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
This week Review and extend mathematical models of Age/Depth relationships and look at their limitations Continue review of basic math relationships, including: linear, quadratic, polynomial, exponential, log and power law models. Introduce you to Waltham s Excel files, e.g.: S_Line.xls, Quadrat.xls, poly.xls, exp.xls, log.xls. Group problems for continued basic review and discussion
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?
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. Age k x Depth 70000 60000 What are the intercepts? 15000 yrs 0 AGE (years) 50000 40000 30000 20000 10000 0-5000 yrs -10000 0 20 40 60 80 100 Dept h (meters) These lines represent cases where the age at 0 depth is different from 0
What are our assumptions No compaction Constant deposition rate (slope is constant) No variation in sediment type (depositional environment) No change in sea level or lake level No tectonic deformation (constant burial rate)
Compaction A kd A 0 Is too simple. 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. we need to use different kinds of functions - non-linear functions. Would a quadratic work? 150000 100000 50000 Age 0-50000 Quadratic vs. Linear Behavior We know the straight line is not a very robust model A 1000D -15,000 and (in red) A 3D 2 1000D -15,000-100000 -50 0 50 100 Depth (meters) The quadratic gives us the increase with depth we might be looking for.
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.
age The quadratic and exponential functions may provide local approximations of age/depth variation. What do you have to do to the coefficients to get a relationship like that at right depth Hint try getting rid of the linear term. Differentiate to locate the minimum and ask how you d shift the minimum to 0.
In Excel (a=2,c=10 and b=0) = ax 2 +10 d ax bx c dx 2 ( ) 2ax b In general x min will be located at b/2a. But if b=0, then the minimum will be at x=0 ar at 0 depth. What s the intercept?
Just retain the positive half So we could use such a model with data support.
Go to the common drive or visit Waltham s site (see link on the class page) http://davidwaltham.com/mathematics-simple-tool-geologists/
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.
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. 11.4 o C/km 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)
Across the Mon river on the NNE site What is the significance of the intercept in this case? T~146 o F at 7852 TD Suggests T~50+1.22d Average temperature near surface is about 50 o F. Slope is about 1.22 o /100ft
Temp Temperature versus depth in region roughly corresponding to upper mantle or asthenosphere 2000 1800 Linear relationship might also work there T The slope tells you 1600 1400 1200 1000 800 600 400 200 y = 1.25x + 1016.7 R² = 0.9985 What does the intercept tell you? Is it meaningful to us in this case? 0 0 100 200 300 400 500 600 700 800 Depth How does temperature vary in the upper mantle, the viscous mechanically weak region?
It s difficult to find one model that fits it all. This quadratic for example misses all the points T (1.537x10 4 ) x 2 1.528x 679.77 5000 4000 Not too good T 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Depth (km)
Try a quadratic for the mantle to core region and ignore the near surface linear lithosphere T (1.537x10 4 ) x 2 1.53x 680 T ( 8.255x10 5 ) x 2 1.05x 1110 5000 5000 4000 4000 T 3000 T 3000 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) Waltham derives this quadratic
Quadratics are better than straight lines, but they don t replicate the variations 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.
We could use a 4 th order polynomial like Waltham to improve the fit. T 1.12x10 12 d 4 2.85x10 8 d 3 0.00031d 2 1.64d 930 5000 4000 T 3000 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 p a0 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. Y 1200 1000 800 600 P ower Laws X -2 X -1.75 2 2 4 What is 0.01 2? See Powers.xls 400 200 X 2 X 1.75 What is 0.01 2? 0 0 1 2 3 4 5 x
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
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 - e z o 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 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 when z=, - -1 0 e 0 e. 368 0 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.
There s a limit to the relationship between porosity and age Exponential decay z o e Exponential growth cd Age ke If we continue the relationship we can see that we get an additional billion years increase in age for and additional 2% decrease in porosity. So, this exponential increase in age with depth also has its limits growth decay
Basic sinusoids
Working with absolute values
Earthquake seismology log-linear relationships The Gutenberg-Richter or frequency magnitude 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?
World seismicity in the last 7 days
Larger number of magnitude 2 and 3 s and many fewer M5 s
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 m and greater.) 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?
On log scale The relationship looks linear and we d refer to this as a log linear relationship 1000 Number of earthquakes per year 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 The dependant variable y islog N What is b? the intercept 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 Where -b is the slope and c is the intercept. c 0.01 5 6 7 8 9 10 Richter Magnitude
Data 2016 to present for M>4
January 12 th Haitian magnitude 7.0 earthquake
If you had observations for 4.5<M<6.2 estimate the frequency of occurrence of M7.0 earthquakes Gutenberg Richter (frequency magnitude) plot N (per year - magnitude m and higher) 100 10 1 0.1 Notice the plot axis formats Haiti (1973-2010) Magnitude 2 and higher 0.01 2 3 4 5 6 7 8 LogN = -0.9258M + 5.3788 Magnitude log( N) bm c?
Look at problem in today s group worksheet)
How do you solve for N? log N 0.926m 5.38 log N 0.926(7.2) 5.38 log N 1.29 What is the quantity log 10 N?
Wrap-up and Status Need more time on group problems 2 next time? Questions about the 1 st set of group problems? 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 Questions on the next set of 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.