Demonstration of sub-meter GPS orbit determination and 1.5 parts in 108 three-dimensional baseline accuracy

Differential tracking of theGPS satellites in high-earth orbit provides a powerful relative positioning capability, even when a relatively small continental U.S. fiducial tracking network is used with less than one-third of the fullGPS constellation. To demonstrate this capability, we have determined baselines of up to2000 km in North America by estimating high-accuracyGPS orbits and ground receiver positions simultaneously. The2000 km baselines agree with very long baseline interferometry(VLBI) solutions at the level of1.5 parts in10 ⁸ and showrms daily repeatability of0.3–2 parts in10 ⁸. The orbits determined for the most thoroughly trackedGPS satellites are accurate to better than1 m. GPS orbit accuracy was assessed from orbit predictions, comparisons with independent data sets, and the accuracy of the continental baselines determined along with the orbits. The bestGPS orbit strategies included data arcs of at least one week, process noise models for tropospheric fluctuations, estimation ofGPS solar pressure coefficients, and combined processing ofGPS carrier phase and pseudorange data. For data arcs of two weeks, constrained process noise models forGPS dynamic parameters significantly improved the solutions.     

GPS inferred geocentric reference frame for satellite positioning and navigation

Accurate geocentric three dimensional positioning is of great importance for various geodetic and oceanographic applications. While relative positioning accuracy of a few centimeters has become a reality using Very Long Baseline Interferometry (VLBI), the uncertainty in the offset of the adopted coordinate system origin from the geocenter is still believed to be of the order of one meter. Satellite Laser Ranging (SLR) is capable of determining this offset to better than10 cm, though, because of the limited number of satellites, this requires a long arc of data. The Global Positioning System (GPS) measurements provide a powerful alternative for an accurate determination of this origin offset in relatively short period of time. Two strategies are discussed, the first utilizes the precise relative positions predetermined byVLBI, where as the second establishes a reference frame by holding only one of the tracking sites longitude fixed. Covariance analysis studies indicate that geocentric positioning to an accuracy of a few centimeters can be achieved with just one day of preciseGPS pseudorange and carrier phase data.     

A unified scheme for processing GPS dual-band phase observations

A unified computational scheme is presented for sequential least-squares processing ofGPS dual-band carrier-beat-phase observations in network-mode positioning with orbit relaxation, and in orbit determination applications. This scheme is applicable to any spatial and temporal distribution of stations and satellites during a particularGPS experiment. Full covariance matrices can be specified for carrier-beat-phases and for weighted constraints on the ionosphere in order to improve phase ambiguity resolution. Physically meaningful choices for these covariance matrices are developed.     

A contribution to the kinematics of plate tectonics

Modern geodetic positioning techniques likeVLBI, SLR, andGPS can be used for plate tectonic studies. A very simple model considers a number of rigid plates rotating individually with a constant rotation rate. An equally simple mathematical model, extended to deformable plates, is presented, based on vector calculus, which relates both absolute and relative point positioning observations to the plate rotation and deformation parameters (rotation rate, direction of rotation vector, deformation rate and direction of deformation center) in a consistent and elementary way. The determination of these plate parameters from various geodetic positioning observations is outlined.     

Application of GPS for transportation related engineering surveys

Summary The Texas State Department of Highways and Public Transportation(SDHPT) has been usingGPS for over two years to establish primary geodetic reference points for engineering projects and mapping control. In accordance with a Five YearGPS Implementation Plant developed in 1982, fourGPS, unmanned, automatic Regional Reference Point(RRP) stations will be installed by September 1, 1986. Five additional stations are planned as justified. EachRRP will consist of a dual frequencyGPS receiver that will ultimately track the satellites continuously. Operation of the receiver, telecommunications and other station keeping chores will be handled by a microcomputer. TheRRP station network will be controlled through another centrally located microcomputer which is also interfaced with a larger mainframe system. EachRRP is designed to service an area bounded by a200 KM radius and will act as the “other” receiver for roving field units operating in aGPS differential measurement mode. In order to meet the installation schedule, early decisions are being made concerning satellite tracking rates, operational scenarios, and telecommunications to facilitate development of the basic hardware and software systems. A period of continual enhancement to hardware, software andRRP operational procedures is expected asGPS technology expands.     

Use of phase data for accurate differential GPS kinematic positioning

DifferentialGPS land kinematic positioning tests conducted at velocities of20 to100 km/h over a baseline of1,000 km using a combination of pseudo-range and phase measurements are described. An algorithm designed for high reliability and accuracy of1 to2 m in real time field operational mode was utilized. The relatively long baseline used for the tests provided valuable information on the effects of broadcast ephemeris errors on the differential results. The tests were conducted with two Texas InstrumentsTI4100 receivers using both theP andC/A codes to assess the effect of both code measurement noise, and ionospheric irregularities on differential positioning over such a baseline. The use of cesium clocks to constrain time was also tested. Accuracies (in terms of repeatabilities) of the order of1 to3 ppm, i.e.,1 to3 m, were obtained.     

