In-Flight Calibration Objectives In synthesis, the objective of the in-flight calibration is to turn the real Gradiometer into an ideal Gradiometer (in the data post processing). In an ideal Gradiometer all accelerometers are identical (with unitary scale factors), linear and perfectly aligned to the Gradiometer reference frame. In-line Ultra-Sensitive (US) axis Transversal Ultra-Sensitive axis Less-Sensitive (LS) axis The GGT diagonal components are obtained from the differential accelerations: GGT = Gravity Gradient Tensor and are affected only by the accelerometer measurement noise.

In-Flight Calibration Objectives In a real Gradiometer the accelerometers have different scale factors, non-linear response, their sensitive axes are non-orthogonal and misaligned. These “defects” couple with the residual linear and angular accelerations of the satellite and give rise to spurious differential accelerations  errors on the GGT. Differential scale factor coupling with in-line linear acceleration Misalignment coupling with transversal linear acceleration Acceleration measured by the accelerometerAiof a real Gradiometer: [K]i = scale factor matrix [dR]i = rotation matrix from accelerometer frame to Gradiometer frame [dS]i = accelerometer inter-axis coupling matrix ; [K2] = quadratic factor matrix

In-Flight Calibration Objectives Scale factors, sensitive axis misalignments and non-orthogonality are collected in three 6x6 Calibration Matrices Mij(one for each Gradiometer arm).To recover the accelerations experienced by the proof masses along the Gradiometer axes from the measured ones, the Inverse Calibration MatricesMIij must be known. (ij = 14, 25, 36) ai= acceleration measured by an ideal accelerometer with unitary scale factor and perfectly aligned to the Gradiometer frame ai= acceleration measured by the real accelerometer Ai on its reference frame • Gradiometer in-flight calibration objectives: • Measure and correct the accelerometer quadratic factors (non linearity). • Measure the elements of the Inverse Calibration Matrices.

Gradiometer In-flight Calibration Method • Satellite elements involved in the Gradiometer in-flight calibration Cold-gas thrusters Cold-gas thrusters Star Tracker Ion Thruster Gradiometer Accelerometer Cold-gas thrusters

Quadratic Factor Calibration Method • Quadratic factor measurement principle • A train of high-frequency (100 Hz) sinusoidal accelerations (Ae = 1e-5 m/s2) is periodically applied (20 s) to the proof mass by the accelerometer controller (no satellite shaking). • In presence of a quadratic non-linearity, a square wave is produced in the accelerometer output, with an amplitude proportional to the quadratic factor. • Overall duration of the proof mass shakings required to measure non-linearity along the two ultra-sensitive axes of the 6 accelerometers: 5 hours. • Quadratic factor correction • The quadratic factor is reduced by displacing the proof mass inside the cage by an amount proportional to the measured K2:

Gradiometer In-flight Calibration Method • Inverse calibration matrices (ICM) determination principle • For one day the satellite is subjected to random, uncorrelated shaking about all the axes, by means of impulsive cold-gas thrusters and of the ion propulsion, operated to get linear and angular accelerations with power spectrum as large as possible between 50100 mHz, and exciting also the angularaccelerationsaround1 mHz • The Gradiometer and star sensor measurements collected during this period are post-processed on the ground by means of an iterative procedure in three macro-steps that provides the inverse calibration matrices.

