Historical Seismograms - Evidence from the AD 2000 Izu Islands ...

3 12 De-yi Feng, Ming-zhou Lin, and Kun-yuan Zhuang2. Fuzzy Assessment of Seismic Intensity of Historical Earthquakes2.1 MethodIn the assessment of seismic intensity of a historical earthquake, we may use amethod based on the concept of degree of approaching for normal fuzzy sets andby means of fuzzy multifactorial evaluation as described by Feng et al. (1984).The damage conditions of buildings and the fissure conditions of the earth's surfaceafter an earthquake may be taken as macroscopic standards for the assessmentof seismic intensity. Let us suppose that the macroscopic standards correspondingto a determinate degree in the seismic intensity scale can be characterized approximatelyby the normal distribution function as:A = -NOwhere x; is the value of i-th sample, a is the average value of xi, b is the standarddeviation, N; is the number of i-th sample with value xi, and No is their totalnumber.According to the statistics obtained on the basis of abundant macroscopic observationaldata, we have compiled the corresponding quantitative comparison tablesfor each standard, as shown in Tables 1 and 2. Table 1 shows a rough comparisonbetween intensities (from VI to XI degrees) and the building damage standard,because the building damage conditions described in the historical materials oftencan not be classified in detail. Table 2 compares intensities with the earth fissurestandard, and it can be used only for assessing the extremely high intensity in theepicentral region of a strong earthquake.Let us suppose that the macroscopic standards for different degrees in the seismicintensity scale form a series of normal fuzzy sets Aj, and the evaluated sample setalso forms a normal fuzzy set A,. Then, we can take the corresponding distributionfunctions (1) as their membership functions with parameters (aj, bj) and (ao, b,)respectively. For two normal fuzzy sets A, and A,, the degree of approaching canbe defined asNi = exp [ - (?>"I,(Aj,Ao) = 1/2{ exp [- ("i-"-)2]bj + bo+ 1)According to the principle of choosing the closest approach, the fuzzy sample setA, is most approaching to the fuzzy model set A; if we have:r, = (A", A;) = max(A,, Aj), j= 1,2,3,.. . (3)The parameters a, and b, for the assessed sample fuzzy set may be obtained fromthe collected macroscopic data, and the parameters aj and bj for different modelfuzzy sets may be taken directly from the corresponding quantitative comparisonvalues in Tables 1 and 2.Then, we may calculate all elements rk,j for different kinds of macroscopic standardsk and different degrees j in the seismic intensity scale, and obtain the matrixof degree of approaching R = (rk,j) which may be reformulated into a normalizedmatrix Q = (qk,j) by means of averaging and normalization.