Vector gravimetry using GPS in free-fall and in an Earth-fixed frame

The Global Positioning System (GPS) is considered in conjunction with a strapdown Inertial Measurement Unit (IMU) for measuring the gravity vector. A comparison of this system in space and on an airborne platform shows the relative importance of each system element in these two different acceleration environments. With currently available instrumentation, the acceleration measurement accuracy is the deciding factor in space, while on an Earth-bound (including airborne) platform, the attitude error of the IMU is most critical. A simulation shows that GPS-derived accelerations in space can be accurate to better than 0.1mgal for a 30s integration time, leading to estimates of 1° mean gravity anomalies on the Earth's surface with an accuracy of 4–5 mgal. On an airborne platform, the horizontal gravity estimation error is tightly coupled to the attitude error of the platform, which can only be bounded by external attitude updates. Horizontal gravity errors of 5mgal are achievable if the attitude is maintained to an accuracy of 1arcsec.     

New results in airborne vector gravimetry using strapdown INS/DGPS

A method for airborne vector gravimetry has been developed. The method is based on developing the error dynamics equations of the INS in the inertial frame where the INS system errors are estimated in a wave estimator using inertial GPS position as update. Then using the error-corrected INS acceleration and the GPS acceleration in the inertial frame, the gravity disturbance vector is extracted. In the paper, the focus is on the improvement of accuracy for the horizontal components of the airborne gravity vector. This is achieved by using a decoupled model in the wave estimator and decorrelating the gravity disturbance from the INS system errors through the estimation process. The results of this method on the real strapdown INS/DGPS data are promising. The internal accuracy of the horizontal components of the estimated gravity disturbance for repeated airborne lines is comparable with the accuracy of the down component and is about 4–8 mGal. Better accuracy (2–4 mGal) is achieved after applying a wave-number correlation filter (WCF) to the parallel lines of the estimated airborne gravity disturbances.     

Response of GPS occultation signals to atmospheric gravity waves and retrieval of gravity wave parameters

We show that the amplitude of the Global Positioning System (GPS) signals in the radio occultation (RO) experiments is an indicator of the activity of the gravity waves (GW) in the atmosphere. The amplitude of the GPS RO signals is more sensitive to the atmospheric wave structures than is the phase. Early investigations used only the phase of the GPS occultation signals for statistical investigation of the GW activity in the height interval 10–40 km on a global scale. In this study, we use the polarization equations and Hilbert transform to find the 1-D GW radio image in the atmosphere by analyzing the amplitude of the RO signal. The radio image, also called the GW portrait, consists of the phase and amplitude of the GW as functions of height. We demonstrate the potential of this method using the amplitude data from GPS/Meteorology (GPS/MET) and satellite mission Challenge Mini-satellite Payload (CHAMP) RO events. The GW activity is nonuniformly distributed with the main contribution associated with the tropopause and the secondary maximums related to the GW breaking regions. Using our method we find the vertical profiles of the horizontal wind perturbations and its vertical gradient associated with the GW influence. The estimated values of the horizontal wind perturbations are in fairly good agreement with radiosonde data. The horizontal wind perturbations v(h) are ±1 to ±5 m s with vertical gradients dv/dh ±0.5 to ±15 m s km at height 10–40 km. The height dependence of the GW vertical wavelength was inferred through the differentiation of the GW phase. Analysis of this dependence using the dispersion relationship for the GW gives the estimation of the projection of the horizontal background wind velocity on the direction of the GW propagation. For the event considered, the magnitude of this projection changes between 1.5 and 10 m s at heights of 10–40 km. We conclude that the amplitude of the GPS occultation signals contain important information about the wave processes in the atmosphere on a global scale.     

Effects of laser beam alignment tolerance on lidar accuracy

One of the major lidar error sources not yet analyzed in the literature is the tolerance of the laser beam alignment with respect to the scanning mirror. In this paper, the problem of quantifying these errors is solved for rotating polygon mirror type lidar systems. An arbitrary deviation of the beam from its design direction–the vector of beam misalignment–can be described by two independent parameters. We choose these as horizontal and vertical components of the misalignment vector in the body frame. Either component affects both, horizontal and vertical lidar accuracy. Horizontal lidar errors appear as scan line distortions—along and across track shifts, rotations and scaling. It is shown that the horizontal component of misalignment results in a scan line first being shifted across the track and then rotated around the vertical at the new center of the scan line. Resulting vertical lidar error, being a linear function of the scan angle, is similar to that produced by a roll bias. The vertical component of the beam misalignment causes scan line scaling and an along track shift. The corresponding vertical error is quadratic with respect to the scan angle. The magnitude of these effects is significant even at tight alignment tolerances and cannot be realistically accounted for in the conventional calibration model, which includes only range, attitude and GPS biases. Therefore, in order to attain better accuracy, this model must be expanded to include the beam misalignment parameters as well. Addition of new parameters into the model raises a question of whether they can be reliably solved for. To give a positive answer to this question, a calibration method must utilize not only ground control information, which is typically very limited, but also the relative accuracy information from the overlapping flight lines.     

