A study of regional gravity field recovery from GOCE vertical gravity gradient data in the Auvergne test area using collocation

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that observes gravity gradients from the space, to be primarily used for the determination of high precision global gravity field models. However, the GOCE gradients, having a dense data distribution, may potentially provide better predictions of the regional gravity field than those obtained using a spherical harmonic Earth Geopotential Model (EGM). This is investigated in Auvergne test area using Least Squares Collocation (LSC) with GOCE vertical gravity gradient anomalies (T_zz), removing the long wavelength part from EGM2008 and the short wavelength part by residual terrain modelling (RTM). The results show that terrain effects on the vertical gravity gradient are significant at satellite altitude, reaching a level of 0.11 E?tv?s unit (E.U.) in the mountainous areas. Removing the RTM effects from GOCE T_zz leads to significant improvements on the LSC predictions of surface gravity anomalies and quasigeoid heights. Comparison with ground truth data shows that using LSC surface free air gravity anomalies and quasi-geoid heights are recovered from GOCE T_zz with standard deviations of 11 mGal and 18 cm, which is better than those obtained by using GOCE EGMs, demonstrating that information beyond the maximal degree of the GOCE EGMs is present. Investigation of using covariance functions created separately from GOCE T_zz and terrestrial free air gravity anomalies, suggests that both covariance functions give almost identical predictions. However, using covariance function obtained from GOCE T_zz has the effect that the predicted formal average error estimates are considerably larger than the standard deviations of predicted minus observed gravity anomalies. Therefore, GOCE T_zz should be used with caution to determine the covariance functions in areas where surface gravity anomalies are not available, if error estimates are needed.     

Using ground gravity to improve ice mass change estimation from GOCE gravity gradients in mid-west Greenland

Vertical gravity gradient anomalies from the Gravity and steady-state Ocean Circulation Explorer (GOCE) DIR-3 model have been used to determine gravity anomalies in mid-west Greenland by using Least-Squares Collocation (LSC) and the Reduced Point Mass (RPM) method. The two methods give nearly identical results. However, compared to LSC, the RPM method needs less computational time as the number of equations to be solved in LSC equals the number of observations. The advantage of the LSC, however, is the acquired error estimates. The observation periods are winter 2009 and summer 2012. In order to enhance the accuracy of the calculated gravity anomalies, ground gravity data from West Greenland is used over locations where the gravity change resulting from ice mass changes is negligible, i.e. over solid rock. In the period considered, the gravity anomaly change due to changes in ice mass varies from ?5 mGal to 4 mGal. It is negative over the outlet glacier Jacobshavn Isbræ, where the mass loss corresponds to a gravity change of approximately ?4 mGal. When using only GOCE vertical gravity gradients, the error estimates range from 5 mGal at the coast to 17 mGal over the ice sheet. Introducing the ground gravity data from West Greenland in the prediction reduces the errors to range from 2 to 10 mGal.     

Impact of GOCE Level 1b data reprocessing on GOCE-only and combined gravity field models

The reprocessing of Gravity field and steady-state Ocean Circulation Explorer (GOCE) Level 1b gradiometer and star tracker data applying upgraded processing methods leads to improved gravity gradient and attitude products. The impact of these enhanced products on GOCE-only and combined GOCE+GRACE (Gravity Recovery and Climate Experiment) gravity field models is analyzed in detail, based on a two-months data period of Nov. and Dec. 2009, and applying a rigorous gravity field solution of full normal equations. Gravity field models that are based only on GOCE gradiometer data benefit most, especially in the low to medium degree range of the harmonic spectrum, but also for specific groups of harmonic coefficients around order 16 and its integer multiples, related to the satellite’s revolution frequency. However, due to the fact that also (near-)sectorial coefficients are significantly improved up to high degrees (which is caused mainly by an enhanced second derivative in Y direction of the gravitational potential — V_YY), also combined gravity field models, including either GOCE orbit information or GRACE data, show improvements of more than 10% compared to the use of original gravity gradient data. Finally, the resulting gradiometry-only, GOCE-only and GOCE+GRACE global gravity field models have been externally validated by independent GPS/levelling observations in selected regions. In conclusion, it can be expected that several applications will benefit from the better quality of data and resulting GOCE and combined gravity field models.     

