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CORRELATIONS OF EARTHQUAKE FOCAL MECHANISMS.
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Y._~ Y._~ Kagan 
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Institute of Geophysics and Planetary Physics,
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University of California, Los Angeles, California 90024-1567, USA;
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E-mail: kagan@@uclaess.bitnet; ibe7yyk@@mvs.oac.ucla.edu
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<ABSTRACT>
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We perform a statistical analysis of the Harvard catalog of
seismic moment tensor solutions. 
We investigate the distribution of hypocenters on focal 
spheres of earthquakes.
The hypocenters are concentrated along fault-~planes; the
hypocenter distribution does not significantly differ for
earthquakes in different depth ranges. 
To study the rotation of focal mechanisms, we have solved an
inverse problem of a 3-D rotation of double-couple earthquake
sources, i.e., for each pair of focal mechanisms we find all
four 3-D rotations which rotate one mechanism into another. 
The stochastic disorientation of focal mechanisms is 
well-~approximated by the rotational Cauchy distribution which
previously has been identified through theoretical arguments
and simulations as the result of stress perturbations caused by
random defects in the medium. 
The Cauchy distribution is characterized by one parameter, &k;
the value of &k is between 0.05 and 0.1 in an earthquake fault
zone, it increases with distance between pairs of events. 
In directions other than a fault-~plane, &k reaches the value
0.5; the Cauchy distribution is then close to completely random
rotations. 
Although &k increases slowly with depth, in general, 
disorientations of focal mechanisms in various depth ranges 
display the same dependence.
Therefore, we can conclude that deep earthquakes are controlled
by the same stress interaction regime as are shallow events. 
The results of these measurements make us question the
suitability of some terms and models that are commonly used in
the theory of an earthquake source. 
For example, we argue that tectonic earthquakes rupture rock 
material which has large-~scale defects (or faults), comparable
in size with tectonic blocks in which they originate.
Properties of the rocks which we call tectonically
self-~organized material, should be significantly different
from those of regular materials where the size of defects is
typically much smaller than the scale of interest. 
These properties should strongly depend on the geometry
(location and orientation) of major defects. 

.sk 2
<Key words:> earthquake focal mechanisms, earthquake source
rotation, fault mechanics, random stress. 
 
.sk 3
RUNNING HEAD: Y._Y._Kagan -- Correlations of focal mechanisms

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1._~ INTRODUCTION

Earthquakes influence each other through a stress
environment; they affect both the time and the mechanism of
later earthquakes. 
.! Direct stress measurements are problematical, but we can infer
.! some important stress field features by the way in which
.! earthquakes relate to strain, and to each other. 
Clearly, understanding the basics of earthquake interaction is
critical if we are to successfully understand and eventually
predict earthquakes. 
Stresses in the Earth's interior are not directly measurable.
However, we can infer the stress pattern on the basis of focal 
mechanism maps.
The stress manifests itself through seismicity by two patterns: 
1) an increased number of earthquakes in places of increased 
deviatoric stress, and 2) the rotation of focal mechanisms
(<cf.> Yoshioka @& Hashimoto 1989). 
Pairwise correlation of focal mechanisms depends on several
types of geometrical patterns: 1) scalar 2-point distribution 
of hypocenters (Kagan 1991a); 2) pairwise correlation of
seismic moment tensors; 3) geometry of earthquake faults; 4)
crustal slab geometry; and 5) the geometrical pattern of
tectonic blocks and plates. 
When we study correlations of focal mechanisms we see only 
the integral effects of all of these patterns.
In another paper (Kagan 1992) we investigate the geometry of 
earthquake fault zone using the correlation tensor; in this 
work we apply a more direct approach to the study of
concentrations of hypocenters related to focal mechanisms of 
earthquakes as well as to 3-D rotations of mechanisms.

We study the geometrical relationship between seismic moment
tensors of successive earthquakes in a given region. 
These relationships can reveal whether simple, elastic stress
increments on known faults trigger earthquakes, or whether
nonlinear effects, complex geometry, and pre-existing
conditions predominate. 
While spatial and temporal clustering have been documented in
several cases, the geometrical relationships have not been
studied except for a few examples (see Kagan 1992). 
Earthquake focal mechanisms depend both on the long-~range 
tectonic stress field, and on its local variations. 
Earthquakes strongly perturb the stress, often causing 
fault-~planes to branch. 
This branching is essential to the triggering of subsequent
earthquakes, and to understanding the observed distribution of
surface faults and earthquake focal mechanisms.
While there have been some studies of branching based on first
principles, we would like to study the process empirically,
deriving a probability distribution for the rotation angles
between pairs of earthquake focal mechanisms. 

We have compiled a computer program which solves an inverse
problem of 3-D rotation of double-~couple
earthquake sources, i.e., for each pair of focal mechanisms or
seismic moment tensor solutions the program determines all four
3-D rotations which rotate one mechanism into another (Kagan
1991c). 
.! Similar programs are now developed for study of disorientations
.! of grains in crystalline materials (Adams, Zhao @& Grimmer
.! 1990, and references therein). 
In most applications only the minimal rotation, i.e., the 
rotation with the smallest rotation angle, is of interest.
If the minimum rotation is small, it can be found by a trial
and error technique: in this case, the ambiguity caused by the
symmetry of the double-~couple source does not interfere with
the interpretation of the stress patterns. 
However, in places where stress is strongly heterogeneous,
all four rotations might have rotation angles which are of
comparable magnitude, hence no rotation is clearly preferable. 
Methods based on geological insight, might not be productive in
those complex environments in deciphering the stress pattern. 
Possibly, these places are of special interest since they might 
correspond to asperities/barriers.

Our program (Kagan 1991c) may be used as a quantitative tool
in studies of stress fields causing earthquakes, investigations
of relations between focal mechanisms and tectonic features of
seismogenic regions, etc. 
In particular, this computer program enables us to study the
stress distribution in rocks as the cause of earthquakes. 
The rotation of focal mechanisms may be represented as a result
of several causes:
1) tectonic long-~range stresses, 
2) internal stresses of previously known earthquakes, 
3) internal stresses of previously unknown earthquakes and 
earthquakes too weak to be registered by a seismographic 
network, and
4) errors in determinations of seismic moment tensor.
The first two of these causes are non-~random and known,
whereas the last two are to be considered random. 
In Kagan (1990) we studied random stresses as the cause of
earthquakes. 
We are studying the stress in order to use the results in 
long-~term and long-~range earthquake prediction.
This ambitious prediction program should aim to reconstruct
the stress regime in a seismogenic region and determine
fracture criteria corresponding to a realistic environment in
an earthquake-~producing fault zone, as well as
to incorporate and integrate earthquake catalog data, geodetic
measurements of soil deformation, as well as results of
geological and paleo-~seismological investigations. 
The results of focal mechanism correlations may also be 
important in modelling of earthquake fracture.
Many models now exist based either on geological insight, or
on the linear theory of elasticity (Scholz 1990, and references
therein); recently there have been efforts, based on ideas of
statistical mechanics (Herrmann @& Roux 1990), to create new
models of fracture and in particular of earthquake rupture. 

Below we use an abbreviation DC to mean a double-~couple
earthquake source. 
In most of our considerations we use the right-~handed
coordinate system centered on each of earthquake hypocenters,
and use the T-, P-, and N-axes of earthquake focal mechanism as
coordinate axes. 
To insure the `right-~handedness' of the system, we preserve
the downward directions of the T- and P-axes, as given in
catalogs of fault-~plane solutions; however, the direction of
the N-axis is chosen according to the right hand rule (<cf.>,
Altmann 1986, p._29). 
We will call this system the TPN-system of coordinates.
We write <p.d.f.> for the probability density function.

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2._~ HYPOCENTER DISTRIBUTION.

In these investigations we have used the Harvard catalog of
moment tensor inversions (Dziewonski <et al.> 1991, and
references therein). 
The catalog starts on January 1, 1977, ends on April 30, 1990, 
and contains close to 9000 events with M]w[_&>_5.0.
(Our experience shows that in order to obtain reliable 
statistical results, one needs at least several hundred events 
in a catalog.)_
We use only those solutions in the catalog which correspond to
a DC source. 
To avoid difficulties in defining the 3-D rotation on the
spherical surface of the Earth, for each pair of events we
have defined a Cartesian coordinate system which is tangent to
the sphere in the middle point of two epicenters; longitude is
multiplied by the value of the cosine of latitude in this
point. 
To insure that this transformation does not yield significant
errors, the maximum distances between pairs of epicenters does
not exceed 500_km. 

