联合多代卫星测高和多源重力数据的局部大地水准面精化方法

本文研究了基于泊松小波径向基函数融合多代卫星测高及多源重力数据精化大地水准面模型的方法.分别以沿轨垂线偏差和大地水准面高高差作为卫星测高观测量,研究了使用不同类型测高数据对于大地水准面建模精度的影响.针对全球潮汐模型在浅水区域及部分开阔海域精度较低的问题,引入局部潮汐模型研究了不同潮汐模型对于大地水准面的影响.数值分析表明:相比于使用沿轨垂线偏差作为测高观测量,基于沿轨大地水准面高高差解算得到的大地水准面模型的精度更高,特别是在海域区域,其精度提高了2.3cm.由于使用沿轨大地水准面高高差作为测高观测量削弱了潮汐模型长波误差的影响,采用不同潮汐模型对大地水准面解算的影响较小.总体而言,船载重力及测高观测数据在海洋重力场的确定中呈现互补性关系,联合两类重力场观测量可以提高局部重力场的建模精度.     

Gravity Anomaly Map over The Black Sea Using Corrected Sea Surface Heights from ERS1, ERS2 and Topex/Poseidon Satellite Altimetry Missions

Approximately nine-year data from ERS1, ERS2 and TOPEX/POSEIDON (T/P) satellite altimetry missions have been used for the recovery of gravity anomalies over the Black Sea. The Corrected Sea Surface Height product of Aviso/Altimetry has been proven to be homogeneous after a cross-over adjustment. The Least Squares Collocation method was applied in a so-called remove-restore procedure. The residual geoid heights, obtained by subtracting EGM96 geoid heights from cross-over adjusted sea surface heights, were inverted to recover residual gravity anomalies in a grid structure over the Black Sea. Finally, EGM96 free air gravity anomalies were added to the predicted residual gravity anomalies to obtain the free air gravity anomalies. In order to check the consistency with respect to an external source, these computed free air gravity anomalies were compared to ship gravity observations, and to alternative satellite altimetry derived gravity anomalies. Comparisons with the observed gravity data yielded that external consistency of the gravity anomalies computed in this study is about 3 mGal for individual ship tracks. Overall external consistency in the test area is 4.8 mGal. Comparison with other satellite altimetry derived gravity anomalies presented a good agreement.     

Optimal Model for Geoid Determination from Airborne Gravity

Two different approaches for transformation of airborne gravity disturbances, derived from gravity observations at low-elevation flying platforms, into geoidal undulations are formulated, tested and discussed in this contribution. Their mathematical models are based on Green's integral equations. They are in these two approaches defined at two different levels and also applied in a mutually reversed order. While one of these approaches corresponds to the classical method commonly applied in processing of ground gravity data, the other approach represents a new method for processing of gravity data in geoid determination that is unique to airborne gravimetry. Although theoretically equivalent in the continuous sense, both approaches are tested numerically for possible numerical advantages, especially due to the inverse of discretized Fredholm's integral equation of the first kind applied on different data. High-frequency synthetic gravity data burdened by the 2-mGal random noise, that are expected from current airborne gravity systems, are used for numerical testing. The results show that both approaches can deliver for the given data a comparable cm-level accuracy of the geoidal undulations. The new approach has, however, significantly higher computational efficiency. It would be thus recommended for real life geoid computations. Additional errors related to regularization of gravity data and the geoid, and to accuracy of the reference field, that would further deteriorate the quality of estimated geoidal undulations, are not considered in this study.     

Modelling the fine-structure of the geoid: Methods,data requirements and some results

The requirements for precise geoid models on local and regional scales have increased in recent years, primarily due to the ongoing developments in height determination by GPS on land, but also due to oceanographic requirements in using satellite altimetry for recovering dynamic sea-surface topography. Suitable methods for geoid computations from gravity data include Stokes integration, FFT methods, and least-squares collocation. Especially the FFT methods are efficient in handling large amounts of gravity data, and new variants of the methods taking earth curvature rigorously into account provide attractive methods for obtaining continental-scale, high-resolution geoid models. The accuracy of such models may be from 2–5 cm locally, to 50–100 cm on regional scales, depending on gravity data coverage, long wave-length gravity field errors, and datum problems. When approaching the cm-level geoid basic geoid definition questions (geoid or quasigeoid?) become very significant, especially in rugged areas. In the paper the geoid modelling methods and problems are reviewed, and some investigations on local data requirements for cm-level geoid prediction are presented. Some actual results are presented from Scandinavia, where a recent regional high-resolution geoid model yields apparent accuracies of 2–10 cm over GPS baselines of 50 to 2000 km.     