Determination of gravity anomaly at sea level from inversion of satellite gravity gradiometric data

Gravity gradients can be used to determine the local gravity field of the Earth. This paper investigates downward continuation of all elements of the disturbing gravitational tensor at satellite level using the second-order partial derivatives of the extended Stokes formula in the local-north oriented frame to determine the gravity anomaly at sea level. It considers the inversion of each gradient separately as well as their joint inversion. Numerical studies show that the gradients T_zz, T_xx, T_yy and T_xz have similar capability of being continued downward to sea level in the presence of white noise, while the gradient T_yz is considerably worse than the others. The bias-corrected joint inversion process shows the possibility of recovering the gravity anomaly with 1 mGal accuracy. Variance component estimation is also tested to update the observation weights in the joint inversion.     

GOCE, Satellite Gravimetry and Antarctic Mass Transports

In 2009 the European Space Agency satellite mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) was launched. Its objectives are the precise and detailed determination of the Earth’s gravity field and geoid. Its core instrument, a three axis gravitational gradiometer, measures the gravity gradient components V _xx , V _yy , V _zz and V _xz (second-order derivatives of the gravity potential V) with high precision and V _xy , V _yz with low precision, all in the instrument reference frame. The long wavelength gravity field is recovered from the orbit, measured by GPS (Global Positioning System). Characteristic elements of the mission are precise star tracking, a Sun-synchronous and very low (260 km) orbit, angular control by magnetic torquing and an extremely stiff and thermally stable instrument environment. GOCE is complementary to GRACE (Gravity Recovery and Climate Experiment), another satellite gravity mission, launched in 2002. While GRACE is designed to measure temporal gravity variations, albeit with limited spatial resolution, GOCE is aiming at maximum spatial resolution, at the expense of accuracy at large spatial scales. Thus, GOCE will not provide temporal variations but is tailored to the recovery of the fine scales of the stationary field. GRACE is very successful in delivering time series of large-scale mass changes of the Antarctic ice sheet, among other things. Currently, emphasis of respective GRACE analyses is on regional refinement and on changes of temporal trends. One of the challenges is the separation of ice mass changes from glacial isostatic adjustment. Already from a few months of GOCE data, detailed gravity gradients can be recovered. They are presented here for the area of Antarctica. As one application, GOCE gravity gradients are an important addition to the sparse gravity data of Antarctica. They will help studies of the crustal and lithospheric field. A second area of application is ocean circulation. The geoid surface from the gravity field model GOCO01S allows us now to generate rather detailed maps of the mean dynamic ocean topography and of geostrophic flow velocities in the region of the Antarctic Circumpolar Current.     

融合多源数据构建区域航空重力梯度扰动全张量

通过联合全球重力位模型(EGM2008)、航空重力扰动数据和剩余地形模型(RTM)数据,基于频谱域(二维FFT变换)和空间域(Stokes数值积分)算法对毛乌素测区GT-2A航空重力测量系统采集的空中测线后处理重力扰动数据进行解算,构建了该地区的航空重力梯度扰动全张量.(1)残余航空重力扰动延拓结果表明:残余航空重力扰动经向下延拓至大地水准面,再向上延拓至航空高度后与原数据差值的标准差为1.0078 mGal,考虑边缘效应后,内缩计算范围得到的差值标准差减小至0.1269 mGal.(2)基于残余重力扰动数据(原航空高度数据及向下延拓数据),通过不同方案解算得到的梯度扰动结果表明:两种方案得到的研究区域重力梯度扰动各分量之差的最大标准差为6.4798E(Γ_(yz)分量),最小标准差为2.6968E(Γ_(xy)分量),内缩计算范围后得到的差值标准差最大值为1.8307E(Γ_(zz)分量),最小值为0.7223E(Γ_(yz)分量).本文的思路和方法可为未来我国自主构建航空重力梯度标定场提供参考.     

