Geol 351 - Geomath

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

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.

Other examples of isostatic computations


Another possibility



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

Some expected differences in the mass balance equations


Island arc systems – isostacy in flux


Geological Survey of Japan

Topographic extremes



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


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

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

Quaternary vertical uplift




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.


Total difference of about 0.1 cm/sec2 from the Alpine region into the Japan Sea

Japan Sea

Mountainous region

Incipient

subduction zone

Schematic representation of subduction zone



The back-arc area in the Japan sea, however, consists predominantly of oceanic crust.

Varying degrees of underplating


Watts, 2001

Seismic profiling provides time-lapse view of coupled loading and deposition


Watts, 2001

Local crustal scale features reflected in the Earth’s gravitational field


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

On Mars too?


http://www.sciencedaily.com/releases/2008/04/080420114718.htm

http://www.nasa.gov/mission_pages/MRO/multimedia/phillips-20080515.html

What forces drive plate motion?


Slab pull and ridge push


http://quakeinfo.ucsd.edu/~gabi/sio15/lectures/Lecture04.html

Slab pull and ridge push relate to isostacy


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

See Excel file RidgePush_SlabPull


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

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.


Recall the Gutenberg-Richter relationship


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.


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


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


m

0 1 2 3 4 5 6 7


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 -



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

Calculating N and 1/N


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.

Observations and predictions


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.

Power laws and fractals


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.


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

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?

Gutenberg-Richter relationshipUsing exponential and linear fitting approaches


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

You can do it either way


Note that b=1.157 and c (the

intercept) = 6.06

In class problems


Practice test to help you review


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!

Geol 351 - Geomath

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