A. Theoretical Fundamentals of Airborne Gravimetry, Parts I and II (Monday, 23 May 2016) B. Theoretical Fundamentals of Gravity

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A. Theoretical Fundamentals of Airborne Gravimetry, Parts I and II (Monday, 23 May 2016) I. Introduction - Airborne Gravity Data Acquisition II. Elemental Review of Physical Geodesy III. Basic Theory of Moving-Base Scalar Gravimetry IV. Overview of Airborne Gravimetry Systems B. Theoretical Fundamentals of Gravity Gradiometry and Inertial Gravimetry (Thursday, 26 May 2016) V. Theoretical Fundamentals of Inertial Gravimetry VI. Theoretical Fundamentals of Airborne Gradiometry

Christopher Jekeli Division of Geodetic Science School of Earth Sciences Ohio State University e-mail: [email protected] National Geodetic Survey

I. Introduction - Airborne Gravity Data Acquisition

• A very brief history of airborne gravimetry

• Why airborne gravimetry?

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.1 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 gravity maps 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 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.2 First Airborne Gravity Test – Air Force Geophysics Lab 1958

Instruction Manual LaCoste Romberg Model “S” Air-Sea Dynamic Gravity Meter, 2002; with permission • The first LaCoste-Romberg • KC-135 jet tanker Model “S” Air-Sea Gravimeter − Doppler navigation system – elevation above mean sea level determined from the tracking range data − flights over an Askania camera tracking range at Edwards Air Force Base

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.3 First Successful Helicopter Airborne Gravimetry Test 1965

• 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 1970s

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.4 Rapid Development with Advent (Brozena 1984) of GPS (1980s and 1990s) GPS • Naval Research Laboratory – John Brozena P3-A Orion aircraft gravimeter system

• National Survey and Cadastre of Olesen (2003) Denmark (DKM) – Rene Forsberg • Academia (in collaboration with industry 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 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.5 Dedicated International Symposia & Workshops

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.6 The Need for Global Gravity Data

• Gravity data until the early 1960s were obtained primarily by 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 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.7 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 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.8 Why Airborne Gravimetry?

Satellite Resolution vs Mission Duration and Integration Time (Jekeli 2004) 10,000 • Satellite-derived gravitational models

are limited in spatial 1,000 measurement resolution because

integration of high inherent time*

satellite speed 100 10 s 5 s • Only airborne resolution [km] resolution 10 gravimetry yields 1 s higher resolution efficiently 1 1 day 1 mo. 6 mo. 1 yr. 5 yr. mission duration * CHAMP: 10 s GOCE: 5 - 10 s GRACE: 5 s Laser-interferometry GRACE follow-on: 1 - 10 s

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.9 Gravity Resolution vs Accuracy Requirements in Geophysics

Satellite Gravimetry/Gradiometry Airborne Gravimetry/Gradiometry

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.10 Geodetic Motivation

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sea River flooding live/photos/000/166/cache/article - - to - coastal flooding from change - - http://images.nationalgeographic.com/wpf/media hurricane (Sandy)

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Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.11 GNSS and Geopotential • Traditional height reference surface: equipotential surface (geoid) − needed for determining and monitoring the flow of water, from flood control to sea level rise − replace arduous spirit leveling with GNSS: H= hN −

ellipsoid height (from GNSS) topographic surface H h orthometric height level, equipotential surface (geoid) N geoid undulation (from gravimetry) reference surface (ellipsoid)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Brief History of Airborne Gravity and Data Needs, C. Jekeli, OSU 1.12 National Geodetic Survey

II. Elemental Review of Physical Geodesy

• Gravitational potential, gravity • Normal gravity • Disturbing potential, gravity anomaly, deflection of the vertical • Geoid determination aspects

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.1 Basic Definitions

• Gravitational potential, V ω − due to mass attraction rotation, E − gravitational acceleration: g =∇ V centrifugal acceleration gravitation gravity • Centrifugal “potential”, φ

− due to Earth’s rotation Earth − centrifugal acceleration: acent =∇φ

• Gravity potential, W = V + φ Physical geodesy makes the distinction between gravitation and gravity, especially in − gravity acceleration: = + g gacent terrestrial gravimetry

