A. Theoretical Fundamentals Of Airborne Gravimetry, Parts .

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A. Theoretical Fundamentals of Airborne Gravimetry,Parts I and II (Monday, 23 May 2016)I. Introduction - Airborne Gravity Data AcquisitionII. Elemental Review of Physical GeodesyIII. Basic Theory of Moving-Base Scalar GravimetryIV. Overview of Airborne Gravimetry SystemsB. Theoretical Fundamentals of Gravity Gradiometry and InertialGravimetry (Thursday, 26 May 2016)V. Theoretical Fundamentals of Inertial GravimetryVI. Theoretical Fundamentals of Airborne GradiometryChristopher JekeliDivision of Geodetic ScienceSchool of Earth SciencesOhio State Universitye-mail: jekeli.1@osu.edu

National Geodetic SurveyI. Introduction - Airborne Gravity Data Acquisition A very brief history of airborne gravimetry Why airborne gravimetry?Airborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.1

A Brief History of Airborne Gravimetry Natural evolution of successes in 1st half of 20th century with ocean-bottom,submarine, and shipboard gravimeters operating in dynamic environments airborne systems promised rapid, if not highly accurate, regional gravitymaps for exploration reconnaissance and military geodetic applications Special challenges critical errors are functions of speed and speed-squared difficulty in accurate altitude & vertical acceleration determination trade accuracy for acquisition speed 1958: First fixed-wing airborne gravimetry test (Thompson and LaCoste 1960) 5-10 minute average, 10 mgal accuracy high altitude, 6-9 km Further tests by exploration concerns LaCoste & Romberg, Austin TX Gravity Meter Exploration Co., Houston, TX 10 mGal accuracy, 3 minute averages (Nettleton et al. 1960)Airborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.2

First Airborne Gravity Test – Air Force Geophysics Lab1958Instruction Manual LaCoste Romberg Model “S” Air-Sea Dynamic Gravity Meter, 2002; with permission The first LaCoste-RombergModel “S” Air-Sea Gravimeter KC-135 jet tanker Doppler navigation system – elevation above mean sea level determinedfrom the tracking range data flights over an Askania camera tracking range at Edwards Air Force BaseAirborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.3

First Successful Helicopter Airborne Gravimetry Test1965 Carson Services, Inc. (Carson Helicopter) gimbal-suspended LaCoste and Romberg Sea gravimeter 5 mGal accuracy, hovering at 15 m altitude (Gumert 1998) Navy sponsored Further tests and development by exploration companies principally, Carson Services throughout the 1960s and 1970sAirborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.4

Rapid Development with Adventof GPS (1980s and 1990s)(Brozena 1984)GPS Naval Research Laboratory– John Brozena National Survey and Cadastre ofDenmark (DKM) – Rene Forsberggravimeter systemP3-A Orion aircraftOlesen (2003) Academia (in collaboration withindustry and government) University of Calgary (K.P. Schwarz) University FAF Munich (G. Hein) Swiss Federal Institute of Technology (E.E. Klingele)Twin-Otter Aircraft Lamont-Doherty Earth (Geological) Observatory (R. Bell) Industry Airborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.5

Dedicated International Symposia& WorkshopsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.6

The Need for Global Gravity Data Gravity data until the early 1960s were obtained primarilyby point measurements on land and along some ship tracks. map of data archive of 1963 (Kaula 1963)Airborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.7

1990s – More Data, Still Many Gaps Greater uniformity, but only at relatively low resolution map of terrestrial 1 1 anomaly archive of 1990 (Rapp and Pavlis 1990)Airborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.8

Why Airborne Gravimetry?Satellite Resolution vs Mission Duration and Integration Time Only airbornegravimetry yieldshigher 000resolution [km] Satellite-derivedgravitational modelsare limited in spatialresolution becauseof high inherentsatellite speed(Jekeli 2004)10,00010010 s5s101s11 day*1 mo. 6 mo. 1 yr.5 yr.mission durationCHAMP: 10 sGRACE: 5 sGOCE: 5 - 10 sLaser-interferometry GRACE follow-on: 1 - 10 sAirborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.9

