GPS-based precise orbit determination and accelerometry for low flying satellites

GPS-based precise orbit determination and

accelerometry for low flying satelllites

GPS-based precise orbit determination and

accelerometry for low flying satellites

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M Luyben,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op woensdag 9 april 2014 om 12:30 uur

door

Jozefina Adriana Aleida VAN DEN IJSSEL

ingenieur luchtvaart- en ruimtevaarttechniek

geboren te Oudewater

Copromotor:

Dr. ir. P.N.A.M. Visser

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. ir. B.A.C. Ambrosius Technische Universiteit Delft, promotor Dr. ir. P.N.A.M. Visser Technische Universiteit Delft, copromotor Prof. dr. E.K.A. Gill Technische Universiteit Delft

Prof. dr. C. Stolle Helmholz Centre Potsdam, GFZ, Duitsland

Prof. dr. A. J¨aggi Astronomisches Institut Universit¨at Bern, Zwitserland Dr. O. Montenbruck Deutsches Zentrum f ¨ur Luft- und Raumfahrt, Duitsland

Ir. R.H.N. Haagmans European Space Agency

Prof. dr. ir. L.L.M. Veldhuis Technische Unversiteit Delft, reservelid

Publicatie van dit proefschrift is mede mogelijk gemaakt door een financi¨ele bij-drage van de vakgroep Astrodynamics and Space Missions, faculteit Luchtvaart-en Ruimtevaarttechniek, Technische Universiteit Delft.

Cover image: The GOCE satellite in orbit around the Earth, courtesy of ESA/AOES Medialab

Printed by Ipskamp Drukkers ISBN 9789462591363

Copyright c 2014 Jose van den IJssel

All rights reserved. No part of the material protected by this copyright may be reproduced, or utilised in any other form of by any means, electronic or mechanical, including photocopying, recording or by any other information storage and retrieval system, without the prior permission of the author.

Contents

1.2 Research objective and motivation . . . 3

1.3 The CHAMP, GRACE and GOCE satellite missions . . . 5

1.4 Outline . . . 8

2 GPS-based precise orbit determination 11 2.1 Introduction . . . 11

2.2 GPS data quality . . . 12

2.3 Data preprocessing strategy . . . 16

2.3.1 GPS data preprocessing . . . 16

2.3.2 Star tracker data preprocessing . . . 21

2.4 Orbit determination scheme . . . 26

2.5 Orbit quality assessment . . . 29

2.5.1 Data residual analysis . . . 30

2.5.2 Orbit overlap analysis . . . 33

2.5.3 Formal orbit errors . . . 35

2.5.4 SLR data analysis . . . 36

2.5.5 K-band ranging analysis . . . 40

2.5.6 External orbit comparisons . . . 42

2.6 Conclusions . . . 47

3 GPS-based accelerometry - simulation experiment 49 3.1 Introduction . . . 49

3.1.1 The principle of GPS-based accelerometry . . . 50

3.1.3 Atmospheric density and wind modeling . . . 51

3.2 Simulation scenario . . . 52

3.3 Model error assessment . . . 55

3.4 Impact of error sources . . . 61

3.5 Impact of accounting for observation correlations . . . 64

3.6 Impact of a priori non-gravitational force models . . . 69

3.7 Impact of data limitations . . . 73

3.8 Optimization strategies . . . 78

3.8.1 Constraining radial accelerations . . . 80

3.8.2 Minimum recovery error . . . 82

3.8.3 Minimum overlap error . . . 89

3.9 Accelerometer calibration assessment . . . 93

3.10 Conclusions . . . 95

4 GPS-based accelerometry - real data experiment 97 4.1 Introduction . . . 97

4.2 Data processing strategy . . . 98

4.3 Results for CHAMP . . . 101

4.4 Results for GRACE . . . 112

4.5 Results for GOCE . . . 122

4.6 Conclusions . . . 135

5 Conclusions and outlook 139 5.1 Conclusions . . . 139

5.2 Outlook . . . 141

Bibliography 145

Acknowledgements

This dissertations reflects the outcome of many years of research and I would like to thank many people for their support during this period. First of all, I would like to thank my daily supervisor and copromotor Pieter Visser. His advice and critical feedback have significantly improved the quality of the research presented here. I am also very grateful to my promotor Boudewijn Ambrosius, for making it possible for me to work in his group and encouraging me to write this dissertation. All my colleagues of the Astrodynamics and Space Missions group deserve to be mentioned, as they contributed to a very pleasant atmosphere to work in. I would like to name particularly, Ron, Relly, Bert, Ejo, Marc, Daphne, Taco, Hermes and Joao. I also want to thank Sander Goossens, who has left our group already for quite some time, but remains a very helpful colleague and a dear friend. Special thanks go to my roommates throughout the years: Hugo Schotman, for being a wonderful roommate and insisting on being my paranymph; Eelco Doornbos, for many interesting discussions on atmospheric density, the careful reading of this dissertation and for supporting me as paranymph; and Wouter van de Wal, for also reading parts of this dissertation and for still being my roommate.

Parts of the research presented in this dissertation were done in the framework of the GOCE High-level Processing Facility (HPF) contract and the Swarm End-to-End Mission Performance Simulator study. I am very grateful to the European Space Agency for the funding of these contracts. I want to thank Roger Haagmans for his guidance of the Swarm study and the interesting discussions. I would also like to thank all my colleagues of the GOCE HPF consortium for the nice coopera-tion during the years. I really enjoyed all the interesting HPF meetings under the guidance of Rune Floberghagen and Reiner Rummel. Special thanks go to Heike Bock and Adrian J¨aggi, for their support and collaboration on POD related topics. I hope that the cooperation with the colleagues of the Swarm Satellite Constellation Application and Research Facility consortium will be just as nice.

Finally, I want to thank my family and friends for their support. Special thanks go to Gerard, Vera and Tom. Your love and support are very much appreciated. Jose van den IJssel

Summary

GPS-based precise orbit determination and accelerometry for low

flying satellites

Atmospheric density models are currently the limiting factor in the accuracy of the dynamic orbit determination and prediction of satellites in a low Earth or-bit. Any improvement in these models would greatly aid in applications such as re-entry prediction, ground-track maintenance of Earth observation satellites and forecast of possible collisions with space debris objects. Because of their use in sci-entific studies, improving these models will also benefit our understanding of the physical processes that occur in the Earth’s upper atmosphere.

State-of-the-art accelerometers onboard of low Earth orbiting (LEO) satel-lites are near perfect instruments for studying density and winds in the upper part of the Earth’s atmosphere. They provide accurate observations of the non-gravitational accelerations acting on a satellite and for low flying satellites atmo-spheric drag is the dominant non-gravitational force. Unfortunately, the number of satellites equipped with such an accelerometer is limited. Therefore, this study investigates the possibility to derive the non-gravitational accelerations acting on LEO satellites from precise GPS tracking observations. The estimation of non-gravitational accelerations using a precise GPS-based orbit determination scheme is referred to as GPS-based accelerometry. With the growing number of satellites equipped with a high-quality GPS receiver, GPS-based accelerometry could be ap-plied to a large range of satellites, orbiting at different altitudes, with different sam-pling of the local solar time and during different solar activity conditions. Such a data set of recovered non-gravitational accelerations offers great potential for the improvement of atmospheric density models.

The objective of this research is to develop, implement and validate a strat-egy to optimally derive non-gravitational accelerations from precise GPS satellite tracking observations of low flying satellites. The focus is on estimating non-gravitational accelerations with high temporal resolution, making optimal use of the dense GPS tracking information.

High-quality satellite orbits are essential for a good GPS-based accelerometry performance. Therefore a data processing strategy is developed to compute pre-cise orbits for LEO satellites from GPS observations. This strategy is based on a

reduced-dynamic orbit determination and uses a Bayesian weighted least-squares estimator and ionosphere-free triple differenced carrier-phase observations. With this strategy, orbits have been computed for the CHAMP, GRACE and GOCE satel-lites with a state-of-the-art accuracy of 2 to 3 cm (1-dimensional).

