Confirming regional 1 cm differential geoid accuracy from airborne gravimetry: the Geoid Slope Validation Survey of 2011

A terrestrial survey, called the Geoid Slope Validation Survey of 2011 (GSVS11), encompassing leveling, GPS, astrogeodetic deflections of the vertical (DOV) and surface gravity was performed in the United States. The general purpose of that survey was to evaluate the current accuracy of gravimetric geoid models, and also to determine the impact of introducing new airborne gravity data from the ‘Gravity for the Redefinition of the American Vertical Datum’ (GRAV-D) project. More specifically, the GSVS11 survey was performed to determine whether or not the GRAV-D airborne gravimetry, flown at 11 km altitude, can reduce differential geoid error to below 1 cm in a low, flat gravimetrically uncomplicated region. GSVS11 comprises a 325 km traverse from Austin to Rockport in Southern Texas, and includes 218 GPS stations ( $\sigma _{\Delta h }= 0.4$ cm over any distance from 0.4 to 325 km) co-located with first-order spirit leveled orthometric heights ( $\sigma _{\Delta H }= 1.3$ cm end-to-end), including new surface gravimetry, and 216 astronomically determined vertical deflections $(\sigma _{\mathrm{DOV}}= 0.1^{\prime \prime })$ . The terrestrial survey data were compared in various ways to specific geoid models, including analysis of RMS residuals between all pairs of points on the line, direct comparison of DOVs to geoid slopes, and a harmonic analysis of the differences between the terrestrial data and various geoid models. These comparisons of the terrestrial survey data with specific geoid models showed conclusively that, in this type of region (low, flat) the geoid models computed using existing terrestrial gravity, combined with digital elevation models (DEMs) and GRACE and GOCE data, differential geoid accuracy of 1 to 3 cm (1 $\sigma )$ over distances from 0.4 to 325 km were currently being achieved. However, the addition of a contemporaneous airborne gravity data set, flown at 11 km altitude, brought the estimated differential geoid accuracy down to 1 cm over nearly all distances from 0.4 to 325 km.     

EGM96,WDM94和GPM98CR高阶地球重力场模型表示深圳局部重力场的比较与评价

为计算深圳精密重力大地水准面，利用62个高精度GPS水准点和4871个实测重力点数据对EGM96，WDM94和GPM98CR全球重力场模型表示深圳局部重力场进行了比较与评价。结果表明，由上述3个重力场模型计算的大地水准面高和重力异常与实测值之间存在明显的系统偏差，当采用GPS水准数据尽可能消除系统偏差以后，大地水准面高的精度得到显著提高，若应用移去-恢复技术确定深圳高精度大地水准面，则WDM94应该是首选的参考重力场模型。     

Minimization and estimation of geoid undulation errors

The objective of this paper is to minimize the geoid undulation errors by focusing on the contribution of the global geopotential model and regional gravity anomalies, and to estimate the accuracy of the predicted gravimetric geoid.The geopotential model's contribution is improved by (a) tailoring it using the regional gravity anomalies and (b) introducing a weighting function to the geopotential coefficients. The tailoring and the weighting function reduced the difference (1) between the geopotential model and the GPS/levelling-derived geoid undulations in British Columbia by about 55% and more than 10%, respectively.Geoid undulations computed in an area of 40° by 120° by Stokes' integral with different kernel functions are analyzed. The use of the approximated kernels results in about 25 cm () and 190 cm (maximum) geoid errors. As compared with the geoid derived by GPS/levelling, the gravimetric geoid gives relative differences of about 0.3 to 1.4 ppm in flat areas, and 1 to 2.5 ppm in mountainous areas for distances of 30 to 200 km, while the absolute difference (1) is about 5 cm and 20 cm, respectively.A optimal Wiener filter is introduced for filtering of the gravity anomaly noise, and the performance is investigated by numerical examples. The internal accuracy of the gravimetric geoid is studied by propagating the errors of the gravity anomalies and the geopotential coefficients into the geoid undulations. Numerical computations indicate that the propagated geoid errors can reasonably reflect the differences between the gravimetric and GPS/levelling-derived geoid undulations in flat areas, such as Alberta, and is over optimistic in the Rocky Mountains of British Columbia.Paper presented at the IAG General Meeting, Beijing, China, August 8–13, 1993.     

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。     

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.     

