1 Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C Dissertationen Heft Nr. 67 Björn Frommknecht Integrated Sensor Analysis of the GRACE Mission München 8 Verlag der Bayerischen Akademie der Wissenschaften in Kommission beim Verlag C. H. Beck ISSN ISBN
3 Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C Dissertationen Heft Nr. 67 Integrated Sensor Analysis of the GRACE Mission Vollständiger Abdruck der von der Fakultät für Bauingenieur- und Vermessungswesen der Technischen Universität München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation vorgelegt von Dipl.-Ing.Univ. Björn Frommknecht München 8 Verlag der Bayerischen Akademie der Wissenschaften in Kommission beim Verlag C. H. Beck ISSN ISBN
4 Adresse der Deutschen Geodätischen Kommission: Deutsche Geodätische Kommission Alfons-Goppel-Straße! D München Telefon ! Telefax / - Vorsitzender: Univ.-Prof. Dr.-Ing. M. Schilcher Prüfer der Dissertation:. Univ.-Prof. Dr.-Ing. Dr.-Ing.h.c. R. Rummel. Univ. Prof. Dr.-Ing. J. Müller, Leinbiz Universität Hannover Die Dissertation wurde am 3..7 bei der Technischen Universität München eingereicht und durch die Fakultät für Bauingenieur- und Vermessungswesen am 6..7 angenommen 8 Deutsche Geodätische Kommission, München Alle Rechte vorbehalten. Ohne Genehmigung der Herausgeber ist es auch nicht gestattet, die Veröffentlichung oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) zu vervielfältigen ISSN ISBN
5 3 Abstract The geodetic twin satellite mission GRACE delivers gravity field models of the Earth of unprecedented accuracy. However, the originally defined mission baseline is not (yet) reached. Among others, deficiencies in the gravity field sensor system or the related signal processing could be responsible for the degraded performance. In this work the GRACE gravity field sensor system is subject to an integrated sensor analysis. First, models of the satellites environment are derived, including models for the direct gravitational accelerations caused by the Earth, the Sun and the Moon and indirect effects like Earth and ocean tides. Among the nongravitational forces air drag, solar radiation pressure and Earth albedo are investigated. The purpose of this part of the work is to model what the gravity sensor system should feel. The second part contains mathematical models of the individual elements of the gravity field sensor system, the star sensor, the accelerometer,the K-band ranging system and the GPS receiver, to derive an understanding of how the sensors detect their environment. The third part consists of the analysis of the raw instrument data. The real measurement signals are analyzed and the measurement performance in terms of a noise level is derived. They are compared to the model output and the derived noise level to the one specified by the instruments manufacturers. Finally the processing of the raw instrument data to the level that is used for the gravity field determination is investigated. A main processing step for all sensors is the application of an anti-aliasing low-pass filter. Alternative filters are tested and evaluated against the filter used for the official data processing. Zusammenfassung Die Satellitenmission GRACE liefert Schwerefeldmodelle bisher nicht verfügbarer Qualität. Aber die vor dem Start anvisierte Genauigkeit ist (noch) nicht erreicht worden. Neben anderen möglichen Ursachen könnten Fehler im Schwerefeldmesssystem und in der dazugehörigen Signalverarbeitung für die reduzierte Genauigkeit verantwortlich sein. In dieser Arbeit wird das GRACE Schwerefeldmesssystem einer integrierten Sensoranalyse unterzogen. Zunächst werden Modelle der Satellitenumgebung vorgestellt. Unter den direkten gravitativen Kräften werden die Erdanziehung sowie die Anziehung von Sonne und Mond modelliert, unter den indirekten gravitativen Kräften werden Erd- und Ozeangezeiten vorgestellt. Unter den nicht-gravitativen Kräften werden der Luftwiderstand, der Strahlungsdruck der Sonne und Erdalbedo untersucht. Diese Modelle geben Auskunft darüber, was das Schwerefeldmesssystem detektieren soll. Im zweiten Teil werden mathematische Modelle der einzelnen Sensoren des Schwerefeldmesssystems erarbeitet. Im einzelnen werden der Beschleunigungsmesser, der Sternsensoren, das K-band Abstandsmesssystem und der GPS-Empfänger modelliert, um eine Vorstellung darüber zu erlangen, wie die Satelliten ihre Umgebung wahrnehmen. Im nächsten Teil werden die rohen Messdaten der Instrumente untersucht. Die Messsignale werden analysiert und die Qualität der Sensoren, wie auch die der Messung, wird über das Fehlerniveau bestimmt. Die gemessenen Signale werden mit den Erwartungen aus den Modellen verglichen, insbesondere wird das aus den Messungen abgeleitete Rauschniveau mit den durch die Intrumentenhersteller gegebenen Spezifikationen verglichen. Schließlich wird die Prozessierung von Rohdaten zu Daten, die für die Schwerefeldbestimmung verwendet werden, untersucht. Ein Hauptschritt der Umwandlung ist die Anwendung eines Tiefpassfilters um die Datenrate ohne das Auftreten von Aliasingeffekten reduzieren zu können. Verschiedene alternative Filter werden diskutiert und miteinander verglichen.
