Download 18 Earthquake Prediction

Forecasting Earthquakes
Lecture 18
Earthquake Prediction
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The meaning of uncertainty
ß Epistemic uncertainty
a Lack of knowledge
z 18th century classical determinism lack of knowledge
was a deficiency which might be remedied by further
learning and experiment
z It is this lack of knowledge which the insurance industry
tries to address
z But we know now there is an intrinsic uncertainty, over
and above our lack of knowledge, e.g. quantum
mechanics, dynamical chaos, etc.
ß Aleatory uncertainty
a Uncertainty associated with randomness
z Named after Latin for dice
z Aleatory uncertainty can be better estimated, but it
cannot be reduced by through advances in theory or
observation
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Different types of probability
ß Our old friend Harold Jeffreys: tossing a coin
a The probability of a head pH depends on the properties of
the coin and is unknown with a prior distribution
a Estimate pH from results of tosses: epistemic probabilities
a For instance they may be an epistemic probability of 0.7
that the aleatory probability pH lies between 0.4 and 0.6
a With repeated tosses the epistemic probability will be
reduced, but the aleatory probability is an inherent
property of the coin can it won’t change
ß For earthquake faults aleatory uncertainty arising
party from the erratic nature of the fault rupture
ß There is an epistemic uncertainty because we don’t
know where all the faults are (yet?)
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Forecasting earthquakes
ß Parkfield project
In 1983 the USGS predicted
that there would a 5.5-6 mag
earthquake at Parkfield in
1988+/- 5 years
ß Loma Prieta earthquake
17.10.89, caused $6bn damage
and killed 68 people
USGS promptly claimed to have
predicted it
ß Forecast map
But the uncertainties in the
estimation of the mean
recurrence time are so large to
make the 1988 map “virtually
meaningless”.
Forecasted probabilities of occurrence of
California earthquakes as endorsed by
the NEPEC IN 1988
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“Forecasted but not predicted”
ß Loma Prieta
Some USGS scientists had published
a “speculation”, not a formal
prediction that an earthquake would
occur at Calaveras Reservoir.
A Loma Prieta foreshock was
ironically found afterwards on the
map containing the flawed prediction.
So the claim that Loma Prieta was
predicted is not true.
The claim that it was forecast in the
statistical sense of the hazard map is
pretty shaky.
But the claim remains that it was
“Forecast but not predicted”.
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Are earthquakes predictable?
ß
ß
ß
ß
ß
Many geophysicists believe that earthquake prediction is
hopeless or plain wrong
These ideas have been jumped on by engineers to ignore
trying to close the knowledge gap
But predicting from local geology that the damage in San
Francisco due to an earthquake in the Marina and at the
Nimitz Freeway is a prediction
So prediction or forecasting must still have an important part
to play in earthquake hazard mitigation: seismologists can and
must predict how earthquakes can affect particular structures
in specific locations
The failure of the Tokai and Parkfield earthquake prediction
programmes clearly has dented or destroyed the old ideas of
predicting earthquakes – but this does not negate the need to
look for what we can predict
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Earthquake prediction
ß We have to answer 4 question:
1.
2.
3.
4.
Where?
How often?
How big?
When?
ß Earthquake prediction can be split into two types:
1.
Statistical prediction (background seismic hazard)
based on previous events and likely future recurrence –
uses instrumental catalogue, archaeological record,
geological (Quaternary) record
2.
Deterministic prediction
the place, magnitude and time of a future event from
observation of earthquake precursors
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Where?
ß Earthquakes occur because of slip
on active faults
These can be found from Quaternary mapping
where faults break the surface of from
seismicity (instrumental or historical)
But note many active faults are only
identified after the earthquake!
N-S normal fault on the Rhine rift
ß Plate tectonics
is only useful on a large
scale
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How often?
ß Palaeoseismology
Geological investigation of active faults (palaeoseismology) can reveal 2
important constraints on average recurrence interval of past events:
ß
Tectonic slip rate
from lithological offset
or plate tectonics (upper bound)
A stream channel offset by the San Andreas fault,
Carrizo Plain, (photo by Robert E. Wallace)
ß
right lateral displacement
Trenching
reveals a section of the recent fault activity
contained in recent sediments (requires
rapid sedimentation from a stream
crossing the fault (e.g., Pallet Creek)
and shows 140 yrs between major
earthquakes on San Andreas – varies
between 50-300 years
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How often?
ß
Trenching
Trenching has revealed that
earthquake recurrence is
irregular. However the average
recurrence can be reconstructed
to evaluate the background
hazard (statistical prediction)
This information can be used to construct:
ß
Hayward Fault, California
Frequency-magnitude statistics
shows synthesis of instrumental, slip rate and
average recurrence from
palaeoseismology for southern California
The slope on the log-linear plot
log N = a – b m
Note how well-correlated the 3 data sets are,
justifying any statistical prediction based
on a continuation of past behaviour
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How often?
ß
Probabilistic prediction
The probability increases with time
most rapidly in subduction zones,
slowest in intraplate zones
probability
for SAF at Pallet Creek
cumualative
discrete
ß
50
140
300
recurrence time
Average recurrence intervals
20-30 yr
Circumpacific subduction
100-200 yr
San Andreas transform
1000-10000 yr Intraplate
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How big?
