GPS . Global Positioning System, the US-operated Global Navigation Satellite System (GNSS) that can be used to determine a user’s position, in terms of latitude, longitude, and height coordinates, anywhere on the globe with a clear view of the sky.
Reference system. A conventionally adopted system that specifies precisely the meaning of a user’s position.
GPS consists of three segments: (1) the space segment of typically 30 satellites that transmit microwave ranging signals and signals containing information on satellite positions and satellite atomic clock time; (2) the control segment of globally distributed stations that monitor, control, and upload data to the space segment; and (3) the user segment consisting of anybody with a GPS receiver who uses the space segment to find their position (Hofmann-Wellenhof et al., 2008).
Geodetic coordinates, such as those obtained by GPS measurements, are generally defined with respect to some reference system that comprises measurement techniques, models, computational formulas, conventions, recommendations, and reference frame coordinates, which together allows users to compute their positions at any time and location within errors and limitations that might be specified (Moritz, 1980). By their very nature, GPS reference systems are global in scale, transcending international boundaries. Although continental or national reference systems are often derived from the global reference systems to satisfy national requirements, the mathematical relationship between the smaller-scale systems and the global systems is precisely specified. Hence, GPS coordinates can be meaningfully compared between any points on or near the Earth’s surface, be they on land, sea, or in the air.
Typical positioning precision is at the level of meters for casual users. For scientific users implementing geodetic technology and methodology, the positioning precision can be improved to the level of millimeters. This improvement is made possible in part through the implementation of geodetic reference systems of a much higher accuracy than the operational reference system implemented by GPS.
To determine the GPS coordinates of any measured point first requires the definition of a reference frame , which is a specially selected global network of accurately measured points with defined Cartesian coordinates. The reference frame is said to “realize” a reference system, in the sense that it provides a tangible basis for the computation of positions in the system (Moritz, 1980). In turn, these GPS reference frame coordinates are derived from the most precise space geodetic measurement systems of the Global Geodetic Observing System (Plag and Pearlman, 2009), including GPS itself, but also satellite laser ranging (SLR) to locate the center of mass of the Earth accurately, and very long baseline interferometry (VLBI) , to determine scale accurately and to determine the orientation of the Earth accurately with respect to the fixed stars. For self-consistency, it is important that the frame itself be determined using the apparatus of the reference system. Hence, when computing a GPS position, a user is leveraging the most accurate scientific geodetic techniques through the reference frame. Reference frames must be updated every once in a while in order to maintain their accuracy, as points on the Earth’s surface have a tendency to move with time, most notably from plate tectonics and large earthquakes.
Once reference frame coordinates are defined, they implicitly specify the location of the reference frame origin, the direction of the right-handed set of Cartesian axes , and the scale by which distance is measured (Torge, 2003). The reference frame coordinates are conventionally defined such that the origin of Cartesian coordinates (0, 0, 0) is most naturally aligned physically with the center of mass of the entire Earth system. The scale is set to be consistent with the conventional speed of light together with the GPS time scale which is driven by atomic frequency standards (BIPM, 1998). The orientation of the frame is physically arbitrary but nevertheless must be defined and is set by historical convention. The z-axis is conventionally selected to point in a fixed direction closely aligned to the Earth’s figure axis. Note that the rotational North Pole wobbles about Earth’s figure axis and so is not suitable as a permanent reference direction. Indeed, it is important for space geodetic techniques to account for the motion of the rotational pole within the adopted reference frame. The conventional equator is defined to be at right angles to the z-axis. Being at right angles to the z-axis, the x-axis and y-axis therefore lie on the conventional equator. The x-axis points to the prime meridian (zero longitude), which is slightly offset from the Greenwich meridian (for historical reasons). The y-axis is then defined to be at right angles to the x-axis, thus forming a right-handed coordinate system.