Evaluation of the third- and fourth-generation GOCE Earth gravity field models with Australian terrestrial gravity data in spherical harmonics

In March 2013, the fourth generation of European Space Agency’s (ESA) global gravity field models, DIR4 (Bruinsma et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010b) and TIM4 (Migliaccio et al. in Proceedings of the ESA living planet symposium, 28 June–2 July, Bergen, ESA, Publication SP-686, 2010), generated from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity observation satellite was released. We evaluate the models using an independent ground truth data set of gravity anomalies over Australia. Combined with Gravity Recovery and Climate Experiment (GRACE) satellite gravity, a new gravity model is obtained that is used to perform comparisons with GOCE models in spherical harmonics. Over Australia, the new gravity model proves to have significantly higher accuracy in the degrees below 120 as compared to EGM2008 and seems to be at least comparable to the accuracy of this model between degree 150 and degree 260. Comparisons in terms of residual quasi-geoid heights, gravity disturbances, and radial gravity gradients evaluated on the ellipsoid and at approximate GOCE mean satellite altitude ( $h=250$  km) show both fourth generation models to improve significantly w.r.t. their predecessors. Relatively, we find a root-mean-square improvement of 39 % for the DIR4 and 23 % for TIM4 over the respective third release models at a spatial scale of 100 km (degree 200). In terms of absolute errors, TIM4 is found to perform slightly better in the bands from degree 120 up to degree 160 and DIR4 is found to perform slightly better than TIM4 from degree 170 up to degree 250. Our analyses cannot confirm the DIR4 formal error of 1 cm geoid height (0.35 mGal in terms of gravity) at degree 200. The formal errors of TIM4, with 3.2 cm geoid height (0.9 mGal in terms of gravity) at degree 200, seem to be realistic. Due to combination with GRACE and SLR data, the DIR models, at satellite altitude, clearly show lower RMS values compared to TIM models in the long wavelength part of the spectrum (below degree and order 120). Our study shows different spectral sensitivity of different functionals at ground level and at GOCE satellite altitude and establishes the link among these findings and the Meissl scheme (Rummel and van Gelderen in Manusrcipta Geodaetica 20:379–385, 1995).     

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.     

Using radial basis functions in airborne gravimetry for local geoid improvement

Radial basis functions (RBFs) have been used extensively in satellite geodetic applications. However, to the author’s knowledge, their role in processing and modeling airborne gravity data has not yet been fully advocated or extensively investigated in detail. Compared with satellite missions, the airborne data are more suitable for these kinds of localized basis functions especially considering the following facts: (1) Unlike the satellite missions that can provide global or near global data coverage, airborne gravity data are usually geographically limited. (2) It is also band limited in the frequency domain. (3) It is straightforward to formulate the RBF observation equations from an airborne gravimetric system. In this study, a set of band-limited RBF is developed to model and downward continue the airborne gravity data for local geoid improvement. First, EIGEN6c4 coefficients are used to simulate a harmonic field to test the performances of RBF on various sampling, noise, and flight height levels, in order to gain certain guidelines for processing the real data. Here, the RBF method not only successfully recovers the harmonic field but also presents filtering properties due to its particular design in the frequency domain. Next, the software was tested for the GSVS14 (Geoid Slope Validation Survey 2014) area in Iowa as well as for the area around Puerto Rico and the US Virgin Islands by use of the real airborne gravity data from the Gravity for the Redefinition of the American Vertical Datum (GRAV-D) project. By fully utilizing the three-dimensional correlation information among the flight tracks, the RBF can also be used as a data cleaning tool for airborne gravity data adjustment and cleaning. This property is further extended to surface gravity data cleaning, where conventional approaches have various limitations. All the related numerical results clearly show the importance and contribution of the use of the RBF for high- resolution local gravity field modeling.     

Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons

Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.     

全国及部分省市地区高精度、高分辨率大地水准面的研究及其实施

结合我国重力和地形资料及国内外较优的重力场模型，研制了适合我国重力场特征的360阶重力场模型WDM94，建立了中国新一代包括全部陆海国土的dm级（似）大地水准面CQG2000，建立了中国以GPS／水准为基础的高程异常控制网，利用海洋卫星测高数据进行我国海洋大地水准面的计算、我国陆地重力（似）大地水准面的研制厦我国陆海（似）大地水准面的拼接；研制了江苏省、海南省、深圳市、大连市、南京市及南水北调西线工程具有cm级精度的省市地区（似）大地水准面模型；结合GPS技术和高精度（似）大地水准面模型，研制了GPS测图软硬件一体化系统。     