In Table_1 we show the distribution of numbers of hypocenters
in a coordinate system formed by the T-, P-, and N-axes of
earthquake focal mechanisms: each quadrant of the focal sphere
is subdivided into 110 spherical triangles and quadrilaterals
with equal area (consequently covering equal solid angles), and
we calculate the number of times a hypocenter in projected into
these cells. 
To make these distributions more amenable to comparison, we
normalize the numbers, so that the total number in each display 
of Table_1 equals 11000.
Therefore, if the distribution of hypocenters in the TPN-system
were completely random, all the numbers in the table would be
approximately equal to 100. 
The entries `Col._Angle' give the colatitude angle &h
corresponding to the lower edge of a segment consisting of one 
or several cells.
The number of pairs for each segment is shown in `Pair number' 
column.
Similar calculations are reported by Frohlich and Willemann
(1987), and Michael (1989) who studied the clustering of
aftershock hypocenters with respect to focal mechanisms of main
shocks. 
In our case we take into account all earthquake pairs 
disregarding whether they are aftershocks or not.

In the last chart (1e) we simulate points around a point
DC source (defect) and measure the second invariant of the
deviatoric stress tensor; if the invariant exceeds a certain
value, an earthquake is considered to occur at that point. 
Concentrations of events are symmetrical in this plot with 
regard to the T-P and the T+P vectors.
This might be explained by the stress concentration of a point 
dislocation: shear stresses increase for both nodal planes:
that phenomenon is called a `fault-~plane ambiguity'. 
For an extended fault, the stress concentration along the
fault-~plane remains strong, although there is a smaller stress
increase perpendicular to the fault (Chinnery, 1963). 
The latter stress concentrations are sometimes invoked as
the explanation of off-~fault aftershocks (Scholz, 1990,
pp._207-~208, and references therein). 
Since, in general, we do not know the sense of direction for
the T- and P-axes, we would expect similar ambiguity in
hypocenters distribution for earthquake catalogs. 
However, the charts `a-d' in Table_1 show strong concentration
of hypocenters near the plane going through the T-P and the
N-vectors, therefore this plane should correspond to the
fault-~plane and the plane going through the T+P and N-vectors 
should usually correspond to the auxiliary plane.
A possible explanation for the preferred concentrations of
hypocenters around the T-P vector is that in the catalog of
fault-~plane solutions (Dziewonski <et al> 1991), both the T-
and P-axes are listed pointing <down>, thus, apparently the
T+P vector is approximately vertical, whereas the T-P vector is
approximately horizontal. 
Since earthquake faults, especially shallow events,
propagate horizontally, the horizontal line has greater chance
to correspond to the fault-~plane. 

We calculate the average angle between the T-P vector and the 
distributions of hypocenters around each segment of a focal 
sphere parallel to the TP plane (entries `Average angle').
Both average angles and the charts in Table_1 suggest 
that for shallow earthquakes hypocenters tend to cluster
between the T-P vector and the P-axis (closer to the former).
For example, for the second chart (`b') we calculate that about
35% (112,894/322,135) of hypocenter pairs are concentrated in a
27&0 cone around the N-axis. 
This is 3.21 times more than expected for a random distribution
of hypocenters. 
For intermediate and deep earthquakes the pattern is slightly 
different: although hypocenters are again close to the T-P 
vector they tend more to the T-axis.

In all charts a significant concentration of hypocenters 
centers around the N-axis.
This concentration is especially strong for shallow and 
intermediate earthquakes separated by 500_km (charts <b> and 
<c>).
This is easy to understand, since seismicity is concentrated in
subducting seismic belts; focal mechanisms of earthquakes in
subducting slabs correspond to a normal thrust on a linear
horizontal fault with the N-axis being parallel to the strike
of the slab (Apperson @& Frohlich 1987; Frohlich 1989). 
Since the depth interval is smaller for shallow and
intermediate earthquakes (0--70_km and 71--300_km) than for
deep shocks (301--700_km), the distribution of hypocenters
should approach the linear distribution at larger distances for
deep events. 

Apperson @& Frohlich (1987) and Frohlich @& Willemann (1987)
used Anderson-~Darling statistics to test for statistical 
significance of the concentration of aftershock hypocenters in
certain focal directions. 
We do not use similar statistical tests here and below in study 
of rotations, for two reasons: 
1) our investigations have an exploratory character;
2) all of the statistical tests are based on independence of 
random variables or, at most, on their short-~range dependence.
However, earthquake hypocenters are strongly clustered and this 
clustering exhibits both short-~range and long-~range 
properties.
There are no appropriate statistical tests for random variables 
having very long-~range correlations (<cf.> Ogata @& Abe 1991),
thus a failure of a statistical test does not indicate that,
for example, the hypothesis of the isotropic distribution of
hypocenters has to be rejected: due to the presence of
hypocenter clusters the resulting distributions may have 
considerably fewer number of degrees of freedom than is implied
in the event number. 
In this work, we will attach more significance to the
consistency of results in different depth ranges or in
different geographic regions, than to formal statistical tests.

.!pg
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3._~ FOCAL MECHANISM ROTATION.

In this section we briefly summarize theoretical statistical
distributions of 3-D rotations. 
For more complete treatment of the subject see Kagan @& 
Knopoff (1985), Altmann (1986), and Kagan (1990; 1991c).
To represent 3-D rotations and products of the rotations, we
use the multiplication of normalized quaternions as a major
technique (see Altmann 1986; Chang, Stock @& Molnar 1990;
Kagan 1991c). 
The quaternion q| is defined as
.tp3

q|_= q^-5&0^5_+ q^-5&1^5i|_+ q^-5&2^5j|_+ q^-5&3^5k|. !(1)
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The first quaternion's component (q^-5&0^5) is its scalar part,
q^-5&1^5, q^-5&2^5, and q^-5&3^5 are vector components of a
`pure' quaternion. 
We take a normalized quaternion as (1), where
.tp3

q^-5&0^5\&2_+ q^-5&1^5\&2_+ q^-5&2^5\&2_+ q^-5&3^5\&2_=_1. 
!(2) 
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The normalized quaternion defines a 3-D rotation, i.e.,
the rotation angle is determined as &F = 2arccos(q^-5&0^5). 
The vector part of a quaternion corresponds to a rotation
axis (Altmann 1986). 
In this paper we consider only counterclockwise rotations 
corresponding to positive angles 0&0_&<_&F_&<_180&0 (Kagan 
1991c).
Spherical coordinates of a rotation pole are
.tp6

&h = arccos@[q^-5&3^5/sin(&F/2)@],
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___&c = arctan(q^-5&2^5/q^-5&1^5), _ if _ &c_&<_0, _ 
then _ &c = 360&0 + &c, !(3)
.sk2
where &c is an azimuth (0_&<_&c_@<360&0), measured clockwise 
from North; and &h is a colatitude (0_&<_&h_&<180&0), &h_=_0
corresponds to the vector pointing down. 

A 3-D random rotation of general objects (with no
symmetry) has a p.d.f.
.tp 3
 
f(&F) = (1/&p)&.(1 - cos&F) _for_ 0&0_&<_&F_&<_180&0, !(4)
.sk1
which is not uniform in &F.
Altmann (1986) discusses why the uniform distribution of &F 
is not the density of the random rotation (see also Kagan
1991c). 
The cumulative angle distribution is
.tp 3
 
F(&F) = (1/&p)&.(&F - sin&F). !(5)
.sk1

The DC earthquake mechanism has a symmetry of a rectangular box
with unequal sides. 
In previous papers (Kagan 1990; 1991c) we discussed in 
detail the influence of the DC symmetry on its rotation 
and on the solution of the inverse problem of rotation, i.e.,
for each pair of focal mechanisms to determine the rotations
which rotate one mechanism into another. 
To make this exposition more self-~contained, we summarize 
briefly the results from the above papers.
As in the above publications, in this paper we consider the
symmetry only with regard to proper rotations (no reflections
are allowed). 
Under such restrictions we can choose four different systems of 
coordinates for a DC source: any system which uses, for
instance, the T-, P-, and N-axes can be transformed by a
rotation around any of these axes by 180&0. 
After the rotation, the axis of rotation keeps its orientation,
whereas two other axes invert their orientation.
The rotations by 180&0 are called <binary> rotations (Altmann
1986, p._209). 
In the quaternion representation, the binary rotation is a pure 
normalized quaternion (see eq._1) with q^-5&1^5, q^-5&2^5, and
q^-5&3^5 being cosines of axis of rotation. 
The binary rotation around the T-, P-, and N-axes corresponds
to multiplication of a quaternion by i|, j|, and k| (Kagan
1991c). 
Thus any quaternion, representing a DC solution can be
multiplied by i|, j|, or k|, and the result, due to the
symmetry properties of a source, is identical. 