Using an extensive simulation study, the developed precise orbit determina-tion (POD) strategy is optimized for GPS-based accelerometry. This study is based on the GRACE mission and to cover the range of possible atmospheric conditions, simulations have been performed for three days with different levels of solar and geomagnetic activity. The resulting optimal processing strategy consists of short 6-hour arcs and uses piecewise linear empirical accelerations to estimate the non-gravitational signal. Using proper constraints for the empirical accelerations sig-nificantly improves the GPS-based accelerometry performance. With overlap anal-ysis and a robust random restart simple hill climbing optimization strategy, near-optimal constraints per orbital direction can be determined. When empirical eration constraints are applied, an additional set of unconstrained constant accel-erations is estimated per arc, to reduce the effect of biased estimation. Relatively short 6-hour orbit arcs are used in order to prevent instabilities in the batch least-squares estimation process and to prevent the build-up of model errors. Especially in along-track direction, the GPS-based accelerometry performance benefits from using such short arcs. In cross-track direction, however, better performance is ob-tained when longer 24-hour arcs are used.

Besides highly accurate satellite orbits, precise gravitational force models are essential for a good GPS-based accelerometry performance. The recent improve-ments in the modeling of the Earth’s gravity field, enabled by data from the CHAMP, GRACE and GOCE missions, have significantly enhanced the GPS-based accelerometry performance. The performance is therefore no longer limited by gravity field model errors and GPS receiver noise is currently the dominant error source. This means that dual-frequency geodetic grade GPS receivers are required for a good GPS-based accelerometry performance.

The optimized GPS-based accelerometry processing strategy is applied to real GPS data from the CHAMP, GRACE and GOCE satellites. These satellites carry high-quality GPS receivers and due to their low altitude, ranging from around 250 to 450 km, they experience relatively large non-gravitational accelerations, which makes them very interesting for atmospheric density and wind modeling. Fur-thermore, these satellites carry electrostatic accelerometer instruments, making it possible to validate the performance of the GPS-based accelerometry experiments. This validation shows that the GPS-based accelerometry performance depends on the signal strength of the estimated non-gravitational signal. Due to the increas-ing effect of drag at lower altitudes, lower flyincreas-ing satellites generally experience stronger non-gravitational accelerations and show better GPS-based accelerometry performance. In general, best GPS-based accelerometry performance is obtained in along-track direction, due to the relatively strong sensitivity to orbit perturbations in this direction. For the low flying GOCE satellite with its drag-free control (DFC) switched off, 10-minute non-gravitational accelerations are obtained with a

rela-Summary xi

tive error of 2.7% compared to the accelerometer observations. For the CHAMP and GRACE satellites the relative errors are, respectively, 8.8% and around 16%. The performance in cross-track direction is noticably worse, with relative errors ranging from 42.7% for GOCE without DFC to 87.9% for GRACE-B. In radial di-rection the GPS-based accelerometry performance is very poor.

A comparison with state-of-the-art non-gravitational models shows that for all selected satellites, the non-gravitational accelerations determined with the GPS-based accelerometry approach have a better agreement with the accelerometer ob-servations than the modeled accelerations in along-track direction. This indicates that the recovered accelerations can be used to improve these models. In cross-track direction, the non-gravitational models perform better, except for bias and scaling errors. Low frequency density variations due to e.g. diurnal temperature variations or fluctuations in solar or geomagnetic activity are well recovered by the GPS-based accelerometry approach. The recovery of high-frequency varia-tions, however, is limited by the achievable temporal resolution of the empirical accelerations. These results show that for satellites equipped with accurate GPS tracking instrumentation, GPS-based accelerometry can contribute to improved atmospheric density modeling. For these satellites, the GPS-based accelerometry technique can be seen as an absolute accelerometer with a reduced temporal reso-lution.

The developed stategy to derive non-gravitational accelerations from GPS ob-servations can be readily applied to GPS data of other LEO satellites that do not carry an accelerometer. Satellites that are suited for GPS-based accelerometry are e.g. the German TerraSAR-X and TanDEM-X satellites, which carry accurate GPS receivers and fly at a relatively low altitude of 514 km. For the recently launced Swarm mission, the developed GPS-based accelerometry strategy will be included in the operational processing of the Swarm data by the Swarm Satellite Constel-lation Application and Research Facility (SCARF). The resulting non-gravitational accelerations will be used for validation of the onboard accelerometer tions. In case of accelerometer anomalies, they will be used as alternative observa-tions for thermospheric density retrievals.

Samenvatting

Precieze GPS baanbepaling en accelerometrie voor laagvliegende

satellieten

Op dit moment zijn modellen van de atmosferische luchtdichtheid de beper-kende factor in de nauwkeurigheid van de dynamische baanbepaling en voor-spelling van satellieten in een lage baan om de aarde. Een verbetering van deze modellen is van grote waarde voor diverse toepassingen, zoals het voorspellen van de terugkeer van ruimteobjecten naar de aarde, het bijsturen van de baan van een satelliet om het gewenste bedekkingspatroon van de aarde te krijgen, en het voorspellen van mogelijke botsingen met stukken ruimtepuin. Vanwege hun ge-bruik in wetenschappelijke studies, bevordert een verbetering van deze modellen ook ons begrip van de fysische processen die in de atmosfeer plaats vinden.

Geavanceerde versnellingsmeters aan boord van satellieten in een lage baan om de aarde zijn ideale instrumenten voor het bestuderen van de luchtdichtheid en wind in de hoogste laag van de atmosfeer. Ze leveren nauwkeurige metin-gen van de niet-gravitationele versnellinmetin-gen die een satelliet ondervindt en voor laagvliegende satellieten is atmosferische luchtweerstand de dominante niet-gravitationele kracht. Helaas beschikken maar weinig satellieten over zo’n ver-snellingsmeter. Daarom wordt in deze studie onderzocht of het mogelijk is de niet-gravitationele versnellingen van deze satellieten te bepalen met behulp van GPS metingen. Het bepalen van niet-gravitationele versnellingen door middel van GPS baanbepaling wordt ook wel GPS accelerometrie genoemd. Doordat steeds meer satellieten beschikken over een geavanceerde GPS ontvanger, kan GPS ac-celerometrie worden toegepast op een groeiende hoeveelheid satellieten. Deze satellieten bevinden zich op verschillende hoogtes, hebben een verschillend be-dekkingspatroon van de aarde en vliegen gedurende verschillende fases van de 11-jaarlijkse zonnecyclus. Zo’n gevarieerde set van niet-gravitationele versnellin-gen is van grote waarde voor het verbeteren van luchtdichtheidsmodellen.

Het doel van dit proefschrift is het ontwikkelen, implementeren en valideren van een optimale strategie om uit de precieze GPS metingen van laagvliegende satellieten de niet-gravitationele versnellingen van deze satellieten te bepalen. De nadruk ligt hierbij op het bepalen van versnellingen met een zo hoog mogelijke

tijdsresolutie, om optimaal te profiteren van de hoge informatiedichtheid van de GPS metingen.

Om met GPS accelerometrie goede resultaten te kunnen behalen is het noodza-kelijk dat de banen voor de betreffende satellieten nauwkeurig bekend zijn. Daarom is een strategie ontwikkeld om gebaseerd op GPS metingen precieze ba-nen te bepalen voor satellieten in een lage baan om de aarde. De ontwikkelde gere-duceerd dynamische baanbepalingsstrategie is gebaseerd op een gewogen Bayesi-aanse kleinste kwadraten schatting en gebruikt ionosfeer-vrije driedubbelverschil GPS fase metingen. Met deze strategie zijn banen berekend voor de CHAMP, GRACE en GOCE satelliet, die een state-of-the-art nauwkeurigheid hebben van 2 tot 3 cm (1-dimensionaal).