Contribution of satellite altimetry data in the geophysical investigation of the Red Sea Region, Egypt

Successful development of geodetic satellite missions has aroused new interest in determining global and regional gravity field based on satellite data. Satellite altimetry data enable direct determination of the geoid over sea regions. In Egypt, where land and marine geophysical data are inadequate because of rough topography and economic reasons, the use of satellite altimetry data is of special importance. The northern Red Sea region has been selected as a site for case study of the current research, after applying spectral analysis to reveal near-surface structure, the residual geoid of the studied region shows a good correlation with the known geologic features. Moreover, satellite-based gravity data enhance small-scale features and agrees well with land and marine gravity data. Thus, geoid undulation and satellite gravity data can be a complementary source of data to determine near-surface and deep structures.     

Ocean Surface Geostrophic Circulation Climatology and Annual Variations Inferred from Satellite Altimetry and GOCE Gravity Data

We have studied, for the first time, variations in absolute surface geostrophic currents (SGC) using satellite data only. The proposed approach combines 18 years’ altimetry data, which provide reliable measurements of absolute sea level (ASL), with a gravity field and steady-state ocean circulation explorer geoid model to obtain dynamic topography, and achieves unprecedented precision and accuracy. Our proposal overcomes the main limitations of existing approaches based solely on altimetry data (which suffer from lack of an independent reference for derivation of ASL maps), and approximations based on in-situ data (which are characterized by a sparse and inhomogeneous coverage in time and space). Features of annual variations of SGC are also addressed. As a result of our study we provide new absolute SGC climatology in the form of a 52-week data set of surface current fields, gridded at quarter degree longitude and latitude resolution and resolving spatial scales as short as 140 km. For presentation, this data set is averaged monthly and the results, presented as monthly climatology, are compared with climatology based on in-situ observations from drifter data.     

应用多种来源重力异常编制中国海陆及邻区空间重力异常图及重力场解读

本文通过分析陆地实测空间重力异常数据、海洋船载测量空间重力异常数据、卫星测高重力异常,布格重力异常数据、EGM2008地球重力模型数据等多种来源数据的性质和精度,并对相关数据进行对比,研究了编制1:500万中国海陆空间重力异常图的数据使用方案和技术方法.在地形较为平坦、实测数据分布均匀的陆区,使用实测数据,在地形复杂,实测数据稀少以及没有实测数据的陆区或岛屿,利用布格重力异常反推空间异常的方法合成平均空间重力数据,西藏地区的数据对比实验证明合成平均空间重力异常数据是一种有效的数据补充.利用三观测列方差分解法在南海地区对船载测量空间重力数据和美国SS系列及丹麦DNSC08GRA卫星重力数据进行了方差分解计算,结果表明不同来源的卫星测高重力数据具有很大的一致性,数据精度较以往有了很大的提高.海区空间重力数据使用原则是在船载重力测量数据校准下,全面使用卫星测高重力数据进行编图.海陆过渡区的异常处理应以EGM2008地球重力模型重力场为基准参考场,实现海陆异常平缓过渡,无缝连接.对中国海陆空间重力异常场进行了小波变换处理,对空间重力异常场进行了解读,勾画出三横四竖的一级重力梯级带及其所围限的8个一级重力异常区,并划分了二级重力异常区和梯级带,为块体构造学体系中大地构造格架的建立提供了地球物理证据.     

基于有限元方法的陆海大地水准面衔接

大陆上用重力数据和GPS水准数据确定(似)大地水准面,海洋上用卫星测高数据确定(似)大地水准面.由于沿海地区和近岸海域往往缺少完好的重力数据,近岸海域卫星测高数据质量相对较差,两类大地水准面在陆海相接区域精度偏低且存在拼合差.纯几何方法拟合陆海局部区域大地水准面,不能顾及大地水准面的物理特性,拟合结果不稳定.顾及到大地水准面的物理特性,依据其在局部所应满足的数学物理方程,拟合陆海局部区域大地水准面问题,转化为Laplace第一边值问题.讨论了有限元法衔接陆海局部区域大地水准面的数学思想,给出了相应的数学模型.     