As a result of the DC symmetry, there are four counterclockwise
rotations with &F_&<_180&0 which transform one mechanism into
another (Kagan 1991c). 
We anticipate that for some problems, a rotation other than 
minimal, for example, corresponding to a certain direction of
a rotation axis, might be chosen, but in all investigations
reported below, we only use the rotation with the minimum
rotation angle. 
As a result of the DC symmetry, the minimum rotation angle
cannot exceed 120&0 (Kagan 1990). 
By comparison, if we add reflections to possible DC
transformations, eight symmetrical positions (two possible
directions for each of three axes) can be found for each DC
source; quaternion representation is no longer useful, since
the quaternions represent only proper rotations. 

The p.d.f. for a random rotation of the DC source is
 
f(&F) = (4/&p)&.(1 - cos&F) ____________for_ 0_&<_&F_&<_{&p}/{2};
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___f(&F) = (4/&p)&.(3 sin&F + 2 cos&F - 2)
_for_ {&p}/{2}_&<_&F_&<_&F^-5&2^5; __and
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___f(&F) = (4/&p)&.&{3sin&F + 2cos&F - 2 -
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__________(6/&p)&.&[2sin&F&.arccos&(({1+cos&F}/{-2cos&F})[1/2]&) -
(1-cos&F)&.arccos{1+cos&F}/{-2cos&F}_&]&}
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_______________________________________for_ &F^-5&2^5_&<_&F_&<_{2&p}/{3},
!(6)
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where &F^-5&2^5_= 2arccos(3[-1/2])_&= 109.47&0.
For the cumulative function we obtain

F(&F) = (4/&p)&.(&F - sin&F) _____________________for_ 0_&<_&F_&<_{&p}/{2}, _and
.sk2
___F(&F) = (4/&p)&.&[2sin&F - 3cos&F - 
2&F + {3}/{2}&.&p - 3&] _for_ {&p}/{2}_&<_&F_&<_&F^-5&2^5.
!(7)
.sk2
For &F^-5&2^5_&<_&F_&<_{2&p}/{3} we compute F(&F) by numerical 
integration of the last equality in (6).

Previously (Kagan 1982) we introduced the rotational Cauchy
distribution to represent rotations of focal mechanisms of 
micro-~dislocations which comprise a focal zone of an 
earthquake.
The one-~parameter Cauchy distribution is especially important
for a representation of earthquake geometry, since it can be
shown by theoretical arguments (Zolotarev 1986, pp._45-~46;
Kagan 1990) and by simulations (Kagan 1990) that the stress
tensor in the medium with defects, has this distribution. 
In our case the defects are not necessarily small entities, 
large earthquake fault systems are in a certain sense 
translational defects (dislocations) in rock material.
The defect is a term used in solid state physics, 
the term <fault> is more familiar to seismologists or
geologists. 
However, to calculate the stress tensor field, we need to know
not only the fault position, but the slip vector as well. 
That is why we prefer to use the term <defect>, which 
combines a fault geometry with displacement on the fault.
Other types of defects, like rotational dislocations or
disclinations (see Kagan 1988), may also contribute to stress
perturbations. 

The Cauchy distribution belongs to a class of so-~called
<stable> distributions which are scale-~invariant, i.e., they
have a power-~law tail.
Thus the Cauchy distribution yields the
fractal geometry of earthquake faults (Kagan 1982; 1990). 
The major property of stable distributions is their invariance 
under addition of random variables: suppose that two
independent stochastic variables X]1[ and X]2[ are distributed
according to the Cauchy distribution with the parameter values
&k^-5&1^5 and &k^-5&2^5. 
The distribution of their sum, <X>, is a convolution of two
distributions 
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F]&k[(X) = F]&k^-5&1^5[(X]1[)*F]&k^-5&2^5[(X]2[), !(8)
.sk1
where &k_=_&k^-5&1^5_+_&k^-5&2^5 (Feller 1966; Zolotarev 1986).
It is difficult to measure the stress tensor itself in the deep
interior of the Earth, but we may infer the stress pattern on 
the basis of stress singularities the sudden onset of which is
registered as earthquakes.
In particular, rotations of earthquake focal mechanisms give us
an indication of the stress redistribution (Yoshioka @&
Hashimoto 1989). 
We argue (Kagan 1990) that the Cauchy distribution of the
stress should produce the rotational Cauchy distribution of DC
sources. 
Then the parameter &k represents the degree of <incoherence> or 
<complexity> of an earthquake fault.

The p.d.f._of the general rotational Cauchy distribution can be
written as (Kagan 1990)
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.nf
.!sp 1,0
.tp 3
f(&F) = {4&k}/{&p}&.(1 - cos&F)~
@[1 + &k&2 + (&k&2 - 1)cos&F@][-2], for 0&0_&<_&F_&<_180&0. !(9)
.sk1
The cumulative rotation angle distribution for (9) is 
.sk2
.nf
.!sp 1,0
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___F(&F) = {2}/{&p} &[arctan(A/&k) - ~
{A&.&k}/{A&2 + &k&2}&] ,               !(10)
.f
.sk2
where A = tan(&F/2).
If we compare (10) with displacement in the direction of the 
Burgers vector due to an edge dislocation (Nabarro, 1967, 
eq._2.15), the similarity of two expressions is obvious.
Actually, if we take the Poisson ratio equal to 1/2, the
formulas differ only by a multiplicative factor. 
This similarity might explain the applicability of the Cauchy
distribution for modelling focal mechanism rotation. 

In Fig._1 we display several (1_-_F) curves corresponding to
various values of &k in a double logarithmic format. 
For comparison the curve for random rotation (5) is also shown.
For small &k, the probability of the angle &F to be greater than
some value <a>, is 
.tp 4

Prob (&F @> a) = 1_-_F(a)_&&_a[-1], !(11)
.sk1
in the interval of rotation angles
a^-5&1^5&.&k_@<_&F_@<_@[&p-(a^-5&2^5&.&k)@], where a^-5&1^5 and
a^-5&2^5 are small numbers (less than 10). 
This means that the rotational Cauchy distribution has a
power-~law form for small &k.

For rotation of double-~couples the Cauchy rotational
distribution (eqs._9 and 10) needs to be modified. 
Unfortunately, no analytic expression is available, hence we
obtain the distribution which we call the <DC rotational Cauchy
distribution>, by simulation. 
In the simulations we rotate a DC using the Cauchy 
distribution (9) and then determine a minimum rotation 
between the initial position of the DC and its rotated 
position; for small rotations the minimum and general 
rotations are the same, but for large &F they often differ.

In Fig._2 we display several curves representing rotation of 
arbitrary objects (with 0&0_&<_&F_&<_180&0) as well as 
rotation of DCs (with 0&0_&<_&F_&<_120&0).
The difference between the general and DC random rotations is 
significant, for example, only 18.2% @[(1/2&p)(&p_-_2)@] of all
random rotations have &F_&<_90&0; or 39.1%
@[(1/6&p)(4&p_-_3&R3)@] of all random rotations have
&F_&<_120&0 (see eq._5). 
The respective values for the DC Cauchy distribution are 72.7% 
@[(2/&p)(&p_-_2)@] and 100%.
A simple argument based on symmetry of the DC, shows that only 
25% of general random rotations correspond to minimum DC 
rotations.
For the other 75% of the rotations, it is possible to 
find a double-~couple rotation with a smaller &F.

For the Cauchy distribution with small value of the parameter
&k (see eqs._9 and 10), the difference between the general
and DC rotations is small for small &F, but increases as the
angle increases. 
This means that for small &k the DC rotational Cauchy 
distribution can also be approximated by the power-~law (11).
For large &k the general Cauchy rotation (9) differs from the
random rotation (4), whereas the Cauchy DC rotations yield a
result similar to the random rotation. 