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.2 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 Earth

ωE boundary = rotational ellipsoid b centrifugal potential on ellipsoid rp φ a 1 V= Ur − ωφ22cos 2 boundary 0 2 ep • boundary function is chosen so that gravity potential on boundary is a

constant, U0

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.3 Normal Gravity Potential

• Expressed as spherical harmonic series in spherical coordinates

Ur()()(),θ= Vr ,, θφθ + r ∞ 21n+ GM a 1 = CPN cosθω+ 22r sin 2 θ ∑ 22nn() e arn=0  2

− closed expression exists in ellipsoidal coordinates

N N − C 2 n depends on only 4 parameters: ωe, C 2 , a, GM − (e.g., WGS84 parameters)

−5 ωe =7.292115 × 10 rad/s N =−×−3 • Normal gravity vector: γ = ∇U C2 0.484166774985 10 a = 6378137. m GM = 3.986004418× 1014 m 3 /s 2

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.4 Gravity Disturbance and Anomaly

• Disturbing potential: TWU= − (W = total gravity potential) • Gravity disturbance vector: δ gg=−=−∇∇WU γ

− gravity disturbance: δ g =g − γ

ggNN−γ  N  δ n =  ≈  − in n-frame (North-East-Down): g ggEE      ggDD−−γγ  DD  ◦ due to symmetry, γE = 0

◦ near Earth’s surface, γN ≈ 0

• Gravity anomaly vector: ∆ggP=−=−∇∇WU P Q PQ γ

− P and Q are points on the ellipsoid normal such that WP = UQ

− gravity anomaly: ∆gPPQ=g − γ

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.5 Deflection of the Vertical P North ξ = north deflection ξ η East η = east deflection

Gravity Deflection of −ξ gg N  the vertical at P vector at P ∇ =δηn ≈−  ≈  Tg  gg E     δγgg DD− 

−η |g| − linear approximation −ξ |g| g − signs agree with convention of astronomic deflection of ellipsoid the vertical

Down

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.6 Geoid Determination 1 • Bruns’s Formula: N= TN + where N0 is a height datum offset PP00γ 0 Q0 • Boundary-value Problem: ∇=2T 0 above geoid (by assumption)

∂∂T 1 γ boundary ∆gTPP′′=−+ ∆∆gg′′, 00 PPa ∂∂hhγ condition PQ00′ Q0 + ∆g gravity reductions Pa′

|| Pa′ flight altitude ∆g ′ P0 ∆gP′ P′

• Stokes’s formula topographic ∆gP′ P′ 0 surface R 0 P0 NNP =0, + ∆ψgSP′′() PP d Ω N geoid 0 4πγ ∫∫ 0 00 P0 Q0 Ω Q0 ellipsoid Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.7 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 wavelengths

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Physical Geodesy Background, C. Jekeli, OSU 2.8 National Geodetic Survey

III. 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 analyses

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.1 Fundamental Physical Laws

• Moving-Base Gravimetry and Gradiometry are based on 3 fundamental laws in physics

− Newton’s Second Law of Motion

Issac Newton − Newton’s Law of Gravitation 1643 - 1727

− Einstein’s Equivalence Principle

• Laws are expressed in an inertial frame

Albert Einstein • General Relativistic effects are not yet needed 1879 – 1955 − however, the interpretation of space in the theory of general relativity is used to distinguish between applied and gravitational forces

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.2 Inertial Frame

i 3 quasar 1

quasar k notation convention: quasar 2 - axis identified by number 2i - superscript identifies frame 1i

• 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 no relative motion on celestial sphere

• International Celestial Reference Frame (ICRF) based on coordinates of 295 stable quasars

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.3 Newton’s Second Law of Motion

• Time-rate of change of linear momentum equals applied force, F

d F mi ()mi x =F dt x  – mi is the inertial mass of the test body (mmii=constant → x=F)

• In the presence of a gravitational field, this law must be modified:

migx=F+ F

– Fg is a force associated with the gravitational acceleration due to a field (or space curvature) generated by all masses in the universe, relative to the freely-falling frame (Earth’s mass and tidal effects due to moon, sun, etc.)

