Tuesday, February 9, 2010

A Mechanism Fitting Earthquake Activity with a Peak

It is possible to design a crude feedback mechanism that gives a good good fit to the earthquake data. The "Poisson fit" to the 2009 earthquake data was used to smooth the data and used to deduce an approximate "gain" function, g_k, for the mechanism. The gain function is the ratio of consecutive occupation numbers for the intervals of the histogram. Knowing the gain function one can compute the fraction of numbers in the states, r, which will result in an earthquake or the fraction, f, which will pass on to the next state.

The values of r and f are all that is needed to describe the mechanism and are the components of the transition matrix. The transition matrix can be used to find the equilibrium values for the occupancy of the states. These values are unchanged when multiplied by the transition matrix. To get the number of earthquakes in each magnitude interval one multiplies the equilibrium values by r_k. In the plot below logN is the logarithm of the number of earthquakes observed and logN_fit is the logarithm of the number computed from the equilibrium values. Note that this mechanism does have a peak unlike the linear fit found previously. The Poisson distribution does not appear to be an equilibrium distribution. k is the value of k used in the Poisson fit and is approximately 140 + M/ΔM with ΔM = 0.1. The peak is at M 1.1 (k = 151) and there are 10 intervals per M 1.0 step.

The gain function equals 1 at the peak and is a steadily decreasing function of k. The values of r and f indicate that the fraction of states experiencing a earthquake decreases with magnitude.

This model is crude but gives a good fit to the data. A more complicated model would allow for transitions to intermediate states instead of completely resetting when an earthquake occurs.