Saturday, February 6, 2010

What Went Wrong with the Poisson Fit?

The Poisson fit resulted with too large a value for λ. How can we explain this result? Let's look at what happens if we try to fit just two points, say, a peak where the slope (s) is 0 at M 1.1 and a slope of -1 at M 5.0. One can derive an approximate formula for the slope. The unknowns are λ and k_0 which can be determined by two points on the curve. The peak tells us that λ approximately equals k_pk. The second point gives us another fix and the intersection of the two lines is the solution we seek. There is only one solution and we can surmise that the problem is that the shape of the distribution is not consistent with the interpretation of λ as a probability.

The derivation of the Poisson distribution indicated that P_k =P^k/k!. If we compare the ratio of the next P_k to P_k we get λ/(k+1). The ratio of succeeding numbers in the fit is the same. This ratio is not constant but steadily decreases with k (or M). At the peak the ratio is 1 and prior to that it is greater than 1 which explains the growth in numbers. So the Poisson fit suggests a mechanism in which there are changes in what happens for each state of the Markov process. In the simple mechanism considered the ratio between succeeding states was 1-r. If r is a fraction this the ratio does not exceed 1 and so there is no peak. Any mechanism would also have to explain the magnitudes as well as how the number of earthquakes depend on the occupancy of the states. Perhaps a delay in advancement or some blockage in the advancement would give a more realistic fit.
The computed distribution gave equilibrium values. Blockage at some magnitude would prevent advancement from one state to the next and one would expect the occupancy of the states below the point of blockage to increase relative to the equilibrium values and those above to decrease somewhat. One needs to consider processes which might compete with the occurrance of earthquakes. The goal is a better explaination of the process behind earthquakes based on evidence. It is a search for knowledge.

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