− action forces, F, and gravitational forces, Fg, are fundamentally different

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.4 Newton’s Law of Gravitation • Gravitational force vector

Mmg mg Fgg=Gm2 ng =  n – G = Newton’s gravitational constant unit vector  attracting – g = gravitational acceleration due to M mass M – mg is the gravitational mass of the test body

− it’s easier to work with field potential, V P g =∇ V V= GM   j • Many mass points Mj M j − law of superposition: VGP = ∑  P j j dM − mass continuum: VG= dM P ∫  M

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.5 Equivalence Principle (1) • A. Einstein (1907): No experiment performed in a closed system can distinguish between an accelerated reference frame or a reference frame at rest in a uniform gravitational field. – consequence: inertial mass equals gravitational mass

mmig= = m • Experimental evidence has not been able to dispute this assumption − violation of the principle may lead to new theories that unify gravitational and other forces − proposed French Space Agency mission, MICROSCOPE*, aims to push the sensitivity by many orders of magnitude

* Micro-Satellite à traînée Compensée pour l’Observation du Principe d’Equivalence (Drag Compensated Micro-satellite to Observe the Equivalence Principle); Berge et al. (2015) http://arxiv.org/abs/1501.01644 Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.6 Equivalence Principle (2)

• Equation of motion in the inertial frame

F i xgii= + m x = acceleration dx2 of rocket ◦  , vector of total x = 2 dt kinematic acceleration

i F i ◦ = a , specific force, or the thrust ⇒ a g = gravity m acceleration resulting from an action force; e.g., thrust of a rocket ag≤⇒no lift-off ! xagiii= +

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.7 What Does an Accelerometer Sense? gravitational field, g gravitational field, g no applied acceleration applied acceleration, a

spring constant, k

force of spring: kz g g za m 0 m 0 g

g input axis g g a

accelerometer indicates: 0 accelerometer indicates: za ~ a • Accelerometer does not sense gravitation, only acceleration due to action force (including reaction forces!)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.8 What Does Accelerometer (or Gravimeter) on Rocket Sense? Rocket: experiences no atmospheric drag engine has constant thrust launches vertically

Accelerometer axis: vertically up

time t0 t1 t2 t3 t4 t5 at rest on rocket fuel is maximum parachute at rest on launch pad ignites gone height deploys launch pad

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.9 Static Gravimetry – Special Case

• Assume non-rotating Earth (for simplicity) g= xa −

• Relative (spring) gravimeter: x=⇒=−0 ag

− it is an accelerometer that senses specific force, a

− with sensitive axis along plumb line, a is the reaction force of Earth’s surface that keeps the gravimeter from falling

• Absolute (ballistic) gravimeter: a=⇒=0  xg

− it tracks a test mass in vacuum (zero spring force)

− indirectly, it senses the reaction force that keeps the reference from falling

• All operational moving-base gravimeters are relative sensors Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.10 Basic Equation for Moving-Base Gravimetry

• In the inertial frame: gi= xa ii −

• Because: – specific forces are measured in a non-inertial frame attached to a rotating body (vehicle) – specific forces and kinematic accelerations refer to different measurement points of the instrument-carrying vehicle – generally, gravitation is desired in a local, Earth-fixed frame x GPS a • Need to introduce: – coordinate frames – rotations and lever-arm effects • Get more complicated expressions for gravimetry equation

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.11 Two Possible Approaches to Determine g (1) • For concepts, consider inertial frame for simplicity: xagiii= + • Position (Tracking) Method to determine the unknown: g − Integrate equations of motion t xxii()t=()()() t + x i ttt −+()()() tt −'agi t ' + i tdt '' 0 00∫ ( ) 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, and was 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 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.12 Two Possible Approaches to Determine g (2) • Accelerometry Method to determine the unknown: g gi= xa ii − − Specific force, a : from accelerometer − Kinematic acceleration, x : by differentiating position from tracking system, 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 requires two independent sensor systems − Tracking system − Accelerometer (gravimeter) − Gravimetry accuracy depends equally on the precision of both systems

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.13 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, 1996

MathCad: example_airborne_INS-GPS.xmcd 2⋅ 104

▬ IMU accelerations

▬ GPS accelerations

[mGal] (offset)

[mGal] Subtract and filter

time [s]

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.14 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 are formulated; usually identified with local North-East-Down (NED) directions (n-frame).