Gravity Resolution vs Accuracy Requirements in GeophysicsSatellite Gravimetry/Gradiometry Airborne Gravimetry/GradiometryAirborne Gravity for Geodesy Summer School , 23-27 May 2016Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.10

sea-level riseRiver 99/homes-flooded-nj-hurricane-sandy-oct-2012 620 422 s a-live/photos/000/166/cache/article-sea-level-rise 16648 600x450.jpgGeodetic Motivationcoastal flooding fromhurricane (Sandy)Airborne Gravity for Geodesy Summer School , 23-27 May mages/missouri%20river%20flooding(1).jpgBrief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.11

GNSS and Geopotential Traditional height reference surface: equipotential surface(geoid) needed for determining and monitoring the flow of water, fromflood control to sea level rise replace arduous spirit leveling with GNSS:H h Nellipsoid height(from GNSS)topographic surfacehHNorthometricheightgeoid undulation(from gravimetry)Airborne Gravity for Geodesy Summer School , 23-27 May 2016level, equipotential surface(geoid)reference surface(ellipsoid)Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU1.12

National Geodetic SurveyII. Elemental Review of Physical Geodesy Gravitational potential, gravity Normal gravity Disturbing potential, gravity anomaly, deflection of thevertical Geoid determination aspectsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.1

Basic Definitions Gravitational potential, Vrotation, ωE due to mass attractioncentrifugalaccelerationgravitation gravity gravitational acceleration: g V Centrifugal “potential”, φ due to Earth’s rotation centrifugal acceleration: acent φ Gravity potential, W V φ gravity acceleration:g g acentAirborne Gravity for Geodesy Summer School , 23-27 May 2016EarthPhysical geodesy makes thedistinction between gravitationand gravity, especially interrestrial gravimetryPhysical Geodesy Background, C. Jekeli, OSU2.2

Normal Gravitational Potential Mathematically simple potential and boundary– approximates Earth’s potential and geoid to about 5 ppm– approximates Earth’s gravity to about 50 ppm– rotates with the EarthωEbrpboundary rotational ellipsoidcentrifugal potential on ellipsoidφaV1 2 22cos U ωrφe p0boundary2 boundary function is chosen so thatgravity potential on boundary is aconstant, U0Airborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.3

Normal Gravity Potential Expressed as spherical harmonic series in spherical coordinatesU ( r ,θ ) V ( r ,θ ) φ ( r ,θ )GMa n 0 a r 2 n 11 2 2 2C P ( cos θ ) ωe r sin θ2N2n 2n closed expression exists in ellipsoidal coordinates C2Nn depends on only 4 parameters: ωe, C2N , a, GM (e.g., WGS84 parameters) Normal gravity vector: γ Uω e 7.292115 10 5 rad/s 0.484166774985 10 3C2N a 6378137. m GM 3.986004418 1014 m3 /s 2Airborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.4

Gravity Disturbance and Anomaly Disturbing potential:T W U(W total gravity potential) W U g γ Gravity disturbance vector: δ g δ g g γ gravity disturbance: gN γ N gN in n-frame (North-East-Down): δ g n g E g E g γ g γ D D D D due to symmetry, γ 0E near Earth’s surface, γN 0 WP U Q gP γ Q Gravity anomaly vector: g P P and Q are points on the ellipsoid normal such that WP UQ gravity anomaly: g gP γ QPAirborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.5

Deflection of the VerticalPNorthEastξηξ north deflectionη east deflectionGravityvector at PDeflection ofthe vertical at P η g ξ g gellipsoid ξ g g N T δ g n η g g E δ g g γ D D linear approximation signs agree with conventionof astronomic deflection ofthe verticalDownAirborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.6

Geoid DeterminationN P0 Bruns’s Formula: 1γQTP0 N 0where N0 is a height datum offset0 Boundary-value Problem: 2T 0 above geoid (by assumption)1 γ T g P′ h P′ γ Q h g P′ , g P′0a0 gravity reductions g P′Pa′00boundaryconditionTP′0Q0 g P′aflight altitude g P′P′ Stokes’s formulaN N0 P0R4πγ Q0 g S (ψ ) d ΩP0′ g P′′P00NPP0 , P0′ΩAirborne Gravity for Geodesy Summer School , 23-27 May 2016P00Q0topographicsurfacegeoidellipsoidPhysical Geodesy Background, C. Jekeli, OSU2.7

