Why does theoretical physics fail to explain and predict earthquake occurrence?

1
Yan Y. Kagan Dept. Earth and Space Sciences,
UCLA, Los Angeles, CA 90095-1567,
ykagan_at_ucla.edu, http//scec.ess.ucla.edu/ykagan.h
tml
WHY DOES THEORETICAL PHYSICS FAIL TO EXPLAIN AND
PREDICT EARTHQUAKE OCCURRENCE?
http//scec.ess.ucla.edu/ykagan/india_index.html
2
Several reasons can be proposed 1.
Multidimensional character of seismicity
earthquake time, space, size, and focal mechanism
need to be modeled. The latter is a symmetric
second-rank tensor of special kind. 2. Intrinsic
randomness of the earthquake occurrence, the need
to apply the theory of stochastic point processes
and appropriate complex statistical techniques.
3
3. Scale-invariant or fractal properties of
earthquake process the theory of random stable
or heavy-tailed variables is significantly more
difficult than that of Gaussian variables and is
only being currently developed. The earthquake
process theory should be renormalizable. 4.
Statistical distributions of earthquake size,
earthquake temporal interaction, their spatial
patterns and focal mechanisms are largely
universal with the values of major parameters
similar for earthquakes in various tectonic
zones. The universality of these distributions
gives us some hope in creating foundations for
earthquake process theory.
4
5. The quality of current earthquake data
statistical analysis is low little or no study
of random and systematic errors is usually
performed, thus most of the published statistical
results are artifacts. 6. Earthquake process
does not operate as an isolated system.
Scale-invariance of the process means that the
largest inhomogeneity of earthquake fault system
(defect) is comparable in size with the
region extent, same for the next largest defect,
etc. Thus, in any seismic region we cannot
separate the largest fault or fault system and
treat the remainder as medium with uniform
properties.
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7. During earthquake rupture propagation focal
mechanisms sometimes undergo large 3-D rotations.
This necessitates applying non-commutative algebra
(e.g., quaternions and gauge theory) to model
earthquake occurrence. Almost all field theories
currently employed in physics are based on
complex numbers, i.e., use commutative
algebra. 8. These phenomenological and
theoretical difficulties are not limited to
earthquakes any fracture of brittle materials,
tensile or shear, would encounter similar
problems.
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Lecture Outline

• 1. Deficiencies of present physical models.
• 2. Earthquake occurrence phenomenology fractal
  distributions of size, time, space, and focal
  mechanisms.
• 3. Fractal model of earthquake process random
  stress interactions.
• 4. Statistical forecasting of earthquakes and its
  testing.

7
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.

8
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
Navier-Stokes equations. (Clay Institute --
prize 1 M).
9
Akiva Yaglom (2001, p. 4) commented that the
turbulence status is different from many other
complex problems that 20-th century physics
solved or have been 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.ed
u/strings2000/millennium.html or 11 problems by
the National Research Council's Board on Physics
and Astronomy (Haseltine, Discover, 2002).
10
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.
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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 much 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.
Kagan, Y. Y., 1992. Seismicity Turbulence of
solids, Nonlinear Science Today, 2, 1-13.
12
Columbia Accident Investigation Board, Report,
Volume I, 2003, p. 82
13
O'Brien, J. F., and Hodgins, J. K.
1999. Graphical modeling and animation of brittle
fracture, Proceedings of Assoc. Computing
Machinery (ACM) SIGGRAPH 99, 137-146.
14
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.

15
Seismicity model

• Since then perhaps hundreds papers have been
  published using this model or its variants. We
  show below that presently much more complicated
  mathematical tools are needed to represent
  brittle shear earthquake fracture.

16
The model must be modernized. Why?

• 1. It is a closed, isolated system, whereas
  tectonic earthquakes occur in an open
  environment. This model justifies spurious
  quasi-periodicity, seismic gaps, and seismic
  cycle models.
• No rigorous observational evidence exists for
  the presence of these features (Rong et al., JGR,
  2003).

17
The model must be modernized. Why?

• 2. Earthquake fault in the model is a
  well-defined geometrical object -- a planar
  surface with dimension 2. In nature only
  earthquake fault system exists as a fractal set.
  This set is not a surface, its dimension is about
  2.2.

18
Kagan, Y. Y., 1982. Stochastic model of
earthquake fault geometry, Geophys. J. R. astr.
Soc., 71, 659-691
19
The model must be modernized. Why?

• 3. Several distinct scales are present in the
  diagram inhomogeneity of planar surface, plates
  size. Earthquakes are scale-invariant. Geometry
  and mechanical properties of the earthquake fault
  zone are the result of self-organization. They
  are fractal.

