Monday, January 25, 2010

Earthquake Probability Distributions

There is a linear fit for the logarithm of number of earthquakes vs magnitude in Modern Global Seismology by Lay and Wallace. Their "b value" is related but not the same as the "base" in the empirical formula for my fit.

If the number of earthquakes in an interval is proportional to b^(-M) as in the empirical linear fit the probability distribution will be proportional to (B-1)B^(-k) where B=b^ΔM (my b equal to 10^s, where s is the absolute value of the slope for log(N) vs M) and k is the number of the histogram interval starting with k=1. Note that k=(M-M_0)/ΔM. This distribution is based of the relative number of events in each interval and is vaguely similar to a Poisson distribution which led me to attempt a fit for that too.

The Poisson distribution is often used in situations where the occurrence of the events is statistically independent but with each event having the same probability of occurrence and one needs the probability of a number of events occurring simultaneously. One can also ask what the probability is for a number of sections of a fault failing at the same time. If there is a key section then its failure could be responsible for the failure of a number of others with the actual number involved depending on the circumstances. The factor k! in the denominator is the number of ways in which the same multiple event can occur. There is only one multiple event for all combinations of the individual events. But how do we explain that a unit step in Mw corresponds to a factor of 30.1 in energy? Mw seems more closely related to the probability of failure while on the other hand M0 is a better measure of energy released. So we have to ask if the probability of a fault failing proportional the area involved. The answer depends on the strength of materials.

One can reject a probability distribution if the data falls too far outside the 3σ bounds where σ is the standard deviation of the expected value. If earthquakes with magnitudes less than M 4.5 were included in the global earthquake data one could more easily tell which distribution was the better fit.

(note on notation: _ is often used to indicate a subscript and ^ a superscript. The square of a can be written as a^2.)

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