National Geodetic Survey

V. Theoretical Fundamentals of Inertial Gravimetry

• Basic gravimetry equation

• Essential IMU data processing for gravimetry

• Kalman filter approaches

• Rudimentary error analysis – spectral window

• Instrumentation

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.1 Inertial Gravimetry • Vector gravimetry, instead of scalar gravimetry

− determine three components of gravity in the n-frame

• Use precision accelerometer triad, instead of gravimeter

− OTF (off-the-shelf) units designed for inertial navigation rather than gravimetry

• Usually consider strapdown mechanization, instead of stabilized platform • Need precision gyroscopes to minimize effect of orientation error on horizontal components • Two documented approaches of data processing

− either integrate accelerometer data or differentiate GPS data

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.2 Direct Method • Recall strapdown mechanization Gravitational Vector n= n i − ib in n-frame gCCib() xa

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

tk

b b δ vk − accelerometer data typically are δ va= ()t dt ⇒≈ak (e.g.) k ∫ δt tk−1 better approximations can be formulated ◦ where δ tt = kk − t − 1 is the data interval, e.g., δt = 1/50 s

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.3 i Determination of C b (1) • Let e be the unit vector about which a rotation ζ ζ by the angle, ζ, rotates the b-frame to the i-frame 3b

cos()ζ 2 q1 b  eζ  bsin()ζ 2 q2 e = c → quaternion vector: q = =  ζ  ζ   csin() 2 q3 d  d sin()ζ 2 q4 2b qqqq2222+−−22()() qqqq + qqqq− 1 2 3 4 23 14 24 13 i = −2222 −+− + 1b Cb 22()qqqq23 14 qqqq 1 2 3 4() qqqq 34 12 2222 22()()qqqq24+ 13 qqqq34−12 qqqq 1−−+ 2 3 4 d 1 • Quaternions satisfy the differential equation: qq= A dt 2

0 ωωω123 ω1 −−ω10 ωω 32 b  − where A = and ω =ω = gyro data ib 2 −−ωω230 ω 1  ω3 −−ωω32 ω 10 − this is a linear D.E. with no singularities

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.4 i Determination of C b (2) • Solution to D.E., if A is assumed constant,

1123 1 qqˆˆ=+++IkΘΘ Θ +− , = 1, 2, k2 kk 8 48 k k1

tk δ = b − using gyro data, typically given as θωk∫ ibdt tk −1

tk − where = ΘAk ∫ dt tk −1 − note: A is assumed constant only in the solution to the D.E., not in using the gyro data (model error, not data error in Θ)

− q ˆ 0 is given by an initialization procedure − solution is a second-order algorithm, neglecting terms of order δt3 − higher-order algorithms are easily developed (Jekeli 2000)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.5 Determination of δgn (1) • Kalman filter approach to minimizing estimation errors

− estimate IMU systematic errors and gravity disturbance vector

n − formulate in i-frame and assume negligible error in Ci − system state updates (observations) are differences in accelerations yag=+−()ii  x i iiiiii  i i i ii =+()aaδδδ +−()() gg −+ xx gx =γ + ΩΩie ie =iδ b −×ψ i ib −−δδ i i CCbba a xg δ ai − δai includes accelerometer errors and orientation errors

− over-script, ~, denotes indicated (measured) quantity

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.6 Determination of δgn (2)

• System state vector, εD d − orientation errors, ψ i; ψωi= −C ibδ dt b ib − IMU biases, scale-factor errors, assumed as random constants − gravity disturbance components, modeled, e.g., as second-order i in Gauss-Markov processes in n-frame, with δδgg= Cn ββ00 2 00 2 NN ddn nn2 δg=−20000 β δg−+ βδgw dt 2 Edt Egδ ββ2 00 DD00

◦ wδg is a white noise vector process with appropriately selected variances

◦ βN , βE , βD are parameters appropriately selected to model the correlation

time of the processes = 2.146 βNED,, • System dynamics equation d εε=FG + w where wD is a vector of white noise processes dt D DD D D

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.7 Determination of δgn (3) • Alternatively, omit gravity disturbance model

− observations assume normal gravitation is correct yag=+−()ii x i i iii  i i i i ii =()()aagx +δδ +− + x gx=γ + ΩΩie ie =iδδ b −×ψ i ib − i CCbba ax δ ai − optimal estimates of IMU systematic errors by Kalman filter yield yˆ

− gravity disturbance estimates: δ gi ≈− yyˆ

◦ assumes residual IMU systematic errors are small and white noise can be filtered

− successfully applied technique (Kwon and Jekeli, 2001)

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.8 Indirect Method • Recall inertial navigation equations in n-frame n dv nb n n n n =Cba −+()ΩΩie in vg + dt dxe = Cenv dt n − integrate (i.e., get IMU navigation solution) and solve for g n using a model and GNSS tracking data − analogous to traditional satellite tracking methods to determine global gravitational field, except gravity model is linear stochastic process, not spherical harmonic model − navigation solution from OTF INS should not be integrated with GNSS! ◦ IMU and GNSS must be treated as separate sensors, just like in scalar gravimetry ◦ use raw accelerometer and gyro data to obtain free-inertial navigation solution - one could pre-process IMU/GNSS data to solve for IMU systematic errors, neglecting gravity