Details Gravity reductions to satisfy the boundary-value conditions re-distribution of topographic mass; consequent indirect effect downward continuation (various methods) Ellipsoidal corrections account for spherical approximation of geoid, boundary condition Include existing spherical harmonic model (satellite-derived) remove-compute-restore techniques Back to Motivation use airborne gravimetry to improve spatial resolution of data(boundary values) – few km to 200 km wavelengthsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Physical Geodesy Background, C. Jekeli, OSU2.8

National Geodetic SurveyIII. Basic Theory of Moving-Base Scalar Gravimetry Fundamental laws of physics and the gravimetry equation Coordinate frames Mechanizations and methods of scalar gravimetry Rudimentary error analysesAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.1

Fundamental Physical Laws Moving-Base Gravimetry and Gradiometry arebased on 3 fundamental laws in physics Newton’s Second Law of Motion Newton’s Law of GravitationIssac Newton1643 - 1727 Einstein’s Equivalence Principle Laws are expressed in an inertial frame General Relativistic effects are not yet neededAlbert Einstein1879 – 1955 however, the interpretation of space in the theory of general relativityis used to distinguish between applied and gravitational forcesAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.2

Inertial Frame3iquasar kquasar 1quasar 22inotation convention:- axis identified by number- superscript identifies frame1i The realization of a system of coordinates that does not rotate(and is in free-fall, e.g., Earth-centered) Modern definition: fixed to quasars – which exhibit norelative motion on celestial sphere International Celestial Reference Frame (ICRF) based oncoordinates of 295 stable quasarsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.3

Newton’s Second Law of Motion Time-rate of change of linear momentum equals applied force, Fd( mi x ) Fdt– mi is the inertial mass of the test bodyFmix ( mi constant mi x F ) In the presence of a gravitational field, this law must be modified:mi x F Fg– Fg is a force associated with the gravitational acceleration due to a field (orspace curvature) generated by all masses in the universe, relative to thefreely-falling frame (Earth’s mass and tidal effects due to moon, sun, etc.) action forces, F, and gravitational forces, Fg, are fundamentally differentAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.4

Newton’s Law of Gravitation Gravitational force vectorMmg Fg G n mg g2 mgn– G Newton’s gravitational constantunit vector – g gravitational acceleration due to M– mg is the gravitational mass of the test body it’s easier to work with field potential, Vg VattractingM massPV GM j Many mass points law of superposition:VP G Mjj mass continuum:Mj dMVP G PjdMMAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.5

Equivalence Principle (1) A. Einstein (1907): No experiment performed in a closed systemcan distinguish between an accelerated reference frame or areference frame at rest in a uniform gravitational field.– consequence: inertial mass equals gravitational massm m mig Experimental evidence has not been ableto dispute this assumption violation of the principle may lead to newtheories that unify gravitational and otherforces proposed French Space Agency mission,MICROSCOPE*, aims to push thesensitivity by many orders of magnitude* Micro-Satellite à traînée Compensée pour l’Observation du Principe d’Equivalence (Drag CompensatedMicro-satellite to Observe the Equivalence Principle); Berge et al. (2015) http://arxiv.org/abs/1501.01644Airborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.6

Equivalence Principle (2) Equation of motion in the inertial frameiF xi gim x accelerationof rocketd 2x x , vector of total2dt kinematic accelerationFi a i , specific force, or thethrust amacceleration resulting fromg gravityan action force; e.g., thrustof a rocketa gi x ai giAirborne Gravity for Geodesy Summer School , 23-27 May 2016 no lift-off !Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.7

What Does an Accelerometer Sense?gravitational field, gno applied accelerationgravitational field, gapplied acceleration, aspring constant, kg0za0mgggforce of spring: kzinput axisaccelerometer indicates: 0mgagaccelerometer indicates: za a Accelerometer does not sense gravitation, only accelerationdue to action force (including reaction forces!)Airborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.8

What Does Accelerometer (or Gravimeter) on Rocket Sense?Rocket: experiences no atmospheric dragengine has constant thrustlaunches verticallyAccelerometer axis:vertically upg0t0at rest onlaunch padt1rocketignitest2fuel isgoneAirborne Gravity for Geodesy Summer School , 23-27 May 2016t3maximumheightt4parachutedeploystimet5at rest onlaunch padTheoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.9