– Earth-centered-Earth-fixed frame: frame with origin at Earth’s center of mass and axes defined by conventional pole and Greenwich meridian (Cartesian or geodetic coordinates) (e-frame).

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.15 Earth-Centered-Earth-Fixed Coordinates

T e eee • Cartesian coordinate vector in e-frame: x = ()xxx1 23

• Geodetic coordinates, latitude, longitude, height: φλ,,h

3e − refer to a particular ellipsoid meridian (assume geocentric) with semi-major e axis, a, and first eccentricity, e x e ellipsoid x3 h

e − are orthogonal curvilinear x e 2 e e-frame coordinates x φ 2 1 λ equator 1e

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.16 Transforming Between Cartesian & Geodetic Coordinates  e 2 e −1 x3 eNsinφ x1 =() Nh + cosφλ cos φ = tan 1+ 22e ee x xx+ 3 ()()12 e x2 =() Nh + cosφλ sin −1 ee λ = tan ()xx21

22 xNehe =1 −+2 sinφ ee e 2 3 ( () ) h=+()() x12 xcosφφ +− x 3 sin aN meridian plane – radius of curvature in prime vertical: prime vertical a plane N = 1− e22 sin φ – radius of curvature in meridian: φ ae()1− 2 M = 3/2 ()1− e22 sin φ

• Cartesian and geodetic coordinates may be used interchangeably Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.17 Rotations and Angular Rates Between Frames • Assume common origin for frames s • C t = matrix that rotates coordinates from t-frame to s-frame s st – vector, x: xx= Ct s stt – matrix, A: A= CACts −1T s ts s − C t is orthogonal: CCst≡=()() C t t • ω st = angular rate vector of t-frame relative to s-frame; components in t-frame

ω1 0 −ωω32 t = ω tt×≡ =ωω− • Let ωst 2 then ωst Ωst 310   ω3 −ωω210 − cross-product is same as multiplication by skew-symmetric matrix  s= sω t ×= st • Time-derivative: CCt t st C tΩ st Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.18 Earth-Fixed vs. Inertial Frames 3ie, • Transformation of coordinates between i-frame ω and e-frame is just a rotation about the 3-axis E i ie i ie  ie ie e xx=Ce ⇒= xx Ce +2 CCee xx +  x e T a e ** cent ωie = ()00ωE ωEt

ωE = Earth’s rotation rate λ 1i 0* = neglect rates of polar motion and 1e precession/nutation

e dωie dt = 0  i iee ee iee ie  x=Ce(Ω ie Ω ie ++ Ω ie ) x2 CΩe ie xx + C e • Extract centrifugal acceleration from other kinematic accelerations ei e ee e eeee Ci x=ΩieΩie x +2Ωie xx +=+ ag e e e ee =−acent + q  defines q ⇒=−g qa

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.19 Navigation Frame • Usually, north-east-down (NED)-frame, or n-frame

− moves with the vehicle – not used for coordinates of the vehicle

− used as reference for velocity and orientation of the vehicle; and, gravity • Alternative: vertical along plumb line, n′-frame 1n − conventional reference for terrestrial gravity 2n − no “horizontal” gravity components h 3n

λ φ deflection of the vertical (DOV) n′ 3 3n Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.20 Body Frame • Axes are defined by principal axes of the vehicle: forward (1), to-the-right (2), and through-the-floor (3)

1b G

• Gravimeter (G) measurements are made either:

− in the b-frame – strapdown system; gyro data provide orientation

− in the n-, n′-frames – platform is stabilized using IMUs* * inertial measurement units

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.21 Moving-Base Gravimetry – Strapdown Mechanization in i-frame

Gravitational Vector gn=CC n() xa i − ib in n-frame ib

accelerometers transformation from GNSS inertial frame to n-frame transformation from body gyros frame to inertial frame ab – inertial accelerations measured by accelerometers in body frame xi – kinematic accelerations obtained from GNSS-derived positions, x, in i-frame

