全球及局部海洋扰动重力反演的快速解析方法

从经典边值问题理论及球谐函数理论出发,在空域推导获得了由大地水准面高以及垂线偏差计算扰动重力的解析计算公式,为利用卫星测高数据反演海洋扰动重力提供了理论基础。针对全球海洋区域和局部海洋区域的扰动重力反演,在前人已有工作基础上,提出了改进的基于一维FFT的精确快速算法,保证了计算结果与原解析方法完全一致,且计算速度提高约20倍。该算法在提高计算效率的同时避免了由于引入FFT而产生的混叠、边缘效应问题,而且对观测数据的序列长度没有硬性要求,使得应用更加灵活。利用EGM2008地球重力场模型分别生成了2.5'分辨率大地水准面高数据和垂线偏差数据,按照本文提出的改进方法(采用全球积分计算)分别反演获得了全球及局部海洋区域的扰动重力。经比较分析,由大地水准面和垂线偏差分别反演获得的扰动重力其差异在0.8×10^-5 m/s²以内,这说明两种反演方法是基本一致的,但在数据包含系统误差的情况下,由垂线偏差反演扰动重力具有一定优势。     

Ellipsoidal geoid computation

Modern geoid computation uses a global gravity model, such as EGM96, as a third component in a remove–restore process. The classical approach uses only two: the reference ellipsoid and a geometrical model representing the topography. The rationale for all three components is reviewed, drawing attention to the much smaller precision now needed when transforming residual gravity anomalies. It is shown that all ellipsoidal effects needed for geoid computation with millimetric accuracy are automatically included provided that the free air anomaly and geoid are calculated correctly from the global model. Both must be consistent with an ellipsoidal Earth and with the treatment of observed gravity data. Further ellipsoidal corrections are then negligible. Precise formulae are developed for the geoid height and the free air anomaly using a global gravity model, given as spherical harmonic coefficients. Although only linear in the anomalous potential, these formulae are otherwise exact for an ellipsoidal reference Earth—they involve closed analytical functions of the eccentricity (and the Earths spin rate), rather than a truncated power series in e². They are evaluated using EGM96 and give ellipsoidal corrections to the conventional free air anomaly ranging from –0.84 to +1.14 mGal, both extremes occurring in Tibet. The geoid error corresponding to these differences is dominated by longer wavelengths, so extrema occur elsewhere, rising to +766 mm south of India and falling to –594 mm over New Guinea. At short wavelengths, the difference between ellipsoidal corrections based only on EGM96 and those derived from detailed local gravity data for the North Sea geoid GEONZ97 has a standard deviation of only 3.3 mm. However, the long-wavelength components missed by the local computation reach 300 mm and have a significant slope. In Australia, for example, such a slope would amount to a 600-mm rise from Perth to Cairns.     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry,altimetry and GOCE data: evaluation by simulation

The determination of local geoid models has traditionally been carried out on land and at sea using gravity anomaly and satellite altimetry data, while it will be aided by the data expected from satellite missions such as those from the Gravity field and steady-state ocean circulation explorer (GOCE). To assess the performance of heterogeneous data combination to local geoid determination, simulated data for the central Mediterranean Sea are analyzed. These data include marine and land gravity anomalies, altimetric sea surface heights, and GOCE observations processed with the space-wise approach. A spectral analysis of the aforementioned data shows their complementary character. GOCE data cover long wavelengths and account for the lack of such information from gravity anomalies. This is exploited for the estimation of local covariance function models, where it is seen that models computed with GOCE data and gravity anomaly empirical covariance functions perform better than models computed without GOCE data. The geoid is estimated by different data combinations and the results show that GOCE data improve the solutions for areas covered poorly with other data types, while also accounting for any long wavelength errors of the adopted reference model that exist even when the ground gravity data are dense. At sea, the altimetric data provide the dominant geoid information. However, the geoid accuracy is sensitive to orbit calibration errors and unmodeled sea surface topography (SST) effects. If such effects are present, the combination of GOCE and gravity anomaly data can improve the geoid accuracy. The present work also presents results from simulations for the recovery of the stationary SST, which show that the combination of geoid heights obtained from a spherical harmonic geopotential model derived from GOCE with satellite altimetry data can provide SST models with some centimeters of error. However, combining data from GOCE with gravity anomalies in a collocation approach can result in the estimation of a higher resolution geoid, more suitable for high resolution mean dynamic SST modeling. Such simulations can be performed toward the development and evaluation of SST recovery methods.     