Met behulp van een uitgebreide simulatie studie is de ontwikkelde baanbepa-lingsstrategie vervolgens geoptimaliseerd voor GPS accelerometrie. Deze studie is gebaseerd op de GRACE satelliet. Om de invloed van mogelijke atmosfe-rische omstandigheden te onderzoeken zijn simulaties gedaan voor drie dagen met grote verschillen in geomagnetische en zonneactiviteit. De resulterende op-timale strategie bestaat uit het bepalen van korte 6-uurlijkse baanoplossingen, waarbij stapsgewijs lineare empirische versnellingen worden gebruikt om het niet-gravitationele signaal te schatten. De behaalde resultaten verbeteren sterk wanneer de a priori variantie van de empirische versnellingen wordt beperkt. Door het analyseren van de geschatte versnellingen van twee elkaar overlap-pende baanoplossingen, kunnen in iedere baanrichting zo goed als optimale waar-den voor de a priori variantie van deze versnellingen worwaar-den bepaald. Wan-neer de a priori variantie van de empirische versnellingen wordt beperkt, dient per baanoplossing een extra set met onbeperkte constante versnellingen te wor-den geschat, om een systematische fout vanwege de mogelijk verkeerde a pri-ori verwachtingswaarde van de versnellingen te reduceren. Het gebruik van relatief korte 6-uurlijkse baanoplossingen voorkomt dat het kleinste kwadraten schattingsproces instabiel wordt en beperkt de modelfout, die toeneemt bij lan-gere baanoplossingen. Met name in de vliegrichting leidt het gebruik van zulke korte baanoplossingen tot betere resultaten. In de richting loodrecht op het baanvlak worden echter betere resultaten behaald wanneer langere 24-uurlijkse baanoplossingen worden gebruikt.

Naast precieze satellietbanen, zijn ook nauwkeurige gravitationele modellen essentieel voor het behalen van goede GPS accelerometrie resultaten. De re-cente verbeteringen van modellen van het gravitatieveld van de aarde, dankzij meetgegevens van de CHAMP, GRACE en GOCE satellieten, hebben de GPS ac-celerometrie resultaten sterk verbeterd. Gravitationele modelfouten zijn hierdoor niet langer de beperkende factor en GPS ontvanger meetruis is tegenwoordig de dominante foutenbron voor GPS accelerometrie. Voor een goed resultaat is het dus noodzakelijk om metingen te gebruiken van nauwkeurige geodetische GPS ontvangers, die het GPS signaal op twee frequenties ontvangen.

De geoptimaliseerde GPS accelerometrie strategie is vervolgens toegepast op echte GPS metingen van de CHAMP, GRACE en GOCE satellieten. Deze

sa-Samenvatting xv

tellieten beschikken over geavanceerde GPS ontvangers. Vanwege hun lage hoogtes, vari¨erend van ongeveer 250 tot 450 km, ondervinden ze relatief grote niet-gravitationele versnellingen, waardoor ze zeer interessant zijn voor het mo-delleren van de atmosferische luchtdichtheid en wind. Deze satellieten zijn ook uitgerust met elektrostatische versnellingsmeters, waardoor het mogelijk is om de behaalde GPS accelerometrie resultaten te valideren. Deze validatie laat zien dat de kwaliteit van het behaalde GPS accelerometrie resultaat afhangt van de sterkte van het te schatten niet-gravitationele signaal. Door de toenemende atmosferi-sche luchtweerstand bij lagere hoogte, ondervinden lager vliegende satellieten in het algemeen grotere niet-gravitationele versnellingen. Hierdoor laten deze sa-tellieten ook betere GPS accelerometrie resultaten zien. In het algemeen worden de beste GPS accelerometrie resultaten behaald in de vliegrichting, vanwege de relatief grote gevoeligheid van de baan voor verstoringen in deze richting. Voor de zeer laagvliegende GOCE satelliet is het mogelijk om, wanneer de atmosferi-sche luchtweerstand van de satelliet niet wordt gecompenseerd door het ’drag-free’ systeem, iedere 10 minuten niet-gravitationele versnellingen te schatten met een relatieve fout van 2.7% ten opzichte van de gemeten versnellingen. Voor de CHAMP en GRACE satellieten zijn de relatieve fouten, respectievelijk 8.8% en ongeveer 16%. In de richting loodrecht op het baanvlak zijn de GPS accelerome-trie resultaten merkbaar slechter, met relatieve fouten vari¨erend van 42.7% voor GOCE, wanneer het drag-free systeem uit staat, tot 87.9% voor GRACE-B. In radi-ale richting levert GPS accelerometrie weinig betekenisvolle resultaten.

De niet-gravitationele versnellingen die een satelliet ondervindt, kunnen ook bepaald worden met behulp van state-of-the-art niet-gravitationele modellen. Een vergelijking met deze gemodelleerde versnellingen laat zien dat voor alle geselecteerde satellieten de geschatte versnellingen in de vliegrichting beter overeenkomen met de gemeten versnellingen dan de gemodelleerde versnellin-gen. Hieruit blijkt dat de geschatte versnellingen gebruikt kunnen worden om deze modellen te verbeteren. In de richting loodrecht op het baanvlak komen de gemodelleerde en gemeten versnellingen beter overeen, op schaalfouten en con-stante fouten na. Laagfrequente variaties in de atmosferische luchtdichtheid van-wege bijv. dagelijkse temperatuur variaties, of schommelingen in geomagnetische en zonneactiviteit, zijn goed terug te vinden in de met behulp van GPS accelerome-trie bepaalde versnellingen. Het schatten van hoogfrequente variaties wordt echter beperkt door de maximaal haalbare temporele resolutie van de empirische versnellingen. Deze resultaten laten zien dat voor satellieten die beschikken over hoogwaardige GPS ontvangers, GPS accelerometrie kan bijdragen aan het ver-beteren van atmosferische luchtdichtheidsmodellen. Voor deze satellieten kan de GPS accelerometrie techniek gezien worden als een absolute versnellingsme-ter met een gereduceerde temporele resolutie.

De ontwikkelde strategie om niet-gravitationele versnellingen te bepalen uit GPS metingen kan zonder problemen worden toegepast op GPS data van andere laagvliegende satellieten die niet beschikken over een versnellingsmeter. Satellie-ten die geschikt zijn voor GPS accelerometrie zijn bijv. de Duitse TerraSAR-X en

TanDEM-X satellieten, die beschikken over hoogwaardige GPS ontvangers en op een relatief lage hoogte van 514 km rondvliegen. Voor de recentelijk gelanceerde Swarm missie zal de ontwikkelde GPS accelerometrie strategie onderdeel zijn van de operationele verwerking van de Swarm data door de Swarm Satellite Constella-tion ApplicaConstella-tion and Research Facility (SCARF). De geschatte versnellingen zullen gebruikt worden om de versnellingsmeters van de Swarm satellieten te valideren. Wanneer deze versnellingsmeters niet goed functioneren, zullen de geschatte ver-snellingen gebruikt worden voor het bepalen van de atmosferische luchtdichtheid.

Chapter

1

Introduction

Satellites in a low Earth orbit (LEO) deliver excellent observations of our planet, which provide valuable information for our understanding of the Earth. To ful-fill their scientific mission requirements, these satellites generally require a precise knowledge of the orbit. To meet these stringent orbit requirements, the satellites are equipped with high-precision tracking systems. Nowadays, Global Position-ing System (GPS) receivers are considered as the primary trackPosition-ing system for pre-cise orbit determination in many satellite missions. These receivers provide near-continuous observations with excellent geometric information, which allow orbit accuracies at the centimeter level for the current satellites missions.

This research focuses on a new space application of GPS, namely its use for deriving precise non-gravitational accelerations of satellites in a low Earth orbit. These non-gravitational accelerations can be measured directly using an onboard accelerometer instrument as well. The estimation of non-gravitational accelera-tions acting on a satellite using a precise GPS-based orbit determination scheme is therefore referred to as GPS-based accelerometry. In the next section, an overview is presented of satellite accelerometers and their applications. This is followed by the objective and motivation of this research. Data from the CHAMP, GRACE and GOCE satellite missions are used extensively in this research, therefore a brief introduction of these missions is given thereafter. Finally, this chapter concludes with a detailed outline of the rest of this dissertation.