7 Contents 5 Contents Abstract Zusammenfassung I Introduction Overview of the GRACE mission Definition of integrated sensor analysis Goals and topics of the work II Force models Gravitational forces Introduction Earth gravity field Gravitational forces of third bodies Direct tides Indirect tides of third bodies Ocean tides Solid Earth pole tides Non-gravitational forces Introduction Macro-model of the GRACE satellites Air drag Introduction Air density models Velocity of the satellite relative to the atmosphere Force models for the derivation of the drag coefficient Solar radiation pressure Introduction Shadow of the Earth Force model
8 6 Contents 5.5 Earth albedo Introduction Force model Comparison of the model results with real data III Sensor models Introduction Overview of the relevant sensor systems Accelerometer Introduction Logical model Mathematical model Measurement model Dynamic measurement model Star sensor Introduction Logical model Measurement model Error model K-band system Introduction Logical model Measurement model Error model GPS Receiver Introduction Measurement models Pseudorange measurements Carrier phase measurements Error model
9 Contents 7 IV Real data processing and analysis Data levels overview Data products overview Science data products Housekeeping data products Data set description Level a data analysis Analysis of the star tracker data Analysis of the accelerometer data Twangs Peaks Thruster events Accelerometer performance estimation Linear acceleration measurement performance estimation Angular accelerations Analysis of the K-band and Ka-band ranging data Level a to level b processing K-band Accelerometer Star Sensor Level b analysis GPS receiver K-band Accelerometer Disturbance effects: peaks, twangs and thruster events Performance assessment Star Sensor Combined analysis Conclusions and discussion Gravitational forces on the satellites Non-gravitational forces on the satellites Gravity field sensor system Outlook Non-gravitational forces on the satellites Level a to level b data processing Acknowledgement
10 8 Annex A Discrete Fourier transform and discrete filters A. Discrete Fourier Transform (DFT) A. Discrete filters A.3 Low-pass filter A.4 High-pass filter A.5 Differentiator A.6 Double differentiator A.7 Filter scaling Annex B Power spectral density (PSD) and standard deviation Annex C Orientation representations and coordinate transforms Annex D Coordinate frames D. Inertial reference frame (IRF) D. Orbit fixed reference frame (ORF) D.3 Earth-fixed reference frame (EFRF) D.4 Satellite fixed reference frame (SRF) D.5 Instrument frames
11 9 Part I. Introduction
12 Overview of the GRACE mission. Overview of the GRACE mission GRACE (Gravity Recovery And Climate Experiment) is a joint project of the National Aeronautics and Space Administration (NASA) and the German Space Center (DLR). The mission has been proposed in 996 jointly by the University of Texas at Austin, Center for Space Research (UTCSR), the GeoForschungsZentrum Potsdam (GFZ), the Jet Propulsion Laboratories (JPL), Space Systems/Loral (SSL), the Deutsches Zentrum für Luftund Raumfahrt e.v. (DLR), and Astrium GmbH. The GRACE mission is successor to the CHAMP mission and was selected as the second mission in NASA s Earth Science System Pathfinder project (ESSP). The Principal Investigator is Prof. Byron Tapley (UTCSR), the Co-Principal Investigator of the mission is Prof. Ch. Reigber of the GeoForschungsZentrum Potsdam. The goal of the mission is to derive highly accurate models of the Earth gravity field for a period of at least five years. As an additional goal, GPS measurements are used to recover profiles of the ionosphere and the troposphere by limb sounding. Unlike the single satellite mission CHAMP, the GRACE mission consists of two identical satellites, GRACE A and