Log relationships, trig functions, earthquakes & computer lab

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1 Log relationships, trig functions, earthquakes & computer lab tom.h.wilson Department of Geoy and Geography West Virginia University Morgantown, WV Logarithms The allometric or exponential functions are in the form y = ab cx y = a cx and b and 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 raised to some power x Thus, given y =0, we know that x must be equal to 2. i. e. y = x

2 x y = By definition, we also say that x is the of y, and can write y = x ( ) = x So the powers of the base are s. can be thought of as an operator like x and which yields a certain result. Unless otherwise noted, the operator is assumed to represent base. So when asked what is y, where y = 45 We assume that we are asking for x such that x = 45 Sometimes you will see specific reference to the base and the question is written as y, where y = 45 y leaves no room for doubt that we are specifically interested in the for a base of. One of the confusing things about arithms is the word itself. What does it mean? You might read y to say - What is the power that must be raised to to get y? How about this operator? - pow The power of base that yields ( ) y y y = 1.653

3 We ve already worked with three bases - 2, and e. Whatever the base, the ging operation is the same. asks what is the power that 5 must be raised to to get. 5 5 =? How do we find these powers? 5 = = = thus = In general, (some number) = base a = b or ( a ) b ( number base Try the following on your own 7 = (7) (3) 3 =? )

4 You will find that is often writ ten as, with no subscript is referred to as the common arithm e is often written as ln. thus 8 = ln8 = e e or ln is referred to as the natural arithm. All other bases are usually specified by a subscript on the, e.g. 5 or 2, etc. Worksheet pb 15: sin(nx) See basics xlsx

5 Graphical sketch What do you think? Are small earthquakes much more common than large ones? 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?

6 m N/year Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year with magnitude greater than m) Number of earthquakes per year Richter Magnitude What would this plot look like if we plotted the of N versus m? 00 Number of earthquakes per year Richter Magnitude This looks almost linear.

7 y = mx + b 00 Number of earthquakes per year In this case y = N Richter Magnitude The Gutenberg-Richter Relation Number of earthquakes per year N = bm + -b is the slope and c is the intercept. c Richter Magnitude

8 The Gutenberg-Richter Relation Number of earthquakes per year Richter Magnitude N = bm + m, the earthquake magnitude also specifies arithmic differences in ground movements. For example - earthquake magnitude of 5 represents ground motion with amplitude times that associated with a magnitude 4 earthquake. c One of the most commonly used Richter magnitude scales determines the magnitude of shallow earthquakes from surface waves according to the following equation m = A Δ + T 3.3 where T is the period in seconds, A the maximum amplitude of ground motion in μm ( -6 meters) and Δ is the epicentral distance in degrees between the earthquake and the observation point.

9 January 12 th Haitian magnitude 7.0 earthquake Mann et al., 1995

10 Shake map USGS NEIC Looking west across Port-au-Prince Mann et al., 1995

11 Looking west across Port-au-Prince suburbs Mann et al., 1995 USGS NEIC

12 ( N) = bm + c Notice the plot axis formats Gutenberg Richter (frequency magnitude) plot N (per year - magnitude m and higher) Total number of earthquakes in the past 36 years ~ 12,000 Haiti ( ) Magnitude 2 and higher Magnitude 7 Earthquake Occurrence present (Haiti and surroundings) 6 Magnitude Year The seismograph network appears to have been upgraded in 1990

13 Large Earthquakes Haiti Region (last century) In the last 1 years there have been 9 magnitude 7 and greater earthquakes in the region Magnitude Year 3 Frequency (N) Magnitude Plot (Haitian Region) 2 Log N Look at Part 2 problem 13 and 4 on today s group worksheet Magnitude

14 Spend a few minutes in group discussion on today s problems See the basics.xls spreadsheet

15 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! Computer lab North Sea Data (Lehman &Keigwan, 1992) AGE(years BP) δo Depth (cm) Lehman and Keigwin undertook one of the first deep sea studies to document changes in sea surface temperature associated with deglaciation

16 Finish reading Chapters 1 and 2 (pages 1 through 38) of Waltham Continue working on this lab for next class Hand in the second set of warm-up questions Everyone have the text?

Basic Review continued

Basic Review continued tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Previously Drew a correlation between basic mathematical representations