Mathematical Methods in GeologyGEOL MA/MAL 595

Text: Mathematics, A Simple Tool for Geologists by David Waltham

Chain of Craters Rd, Hawaii, Feb 13, 2003.Road built on older flows 1960's & several 100 yrs older

Klyuchevskoy VolcanoKamchatka PeninsulaMay 31, 2007

What is Quantitative About a Volcanic Eruption ?

Course Goals

* To improve simple math concepts and tools by using them with geological examples

* Improve your ability to apply math to geological problems

* Think of mathematics as a language- method of communication- must take each lesson one at a time- skipping lessons can lead to frustration

Traditional Geology is often “Qualitative”

- What happened ? - What is the order or sequence of events ?

Let's Look at This Problem “Quanitatively”

What time did the man die ?

What is his temperature now ?

What was the location of the man when he was killed ?

Example of a Sedimentary Basin

What can we say about this sedimentary basin?

How is sediment traveling in this lake bed ?

What is the order of deposition ?

Can we say anything else ?

Sedimentary Basin: A More Quantitative Approach

Which sediment layer is oldest, youngest ?

Does this suggest a relationship between depth and age ?

Assuming the sedimentation rate is constant, how could we write this relationship ?

Sedimentary Basin: A More Quantitative Approach

Which sediment layer is oldest, youngest ?

Does this suggest a relationship between depth and age ?

Assuming the sedimentation rate is constant, how could we write this relationship ?

Depth = Age

Depth = Age

Proportionality Constant

If Depth increases, how does Age change ?

Depth is proportional to Age if both increase.

Let Depth be described in meters and Age in years.

If one meter of sediment is deposited in one year, what is the constant of proportionality ?

If two meters of sediment is deposited in one year, what is the constant of proportionality ?

Proportionality Constant

We can replace the words with letters:Depth = DAge = A

Depth = k x AgeDepth = k x Age

D = k A

What are the units of k ?The constant, k, tells us how rapidly sediments accumulate.What does a large value of k tell us about accumulation ?What does a small value of k tell us ?

Mono Lake, California:

If 4 centimeters (cm) were deposited in one yr,what was the rate of sedimentation ?

Pyramid Lake, Nevada:

The sediment layer of magnetization is much thinner in Lake Pyramid than observed in Lake Mono.

What may have caused this ?

Magnetic Reversals Recorded in Lake Sediments

If 2 cm were deposited in one yr,what was the rate of sedimentation in Pyramid Lake ?

(Actual estimated rates are much slower 25 cm/kyr and 12 cm/kyr)

Quantifying Geological Processes

You have just produced mathematical expressions relating geological variables.

Did we gain anything from this effort ?

Was the accuracy of our estimates improved ?

Only SOMETIMES !

Sometimes a mathematical expression won't tell you anythingyou don't already know.

But the ease of manipulating or re-arranging an equation can oftengive us new insight into geological processes.

Mathematical equations are also very consistent. (Same data –> same answer)

D = k A

Quantifying Geological Processes

D = k AMathematical expressions can be tested.

Expressions developed from known data can be used to make predictions of data we don't have yet – that are difficult to measure.

In the previous example, you could predict the age of a particulardeposit in Mono Lake. You could then test this prediction byobtaining a geochemical dating method.

Example – earthquake hazard – Chino Earthquake - movie.

Can Mathematics be Wrong ?

D = k A

What could we have overlooked in this example ?

Are there other physical properties of this problem which couldchange our results or calculations ?

Mathematical equations are rarely 100% correct (almost never)!

We hope that if the expressions represent the major physicalproperties of the system, that the results will be close.

How to Solve ProblemsWhat is a problem ?

- By definition, “an obstacle which makes it difficult to achieve a desired goal, objective, or purpose”

(Wikipedia)literally: “obstacle”Etymology: from proballein to throw forward

from pro- forward and ballein to throw(Merriam-Webster)

A “Little Problem” - just a little difficultA “Great Problem” - very difficult

- Some degree of difficulty belongs to every notion of a problem

- Where there is no difficulty, there is no problem.

How to Solve Problems

Read / Understand the problem- “Know-how” (practical experience) is most valuable tool- Write down/discuss what (anything) you know about the problem- Draw a picture or sketch- Write down what you don't know- Write down what you want to find

Devise a Plan

Carry out the Plan- Write explanation of everything you do- Use written explanations of equations- Should have more text than math

Look back at your work

Four Step approach:

A Few Example Problems:

1. A bird trying to find his partner, starting walking from point P, and walked one meter due north. Then he changed direction and walked one meter due east. Then he turned again to the right and walked one meter due south, and arrived exactly at the point P he started from. What color was the bird ? Why didn't he fly the route ?

2. Paul wants a piece of land, exactly level, which has north-south, and the two others exactly east-west, and each boundary line measures exactly 100 ft. Can Paul buysuch a piece of land in the U.S. ?

To solve a problem use the steps we discussed:

1. Make a list of anything you know about this problem2. Make a list of what you do not know, what you need to find3. DRAW a simple picture of the problem!

then...

4. Devise a plan to solve the problem5. Carry out your plan6. Look back at your work.

A Few Example Problems:

2. Paul wants a piece of land, exactly level, which has four boundary lines. Two boundary lines run exactlynorth-south, and the two others exactly east-west, and each boundary line measures exactly 100 ft. Can Paul buysuch a piece of land in the U.S. ?