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 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 Tom Wilson, Department of Geology and Geography
This week 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 week) Group problems for continued basic review and discussion Tom Wilson, Department of Geology and Geography
Common relationships between geologic variables. What kind of mathematical model can you use to represent different processes? Age k x Depth A linear relationship what would it look like Whether it represents the geologic process adequately is an assumption we make?
Assume a linear relationship? How thick was it originally? Over what length of time was it deposited?
Consider another area an alien but very interesting landscape
510,000 years 0.5 to 2.1 million years ago 2.1 to 2.7 million years ago
Milankovitch cycles
Astronomical forcing of global climate: Milankovitch Cycles http://www.sciencecourseware.org/eec/globalwarming/tutorials/milankovitch/ Take the quiz
Solar insolation http://pveducation.org/pvcdrom/properties-of-sunlight/calculation-of-solar-insolation
http://www.nasa.gov/mission_pages/mro/multimedia/phillips-20080515.html http://www.sciencedaily.com/releases/2008/04/080420114718.htm
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. 70000 60000 AGE (years) 50000 40000 30000 20000 10000 What are the intercepts? 0-10000 0 20 40 60 80 100 Dept h (meters) These lines represent cases where the age at 0 depth is different from 0
A kd A 0 Should we expect age depth relationships to be linear? 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.
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.
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.
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. 150000 Quadratic vs. Linear Behavior Here are two different possible representations of age depth data 100000 50000 Age 0-50000 A 1000D -15,000 and (in red) A 3D 2 1000D -15,000-100000 -50 0 50 100 Depth (meters) What kind of equation is this?
Quadratics The general form of a quadratic equation is The roots x 2 b b 4ac 2a Quadratics y ax 2 bx c 125 y 2x 2 10x 20 75 Y 25 roots -25-75 -6-4 -2 0 2 4 6 X 2 y 2x y 2 3x 60 Similar examples are presented in the text.
Go to the common drive or visit Waltham s site (see link on the class page) http://davidwaltham.com/mathematics-simple-tool-geologists/ Tom Wilson, Department of Geology and Geography
age But that doesn t provide us with the model we have in mind. Open quadrat.xls and explore What do you have to do to the coefficients to get a relationship like that at right Hint try getting rid of the linear term depth Tom Wilson, Department of Geology and Geography
In Excel (=xa=1,c=10 and b=0) = x 2 +10 Tom Wilson, Department of Geology and Geography
Age Well we only wanted the positive half Depth So we could use such a model with data support. Tom Wilson, Department of Geology and Geography
Waltham s excel files have been placed on the common drive. Copy them to your network drive (G, N ) Have a look at a few - S_Line.xls, Quadrat.xls, poly.xls, exp.xls, log.xls. Tom Wilson, Department of Geology and Geography
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 Depth (km) Temperature ( o C) 0 10 100 1150 400 1500 700 1900 2800 3700 5100 4300 6360 4300 T 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km) See http://www.ucl.ac.uk/mathematics/geomath/powcontext/poly.html
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 T 11.4x 10 provides a good approximation. 5000 From 100-700km the relationship T 1.25x 1017 works well. T 4000 3000 Can we come up with an equation that will fit the variations of temperature with depth - for all depths? See text. 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km)
Temp We ll show you how to do this in Excel 2000 1800 T The slope tells you 1600 1400 1200 1000 800 600 400 y = 1.25x + 1016.7 R² = 0.9985 What does the intercept tell you? Is it meaningful to us in this case? 200 0 0 100 200 300 400 500 600 700 800 Depth Tom Wilson, Department of Geology and Geography
The quadratic relationship plotted below approximates temperature depth variations. T (1.537x10 4 ) x 2 1.528x 679.77 5000 4000 3000 T 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km)
The formula - below right - is presented by Waltham. In his estimate, he does not try to fit temperature variations in the upper 100km. T (1.537x10 4 ) x 2 1.53x 680 T ( 8.255x10 5 ) x 2 1.05x 1110 5000 5000 4000 4000 3000 3000 T T 2000 2000 1000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km) 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km)
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. 5000 5000 4000 4000 T( O C) 3000 2000 T 3000 2000 1000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km) 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km) Can we do better?
The general class of functions referred to as polynomials include x to the power 0, 1, 2, 3, etc. The straight line y mx b 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 quadratic y ax bx Polynomial, and the equation 2 c is a second order y a x a x a x... a n n 1 n 2 n n 1 n 2 0 is an nth order polynomial.
y a x a x a x... a n n 1 n 2 n n 1 n 2 0 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.
Waltham offers the following 4 th order polynomial as a better estimate of temperature variations with depth. T 1.12x10 12 d 4 2.85x10 8 d 3 0.00031d 2 1.64d 930 5000 4000 3000 T 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km)
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 y ax 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. 2 2 4 What is What is 0.01 0.01 2? 2? See Powers.xls p Y 1200 1000 800 600 400 200 a 0 0 P ower Laws X -2 X -1.75 0 1 2 3 4 5 x X 2 X 1.75
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). 1/ 2 d ax d D (km) 0 1 2 3 4 Ocean Floor Depth Spreading Ridge 0 Like a quadratic but not, since the exponent is not 2. 5 0 200 400 600 800 1000 X (km) What physical process do you think might be responsible for this pattern of seafloor subsidence away from the spreading ridges?
