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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 mid1800’s a largescale 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 higherdensity (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 higherdensity material (Earth’s mantle), similar to (lowdensity) icebergs
ﬂoating on (higherdensity) water. The base of the crust is thus an exaggerated, mir
ror image of the topography. Areas of high elevation have lowdensity “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.‘...,'yv,.... _ 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
IIIIIIII 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 andAiry 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’mra'.°u,.'4¢w4  l
Mantle
b) Airy Model
Sea Level   ' ' ' ' ' ' '"
3 ’72
Mantle [12' = IIIIIIIlIIIIIIIIIIIIIIIII 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 twodimensional 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 twodimensional 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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 Fall '07
 ANANDAKRISHNAN
 Plate Tectonics, Structure of the Earth, Isostasy, Pratt Model, airy models

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