Airborne gravimetry used in precise geoid computations by ring integration

Two detailed geoids have been computed in the region of North Jutland. The first computation used marine data in the offshore areas. For the second computation the marine data set was replaced by the sparser airborne gravity data resulting from the AGMASCO campaign of September 1996. The results of comparisons of the geoid heights at on-shore geometric control showed that the geoid heights computed from the airborne gravity data matched in precision those computed using the marine data, supporting the view that airborne techniques have enormous potential for mapping those unsurveyed areas between the land-based data and the off-shore marine or altimetrically derived data. Received: 7 July 1997 / Accepted: 22 April 1998     

航空重力测量技术及其应用

近十几年来,航空重力测量技术的研究和应用日趋活跃,业已成为地球重力场研究中最为热门的领域之一。本文介绍了国外航空重力测量技术及其应用情况,概述了我国首套航空重力测量系统CHAGS的试验、作业和应用现状,对航空重力测量发展趋势作了展望。     

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     

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.     

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.     

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     

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.     

Airborne LaCoste &; Romberg gravimetry: a space domain approach

This paper introduces a new approach to reduce the airborne gravity data acquired by a LaCoste &; Romberg (L&;R) air/sea gravimeter, or other similar gravimeters. The acceleration exerted on the gravimeter is the sum of gravity and the vertical and Eötvös accelerations of the aircraft. The L&;R gravimeter outputs are: (1) the beam position, (2) the spring tension and (3) the cross coupling. Vertical and Eötvös accelerations are computed from GPS-derived aircraft positions. However, the vertical perturbing acceleration sensed by the gravimeter is not the same as the one sensed by the aircraft (via GPS). A determination of the aircraft-to-sensor transfer function is necessary. The second-order differential equation of the motion of the gravimeter’s beam mixes all the input and output parameters of the gravimeter. Conventionally, low-pass filtering in the frequency domain is used to extract the gravity signal, the filter being applied to each flight-line individually. By transforming the differential equation into an integral equation and by introducing related covariance matrices, we develop a new filtering method based on a least-squares approach that is able to take into account, in one stage, the data corresponding to all flight-lines. The a posteriori covariance matrix of the estimated gravity signal is an internal criterion of the precision of the method. As an example, we estimate the gravity values along the flight-lines from an airborne gravity survey over the Alps and introduce an a priori covariance matrix of the gravity disturbances from a global geopotential model. This matrix is used to regularize the ill-posed Fredholm integral equation introduced in this paper.     

A comparison and analysis of airborne gravimetry results from two strapdown inertial/DGPS systems

In September 1996 the University of Calgary tested a combination of strapdown inertial navigation systems and differential global positioning system (DGPS) receivers for their suitability to determine gravity at aircraft flying altitudes. The purpose of this test was to investigate the long-term accuracy and repeatability of the system, as well as its potential for geoid and vertical gradient of gravity determination. The test took place during a 3-day period in the Canadian Rocky Mountains over a single 100 × 100 km area which was flown with 10-km line spacing. Two flights were done at 4350 m in E–W and N–S profile directions, respectively, and one at 7300 m with E–W profiles. Two strapdown inertial systems, the Honeywell LASEREF III and the Litton-101 Flagship, were flown side by side. Comparison of the system estimates with an upward-continued reference showed root-mean-square (RMS) agreement at the level of 3.5 mGal for 90- and 120-s filter lengths. The LASEREF III, however, performed significantly better than the Litton 101 for shorter filtering periods of 30 and 60 s. A comparison between the two systems results in an RMS agreement of 2.8 and 2.3 mGal for the 90- and 120-s filters. The better agreement between the two systems is mainly due to the fact that the upward-continued reference has not been filtered identically to the system gravity disturbance estimates. Additional low-frequency differences seem to point to an error in the upward-continued reference. Finally, an analysis of crossover points between flight days for the LASEREF III shows a standard deviation of 1.6 mGal, which is near the noise level of the INS and GPS data. Further improvements to the system are possible, and some ideas for future work are briefly presented. Received: 17 March 1998 / Accepted: 1 February 1999     

高动态GPS观测数据的仿真研究

主要介绍了高动态GPS观测数据的仿真情况,分析了GPS观测值仿真的意义、原理、及流程,重点讨论了各个模块系统的功能及采用的数学模型。利用编制的仿真软件仿真了武汉地区的GPS观测值,并与仿真的导航电文进行单点定位计算,算例结果表明本文方法的正确性和可靠性,能满足不同层次人员对GPS仿真观测值的需要。     