For the purpose of comparison we also calculate a distribution
which is an analog to the Gaussian distribution for the 3-D
rotational data (Kagan @& Knopoff 1985): this is the von
Mises-~Fisher-~type rotational distribution which may model
errors in determination of focal mechanisms. 
To obtain this distribution we generate a 3-D normally
distributed random variable u| (u^-4&1^4, u^-4&2^4, u^-4&3^4)
with the standard deviation &s]u[ and then calculate the unit
quaternion 
.tp 6  

q^-5&0^5 = (1 + u^-5&1^5\&2 + u^-5&2^5\&2 + u^-5&3^5\&2)[-1/2]
__and 

q]i[ = u]i[&.(1 + u^-5&1^5\&2 + u^-5&2^5\&2 +
u^-5&3^5\&2)[-1/2] __for i = 1,2,3.               !(12) 
.sk1
The orthogonal matrix which corresponds to this quaternion,
simulates a von Mises-~Fisher-~type rotational distribution 
(<ibid>).
 
We display several cumulative curves for distribution (12) in 
Fig._3c, for comparison purposes one curve for the Cauchy 
distribution (10) with &k_=_0.1, and the random curve (7) are
also included. 
The curves behaviour is significantly different from that of 
the Cauchy distribution (Fig._3a,b), this difference is
especially obvious for small values of the distribution
parameter: whereas the Cauchy distribution has a long 
power-~law tail, the von Mises-~Fisher distribution displays
more Gaussian type behaviour. 
Again for large values of the parameters, both distributions 
approach the random distribution, hence they become
almost identical.

Following Kagan (1990) we also produced several sets of 
random stress simulations.
For the purpose of illustration we assume that the initial
stress condition is one of pure shear stress, i.e., only 
&s^-5&1^5^-5&1^5_=_-_&s^-5&2^5^-5&2^5 are non-~zero.
In these simulations the stress in medium is taken to be 
slightly lower (by a multiplicate constant 1+&d) than the
critical fracture stress. 
A reference point in medium is surrounded by randomly 
distributed 100 point defects.
As the defects we use DC sources with three different 
orientations: 1) m^-5&1^5^-5&1^5_=_-_m^-5&2^5^-5&2^5, i.e, the 
moment tensor of the defects is parallel to the stress tensor;
2) m^-5&1^5^-5&2^5_=_m^-5&2^5^-5&1^5; and 3) each defect is 
randomly rotated.
We calculate the resulting stress tensor, i.e., the sum of the
initial stress and the internal stresses caused by these
defects. 
If the second invariant J^-5&2^5, of the resulting deviatoric
stress tensor, is greater than the critical value, we calculate
the 3-D rotation from the initial principal axes of the stress
to their new position. 
We assume that the fracture surface corresponds to the plane 
bisecting the maximum and minimum principal stresses.

The value of &d is taken to be 1/30, 1/10, and 1/3.
In all of the simulations described here, we have analyzed
10[7] cases of fracture.
The resulting histograms of the stress rotation are shown in 
Fig._3d.
For the purpose of comparison, one curve for the Cauchy
distribution (10) with &k_=_0.1, and the random curve (7) are
also included. 
Simulations with &d_=_0.1 are well approximated by the Cauchy 
rotational distribution (see also Kagan 1990).
These plots also demonstrate that the rupture rotation depends
not only on the value of &d, but on orientation of defects. 
As mentioned earlier (Kagan 1990) in this simulation we use 
a very simple fracture criterion; in reality the fracture
should depend on the value of all three stress invariants, this
might significantly modify the results of the simulations. 
Therefore, we use curves in Fig._3d only as a preliminary
indication of fracture behaviour. 

In Fig._4 we combine the results of rotation simulations for 
comparison with the measurements of rotation in a
catalog of earthquake fault mechanism solutions.
All rotational distributions studied display a similar
behaviour: with increase of a simulation parameter they all
approach the random distribution. 
There is little difference in the plot between the Cauchy 
distribution which has a power-~law form and the von 
Mises-~Fisher distribution, thus both the median and the
average are poor discriminants for these distributions. 

.!pg
.!4
.sk 2
.tp 6
.c
4._~ RESULTS.

Non-~statisticians may not be familiar with certain
statistical properties of the Cauchy distribution, so we will
comment briefly on their peculiarities. 
All of the statistical moments are infinite for the Cauchy 
distribution, which signifies that neither the mean, nor the
standard deviation exist for the distribution; an average of
<N> observations of a random Cauchy variable has the same
distribution as each variable itself (Feller 1966, p._497). 
Heuristically, in a sample of the Cauchy variables, the sum of
observed values is usually dominated by one value which is so
large that it completely overwhelms the others. 
Therefore, the sample average cannot represent a Cauchy random
variable, the median or quartiles of a histogram are more
appropriate. 

The situation is a little different for the rotational Cauchy 
distribution: since the space of all rotations is limited, all 
the moments are finite.
However, for the small value of the parameter &k in (9) the 
behaviour of a sum of random variables may strongly depend on 
one or a few `outliers'.
In such a case, the median of a histogram might be a more
effective way to describe statistical properties of a
variable. 
In most of our measurements, the values of &k are not very
small, so in statistical investigations of DC rotations we use
both averages and medians of obtained distributions. 

To illustrate the determination of the DC rotation, in Table_2 
and Fig._5 we display five focal mechanisms of earthquakes 
which occurred in the Los Angeles area in 1977-1990.
In Table_3 we list vectors connecting all hypocenters and 
pairwise 3-D rotations, by which the first focal mechanism in a 
pair can be transformed into the second mechanism.
The vectors are displayed in the right-~handed geographical
system of coordinates (North-~East-~Down), as well as in the
TPN system (i.e., formed by axes of the first mechanism). 
For all pairs we list the minimum rotation angle and the
TPN spherical coordinates of a rotation pole on a reference
sphere. 
For three pairs we also display all four possible rotations: in
this case, rotation poles are shown in the spherical
geographical system. 
We choose only three pairs for illustration only: they 
demonstrate the rotations in the simplest case (pair 1-4), 
these rotations can easily be obtained by inspection, and in
a more complex case (pair 2-3), where the rotations are not
so obvious. 
In our statistical analysis, we determine the minimum rotation
angles for many thousands of earthquake pairs (for shallow
earthquakes, the total number of pairs separated by less than
500_km is over 300,000), and make histograms of these angles. 

In Fig._6 we display histograms for distribution of &F for 
shallow earthquake pairs which are separated by a distance 
of 42_km to 59_km.
We study whether the rotation of focal mechanisms depends on 
where the second earthquake of a pair is situated with regard
to the first event. 
Thus we measure the rotation angle for hypocenters located in
30&0 cones around each principal axis (see curves, marked the
T-, P-, and N-axes) of the first event. 
For two curves (marked the T-P and T+P vectors) hypocenters 
are located in a 15.3&0 segment perpendicular to the
vector; the total area of such a segment is equal to the area
of two spherical caps around each axis. 
The curves in Fig._6 are narrowly clustered, and are obviously
well approximated by the DC rotational Cauchy distribution. 
Thus all earthquakes, regardless of their spatial orientation,
have focal mechanisms similar to a previous event. 
Comparing the curves with Fig._3a,b shows that earthquakes in 
the cone around the N-axis correspond to a smaller &k-value
than events near the T-axis. 
The rotations in Figs._1-4 are calculated for a single
reference point, in Fig._6, on the other hand, we measure the
rotation of one focal mechanism into another. 
However, if the rotation angle follows the Cauchy distribution
with parameter &k, the sum of two variables is again 
distributed (see eq._8) according to the Cauchy rotation with 
the value of the parameter 2&k.

In Table_4 to show how the &F distribution depends on the cone
angle, we have calculated histograms for various angles. 
Both average and median rotation angles indicate that if the 
cone is narrowed, the difference between the T- and N-axis 
distributions increases.
Thus focal mechanisms near a fault-~plane are more coherent
than those outside the plane. 
We use the cone angle of 30&0 in all following computations to
insure sufficient statistical stability of the &F estimates.

The distribution of &F for intermediate earthquakes (Fig._7)
follows in general the same pattern as the distribution for
shallow events (Fig._6): it is also reasonably well
approximated by the DC Cauchy distribution. 
The parameter &k of the Cauchy distribution is 0.2, i.e., much 
larger than the parameter for shallow earthquakes.
Although mechanisms near a fault-~plane are more coherent, a 
difference between these events and `off-~plane' mechanisms is 
not as large as in the case of shallow seismicity.
Similar behaviour is observed for deep earthquakes.

