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     

Flight test results from a strapdown airborne gravity system

In June 1995, a flight test was carried out over the Rocky Mountains to assess the accuracy of airborne gravity for geoid determination. The gravity system consisted of a strapdown inertial navigation system (INS), two GPS receivers with zero baseline on the airplane and multiple GPS master stations on the ground, and a data logging system. To the best of our knowledge, this was the first time that a strapdown INS has been used for airborne gravimetry. The test was designed to assess repeatability as well as accuracy of airborne gravimetry in a highly variable gravity field. An east-west profile of 250 km across the Rocky Mountains was chosen and four flights over the same ground track were made. The flying altitude was about 5.5km, i.e., between 2.5 and 5.0km above ground, and the average flying speed was about 430km/h. This corresponds to a spatial resolution (half wavelength of cutoff frequency) of 5.07.0km when using filter lengths between 90 and 120s. This resolution is sufficient for geoid determination, but may not satisfy other applications of airborne gravimetry. The evaluation of the internal and external accuracy is based on repeated flights and comparison with upward continued ground gravity using a detailed terrain model. Gravity results from repeated flight lines show that the standard deviation between flights is about 2mGal for a single profile and a filter length of 120s, and about 3mGal for a filter length of 90s. The standard deviation of the difference between airborne gravity upward continued ground gravity is about 3mGal for both filter lengths. A critical discussion of these results and how they relate to the different transfer functions applied, is given in the paper. Two different mathematical approaches to airborne scalar gravimetry are applied and compared, namely strapdown inertial scalar gravimetry (SISG) and rotation invariant scalar gravimetry (RISG). Results show a significantly better performance of the SISG approach for a strapdown INS of this accuracy class. Because of major differences in the error model of the two approaches, the RISG method can be used as an effective reliability check of the SISG method. A spectral analysis of the residual errors of the flight profiles indicates that a relative geoid accuracy of 23cm over distances of 200km (0.1 ppm) can be achieved by this method. Since these results present a first data analysis, it is expected that further improvements are possible as more refined modelling is applied. Received: 19 August 1996 / Accepted: 12 May 1997     

GAINS中重力传感器信息的扰动改正

重力辅助惯性导航系统GAINS(Gravity Aided Inertial Navigation System)是利用地球物理特征信息──重力来完成水下运动载体的辅助惯性导航与定位。为实现重力匹配以校正惯性导航随时间累积的误差,首先必须对重力传感器输出信息进行扰动改正。分析了水下运动状态下重力传感器受到的各种重力扰动,如垂直扰动加速度、水平扰动加速度以及厄特弗斯效应影响所产生的原因,研究了扰动误差模型与INS导航精度之间的关系,并通过计算,提出了可直接以INS输出数据而无需其它外部有源导航信息进行扰动分离的方法。     

GAINS中重力传感器信息的扰动改正

重力辅助惯性导航系统GAINS(Gravity Aided Inertial Navigation System)是利用地球物理特征信息重力来完成水下运动载体的辅助惯性导航与定位.为实现重力匹配以校正惯性导航随时间累积的误差,首先必须对重力传感器输出信息进行扰动改正.分析了水下运动状态下重力传感器受到的各种重力扰动,如垂直扰动加速度、水平扰动加速度以及厄特弗斯效应影响所产生的原因,研究了扰动误差模型与INS导航精度之间的关系,并通过计算,提出了可直接以INS输出数据而无需其它外部有源导航信息进行扰动分离的方法.     

Strapdown INS/DGPS airborne gravimetry tests in the Gulf of Mexico

Combining data from a Strapdown Inertial Navigation System and a Differential Global Positioning System (SINS/DGPS) has shown great promise in estimating gravity on moving platforms. Previous studies on a ground-vehicle system obtained 1–3 mGal precision with 2 km spatial resolution. High-accuracy Inertial Measurement Units (IMU) and cm-level positioning solutions are very important in obtaining mGal-level gravity disturbance estimates. However, these ideal configurations are not always available or achievable. Because the noise level in the SINS/DGPS gravimetric system generally decreases with an increase of speed and altitude of the platform, the stringent constraints on the IMU and GPS may be relieved in the airborne scenario. This paper presents an investigation of one navigation-grade and one tactical-grade IMU for the possibility of low-cost INS/GPS airborne gravimetry. We use the data collected during the Gravity-Lidar Study of 2006 (GLS06), which contains aerogravity, GPS, and INS along the northern coastline of the Gulf of Mexico. The gravity disturbance estimates from the navigation-grade IMU show 0.5–3.2 mGal precision compared with the onboard gravimeter’s measurements and better than 3 mGal precision compared with the upward continued surface control data. Due to relatively large (240 s) smoothing window, the results have about 34 km along-track resolution. But the gravity estimates from the tactical-grade IMU have much poorer precisions. Nonetheless, useful contributions from the tactical-grade IMU could be extracted for longer wavelengths.     