−sinφ cos()() λω +−EEttsin φ sin λω + cos φ  n =−+λω λω+ – transformation obtained from Ci sin()()EEttcos 0 GNSS-derived positions, φ, λ −cosφ cos()() λω +−EEttcos φ sin λω +−sin φ − lever-arm effects are assumed to be applied

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.22 Moving-Base Gravimetry – Stabilized Mechanization gyros platform (p-) frame Gravitational Vector gn=CC ni xa − n p in n-frame ip accelerometers: vertical plus transformation from GNSS 2 horizontal inertial frame to n-frame tilt of platform relative to n-frame • Inertial accelerations, a p , from accelerometers in platform frame • Mechanizations − two-axis damped platform – level (n′-frame) alignment using gyro-driven gimballed platform in the short term and mean zero output of horizontal accelerometers in the long term ◦ adequate for benign dynamics − Schuler-tuned inertial stabilized platform – alignment to n-frame based on inertial /GNSS navigation solution and gyro-driven platform stabilization n ◦ ideally, CI p = ; but note, n-frame differs from n′-frame by deflection of the vertical ◦ better for more dynamic environments

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.23 Two-Axis Stabilized Platform • Schematic for one axis

− gryoscope maintains direction in space, and motor commands torque motor to correct deviation of platform orientation due to non-level vehicle processor

− horizontal accelerometer, through processor, ensures that gryoscope reference direction is precessed to account for Earth rotation and curvature

◦ zero acceleration implies level orientation (without horizontal specific forces!)

◦ ad hoc damping of platform by processor

◦ corrects gyro drift, but is subject to accelerometer bias • Schuler-tuned three-axis stabilization: more accurate IMUs and n- frame stabilization (using navigation solution velocity in n-frame) Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.24 Scalar Moving-Base Gravimetry • Determine the magnitude of gravity – the plumb line component − consistent with ground-based measurement (recall gravimeter is leveled) n n i nb n n n − gravitation vector: g=−=CCib x a()−acent +− qa n nnn a − gravity vector: gg= + acent gn= qa nn − GNSS  qn ′ ′′ − this holds in any frame! e.g., gn= qa nn − qn − note: straight and level flight → qann deflection of the vertical n n (DOV) − thus: g = g , but note: gg≠ 3 h ◦ n-frame mechanization does not account g n for the deflection of the vertical along plumb line − since n′-frame is aligned to plumb line, n′ n ellipsoid gg= 3 3 Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.25 Unconstrained Scalar Gravimetry

• One Option: g =gn = qa nn −

◦ unconstrained in the sense that the frame for vectors is arbitrary (n-frame is used for illustration)

◦ also known as strapdown inertial scalar gravimetry (SISG)

n ni n ◦ q=Ci  xa + cent obtained exclusively from GNSS

n nb ◦ aa= Cb requires orientation of b-frame (relative to n-frame)

• Requires comparable accuracy in all accelerometers and precision gyros if platform is arbitrary (e.g., strapdown)

• Calgary group demonstrated good results (e.g., Glennie and Schwarz 1999); see also (Czompo and Ferguson 1995)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.26 Rotation-Invariant Scalar Gravimetry (RISG) • Another option to get g : based on total specific force from gravimeter and orthogonal accelerometers

222 222 aaaa2 =++ppp =++ aaa nnn′′′ ()()()()()()1 23( 1 23)

22 n′2 nn ′′ • Then a3=−− aa()() 12 a

22 2 nn′′ nn ′′ =a −−() qg11 −−() qg 22

22 2 nn′′ nn′′ =−−aq()()12 q since gg12=0 =

22 nn'2 n n n′ n gg≡33 ≈− q a −()() q 1 − q 2 neglecting the DOV, q = q • Platform orientation is not specifically needed for a2 − however, errors in qn, being squared, tend to bias the result (Olesen 2003)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.27 RISG Approach in More Detail (1)

• Define Earth-fixed velocity vector in the n-frame  vN φ () Mh+ −−φ λ φλ φ sin cos sin sin cos n ne   n = − λλ vx=CeE =v =λφ() Nh + cos Ce sin cos 0  −  −cosφ cos λ −− cos φλ sin sin φ vhD  • It can be shown (Appendix A) that