20
The model must be modernized. Why?

• 4. Incompatibility problem is circumvented
  because of flat plate boundaries. Real earthquake
  faults always contain triple junctions further
  deformation is impossible without creating new
  fractures and rotational defects (disclinations).

21
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).
22
The model must be modernized. Why?

• 5. Because of planar boundaries, stress
  concentrations are practically absent after a
  major earthquake. Hence few or no aftershocks.

Kagan and Knopoff (JGR,1981) Kagan (GJRAS, 1982)
proposed a new model of earthquake occurrence,
based on fractal geometry and 3-D rotation of
mechanisms.
23
The model must be modernized. Why?

• 6. All earthquakes in the model have the same
  focal mechanism. Variations of mechanisms that
  are obvious even during a cursory inspection of
  focal mechanism maps are not taken into account.

Kagan (2000). Temporal correlations of earthquake
focal mechanisms, Geophys. J. Int., 143, 881-897.
24
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Schwartz and Coppersmith (1984) proposed the
  characteristic model. McCann et al. (1979) and
  Nishenko (1991) formulated testable hypothesis --
  about 100 zones in circum-Pacific belt. Kagan
  Jackson (JGR, 1991, 1995), and Rong et al. (JGR,
  2003) tested these predictions and found that
  earthquakes after 1979 or 1989, respectively, do
  not support the model.

25
McCann et al. (1979) The map of seismic gap zones
-- compare Sumatra 2004 rupture. Kagan Jackson
(1991) tested the map the result is negative.
26
Characteristic Earthquakes, Seismic gaps,
Quasi-periodicity

• Bakun Lindh (Science, 1985) -- Parkfield
  prediction, 95 probability of M6 event in
  1985-1993. No earthquake occurred. In 2004 M6
  event in the Parkfield area. Only few of the
  predicted features were observed. Bakun et al.
  (Nature, 2005) review the experiment results --
  no new prediction is issued. See Jackson Kagan
  (2006) http//scec.ess.ucla.edu/
  ykagan/parkf2004_index.html.

27
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Chris Scholz in the 1999 Nature Debate
• In their Kagan Jackson more recent study,
  they found, in contrast, less events than
  predicted by Nishenko 1991. But here the
  failure was in a different part of the physics
  the assumptions of recurrence times made by
  Nishenko. These recurrence times are based on
  very little data, no theory, and are
  unquestionably suspect.

28
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Bakun et al., Nature, 2005
• (The characteristic earthquake model can also
• be tested using global data sets. Kagan and
  Jackson 1995 concluded that too few of
  Nishenkos 1991 predicted gap-filling
  circum-Pacific earthquakes occurred in the first
  5 yr.)

29
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Despite the failure of these predictions, this
  model was employed in San Francisco Bay area
  (Working Group, 2003) -- "there is a 0.62
  0.38--0.85 probability of a major, damaging M
  gt 6.7 earthquake striking the greater San
  Francisco Bay Region over the next 30 years
  (2002--2031)."

30
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Stark and Freedman (2003) argue that the
  probabilities defined in such a prediction are
  meaningless because they cannot be validated.
  They suggest that the reader "should largely
  ignore the USGS probability forecast."
• See more detail in Jackson Kagan (2006)
  http//scec.ess.ucla.edu/ykagan/parkf2004_index.h
  tml

31
Characteristic Earthquakes, Seismic Gaps,
Quasi-periodicity

• Thomas Kuhn (1965) questioned how one can
  distinguish scientific and non-scientific
  predictions. As an example, he used astronomy
  versus astrology -- both issue predictions that
  sometimes fail. However, astronomers learn from
  these mistakes, modify and update their models,
  whereas astrologers do not.

32
Current Physical Models of Seismicity

• 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.

33
Current Physical Models of Seismicity

• 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 the major idea of the plate tectonics.
  However, if blocks are rigid, stress
  concentrations at other block boundaries and
  blocks triple junctions should influence the
  earthquake pattern at any particular boundary.

34
Current Physical Models of Seismicity

• 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. Examples are selectively
  chosen data. These models have a large number of
  adjustable parameters, both obvious and hidden,
  to simulate seismic activity.

35
Current seismicity physical models Enrico
Fermi I remember my friend Johnny von Neumann
used to say, with four parameters I can fit an
elephant, and with five I can make him wiggle his
trunk. Dyson, 2004. Nature, 427(6972), 297.
36
Current Physical Models of Seismicity

• The models 'successfully' claim as their
  consequence some artifacts or non-existent
  phenomena, such as characteristic earthquakes or
  non-zero value for the c-coefficient in Omori's
  law.
• They are less successful in solving
  pseudo-problems of their own creation, such as
  heat flow paradox or strength of earthquake
  faults.

