Application of fuzzy logics in determination of soil parameters.
Prskalo, Maja ; Prskalo, Zoran
1. INTRODUCTION
Grain size, as the most widely used term in sedimentology, is not
uniquely defined except perhaps only for simplest geometrical objects
such as sphere (diameter) or cube (edge length). For irregular particles
like sand grains, which can be quantified in terms of similarity with
easily recognizable geometrical shapes, size generally depends on the
measurement method, which in turn depends on the subject of study. Such
descriptive terms are subjective and of little assistance when grains do
not have a clearly identifiable shape. Quantitative description and
statistical comparison of shapes of grain populations can only be
achieved by the use of numerical parameters of shape (Santamarina,
2001). The paper uses the fuzzy logics, i.e. fuzzy sets, in the defining
and determining grain shapes though the coefficient of grain shape.
2. PREVIOUS RESEARCH
It is possible to distinguish four basic grain shapes on the basis
of relations of their axes (Prskalo, 2008). From the ratios of
intermediate axis to long axis (b/a) and short axis to intermediate axis
(c/b), discoidal, spheroidal, bladed and rod-shaped grain forms are
obtained. For description purposes, it is possible to take the entire
range of roundness and divide it into a small number of divisions, each
of which being designated as a roundness class, and measurement value as
the coefficient of grain shape.
Laboratory studies and measurements of grain roundness in
present-day sediments have shown that grain rounding is a very slow
process, which quickly becomes much slower as size decreases.
Kuenen's experiment (Pettijohn et al., 1987) proved that 20,000 km
of transport would cause less than 1% of weight loss in the angular,
medium grained sand of quartz, thus confirming earlier test studies.
Although it would be desired to assess quantitatively the distance of
travel for sand from its average roundness or percentage of angular
grains, this is still impossible because the process of rounding is
still inadequately comprehended.
3. MAMDANI'S DIRECT METHOD
By using a simple example with two input variables A and B and one
output variable C with application of the IF--THEN rule, one obtains:
Rule 1: IF x is [A.sub.1] and y is [B.sub.1] THEN z is [C.sub.1]
Rule 2: IF x is [A.sub.2] and y is [B.sub.2] THEN z is [C.sub.2]
(1)
where [A.sub.1], [A.sub.2], [B.sub.1], [B.sub.2], [C.sub.1] and
[C.sub.2] are fuzzy sets. Figure 1 shows the process of reasoning of
Mamdani's direct method (Tanaka, 1997). Input values, i.e. fuzzy
sets A and B give the descriptive values of grain sorting and grain size
with the coefficient of grain shape C as the linguistic variable:
A = {extremely poorly sorted to very poorly sorted, poorly sorted
to moderately sorted, moderately well sorted to well sorted, very well
sorted}
B = {very coarse or coarser grains, coarse grains, medium grains} C
= {very angular, angular, semiangular, rounded, very rounded}
[FIGURE 1 OMITTED]
After the degrees of membership function are determined in the
fuzzification stage, the next step is to use linguistic rules to decide
what action it is necessary to take as a response to given set of
membership function degree. To calculate numerical results of linguistic
rules based on the system input values, a technique called min-max
inference is used.