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.9 Determination of δgn (1) • Solve for gravity disturbance treated as an error state (among many others) in the linear perturbation of navigation equations

− typical error states collected in state vector, εΙ , include: ◦ position errors, velocity errors, orientation errors ◦ IMU systematic errors (biases, etc.) ◦ gravity disturbance components d εε=FG + w where wI is a vector of white noise processes dt I II I I − integration is done numerically (e.g., using linear finite differences)

εεIk()()()()t= ΦG Ikk tt, −−11 Ik t+ Ik tw Ik() t where Φ = state transition matrix

− observations are differences, IMU-indicated minus GNSS positions, treated as updates to the corresponding system states

y()()()()tk= H tt kεv Ik+ t kwhere v is a vector of discrete white noise processes

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.10 Determination of δgn (2) • Gravity disturbance model − stochastic process; e.g., second-order Gauss-Markov process, ββ00 2 00 2 NN ddn nn2 δg=−20000 β δg −+ βδgw dt 2 Edt Egδ ββ2 00 DD00

◦ wδg is a white noise vector process with appropriately selected variances

◦ βN , βE , βD are parameters appropriately selected to model the correlation

time of the processes = 2.146 βNED,, • Kalman filter/smoother estimate, is optimal in the sense of minimum mean square error − theoretically the gravity model is an approximation since the gravity field is not a linear, finite-dimensional, set of independent along-track signals as required/modeled in the system state formalism − successful estimation depends on stochastic separability of gravity disturbance from accelerometer errors and coupled gyro errors

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.11 Rudimentary Error analysis • Assume that n- and i-frames coincide (approx. valid for < 1 hour) negligible n nn n n n nb n b nn g≈−⇒ xa δδgx=+−− ΨCbb a Cδ aΓ δ p

gravitation errors in errors in errors in gravitational position errors kinematic sensor inertial gradients errors acceleration orientation acceleration

• δg PSD is obtained from models of IMU and position error PSDs and PSD of vehicle acceleration

ΦΦΦΦΦ=+++ δg1 δψ x1 32 a ψ 23 aa δ1

ΦΦΦΦΦ=+++ δg3 δψ x3 21 a ψ 12 aa δ3

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.12 PSD Models

Gravitational Field Accelerations* MathCad: accpsdfit.mcd MathCad: degvarfit4.mcd EGM96 dashed-line: psd model Φ a

] 2 2 1'x1' ∆g /(cy/m) /Hz] 2 2 ) 2 model Φ a3 [(m/s f ) [mgal ( g δ Φ Φ a1

frequency [cy/m] frequency [Hz] * data from Twin Otter aircraft – Accelerometers: white noise (±25 mGal) – Gyros: rate bias (± 0.003°/hr) plus white noise (±0.06 °/hr/√Hz) approximated by simple PSD ° – Orientation: initial bias (0.005 ) models − Position: white noise (±0.1 m) − Aircraft speed = 250 km/hr; altitude = 1000 m

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.13 PSDs of Gravity Errors, δg1, δg3, Versus Along-Track PSDs of Signals, g1, g3

0.01

3 1 10 Spectral Window: 4 S 1 10 g3 Φ δ x1,3

5 S 1 10 g1 signal-to-noise ratio > 1 ]/Hz ]/Hz 4

/s 6 2 1 10 g : [m 1 7 1 10 Φ δg1 8 1 10 g3 : + Φδg 9 3 1 10 4 3 1 10 1 10 0.01 0.1 Frequency [Hz]

• Signal-to-noise ratio > 1 over larger bandwidth for g3

• Significant error source is orientation bias, especially for g1, g2

• Signal PSD’s move to right and down with increased velocity

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.14 Pendulous, Force-Rebalance Accelerometer (e.g.) • Schematic: Pulse Rebalance pulse train Amplifier Electronics input axis plan view Forcer Proof 1c Direction Mass Shadow Arm 3c

Case Hinge Permanent Photo Detector Magnet Forcer Coil

side view 2c 3c Light Emitting Diode

• Torque needed to keep proof mass in equilibrium is a measure of acceleration

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.15 Pendulous, Force-Rebalance Accelerometer (e.g.) • Honeywell (Allied Signal, Sundstrandt) QA3000

• Litton (Northrop Grumman) A-4 Miniature Accelerometer Triad

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.16 INS Used for Airborne Gravimetry

Honeywell Laseref III Honeywell H-770 Litton LN100

University of Calgary Intermap Technologies Ohio State University

ITC-2 Inertial Survey System Strapdown Platforms

Stabilized Platform

Bauman Moscow State Technical University iMAR RQH 1003

Airborne Gravity for Geodesy Summer School , 23-27 May 2016 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU 5.17