Practical field surveying and computations

Sans résumé

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Practical field surveying and computations A.L. Allan J.R. Hollwey J.H.B. Maynes

     

A bilinear approximation of the surface relief in terrain correction computations

A new method is presented for the computation of the gravitational attraction of topographic masses when their height information is given on a regular grid. It is shown that the representation of the terrain relief by means of a bilinear surface not only offers a serious alternative to the polyhedra modeling, but also approaches even more smoothly the continuous reality. Inserting a bilinear approximation into the known scheme of deriving closed analytical expressions for the potential and its first-order derivatives for an arbitrarily shaped polyhedron leads to a one-dimensional integration with – apparently – no analytical solution. However, due to the high degree of smoothness of the integrand function, the numerical computation of this integral is very efficient. Numerical tests using synthetic data and a densely sampled digital terrain model in the Bavarian Alps prove that the new method is comparable to or even faster than a terrain modeling using polyhedra.     

重力梯度向量化调和分析的FFT算法

在利用卫星重力梯度数据求解地球重力场模型位系数时,为进一步提高调和分析法的计算效率,推导出了利用5种组合{Tzz}、{Txx+Tyy}、{Tzz-Txx-Tyy}、{Txz,Tyz}、{Txx,Tyy,Txy}构建重力场模型位系数的向量化调和分析法和向量化调和分析的FFT算法.实验表明,利用30′×30′格网Tzz数据...     

Gravity field determination from satellite gradiometry

An orbiting gradiometer measures simultaneously several gravity quantities, ideally all six second-order derivatives of the gravitational potential. These contain information on the orbit, on the structure of the gravity field, and on the attitude of the space-craft. Due to the availability of several components simultaneously it is possible to separate orbit determination from attitude or gravity field recovery. This facilitates the analysis of the gradiometer measurements and allows the use of the principles of fast spherical harmonic analysis. The separation of gravity field recovery and orbit determination is tested numerically with a simplified gravity field (with a purely zonal spherical harmonic expansion) up to degree 300. For both the potential coefficients and for the orbit an almost exact recovery is attained after two iteration steps.     

Gravity field error analysis for pendulum formations by a semi-analytical approach

Many geoscience disciplines push for ever higher requirements on accuracy, homogeneity and time- and space-resolution of the Earth’s gravity field. Apart from better instruments or new observables, alternative satellite formations could improve the signal and error structure compared to Grace. One possibility to increase the sensitivity and isotropy by adding cross-track information is a pair of satellites flying in a pendulum formation. This formation contains two satellites which have different ascending nodes and arguments of latitude, but have the same orbital height and inclination. In this study, the semi-analytical approach for efficient pre-mission error assessment is presented, and the transfer coefficients of range, range-rate and range-acceleration gravitational perturbations are derived analytically for the pendulum formation considering a set of opening angles. The new challenge is the time variations of the opening angle and the range, leading to temporally variable transfer coefficients. This is solved by Fourier expansion of the sine/cosine of the opening angle and the central angle. The transfer coefficients are further applied to assess the error patterns which are caused by different orbital parameters. The simulation results indicate that a significant improvement in accuracy and isotropy is obtained for small and medium initial opening angles of single polar pendulums, compared to Grace. The optimal initial opening angles are \(45^\circ \) and \(15^\circ \) for accuracy and isotropy, respectively. For a Bender configuration, which is constituted by a polar Grace and an inclined pendulum in this paper, the behaviour of results is dependent on the inclination (prograde vs. retrograde) and on the relative baseline orientation (left or right leading). The simulation for a sun-synchronous orbit shows better results for the left leading case.     

Gravity field convolutions without windowing and edge effects

A new set of formulas has been developed for the computation of geoid undulations and terrain corrections by FFT when the input gravity anomalies and heights are mean gridded values. The effects of the analytical and the discrete spectra of kernel functions and that of zero-padding on the computation of geoid undulations and terrain corrections are studied in detail.Numerical examples show that the discrete spectrum is superior to the analytically-defined one. By using the discrete spectrum and 100% zero-padding, the RMS differences are 0.000 m for the FFT geoid undulations and 0.200 to 0.000 mGal for the FFT terrain corrections compared with results obtained by numerical integration.     

Prevalence of the terrain reversal effect in satellite imagery

The terrain reversal effect is a perceptual phenomenon which causes an illusion in various 3D geographic visualizations where landforms appear inverted, e.g. we perceive valleys as ridges and vice versa. Given that such displays are important for spatio-visual analysis, this illusion can lead to critical mistakes in interpreting the terrain. However, it is currently undocumented how commonly this effect is experienced. In this paper, we study the prevalence of the terrain reversal effect in satellite imagery through a two-stage online user experiment. The experiment was conducted with the participation of a diverse and relatively large population (n = 535). Participants were asked to identify landforms (valley or ridge?) or judge a 3D spatial relationship (is A higher than B?). When the images were rotated by 180°, the results were reversed. In a control task with ‘illusion-free’ original images, people were successful in identifying landforms, yet a very strong illusion occurred when these images were rotated 180°. Our findings demonstrate that the illusion is acutely present; thus, we need a better understanding of the problem and its solutions. Additionally, the results caution us that in an interactive environment where people can rotate the display, we might be introducing a severe perceptual problem.     