Application of Gravity Gradients in the Process of GOCE Orbit Determination

The possibility of improving the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) mission satellite orbit using gravity gradient observations was investigated. The orbit improvement is performed by a dedicated software package, called the Orbital Computation System (OCS), which is based on the classical least squares method. The corrections to the initial satellite state vector components are estimated in an iterative process, using dynamic models describing gravitational perturbations. An important component implemented in the OCS package is the 8th order Cowell numerical integration procedure, which directly generates the satellite orbit. Taking into account the real and simulated GOCE gravity gradients, different variants of the solution of the orbit improvement process were obtained. The improved orbits were compared to the GOCE reference orbits (Precise Science Orbits for the GOCE satellite provided by the European Space Agency) using the root mean squares (RMS) of the differences between the satellite positions in these orbits. The comparison between the improved orbits and the reference orbits was performed with respect to the inertial reference frame (IRF) at J2000.0 epoch. The RMS values for the solutions based on the real gravity gradient measurements are at a level of hundreds of kilometers and more. This means that orbit improvement using the real gravity gradients is ineffective. However, all solutions using simulated gravity gradients have RMS values below the threshold determined by the RMS values for the computed orbits (without the improvement). The most promising results were achieved when short orbital arcs with lengths up to tens of minutes were improved. For these short arcs, the RMS values reach the level of centimeters, which is close to the accuracy of the Precise Science Orbit for the GOCE satellite. Additional research has provided requirements for efficient orbit improvement in terms of the accuracy and spectral content of the measured gravity gradients.     

基于时空域混合法利用Kaula正则化精确和快速解算GOCE地球重力场

为了研究卫星重力梯度技术对中高频地球重力场反演精度的影响,本文基于时空域混合法,利用Kaula正则化反演了250阶GOCE地球重力场.模拟结果表明:第一,时空域混合法是精确和快速求解高阶地球重力场的有效方法;第二,Kaula正则化是降低正规阵病态性的重要方法;第三,基于改进的预处理共轭梯度迭代法可快速求解大型线性方程组...     

Conforming Falcon‡ gravity and the global gravity anomaly

Gravity derived only from airborne gravity gradient measurements with a normal error distribution will have an error that increases with wavelength. It is straightforward in principle to use sparsely sampled regional gravimeter data to provide the long wavelength information, thereby conforming the derived gravity to the regional gravity. Regional surface or airborne gravimeter data are not always available and can be difficult and expensive to collect in many of the areas where an airborne gravity gradiometer survey is flown. However the recent release by the Danish National Space Centre of the DNSC08 global gravity anomaly data has provided regional gravity data for the entire earth of adequate quality for this purpose. Studies over three areas, including comparisons with ground, marine and airborne gravimetry, demonstrate the validity of this approach. Future improvements in global gravity anomaly data are expected, particularly as the product from the recently launched Gravity field and steady‐state Ocean Circulation Explorer (GOCE) satellite becomes available and these will lead directly to an improvement in the very wide bandwidth gravity available after conforming gravity derived from gravity gradiometry with the global gravity.     

Structure and State of Stress of the Chilean Subduction Zone from Terrestrial and Satellite-Derived Gravity and Gravity Gradient Data

