tom.h.wilson Department of Geology and Geography West Virginia University Morgantown, WV

Practice questions and discussion Problem 1: Given the velocity determine the distance from the ridge.

Velocity (spreading rate) = 3 cm/year, Age = ~31My Distance to ridge = age.velocity= 930km

Given > Evaluate logM Problem 2: Refer to pages for basic discussions of mathematical manipulations of logs and exponentials.

Problem 2:

Evaluate the log 2 5. Show work on attached paper. Problem 3: See page 39

We’ve already worked with three bases - 2, 10 and e. Whatever the base, the logging operation is the same. How do we find these powers? 31

In general, or Try the following on your own 32

log 10 is referred to as the common logarithm thus log e or ln is referred to as the natural logarithm. All other bases are usually specified by a subscript on the log, e.g. 33

Sediment grain size classification using the phi scale -

Porosity as a function of depth is expressed as, where  (z) is the porosity at depth z, is a constant term,  0 is the porosity at z = 0, e is the natural base and z is the depth and has units of kilometers. Given that  0 is 0.6 and that  (100meters) =0.57, a) determine and b) determine the porosity at a depth of 1 kilometer. Show your work on the attached paper. Problem 4:

See pages 56 and 57

In problem 4 you have to solve for rather than z. Take ln to get > Then do algebra to solve for.

Can you evaluate the natural log of Recall some slides from earlier lectures

is a straight line. Power law relationships end up being straight lines when the log of the relationships is taken. In our next computer lab we’ll determine the coefficients c (or ) and ln(  0 ) that define the straight line relationship above between ln(  ) and z. We will also estimate power law and general polynomial interrelationships using PsiPlot.

8 kilometer crustal root 1 km topo relief  Crust = 2.67 g/cm 3 moho  Mantle Lithosphere = 3.3 g/cm 3 Mountain 5. The figure on the next page shows a highly simplified mountain range having 1 km of relief above the surrounding area. A crustal root extends 8km into the mantle lithosphere. Assume the crust has a density of 2.67 g/cm 3 and that the mantle lithosphere has a density of 3.3 g/cm 3. Determine whether the kilometer topographic relief is compensated. If it is not, how deep should the crustal root be to compensate the mountain?

Back to isostacy- The ideas we’ve been playing around with must have occurred to Airy. You can see the analogy between ice and water in his conceptualization of mountain highlands being compensated by deep mountain roots shown below.

In the diagram below left we have an equilibrium condition. In the diagram below right, we have upset this equilibrium. How deep must the mountain root be to stabilize a mountain with elevation e ?

In the diagram below we refer to the compensation depth. This depth is the depth above which the combined weight of a column of mantle and crust of unit horizontal cross section is constant. Regardless of where you are, the total mass of material overlying the compensation depth will be constant.

If the weight of material above a reference depth is not constant then the crust is not in equilibrium or crustal roots will have to extend below that depth to compensate for the mass excess. The relationship that must hold for the combined weight of crust and mantle above the compensation depth allows us to solve for r (see below)...

Again we have simplified the equation by assuming that the horizontal cross section of these vertical columns has equal area in all cases, hence the areas cancel out and the mass equivalence relationship reduces to the product of the density and thickness (l, d, L or D).

Take a few moments and verify that

8 kilometer crustal root 1 km topo relief  Crust = 2.67 g/cm 3 moho  Mantle Lithosphere = 3.3 g/cm 3 Mountain AB The mass between the surface and the compensation depth at A must be equal to that at point be from the surface to the compensation depth. The basic equation to solve is Will work in class -

An example Computer exam -

Statement of the problem- Calculate the depth to water table as a function of distance from the well for cones of depression having radii of 300 meters and 2000 meters.

Basic PsiPlot Setup-

Summary of results- 1. The computations reveal that the cone of depression rises sharply in the vicinity of the well. A rise of nearly 1.5 meters occurs within a distance of 5 meters from the well center. At distances of 5 to 25 meters from the well, the water table rises much more gradually by only about 1/2 meter over the 20 meter range.

Summary of results (cont.)- 2. The water level drop associated with the 2km radius cone of depression is greater than that associated with the 300 meter radius cone of depression by about 0.75 meters. Otherwise the two curves are very similar in shape.

Using Excell -

It’s always a good idea to make one computation by hand!

Prolem 2 -PsiPlot

Statement of the problem - Compute the subsidence of the seafloor relative to the ridge crest for a 100 million year time span. Also compute the change in depth per million year time intervals during that 100 million year time frame.

Summary of the results- Ocean crust drops rapidly from the ridge crest by about 1000 meters during the first 10 million years after its formation. Thereafter, it drops more gradually at the rate of about 1000 meters per 40 million years.

The plot of change in depth per million year time interval provides a nice illustration of the variations in subsidence rate. Initially the subsidence rate exceeds 350 meters per million years and drops quickly - within about 10 million years to subsidence rates of meters per million years.

Problem 2 - EXCEL