航空重力测量数据向下延拓中空间协方差函数特性研究

本文通过对航空重力测量数据的分析,建立起具有空间相关特性的空间协方差函数模型。利用所建立的模型,将航空重力测量试验数据采用最小二乘配置的方法向下延拓到地面,与地面已知重力数据进行比较。由比较结果,分析了空间协方差函数模型对航空重力测量数据向下延拓结果的影响,并表明所建立的空间协方差函数模型,满足协方差函数特性,且在局部区域使用能够达到满意的精度要求。     

Solution of the downward continuation problem by collocation

Summary The modern satellite techniques enable to measure the gravity acceleration in space. The downward continuation of this measurements to the earth's surface is by the closed theory practically impossible. Using the method of collocation we can estimate the values of Δ g in the limited region underneath the observation with a little loss of accuracy. Results of some numerical tests for elevation of 300 km are presented.     

重力异常向上延拓严密改化模型及向下延拓应用

重力异常向上延拓全球积分模型在航空重力测量数据质量评估和向下延拓迭代计算等领域具有广泛的应用。为了消除积分核函数奇异性影响,需要对该模型进行基于积分恒等式的移去-恢复转换及全球积分域的分区改化处理。在此过程中,传统改化处理方法往往忽略了全球积分过渡到局域积分引起的积分恒等式偏差影响,从而导致不必要的计算模型误差,最终影响向上延拓计算结果的可靠性,甚至影响向下延拓迭代解算结果的稳定性。针对此问题,本文开展了重力异常向上延拓积分模型改化及向下延拓应用分析研究,依据实测数据保障条件和积分恒等式适用条件要求,导出了重力异常向上延拓积分模型的分步改化公式,提出了补偿传统改化模型缺陷的修正公式,并将最终的严密改化模型应用于重力异常向下延拓迭代解算。使用超高阶地球位模型EGM2008作为标准位场开展数值计算检验,分别对重力异常向上延拓分步改化模型的计算精度及在向下延拓迭代解算中的应用效果进行了检核评估,验证了采用严密改化模型的必要性和有效性。     

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.     

Spherical integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients

New integral formulas for upward/downward continuation of gravitational gradients onto gravitational gradients are derived in this article. They provide more options for continuation of gravitational gradient combinations and extend available mathematical apparatus formulated for this purpose up to now. The starting point represents the analytical solution of the spherical gradiometric boundary value problem in the spatial domain. Applying corresponding differential operators on the analytical solution of the spherical gradiometric boundary value problem, a total of 18 integral formulas are provided. Spatial and spectral forms of isotropic kernels are given and their behaviour for parameters of a GOCE-like satellite is investigated. Correctness of the new integral formulas and the isotropic kernels is tested in a closed-loop simulation. The derived integral formulas and the isotropic kernels form a theoretical basis for validation purposes and geophysical applications of satellite gradiometric data as provided currently by the GOCE mission. They also extend the well-known Meissl scheme.     

Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains

We investigate the stability of a discrete downward continuation problem for geoid determination when the surface gravity observations are harmonically continued from the Earth's surface to the geoid. The discrete form of Poisson's integral is used to set up the system of linear algebraic equations describing the problem. The posedness of the downward continuation problem is then expressed by means of the conditionality of the matrix of a system of linear equations. The eigenvalue analysis of this matrix for a particularly rugged region of the Canadian Rocky Mountains shows that the discrete downward continuation problem is stable once the topographical heights are discretized with a grid step of size 5 arcmin or larger. We derive two simplified criteria for analysing the conditionality of the discrete downward continuation problem. A comparison with the proper eigenvalue analysis shows that these criteria provide a fairly reliable view into the conditionality of the problem.The compensation of topographical masses is a possible way how to stabilize the problem as the spectral contents of the gravity anomalies of compensated topographical masses may significantly differ from those of the original free-air gravity anomalies. Using surface gravity data from the Canadian Rocky Mountains, we investigate the efficiency of highly idealized compensation models, namely the Airy-Heiskanen model, the Pratt-Hayford model, and Helmert's 2nd condensation technique, to dampen high-frequency oscillations of the free-air gravity anomalies. We show that the Airy-Heiskanen model reduces high-frequencies of the data in the most efficient way, whereas Helmert's 2nd condensation technique in the least efficient way. We have found areas where a high-frequency part of the surface gravity data has been completely removed by adopting the Airy-Heiskanen model which is in contrast to the nearly negligible dampening effect of Helmert's 2nd condensation technique. Hence, for computation of the geoid over the Canadian Rocky Mountains, we recommend the use of the Airy-Heiskanen compensation model to reduce the gravitational effect of topographical masses.In addition, we propose to solve the discrete downward continuation problem by means of a simple Jacobi's iterative scheme which finds the solution without determining and storing the matrix of a system of equations. By computing the spectral norm of the matrix of a system of equations for the topographical 5 × 5 heights from a region of the Canadian Rocky Mountains, we rigorously show that Jacobi's iterations converge to the solution; that the problem was well posed then ensures that the solution is not contaminated by large roundoff errors. On the other hand, we demonstrate that for a rugged mountainous region of the Rocky Mountains the discrete downward continuation problem becomes ill-conditioned once the grid step size of both the surface observations and the solution is smaller than 1 arcmin. In this case, Jacobi's iterations converge very slowly which prevents their use for searching the solution due to accumulating roundoff errors.     

