APPLICATIONS OF FUZZY MATHEMATICS IN EARTHQUAKE PREDICTION

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1 J. Phys. Earth, 34, Suppl., S213-S221, 1986 APPLICATIONS OF FUZZY MATHEMATICS IN EARTHQUAKE PREDICTION Deyi FENG,* Mingzhou LIN,** Chun JIANG,* and Ronghua CHENG*** * Seismological Bureau of Tianjin, Tianjin, China ** Seismological Bureau of Shanghai, Shanghai, China *** Center for Analysis and Prediction, State Seismological Bureau, Beijing, China (Received June 22, 1985; Revised December 22, 1986) This paper presents a direct method for fuzzy recognition of earthquake precursors and a method of fuzzy clustering analysis of temporal seismicity patterns for earthquake prediction. The direct method for fuzzy recognition of earthquake precursors is based on their fuzzy description by use of membership functions. If we use a suitable membership function, the abnormal features of a precursor can be more clearly recognized. The membership function proposed by Feng et al. in 1981 is applied to reported changes in VP/VS, resistivity, and radon content. The basic idea of application of fuzzy clustering analysis to earthquake prediction is that time intervals with different sizes of the largest earthquake can be classified on the basis of fuzzy similar relations according to the various kinds of statistical indices of seismicity. The method based on the fuzzy equivalent relation includes the following steps: obtaining a fuzzy resemblance relation between samples, transforming it into a fuzzy equivalent relation, and classifying the original samples. The statistical indices of seismicity used by Feng et al. in 1984 is employed for clustering analysis of seismicity in the North-South Seismic Belt of China. 1. Introduction In many problems of earthquake researches, especially the problems on earthquake prediction, we cannot avoid staying at the stage of experiences and using a series of fuzzy factors or indices. In this case, some methods of fuzzy mathematics can play an important role. These methods are based on the concept and theory of fuzzy sets (ZADEH, 1965), and they attract much attention in various scientific fields (DUBOIS and PRADE, 1980; FENG et al., 1983; WANG and CHANG, 1980; FENG et al., 1984/85). Particularly, the applications of fuzzy sets theory to many fields in earthquake researches, including the earthquake hazard assessment, earthquake prediction, and earthquake engineering, are receiving increasing attention among scientists in seismology, engineering, and mathematics. In recent years, Chinese seismologists and mathematicians have been studying the applications of fuzzy mathematics in earthquake researches. The various methods of fuzzy mathematics have been applied to a series of problems in earthquake researches, as shown in Table 1. This paper presents a direct method for fuzzy recognition of earthquake precursors S213

2 S214D. FENG et al. Table 1. The topics in earthquake researches and applied methods of fuzzy mathematics. FPR, Fuzzy pattern recognition; FCA, fuzzy clustering analysis; FIA, fuzzy information retrieval; FSC, fuzzy similarity choice; FME, fuzzy multifactorial evaluation; FMDA, fuzzy method of degree of approaching; FDM, fuzzy decision-making; FR, fuzzy reasoning. and a method of fuzzy clustering analysis of temporal seismicity patterns for earthquake prediction, which is based on the fuzzy equivalent relation. To illustrate this method, some preliminary results obtained for large and moderate earthquakes occurring on the North- South Seismic Belt of China will be shown. 2. The Direct Method for Fuzzy Recognition of Earthquake Precursors Generally, we can consider any set of precursor data to be a fuzzy set characterized by some membership function. The direct method for fuzzy recognition of earthquake precursors is simply based on the membership functions of different precursors. Therefore, its effectiveness depends on how we can construct these membership functions appropriately. Quite a number of precursor data, as is well-known, show two types of anomaly before a large earthquake: so called "bay-type" anomaly and "inverse bay-type" anomaly. The bay-type anomaly is characterized by a time sequence of decrease, stationary state, and increase of observed quantity prior to an earthquake occurrence. The inverse bay-type anomaly is characterized by a sequence of increase, stationary state, and decrease. A distinction between normal and abnormal periods is mainly made by a change in observed quantity i.e., rate y.=a.(t) rather than by a value of the quantity itself yi at time t=ti Thus the corresponding membership functions may be constructed by employing mainly the absolute value of the rate of observed quantity. We introduce Ki, the value of average slope or rate of observed quantity y=a(t) for the i-th time interval, defined by

3 6.7 eathquake. Applications of Fuzzy Mathematics S215 where l may be taken as 1 or 2. Also the consistency of the change is important. As a measure of the consistency we use ri, the coefficient of correlation between the observed quantity and elapsed time, which is defined by By using Ki and ri, we may write the membership function of a single precursor in the form of (FENG et al., 1981)(1) Fig. 1. Temporal variations of VP/VS (a) and its membership function (b) (FENG et al., data are obtained at Janchuan station before the Yan-yuan M= 1984). The VP/VS