1.1

Spaceborne accelerometry

The principle of an electrostatic accelerometer is based on electrostatic levitation to suspend a proof mass in a cage. When the center of mass (COM) of the proof mass is located at the COM of the satellite, the proof mass and the satellite are subject to the same gravitational forces. The non-gravitational forces however, act only on the satellite and not on the proof mass. When the cage of the proof mass is rigidly

attached to the satellite, this induces a movement of the proof mass with respect to the cage. The electrostatic forces that are required to keep the proof mass in its cage are a measure of the observed non-gravitational accelerations. By measuring the total effect of all non-gravitational accelerations acting on the satellite with an accelerometer, a separation of the gravitational and non-gravitational forces is pos-sible. This information can be used to improve the gravitational force modeling of the Earth, as well as the precise orbit determination of the satellite. Because atmo-spheric drag is the dominant non-gravitational force for satellites in a low Earth orbit, the accelerometer is also a near perfect instrument for studying atmospheric density and winds.

From around 1960 to 1980, already several accelerometer instruments were flown on dedicated thermosphere missions. The MESA (Miniature Electrostatic Single-axis Accelerometer) instrument was flown on several Atmospheric Explorer satellites. A follow-on to MESA, the SETA (Satellite Electrostatic Triaxial Ac-celerometer) instrument flew on several US missions. The Drag Balance Instru-ment was flown on the Italian San Marco satellites and the French CASTOR satel-lite carried the CACTUS (Capteur Acc´el´erom`etrique Capacitif Triaxial Ultra Sen-sible) instrument. These accelerometers were generally operated on satellites in elliptical orbits and, owing to sensitivity limitations, their measurements were mainly confined to relatively short time spans at very low altitudes [Doornbos, 2011]. More recently, the ASTRE (Acc´el´erom`etre Spatial Triaxial Electrostatique) instrument was developed to monitor the residual microgravity disturbance level of manned space laboratories. This accelerometer flew on the Columbia shuttle in 1996 [Touboul et al., 1999].

Since 2000, several geopotential satellites have been equipped with ultra sen-sitive onboard accelerometer instruments. The first mission to carry such a high class accelerometer is the CHAMP satellite, launched in July 2000. Its STAR (Space Triaxial Accelerometer for Research) accelerometer was specifically designed for the CHAMP mission in order to separate the non-gravitational forces from the gravity signal. Information about the STAR accelerometer and its performance is given by Grunwaldt and Meehan [2003] and Perosanz et al. [2003]. The twin GRACE satellites, launched in March 2002, are equipped with superSTAR accelerometers, which are an updated version of the STAR accelerometer. These accelerometers are also used to separate the non-gravitional force contributions from the gravita-tional ones. Specifications of the superSTAR performance can be found in Tapley et al. [2007]. Although not included in the main scientific goals of the CHAMP and GRACE missions, the nearly continuous observations of these highly sensitive ac-celerometers provide the opportunity to derive data on atmospheric density and winds with high temporal resolution. Several authors have computed atmospheric densities from CHAMP accelerometer data [Bruinsma et al., 2004; Liu et al., 2005] or using GRACE accelerometer data [Tapley et al., 2007]. A combined estimation of at-mospheric densities and wind from the CHAMP accelerometer data is performed by Liu et al. [2006], Sutton et al. [2007] and Doornbos et al. [2010].

1.2 Research objective and motivation 3

a gravitational gradiometer [Rummel et al., 2011]. This instrument consists of three orthogonally mounted pairs of three-axis accelerometers. The differential gradiometer observations are used to measure the gravitational gradients and ro-tational accelerations and the common-mode gradiometer observations measure the gravitational accelerations. Using closed-loop controls, the observed non-gravitational accelerations are partially compensated by an ion thruster, resulting in the drag-free flight of the satellite.

The Swarm mission, recently launched on 22 November 2013, is also equipped with accelerometers [Friis-Christensen et al., 2008]. This constellation of three satel-lites is designed for studies of the Earth’s magnetic field. Their onboard MAC (Micro Accelerometer) instruments [Fedosov and Peˇrest´y, 2011] will be used for at-mospheric density and wind retrieval.

1.2

Research objective and motivation

This research was initiated by an European Space Agency (ESA) study in prepa-ration for the Swarm mission [Van den IJssel and Visser, 2004]. In this study the possibility to use GPS-based accelerometry as an alternative for an onboard ac-celerometer instrument was investigated, taking into account the observational requirements of the Swarm mission. The work presented here is a continuation of this study and assesses the feasibility of the GPS-based accelerometry concept for all LEO satellites equipped with a GPS receiver. With this concept, the data from these satellites could be used for atmospheric density modeling purposes in a similar way as satellites equipped with an accelerometer.

Atmospheric density models are currently the limiting factor in the accuracy of orbit determination and prediction of satellites in a low Earth orbit. According to Marcos [1990] errors of 20% or more are common for these models. Any improve-ment in these models would greatly aid in applications such as re-entry prediction, ground-track maintenance of Earth observation satellites and forecast of possible collisions with space debris objects. Because of their use in atmospheric studies, improving these models will also benefit our understanding of the thermosphere. As mentioned in the previous section, the current spaceborne accelerometers pro-vide very accurate non-gravitational observations with high temporal resolutions of 0.1 to 1 Hz, which makes them very suitable for atmospheric density and wind data retrieval. However, because only a limited number of satellites are equipped with an accelerometer, the spatial resolution of these data are limited. The growing amount of satellites that carry precise GPS receivers, which are orbiting at differ-ent altitudes, local solar time and during differdiffer-ent parts of the solar cycle, there-fore offers great potential for the improvement of atmospheric density models. Furthermore, for satellites that are equipped with an accelerometer, GPS-based ac-celerometry could be used to indirectly calibrate and validate the accelerometer observations.

a strategy to optimally derive non-gravitational accelerations from precise GPS satellite tracking observations of low flying satellites. The focus of this study is on estimating non-gravitational accelerations with high temporal resolution, mak-ing optimal use of the dense GPS trackmak-ing information. This allows to observe short-period variations in the density, due to e.g. geomagnetic storms. The opti-mal GPS-based accelerometry strategy developed in this dissertation is tested with actual GPS data from the CHAMP, GRACE and GOCE missions. This allows a pre-cise validation of the resulting non-gravitational accelerations with the accelerom-eter or gradiomaccelerom-eter data from these missions. To daccelerom-etermine if the GPS-based ac-celerometry concept can be used to improve the current non-gravitational force models, the GPS-based accelerometry performance of these satellites is compared with the state-of-the-art non-gravitational force modeling. In addition, an assess-ment is made of the calibration of the accelerometer observations using the non-gravitational accelerations resulting from the GPS-based accelerometry approach. Several methods are available to acquire density information from satellite tracking data or orbits. An overview of these methods is given in Doornbos [2011] and a brief summary is given below. For a large number of space objects, orbital information is provided by the US Space Surveillance Network in the form of two-line element (TLE) data. These data can be processed to density using an efficient algoritm designed by Picone et al. [2005]. The main advantage of this technique is that it is applicable to many space objects. A major disadvantage is the coarse tem-poral resolution and the limited accuracy of the TLE data. For geodetic satellites equipped with precise tracking systems, such as the Satellite Laser Ranging (SLR) system [Pearlman et al., 2002], the French Doppler Orbitography and Radioposi-tioning Integrated by Satellite (DORIS) system [Willis et al., 2010] or the GPS sys-tem [Dow et al., 2009], precise orbit determination schemes can be used to derive density data. Generally, these dynamic orbit computation schemes include aero-dynamic scale factors as part of the adjusted force model parameters. By applying these scale factors as a correction to the density model, the model can be improved. Although the number of available satellites is limited compared to the TLE space objects, the accuracy of these tracking data is much better. In principle, the number of force model parameters that can be estimated is limited by the coverage of the tracking observations. For the GPS system, with its continuous tracking and ex-cellent geometric information, this number can be very large, while for the sparse SLR tracking, the number will be limited.