Gradiometer In-flight Calibration Method Step 3 Step 3 Step 3 Step 3 Update the Update the Step 2 Step 2 Step 1 Step 1 Update the Update the Step 2 Step 2 Step 1 Step 1 determination of the determination of the determination of the determination of the Determination of the Determination of the Determination of the Determination of the Determination of the Determination of the Determination of the Determination of the Inverse Calibration Inverse Calibration Inverse Calibration Inverse Calibration transverse absolute transversal absolute Inverse Calibration Inverse Calibration transverse absolute transverse absolute Inverse Calibration Inverse Calibration Matrices, starting Matrices, starting Matrices, starting Matrices, starting scale factors, using scale factors, using Matrices, assuming Matrices, assuming scale factors, using scale factors, using Matrices, assuming Matrices, assuming from the from the ICMs ICMs from the from the ICMs ICMs the the ICMs ICMs determined determined unitary absolute scale unitary absolute scale the the ICMs ICMs determined determined unitary absolute scale unitary absolute scale determined at Step 1 determined at Step 1 determined at Step 1 determined at Step 1 at Step 1 at Step 1 factors factors at Step 1 at Step 1 factors factors and the transversal and the transversal and the transversal and the transversal absolute scale factors absolute scale factors absolute scale factors absolute scale factors determined at Step 2 determined at Step 2 determined at Step 2 determined at Step 2 Step 1 Step 1 and Step 3 are in and Step 3 are in Repetition of Repetition of turn iterative procedures turn iterative procedures Repetition of Repetition of NO NO Step 2 with Step 2 with which normally converge which normally converge Step 2 with Step 2 with the the ICMs ICMs Convergence? Convergence? after after ~10 iterations. The ~10 iterations. The the the ICMs ICMs determined determined overall process converges overall process converges determined determined at Step 3 at Step 3 normally after 2 normally after 2 - - 3 iterations 3 iterations YES YES at Step 3 at Step 3 « « of the Step 2 of the Step 2 Step 3 loop Step 3 loop. STOP STOP • Scheme of the iterative process for the determination of the ICM Collection of gradiometer and star tracker measurements during a 1-day period in which the satellite is subjected to random uncorrelated shaking about all the axes, by means of impulsive cold-gas thrusters and of the ion thruster.

Gradiometer In-flight Calibration Method • Basic equation set for ICM determination in calibration Step 1 • Assuming that between 50 and 100 mHz (upper part of the MBW) the GG signal is weak, the (unknown) GGT components can be removed from the above equations:

Gradiometer In-flight Calibration Method • Basic equation set for ICM determination in calibration Step 1 • The differential accelerations at the accelerometer locations can be expressed as function of the measured common and differential accelerations via the ICM elements: ……………………. • By measuring for N times the common and differential accelerations and the angular rates, we can build 9 sets of linear equations where the unknowns are the elements of the last three rows of MI14, MI25, MI36. The equations can be solved with the least-squares method. • These 9 sets of linear equations are solved iteratively, starting from assuming unitary ICM (1 for scale factors, 0 for misalignments) in order to compute the right-hand sides.

Gradiometer In-flight Calibration Method Calibration Step 1: scheme of the iterative process for the determination of the ICM in the first part of the calibration.

Gradiometer In-flight Calibration Method • Basic equation set for the ICM determination in the calibration Step 2 • Angular accelerations obtained from the Gradiometer (by linear accelerations difference) • Angular accelerations obtained from the star trackers (by double derivative of the attitude angles) • Common scale factors (SF) of the accelerometer pairs along their transversal axes. The in-line common SF are then obtained from transversal SF through numerical relationships.

Gradiometer In-flight Calibration Results • Gradiometer in-flight calibrations since the beginning of the mission: • First calibration (June 18th - 19th 2009) Accelerometer proof mass rotation around the LS axis () controlled by 4 electrodes. Star tracker in the attitude control loop: STR1. Force bias applied. • Second calibration (Sept 28th- 29th 2009) Proof mass  rotation controlled by 4 electrodes. Star tracker in the loop: STR1. • Third calibration (Oct 8th - 9th 2009) Proof mass  rotation controlled by 2 electrodes. Star tracker in the loop: STR1. • Fourth calibration (January 11th- 12th 2010) Proof mass  rotation controlled by 2 electrodes. Star tracker in the loop: STR1. • Fifth calibration (March 4th - 5th 2010) Proof mass  rotation controlled by 2 electrodes. Star tracker in the loop: STR1. • Sixth calibration (May 6th - 7th 2010) Proof mass  rotation controlled by 2 electrodes. Star tracker in the loop: STR1.