Table_5 lists the average values of &F with increasing depth 
range.
In general, the rotation angle increases with depth, although
the values for the T- and P-axes do not display significant
variation as depth exceeds 10_km. 
For almost all depth intervals, rotations of earthquake focal
mechanisms contained near the T-axis are larger than for 
events in other directions.
Only for intermediate earthquakes does this pattern seem to be
replaced by events close to the P-axis, still the distribution
difference between the T- and P-axes is not large. 
The strongest depth dependence is for the N-axis, with &F
increasing by about 50% near the 70_km boundary. 
A generally higher coherence of earthquake focal parameters in
the neighbourhood of the N-axis might be explained by the
concentration of seismicity in subducting seismic belts: the
N-axis is more likely to point to other events situated on the
same lithospheric slab. 
The larger coherence of shallow earthquakes might then be
connected to a greater rigidity of the slab in the upper
70_km; below this boundary the slab is more likely to be bent
and contorted. 

All the previous plots are calculated for the seismic moment
cutoff log_M^-5&0^5_=_16.5 (M]w[_&>_5.0). 
We calculate similar curves for various cutoffs (Table_6).
The values of the rotation angles (average and median) suggest
that there is a weak (or no) dependence of source coherence on
the size of an earthquake. 
In principle, we expect that weak events might follow minor
perturbations of stress and be less coherent. 
We need to study regional catalogs of fault-~plane solutions to
investigate this pattern. 

Fig._8 shows histograms for shallow earthquakes separated by
about 320_km. 
Here the curves corresponding to fault-~planes (the N-, T-P,
and T+P vectors) are clearly separated from the histograms
connected with the T- and P-axes: whereas the rotation near the
fault-~plane is relatively small (&k_&=_0.15), the earthquakes
which are situated in cones around the T- and P-axes have focal
mechanisms which are essentially uncorrelated with the primary
event. 
The degree of mechanism incoherence in the neighbourhood of a
fault-~plane increases with distance (from about &k_=_0.07 for
small distances to &k_=_0.15 for long-~range interaction). 

In Fig._9 we test whether the &F histograms are well
approximated by the Cauchy distribution for various seismogenic
regions. 
We subdivide the Earth's surface into four quadrants and 
calculate the histograms.
The curves comparison with each other as well as with the 
Cauchy curves in Fig._3a, show that the histograms differ from 
the Cauchy distribution only by a few percent.
Further accumulation of focal mechanism data should allow a
more detailed study of regional distributions of the rotation
angle. 

Since the &F distribution is reasonably well approximated by
the DC rotational Cauchy distribution, we can characterize the
angle histogram by its average (&F]a[) or median angle
(&F]m[). 
In Fig._10, we display the dependence of &F]m[ on distance and
depth range. 
All curves show an increase of focal mechanism incoherence
when distance increases; for shallow events in a fault zone
(Fig._10a, curves corresponding to the N-axis, T-P and T+P
vectors) the incoherence is small and relatively stable in the
distance range 0-100_km. 
The latter distance value seems to correspond to an average
width of subduction belts.
The conformity of all curves in Fig._6 can be explained,
by all earthquake pairs belonging to the same seismic belt. 
Earthquakes situated in the T- and P-cones, which do not belong
to the same fault-~plane, still display a significant coherence
at small distances. 
However, for the T- and P-curves in Fig._8 this coherence is
lost; at distances of 320_km a second earthquake should
belong on the average to a completely different fault system. 

Shallow earthquakes (Fig._10a) differ from intermediate
(Fig._10c) and deep (Fig._10d) events in two respects: the 
range of &F]m[ change is much larger for shallow seismicity.
The same applies to the difference between directions lying in
the fault-~plane and cones around the T- and P-axes. 
The larger &F]m[ values for deeper events can be explained by
an older age of the crustal slab responsible for these
earthquakes: according to (8) the value of &k and &F]m[ should
increase with the age of the slab. 

To test whether the difference between shallow and deeper
seismicity is due to the presence of aftershocks and other
dependent events in shallow sequences, we create the residual
(declustered) catalog (Kagan, 1991b), from which we effectively
delete aftershocks. 
We repeat our calculations with the declustered catalog (the 
results are shown in Fig._10b).
The difference between the N-axis, on the one hand, and the T-
and P-axes, on the other hand, has decreased, but it is still
significantly larger than the difference for intermediate and
deep events. 
The reason for this difference might be a greater plasticity
and deformation of the lithospheric slab at depths above 70_km
(see discussion for Table_6). 

We have investigated the distribution of rotation axes to see
whether the poles are concentrated on a reference sphere and
how these concentrations depend on the direction of the vectors
connecting two hypocenters or on the depth of earthquakes. 
The study of rotation pole concentrations increases
significantly the dimensionality of statistical distributions:
unfortunately, the available data does not yield reliable
conclusions. 
We find that the distributions appear to be different for
various vector directions but we could not find any consistent
picture. 
In Table_7 we list several distributions of rotation poles in a 
format similar to that used in Tables_1 and 2 in Kagan (1990).
In Table_7, similarly to Table_1, we subdivide the positive
octant of the sphere into 55 spherical triangles and
quadrilaterals with equal area and then calculate the number of
times the rotation axis intersects each of these cells. 

The upper chart in Table_7 shows the distribution of axes 
for rotation angles of less than 15 degrees.
The distribution is randomly uniform as it should be, because
these rotations are caused, most probably, by random errors;
for larger angles (the second chart) the distribution is closer
to that obtained through simulation, showing that stresses of
the first earthquake influence the focal mechanism of the
second earthquake in a pair. 
The last chart in the table shows the rotation of the shear
stress tensor by a point DC source. 
We place a source uniformly randomly on a surface of a sphere 
surrounding the reference point.
At this point the initial simple shear stress is augmented by 
the stress of the DC; if the stress invariant J^-5&2^5
increases as a result of the additional stress, we determine
the rotation of the principal stress axes. 
The chart exhibits a strong distribution inhomogeneity of 
rotation poles: however, our simulations with many defects 
(Kagan 1990) does not show a significant difference between
various directions. 
The distribution of rotation axes should be studied in future,
when more information on focal mechanisms will be available. 

.!pg
.!5
.sk 2
.tp 6
.c
5._~ DISCUSSION.

The results presented above, suggest that rotations of focal 
mechanisms follow the rotational Cauchy distribution; by 
implication tectonic stress which causes earthquakes, is
distributed according to the Cauchy distribution.
This conclusion has been proposed earlier (Zolotarev 
1986, pp._45-~46; Kagan 1990) as a necessary consequence of the
presence of defects in rock material.
Since the Cauchy distribution is stable with a power-~law 
tail, its control of stress means that the earthquake
rupture has to produce the fractal geometry. 
Thus our results are a quantitative confirmation of 
a general model of earthquake rupture which assumes that 
an earthquake occurs on previously created complex faults.
However, such a model should also allow for evolution of a
fault system and creation of new fractures. 
The above theoretical considerations (<ibid>) assume that
linear elasticity is valid everywhere outside defects, and yet
their predictions have been validated in our measurements. 
Therefore, if we know the geometry of defects, we should be 
able to calculate the stress tensor field, and, if we know 
fracture criteria, to predict when and where the next ruptures
are going to occur. 
Possibly, we will never have a complete geometrical description
of earthquake faults, but the knowledge of the statistical
properties of defects geometry should help us to make
quantitative probabilistic predictions of future earthquakes. 

Previously Kagan (1982) found that to explain the complex
geometry of the San Andreas fault we need the value of &k for
the rotational Cauchy distribution of the order 2&*10[-6] to 
2&*10[-5]. 
These values correspond roughly to the ratio of the average
slip to the length of rupture during an earthquake (Scholz,
1990, pp._183-~186). 
For earthquakes occurring on a complex fault system, we obtain 
the rotations of focal mechanisms which are approximated by the 
Cauchy distribution with the value of &k 0.05--0.1 (see Figs._4 
and 10).
It is possible that `single' faults in a fault zone are 
much more coherent, with &k being of the order 10[-5] to 
10[-6] (Kagan 1982; 1990).
Thus, we might hypothesize that an earthquake fault system 
starts with a relatively simple fault which exhibits only 
a slight complexity, like branching, bending, and rotation.
For such a fault the value of &k should be small, of the order 
10[-5].
In the course of its tectonic history, the complexity of a
fault system increases, in terms of &k values this increase 
being reflected in a gradual accumulation of rotations
according to (8). 
Therefore, we estimate that to reach &k_=_0.1 the fault system
should require about 10[4] to 10[5] large earthquakes to occur
in a fault zone. 