Several authors have presented improved densities from precise satellite track-ing observations. Doornbos [2011] shows time series of 6-hourly scale factors es-timated from SLR and DORIS tracking data of the Envisat satellite and from SLR tracking of the ERS-2 satellite. Willis et al. [2005] also estimate 6-hourly scale fac-tors from DORIS tracking of several satellites. Furthermore, they also show re-sults obtained with tightly constrained 1-minute scale factors. Instead of using actual tracking observations, McLaughlin et al. [2011] use precise CHAMP orbits as pseudo-tracking data to estimate density model corrections. This has the

advan-1.3 The CHAMP, GRACE and GOCE satellite missions 5

tage that the orbit determination scheme does not have to be adjusted for different types of tracking data.

In this study, instead of density model corrections, the total non-gravitational signal is recovered in a precise GPS-based orbit determination scheme. This has the advantage that the recovered non-gravitational accelerations can be easily compared and combined with accelerometer measurements. Furthermore, the same processing strategy to derive density and winds can be applied to satel-lites with and without an accelerometer. The processing of the recovered non-gravitational accelerations to density and winds is beyond the scope of this disser-tation. Strategies for deriving density and winds from accelerometer observations can be found in Doornbos et al. [2010], Sutton et al. [2007] and Bruinsma et al. [2004].

1.3

The CHAMP, GRACE and GOCE satellite

missions

Throughout this research extensive use is made of data from the CHAMP, GRACE and GOCE satellite missions. A brief overview of these missions is therefore pre-sented here.

The Challenging Minisatellite Payload (CHAMP) satellite, illustrated in fig-ure 1.1, is a German small satellite mission for geoscientific and atmospheric re-search and applications. The CHAMP satellite was launched on 15 July 2000 into an almost circular, near polar orbit with an initial altitude of approximately 450 km and an inclination of 87◦_{. The scheduled mission life time was five years.}

How-ever, the satellite has been in orbit for slightly more than 10 years, before finally re-entering on 19 September 2010. Its primary mission objectives were the accu-rate determination of the Earth’s gravity field, the estimation of the magnetic field including its spatial and temporal variations, as well as the high resolution profil-ing of the Earth’s atmosphere and ionosphere. To achieve these science goals, the satellite was equipped with a number of instruments, such as the BlackJack GPS receiver, the STAR accelerometer, multiple magnetometers, a laser retro-reflector and fully autonomous star sensors. At the time of launch, it was the lowest known orbiting satellite to use the GPS constellation for precise orbit determination. A more detailed description of the CHAMP mission can be found in Reigber et al. [1996].

The Gravity Recovery and Climate Experiment (GRACE) mission, also illus-trated in figure 1.1, is a joint US-German mission which consists of two identi-cal formation flying spacecraft following each other with a nominal distance of 220 km on the same orbital trajectory. The GRACE satellites were launched on 17 March 2002 in a near polar, almost circular orbit with an initial altitude of approx-imately 500 km and an inclination of 89◦. The scheduled lifetime was five years, but the mission is still operational and the end of the mission is not expected be-fore 2015. The primary mission objective is to measure the time varying changes in the Earth’s gravity field with unprecedented accuracy [Tapley et al., 2004c], which

Figure 1.1 Artist’s impressions of CHAMP (top), GRACE (center) and GOCE (bottom). Courtesy of Astrium, NASA/JPL and ESA.

1.3 The CHAMP, GRACE and GOCE satellite missions 7

is enabled by the mission’s key instrument, the Ka-Band Ranging System (KBR). This new instrument measures the distance variation between the two satellites with a very high precision. In addition, the GRACE satellites carry SuperSTAR accelerometers to measure the non-gravitational accelerations, BlackJack GPS re-ceivers to obtain the position and velocity of the satellites, laser retro-reflectors for orbit validation purposes and star cameras for attitude determination. More de-tailed information about the GRACE mission can be found in Tapley et al. [2004b].

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite, also depicted in figure 1.1, is the first Earth explorer core mission of the European Space Agency. The GOCE satellite was launched on 17 March 2009 into a sun-synchronous dusk-dawn orbit with an exceptionally low initial altitude of about 280 km and an inclination of 96.5◦. Due to fuel depletion, the satellite has reentered the Earth’s atmosphere on 11 November 2013, which means that the mission life-time has exceeded the scheduled operation life-time of 20 months. The main mission purpose was the determination of the stationary part of the Earth’s gravity field and geoid with the highest possible accuracy and spatial resolution from gravity gradiometer and GPS measurements. It is the first satellite to apply gradiometry in space, carrying six highly accurate accelerometers, arranged in pairs along three orthogonal axes. The low orbit altitude minimizes the attenuation of the gravity field signal, which makes the orbit more sensitive to the higher degrees of the grav-ity potential. However, the atmospheric drag forces are much stronger at lower altitudes, which would result in a rapid orbit descent. To maintain the low Earth altitude, the satellite is therefore equipped with a drag-free and attitude control system (DFACS), which uses an electric ion propulsion assembly (IPA) to compen-sate the relatively large atmospheric drag in the flight direction of the compen-satellite and magnetotorquers for attitude stabilization. In addition, the GOCE satellite carries laser retro-reflectors for SLR tracking and star cameras for attitude determination. More information about the GOCE mission can be found in Drinkwater et al. [2003]. Figure 1.2 gives an overview of the evolution of the orbits of the different satel-lites. The trajectories of the CHAMP and GRACE satellites are spiraling down-wards, due to the effect of atmospheric drag. The slope of the orbit decay is corre-lated with the solar activity, with larger decay rates around the solar maximum at the beginning of 2002 and again with the onset of high solar activity in 2012. Due to the higher altitude, the GRACE satellites experience considerably lower drag acceleration and during the unusual deep solar minimum of 2008/2009 they have remained at virtually the same height. Four orbit raising manoeuvres can be iden-tified for CHAMP, occurring in respectively, June and December 2002, March 2006 and March 2009, which have been performed to prolong the mission lifetime. Dur-ing the commissionDur-ing phase at the beginnDur-ing of the mission, the GOCE satellite was lowered to the designed measurement altitude of about 260 km. The satellite remained at this controlled altitude due to its onboard drag-free control system. However, several deviations from this altitude are visible, which are caused by satellite anomalies. An overview of the GOCE satellite anomalies is presented in Kuijper and Matatoros [2012]. During the final stages of the GOCE mission the

or-2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 200 250 300 350 400 450 500 550 altitude (km) date (years) CHAMP GRACE GOCE

Figure 1.2 Evolution of the orbits of the CHAMP, GRACE and GOCE satellites. For each satellite, the minimum, maximum and average altitude above the GRS80 reference ellipsoid is depicted.

bit has been lowered several times. On 21 October, the mission came to a natural end when it ran out of fuel. In the following three weeks the satellite gradually descended, before finally disintegrating in the atmosphere.

1.4

Outline

High-quality satellite orbits are essential for a good GPS-based accelerometry per-formance, therefore chapter 2 describes the precise orbit determination (POD) strategy that has been developed to compute orbits for LEO satellites based on GPS observations. Since the quality of the GPS observations is a crucial factor of the POD, the chapter begins with an overview of the GPS data collected by re-ceivers from the CHAMP, GRACE and GOCE satellites. For each satellite, one year of GPS data is analysed and used for POD. Various orbit evaluation methods are introduced and used to assess the accuracy of the resulting orbits.

In chapter 3 the GPS-based accelerometry concept is explained in detail. This is followed by an extensive simulation study that has been conducted to determine

1.4 Outline 9

the optimal processing strategy and to assess the performance of the GPS-based accelerometry approach. The simulation study is based on real accelerometer data from the GRACE-A satellite and the processing strategy is developed from the regular GPS-based POD strategy for LEO satellites described in chapter 2. In the simulation study the impact of different error sources is analyzed and several op-timization schemes are investigated to improve the GPS-based accelerometry per-formance.

