Lecture 2 Gravity Measument and Reduction PDF

GSS611 Physical Geodesy Gravity Measurement and Reduction  SI Unit For Gravity Unit for gravity is Gal or cms-2, and named after Galileo Galilee 1 Gal = 1 cms-2 = 10-2 ms-2 1 mgal = 10-3 gal = 10-3 cms-2 = 10-5 ms-2...

GSS611 Physical Geodesy Gravity Measurement and Reduction  SI Unit For Gravity Unit for gravity is Gal or cms-2, and named after Galileo Galilee 1 Gal = 1 cms-2 = 10-2 ms-2 1 mgal = 10-3 gal = 10-3 cms-2 = 10-5 ms-2 1 µgal = 10-3 mgal = 10-6 gal = 10-6 cms-2 = 10-8 ms-2 Gravity at equator ≈ 978 cms-2 = 978 Gal = 9.78 ms-2 = 978,000 mgal  Gravity Datum Earliest gravity datum — Potsdam system (in Germany) Established in 1906, using reversible pendulums to established absolute gravity value ▪ gp =981.274 ± 0.003gal Found to be in error and replaced by International Gravity Standardization Net 1971 (IGSN71) ▪ gp = 981.2609 ± 0.000017gal IGSN71 Contains 1854 re-occupiable stations distributed worldwide Each gravity acceleration value was determined by least squares adjustment Established the basic gravity datum for today's relative gravity survey  Gravity Datum Peninsular Malaysia Gravity Base Network Total station = 33 Accuracy = ±0.005 mGal IGSN station = 4 (KL, Penang, Malacca, Singapore) Accuracy =0.05 mGal — 0.1 mGal  Gravity Observation Technique Gravity observation is used to establish the value of gravity acceleration at the desired point Absolute observation : observing directly the value of g at a point Relative measurement: observing the difference in g between two point (known and unknown) Gravity Apparatus i. Pendulum – Absolute observation ii. Free Fall Device-Absolute observation iii. Gravimeters – Absolute and Relative observation  Gravity Observation Technique Pendulum Simple pendulum use dimensionless material suspended on a perfectly flexible, unstretchable massless string of length, L Swing the pendulum and measure the oscillation period, where , 𝑙 𝑇 = 2𝜋 𝑔 Simple theory, but difficult to implement because: What is the ideal pendulum? What is the ideal length? Timing accuracy? To achieve 1mGal accuracy, length has to be measured to an accuracy of 1 micron (10-3mm) and time to an accuracy of 2.5 x 10-7sec  Gravity Observation Technique Errors Using Pendulum Air friction dampens the pendulum swing and also causing buoyancy effect, which tend to support the pendulum by the amount that is displaced Molecule of air are absorbed on the shark of the pendulum and raises the center of gravity, thereby changing the length Vibration on the foundation in which the pendulum rest Reversible Pendulum Developed by Kater, 1818 Swing two pendulum on a fix axis but on a different direction with same oscillation period Has different distances from centre of gravity, known as h1,h2 Removes the impact of vibration 2 2 2 2 2 2 4𝜋 𝑇1 + 𝑇2 𝑇1 − 𝑇2 ℎ1 + ℎ2 𝑃𝑒𝑟𝑖𝑜𝑑, 𝑇 = ℎ1 + ℎ2 = + 𝑔 2 2 ℎ1 + ℎ2 Gravity Observation Technique Reversible Pendulum  Gravity Observation Technique Falling Body Device Observing the motion of falling body over a distance of 1 or 2 meters 1 Position, 𝑧 of a body: 𝑧 = 𝑧𝑜 + 𝑣𝑜 𝑡 + 𝑔𝑡 2 2 For 3 position of z, we have: 1 ▪ 𝑧1 = 𝑧𝑜 + 𝑣𝑜 𝑡1 + 𝑔𝑡12 2 1 ▪ 𝑧2 = 𝑧𝑜 + 𝑣𝑜 𝑡2 + 𝑔𝑡22 2 1 ▪ 𝑧3 = 𝑧𝑜 + 𝑣𝑜 𝑡3 + 𝑔𝑡32 2 Eliminating 𝑧𝑜 and 𝑣𝑜 , we get: 2[ 𝑧1 − 𝑧2 (𝑡1 − 𝑡3 ) − (𝑧1 − 𝑧3 )(𝑡1 − 𝑡2 )] 𝑔= (𝑡1 − 𝑡2 )(𝑡1 − 𝑡3 )(𝑡2 − 𝑡3 )  Gravity Observation Technique Problems Value of g at higher level and lower levels of instruments differs a little ▪ Due to difference in distance from centre of the earth Rising and Falling Body Improvement over failing body method Body is thrown upward and then fall freely – Body will pass the same level twice – 𝑧1 , 𝑧4 = 𝑡1 , 𝑡4 and 𝑧2 , 𝑧3 = 𝑡2 , 𝑡3 Hence, we have four equations: 1 1 – 𝑧1 = 𝑧𝑜 + 𝑣𝑜 𝑡1 + 𝑔𝑡12 , 𝑧2 = 𝑧𝑜 + 𝑣𝑜 𝑡2 + 𝑔𝑡22 2 2 1 1 – 𝑧4 = 𝑧𝑜 + 𝑣𝑜 𝑡4 + 𝑔𝑡42 , 𝑧3 = 𝑧𝑜 + 𝑣𝑜 𝑡3 + 𝑔𝑡32 2 2  Gravity Observation Technique Rising and Falling Body Eliminating 𝑧𝑜 and 𝑣𝑜 , and noting 𝑧1 = 𝑧4 and 𝑧2 = 𝑧3 8 𝑧2 − 𝑧1 ▪ We get: 𝑔 = 𝑡4 − 𝑡1 2 − (𝑡3 − 𝑡2 )2 Accuracy: 0.006 mGal Error - Friction, temperature Problem with Pendulum and Falling Body Device Bulky equipment Measurement were time consuming Costly Need better accuracy  Gravity Observation Technique Gravimeter (Gravity Meter) Basic principle Static method. No parts in the measurement moving Use elastic spring with a mass suspended down, the elastic force produce by the elongation is equal to the gravity force acting on the mass Hook’s Law : 𝑚𝑔 = 𝑘(𝑙 − 𝑙𝑜 ) Where: 𝑘 = Spring elasticity constant 𝑙𝑜 , 𝑙 = length of spring before and after mass is suspended  Gravity Observation Technique At another place where value is (𝑔 + 𝛿𝑔), the spring elongation will change accordingly to: 𝛿(𝑙−𝑙𝑜 ) 𝛿𝑔 = 𝑙−𝑙𝑜 𝑔 Problem: small changes in 𝛿𝑔 (1 mGal), for (𝑙 − 𝑙𝑜 ) = 1m, gives 𝛿 𝑙 − 𝑙𝑜 = 10−3 𝑚𝑚 = 1 𝑚𝑖𝑐𝑟𝑜𝑛 Therefore, requires method of recording (optical) and magnification (capacitative photo electric)  Gravity Observation Technique Commercial Gravimeters: LaCoste and Romberg, Texas Instruments (Worden Gravity meter), and Scintex Accuracy: 0.01 mGal Resolution: 0.005 mGal Errors - Spring drift and temperature Schematic-of-a-La-Coste- Romberg-gravimeter  Field Observation Gravity loop should never exceed 72 hours. Observation corrected for static and dynamic drift, earth tide and calibration There are three (3) method i. Profile method ii. Star method iii. Step method  Field Observation Profile method Starting at point 1, traverse through to point 2,3,..n and return through n, n-1,n-2..and back to 1 Used to determine gravity and drift correction 1 2 3 n-1 n 3 Star method Start from point 1, to 2 and back to 1 Then 1 to 3 and back to a and so on 1 (base) Suitable for tying points surrounding the base 2 4  Field Observation Step method Start from 1, to 2, back to 1, back to 2, to 3, back to 2, back to 3 ,to 4….and so on used when the survey cannot be completed in days work Every station will have 3 readings except station 1 1 2 3 4  Field Gravimeter Calibration Measurement at two known gravity station (g1 and g2) using same instrument. Let A1 and A2 is measured gravity at Stn 1 and Stn 2 Find the mean value (g1+g2) /2 = k (coefficient) g1 A1 1 2 g2 A2 compare with known k value. If the value of K’ is K’ = A2-A1 = ∆ A smaller than k value, show the instrument in good g2-g1 ∆g condition  Field Gravimeter Calibration Malaysia approach calibration KPUP 9/2021  Gravity Correction Purpose of gravity correction Must be applied to observation of gravitational acceleration to isolate the effect cause by geologic structure Types of correction 1. Observed Gravity (gobs) –Gravity readings observed at each gravity station after corrections have been applied for instrument drift and tides, etc 2. Latitude correction (gn)- correction subtracted from gobs that accounts for the earth’s elliptical shape and rotation. The gravity value that would be observed if the earth were a perfect , rotating