GRACE B, resp. Tom and Jerry. The satellite constellation is depicted in figure.. Figure.: Overview of the GRACE satellites configuration. Considering the satellite in the front, the GPS antenna and the star sensor baffles are visible. The K-band microwave link is indicated as a blue ray. On the back side of the satellite, the ion thruster used for orbit maintenance is visible. Each satellite is equipped with a GPS Turbo Rogue Space Receiver that allows precise orbit determination. In order to be able to take into account the effect of non-gravitational forces, both satellites are equipped with ultra-sensitive Super STAR capacitive accelerometers.
13 Overview of the GRACE mission An absolute novelty is the use of a K-band microwave link between the satellites to measure the inter-satellite distance with micrometer accuracy. To guarantee sufficient power supply and to enable operation of the K-band system, the attitude of the satellites is observed by star sensors and controlled by cold gas attitude thrusters and magnetic torque rods. The GRACE satellites were successfully launched on March 7, from Plesetsk Cosmodrome in Russia. The inclination of the satellites orbits is about 89, the semi-major axis was about 687 km, the eccentricity was almost zero:.. Since then the orbit decayed to an altitude of about 46 km. The nominal distance between the two satellites is ±5 km.
14 Definition of integrated sensor analysis. Definition of integrated sensor analysis The subject to the integrated sensor analysis is a complex system consisting of several individual sensors. The purpose of the integrated sensor analysis is to understand the functionality of each individual sensor, the interaction between the individual sensors and their operation as a sensor system. The former is achieved by deriving mathematical models of the measurement process of each sensor. A common way is to model the signal that is to be sensed and then to model the error behavior of the sensor. The error behavior of the sensor is usually given by the manufacturer. The combination of the simulated signal and simulated error is a good approximation of the real measurement behavior. The latter is reached by combining the individual models to form the integrated model: The signal flow in the sensor systems is derived, thus the dependencies can be modeled. Also the processing steps necessary to transform between the individual data levels are investigated.
15 3 3. Goals and topics of the work This work consists of three parts: The description of the forces acting on the GRACE satellites, the description of the GRACE system simulator and the description of the real data processing and analysis. The first part describes models for volume forces and surface forces that act on the GRACE satellites including models for gravitational forces like the gravity of the Earth or the effects caused by third bodies like the Sun or the Moon. Among the surface forces atmospheric drag models, models for solar radiation pressure and for Earth albedo are analyzed. Special emphasis is put on the analysis of the atmospheric drag models, as at the orbit height of GRACE air drag is the dominant surface force. The second part describes how the GRACE gravity field sensor system feels its environment. Measurement models for the sensors that constitute the gravity field sensor system, namely the star sensor, the accelerometer, the K-band ranging system and the GPS receiver are derived as mathematical models. The third part is devoted to the analysis of the real satellite data. Performance estimates for the individual instruments are derived and compared to the theoretical performance estimated in the second part. Also the transformation from raw satellite data to the data level that is used for the gravity field determination is investigated.