Numerical Studies on the Harmonic Downward Continuation of Band-Limited Airborne Gravity

In this paper, the numerical stability and efficiency of methods of harmonic downward continuation from flying altitudes are treated for sampled gravity field data. The problem is first formulated in its continuous form, i.e. as the inverse solution of the spherical Dirichlet problem, and is then approximated by Gaussian quadrature to yield a finite system of linear equations. The numerical stability of this system is investigated for both error-free gravity data and for the noisy and band-limited gravity measurements usually obtained from airborne gravity surveys. It can be shown that the system becomes ill-conditioned, once the ratio between flying altitude and data sampling rate exceeds a certain limit. It can also be shown that noisy measurements tend to generate a solution that is practically useless, long before the system becomes ill-conditioned. Therefore, instead of treating the general solution of the discrete downward continuation problem, the more modest question is studied, for which range of flying altitudes and sampling rates, the numerical solution of the discrete linear system can be considered as practically useful. Practically useful will be defined heuristically as of sufficient accuracy and stability to satisfy the requirements of the user. The question will be investigated for the specific application of geoid computation from gravity data sampled at flying altitudes. In this case, a stable solution with a standard deviation of a few centimeters is required. Typical flight parameters are heights of 2–6 km, a minimum half-wavelength resolution of 2 km, and data noise between 0.5 and 1.5 mGal. Different methods of geoid determination, different solution techniques for the resulting systems of linear equations, and different minimization principles will be compared. As a result operational parameters will be defined which, for a given noise level, will result in a geoid accuracy of a few centimeters for the estimated band-limited gravity field spectrum.     

21世纪重力与磁法勘探的展望

对21世纪重力与磁法勘探的仪器，数据处理技术，解释理论与方法，应用领域等方面的发展方向进行了分析与展望，发展航空标量，矢量，梯度重力测量和航空全梯度磁力测量，三分量磁力测量，提高综合信息采集能力；开展卫星重磁测量，综合卫星，航空，地面重磁测量资料研究地球结构与构造；发展高精度数据处理技术；重磁异常弱信号的提取，不同深度重磁异常的划分，低纬底变倾角化磁极以及位场面延拓；发展复杂条件下三维重磁场多参数综合反演可视化技术以及快速自动反演技术；探索磁性多参数的应用新领域，充分发挥磁法在环境污染调查中的作用并开拓应用新领域。     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

中国海及邻域重力场特征及其构造解释

利用中国海及邻域 2’× 2’由Seaset,Geosat,ERS 1及TOPEX/POSEIDON等卫星测高资料解算的自由空气重力异常 ,结合ETOPO5全球海底地形资料计算了中国海及邻域布格重力异常 ,并反演了得到了中国海及邻域地壳厚度 .经对重力异常特征进行分析 ,讨论了重力异常与大地构造及其活动的相关性 .     

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.     

Recovery of marine gravity anomalies from ERS1, ERS2 and ENVISAT satellite altimetry data for geoid computations over Norway

Free-Air Anomalies (FAA) for the Norwegian marine area including some parts of the North Sea, the Norwegian Sea and the Barents Sea are computed from satellite altimetry data. A total of 84 cycles of ERS2 along-track data, 25 cycles of ENVISAT along-track data and high density ERS1 data during its geodetic mission are used. The new geopotential model from the Gravity Recovery and Climate Experiment (GRACE) mission, GGM02S (Tapely et al., 2005) is used to compute the long wavelength contributions of the geoid and the FAA. To correct data for mean dynamic topography, the available Levitus climatology model (Levitus and Boyer, 1994) is used. Corrected data are then used to compute along-track gradients in each cycle-pass to suppress the orbital and the atmospheric errors below the noise level of the altimeter. Resulted gradients are then stacked and the east-west and the north-south components of the deflection of verticals are computed where ascending and descending tracks meet each other. Finally, the inverse Vening-Meinesz formula is implemented on the gridded deflections to compute FAA. Results are then compared with available marine and airborne data. Standard deviations of ± 4.301 and ± 6.159 mGal in comparison with airborne and marine FAA were achieved. Thereafter, the derived anomalies are combined with marine and airborne FAA together with the land FAA to compute a fine resolution geoid for Norway and the surrounding marine areas. This geoid is evaluated over sea and land with the synthetic geoid (the geoid derived from the mean sea surface by subtracting the mean dynamic topography) and Global Positioning System (GPS)-levelling and the standard deviations of the differences are ± 20.9 and ± 12.8 cm respectively. ali.soltanpour@ntnu.no, hossein.nahavandchi@ntnu.no, kourosh.ghazavi@ntnu.no     

海域航空重力快速构建区域大地水准面

我国在海域开展了大规模的航空重力勘探,这些资料对构建高精度大地水准面具有重要价值.基于此,本文提出一种利用海域航空重力测量数据快速构建大地水准面的方法.该方法基于移去-恢复法思想,利用位场最小曲率方法对航空重力数据进行高精度向下延拓并获取相应的扰动位,实现航空重力测量快速构建海域大地水准面.与斯托克斯积分计算相比,采用了处理效率更高的频率域位场转换,解决了向下延拓及垂向积分时航空重力异常数据空白及扩边问题,具有较高的位场转换精度.本文应用EGM2008模拟航空重力数据进行模型验证,计算结果与其给出的水准面的精度相当;同时,也选取GRAV-D计划的航空重力数据进行实际验证,计算结果与xGEOID18B水准面模型精度基本一致.模型验证和实际应用验证了本方法的实用性.     