Stochastic estimation of tropospheric path delays in global positioning system geodetic measurements

Water vapor radiometric (WVR) and surface meteorological (SM) measurements taken during three Global Positioning System (GPS) geodetic experiments are used to calculate process noise levels for random walk and first-order Gauss-Markov temporal models of tropospheric path delays. Entire wet and combined wet and dry zenith delays at each network site then are estimated simultaneously with the geodetic parameters without prior calibration. The path delays and corresponding baseline estimates are compared to those obtained with calibrated data and stochastic residual delays. In this manner, the marginal utility of a priori tropospheric calibration is assessed given the ability to estimate the path delays directly using only theGPS data. Estimation of total zenith path delays with appropriate random walk or Gauss-Markov models yields baseline repeatabilities of a few parts in 10⁸. This level of geodetic precision, and accuracy as suggested by analyses on collocated baselines estimated independently by very long baseline interferometry, is comparable to or better than that obtained after path delay calibration usingWVR and/orSM measurements. Results suggest thatGPS data alone have sufficient strength to resolve centimeter-level zenith path delay fluctuations over periods of a few minutes.     

Local geoid determination and comparison with GPS results

Ellipsoidal heights have been determined for a test network in Lower Saxony withGPS. TheGPS results, with a relative precision of a few centimeters, have been used to compute quasigeoid heights by substracting leveling heights. This data set is compared to mainly gravimetrically determined quasigeoid heights using least squares collocation techniques. The discrepancies between the two data sets amount to about ±2cm, the maximum interstation distance is about50 km. This agreement shows, thatGPS can be used in combination with gravity information to obtain normal heights withcm-precision.     

Global positioning system carrier phase: Description and use

After removing the modulation from the Global Positioning System (GPS) signal (L ₁ orL ₂) a pure carrier signal remains. Suppose this carrier is continuously and precisely tracked by aGPS receiver. Furthermore, suppose the phase of the carrier is periodically measured and recorded (nearly simultaneously at two or more locations) with respect to receiver oscillators having the same nominal frequency as theGPS carrier. This paper first considers alternative modeling and processing approaches to these observational data for static operations. Then an approach to dynamic relative positioning using triple differences is presented. This approach should lend itself to performing centimeter accuracy relative surveys in seconds rather than hours. An approach to fixing cycle slips, automatically, is included.     

A rational approach to the use of a large number of GPS receivers

We consider the problem of designing a geodetic survey when using more than twoGPS receivers.R.A. Snay previously published a solution to the problem for the case of 3 and 4 receivers. We present a solution for the case of 5, 6, 7 and 8 receivers. Finally we show an application of the method proposed.     

Network design strategies applicable to GPS surveys using three or four receivers

Strategies applicable to the design ofGPS surveys involving deployment of either three or four compatible receivers are presented. During aGPS observing session, the receivers operate simultaneously, producing three-dimensional cartesian coordinate differences for the lines interconnecting the receivers. Different strategies provide the network designer with several options for planning the survey. The designer may opt for a survey in which each mark is occupied three times, that is, during three separate observing sessions, or he may elect a more economical survey in which each mark is occupied only twice. The designer may also choose between two fundamentally different network geometries (a loop geometry or an areal geometry) to design a survey compatible with the spatial distribution of network marks. The strategies can be extended to other geometries. The strategies produce efficient networks in that no two marks are jointly occupied for more than one observing session. This feature produces the maximum number of distinct, directly observed lines for the given number of receivers and observing sessions. The strategies also favor observations over those lines connecting marks near one another. This feature helps survey logistics by reducing travel time between observing sessions.     

Improving the computational efficiency of the ambiguity function algorithm

Techniques are described in this paper for improving the Ambiguity Function Method (AFM) for differential GPS positioning using phase observations, (a) that take advantage of optimal dual-frequency observable combinations to improve thereliability of the AFM, and (b) that significantly shorten the computation time necessary for the AFM. The procedure can be used for kinematic positioning applications if a Kalman filter predicted position is accurate enough as an initial position for the suggested AFM searching procedure, or pseudokinematic mode using say a triple-difference solution as an initial position for static positioning if the baseline length is short (typically <5km).     

Accuracy analysis for the NAD 83 geodetic reference system

The North American Datum of 1983 (NAD 83) provides horizontal coordinates for more than 250,000 geodetic stations. These coordinates were derived by a least squares adjustment of existing terrestrial and space-based geodetic data. For pairs of first order stations with interstation distances between 10km and 100km, therms discrepancy between distances derived fromNAD 83 coordinates and distances derived from independentGPS data may be suitably approximated by the empirical rulee=0.008 K^0.7 where e denotes therms discrepancy in meters and K denotes interstation distance in kilometers. For the same station pairs, therms discrepancy in azimuth may be approximated by the empirical rule e=0.020 K^0.5. Similar formulas characterize therms discrepancies for pairs involving second and third order stations. Distance and orientation accuracies, moreover, are well within adopted standards. While these expressions indicate that the magnitudes of relative positional accuracies depend on station order, absolute positional accuracies are similar in magnitude for first, second, and third order stations. Adjustment residuals reveal a few local problems with theNAD 83 coordinates and with the weights assigned to certain classes of observations.     