16 4 Part II. Force models
17 4 Gravitational forces 5 4. Gravitational forces 4.. Introduction The gravitational forces are by far the strongest forces that act on the satellites and mainly determine their orbits and relative motion. In the following sections the different kinds of gravitational forces will be briefly introduced. In terms of magnitude one expects that the gravitational forces of the Earth are strongest, followed by the forces caused by Moon, Sun and the solid Earth and ocean tides. 4.. Earth gravity field The gravitational accelerations are the first derivative of the gravitational potential of the Earth. The potential and its first order derivatives can be expressed in the following way: V i = GM r ( ) R l l λ i p (αcos(mλ) + β sin(mλ)), (4.) r l= m= l max with differentiation w.r.t λ i p α β none Plm Clm Slm r (l+) r P lm C lm S lm θ P lm C lm S lm λ m P lm S lm C lm The normalized associated Legendre functions and their derivatives are computed by recursive formulas. For convenience, the following substitutions were made: P lm = P lm (cos θ), P lm = P lm (cos θ) θ. P lm = η cos θp l,m σp l,m, (4.) P l,l = τ cos θp l,l, (4.3) P ll = ν sinθp l,l, (4.4) P lm = η(cos θp l,m sinθp l,m) σp l,m, (4.5) P l,l = τ(cos θp l,l sin θp l,l ), (4.6) P ll = ν(sin θp l,l + cos θp l,l ), (4.7) with the normalization factors: η = (l + )(l ) (l + m)(l m), σ = (l + m)(l m) (l + )(l ), τ = l + l +, ν =. (4.8) l
18 6 4 Gravitational forces The initial values are: P, =, P, = 3cosθ, P, = 3 sin θ, P, =, P, = 3sin θ, P, = 3cosθ. To obtain the accelerations in cartesian coordinates, the first order derivatives in spherical coordinates have to be transformed using the chain rule: ẍ e = V x ÿ e = V y z e = V z = V r r = V r = V r x + V θ θ x + V λ λ x, (4.9) r y + V θ θ y + V λ λ y, (4.) r z + V θ θ z, (4.) where the partial derivations are simply derived from geometry: r/ x = sin θ cos λ, (4.) r/ y = sin θ sin λ, (4.3) r/ z = cos θ, (4.4) θ/ x = cos θ cos λ/r, (4.5) θ/ y = cos θ sin λ/r, (4.6) θ/ z = sin θ/r, (4.7) λ/ x = sinλ/(r sinθ), (4.8) λ/ y = cos λ/(r sin θ). (4.9) For simulation purposes or for orbit integration it is necessary to transform these accelerations from the earthfixed system to the inertial system. Figure 4. shows the resulting accelerations in the Inertial Reference Frame (IRF). The magnitude is about 8.4 m/s or about 84% of the acceleration on the Earth s surface. We notice a modulation with the orbit frequency, as the distance from the satellite to the Earth and its velocity vary during a revolution. The first term of the spherical harmonic expansion of the gravity field, i.e. l=, accounts for 99% of the gravitational acceleration, a expansion up to degree, i.e. including the effect of the Earth s oblateness, represents about 99.99% of the characteristics of the gravitational acceleration of the Earth. Thus an expansion up to degree seems to be sufficient for most purposes except gravity field determination studies of course.
19 4 Gravitational forces deg deg deg 5 deg [m/s ] time [h] deg deg 5 deg [m/s ] time [h] Figure 4.: The top panel shows the gravitational accelerations induced by the Earth. The different lines show the accelerations resulting from an evaluation of the spherical harmonic series up to a certain degree. The bottom panel shows the change in acceleration from one evaluation level to the other. It seems that it is sufficient to evaluate the Legendre series up to degree for simulations, as the change for higher degrees is very small.