Note that GPS Cartesian coordinates are more fundamental than GPS latitude, longitude, and height, which are derived quantities. Definition of GPS latitude, longitude, and height requires the notion of an ellipsoid that has a similar shape to the physical Earth, with its surface closely coinciding with the equipotential (physically level) surface of mean sea level, that is, the “geoid. ” The ellipsoid of the GPS reference system must have a defined equatorial radius, and a defined flattening factor , from which polar radius can be computed. Fundamentally, a user’s GPS receiver will compute the user’s Cartesian coordinates first and then compute the latitude, longitude, and height (Torge, 2003).
Note that the height defined in this fashion is geometrically the distance normal to the surface of the ellipse. This is not generally the same thing as physical height above mean sea level. To compute a physical height, it is further necessary to correct for the height of the ellipsoid above the geoid (Torge, 2003). Such corrections are provided by models as a function of latitude and longitude. Deviations between the geometric ellipsoid and the physical geoid can exceed 100 m in some locations. Given that uncertainty in geoid models can far exceed the uncertainty in GPS Cartesian position, it is usually preferable to use GPS height above the ellipsoid unless the application demands a physical height. For example, it is typically sufficient to use ellipsoidal height for monitoring height variation in time. However, physical height may be needed for large-scale engineering projects involving the flow of water.
GPS Reference System WGS 84
The operational GPS reference system is known as WGS 84 (“World Geodetic System 1984”). Given GPS ranging data recorded by a user’s receiver, the receiver’s coordinates (longitude, latitude, and height) can be computed in WGS 84 using a conventional set of equations specified by the official GPS Interface Specification document, which can be considered an integral part of the reference system. User access to the WGS 84 reference system is enabled by the transmission of data to the user on the orbits and atomic clock times of the GPS satellites. These data are transmitted in the “navigation message ” and can be used to compute the satellite positions in WGS 84 by the “ephemeris algorithm ” defined by GPS Interface Specification (Hofmann-Wellenhof et al., 2008). The navigation message is uploaded to the GPS satellites by the GPS control segment. Data in the navigation message is computed by the GPS control segment by least-squares estimation of the GPS orbit trajectories and atomic clock times while holding fixed the WGS 84 reference frame coordinates of official GPS tracking stations.
While the WGS 84 reference frame coordinates are improved from time to time, the reference system WGS 84 maintains its name. Since its initial development, the WGS 84 reference frame has become considerably improved by making it consistent with a much higher accuracy reference system ITRF discussed in the next section. Therefore coordinates that are specified in WGS 84 are now consistent with ITRF, though the degree of coordinate accuracy in either frame is another issue. Moreover, the ellipsoid specified in WGS 84 is so close in definition to the ITRF that it makes no practical difference to the user’s computed longitude, latitude, and height.
International Terrestrial Reference System ITRS
For the most demanding geodetic applications, scientists use the International Terrestrial Reference System (ITRS). The reference frame of ITRS is known as the International Terrestrial Reference Frame (ITRF) and is the responsibility of the International Earth Rotation and Reference Systems Service (IERS). The ITRS consists of sophisticated, accurate models and the ITRF, that together enable the computation of coordinates with the precision of millimeters (IERS Conventions, 2010). ITRF is updated from time to time and is labeled by the year of the last data that contributed to the frame definition. For example, at the time of writing, the most recent version was ITRF2008 (Altamimi et al., 2011), to be upgraded to ITRF2014 in 2016. Specifically for geodetic GPS, the reference frame that realizes ITRF2008 is “IGS08,” which is maintained by the International GNSS Service (IGS).
In order to accommodate plate tectonic motion, ITRF is defined not only by position coordinates of reference frame stations but also by velocity coordinates. Thus, given the position coordinates at the reference date (“epoch”) specified by ITRF, the position coordinates can be computed at any arbitrary date. Such a scheme is said to define a “secular frame, ” as it accommodates station motion that is linear in time (Altamimi et al., 2011).