On the error of analytical downward continuation of the earth's external gravitational potential on and inside the earth's surface

The error of the analytical downward continuation was studied under planar approximation in a previous paper by the author (1994). This error is studied further by taking the earth's curvature into account in this paper. It is shown that if a smoothing procedure is applied, the analytical downward continuation can be used to determine the earth's gravitational potential on and above the earth's surface to any required accuracy. For the potential inside the earth's topographic mass the error can also be controlled and corrected. Received 1 August 1995; Accepted 13 September 1996     

The geopotential to (14, 14) from a combination of satellite and gravimetric data

A set of2261 5°×5° mean anomalies were used alone and with satellite determined harmonic coefficients of the Smithsonian' Institution to determine the geopotential expansion to various degrees. The basic adjustment was carried out by comparing a terrestrial anomaly to an anomaly determined from an assumed set of coefficients. The (14, 14) solution was found to agree within ±3 m of a detailed geoid in the United States computed using1°×1° anomalies for an inner area and satellite determined anomalies in an outer area. Additional comparisons were made to the input anomaly field to consider the accuracy of various harmonic coefficient solutions. A by-product of this investigation was a new γ_E=978.0463 gals in the Potsdam system or978.0326 gals in an absolute system if −13.7 mgals is taken as the Potsdam correction. Combining this value of γ_E withf=1/298.25, KM=3.9860122·10 ²² cm ³ /sec ² , the consistent equatorial radius was found to be6378143 m.     

The analytical continuation bias in geoid determination using potential coefficients and terrestrial gravity data

One important application of an Earth Gravity Model (EGM) is to determine the geoid. Since an EGM is represented by an external-type series of spherical harmonics, a biased geoid model is obtained when the EGM is applied inside the masses in continental regions. In order to convert the downward-continued height anomaly to the corresponding geoid undulation, a correction has to be applied for the analytical continuation bias of the geoid height. This technique is here called the geoid bias method. A correction for the geoid bias can also be utilised when an EGM is combined with terrestrial gravity data, using the combined approach to topographic corrections. The geoid bias can be computed either by a strict integral formula, or by means of one or more terms in a binomial expansion. The accuracy of the lowest binomial terms is studied numerically. It is concluded that the first term (of power H²) can be used with high accuracy up to degree 360 everywhere on Earth. If very high mountains are disregarded, then the use of the H² term can be extended up to maximum degrees as high as 1800. It is also shown that the geoid bias method is practically equal to the technique applied by Rapp, which utilises the quasigeoid-to-geoid separation. Another objective is to carefully consider how the combined approach to topographic corrections should be interpreted. This includes investigations of how the above-mentioned H² term should be computed, as well as how it can be improved by a correction for the residual geoid bias. It is concluded that the computation of the combined topographic effect is efficient in the case that the residual geoid bias can be neglected, since the computation of the latter is very time consuming. It is nevertheless important to be able to compute the residual bias for individual stations. For reasonable maximum degrees, this can be used to check the quality of the H² approximation in different situations.Acknowledgement The author would like to thank Prof. L.E. Sjöberg for several ideas and for reading two draft versions of the paper. His support and constructive remarks have improved its quality considerably. The valuable suggestions from three unknown reviewers are also appreciated.     

The combination of GNSS-levelling data and gravimetric (quasi-) geoid heights in the presence of noise

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.     

