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Space geodetic techniques and their data applications
Introduction
The launch of artificial satellites as early as 1957 has presented an unprecedented prospect of using the long period of available satellite data to study the size, shape and rotation of Earth as well as the variations in the Earth’s gravity field. This is known as the three pillars of geodesy1 (geokinematics, rotation and gravity field) in modern geodesy. In particular, the use of satellites for geodetic applications led to the development of satellite geodesy2 . Satellite geodesy observations are achieved through space based techniques, particularly SLR and satellite positioning (e.g., Global Navigation Satellite Systems (GNSS)). The SLR technique measures the travel time (converted to range and corrected for a number of range delay parameters) of a transmitted laser pulse from the ground tracking station to the orbiting satellite in space and back to the ground station with an accuracy of approximately a centimetre. Applications of SLR measurements include the determination of the Earth’s gravity field, monitoring of motion of the tracking station network with respect to the geocentre as well as calibration of geodetic microwave techniques (e.g. calibration of satellite orbits where the satellites are equipped with radar altimeters). On the other hand satellite based positioning and navigation systems, in particular the Global Positioning System (GPS), have opened unlimited possibilities for its use e.g. in geodetic control surveying and navigations (Seeber, 2003). For example, GPS data have been used for precise land navigation and have contributed to the establishment of precise geodetic control and the determination of GPS elevations above sea level.Very Long Baseline Interferometry (VLBI), a technique which was developed in the late 1960’s, has been broadly used in various fields of geodynamics such as global plate tectonic measurements and studies of variations in the Earth’s rotation (Ryan et al., 1993; Eubanks et al.,1993). This technique has also resulted in the establishment and maintenance of an accurate and stable inertial (celestial) reference system which replaced the fundamental star catalogues. Here,a catalogue of Quasars (stable and distant radio sources) is used for defining the International Celestial Reference Frame (ICRF). The three techniques (SLR, GNSS and VLBI) together with Doppler Orbitography and Radio positioning by Satellites (DORIS) and even the Lunar Laser Ranging (LLR) technique, are the precise geodetic measurement methods and are often referred to as space geodetic methods or techniques (Koyama et al., 1998). Space geodetic technique are the fundamental tools for modern geodesy whose scope encompasses the provision of services to both society and the scientific community. Since these techniques have different characteristics in many aspects, it is preferred to collocate them (locate them on the same site) in order to compare the different and independently obtained results with each other thus improving their individual reliability. In this chapter the key space geodetic milestones are described and then a brief discussion of the principle and the main observables of the three space geodetic techniques (i.e., SLR, VLBI and GNSS) follow. A detailed focus is given to the SLR technique since it is used in this study. Here the discussion includes the properties of SLR,modelling factors that affect the accuracy of SLR measurements and some applications.
Milestones in space geodesy
Going back in history, geodesy together with its counterparts e.g. surveying, positioning and navigation merely meant measuring of angles as shown in Figure 1. To achieve such measurements the scale was roughly introduced by known distances between two sides of interest. Measurements using cross-staffs were often used to perform relative, local and absolute positioning (Beutler, 2004). A cross-staff is a mechanical device used to measure the angle between two objects (e.g., stars), see for example Figure 2. Historically the cross-staff was used in navigation to help sailors orient themselves, astronomers to study the sky, and by surveyors interested in taking accurate measurements. The cross-staff consists of a long pole with a series of markings and a sliding bar mounted at a perpendicular angle called a transom. To use a crossstaff, the navigator would position the end of the pole on the cheek just below the eye, and pick two objects to sight to, such as the horizon and the Sun. The navigator would then slide the transom along the cross-staff until one end lines up with one object and the opposite end lines up with the other object. Once the transom is in position, the marking covered by the transom indicates the angle between the two objects, which can be used to calculate latitude and to collect other information.The geographical latitude of a site could be established by determining the elevation of the Sun or the polar star Polaris3 and then looking up the latitude from a pre-calculated table. On the other hand, the longitude was determined by calculating the time difference between the unknown site and