3.1 Application of rules on the proposed model of fuzzy logics
The twelve important rules governing in this system are:
If x is from [A.sub.1] and y from [B.sub.1] then z is from
[C.sub.1]
If x is from [A.sub.1] and y from [B.sub.2] then z is from
[C.sub.1]
If x is from [A.sub.1] and y from [B.sub.3] then z is from
[C.sub.2]
If x is from [A.sub.2] and y from [B.sub.1] then z is from
[C.sub.2]
If x is from [A.sub.2] and y from [B.sub.2] then z is from
[C.sub.2]
If x is from [A.sub.2] and y from [B.sub.3] then z is from
[C.sub.3]
If x is from [A.sub.3] and y from [B.sub.1] then z is from
[C.sub.3]
If x is from [A.sub.3] and y from [B.sub.2] then z is from
[C.sub.3]
If x is from [A.sub.3] and y from [B.sub.3] then z is from
[C.sub.4]
If x is from [A.sub.4] and y from [B.sub.1] then z is from
[C.sub.4]
If x is from [A.sub.4] and y from [B.sub.2] then z is from
[C.sub.4]
If x is from [A.sub.4] and y from [B.sub.3] then z is from
[C.sub.5] (2)
Reasoning each of the rules, it follows that:
* If sediment is extremely poorly sorted to very poorly sorted, and
grain size very coarse or coarser then the grain shape is very angular;
* If sediment is extremely poorly sorted to very poorly sorted, and
grain size coarse then the grain shape is very angular;
* If sediment is extremely poorly sorted to very poorly sorted, and
grain size medium then the grain shape is angular;
* If sediment is poorly sorted to moderately sorted, and grain size
very coarse or coarser then the grain shape is angular;
* If sediment is poorly sorted to moderately sorted, and grain size
coarse then the grain shape is angular;
* If sediment is poorly sorted to moderately sorted, and grain size
medium then the grain shape is semiangular to semirounded;
* If sediment is moderately well sorted to well sorted, and grain
size very coarse or coarser then the grain shape is semiangular to
semirounded;
* If sediment is moderately well sorted to well sorted, and grain
size coarse then the grain shape is semiangular to semirounded;
* If sediment is moderately well sorted to well sorted, and grain
size medium then the grain shape is rounded;
* If sediment is very well sorted, and grain size very coarse or
coarser then the grain shape is rounded;
* If sediment is very well sorted, and grain size coarse then the
grain shape is rounded;
* If sediment is very well sorted, and grain size medium then the
grain shape is very rounded.
The fuzzy set has to be converted to a definite value, and the
operation of conversion of fuzzy set is defuzzification by which the
coefficient of grain shape is obtained as the final solution.
4. SOLUTIONS OBTAINED FROM THE PROPOSED MODEL OF FUZZY LOGICS
A system with two inputs, one having four linguistic terms and the
other having three, and one output with five linguistic terms, has the
total of 4 x 3 x 5 = 60 different rules that can be used to describe the
full strategy of fuzzy control. Although there are totally 60 possible
rules of assessment, in the description in this case it is enough to use
only twelve of them (which are consequently merged into five rules). One
way to reduce the number of rules is solved by the use of fuzzy union
among antecedent variables. This will make it possible for the fuzzy
control system to be able to roughly express human thinking using a
simple description of system behavior, Figure 2 (Demicco, 2004). If
input values from Figure 2 are taken, namely the grain size 0.5 and
sorting 0.4, the coefficient of grain shape of 0.312 is obtained as the
solution. For the purpose of verification and comparison of results, in
the Matlab software a model of relations of grain size, sorting and
coefficients of grain shape was made, Figure 3. With application of the
method of gravity center defuzzification, the following membership
functions were selected: triangular and trapezoidal.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
5. CONCLUSION
Application of fuzzy logics in geological problems is a world
trend. The model used in this paper for the coefficient of grain shape
is a particular contribution to these investigations. Results obtained
by the application of fuzzy logics are almost identical to actually
measured values of the coefficient of grain shape in laboratory
conditions which is kf = 0.32, while the same value when applying the
model of fuzzy logics is kf=0.314-0.35, Figure 4. Similarly, application
of this methodological approach can be proposed for application in many
scientific disciplines, especially those that do not use a sufficient
number of verified and precise information in their work.
6. REFERENCES
Prskalo, M. (2008). Geomehanicke odlike blidinjske sinklinale u
funkciji geoloskog nastanka prostora, Dissertation, Faculty of Civil
Engineering, University of Mostar, ISBN 978-9958-9170-5-9, Mostar,
B&H
Tanaka, K. (1997). An Introduction to Fuzzy Logic for Practical
Applications, Springer, pp 86-87
Pettijohn, F.J., Potter, P.E. & Siever, R. (1987). Sand and
sandstone, Second Edition, Springer-Verlag, pp 74-78, Table 3.1, Figure
3.1
Demicco, R.V. & Klir, G.J. (2004). Fuzzy logic in Geology,
Elsevier, pp 98-99
Santamarina, J.C., Klein, K. A. & Fam, M. A. (2001). Soil and
waves, Wiley & Sons, New York, USA, pg.34