If the return time for large earthquakes in a fault zone is of
an order of a few hundred years, the age of the zone would
reach several millions or tens of millions of years, the value
which corresponds to standard views of plate tectonics. 
This might mean that instrumental earthquake catalogs which 
span several decades, might have relatively little information
on the geometry of defects.
This geometry, as we have suggested earlier, determines the
future developments of the fault system. 
A similar conclusion may be inferred for regional geodetic
data or measurements of regional tectonic stresses, which also
reveal only the recent deformation of tectonic blocks. 
Earthquakes in tectonic blocks might depend weakly on the
present deformation, and more strongly on the geometry of known
and hidden defects in rock medium. 
These hidden faults might explain why many earthquake focal 
mechanisms disagree with directions of principal stress.
These stress directions are inferred from known faults or from
stress tensor measurements close to the surface of the Earth. 
However, additional geological and paleo-~seismological
investigations might supply the necessary ingredients for
evaluation of a stress pattern in a region and of future
seismic risk. 

From the point of view of the focal mechanism interaction, and,
consequently, stress influence on the earthquake rupture
process, intermediate and deep events do not differ
significantly from shallow earthquakes. 
This provides strong evidence that the mechanism producing 
all of these earthquakes is essentially the same.
This conclusion might bear on the discussion of the nature of
deep earthquakes (see Frohlich 1989; Kirby, Durham @& Stern
1991; Meade @& Jeanloz 1991, and references therein). 
The above authors argue that the mechanism responsible for 
shallow seismicity (shear failure) cannot be applied to deeper 
earthquakes, hence some form of phase transformation is
called for as an explanation. 
Our results indicate that even if phase transformations are
responsible for deep earthquakes, the distribution of
hypocenters with regard to focal mechanisms of earthquakes and
the rotation of focal mechanisms for deep events, follow almost
the same pattern as shallow seismicity: this imposes certain
constraints on models for both shallow and deep earthquake
fracture. 

What is the relation between the rotation of tectonic
blocks as measured in tectonophysics investigations
(Molnar 1983; Jackson @& McKenzie 1988; England @& Jackson
1989; Molnar @& Lyon-Caen 1989; Chang <et al.> 1990),
and our results on focal mechanism rotations?_
Motions considered in the above papers, are predominantly 2-D:
they comprise the 2-D rotations and translations of tectonic
blocks over the surface of the Earth. 
The 2-D motions can be represented by 3 parameters (one for 
rotation and two for translation); hence the 3-D rotation which
is also characterized by 3 degrees of freedom, is an
appropriate tool for their representation (Chang <et al.> 1990).
In principle, the 2-D consideration should eventually be
extended to the 3-D deformation of blocks and microblocks of
rock material. 
A motion in 3-D requires 6 parameters for its characterization:
3 degrees of freedom for 3-D rotations and a similar number for
translations. 
Unlike the above studies, we investigate the rotations of
earthquake focal <mechanisms> which are sources of block
deformation and displacement. 
We always consider the sources in 3-D.
Another difference between our approach and that of the above
papers, is that they study regional properties of focal
mechanisms to infer an <average> deformation of tectonic
blocks. 
In this work we concentrate on studies of local and regional
stress <variations> as they are reflected in mutual rotations
of earthquake focal mechanisms. 

However, there is a connection between both of the above
rotations: if all focal mechanisms are coherent (there is no
rotation), there are no disclinations (rotational defects) in
the cumulative sum of moment tensors of these earthquakes
(Kagan 1988, and references therein). 
The disclinations which are the third-~rank seismic moment
tensors, provide for the rotational deformation of medium:
although one can represent centers of rotation through the
asymmetric second-~rank tensors (Molnar 1983; Jackson @&
McKenzie 1988), these second-~rank sources do not have their
angular momentum compensated, hence they do not properly
represent <internal> sources. 
The lowest order representation of rotational internal seismic
sources are disclinations which can be thought as compensated
pairs of rotation centers (see more in Kagan 1988). 
Therefore, the rotation of focal mechanisms studied in this 
paper, is a necessary condition for rotational deformation of 
rock medium and tectonic blocks.

In a recent paper (Kagan 1991a) we estimated the fractal 
dimension of a set of earthquake hypocenters.
For shallow earthquakes this dimension is found to be
2.20&+0.05. 
Is this fractal dimension connected in any way with our present 
results?_
An ideal solid crystal (without defects) should fail along a
planar dislocation, hence the dimension of fracture should be 
2.0; for such crystal the Cauchy distribution (9) has &k_=_0.
However, the results of our measurements of the focal mechanism
rotation indicate that due to the tectonic evolution, focal
zones of earthquakes contain partially incoherent defects
distributed according to the Cauchy distribution with &k as
large as 0.2. 
It would be of great interest to know whether &k_=_0.2 implies 
the hypocentral fractal dimension of 2.2.

On the basis of the above discussion we propose a new framework
for geometry and mechanics of earthquake fault zones. 
Theoretical strength of materials is two to three orders of 
magnitude larger than their observed strength (Lardner 1974, 
pp._6, 15; Scholz, 1990, pp._2, 3).
The difference is thought to be due to defects such as
dislocations, inclusions, grain interfaces, etc. 
The size of the defects is usually much smaller than the size 
of an object to be tested.
For the purpose of the discussion below we call the former
materials `ideal' and latter materials `regular'.
The shear strength of ideal materials is of the order &m to 
&m/10, where &m is the shear modulus; for regular materials the
strength is &m/100 to &m/1000 (Lardner 1974).

In earthquake studies we consider the propagation of a rupture 
through rock material which was subject, during millions 
of years of its tectonic history, to repeated earthquake 
deformation.
As the result of this process, the largest defects which we
identify with the fault systems, have approximately the same
size as tectonic blocks. 
These defects are therefore critically self-~organized (Chen,
Bak @& Obukhov 1991) coherent assemblages (aggregates) of many
small earthquake sources. 
Below we call such materials <tectonically self-~organized 
rocks>.
The apparent strength of such materials is strongly dependent
on the orientation of the rupture, and might again be 
diminished by the factors of about 100 or 1000 compared to
regular materials (the effective strength is &m/10[4] to
&m/10[5]). 
In a certain sense, such rock material, deformed under the
tectonic stress, has a zero strength: if we define the strength
of material by the onset of the nonlinear response, tectonic
blocks always have some small earthquakes occurring in them,
hence they always have subvolumes where the stress is at its
critical value. 

In another sense, we hypothesize that the strength of rocks is
essentially the same everywhere: if some blocks of material
show little internal deformation, it is not due to their
greater intrinsic strength, but conversely, due to the absence
of large defects in these blocks: hence the seeming strength is
a consequence of previous tectonic history of these blocks
(<cf.> Davy, Sornette @& Sornette 1990). 
Thus we propose that earthquakes occur on faults not because 
these faults are <weak> or have inferior strength, but because 
previous earthquakes have left very strong internal stresses in
their wake, so that relatively small stress increments are
needed to fracture the area again (Kagan 1990). 
Whereas the fault strength computations are based on various,
sometimes arbitrary, assumptions (Scholz 1990), calculations of
internal stresses caused by defects is a straightforward
outcome of the elasticity theory. 
For example, after an earthquake stress redistributes itself, 
decreasing in the focal zone and increasing outside
the zone: again this redistribution is readily calculated,
whereas we need additional assumptions to compute apparent
strength variations.

In specimens or objects made of regular material two scales
are distinguished: a small scale of defects and a large scale
of the object itself. 
Therefore, in most cases we can separate the small scale
effects by introducing effective properties of the material.
These properties can be either measured in special tests or
calculated using known engineering formulas. 
In the tectonically self-~organized blocks of material, defects 
are almost always large scale, hence no scale separation is 
possible.
Thus our results imply that there are no average or effective
properties of tectonically self-~organized rocks. 