20 8 4 Gravitational forces 4.3. Gravitational forces of third bodies Gravitational forces caused by third bodies the gravitational forces caused by all other celestial bodies than the Earth. In terms of magnitude the tidal forces of Sun and Moon are dominating Direct tides The gravitational acceleration of a third body can be described as the acceleration of a point mass M, cf. Montenbruck and Gill (): r sat,i = GM r body r sat r body r sat 3, (4.) with r body r sat GM the geocentric position vector of the body, the geocentric position vector of the satellite, the gravitational constant times the mass of the attracting body. In an earth-fixed coordinate system the acceleration of the satellite is derived as the difference between the acceleration of the satellite caused by the body and the acceleration of the Earth caused by the body: r sat,e = r sat,i r Earth,i ( rbody r sat = GM r body r sat 3 r ) body r body 3. (4.) r-.. sat,e r-.. sat,i r-.. sat,e r-.. Earth,i r-.. sat,e Earth r-.. sat,e 3rd body Figure 4.: Tidal ellipse of relative acceleration with respect to the center of an accelerated coordinate system. In this example the resulting acceleration on a satellite caused by a third body like the Sun or the Moon in the earth-fixed coordinate system is shown. Figure 4. shows the tidal acceleration of the satellite relative to the Earth -two body system due to a third body. It has the form of a tidal ellipse. We realize that the satellite is accelerated away from the Earth if it is in front of or behind the Earth and that the resulting acceleration is directed towards the Earth if the satellite is above or below the Earth. This equation can not only be applied to a satellite orbiting the Earth, it also describes the relative acceleration of mass elements of a body with respect to its center of mass. Equation (4.) therefore also describes the solid Earth tidal deformation due to Sun and Moon.
21 4 Gravitational forces Indirect tides of third bodies As mentioned above, the tidal forces of the other planets of our solar system deform the Earth. The deformation depends on the local properties of the Earth s crust. The so called Love numbers reflect the elasticity of the Earth s crust. The deformation leads to a change of the mass distribution of the Earth in an earth-fixed coordinate system and therefore has to be taken into account when modeling the forces acting on a satellite. As this effect is not caused directly by the gravitating body, it is called indirect. In principle not only the Earth is deformed by this effect, but also all the other planets. For satellites orbiting the Earth this effect is neglected, as the other planets are assumed to be point masses. The change in the mass distribution of the Earth causes a change of the geopotential. It is expanded in a series of spherical harmonic coefficients and described in more detail e.g. in McCarthy (6). We will give here only a brief outline: C nm i S nm = k nm 3 ( ) GM j RE P nm(sinφ j )e imλ j, (4.) n + GM j= E R j where k nm is the nominal Love number for degree n and order m, R E is the equatorial radius of the Earth, GM j is the gravitational parameter for the Moon (j = ) and the Sun (j=3), GM E is the gravitational parameter for the Earth, R j is the distance from geocenter to Moon or Sun, Φ j is the body fixed geocentric latitude of the Moon or the Sun, λ j is the body fixed geocentric longitude of the Moon or the Sun, and P nm are the normalized associated Legendre functions. The redistribution of the masses depends also on the frequency band affected by the different tidal effects, therefore a frequency dependent correction to the equation above has to be derived: C nm i S nm = f(n,m) A(f(n, m)) e iθ f(n,m), (4.3) where f(n, m) A(f(n, m)) θ f(n,m) is the frequency of the excitation, is the amplitude of the excitation, accounts for the Earth rotation and the type of the tide. The needed amplitudes and frequencies are tabulated in McCarthy (6) Ocean tides The oceans react to the tidal forces as well. There are two effects:. the change in the mass distribution of the Earth due to the movement of the water,
