Sunday, January 31, 2010
It is not at all certain that the probability distribution associated with earthquakes is a Poisson distribution. It seems to fit the observations with a similar peak and trailing edge. The same could be said about a binomial distribution. Even if it is a Poisson distribution there is no guarantee that the total number of earthquakes is constant over time. One point in favor of a Poisson distribution is that whole numbers are involved and the number of events is never negative.
And, I am approaching the limit of what I know or can surmise about earthquakes. I did not know about the peak in magnitude when I started. So I am still learning. The Poisson distribution involves multiplicities but the probability involved may be that something will not happen.
So to avoid babbling I will probably have to take a little more time to think about what I am doing before making new posts.
Thursday, January 28, 2010
If we assume a Poisson distribution then the expected number of all earthquakes for the intervals involved would be some factor N0 = R0 ΔT p(M, ΔM) or,
where k=M/ΔM. More generally, one can assign intervals for the numbers themselves.
Whatever the intervals involved, one can consider a particular partition of the intervals and ask the probability or rate at which earthquakes will fall within the partition. One can also consider partitions of different sizes and compare rates and probabilities. For example, if the probability that an earthquake will occur in an interval of time of width Δt is rΔt one can ask the probability that it will occur in some larger interval ΔT. The probability that it will occur in the first Δt subinterval of ΔT is just p. For the second Δt the probability is the probability that it will not occur in the first interval, 1-p, times the probability that it will occur in the second, also p, or (1-p)p. For the third interval we have (1-p)(1-p)p and so on for all of ΔT. This is a partial sum of a geometric series and the total is the difference between two infinite sums.
The subintervals of time do not combine linearly. The total probability, P, is not a simple sum. And, since 1-p is less than one, for large values of m the total probability will be very near to 1 and the circumstances are likely to happen. So we have verified a form of Murphy's Law that if something can possibly go wrong it will.
Wednesday, January 27, 2010
At first glance it seems that there are many more points of the histogram below the fitted curve than above it but it should be remembered that the number of earthquakes associated with each point is N_k = 10^logN_k.
(edit: The rms error indicated was based on the deviations for ln(N) so the rms error for log(N) is 0.163.)
Monday, January 25, 2010
If the number of earthquakes in an interval is proportional to b^(-M) as in the empirical linear fit the probability distribution will be proportional to (B-1)B^(-k) where B=b^ΔM (my b equal to 10^s, where s is the absolute value of the slope for log(N) vs M) and k is the number of the histogram interval starting with k=1. Note that k=(M-M_0)/ΔM. This distribution is based of the relative number of events in each interval and is vaguely similar to a Poisson distribution which led me to attempt a fit for that too.
The Poisson distribution is often used in situations where the occurrence of the events is statistically independent but with each event having the same probability of occurrence and one needs the probability of a number of events occurring simultaneously. One can also ask what the probability is for a number of sections of a fault failing at the same time. If there is a key section then its failure could be responsible for the failure of a number of others with the actual number involved depending on the circumstances. The factor k! in the denominator is the number of ways in which the same multiple event can occur. There is only one multiple event for all combinations of the individual events. But how do we explain that a unit step in Mw corresponds to a factor of 30.1 in energy? Mw seems more closely related to the probability of failure while on the other hand M0 is a better measure of energy released. So we have to ask if the probability of a fault failing proportional the area involved. The answer depends on the strength of materials.
One can reject a probability distribution if the data falls too far outside the 3σ bounds where σ is the standard deviation of the expected value. If earthquakes with magnitudes less than M 4.5 were included in the global earthquake data one could more easily tell which distribution was the better fit.
(note on notation: _ is often used to indicate a subscript and ^ a superscript. The square of a can be written as a^2.)
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.)
Thursday, January 21, 2010
On can find the number of events greater than or equal to a given magnitude by setting ΔM equal to infinity in the first set of formulas in the last blog which would yield an "M+" function. The number in each interval would be the difference of the function values for its bounds. The differential rate is more useful since it allows one to use different time periods.
R0 = 2.033 x 10^8 per year assuming the interval if from M to M+ΔM.
If instead the interval is from M-ΔM/2 to M+ΔM/2, i. e., M is a central value, the following formulas should be used and again R0 = 2.033 x 10^8 per year.
The fit in the last blog assumed that M was a central value so the second formula is required to find R0.
Wednesday, January 20, 2010
I did a fit for the histogram based on the data above for the period from the beginning of 2009 to the present and when logs of the counts are used the curve is fairly linear. The fit yields an empirical equation for the annual number of earthquakes in M 0.25 intervals.
Tuesday, January 19, 2010
Another way of looking at the data is with a histogram which plots counts for equal intervals of magnitude. More data might give improved estimates of the probabilites of an earthquake of a given magnitude. That's one reason for studying all the earthquakes at this time. The accuracy of results depends on the number of events in a particular interval so one can't really trust the curve at higher magnitudes. There is no doubt that there are fewer 4.0s present.
Monday, January 18, 2010
One can smooth the data by averaging over time to give a plot similar to the following. The boxy nature of the plot is due to major events contributing to the averaging time of 1/5th of a day. The blue dots are the major earthquakes that occurred in the last week and give an impression of what the peak magnitudes are. The average itself seems to be unstable.
Friday, January 8, 2010
"The wave-motion in a ray of light can be compared to a succession of long straight waves rolling onward in the sea. If the motion of the waves is slower at one end that the other, the whole wave-front must gradually slew round, and the direction in which it is rolling must change. In the sea this happens when one end of the wave reaches shallow water before the other, because the speed in shallow water is slower. It is well known that this causes waves proceeding diagonally across a bay to slew round and come in parallel to the shore; the advanced end is delayed in the shallow water and waits for the other. In the same way when the light waves pass near the sun, the end nearest the sun has the smaller velocity and the wave-front slews round; thus the course of the waves is bent.
"Light moves more slowly in a material medium than in vacuum, the velocity being inversely proportional to the refractive index of the medium. The phenomenon of refraction is in fact caused by a slewing of the wave-front in passing into a region of smaller velocity. We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light."
-Sir Arthur Stanley Eddington, Space, Time and Gravitation, 1921