单站GPS确定航空重力测量载体运动加速度的方法与结果分析

飞机运动加速度的测量精度是制约航空重力测量技术发展的主要障碍之一。相较于传统动态差分GPS（differential GPS,DGPS）技术,所提方法采用单站测量模式,无需布设地面基准站。首先通过相位历元间差分解得高精度历元间位移序列,然后结合泰勒一阶中心差分获得载体加速度,重点分析了卫星轨道和卫星钟差对加速度估计的影响,结果表明,不同卫星轨道产品对加速度估计影响较小,而卫星钟差采样率对加速度估计的影响很大。结合中国陕西省境内的GT-2A航空重力测量系统飞行实测数据,利用单站法解算的加速度联合重力和姿态数据解算重力扰动结果与DGPS解算的重力扰动符合较好,当滤波长度为100 s时,两者互差优于1.0 mGal。重力扰动交叉点不符值网平差后,均方根（root mean square,RMS）为1.13 mGal。与地面重力实测值比较的结果表明,所提方法与DGPS方法在精度上基本一致,说明单站法标量航空重力测量是可行的。     

A comparison of stable platform and strapdown airborne gravity

To date, operational airborne gravity results have been obtained using either a damped two-axis stable platform gravimeter system such as the LaCoste and Romberg (LCR) S-model marine gravimeter or a strapdown inertial navigation system (INS), showing comparable accuracies. In June 1998 three flight tests were undertaken which tested an LCR gravimeter and a strapdown INS gravity system side by side. To the authors' knowledge, this was the first time such a comparison flight was undertaken. The flights occurred in Disko Bay, off the west coast of Greenland. Several of the flight lines were partly flown along existing shipborne gravity profiles to allow for an independent source of comparison of the results. The results and analysis of these flight tests are presented. The measurement method and error models for both the stable platform and strapdown INS gravity systems are presented and contrasted. An intercomparison of gravity estimates from both systems is given, along with a comparison of the individual estimates with existing shipborne gravity profiles. The results of the flight tests show that the gravity estimates from the two systems agree at the 2–3 mGal level, after the removal of a linear bias. This is near the combined noise level of the two systems. It appears that a combination of both systems would provide an ideal airborne gravity survey system, combining the excellent bias stability of the LCR gravimeter with the higher dynamic range and increased spatial resolution of the strapdown INS. Received: 3 June 1999 / Accepted: 30 November 1999     

Airborne vector gravimetry using precise,position-aided inertial measurement units

Vector gravimetry using a precise inertial navigation system continually updated with external position data, for example using GPS, is studied with respect to two problems. The first concerns the attitude accuracy requirement for horizontal gravity component estimation. With covariance analyses in the space and frequency domains it is argued that with relatively stable uncompensated gyro drift, the short-wavelength gravity vector can be estimated without the aid of external attitude updates. The second problem concerns the state-space estimation of the gravity signal where considerable approximations must be assumed in the gravity model in order to take advantage of the ensemble error estimation afforded by the Kalman filter technique. Gauss-Markov models for the gravity field are specially designed to reflect the attenuation of the signal at a specific altitude and the omission of the long-wavelength components from the estimation. With medium accuracy INS/GPS systems, the horizontal components of gravity with wavelengths shorter than 250 km should be estimable to an accuracy of 4–6 mgal (µg); while high accuracy systems should yield an improvement to 1–2 mgal.     

