Lillie223-237

Lillie223-237 - CHAPTERS Gravity and Isostasy gravity...

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Unformatted text preview: CHAPTERS Gravity and Isostasy gravity (grav' 9 re) n., [ < L. gravis, heavy], I. the force of attraction between masses 2. the force that tends to draw bodies in Earth’s sphere toward Earth ’s center. isostasy (i sit ste' sé) n., [ < Gr. isos, equal; < Gr. stasis, standing], a state of balance whereby columns of material exert equal pressure at and below a compensating depth. gravity and isostasy (grav’ a té end isti ste' se) n., the study of spatial variations in Earth’s gravitational field and their relationship to the distribution of mass within the Earth. Earth’s gravity and magnetic forces are potential fields that provide information on the nature of materials within the Earth. Potential fields are those in which the strength and direction of the field depend on the position of observation within the field; the strength of a potential field decreases with distance from the source. Compared to the magnetic field, Earth’s gravity field is simple. Lines of force for the gravity field are directed toward the center of the Earth, While magnetic field strength and direction depend on Earth’s positive and negative poles (Fig. 8.1). a) Gravity Field b) Magnetic Field / FIGURE 8.1 Earth’s potential fields. a) The gravity field is symmetric. Force vectors (arrows) have approximately equal magnitude and point toward the center of the Earth. b) The magnitude and direction of the magnetic field is governed by positive (south) and negative (north) poles. Magnitude varies by a factor of two from equator to pole. 223 224 Chapter 8 Gravity and Isostasy EARTH'S GRAVITY FIELD Gravity is the attraction on one body due to the mass of another body. The force of one body acting on another is given by Newton’s Law of Gravitation (Fig. 8.23): mimz r2 F=G where: F = force of attraction between the two objects (N) G = Universal Gravitational Constant (6.67 X 10"11 NmZ/kgz) m1, m2 = mass of the two objects (kg) r = distance between the centers of mass of the objects (in). The force (F) exerted on the object with mass m1 by the body with mass m2, is given by Newton’s Second Law of Motion (Fig. 82b): F = mla where: a = acceleration of object of mass m1 due to the gravitational attraction of the object with mass m2 (m/sz). Solving for the acceleration, then combining the two equations (Fig. 8.2c): F 1 Gmlm2 a : — : — 2 m1 In1 r Grn2 a = 2 I For Earth’s gravity field (Fig. 8.3a), let: a = g = gravitational acceleration observed on or above Earth’s surface; m2 = M = mass of the Earth; r = R = distance from the observation point to Earth’s center of mass; so that: FIGURE 8.2 a) The gravitational force between two objects is directly proportional to their masses (m1, m2), and inversely proportional to the square of their distance (r). b) The mass (m1), times the acceleration (a) due to mass (m2), determines the gravitational force (F). c) The acceleration due to gravity (a) of a body depends only on the mass of the attracting body (m2) and the distance to the center of that mass (r). Gravity Anomalies 225 FIGURE 8.3 a) The mass (M) of the Earth and radius (R) to Earth’s center determine the gravitational acceleration (g) of objects at and above Earth’s surface. b) The acceleration is the same (g), regardless of the mass of the object. c) Objects at Earth’s surface (radius R1) have greater acceleration than objects some distance above the surface (radius R2). The above equation illustrates two fundamental properties of gravity. 1) Acceleration due to gravity (g) does not depend on the mass (m1) attracted to the Earth (Fig. 8.3b); in the absence of air resistance, a small mass (feather) will accelerate toward Earth’s surface at the same rate as a large mass (safe). 2) The farther from Earth’s center of mass (that is, the greater the R), the smaller the gravitational acceleration (Fig. 8.3c); as a potential field, gravity thus obeys an inverse square law. The value of the gravitational acceleration on Earth’s surface varies from about 9. 78 m/s2 at the equator to about 9.83 m/32 at the poles (Fig. 8.4a).The smaller acceleration at the equator, compared to the poles, is because of the combination of three factors. 1) There is less inward acceleration because of outward acceleration * caused by the spin of the Earth; the spin (rotation) is greatest at the equator but reduces to zero at the poles. 