Forecasting Earthquakes

Lecture 18

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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?)

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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 Forecasted probabilities of occurrence of recurrence time are so large to California earthquakes as endorsed by make the 1988 map “virtually the NEPEC IN 1988 meaningless”.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD “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”.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Are earthquakes predictable?

ß Many geophysicists believe that earthquake prediction is hopeless or plain wrong ß These ideas have been jumped on by engineers to ignore trying to close the knowledge gap ß 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 ß 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 ß 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Earthquake prediction

ß We have to answer 4 question: 1. Where? 2. How often? 3. How big? 4. 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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 GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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) right lateral displacement ß 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 GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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: Hayward Fault, California

ß 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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 Average recurrence intervals 20-30 yr Circumpacific subduction 50140 300 100-200 yr San Andreas transform recurrence time 1000-10000 yr Intraplate

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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)

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD ß Fault segmentation How big? The Anatolian fault has ruptured this century in well- documented 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD How big? ß Faults are segmented by bends,en-echelon offset, and variations in frictional strength. ß There lead to zones of local compression (asperities) and tension (fault jogs).

Parkfield GEOMETRY

(Exercise: 20km x asperity jog 10km x 10-4 [shear stress] ≈ 6.5 mag) 20 km SHEAR STRESS

The asperity represents an increase in rock strength and must be broken before slip can occur on the segment

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD How big? ß The fault jog represents a ‘shatter zone’ of dispersed fracture which stops the earthquake by absorbing energy

further extension resisted by: a) suction of fluids filling fractures (e.g., capillary force) b) further distributed cracking ß 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD How big?

ß Summary – earthquake magnitude

Subduction 8< Mw <10 (e.g, Chile 1960)

Cont. transform Mw ∼ 8 (e.g. San Andreas )

Active intraplate Mw ∼ 7 (e.g. New Madrid)

Oceanic ridge Mw ∼ 6

Moderate intraplate Mw ∼ 5-6 (UK, N. Sea)

Continental cratons Mw ∼ 5 (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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Statistical distribution

Gutenberg-Richter magnitude- 100000 frequency distribution: 10000

1000 a log10 N = a - b M z N - number of 100 earthquakes in 10 magnitude range Number of Events z M - earthquake 1 345678 magnitude Magnitude z Seismic b-value defines log-linear Seismic b-value distribution

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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 .

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD (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.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD (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.)

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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.

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD 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 Pr[N = x] = p (x) = e−λt x = 0,1,2,... N x!

N is a Poisson random variable with parameter, µ = λt. Poisson distribution λ=3 E[N] = µ Number of earthquakes in time t

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Poisson model - continuous If T is elapsed time till the first event occurs then T has exponential probability density function:

−λt fT (t) = λe , t > 0

E[T] = 1/λ Mean interval between earthquakes

Continuous Poisson distribution

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Gutenberg-Richter magnitude-frequency distribution 100000

10000 log10 N = a - b M 1000 a Empirical distribution 100 10 a Set β = b ln 10 ≅ 2.3 b Number of Events 1 345678 can re-write as: Magnitude -βx fM(x) = β e

Exercise for the student This is exponential pdf for Poisson process

β-1 estimates mean magnitude

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD Scale invariance of nature

GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD