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--- gibson:teaching:fall-2013:math445:lab8 [2013/11/04 13:52]
gibson
+++ gibson:teaching:fall-2013:math445:lab8 [2013/11/04 19:04] (current)
gibson
@@ Line 21: / Line 21: @@
 For each data set, turn in your plots and your estimate of the function //y(x)//.
-**Problem 1: The distribution of earthquake magnitudes.** Big earthquakes are rare, and little earthquakes are frequent. In fact, there is a very
+**Problem 1: The distribution of earthquake magnitudes, by Moment Magnitude scale.** Big earthquakes are rare, and
-clean empirical law that governs how many earthquakes of a given magnitude typically occur world-wide in a given year. Your job is to deduce
+little earthquakes are frequent. In fact, there is a very clean empirical law that governs how many earthquakes of a
-that law from the following historical data.
+given magnitude typically occur world-wide in a given year. Your job is to deduce that law from the following
+historical data.
 <code>
-% magnitude number/year
+% M N
 2
 18
@@ Line 37: / Line 38: @@
 </code>
-The first column is the Richter magnitude //R//, and the second column is the number of earthquakes //N// of that magnitude that occur, on average, in a year. (The last two entries are estimates, since it's impossible to detect every small earthquake around the world.)
+The first column is the [[http://en.wikipedia.org/wiki/Moment_magnitude_scale | moment magnitude]] //M//, and the second column is the number of earthquakes //N// of that magnitude that occur, on average, in a year. The last two entries are estimates, since it's impossible to detect every small earthquake around the world. The data are obtained from [[http://www.earthquake.ethz.ch/education/NDK/NDK|Earthquake Statistics and Earthquake Prediction Research]] by Stefan Wiemer, Institute of Geophysics, Zurich.
-Deduce a functional relationship //N(R)//.
+Using Matlab plotting commands, deduce the form of the functional relationship //N(M)//. Estimate  the constants in the relationship by estimating the slope and the //y//-intercept, and then fine-tuning by matching the plot of your estimate against the plot of the data.
+**Problem 2: The distribution of earthquake magnitudes, by energy.** The moment magnitude scale is logarithmic, in that an earthquake of magnitude //M+1// releases about 32 times energy than an earthquake of magnitude //M//. The following data
+set gives the number //N// of earthquakes in a given year of energy //E// measured in Joules.
+<code>
+% E  N
+e16 2
+e15 18
+e13 120
+e12 800
+e10 6200
+e09 49000
+e07 365000
+e06 2920000
+</code>
+Deduce the form of the functional relation //E(N)// using Matlab plotting, then estimate and fine-tune the constants
+in the relation, just as in problem 1.
+**Problem 3: World population.** The following data set provides the human population //P// of the earth at a given
+time //t//, measured in years A.D.
+<code>
+% t   P
+2e09
+3e09
+4e09
+5e09
+6e09
+7e09
+</code>
+Deduce the form of the functional relation //P(t)// and determine the constants graphically.
+Assume that the formula you derived for //P(t)// is valid indefinitely into the future and the past. What year will
+the population of the earth reach one trillion? What year were the first humans born? Do you believe these answers?
+If not, why not?

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