GPS phase accelerations for moving-base vector gravimetry

For airborne gravimetry using INS and GPS, the accelerations from both systems are differenced to yield the gravity acceleration. Usually, the GPS acceleration is determined by first solving for the position of the vehicle relative to a base station and subsequently taking two time derivatives of the vertical component. An alternative method is to time-differentiate the observed phases directly, thus avoiding the cycle ambiguity problem that must be solved for positioning and that is fraught with (certainly not insurmountable) difficulties in the event of a cycle slip. Due to the largely unpredictable receiver-clock errors and the imposition of the Selective Availability degradation, doubly differenced (in space) phase accelerations are used to obtain the relative vehicle accelerations. Test results for stationary receivers show that the acceleration vector can be determined from phase accelerations to an accuracy of 1 mgal for 40-s averages. The mathematical formulation of the acceleration determination also highlights certain other advantages over traditional methods, such as the avoidance of the E?tv?s correction, although a similar kind of velocity effect must be determined. Received: 9 September 1996 / Accepted: 14 April 1997     

Impact of GPS satellite and pseudolite geometry on structural deformation monitoring: analytical and empirical studies

Global positioning systems (GPS) have in recent years been increasingly used to monitor the deformations of large structures, particularly the deflections of long suspension bridges. When appropriately employed, and with the presence of a strong satellite geometry, GPS can supply timely and accurate structural deformation information. However, the three-dimensional (3-D) positioning accuracies in a local coordinate system are uneven. For instance, the vertical component of 3-D coordinates is less accurate than the horizontal component. In addition, GPS satellite availability tends to be a function of the latitude of the observation site and its surrounding obstructions. As a consequence, the accuracy of the north–south component is typically worse than that of the east–west component in mid-latitude areas (>45), and in some of the worst situations the horizontal positioning accuracy could even degrade to the same level as that of the vertical component. With such measurements it might not be possible to correctly interpret the real structural deformations. Furthermore, an insufficient number of satellites, caused by signal obstruction, could make it impossible to use GPS alone for kinematic positioning, even when integrated with other sensors such as triaxial accelerometers. With the aim of improving 3-D positioning accuracies for the monitoring of structural deflections, especially in vertical and northern directions, the optimal location selection of an array of ground-based pseudolites to augment GPS satellite geometry using an analytical simulation technique proposed by the authors is considered. Achievable 3-D positioning accuracies are estimated by simulating a real bridge deformation scenario using augmented transmitter geometry and compared with actual positioning accuracies calculated from the measurements gathered from a bridge trial. The results show that with an augmented satellite geometry and multipath mitigation it is possible to achieve uniform 3-D positioning accuracies of a few millimetres.     

关于我国1957年旧重力基本网的精度与系统转换

简述了原国家重力基本网(57网)的历史和技术要点,详细地分析了57网的各类误差和产生原因,提出了新、旧系统转换的方法。本文研究表明,57网基本点相对观测的实际精度约为±0.06mgal,仪器平均值误差为1.6×10~(-4)。基本点重力值含-13.56mgal的基准误差和1.9×10~(-4)的尺度差,并含有±0.1～0.2mgal的非线性系统误差。对基本点实行新、旧系统转换误差为±0.05mgal,基本不损失其观测精度。     

TOPEX/Poseidon orbit error assessment

This paper discusses the accuracy of TOPEX/Poseidon orbits computed at Delft University, Section Space Research & Technology (DUT/SSR&T), from several types of tracking data,i.e. SLR, DORIS, and GPS. To quantify the orbit error, three schemes are presented. The first scheme relies on the direct altimeter observations and the covariance of the JGM-2 gravity field. The second scheme is based on crossover difference residuals while the third scheme uses the differences of dynamic orbit solutions with the GPS reduced-dynamic orbit. All three schemes give comparable results and indicate that the radial orbit error of TOPEX/Poseidon is 3–4 cm. From the orbit comparisons with GPS reduced dynamic, both the along-track and cross-track errors of the dynamic orbit solutions were found to be within 10–15 cm.     

