VẬT lý địa CHẤN grativity tt

Some Gravity Basics Newton’s Law of Gravitation states that the force of attraction F between two masses m1 and m2 whose dimensions are small with respect to the distance r between them is given by the equation below, where G is the Gravitational Constant (6.67x10-11 m3kg-1s-2) Gm1m2 F= r r m1 m2 •The attractive force between two small masses is proportional to the product of their masses – doubling either mass doubles the force, doubling both masses quadruples the force •The attractive force is inversely proportional to the square of the separation – doubling the separation reduces the attraction by a quarter Some Gravity Basics For the case of a mass above the earth we can re-write the previous equation using ME as the mass of the Earth, ms as the mass of the object, and RE as the distance between the centers of the objects GM E F= ms = ms g RE GM E g= RE2 RE ms = RE ME ms ME •Force is related to mass by acceleration, and the term g is known as the gravitational acceleration •In the case of a uniform sphere, it can be approximated to a “point mass” of smaller radius but equal mass •The attractive force of an irregular body can be calculated by approximating it as a series of smaller bodies Some Gravity Basics GM E g= RE2 RE ms ME •On a spherically symmetric earth the value of g would be the same everywhere • We would not know anything about the composition of the earth – it could be a hollow shell, have a hollow center, etc •Luckily the earth is highly heterogeneous What is the mass of the earth? Radius at equator = 6378160 m G = 6.672x10-11m3kg-1s-2 g = 9.78 ms-2 The Mass of the Earth GM E gR 9.78 ms −2 * (6378160 m) 24 g=  → M =  → M = ≈ 96 * 10 kg E E −11 −1 − RE G 6.67 *10 m kg s What is wrong with this number given the values in the adjacent table? From Kearey et al., 2002 What assumptions are made?? Using this mass, we can calculate that the average density of the earth is about 5500 kg m-3 (5.5 Mg m-3) Rock Densities •In the case of a homogeneous, spherically symmetric earth, gravity is constant This is obviously not the case in reality Gravity varies over the earth’s surface for a number of reasons, one of which is variations in density •Density is typically measured in units of kg m-3 – how we measure density? •Most common rocks range in density from 1500-3000 kg m-3 (e.g granite = 25002800 kg m-3, basalt = 2700-3000 kg m-3) A few metalliferous ores have densities well above 3000 kg m-3 (e.g magnetite ore = 4900-5300 kg m-3) •Densities of common rocks overlap, so they have limited use for lithological identification •Densities of weathered and/or exposed rocks tends to be less than their buried counterparts, so care has to be taken when collecting samples for density measurements and when interpreting gravity surveys WHY?? •Gravity anomalies are the result of density contrasts •In this example, the density contrast, Δρ, = 300 kg m-3 What might these rock types be?? 2700 kg m-3 2400 kg m-3 Gravity Units Gravity variations are very small Suppose that in the previous example the center of the sphere is at a depth of 100 m and it has a radius of 50 m Δρ = 300 kg m-3 •If an object is dropped on the ground above the sphere it will fall with a little greater acceleration than an object dropped to the side The excess gravity due to the sphere is: d = 100 m r = 50 m G∆m G = πr ∆ρ d d G −6 -2 ∆g = π 50 * 300 = 048 * 10 m s 100 ∆g = •As would be expected, this is a very small value, and rather inconvenient to use To make things easier, we use mGal (note that G is capitalized to honor Galileo) mGal = 10-5 m s-2 •The buried sphere thus produces an anomaly of 0.1048 mGal Measuring Gravity Time the fall of an object – this can be done accurately in the lab, but it is not practical for the field Absolute gravity Measure the period of a pendulum – the period depends on the dimensions of the pendulum and the value of gravity Same problems as Accuracy of 0.001 mGal Absolute gravity Measure relative gravity using the principles of a spring balance This can be converted to absolute gravity by taking a measurement at a point where absolute gravity is known Here, an increase in gravity has caused an increase in the weight of a mass, resulting in a spring of length s being stretched by an amount Δs The extension of the spring is proportional to the extending force (Hooke’s Law) m m∆g = k∆s and ∆s = ∆g k k is the elastic spring constant Measuring Gravity In this example, the fact that the spring acts as a measuring device and a support for the mass restricts the accuracy of the instrument This was however the basis of early instruments known as stable gravimeters From Kearey et al., 2002 