Tutorial Ellipsoid, Geoid, Gravity, Geodesy, and Geophysics
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A Fast Method for Calculation of Marine Gravity Anomaly
applied sciences Article A Fast Method for Calculation of Marine Gravity Anomaly Yuan Fang 1, Shuiyuan He 2,*, Xiaohong Meng 1,*, Jun Wang 1, Yongkang Gan 1 and Hanhan Tang 1 1 School of Geophysics and Information Technology, China University of Geosciences, Beijing 100083, China; [email protected] (Y.F.); [email protected] (J.W.); [email protected] (Y.G.); [email protected] (H.T.) 2 Guangzhou Marine Geological Survey, China Geological Survey, Ministry of Land and Resources, Guangzhou 510760, China * Correspondence: [email protected] (S.H.); [email protected] (X.M.); Tel.: +86-136-2285-1110 (S.H.); +86-136-9129-3267 (X.M.) Abstract: Gravity data have been playing an important role in marine exploration and research. However, obtaining gravity data over an extensive marine area is expensive and inefficient. In reality, marine gravity anomalies are usually calculated from satellite altimetry data. Over the years, numer- ous methods have been presented for achieving this purpose, most of which are time-consuming due to the integral calculation over a global region and the singularity problem. This paper proposes a fast method for the calculation of marine gravity anomalies. The proposed method introduces a novel scheme to solve the singularity problem and implements the parallel technique based on a graphics processing unit (GPU) for fast calculation. The details for the implementation of the proposed method are described, and it is tested using the geoid height undulation from the Earth Gravitational Model 2008 (EGM2008). The accuracy of the presented method is evaluated by comparing it with marine shipboard gravity data. -
THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P. -
B3 : Variation of Gravity with Latitude and Elevation
Geophysics 210 September 2008 B3 Variation of gravity with latitude and elevation By measuring the subtle changes in the acceleration of gravity from one place to another, it is possible to learn about changes in subsurface density. However, other factors can cause gravity to vary with position on the Earth. These effects must be removed from measurements in order to use gravity data to study the interior of the Earth. B3.1 Variation of gravity with latitude It is observed that at the Equator, g E = 978,033 mgal while at the poles g P = 983,219 mgal This difference is 5186 mgal, which is a lot larger than changes in gravity because of subsurface density. Can this observation be explained by the fact that the Earth is a rotating ellipsoid? (A)The Earth is distorted by rotation The Earth is an oblate spheroid. R E = 6378 km R P = 6357 km. Qualitative answer Since a point on the Equator is further from the centre of the Earth than the poles, gravity will be weaker at the Equator and g E < g P Quantitative answer GM E 24 For a sphere g (r) = where the mass of the Earth, ME = 5.957 10 kg. r 2 At the North Pole, RP = 6357 km and g P = 983,219 mgal. If we move up 21 km to the equator, the decrease in gravity will be 6467 mgal Thus g E = g P - 6467 mgal, which is too much to explain the observed difference between the Equator and the Poles. 1 Geophysics 210 September 2008 (B) - Centrifugal forces vary with latitude The rotation of the Earth also causes gravity to vary with latitude. -
Detection of Caves by Gravimetry
Detection of Caves by Gravimetry By HAnlUi'DO J. Cmco1) lVi/h plates 18 (1)-21 (4) Illtroduction A growing interest in locating caves - largely among non-speleolo- gists - has developed within the last decade, arising from industrial or military needs, such as: (1) analyzing subsUl'face characteristics for building sites or highway projects in karst areas; (2) locating shallow caves under airport runways constructed on karst terrain covered by a thin residual soil; and (3) finding strategic shelters of tactical significance. As a result, geologists and geophysicists have been experimenting with the possibility of applying standard geophysical methods toward void detection at shallow depths. Pioneering work along this line was accomplishecl by the U.S. Geological Survey illilitary Geology teams dUl'ing World War II on Okinawan airfields. Nicol (1951) reported that the residual soil covel' of these runways frequently indicated subsi- dence due to the collapse of the rooves of caves in an underlying coralline-limestone formation (partially detected by seismic methods). In spite of the wide application of geophysics to exploration, not much has been published regarding subsUl'face interpretation of ground conditions within the upper 50 feet of the earth's surface. Recently, however, Homberg (1962) and Colley (1962) did report some encoUl'ag- ing data using the gravity technique for void detection. This led to the present field study into the practical means of how this complex method can be simplified, and to a use-and-limitations appraisal of gravimetric techniques for speleologic research. Principles all(1 Correctiolls The fundamentals of gravimetry are based on the fact that natUl'al 01' artificial voids within the earth's sUl'face - which are filled with ail' 3 (negligible density) 01' water (density about 1 gmjcm ) - have a remark- able density contrast with the sUl'roun<ling rocks (density 2.0 to 1) 4609 Keswick Hoad, Baltimore 10, Maryland, U.S.A. -
