Log relationships, trig functions, earthquakes & computer lab tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV

Logarithms The allometric or exponential functions are in the form 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.

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 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

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 log 10 y to say - ”What is the power that 10 must be raised to to get y?” How about this operator? - The power of base 10 that yields (  ) y

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?

In general, or Try the following on your own

log 10 is referred to as the common logarithm thus log e or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g.

Worksheet – pb 15: sin(nx) See basics xlsx

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?

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

This looks almost linear.

In this case …

The Gutenberg-Richter Relation -b is the slope and c is the intercept.

The Gutenberg-Richter Relation m, the earthquake magnitude also specifies logarithmic differences in ground movements. For example - earthquake magnitude of 5 represents ground motion with amplitude 10 times that associated with a magnitude 4 earthquake.

One of the most commonly used “Richter magnitude” scales determines the magnitude of shallow earthquakes from surface waves according to the following equation where T is the period in seconds, A the maximum amplitude of ground motion in  m (10 -6 meters) and  is the epicentral distance in degrees between the earthquake and the observation point.

January 12 th Haitian magnitude 7.0 earthquake

Mann et al., 1995

Shake map USGS NEIC

Looking west across Port-au-Prince Mann et al., 1995

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

USGS NEIC

Notice the plot axis formats Total number of earthquakes in the past 36 years ~ 12,000

The seismograph network appears to have been upgraded in 1990

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

Look at Part 2 problem 13 and 4 on today’s group worksheet

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

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!

Lehman and Keigwin undertook one of the first deep sea studies to document changes in sea surface temperature associated with deglaciation Computer lab …

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?