GPS/Pseudolite/INS integration: concept and first tests

This paper discusses the introduction of pseudolites (ground-based GPS-like signal transmitters) into existing integrated GPS/INS systems in order to provide higher availability, integrity, and accuracy in a local area. Even though integrated GPS/INS systems can overcome inherent drawbacks of each component system (line-of-sight requirement for GPS, and INS errors that grow with time), performance is nevertheless degraded under adverse operational circumstances. Some typical examples are when the duration of satellite signal blockage exceeds an INS bridging level, resulting in large accumulated INS errors that cannot be calibrated by GPS. Such a scenario, unfortunately, is a common occurrence for certain kinematic applications. To address such shortcomings, both pseudolite/INS and GPS/pseudolite/INS integration schemes are proposed here. Typically, the former is applicable for indoor positioning where the GPS signal is unavailable for use. The latter would be appropriate for system augmentation when the number and geometry of visible satellites is not sufficient for accurate positioning or attitude determination. In this paper, some technical issues concerned with implementing these two integration schemes are described, including the measurement model, and the appropriate integration filter for INS error estimation and correction through GPS and pseudolite (PL) carrier phase measurements. In addition, the results from the processing of simulated measurements, as well as field experiments, are presented in order to characterize the system performance. As a result, it has been established that the GPS/PL/INS and PL/INS integration schemes would make it possible to ensure centimeter-level positioning accuracy even if the number of GPS signals is insufficient, or completely unavailable. Electronic Publication     

The ADS40 Vaihingen/Enz geometric performance test

This paper presents the main results of a comprehensive ADS40 performance analysis conducted at the Vaihingen/Enz test field. As such it represents one example of an independent in-flight performance study for one of the new and commercially operational digital airborne camera systems. Based on a large number of well coordinated and defined object points, which served as independent check points, the absolute geometric accuracy of the ADS40 from true operational data has been verified. Empirical analysis of data from flying heights ranging from 1500 m to 4000 m proved the ADS40 geometric accuracy to be in the range of 1–2 μm at image scale for horizontal coordinates and 0.03–0.05‰ of the flying height for vertical components. This is fully within specification for airborne imaging.     

A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery

In the recent design of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission, the gravity gradients are defined in the gradiometer reference frame (GRF), which deviates from the actual flight direction (local orbit reference frame, LORF) by up to 3–4°. The main objective of this paper is to investigate the effect of uncertainties in the knowledge of the gradiometer orientation due to attitude reconstitution errors on the gravity field solution. In the framework of several numerical simulations, which are based on a realistic mission configuration, different scenarios are investigated, to provide the accuracy requirements of the orientation information. It turns out that orientation errors have to be seriously considered, because they may represent a significant error component of the gravity field solution. While in a realistic mission scenario (colored gradiometer noise) the gravity field solutions are quite insensitive to small orientation biases, random noise applied to the attitude information can have a considerable impact on the accuracy of the resolved gravity field models.     

A new approach for airborne vector gravimetry using GPS/INS

A new method for airborne vector gravimetry using GPS/INS has been developed and the results are presented. The new algorithm uses kinematic accelerations as updates instead of positions or velocities, and all calculations are performed in the inertial frame. Therefore, it is conceptually simpler, easier, more straightforward and computationally less expensive compared to the traditional approach in which the complex navigation equations should be integrated. Moreover, it is a unified approach for determining all three vector components, and no stochastic gravity modeling is required. This approach is based on analyzing the residuals from the Kalman filter of sensor errors, and further processing with wavenumber coefficient filterings is applied in case closely parallel tracks of data are available. An application to actual test-flight data is performed to test the validity of the new algorithm. The results yield an accuracy in the down component of about 3–4 mGal. Also, comparable results are obtained for the horizontal components with accuracies of about 6 mGal. The gravity modeling issue is discussed and alternative methods are presented, none of which improves on the original approach. Received: 18 April 2000 / Accepted: 14 August 2000     

GNSS相应坐标系之间的关系及其影响

不同的GNSS采用的坐标系定义几乎相近,但参考椭球及其坐标实现不同,这将影响多GNSS融合导航定位效果。根据各GNSS坐标系所采用参考椭球的基本常数,计算比较了不同坐标系参考椭球参数的差异;导出了相应的正常重力公式,比较了这些正常重力公式确定的正常重力值差异;最后分别从坐标系统的定义与实现两个方面分析了其对定位结果的影响。结果表明:1)GPS(BDS)与Galileo和GLONASS所使用的参考椭球引起正常重力差约为0.15和0.30 mgal;2)GPS与BDS,Galileo及GLONASS所使用参考椭球引起纬度分量最大差异约为0.1 mm,3 cm和3 cm,高程分量约为0.1 mm,0.5 m和1 m;3)各GNSS所使用坐标框架间转换参数引起的坐标变化达到厘米级。     