Fracture mechanics studies initial formation in a regular, 
previously unbroken material of a defect which is comparable
in size with the object.
In most cases this defect is a tensile crack.
In the fracture mechanics, geometrical models of a defect 
are usually simple, and the defect occupies a small part of
the object, even at the more advanced stages of a fracture. 
Many of these ideas have been incorporated into the models of 
earthquake rupture (Scholz 1990).
We see the following difficulties with the standard models of
an earthquake source:
.sk1
.lm 15
.p -3,0

1) the source is usually assumed to be a rupture on a single
<planar> surface, or on a surface, at least described by a
simple Euclidean geometry. 
Our results indicate that the surface is not planar, but has a
fractal dimension of about 2.2 for shallow faults (Kagan
1991a). 
(A plane has the dimension 2.0.)_
These fractal properties of earthquake faults extend from
relatively small to very large distances of hundreds and
thousands km. 
Scholz (1990, pp._50, 51) recognizes that the faults'
scale-~invariant character differentiates them not only from
theoretical planar faults but also from laboratory fault models
which have a clear scaling cutoff. 
Moreover, analysis of fault geometry (Kagan 1982) indicates 
that earthquakes do not occur on a single (possibly 
wrinkled or even fractal) surface, but on a fractal structure
of many closely correlated faults. 
The total number of these infinitesimal faults might be 
infinite.
.! In this circumstance, the yield strength of a fault system is
.! not directly connected with the usual picture of the static
.! friction along a fault, but is instead connected with the
.! ability of a fracture on one segment or collection of segments
.! to trigger additional motions on other segments possibly remote
.! from the ruptured parts. 

2) the geometry of a source is usually assumed to be fixed: it
is not clear how such a pattern has been formed, what its
evolution history is, whether the proposed geometry is stable
with regard to perturbations, etc. 
For example, crustal deformation is sometimes described in 
terms of tectonic block displacement.
However, unless we consider breaking up the constituent blocks,
this approach has also a disadvantage of a `frozen' geometry:
earthquake ruptures are not allowed to penetrate the blocks,
even if stresses on these blocks may be exceedingly high. 
.lm 8
.p 3,1

It will take some time and effort to develop all details of a
new framework, but we hope that it will allow us to study
stresses in rocks in order to use the results in long-term
and long-range earthquake prediction. 
Comparing the distribution of rotation angles with those
expected from known earthquakes, and studying the distribution
as it depends on distance between earthquakes, we evaluate the
ratio of deterministic stress to random stress. 
In principle, this knowledge should allow us to use a 3-D
stress pattern to calculate long-term and long-range
probabilities of earthquakes with a certain focal mechanism 
occurring in seismogenic regions. 
It should also allow us to predict where `hidden' faults may be
situated and what the probability of them generating
earthquakes is.

Investigations of rotations of focal mechanisms should also 
help us to study the fault zone, to subdivide the zone into 
separate faults, to subdivide focal mechanism maps into 
zones of various mechanism incoherence, etc.
In all of these statistical studies it is important to know the 
distribution of the focal mechanism rotation.
If the rotation follows the Cauchy distribution, we can
characterize all of these patterns by just one parameter value.
Proposed methods present a quantitative tool to study the above 
problems.

.!pg
.!6
.sk 2
.tp 6
.c
6._~ CONCLUSIONS.

1. We investigate the distribution of hypocenters on focal
spheres of earthquakes. 
The hypocenters are concentrated along the fault-~planes; the
distribution does not significantly differ for earthquakes in
different depth ranges. 

2. The pairwise disorientation (rotation) of focal mechanisms
is well approximated by the rotational Cauchy distribution
which previously have been identified by theoretical arguments
and simulations as the result of stress perturbations caused by
random defects in medium. 

3. For shallow seismicity, the parameter of the Cauchy 
distribution &k is 0.05 to 0.1 in the neighbourhood of a 
fault-~plane, increases with distance between pairs of
events. 
In directions other than a fault-~plane, &k approaches the
value 0.5, the Cauchy distribution then does not differ
significantly from the completely random pattern of rotations. 

4. Although &k increases slowly with depth, disorientations of
focal mechanisms in various depth ranges display a similar
dependence on distance and magnitude. 
Therefore, we may conclude that deep earthquakes are controlled
by the same stress interaction regime as are shallow events. 

5. The results of these measurements make us question the
suitability of some terms and models that are
commonly used in the theory of an earthquake source. 
For example, we argue that tectonic earthquakes rupture rock 
material which has large-~scale defects, comparable in size 
with any volume considered.
Properties of the rocks which we call tectonically
self-~organized material, should be significantly different from
that of regular materials where the size of defects is
typically considerably smaller than the scale of interest. 
These properties should strongly depend on the geometry
(location and orientation) of major defects. 

.!pg
.!7
.sk1
.tp6
.c
ACKNOWLEDGMENTS
.sk1
I am grateful to L._Knopoff, and D._D._Jackson of UCLA for
their valuable comments, as well as F._Leader of UCLA for his
help in computer plotting. 
I thank Dr._J._Dunphy of the USGS for sending a catalog of
seismic moment tensor solutions in computer-~readable form. 
 
.!Ref
.pg
.sp 1,0
.lm 15
.p -7,1
.c
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Apperson, K._D. @& C._Frohlich, 1987.
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and B axes of intermediate and deep focus 
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Chang, T., J._Stock @& P._Molnar, 1990.
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<Geophys._J._Int.>, <101>, 649-~661.
 
Chen, K., P._Bak @& S._P._Obukhov, 1991.
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Chinnery, M._A., 1963.
The stress changes that accompany strike-~slip faulting,
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Davy P., A._Sornette @& D._Sornette, 1990. 
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Dziewonski, A._M., G._Ekstrom, J._H._Woodhouse @& 
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England, P. @& J._Jackson, 1989.
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Feller, W., 1966.
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Statistical methods for comparing directions to the 
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Kagan, Y. Y., 1990.
Random stress and earthquake statistics: spatial dependence,
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Kagan, Y._Y., 1991b.
Likelihood analysis of earthquake catalogs,
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Deep-focus earthquakes and recycling of water into the Earth's
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Spatial patterns of aftershocks of shallow earthquakes in 
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Fault-~plane solutions of earthquakes and active tectonics of 
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Spatial patterns of aftershocks of deep focus earthquakes,
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Yoshioka, S. @& M._Hashimoto, 1989.
The stress field induced from the occurrence of the 1944 
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.pg
.sp 1,1
.!9
.c
<Figure captions>.
.lm 24
.p -16,1
.sk1

Fig._1._________Dependence of the Cauchy distribution on the
rotation angle &F, for several &k values: 
Left solid line_-- &k_=_10[-4]; 
dashed line_-- &k_=_10[-3]; 
dashdot line_-- &k_=_0.01; 
dotted line_-- &k_=_0.1. 
Right solid line is for random rotation.

Fig._2a,b.______Distributions of rotation angles for the Cauchy 
rotation and random rotation of arbitrary objects and 
DCs.
Solid lines_-- random rotation; dashed lines_-- Cauchy rotation
with &k_=_0.1; dashdot lines_-- Cauchy rotation with &k_=_0.5. 
Arbitrary rotation has a domain 0-180&0, rotation of
DCs is defined for 0&0_&<_&F_&<_120&0.
.br
a) Cumulative distributions.
.br
b) Density curves.

Fig._3a,b,c,d___Distributions of rotation angles for 
DCs.
.br
a) Cumulative distributions for Cauchy rotation: values of &k
are 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5 from left
to right (dashdot lines).
.br
b) Density curves for Cauchy rotation (dashdot lines).
.br
c) Cumulative distributions for von Mises-Fisher rotation:
values of &k are 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 from left to
right (dashdot lines).
.br
d) Cumulative distributions for simulation of stress rotation:
values of &d are 0.033, 0.1, 0.33 from left to right
(dashdot lines).
.br
Solid lines are for random rotation, dashed lines represent the 
Cauchy rotation (1) with &k_=_0.1.

Fig._4a,b_______Rotation angles for several models:
stars_-- Cauchy rotation; 
circles_-- von Mises-Fisher rotation;
x-marks_-- stress simulation, random rotation of defects;
plusses_-- stress simulation, defects are diag@[1,-1,0@];
points_-- stress simulation, defects are M]12[_=_M]21[_=_1.
Solid line is for random rotation.