Greenwich ( 0 longitude). The time difference parameter was normally derived either by observing the Sun and measuring the local solar time or by observing certain stars and measuring their sidereal time. Problems related to the realisation of Greenwich time at the unknown observing time were solved by measuring lunar angular distances, lunar distances and angles between bright stars and the Moon. Increased accuracy in lunar orbit prediction allowed angular distances between the Moon and stars to be precisely predicted and tabulated in astronomical and nautical almanacs as a function of Greenwich local time (Beutler, 2004). The development of new instruments such as marine chronometers (these are highly accurate clocks kept aboard ships and used to determine longitude through celestial navigation) resulted in dramatic improvement in navigation accuracy (Beutler, 2004). For instance, the cross-staff method was quickly replaced by more sophisticated optical devices which included telescopes. This innovative development allowed the determination of more accurate star fundamental catalogues and improvement in predicting motion of planets. Disciplines of fundamental astronomy emerged from the interaction between positioning, navigation, geodesy and surveying. Under such disciplines the terrestrial reference system was realized based on the geographical coordinates of a network of astronomical observatories with an accuracy of about 100 m (Beutler 2004). On the other hand the celestial reference frame was realized by using the derived-fundamental catalogues of stars. The transformation between the terrestrial reference and celestial reference frames enabled the monitoring of Earth rotation in inertial space and on the Earth’s surface. Such monitoring revealed that the motion of the Earth’s rotation exhibited short periodic variations. For example the Length-Of-Day (LOD) was noticed to slowly increase by about 2 ms per century. In addition, discoveries of the Earth’s rotation axis moving on the Earth’s surface (these are polar motion effects) were also reported in the literature. Historically, surveying and navigational equipment were too inaccurate to measure observables such as changes in LOD or polar motion, but as equipment and techniques improved, it was quickly determined that the dynamics of the rotation of the Earth was not simply a case of undisturbed slow and predictable rotation.
The determination of the Earth’s gravitational field also plays an essential role in geodesy and surveying. In the pre-space geodesy era, the gravity field of Earth was determined solely by in situ measurements on or near the surface of the Earth. Terrestrial instruments, which included gravimeters4 and zenith cameras, were developed for gravity field measurements. These instruments however were suited for modelling the local as well as regional properties of the Earth’s gravitational field. The desire to model the global properties of the gravity field of the Earth resulted in the development of satellite gravity missions. The use of artificial satellite missions led to the development of satellite geodesy (Kaula, 1966). Today there are four primary techniques, namely SLR, VLBI, GNSS and DORIS that are used in space geodetic observations for the purpose of studying the size, figure and deformation of the Earth and determination of its gravity field and the field’s spatial and temporal variations. Apart from scientific interest, contributions from space geodetic techniques may also be applied in most societal areas ranging from disaster prevention and mitigation, to the provision of resources such as energy, water and food and also gaining an understanding of climate change.
Modern space geodetic techniques
Space geodetic techniques which include SLR, GNSS and VLBI and DORIS are fundamental tools of geodesy. Principles and properties of GNSS, VLBI and SLR methods are briefly reviewed in the following sections. For more information on DORIS the reader is referred to the following published literatures, Gambis (2004), Willis et al. (2006) and Coulot et al. (2007).
GNSS observable
Global Navigation Satellite System is a term used to describe a group of satellite based navigation systems that allow for the determination of positions anywhere on Earth. Currently the most commonly used GNSS consist of three main satellite technologies: the American controlled GPS, the Russian controlled Global Orbiting Navigation Satellite System (GLONASS) and the European GALILEO system. Each of these satellite systems consists mainly of three segments: (a) space segment, (b) control segment and (c) user segment (Aerospace Corporation, 2003). The GPS is the most utilized system for positioning, navigation and timing purposes and GPS satellites act as reference points from which receivers on the ground determine their positions. The navigation principle is based on the measurement of pseudo ranges between the user and at least three satellites. Ground stations precisely monitor the orbit of every satellite by measuring the travel time of the signals transmitted from the satellite distances between receiver and satellites. Resulting measurements include position, direction and speed.