似大地水准面的误差分析与抑制技术

大地水准面误差分析与精度评定是局部重力场逼近技术的重要组成部分,是大地水准面精化工程外业方案优化、算法设计和工程质量评价的基本依据。本文分别从地面重力数据误差和局部重力场算法两个方面,分析cm级大地水准面误差的影响特性,提出重力数据误差与大地水准面精度之间普遍适用的规律,推荐一种GPS水准和地面重力数据联合平差的精度评定方法,结合实例和模拟计算分析,介绍大地水准面误差分析与误差抑制方法。     

关于重力大地水准面计算精度问题

在众多影响重力大地水准面计算精度的因素中,计算模型误差及地面观测数据误差是其中的两个主要误差源。本文从实用角度出发,详细探讨了重力大地水准面计算公式选择问题;并通过实际数值计算,研究确定了地面观测数据计算精度、密度及覆盖范围等参数指标;最后以180阶次位系数定义地球重力场,对重力大地水准面计算精度进行了实际检验。     

A strategy for determining the regional geoid by combining limited ground data with satellite-based global geopotential and topographical models: a case study of Iran

The computation of regional gravimetric geoid models with reasonable accuracy, in developing countries, with sparse data is a difficult task that needs great care. Here we investigate the procedure for gathering, evaluating and combining different data for the determination of a gravimetric geoid model for Iran, where limited ground gravity data are available. Heterogeneous data, including gravity anomalies, the high-resolution Shuttle Radar Topography Mission global digital terrain model and different global geopotential models including recently published Gravity Recovery and Climate Experiment models, are combined through least-squares modification of the Stokes formula. The new gravimetric geoid model, IRG04, agrees considerably better with GPS/levelling than any of the other recent local geoid model in the area. Its RMS fit with GPS/levelling is 0.27 m and 3.8 ppm in the absolute and relative view, respectively. The relative accuracy of IRG04 is four times better than the most recently published global and regional geoid models available in this area. This progress shows the practical potential of the method of least-squares modification of Stokes’s formula in combination with heterogeneous data for regional geoid determination     

利用GOCE模拟观测反演重力场的Torus法

在介绍Torus方法反演地球重力场模型的基本原理和方法的基础上,基于圆环面上均匀分布的卫星引力梯度模拟观测值解算了200阶次的地球重力场模型,在无误差情况下,Torus方法解算模型的阶误差RMS小于10-16,验证了该方法的严密性。利用61dGOCE卫星轨道上无误差的模拟引力梯度观测值解算了200阶次的地球重力场模型,分析了格网化误差、极空白对解算精度的影响,迭代3次后,在不考虑低次系数情况下,模型的大地水准面阶误差和累积误差均较小,最大值仅为0.022mm和0.099mm。在沿轨卫星引力梯度模拟数据中加入5mE/Hz1/2的白噪声,基于Torus方法和空域最小二乘法解算了200阶次的地球重力场模型,Torus方法的精度略低于空域最小二乘法的精度,在不考虑低次项的情况下,两种方法解算模型的大地水准面阶误差最大值分别为1.58cm和1.45cm,累积误差最大值分别为6.37cm和5.55cm。但由于采用了二维快速傅里叶技术和块对角最小二乘法,极大地提高了计算效率。本文数值结果说明Torus方法是一种独立有效的方法,可用于GOCE任务海量卫星引力梯度观测值反演重力场的快速解算。     

On the combination of gravity anomalies and gravity disturbances for geoid determination in Western Australia

The geoid gradient over the Darling Fault in Western Australia is extremely high, rising by as much as 38 cm over only 2 km. This poses problems for gravimetric-only geoid models of the area, whose frequency content is limited by the spatial distribution of the gravity data. The gravimetric-only version of AUSGeoid98, for instance, is only able to resolve 46% of the gradient across the fault. Hence, the ability of GPS surveys to obtain accurate orthometric heights is reduced. It is described how further gravity data were collected over the Darling Fault, augmenting the existing gravity observations at key locations so as to obtain a more representative geoid gradient. As many of the gravity observations were collected at stations with a well-known GRS80 ellipsoidal height, the opportunity arose to compute a geoid model via both the Stokes and the Hotine approaches. A scheme was devised to convert free-air anomaly data to gravity disturbances using existing geoid models, followed by a Hotine integration to geoid heights. Interestingly, these results depended very weakly upon the choice of input geoid model. The extra gravity data did indeed improve the fit of the computed geoid to local GPS/Australian Height Datum (AHD) observations by 58% over the gravimetric-only AUSGeoid98. While the conventional Stokesian approach to geoid determination proved to be slightly better than the Hotine method, the latter still improved upon the gravimetric-only AUSGeoid98 solution, supporting the viability of conducting gravity surveys with GPS control for the purposes of geoid determination. AcknowledgementsThe author would like to thank Will Featherstone, Ron Gower, Ron Hackney, Linda Morgan, Geoscience Australia, Scripps Oceanographic Institute and the three anonymous reviewers of this paper. This research was funded by the Australian Research Council.     

Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

The height datum problem and the role of satellite gravity models

Regional height systems do not refer to a common equipotential surface, such as the geoid. They are usually referred to the mean sea level at a reference tide gauge. As mean sea level varies (by ±1 to 2 m) from place to place and from continent to continent each tide gauge has an unknown bias with respect to a common reference surface, whose determination is what the height datum problem is concerned with. This paper deals with this problem, in connection to the availability of satellite gravity missions data. Since biased heights enter into the computation of terrestrial gravity anomalies, which in turn are used for geoid determination, the biases enter as secondary or indirect effect also in such a geoid model. In contrast to terrestrial gravity anomalies, gravity and geoid models derived from satellite gravity missions, and in particular GRACE and GOCE, do not suffer from those inconsistencies. Those models can be regarded as unbiased. After a review of the mathematical formulation of the problem, the paper examines two alternative approaches to its solution. The first one compares the gravity potential coefficients in the range of degrees from 100 to 200 of an unbiased gravity field from GOCE with those of the combined model EGM2008, that in this range is affected by the height biases. This first proposal yields a solution too inaccurate to be useful. The second approach compares height anomalies derived from GNSS ellipsoidal heights and biased normal heights, with anomalies derived from an anomalous potential which combines a satellite-only model up to degree 200 and a high-resolution global model above 200. The point is to show that in this last combination the indirect effects of the height biases are negligible. To this aim, an error budget analysis is performed. The biases of the high frequency part are proved to be irrelevant, so that an accuracy of 5 cm per individual GNSS station is found. This seems to be a promising practical method to solve the problem.     

利用引力梯度数据计算径向梯度的优化方法

根据重力梯度观测各分量的方差及协方差信息,提出了利用GOCE梯度数据计算径向重力梯度的优化方法。首先给出了径向重力梯度的计算方法,并深入分析了误差传播规律,通过建立相应的条件极值问题,给出了计算径向重力梯度最优组合因子的方法;通过模拟数据验证了本文所提出的优化因子的优越性。实际数据计算表明:相对于传统方法,采用优化组合因子可使反演所得引力位模型的累积大地水准面精度在250阶时提高约2 cm。由于径向重力梯度不仅可以用于地球引力场模型的求解,也可直接应用于地球物理问题的讨论,因此本文所提出的优化方法也可对部分地球动力学问题的讨论提供方便。     

Truncated geoid and gravity inversion for one point-mass anomaly

The truncated geoid, defined by the truncated Stokes' integral transform, an integral convolution of gravity anomalies with the Stokes' function on a spherical cap, is often used as a mathematical tool in geoid computations via Stokes' integral to overcome computational difficulties, particularly the need to integrate over the entire boundary spheroid. The objective of this paper is to demonstrate that the truncated geoid does, besides having mathematical applications, have physical interpretation, and thus may be used in gravity inversion. A very simple model of one point-mass anomaly is chosen and a method for inverting its synthetic gravity field with the use of the truncated geoid is presented. The method of inverting the synthetic field generated by one point-mass anomaly has become fundamental for the authors' inversion studies for sets of point-mass anomalies, which are published in a separate paper. More general applications are currently under investigation. Since an inversion technique for physically meaningful mass distributions based on the truncated geoid has not yet been developed, this work is not related to any of the existing gravity inversion techniques. The inversion for one point mass is based on the onset of the so-called dimple event, which occurs in the sequence of surfaces (or profiles) of the first derivative of the truncated geoid with respect to the truncation parameter (radius of the integration cap), its only free parameter. Computing the truncated geoid at various values of the truncation parameter may be understood as spatial filtering of surface gravity data, a type of weighted spherical windowing method. Studying the change of the truncated geoid represented by its first derivative may be understood as a data enhancement method. The instant of the dimple onset is practically independent of the mass of the point anomaly and linearly dependent on its depth. Received: 26 September 1996 /Accepted: 28 September 1998     