Position coordinates in ITRF are regularized coordinates, in the sense that they do not represent the actual current position of a point, but rather the position after correcting for all physical effects specified by ITRS (IERS Conventions, 2010). During any given day, large nonlinear displacements of the Earth’s surface of up to 0.5 m result from the sum of many geophysical effects. The largest effect is caused by solid Earth tides , which are the deformations of the solid Earth caused by the gravitational pull of the moon and sun. A lesser effect which is important nearer the coastlines is ocean tidal loading. The ITRS convention document (IERS Conventions, 2010) specifies in great detail how to compute such displacements. Models must be specified in ITRS to be consistent with the definition that the origin is at the center of mass of the entire Earth system, including the ocean and atmosphere.
Except for predictable tidal motions, nonlinear motions of time constants greater than 1 day are typically not modeled as part of a station’s regularized coordinate. Instead, time series of coordinates are used by scientists to investigate such effects, which can be difficult to model accurately. Examples of such effects include the Earth’s elastic and viscoelastic response surface mass loading from the atmosphere, groundwater, nontidal ocean, snow, and ice.
The ITRS also specifies fundamental constants and recommended models for the geodetic techniques that contribute to the definition of ITRF . This ensures a level of consistency between the various techniques when combining the data to define ITRF. Moreover, the ITRS convention document (currently IERS Conventions, 2010) continues to be improved and thus represents the best working practices at any given time. Techniques that contribute to ITRF currently include GPS, VLBI, SLR, and the French satellite system “DORIS ” (Doppler Orbitography and Radiopositioning Integrated by Satellite).
One of the largest effects that requires consistent modeling in ITRS is relativity . Einstein’s theory of relativity predicts that the rate of time between two ideal clocks is a function of both relative speed (special relativity) and difference in gravitational potential (general relativity). For GPS clocks, these effects are of opposite sign, but general relativity is the larger effect (Hofmann-Wellenhof et al., 2008). The net effect on the average rate of GPS atomic clock time is mitigated in the satellite hardware by setting the frequency of the clocks lower as a function of the semimajor axis of the orbit. The small variation in satellite speed and gravitational potential as it orbits the Earth can produce a 30 ns (billionth of a second) variation of GPS atomic clock time, which if uncorrected would result in 10 m level positioning errors.
For modeling purposes, therefore coordinate time must be defined by ITRS a framework consistent with Einstein’s theory of relativity. For this purpose, one possible choice of coordinate time is that of a perfect clock following Earth’s trajectory in space, but without the Earth’s mass being present. For an ideal clock located at the Earth’s center of mass , the effect of Earth’s mass is simply to shift the rate of time by a constant. Similarly, it turns out that an ideal clock co-rotating with the Earth on the geoid would have its clock time shifted by a constant, no matter where the clock may be geographically. This choice is called “Terrestrial Time ” (TT). The rates between these three different types of coordinate time are specified by the ITRS; hence, it becomes straightforward to select a coordinate time convenient for modeling purposes. However, each choice of coordinate time necessarily changes the spatial scale by the same relative factor as the constant clock rate differences. These scale factors are specified by ITRS. The ITRF adopts the Scale . In addition to accounting for these timing effects, for the most precise applications, it is necessary to account for the so-called Shapiro effect of general relativity, which predicts that the distance between a satellite and the Earth can be more than 1 cm longer than in flat space (the simple geometrical computation using Pythagoras’ theorem).
Other effects with models specified by ITRS include the Earth’s gravity field, orbit models, and propagation delay in the atmosphere at radio and optical frequencies. Errors in these models could in principle lead to differences in scale between different techniques and errors in the realization of the Earth’s center of mass. Therefore, these are topics of ongoing research to ensure the most accurate specification of ITRS, the best achievable consistency between the techniques, and best collective accuracy of all techniques as they contribute to ITRF.
Despite consistency being imposed by ITRS in a relativistic, multi-technique framework, results show that there are significant scale differences between the different geodetic techniques that cannot yet be explained. In the case of GPS, it is understood that there is a fundamental problem of defining scale when it is not perfectly known where the electrical phase center of the satellite transmitting antenna is with respect to the satellite center of mass. Therefore SLR is used to calibrate GPS scale. However the difference in scale between SLR and VLBI (at the level of a centimeter over one Earth radius) remains an unsolved problem. For this reason, empirical scale differences between the techniques are estimated when combining solutions to generate ITRF, even though in principle these scale differences should not exist.
