Earthquake Time Series Analysis by Cantor Set Model in Taiwan

The Cantor set fractal distribution and probability model are com­ bined to analyze earthquake time series of two seismogenic zones of Tai­ wan from 1900 to 1999. One-dimensional box counting method is adopted to measure the occurrence probability of earthquakes equal to or larger than a certain magnitude threshold with different time scales. The results show that the fractal dimension has a negative correlation with the magni­ tude threshold, and the ideal critical time scale has an exponential correla­ tion with the magnitude threshold. The ideal critical time scale can be deemed as the "lower limit" of recurrence period of earthquake clustering that can be referenced in designing earthquake resistant construction. Due to frequent and continuous clustering of earthquakes occurred in the Chianan zone, we suggest that the fault activities in the Chianan zone are more active than that in the Center zone of Taiwan.


INTRODUCTION
It is important to understand the temporal fractal behavior of earthquake occurrence when analyzing earthquake time series (Smalley et al. 1987;Ogata 1988; Turcotte 1992; Papadopoulos and Dedousis 1992; Chen et al. 1998;Tsai 1999).Gutenberg and Richter (1944) first noticed that the relation between cumulative earthquake frequency (N) and magnitude (M) is logN = a b X M. Studies show that the b value is usually between 0.8-1.2,whereas the a value is a region-dependent measurement (Evemden 1970;Wang 1991).After Mandelbrot ( 1967) intro TAO, Vol. 14, No. I, March 2003 multifractality of earthquake time series after analyzing data from mainland China and Tai wan using a generalized correlation time integral [ Cq (t)].The Cantor-set type clustering is characteristic to many natural phenomena.Smalley et al. (1987) concluded that earthquake clustering in time is a scale-invariant process, and that the earthquake time distribution is certainly different from Poisson distribution.Mayer ( 1992) analyzed desert storm sequence and found that storm data is self-similar with different rainfall scales and that the fractality of storm clustering prevails in scale of decades.Olsson et al. (1992) and Lin (1998) confirmed that rainfall time series are scale-invariant in time, and Mazzarella (1998) obtained a similar result when analyzing sea flooding's catalog in Venice.Chen et al. (1998) and Tsai (1999) provided a sieve (or threshold) method for describing fractal-clustering variation of earthquakes.Lin (1998) and Tsai (1999) further developed probabilistic concept to construct the recurrence period of natural events.
Recurrence period is a very important parameter, especially for natural catastrophes.In seismic hazard evaluation, it can be a critical reference value in designing the aseismatic strength of buildings or civil constructions during their designed life.This study applies threshold and probability methods to compute the fractal dimension and the recurrence periods of earth quake clustering in time domain for two seismogenic zones in Taiwan.Further more, a new effective method for measuring the recurrence period will also be given and discussed in details.

Similarity Dimension and Box-Counting
Self-similarity and scale-invariance are the basic concepts of fractal geometry.Suppose a D-dimensional cube in Euclidean space is divided into N equal small cubes.In such a way, every side of the original cube is divided into m divisions.As a ratio scale set to be r = 1 I m, then: We define the similarity dimension (D) as: The similarity dimension D can be either an integer or a fraction.

(la) (lb)
In practical computation, box-counting method is usually used for obtaining D values (Rasband 1900;Peitgen et al. 1992;Falconer 1993;Isham 1993;Stoyan and Stoyan 1994;Lee 1995;Cheng et al. 1999).Let A be a set of finite points within a D-dimensional Euclidean Space, and let N( e) be the fewest D-dimensional cubes of size £ in every side, which can encompass all points of set A. The cube size (£)here would be similar to the ratio scale (r) , and N( £)increases as e approaches to zero.As £ approaching zero, N( £) would gradually become close to the exact number of points for set A. So the relationship between small £ and N( £)is: or DB = lim l n( N (e)) .
e�o ln(l/e) (2) D B is referred as the "Box-counting Dimension" or "Box-Dimension" and is also generated under the self-similarity concept.It is also easily to understand that by Eq. (lb) and Eq. ( 2), D and D B of an object with self-similarity would be the same while the ratio scale (r) and the cube size ( £) are infinitesimal.The box-counting method has been applied to compute fractal di mensions of natural features in two and three dimensions (Lee 1995;Cheng et al. 1999) .In this study, the method is applied to one-dimensional analysis.

