Die Satellitenmission GOCE der ESA Eine Herausforderung an Mathematik, Numerik und Informatik H. Sünkel Institut für Geodäsie Technische Universität Graz und Institut für Weltraumforschung Österreichische Akademie der Wissenschaften

Galilei - Newton - Einstein Gravitation Space-time „Mass tells space-time how to curve and space-time tells mass how to move“. Mass Gravity = Gravitation + Rotation

Gravity in control of our daily life Shape of the Earth‘s surface Distribution of land and water on Earth Speed of processes inside, on and outside the Earth Density and constitution of the Earth‘s atmosphere Biological processes Growth of plants Anatomy and physiology of men and animals Motion of living organism Architecture of buildings Mechanical design and structure of machines and vehicles Gravity scales life in space and time

A primitive Earth model Crust Radial-symmetric, not rotating, static mass distribution Upper mantle Lower mantle Core Radial-symmetric, static gravity field

Gravity anomalies due to mass anomalies Tectonic processes (Lithosphere) Coupling Mantle - Core Rheology Mantle convection

The dual role of the gravity field 108 - 106 yr Time Scales 102 - 100 yr Volcanic activities Ice Sheet Melting Ocean circulation & heat transport Post-glacial rebound Sea level change

Gravitational potential Turning inside out mass Gravitational potential shape ? Geoid

The gravitational potential Properties of the gravitational potential: 1. is harmonic outside the Earth: 2. decreases to zero towards infinity 3. belongs to an infinite dimensional space Consequence: is represented by a linear combination of harmonic functions (= solutions of )

The gravity potential Gravitational potential ( ) Rotational potential ( ) Gravity potential ( ) Unique „global horizontal“ surface of constant gravity potential ( ) at „mean“ sea level: geoid Global reference surface for „orthometric height“ Unique „local vertical“ Reference direction for „local-horizontal reference system“

Surface gravity: incomplete data coverage gravdensBGI.gif Surface gravity: incomplete data coverage

POD for gravity field recovery Satellite orbit The idea: ? Mass distribution Gravitational field

POD for gravity field recovery Equation of motion, defined in a space-fixed geocentric reference system free fall (around the Earth) = Satellite motion due to surface forces Satellite orbit as a function of gravitational field parameters Reference gravitational field controlled by parameters Reference satellite orbit as a function of gravitational field parameters

POD for gravity field recovery Principle: Real orbit from satellite tracking Reference orbit based on a priori gravitational field Residual harmonic coefficients unknowns Functional relation

POD for gravity field recovery Pseudo-observations Design matrix from partials LSA Observation residuals Harmonic coefficient (parameter) residuals

The Earth’s gravitational field: models lmax JGM3 lmax = 70 OSU91a lmax = 360 EGM96 lmax = 360 l m 0 0 0. 0. 1 0 0. 0. 1 1 0. 0. 2 0 -0.48417e-03 0. 2 1 0.85718e-12 0.28961e-11 2 2 0.24382e-05 -0.13999e-05 3 0 0.95714e-06 0. 3 1 0.20297e-05 0.24943e-06 3 2 0.90465e-06 -0.62044e-06 3 3 0.72030e-06 0.14147e-05 ... ... ... ...

Open problems Solid Earth Physics anomalous density structure of lithosphere and upper mantle Oceanography quasi-stationary dynamic ocean topography Sea Level Change Glaciology ice sheet balance Geodesy unification of height systems, levelling by GPS, inertial navigation, orbit prediction

Accuracy Now: Required: Gravity field: current knowledge The geoid: a surface of constant gravity potential at zero level Offset from reference ellipsoid: Accuracy Now: + 100 m Required: 0 m - 100 m

Gravity field exploration from space 3 Mission scenarios: 1. Satellite-to-Satellite Tracking in high-low mode SST - hl 2. Satellite-to-Satellite Tracking in low-low mode SST - ll 3. Satellite Gravity Gradiometry SGG

Gravity Field Satellite Missions

Love affairs with body Earth (... “move your body close to mine”) CHAMP (2000) GRACE (2002) GOCE (2006)

