Monday, January 25, 2010

A Poisson Fit to 2009 Earthquake Data

The histogram containing the number of earthquakes from the beginning of 2009 to mid January 2010 fall primarily within the 3σ error bounds of a Poisson Distribution as seen in the plot below. The red dots are the number of earthquakes for each M 0.1 interval. The magnitude assigned to an interval was the magnitude of its lowest bound. Logarithms were used to facilitate the fit which was complicated by the fact that k_0 was difficult to determine and the peak magnitude was also needed. The method of least squares was used to determine the values of M_peak, N_0, λ, and k_0. Each data point was weighted by the number of earthquakes in the interval.

The equation for N(k) is the Poisson Distribution. Stirling's Formula was used to simplify the computations. The deviations of the data points from the solid blue line appear to statistical in nature. The largest earthquakes were omitted because of their low probabilities. The fit indicates the the peak magnitude is at approximately M 1.175.

(edit: In the plot what was referred to as the variance (Var) was not the average but the weighted sum of the square of the deviations of the natural logarithms of the number of earthquakes in each interval. It is the sum of the square of the errors for all the events which was the function that was optimized in the fit.)

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