1. Basic Review continued tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

2. Previously Tom Wilson, Department of Geology and Geography Drew a correlation between basic mathematical representations of lines and curves and geologic models Reviewed use of subscripts and exponents, scientific notation Emphasized attention to units conversion issues Highlight limitations inherent in the underlying assumptions of any given mathematical model of a geological process

3. This week Tom Wilson, Department of Geology and Geography Consider mathematical models of Age/Depth relationships/limitations Continue review of basic math relationships, including: linear, quadratic, polynomial, exponential, log and power law behavior. Introduce Waltham’s Excel files, e.g.: S_Line.xls, Quadrat.xls, poly.xls, exp.xls, log.xls. Introduce basic Excel file structure (next time) Group problems for continued basic review and discussion

4. A linear relationship – what would it look like Common relationships between geologic variables. What kind of mathematical model can you use to represent different processes? Whether it represents the geologic process adequately is an assumption we make?

5. How thick was it originally? Over what length of time was it deposited? Assume a linear relationship?

6. Consider another area – an alien but very interesting landscape

8. 510,000 years 2.1 to 2.7 million years ago 0.5 to 2.1 million years ago

9. Milankovitch cycles …

10. Astronomical forcing of global climate: Milankovitch Cycles Take the quiz http://www.sciencecourseware.org/eec/GlobalWarming/Tutorials/Milankovitch/

11. http://www.sciencedaily.com/releases/2008/04/080420114718.htm http://www.nasa.gov/mission_pages/MRO/multimedia/phillips-20080515.html

12. The previous equation assumes that the age of the sediments at depth =0 are always 0. Thus the intercept is 0 and we ignore it. These lines represent cases where the age at 0 depth is different from 0 What are the intercepts?

13. … we would guess that the increased weight of the overburden would squeeze water from the formation and actually cause grains to be packed together more closely. Thus meter thick intervals would not correspond to the same interval of time. Meter-thick intervals at greater depths would correspond to greater intervals of time. Should we expect age depth relationships to be linear?

14. We might also guess that at greater and greater depths the grains themselves would deform in response to the large weight of the overburden pushing down on each grain.

15. These compaction effects make the age-depth relationship non-linear. The same interval of depth  D at large depths will include sediments deposited over a much longer period of time than will a shallower interval of the same thickness.

16. The relationship becomes non-linear. The line y=mx+b really isn’t a very good approximation of this age depth relationship. To characterize it more accurately we need to use different kinds of functions - non-linear functions. Here are two different possible representations of age depth data and (in red) What kind of equation is this?

17. Quadratics The general form of a quadratic equation is Similar examples are presented in the text. roots

18. Waltham’s excel files have been placed on the common drive. Copy them to your network drive (G, N …) Tom Wilson, Department of Geology and Geography Have a look at a few - S_Line.xls, Quadrat.xls, poly.xls, exp.xls, log.xls.

19. The increase of temperature with depth beneath the earth’s surface (taken as a whole) is a non-linear process. Waltham presents the following table See http://www.ucl.ac.uk/Mathematics/geomath/powcontext/poly.html

20. We see that the variations of T with Depth are nearly linear in certain regions of the subsurface. In the upper 100 km the relationship Can we come up with an equation that will fit the variations of temperature with depth - for all depths? See text. From 100-700km the relationship provides a good approximation. works well.

21. The quadratic relationship plotted below approximates temperature depth variations.

22. The formula - below right - is presented by Waltham. In his estimate, he does not try to fit temperature variations in the upper 100km.

23. Either way, the quadratic approximations do a much better job than the linear ones, but, there is still significant error in the estimate of T for a given depth. Can we do better?

24. The general class of functions referred to as polynomials include x to the power 0, 1, 2, 3, etc. The straight line is just a first order polynomial. The order corresponds to the highest power of x in the equation - in the above case the highest power is 1. The quadraticis a second order Polynomial, and the equation is an nth order polynomial.

