Geol 351 - Geomath Recap some ideas associated with isostacy and curve fitting tom.h.wilson tom. wilson@mail.wvu.edu 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. 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

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

Another possibility Tom Wilson, Department of Geology and Geography

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

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

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

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

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

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

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

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

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

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 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 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

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

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

"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

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

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

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

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

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

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

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