Δs, the extension of the spring, must be measured to an accuracy of 1:10 for gravity surveying on land This is the same as, measuring the distance from here to Birmingham (assuming 80 km) to an accuracy of 0.8 mm The problem of a dual role spring is overcome in an unstable gravimeter, an example of which is the LaCoste and Romberg gravimeter •The meter consists of a hinged beam carrying a mass supported by a spring attached immediately above the hinge •The magnitude of the moment exerted by the spring on the beam is dependant upon the extension of the spring and the sine of the angle θ •By suitable design of the spring and beam geometry the magnitude of the increase in restoring force with increasing gravity can be made as small as possible •A zero-length spring is used pre-tensioned in manufacture so that the restoring force is proportional to the physical length of the spring rather than the extension •The instrument is read by restoring the beam to horizontal by altering the vertical location of the spring attachment The number of screw turns is the reading From Kearey et al., 2002 Unstable Gravimeters Unstable Gravimeters As changes in temperature can cause changes in the length of the spring, a thermostat keeps the instrument at a constant temperature Other things to worry about: •Make sure the meter is perfectly level •Check the meter temperature •In a micro-gravity survey make sure that you always take the measurements from the same side of the meter •Take notes about the terrain immediately surrounding the meter (e.g are you next to a boulder) •Don’t take readings following an earthquake •Don’t bump the meter •Capable of measurements to an accuracy of 0.01 mGal across a 5000 mGal range Gravity Reduction Latitude Correction • Gravity varies with latitude because: The Earth is not spherical Centrifugal force has given it an equatorial bulge Thus a point on the equator is further away from the center of the earth This acts to reduce gravity The amplitude of this effect is reduced by the fact that the mass underlying equatorial regions is greater than that underlying polar regions The angular velocity of a point on the earth’s surface decreases from a maximum at the equator to zero at the poles This acts to fling bodies away from its rotation axis Gravity at the poles exceeds gravity at the equator by ~5,186 mGal From Kearey et al., 2002 • Gravity Reduction Latitude Correction (continued) • As both effects vary with latitude they can be combined into a single formula, Caliraut’s formula gθ = 978031.8(1 + 0.0053924 sin θ − 0.0000059 sin 2θ ) Where gθ is the predicted value of gravity at latitude θ and 978031.8 is the value of gravity at the equator The equation with the above constants is also know as the International Gravity Formula 1967 The value gθ gives the predicted value of gravity at sea-level at any point on the Earth’s surface, and is subtracted from the observed gravity to correct for latitude variation Gravity Reduction Free-Air Correction •The free-air correction is one of three topographic corrections Two more will follow g= GM E RE2 •To reduce to datum an observation taken at height h: FAC = 0.3086h mGal (h in meters) •The FAC is positive for an observation point above the datum to correct for the decrease in gravity with elevation From Kearey et al., 2002 •The free-air correction (FAC) corrects for the decrease in gravity with height that is due to an increased distance from the center of the earth – remember: Bouguer Correction •The second topographic correction is the Bouguer correction The FAC takes into account the height of the station – the gravitational attraction of the rock between the observation point and the station is ignored •The Bouguer correction removes the effect by approximating the rock layer beneath the observation point as an infinite horizontal slab with a thickness equal to the elevation of the observation above the datum BC = 2πGρh = 0.04193 ρh mGal, ρ in Mg m -3 •On land the Bouguer correction must be subtracted, as the gravitational attraction of the slab needs to be removed •At sea the correction is positive to account for the lack of rock between the surface and the sea bead – the water is replaced by rock The above equation becomes: BC = 2πG ( ρ r − ρ w ) z Where z is the water depth, ρw is the density of water, and ρr is the density of rock From Kearey et al., 2002 Gravity Reduction Gravity Reduction Terrain Correction •The final topographic correction The Bouguer correction make the assumption that the topography around the observation point is flat – this is rarely the case •This correction is always positive •The regions designated A forms part of the Bouguer