A new technique to determine geoid and orthometric heights from satellite positioning and geopotential numbers

This paper takes advantage of space-technique-derived positions on the Earth’s surface and the known normal gravity field to determine the height anomaly from geopotential numbers. A new method is also presented to downward-continue the height anomaly to the geoid height. The orthometric height is determined as the difference between the geodetic (ellipsoidal) height derived by space-geodetic techniques and the geoid height. It is shown that, due to the very high correlation between the geodetic height and the computed geoid height, the error of the orthometric height determined by this method is usually much smaller than that provided by standard GPS/levelling. Also included is a practical formula to correct the Helmert orthometric height by adding two correction terms: a topographic roughness term and a correction term for lateral topographic mass–density variations.     

The Effects of Stokes’s Formula for an Ellipsoidal Layering of the Earth’s Atmosphere

The application of Stokes’s formula to determine the geoid height requires that topographic and atmospheric masses be mathematically removed prior to Stokes integration. This corresponds to the applications of the direct topographic and atmospheric effects. For a proper geoid determination, the external masses must then be restored, yielding the indirect effects. Assuming an ellipsoidal layering of the atmosphere with 15% increase in its density towards the poles, the direct atmospheric effect on the geoid height is estimated to be −5.51 m plus a second-degree zonal harmonic term with an amplitude of 1.1 cm. The indirect effect is +5.50 m and the total geoid correction thus varies between −1.2 cm at the equator to 1.9 cm at the poles. Finally, the correction needed to the atmospheric effect if Stokes’s formula is used in a spherical approximation, rather than an ellipsoidal approximation, of the Earth varies between 0.3 cm and 4.0 cm at the equator and pole, respectively.     

地形改正与地形直接影响的转化关系

传统的第三边值问题的解算方法有Molodensky算法和Stokes-Helmert算法两种。在Molodensky算法中使用的地形改正和Stokes-Helmert算法中使用的直接影响均由大地水准面外地形产生,因而必然存在关系。本文通过推导给出了直接影响是地形改正、层间改正与压缩地形影响3项之和的结论。在此基础上,给出了直接影响的质量线平面积分算法、质量棱柱平面积分算法和地形改正的球面积分算法。此外本文还推导了布格球冠层间改正算法。通过实验得出,直接影响的质量线平面积分算法和质量棱柱平面积分算法与传统球面积分算法的差异分别为3.81和1.64 m Gal;地形改正球面积分算法与传统质量线、质量棱柱平面积分的差异分别为3.92和1.69 m Gal。该结果说明,本文推导的直接影响与地形改正的关系式是正确有效且实用的。     

局部地形改正快速计算的GPU并行的棱柱法

在重力归算中,局部地形改正在重力勘探、地壳结构分析和大地水准面计算等领域有着重要意义,但严格棱柱体积分公式计算效率低,而快速计算公式则会降低计算精度。本文利用CUDA并行编程平台,提出一种地形格网重新编码和严格棱柱体积分八分量拆解方法,实现了基于CPU+GPU异构并行技术的严格棱柱体积分计算地形改正快速并行算法,克服了GPU各个线程计算任务分配和线程计算超载问题,解决了局部地形改正的高分辨率、高精度严密公式的快速计算难题。通过试验,在显卡型号为Tesla V100的计算机上进行4°×6°范围,积分半径40'和分辨率1'的局部地形改正计算仅需1.5 s;分辨率10″的局部地形改正计算仅需14.6 min;进行分辨率3″的地形改正计算耗时45.7 h,而传统串行算法则难以完成计算。在保证微伽级以上计算精度的条件下,计算加速比最高达到850倍以上,有效缩短了计算耗时,提高了计算效率。本文还依据上述并行算法对全国范围地形改正量进行计算。结果表明,我国地形改正量普遍低于80 mGal（1 Gal=10^-2 m/s²）,平均值1.83 mGal,最大值达到196 mGal。     

Topographic effects by the Stokes–Helmert method of geoid and quasi-geoid determinations

The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H ². The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is shown that the spherical approximation to power H ² leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜² (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed. The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height). Received: 18 August 1998 / Accepted: 4 October 1999     

有限范围的重力层间改正算法

层间改正是重力归算的一项重要内容,传统的平面层层间改正、球面层层间改正与地形改正的范围不一致,因此均存在远区虚拟地形引入的近似误差,且计算点高度越高,此误差越大。本文提出使用有限范围的层间改正进行重力归算的方法,使其区域范围与地形改正的范围一致。然后给出了有限范围层间改正的简便计算方法,该算法与通过地形改正严密积分法演化来的算法具有较好的一致性。内插试验说明当计算点地形高于1000m时,内插应使用基于有限范围层间改正的重力归算方法。     