An evaluation of orthometric height accuracy using bore hole gravimetry

Errors are introduced in orthometric height computations by the use of standard formulas to estimate mean gravity along the plumb line. Direct measurements of gravity between the Earth’s surface and sea level from bore hole gravimetry were used to determine the magnitude of these errors. For the seven cases studied, errors in orthometric height, due to the use of the Helmert method for computing mean gravity along the plumb line, were generally small (<2 cm). However, in one instance the error was substantial, being9.6 cm. The results verified the general validity of the Poincaré-Prey approach to estimation of gravity along the plumb line and demonstrated that the suggestion byVanicek (1980) that the air gradient is more appropriate is incorrect. With sufficient topographic information to compute terrain corrections, and density estimates from surface gravity, errors in mean gravity along the plumb line should contribute no more than 3cm to orthometric height computation.     

多系统融合精密单点定位在机载雷达电力线巡检中的应用

通过无人机搭载激光雷达(LiDAR)测量系统进行电力线巡检实验,提出了一种基于多系统精密单点定位(PPP)的机载LiDAR电力线巡检方法,重点分析了基站差分、单系统PPP、多系统融合PPP三种处理策略生成POS数据的方法,并利用江苏省中北部采集的机载雷达数据及人工布设的控制点结果进行验证. 实验结果表明:三种方法得到的POS数据均方根误差(RMSE)均在0.016 m以内,与单系统PPP相比,多系统PPP在高程方向提高了12%,在东(E)方向提高了21%,在北(N)方向提高了17%,文中提出的方法生成三维(3D)点云数据精度较高,生成的点云数据精度满足精细电力线巡检的要求.      

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.     

Regional adjustment of inertial gravity disturbance vector measurements by optimal two-dimensional smoothing

A two-dimensional signal processing algorithm is developed to obtain smoothed estimates of the gravity disturbance vector from vector measurements obtained by an inertial surveying system. The method differs from a conventional least squares regional adjustment of such measurements in that it accommodates a signal model in the smoothing process. Using principles from the physical theory of geodesy, it is shown that for a local region on the surface of the earth, an appropriate signal model is obtained by applying the two-dimensional Laplacian operator to a function representing the surface disturbance potential and equating the result to spatial white noise. The model of the vector measurement is the three orthogonal spatial derivatives of a three dimensional disturbance potential evaluated at the surface contaminated by additive white noise. The problem of simultaneous smoothing of all the gravity disturbance measurements from all survey traverses in the region is solved by representing the surface disturbance potential by a two-dimensional Karhunen-Loeve expansion that makes no specific reference to either the geometry or the ordering of the parameter space, thereby making no assumptions of causality, stationarity or isotropy. The problem of estimating the gravity anomaly and the two vertical deflection components reduces to estimating the Karhunen-Loeve coefficients which are uncorrelated and rapidly converging. Simulation results as well as smoothing of actual gravity disturbance vector measurements obtained by the U.S. Army Engineer Topographic Laboratories (USAETL) with the Rapid Geodetic Survey System (RGSS) at the White Sands Missile Range (WSMR) are presented in the paper. An analysis of these results shows that the optimal two-dimensional smoother obtains a performance benefit relative to conventional regional least squares by a factor of 2 and a benefit relative to single-traverse smoothed results by a factor of 4.

---

Sommaire Un algorithme de traitement du signal en deux dimensions est développé pour obtenir une estimation lissée du vecteur de la perturbation de la pesanteur à partir des mesures de vecteur obtenues avec un système d’arpentage inertiel. La méthode diffère d’un compensation régionale conventionnelle par moindres carrés de telles mesures, par le fait qu’elle contient un modèle du signal dans le processus de compensation. En s’appuyant sur les principes de la géodésie physique, il est montré que pour une région locale de la surface de la terre, un modèle approprié du signal est obtenu en appliquant l’opérateur à deux dimensions de Laplace à une fonction représentant le champ perturbateur à la surface de la terre et égalisant le résultat à un bruit blanc spatial. Le modèle du vecteur de mesures est défini par les trois dérivées spatiales de la fonction tridimensionnelle du potentiel perturbateur évaluées à la surface et contaminées par un bruit blanc. Le problème du lissage de toutes les mesures de gravité perturbatrice obtenues à partir des polygonales effectuées est résolu en représentant le potentiel perturbateur à la surface à l’aide d’un développement Karhunen-Loeve à deux dimensions qui ne fait aucunement référence à la géométrie ou à l’ordre des paramètres; ceci prévient toute dépendance spatiale des points adjacents. Le problème de l’estimation de l’anomalie de la gravité et des deux composantes de la déviation de la verticale se réduit à celle des coefficients Karhunen-Loeve qui sont non-corrélés et convergent rapidement. Les résultats de simulation aussi bien que le lissage des données du vecteur de perturbation de la pesanteur foumi par l’U.S. Army Engineer Topographics Labs (USAETL) sont présentés. L’analyse de ces résultats montre que le lissage optimal à deux dimensions améliore les résultats par un facteur 2 comparés aux résultats d’une compensation régionale par moindres carrés, et par un facteur 4 comparés aux résultats lissés d’une simple traverse.