Local geoid computation from gravity using the fast fourier transform technique

Model computations were performed for the study of numerical errors which are interjected into local geoid computations byFFT. The gravity field model was generated through the attractions of granitic prisms derived from actual geology. Changes in sampling interval introduced only0.3 cm variation in geoid heights. Although zero padding alone provided an improvement of more than5 cm in theFFT generated geoid, the combination of spectral windowing (tapering) and padding further reduced numerical errors. For theGPS survey of Franklin County, Ohio, the parameters selected as a result of model computations, allow large reduction in local data requirements while still retaining the centimeter accuracy when tapering and padding is applied.     

Establishment of a 3-dimensional geodetic network using the MACROMETER IItm dual-band surveyor

Summary During July of 1985, Aero Service conducted aGPS research project over a900 square km area in the Sacramento Valley of California. The project was partially funded by the California Department of Water Resources and was coordinated with the Sacramento office of U.S.G.S. The survey was designed to evaluate the accuracy, efficiency, and reliability ofMACROMETER II technology, as an alternative to conventional leveling techniques, for the monitoring of land subsidence. Thirty independent baseline vectors were determined between21 pre-existing benchmark locations. The majority of baseline vectors measured approximately10 km in length and were observed in a highly productive mode of three,1-hour observations per day. Six baseline vectors ranging from34 to56 km in length were observed as single day, 3.5 hour observations. In all cases the integer values ofL ₁ andL ₂ double-differenced phase biases were determined. The relative positions of stations in the network were determined to within1 part per million(1 ppm) in both horizontal coordinates, and about1.6 ppm in the vertical. Operational aspects of the project are described. Project results are examined with emphasis on the added benefits of dual-frequency measurements; the repeatability of interferometric determinations of individual baseline vectors; and the three-dimensional vector closure of the networks as a whole.     

Observable quantities in satellite gradiometry

Deriving the observables for satellite gravity gradiometry, several workers have identified the invariants under spatial rotation of the gravitation gradient tensor for obtaining quantities insensitive to the precise (unrecoverable) attitude of the satellite. Extending this work we show:

1. Considering that an approximate (not precise) attitude recovery for these, three-axes-stabilised, satellites is to be expected, one can identifythree independent invariants instead of two.
2. Besides studying gradient tensor invariants for one observation time, one should also study (as withGPS observables) first and seconddifferences between successive tensor component values in time. Bias and trend patterns in the measured tensor components caused by satellite rotation uncertainty, and by attitude uncertainty in some cross components, are shown to cancel. Information thus obtained is exclusively high-frequency, however.

Observation equations for gradiometry are derived taking three satellite attitude angles into account. Various alternatives for the satellite’s nominalattitude law are discussed.     

Geocenter variations derived from GPS tracking of the GRACE satellites

Two 4.5-year sets of daily geocenter variations have been derived from GPS-LEO (Low-Earth Orbiter) tracking of the GRACE (Gravity Recovery And Climate Experiment) satellites. The twin GRACE satellites, launched in March 2002, are each equipped with a BlackJack global positioning system (GPS) receiver for precise orbit determination and gravity recovery. Since launch, there have been significant improvements in the background force models used for satellite orbit determination, most notably the model for the geopotential, which has resulted in significant improvements to the orbit determination accuracy. The purpose of this paper is to investigate the potential for determining seasonal (annual and semiannual) geocenter variations using GPS-LEO tracking data from the GRACE twin satellites. Internal comparison between the GRACE-A and GRACE-B derived geocenter variations shows good agreement. In addition, the annual and semiannual variations of geocenter motions determined from this study have been compared with other space geodetic solutions and predictions from geophysical models. The comparisons show good agreement except for the phase of the z-translation component.     

On the internal stability of GPS solutions

Internal stability is referred to as the uniform asymptotic stability of Kalman filtering solutions, addressing the effects of initial-state errors, observation sample rates, lengths of observation sessions and so on. This paper presents the theory of the internal stability of GPS solutions, including mathematical definitions, properties and theorems addressing the dependence of internal stability of the filtering solutions on the controllability and observability of a GPS dynamic positioning system. The theory has been applied in analysing the stability performance of different GPS positioning modes, and the actual GPS positioning results have demonstrated graphically the asymptotic convergence of those stable solutions versus observation tracking time/filtering time-steps. The internal stability is therefore considered as an important performance criterion to evaluate different GPS positioning solutions. Received: 4 June 1993 / Accepted: 13 September 1996     

Impacts of geodynamic phenomena on systems for height and gravity

Geodynamic phenomena of permanent or secular characters play a significant role when defining height systems and gravity systems. A treatment is here given of the permanent earth tide, postglacial land uplift, sea level changes and polar drift from this point of view.