25. In general the order of the polynomial tells you that there are n-1 bends in the data or n-1 bends along the curve. The quadratic, for example is a second order polynomial and it has only one bend. However, a curve needn’t have all the bends it is permitted! Higher order generally permits better fit of the curve to the observations.

26. Waltham offers the following 4 th order polynomial as a better estimate of temperature variations with depth.

27. In sections 2.5 and 2.6 Waltham reviews negative and fractional powers. The graph below illustrates the set of curves that result as the exponent p in is varied from 2 to -2 in -0.25 steps, and a 0 equals 0. Note that the negative powers rise quickly up along the y axis for values of x less than 1 and that y rises quickly with increasing x for p greater than 1. See Powers.xls

28. Power Laws - A power law relationship relevant to geology describes the variations of ocean floor depth as a function of distance from a spreading ridge (x). What physical process do you think might be responsible for this pattern of seafloor subsidence away from the spreading ridges?

29. Section 2.7 Allometric Growth and Exponential Functions Allometric - differential rates of growth of two measurable quantities or attributes, such as Y and X, related through the equation Y=ab cX - This topic brings us back to the age/depth relationship. Earlier we assumed that the length of time represented by a certain thickness of a rock unit, say 1 meter, was a constant for all depths. However, intuitively we argued that as a layer of sediment is buried it will be compacted - water will be squeezed out and the grains themselves may be deformed. The open space or porosity will decrease.

30. Waltham presents us with the following data table - Over the range of depth 0-4 km, the porosity decreases from 60% to 3.75%!

31. This relationship is not linear. A straight line does a poor job of passing through the data points. The slope (gradient or rate of change) decreases with increased depth. Waltham generates this data using the following relationship. It’s not constant and it does not decrease linearly

32. This equation assumes that the initial porosity (0.6) decreases by 1/2 from one kilometer of depth to the next. Thus the porosity (  ) at 1 kilometer is 2 -1 or 1/2 that at the surface (i.e. 0.3),  (2)=1/2 of  (1)=0.15 (i.e.  =0.6 x 2 -2 or 1/4th of the initial porosity of 0.6. Equations of the type are referred to as allometric growth laws or exponential functions.

33. The porosity-depth relationship is often stated using a base different than 2. The base which is most often used is the natural base e where e equals 2.71828... In the geologic literature you will often see the porosity depth relationship written as  0 is the initial porosity, c is a compaction factor and z - the depth. Sometimes you will see such exponential functions written as In both cases, e=exp=2.71828

34. Waltham writes the porosity-depth relationship as Note that since z has units of kilometers (km) that c must have units of km -1 and must have units of km. Note that in the above form when z=, represents the depth at which the porosity drops to 1/e or 0.3678 of its initial value. In the formc is the reciprocal of that depth.

35. 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?

36. World seismicity in the last 7 days Tom Wilson, Department of Geology and Geography

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

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

39. Looks almost like a straight line. Recall the formula for a straight line? On log scale

40. What does y represent in this case? What is b? the intercept

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

42. January 12 th Haitian magnitude 7.0 earthquake

43. Shake map USGS NEIC

45. Notice the plot axis formats

46. The seismograph network appears to have been upgraded in 1990.

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

48. Look at problem 19 (see also 4 on today’s group worksheet)

49. With what frequency should we expect a magnitude 7 earthquake in the Haiti area?

50. How do you solve for N? We will return to this class of problems for further discussion on Thursday.

51. For a really cool and informative movie have a look at http://usgsprojects.org/fragment/ Tom Wilson, Department of Geology and Geography

52. Wrap-up and Status Complete and hand in answers to today’s group problems Warm-up problems 1-20 will be due NEXT Thursday. We will continue our review of these problems and their solution next Tuesday. Also have a look at problems 2.11, 2.12, 2.13 and 2.15 (Chapter 2 of Waltham)

53. Finish reading Chapters 1 and 2 (pages 1 through 38) of Waltham In the next class we will spend some time reviewing logs and trig functions. Following that, we will focus on some of the problems related to the material covered in Chapters 1 and 2. We’re still going over background information related to the intro problems (1-20).