basic review continued - west virginia university

Basic Review continued

tom.h.wilson

[email protected]

Department of Geology and Geography

West Virginia University

Morgantown, WV

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

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

Depth x kAgeA 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?

How thick

was it

originally?

Over what

length of

time was it

deposited?

Assume a linear relationship?

Consider another area – an alien but very

interesting landscape

510,000 years

2.1 to 2.7 million

years ago

0.5 to 2.1 million

years ago

Milankovitch cycles …

Astronomical forcing of global climate:

Milankovitch Cycles

Take the quiz

http://www.sciencecourseware.org/eec/GlobalWarming/Tutorials/Milankovitch/

Solar insolation

http://pveducation.org/pvcdrom/properties-of-sunlight/calculation-of-solar-insolation

http://www.sciencedaily.com/releases/2008/04/080420114718.htm

http://www.nasa.gov/mission_pages/MRO/multimedia/phillips-20080515.html

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.

-10000

0

10000

20000

30000

40000

AG

E (

years

) 50000

60000

70000

0 20 40 60 80 100

Depth (meters)

These lines represent cases where the age

at 0 depth is different from 0

What are the

intercepts?

… 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.

0AD kA

Should we expect age

depth relationships

to be linear?

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.

Quadratic vs. Linear Behavior

-50 0 50 100

Depth (meters)

-100000

-50000

0

50000

100000

150000

Age

Here are two different possible

representations of age depth data

15,000-D1000A

and (in red)

15,000-D10003 2 DA

What kind of equation is this?

QuadraticsThe general form of a quadratic equation is

cbxaxy 2

-6 -4 -2 0 2 4 6

X

-75

-25

25

75

125

Y

Quadratics

23 60y x

22xy

20102 2 xxy

Similar examples

are presented in the

text.

roots

2 4

2

b b acx

a

The roots

Go to the common drive or visit Waltham’s site

(see link on the class page)


http://davidwaltham.com/mathematics-simple-tool-geologists/

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 ag

e

depth

Hint – try getting rid of the

linear term

In Excel (=xa=1,c=10 and b=0) = x2+10


Well we only wanted the positive half


Depth

Age

So we could use such a model with data support.

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.

Depth (km) Temperature (oC)

0 10

100 1150

400 1500

700 1900

2800 3700

5100 4300

6360 4300

Waltham presents the

following table

0

1000

2000

3000

4000

5000

T

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

0

1000

2000

3000

4000

5000

T

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

Can we come up with an

equation that will fit the

variations of temperature with

depth - for all depths?

See text.

11.4 10T x

101725.1 xT

From 100-700km the

relationship

provides a good approximation.

works well.

We’ll show you how to do this in Excel


y = 1.25x + 1016.7R² = 0.9985

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 100 200 300 400 500 600 700 800

Tem

p

Depth

T The slope tells you …

What does the intercept tell you?

Is it meaningful to us in this case?

0

1000

2000

3000

4000

5000

T

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

The quadratic relationship plotted below approximates

temperature depth variations.

77.679528.1)10537.1( 24 xxxT

5000

3000

1000

T

0

2000

4000

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

0

1000

2000

3000

4000

5000

T

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

111005.1)10255.8( 25 xxxT68053.1)10537.1( 24 xxxT

The formula - below right - is presented by Waltham. In

his estimate, he does not try to fit temperature variations

in the upper 100km.

5000

3000

1000

T

0

2000

4000

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.

Can we do better?

0

1000

2000

3000

4000

5000

T(O

C)

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

The general class of functions referred to as

polynomials include x to the power 0, 1, 2, 3, etc.

The straight line

cbxaxy 2

bmxy

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 is a second order

Polynomial, and the equation

1 2

1 2 0...n n n

n n ny a x a x a x a

is an nth order polynomial.

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.

1 2

1 2 0...n n n

n n ny a x a x a x a

5000

3000

1000

T

0

2000

4000

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

93064.100031.01085.21012.1 238412 dddxdxT

Waltham offers the following 4th order polynomial as a better

estimate of temperature variations with depth.