Another feature of ITRF is the need to specify “site ties, ” which are the relative coordinates between points associated with different techniques at the same site, so-called fundamental geodetic stations. Site ties are necessary to put the various techniques into the same reference frame. However, site ties turn out to be difficult to measure accurately and are a weak link in the construction of an ITRF. Methods to improve site tie measurements are an ongoing topic of research.
Scientific Requirements of GPS Reference Systems
The most stringent accuracy requirements on space geodetic reference systems are driven by scientific applications (Plag and Pearlman, 2009). The most demanding scientific application that has been identified by the US National Research Council ( 2010) is the monitoring of global sea level change over decades. In particular, it is important for projections and models of global climate change that we be able to detect any acceleration in the rate of global sea level change. Measuring global sea level change requires accurately locating the Earth system center of mass, at the level of 1 mm per decade or 0.1 mm/year. Systems to measure sea level include satellite radar altimeters , which themselves must be positioned accurately within ITRF using GPS and SLR. Therefore both GPS and SLR positioning of the satellites must be performed as consistently as possible in ITRF, with the full framework of ITRS.
Achieving the goal of 0.1 mm/year accuracy of the reference frame origin and scale is a formidable task that will require international support, upgraded observation systems, and collaborative research in the fields of geodesy and disciplines that can help improve models of geodetic observations, such as tectonics, rheology, oceanography, hydrology, atmospheric science, and cryospheric science. As identified by National Research Council ( 2010), this in turn will require us to educate the next generation of students in the field of geodesy and related disciplines, to ensure that progress continues toward the goal of being able to meet the most demanding scientific requirements .
GPS reference systems allow users of GPS to position themselves with coordinates that are well defined, such that it is meaningful to compare coordinates and their change in time between points anywhere in the world, on or near the Earth’s surface. The most accurate reference system is the ITRS, designed to support scientific applications. ITRF is the reference frame of ITRS, which is realized using space geodetic techniques including SLR, VLBI, GPS, and DORIS. ITRF station coordinates are thought to be accurate at the few millimeter level. The origin of ITRS is ideally defined to lie at the center of mass of the Earth system, as this is the natural definition of the origin of the gravity field, which determines the orbits of space geodetic satellites. The drift of the ITRF origin with respect to the true center of mass of the Earth system is thought to be no larger than 0.5 mm/year. Much research and investment in geodetic infrastructure are required to push this drift down to the level of 0.1 mm/year, to support scientific monitoring of global sea level rise.
References and Reading
Altamimi, Z., Collilieux, X., and Métivier, L., 2011. ITRF2008: an improved solution of the international terrestrial reference frame. Journal of Geodesy, 85(8), 457–473, doi:10.1007/s00190-011-0444-4. CrossRef
BIPM, 1998. The International System of Units (SI), 7th edn. Paris: International Bureau of Weights and Measures.
IERS Conventions, 2010. In Petit, G., and Luzum, B. (eds.), IERS Technical Note 36. Frankfurt am Main: Verlag des Bundesamts für Kartographie und Geodäsie. 179 pp., ISBN 3-89888-989-6 (print version) http://www.iers.org/IERS/EN/Publications/TechnicalNotes/tn36.html
Hofmann-Wellenhof, B., Lichtenegger, H., and Wasle, E., 2008. GNSS- Global Navigation Satellite Systems: GPS, GLONASS, Galileo, and More. Wein/New York: Springer. ISBN 978-3-211-73012-6.
Moritz, H., 1980. Geodetic reference system 1980. Bulletin Géodésique, 54(3), 395–405. CrossRef
National Research Council, 2010. Precise Geodetic Infrastructure: National Requirements for a Shared Resource. Washington, DC: The National Academies Press.
Plag, H.-P., and Pearlman, M., 2009. Global Geodetic Observing System. Berlin: Springer. CrossRef
Torge, W., 2003. Geodesy. Berlin: Walter de Gruyter.