Lillie237-243 - Isostasy 237 (that is, the initial length...

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Unformatted text preview: Isostasy 237 (that is, the initial length of the spring) measured at the base station represents the absolute gravity at that point. Measurements are then taken at other stations, with the changes in length of the spring recorded. The gravimeter is calibrated so that a given change in spring length (AL) represents a change in gravity (Ag) by a certain amount (in mGal). The acceleration (g) can then be computed by adding the value of Ag to the absolute gravity of the base station. At sea, gravity surveying is complicated by the fact that the measurement plat- form is unstable. Waves move the ship up and down, causing accelerations that add or subtract from the gravity. Also, like Earth’s rotation, the speed of the ship over the water results in an outward acceleration; in other words, the ship’s velocity adds to the velocity of Earth’s rotation. An additional correction, known as the Eotvo's correction, is therefore added to marine gravity measurements (Telford et al.,1976): lVEC = 7.503 V com) sina + 0.004154 VTi where: EC = Eotvos correction (mGal) V = speed of ship (knots; 1 knot = 1.852 km/hr = 0.5144 m/s) 4) = latitude of the observation point (degrees) or = course direction of ship (azimuth, in degrees). ISOSTASY Until quite recently, surveyors leveled their instruments by suspending a lead weight (plumb bob) on a string. In the vicinity of large mountains, it was recognized that a correction must be made because the mass excess of the mountains standing high above the surveyor’s location made the plumb bob deviate slightly from the vertical (Fig. 8.16a). In the mid-1800’s a large-scale survey of India was undertaken. Approaching the Himalaya Mountains from the plains to the south, the correction was calculated and applied. A systematic error was later recognized, however, as the plumb bob was not deviated toward the mountains as much as it should have been (Fig. 8.16b). This difference was attributed to mass deficiency within the Earth, beneath the excess mass of the mountains. Pratt and Airy Models (Local Isostasy) Scientists proposed two models to explain how the mass deficiency relates to the topography of the Himalayas. Pratt assumed that the crust of the Earth comprised blocks of different density; blocks of lower density need to extend farther into the air in order to exert the same pressure as thinner blocks of higher-density (Fig. 8.17a). The situation is analogous to blocks of wood, each of different density, floating on water. By the Pratt model, the base of the crust is flat, so that the surface of equal pressure (depth of compensation) is essentially a flat crust/mantle boundary. In the model of Airy (Fig. 8.17b), crustal blocks have equal density, but they float on higher-density material (Earth’s mantle), similar to (low-density) icebergs floating on (higher-density) water. The base of the crust is thus an exaggerated, mir- ror image of the topography. Areas of high elevation have low-density “crustal roots” supporting their weight, much like a beach ball lifting part of a swimmer’s body out of the water. "w my" 238 Chapter 8 Gravity and lsostasy FIGURE 8.16 a) Expected deflection ( a plumb bob (highly exaggerated), due the attraction of the mass of a mountaii range. b) The actual deflection for the Himalayas was less than expected, due a deficiency of mass beneath the mountains. 0 = Angle of Deflection Plumb Bob B = Expected Deflection ¢ = Actual Deflection Mass Dewey Beneath Mountains FIGURE 8.17 Pratt and Airy models of local isostatic compensation. In both models, pressure exerted by crustal a) Pratt Model H . hast Density . v2.,,....,.,r.‘...,'y-v,.... _ I _ vo~~vovovtv~v6 :t’: 0‘o’e‘o‘o‘o’o’o’o’o‘o‘o‘o’ o'v’3’:‘:’:':‘:’:’:‘:’:‘:‘:‘:‘:’: celumns ls equal on honzontal planes at o‘e’o‘o’o‘o‘o‘c’o‘o‘o'o’o‘o‘o'o‘ o‘c’o'o‘o‘o‘o‘o‘e‘o‘o’o’o‘o’o°o‘¢ ‘ 5.,o....,o.5......‘955...’ .o.o.o.m....o.o.o¢.§.o.o.o,o. and below the depth of compensatlon. 'o‘o’0‘o‘o’0’9‘0’30‘0‘0'0’0‘0‘0' 0'.'o‘o‘o‘o’o‘o‘o‘o‘o‘o°¢‘o’o‘o’c 09000000600000 $fi$fi$fi$$$$fi~5$fig Depth of Compensation 0009000000990... 