Yan Y. Kagan
Dept. Earth and Space Sciences, UCLA, Los Angeles,
CA 90095-1567, ykagan@ucla.edu,
http://scec.ess.ucla.edu/ykagan.html
Earthquakes and Fractures in Solids:
Why do we fail to understand them and
what can be done?
http://scec.ess.ucla.edu/~ykagan/india_index.html
Outline
1. Fracture and turbulence -- no significant
theoretical progress.
2. Deficiencies of present physical models for
earthquake occurrence.
3. Phenomenology: fractal distributions of size, time,
space, and focal mechanisms.
4. Fractal model of earthquake process: random
stress interactions.
5. Statistical forecasting earthquakes and its testing
(more tomorrow at 12:00 in room 1707).
Two Major Unsolved Problems of
Modern Science
1. Turbulent flow of fluids (Navier-Stocks equations).
2. Brittle fracture of solids.
Plastic deformation of materials is an intermediate
case: it behaves as a solid for short-term interaction
and as a liquid for long-term interaction.
Kagan, Y. Y., 1992. Seismicity: Turbulence of solids,
Nonlinear Science Today, 2, 1-13.
Navier-Stokes Equation
“Waves follow our boat as we meander
across the lake, and turbulent air currents
follow our flight in a modern jet.
Mathematicians and physicists believe that an
explanation for and the prediction of both the
breeze and the turbulence can be found
through an understanding of solutions to the
Navier-Stokes equations. Although these
equations were written down in the 19-th
Century, our understanding of them remains
minimal. The challenge is to make substantial
progress toward a mathematical theory which
will unlock the secrets hidden in the NavierStokes equations” (Clay Institute -- one of
seven math millennium problems -- prize
$1,000,000).
Akiva Yaglom (2001, p. 4) commented that the turbulence
status is different from many other complex problems that
20-th century physics solved or was trying to solve:
"However, turbulence theory deals with the most ordinary
and simple realities of the everyday life such as, e.g., the jet
of water spurting from the kitchen tap."
Nevertheless, the turbulence problem is not among the ten
millennium problems in physics presented by University of
Michigan Ann Arbor, see
http://feynman.physics.lsa.umich.edu/strings2000/millennium.html
or 11 problems by the National Research Council's board on
physics and astronomy (Haseltine, Discover, 2002).
Horace Lamb on turbulence (1932):
"I am an old man now, and when I die and go to
Heaven there are two matters on which I hope for
enlightenment. One is quantum electrodynamics,
and the other is the turbulent motion of fluids. And
about the former I am really rather optimistic."
Goldstein, S., 1969. Fluid mechanics in the first half
of this century, Annual Rev. Fluid Mech., 1, p. 23.
This story is apocryphally repeated with Einstein,
von Neumann, Heisenberg, Feynman, and others.
Brittle Fracture of Solids
Similarly, brittle fracture of solids is commonly
encountered in everyday life, and still there is no
real theory explaining its properties or predicting
the outcome of the simplest occurrences, like
breaking a glass. It is certainly a more difficult
scientific problem than turbulence, and while the
turbulence attracted first-class mathematicians and
physicists, no such interest has been shown in
mathematical theory of fracture and large-scale
deformation of solids.
Seismicity model
This picture represent a paradigm
of the current earthquake physics.
Originally, when Burridge and
Knopoff proposed this model in
1967, this was the first
mathematical treatment of
earthquake rupture, a very
important development.
Since then perhaps hundreds
papers have been published using
this model or its variants.
Kagan, Y. Y., 1982.
Stochastic model
of earthquake fault
geometry,
Geophys. J. R. astr.
Soc., 71, 659-691
Current seismicity physical models
• Dieterich, JGR, 1994; Rice and Ben-Zion,
Proc. Nat. Acad., 1996; Langer et al., Proc. Nat.
Acad., 1996, see also review by Kanamori and
Brodsky, Rep. Prog. Phys., 2004 -- their major
paradigm: two blocks separated by a planar
boundary with friction.