CHAllenging Minisatellite Payload The CHAMP mission: spacecraft CHAllenging Minisatellite Payload

The CHAMP Mission: SST - hl sst_hl.eps The CHAMP Mission: SST - hl GPS - satellites SST - hl CHAMP 2000 3-D accelerometer Earth mass anomaly

The CHAMP mission: objectives Earth‘s gravity field and its temporal variations Gravity Field Model „Eigen 1S“ Geostrophic currents

The CHAMP mission: objectives Earth‘s magnetic field and its temporal variations

The CHAMP mission: objectives Earth‘s atmosphere and ionosphere and temporal variations

The CHAMP mission: payload STAR accelerometer Laser retro-reflector Electrostatic STAR Accelerometer GPS Receiver TRSR-2 Laser Retro Reflector Fluxgate Magnetometer Overhauser Magnetometer Advanced Stellar Compass Digital Ion Drift Meter Fluxgate Magnetometer Overhauser Magnetometer

The CHAMP mission:spacecraft front side view rear side view

The CHAMP mission: launch & orbit Launch: July 15, 2000, Cosmodrome Plesetsk almost circular orbit: e = 0.004 near polar orbit: i = 87° initial altitude: 454 km satellite lifetime: 5 years COSMOS launch vehicle

The CHAMP mission: altitude

The GRACE Mission

The GRACE Mission: SST-hl and SST-ll combined sst_ll.eps The GRACE Mission: SST-hl and SST-ll combined GPS - satellites SST - ll SST - hl GRACE 2002 Earth mass anomaly

The GRACE Mission: constellation Two satellites following each other on the same orbital track Position and velocity of the satellites are measured using onboard GPS antennae Interconnected by a K-band microwave link S-band radio frequencies used for communication with ground stations

The GRACE Mission: constellation

The GRACE Mission: payload K-Band Ranging System (KBR) Accelerometer (ACC) GPS Space Receiver (GPS) Laser Retro-Reflector (LRR) Star Camera Assembly (SCA) Coarse Earth and Sun Sensor (CES) Ultra Stable Oscillator (USO) Center of Mass Trim Assembly (CMT)

The GRACE Mission: spacecraft structure

The GRACE Mission: launch & orbit Launch: March 17, 2002, Cosmodrome Plesetsk almost circular orbit: e < 0.005 near polar orbit: i = 89° initial altitude: 485 - 500 km satellite system lifetime: 5 years

The CHAMP mission: altitude

Gravity Field and Steady-state The GOCE Mission Gravity Field and Steady-state Ocean Circulation Explorer

Project scientist (ESA) and The GOCE mission: team structure GOCE Project Team GOCE Industry Team GOCE - MAG Project scientist (ESA) and members European GOCE Gravity Consortium (EGG-C) Team leader and Task leaders Science Data Use: Solid Earth Science Data Use: Oceanography Science Data Use: Ice Science Data Use: Geodesy Science Data Use: Sea Level

The GOCE Mission: SST-hl and SGG combined gradiometry.eps The GOCE Mission: SST-hl and SGG combined GPS - satellites SST - hl GOCE 2006 SGG Earth mass anomaly

The GOCE mission: launch & orbit Launch: Feb. 2006, Cosmodrome Plesetsk (62.7° N, 40.3° E) Total satellite mass: 800 kg Orbit inclination: i = 97° Injection altitude: 270 km Passive descent from 270 km to 250 km Altitude controlled by ion thrusters Satellite lifetime: 2 years

GOCE: mission parameters Mission duration: 20 months Commission phase: 3 months Phase 1: 6 months Hibernation phase: 5 months Phase 2: 6 months Design orbit altitude: Phase 1: 250 km Phase 2: 240 km Orbit characteristics: Dawn-dusk sun-synchronous, Inclination: 96.5° Injection eccentricity: 0.000 phase 1: two months repeat, phase 2: eventually drifting ground track Maximum air drag during gradiometer operation: 0.3 mGal / in mbw

The GOCE mission: timeline

The GOCE mission: spacecraft design Scientific payload: 3-axis gravity gradiometer GPS receiver SLR retroreflector Star tracker center of mass Auxiliary equipment: Ion thrusters FEEPS Solar panels (8 m²) On board computer Telemetry S 44

The GOCE mission: gravity gradiometer Z Gravity gradiometer: 3-axis mbw: 5 - 100 mHz Precision: < 1 mE Y X 1 mE: curvature radius of equipotential surface of 10 Mill. km !

