1. Geol Geomath
Recap some ideas associated with isostacy and curve fitting
tom.h.wilson
tom.
Department of Geology and Geography
West Virginia University
Morgantown, WV
Tom Wilson, Department of Geology and Geography

2. 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.
Lid refers to the mantle part of the lithosphere; lid is the brittle part of the asthenosphere
Lithosphere includes the crust and upper part of the mantle (the mantle lid)
Asthenosphere is upper mantle
Tom Wilson, Department of Geology and Geography

3. Other examples of isostatic computations
Tom Wilson, Department of Geology and Geography

4. Another possibility
Tom Wilson, Department of Geology and Geography

5. C B A
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
C=2.76
At C C x 50 = 116
C=2.32
Tom Wilson, Department of Geology and Geography

6. Some expected differences in the mass balance equations
Tom Wilson, Department of Geology and Geography

7. Island arc systems – isostacy in flux
To illustrate the variety of quantitative applications brought to bear on geological problems
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan

8. Topographic extremes Japan Archipelago North American Plate
Kuril Trench
Eurasian Plate
Japan Trench
Japan Trench
Pacific Plate
Nankai Trough
Izu-Bonin Arc
Izu-Bonin Trench
Philippine Sea Plate
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan

9. In the red areas you weigh more and in the blue areas you weigh less.
North American Plate
The Earth’s gravitational field
In the red areas you weigh more and in the blue areas you weigh less.
Kuril Trench
g ~0.6 cm/sec2
Eurasian Plate
Japan Trench
Pacific Plate
Nankai Trough
Izu-Bonin Trench
Izu-Bonin Arc
Philippine Sea Plate
Tom Wilson, Department of Geology and Geography
Geological Survey of Japan

10. Quaternary vertical uplift
Geological Survey of Japan
Tom Wilson, Department of Geology and Geography

11. Incipient subduction zone
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.
Japan Sea
Incipient subduction zone
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

12. 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

13. Varying degrees of underplating
Marquesas in the southern pacific - part of french polynesia
Canary offshore northern africa
Watts, 2001
Tom Wilson, Department of Geology and Geography

14. Seismic profiling provides time-lapse view of coupled loading and deposition
Onlapping suggests deposits originally sloping down onto ocean floor. Off lapping suggests sediments deposited in a crustal depression surrounding the volcanic island is it continued to develop. The weight of the volcanic deposits began to deform the lithosphere elastically.
Watts, 2001
Tom Wilson, Department of Geology and Geography

15. Local crustal scale features reflected in the Earth’s gravitational field
Tom Wilson, Department of Geology and Geography

16. Gravity models reveal changes in crustal thickness
Crustal thickness in WV Derived from Gravity Model Studies
Tom Wilson, Department of Geology and Geography

17. On Mars too?
Tom Wilson, Department of Geology and Geography

18. What forces drive plate motion?
Tom Wilson, Department of Geology and Geography

19. Slab pull and ridge push
Tom Wilson, Department of Geology and Geography

20. 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
A more accurate formulation takes into account the temperature dependence of density, the diffusion of heat, and the velocity of the subducting slab.
Tom Wilson, Department of Geology and Geography

21. See Excel file RidgePush_SlabPull
Tom Wilson, Department of Geology and Geography

22. The weight of the mountains exerts a force on adjacent oceanic plates and mantle
Tom Wilson, Department of Geology and Geography

23. 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

24. Recall the Gutenberg-Richter relationship
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.
Tom Wilson, Department of Geology and Geography

25. However, the relationship
indicates that log N will also vary in proportion to the log of the fault surface area. Hence, we could also
Tom Wilson, Department of Geology and Geography

26. Gutenberg Richter relation in Japan
Tom Wilson, Department of Geology and Geography

27. "Best fit" line
In this fitting lab you’ll calculate the slope and intercept for the “best-fit” line
In this example -
Slope = b =-1.16
intercept = 6.06
Tom Wilson, Department of Geology and Geography

28. 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.
Tom Wilson, Department of Geology and Geography

29. Calculating N and 1/N
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

30. Observations and predictions
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.
Tom Wilson, Department of Geology and Geography

31. 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.
Tom Wilson, Department of Geology and Geography

32. Tom Wilson, Department of Geology and Geography

33. 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

34. 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

35. Gutenberg-Richter relationship Using exponential and linear fitting approaches
Tom Wilson, Department of Geology and Geography

36. Show that these two forms are equivalent
Note that log 1,151, =
Also note that log(e-2.66x) = -2.66log(e) =
-1.155
Tom Wilson, Department of Geology and Geography

37. Note that b=1.157 and c (the intercept) = 6.06
You can do it either way
Note that b=1.157 and c (the intercept) = 6.06
Tom Wilson, Department of Geology and Geography

38. In class problems
Tom Wilson, Department of Geology and Geography

39. Practice test to help you review
Tom Wilson, Department of Geology and Geography

40. 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