Comparison of the Graf sea gravimeter with the Vening Meinesz apparatus on board the submarineU.S.S. Becuna

Measurements of gravity were made on boardU.S.S. Becuna (SS 319) with a Vening Meinesz pendulum apparatus and a Graf sea gravimeter at approximately the same time. Comparison of data uncorrected for depth of submergence, E?tvos correction and second order effects of horizontal acceleration showed that there was a change related to time of observation. These corrections were not made as they would be the same for both sets of observations, and no provision had yet been made to take care of the horizontal accelerations for the Graf sea gravimeter. The variation with otime could be caused by instrumental drift or scale calibration. After removal of this effect by visually fitting the data trend with a straight line, there were three observations with large discrepancies, seventeen with discrepancies of 3–9 mgal and thirty-nine with discrepancies of 0–3 mgal, showing close approach to a statistical distribution. The three large discrepancies may be dismissed because of very bad depth control during the observations. The discrepancies 3–9 mgals are larger than expected and perhaps are attributable to depth control and inadequate observation time for the Graf sea gravimeter. It is concluded that the Graf sea gravimeter shows great promise for use on a submarine. An apparatus to take into account the horizontal acceleration effects must be added, and suitable drift characteristic obtained. Despite its advantages of ruggedness, ease of operation, ease of data reduction, reduced size and weight, many more comparisons of the Graf sea gravimeter with the Vening Meinesz pendulum equipment should be made before the latter are displaced Probably the Graf sea gravimeter will be usable on a stable platform on a surface vessel at least in calm sea states.     

Comparison of undulation difference accuracies using gravity anomalies and gravity disturbances

Errors are considered in the outer zone contribution to oceanic undulation differences as obtained from a set of potential coefficients complete to degree 180. It is assumed that the gravity data of the inner zone (a spherical cap), consisting of either gravity anomalies or gravity disturbances, has negligible error. This implies that error estimates of the total undulation difference are analyzed. If the potential coefficients are derived from a global field of 1°×1° mean anomalies accurate to ε_Δg=10 mgal, then for a cap radius of 10°, the undulation difference error (for separations between 100 km and 2000 km) ranges from 13 cm to 55 cm in the gravity anomaly case and from 6 cm to 36 cm in the gravity disturbance case. If ε_Δg is reduced to 1 mgal, these errors in both cases are less than 10 cm. In the absence of a spherical cap, both cases yield identical error estimates: about 68 cm if ε_Δg=1 mgal (for most separations) and ranging from 93 cm to 160 cm if ε_Δg=10 mgal. Introducing a perfect 30-degree reference field, the latter errors are reduced to about 110 cm for most separations.     

High-rate precise point positioning (PPP) to measure seismic wave motions: an experimental comparison of GPS PPP with inertial measurement units

High-rate GPS has been widely used to construct displacement waveforms and to invert for source parameters of earthquakes. Almost all works on internal and external evaluation of high-rate GPS accuracy are based on GPS relative positioning. We build an experimental platform to externally evaluate the accuracy of 50-Hz PPP displacement waveforms. Since the shake table allows motion in any of six degrees of freedom, we install an inertial measurement unit (IMU) to measure the attitude of the platform and transform the IMU displacements into the GPS coordinate system. The experimental results have shown that high-rate PPP can produce absolute horizontal displacement waveforms at the accuracy of 2–4 mm and absolute vertical displacement waveforms at the sub-centimeter level of accuracy within a short period of time. The significance of the experiments indicates that high-rate PPP is capable of detecting absolute seismic displacement waveforms at the same high accuracy as GPS relative positioning techniques, but requires no fixed datum station. We have also found a small scaling error of IMU and a small time offset of misalignment between high-rate PPP and IMU displacement waveforms by comparing the amplitudes of and cross-correlating both the displacement waveforms.     