从物理大地测量角度分析重力异常用于重力反演时所存在的误差及其改正

从重力异常△g的原始定义出发，对将其用于反演地球内部密度结构时，本身所存在的不合理性及其造成的误差进行分析和估算，并给出相应的公式。主要内容包括以下3个方面。1．重力异常中所包含的两个界面所造成的物理上的不确定性分析及其所造成的几何差异估计。2．当将实际重力与模型正常重力严格地视作为矢量，传统的重力异常或重力扰动由于未考虑垂线偏差所造成的误差分析和计算。3．对试图以具有确定物理意义的地球物理模型（如PREM）代替大地测量中的正常椭球作为正常场源时，共合理性及存在的问题进行分析，并分析对其对应的正常重力之间的差异进行估算。     

地形效应校正的遥感地热异常提取

在山地复杂地形条件下,利用热红外遥感获得的地表温度分布显著受到地形的影响,真实的地热异常信息往往难以识别,热红外遥感应用于山区地热勘探受到极大限制。以广东龙川地热勘查区为研究区,初步探讨了山地环境中如何抑制地形效应,以有效提取地热异常。首先,基于Landsat ETM+遥感数据反演地表温度,分析坡向和坡度两个地形因子与地表温度的关系;然后,在此基础上,将研究区的地表温度按坡向分成3个子区(阳坡、过渡坡和阴坡),根据阳坡地表温度与坡向的线性拟合关系将其校正到水平坡度上;最后,结合地质构造分布和地表覆被情况,在3个子区识别了4处地热异常,并与已知地热点进行比较验证。结果表明:坡向分区和阳坡坡度校正能够有效抑制地形效应,提高遥感地热异常识别精度,为山区地热资源的预测评价提供新思路。     

Simulation of regional gravity field recovery from satellite gravity gradiometer data using collocation and FFT

When planning a satellite gravity gradiometer (SGG) mission, it is important to know the quality of the quantities to be recovered at ground level as a function of e.g. satellite altitude, data type and sampling rate, and signal variance and noise. This kind of knowledge may be provided either using the formal error estimates of wanted quantities using least-squares collocation (LSC) or by comparing simulated data at ground level with results computed by methods like LSC or Fast Fourier Transform (FFT). Results of a regional gravity field recovery in a 10^o×20^o area surrounding the Alps using LSC and FFT are reported. Data used as observations in satellite altitude (202 or161 km) and for comparison at ground level were generated using theOSU86F coefficient set, complete to degree 360. These observations are referred to points across simulated orbits. The simulated quantities were computed for a 45 days mission period and 4 s sampling. A covariance function which also included terms above degree 360 was used for prediction and error estimation. This had the effect that the formal error standard deviation for gravity anomalies were considerably larger than the standard deviations of predicted minus simulated quantities. This shows the importance of using data with frequency content above degree 360 in simulation studies. Using data at202 km altitude the standard deviation of the predicted minus simulated data was equal to8.3 mgal for gravity and0.33 m for geoid heights.     

Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data

A global geopotential model, like EGM2008, is not capable of representing the high-frequency components of Earth’s gravity field. This is known as the omission error. In mountainous terrain, omission errors in EGM2008, even when expanded to degree 2,190, may reach amplitudes of 10 cm and more for height anomalies. The present paper proposes the utilisation of high-resolution residual terrain model (RTM) data for computing estimates of the omission error in rugged terrain. RTM elevations may be constructed as the difference between the SRTM (Shuttle Radar Topography Mission) elevation model and the DTM2006.0 spherical harmonic topographic expansion. Numerical tests, carried out in the German Alps with a precise gravimetric quasigeoid model (GCG05) and GPS/levelling data as references, demonstrate that RTM-based omission error estimates improve EGM2008 height anomaly differences by 10 cm in many cases. The comparisons of EGM2008-only height anomalies and the GCG05 model showed 3.7 cm standard deviation after a bias-fit. Applying RTM omission error estimates to EGM2008 reduces the standard deviation to 1.9 cm which equates to a significant improvement rate of 47%. Using GPS/levelling data strongly corroborates these findings with an improvement rate of 49%. The proposed RTM approach may be of practical value to improve quasigeoid determination in mountainous areas without sufficient regional gravity data coverage, e.g., in parts of Asia, South America or Africa. As a further application, RTM omission error estimates will allow refined validation of global gravity field models like EGM2008 from GPS/levelling data.     

Downward continuation of the free-air gravity anomalies to the ellipsoid using the gradient solution and terrain correction—An attempt of global numerical computations

The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models. The basic method for the downward continuation is the gradient solution (theg ₁ term). The terrain correction has also been computed because of the role it can play as a correction term when calculating harmonic coefficients from surface gravity data. Theg ₁ term and the terrain correction were expanded into the spherical harmonics up to180 ^th order. The corrections (theg ₁ term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg ₁ term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical).     