ellipsoid is referred to as the normal gravity. – Gravity INCREASES with increasing latitude. Correction is ADDED as we move toward the equator. – gn = 978031.85 (1.0 + 0.005278895 sin2(lat) + 0.000023462 sin4(lat)) (mgal)  Gravity Reduction Gravity measured on the earth’s surface cannot be directly compared with normal gravity 𝛾, at reference ellipsoid for obtaining gravity anomaly A proper comparison can only made after observed gravity g is reduced to a value go , referred to the geoid-the main objective of gravity reduction P gp go Topography Geoid ϒ Since there are mass between the terrestialEllipsoid surface and the geoid, reduction methods differ depending on the way in which the topographic masses are handled  Normal gravity at ellipsoid P gp go Topography Geoid ϒ Ellipsoid  Gravity Reduction Gravity reduction is required for two main purposes: 1. Geodetic ▪ Determination of the geoid ▪ Interpolation and extrapolation of gravity 2. Geophysical The use Stoke’s formula for determining of geiod requires that gravity anomalies ∆g , represent the boundary value at the geoid which implies two conditions: ▪ Gravity must refer to geoid ▪ There must no masses outside the geoid Gravity anomaly Gravity anomaly-Free air Gravity anomaly-Free air  Gravity anomaly-Free air The gravity reduction consists of the following steps: I. The topographic masses outside the geoid are completely removed or shifted below sea level II. The gravity station is then lowered from earth’s surface to geoid Free Air Reduction Purpose - Reduces the observed value of gravity to the geoid by assuming no masses between the terrestrial and geoidal surface In effect , the observation points is assumed suspended in mid air H meters above the geoid  Gravity anomaly-Free air Free Air Reduction 𝜕𝑔 Given as: 𝛿𝑔𝑓 = − 𝐻 𝜕𝐻 𝜕𝑔 Where: = vertical gradient of gravity and given by Brun’s 𝜕𝐻 equation 𝜕𝑔 Brun’s equation: = −2𝑔𝐽 + 4𝜋𝐺𝜌 − 2𝜔2 , which can be 𝜕𝐻 𝜕𝛾 approximated by the vertical gradient of normal gravity 𝜕ℎ  Gravity anomaly-Free air Free Air Reduction 𝜕𝑔 𝜕𝛾 Hence: 𝛿𝑔𝑓 = − 𝐻 ≅ − 𝐻 𝜕𝐻 𝜕ℎ This leads to (from gravity above ellipsoid, to a second approximation): 𝛿𝑔𝑓 = 3.0877𝑥10−6 − 4.3977𝑥10−19 𝑠𝑖𝑛2 𝜙 𝐻 − 7.2125𝑥10−13 𝐻2 For an average value of the equation become: 𝛿𝑔𝑓 = 3.0855𝑥10−6 𝐻 − 7.2125𝑥10−13 𝐻2 Ignoring the H terms, the free air correction is given as: 𝛿𝑔𝑓 = 3.0855𝑥10−6 𝐻 [𝑚𝑠 −2 ] / 𝟎. 𝟑𝟎𝟖𝟔𝑯 mGal  Gravity anomaly-Free air Free Air reduction (gfa) –The free-air reduction accounts for gravity variations caused by elevation differences in the observation locations. (does NOT include the effect for mass between observed point and the datum) – Δgfa = 0.3086h (mgal) – where h is the elevation (in meters) at which the gravity station is above/below the datum – Correction is ADDED for stations ABOVE the datum – Correction is SUBTRACTED for stations BELOW the datum Gravity anomaly-Bouguer  Gravity anomaly-Bouguer Bouguer Plate effect The purpose of Bouguer reduction is to completely remove the effect of the topographic Bouguer correction is given as : ∆𝑔𝐵 = −2𝜋𝐺𝜌𝐻 Where 𝐺 is the constant of gravitation and 𝜌 is the density of the material 𝑖𝑓 𝐺 = 6.673𝑥10−11 ; 𝜌 = 2670𝑘𝑔𝑚−3 ∆𝑔𝐵 = −0.1119 𝐻 Gravity anomaly-Bouguer  Gravity Reduction Bouguer plate reduction (gb)- The Bouguer reduction is a first-order correction to account for the excess mass underlying observation points located at elevations higher than the elevation datum (sea level or