In chapter 4 the optimal GPS-based accelerometry processing strategy resulting from the analysis of chapter 3 is applied to about two months of GPS data from the CHAMP, GRACE and GOCE satellites. For GOCE, also a second data set has been processed, consisting of a 9-day period when the nominal drag-free control of the satellite was switched off. The resulting non-gravitational accelerations have been validated with existing satellite accelerometer or gradiometer observations.

The last chapter of this dissertation presents conclusions and gives recommen-dations for future study.

Chapter

2

GPS-based precise orbit

determination

2.1

Introduction

Nowadays, most Earth observation satellites require a precise knowledge of the orbit in order to fulfill their scientific mission requirements. To accomplish this, several satellite tracking systems are available. Most of these systems, like e.g. the SLR, PRARE and DORIS systems, rely on tracking stations on ground. These ground station networks cannot deliver continuous satellite tracking data, and usually only a limited number of stations can be used simultaneously. The GPS system, on the other hand, relies on the GPS satellite constellation, which can pro-vide continuous tracking of the spacecraft and gives excellent geometric informa-tion. This makes the GPS system very well suited for the precise orbit determina-tion of Earth orbiting satellites.

The concept of using GPS for POD at the decimeter accuracy level has been demonstrated for the first time by the TOPEX/POSEIDON onboard GPS receiver [Bertiger et al., 1994; Tapley et al., 1994; Perosanz et al., 1997]. However, for low Earth orbiting satellites like the CHAMP, GRACE and GOCE satellites, the POD has been challenged by the very low altitude of these satellites. At altitudes ranging from 250 to 500 km, these satellites experience large geopotential and atmospheric drag perturbations that affect the satellite orbit. Also due to the low orbit altitude, the rapid satellite motion causes rapid changes in tracking geometry and this results in comparatively short passes of continuous observations. If a purely dynamic orbit determination strategy is used, this requires a very precise modeling of the geopotential and drag perturbations as the uncertainties in the models will have a large effect on the orbit accuracy. On the other hand, if a kinematic strategy is used, the short tracking passes will affect the achievable orbit accuracy as well. To make optimal use of both the data strength and the orbit dynamics, a so-called reduced-dynamic orbit determination strategy looks promising, as this technique balances

the contributions from the dynamic models and the geometric information [Wu et al., 1991].

The GPS data processing procedures for LEO satellites can be further distin-guished by their differencing level, which ranges from zero differences to triple differences [Tapley et al., 2004a]. Single differenced GPS observations between a LEO receiver and two GPS satellites have the advantage that the clock error of the LEO receiver is removed, while the clock error of the GPS satellites can be removed by forming single differenced observations between a GPS satellite, the LEO receiver and another LEO or ground receiver. Both clock errors are eliminated with double differenced GPS observations. The use of triple differenced GPS ob-servations furthermore eliminates the carrier-phase ambiguities. In addition, the effect of other error sources, like GPS satellite ephemeris errors or atmospheric er-rors, are generally substantially reduced by differencing. The elimination of such a large number of error sources makes the triple differences approach very effec-tive. Unfortunately, these advantages come at the expense of an increased obser-vation noise level. Furthermore, double or triple differencing strategies require that observations from a GPS ground station network are included in the process-ing, which significantly increases the amount as well as the complexity of the data processing.

This chapter presents precise GPS-derived orbits for the CHAMP, GRACE and GOCE satellites using a batch least-squares reduced-dynamic triple differenced technique. To obtain orbits with an accuracy of a few cm, it is crucial that high-quality continuous GPS tracking observations are available. Therefore, the first section of this chapter describes the GPS receivers carried by the different satellites and gives an assessment of the quality of GPS data collected with these receivers. Section 2.3 gives for each satellite an overview of the preprocessing strategy that has been applied for the GPS data as well as for the star tracker data, which are used to reconstruct the attitude of the satellite. An overview of the orbit determi-nation scheme is presented in section 2.4. For each satellite, a one-year data period has been processed and the resulting orbits are described in section 2.5, together with the various evaluation methods that have been used to assess the accuracy of these orbits. Finally, the chapter concludes with a short summary of the results.

2.2

GPS data quality

A crucial factor of GPS-based precise orbit determination is the quality of the GPS data. This section gives an overview of the different GPS receivers of the three gravity field missions and describes the GPS data collected with these onboard receivers. In general, the data quantity of the GPS tracking instrument is a good indication for the performance of a GPS receiver. For each satellite, the GPS instru-ment tracking performance is assessed based on the number of available tracking channels, the data rate and the occurrence of data gaps, as well as the elevation threshold.

2.2 GPS data quality 13 0 5 10 0 20 40 60

nr of GPS observations per epoch

time (%) CHAMP 0 5 10 0 20 40 60

nr of GPS observations per epoch GRACE−A GRACE−B 0 5 10 0 20 40 60

nr of GPS observations per epoch GOCE

GOCE L1+L2

Figure 2.1 Overview of the average GPS tracking performance for the different satellites based on one year of data for 2003 (CHAMP and GRACE) or 2010 (GOCE).

The BlackJack GPS receiver onboard of CHAMP is developed by the Jet Propul-sion Laboratory (JPL) and is a second generation TurboRogue Space Receiver. The receiver is able to collect dual-frequency phase and pseudo-range measure-ments through four different antenna’s. For POD, data are used from the main up-looking antenna, which is equipped with a choke ring. From this antenna, the receiver generates GPS observations with a frequency of 0.1 Hz. Although the CHAMP receiver has 12 channels available for the POD antenna, the receiver has not been able to track this maximum number of satellites. At the beginning of the mission, the tracking capability was limited to seven satellites. On March 22, 2001, the satellite was commanded to track up to no more than eight satellites simul-taneously to keep the number of resets low [Grunwaldt and Meehan, 2003]. After a software upload on March 5, 2002, the satellite was eventually able to track up to ten satellites. This is illustrated in figure 2.1, which shows the average track-ing performance of the CHAMP GPS receiver based on all available GPS data for 2003. Usually, the CHAMP GPS receiver tracks between 8 to 10 GPS satellites and on average the receiver collects observations from 8.49 satellites. Periods where not a single GPS satellite is tracked are limited to 0.31% of the time. CHAMP GPS data can be obtained from the Information System and Data Center (ISDC) at the GeoForschungsZentrum (GFZ), available at http://isdc.gfz-potsdam.de.

Both GRACE satellites are equipped with the Instrument Processing Unit (IPU), which is a modified version of the JPL BlackJack receiver. In addition to making GPS observations, the IPU also processes the Star Camera and K-Band Ranging (KBR) signals. The GPS receiver collects dual-frequency phase and pseudo-range observations from three antenna’s. For POD only data are used from the zenith-viewing POD antenna, which is also equipped with a choke ring. The GPS pseudo-range data are sampled every 10 seconds and the phase data are recorded at 1 Hz. The receiver firmware allows for tracking of up to 12 GPS satellites for the POD an-tenna, but was set to no more than ten [Dunn et al., 2003]. This is again illustrated in figure 2.1, which shows for both GRACE satellites the average GPS tracking perfor-mance based on all available GPS data for 2003. For GRACE-A, the GPS tracking performance is comparable to the performance obtained with the CHAMP GPS

0 100 200 300 0 2 4 6 8 10 GPS data gaps (%) time in 2003 (days) CHAMP 0 100 200 300 0 2 4 6 8 10 time in 2003 (days) GRACE−A GRACE−B 0 100 200 300 0 2 4 6 8 10 time in 2010 (days) GOCE

Figure 2.2 Overview of the daily percentage of GPS tracking data gaps for the different satellites. Occasional data gaps larger than 10% are not included in this figure.

receiver, with an average of 8.50 tracked GPS satellites per epoch. The GRACE-B satellite has a slightly worse tracking performance compared to GRACE-A. On average the receiver collects observations from 7.49 satellites and this results in around 12% less GPS data for GRACE-B. Data gaps are also more common for GRACE-B than for GRACE-A, occurring respectively 1.14 and 0.54% of the time. Figure 2.2 shows that the occurrence of data gaps for the GRACE satellites shows a relatively systematic pattern. A further analysis of the GRACE tracking observa-tions shows that most gaps occur in multiples of exactly 5 minutes.