Another relationship There is also a relationship between the age of the oceanic crust and its depth such that d d at o 1 2 Visit http://oceansjsu.com/105d/exped_boundaries/9.html Tom Wilson, Department of Geology and Geography
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.
Waltham presents us with the following data table - Depth (km) Porosity ( ) 0 0.6 1 0.3 2 0.15 3 0.075 4 0.0375 Over the range of depth 0-4 km, the porosity decreases from 60% to 3.75%! Porosity 0.6 0.5 0.4 0.3 1 or 2 0 d 0 2 d 0.2 0.1 0.0 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 Depth
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. 0.6 Porosity 0.5 0.4 0.3 0.2 It s not constant and it does not decrease linearly 0.1 0.0-0.1 0 1 2 3 4 5 Depth Waltham generates this data using the following relationship. 1 d 0 d or 02 ; in this case =0.6*2 2 d
z 0.6 x 2 where z = d = depth 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 cx y ab are referred to as allometric growth laws or exponential functions.
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 -cz 0 e 0 is the initial porosity, c is a compaction factor and z - the depth. Sometimes you will see such exponential functions written as -cz 0 exp In both cases, e=exp=2.71828
Waltham writes the porosity-depth relationship as z - 0 e 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 z - 0 e - -1 0 e 0 e. 368 0 when z=, represents the depth at which the porosity drops to 1/e or 0.3678 of its initial value. 0 In the form -cz 0 e c is the reciprocal of that depth.
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?
Tom Wilson, Department of Geology and Geography World seismicity in the last 7 days
Larger number of magnitude 2 and 3 s and many fewer M5 s Tom Wilson, Department of Geology and Geography
m N/year 5.25 537.03 5.46 389.04 5.7 218.77 5.91 134.89 6.1 91.20 6.39 46.77 6.6 25.70 6.79 16.21 7.07 8.12 7.26 4.67 7.47 2.63 7.7 0.81 7.92 0.66 7.25 2.08 7.48 1.65 7.7 1.09 8.11 0.39 8.38 0.23 8.59 0.15 8.79 0.12 9.07 0.08 9.27 0.04 9.47 0.03 Observational data for earthquake magnitude (m) and frequency (N, number of earthquakes per year (worldwide) with magnitude greater than m) Number of earthquakes per year 600 500 400 300 200 100 0 5 6 7 8 9 10 Richter Magnitude What would this plot look like if we plotted the log of N versus m?
1000 Number of earthquakes per year On log scale 100 10 1 0.1 0.01 5 6 7 8 9 10 Richter Magnitude Looks almost like a straight line. Recall the formula for a straight line?
y mx b What does y represent in this case? Number of earthquakes per year 1000 100 10 1 0.1 What is b? the intercept y log N 0.01 5 6 7 8 9 10 Richter Magnitude
The Gutenberg-Richter Relationship or frequency-magnitude relationship Number of earthquakes per year 1000 100 10 1 0.1 log N bm -b is the slope and c is the intercept. c 0.01 5 6 7 8 9 10 Richter Magnitude
January 12 th Haitian magnitude 7.0 earthquake
Shake map USGS NEIC
USGS NEIC
Notice the plot axis formats log( N) bm c Gutenberg Richter (frequency magnitude) plot N (per year - magnitude m and higher) 100 10 1 0.1 Haiti (1973-2010) Magnitude 2 and higher 0.01 2 3 4 5 6 7 8 Magnitude
7 Earthquake Occurrence 1973- present (Haiti and surroundings) 6 Magnitude 5 4 3 2 1975 1980 1985 1990 1995 2000 2005 2010 Year The seismograph network appears to have been upgraded in 1990.
8.0 7.5 Large Earthquakes Haiti Region (last century) In the last 110 years there have been 9 magnitude 7 and greater earthquakes in the region Magnitude 7.0 6.5 6.0 1920 1940 1960 1980 2000 Year
3 Frequency (log10n) Magnitude Plot (Haitian Region) 2 Log10 N 1 0-1 Look at problem 19 (see also 4 on today s group worksheet) -2 2 3 4 5 6 7 Magnitude
3 Frequency (log10n) Magnitude Plot (Haitian Region) 2 logn=-0.935 m + 5.21 Log 10 N 1 0-1 With what frequency should we expect a magnitude 7 earthquake in the Haiti area? -2 2 3 4 5 6 7 Magnitude
How do you solve for N? 3 Frequency (log10n) Magnitude Plot (Haitian Region) Log 10 N 2 1 logn=-0.935 m + 5.21 log N 0.935m 5.21 log N 0.935(7.2) 5.21 log N 1.52 0-1 -2 2 3 4 5 6 7 Magnitude We will return to this class of problems for further discussion on Thursday.
For a really cool and informative movie have a look at http://usgsprojects.org/fragment/ Tom Wilson, Department of Geology and Geography
Wrap-up and Status Complete and hand in answers to Group Problems 2 (not today s worksheet) next time Also have a look at problems 2.11, 2.12, 2.13 and 2.15 (Chapter 2 of Waltham)
Finish reading Chapters 1 and 2 (pages 1 through 38) of Waltham We re still going over the intro group problems 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.