2) There is less acceleration at the equator because of the Earth’s outward bulging, thereby increasing the radius (R) to the center of mass. 3) The added mass of the bulge creates more acceleration. Notice that the first two factors lessen the acceleration at the equator, while the third increases it. The net effect is the observed ~0.05 m/s2 difference. Gravitational acceleration (gravity) is commonly expressed in units of milli- gals (mGal), where: 1 Gal = 1 cm/s2 = 0.01 m/s2 so that: 1 mGal = 10‘3 Gal = 10‘3 cm/s2 = 10‘5 m/sz. Gravity, therefore, varies by about 5000 mGal from equator to pole (Fig. 8.4b). GRAVITY ANOMALIES Gravity observations can be used to interpret changes in mass below different regions of the Earth. To see the mass differences, the broad changes in gravity from equator to pole must be subtracted from station observations. This is accomplished 226 Chapter 8 Gravity and lsostasy a g z 9.83 m/s2 Increased Radius '49 ':(> z 9, 78 m 32 g Earth’s Rotation g / _ .49 Excess Mass <Lr:' +49 b g a 933,000 mGal , l Equator (¢ - 0‘) g, - 978,031.85 mGaI Pale (¢ - 90°) = 983,217. 72 mGaI l 7 Us FIGURE 8.4 a) Three main factors responsible for the difference in gravitational acceleration at the equator compared to the poles. b) Gravity increases from about 978,000 mGai at the equator, to about 983,000 mGal at the poles. 0) Variation in gravity from equator to pole, according to 1967 Reference Gravity Formula. by predicting the gravity value for a station’s latitude (theoretical gravity), then sub- tracting that value from the actual value at the station (observed gravity), yielding a gravity anomaly. Theoretical Gravity The average value of gravity for a given latitude is approximated by the 1967 Reference Gravity Formula, adopted by the International Association of Geodesy: lg, = ge (1 + 0.005278895sin2¢ + 0.000023462sin4¢)] where: g‘ = theoretical gravity for the latitude of the observation point (mGal) ge : theoretical gravity at the equator (978,031.85 mGal) (i) : latitude of the observation point (degrees). The equation takes into account the fact that the Earth is an imperfect sphere, bulging out at the equator and rotating about an axis through the poles (Fig. 8.43). Gravity Anomalies 227 For such an oblate spheroid (Fig. 8.4c), it estimates that gravitational acceleration at the equator (cl) = 0°) would be 978,031.85 mGal, gradually increasing with latitude to 983,217.72 mGal at the poles (d2 = 90°). Free Air Gravity Anomaly Gravity observed at a specific location on Earth’s surface can be Viewed as a func— tion of three main components (Fig. 8.5): 1) the latitude (d3) of the observation point, accounted for by the theoretical gravity formula; 2) the elevation (AR) of the sta- tion, which changes the radius (R) from the observation point to the center of the Earth; and 3) the mass distribution (M) in the subsurface, relative to the observation point. The free air correction accounts for the second effect, the local change in grav— ity due to elevation.That deviation can be approximated by considering how gravity changes as a function of increasing distance of the observation point from the center of mass of the Earth (Fig. 8.6a). Consider the equation for the gravitational acceler— ation (g) as a function of radius (R): GM F? The first derivative of g, with respect to R, gives the change in gravity (Ag) with increasing distance from the center of the Earth (that is, increasing elevation, AR): 1' _____2(@4_>_:2_(fl)_;2() Afi’flo AR dR R3 R R2 R g Assuming average values of g 3 980,625 mGal and R z 6,367 km = 6,367,000 In (Fig. 8.4c): dg/dR z -0.308 mGal/m dg/dR = average value for the change in gravity with increasing elevation. where: A R FIGURE 8.5 Three factors determining gravity at an observation point: a) latitude on (4)); b) distance from sea—level datum to observation point (AR); c) Earth’s mass Topography distribution (M), relative to the station / location (M includes material above as well as below sea level). (i) is accounted for by subtracting the theoretical gravity from the observed gravity, and AR by the free air correction.T'he remaining value (free air anomaly) is thus a function of M. Sea Level all m n km...