The GPS data of the GRACE satellites are publicly available as part of the so-called GRACE Level 1B products [Case et al., 2002]. These products can be retrieved from the NASA JPL Physical Oceanography Distributed Active Archive Center (PO.DAAC), available at http://podaac.jpl.nasa.gov, and are also available at the ISDC. The GRACE level 1B GPS data are already preprocessed. During this pre-processing the GPS observations are flagged for phase discontinuities and time tag corrected to GPS time. The resulting pseudo-range measurements are re-sampled to the nominal 0.1 Hz data rate using linear interpolation. To obtain highest ac-curacy, a remove-restore procedure is used, where the large dynamic range is re-moved from the absolute range bias before interpolation. The time tag corrected 1 Hz phase observations are re-sampled to 10-second intervals using cubic interpo-lation over a 10-second data span [Case et al., 2002].

The GOCE satellite carries two dual-frequency space-qualified Lagrange GPS receivers, manufactured by Alcatel Alenia Space (formerly Laben). Both receivers are connected to independent helix antennas and collect dual-frequency phase and pseudo-range measurements on 12 channels with a data rate of 1 Hz. The receivers are assigned as main and redundant receiver, which means that nominally only the main receiver is used and the redundant one serves as a back-up. Figure 2.1 confirms that the main GOCE GPS receiver indeed tracks up to 12 GPS satellites. The average number of tracked GPS satellites is 11.11, which is significantly larger than for CHAMP and GRACE. However, it is known that for the GOCE GPS re-ceiver it takes longer for the second frequency to acquire lock compared to the first

2.2 GPS data quality 15

CHAMP GRACE-A GRACE-B GOCE

nr of GPS observations in 1º x 1º bins

0 500 1000

Figure 2.3 Azimuth-elevation diagrams of the distribution of GPS observations in the antenna-fixed frame for the different satellites based on one year of data for 2003 (CHAMP and GRACE) or 2010 (GOCE). The flight direction is upwards.

frequency, due to receiver tracking loops and a lower carrier-to-noise ratio. The second frequency also loses lock earlier than the first frequency. Furthermore, the GPS receiver shows occasional unexpected losses on the second frequency during a satellite pass [Van den IJssel et al., 2011]. Without observations on the second fre-quency it is not possible to make the ionosphere-free combination of observations, which is used for all POD computations in this dissertation. Therefore, figure 2.1 also shows the GPS tracking performance in case observations with losses on the second frequency are not included. In that case the average number of tracked GPS satellites reduces to 10.17, which is still significantly larger than for CHAMP and GRACE. It has to be stressed that the 56-day period during the summer of 2010 when the GOCE satellite suffered from a serious anomaly is not included in the GPS tracking performance shown in figure 2.1. Data gaps occur around 0.71% of the time, however, these gaps are mostly limited to a few days with satellite anomalies which took place in the second half of February 2010. Dur-ing nominal satellite operations the number of data gaps is extremely low and most days are completely free from data gaps. GOCE data are publicly available through the GOCE Virtual Online Archive (VOA), available at http://eo-virtual-archive1.esa.int/Index.html.

Figure 2.3 shows the number of GPS observations collected by the POD an-tennas of the different satellites in antenna-fixed azimuth-elevation diagrams. All results are given in 1◦_{× 1}◦ _{bins and again based on one year of GPS data. The}

diagram of the GOCE satellite is determined using only GPS data without losses on the second frequency. For each satellite, the GPS data are down-sampled to a 30-second time interval, which is the data rate used for the POD computations. Differences between the nominal GPS data rate for the different satellites are there-fore not visible in this figure. The azimuth is counted clockwise, with the flight direction pointing upwards. The elevation is 0◦at the outer border and 90◦in the center of the plot. GPS observations collected at negative elevation angles are not

taken into account in this figure. For CHAMP this means that 0.15% of the GPS tracking data are ignored, while for GRACE-A and GRACE-B, respectively, 1.91 and 0.03% of the GPS data are collected at negative elevation angles. For GOCE around 0.20% of the GPS tracking data are ignored. When GPS observations with losses on the second frequency are also taken into account, 0.54% of the GPS data are collected at negative elevation angles, which confirms that most losses on the second frequency occur at low elevations at the beginning and end of a satellite tracking pass.

Figure 2.3 shows a tracking coverage that is typical for near polar satellites in a low Earth orbit with a zenith-pointing GPS antenna. The vertical stripes are due to the different orbital planes of the GPS satellites. Less observations are collected at the highest elevations, because at the poles there are no GPS observations at high elevations, due to the orbit geometry of the GPS constellation. Both CHAMP and GRACE-A show an elevation threshold for signal acquisition of around 10◦_{. For}

GRACE-B this elevation threshold is slightly higher and GOCE shows the lowest elevation threshold for signal acquisition. GRACE-A and GOCE generally are able to track setting GPS satellites down to 0◦_{or even below. The receivers on CHAMP}

and GRACE-B both lose lock earlier at the end of a tracking pass.

2.3

Data preprocessing strategy

An efficient preprocessing and data screening of the GPS observations is essential for the performance of GPS-based POD, and these steps are explained in detail in section 2.3.1. In addition to GPS data, the CHAMP, GRACE and GOCE satel-lites also deliver attitude data from onboard star cameras. These data are used for an accurate attitude reconstruction, which is needed for the POD computations. Section 2.3.2 describes the attitude data for the different satellites, as well as the preprocessing steps that are applied to these data.

2.3.1

GPS data preprocessing

It is of crucial importance for GPS-based orbit determination to have GPS data of good quality. However, bad measurements are regularly encountered, even in data obtained from geodetic-type receivers. Therefore, the first step in the preprocess-ing of the GPS data consists of a careful screenpreprocess-ing of the GPS observations to reli-ably detect and reject bad measurements. To clean the GPS data, all observations are screened using the linear Melbourne-W ¨ubbena combination LMW [Melbourne,

1985; W ¨ubbena, 1985]. This combination is given by LMW = 1 f1− f2 (f1λ1Φ1− f2λ2Φ2) − 1 f1+ f2 (f1P1+ f2P2) (2.1)

where f1and f2are the GPS carrier frequencies with corresponding wavelengths

λ1 (≈ 19 cm) and λ2 (≈ 24 cm). P1 and P2 are the code observations at f1 and

2.3 Data preprocessing strategy 17 0 100 200 300 0 5 10 15 20 25 rejected GPS data (%) time in 2003 (days) CHAMP 0 100 200 300 0 5 10 15 20 25 time in 2003 (days) GRACE−A GRACE−B 0 100 200 300 0 5 10 15 20 25 time in 2010 (days) GOCE

Figure 2.4 Overview of the daily percentage of rejected GPS data for the different satellites using Melbourne-W ¨ubbena editing.

This combination of both phase and code measurements eliminates the effect of the ionosphere, the geometry, the clocks and the troposphere. For undifferenced observations this means that the combination can be used to detect outliers and cycle slips.

The screening procedure is implemented in an iterative way. During each it-eration, the data per observation pass is checked for outliers in the Melbourne-W ¨ubbena combination. Observations that differ more than a user specified edit level from the mean value of the Melbourne-W ¨ubbena combination are removed. The value of the mean Melbourne-W ¨ubbena combination is updated during the pass using a moving window of five measurements. If the pass consists of less than five measurements the pass is removed. The iteration stops when observa-tions are no longer removed. In general, a maximum of four iteraobserva-tions is needed to clean the data. When a cycle slip occurs the Melbourne-W ¨ubbena combination usually changes significantly, therefore a new mean value is determined for the Melbourne-W ¨ubbena combination after each detected cycle slip. For all ground stations and the different satellites, the nominal edit level is 1 m. Assuming that the code observation noise is around a few dm for all selected satellites and ground stations, this edit level is well above the noise level of the LMW combination.