Comparing the Kalman filter with a Monte Carlo-based artificial neural network in the INS/GPS vector gravimetric system

Rigorous physical and mathematical analysis has been intensively developed to obtain the gravity disturbance vector from the inertial navigation system and the global positioning system. However, the combination of the observation noise and the systematic INS errors make it very challenging to accurately and efficiently describe the dynamics of the system with rigorous equations. Thus, the accuracy of the gravity disturbance estimates, especially in the horizontal components, is limited by the insufficient error models. To overcome the difficulty of directly modeling the systematic errors with exact mathematical equations, a Monte Carlo based artificial neural network is successfully applied in the moving base gravimetric system. The computation results show significant improvement in the precision of all components of the gravity disturbance estimates.     

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     

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     

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.     

扰动重力水平分量对惯导系统的位置误差影响

研究了不同运动状态下扰动重力水平分量（HDG）对高精度惯导系统（inertial navigation system,INS）的位置误差影响。首先推导了HDG对INS误差影响的状态空间方程,进而推导出3种运动条件下INS位置误差与HDG之间的解析关系式,设计了基于惯导解算求解上述影响的方法。在匀速运动条件下,分别通过解析式与惯导解算两种方法计算了相同HDG引起的INS位置误差。解析式计算结果表明,±80 mGal（1 mGal=10-5 m/s2）范围内变化的HDG约可引起最大约3 000 m的INS位置误差;对两种方法计算结果的比较显示,所得INS位置误差的量级与变化情况基本一致,两组结果验证了各自方法的有效性。     

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.     

GPS单点测速的误差分析及精度评价

首先从理论和实测数据模拟两方面分析了SA取消后各类误差源对GPS测速的影响,推导并计算了GPS单点测速可能达到的精度水平。然后用静态数据模拟动态测速试验和实测动态数据测速与同步高精度惯导测速的动态试验进行验证。结果表明,采用载波相位导出的多普勒观测值使用静态数据模拟动态测速,其精度可以达到mm/s级;用接收机输出的多普勒观测值进行测速时,其精度为cm/s级。在动态测速试验中,GPS单点测速方法（即多普勒观测值测速与导出多普勒观测值测速）间的符合精度达到cm/s级,与高精度的惯导测速结果的符合精度为dm/s级,而且和运动载体的动态条件（如加速度和加速度变化率的大小）具有很强的相关性。     

A combined algorithm of improving INS error modeling and sensor measurements for accurate INS/GPS navigation

Although the integrated system of a differential global positioning system (DGPS) and an inertial navigation system (INS) had been widely used in many geodetic navigation applications, it has sometimes a major limitation. This limitation is associated with the frequent occurrence of DGPS outages caused by GPS signal blockages in certain situations (urban areas, high trees, tunnels, etc.). In the standard mechanization of INS/DGPS navigation, the DGPS is used for positioning while the INS is used for attitude determination. In case of GPS signal blockages, positioning is provided using the INS instead of the GPS until satellite signals are obtained again with sufficient accuracy. Since the INS has a very short-time accuracy, the accuracy of the provided INS navigation parameters during these periods decreases with time. However, the obtained accuracy in these cases is totally dependent on the INS error model and on the quality of the INS sensor data. Therefore, enhanced navigation parameters could be obtained during DGPS outages if better inertial error models are implemented and better quality inertial measurements are used. In this paper, it will be shown that better INS error models are obtained using autoregressive processes for modeling inertial sensor errors instead of Gauss–Markov processes that are implemented in most of the current inertial systems and, on the other hand, that the quality of inertial data is improved using wavelet multi-resolution techniques. The above two methods are discussed and then a combined algorithm of both techniques is applied. The performance of each method as well as of the combined algorithm is analyzed using land-vehicle INS/DGPS data with induced DGPS outage periods. In addition to the considerable navigation accuracy improvement obtained from each single method, the results showed that the combined algorithm is better than both methods by more than 30%.     