correction slab although they not exist The Bouguer correction has hence overcorrected and their effect must be restored From Kearey et al., 2002 •Region B consists of rock that has been excluded from the Bouguer correction It exerts an upward attraction at the observation point causing gravity to decrease Its attraction thus must be added Gravity Reduction Terrain Correction (cntd) •Terrain corrections are classically carried out using a Hammer chart •Divided by radial and concentric lines into 130 compartments •Outermost zone extends to almost 22 km – beyond this topographic effects are usually negligible •Chart is laid on a map with the observation point at its center – the average elevation of each compartment is calculated and the elevation of the observation point subtracted •Gravitational attraction of each compartment is determined by reference to tables •The terrain correction is the sum of all the compartments •It is difficult to fully automate this step, especially for the inner rings of the hammer chart Gravity Reduction Tidal Correction •Gravity measured at a fixed location will vary with time because of periodic variations in the gravitational effects of the Sun and Moon •These gravitational effects also cause the shape of the earth to vary in much the same way that the celestial attractions cause tides in the sea •Solid earth tides are smaller than oceanic tides and lag further behind lunar motion (the effect of the moon is greater than that of the sun) •Solid earth tides cause the elevation of a point to vary by a few cm, thus varying its distance from the center of the earth •Maximum amplitude of 0.3 mGal •Tidal variations are predictable and can be easily calculated Gravity Reduction Eötvös Correction •Applied to gravity measurements taken on a moving vehicle, e.g., a ship or a helicopter •Depending on the direction of travel of the vehicle, the motion will generate a centripetal acceleration which either reinforces or opposes gravity •The correction is: EC = 4.040v sin α cos θ + 0.001211v mGal Where v is the speed in km hr-1, α is the heading, and θ is the latitude •The correction is positive for motion from east to west •Approximately 2.5 mGal for each kph in an east-west direction at 55 oN Free-air and Bouguer Anomalies FAA = g obs − gθ + FAC (± EC ) BA = g obs − gθ + FAC ± BC + TC (± EC ) The Bouguer anomaly forms the basis for the interpretation of land gravity data Typically the Free-Air anomaly is used at sea More advanced Fourier methods for the calculation of Bouguer anomalies at sea are available Gravity Anomalies The gravitational attraction in the direction of the mass is given by: ∆g r = Gm r2 Since a gravity meter only measures the vertical component of the attraction Δgz, the gravity anomaly Δg caused by the mass is: Gm Gmz cos θ or ∆ g = r2 r3 From Kearey et al., 2002 ∆g = Gravity Anomalies Gmz ∆g = r This equation can be used to build up the gravity anomaly of many simple geometric shapes by constructing them from a suite of small elements which correspond to point masses and then summing (integrating) the attractions of these elements to derive the anomaly of the whole body Integration of the above equation in the horizontal direction provides the equation for a line mass extending to infinity in that direction: ∆g = 2Gmz r2 Indirect Density Determination Nettleton’s Method Now that we have defined the Bouguer anomaly we can discuss an alternative method for regional density determination •The density that yields a Bouguer anomaly with the least correlation (+ve or –ve) with the topography is taken to represent the average density of the prominence From Kearey et al., 2002 •Gravity data over an isolated prominence are reduced using a series of different densities for the Bouguer and terrain corrections Direct Interpretation The shape and size of the anomaly provides some information on the anomalous body independent of the true shape of the body Half-width method •The half-width of the anomaly (x1/2) is the horizontal distance from the anomaly maximum to the point at which the anomaly has reduced to half its maximum value •If the anomaly results from a point mass the depth is: −1 •As this is the depth to the center of mass it is an overestimate of the depth to the top of the body From Kearey et al., 2002 z= x1 Direct Interpretation Excess Mass •The excess mass (the difference in mass between the body and the mass of the country rock that would otherwise occupy the space) can be uniquely determined •The survey is divided into n grid squares of area Δa and the mean residual anomaly Δg is found for each The excess mass is then given by: •To compute the actual mass M of the