Y. Y. KAGAN 

                IS AN EARTHQUAKE THE PHYSICAL ENTITY? 

When reading many papers on statistical treatment of seismicity, I am
often taken aback by lack of distinction between, on the one hand,
statistical empirical or phenomenological and, on the other hand,
physical models. In phenomenological treatment of earthquake occurrence
we can take earthquake data (usually catalogs) as given and try to
approximate them by a stochastic model. The model can be used to
extrapolate catalogs: i.e., statistically forecast earthquakes, using
the same catalogs as input data. This is what some of our papers (Kagan
and Knopoff, 1987b; Kagan, 1991, Kagan and Jackson, 2000, etc.) have
been trying to accomplish. 

However, identification of an earthquake is a partially human
enterprise, one can understand it if one compares various earthquake
catalogs. Let us consider aftershock numbers for the 1999 Chi-Chi
(Taiwan) earthquake: the local catalog (Shin and Teng, 2001, p. 902)
lists 17 M_L>=5.0 events in the first hour of the sequence, some of
them quite large, including four with M_L>=6.4. For the same time
period, the PDE catalog shows six m_b>=4.5 aftershocks. Finally, the
first aftershock in the Harvard CMT dataset comes only four hours after
the mainshock. Therefore, depending on the frequency characteristics of
a seismographic network, the number of stations, and the seismogram
processing technique, the same earthquake sequence could be identified
in different catalogs as one complex earthquake with some subevents, or
as a foreshock/mainshock/aftershock sequence with many `individual'
earthquakes (Kagan, 2003; 2004). 

We will call the aftershocks, included in earthquake catalogs and
discussed above, as REGULAR aftershocks. Vidale et al. (2003; 2004) and
Peng and Vidale (2004) note that the number of aftershocks in the first
few minutes of an aftershock sequence, observed on high-pass filtered
seismograms, is several times higher than the aftershock numbers
recorded even in local catalogs. These CLOSE aftershocks are usually
`buried' in the coda waves of a mainshock. 

Chen et al. (2004; 2005) find that in the 1999 Chi-Chi, Taiwan
earthquake, for instance, apparent aftershocks, identified through
high-frequency strong-motion seismograms (10-50 hz), start after
passing the rupture front. Then these shocks decay locally according to
Omori's law even when the rupture continues in more distant parts of
earthquake fault. Chen et al. (2004; 2005) identified several hundreds
of these SIMULTANEOUS aftershocks in the first 25 sec after the
mainshock origin time. Therefore, depending on the seismogram
frequency, the availability of seismic stations in a earthquake focal
zone, and the interpretation efforts, the number of earthquake events
may increase by orders of magnitude. Finally, the mainshock rupture
process is not smooth, we can also consider it as consisting of many
subevents (Kagan, 2004). 

The most of the statistical models for earthquake occurrence (like
Ogata, 1988; Kagan, 1991, etc., see more in
http://moho.ess.ucla.edu/~kagan/clust.txt ) treat earthquake catalogs
as a population set, with earthquakes considered as individual
entities. As I explained above, the results of such statistical
analysis are appropriate for catalog extrapolation (earthquake
forecasting). Some consequences for physical properties of the
earthquake process can be obtained, but one has to be very careful, 
first, about the errors and distortions in earthquake catalogs and,
second, about extrapolating the obtained model beyond its
phenomenological boundaries. Such extrapolations toward smaller 
inter-earthquake time intervals, smaller size earthquakes, etc., may 
see a model breakdown, caused not by physical properties of earthquake 
occurrence, but by peculiarities of earthquake identification
technique. 

Therefore, the situation changes drastically, if we try to infer
physical properties of earthquake process from catalog phenomenology.
First, there is no such physical entity as AN EARTHQUAKE, it has no
meaning as, for instance, do concepts A MOUNTAIN, A CLOUD, etc. So when
papers discuss properties of INDIVIDUAL earthquakes, this looks to me
like biological research where individuals are easily distinguished
rather than physical one where continuum theories are normally used. 

Similarly, consideration of AN EARTHQUAKE FAULT is also suspect.
Analysis of earthquake hypocenters distribution and simulation of
earthquake fracture surfaces (Kagan, 1982; Libicki and Ben-Zion, 2005)
suggest that earthquake faults constitute a complicated fractal system.
An ideal solid crystal (without defects) should fail along a planar
dislocation; however since all natural materials and rocks have
defects, this would cause branching and bending of earthquake faults
due to fault displacement incompatibility (King, 1983; Gabrielov et
al., 1996), hence a fractal fault system would form which cannot be
represented by one or a few individual faults. 

A common idea that I disagree with about earthquake faults development
is that new or geologically young faults are considered to be complex,
disordered, whereas mature or old faults have a simpler planar pattern
(Ben-Zion and Sammis, 2003). Geometrical and mechanical fault
complexity which is due to displacement incompatibility, can only
increase with time, and this complexity has a wide-range, global
character. Fault healing, on the other hand, which is caused by
geochemical or diffusion processes is local, hence it cannot
significantly modify the grand-scale, self-similar properties of the
earthquake fault system. The difference between new, immature and
mature, large-displacement faults is their surface manifestation: in
old fault systems it should be easier to find relatively planar
segments and these segments are more easily seen at the Earth surface.
For example, the San Andreas fault system occupies hundreds km, and one
can find both simple and complex surface expressions. What matters for
the physical picture is the hypocentral patterns at seismogenic depths
and it should be studied. 