37
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.
38
Southern California earthquake epicenters black
-- wave-correlation method color standard
(first arrival) method. Accuracy of the first
method is tens of m, of the second about 1 km.
39
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).

40
Southern California earthquakes 1800-2005
Blue -- focal mechanisms determined. Orange --
Mechanisms estimated through interpolation
41
Statistical Studies of Earthquake Catalogs
moment tensor

• Kostrov (1974) proposed that earthquake can be
  described by a second-rank tensor. Gilbert
  Dziewonski (1975) first obtained the tensor
  solution from seismograms.
• However, statistical investigations even now
  remain largely restricted to time-size-space
  regularities.
• Why? Statistical tensor analysis requires entry
  to truly modern mathematics.

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43
Observational Results
Earthquake process exhibits scale-invariant,
fractal properties

• (1) Earthquake size distribution is a power-law
  (Gutenberg-Richter) 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.

44
Using the Harvard CMT catalog of 15,015 shallow
events
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Review of results on spectral slope, b
Although there are variations, none is
significant with 95-confidence. Kagans 1999
hypothesis of uniform b still stands.
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49
IMPLICATIONS

1. Now that we know the coupled thickness of the
   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.

Long-term-average (Poissonian) seismicity maps
Kinematic Model
Moment Rates
50
Relation Between Moment Sums and Tectonic
Deformation

• From the beginning of the plate tectonics
  hypothesis it was assumed that earthquakes are
  due to plate boundaries deformation.
• Calculations for global tectonics and large
  seismic regions justified such approach.
• However, application of this assumption to
  smaller regions has usually been inconclusive due
  to high variability of seismic moment sums.

51
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.
52
Zaliapin, I. V., Y. Y. Kagan, and F. Schoenberg,
2005. Approximating the distribution of Pareto
sums, Pure Appl. Geoph. (PAGEOPH), 162(6-7),
1187-1228.
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54
Non-linear increase of sums of heavy-tailed
distributions may explain accelerated moment or
Benioff strain release (Bufe Varnes, 1993, and
many others)
.
55
Sumatra M 9.1 earthquake
56
Cross -- Harvard catalog Blue stars -- PDE
catalog Green dots -- CalTech catalog.
57
Kagan, Y. Y., and H. Houston, 2005. Relation
between mainshock rupture process and Omori's law
for aftershock moment release rate, Geophys. J.
Int., 163(3), 1039-1048
58
Earthquake Scaling M L3

• It is commonly believed that the earthquake focal
  size scaling (i.e., dependence of the size on
  seismic moment) is different for events of
  various focal mechanisms.
• In particular, strike-slip earthquakes which
  occur on vertical faults are considered to have
  two power-law dependencies break occurring
  around 15-20 km (corresponding to M6 event).
• Long debate between Scholz and Romanowicz.

59
Aftershock map for 1997/12/5 Kamchatka
earthquake (M 5.321020 Nm, m 7.82). Stars --
aftershock epicenters, cross -- average
aftershock position, circle -- centroid vertical
projection onto the Earth surface, diamond --
NEIC mainshock epicenter. Two-dimensional
Gaussian approximation of aftershock distribution
is shown.
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Earthquakes 1977-2005 Mw (CMT), aftershocks (PDE)
62
Earthquakes 1977-2004
63
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
  very difficult.

64
Spatial Moments Two- Three- 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.
65
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).
66
New ms -- http//scec.ess.ucla.edu/ykagan/p2rev_i
ndex.html
67
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).

68

• 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.

69
Kagan, Y. Y., 2000. Temporal correlations of
earthquake focal mechanisms, Geophys. J. Int.,
143, 881-897.
70
Stress and Earthquakes

• Stress due to past earthquakes can be calculated
  with reasonable accuracy.
• Evaluation of tectonic stress is more difficult,
  especially for smaller scales.
• Incremental stress is found to be distributed
  according to Cauchy distribution, the behavior
  predicted by Zolotarev (1983).
• Rotation of focal mechanisms is governed by
  Cauchy distribution.

71
Focal mechanisms of earthquakes for 1850-2003 in
the southern California area and major surface
faults. Lower hemisphere diagrams of focal
spheres are shown the diagrams can be thought of
as 3-D rotations of the mechanism. All events
with magnitude m gt 6.5 are replaced by extended
sources, containing several smaller rectangular
dislocation patches matching total earthquake
moment. Kagan, Jackson, and Liu, 2005. Stress and
earthquakes in southern California, 1850-2004, J.
Geophys. Res., 110(5), B05S14
72
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 a 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?