body the densities of both the anomalous body (ρ1) and the country rock (ρ2) must be known: ρ1M c M= ( ρ1 − ρ ) •This method can be used to estimate, for example, the tonnage of an ore body From Mussett and Khan, 2000 n Me = ∆g i ∆ai ∑ 2πG i =1 Indirect Interpretation Types of forward gravity modeling: •2-D: The body being modeled extends to infinity in and out (at a right angle) of the plane of the model •2.5-D: As 2-D, but the extent of the body in and out of the plain can be limited •2.75-D: As 2.5-D but the angle of the body with respect to the plain of the model can be varied •3-D: Modeling the full three-dimensional extent of the body Inverse modeling •Using the supplied anomaly the computer automatically constructs a model that fits Hopefully you can constrain some of the parameters [...]... overcorrected and their effect must be restored From Kearey et al., 2002 •Region B consists of rock that has been excluded from the Bouguer correction It exerts an upward attraction at the observation point causing gravity to decrease Its attraction thus must be added Gravity Reduction Terrain Correction (cntd) •Terrain corrections are classically carried out using a Hammer chart •Divided by radial and... sea More advanced Fourier methods for the calculation of Bouguer anomalies at sea are available Gravity Anomalies The gravitational attraction in the direction of the mass is given by: ∆g r = Gm r2 Since a gravity meter only measures the vertical component of the attraction Δgz, the gravity anomaly Δg caused by the mass is: Gm Gmz cos θ or ∆ g = r2 r3 From Kearey et al., 2002 ∆g = Gravity Anomalies... elements which correspond to point masses and then summing (integrating) the attractions of these elements to derive the anomaly of the whole body Integration of the above equation in the horizontal direction provides the equation for a line mass extending to infinity in that direction: ∆g = 2Gmz r2 Indirect Density Determination Nettleton’s Method Now that we have defined the Bouguer anomaly we can discuss... negligible •Chart is laid on a map with the observation point at its center – the average elevation of each compartment is calculated and the elevation of the observation point subtracted •Gravitational attraction of each compartment is determined by reference to tables •The terrain correction is the sum of all the compartments •It is difficult to fully automate this step, especially for the inner rings... with time because of periodic variations in the gravitational effects of the Sun and Moon •These gravitational effects also cause the shape of the earth to vary in much the same way that the celestial attractions cause tides in the sea •Solid earth tides are smaller than oceanic tides and lag further behind lunar motion (the effect of the moon is greater than that of the sun) •Solid earth tides cause... horizontal slab with a thickness equal to the elevation of the observation above the datum BC = 2πGρh = 0.04193 ρh mGal, ρ in Mg m -3 •On land the Bouguer correction must be subtracted, as the gravitational attraction of the slab needs to be removed •At sea the correction is positive to account for the lack of rock between the surface and the sea bead – the water is replaced by rock The above equation becomes:... distance from the center of the earth – remember: Bouguer Correction •The second topographic correction is the Bouguer correction The FAC takes into account the height of the station – the gravitational attraction of the rock between the observation point and the station is ignored •The Bouguer correction removes the effect by approximating the rock layer beneath the observation point as an infinite horizontal... body the densities of both the anomalous body (ρ1) and the country rock (ρ2) must be known: ρ1M c M= ( ρ1 − ρ 2 ) •This method can be used to estimate, for example, the tonnage of an ore body From Mussett and Khan, 2000 1 n Me = ∆g i ∆ai ∑ 2πG i =1 Indirect Interpretation Types of forward gravity modeling: •2-D: The body being modeled extends to infinity in and out (at a right angle) of the plane of ... measured in units of kg m-3 – how we measure density? •Most common rocks range in density from 150 0-3 000 kg m-3 (e.g granite = 25002800 kg m-3, basalt = 270 0-3 000 kg m-3) A few metalliferous ores... model •2.5-D: As 2-D, but the extent of the body in and out of the plain can be limited •2.75-D: As 2.5-D but the angle of the body with respect to the plain of the model can be varied •3-D: Modeling... heterogeneous What is the mass of the earth? Radius at equator = 6378160 m G = 6.672x1 0-1 1m3kg-1s-2 g = 9.78 ms-2 The Mass of the Earth GM E gR 9.78 ms −2 * (6378160 m) 24 g=  → M =  → M = ≈