Saturday, February 20, 2010

A New Fit for the Southern California Earthquake Data

I was able to find a approximate fit for the Southern California earthquake data but had to fit by eye again. This time I assumed that log(g) was a polynomial and used the values of g computed from the polynomial to compute the distribution for the expected number of earthquakes to avoid repeating my mistake. The program that I wrote to improve on the fit didn't work this time. The feedback mechanism appears to work.

The gain function is what one would expect being greater than one to the left of the peak and less than one on the right where the distribution steadily decreases. Where the slope of the plot of logN approximately a constant the ratio of the consecutive number of earthquakes would be the same and one would expect g to be a flat line for these magnitudes. In the plot below the solid line represents the g values computed from the polynomial and the points are the computed ratios for the consecutive number of earthquakes and are a check on the results.

The fraction of states, r, for which there is an earthquake is 1/2 at the peak as one would expect since g = 1 at the peak. r(1+g)=1. I borrowed the term gain for consecutive ratios from the analogy with the ratio of the input and output of a feedback amplifier. An amplifier is stable if the gain is not -1 which is the case for earthquakes being whole numbers. So the earthquake mechanism is probably stable too. One would expect any fluctuations to be damped out.

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