 0 ()λω+−2e sin φ φ n  ndv n nn ΩΩnn+ =−+λω2 sin φ 0 −+λω2 cosφ qv=++()ΩΩie in ie in ()e ()e dt  φλω+ φ ()2e cos 0 • Thus, strictly from GNSS, the third component is v2 v2 qhn =−+ 2ωφ v cos +N + E 3 eE MhNh++

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.28 RISG Approach in More Detail (2)

• Inertial acceleration includes tilt error if platform is not level

22 n′′ np p 2 n′′n a3 =()C p a =−=−− a3 δ atilt  aq()()1 q 2 3  • Kinematic acceleration (from GNSS) includes neglect of DOV

qn′′=C nnq = qq n −δ 3 ()n 3 3 DOV

• Third component of g n ' = qa nn ′′ − , along plumb line,

v2 v2 g=−−+ ahp  2ω v cos φ +N +E +δδa − q 3 EE Mh++() Nh tilt DOV gravimeter

δ gEötvös

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.29 Eötvös Effect

• Exact in n-frame (note: vN,E at altitude!) Loránd Eötvös 2 2 1848 - 1919 vN vE δgvEötvös =2 ωφEE cos ++ heading = 45°, latitude = 45° Mh++() Nh MathCad: EotvosEffect.xmcd

• Approximations

− spherical:

δgv≈+2 ωφ cos Eötvös [mGal] Effect Eötvös EE Rh+

− first-order ellipsoidal (Harlan 1968): total velocity [km/hr] vh2   h δ ≈ +− −22φ − α + ω αφ+ + 2 gEötvös 1f( 1cos() 32sin) 2vE sincos 1 Of() aa  a

a = ellipsoid semi-major axis; α = azimuth; v = ground speed!

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.30 DOV Error in Kinematic Acceleration • DOV components define the small angles between the n- and n′-frames geodetic astronomic 10−ξ meridian meridian n′ =−=−ηξ η CRn 12()() R 01  north pole ξη 1 geodetic DOV − ignore rotation about 3-axis zenith east neg. plumb ξ line η • DOV error δq=− qnC nn′q =−−ξη qq n n horizon DOV 3 ()n 3 1 2 4 • Assume rms(DOV) = 10 arcsec, q1,2 = 10 mGal

rms()δ qDOV = 0.7 mGal • This error is correctable, e.g., using EGM2008 deflection model

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.31 Tilt Error (1) • One way to compute tilt error (p. 3.27)

22 p2 nn δ aatilt=−− 3 aq()()1 − q2 (neglecting DOV)

nn − random errors in qq 12 , are squared and can cause bias (rectification error) • Better model for the tilt error (Olesen 2003) 2 p χ − define orientation angles, ν, χ, α ν p − assume ν, χ are small; α is arbitrary 1 cosα−+ sin α χ cos αν sin α n  ≈−α α χ αν α α C p sin cos sin cos  −χν 1 3p

− thus, approximate platform stabilization is required!

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.32 Tilt Error (2)

n n np • From gq= − C p a,

n np p p g qa11−+−cosα a 2 sin α( χ cos αν + sin α)a3 1 =  n np−−α p α −− χ αν α p g2 qa21sin a 2 cos( sin cos )a3

nn • If gg12=0 = (neglecting DOV is second-order effect on tilt error)

1 n np χ=p (q1cos αα +− qa 21 sin ) a3 Tilt angles are computed from accelerometers, GNSS, 1 nn pand azimuth ν=p (qq12sin αα −+ cos a 2) a3

p np • Third component of δ aatilt = − C p a

pp δaatilt≈− 1 χν a 2 (first-order approximation)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.33 Tilt Error (3) • Tilt error can be written as

wp wp wn n qa−− qa qq11=cosαα + q 2 sin δ a≈+11 aapp 22 tilt pp1 2 wn n aa33 qq21=−+sinαα q 2 cos

