Thursday, January 28, 2010

Earthquake Probabilities vs Probability Distributions

There is a potential source of confusion over what we mean by the probability of an earthquake. One can ask what the probability is that an earthquake will fall within a given area, interval of time and magnitude interval. But this probability can be broken down into two parts. The first is the probability that an earthquake of any magnitude will occur and then the probability that it will then fall within a given magnitude range. The second part is associated with the distribution of earthquakes. It is a conditional probability since it assumes that the earthquakes have occurred. It is also a relative measure of the number of earthquakes that will occur in the possible magnitude ranges.

If we assume a Poisson distribution then the expected number of all earthquakes for the intervals involved would be some factor N0 = R0 ΔT p(M, ΔM) or,

where k=M/ΔM. More generally, one can assign intervals for the numbers themselves.

Whatever the intervals involved, one can consider a particular partition of the intervals and ask the probability or rate at which earthquakes will fall within the partition. One can also consider partitions of different sizes and compare rates and probabilities. For example, if the probability that an earthquake will occur in an interval of time of width Δt is rΔt one can ask the probability that it will occur in some larger interval ΔT. The probability that it will occur in the first Δt subinterval of ΔT is just p. For the second Δt the probability is the probability that it will not occur in the first interval, 1-p, times the probability that it will occur in the second, also p, or (1-p)p. For the third interval we have (1-p)(1-p)p and so on for all of ΔT. This is a partial sum of a geometric series and the total is the difference between two infinite sums.

The subintervals of time do not combine linearly. The total probability, P, is not a simple sum. And, since 1-p is less than one, for large values of m the total probability will be very near to 1 and the circumstances are likely to happen. So we have verified a form of Murphy's Law that if something can possibly go wrong it will.

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