al Then B is bl" is true is represented by bpa assigned to the
fuzzy subset of hypotheses induced by the proposition. The
compliment of this probability is assigned to the degree of ignorance
associated with the proposition, since our system generates only one
fuzzy subset of hypotheses for each evidence.
(B) Test Result
In order to validate the system, thirty interpretations were tested to
see if the system's interpretations conformed to those of the expert.
With reference to the test results listed in Table 1, one can notice
that the system interpreted maturities are biased towards higher
LOM. However, these errors are all higher than they should be and
consistent by itself, and can be traced to the membership function
definitions. We are currently fine tuning these membership functions
to correct the problem and plan to test the system with additional
field data..
V. CONCLUSIONS
We presented a knowledge-based system in which linguistic
imprecisions and uncertainties associated with fuzzy rules are
modeled in the frame work of a generalized Dempster-Shafer Theory.
This development is significant in that many application problems in
oil exploration requires a mechanism of combining fuzzy information
from various sources.
Even though the current biomarker interpretation system has been
tested on only 30 data sets, the system will be further tested with
additional field data and expanded to handle interpretations for
other characteristics such as source facies, depositional
environments, and the degree of biodegradation.
278
REFERENCES
1. Yen, J., Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE
Trans., Sys., Man, & Cyb., Vol. 20, No, 3, May/June 1990
2. Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility,
Fuzzy Sets & Systems 1(1978) 3-28, North-Holland Publishing
Co., 1978
3. Dempster, A.P., Upper and lower probabilities induced by a
multivalued mapping, Annals Math. Statistics, Vol. 38, No. 2,
1967, pp.325-339
4 Shafer, G., A mathematical theory of evidence, Princeton Univ.
Press, Princeton, N.J., 1976
5. lshizuka, M., K.S. Fu, and J.T.P. Yao, Inference procedures and
uncertainty for the problem-reduction method, Inform. Sci.,
Vol. 28, 1982, pp. 179-206
6. Yager, R., Generalized probabilities of fuzzy events from fuzzy
belief structure, Inform. Sci., Vol. 28, 1982, pp. 45- 62
279
Data
Table 1.
Set Number
1
2
3
4
5
6
7
8
9
I0
Ii
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
3O
Comparison of interpretations
Interpreted LOM System Generated
LOM
8-9
9
9
9
9
9
8.5-9
>I0
9
9
9
7.5-8
>i0
>i0
I0-Ii
ii
9
7.5-8
8
10
10
10
10
9
9
10
9-10
9
10-11
10-11
9-10
I0
I0
I0-ii
i0
I0-Ii
9-10
Ii
9-10
9-10
9-10
7
11-11.5
11-11.5
ii
Ii
i0
8-9
9-10
ii-ii .5
11-11.5
Ii
Ii-II .5
9
9.5-10
Ii-Ii .5
ii
9.5-10
Ii
I0-II
280
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281


Application of fuzzy set and Dempster-Shafer theory to organic geochemistry interpretation

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