轻小型无人机飞控测姿数据辅助测图精度分析

针对轻小型无人机系统难以集成重量、尺寸较大的定姿定位系统,造成高精度传感器定姿定位数据缺失的问题,该文研究了基于无人机飞控系统定姿定位数据辅助测图的方法,统计分析了飞控系统姿态测量精度以及基于姿态数据辅助的测图精度。YS09无人机飞控系统姿态测量误差较大,姿态中误差低于±3°,航向中误差低于±5°。在直接地理定向的情况下,姿态角影响水平和高程测图精度,其中对平面精度影响小于0.1 H,对高程精度影响小于0.3 H;偏航角影响平面测图精度,影响约为0.098 H。结果表明:基于飞控姿态数据辅助测图精度较低,难以满足大比例测图的精度需求,仅适用于应急条件下对精度要求较低的测图需求。     

城区机载LiDAR数据与航空影像的自动配准

为解决机载LiDAR数据与航空影像集成应用中二者的配准问题,提出了一种机载LiDAR数据与航空影像配准的方法。首先,直接在LiDAR点云中提取建筑物3维轮廓线,通过将轮廓线规则化得到由两条相互垂直的直线段组成的建筑物角特征,并在航空影像上提取直线特征;然后,根据影像初始外方位元素将建筑物角特征投影到航空影像上,并采用一定的相似性测度在影像上寻找同名的影像角特征;最后,将角特征的角点当作控制点,利用传统的摄影测量光束法区域网平差解求影像新的外方位元素。解算过程中采用循环迭代策略。本方法的主要特点是,直接从LiDAR点云中提取线特征,避免了常规方法从距离图(或强度图)中提取线特征所产生的内插误差。通过与现有基于点云强度图的配准方法的对比实验表明,在低精度初始外方位元素的辅助下,本文方法能够达到较高的配准精度。     

Acquisition of directional parametes in aerial images based on DEM data

This paper develops a method which can be used to assist aerial navigation by determining the spatial position and posture of the aerial photographic plane. After the method, aerial images match known DEM to capture the spatial position and posture. Some aerial images and terrain data are used to testify our method. Compared with those of analytic and stereo mappers, the results by our method are correspondent to real measurements well.     

Acquisition of Directional Parameters in Aerial Images Based on DEM Data

This paper develops a method which can be used to assist aerial navigation by determining the spatial position and posture of the aerial photographic plane. After the method, aerial images match known DEM to capture the spatial position and posture. Some aerial images and terrain data are used to testify our method. Compared with those of analytic and stereo mappers, the results by our method are correspondent to real measurements well.     

A general method for an explicit determination of the shape of the earth from gravimetric data

Summary The principal formulae of the geophysical geodesy are based on the famous explicit expression of Stokes (1849). Up to now, there has been no method for a computation of the corresponding explicit expression of a (non-spherical) surface with masses outside the geoid. In this paper there is a solution of this problem. Another paper on this subject was presented to “Nordiska geodetm?tet” in Copenhagen May 1959 (in Swedish).     

Performance evaluation of automated approaches to building detection in multi-source aerial data

Automated approaches to building detection in multi-source aerial data are important in many applications, including map updating, city modeling, urban growth analysis and monitoring of informal settlements. This paper presents a comparative analysis of different methods for automated building detection in aerial images and laser data at different spatial resolutions. Five methods are tested in two study areas using features extracted at both pixel level and object level, but with the strong prerequisite of using the same training set for all methods. The evaluation of the methods is based on error measures obtained by superimposing the results on a manually generated reference map of each area. The results in both study areas show a better performance of the Dempster-Shafer and the AdaBoost methods, although these two methods also yield a number of unclassified pixels. The method of thresholding a normalized DSM performs well in terms of the detection rate and reliability in the less vegetated Mannheim study area, but also yields a high rate of false positive errors. The Bayesian methods perform better in the Memmingen study area where buildings have more or less the same heights.     

无重力数据情况下阜新似大地水准面的确定

将地球位模型计算的阜新似大地水准面拟合于GPS水准实测的似大地水准面。其方法是对每一个GPS水准点,以选定的地球位模型计算其高程异常值,并与其GPS水准实测高程异常值进行比较,得到高程异常的改正值,以此改正附近2.5′×2.5′格网点上用地球位模型计算的高程异常。     

Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data

The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness.     

无人机红外传感器改制及数据获取研究

针对云雾复杂气候条件下无人机可见光影像难以获取的问题,文章在分析了地质灾害应急遥感监测伴随的复杂气候条件的基础上,根据目前无人机传感器的构造原理,通过对各种红外滤片的分析和比较,研究基于红外滤片元器件的无人机传感器红外改制技术和红外影像数据增强方法,对航摄传感器进行红外改装。实验结果表明,无人机红外传感器在多云雾条件下能够获取高精度清晰影像,有效解决云雾条件下地质灾害应急数据获取难题,为地质灾害应急决策提供参考。