“ A .m 228 Chapter 8 Gravity and Isostasy a A: = g - 8 AR = R: - 123, g, = 980,625 mGaI H, ~ 6,367.000m Ag/AF! ~ dg/dR - -2g, /H, - aaoa mearm " Cal Center of Earth FAC - h x (0.308 mearm) Sea Level Datum FIGURE 8.6 Free air correction. a) Rising upward from Earth’s surface, gravitational acceleration decreases by about 0.308 mGal for every meter of height. h) A gravity station at high elevation tends to have a lower gravitational acceleration (g) than a station at lower elevation. c) The free air correction (FAC) accounts for the extended radius to an observation point, elevated h meters above a sea level datum. The above equation illustrates that, for every 3 m (about 10 feet) upward from the surface of the Earth, the acceleration due to gravity decreases by about 1 mGal. Stations at elevations high above sea level therefore have lower gravity readings than those near sea level (Fig. 8.6b). To compare gravity observations for stations with different elevations, a free air correction must be added back to the observed values (Fig. 8.60). I ’ FAC = h x (0.308 mGal/m) where: FAC = free air correction (mGal) h = elevation of the station above a sea level datum (In). The free air gravity anomaly is the observed gravity, corrected for the latitude and elevation of the station: Agra = g — g. + FAC Agfa = free air gravity anomaly g = gravitational acceleration observed at the station. where: Notice in the above equation that: 1) subtracting the theoretical gravity (gt) from the observed gravity (g) corrects for the latitude, thus accounting for the spin and Gravity Anomalies 229 bulge of the Earth; and 2) adding the free air correction (FAC) puts back the gravity 10st to elevation, thereby correcting for the increased radius (R) to Earth’s center. 5 ‘5- The free air gravity anomaly is a function of lateral mass variations (M in _' i Fig. 8.5), because the latitude and elevation effects (<1) and AR in Fig. 8.5) have been ‘ l corrected. Fig. 8.7 shows what a profile of changing free air anomalies might look - ; like across bodies of excess and deficient mass. Notice that the anomaly shows rela- tively high readings near the mass excess, low readings near the mass deficiency; there are also abrupt changes that mimic sharp topographic features. J Bouguer Gravity Anomaly is Even after elevation corrections, gravity can vary from station to station because of differences in mass between the observation points and the sea—level datum. 3 ' Relative to areas near sea level, mountainous areas would have extra mass, tending to increase the gravity (Fig. 8.8a). " The Bouguer correction accounts for the gravitational attraction of the mass “ above the sea-level datum. This is done by approximating the mass as an infinite slab, with thickness (h) equal to the elevation of the station (Fig. 8.8b). The attrac— tion of such a slab is: ‘ BC = 211th BC = Bouguer correction p = density’of the slab t G = Universal Gravitational Constant h = thickness of the slab (station elevation). where: Substituting the values of G and 211 yields: ‘l-‘l BC = 0.0419ph u where BC is in mGal (10‘5 m/sz); p in g/Cm3 (103 kg/ms); h in m. H .3 + _ FIGURE 8.7 General form of free air E ‘ ,l"\‘ ' gravity anomaly profile across areas of I.’ \I / Air Anomaly mass excess and mass deficiency. g ix. ~ I‘": '1". ‘w"'~. ,fN‘ ‘éhl E -’ ‘-. ’ “‘1 Stations 4, / / \ Topography\ Sea Level Datum Mass Def/Clancy 230 Chapter 8 Gravity and lsostasy High g a n Lowg EL... FIGURE 8.8 Bouguer correction. a) The extra mass of mountains results in higher gravity relative to areas near sea level. b) To account for the excess mass above a sea level datum, the Bouguer correction assumes an infinite slab of density (p), with thickness (h) equal to the station’s elevation. a) On Land Station /Topography Sea Level Datum b) At Sea Water Station infinite Slab Rock pa - 2.67 g/cms FIGURE 8.9 Standard Bouguer correction values. a) On land, the reduction density (p) is commonly taken as +2.67 g/cm3.The thickness of the infinite slab is equal to the station elevation (h). b) At sea, the reduction density (~1.64 g/cm3) is the difference between that of sea water (1.03 g/cm3) and underlying rock (2.67 g/cm3).The thickness of the slab is equal to the water depth (hw). Bouguer Gravity Anomaly on Land For regions above sea level (Fig. 8.9a), the simple Bouguer gravity anomaly (AgB) results from subtracting the effect of the infinite slab (BC) from the free air gravity anomaly: Ag]; : Agfa _ BC To determine the Bouguer correction, the density of the infinite slab (p) must be assumed (the reduction density). The reduction density is commonly taken as 2.67 g/cm3, a typical density of granite (Figs. 3.9, 3.10). Gravity Anomaly (meal) Gravity Anomalies 231 profile in Fig. 8.7. Bouguer Anomaly/ Stations Topography\ \ .... .-._.-...-§?§..L.9.!.