Current seismicity physical models
• These models describe only one boundary between
blocks, they do not account for a complex interaction
of other block boundaries and, in particular, its triple
junctions. Seismic maps convincingly demonstrate
that earthquakes occur mostly at boundaries of
relatively rigid blocks. This is a major idea of the
plate tectonic. However, if blocks are rigid, stress
concentrations at other block boundaries and block's
triple junctions should influence earthquake pattern at
any particular boundary. Geometric strain
incompatibility is ignored.
Example of geometric incompatibility near fault junction. Corners A and
C are either converging and would overlap or are diverging; this
indicates that the movement cannot be realized without the change of the
fault geometry (Gabrielov, A., Keilis-Borok, V., and Jackson, D. D.,
1996. Geometric incompatibility in a fault system, P. Natl. Acad. Sci.
USA, 93, 3838-3842).
Current seismicity physical models
• No rigorous testing of these models is
performed. At the present time, numerical
earthquake models have shown no predictive
capability exceeding or comparable to the
empirical prediction based on earthquake
statistics. Confirming examples are selectively
chosen data. These models have a large
number of adjustable parameters, both obvious
and hidden, to simulate seismic activity. Math
used is at least 150 years old.
Earthquake Phenomenology
Modern earthquake catalogs include origin
time, hypocenter location, and second-rank
seismic moment tensor for each earthquake.
The tensor is symmetric, traceless, with zero
determinant: hence it has only four degrees of
freedom -- one for the norm of the tensor and
three for the 3-D orientation of the earthquake
focal mechanism. An earthquake occurrence is
considered to be a stochastic, tensor-valued,
multidimensional, point process.
Statistical studies of earthquake
catalogs -- time, size, space
• Catalogs are a major source of information on
earthquake occurrence.
• Since late 19-th century certain statistical
features were established: Omori (1894)
studied temporal distribution; Gutenberg &
Richter (1941; 1944) -- size distribution.
• Quantitative investigations of spatial patterns
started late (Kagan & Knopoff, 1980).
Statistical studies of earthquake
catalogs -- moment tensor
• Kostrov (1974) proposed that earthquake is
described by a second-rank tensor. Gilbert &
Dziewonski (1975) first obtained tensor
solution from seismograms.
• However, statistical investigations even now
remained largely restricted to time-size-space
regularities.
• Why? Statistical tensor analysis requires entry
to really modern mathematics.
(a) Fault-plane trace on a
surface. Earthquake rupture
starts at the hypocenter
(epicenter is the projection
of a hypocenter on the
Earth's surface), and
propagates with velocity
close to that of shear waves
(2.5--3.5 km/s).
(b) Double-couple source,
equivalent forces yield the
same displacement as the
extended fault rupture in a
far-field.
(c) Equal-area projection of
quadrupole radiation
patterns.
Earthquake Focal Mechanism
Double-couple tensor M = M diag [1, -1, 0] has 4
degrees of freedom, since its 1st and 3rd
invariants are zero. The normalized tensor
corresponds to a normalized quaternion
q = (0, 0, 0, 1). Arbitrary double-couple source
is obtained by multiplying the initial
quaternion by a quaternion representing a 3-D
rotation (see Kagan, GJI, 163(3), 1065-1072,
2005).
Using the Harvard CMT catalog of 15,015 shallow events:

( M )  ( M t / M ) exp[( M t  M ) / M c ]
Review of results on spectral slope, :
Although there are variations, none is significant with 95%-confidence.
Kagan’s [1999] hypothesis of uniform  still stands.
Relation between moment sums
and tectonic deformation
1. Now that we know the coupled thickness of
seismogenic lithosphere in each tectonic
setting, we can convert surface velocity
gradients to seismic moment rates.
2. Now that we know the frequency/magnitude
distribution in each tectonic setting, we can
convert seismic moment rates to earthquake
rate densities at any desired magnitude.
Kinematic
Model
Moment
Rates
Long-term-average
(Poissonian)
seismicity maps
Moment rate vs. tectonic rate
• Tapered Gutenberg-Richter distribution of
scalar seismic moment, survival function

( M )  ( M t / M ) exp[( M t  M ) / M c ]
By integrating the distribution of seismic moment
we obtain relation between seismic moment rate,
seismic activity rate, beta, and corner moment:


M s  0M 0  M
1 
c
(2   ) /(1   )
Kagan, GJI, 149, 731-754, 2002
Naïve summation of seismic
moment
If the exponent is less than 2.0, the sum of
power-law distributed variables
( M )  M  
converges to a stable distribution with pdf:
 (M ,  ,  ,  , )
where  is symmetry parameter,  ,  are shift
and width parameters, in the Gaussian distribution
they are only valid parameters.