---

---

Presented at the Second International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

Multi-GNSS precise point positioning with raw single-frequency and dual-frequency measurement models

GPS Solutions - The emergence of multiple satellite navigation systems, including BDS, Galileo, modernized GPS, and GLONASS, brings great opportunities and challenges for precise point positioning...     

轨道几何状态惯性检测系统零速姿态修正方法

为提高轨道几何状态惯性检测系统姿态精度,提出一种零速姿态修正方法,利用轨道几何状态惯性检测系统载体系横向及垂向速度为0的先验条件,构造Kalman滤波横向及垂向观测量。选用HT-50M型激光陀螺仪和JN-06A型石英挠性加速度计组成惯性导航系统,并进行现场检测试验。结果表明:在轨检仪连续工作300m情况下,零速姿态修正方法能够有效抑制惯性导航系统姿态误差增长,轨道检测精度提高2倍。     

Real time estimation of position and the gravity vector with an inertial survey system

Based on sine functions for fitting horizontal velocity errors in an inertial survey system and a step or slope function approximation of the vertical deflection between stops, a positioning accuracy of 10 m (CEP) can be achieved with a car-borne system – a modified -21 aircraft navigation system, with a 10 min travel period between Zero Velocity Update Times (ZUPTs) and a total 1.5 hour survey time. A rough estimation of the vertical deflection can also be expected in real time out of the approach. Also a forward-backward-forward run is presented which demonstrates determination of the gravity anomaly. Received 10 March 1995; Accepted 8 July 1996     

Different units of measurement of carotenoids estimation in cotton using hyperspectral indices and partial least square regression

This paper examines the use of canopy reflectance for different units of measurements of carotenoids estimation. Field spectral measurements were collected over cotton in different intensive field campaigns organized during the growing seasons of 2010 and 2011. Three units of measurement were evaluated carotenoids expressed as a mass per unit soil surface area (g/m²), a mass per unit leaf area (μg/cm²), and a mass per unit fresh leaf weight (mg/g), respectively. Four methods were compared to retrieve amount of carotenoids: stepwise multiple linear regression (SMLR), published spectral indices, band combination indices, and partial least square regression (PLSR). Results show that maximum sensitivity of reflectance to variation in different units of measurement of carotenoids was found in the green region at 515–550 nm, and at 715 nm and 750 nm regions in the far-red wavelengths. The predictive accuracies of Car (g/m²), Car (μg/cm²) and Car (mg/g) were tested on a validation data set and the results show that the highest R² values between estimations and observations were 0.468 for Car (g/m²), 0.563 for Car (μg/cm²), and 0.456 for Car (mg/g), with relative root mean square error (RMSE%, RMSE/mean) of 48.72%, 22.07% and 21.07%, respectively. Compared to Car (g/m²) and Car (mg/g), the model performance indices for Car (μg/cm²) show a high degree of consistency among the R² values and RMSE% and MAE% values. Further comparison were performed among the estimation accuracies of different unit carotenoids and among the different approaches used in the study by a paired-t-test. The results indicate that although the best estimation results for Car (μg/cm²) and Car (mg/g) were both obtained based on PLSR, they can be estimated by all four adopted methods without significant differences (P > 0.1). Whereas for Car (g/m²), the best estimation results were obtained based on published vegetation indices CIred-edge, which were significantly better than the estimation results based on SMLR (P < 0.000). In summary, the results of this study show that even the carotenoids expressed on concentration (mg/g) or content (μg/cm²) basis at leaf level can be estimated with the same prediction accuracies to the carotenoids expressed as a mass per unit surface area (g/m²) at canopy level using reflectance measurement at canopy level.