Cantor Set
The German mathematician G. Cantor introduced this set in 1883.The Cantor set (also called as Cantor dust) is an infinite point set in the unit interval [0,1].The set is constructed by initially dividing a line of length 1 into three segments.The middle third is then removed, leaving the two side segments.Divide each remaining segment into three and remove the middle third again.Repeat the process infinite times and the limit will form an irfinite number of points while the total length of the undeleted line segments approaching zero.This kind of Cantor set is referred as the deterministic Cantor set (Fig. la) due to its deterministic genera tion process.If in each step we randomly select one of the three segments to be removed, the resulting set is referred as the random Cantor set (Fig. I b).For both the deterministic and random Cantor sets, D = ln2 I ln3 :::: : 0.6309 by Eq .(lb) and Eq.(2) while the measuring size approaches infinitesimal.In practice, the generation process of Cantor set would be finite between 0 and 1 because natural point events in a certain time period are always finite.The random Cantor set produces a distribution of clustering which is usually applied to describe many natural point events clustered irregularly in time, such as rainfall (Lin 1998) or earth quakes (Smalley et al. 1987;Tsai 1999) .

Fractal Distribution Probability of Point Events
In this section, we relate fractal distribution to probability (Turcotte 1992).This can be done using the sequence of one-dimensional line segments of the Cantor set illustrated in Fig.
1.The objective is to determine the probability [P(r)] that a step of ratio scale r will include a line segment.At step zero, the number of line segment is N = 1 and the probability that a set of ratio scale r = 1 will include a line segment is P(r) = 1; at stepl, we have N = 2, r = 1/3 and P (r) = 2/3, and at step2, N = 4, r = 1/9 and P(r) = 4/9 (Figs.la, b) .So the probability [P(r)] can be generalized to: ( 3) In Eq. (la), N is related to r, so that we obtain: By applying one-dimensional box-counting, we can derive a relationship similar to Eq.
In the range of total time length (Q), we apply several different smaller time length as time scales E K (K;::: 1,2,3, ... ,L).For one-dimensional box-counting, the time scale (E K ) can be seen as the "line length", which is similar to the concept of the cube size (£)and the ratio scale (r) in section 2.1.By each E K ' the range of the total time length would be divided into small time units (Q/E K ).Regardless to the actual events [T/1il] occurred in a certain time unit, the time unit would only be counted once if there is an event occurred in it.Thus, we can compute the cumulative frequency N(Jjl ( E K ) of time unit that contains at least one event.Finally, we estab lish the occurrence probability of point events in the form as follows: By Eq. (4), we obtain: In some practical cases, the straight line becomes curved when approaching probabil ity 1 and is often interpreted as a combination of scale-invariant clustering and random events (Smalley et al. 1987).In order to define E* more precisely, we extrapolate the regression-line with probability equal to 1 to replace E* and define it as the critical scale, E**.Therefore, the E** is an ideal critical time scale when fractal structure remained unchanged through the time axis.In this study, E** is a very important physical concept and can be applied as a reference for aseismatic evaluation.

Seismic Background 91
Taiwan is situated in the boundary between the Euroasia plate and the Philippine Sea plate.It is an area with a high level of seismicity where global and local seismic networks and several previous instrumental observations (Hsu 1961) have provided a continuous record of events since 1900.Due to the difference between magnitudes recorded by TTSN (Taiwan Telemetered Seismographic Network; duration time magnitude, M0 from 1973 to 1990) and CWBSN (Central Weather Bureau Seismic Network; local magnitude, M L since 199 1), the earthquake catalog has been calibrated by applying the local magnitude (M L ).According to the magnitude resolution of seismic networks, the earthquake catalog can be mainly divided into two periods: from 1900 to 197 2 and from 1973 till today (Wang and Shin 1998).Catalog from 1900 to 1972 only contains large earthquakes with M L � 4 and the locations are assigned at 1 0 X 10 km grid positions due to the limited hypocenter determination ability.Catalog from 1973 till today contains earthquakes with ML � 2 and the locations determined by the CWBSN can be constrained to an accuracy within 2-km inland and 5-km offshore by using widespread stations and modern instruments.This study is utilized earthquake data of M L � 4 from 1900 to 1999.
We consider two seismogenic zones defined by Cheng (1997) based on regional tectonic framework, distribution of active faults, seismicity and earthquake characteristics.They con tain intra-plate earthquakes shallower than 35-km in depth and are renamed as the Center zone and the Chianan zone in this study (Fig. 4).The Center zone contains a Sanyi-Puli seismic 1.E+02 '1------"""" "-----:-"2""'*'"'*"* ... ��""'1 ';;'!.Table 1.Summarized results of threshold-fractal analysis of the Center and Chianan zones.Dis fractal dimension; £'"is the critical time scale considered here as the "lower limit" of recurrence period and Pen is the value of intercept probability.Richter relationship is logN = 3. 107 -1.117 X M and the selected earthquake ranges from M L = 4 to M L = 7.These two zones are located at mountain front with high ground deformation according to OPS surveyed near Chiayi and Tainan (Yu 1997).
Based on time resolution of the earthquake catalog, the event number and the total time length of the data set, we chose 1 minute as the smallest time scale.The time scale is increased by a factor of 3 up to 317(129140163 minutes or 245.53 years) during the computing process.