In the physical model of earthquakes (Kagan and Knopoff, 1981; Kagan,
1982), earthquakes and faults are defined as clusters of infinitesimal
dislocations, and their identification as earthquakes depends on
extraneous circumstances. In this model, for example, the earthquake
size distribution (the Gutenberg-Richter law) is simply a consequence
of the branching property of the stochastic model. Geometrical
branching of earthquake faults is modeled by a simple rotational Cauchy
distribution. The appropriate mathematical model for this is a
stochastic branching process with continuous state space (Jirina, 1958;
Athreya and Ney, 1972, pp. 257-262; for new developments in this area
see, for instance, Birkner et al., 2005, and references therein). The
theory of such processes is much more difficult (it is continuum) than
usual population processes. 

This scale-invariant physical model of earthquake occurrence can be
justified by a mathematical model of random stress evolution and
redistribution. In particular, Omori's law can be explained as the
result of random stress following a Brownian motion trajectory and
reaching the critical value (Kagan and Knopoff, 1987a). Theoretical
calculations, simulations and measurements of the rotation of
earthquake focal mechanisms suggest that the stress in the earthquake
focal zones follows the Cauchy distribution which is one of the stable
probability distributions (Kagan, 1990). 

Therefore, when one tries to build up a physical model of earthquake
occurrence based on individual earthquakes or faults, it is not clear
how to understand many of these statements. Earthquake catalogs are a
very distorted representation of the earthquake process, one needs to
study their errors and biases (Kagan, 2003; 2004) to understand what
constitutes a physical effect and what is an artifact. Then, when
results of catalog statistical analysis are obtained, one needs to see
whether they are reproduced in other catalogs. In principle, one should
model not catalogs but the underlying physical process and then try to
understand what phenomena should be expected and try to see them in
real earthquake data. 

REFERENCES:

Athreya, K. B., and P. E. Ney, 1972. Branching Processes,
Springer-Verlag, New York, 287 pp. 

Ben-Zion, Y. and C. G. Sammis, 2003. Characterization of Fault Zones,
Pure Appl. Geophys., 160, 677-715. 

Birkner M, Blath J, Capaldo M, et al., 2005. Alpha-stable branching and
beta-coalescents, Electronic Journal of Probability, 10, 303-325.

Chen, Y.-L., C. G. Sammis, and T.-L. Teng, 2004. A high frequency view
of 1999 Chi-Chi, Taiwan, source rupture and fault mechanics, Eos Trans.
AGU, 85(47), Fall Meet. Suppl., Abstract S33C-07. 

Chen, Y.-L., C. G. Sammis, and T.-L. Teng, 2005. A high frequency view
of 1999 Chi-Chi, Taiwan, source rupture and fault mechanics, Bull.
Seismol. Soc. Amer., submitted. 

Jirina, M., 1958. Stochastic branching processes with continuous state
space, Czech. Math. J., 8, 292-313. 

Kagan, Y. Y., 1982. Stochastic model of earthquake fault geometry,
Geophys. J. R. astr. Soc., 71, 659-691. 

Kagan, Y. Y., 1990. Random stress and earthquake statistics: Spatial
dependence, Geophys. J. Int., 102, 573-583.

Kagan, Y. Y., 1991. Likelihood analysis of earthquake catalogues,
Geophys. J. Int., 106, 135-148. 

Kagan, Y. Y., 2003. Accuracy of modern global earthquake catalogs,
Phys. Earth Planet. Inter., 135(2-3), 173-209. 

Kagan, Y. Y., 2004. Short-term properties of earthquake catalogs and
models of earthquake source, Bull. Seismol. Soc. Amer., 94(4),
1207-1228. 

Kagan, Y. Y., and D. D. Jackson, 2000. Probabilistic forecasting of
earthquakes, Geophys. J. Int., 143, 438-453. 

Kagan, Y. Y., and Knopoff, L., 1981. Stochastic synthesis of earthquake
catalogs, J. Geophys. Res., 86, 2853-2862.

Kagan, Y. Y., and Knopoff, L., 1987a. Random stress and earthquake statistics:
Time dependence, Geophys. J. R. astr. Soc., 88, 723-731.

Kagan, Y. Y., and Knopoff, L., 1987b. Statistical short-term earthquake
prediction, Science, 236, 1563-1567. 

Libicki, E., and Y. Ben-Zion, 2005. Stochastic Branching Models of
Fault Surfaces and Estimated Fractal Dimension, Pure Appl. Geophys.,
162(6-7), 1077-1111. 

Ogata, Y., 1988. Statistical models for earthquake occurrence and
residual analysis for point processes, J. Amer. Statist. Assoc., 83,
9-27. 

Peng, Z., and Vidale, J. E., 2004 Early aftershock decay rate of the M6
Parkfield earthquake, Eos Trans. AGU, 85(47), Fall Meet. Suppl.,
Abstract S51C-0170X. 

Shin, T. C., and T. L. Teng, 2001. An overview of the 1999 Chi-Chi,
Taiwan, earthquake, Bull. Seismol. Soc. Amer., 91(5): 895-913. 

Vidale, J. E., E. S. Cochran, H. Kanamori, and R. W. Clayton, 2003.
After the lightning and before the thunder; non-Omori behavior of early
aftershocks? Eos Trans. AGU 84(46), Fall Meet. Suppl., Abstract
S31A-08. 

Vidale, J. E., Z. Peng, and M. Ishii, 2004. Anomalous aftershock decay
rates in the first hundred seconds revealed from the Hi-net borehole
data, Eos Trans. AGU, 85(47), Fall Meet. Suppl., Abstract S23C-07.