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

0aaxy p

is varied from 2 to -2 in -0.25

steps, and a0 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.

X2

X1.75

X-2

X-1.75

x

0 1 2 3 4 5

0

200

400

600

800

1000

1200

Y

Power Laws

?01.0 isWhat

?01.0 isWhat

42

2

2

2

See Powers.xls

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).

02/1 daxd

Spreading Ridge

0 200 400 600 800 1000

X (km)

0

1

2

3

4

5

D (km)

Ocean Floor Depth

What physical process do you think might be responsible for this pattern

of seafloor subsidence away from the spreading ridges?

Like a quadratic but not, since the exponent is not 2.

Another relationship


There is also a relationship between the age of the oceanic crust and its depth such that

12

od d aT

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 -

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%!

Depth

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Poro

sity

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 01 or 2

2d

d

Depth

0 1 2 3 4 5

Poro

sity

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

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.

0 01 or 2 ; in this case =0.6*2

2d d

d

It’s not constant and it

does not decrease linearly

0.6 2 where z = d = depthzx

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

cxaby

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

-cz0 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

z-

0 e

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.

z-

0 eNote that in the above form when z=,

01-

0

-

0 368.0

ee

represents the depth at which the porosity drops

to 1/e or 0.3678 of its initial value.

-cz0 e In the form 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?

World seismicity in the last 7 days


Larger number of magnitude 2 and 3’s and

many fewer M5’s


5 6 7 8 9 10

Richter Magnitude

0

100

200

300

400

500

600

Num

ber

of

eart

hquakes

per

year

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)

What would this plot look like if we plotted

the log of N versus m?

0.01

0.1

1

10

100

1000

Num

ber

of

eart

hquakes

per

year

5 6 7 8 9 10

Richter Magnitude

Looks almost like a straight

line. Recall the formula for a

straight line?

On

log s

cale

bmxy

0.01

0.1

1

10

100

1000

Num

ber

of

eart

hquakes

per

year

5 6 7 8 9 10

Richter Magnitude

What does y represent in this case?

Ny log

What is b?

the intercept

5 6 7 8 9 10

Richter Magnitude

0.01

0.1

1

10

100

1000

Num

ber

of

eart

hquakes

per

year

cbmN log

The Gutenberg-Richter Relationship

or frequency-magnitude relationship

-b is the slope

and c is the

intercept.

January 12th Haitian magnitude 7.0 earthquake

Shake map

USGS NEIC

USGS NEIC

Magnitude

2 3 4 5 6 7 8

N (

pe

r y

ea

r -

ma

gn

itu

de

m a

nd

hig

he

r)

0.01

0.1

1

10

100

Gutenberg Richter (frequency magnitude) plot

Haiti (1973-2010) Magnitude 2 and higher

log( )N bm c Notice the plot axis formats

Year

1975 1980 1985 1990 1995 2000 2005 2010

Ma

gn

itu

de

2

3

4

5

6

7Earthquake Occurrence 1973- present (Haiti and surroundings)

The seismograph network appears to have been upgraded in 1990.

Year

1920 1940 1960 1980 2000

Ma

gn

itu

de

6.0

6.5

7.0

7.5

8.0Large Earthquakes Haiti Region (last century)

In the last 110 years there

have been 9 magnitude 7

and greater earthquakes in

the region

Magnitude

2 3 4 5 6 7

Lo

g10 N

-2

-1

0

1

2

3Frequency (log10N) Magnitude Plot (Haitian Region)

Look at problem 19 (see

also 4 on today’s group

worksheet)

Magnitude

2 3 4 5 6 7

Lo

g10 N

-2

-1

0

1

2


logN=-0.935 m + 5.21

With what frequency should

we expect a magnitude 7

earthquake in the Haiti area?

log 0.935 5.21

log 0.935(7.2) 5.21

log 1.52

N m

N

N

Magnitude

2 3 4 5 6 7

Lo

g10 N

-2

-1

0

1

2


logN=-0.935 m + 5.21

How do you solve for N?

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/


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

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 the intro group problems

basic review continued - west virginia university

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