22 4 Gravitational forces. the change in the mass distribution of the Earth due to the changed load on the Earth s crust caused by the changing water load. The effect on the geopotential is again formulated as a change in the spherical harmonic coefficients: C nm i S nm = F nm s(n,m) (C snm ± S snm)e ± ±iθsnm ), (4.4) + where F nm accounts for the changed load on the crust, C snm, ± Ssnm pm are the ocean tide coefficients for the tide type s, θ snm is the argument of the tide type s. The summation over + and - denotes the respective addition of the retrograde and prograde waves. The interested reader may consult McCarthy (6) for a more detailed description Solid Earth pole tides The pole tides are generated by the centrifugal effect of the polar motion. They affect only the geopotential coefficients C and S : C = C (m C m ), (4.5) S = C (m + C m ), (4.6) where C =.333 9, C =.5, m, m are the wobble variables directly related the polar motion variables (x p, y p). A detailed description is to be found in McCarthy (6). Figure 4.3 shows the tidal accelerations on a GRACE satellite caused by Sun and Moon in an earth-fixed inertial reference frame over one day. We notice a strong signal on twice per revolution for all constituents. The direct tidal acceleration caused by the Moon is strongest at a level of 7 7 m/s followed by the direct tidal acceleration caused by the Sun at a level of about 3 7 m/s, i.e. the tidal force of the Moon is about three times stronger than the tidal force caused by the Sun. The ratio between the forces of Sun and Moon depends on the orientation of the orbit with respect to the positions of Sun and Moon. Here the orbit normal points towards the Sun, i.e. the distance between the satellite and the Sun is almost constant, therefore the force is also almost constant. The orbit plane is aligned with the direction to the Moon, therefore the exerted force by the Moon varies strongly. The magnitude of the indirect tidal accelerations caused by the Moon are at level of about 7 m/s, those caused by the Sun are at level of about 8 8 m/s. The accelerations caused by the ocean tides vary between
23 4 Gravitational forces 6 acceleration [m/s ] time [h] sun moon indirect moon indirect sun freq. dep. ocean pole tide Figure 4.3: Tidal accelerations due to Sun and Moon over one day for a GRACE satellite. acceleration [m/s /srqt(hz)] sun moon indirect moon indirect sun freq. dep. ocean pole tide frequency [Hz] Figure 4.4: Power spectral density of the tidal accelerations caused to Sun and Moon for a GRACE satellite. 8 m/s and 7 m/s. The magnitude of the frequency dependent corrections is about.5 8 m/s, the pole tide varies between 3 9 m/s and 8 m/s and shows also a semidiurnal period. All effects can be detected by the GRACE sensor system, as its sensitivity is about 3 m/s.
24 4 Gravitational forces In order to compare the variations of the different effects, we consider figure 4.4, where the tidal accelerations are displayed in the frequency domain as a power spectral density. We notice that the dominant frequencies are harmonics of the orbit frequency. The dominant amplitudes are found on twice per revolution. Concerning the magnitude of the different effects, the direct tides of the Moon are strongest, followed by the ocean tides, the indirect tides of the Moon, the direct tides of the Sun, the indirect tides of the Sun, the pole tides and the frequency dependent corrections. The ocean tides show a less strong correlation with harmonics of the orbit frequency, they seem to be rather independent from the satellite s position.
25 5 Non-gravitational forces 3 5. Non-gravitational forces 5.. Introduction The non-gravitational forces acting on the satellites are significantly smaller than the major gravitational ones. Depending on the orbit height, either the solar radiation pressure (as for high orbiting GPS satellites) or the atmospheric drag and earth albedo (as for low orbiting satellites) have significant influence on the forces acting on the satellites. As the GRACE satellites are low orbiting satellites, we expect dominant influence of the atmospheric drag, followed by the earth albedo and the solar radiation pressure. We did not investigate the effects of anisotropic thermal emission, the recoil from the radiowave beams used for the data downlink, electrostatic effects and others that are considered negligible. The main emphasis was put on the analysis of different air drag models, as the air drag is the dominant nongravitational force. A general idea was to assess if the accelerometer measurements can be replaced by model outputs. In the following sections we will first present a macro model of the GRACE satellites, that is a prerequisite for all force models, and then present the individual models. 5.. Macro-model of the GRACE satellites Figure 5. shows an engineering drawing of the view from the front and figure 5. shows the view from the side of a GRACE satellite. Top Port outer Starboard outer Port inner Bottom Starboard inner Boom Figure 5.: Engineering drawing of the front of a GRACE satellite. Image from Bettadpur (7). Table 5. shows the properties of the individual surface elements of the used macro-model of the GRACE satellites. The surface element location, area, normal vector in the SRF, the coefficients describing the interaction with visible light and infra-red radiation and the type of material is given.
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