It is well known that the quality of gravity modelling of the Earth’s lithosphere is heavily dependent on the limited number of available terrestrial gravity data. More recently, however, interest has grown within the geoscientific community to utilise the homogeneously measured satellite gravity and gravity gradient data for lithospheric scale modelling. Here, we present an interdisciplinary approach to determine the state of stress and rate of deformation in the Central Andean subduction system. We employed gravity data from terrestrial, satellite-based and combined sources using multiple methods to constrain stress, strain and gravitational potential energy (GPE). Well-constrained 3D density models, which were partly optimised using the combined regional gravity model IMOSAGA01C (Hosse et al. in Surv Geophys, 2014, this issue), were used as bases for the computation of stress anomalies on the top of the subducting oceanic Nazca plate and GPE relative to the base of the lithosphere. The geometries and physical parameters of the 3D density models were used for the computation of stresses and uplift rates in the dynamic modelling. The stress distributions, as derived from the static and dynamic modelling, reveal distinct positive anomalies of up to 80 MPa along the coastal Jurassic batholith belt. The anomalies correlate well with major seismicity in the shallow parts of the subduction system. Moreover, the pattern of stress distributions in the Andean convergent zone varies both along the north–south and west–east directions, suggesting that the continental fore-arc is highly segmented. Estimates of GPE show that the high Central Andes might be in a state of horizontal deviatoric tension. Models of gravity gradients from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite mission were used to compute Bouguer-like gradient anomalies at 8 km above sea level. The analysis suggests that data from GOCE add significant value to the interpretation of lithospheric structures, given that the appropriate topographic correction is applied.     

Combined Regional Gravity Model of the Andean Convergent Subduction Zone and Its Application to Crustal Density Modelling in Active Plate Margins

The Central Andean subduction system is one of the most active geological structures on Earth. Although there have been a few previous studies, the structure and dynamics of the system are still not well understood. In the present study, we determine a combined regional gravity model of the Andean convergent subduction region for constraining lithospheric models. After a thorough validation and cleaning of the terrestrial gravity and height databases, the method of Least Squares Collocation was applied to consistently combine terrestrial and satellite gravity data, putting much emphasis on the stochastic modelling of the individual data components. As a result, we computed the first high-resolution regional gravity model of the study region that includes GOCE satellite gravity information. The inclusion of GOCE is an essential distinction from the independent global gravity model EGM2008. Validation against EGM2008 reveals that our regional solution is very consistent in regions where terrestrial gravity data are available, but shows systematic differences in areas with terrestrial data gaps. Artefacts in the EGM2008 of up to 150 mGal could be identified. The new combined regional model benefits from the very homogeneous error characteristics and accuracy of GOCE gravity data in the long-to-medium wavelengths down to 80–100 km. Reliable density modelling became possible also in the region of Central Andes, which lacks terrestrial gravity data. Finally, density models were adapted to fit the new regional gravity field solution. The results clearly demonstrate the capabilities of GOCE to better constrain lithospheric models.     

SUBSURFACE GRAVITY MEASUREMENTS AND DENSITY MAPPING IN THE MAILARAM COPPER MINES,ANDHRA PRADESH,INDIA*

Gravity measurements were made in the Mailaram copper mines, Andhra Pradesh. The observations were distributed between the two shafts situated about 220 m apart and in the three levels up to a maximum depth of 100 m. Assuming a normal free-air gradient, average densities for the three layers were determined as 2.631, 2.604, and 2.823 g cm^-3, respectively. Upon incorporating the weighted mean density values from measurements on samples, the free-air gradients were found to be 0.315 mGal m^-1 for the second layer (i.e. between the first and second levels) and 0.2978 mGal m^-1 for the third layer (i.e. between the second and third levels). The density variation map obtained from the gravity data, the deduced anomalies, the weighted mean density values from measurements on rock samples, and the varying free-air coefficients all suggest correspondence with the concentration of ore lodes.     

捷联式航空重力测量算法比较

捷联式航空重力测量系统与平台式系统相比具有体积小、重量轻、功耗低等许多优点,近些年来取得了显著的研究进展.本文给出了捷联式航空重力测量的两种算法模型:捷联式惯性标量重力测量(SISG)和旋转不变式标量重力测量(RISG)模型,并对其误差模型作了初步讨论.利用我国首套捷联式航空重力仪SGA-WZ01在某海域的部分试验数据,对两种算法模型进行了比较分析,表明其差值之标准差对于200s的滤波长度小于0.5mGal.同时,利用两组重复测线数据估算了不同滤波尺度下的两种算法的内符合精度,表明SISG算法略优于RISG算法.对于200s和300s的滤波长度,SISG的内符合精度分别为1.06mGal和0.80mGal.     