4 S216D. FENG et al. where the subscript i denotes the i-th time interval. The constant a in Eq. (1) needs to be determined empirically. It may be determined so as to make Đi greater than 0.5 when precursory anomaly related to a large earthquake appears. In general, the value of Đi may vary from 0 to 1, and when the value of Đi is greater, the observation shows more abnormal state, i.e. the probability for large earthquake occurrence is greater. Figure 1 shows the variations of seismic velocity ratio VP/VS observed at Janchuan station before the Yan-yuan M=6.7 earthquake and the corresponding membership function Đ. The function was calculated by Eq. (1) and the constant a was taken as 0.8. From Fig. 1 we can see that the beginning and recovery phase of VP/VS anomaly are clearly distinguished by this membership function. When a fuzzy set is composed of several kinds of precursors, the total membership Fig. 2. (a) Temporal variations of earth resistivity and radon content obtained attianshui station in (b) Temporal variations of the membership function Đĕ R

5 Applications of Fuzzy Mathematics S217 function can be written as (2) where signs É and È denote the disjunction and conjunction respectively, namely (3) The constants n1, n2, c, nm must be determined empirically. Figure 2 gives the variations of earth resistivity and radon content observed at Tianshui station between June and December 1970 and the total membership function R=ƒÊƒÏ ɃÊR. The single membership functions ƒêƒï and ƒêr are calculated by using ƒêƒï Eq. (1) and taking ƒêƒï=0.5 and ƒ R=0.6. From each temporal variation, it is difficult to pick out the beginning of precursory anomalies. But on the basis of the variations of membership function we can determine the precursory times of two earthquakes of M>4 which occurred about 150 km away from the station. Finally, in some cases it may be difficult to get the mathematical expression for membership function, but the precursors can be indicated by fuzzy description according to the direct experiences. Some examples were given by FENG et al. (1983). 3. The Application of Fuzzy Clustering Analysis to Multi-approach Earthquake Prediction The fuzzy clustering analysis was introduced by RUSPINI (1969) to solve the representation problem of conventional clustering. By using the method of fuzzy clustering, samples can be classified appropriately into groups with different fuzzy multifactors. The basic idea of application of fuzzy clustering analysis to earthquake prediction is that sequences of earthquakes with different maximum magnitudes can be classified on the basis of fuzzy similar relations according to the various kinds of statistical indices of seismicity. Several methods of fuzzy clustering analysis have been suggested. One of them is based on the fuzzy equivalent relation (FENG et al., 1984). It includes mainly the following three steps: (1) Obtaining the fuzzy resemblance relation between samples Supposing that we have n sample vectors X1, X2, c Xn, then we can obtain the fuzzy resemblance relation as (4) where axk a is the modulus of vector Xk, and (Xi, Xj) the scalar product of Xi and Xj. The general fuzzy relation can be represented by a fuzzy n ~n matrix R=[rij]. Obviously, the relation R is resemblance, because it is reflexive (rij<1 when i j, ri,j= 1) and symmetric (rij=rji) (DUBOIS and PRADE, 1980; FENG et al., 1983). (2) Transforming the fuzzy resemblance relation into a fuzzy equivalent (similarity) relation The relation is said to be an equivalent or similarity relation if it is reflexive, symmetric and has max-min transitivity (GUPTA et al., 1977). According to the corresponding

6 S218 D. FENG et al. theorem (FENG et al., 1983), the fuzzy resemblance relation can be transformed into a fuzzy equivalent relation by use of compositional operation on matrix R: where qij is an element of matrix Q and is defined as (5) i.e. Using the multiple compositional operation n-1 times, we can obtain the fuzzy equivalent matrix Rn-1=R R R c R. (3) Classifying the original samples by use of threshold parameter Selecting a suitable value for threshold parameter ƒé, we can classify the original samples into several different groups. When samples Xj and X; satisfy the following condition, these samples belong to the same group. where URn-1(Xi, Xj) denotes the element (1,1) of fuzzy equivalent matrix Rn-1. Every group must include all samples satisfying the above conditions. The value of threshold parameter (6) must be determined empirically. ƒé Taking some statistical indices of seismicity in a given region, the multi-approach earthquake prediction may be made by using the method mentioned above. In illustration of this method, we take four fundamental statistical indices of seismicity, namely, the earthquake frequency N, the maximum magnitude M, and their differences between two adjacent time intervals N.=ƒ N and M.=ƒ M, in the adjacent three time intervals (FENG et al., 1984/85). Then we have 12 statistical indices together for evaluating the seismic risk in the j-th time interval. They are Nj-i, N.j-i, Mj-i, M.j-i (i=1, 2, and 3). These statistical indices must be normalized, and they can be considered as the components X1i, X2i, c X12i of vector Xi in Eq. (4). After fuzzy clustering analysis all the time intervals characterized by vectors Xi, i=1, 2, c, n, can be classified into several groups or clusters of different maximum earthquake magnitude. Classification of the last time interval can be used for a prediction of the maximum magnitude of earthquake. The selection of the value 2 should ensure that the time intervals with different ranges of maximum magnitude should be separated completely or, in any case, by least confused. Let us give an actual example and try to evaluate the maximum, magnitude of an. earthquake in the next year on North-South Seismic Belt of China from the past adjacent three years. We use the seismicity data of M_??_4.7 in two regions 102 E E, 23 N- 32 N and 104 E E, 32 N-39 N in The earthquake catalogues of China compiled by Institute of Geophysics, State Seismological Bureau are used. The sample vector of the year 1964 is composed of statistical indices of seismicity in years 1961, 1962, and The seismicity data in 1960 is necessary to calculate N. and M. in Thus 17 sample vectors are obtained. The matrices R and Rn -1 calculated by the above-mentioned method are listed in Table 2.