Naïve summation of seismic
moment
• For small values of moment (M) in the G-R
tapered distribution, it behaves as a pure
power-law (Pareto) distribution
( M )  ( M t / M )

Then median (or any quantile) is proportional to
(N )  N
1/ 
hence
 (40)  2.8   (20)
Zaliapin, Kagan, and Schoenberg, PAGEOPH,
162(6-7), 1187-1228, 2005
Holt, W. E., Chamot-Rooke, N., Le Pichon, X., Haines, A. J.,
Shen-Tu, B., and Ren, J., 2000. Velocity field in Asia inferred
from Quaternary fault slip rates and Global Positioning
System observations, J. Geophys. Res., 105, 19,185-19,209.


  M seismic/ M tectonic
Sumatra M 9.1 earthquake
Temporal Earthquake Distribution
1
• Omori's (1894) law: n(t )  (t  c)
• Time shift c-coefficient is the result of
overlapping seismic records after large
earthquake and its strong aftershocks.
• Singularity at t=0 means that earthquake is a
cluster of events, these events resolution
depends on quality of seismographic network
and interpretation technique -- there is no
individual earthquake!
Spatial Distribution of Earthquakes
• We measure distances between pairs, triplets,
and quadruplets of events.
• The distribution of distances, triangle areas,
and tetrahedron volumes turns out to be fractal,
i.e., power-law.
• The power-law exponent depends on catalog
length, location errors, depth distribution of
earthquakes. All this makes statistical analysis
difficult.
Spatial moments:
TwoThree- and
Four-point functions;
Distribution of
distances (D), surface
areas (S), and volumes
(V) of point simplexes
is studied. The
probabilities are
approximately 1/D,
1/S, and 1/V.
New ms -- http://scec.ess.ucla.edu/~ykagan/p2rev_index.html
Kagan, Y. Y., 1992.
Correlations of
earthquake focal
mechanisms,
Geophys. J. Int., 110,
305-320.
• Upper picture -distance 0-50 km.
• Lower picture -distance 400-500 km.
Upper solid line -Cauchy distribution;
Dashed line - random
rotation.
Kagan, Y. Y., 2000. Temporal correlations of earthquake focal
mechanisms, Geophys. J. Int., 143, 881-897.
Branching model for dislocations
(Kagan and Knopoff, JGR,1981;
Kagan, GJRAS, 1982)
• Predates use of self-exciting, ETAS models
which also have branching structure.
• A more complex model, exists on more
fundamental level.
• Continuum-state critical branching random
walk in T x R3 x SO(3).
• Many unresolved claims, mathematical issues:
is the synthetic earthquake set scale-invariant?
Critical
branching
process -genealogical
tree of
simulations
(a) Pareto
distribution
of time
intervals
time^(1-u)
(b) Rotation of
focal
mechanisms
follows a
Cauchy
distribution
Simulated source-time functions and seismograms for shallow earthquake
sources. The upper trace is a synthetic cumulative source-time function. The
middle plot is a theoretical seismogram, and the lower trace is a convolution of
the derivative of source-time function with the theoretical seismogram.
Kagan, Y. Y., and Knopoff, L., 1981. Stochastic synthesis of
earthquake catalogs, J. Geophys. Res., 86, 2853-2862.
Kagan, Y. Y.,
and Knopoff,
L., 1987.
Random
stress and
earthquake
statistics:
Time
dependence,
Geophys. J. R.
astr. Soc., 88,
723-731.
Snapshots of fault
propagation. Rotation of
focal mechanisms is
modeled by the Cauchy
distribution. Integers in the
frames # indicate the
numbers of elementary
events to which these
frames correspond. Frames
show the development of
an earthquake sequence.
Normalized quaternions represent SO(3)
group of 3-D rotations, their multiplication
is non-commutative
q1  q2  q2  q1
Non-commutability of 3-D rotations presents a
major difficulty in creating probabilistic theory of
earthquake rupture propagation.