0.00.09.09.09... lopOp&pOfifipopJp&p III-IIIII b) Airy Model Sea Level High Density High Density Depth of Compensation Hydrostatic pressure is the pressure exerted on a point within a body of water. Similarly, pressure at a given depth within the Earth (Fig. 8.1821) can be viewed as / lithostatic pressure, according to: P=pgz . where: ‘, P 2 pressure at the point within the Earth p = average density of the material above the point “ ’\_‘ a Isostasy 239 FIGURE 8.18 a) Pressure (P) at depth . I (z) is a function of the density (p) of the material above a point within the Earth. I I V I _ « §:1:;::2_3:::_::32§$2§ b) For the Pratt and Airy models, the P - Constant Depth ofOomponsafian pressure depends on the density and thickness (11) of crustal blocks. In both models, pressure equalizes at the depth of compensation. Airy Model P - Constant Depth of Compensation g = acceleration due to gravity (z 9.8 m/sz) z = depth to the point. For the Pratt and-Airy models (Fig. 8.18b), the pressure exerted by a crustal block can be expressed as: P = pgh where: P = pressure exerted by the crustal block p = density of the crustal block h = thickness of the crustal block. In both the Pratt and Airy models, the pressure must be the same everywhere at the depth of compensation. For the Pratt model, the base of each block is at the exact depth of compensation, so that: P = Pzghz : [333113 2 943114 = Psghs where: p2, p3, p4, p5 = density of each block h2, h3, h4, h5 = thickness of each block. Dividing out a constant gravitational acceleration (g): P/g = chz = 93113 = P4114 = Pshs In the particular Pratt model shown in Fig. 8.1%, p5 < p4 < p3 < pZ < p1, where p1 is the density of the substratum (Earth’s mantle). In an Airy model the crustal density (p2) is constant and less than the mantle density (p1). Only the thickest crustal block extends to the depth of compensation. For the Airy isostatic model in Fig. 8.1%, the pressure exerted at the depth of com- pensation (divided by g) is: P/g = chs = (92114 + Plhci) = (92113 + Pr 3, = (92112 + Pl 2’ ) where: hz’, h3’, h4’ = thickness of mantle column from the base of each crustal block to the depth of compensation. 240 Chapter 8 Gravity and Isostasy p,- Model FIGURE 8.19 Density (p) and thickness 3) a” (h, h') relationships for Pratt and Airy {H isostatic models. P = pressure; Sea Leve -- _‘ "‘;:’;;;;:;;;, ‘ ,1;;;;;;;;;;;;;;;;;;;;;;;s"t' ' ' :zw ‘ g = gravitational acceleration. 9.. 0. 9.0.59 00.. o .0 ’ i 0009 0.0 "O. .. coo. coo l'o‘e‘o‘o 0'... ‘ woo cool. oo.o.oeoo.0.¢eeoooo P/g a: Constant L- . . . - - - - V z’m-ra'-.°u,.'4¢-w4 - l Mantle b) Airy Model Sea Level-- - - ' ' ' ' ' ' '" 3 ’72 Mantle [12' = IIIIIIIl-IIIIIIIIIIIIIII-II Ill. IIIIIIIIIIIIIIIIII III Depth of Compensation Ain Isostatic Model Oceanh M (p z 0) Continent Mountains Water (p z 1.0 g/cms) -- \ Mantle Depth of Compensation Pressure - Constant FIGURE 8.20 Airy isostatic model. Oceanic regions have thin crust, relative to continental regions. The weight of extra mantle material beneath the thin oceanic crust pulls downward until just enough depth of water fills the basin to achieve isostatic equilibrium. Mountainous regions have thick crust, relative to normal continental regions. The crustal root exerts an upward force until it is balanced by the appropriate weight of mountains. While regions often exhibit components of both hypotheses, isostatic compen— sation is generally closer to the Airy than the Pratt model. Pure Airy isostatic com- pensation for regions with oceanic and continental crust, as well as thickened crust weighted down by mountains, might exhibit the form illustrated in Fig. 8.20. Notice that the crustal root beneath elevated regions is typically 5 to 8 times the height of the topographic relief. At the depth ofcompensation beneath each region, two equa— tions hold true. 1) The total pressure (P) exerted by each vertical column, divided by the gravitational acceleration (g), is constant: lsostasy 241 P/ g = paha + pwhw + pchc + pmhm = Constant where: pa = density of the air (p21 as 0) ha = thickness of the air column, up to the level of the highest topography pw = density of the water hW = thickness of the water column pc = density of the crust h0 = thickness of the crust pm = density of the mantle h,rn = thickness of the mantle column, down to the depth of compensation. 2) The total thickness (T) of each vertical column is constant: T = ha + hw + hc + hm = Constant If the isostatic column (P/ g) can be determined or assumed for one area, then solv- ing the two equations simultaneously can be used to estimate thicknesses (h) and/or densities (p) for vertical columns beneath other areas. Lithospheric Flexure (Regional lsostasy) Both the Pratt and Airy models assume local isostasy, whereby compensation occurs directly below a load (Fig. 8.21a); supporting materials behave like liquids, flowing to accommodate the load. In other words, the materials are assumed to have no rigidity. Most Earth materials, however, are somewhat rigid; the effect of a load is distributed over a broad area, depending