the geoid). Conversely, it accounts for a mass deficiency at observation points located below the elevation datum – Δgb = - 0.04193 ρ h (mgal) ρ is the average density of the rocks underlying the survey area in g/cm3 – Correction is SUBTRACTED for stations ABOVE the datum – Correction is ADDED for stations BELOW the datum Gravity anomaly-Complete Bouguer  Gravity anomaly-Complete Bouguer Terrain reduction (gt)- The Terrain reduction accounts for variations in the observed gravitational acceleration caused by variations in topography near each observation point. Because of the assumptions made during the Bouguer plate correction, the terrain correction is positive regardless of whether the local topography consists of a mountain or a valley. – Correction is ADDED whether local terrain is a mountain or valley – Computed by tables, templates or computers Gravity Corrections and Reductions Classical Definition of gravity anomaly  Classical Definition of gravity anomaly The classical gravity anomaly (∆𝑔) for a point P on the Earth’s surface can be defined as the difference between the gravity at point on the geoid ( 𝑔𝑝0 ) and the normal gravity at point Q on the reference ellipsoid 𝛾𝑄  Modern Definition of gravity anomaly classical gravity anomaly depends on the density of the masses between the Earth’s surface and the geoid and the height of the surface point above the geoid involves making assumptions concerning the density of the masses above the geoid Molodensky in 1945 (Molodensky et al., 1962) formulated the Molodensky’s boundary value problem (BVP) The gravity anomaly at a point P on the Earth’s surface is now defined as the difference between the actual gravity measured at point P and the normal gravity on the telluroid Modern Definition of gravity anomaly Geoid and Quasigeoid  Type of Height Geopotential number Potential difference between the geoid level and the geopotential surface through a point on the Earth surface Geopotential numbers can be used to define height and are considered a natural measure for height Height = C / gravity  Type of Height Orthometric Height Height measured along the curved plumb line with respect to geoid level 𝐶 Use mean gravity 𝐻= 𝒈𝒎 Normal Height Height measured along the normal plumb line Use normal gravity 𝐶 𝐻= 𝜸𝒎 Dynamic Height Use normal gravity at latitude defined on the ellipsoid at 45 degree latitude 𝐶 𝐻= 𝜸𝟒𝟓𝒐  Type of Height Normal Height Orthometric Height  Example Based on the information, compute the free air anomaly and bouguer anomaly at point A Coordinates = 1˚ 10’ 10.132” N , 100˚25’ 32.126” E Height of point A = 18.00m Observed Gravity = 978049.70 m/s2 Normal Gravity = 978038.14 m/s2 Density = 2650 kg/m3 Gravitational constant, G = 6.674215 x 10-11 Nm2kg-2 Free air anomaly, 𝐹𝐴 𝐹𝐴 = 𝑔 − 𝛾0 + 0.3086 𝐻 = 978049.70 -978038.14+ 0.3086 x 61.42 = 30.5142 miliGals Bouguer anomaly ,BA 𝐵𝐴 = 𝐹𝐴 − 𝐵𝑜𝑢𝑔𝑒𝑢𝑟 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝐵𝑜𝑢𝑔𝑒𝑢𝑟 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = 2𝜋𝐺𝜌𝐻 = 2𝜋 𝑥 6.674215 x 10-11x2670 x 61.42 = 1.11122 x 10-6 ms-2 Convert to miliGals = 1.11122 x 10-6 / (1x10-5) Bouguer anomaly ,BA 𝐵𝐴 = 𝐹𝐴 − 𝐵𝑜𝑢𝑔𝑒𝑢𝑟 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝐵𝑜𝑢𝑔𝑒𝑢𝑟 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = 2𝜋𝐺𝜌𝐻 = 2𝜋 𝑥 6.674215 x 10−11x2670 x 61.42 = 1.11122 x 10-6 ms-2 Convert to miliGals = 1.11122 x 10-6 / (1x10-5) =0.111122286 BA= 30.5142 - 0.111122286 = 30.40307771miliGals