The ellipsoidal corrections to the topographic geoid effects

The topographic effects by Stokes formula are typically considered for a spherical approximation of sea level. For more precise determination of the geoid, sea level is better approximated by an ellipsoid, which justifies the consideration of the ellipsoidal corrections of topographic effects for improved geoid solutions. The aim of this study is to estimate the ellipsoidal effects of the combined topographic correction (direct plus indirect topographic effects) and the downward continuation effect. It is concluded that the ellipsoidal correction to the combined topographic effect on the geoid height is far less than 1 mm. On the contrary, the ellipsoidal correction to the effect of downward continuation of gravity anomaly to sea level may be significant at the 1-cm level in mountainous regions. Nevertheless, if Stokes formula is modified and the integration of gravity anomalies is limited to a cap of a few degrees radius around the computation point, nor this effect is likely to be significant.AcknowledgementsThe author is grateful for constructive remarks by J Ågren and the three reviewers.     

Assessment of different topographic correction methods in ALOS AVNIR-2 data over a forest area

Abstract

Because the removal of topographic effects is one the most important pre-processing steps when extracting information from satellite images in digital Earth applications, the problem of differential terrain illumination on satellite imagery has been investigated for at least 20 years. As there is no superior topographic correction method applicable to all areas and all images, a comparison of topographic normalization methods in different regions and images is necessary. In this study, common topographic correction methods were applied on an ALOS AVNIR-2 image of a rugged forest area, and the results were evaluated through different criteria. The results show that the simple correction methods [Cosine, Sun-Canopy-sensor (SCS), and Minnaert correction] are inefficient in exceptionally rough forests. Among the improved correction methods (SCS+C, modified Minnaert, and pixel-based Minnaert), the best result was achieved using a pixel-based Minnaert approach in which a separate correction factor in various slope angles is used. Thus, this method should be considered for topographic correction, especially in forests with severe topography.     

利用地形正规化校正反立体现象

影像目视判读常会遇到山脊与沟谷的凹凸感与现实相反的反立体现象。消除反立体现象,能有效提高非专业人员对遥感影像的正确使用。立足于反立体现象的成因,本文采用地形正规化模型来校正影像的反立体现象,推导出Lambertian、Cosine-Civco、c校正、b校正这4种地形正规化模型的反立体校正式;对这4种地形正规化模型的反立体校正效果进行了对比,并且与其他5种校正法也进行了对比。通过3个实验区的校正发现,这4种地形正规化模型均能校正反立体现象,但校正影像存在色调偏差;Lambertian、Cosine-Civco的反立体校正影像立体感较强,但影像色调改变较大,视觉效果偏差;c校正、b校正的校正影像在视觉效果和定量指标上都比较接近,基本保持地物光谱信息,校正效果相对较好。从定量指标来看,b校正的反立体校正影像的各指标值整体最小,一定程度代表b校正能取得相对较好的反立体校正效果。与其他方法的对比表明,c校正和b校正的反立体校正不局限于波段个数,在有效消除反立体现象的同时,能相对较好的保留地物光谱信息,有利于影像的定量应用。     

A comparison of the tesseroid,prism and point-mass approaches for mass reductions in gravity field modelling

The calculation of topographic (and iso- static) reductions is one of the most time-consuming operations in gravity field modelling. For this calculation, the topographic surface of the Earth is often divided with respect to geographical or map-grid lines, and the topographic heights are averaged over the respective grid elements. The bodies bounded by surfaces of constant (ellipsoidal) heights and geographical grid lines are denoted as tesseroids. Usually these ellipsoidal (or spherical) tesseroids are replaced by “equivalent” vertical rectangular prisms of the same mass. This approximation is motivated by the fact that the volume integrals for the calculation of the potential and its derivatives can be exactly solved for rectangular prisms, but not for the tesseroids. In this paper, an approximate solution of the spherical tesseroid integrals is provided based on series expansions including third-order terms. By choosing the geometrical centre of the tesseroid as the Taylor expansion point, the number of non-vanishing series terms can be greatly reduced. The zero-order term is equivalent to the point-mass formula. Test computations show the high numerical efficiency of the tesseroid method versus the prism approach, both regarding computation time and accuracy. Since the approximation errors due to the truncation of the Taylor series decrease very quickly with increasing distance of the tesseroid from the computation point, only the elements in the direct vicinity of the computation point have to be separately evaluated, e.g. by the prism formulas. The results are also compared with the point-mass formula. Further potential refinements of the tesseroid approach, such as considering ellipsoidal tesseroids, are indicated.     