Figure 2.4 gives an overview of the amount of GPS data that is rejected for the different satellites during this screening procedure. In most cases, data at low el-evations at the beginning and end of a pass are removed. Significantly less data are removed for GRACE-B compared to GRACE-A, due to the fact that GRACE-B tracks less data at the lower elevations. Two large discontinuities are visible in the amount of rejected GPS data for GRACE-A, occurring around day 129 and 319. The same discontinuities can also be found in the total number of collected GPS observations, as well as in the number of GPS observations collected at negative elevation angles. After day 129, the number of GPS observations collected at nega-tive elevation angles by the GRACE-A GPS receiver greatly increases from almost no observations to more than 500 observations per day. This seems to suggest up-dates were made in the onboard software controlling the GPS tracking loops. For GOCE, the number of rejected observations is very small, even though the receiver

0 100 200 300 −2 0 2 clock drift (ns/s) time in 2010 (days) 0 100 200 300 −10 −5 0 5 10

clock bias (msec)

time in 2010 (days)

Figure 2.5 Overview of the GOCE clock behavior for 2010.

collects GPS observations at relatively low elevations, as shown in figure 2.3. GPS observations with losses on the second frequency are not included in these screen-ing statistics. On average, around 7 to 10% of the GOCE GPS data are affected by losses on the second frequency and these data are also rejected for POD computa-tions. It has to be mentioned that next to this data screening, the quality of the GPS phase observations is also assessed during the orbit determination process, when outliers are identified and removed as well.

For CHAMP, the specification for the accuracy of the time calibration of the GPS receiver is better than 1 µs from GPS time. A clock steering loop is used to update the clock bias when it exceeds the 1 µs and there is no long term drift of the CHAMP onboard clock. K¨onig et al. [2003, 2005a] have shown that, despite some large exceptional deviations during software uploads and onboard receiver reboots, the nominal clock error is around this 1 µs level. The GRACE onboard clock is not closely steered to GPS time, instead it is free running with only a lim-ited amount of clock steering. Kim and Lee [2009] have shown that clock offsets of up to 0.1 seconds can occur. However, as mentioned before, the publicly avail-able level 1B GPS data are preprocessed to correct for this clock offset. Due to the ultra-stable oscillator onboard of GRACE, the absolute time is determined relative to a ground reference to less than 1 ns [Bertiger et al., 2003] and no further time tag corrections are needed for GRACE level 1B products.

The GOCE onboard clock is also not closely steered to GPS time. Figure 2.5 shows the estimated GOCE GPS receiver clock drift and bias for the selected 1-year period. This figure shows that the clock offset can be as large as 10 ms. Sev-eral discontinuities are visible in the clock drift and bias around the second half of February 2010, due to a few days with GOCE satellite anomalies and a switch from the main onboard computer to the redundant one. Although the receiver clock er-ror cancels out in the differencing, clock erer-rors of up to 10 ms cause non-negligible time tag errors. Therefore, a clock-correction step is included in the preprocessing of the GOCE GPS data. In this step, a simple Single Point Positioning (SPP) solu-tion [Leick, 2004] is computed and the resulting receiver clock errors are used for a least-squares estimation of the clock drift and bias, taking possible clock jumps into account. Using an iterative procedure, the SPP clock offsets are compared with

2.3 Data preprocessing strategy 19

x (m) y (m) z (m) block II/IIA 0.279 0.000 1.023

block IIR 0.000 0.000 0.000

Table 2.1 GPS satellite relative phase center offsets in the satellite body frame.

the estimated clock bias and drift values, and outliers are removed. The resulting clock and drift values are used to correct the observation time tags.

Because the internal clock is not steered to integer seconds, the observation epochs have fractional offsets. This means that the GPS observations have to be interpolated when differencing schemes are applied in the POD in order to syn-chronize the observations with the data from the ground stations. The preprocess-ing of the GOCE data, therefore, also includes an interpolation step. In this step, the GPS phase observations are re-sampled to integer seconds using a cubic splines interpolation scheme [Press et al., 2007]. This is similar to the cubic interpolation used to generate the Level 1B GRACE phase observations [Case et al., 2002]. No pseudo-range observations are used in the POD computations presented in this dissertation, therefore no interpolation procedure is required for these observa-tions.

Any type of precise positioning application using GPS data requires an accu-rate knowledge of the GPS satellite positions. These GPS ephemeris are provided by the International GNSS service (IGS) [Dow et al., 2005]. The IGS provides three types of GPS ephemeris: the final, the rapid and the ultra-rapid orbits. All results presented in this dissertation are obtained using the IGS final orbits, which have a reported precision of 2 to 3 cm and are released with a latency of about 2 weeks. When using IGS products, the corresponding IGS antenna phase center correc-tions used for the generation of the product have to be applied. Table 2.1 gives an overview of the relative GPS antenna phase center offset (PCO) values that are applied by the IGS before the switch to absolute antenna phase center modeling in November 2006. The relative PCOs are GPS block dependent and defined in the GPS satellite body coordinate system. In this research, the POD computations for CHAMP and GRACE are based on 1 year of GPS data collected in 2003, and therefore relative offsets are used for the POD computations of these satellites. It is assumed that these relative offsets are constant and phase center variations (PCV) are ignored when relative PCOs are used.

The GOCE POD computations in this research are based on GPS data from 2010, therefore these computations require the use of absolute GPS antenna phase center modeling. The absolute GPS antenna phase center offsets adopted by the IGS after the switch are modeled by a consistent set of PCO and PCV values and are provided in the antenna exchange (ANTEX) format [Rothacher and Schmid, 2010]. The IGS05.atx model is described in Schmid et al. [2007] and contains antenna phase center information for the entire GPS constellation as well as for a large num-ber of different GPS receiver antenna types. The antenna PCV maps for the GPS

ALBH ANKR AOML AREQ ARTU BAHR BOGT CHAT CHPI CHUR COCO CRO1 DARW DGAR EISL GLPS HOB2 HRAO IISC KERG KOUR LHAZ LPGS MADR MALI MAS1 MAUI MAW1 MCM4 MDO1 NKLG NLIB NTUS OHI2 PERT POL2 POTS REUN REYK RIOG SANT STJO SUTH THTI TIXI TNML TOW2 ULAB USUD WHIT

Figure 2.6 Selected ground station network for CHAMP and GRACE processing.

ground receivers can be either azimuth/elevation dependent, or purely elevation dependent. The antenna PCV maps for the GPS satellites are block-specific and purely nadir dependent. Each GPS satellite has a satellite-specific z-offset, while the x- and y-offsets are fixed to manufacturer’s values. The antenna PCV maps for the GPS satellites are restricted to boresight angles of 14◦, due to the fact that these PCV values are calibrated using terrestrial measurements. However, in the processing of GPS measurements from the GOCE satellite, slightly larger boresight angles of 15◦to 16◦are also encountered. To accommodate this, GPS antenna PCV values for boresight angles above 14◦ _{have been taken equal to the value at 14}◦_.

Since June 20, 2013, the IGS absolute GPS antenna PCV maps include values for boresight angles from 14◦ _{to 17}◦_{, based on data from the LEO missions Jason-2,}

GRACE, GOCE and MetOp-A. However, to be consistent with the IGS processing standards in 2010, this map has not been used in the processing.

In addition to the PCV maps for the GPS satellites and ground stations, it is also possible to include a PCV map for the GOCE satellite. Bock et al. [2011a] have empirically determined an antenna PCV map for the GOCE satellite, based on binned postfit GPS observation residuals, and this map has also been included in the processing of the GOCE GPS data. Elevation and/or azimuth dependent PCV maps are generally taken into account during the actual POD computations. However, if a reasonably accurate orbit solution is available, it is also possible to incorporate the PCV map corrections already during the preprocessing of the GPS data. In this research, an additional phase map correction step is therefore included in the preprocessing scheme for the GOCE satellite.

In the next preprocessing step, the GPS data of the LEO satellite is sampled to a 30-second interval, which is the data interval for the network of GPS ground stations, and ionosphere-free triple differences are formed using the carrier-phase