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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 102.06 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