The effect on the geoid of lateral topographic density variations

Gravimetric geoid determination by Stokes formula requires that the effects of topographic masses be removed prior to Stokes integration. This step includes the direct topographic and the downward continuation (DWC) effects on gravity anomaly, and the computations yield the co-geoid height. By adding the effect of restoration of the topography, the indirect effect on the geoid, the geoid height is obtained. Unfortunately, the computations of all these topographic effects are hampered by the uncertainty of the density distribution of the topography. Usually the computations are limited to a constant topographic density, but recently the effects of lateral density variations have been studied for their direct and indirect effects on the geoid. It is emphasised that the DWC effect might also be significantly affected by a lateral density variation. However, instead of computing separate effects of lateral density variation for direct, DWC and indirect effects, it is shown in two independent ways that the total geoid effect due to the lateral density anomaly can be represented as a simple correction proportional to the lateral density anomaly and the elevation squared of the computation point. This simple formula stems from the fact that the significant long-wavelength contributions to the various topographic effects cancel in their sum. Assuming that the lateral density anomaly is within 20% of the standard topographic density, the derived formula implies that the total effect on the geoid is significant at the centimetre level for topographic elevations above 0.66 km. For elevations of 1000, 2000 and 5000 m the effect is within ± 2.2, ± 8.8 and ± 56.8 cm, respectively. For the elevation of Mt. Everest the effect is within ± 1.78 m.     

The topographic bias by analytical continuation in physical geodesy

This study emphasizes that the harmonic downward continuation of an external representation of the Earth’s gravity potential to sea level through the topographic masses implies a topographic bias. It is shown that the bias is only dependent on the topographic density along the geocentric radius at the computation point. The bias corresponds to the combined topographic geoid effect, i.e., the sum of the direct and indirect topographic effects. For a laterally variable topographic density function, the combined geoid effect is proportional to terms of powers two and three of the topographic height, while all higher order terms vanish. The result is useful in geoid determination by analytical continuation, e.g., from an Earth gravity model, Stokes’s formula or a combination thereof.     

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

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

The effect of atmospheric and topographic correction methods on land cover classification accuracy

Mapping of vegetation in mountain areas based on remote sensing is obstructed by atmospheric and topographic distortions. A variety of atmospheric and topographic correction methods has been proposed to minimize atmospheric and topographic effects and should in principle lead to a better land cover classification. Only a limited number of atmospheric and topographic combinations has been tested and the effect on class accuracy and on different illumination conditions is not yet researched extensively. The purpose of this study was to evaluate the effect of coupled correction methods on land cover classification accuracy. Therefore, all combinations of three atmospheric (no atmospheric correction, dark object subtraction and correction based on transmittance functions) and five topographic corrections (no topographic correction, band ratioing, cosine correction, pixel-based Minnaert and pixel-based C-correction) were applied on two acquisitions (2009 and 2010) of a Landsat image in the Romanian Carpathian mountains. The accuracies of the fifteen resulting land cover maps were evaluated statistically based on two validation sets: a random validation set and a validation subset containing pixels present in the difference area between the uncorrected classification and one of the fourteen corrected classifications. New insights into the differences in classification accuracy were obtained. First, results showed that all corrected images resulted in higher overall classification accuracies than the uncorrected images. The highest accuracy for the full validation set was achieved after combination of an atmospheric correction based on transmittance functions and a pixel-based Minnaert topographic correction. Secondly, class accuracies of especially the coniferous and mixed forest classes were enhanced after correction. There was only a minor improvement for the other land cover classes (broadleaved forest, bare soil, grass and water). This was explained by the position of different land cover types in the landscape. Finally, coupled correction methods showed most efficient on weakly illuminated slopes. After correction, accuracies in the low illumination zone (cos β ≤ 0.65) were improved more than in the moderate and high illumination zones. Considering all results, best overall classification results were achieved after combination of the transmittance function correction with pixel-based Minnaert or pixel-based C-topographic correction. Furthermore, results of this bi-temporal study indicated that the topographic component had a higher influence on classification accuracy than the atmospheric component and that it is worthwhile to invest in both atmospheric and topographic corrections in a multi-temporal study.     