Airborne gravimetry using a strapped-down LaCoste and Romberg air/sea gravity meter system: a feasibility study

Marine gravimeters mounted on stabilized platforms are commonly used in aircraft to perform airborne gravity measurements. The role of the stabilized platform is to level the sensor mechanically, whatever the aircraft attitude. However, this compensation is generally insufficient due to the sensitivity of modern gravity sensors. Correcting the offlevel error requires that an offlevel correction calculated from positioning data be added to gravimeter measurements, which complicates not only the processing, but also the assessment of precision and resolution. This paper is a feasibility study describing the levelling of a completely strapped‐down LaCoste and Romberg gravimeter for airborne gravimetry operation, by means of GPS positioning data. It focuses on the calculation of the sensor offlevel correction needed for the complete gravity data processing. The precision of the offlevel correction that can be achieved, in terms of GPS data precision and gravity wavelengths, is theoretically studied and estimated using the gravity and GPS data acquired during the Alpine Swiss French airborne gravity survey carried out in 1998 over the French Western Alps. While a 1 cm precision of GPS‐determined baseline coordinates is sufficient to achieve a 5 mGal precision of the offlevel correction, we maintain that this precision has to reach 1 mm to ensure a 1 mGal precision of the offlevel correction at any wavelength. Without a stabilized platform, the onboard instrumentation becomes significantly lighter. Furthermore, the correction for the offlevel error is straightforward and calculated only from GPS data. Thus, the precision and the resolution of airborne gravity surveys should be estimated with a better accuracy.     

A Wavelet-Based Assessment of Topographic-Isostatic Reductions for GOCE Gravity Gradients

Gravity gradient measurements from ESA’s satellite mission Gravity field and steady-state Ocean Circulation Explorer (GOCE) contain significant high- and mid-frequency signal components, which are primarily caused by the attraction of the Earth’s topographic and isostatic masses. In order to mitigate the resulting numerical instability of a harmonic downward continuation, the observed gradients can be smoothed with respect to topographic-isostatic effects using a remove–compute–restore technique. For this reason, topographic-isostatic reductions are calculated by forward modeling that employs the advanced Rock–Water–Ice methodology. The basis of this approach is a three-layer decomposition of the topography with variable density values and a modified Airy–Heiskanen isostatic concept incorporating a depth model of the Mohorovi?i? discontinuity. Moreover, tesseroid bodies are utilized for mass discretization and arranged on an ellipsoidal reference surface. To evaluate the degree of smoothing via topographic-isostatic reduction of GOCE gravity gradients, a wavelet-based assessment is presented in this paper and compared with statistical inferences in the space domain. Using the Morlet wavelet, continuous wavelet transforms are applied to measured GOCE gravity gradients before and after reducing topographic-isostatic signals. By analyzing a representative data set in the Himalayan region, an employment of the reductions leads to significantly smoothed gradients. In addition, smoothing effects that are invisible in the space domain can be detected in wavelet scalograms, making a wavelet-based spectral analysis a powerful tool.     

旋转式航空重力梯度仪动态测量误差传递模型与事后误差补偿

航空重力梯度测量技术可快速、高效地完成面积性重力梯度数据采集工作,在矿产资源勘查、军事目标探测等诸多科学领域具有广泛的应用.而航空重力梯度动态测量误差补偿方法是重力梯度动态测量数据处理中的一项重要工作.本文首先对旋转式重力梯度仪误差传递机理进行了定量分析,在综合考虑重力梯度仪系统非理想因素相互作用的情况下,建立了多种非...     

