geol 351 - geomath
DESCRIPTION
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. Explanations for lowered gravity over mountain belts. - PowerPoint PPT PresentationTRANSCRIPT
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Geol 351 - Geomath
Tom Wilson, Department of Geology and Geography
tom.h.wilsontom. [email protected]
Department of Geology and GeographyWest Virginia University
Morgantown, WV
Recap some ideas associated with isostacy and curve fitting
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Explanations for lowered gravity over mountain belts
Tom Wilson, Department of Geology and Geography
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.
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Other examples of isostatic computations
Tom Wilson, Department of Geology and Geography
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Another possibility
Tom Wilson, Department of Geology and Geography
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Tom Wilson, Department of Geology and Geography
At A 2.9 x 40 = 116
The product of density and thickness must remain constant in the Pratt model.
ACB
At B C x 42 = 116
At C C x 50 = 116 C=2.32
C=2.76
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Some expected differences in the mass balance equations
Tom Wilson, Department of Geology and Geography
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Island arc systems – isostacy in flux
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
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Topographic extremes
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
Japan Archipelago North American Plate
Philippine Sea Plate
Pacific Plate
Eurasian Plate
Jap
an T
ren
chJa
pan
Tre
nch
Kuril Trench
Izu-B
on
in T
rench
Nankai Trough
Izu-B
on
in A
rc
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Tom Wilson, Department of Geology and Geography
The Earth’s gravitational field
In the red areas you weigh more and in the blue areas you weigh less.
Geological Survey of Japan
g ~0.6 cm/sec2
North American Plate
Philippine Sea Plate
Pacific Plate
Eurasian Plate
Jap
an T
ren
ch
Kuril T
rench
Izu-B
on
in T
rench
Nankai Trough
Izu-B
on
in A
rc
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Quaternary vertical uplift
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
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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.
Geological Survey of Japan
Total difference of about 0.1 cm/sec2 from the Alpine region into the Japan Sea
Japan Sea
Mountainous region
Incipient
subduction zone
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Schematic representation of subduction zone
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan
The back-arc area in the Japan sea, however, consists predominantly of oceanic crust.
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Varying degrees of underplating
Tom Wilson, Department of Geology and Geography
Watts, 2001
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Seismic profiling provides time-lapse view of coupled loading and deposition
Tom Wilson, Department of Geology and Geography
Watts, 2001
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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
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Gravity models reveal changes in crustal thickness
Tom Wilson, Department of Geology and Geography
Crustal thickness in WV Derived from Gravity Model Studies
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On Mars too?
Tom Wilson, Department of Geology and Geography
http://www.sciencedaily.com/releases/2008/04/080420114718.htm
http://www.nasa.gov/mission_pages/MRO/multimedia/phillips-20080515.html
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What forces drive plate motion?
Tom Wilson, Department of Geology and Geography
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Slab pull and ridge push
Tom Wilson, Department of Geology and Geography
http://quakeinfo.ucsd.edu/~gabi/sio15/lectures/Lecture04.html
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Slab pull and ridge push relate to isostacy
Tom Wilson, Department of Geology and Geography
The ridge push force
The slab pull forceA simple formulation for the slab pull per unit length is
sp slabF V 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
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See Excel file RidgePush_SlabPull
Tom Wilson, Department of Geology and Geography
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The weight of the mountains exerts a force on adjacent oceanic plates and mantle
Tom Wilson, Department of Geology and Geography
http://www.geosci.usyd.edu.au/users/prey/Teaching/Geos-3003/Lectures/geos3003_IsostasySld1.html
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Island arc seismicity
Tom Wilson, Department of Geology and Geography
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.
Geological Survey of Japan
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Recall the Gutenberg-Richter relationship
Tom Wilson, Department of Geology and Geography
5 6 7 8 9 10
Richter Magnitude
0.01
0.1
1
10
100
1000
Num
ber
of e
arth
quak
es p
er y
ear
abmN log
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.
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Tom Wilson, Department of Geology and Geography
1/ 2log 2 log( )N b A However, the relationship
indicates that log N will also vary in proportion to the log of the fault surface area. Hence, we could also
-2
-1
0
1
2
3
Log
of
the
Num
ber
of E
arth
quak
es p
er Y
ear
1 10 100 1000
Square Root of Fault Plane Area (kilometers) (Characteristic Linear Dimension)
1/ 2 )where r A
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Tom Wilson, Department of Geology and Geography
Gutenberg Richter relation in Japan
m
0 1 2 3 4 5 6 7
Frequency-Magnitude data (west-central Japan)
1
10
100
1000
N
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Tom Wilson, Department of Geology and Geography
m
0 1 2 3 4 5 6 7
Frequency-Magnitude data (west-central Japan)
1
10
100
1000
N
Slope = b =-1.16
intercept = 6.06
In this fitting lab you’ll calculate the slope and
intercept for the “best-fit” line
In this example -
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Tom Wilson, Department of Geology and Geography
Frequency-Magnitude data (west-central Japan)
m
0 1 2 3 4 5 6 7 8
0.01
0.1
1
10
100
1000
N
?
06.616.1log mN
Recall that once we know the slope and intercept of the Gutenberg-Richter relationship, e.g. As in -
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.
Doing this yields the prediction that in this region of Japan there will be 1 earthquake with magnitude 7 or greater every 115 years.
log 8.12 6.06N
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Calculating N and 1/N
Tom Wilson, Department of Geology and Geography
06.616.1log mNlog 8.12 6.06N
log 2.06N
log 2.0610 10
7 & 0.00871
1 114.87 &
N
m greateror N
year
yearsor N m 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.
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Observations and predictions
Tom Wilson, Department of Geology and Geography
3 1
34
37
40
129 132 135 138 14 1 144
JA PA N SEA
PA CIFIC PLA TE
M 7 .0 , 16 0 0 - 19 9 7
5 .7 M 6 .9 , 18 8 5 - 19 9 7
4 . 1 M 5 .6 , 19 2 6 - 19 9 7
PHILIPPINE SEA PLA TE
IST L
MT L
Izu Peninsula
C o m puta t io n po ints
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.
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Power laws and fractals
Tom Wilson, Department of Geology and Geography
DCrN
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.
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Tom Wilson, Department of Geology and Geography
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Tom Wilson, Department of Geology and Geography
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.
Relationship described by power laws
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Let’s look at the power law and GR problem in Excel
Tom Wilson, Department of Geology and Geography
What do you get when you take the log of N=Cr-D?
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Gutenberg-Richter relationshipUsing exponential and linear fitting approaches
Tom Wilson, Department of Geology and Geography
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Show that these two forms are equivalent
Tom Wilson, Department of Geology and Geography
Note that log 1,151,607.06 = 6.0613
Also note that log(e-2.66x) = -2.66log(e) =
-1.155
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You can do it either way
Tom Wilson, Department of Geology and Geography
Note that b=1.157 and c (the
intercept) = 6.06
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In class problems
Tom Wilson, Department of Geology and Geography
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Practice test to help you review
Tom Wilson, Department of Geology and Geography
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Currently with a look ahead
Tom Wilson, Department of Geology and Geography
• 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!