The First Measurement of the Deflection of the Vertical in Longitude
Eur. Phys. J. H. DOI: 10.1140/epjh/e2014-40055-2 The first measurement of the deflection of the vertical in longitude The figure of the earth in the early 19th century Andreas Schrimpfa Philipps-Universit¨atMarburg, Fachbereich Physik, Renthof 5, D-35032 Marburg, Germany Abstract. During the summer of 1837 Christian Ludwig Gerling, a for- mer student of Carl Friedrich Gauß’s, organized the world wide first de- termination of the deflection of the vertical in longitude. From a mobile observatory at the Frauenberg near Marburg (Hesse) he measured the astronomical longitude difference between C.F. Gauß’s observatory at G¨ottingenand F.G.B. Nicolai's observatory at Mannheim within an er- ror of 000: 4. To achieve this precision he first used a series of light signals for synchronizing the observatory clocks and, second, he very carefully corrected for the varying reaction time of the observers. By comparing these astronomical results with the geodetic{determined longitude dif- ferences he had recently measured for the triangulation of Kurhessen, he was able to extract a combined value of the deflection of the vertical in longitude of G¨ottingenand Mannheim. His results closely agree with modern vertical deflection data. 1 Introduction The discussion about the figure of the earth and its determination was an open ques- tion for almost two thousand years, the sciences involved were geodesy, geography and astronomy. Without precise instruments the everyday experience suggested a flat, plane world, although ideas of a spherically shaped earth were known and ac- cepted even in the ancient world. Assuming that the easily observable daily motion of the stars is due to the rotation of the earth, the rotational axis can be used to define a celestial sphere; a coordinate system, where the stars' position is given by two angles. -
Gravity Anomaly Measuring Gravity
Gravity Newton’s Law of Gravity (1665) 2 F = G (m1m2) / r F = force of gravitational attraction m1 and m2 = mass of 2 attracting objects r = distance between the two objects G -- ? Earth dimensions: 3 rearth = 6.378139 x 10 km (equator) 27 mearth = 5.976 x 10 g 2 F = G (m1m2) / r ) 2 F = mtest (G mEarth / rEarth ) Newton’s Second Law: F = ma implies that the acceleration at the surface of the Earth is: 2 (G mEarth / rEarth ) = ~ 981 cm/sec2 = 981 “Gals” variations are on the order of (0.1 mgal) Precision requires 2 spatially based corrections must account for 2 facts about the Earth it is not round (.... it’s flattened at the poles) it is not stationary ( .... it’s spinning) Earth is an ellipsoid first detected by Newton in 1687 clocks 2 min/day slower at equator than in England concluded regional change in "g" controlling pendulum translated this 2 min into different values of Earth radii radiusequator > radiuspole requator / rpole = 6378/6357 km = flattening of ~1/298 two consequences equator farther from center than poles hence gravity is 6.6 Gals LESS at the equator equator has more mass near it than do the poles hence gravity is 4.8 Gals MORE at the equator result gravity is 1.8 Gals LESS at the equator Earth is a rotating object circumference at equator ~ 40,000 km; at poles 0 km all parts of planet revolve about axis once in 24 hrs hence equator spins at 40,000 =1667 km/hr; at poles 0 km/hr 24 outward centrifugal force at equator; at poles 0 result .... -
Instruments and Gravity Processing Drift and Tides