加入地形改正的卫星影像多项式模型研究

本文针对一般多项式只适用于平坦地区或垂直成像卫星影像正射纠正的不足,根据摄影测量理论中地形起伏产生的像方位移可以用特定的数学公式表达的特点,在多项式纠正方法中引入投影差改正,实现了更精确的微分多项式纠正。文章首先分析了地形起伏引起像点位移原理,在此基础上建立了高程改正的卫星影像多项式模型,设计并实现了修正后的模型的定向方法及正反算算法。实验表明该模型较一般多项式模型X方向上精度有很大提高,适用于山区卫星影像纠正。     

陡峭山区影像的半经验地形校正

本文提出一种新的半经验地形校正模型SCEDIL(Simple topographic Correction using Estimation of Diffuse Light),该模型通过结合DEM与光学影像数据寻找局部区域内完全光照和阴影的水平像元,并以光照、阴影水平像元的平均反射率值估算局部区域散射辐射比,提高了陡峭山区影像的地形校正精度。以高分一号卫星和Landsat ETM+影像为例,从目视判读和定量分析两个方面,比较分析该算法与传统半经验地形校正算法(C、SCS+C)的校正结果。结果表明:(1)对较为平坦的地形,SCEDIL和C、SCS+C校正都有较好的目视结果;对地面起伏较大的陡峭地形,C、SCS+C校正后,原阴影区域易呈现破碎化特征,SCEDIL校正后,原阴影区域过渡较为平滑。(2)SCEDIL校正后,各波段反射率的均值和标准差优于C、SCS+C校正,SCEDIL校正后,影像总分类精度与同类地物光谱信息均一性均优于C和SCS+C校正。SCEDIL半经验地形校正方法能有效地去除影像中的地形干扰,尤其对陡峭地形的校正效果,优于常规地形校正模型。     

The effect of topography on the determination of the geoid using analytical downward continuation

The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth, too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination of the geoid) by using the method of analytical downward continuation. It is shown that the disturbing potential obtained by using the analytical downward continuation is different from the true disturbing potential on the sea level mostly by a −2πGρh 2/p. This correction is important and it is very easy to compute and add to the final results. A terrain effect (effect of the topography from the Bouguer plate) is found to be much smaller than the correction of the Bouguer plate and can be neglected in most cases. It is also shown that the geoid determined by using the Helmert’s second condensation (including the indirect effect) and using the analytical downward continuation procedure (including the topographic effect) are identical. They are different procedures and may be used in different environments, e.g., the analytical downward continuation procedure is also more convenient for processing the aerial gravity gradient data. A numerical test was completed in a rough mountain area, 35°<ϕ<38°, 240°<λ<243°. A digital height model in 30″×30″ point value was used. The test indicated that the terrain effect in the test area has theRMS value ±0.2−0.3 cm for geoid. The topographic effect on the deflections of the vertical is around1 arc second.     

GPS高程转换中考虑地形改正的几何方法

讨论了出区地形变化较大,在进行ＧＰＳ高程转换时所应加入的地形改正,给出了改正公式。     

Topographic and atmospheric corrections of gravimetric geoid determination with special emphasis on the effects of harmonics of degrees zero and one

 The topographic and atmospheric effects of gravimetric geoid determination by the modified Stokes formula, which combines terrestrial gravity and a global geopotential model, are presented. Special emphasis is given to the zero- and first-degree effects. The normal potential is defined in the traditional way, such that the disturbing potential in the exterior of the masses contains no zero- and first-degree harmonics. In contrast, it is shown that, as a result of the topographic masses, the gravimetric geoid includes such harmonics of the order of several centimetres. In addition, the atmosphere contributes with a zero-degree harmonic of magnitude within 1 cm. Received: 5 November 1999 / Accepted: 22 January 2001     

The IAG approach to the atmospheric geoid correction in Stokes' formula and a new strategy

The well-known International Association of Geodesy (IAG) approach to the atmospheric geoid correction in connection with Stokes' integral formula leads to a very significant bias, of the order of 3.2 m, if Stokes' integral is truncated to a limited region around the computation point. The derived truncation error can be used to correct old results. For future applications a new strategy is recommended, where the total atmospheric geoid correction is estimated as the sum of the direct and indirect effects. This strategy implies computational gains as it avoids the correction of direct effect for each gravity observation, and it does not suffer from the truncation bias mentioned above. It can also easily be used to add the atmospheric correction to old geoid estimates, where this correction was omitted. In contrast to the terrain correction, it is shown that the atmospheric geoid correction is mainly of order H of terrain elevation, while the term of order H ² is within a few millimetres. Received: 20 May 1998 / Accepted: 19 April 1999