ww − where qq 12 , are kinematic accelerations in the “wander-azimuth” frame • Taking differentials of the specific force components, wp−− p wp −− p wppwpp − +− ()qa11 a 1pp() qa 22 a 2()() qaaqaa 111 222p δδ()atilt =+−ppδ aa1 δ2 2 δa3 p aa33 ()a3

wp p 46p p 4 • Assume ()q1,2−=⋅ a 1,2 aO 3 ()νχ, 10 mGal,a3 ≈ 10 mGal,a1,2 = O() 10 mGal − error in vertical accelerometer (gravimeter) is second-order for tilt correction − error in horizontal accels. can be 100 worse than tilt correction accuracy • In practice, tilt correction is subjected to appropriate filters; see (Olesen 2003)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.34 Lever-Arm Effect

• Apply to kinematic acceleration derived from GNSS tracking

− assume gravimeter is at center of mass of vehicle

ii i • In the inertial frame: xxantenna= gravimeter + b

i ib b − where bb= Cb , b = fixed antenna offset relative to gravimeter

i Cb− obtained from gyro data

2  id ii • Numerical differentiation: xgravimeter =x()antenna − b dt 2

− extract relevant component in particular frame

ni  ◦ n-frame: vertical component of C i x gravimeter  h

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.35 Scalar Gravimetry Equation

• Final equation for the gravity anomaly at altitude point, Pa′  ∆g′′= fh −+ δ g + δδ a − q − γ −() f − g PQaaEötvös tilt DOV 0 0

p − where fa = − 3 is the gravity meter reading

− where γ is normal gravity at the normal height of P ′ above the ellipsoid Qa′ a

− where f0 – g0 is the initial offset of the gravimeter reading from true gravity

− gravimeter measurement, f , includes various inherent instrument corrections

− tilt error depends on accuracy of platform accelerometers

− exact and sufficiently approximate formulas exist for normal gravity at Qa′

− accuracy in h must be commensurate with gravimeter accuracy

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Airborne Gravimetry, C. Jekeli, OSU 3.36 National Geodetic Survey

IV. Overview of Airborne Gravimetry Systems

• LaCoste/Romberg sea-air gravimeters

• Other Airborne gravimeters

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.1 Airborne Gravimeters

• Instrumentation overview of scalar airborne gravimetry

− LaCoste-Romberg instruments dominate the field

− many other instrument types in operation or being tested

• All gravimeters are single-axis accelerometers

− mechanical spring accelerometers (vertical spring, horizontal beam)

◦ manual or automatic (force-rebalance) nulling

− torsion wire (horizontal beam, no nulling)

− electromagnetic spring (force-rebalance)

− vibrating string accelerometers

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.2 Lucien J.B. LaCoste (1908-1995) “The gravity meters Lucien B. LaCoste invented revolutionized geodesy and gave scientists the ability to precisely measure variations in Earth's gravity from land, water, and space” J.C. Harrison (1996)*

• Inventor • Scientist • Teacher • Entrepreneur

At University of Texas, Austin

*https://web.archive.org/web/20080527061634/http://www.agu.org/sci_soc/lacoste.html Earth in Space Vol. 8, No. 9, May 1996, pp. 12-13. © 1996 American Geophysical Union; see also (Harrison 1995). Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.3 LaCoste-Romberg Air-Sea Model S Gravimeter • Beam is in equilibrium if torque(spring) = torque(mg)

k()−=0 bsinβα mga sin exactly vertical − law of sines: sinβα= d sin A k − − zero-length spring:  0 = 0 ()0 ⇒=kbd mga d  − independent of α → equilibrium at any beam position for a given g β α b − independent of  → no change in spring O a length could accommodate a change in g damper mg • From (Valliant 1992): finite sensitivity infinite sensitivity g with damper gg+ ∆  measure beam velocity! Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.4 (Inherent) Cross-Coupling Effect • Horizontal accelerations couple into the vertical movement of horizontal beam gravimeters that are not nulled − total torque on beam due to external z accelerations: x T= ma( xsinθθ ++() g z cos ) − it can be shown (LaCoste and Harrison 1961) that the cross-coupling error is θ 1 ε= x θψcos a mx 2 11

where x 11 , θ are amplitudes of components of x and θ , respectively, that have the same period and phase difference, ψ mg()+  z 2 − e.g., θε11=°=1 ,x 0.1 m/s ⇒= 90 mGal • There is no cross-coupling effect for − force-rebalance gravimeters − vertical-spring gravimeters

