Transcript No Slide Title - West Virginia University

Geol 351 - Geomath
Recap some ideas associated with isostacy
and curve fitting
tom.h.wilson
tom. [email protected]
Department of Geology and Geography
West Virginia University
Morgantown, WV
Tom Wilson, Department of Geology and Geography
Explanations for lowered gravity over
mountain belts
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.
Tom Wilson, Department of Geology and Geography
Other examples of isostatic computations
Tom Wilson, Department of Geology and Geography
Another possibility
Tom Wilson, Department of Geology and Geography
A
B
C
The product of density and thickness must
remain constant in the Pratt model.
At A 2.9 x 40 = 116
At B C x 42 = 116
At C C x 50 = 116
Tom Wilson, Department of Geology and Geography
C=2.76
C=2.32
Some expected differences in the mass
balance equations
Tom Wilson, Department of Geology and Geography
Island arc systems – isostacy in flux
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Topographic extremes
Japan Archipelago
North
American Plate
Eurasian Plate
Pacific Plate
Philippine
Sea Plate
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
North
American Plate
The Earth’s gravitational field
In the red areas you weigh more and
in the blue areas you weigh less.
g ~0.6 cm/sec2
Eurasian Plate
Pacific Plate
Philippine
Sea Plate
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Quaternary vertical uplift
Geological Survey of Japan
Tom Wilson, Department of Geology and Geography
The gravity anomaly map shown here indicates that the mountainous region is associated with an
extensive negative gravity anomaly (deep blue colors). This large regional scale gravity anomaly
is believed to be associated with thickening of the crust beneath the area. The low density crustal
root compensates for the mass of extensive mountain ranges that cover this region. Isostatic
equilibrium is achieved through thickening of the low-density mountain root.
Mountainous
region
Total difference of about 0.1 cm/sec2 from
the Alpine region into the Japan Sea
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Schematic representation of subduction zone
The back-arc area in the Japan sea, however,
consists predominantly of oceanic crust.
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Varying degrees of underplating
Watts, 2001
Tom Wilson, Department of Geology and Geography
Seismic profiling provides time-lapse view of
coupled loading and deposition
Watts, 2001
Tom Wilson, Department of Geology and Geography
Local crustal scale features reflected in the
Earth’s gravitational field
Tom Wilson, Department of Geology and Geography
http://pubs.usgs.gov/imap/i-2364-h/right.pdf
Gravity models reveal changes in crustal
thickness
Crustal thickness in WV Derived from Gravity Model Studies
Tom Wilson, Department of Geology and Geography
On Mars too?
http://www.nasa.gov/mission_pages/MRO/multimedia/phillips-20080515.html
http://www.sciencedaily.com/releases/2008/04/080420114718.htm
Tom Wilson, Department of Geology and Geography
What forces drive plate motion?
Tom Wilson, Department of Geology and Geography
Slab pull and ridge push
http://quakeinfo.ucsd.edu/~gabi/sio15/lectures/Lecture04.html
Tom Wilson, Department of Geology and Geography
Slab pull and ridge push relate to isostacy
The ridge push force
The slab pull force
A simple formulation for the slab pull per unit length is
Fsp  Vslab  g
A more accurate formulation takes into account the temperature dependence of
density, the diffusion of heat, and the velocity of the subducting slab.
http://www.geosci.usyd.edu.au/users/prey/Teaching/Geos-3003/Lectures/geos3003_ForcesSld5.html
Tom Wilson, Department of Geology and Geography
See Excel file RidgePush_SlabPull
Tom Wilson, Department of Geology and Geography
The weight of the mountains exerts a force on
adjacent oceanic plates and mantle
http://www.geosci.usyd.edu.au/users/prey/Teaching/Geos-3003/Lectures/geos3003_IsostasySld1.html
Tom Wilson, Department of Geology and Geography
Island arc seismicity
The problem assignment (see last page of
exercise), will be due next week. The
exercise requires that you derive a
relationship for specific frequency magnitude
data to estimate coefficients, and predict the
frequency of occurrence of magnitude 6 and
greater earthquakes in that area.
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Recall the Gutenberg-Richter relationship
Number of earthquakes per year
1000
log N  bm  a
100
10
1
0.1
0.01
5
6
7
8
Richter Magnitude
Tom Wilson, Department of Geology and Geography
9
10
we have the variables m vs N
plotted, where N is plotted on
an axis that is
logarithmically scaled. -b is
the slope and a is the
intercept.
