Geology 351 - geomathematics

1. Geology 351 - geomathematics Earthquakes, log relationships, trig functions tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

2. Objectives for the day • Explore the use of earthquake frequency-magnitude relations in seismology • Learn to use the frequency-magnitude model to estimate recurrence intervals for earthquakes of specified magnitude and greater. • Learn how to express exponential functions in logarithmic form (and logarithmic functions in exponential form). • Review graphical representations of trig functions and absolute value of simple algebraic expressions Tom Wilson, Department of Geology and Geography

3. related materials that may be of interest some interesting seismotectonics @ http://usgsprojects.org/fragment/index.html Tom Wilson, Department of Geology and Geography

4. http://usgsprojects.org/fragment/download.html Tom Wilson, Department of Geology and Geography

5. Oceanic crustal fragment underlies complex sea bottom bathymetry Tom Wilson, Department of Geology and Geography

6. Life over a subducting oceanic place zone can be exciting Tom Wilson, Department of Geology and Geography

7. A useful log relationship in seismology The Gutenberg- Richter relationship Are small earthquakes much more common than large ones? Is there a relationship between frequency of occurrence and magnitude? Fortunately, the answer to this question is yes, but is there a relationship between the size of an earthquake and the number of such earthquakes?

8. World seismicity – Jan 9 to jan16, 2014 Tom Wilson, Department of Geology and Geography

9. IRIS Seismic Monitorhttp://www.iris.edu/seismon/ Tom Wilson, Department of Geology and Geography

10. Larger number of magnitude 2 and 3’s and many fewer M5’s Tom Wilson, Department of Geology and Geography

11. Magnitude distribution Tom Wilson, Department of Geology and Geography

12. Some worldwide data Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year (worldwide) with magnitude m and greater) ? Number of earthquakes per year of Magnitude m and greater What would this plot look like if we plotted the log of N versus m?

13. On log scale Number of earthquakes per year of Magnitude m and greater Looks almost like a straight line. Recall the formula for a straight line?

14. Here is our formula for a straight line … What does y represent in this case? What is b? the intercept

15. The Gutenberg-Richter Relationship or frequency-magnitude relationship -b is the slope and c is the intercept.

16. January 12th, 2010 Haitian magnitude 7.0 earthquake

17. Shake map USGS NEIC

18. USGS NEIC

19. Notice the plot axis formats Limited observations

20. Low magnitude seismicity The seismograph network appears to have been upgraded in 1990.

21. In the last 110 years there have been 9 magnitude 7 and greater earthquakes in the region

22. Magnitude 7 earthquakes are predicted from this relationship to occur about once every 20 years. Let’s work through an example using a magnitude of 7.2

23. Let’s determine N for a magnitude 7.2 quake.

24. How do you solve for N? What is N? Let’s discuss logarithms for a few minutes and come back to this later.

25. Logarithms Logarithms are based (initially) on powers of 10. We know for example that 100=1, 101=10 102=100 103=1000 And negative powers give us 10-1=0.1 10-2=0.01 10-3=0.001, etc. Tom Wilson, Department of Geology and Geography

26. General definition of a log The logarithm of x, denoted log x solves the equation 10logx =x The logarithm of x is the exponent we have to raise 10 to - to get x. So log 1000 = 3 since 103 = 1000 & Log 10y =y since Tom Wilson, Department of Geology and Geography

27. Some more review examples What is log 10? We rewrite this as log (10)1/2. Since we have to raise 10 to the power ½ to get 10, the log is just ½. • Some other general rules to keep in mind are that • log (xy)=log x + log y • log (x/y)= log x – log y • log xn =n log x Tom Wilson, Department of Geology and Geography

28. Take a look at exponential (allometric) functions and b and 10 are the bases. These are constants and we can define any other number in terms of these constants raised to a certain power. Given any number y, we can express y as 10 raised to some power x Thus, given y =100, we know that x must be equal to 2.

29. By definition, we also say that x is the log of y, and can write So the powers of the base are logs. “log” can be thought of as an operator like x (multiplication) and  which yields a certain result. Unless otherwise noted, the operator “log” is assumed to represent log base 10. So when asked what is We assume that we are asking for x such that

30. Sometimes you will see specific reference to the base and the question is written as leaves no room for doubt that we are specifically interested in the log for a base of 10. One of the confusing things about logarithms is the word itself. What does it mean? You might read log10y to say -”What is the power that 10 must be raised to to get y?” How about this operator? -

31. The power of base 10 that yields ()y What power do we have to raise the base 10 to, to get 45 Tom Wilson, Department of Geology and Geography

32. We’ve already worked with three bases: 2, 10 and e. Whatever the base, the logging operation is the same. How do we find these powers?

33. In general, or Try the following on your own

34. log10 is referred to as the common logarithm thus loge or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g.

35. Return to the problem developed earlier Where N, in this case, is the number of earthquakes of magnitude 7.2 and greater per year that occur in this area. What is N? You have the power! Call on your base!

36. Base 10 to the power Since -1.53 is the power you have to raise 10 to to get N. Take another example: given b = 1.25 and c=7, how often can a magnitude 8 and greater earthquake be expected? (don’t forget to put the minus sign in front of b!) log N = …. Tom Wilson, Department of Geology and Geography

37. Seismic energy-magnitude relationshipsmore logs What energy is released by a magnitude 4 earthquake? A magnitude 5? More logs and exponents!

38. See http://www.cspg.org/documents/Conventions/Archives/Annual/2012/313_GC2012_Comparing_Energy_Calculations.pdf For applications to microseismic events produced during frac’ing. Tom Wilson, Department of Geology and Geography

39. Try your hand at these questions.

40. Review: Here’s a problem similar to the inclass problem from last time. (see handout) e.g. Worksheet – pbs 16 & 17: sin(nx) … and basics.xls

41. A review of the problems from last time Tom Wilson, Department of Geology and Geography

42. Try another: sin(4x) Tom Wilson, Department of Geology and Geography

43. Graphical sketch problem similar to problem 18 What approach could you use to graph this function? Really only need three points: y (x=0), x(y=0) and one other.

44. Have a look at the basics.xlsx file Some of the worksheets are interactive allowing you to get answers to specific questions. Plots are automatically adjusted to display the effect of changing variables and constants Just be sure you can do it on your own!

45. Spend the remainder of the class working on Discussion group problems. The one below is all that will be due today Tom Wilson, Department of Geology and Geography

46. Warm-up problems 1-20 will be due next Tuesday. Bring any remaining questions to class on Thursday Tom Wilson, Department of Geology and Geography

47. In the next class, we will spend some time working with Excel. Tom Wilson, Department of Geology and Geography

48. Next Time • Hand in group problems before leaving today • Look over problems 2.11 through 2.13 • Continue your reading • We examine the solutions to 2.11 and 2.13 using Excel next time.