A new,high-resolution geoid for Canada and part of the U.S. by the 1D-FFT method

A new, high-resolution and high-precision geoid has been computed for the whole of Canada and part of the U.S., ranging from 35°N to about 90°N in latitude and 210°E to 320°E in longitude. The OSU91A geopotential model complete to degree and order 360 was combined with a 5 × 5 mean gravity anomaly grid and 1km × 1km topographical information to generate the geoid file. The remove-restore technique was adopted for the computation of terrain effects by Helmert's condensation reduction. The contribution of the local gravity data to the geoid was computed strictly by the 1D-FFT technique, which allows for the evaluation of the discrete spherical Stokes integral without any approximation, parallel by parallel. The indirect effects of up to second order were considered. The internal precision of the geoid, i.e. the contribution of the gravity data and the model coefficients noise, was also evaluated through error propagation by FFT. In a relative sense, these errors seem to agree quite well with the external errors and show clearly the weak areas of the geoid which are mostly due to insufficient gravity data coverage. Comparison of the gravimetric geoid with the GPS/levelling-derived geoidal heights of eight local GPS networks with a total of about 900 stations shows that the absolute agreement with respect to the GPS/levelling datum is generally better than 10 cm RMS and the relative agreement ranges, in most cases, from 4 to 1 ppm over short distances of about 20 to 100km, 1 to 0.5 ppm over distances of about 100 to 200 km, and 0.5 to 0.1 ppm for baselines of 200 to over 1000 km. Other existing geoids, such as UNB90, GEOID90 and GSD91, were also included in the comparison, showing that the new geoid achieves the best agreement with the GPS/levelling data.Presented at theIAG General Meeting, Beijing, P.R. China, Aug. 6–13, 1993     

陆海交界区域厘米级精度似大地水准面的确定

为了得到我国某陆海交界区厘米级精度的区域(似)大地水准面,利用43个高精度GPS/水准点和1 045个实测重力点数据对EGM96,WDM94和GFZ计算的局部重力(似)大地水准面进行了比较与评价。结果表明,在该测区用移去-恢复法确定重力(似)大地水准面时,EGM96应该是首选参考重力场模型。该测区处在陆海交界处,海域无GPS/水准数据。经比较发现,采用距离倒数加权平均法将该区重力似大地水准面拟合于GPS/水准数据比在大范围使用的多项式法效果更好。采用该方法计算的测区(似)大地水准面精度优于3cm。     

Error analysis of the NGS’ surface gravity database

Are the National Geodetic Survey’s surface gravity data sufficient for supporting the computation of a 1 cm-accurate geoid? This paper attempts to answer this question by deriving a few measures of accuracy for this data and estimating their effects on the US geoid. We use a data set which comprises ${\sim }1.4$ million gravity observations collected in 1,489 surveys. Comparisons to GRACE-derived gravity and geoid are made to estimate the long-wavelength errors. Crossover analysis and $K$ -nearest neighbor predictions are used for estimating local gravity biases and high-frequency gravity errors, and the corresponding geoid biases and high-frequency geoid errors are evaluated. Results indicate that 244 of all 1,489 surface gravity surveys have significant biases ${>}2$  mGal, with geoid implications that reach 20 cm. Some of the biased surveys are large enough in horizontal extent to be reliably corrected by satellite-derived gravity models, but many others are not. In addition, the results suggest that the data are contaminated by high-frequency errors with an RMS of ${\sim }2.2$  mGal. This causes high-frequency geoid errors of a few centimeters in and to the west of the Rocky Mountains and in the Appalachians and a few millimeters or less everywhere else. Finally, long-wavelength ( ${>}3^{\circ }$ ) surface gravity errors on the sub-mGal level but with large horizontal extent are found. All of the south and southeast of the USA is biased by +0.3 to +0.8 mGal and the Rocky Mountains by $-0.1$ to $-0.3$  mGal. These small but extensive gravity errors lead to long-wavelength geoid errors that reach 60 cm in the interior of the USA.     

Estimating geoid height change in North America: past, present and future

The forthcoming GRAV-D gravimetric geoid model over the United States is to be updated regularly to account for changes in geoid height. Its baseline precision is to be at the 10–20 mm level over non-mountainous regions. The aim of this study is to provide an estimate of the magnitude, time scale, and spatial footprint of geoid height change over North America, from mass redistribution processes of hydrologic, cryospheric and solid Earth nature. Geoid height changes from continental water storage changes over the past 50 years and predicted over the next century are evaluated and are highly dependent on the used model. Groundwater depletion from anthropogenic pumping in regional scale aquifers may lead to geoid changes of 10 mm magnitude every 50–100 years. The GRACE time varying gravity fields are used to (i) assess the errors in a glacial isostatic adjustment model, which, if used to correct the GRAV-D model, may induce errors at the 10 mm geoid height level after ~20 years, (ii), evaluate geoid height change over ice mass loss regions of North America, which, if they remain unchanged in the future, may lead to geoid height changes at the 10 mm level in under a decade and (iii), compute sea level rise and its effect on the geoid, which is found to be negligible. Coseismic gravitational changes from past North American earthquakes are evaluated, and lead to geoid change at the 10-mm level for only the largest thrust earthquakes. Finally, geoid change from volcanic processes are assessed and found to be significant with respect to the GRAV-D geoid model baseline precision for cataclysmic events, such as that of the 1980 Mt. St. Helens eruption. Recommendations on how to best monitor geoid change in the future are given.     