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.5 LaCoste-Romberg Model S Sensor and Platform Interior Side View

Stabilized Platform

Dampers

Outer Frame

Accelerometers

Gyroscopes

view of top lid From: Instruction Manual, LaCoste and Romberg Model “S” Air-Sea Dynamic Gravimeter, 1998; with permission Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.6 LaCoste/Romberg TAGS-6 (Turn-key Airborne Gravity System)

http://www.microglacoste.com/tags-6.php

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.7 BGM-3 Gravimeter induction • Bell Aerospace (now Lockheed Martin) − Model XI pendulous force-rebalance accelerometer − current needed to keep test mass in null position is proportional to acceleration

(Seiff and Knight 1992); see also (Bell and Watts 1986)

Fugro  WHOI Two BGM-3 gravimeters installed https://www.unols.org/sites/default/files/Gravimeter_Kinsey.pdf on the USCG ship Healy

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.8 Sea Gravimeter KSS31, Bodenseewerk Geosystem GmbH • Gravity sensor based on Askania vertical-spring gravimeter − Federal Institute for Geosciences and Natural Resources (BGR) − force rebalance feedback system capacitive − highly damped output, ~ 3 minute average transducer − sensor on a gyro-stabilized platform spring

− also used for fixed wing and helicopter gravimetry tube

Heyde, J. (2010) thread

damping unit permanent magnet

•http://www.bgr.bund.de/DE/Themen/MarineRohstoffforschung/Meeresforschung/Geraete/Gravimeter/gravimeter_inhalt.html •http://www.bgr.bund.de/EN/Themen/GG_Geophysik/Aerogeophysik/Aerogravimetrie/aerogravimetrie_node_en.html Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.9 Chekan-A Gravimeter • Air-Sea gravimeter; CSRI* Elektropribor, St. Petersburg, Russia − gravity sensed by deflection of pendulum hinged on quartz torsion wire in viscous fluid − pendulum deflection: 0.3–1.5 ″/mGal; e.g., ±1° → ±10 Gal total range (0.36 ″/mGal) − cross-coupling effect minimized by double-beam reverted pendulums

− evolutionary modifications: Chekan AM, “Shelf” (Krasnov et al. 2014)

*Central Scientific & Research Institute

Installation in test aircraft http://www.gravionic.com/gravimetry.html

(Stelkens-Kobsch 2005) (Krasnov et al. 2008) (Hinze et al. 2013)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.10 Airborne Gravimetry on Airship Platform

Airship AU-30

Relative gravimeter Chekan in a cabin of AU

(http://rosaerosystems.com/airships/obj17)

− test flight January 2014 − reported in IAG Commission 2 Travaux 2015

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.11 Airborne Gravimeter GT-2A • Gravimeter system designed by Gravimetric Technologies (Russia) − vertical accelerometer of axial design with a test mass on spring suspension − photoelectric position pickup − moving-coil force feedback transducer − three-axis gyro-stabilized platform − large dynamic range

√

(Gabell et al. 2004)

http://eongeosciences.com/wp-content/uploads/2015/01/GT_2A.pdf (Canadian Micro Gravity)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.12 Sander Geophysics Ltd. AirGrav System • “Purpose-built” airborne gravimeter designed for airborne environment, not modified sea gravimeter − Honeywell inertial navigation grade accelerometers (Annecchione et al. 2006, Sinkiewicz et al. 1997) − Schuler-tuned (three axis) inertially stabilized platform − three-accelerometer system; vertical accelerometer used as gravimeter − advertize gravimetry on topography-draped profiles − demonstrated success in vector gravimetry − comparison tests over Canadian Rockies between AirGrav and GT-1A (Studinger et al. 2008) http://www.sgl.com/news/Sander%20Geophysics%20-%20Antarctica.pdf Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Overview of Airborne Gravimetry Systems, C. Jekeli, OSU 4.13