Friday, February 19, 2010

A Review of the Earthquake Mechanism's Fit to the Observed Earthquake Numbers

I should probably attempt to clarify the interrelations between the quantities involved in the earthquake mechanism. Of primary importance is the number of earthquakes that occur in a given time. There is also the number occupying each state which needs to be determined. For each set of numbers there is a distribution and therefore a set of ratios of consecutive terms, g, associated with it which was referred to as the gain or g function. The g function searched for in the fit was for the expected number of earthquakes* and so was unknown. The valuses of g do not contain enough information to determine the actual numbers observed but are more closely related to the relative probabilities of the earthquakes*. To determine the expected values for each interval of the histogram one needs to know the total number of earthquakes as well as the g values. From the g values one can determine the fraction of earthquakes, r, that will happen for each histogram interval and the fraction, f, that will not happen. From r and f one gets the transition matrix which in turn gives the relative probability of an earthquake being in a particular interval which is assumed to be the distribution which is not changed under multiplication by the transition matrix. Multiplying the relative probabilities of the states by the r values and the total number of earthquakes gives the expected number of earthquakes for each interval. The expected number of earthquakes is then compared with the observed distribution and the search for the coefficients of the log-log g polynominal is performed to minimize the root mean square error between the two distributions. The result is the fit for the observed number of earthquakes.

For different intervals of time one can estimate the total number of earthquakes in a particular interval as N = R ΔT. This number is then multiplied by the relative probabilities to estimate the number of earthquakes for each histogram interval.

*edit The g function was assumed to be for the number of earthquakes. The appears to be a discrepancy between the fit g function and the ratio of consecutive number of expected earthquakes. They should agree and I don't know what went wrong. The fit may have compensated for an error so one cannot assume that the earthquake mechanism works properly.

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