Gravity 5 Objectives Instruments Gravity Gravity 5 Corrections Instruments and gravity processing Drift and Tides Latitude Free Air Chuck Connor, Laura Connor Atmosphere Simple Bouguer Summary Potential Fields Geophysics: Week 5 Further Reading EOMA Gravity 5 Objectives for Week 5 Gravity 5 Objectives Instruments • Gravity Learn about gravity Corrections instruments Drift and Tides • Learn about Latitude processing of gravity Free Air data Atmosphere • Make the Simple Bouguer corrections to Summary calculate a simple Further Bouguer anomaly Reading EOMA Gravity 5 The pendulum The first gravity data collected in the US were Gravity 5 obtained by G. Putman working for the Coast and Geodetic survey, around 1890. These gravity data comprised a set of 26 measurements made along a roughly E-W transect across the entire continental Objectives US. The survey took about 6 months to complete and was designed primarily to investigate isostatic Instruments compensation across the continent. Gravity Putnam used a pendulum gravity meter, based on the Corrections relation between gravity and the period of a pendulum: Drift and Tides 4π2l g = Latitude T 2 Free Air where: l is the length of the pendulum Atmosphere T is the pendulum period Simple Example Seems easy enough to obtain an absolute gravity Bouguer reading, but in practice pendulum gravity meters are Assuming the period of a pendulum is known to be 1 s Summary problematic. The length of the pendulum can change exactly, how well must the length of the pendulum be with temperature, the pendulum stand tends to sway, known to measure gravity to 10 mGal precision? Further air density effects the measurements, etc. -
The Meridian Arc Measurement in Peru 1735 – 1745
The Meridian Arc Measurement in Peru 1735 – 1745 Jim R. SMITH, United Kingdom Key words: Peru. Meridian. Arc. Triangulation. ABSTRACT: In the early 18th century the earth was recognised as having some ellipsoidal shape rather than a true sphere. Experts differed as to whether the ellipsoid was flattened at the Poles or the Equator. The French Academy of Sciences decided to settle the argument once and for all by sending one expedition to Lapland- as near to the Pole as possible; and another to Peru- as near to the Equator as possible. The result supported the view held by Newton in England rather than that of the Cassinis in Paris. CONTACT Jim R. Smith, Secretary to International Institution for History of Surveying & Measurement 24 Woodbury Ave, Petersfield Hants GU32 2EE UNITED KINGDOM Tel. & fax + 44 1730 262 619 E-mail: [email protected] Website: http://www.ddl.org/figtree/hsm/index.htm HS4 Surveying and Mapping the Americas – In the Andes of South America 1/12 Jim R. Smith The Meridian Arc Measurement in Peru 1735-1745 FIG XXII International Congress Washington, D.C. USA, April 19-26 2002 THE MERIDIAN ARC MEASUREMENT IN PERU 1735 – 1745 Jim R SMITH, United Kingdom 1. BACKGROUND The story might be said to begin just after the mid 17th century when Jean Richer was sent to Cayenne, S. America, to carry out a range of scientific experiments that included the determination of the length of a seconds pendulum. He returned to Paris convinced that in Cayenne the pendulum needed to be 11 lines (2.8 mm) shorter there than in Paris to keep the same time. -
Interpreting Gravity Anomalies in South Cameroon, Central Africa
EARTH SCIENCES RESEARCH JOURNAL Earth Sci Res SJ Vol 16, No 1 (June, 2012): 5 - 9 GRAVITY EXPLORATION METHODS Interpreting gravity anomalies in south Cameroon, central Africa Yves Shandini1 and Jean Marie Tadjou2* 1 Physics Department, Science Faculty, University of Yaoundé I, Cameroon 2 National Institute of Cartography- NIC, Yaoundé, Cameroon * Corresponding author E-mail: shandiniyves@yahoo fr Abstract Keywords: fault, gravity data, isostatic residual gravity, south Cameroon, total horizontal derivative The area involved in this study is the northern part of the Congo craton, located in south Cameroon, (2 5°N - 4 5°N, 11°E - 13°E) The study involved analysing gravity data to delineate major structures and faults in south Cameroon The region’s Bouguer gravity is characterised by elongated SW-NE negative gravity anomaly corresponding to a collapsed structure associated with a granitic intrusion beneath the region, limited by fault systems; this was clearly evident on an isostatic residual gravity map High grav- ity anomaly within the northern part of the area was interpreted as a result of dense bodies put in place at the root of the crust Positive anomalies in the northern part of the area were separated from southern negative anomalies by a prominent E-W lineament; this was interpreted on the gravity maps as a suture zone between the south Congo craton and the Pan-African formations Gravity anomalies’ total horizontal derivatives generally reflect faults or compositional changes which can describe structural trends The lo- cal maxima -
Alfonsi-Article Marine V5
