You should be able to show that A0 is -15,000 years. That means it will take 15,000 years for the lake to fill up.

Age =?

-1x105

-5x104

0

5x104

1x105

Age

(ye

ars)

-50 0 50 100

Depth (meters)

-15,000

present day depth at age = 0.

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 have to introduce non-linearity into the formulation. So let’s start looking at some non-linear functions.

Quadratic vs. Linear Behavior

-50 0 50 100

Depth (meters)

-100000

-50000

0

50000

100000

150000

Age

Compare the functions

15,000-D1000Aand (in red)

15,000-D10003 2 DA

What kind of equation is this?

The increase of temperature with depth beneath the earth’s surface is a non-linear process.

Waltham presents the following table

0

1000

2000

3000

4000

5000

T

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

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?

Let’s try a quadratic.

1020 xT

101725.1 xTFrom 100-700km the relationship

provides a good approximation.

works well.

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 temperature variations rise non-linearly toward a maximum value (there is one bend in the curve), however, the quadratic equation (second order polynomial) does not do an adequate job of defining these variations with depth.

Noting the number of bends in the curve might provide you with a good starting point. You could then increase the order to obtain further improvements.

5000

3000

1000

T

0

2000

4000

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

5000

3000

1000

T

0

2000

4000

0 1000 2000 3000 4000 5000 6000 7000

Depth (km)

93064.100031.01085.21012.1 238412 xxxxdxT

Waltham offers the following 4th order polynomial as a better estimate of temperature

variations with depth.

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?

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 and 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 0.368 of its initial value.

-cz0 e In the form c is the reciprocal of that depth.

5 6 7 8 9 10

Richter Magnitude

0

100

200

300

400

500

600

Num

ber

of e

arth

quak

es p

er y

ear

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?

0.01

0.1

1

10

100

1000

Num

ber

of e

arth

quak

es p

er y

ear

5 6 7 8 9 10

Richter Magnitude

This looks like a linear relationship. Recall the formula for a straight line?

5 6 7 8 9 10

Richter Magnitude

0.01

0.1

1

10

100

1000

Num

ber

of e

arth

quak

es p

er y

ear

cbmN log

The Gutenberg-Richter Relation

-b is the slope and c is the intercept.

Conceptual Age-Depth Relationship

0

2000

4000

6000

8000

10000

12000

14000

16000

0 500 1000 1500 2000 2500

Depth (cm)

Ag

e (y

ears

)Age

Depth (cm)

0 500 1000 1500 2000 2500

Ag

e (

yea

rs)

0

2000

4000

6000

8000

10000

12000

14000

16000Recent Sedimentation Record - North Sea

Conceptual Age-Depth Relationship

0

2000

4000

6000

8000

10000

12000

14000

16000

0 500 1000 1500 2000 2500

Depth (cm)

Ag

e (y

ears

)Age

Finish reading Chapters 1 and 2 (pages 1 through 38) of Waltham

Tuesday, 29th - Hand in the basic review problems