Basic Review continued

1
Geology 351 - Geomath
Basic Review continued
tom.h.wilson tom.wilson_at_mail.wvu.edu
Department of Geology and Geography West Virginia
University Morgantown, WV
2
Week 1

• We introduced the idea that quantitative
  representations of data serve as models that can
  be used to predict geological relationships. The
  simple example of linear age/depth relationship
  was used.
• We also examined mathematical models used to
  predict sea level rise and the decline in area of
  arctic ice coverage.
• Reviewed use of subscripts and exponents,
  scientific notation
• Emphasized attention to units conversion issues.
• Introduced spreadsheets as a tool for
  visualizing data and developing and testing
  mathematical models.
• Illustrated how calculations are sensitive to
  precision.

3
This week

• Examine mathematical models in more detail and
  note limitations inherent in the underlying
  assumptions of any given mathematical model.
• continue our review of basic math relationships,
  including linear, quadratic, polynomial,
  exponential, log and power law behavior.
• Introduce Walthams Excel files, e.g.
  S_Line.xls, Quadrat.xls, poly.xls, exp.xls,
  log.xls.
• Introduce basic Excel file structure.
• Work in groups on problems requiring use of these
  math tools.

4
Objectives for the day

• Review some familiar quantitative relationships
  used to model geological data
• Give you additional experience solving problems
  using these relationships

• Note
• Today, you will be asked to hand in answers to
  three questions related to group problems 1 3.
• Some additional group problems may be handed out
  in class today, but time may not permit us to
  complete these during class today. Additional
  time and review will be spent on Thursday.
• Warm-up problems 1-19 will be due NEXT Tuesday.
  We will continue our review of these problems and
  their solution through the week.

5
Chapter 2 -
Common relationships between geologic variables.
What kind of mathematical model can you use to
represent different processes?
We discussed this simple linear relationship last
time
This is a linear relationship
Whether it represents the geologic process
adequately is an assumption we make?
6
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.
What are the intercepts?
These lines represent cases where the age at 0
depth is different from 0
7
Should we expect age depth relationships to be
linear?
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.
8
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.
9
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.
10
The relationship becomes non-linear. The line
ymxb really isnt 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.
Here are two different possible representations
of age depth data
and (in red)
What kind of equation is this?
11
Quadratics The general form of a quadratic
equation is
Similar examples are presented in the text.
12
The increase of temperature with depth beneath
the earths surface (taken as a whole) is a
non-linear process.
Waltham presents the following table
?
See http//www.ucl.ac.uk/Mathematics/geomath/powco
ntext/poly.html
13
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
provides a good approximation.
From 100-700km the relationship
?
works well.
Can we come up with an equation that will fit the
variations of temperature with depth - for all
depths? See text.
14
The quadratic relationship plotted below
approximates temperature depth variations.
15
The formula - below right - is presented by
Waltham. In his estimate, he does not try to fit
temperature variations in the upper 100km.
16
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?
17
The general class of functions referred to as
polynomials include x to the power 0, 1, 2, 3,
etc. The straight line
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
is an nth order polynomial.
18
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 neednt have all the bends it is
permitted! Higher order generally permits better
fit of the curve to the observations.
19
Waltham offers the following 4th order polynomial
as a better estimate of temperature variations
with depth.
20
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
Powers
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.
See Powers.xls
21
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).
What physical process do you think might be
responsible for this pattern of seafloor
subsidence away from the spreading ridges?
22
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 Yab 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.
23
Waltham presents us with the following data table
-
Over the range of depth 0-4 km, the porosity
decreases from 60 to 3.75!
24
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.
25
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
are referred to as allometric growth laws or
exponential functions.
26
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
?0 is the initial porosity, c is a compaction
factor and z - the depth. Sometimes you will see
such exponential functions written as
In both cases, eexp2.71828
27
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.
Note that in the above form
when z?,
? represents the depth at which the porosity
drops to 1/e or 0.368 of its initial value.
In the form
c is the reciprocal of that depth.
28
Earthquake Seismology Application
The Gutenberg-Richter Relation
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?
29
World seismicity in the last 7 days
30
Larger number of magnitude 2 and 3s and many
fewer M5s
31
When in doubt -collect data.
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?
32
On log scale
Looks almost like a straight line. Recall the
formula for a straight line?
33
What does y represent in this case?
What is b?
the intercept
34
The Gutenberg-Richter Relationship or
frequency-magnitude relationship
-b is the slope and c is the intercept.
35
January 12th Haitian magnitude 7.0 earthquake
36
Shake map
USGS NEIC
37
USGS NEIC
38
Notice the plot axis formats
39
The seismograph network appears to have been
upgraded in 1990.
40
In the last 110 years there have been 9 magnitude
7 and greater earthquakes in the region
41
Look at problem 19 (see also 4 on todays group
worksheet)
42
With what frequency should we expect a magnitude
7 earthquake in the Haiti area?
43
How do you solve for N?
We will return to this class of problems for
further discussion on Thursday.
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Wrap-up and Status

• Complete and hand in answers to three questions
  related to group problems 1-3.
• Warm-up problems 1-19 will be due NEXT Tuesday.
  We will continue our review of these problems and
  their solution this Thursday.

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For Next Time
Finish reading Chapters 1 and 2 (pages 1 through
38) of Waltham
Were still going over background information
related to those 19 intro in-class problems.
Additional group/in-class problems focus your
attention on problems 15 through 18.
In the next class we will spend some time
reviewing logs and trig functions. Following
that, we will begin learning how to use to solve
some problems related to the material covered in
Chapters 1 and 2.