Fractal Characterization of Seismic Networks in Taiwan

The fractal dimension is calculated for the geometrical distributions of the seismic stations of three networks (the old CWB seismic network, the TTSN and the CWBSN) in Taiwan based on the correlation integral algorithm proposed by Hirata et aL (1987). Results show that the distribution of the data points of correlation integral for the old CWB seismic network distribute very irregularly and cannot be approximated by a fractal set point. The fractal dimesion value for the TTSN (1.18±0.02) is less than that for the CWBSN (1.56±0.01). This indicates that the dimension resolution and the detectability of sparse phenomena are lower for the former than the latter. (

detectability of sparse phenomena are lower for the former than the latter.
Fractal properties are commonly found with natural phenomena (Mendelbrot, 1983). In 1986 Lovejoy and his associates (Lovejoy et al., 1986a, b;Lovejoy and Schertzer, 1986) reported that the World Meteorological Station Network (9563 stations Wang ( 1989), and those of the CWBSN stations can be found in each volume of the Seismological Bulletin published by the CWB. All the stations of the CWBSN are plotted in Since the distance between any two stations is generally greater than 10 km and there are so few stations with an elevation greater than 1 km, the possible effect due to elevation is not taken into account. In this study, the fractal dimension is measured for the old CWB seismic network, the TTSN, and the CWBSN as detennined by correlation integrals.  The correlation integrals C(r) for the location distributions (h1, h2, h3, ... , hN) are calculated with the following for1nula (cf. Hirata et al., 1987): where Nr(R<r) is the number of pairs (hi, hj) with a distance smaller than t, and N is the number of stations used. If the distribution has a fractal structure, C(r) is expressed by: (2) where D is the correlation fractal dimension . The solid lines represent the regression lines of the data points with r less than rc=lOO km (or log rc=2).
The crosses, open circles and open triangles denote the data points for the old CWB seismic network, the TTSN and the CWBSN, respectively. It can be seen that for the TTSN and CWBSN when the distance is less than a certain critical value re, the data points mostly distribute along a straight line; in contrast, when the distance is greater than that a value, the two patterns of data points bend downward. Such a critical value is about 100 km, or log(rc)=2.0. For r<rc, the data points of the three networks separate remarkably; while for r>r c all data points are close to one another. However, the data points for the old CWB seismic network distribute very irregularly and cannot be fitted by a straight line. Consequently, the distribution of stations of the old CWB seismic network cannot be approximated by a fractal point set. The bending of the pattern of data points for the TTSN and CWBSN indicates that the C(r) value for r>rc is less than the value estimated from the regression equation deduced from the data points with r<rc. From Figure 1, it can be seen that the re value is almost equal to the size of Taiwan Island in the east-west direction. The correlation integral algorithm is based on a circle in two-dimensional space as in the present study or a sphere in three-dimensional space. The length (along the north-south direction) and the largest width (along the east-west direction) of Taiwan Island are about 400 km and 100 km, respectively.
Hence, almost all of the stations used are located on a rectangular surface of 400 km length and 100 km width. When the r value is greater than the width of the rectangle, the number of the pairs of stations counted from the rectangle must be less than the expected value based on a circle with a radius of r. The ref ore, the size of the width of the rectangle would cause a so-called finite-size effect on the computed results. For the present situation, the critical size is the width of Taiwan Island. The use of a non-circle distribution of stations limits the reliability of the results. In other words, the fractal dimension can only be estimated from the data points with r<r c. For the data points with r<r c, the slope values (i.e. the D. values) inferred from the data points are 1.18±0.01 and 1.56±0.01 for the TISN an · d the CWBSN, respectively.
A non-integer value of fractal dimension represents the existence of voids in the object , or set. From the viewpoint of the distribution of the seismic stations, the existence of voids indicates the existence of areas where no station is installed. The smaller the value of fractal dimension, the larger the number of voids or the higher the degree of heterogeneity of the object. Hence, the fact that the D value for the ITSN is smaller than that for the CWBSN displays a more heterogeneous distribution of stations for the f ortner than for the latter. From Figure I, it can be seen that most of the ITSN stations distribute along the two sides of the Central Range, and only a · few stations are located on the Central Range. On the other hand, although most of the CWBSN stations distribut along the two sides of the Central Range, a large number of the stations is at the Central Range. Hence, there is a less homogeneous distribution of stations for the ITSN than for the CWBSN over Taiwan Island. In other words, the distribution of the CWBSN stations is more two-dimensional than that of the ITSN stations. The ref ore, the fractal dimension is larger for the f or1ner than the latter. In addition, according to the concept proposed by Lovejoy and associates, the dimension resolution and the detectability of sparse phenomena of the CWBSN arC? higher than those of the TISN. .