on the flexaral rigidity of the supporting material. Models of regional isostasy therefore take lithospheric strength into account (Fig. 8.21b). A common model of regional isostatic compensation is that of an elastic plate that is bent by topographic and subsurface loads. The flexural rigidity (D) of the plate determines the degree to which the plate supports the load. The elastic plate model is analogous to a diving board, the load being the diver standing near the end of the board (Fig. 8.22). A thin, weak board (small D) bends greatly, especially near the diver. A thicker board of the same material behaves more rigidly; the diver causes a smaller deflection. The flexural rigidity (resistance to bending) thus depends on the elastic thickness of each board. a) Local lsostasy b) Regional Isostasy Load Load isostasy.Where there is no rigidity, compensation is directly below the load. b) Regional isostasy. Materials with rigidity are flexed, distributing the load over a broader region. FIGURE 8.21 The type of isostatic compensation depends on the flexural rigidity of the supporting material. a) Local 242 Chapter 8 Gravity and Isostasy Elastic Thickness FIGURE 8.22 Flexural rigidity. a) A thin diving board (small elastic thickness) has low flexural rigidity. b) A thick board (large elastic thickness) has high flexural rigidity. Strong (Th/ck) Plate I b Load DGPI'BSSIOD hfigfi’g’ Weak (Thin) Plate _ Pen. Elastic La“ EEW‘ESS'O” / U199 Thickness 7 Plate With No Strength FIGURE 8.23 a) Parameters for two-dimensional model of a plate flexed by a linear load. Both the plate and load extend infinitely in and out of the page. See text for definition of variables. b) Positions of depressions and bulges formed on the surface of a flexed plate. A strong plate has shallow but wide depressions. The depressions and peripheral bulges have larger amplitudes on a weak plate, but are closer to the load. A very weak plate collapses into local isostatic equilibrium. The deflection of a two-dimensional plate, due to a linear load depressing the plate’s surface, is developed by Turcotte and Schubert (1982).The model (Fig. 8.23a) assumes that material below the plate is fluid. The vertical deflection of points along the surface of the plate can be computed according to: D(d“w/d“x) + (9., - pa)gw = q(x) where: D = flexural rigidity of the plate w vertical deflection of the plate at x H nd lsostasy 243 X = horizontal distance from the load to a point on the surface of the plate pa = density of the material above the plate pb 2 density of the material below the plate g = gravitational acceleration q(x) = load applied to the top of the plate at x. Four important concepts are illustrated by solutions to the above equation (Fig. 8.23b): 1) a strong lithospheric plate (large D) will have a small amplitude deflection (small w), spread over a long wavelength; 2) a weak lithospheric plate (small D) has large deflection (large w), but over a smaller wavelength; 3) where plates have significant strength, an upward deflection (“ peripheral” or “flexural” bulge) develops some distance from the load, separated by a depression; 4) plates with no strength collapse into local isostatic equilibrium. Two simplified examples of lithospheric flexure are shown in Fig. 8.24. At a subduction zone (Fig. 8.24a), flexure is analogous to the bending at the edge of a diving board (Fig. 8.22). The load is primarily the topography of the accretionary wedge and volcanic are on the overriding plate. Flexure of the downgoing plate results in a depression (trench) and, farther out to sea, a bulge on the oceanic crust. The mass of high mountains puts a load on a plate that can be expressed in both directions (Fig. 8.24b). Depressions between the mountains and flexural bulges (“foreland basins”) can fill with sediment to considerable thickness. ' FIGURE 8.24 Examples of lithospheric a) ‘9 flexure. a) A flexural bulge and QQ 465$!) depression (trench) develop as the (“#6) é‘a Y. *3) downgoing plate is flexed at a subduction & Zone. b) The weight of a mountain range causes adjacent depressions that fill with sediment (foreland basins). Lithosphere Asthenosphere b) Mountain Range Load Fore/and (Mountains) Basin Flexural Bulge Elastic mlbkness ...
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This note was uploaded on 07/23/2008 for the course GEOSC 203 taught by Professor Anandakrishnan during the Fall '07 term at Pennsylvania State University, University Park.

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Lillie237-243 - Isostasy 237 (that is, the initial length...

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