Download Global earthquake prediction experiments

Yan Y. Kagan
Dept. Earth and Space Sciences, UCLA, Los Angeles,
CA 90095-1567, [email protected],
http://scec.ess.ucla.edu/ykagan.html
GLOBAL EARTHQUAKE PREDICTION
EXPERIMENTS
http://moho.ess.ucla.edu/~kagan/csep06.ppt
Earthquake Occurrence –
Stochastic Process
• Multidimensional marked (seismic moment, M)
(or tensor-valued) point process in time, space,
and 3-D orientation of double-couple focal
mechanism -T x R3 x SO(3).
• All the marginal distributions are power-law, i.e.,
they are fractal, heavy-tailed, stable (L-stable)
distributions.
NEW STATISTICS – STABLE
DISTRIBUTIONS
Statistics development over last two centuries has been
dominated by the Gaussian distribution – a special
case of stable laws. Other stable distributions have
completely different properties: mean, standard
deviation, coefficient of correlation either does not
exist or behaves erratically. The stable distributions
theory started to be developed only 20 years ago and
there are not enough statisticians who sufficiently
know it.
Zaliapin, Kagan, Schoenberg, PAGEOPH, 162, 2005.
DIFFERENTIAL GEOMETRY,
GROUP THEORY
• Earthquake focal mechanisms sometimes undergo
large random 3-D rotations. To describe a set of focal
mechanisms we need to have a statistical theory of 3D rotations. The group SO(3) is non-commutative
(the sum of two rotations depends on its order), its
theory is very complicated. We need to involve
mathematicians to develop methods appropriate to
handling random rotations and random tensors.
See more in
http://scec.ess.ucla.edu/~ykagan/india_index.html
Earthquake size distribution
Tapered Gutenberg-Richter distribution of
scalar seismic moment, survival function

( M )  ( M t / M ) exp[( M t  M ) / M c ]
M t Moment threshold
 Exponent, similar to b-value   b / 1.5
M c Corner moment
Bird and Kagan, BSSA, 94(6), 2380-2399, 2004.
Review of results on spectral slope, :
Although there are variations, none is significant with 95%-confidence.
Kagan’s [1999] hypothesis of uniform  still stands.
Moment rate vs. tectonic rate
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;
Bird and Kagan, BSSA, 94(6), 2380-2399, 2004;
Boettcher & Jordan, JGR, 109(B12), B12302, 2004.
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
De Mare, J., Optimal
prediction of catastrophes
with applications to
Gaussian processes,
Ann. Probability, 8,
841-850, 1980.
Molchan, G. M., and
Y. Y. Kagan, 1992.
Earthquake prediction
and its optimization,
J. Geophys. Res., 97,
4823-4838.
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.
(a) Earthquake
catalog data
Phenomenological models:
(b) Point process
branching along
magnitude axis,
introduced by
Kagan (1973a;b)
(c) Point process
branching along time
axis (Hawkes, 1971;
Kagan & Knopoff,
1987; Ogata, 1988)
Long-term
forecast:
1977-today
Spatial smoothing
kernel is optimized
by using the first,
temporal, part of a
catalog to forecast
its second part.
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 longterm forecast;
red line is
the hybrid
forecast.
Short-term
forecast uses
Omori's law to
extrapolate
present seismicity.
Red spot north of
Kamchatka is a
recent (2006/4/20)
M7.6 earthquake
Kagan, Y. Y.,
D. D. Jackson,
and Y. F.
Rong, 2006.
A testable
five-year
forecast of
moderate and
large
earthquakes in
southern
California
based on
smoothed
seismicity,
Seism. Res.
Lett.,
submitted.
Helmstetter, A., Y. Y. Kagan, and D. D. Jackson, 2006.
Comparison of short-term and time-independent earthquake
forecast models for southern California, Bull. Seismol. Soc.
Amer., 96(1), 90-106.
Here we
demonstrate
forecast
effectiveness:
displayed
earthquakes
occurred after
smoothed
seismicity
forecast
was calculated.
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 2.5—
97.5% 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.
Molchan, G. M., and Y. Y. Kagan, 1992.
Earthquake prediction and its optimization,
J. Geophys. Res., 97, 4823-4838.
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,
Tectonophysics, pp.
33-38.
Likelihood ratio – information/eq
We approximate earthquake occurrence by Poisson
cluster process and calculate the earthquake rate


 
( Mi )
(t , x, M )   ( x, M )   (t  ui , x  yi , M )
Likelihood function is
i


l     (t , x , M  )dtdx dM 

t , x ,M


ln

(
t
,
x
,
M

)
dN
(
t
,
x
,
M
)


t , x ,M
Likelihood ratio – information/eq
Similarly we obtain likelihood function for the null
hypothesis model (Poisson process in time) l0
Information content of a catalog :
I  (l  l0 ) /( N ln 2)
(bits/earthquake)
characterizes uncertainty reduction by use of a
particular model.
Kagan and Knopoff, PEPI, 1977; Kagan, GJI, 1991;
Kagan and Jackson, GJI, 2000; Helmstetter, Kagan
and Jackson, BSSA, 2006
Likelihood ratio – information/eq
• Because of power-law dependence of earthquake rate
on time (Omori’s law), the likelihood function (l)
approaches infinity when the prediction horizon is
close to zero (i.e., for real-time earthquake
prediction).
• Similarly, when spatial resolution of earthquake data
increases, (l) again goes to infinity.
• Molchan diagram also approaches the ideal state
(  0;  0) for the prediction horizon close to zero,
or if the location accuracy significantly increases.
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
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).
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.
Snapshots of fault
propagation (intersection
with a plane). 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.
Kagan, Y. Y., and Knopoff, L., 1984. A stochastic
model of earthquake occurrence, Proc. 8-th Int.
Conf. Earthq. Eng., 1, 295-302.
Conclusions
• Using universality of the distribution of
earthquake size (seismic moment) we
quantitatively describe plate tectonic
deformation end evaluate long-term earthquake
rate.
• Time-independent seismic hazard is evaluated
by optimized smoothing of past seismicity.
Conclusions
• Temporal earthquake occurrence is modeled
and forecasted by a multidimensional branching
process.
• Finally, kinematic continuum-state critical
branching process can be used to generate
whole earthquake rupture process and obtain a
randomized set of seismograms – extrapolation
of the process.
End
Thank you
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.
No probability – no prediction