Simulation results:
• A model of random defect interaction in a
critical stress environment explains most of the
available empirical statistical results.
• Omori's law is a consequence of a Brownian
motion-like behavior of random stress due to
defect dynamics.
• The evolution and self-organization of defects
in the rock medium are responsible for the
fractal spatial patterns of earthquake faults
(Zolotarev, 1986; Kagan, 1990; 1994).
Earthquake Probability
Forecasting
• The fractal dimension of earthquake process is
lower than the embedding dimension:
• Time – 0.5 in 1D
• Space – 2.2 in 3D
• Focal mechanisms – Cauchy distribution
• This allows us to forecast probability of earthquake
occurrence – specify regions of high probability, use
temporal clustering for evaluating possibility of new
event and predict its focal mechanism.
Forecast example:
displayed
earthquakes
occurred after
smoothed
seismicity forecast
was calculated.
Forecast
effectiveness can be
evaluated by the
likelihood method
(Kagan and Jackson,
GJI, 143, 438-453,
2000).
Time history
of long-term
and shortterm forecast
for a point at
latitude
39.47 N.,
143.54 E.
northwest of
Honshu
Island, Japan.
Blue line is
the longterm forecast;
red line is
the shortterm forecast
(Jackson and
Kagan, SRL,
70, 393-403,
1999).
Kagan, Y. Y., and Knopoff, L., 1984. A stochastic
model of earthquake occurrence, Proc. 8-th Int.
Conf. Earthq. Eng., 1, 295-302.
WHY DOES THEORETICAL PHYSICS FAIL
TO EXPLAIN AND PREDICT EARTHQUAKE
OCCURRENCE?
•
1. There are major, perhaps fundamental difficulties in
creating a comprehensive physical/mathematical theory of
brittle fracture and earthquake rupture process.
•
2. However, the development of quantitative models of
earthquake occurrence needed to evaluate probabilistic
seismic hazard is within our reach.
•
3. It will require a combined effort of Earth scientists,
physicists, statisticians, as well as pure and applied
mathematicians.
End
Thank you
Conclusions
• The major theoretical challenge in describing
earthquake occurrence is to create scale-invariant
models of stochastic processes, and to describe
geometrical/topological and group-theoretical
properties of stochastic fractal tensor-valued fields
(stress/strain, earthquake focal mechanisms).
• It needs to be done in order to connect
phenomenological statistical results and attempts of
earthquake occurrence modeling with a non-linear
theory appropriate for large deformations.
• The statistical results can also be used to evaluate
seismic hazard and to reprocess earthquake catalog
data in order to decrease their uncertainties.
Observational results:
Earthquake process exhibits scale-invariant, fractal properties:
• (1) Earthquake size distribution is a power-law (GutenbergRichter) with an exponential tail. The power-law exponent has
a universal value for all earthquakes. The maximum (corner)
magnitude values are determined for major tectonic provinces.
• (2) The temporal fractal pattern is power-law decay of the rate
of the aftershock and foreshock occurrence (Omori's law).
Power-law time pattern can be extended to small time intervals
explaining the complex structure of the earthquake rupture
process.
• (3) Spatial distribution of earthquakes is fractal; the correlation
dimension of earthquake hypocenters is about 2.2 for shallow
earthquakes.
• (4) Disorientation of earthquake focal mechanisms is
approximated by the rotational 3-D Cauchy distribution.
Southern California earthquakes
1800-2005
Blue -- focal
mechanisms
determined.
Orange -estimated
through
interpolation
Simulation results:
• The Cauchy and other symmetric stable
distributions govern the stress caused by these
defects (Zolotarev, 1986; Kagan, 1990; 1994).
• Random rotation of focal mechanisms is
controlled by the rotational Cauchy and other
stable distributions.
Distribution of distances between hypocenters N(R,t) for the Hauksson & Shearer
(2005) catalog, using only earthquake pairs with inter-event times in the range
[t, 1.25t]. Time interval t increases between 1.4 minutes (blue curve) to 2500 days
(red curve). See Helmstetter, Kagan & Jackson (JGR, 2005).