1/ 2
log
N


2
b
log(
A
)
However, the relationship
where r  A1/ 2 )
indicates that log N will also vary in proportion to the
log of the fault surface area. Hence, we could also
Log of the Number of Earthquakes per Year
3
2
1
0
-1
-2
1
10
100
1000
Square Root of Fault Plane Area (kilometers)
(Characteristic Linear Dimension)
Tom Wilson, Department of Geology and Geography
Gutenberg Richter relation in Japan
Frequency-Magnitude data (west-central Japan)
1000
N
100
10
1
0
1
2
3
4
m
Tom Wilson, Department of Geology and Geography
5
6
7
Frequency-Magnitude data (west-central Japan)
1000
N
100
In this fitting lab you’ll
calculate the slope and
intercept for the “best-fit” line
10
1
0
1
2
3
4
5
6
7
m
In this example -
Slope = b =-1.16
intercept = 6.06
Tom Wilson, Department of Geology and Geography
Recall that once we know the slope and intercept
of the Gutenberg-Richter relationship, e.g. As in Frequency-Magnitude data (west-central Japan)
we can estimate the probability
or frequency of occurrence of
an earthquake with magnitude
7.0 or greater by substituting
m=7 in the above equation.
1000
100
N
log N  1.16m  6.06
10
1
0.1
log N  8.12  6.06
0.01
0
1
2
3
4
m
Tom Wilson, Department of Geology and Geography
5
6
7
?8
Doing this yields the prediction
that in this region of Japan
there will be 1 earthquake with
magnitude 7 or greater every
115 years.
Calculating N and 1/N
log N  1.16m  6.06
log N  8.12  6.06
log N  2.06
10log N  102.06
m7 & greater
or N  0.00871
year
years
or 1  114.8
N
m7 & greater
There’s about a one in a hundred chance of having a
magnitude 7 or greater earthquake in any given year,
but over a 115 year time period the odds are close to 1
that a magnitude 7 earthquake will occur in this area.
Tom Wilson, Department of Geology and Geography
Observations and predictions
M 7.0, 1600-1997
5.7 M 6.9, 1885-1997
4.1 M 5.6, 1926-1997
Computation points
Historical activity in the
surrounding area over the past
400 years reveals the presence
of 3 earthquakes with
magnitude 7 and greater in this
region in good agreement with
the predictions from the
Gutenberg-Richter relation.
40
JAPAN SEA
ISTL
37
34
Izu
Peninsula
MTL
PACIFIC
PLATE
31
129
PHILIPPINE SEA PLATE
132
135
Tom Wilson, Department of Geology and Geography
138
141
144
Power laws and fractals
Another way to look at this relationship is to say that
it states that the number of breaks (N) is inversely
proportional to fragment size (r). Power law
fragmentation relationships have long been
recognized in geologic applications.
N  Cr
Tom Wilson, Department of Geology and Geography
D
Tom Wilson, Department of Geology and Geography
Relationship described by power laws
Box counting is a method used to determine the fractal dimension. The process
begins by dividing an area into a few large boxes or square subdivisions and
then counting the number of boxes that contain parts of the pattern. One then
decreases the box size and then counts again. The process is repeated for
successively smaller and smaller boxes and the results are plotted in a logN vs
logr or log of number of boxes on a side as shown above. The slope of that line
is the fractal dimension.
Tom Wilson, Department of Geology and Geography
Let’s look at the power law and GR problem in
Excel
What do you get when you take the
log of N=Cr-D?
Tom Wilson, Department of Geology and Geography
Gutenberg-Richter relationship
Using exponential and linear fitting approaches
Tom Wilson, Department of Geology and Geography
Show that these two forms are equivalent
Note that log
1,151,607.06 = 6.0613
Also note that log(e-2.66x)
= -2.66log(e) =
-1.155
Tom Wilson, Department of Geology and Geography
You can do it either way
Note that b=1.157
and c (the
intercept) = 6.06
Tom Wilson, Department of Geology and Geography
In class problems
Tom Wilson, Department of Geology and Geography
Practice test to help you review
Tom Wilson, Department of Geology and Geography
Currently with a look ahead
• All recent work (isostacy, 3.10, 3.11 & settling
velocity problem) should have been turned in no
later than yesterday.
• All work turned in has been graded and returned
• There may be some in class work undertaken as
part of the mid term review, but nothing else is due
till after the mid term.
• Problems due after the mid term include book
problems 4.7 and 4.10 and the fitting lab problem
(either option I or II).
• Spend your time reviewing and getting ready for
next Thursday’s mid term exam!
Tom Wilson, Department of Geology and Geography