Regional gravity field modeling based on rectangular harmonic analysis

Regional gravity field modeling with high-precision and high-resolution is one of the most important scientific objectives in geodesy,and can provide fundamental information for geophysics,geodynamics,seismology,and mineral exploration.Rectangular harmonic analysis(RHA)is proposed for regional gravity field modeling in this paper.By solving the Laplace’s equation of gravitational potential in local Cartesian coordinate system,the rectangular harmonic expansions of disturbing potential,gravity anomaly,gravity disturbance,geoid undulation and deflection of the vertical are derived,and so are the formula for signal degree variance and error degree variance of the rectangular harmonic coefficients(RHC).We also present the mathematical model and detailed algorithm for the solution of RHC using RHA from gravity observations.In order to reduce the edge effects caused by periodic continuation in RHA,we propose the strategy of extending the size of computation domain.The RHA-based modeling method is validated by conducting numerical experiments based on simulated ground and airborne gravity data that are generated from geopotential model EGM2008 and contaminated by Gauss white noise with standard deviation of 2 mGal.The accuracy of the 2.5′×2.5′geoid undulations computed from ground and airborne gravity data is 1 and 1.4cm,respectively.The standard error of the gravity disturbances that downward continued from the flight height of 4 km to the geoid is only 3.1 mGal.Numerical results confirm that RHA is able to provide a reliable and accurate regional gravity field model,which may be a new option for the representation of the fine structure of regional gravity field.     

用GRACE卫星跟踪数据反演地球重力场

利用141天GRACE卫星观测资料,包括K波段、星载加速度和卫星轨道数据,反演了80阶地球重力场模型IGGGRACE01S,该模型在半波长为500km的空间分辨率上,确定大地水准面的精度约为0012m,中长波(<80阶)精度优于重力卫星发射以前研制的重力场模型. 与EIGEN_GRACE02S、EIGEN_CHAMP03S和EGM96模型的位系数相比,该模型系数最接近于EIGEN_GRACE02S,与另两个模型差异较大. 比较几种模型确定的全球重力异常和大地水准面起伏,结果发现IGGGRACE01S与EIGEN_GRACE02S模型的计算结果比较接近,与EGM96模型结果差异较大,差别较大地区主要在南极等地区. 对于中国大陆,比较IGGGRACE01S模型（前72阶）计算的重力异常和NIMA重力异常数据（25°×25°网格）,两者之间的标准偏差为48mGal.     

Monitoring Sea Level in the Coastal Zone with Satellite Altimetry and Tide Gauges

We examine the issue of sustained measurements of sea level in the coastal zone, first by summarizing the long-term observations from tide gauges, then showing how those are now complemented by improved satellite altimetry products in the coastal ocean. We present some of the progresses in coastal altimetry, both from dedicated reprocessing of the radar waveforms and from the development of improved corrections for the atmospheric effects. This trend towards better altimetric data at the coast comes also from technological innovations such as Ka-band altimetry and SAR altimetry, and we discuss the advantages deriving from the AltiKa Ka-band altimeter and the SIRAL altimeter on CryoSat-2 that can be operated in SAR mode. A case study along the UK coast demonstrates the good agreement between coastal altimetry and tide gauge observations, with root mean square differences as low as 4 cm at many stations, allowing the characterization of the annual cycle of sea level along the UK coasts. Finally, we examine the evolution of the sea level trend from the open to the coastal ocean along the western coast of Africa, comparing standard and coastally improved products. Different products give different sea level trend profiles, so the recommendation is that additional efforts are needed to study sea level trends in the coastal zone from past and present satellite altimeters. Further improvements are expected from more refined processing and screening of data, but in particular from the constant improvements in the geophysical corrections.     

Geoid and Sea Surface Topography Derived from ERS-1 Altimeter Data by the Adjoint Method

A method for splitting sea surface height measurements from satellite altimetry into geoid undulations and sea surface topography is presented. The method is based on a combination of the information from altimeter data and a dynamic sea surface height model. The model consists of geoid undulations and a quasi-geostrophic model for expressing the sea surface topography. The goal is the estimation of those values of the parameters of the sea surface height model that provide a least-squares fit of the model to the data. The solution is accomplished by the adjoint method which makes use of the adjoint model for computing the gradient of the cost function of the least-squares adjustment and an optimization algorithm for obtaining improved parameters. The estimation is applied to the North Atlantic. ERS-1 altimeter data of the year 1993 are used. The resulting geoid agrees well with the geoid of the EGM96 gravity model.