€LQ€'Z‘.L’ZZ Mass ; Deficiency The standard Bouguer correction for areas above sea level is thus: BC = 0.0419ph = (0.0419)(2.67g/cm3)h = (0.112 mGal/m) x h where h is in m. The equation illustrates that, for about every 9 m of surface eleva- tion, the increased mass below the observation point adds about 1 mGal to the observed gravity. Using the standard correction, the simple Bouguer gravity anom- aly on land is computed from the free air gravity anomaly according to the formula: AgB = Agfa — (0.112 mGal/m) h (h in meters). Like the free air gravity anomaly, the Bouguer gravity anomaly reflects changes in mass distribution below the surface. The Bouguer anomaly, however, has had an additional correction, removing most of the effect of mass excess above a sea level datum (on land). Bouguer Corrections applied to the free air gravity profile (Fig. 8.7) would therefore yield a Bouguer gravity profile illustrated in Fig. 8.10.'Ihe two profiles illustrate three general properties of gravity anomalies. 1) For stations above sea level, the Bouguer anomaly is always less than the free air anomaly (the approximate attraction of the mass above sea level has been removed from the free air anomaly). 2) Short-wavelength changes in the free air anomaly, due to abrupt topographic changes, have been removed by the Bouguer correction; the Bouguer anomaly is therefore smoother than the free air anomaly. 3) Mass excesses result in positive changes in gravity anomalies; mass deficiencies cause negative changes. Bouguer Gravity Anomaly at Sea In areas covered by the sea, gravity is gen- erally measured on the surface of the water (Fig. 8.9b). In the strictest sense, FIGURE 8.10 Bouguer correction applied to the free air gravity anomaly 232 Chapter 8 Gravity and Isostasy Bouguer anomalies at sea are exactly the same as free air anomalies, because station elevations (h) are zero: AgB = Agfa — 0.0419ph; h = 0, so that: AgB = AgfEl A type of Bouguer correction can be applied, however, because the density and depth of the water are well known. Instead of stripping the topographic mass away, as is done on land, the effect can be thought of as “pouring concrete to fill the ocean. Thus, the Bouguer correction at sea can be envisioned as an infinite slab, equal to the depth of the water and with density equalling the difference between that of water and “concrete”: BCS = 0.0419ph = 0.0419(pW — pc)hw where: BCS = Bouguer correction at sea pW = density of sea water pc = density of “concrete” hw = water depth below the observation point. Assuming pW = 1.03 g/cm3 and pC = 2.67 g/cm3: BCS = 0.0419 (—1.64 g/cm3) hw = —0.0687 (mGal/m) x 11w where BCS is in mGal and hw in m. Retaining the convention defined above, the Bouguer correction at sea is sub- tracted from the free air anomaly to yield the Bouguer gravity anomaly at sea (Ang): Ang = Agfa _ BCs Notice that the water is a mass deficit when compared to adjacent landmasses of rock; the negative Bouguer correction at sea thus means that some value must be added to the free air anomaly to compute the Bouguer anomaly at sea: Ang = Agfa + (0.0687 mGal/m) hw (hw in meters). Complete Bouguer Gravity Anomaly The infinite slab correction described above yields a simple Bouguer anomaly. That correction is normally sufficient to approximate mass above the datum in the vicinity of the station (Fig. 8.11a). In rugged areas, however, there may be significant effects due to nearby mountains pulling upward on the station, or valleys that do not contain mass that was sub- tracted (Fig. 8.11b). For such stations, additional terrain corrections (TC; see Telford et al., 1976) are applied to the simple Bouguer anomaly (AgB), yielding the complete Bouguer gravity anomaly (Ach): Ach : Agra + TC Summary of Equations for Free Air and Bouguer Gravity Anomalies Fig. 8.12 illustrates parameters used to determine free air and Bouguer gravity anomalies. The formulas below yield standard versions of the anomalies. Gravity Anomalies 233 a Station Topography Station FIGURE 8.11 Terrain correction. a) In areas of low relief, the Bouguer slab approximation is adequate; terrain correction is unnecessary. b) High relief areas require terrain correction, to account for lessening of observed gravity due to mass of mountains above the slab (1), and overcorrection due to valleys (2). For both situations, the terrain correction is positive, making the complete Bouguer anomaly higher than the simple Bouguer anomaly. ON LAND FAC = h (0.308 mGal/m) BC = h (0. 