Gravity field terrain effect computations by FFT

The widespread availability of detailed gridded topographic and bathymetric data for many areas of the earth has resulted in a need for efficient terrain effect computation techniques, especially for applications in gravity field modelling. Compared to conventional integration techniques, Fourier transform methods provide extremely efficient computations due to the speed of the Fast Fourier Transform (FFT. The Fourier techniques rely on linearization and series expansions of the basically unlinear terrain effect integrals, typically involving transformation of the heights/depths and their squares. TheFFT methods will especially be suited for terrain reduction of land gravity data and satellite altimetry geoid data. In the paper the basic formulas will be outlined, and special emphasis will be put on the practial implementation, where a special coarse/detailed grid pair formulation must be used in order to minimize the unavoidable edge effects ofFFT, and the special properties ofFFT are utilized to limit the actual number of data transformations needed. Actual results are presented for gravity and geoid terrain effects in test areas of the USA, Greenland and the North Atlantic. The results are evaluated against a conventional integration program: thus, e.g., in an area of East Greenland (with terrain corrections up to10 mgal), the accuracy ofFFT-computed terrain corrections in actual gravity stations showed anr.m.s. error of0.25 mgal, using height data from a detailed photogrammetric digital terrain model. Similarly, isostatic ocean geoid effects in the Faeroe Islands region were found to be computed withr.m.s. errors around0.03 m     

A method to validate gravimetric-geoid computation software based on Stokes's integral formula

A method is presented with which to verify that the computer software used to compute a gravimetric geoid is capable of producing the correct results, assuming accurate input data. The Stokes, gravimetric terrain correction and indirect effect formulae are integrated analytically after applying a transformation to surface spherical coordinates centred on each computation point. These analytical results can be compared with those from geoid computation software using constant gravity data in order to verify its integrity. Results of tests conducted with geoid computation software are presented which illustrate the need for integration weighting factors, especially for those compartments close to the computation point. Received: 6 February 1996 / Accepted: 19 April 1997     

A synthetic Earth Gravity Model Designed Specifically for Testing Regional Gravimetric Geoid Determination Algorithms

A synthetic [simulated] Earth gravity model (SEGM) of the geoid, gravity and topography has been constructed over Australia specifically for validating regional gravimetric geoid determination theories, techniques and computer software. This regional high-resolution (1-arc-min by 1-arc-min) Australian SEGM (AusSEGM) is a combined source and effect model. The long-wavelength effect part (up to and including spherical harmonic degree and order 360) is taken from an assumed errorless EGM96 global geopotential model. Using forward modelling via numerical Newtonian integration, the short-wavelength source part is computed from a high-resolution (3-arc-sec by 3-arc-sec) synthetic digital elevation model (SDEM), which is a fractal surface based on the GLOBE v1 DEM. All topographic masses are modelled with a constant mass-density of 2,670 kg/m³. Based on these input data, gravity values on the synthetic topography (on a grid and at arbitrarily distributed discrete points) and consistent geoidal heights at regular 1-arc-min geographical grid nodes have been computed. The precision of the synthetic gravity and geoid data (after a first iteration) is estimated to be better than 30 μ Gal and 3 mm, respectively, which reduces to 1 μ Gal and 1 mm after a second iteration. The second iteration accounts for the changes in the geoid due to the superposed synthetic topographic mass distribution. The first iteration of AusSEGM is compared with Australian gravity and GPS-levelling data to verify that it gives a realistic representation of the Earth’s gravity field. As a by-product of this comparison, AusSEGM gives further evidence of the north–south-trending error in the Australian Height Datum. The freely available AusSEGM-derived gravity and SDEM data, included as Electronic Supplementary Material (ESM) with this paper, can be used to compute a geoid model that, if correct, will agree to in 3 mm with the AusSEGM geoidal heights, thus offering independent verification of theories and numerical techniques used for regional geoid modelling.Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00190-005-0002-z     