Geometric test field calibration of digital photogrammetric sensors

Test field system calibration will be a fundamental part of the future photogrammetric production line. Accurate calibration and performance evaluations are necessary for fully assessing the stability and accuracy of digital sensing techniques. In this paper, a method of comprehensive geometric calibration in a test field has been developed and empirically tested using eight image blocks collected with three UltraCamD digital large format photogrammetric cameras. Permanent photogrammetric test fields form the basis of the method. Important components of the method are determination of system parameters, evaluation of systematic errors, and assessment of geometric accuracy. The results showed that UltraCamD images contained systematic deformations that could not be modeled with single lens additional parameter models. Good point determination accuracy was obtained despite the systematic errors; the typical accuracy was 2–3 μm in image space in the horizontal coordinates and 0.05–0.09‰ of the object distance in height. One of the cameras had significantly poorer performance. In the worst cases, the horizontal accuracy was 5 μm in image space and the height accuracy was 0.18‰ of the object distance. The analog cameras gave better results than the UltraCamD, but the development of appropriate mathematical models for UltraCamD as well as improvements in digital sensors may change the situation in the near future.     

Local geoid determination from airborne vector gravimetry

Methods are illustrated to compute the local geoid using the vertical and horizontal components of the gravity disturbance vector derived from an airborne GPS/inertial navigation system. The data were collected by the University of Calgary in a test area of the Canadian Rocky Mountains and consist of multiple parallel tracks and two crossing tracks of accelerometer and gyro measurements, as well as precise GPS positions. Both the boundary-value problem approach (Hotines integral) and the profiling approach (line integral) were applied to compute the disturbing potential at flight altitude. Cross-over adjustments with minimal control were investigated and utilized to remove error biases and trends in the estimated gravity disturbance components. Final estimation of the geoid from the vertical gravity disturbance included downward continuation of the disturbing potential with correction for intervening terrain masses. A comparison of geoid estimates to the Canadian Geoid 2000 (CGG2000) yielded an average standard deviation per track of 14 cm if they were derived from the vertical gravity disturbance (minimally controlled with a cross-over adjustment), and 10 cm if derived from the horizontal components (minimally controlled in part with a simulated cross-over adjustment). Downward continuation improved the estimates slightly by decreasing the average standard deviation by about 0.5 cm. The application of a wave correlation filter to both types of geoid estimates yielded significant improvement by decreasing the average standard deviation per track to 7.6 cm.     

The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram

 In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced. Received: 10 May 2000 / Accepted: 26 February 2001     

Interferometric Baseline Performance Estimations for Multistatic Synthetic Aperture Radar Configurations Derived from GRACE GPS Observations

Recent studies have demonstrated the usefulness of global positioning system (GPS) receivers for relative positioning of formation-flying satellites using dual-frequency carrier-phase observations. The accurate determination of distances or baselines between satellites flying in formation can provide significant benefits to a wide area of geodetic studies. For spaceborne radar interferometry in particular, such measurements will improve the accuracy of interferometric products such as digital elevation models (DEM) or surface deformation maps. The aim of this study is to analyze the impact of relative position errors on the interferometric baseline performance of multistatic synthetic aperture radar (SAR) satellites flying in such a formation. Based on accuracy results obtained from differential GPS (DGPS) observations between the twin gravity recovery and climate experiment (GRACE) satellites, baseline uncertainties are derived for three interferometric scenarios of a dedicated SAR mission. For cross-track interferometry in a bistatic operational mode, a mean 2D baseline error (1σ) of 1.4 mm is derived, whereas baseline estimates necessary for a monostatic acquisition mode with a 50 km along-track separation reveal a 2D uncertainty of approximately 1.7 mm. Absolute orbit solutions based on reduced dynamic orbit determination techniques using GRACE GPS code and carrier-phase data allows a repeat-pass baseline estimation with an accuracy down to 4 cm (2D 1σ). To assess the accuracy with respect to quality requirements of high-resolution DEMs, topographic height errors are derived from the estimated baseline uncertainties. Taking the monostatic pursuit flight configuration as the worst case for baseline performance, the analysis reveals that the induced low-frequency modulation (height bias) fulfills the relative vertical accuracy requirement (σ<1 m linear point-to-point error) according to the digital terrain elevation data level 3 (DTED-3) specifications for most of the baseline constellations. The use of a GPS-based reduced dynamic orbit determination technique improves the baseline performance for repeat-pass interferometry. The problem of fulfilling the DTED-3 horizontal accuracy requirements is still an issue to be investigated. DGPS can be used as an operational navigation tool for high-precision baseline estimation if a geodetic-grade dual-frequency spaceborne GPS receiver is assumed to be the primary instrument onboard the SAR satellites. The possibility of using only single-frequency receivers, however, requires further research effort.Deutsche Forschungsgemeinschaft (DFG) research fellow until Sept. 2004 at the Microwaves and Radar Institute, Deutsche Zentrum für Luft- und Raumfahrt (DLR) e.V., 82234 Weßling, Germany