Mathematical Methods in Geology

1
Mathematical Methods in Geology GEOL MA/MAL 595
Text Mathematics, A Simple Tool for Geologists
by David Waltham
2
What is Quantitative About a Volcanic Eruption ?
3
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
4
Traditional Geology is often Qualitative
- What happened ? - What is the order or
sequence of events ?
5
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 ?
6
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 ?
7
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 ?
8
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 ?
9
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 ?
10
Proportionality Constant
We can replace the words with letters Depth
D Age 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 ?
11
Magnetic Reversals Recorded in Lake Sediments
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 ?
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)
12
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 ?
D k A
Only SOMETIMES !
Sometimes a mathematical expression won't tell
you anything you don't already know. But the
ease of manipulating or re-arranging an equation
can often give us new insight into geological
processes. Mathematical equations are also very
consistent. (Same data gt same answer)
13
Quantifying Geological Processes
Mathematical 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 particular deposit in Mono
Lake. You could then test this prediction
by obtaining a geochemical dating
method. Example earthquake hazard Chino
Earthquake - movie.
D k A
14
Can Mathematics be Wrong ?
D k A
What could we have overlooked in this example
? Are there other physical properties of this
problem which could change our results or
calculations ?
Mathematical equations are rarely 100 correct
(almost never)!
We hope that if the expressions represent the
major physical properties of the system, that the
results will be close.
15
How to Solve Problems
What 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
difficult A Great Problem - very difficult -
Some degree of difficulty belongs to every notion
of a problem - Where there is no difficulty,
there is no problem.
16
How to Solve Problems
Four Step approach

• 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

17
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
?
To solve a problem use the steps we
discussed 1. Make a list of anything you know
about this problem 2. Make a list of what you do
not know, what you need to find 3. DRAW a simple
picture of the problem! then...
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 buy such a
piece of land in the U.S. ?
4. Devise a plan to solve the problem 5. Carry
out your plan 6. Look back at your work.
18
A Few Example Problems
2. Paul wants a piece of land, exactly level,
which has four boundary lines. Two boundary
lines run exactly north-south, and the two others
exactly east-west, and each boundary line
measures exactly 100 ft. Can Paul buy such a
piece of land in the U.S. ?