7 Applications of Fuzzy Mathematics S219 Table 2. Matrices R and Rn-1 Taking Ď= for fuzzy clustering analysis, we can classify 17 samples (years from 1964 to 1980) into 7 groups: (1) i=1 (1967, maximum magnitude M=6.8), 4 (1971, M=6.7), 7 (1974, M=7.1), and 15 (1977, M=5.4). Excepting 1977, the mean magnitude of the largest earthquakes in each year of this group is 6.9 } 0.2. (2) i=3 (1973, M=7.6), 5 (1976, M=7.3), and 6 (1979, M=6.8). The mean magnitude of the largest earthquakes of this group is 7.2 } 0.4. (3) i= 2 (1964, M=5.4), 8 (1965, M.6.1), 9 (1966, M=6.5), 10 (1969, M=5.4), and 12 (1970, M=7.7). Excepting 1970, the mean magnitude of the largest earthquakes in the other four years is 5.9 } 0.6.

8 S220 D. FENG et al. (4) i= 13 (1972, M=5.7), 14 (1975, M=6.2), and 16 (1978, M= 5.8). The mean magnitude of the largest earthquakes in these years is 5.9 } 0.3. (5) i=11 (1968, M=5.7). (6) i=17 (1980, M=5.8). We attempt to repeat the same analysis with the data set in years As a result, we obtain the same six groups after fuzzy clustering analysis with parameter Ď taken as All the new samples, years of 1981, 1982, 1983, and 1984 belong to the third group (the maximum magnitude = 5.9 } 0.6). The maximum magnitudes in these years are in fact M=6.9, 6.0, 5.4, and 5.3. Obviously, if a sample belongs to the first or the second group, a strong earthquake of M_??_6.7 may occur in that time interval with a large probability. If a time interval in the next year belongs to one of these two types after the fuzzy clustering analysis, then a prediction of an earthquake of M_??_6.7 may be made. 4. Conclusion In this paper, the direct and indirect methods for fuzzy recognition developed in fuzzy mathematics are applied to the earthquake prediction researches, namely, the studies of earthquake precursors, and multi-approach earthquake prediction. The direct method for fuzzy recognition is based on the construction of membership functions of different precursors. Some concrete methods and formulae for constructing the corresponding membership functions have been suggested. The indirect method for fuzzy recognition, presented in this paper is a method of fuzzy clustering analysis. It is based on the fuzzy equivalent relation. It was used for earthquake hazard assessment and multi-approach prediction. From the results obtained we can see a good prospect of using the methods of fuzzy mathematics for earthquake prediction researches. Of course, these methods, especially the examples for illustrating them need to be further examined, tested, and developed continually in the future. manuscript. The authors are grateful to Dr. Kunihiko Shimazaki for his useful suggestions in reviewing the REFERENCES DUBOIS, D. and H. PRADE, Fuzzy Sets and Systems: Theory and Application, Academic Press, New York, FENG, D., J. GU, M. LIN, S. XU, and X. YU, Assessment of earthquake hazard by simultaneous use of the statistical method and the method of fuzzy mathematics, Pure Appl. Geophys., 122, , 1984/85. FENG, D., M. LIN, J. GU, S. LOU, and H. CHEN, Applications of fuzzy mathematics to earthquake. prediction and investigation of the preparatory model of earthquake source, Northwest. Seismol. J., 3, No. 3, 1-8, 1981 (in Chinese). FENG, D., S. LOU, J. GU, T. ZHONG, M. LIN, and H. CHEN, The application of fuzzy mathematics in multi-approach earthquake prediction, Acta Seismol. Sin., 6, , 1984 (in Chinese).

9 Applications of Fuzzy MathematicsS221 FENG, D., S. LOU, M. LIN, H. CHEN, J. GU, and T. ZHONG, The Methods of Fuzzy Mathematics and Their Applications, Seismological Press, Beijing, 1983 (in Chinese). GUPTA, M. M., G. N. SARIDIȘ and B. R. GAINES (ed.), Fuzzy Automata and Decision Processes, Elsevier North-Holland, Inc., New York, RUSPINI, E. H., A new approach to clustering, Inf. Control, 15, 22-32, WANG, P. P. and S. K. CHANG, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, Plenum, New York, ZADEH, L. A., Fuzzy sets, Inf. Control, 8, , 1965.