1 12 mGaI/m) A T SEA FAC = o (h = a) 1903 = -hw (0.0687 mGaI/m) /Topography p = +2.67 g/cm3 h Sea Level Datum' FIGURE 8.12 Standard parameters used to compute gravity anomalies on land and at sea. FAC = free air correction; BC = Bouguer correction; BCS = Bouguer correction at sea; p = reduction density; h (elevation) and hW (water depth) in meters. Theoretical Gravity gt = ge (1 + 0.0052788953in2rb + 0.000023462sin4¢) gt = theoretical gravity for the latitude of the observation point (mGal) ge = theoretical gravity at the equator (978,031.85 mGal) d) = latitude of the observation point (degrees). Free Air Gravity Anomaly Agfa = (g — g) + h(0.308 mGal/m) Agfa = free air gravity anomaly (mGal) ms {9” 3 234 Chapter 8 Gravity and Isostasy % ' g = observed gravity (mGal) gt = theoretical gravity (mGal) h = elevation above sea level datum (m). I Bouguer Gravity Anomaly Ag]; 2 Agfa _ BC = Agfa — 0.04l9ph I 1‘; BC = Bouguer correction (mGal) ii 4 p = reduction density (g/Cm3) a) On Land AgB = Agfa — (0.112 mGal/m) h (for p = +2.67) AgB = simple Bouguer gravity anomaly (mGal) ’1 h = elevation above sea-level datum (m). ' i‘, . b) At Sea I:ng = Agfa + (0.0687 mGal/m) 11w] (for p = —1.64) Ang = Bouguer gravity anomaly at sea (mGal) hw = water depth below observation point (m). Ach = Ag]; + TC Ach = complete Bouguer gravity anomaly (mGal) TC = terrain correction (mGal). c) In Rugged Terrain: MEASUREMENT OF GRAVITY M l l Gravitational acceleration on Earth’s surface can be measured in absolute and rela— ' j " tive senses (Fig. 8.13). Absolute gravity reflects the actual acceleration of an object ‘ I as it falls toward Earth’s surface, while relative gravity is the difference in gravita- tional acceleration at one station compared to another. 5. I a) Absolute Gravity FIGURE 8.13 21) Absolute gravity is the true gravitational acceleration (g). b) Relative gravity reflects the difference in gravitational acceleration (Ag) at one station (g1) compared to another (g2). ' Measurement of Gravity 235 FIGURE 8.14 Measurement of absolute gravity. a) Weight drop.The object accelerates from an initial velocity of V0 at time (T = 0), to a velocity of VI at time (T = t), as it falls a distance (2). b) Pendulum. Gravitational acceleration is a function of the pendulum’s length (L) and period of oscillation (T). a) Weight Drop b) Pendulum Absolute Gravity There are two basic ways to measure absolute gravity. In the weight drop method (Fig. 8.14a), the velocity and displacement are measured for an object in free fall. The absolute gravity is computed according to: z = vot + §gt2 where: z = distance the object falls t = time to fall the distance z v0 = initial velocity of the object g = absolute gravity. The absolute gravity is thus: Using the second method (Fig. 8.14b), a pendulum oscillates according to: T = 211' V L/ g where: T = period of swing of the pendulum L = length of the pendulum. The absolute gravity is computed according to: Relative Gravity The precision necessary to obtain reliable, absolute gravity observations makes those measurements expensive and time consuming. Relative gravity measure- ments, however, can be done easily, with an instrument ( gravimeter) that essentially measures the length of a spring (L; Fig. 8.15a). The mass of an object suspended from the spring remains constant. When the gravimeter is taken from one station location to another, however, the force (F) that the mass (m) exerts on the spring varies with the local gravitational acceleration (g): F=mg 7 .1 V t! i 236 Chapter 8 Gravity and lsostasy I; a -. ..... v g, Spring L L a g I { Mass ! i . b Station 1 Station 2 I. "I" AL' = L2 - L1 L2 Ag 0: AL 33322....” AL F, , I; - Weight of Mass at Stations 1 and 2 ll 5 (It . I .1 ,x.._______. 1 . ‘ I .18“. FIGURE 8.15 Measurement 0 f relative gravity. a) A gravimeter measures the length of a spring (L), which is proportional to the gravitational acoeleration (g). b) A force (F1) at one station results in a spring length (L1). The length may change to L2 because of a different force (F2) at another station. The force exerted by the mass is a function of g; the change in length of the spring (AL) is thus proportional to the change in gravitational acceleration (Ag). c) Map of relative gravity survey. The traverse starts with a measurement at the base station, then each of the 16 stations, followed by a re-measurement at the base station. ‘: l i so that: p f i ll ‘ g = F/m _ p it In other words, the mass will weigh more or less (exert more or less force), depend- } ' : 1 ing on the pull of gravity (g) at the station.A gravimeter is simply weighing the mass ‘ at different stations; the spring stretches (+AL) where there is more gravity and contracts (—AL) when gravity is less (Fig. 8.15b). If we know the absolute gravity at a starting point (base station), we can use a ‘ gravimeter to measure points relative to that station (Fig. 8.150). The initial reading 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" ...
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Lillie223-237 - CHAPTERS Gravity and Isostasy gravity...

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