Data requirements for a 5-mm quasi-geoid in the Netherlands

We assess the surface gravity data requirements for a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf in terms of omission and commission errors. The omission error critically depends on the roughness of the topography and bathymetry. For the Netherlands continental shelf, Central and Northern Netherlands, the omission error is well described by the model 0.32d mm, where d is the data spacing in km. For the more hilly Southern Netherlands, the omission error model is 0.92d mm. The commission error depends on the kernel modification, the data spacing, and the data accuracy. When using the spheroidal Stokes kernel, it is well described by 0.277 d σ_Δg mm, where σ_Δg is the noise standard deviation of surface gravity data in mGal. An upper bound of the commission error of the state-of-the-art satellite-only gravity model GOCO05S over the Netherlands is e^{0.03676L–11.419} mm, where L is the maximum degree up to which this model is used. Only if this model is truncated at a sufficiently low degree, e.g., at degree 100, its contribution to the total commission error can be neglected. We determine the total error as the sum of commission and omission errors. Hence, to realize a 5-mm quasi-geoid model for the Netherlands mainland and continental shelf, a data spacing of 3.5 km is needed when assuming a noise standard deviation of 1.5 mGal for surface gravity data. The currently available land-based gravity data fulfill this requirement. This does not apply to the situation at sea, where the density of the shipboard gravity data and the accuracy of the radar altimeter-derived data do not allow the realization of a 5-mm quasi-geoid model.     

Accurate Establishment of Error Models for the Satellite Gravity Gradiometry Recovery and Requirements Analysis for the Future GOCE Follow-On Mission

Firstly, the new single and combined error models applied to estimate the cumulative geoid height error are efficiently produced by the dominating error sources consisting of the gravity gradient of the satellite-equipped gradiometer and the orbital position of the space-borne GPS/GLONASS receiver using the power spectral principle. At degree 250, the cumulative geoid height error is 1.769 × 10^?1 m based on the new combined error model, which preferably accords with a recovery accuracy of 1.760 ×10^?1 m from the GOCE-only Earth gravity field model GO_CONS_GCF_2_TIM_R2 released in Germany. Therefore, the new combined error model of the cumulative geoid height is correct and reliable in this study. Secondly, the requirements analysis for the future GOCE Follow-On satellite system is carried out in respect of the preferred design of the matching measurement accuracy of key payloads comprising the gravity gradient and orbital position and the optimal selection of the orbital altitude of the satellite. We recommend the gravity gradient with an accuracy of 10^?13?10^?15 /s², the orbital position with a precision of 1-0.1 cm and the orbital altitude of 200-250 km in the future GOCE Follow-On mission.     

巴颜喀拉地块东部及邻区重力均衡与岩石圈有效弹性厚度

基于SIO(Scripps Institute of Oceanography)最新全球重力和高程模型,计算了巴颜喀拉地块东部及邻区的布格重力异常、均衡重力异常、岩石圈有效弹性厚度及荷载比.结合大地热流、地震速度结构、地震活动和断裂构造分布等,分析了地壳均衡状态和岩石圈有效弹性厚度、地质构造单元间的差异及与地震活动的相关性特征.研究结果表明,该区域布格重力变化范围约为-500～0mGal(1mGal=10~(-5)m·s~(-2),下同),在巴颜喀拉块体东部区域形成弧形重力梯度带,近年来的中强地震活动频发于该梯度带不同部位,应与其应力依次释放有关;均衡重力异常结果表明,其变化范围约为-80～+100mGal,且大部分区域处于±20mGal以内的被认为处于重力均衡的状态,重力非均衡(正或负)多出现于块体边界带附近,地震多发生在靠近块体边界的均衡重力异常(正或负,主要为正)区域内;巴颜喀拉地块东部及邻区岩石圈有效弹性厚度(T_e)为10～65km,不同构造单元之间T_e空间分布差异明显,较低的T_e值出现在龙门山构造带附近,T_e值为20km左右,岩石圈荷载加载比为0.5～0.8,表明现今的岩石圈挠曲状态主要由莫霍面加载形成.进一步分析表明,巴颜喀拉地块东部挤压增生与横向流动同时发生,是造成该区域地震发生与重力均衡异常高值重合、岩石圈有效弹性厚度和大地热流值较低的主要原因.本文获得的地壳均衡特征及岩石圈有效弹性强度结果,加深了对巴颜喀拉东部及邻区岩石圈构造演化过程的认识.