Exploring gravity field determination from orbit perturbations of the European Gravity Mission GOCE

 A comparison was made between two methods for gravity field recovery from orbit perturbations that can be derived from global positioning system satellite-to-satellite tracking observations of the future European gravity field mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer). The first method is based on the analytical linear orbit perturbation theory that leads under certain conditions to a block-diagonal normal matrix for the gravity unknowns, significantly reducing the required computation time. The second method makes use of numerical integration to derive the observation equations, leading to a full set of normal equations requiring powerful computer facilities. Simulations were carried out for gravity field recovery experiments up to spherical harmonic degree and order 80 from 10 days of observation. It was found that the first method leads to large approximation errors as soon as the maximum degree surpasses the first resonance orders and great care has to be taken with modeling resonance orbit perturbations, thereby loosing the block-diagonal structure. The second method proved to be successful, provided a proper division of the data period into orbital arcs that are not too long. Received: 28 April 2000 / Accepted: 6 November 2000     

Gravity field approximation in the area of Greece with emphasis on local characteristics

The accuracy of the gravity field approximation depends on the amount of the available data and their distribution as well as on the variation of the gravity field. The variation of the gravity field in the Greek mainland, which is the test area in this study, is very high (the variance of point free air gravity anomalies is 3191.5mgal ²). Among well known reductions used to smooth the gravity field, the complete isostatic reduction causes the best possible smoothing, however remain strong local anomalies which disturb the homogeneity of the gravity field in this area. The prediction of free air gravity anomalies using least squares collocation and regional covariance function is obtained within a ±4 ... ±19mgal accuracy depending on the local peculiarities of the free air gravity field. By taking into account the topography and its isostatic compensation with the usual remove-restore technique, the accuracy of the prediction mentioned obove was increased by about a factor of 4 and the prediction results become quite insensitive to the covariance function used (local or regional). But when predicting geoidal heights, in spite of using the smoothed field, the prediction results remain still depend on the covariance function used in such a way that differences up to about 50cm/100km result between relative geoidal heights computed with regional or local covariance functions.     

Airborne gravimetry used in precise geoid computations by ring integration

Two detailed geoids have been computed in the region of North Jutland. The first computation used marine data in the offshore areas. For the second computation the marine data set was replaced by the sparser airborne gravity data resulting from the AGMASCO campaign of September 1996. The results of comparisons of the geoid heights at on-shore geometric control showed that the geoid heights computed from the airborne gravity data matched in precision those computed using the marine data, supporting the view that airborne techniques have enormous potential for mapping those unsurveyed areas between the land-based data and the off-shore marine or altimetrically derived data. Received: 7 July 1997 / Accepted: 22 April 1998     

用近邻权法推估重力异常

本文根据非参数回归理论提出了一种推估空间重力异常的方法,此方法不需要任何假定,计算简单。试算结果表明,其精度与现行的其它方法相符。     

Computing components of the gravity field induced by distant topographic masses and condensed masses over the entire Earth using the 1-D FFT approach

 A new method for computing gravitational potential and attraction induced by distant, global masses on a global scale has been developed. The method uses series expansions and the well known one-dimensional fast Fourier transform (1-D FFT) method. It has been proven to be significantly faster than quadrature while being equally accurate. Various quantities were studied to cover the two primary applications of the Stokes–Helmert scheme of modeling effects. These two applications (or paths), given the names R/r/D and R/D/r, are briefly discussed, although the primary objective of the paper is to provide computational information to either path, rather than choosing one path as preferable to the other. It is further shown that the impact of masses outside a 4-degree cap can impact the absolute computation of the geoid at more than 1 cm, and should therefore be included in all local geoid computations seeking that accuracy. Received: 13 December 2000 / Accepted: 3 September 2001     

联合GOCE卫星轨道和重力梯度数据严密求解重力场的模拟研究

论述了联合卫星轨道和重力梯度数据严密求解重力场的方法及数据处理方案,研究了GOCE重力场反演中有色噪声的AR去相关滤波、病态法方程的Kaula正则化和观测值最优加权的方差分量估计等关键问题。模拟结果表明:①极空白问题会降低法方程求解的稳定性,导致低次位系数的求解精度较低,而Kaula正则化可有效用于GOCE病态法方程的求解,并得到合理稳定的解;②重力梯度有色噪声会降低GOCE重力场求解的整体精度,特别是对低阶位系数的影响最为明显,而AR去相关滤波法可有效处理有色噪声,但解算结果仍含有低频误差;③方差分量估计可有效确定SST和SGG两类观测值的最优权比,并且有色噪声造成的低频误差经过联合求解后得到了抑制;④利用30d、5s采样的GOCE模拟数据恢复200阶次的重力场模型,其大地水准面和重力异常精度在纬度±83°范围内分别为±3.81cm和±1.056mGal。