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V. Theoretical Fundamentals of Inertial Gravimetry

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V. Theoretical Fundamentals of Inertial Gravimetry 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, k2 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 nn2 δg=−20000 β δg−+ βδgw dt 2 Edt Egδ ββ2 00 DD00 ◦ 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 nn2 δg=−20000 β δg −+ βδgw dt 2 Edt Egδ ββ2 00 DD00 ◦ 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 ) [mgal f ( 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.
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