Computation and Discussion
Within this catalog, there are 158 events selected in the Center zone and 396 events in the Chianan zone.Results of the threshold-fractal analysis of the earthquake catalog of the Center  !.
The fractal dimension D decreases as the magnitude threshold (JJ) increases (Fig. 6a).It implies that earthquake series is multifractal with respect to magnitude and the relationship is a negative linear in terms of the magnitude threshold.This result is different from the expo nential decreasing ofrainfall sequences (Lin 1998) and discloses some basic differences in the generation of these natural events.On the contrary, the critical scale £**,increases with JJ in an exponential manner (Fig. 6b ).
Let the number of clusters (Cn) be the frequency computed by time scale of 1 minute.It is usually the same as the total number of events while time resolution is 1 minute.Then we can compute the probability (Pen) by (Cn IQ), where Q is the total time length of 1-minute resolution.As shown in Fig. 3, P en equals to the intercept probability where Log(l):::: : 0. Fig. 7a plots fractal dimension verse Pen and the trends show different logarithmic increases with different time intervals and spatial zones.Smalley et al. (1987) indicated that the more isolated are the clusters, the smaller is the value of D. In other words, if Cn is relatively smaller (or P e n is smaller) and more isolated in a certain time interval, it would form sparser clusters, and the recurrence periods of clusters would become longer with relative smaller fractal dimension.As Jj becomes large, events will then occur sparsely, forming isolated clusters with large £" along the time axis with a smaller fractal dimension.

Conclusion
In this study, we found that when Jj = 4, the D value of the Chianan zone is much larger than that of the Center zone.D values of the two zones become almost equal when ]j = 7 (Fig. 6a).Thus, we may predict that there is no earthquake clustering in these two zones when magnitude ]j is equal to or larger than 7 (e.g. the Chi-Chi earthquake).In Fig. 6b, all of the E** values in the Chianan zone are smaller than those in the Center zone and two regression lines are almost parallel.These results indicate that earthquake clustered more frequently and con tinuously in the Chianan zone than in the Center zone, especially for earthquakes with smaller magnitude.For larger earthquakes, the temporal behaviors of these two zones are similar.We may conclude that fault activities are more active in the Chianan zone than in the Center zone.
The accumulated tectonic stress may be released through seismic activities more rapidly in the Chianan zone than in the Center zone. (8) Since E K = 1, the value of E" does not change with D. This is the most significant result of this study in that the E ** can be found if we only know how many events with magnitude equal to or larger than a critical Jj had occurred, and the duration of total time length we want to include.Using these equations, we can easily extend the application into higher or lower magnitudes, such as M L :::;-; 1 or M L ;::: : 8, when historical earthquake records were not complete.
Finally, E .. can be interpreted as the "lower limit" of seismic recurrence period when designing the aseismatic strength for buildings or civil constructions in their designed life.For an example, in the Chianan zone, earthquakes with M L ;::: : 7 will occur when the time scale is equal to or larger than 50.7 years (Table 1).If we are to design a construction for more than 50 years in the Chianan zone, the aseismatic strength should be "at least" able to withstand vibra tions caused by earthquakes with ML = 7.
. I. Three segments Cantor set produced after four repeating processing of "Generator (Stepl)" from "Initiator (StepO)".(a) Deterministic Cantor set and (b) Random Cantor set.The length of the undeleted line seg ments is just for enhancing the point set.The fractal dimensions for both sets are the same ( ""'0.6309).

Fig. 2 .Figure 3
Fig.2.Generation of point-process [T; r1n] having magnitudes equal to or larger than some threshold (Jj).+ expresses events with a given magnitude threshold.T1(Ji) can be seemed as a smaller-quake point-process and T }h l as a larger-quake point-process.

Fig. 4 .
Fig. 4. Map showing the two seismogenic zones of Taiwan.The two gray areas, (a) Sanyi-Puli seismic belt and (b) Chianan seismic belt, are the belts concentrated in seismic activities.

Fig. 5 .
Fig. 5. Threshold fractal analysis of the Center and Chianan zones.Labels I, II, III and IV correspond to magnitude thresholds Jj = 4, 5, 6 and 7 respectively.
It's Gutenberg-Richter relationship is logN = 2.553 � 1.087 X M, and the magni-.tude ranges from M L = 4 to M L = 7.In this zone, the Chi-Chi earthquake of M L = 7.3 occurred in 21 September 1999 with the surface ruptures extended more than 105-km, the fa ult cliffs elevated several meters high, and the maximum PGA (Peak Ground Acceleration) greater than 1G(COS 1999).The Chianan zone contains a Chianan seismic active zone.It's Gutenberg

Figure
Figure 7b is the log-log plot of E*' versus Pen' and the entire data set is regressed by a power function: E** = 252 73 x p •0•9288 ' Cn