Static Gravimetry – Special Caseg x a Assume non-rotating Earth (for simplicity)x 0 a g Relative (spring) gravimeter: it is an accelerometer that senses specific force, a with sensitive axis along plumb line, a is the reaction forceof Earth’s surface that keeps the gravimeter from falling Absolute (ballistic) gravimeter: a 0 x g it tracks a test mass in vacuum (zero spring force) indirectly, it senses the reaction force that keeps thereference from falling All operational moving-base gravimeters are relativesensorsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.10

Basic Equation for Moving-Base Gravimetry In the inertial frame:i g xi ai Because:– specific forces are measured in a non-inertial frame attached toa rotating body (vehicle)– specific forces and kinematic accelerations refer to differentmeasurement points of the instrument-carrying vehicle– generally, gravitation is desired in alocal, Earth-fixed frame Need to introduce:a x GPS– coordinate frames– rotations and lever-arm effects Get more complicated expressions for gravimetry equationAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.11

Two Possible Approaches to Determine g (1)ix ai gi For concepts, consider inertial frame for simplicity: Position (Tracking) Method to determine the unknown: g Integrate equations of motiontx i ( t ) x i ( t0 ) x i ( t0 )( t t0 ) t t ') ( a i ( t ') g i ( t ') ) dt '( t0 Positions, x: from tracking system, like GPS or other GNSS Specific forces, a: from accelerometer method is used for geopotential determination with satellite tracking, andwas used also with ground-based inertial positioning systems Advantage: do not need to differentiate x to get x Disadvantage: g must be modeled in some way to perform the integration(e.g., spherical harmonics in satellite tracking, with statistical constraint) Not used for scalar airborne gravimetry due to vertical instability of integral but can be (is) used for horizontal components of gravity!Airborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.12

Two Possible Approaches to Determine g (2) Accelerometry Method to determine the unknown: gi g xi ai Specific force, a : from accelerometer Kinematic acceleration, x : by differentiating position from trackingsystem, like GPS (GNSS) Advantage: g does not need to be modeled Disadvantage: positions are processed with two numerical differentiations advanced numerical techniques may be less serious than gravity modeling problem Either position method or accelerometry method requirestwo independent sensor systems Tracking system Accelerometer (gravimeter) Gravimetry accuracy depends equally on the precision of both systemsAirborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.13

The Challenge of Airborne Gravimetry Both systems measure large signals e.g., ( 10000 mGal) Desired gravity disturbance is orders of magnitude smaller signal-to-noise ratio may be very small, depending on system accuracies e.g., INS/GPS system – data from University of Calgary, 1996MathCad: example airborne INS-GPS.xmcd IMU accelerations[mGal]2 104[mGal] GPS accelerations(offset)Subtract and filtertime [s]Airborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.14

Coordinate Frames Other coordinate frames– rotating with respect to inertial frame,– may have different origin point,– have different form of Newton’s law of motion,– all defined by three mutually orthogonal, usually right-handed axes(Cartesian coordinates). Specific frames to be considered:– navigation frame: frame in which navigation equations areformulated; usually identified with local North-East-Down (NED)directions (n-frame).– Earth-centered-Earth-fixed frame: frame with origin at Earth’scenter of mass and axes defined by conventional pole and Greenwichmeridian (Cartesian or geodetic coordinates) (e-frame).Airborne Gravity for Geodesy Summer School , 23-27 May 2016Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU3.15

Earth-Centered-Earth-Fixed Coordinates( Cartesian coordinate vector in e-frame: x xee1e2xe3x)T Geodetic coordinates, latitude, longitude, height: φ , λ , h3e refer to a particular ellipsoid(assume geocentric) with semi-majoraxis, a, and first eccentricity, emeridianxee3x are orthogonal curvilineare-frame coordinatese1x1eAirborne Gravity for Geodesy Summer School , 23-27 May 2016λellipsoidhφx2e2eequatorTheoretical Fundamentals of Airborne Gra

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.2 Moving-Base Gravimetry and Gradiometry are based on 3 fundamental laws in physics Fundamental Physical Laws Issac Newton 1643 - 1727 Newton’s Second L

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