ß Size or previous event
a Magnitude, intensity, extent of fault breach
a Particularly “characteristic earthquake”
Most predictions of the size of a future event are based on
past observation
ß Seismic gap
Subduction earthquakes gradually filled the
Japan-Kurile arc with aftershocks, leaving
gap which was filled by 1973 earthquake
The earthquake magnitude was predicted by the
size of the gap
Mw = 2/3 (log10 M0 – 9)
[M0 in Nm]
M0 = µ A s
A= l x w
s/w = 10-4 (strain drop equivalent to 30
bar stress drop)
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ß
How big?
Fault segmentation
The Anatolian fault has ruptured this century in welldocumented segments, like the San Andreas
Not only that the individual fault breaks migrate along the
fault, so that the whole fault is eventually broken in
sequence
NB seismic gap south of Istanbul
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How big?
ß
ß
Faults are segmented by bends,en-echelon offset, and
variations in frictional strength.
There lead to zones of local compression (asperities) and
tension (fault jogs).
Parkfield
GEOMETRY
asperity
jog
20 km
(Exercise: 20km x
10km x 10-4 [shear
stress] ≈ 6.5 mag)
SHEAR STRESS
The asperity represents an increase in rock strength and
must be broken before slip can occur on the segment
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How big?
ß
The fault jog represents a ‘shatter zone’ of dispersed fracture which
stops the earthquake by absorbing energy
further extension resisted by:
ß
a)
suction of fluids filling fractures (e.g.,
capillary force)
b)
further distributed cracking
The fault jog may not be observable if the fracture at depth does not
reach the surface, but may see:
EN-ECHELON OFFSET
zones of distributed deformation
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How big?
ß Summary – earthquake magnitude
Subduction
Cont. transform
Active intraplate
Oceanic ridge
Moderate intraplate
Continental cratons
8< Mw <10
Mw ∼ 8
Mw ∼ 7
Mw ∼ 6
Mw ∼ 5-6
Mw ∼ 5
(e.g, Chile 1960)
(e.g. San Andreas )
(e.g. New Madrid)
(UK, N. Sea)
(Antarctic 4.5)
N.B. These are related to (a) the width of the seismogenic zone, and (b)
the rate of tectonic activity
The smallest fault capable of breaking the surface is about M6
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Statistical distribution
a
log10 N = a - b M
z N - number of
earthquakes in
magnitude range
z M - earthquake
magnitude
z Seismic b-value
defines log-linear
distribution
100000
Number of Events
Gutenberg-Richter magnitudefrequency distribution:
10000
1000
100
10
1
3
4
5
6
7
Magnitude
Seismic b-value
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8
Poisson statistics
•Some events are rather rare , they don't happen that often. For instance,
car accidents are the exception rather than the rule. Still, over a period of
time, we can say something about the nature of rare events.
The Poisson distribution was derived by the French mathematician
Poisson in 1837, and the first application was the description of the
number of death by horse kicking in the Prussian army.
The Poisson distribution is a mathematical rule that assigns probabilities
to the number occurrences. The only thing we have to know to specify
the Poisson distribution is the mean number of occurrences.
The Poisson distribution resembles the binomial distribution if the
probability of an accident is very small .
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Earthquakes as Poisson process
Basis of linear treatment of earthquake risk as
stochastic process - randomness
a Earthquakes are independent
a Seismicity is stationary
a Earthquakes can’t be simultaneous
ß Use instrumental / historic catalogues
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(i) Independence
ß
Pr[A|B] = Pr[A]
where A and B are any two events in the process.
That is to say the probability of A occurring given B occurring
is equal to the probability of just A occurring.
In other words it makes no difference whether any other
event B occurs or not – much less when it occurs, how
large it is and so on.
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(ii) Stationarity
The probability of exactly 1 event occurring in this short interval of
length ∆t is equal to λ.∆t, proportional to the length of the
interval.
λ is the rate of the process.
(iii) Orderliness
In a sufficiently short length of time, ∆t, only 0 or 1 event can occur.
(Simultaneous events are impossible.)
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Poisson Process
Any process showing independence, stationarity &
orderliness is a Poisson process.
But any Poisson-distribution has not necessarily been
generated by a Poisson process.
A Poisson process can result from random operations
performed on a set of non-Poisson processes. It is a
limiting case to which other point processes
converge in a statistical sense. Palm-Khinchin
Theorem.
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Poisson model - discrete case
If have a Poisson process, N is the number of events in
time, t, λ is the rate, then the probability function for
N is:
(
λt )
( x) =
x
Pr[ N = x ] = p N
x!
e
− λt
x = 0,1, 2,...
N is a Poisson random variable with parameter,
Poisson distribution λ=3
µ = λt.
E[N] = µ
Number of earthquakes in time t
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Poisson model - continuous
If T is elapsed time till the first event occurs then T has exponential
probability density function:
f T (t ) = λ e
E[T] = 1/λ
− λt
,
t>0
Mean interval between earthquakes
Continuous Poisson distribution
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Gutenberg-Richter magnitude-frequency
distribution
log10 N = a - b M
a Empirical distribution
a Set β = b ln 10 ≅ 2.3 b
Number of Events
100000
10000
1000
100
10
1
3
can re-write as:
4
5
6
7
Magnitude
fM(x) = β e-βx
Exercise for the student
This is exponential pdf for Poisson process
β-1 estimates mean magnitude
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8
Scale invariance of nature
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