Gradiometer In-flight Calibration Results • Values of the ICM elements measured in the six calibrations In-line, common SF Transversal, common SF (LS) Transversal, common SF (US) Transversal, diff. SF (LS) Transversal, diff. SF (US) In-line, differential SF

Gradiometer In-flight Calibration Results • Values of the ICM elements measured in the six calibrations Common misalignment (, Y) Common misalignment (, X) Common misalignment (, Z) Diff. misalignment (, X) Diff. misalignment (, Y) Diff. misalignment (, Z)

Gradiometer In-flight Calibration Results • Variations of the ICM elements through the six calibrations • No clear, large linear trends have been identified in the variation of the ICM elements throughout the six calibrations, apart for the in-line common SF of the accelerometer pair 25 (~1200 ppm/month) and the in-line differential SFs of the accelerometer pairs 14 (~110 ppm/month) and 25 (~35 ppm/month). • Largest variations between successive calibrations: • Jumps in the common rotations of accelerometer pairs 14, 36 around Y and of accelerometer pair 25 around Z between September 2009 and October 2009, associated to modification of the rotation control of the accelerometer proof mass around the LS axis (). • Discontinuities of the common scale factors along the in-line and transversal US axes of accelerometer pairs 14, 36, part of which can be attributed to the measurement error of the star trackers (utilized to determine these parameters). The correlation in the variation of these common SFs is due to the relationships utilised for their determination in the TAS-I method. • Most stable parameters: alignments of the accelerometer axes to the Gradiometer reference frame.

Calibration Impact on Gradiometric Performance GGT traces computed with non calibrated measurements • Spectral density of the GGT trace computed on the measurements of June 2009, September 2009, October 2009, January 2010, March 2010, May 2010, calibrated with the ICMs of the six calibrations performed so far (each data set processed with the closest ICMs). Calibration impact on performance GGT traces computed with calibrated measurements

Calibration Impact on Gradiometric Performance • Spectral density of the GGT trace computed on the measurements of March 24th-31st 2010, calibrated with the ICMs of the six calibrations performed so far. • Periodic calibrations (every ~2 months) are needed to correct the temporal variation of the Gradiometer parameters (scale factors, misalignments). Effect on performance of the temporal variation of the Gradiometer parameters Calibration impact on performance

Calibration Impact on Gradiometric Performance In-line differential scale factors of all pairs calibrated • The largest impact on the performance is produced by the coupling of the drag acceleration along the Gradiometer axes with the difference of the scale factors of the accelerometer axes aligned to Gradio axes (in-line differential scale factors). Spectral density of the acceleration along Y, Z (not subject to drag-free control) In-line differential scale factor of pair 14 calibrated No calibration Spectral density of the residual acceleration along X (flight direction) under the drag-free control action (10x below the requirement) Full calibration Drag accelerations measured by the Gradiometer (the drag-free control acts along X only). Impact on GGT trace of the differential scale factors coupling with in-line linear acceleration

Calibration Impact on Gradiometric Performance In-line differential scale factors of all pairs calibrated • At low frequencies (< 5 mHz), the coupling of the purely differential accelerations (gravity gradients, centrifugal accelerations) with the absolute scale factors of the accelerometer pairs along all axes (common scale factors) becomes significant. In-line differential scale factor of pair 14 calibrated No calibration Differential scale factors and differential misalignments of all pairs calibrated Differential scale factors, differential misalignments and common scale factors of all pairs calibrated Full calibration

Calibration Impact on Gradiometric Performance • Mean value of the GGT trace between 10100 mHz obtained by keeping fixed (as provided by the calibration) all ICM elements but the in-line differential scale factors (varied in a wide range of values). • The minimum value of the trace is obtained using the values of the differential scale factors provided by the calibration method. The calibration works! Calibration value Calibration value Calibration value GGT trace versus in-line differential scale factor of the acceler. pair 36 GGT trace versus in-line differential scale factor of the acceler. pair 14 GGT trace versus in-line differential scale factor of the acceler. pair 25

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