格尔木市高精度数字高程基准模型研究

大地水准面（数字高程基准）为国家高程基准的建立与维持提供了全新的思路。然而，受限于地形、重力数据等原因，高原地区高精度数字高程基准模型的建立一直是大地测量领域的难题。本文以格尔木地区为例，探讨了高原地区高精度数字高程基准模型的建立方法。首先，基于重力和地形数据，由第二类Helmert凝集法计算了格尔木重力似大地水准面。在计算中，考虑到高原地形对大地水准面模型的影响，采用了7.5″×7.5″分辨率和高精度的地形数据来恢复大地水准面短波部分的方法，以提高似大地水准面的精度。然后，利用球冠谐调和分析方法将GNSS水准与重力似大地水准面联合，建立了格尔木高精度数字高程基准模型。与实测的67个高精度GNSS水准资料比较，重力似大地水准面的外符合精度为3.0 cm，数字高程基准模型的内符合精度为2.0 cm。     

Regional geoid of the Weddell Sea,Antarctica, from heterogeneous ground-based gravity data

We present a geoid solution for the Weddell Sea and adjacent continental Antarctic regions. There, a refined geoid is of interest, especially for oceanographic and glaciological applications. For example, to investigate the Weddell Gyre as a part of the Antarctic Circumpolar Current and, thus, of the global ocean circulation, the mean dynamic topography (MDT) is needed. These days, the marine gravity field can be inferred with high and homogeneous resolution from altimetric height profiles of the mean sea surface. However, in areas permanently covered by sea ice as well as in coastal regions, satellite altimetry features deficiencies. Focussing on the Weddell Sea, these aspects are investigated in detail. In these areas, ground-based data that have not been used for geoid computation so far provide additional information in comparison with the existing high-resolution global gravity field models such as EGM2008. The geoid computation is based on the remove–compute–restore approach making use of least-squares collocation. The residual geoid with respect to a release 4 GOCE model adds up to two meters and more in the near-coastal and continental areas of the Weddell Sea region, also in comparison with EGM2008. Consequently, the thus refined geoid serves to compute new estimates of the regional MDT and geostrophic currents.     

湖南地区似大地水准面模型再精化研究

湖南地区2007版似大地水准面模型存在数据源质量相对不高、局部区域存在粗差等问题。通过引入EGM2008地球重力场模型、高分辨率DEM数据、高精度GPS/水准数据等新数据源,采用最小二乘配置法与格林基函数样条插值算法,在原有模型的基础上得到1′×1′分辨率的再精化新模型。精度检测结果表明,新模型的内、外符合精度相较2007模型提升80.7%和22.9%,有效降低汉寿等地区的残差,对湖南省地区的复杂地形具有更好的适用性。     

Canadian gravimetric geoid model 2010

A new gravimetric geoid model, Canadian Gravimetric Geoid 2010 (CGG2010), has been developed to upgrade the previous geoid model CGG2005. CGG2010 represents the separation between the reference ellipsoid of GRS80 and the Earth’s equipotential surface of $W_0=62{,}636{,}855.69~\mathrm{m}^2\mathrm{s}^{-2}$ W 0 = 62 , 636 , 855.69 m 2 s ? 2 . The Stokes–Helmert method has been re-formulated for the determination of CGG2010 by a new Stokes kernel modification. It reduces the effect of the systematic error in the Canadian terrestrial gravity data on the geoid to the level below 2 cm from about 20 cm using other existing modification techniques, and renders a smooth spectral combination of the satellite and terrestrial gravity data. The long wavelength components of CGG2010 include the GOCE contribution contained in a combined GRACE and GOCE geopotential model: GOCO01S, which ranges from $-20.1$ ? 20.1 to 16.7 cm with an RMS of 2.9 cm. Improvement has been also achieved through the refinement of geoid modelling procedure and the use of new data. (1) The downward continuation effect has been accounted accurately ranging from $-22.1$ ? 22.1 to 16.5 cm with an RMS of 0.9 cm. (2) The geoid residual from the Stokes integral is reduced to 4 cm in RMS by the use of an ultra-high degree spherical harmonic representation of global elevation model for deriving the reference Helmert field in conjunction with a derived global geopotential model. (3) The Canadian gravimetric geoid model is published for the first time with associated error estimates. In addition, CGG2010 includes the new marine gravity data, ArcGP gravity grids, and the new Canadian Digital Elevation Data (CDED) 1:50K. CGG2010 is compared to GPS-levelling data in Canada. The standard deviations are estimated to vary from 2 to 10 cm with the largest error in the mountainous areas of western Canada. We demonstrate its improvement over the previous models CGG2005 and EGM2008.