The GOCE mission: gradiometer noise 5 mHz mbw 100 mHz Specified noise psd 3 mE / Hz specified (Predicted per- formance curve derived from a combination of analysis and test) Predicted noise psd

The GOCE mission: gravity gradiometry Accelerometer equations: Common mode: Differential mode: V ... gravitational tensor

The GOCE mission: gravity gradiometry Measurement tensor: Attitude: Gravitational tensor:

The GOCE mission: attitude & drag control Ion thruster (for drag compensation) Micro thruster (for attitude control) Field emission electric propulsion system (FEEPS) Gas: Indium (Austrian system) Caesium (Italian system) Gas: Xenon thrust level: 1 - 20 mN thrust level: 0.0001 - 0.1 mN

The GOCE mission: observation sensitivities SST (hi-lo): SGG: Orbit perturbations Gradiometer data Sat. alt. smoother Sat. alt. smoother SST amplifier SGG amplifier

The GOCE mission: data processing Gravitational potential as a function of position P: 100 000 parameters Gravity gradiometer observations: 100 000 000 observations LSA

The GOCE mission: gravity field recovery vik.bw.eps The GOCE mission: gravity field recovery 100 000 000 observations 100 000 parameters

The GOCE mission: processing methods Spacewise Timewise Otherwise

The GOCE mission: processing methods The Runner: Semi-Analytic approach (SA) Approximative, iterative algorithm: works partially in the frequency domain (FFT) and uses the dominant block-diagonal structure of the normal equations The Progressive Worker: Parallel pcgma package Approximative algorithm, using a block-diagonal preconditioner as a first guess (initialization step) and applying iteratively a conjugate gradient method The Real Life: Distributed Non-approximative Adjustment (DNA) Strict solution of the normal equation system Flexible, but enormous computation requirements

The GOCE mission: spacewise processing Potential regarded as a function of space position P SGG observations regarded as a function of space position P

The GOCE mission: spacewise processing LSA SGG observ. Harmonic coeff. Iterative Direct serial parallel serial parallel

The GOCE mission: spacewise processing Preconditioned Conjugated Gradient Method (PCGM) Iterative solution Preconditioning by appropriate representative matrix Inclusion of prior information possible Extended by frequency selective filters to account for coloured noise SGG data Serial and parallel versions operational Successfully tested up to degree 300 on different platforms (DEC-Alpha, SGI Origin, CRAY T3E, Graz Beowulf Cluster)

The GOCE mission: spacewise processing Distributed Non-approximative Adjustment (DNA) Direct solution Inclusion of prior information possible Inclusion of non-gravitational parameters (non-conservative force model, calibration parameters, ...) Extended by frequency selective filters to account for coloured noise SGG data Parallel versions successfully tested up to degree 180 on the Graz Beowulf Cluster

The GOCE mission: timewise processing Potential regarded as a function of time t SGG observations regarded as a function of time

The GOCE mission: timewise processing Sensitivity coeff. Harmonic coeff. Lumped coeff. In case of a periodic SGG time series: Lumped coefficients = Fourier coefficients of SGG time series Lumped coefficients considered as pseudo observations in LSA for the estimation of the harmonic coefficients Ideally the normal matrix has block diagonal structure In reality it is (strongly) block diagonally dominant Iteration required due to sun-synchronous orbit and varying orbit parameters

The gravitational potential and derived quantities 100 km resolution requires L = 20000 km / 100 km = 200 Geoid: Resolution: 100 km Accuracy: 1 cm Gravity anomaly: Resolution: 100 km Accuracy: 1 mGal