一种有效的区域似大地水准面精度检测方法

利用大地水准面逼近严密理论Molodenskii级数解,结合GPS、水准、重力和地形资料精化区域似大地水准面,其精度已从分米级向厘米级、高分辨率方向发展。如何客观、高效地评价区域似大地水准面精度是本文研究的主要目的。作者以广西似大地水准面精度检测为例子,给出了一种综合考虑GPS/水准检测点GPS大地高和水准测量误差的区域似大地水准面精度检测方法。利用覆盖广西境内的446个C级GPS控制网和航控GPS控制网GPS/水准检测点,点间距约25km,采用加权平均法计算广西似大地水准面外符合检测精度为±0.041m。     

Regional geoid computation by least squares modified Hotine’s formula with additive corrections

Geoid and quasigeoid modelling from gravity anomalies by the method of least squares modification of Stokes’s formula with additive corrections is adapted for the usage with gravity disturbances and Hotine’s formula. The biased, unbiased and optimum versions of least squares modification are considered. Equations are presented for the four additive corrections that account for the combined (direct plus indirect) effect of downward continuation (DWC), topographic, atmospheric and ellipsoidal corrections in geoid or quasigeoid modelling. The geoid or quasigeoid modelling scheme by the least squares modified Hotine formula is numerically verified, analysed and compared to the Stokes counterpart in a heterogeneous study area. The resulting geoid models and the additive corrections computed both for use with Stokes’s or Hotine’s formula differ most in high topography areas. Over the study area (reaching almost 2 km in altitude), the approximate geoid models (before the additive corrections) differ by 7 mm on average with a 3 mm standard deviation (SD) and a maximum of 1.3 cm. The additive corrections, out of which only the DWC correction has a numerically significant difference, improve the agreement between respective geoid or quasigeoid models to an average difference of 5 mm with a 1 mm SD and a maximum of 8 mm.     

A numerical investigation on height anomaly prediction in mountainous areas

This paper provides numerical examples for the prediction of height anomalies by the solution of Molodensky's boundary value problem. Computations are done within two areas in the Canadian Rockies. The data used are on a grid with various grid spacings from 100 m to 5 arc-minutes. Numerical results indicate that the Bouguer or the topographicisostatic gravity anomalies should be used in gravity interpolation. It is feasible to predict height anomalies in mountainous areas with an accuracy of 10 cm (1) if sufficiently dense data grids are used. After removing the systematic bias, the differences between the geoid undulations converted from height anomalies and those derived from GPS/levelling on 50 benchmarks is 12 cm (1) when the grid spacing is 1km, and 50 cm (1) when the grid spacing is 5. It is not necessary, in most cases, to require a grid spacing finer than 1 km, because the height anomaly changes only by 3 cm (1) when the grid spacing is increased from 100 m to 1000 m. Numerical results also indicate that, only the first two terms of the Molodensky series have to be evaluated in all but the extreme cases, since the contributions of the higher order terms are negligible compared to the objective accuracy.     

GPS-gravimetric geoid determination in Egypt

The main objective of this study is to improve the geoid by GPS/leveling data in Egypt. Comparisons of the gravimetric geoid with GPS/leveling data have been performed. On the basis of a gravimetric geoid fitted to GPS/leveling by the least square method, a smoothed geoid was obtained. A high-resolution geoid in Egypt was computed with a 2.5′×2.5′ grid by combining the data set of 2600 original point gravity values, 20″×30″ resolution Digital Terrain Model (DTM) grid and the spherical harmonic model EGM96. The method of computation involved the strict evaluation of the Stokes integral with 1D-FFT. The standard deviation of the difference between the gravimetric and the GPS/leveling geoid heights is ±0.47 m. The standard deviation after fitting of the gravimetric geoid to the GPS/leveling points is better than ±13 cm. In the future we will try to improve our geoid results in Egypt by increasing the density of gravimetric coverage.     

GPS-GRAVIMETRIC GEOID DETERMINATION IN EGYPT

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模拟大地水准面应用于GPS水准内插的研究

采用移去-恢复技术和多面函数法对已知GPS水准数据进行了内插处理。试验结果表明．利用实际地球物理资料得到的模拟大地水准面精度最高。