73
Critical branching process --genealogical tree of
simulations
74
(a) Pareto distribution of time intervals
time(1-u) (b) Rotation of focal mechanisms
follows a Cauchy distribution
75
Simulated source-time functions and seismograms
for shallow earthquake sources. The upper trace
is a synthetic 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.
76
Kagan, Y. Y., and Knopoff, L., 1987. Random
stress and earthquake statistics Time
dependence, Geophys. J. R. astr. Soc., 88,
723-731.
77
Snapshots of a fault propagation, simulated
fault. Integers in the frames indicate the
numbers of elementary events to which these
frames correspond. Ten frames on the left show
the development of an earthquake sequence in the
whole area. The frames on the right are
magnifications of the parts enclosed in
rectangles.
78
Non-commutability of 3-D rotations presents a
major difficulty in creating probabilistic theory
of earthquake rupture propagation
79
Hamilton's letter to his son

• Every morning in the early part of the
  above-cited month October 1843, on my coming
  down to breakfast, your (then) little brother
  William Edwin, and yourself, used to ask me
  "Well, Papa, can you multiply triplets?" Whereto
  I was always obliged to reply, with a sad shake
  of the head No, I can only add and subtract
  them".

80
Brougham Bridge, Dublin
Here as he walked by on the 16th of October
1843Sir William Rowan Hamilton in a flash of
genius discoveredthe fundamental formula for
quaternion multiplication cut it on a stone
of this bridge.
81
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.
82
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.

83
Simulation Results

• The Cauchy and other symmetric stable
  distributions govern the stress caused by these
  defects.
• Random rotation of focal mechanisms is
  controlled by the rotational Cauchy and other
  stable distributions.
• Orientation of these dislocations is defined by
  a normalized quaternion (3 DF), but they
  determine the random stress (6 DF).

84
Probabilistic vs. alarm forecasts

• A modern scientific earthquake forecast should
  be quantitatively probabilistic. In 1654 Pierre
  de Fermat and Blaise Pascal exchanged letters in
  which they founded the quantitative probability
  theory. Now more than 350 years later, any
  earthquake forecast without direct use of
  probability has a medieval flavor. This is
  perhaps the reason the general public and media
  are so attracted to yes/no forecasts.

85
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.

86
(a) Earthquake catalog data
(b) Point process branching along magnitude axis,
introduced by Kagan (1973ab)
(c) Point process branching along time axis
(Hawkes, 1971 Kagan Knopoff, 1987 Ogata, 1988)
87
Here we demonstrate forecast effectiveness dis
played earthquakes occurred after smoothed
seismicity forecast was calculated.
88
Time history of long-term and hybrid (short-term
plus 0.8 long-term) forecast for a point at
latitude 39.47 N., 143.54 E. northwest of Honshu
Island, Japan. Blue line is the long-term
forecast red line is the hybrid forecast.
89
Short-term forecast uses Omori's law
to extrapolate present seismicity. Red spot east
of Honshu Island is the consequence of the M7
2005/11/14 earthquake
90
Forecast Efficiency Evaluation

• We simulate synthetic catalogs using smoothed
  seismicity map.
• Likelihood function for simulated catalogs and
  for real earthquakes in the time period of
  forecast is computed.
• If the real earthquakes likelihood value is
  within 2.597.5 of synthetic distribution, the
  forecast is considered successful.
• Kagan, Y. Y., and D. D. Jackson, 2000.
  Probabilistic forecasting of earthquakes, (Leon
  Knopoff's Festschrift), Geophys. J. Int., 143,
  438-453.

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92
Kossobokov, 2006. Testing earthquake prediction
methods The West Pacific short-term forecast
of earthquakes with magnitude MwHRV \ge 5.8",
Tectonophysics, 413(1-2), 25-31. See also Kagan
Jackson, pp. 33-38.
93
Molchan, G. M., and Y. Y. Kagan, 1992.
Earthquake prediction and its optimization, J.
Geophys. Res., 97, 4823-4838.
94
Kagan, Y. Y., and Knopoff, L., 1984. A stochastic
model of earthquake occurrence, Proc. 8-th Int.
Conf. Earthq. Eng., 1, 295-302.
95
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
  at 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.

96
End Thank you
97
Earthquake Probability Forecasting

• The fractal dimension of earthquake process is
  lower than the embedding dimension
• Space 2.2 in 3D
• Time 0.5 in 1D
• This allows us to forecast probability of
  earthquake occurrence specify regions of high
  probability and use temporal clustering to
  evaluate the possibility of a new event.