WEBVTT
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Shown on this map of the upper and the surrounding regions
are earthquakes that have happened in these areas since
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nineteen. Eighty two large circles indicate strong quakes and
small circles, weak ones. The apparing grub is marked in
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white. What do you notice? first of all, it appears that
there are more small quakes than big ones. Secondly, the
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ethnicity has quite a rendered distribution over time.
Thirdly, small clusters of aftershock sometimes occur around
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larger earthquakes. The statistic description of these three
phenomena is an important foundation for being able to
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estimate seismic hazards @unoise@ hello. And welcome in
this video, you will become acquainted with the three most
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important statistical distribution models used in cosmology.
You will learn about their significance for estimating
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seismic hazards, and learn how you can assess data sets,
specifically with regard to these statistical properties, as in
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the introduction. I would like to show you these distributions
using the earthquakes in the upper and the
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surrounding regions. Firstly, I will be introducing the
distribution, which describes the fact that earthquakes
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occur independently in time. That means that I have a random
distribution. This is an important prerequisite for many
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size analyses and greatly simplifies more advanced
calculations to do so. I will now transfer the earthquakes from
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the upper from the map we saw earlier to a diagram, and
at the same time limit myself to quakes with a magnitude
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greater than two. We will also ignore four shocks and
aftershocks. The diagram now shows how many main shocks occurred
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in each two month time window. Once again, it appears that
the occurrence of a time is random in order to confirm this
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impression, we test the data for process distribution and ask
how often do five events occur every two months, and how
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often? four, three, two, one or none at all. This is what
this histogram shows, which indicates the number of time
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windows, and and the number of the corresponding quakes K in
it. Hence we see that no main shock occurred in forty three
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time windows, and exactly one and fifty seven of them. A
distribution is considered to exist when n equals and
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zero times slumped into the power of K, divided by Kay fictorial
times, aid to the power of negative Lantern and zero is
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the number of time windows, in this case thirty one years times
six corresponding windows, giving one hundred eighty two
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lampter is easy to calculate for the first bar, K equal zero.
Hence the following applies, and of K equal zero equals
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and zero times E to the power of negative lampsa. If we
solve for lumber, we get lumber equals minus the natural
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logarithm of n of zero divided by equals negative l n
of forty three divided by one hundred eighty, two equals one
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point four, four. With this, we can calculate the theoretical
process distribution according to the formula. This is
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what it looks like, and is very close to the observed
distribution of main shocks in the upperrangram. Hence, the main
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shocks follow a person distribution and are therefore
independent in time. This means that the seismic history is
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irrelevant when determining if an earthquake will happen or
not. If I had not filtered out the foreign . This would
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have been not the case. After all the definition, they are
dependent on the currents of the main shock. The second
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distribution that I will be introducing was first described
by Gutenberg and Rita for California. These two scientists
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found the mathematical description for the fact that smaller
quakes occur more frequently than large ones. As you will
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soon see the same observation can also be made in the upper
line ground. Qualitatively, this is already apparent on the
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map. First, let me show the magnitude distribution of a time.
We see that only two earthquakes, with a magnitude equal
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or greater than four have occurred since nineteen. Eighty
two. By the way, the mean value is two point three, five. We
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will need it in the next step, as we did previously, we now
create a histogram to investigate the distribution and more
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detail for this purpose, we count how often each year earthquakes
occur, which have a magnitude greater than a certain
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value, and enter this in a histogram. We speak of a
researcher distribution when the data forms a line in the
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logarithmic plot, which is the case here. The corresponding
formula for this is the decayed, global rhythm of the number
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of quakes N equals A minus B times the magnitude M. Hence, the
B value is a negative line gradient and can be calculated
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using a maximum likelihood estimation. B equals the decayed
logorism of E, divided by the mean magnitude and bar minus
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the smallest, completely used magnitude M, C, substituting
the mean magnitude two point three, five from before, and M,
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C equals one point nine five. We get a B value of one point
zero nine. This is very close to a value of one, which in
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this case says that ten times less, making two, three
quakes occur than make me two, two quakes. The a value is
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calculated by substituting the number of quakes greater than
M, C. And in the example gives three point zero, seven.
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This allows for a simplified estimation of how frequent a
strong quake of magnitude five would be lock. N equals three
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point zero, seven minus one point zero, nine times five
equals minus two point three, eight. That means n equals zero
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point zero, zero, four per year, or once every two hundred
fifty years, the reached the distribution is also valid
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globally. The number of earthquakes falls with
increasing magnitude worldwide with B value of one. This
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distribution is useful in order to make predictions about
the probability of strong quakes which have not yet or have
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only been observed in historic times. To do so, we extrapolate
the estimate up to a maximum expected magnitude and
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beyond, and read of its frequency @unoise@ the third
distribution I would like to introduce to you is Maurice Law. It
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describes the decrease in the number of aftershocks of a time
for this purpose, we will focus on the earthquake of
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in the Vash Mountains on February. Twenty two, two thousand
three, which had a magnitude of five point four, and
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resulted in minor damage to building. Shown here is the
spatial distribution of the aftershocks. They form a cluster
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around the main chuck. The depict of the number of quakes in
two day intervals shows the decline on earthquake activity
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on Maurice law specifically adapts this decline in order
to use it formally. We retreat the coordinate system and
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multiply the number of days by the number of the
most commonly used form of law is rate of occurrence
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and equal C divided by K plus time T to the power of P. The
is typically assumed to be approximately one such that
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the equation is simplified, and only the values of C and K
need to be determined. We now multiply by tea and
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obtain and K plus and tea equals. See, rearranging the
equation gives anti equal C minus and K. This now corresponds
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exactly to our diagram. Number of aftershocks. N time stays
T on the Y axis and N on the X axis. If we now fit the
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distribution of point to a line, the gradient of the line
equals minus K, and the intercept equals C. K is therefore
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proximately minus one point four and C equals forty. If we
substitute that into Maurice law, we can now more or less
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adapt the original data of the previous diagram. Hence,
law provides information about the decline and the number
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of aftershocks after a large earthquake. If additionally we
wish to make predictions regarding the expected magnitudes.
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We can utilize the empirical observation that the largest
aftershock magnitude is smaller than that of the main shark by
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round one. If we then also assume Gutenberg reach the
distribution for the aftershocks, it is possible to roughly
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estimate the expected aftershock magnitudes. You have now
learned about the three most important distribution models in
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Cesmology. First, the person distribution, which describes
the rendered distribution of main shocks that is independent
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over time. Secondly, the reached the distribution,
which indicates the ratio between the frequency of strong
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versus weak quakes. The decrease here is logarithmic. And
third of all, a Maurice law, which shows how the number of
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aftershocks decreases over time, or how long after a strong
earthquake after shocks can still be expected in seismic
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hazard assessment. These three concepts are often combined
with each other in order to be able to make predictions
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regarding the probability of future earthquakes.