The GOCE mission: performance ( cumulative error ) Goal: < 10 mm < 1 mGal

The GOCE mission: error spectrum Slm  order  Clm EGM spherical harmonic error spectrum

Closed loop simulation Gravity field Harmonic coeff. Gradiometer signal Harmonic analysis Coloured noise Observations Filtering

The GOCE mission: closed loop simulation “Delft data-set”: 29 days repeat orbit ~ 1.5 million observations (SGG diagonal components) lmax = 180 Size of full normal equation matrix 4.1 GByte Very different time behaviour of the three methods

The GOCE mission: solution time Near linear speed-up with additional processing elements (PE)

The GOCE mission: test performance Solution time:

The TU Graz cluster Components and performance March 2001: TU Graz started to build up a Beowulf cluster 24 Dual-Pentium (866 MHz) PC’s (computing nodes) 1GB RAM / 18 GB local HD 1 Master server PC 1.5 GB RAM 5*73 GB RAID 5 Linux (Redhat, Kernel 2.4.19) , MPICH, ... TU Graz cluster outperforms Columbus / Ohio cluster by a factor of 2 Same performance as the CRAY T3E in Columbus / Ohio

The TU Graz cluster: future plans (1/2) HP SC-45 cluster 10 ES45 Alphaserver 4 CPUs each / 1.25 GHz 2 ES45 with 32 GB RAM 8 ES45 with 16 GB RAM 120 GB local hard disks connected via 16 ports Quadrics switch Realization: October 2002

The TU Graz cluster: future plans (2/2) System based on Linux connection between the nodes via Myrinet (Cluster 1) connection between the nodes via GigaBit Ethernet (Cluster 2) 24 nodes (Cluster 1) 36 nodes (Cluster 2) Intel / 2.6 GHz CPUs 0.5 GB RAM per CPU 5*73 GB SCSI RAID5 Realization: January 2003

Gravity satellite missions: comparison 2 years 5 years duration 250 km ~450 km 454 km altitude 96.5° 89.0° 87.3° inclination SST-hl/SGG SST-hl / ll SST-hl mode 2006 2002 2000 start GOCE GRACE CHAMP

Gravity satellite missions: comparison Accuracy and resolution 70 km 250 km 650 km resolution (half wavelength) < 1 cm ~ 1 cm accuracy GOCE GRACE CHAMP Harmonic degree 30 80 300

Gravity satellite missions Accuracy and resolution Kaula EGM 96 SST - hl SGG SST - ll

The GOCE mission Ocean topography

The GOCE mission Dynamic ocean topography Navier-Stokes equation Steady - state flow: H ... Sea surface height (SSH) relative to geoid x, y ... Local cartesian coordinates (W-E, S-N) u, v ... Surface velocity (W-E, S-N) g ... Gravity f ... Coriolis term

Gravity satellite missions Ocean topography signal Range: -75 to +75 cm Signal l = 20 l = 200 l = 80

The GOCE mission Mean dynamic ocean topography

The GOCE mission Geostrophic current

The GOCE mission: geotomography Propagation of seismic waves Problem: conversion velocity  density

The GOCE mission: solid Earth Volcanoes, Hot Spots Spreading Rifting Subduction Orogeny

The GOCE mission: surveying

Gravity satellite missions: summary CHAMP magnetic field determination temporal variations of the gravity field GRACE improved knowledge of the geoid estimates of time variable components GOCE highly precise static geoid determination

Gravity satellite missions: benefits Oceanography: Absolute ocean circulation Sea level changes Ice mass balance Solid Earth Physics: Geotomography Processes in the deep Earth‘s interior Earthquake prediction Geodesy: Unified height datum GPS levelling Orbit prediction Inertial navigation

Gravity satellite missions Benefits for geosciences Ice Sheet Mass Balance Geodesy Absolute Ocean Circulation Sea Level Change Studies Solid Earth Physics GOCE will cover Theme 1 of ESA‘s Living Planet Programme (with the exception of the magnetic field part)

The End