航空重力向上延拓的外部精度评估

分别采用基于梯度、基于泊松积分和基于快速傅里叶变换（FFT）的地面重力向上延拓方案,并提出交叉检验方法估计地面重力数据误差及其空中误差传播,对毛乌素测区GT-2A航空重力测量系统采集的空中测线数据进行外符合精度评价。对比结果表明：地面重力格网插值误差和代表性误差对空中点的影响达到0.66~0.92 mGal（1 Gal=1×10^-2 m/s²）,航空重力数据误差估计必须扣除这一影响;基于泊松积分和基于FFT的地面重力向上延拓方法能够客观评价航空重力观测值的外符合精度,二者表现相当;扣除地面重力误差影响后,在包含残余边界效应的情况下,毛乌素测区GT-2A航空重力空中测线重力扰动的外符合精度优于1.42 mGal。     

Effect of the SRTM global DEM on the determination of a high-resolution geoid model: a case study in Iran

Any errors in digital elevation models (DEMs) will introduce errors directly in gravity anomalies and geoid models when used in interpolating Bouguer gravity anomalies. Errors are also propagated into the geoid model by the topographic and downward continuation (DWC) corrections in the application of Stokes’s formula. The effects of these errors are assessed by the evaluation of the absolute accuracy of nine independent DEMs for the Iran region. It is shown that the improvement in using the high-resolution Shuttle Radar Topography Mission (SRTM) data versus previously available DEMs in gridding of gravity anomalies, terrain corrections and DWC effects for the geoid model are significant. Based on the Iranian GPS/levelling network data, we estimate the absolute vertical accuracy of the SRTM in Iran to be 6.5 m, which is much better than the estimated global accuracy of the SRTM (say 16 m). Hence, this DEM has a comparable accuracy to a current photogrammetric high-resolution DEM of Iran under development. We also found very large differences between the GLOBE and SRTM models on the range of −750 to 550 m. This difference causes an error in the range of −160 to 140 mGal in interpolating surface gravity anomalies and −60 to 60 mGal in simple Bouguer anomaly correction terms. In the view of geoid heights, we found large differences between the use of GLOBE and SRTM DEMs, in the range of −1.1 to 1 m for the study area. The terrain correction of the geoid model at selected GPS/levelling points only differs by 3 cm for these two DEMs.     

Differentiation for High-Precision GPS Velocity and Acceleration Determination

Accurate estimates of the velocity and acceleration of a platform are often needed in high dynamic positioning, airborne gravimetry, and geophysics. In turn, differentiation of GPS signals is a crucial process for obtaining these estimates. It is important in the measurement domain where, for example, the phase measurements are used along with their instantaneous derivative (Doppler) to estimate position and velocity. It is also important in postprocessing, where acceleration is usually estimated by differentiating estimates of position and velocity. Various methods of differentiating a signal can have very different effects on the resulting derivative, and their suitability varies from situation to situation. These comments set the stage for the investigations in this article. The objective is twofold: (1) to carry out a comprehensive study of possible differentiation methods, characterizing each in the frequency domain; and (2) to use real data to demonstrate each of these methods in both of the measurement and position domains, in conditions of variable, high, or unknown dynamics. Examples are given using real GPS data in both the measurement domain and in the position and velocity domain. The appropriate differentiator is used in several cases of varying dynamics to derive a Doppler signal from carrier phase measurements (rather than using the raw Doppler generated by the receiver). In the statistic case, it is seen that the accuracy of velocity estimates can be improved from 4.0 mm/s to 0.7 mm/s by using the correct filter. In conditions of medium dynamics experienced in an airborne gravity survey, it is demonstrated that accelerations as the 2–4 mGal level (1 mGal = 0.00001 m/s²) can be obtained at the required filtering periods. Finally, a precision motion table is used to show that when using the correct filter, velocity estimates under high dynamics can be improved by an order of magnitude to 27.0 mm/s. ? 1999 John Wiley & Sons, Inc.     

Cross-Coupling Correction for LaCoste ＆ Romberg Airborne Gravimeter

Abstract The cross-coupling corrections for the LaCoste ＆ Romberg airborne gravimeter are computed as a linear combination of 5 so-called cross-coupling monitors. The weight factors （coefficients） determined from marine gravity data by the factory are obviously not optimal for airborne application. These coefficients are recalibrated by minimizing the difference between airborne data and upward continued surface data （external calibration） and by minimizing the errors at line crossings （internal calibration） respectively. An integrating method to recalibrate the above-mentioned coefficients and the beam scale factor simultaneously is also presented. Experimental results show that the systemic errors in the airborne gravity anomalies can be greatly reduced by using any of the recalibrated coefficients. For example, the systemic error is reduced from 4.8 mGal to 1.8 mGal in Datong test.