Fig._5._________Focal mechanisms of earthquakes from the
Harvard list in the Los Angeles area, 1977-1990: latitude
limits 33.0-35.0N, longitude limits 116.25-118.75W. 
Lower hemisphere diagrams of focal spheres are shown, 
compressional quadrants (around the T-axis) are shaded.
The numbers near diagrams correspond to that of Table_2.

Fig._6a,b_______Distributions of rotation angles for pairs of 
focal mechanisms of shallow earthquakes; hypocenters are
separated by distances between 50&*2[-0.25] (@~42)
and 50&*2[0.25] (@~59) km;
dots_-- all hypocenters;
circles_-- hypocenters in 30&0 cones around the T-axis;
plusses_-- hypocenters in 30&0 cones around the P-axis;
stars_-- hypocenters in 30&0 cones around the N-axis;
dashed line_-- hypocenters in 15.3&0 segment 
perpendicular to the T-P vector;
dashdot line_-- hypocenters in 15.3&0 segment 
perpendicular to the T+P vector.
Left solid line is for the Cauchy rotation with &k_=_0.1;
Right solid line is for the random rotation.

Fig._7a,b_______Distributions of rotation angles for pairs of 
focal mechanisms of intermediate earthquakes; hypocenters are
separated by distances between 50&*2[-0.25] (@~42)
and 50&*2[0.25] (@~59) km.
Line-~types are the same as in Fig._6.
The Cauchy distribution has &k_=_0.2.

Fig._8a,b_______Distributions of rotation angles for pairs of 
focal mechanisms of shallow earthquakes; hypocenters are
separated by distances between 320&*2[-0.25] (@~269) and 
320&*2[0.25] (@~381) km. 
Line-~types are the same as in Fig._6. 
The Cauchy distribution has &k_=_0.15.

Fig._9__________Distributions of rotation angles for pairs of 
focal mechanisms of shallow earthquakes; hypocenters are
separated by distances between 0 and 80_km;
dots_-- all hypocenters, N_=_6392;
circles_-- hypocenters in south-~western quadrant, N_=_1462;
plusses_-- hypocenters in south-~eastern quadrant, N_=_1801;
stars_-- hypocenters in north-~western quadrant, N_=_1080;
dashed line_-- hypocenters in north-~eastern quadrant, N_=_2049.

Fig._10a,b,c,d__Dependence of median angle on distance.
.br
a) shallow earthquakes;
.br
b) shallow earthquakes, residual catalog;
.br
c) intermediate earthquakes;
.br
d) deep earthquakes.
.br
Line-~types are the same as in Fig._6.

.pg 35
.sk1,,1
.sp 1,1
.lm 18
.rm 78
.c
Table_2.
 
.c
Shallow earthquakes from the Harvard list in the Los Angeles area, 1977-1990
.sk1
.sp 1,0
.nf

No   Year mo day h  m   Centroid coord    M^-5&0^5     T-    P-axis 
]                        lat   long  dep.  exp  pl azm  pl azm[


 1   1979  3 15 21  7  34.34 -117.03 10  17.4   0 122   0  32  
 2   1986  7  8  9 20  33.63 -116.57 15  18.1  47 277  23 160  
 3   1987 10  1 14 42  33.99 -118.25 15  17.9  76 337  14 175  
 4   1987 10  4 10 59  34.24 -117.76 15  16.9   0 103   0  13  
 5   1990  2 28 23 43  34.11 -118.06 15  17.6  19 264   5 172  

.f

.!pg
.sk1,,1
.sp 1,1
.lm 13
.rm 83
.c
Table_3.
 
.c
Earthquakes from the Harvard list in the Los Angeles area, 1977-1990:
.c
Rotation of focal mechanisms for any pair of earthquakes in Table_2.
.sk1
.sp 1,0
.nf

Event  Dist    Geogr. cosines      TPN cosines     Angle Colat. Azm.
] pair   km                                            &F     &h     &c[

 1 2   89.7  -0.88  0.47  0.06   0.87 -0.50 -0.06   68.2  56.9 131.6
                                                    68.2 123.1 163.6
                                                   138.9  70.6 281.7
                                                   143.7 111.7  11.7
                                                   144.4  29.5  73.6
 1 3  118.8  -0.33 -0.94  0.04  -0.63 -0.78 -0.04   88.6  77.2 154.1
 1 4   68.1  -0.16 -0.98  0.07  -0.75 -0.66 -0.07   19.0   0.0   0.0
                                                    19.0 180.0   0.0
                                                   161.0   0.0   0.0
                                                   180.0  90.0  22.5
                                                   180.0  90.0 112.5
 1 5   98.1  -0.26 -0.96  0.05  -0.68 -0.73 -0.05   43.6  27.5 145.4
 2 3  160.2   0.25 -0.97  0.00   0.68 -0.52  0.52   41.4  84.9  24.2
                                                    41.4  52.8 183.5
                                                   142.5 106.0 346.0
                                                   163.4 152.3 117.1
                                                   176.4  69.8  70.6
 2 4  129.0   0.53 -0.85  0.00   0.62 -0.72  0.31   59.7 109.4 141.1
 2 5  147.4   0.36 -0.93  0.00   0.66 -0.61  0.44   37.1  88.6 144.3
 3 4   52.9   0.52  0.85  0.00   0.04 -0.44 -0.90   86.0  92.9 163.9
 3 5   22.0   0.61  0.80  0.00   0.06 -0.52 -0.85   67.6  82.3 177.6
 4 5   31.1  -0.46 -0.89  0.00  -0.76 -0.65  0.00   28.0  45.0 154.9

.! See (3.2) for notation.
.f
.pg
.sk2,,1
.sp 1,1
.lm 23
.rm 73
.c
Table_4.
.sk1
 
.c
Values of rotation angles for shallow earthquake focal mechanisms 
.c
situated in a cone around principal axes of primary event, distance between 
.c
hypocenters is between 80&*2[-0.25] (@~67.3) and 80&*2[0.25] (@~95.1) km 
.sk1
.sp 1,0
.nf

 Cone Angle    T-      P-    N-axis    

             Average rotation angle

    10        63.9    53.7    31.7
    15        62.7    54.3    31.6
    20        62.5    52.3    32.2
    25        62.5    51.0    33.2
    30        61.9    50.1    34.2

              Median rotation angle

    10        69.7    57.3    20.8
    15        67.7    54.8    21.4
    20        66.9    51.0    22.3
    25        67.3    48.4    23.1
    30        66.6    47.8    24.2
.f 

.pg
.sk2,,1
.sp 1,1
.lm 23
.rm 73
.c
Table_5.
.sk
.c
Values of rotation angles for earthquake focal mechanisms 
.c
for different depth ranges, distance between 
.c
hypocenters is between 80&*2[-0.25] (@~67.3) and 80&*2[0.25] (@~95.1) km 
.sk1
.sp 1,0
.nf

   Depth    Event   Average rotation angle:
  range km  number
                      T-      P-    N-axis

   0 -  10   732     37.9    34.6    24.4
  11 -  20  3137     56.0    48.2    35.3
  21 -  30   667     64.4    51.3    28.2
  31 -  40   783     69.2    48.7    29.0
  41 -  50   484     74.2    51.3    27.5
  51 -  70   589     66.6    52.3    32.3
  71 - 140   885     60.9    63.6    48.7
 141 - 300   645     55.0    60.0    44.3
 301 - 700   636     63.9    59.4    54.6
.f 

.pg
.sk2,,1
.sp 1,1
.lm 23
.rm 73
.c
Table_6.
.sk1
 
.c
Values of rotation angles for earthquake focal mechanisms 
.c
for different magnitudes and depth ranges, distance between 
.c
hypocenters is between 80&*2[-0.25] (@~67.3) and 80&*2[0.25] (@~95.1) km 
.sk1
.sp 1,0
.nf

  Magnitude      shal.   intr.   deep 
     M]w[
                Average rotation angle

     5.0         44.2    55.3    57.5
     5.2         44.2    55.0    57.5
     5.4         44.0    53.8    55.0
     5.6         43.8    54.8    53.6
     5.8         42.3    53.4    53.7
     6.0         40.9    51.9    58.0

                 Median rotation angle

     5.0         36.8    53.4    57.8
     5.2         36.7    52.9    57.8
     5.4         36.7    51.0    55.1
     5.6         36.4    52.1    52.5
     5.8         34.5    50.0    52.5
     6.0         31.3    46.7    59.5
.f