1 Liliane ALFONSI Groupe d’Histoire et de Diffusion des Sciences d’Orsay (GHDSO ) Bât. 407. Université Paris Sud 11 91405 ORSAY Cédex [email protected] Un successeur de Bouguer : Étienne Bézout (1730–1783) commissaire et expert pour la marine Résumé : Entré à l’Académie Royale des Sciences dès 1758, Étienne Bézout, compte parmi ses tâches, l’obligation d’étudier différents ouvrages et outils présentés à l’Académie. Si dès le début il se voit confier l’étude d’instruments relatifs à la navigation, cela s’amplifiera à partir de 1764, date à laquelle Choiseul, ministre de la Marine, charge Bézout de réorganiser la formation des officiers de ce corps. Ses visites dans les ports pour faire passer les examens de la Marine, l‘amènent à appréhender concrètement les problèmes de la navigation. Cela lui vaudra d’être désigné pour juger la validité des critiques que l’hydrographe Blondeau énonce à l’encontre du traité de Pilotage de Bouguer. Par ailleurs cette situation amène Bézout à être membre de l’Académie de Marine de Brest et à être un des responsables de son association avec l’Académie des Sciences. Nous étudierons ces divers points. Enfin, Bézout écrit en 1769 un Traité de navigation que nous pourrons comparer à l’ouvrage de Bouguer sur le même sujet. Mots clés : Académie des Sciences, Marine, Académie de Marine, Traité de Navigation, Bézout, Bouguer. Abstract : Étienne Bézout, member of the Académie Royale des Sciences, have to study some works and books sended to this institution. In this article, we will look at his responsibility for Navy, before and after 1764, which is the year of Bézout’s nomination at the charge of Examinateur des Gardes du Pavillon et de la Marine. -
The Deflection of the Vertical, from Bouguer to Vening-Meinesz, and Beyond – the Unsung Hero of Geodesy and Geophysics
EGU21-596 https://doi.org/10.5194/egusphere-egu21-596 EGU General Assembly 2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. The Deflection of the Vertical, from Bouguer to Vening-Meinesz, and Beyond – the unsung hero of geodesy and geophysics Christopher Jekeli Ohio State University, School of Earth Sciences, Division of Geodetic Science, United States of America ([email protected]) When thinking of gravity in geodesy and geophysics, one usually thinks of its magnitude, often referred to a reference field, the normal gravity. It is, after all, the free-air gravity anomaly that plays the significant role in terrestrial data bases that lead to Earth Gravitational Models (such as EGM96 or EGM2008) for a multitude of geodetic and geophysical applications. It is the Bouguer anomaly that geologists and exploration geophysicists use to infer deep crustal density anomalies. Yet, it was also Pierre Bouguer (1698-1758) who, using the measured direction of gravity, was the first to endeavor a determination of Earth’s mean density (to “weigh the Earth”), that is, by observing the deflection of the vertical due to Mount Chimborazo in Ecuador. Bouguer’s results, moreover, sowed initial seeds for the theories of isostasy. With these auspicious beginnings, the deflection of the vertical has been an important, if not illustrious, player in geodetic history that continues to the present day. Neglecting the vertical deflection in fundamental surveying campaigns in the mid to late 18th century (e.g., Lacaille in South Africa and Méchain and Delambre in France) led to errors in the perceived shape of the Earth, as well as its scale that influenced the definition of the length of a meter. -
Bouguer Gravity Anomaly
FS–239–95 OCTOBER 1997 Introduction to Potential Fields: Gravity Introduction acceleration, g, or gravity. The unit of gravity is the Gravity and magnetic exploration, also referred to Gal (in honor of Galileo). One Gal equals 1 cm/sec2. as “potential fields” exploration, is used to give geo- Gravity is not the same everywhere on Earth, scientists an indirect way to “see” beneath the Earth’s but changes with many known and measurable fac- surface by sensing different physical properties of tors, such as tidal forces. Gravity surveys exploit the rocks (density and magnetization, respectively). Grav- very small changes in gravity from place to place ity and magnetic exploration can help locate faults, that are caused by changes in subsurface rock dens- mineral or petroleum resources, and ground-water res- ity. Higher gravity values are found over rocks that ervoirs. Potential-field surveys are relatively inexpen- are more dense, and lower gravity values are found sive and can quickly cover large areas of ground. over rocks that are less dense. What is gravity? How do scientists measure gravity? Gravitation is the force of attraction between two Scientists measure the gravitational acceleration, bodies, such as the Earth and our body. The strength g, using one of two kinds of gravity meters. An of this attraction depends on the mass of the two bod- absolute gravimeter measures the actual value of g ies and the distance between them. by measuring the speed of a falling mass using a A mass falls to the ground with increasing veloc- laser beam. Although this meter achieves precisions ity, and the rate of increase is called gravitational of 0.01 to 0.001 mGal (milliGals, or 1/1000 Gal), they are expensive, heavy, and bulky.