Vladimir N. Troyan and Vadim M.Uritsky

Department of Geophysics, St.Petersburg State University

Petrodvoretz, St.Petersburg 198904, Russia

e-mail: troyan@hq.pu.ru, uritsky@snoopy.phys.spbu.ru

It is known that for the majority of geophysical phenomena, only statistical bounds on the probability of some characteristics are related. This is especially true for the seismic activity, which is basically chaotic and non-predictable. In this case, the statistical approaches give an opportunity to find hints even in those seismic processes that are normally thought of as being completely random.

For many active seismic regions, earthquakes display scale-invariant (fractal) geometrical properties in a number of ways [3,6,8,9]. The basic quantitative characteristics of scale-invariant patterns is known to be fractal dimension D. Fractal analysis supplies an adequate framework for investigating clustering aspects of the seismic activity. Within a limited range of magnitude, temporal, and spatial scales satisfying scale-invariant behavior of seismic data, the fractal approach allows to quantify a degree of clustering of earthquakes by a single value D.

During the recent years, fractal analysis methods were successfully applied to studying earthquake clustering in a number of seismic-active regions. The studies of fractal characteristics in eastern areas of the circum-Pacific belt were mainly concentrated on North-American tectonic systems, in particular, the San Andreas Fault zone [7]. However, the fractal behavior of South-American tectonic systems has not been investigated in detail as yet.

In the present paper, we analyzed fractal clustering of the regional seismicity in Northern Chile, one of most active seismic zones on Earth situated at the convergence zone between the subducting oceanic Nazca-plate and the continental South American plate.

We used the data of two temporary seismological networks (PISCO'94 and CINCA'95) that were laid out in northern Chile within the frame of the German research project SFB 267 "Deformation Processes in the Andes" [4] to obtain high-quality data of the interesting zone.

The distribution of the seismicity in area around Antofagasta is highly non-uniform and constitutes three clusters of hypocenters at different range of depth (Fig.1). The shallow seismicity down to about 50-70 km depth (cluster 1) is caused by the interaction along the seismogenic zone between the downgoing slab and the overlying plate. The pattern of intermediate depth seismicity (clusters 2 and 3) is concentrated in the depth range between 80 and 250 km. This seismicity is generally associated with dehydration processes in the downgoing oceanic lithosphere [5] and explained physically as an effect of natural hydraulic fracturing.

The size of investigated area covered with continuous recording and the resolution of the event location provided a wide range of spatial scales. In order to quantify the degree of multi-scale clustering of earthquake epicenters, we used a pair-correlation technique. For obtaining the fractal dimension, one can count the number N(r) of pairs of seismic epicenters separated by a distance less than r in accordance with to the well-known Grassberger-Procaccia relation C(r) = 2N(r)/N_t(N_t-1), where C(r) is the correlation integral and N_t is total number of events [6]. The case of fractally distributed points corresponds to a power-law behavior of the corelation integral: C(r)~r^D. This relation allows to obtain the value of D directly from the least-square fitting of the C(r) curve in log-log coordinates.

We applied the described above method both to the 2D analysis of the epicenter distribution and 3D analysis of the hypocenters. The main results of the study are presented in Fig. 2.

As can be seen, the data sets are consistent with fractal model in the range of distances r of 5-80 km. The C(r) functions indeed fit a power law, so all studied seismic subsystems can be considered as spatial fractals. However, the values of power exponent yielding fractal dimension differ: the dimension of shallow cluster is essentially smaller than D-value of clusters 2 and 3 showing that the first cluster has a more complicated and correlated spatial organization. This difference was shown to be statistically significant (the standard error of D estimation is of the order of 0.01).

It has been also of interest to correlate fractal dimensions with the Guttenberg-Richter exponent b obtained for the same clusters. Aki and Turcotte showed that the spatial dimension is twice the b-value (D=2b) as a necessary consequence of the scale-invariant distribution of fault areas. However, our results indicate that only cluster 1 satisfies approximately this theoretical relation and so it can in some sense be considered as a "classic" fractal tectonic systems (Fig.3). The data for clusters 2 and 3 do not satisfy this law.

The result obtained indicates the existence of a specific mechanism responsible for the clustering of shallow earthquake epicenters. This suggestion agrees well with accepted view on the shallow seismicity origin in area of study. It is known in the western part of the Antofagasta area, where small depths events dominate, the motion of the tectonic plates is controlled by so-called stick-slip process. Tectonic systems of this type have many spatially distributed degrees of freedom and so belong to the class of large interactive nonlinear systems displaying the effect of self-organized criticality (SOC) [1,2]. The state of SOC appears to be a stationary point of this kind of dynamics which implies the superposition of tectonic instabilities of all possible scales and produces scale-invariant statistical distributions of seismic activity over released energy as well as its spatial-temporal fractal patterns. A possible reasons for the scale-invariant statistics of intermediate depth events (clusters 2 and 3) are multiple phase transformations and fluids release triggering tectonic instabilities which can also be described on the base of the self-organized criticality approach. However it is obvious that this (or some other) mechanism for deep earthquakes should differ greatly from that of shallow seismic activity, in accordance with the results obtained in our study.

From practical point of view it is important to note that the fractal behavior of dynamic systems can often be described and modeled in terms of a limited number of internal parameters determining its external characteristics, including the dynamics of catastrophes. So, one can expect that further investigation of fractal clustering of seismic events in the area of study may be helpful for predicting strong earthquakes.

1. The clusters of seismic epicenters in studied seismic active area referring to three depth ranges possess different fractal characteristics. The most pronounced difference was shown to exist between the cluster 1 (shallow earthquakes) and two other clusters.

2. A specific seismological regime of cluster 1 can be conditioned by the effect of self-organized criticality arising naturally in seismic systems with many coupled degrees of freedom controlled by slip-stick mechanism.

3. There should in principal be possible a compact description and statistical prognosis of the shallow cluster's seismicity based on its fractal characteristics.

This work was supported by the DAAD grant within the Leonhard Euler cooperation program held by the Free University of Berlin. The authors thank Prof.P.Giese and Dr.C.Haberland for useful discussions.

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