The VLBI observable
Very Long Baseline Interferometry (VLBI) as a technique measures the delay in the arrival times of radio signals produced by a distant source being monitored simultaneously at two terrestrial antennas; see for example schematic representation in Figure 3. The time difference between the arrivals of the signal at each radio telescope is derived by correlation (at the correlator). These time delays and/or its derivative are used to calculate precisely the distance and direction of the baselines between the telescopes. Extragalactic objects that generate radio signals are often considered as point sources due to their great distance. In practice, for the purpose of geodetic VLBI, these sources (quasars) are carefully selected to ensure that they exhibit low proper motion and minimal source structure, so as to appear fixed and point-like. When this happens the time dependence of the time delay is generated via the Earth’s motion, although it is dependent on the source location and the baseline vector between the two antennas.In VLBI measurements the main observed quantities include the geometric delay, phase delay, group delay, the delay rate, and correlated amplitude. The geometric delay is directly related to the fringe phase as a function of frequency. It is as a result of the combination of the geometry of baseline and the direction to the radio source. Mathematically this delay observable can be described as in Tanir et al. (2006) and is expressed in Equation (11) where c is the speed of light, B is the baseline vector between two stations and k is the unit vector towards the observed source. The baseline vector B can be transformed between the terrestrial geocentric system and celestial geocentric system. Such a transformation may be
formulated as in Tanir et al. (2006) and is described as per Equation (12).
SLR observable
Satellite Laser Ranging (SLR) is a technique that measures the two-way travel time of a short laser pulse which is reflected by an orbiting satellite. This method of measurement is applied to
orbiting satellites equipped with special mirrors known as retro-reflectors (which are made from glass prisms). A schematic diagram illustrating the operation of a typical SLR system is
presented in Figure 4. In a typical SLR system, a transmitting telescope emits short laser pulses with energy between 10 and 100 mJ at a pulse repetition frequency ranging between 5 and 20
Hz. Some modern systems have lower power levels and higher firing rates up to 2 kHz. The emitted laser pulse has a typical duration of two hundred or less picoseconds, most often specified by the Full Width Half Maximum (FWHM) of the pulse.
Abstract
Declaration
Dedication
Acknowledgements
Table of contents
List of Tables
List of Figures
Acronyms
1. Introduction
1.1. Background
1.2. Significance of the research
1.3. Aim and objectives
1.4. Outline of the thesis
2. Space geodetic techniques and their data applications
2.1. Introduction
2.2. Milestones in space geodesy
2.3. Modern space geodetic techniques
2.3.1. GNSS observable
2.3.2. The VLBI observable
2.3.3. SLR observable
2.4. Modelling strategies in SLR
2.4.1. Forces acting on an orbiting satellite
2.4.2. Tropospheric delay modelling
2.5. Applications of SLR measurements
2.5.1. International Terrestrial Reference Frame (ITRF)
2.5.2. Gravity field
2.5.3. Determination of the geoid
2.5.4. Precise satellite orbit determination
2.6. Global geopotential models
2.6.1. Satellite-only GGMs
2.6.2. Combined GGMs
2.6.3. Tailored GGMs
2.6.4. Some remarks on the classification of gravity field models
2.7. Concluding remarks
3. Data and analysis
3.1. Introduction
3.2. Data
3.3. Satellites
3.4. SLR analysis software
3.4.1. Software parameterization
3.5. Data analysis
3.6. Concluding remarks
4. Investigating the accuracy of gravity field models using satellite laser ranging data
4.1. Introduction
4.2. Background
4.2. Analysis of gravity field models
4.2.1. Improvements in gravity field modelling
4.2.2. Trends in O-C residuals based on developments in gravity field modelling
4.3. Investigating possible improvements in the SDAS package
4.4. Concluding remarks
5. Analysis of the effect of tide parameterization on the accuracy of gravity field models
5.1. Introduction
5.2. Background
5.2.1. Solid Earth tides
5.2.2. Pole tides
5.3. Parameterization
5.4. Models Evaluated
5.5. Statistical analysis of O-C residuals
5.6. Statistical significance of the variations in the standard deviation of O-C residuals between models
5.7. Concluding remarks
6. Geophysical applications of Earth’s oblateness parameter J2
6.1. Introduction
6.2. Background
6.3. Inter-comparisons between SDAS estimated J2 and a priori J2 of EGM96, GRIM5C1,GGM03C and AIUB-GRACE01S models.
6.4. Geophysical modes of oscillation inherent in LOD, AAM and J2
6.4.1. Analysis of phase synchrony
6.5. Concluding remarks
7. Conclusion and recommendations for future research
7.1. Summary
7.2. Concluding remarks
7.3. Recommendations
7.3.1. Assessment of additional SLR LAGEOS data
7.3.2. Probing the significance of SLR parameterization
7.3.3. Additional satellites
7.3.4. Technical issues
References
Appendix A5
Appendix A6
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Evaluation of Earth gravity field models used for precise satellite orbit determination through applications of Satellite Laser Ranging data
