Error-covariances of the estimates of spherical harmonic coefficients computed by LSC,using second-order radial derivative functionals associated with realistic GOCE orbits

Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC.     

A comparison of Stokes' numerical integration and collocation,and a new combination technique

Summary Basically two different evaluation methods are available to compute geoid heights from residual gravity anomalies in the inner zone: numerical integration and least squares collocation.If collocation is not applied to a global gravity data set, as is usually the case in practice, its result will not be equal to the numerical integration result. However, the cross covariance function between geoid heights and gravity anomalies can be adapted such that the geoid contribution is computed only from a small gravity area up to a certain distance _o from the computation point. Using this modification, identical results are obtained as from numerical integration.Applying this modification makes the results less dependent on the covariance function used. The difference between numerical integration and collocation is mainly caused by the implicitly extrapolated residual gravity anomaly values, outside the original data area. This extrapolated signal depends very much on the covariance function used, while the interpolated values within the original data area depend much less on it.As a sort of by-product, this modified collocation formula also leads to a new combination technique of numerical integration and collocation, in which the optimizing practical properties of both methods are fully exploited.Numerical examples are added as illustration.     

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.     

Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data

The estimation of a local empirical covariance function from a set of observations was done in the Faeroe Islands region. Gravity and adjusted Seasat altimeter data relative to theGPM2 spherical harmonic approximation were selected holding one value in celles of1/8°×1/4° covering the area. In order to center the observations they were transformed into a locally best fitting reference system having a semimajor axis1.8 m smaller than the one ofGRS80. The variance of the data then was273 mgal ² and0.12 m ² respectively. In the calculations both the space domain method and the frequency domain method were used. Using the space domain method the auto-covariances for gravity anomalies and geoid heights and the cross-covariances between the quantities were estimated. Furthermore an empirical error estimate was derived. Using the frequency domain method the auto-covariances of gridded gravity anomalies was estimated. The gridding procedure was found to have a considerable smoothing effect, but a deconvolution made the results of the two methods to agree. The local covariance function model was represented by a Tscherning/Rapp degree-variance model,A/((i−1)(i−2)(i+24))(R _B /R _E )²ⁱ⁺², and the error degree-variances related to the potential coefficient setGPM2. This covariance function was adjusted to fit the empirical values using an iterative least squares inversion procedure adjusting the factor A, the depth to the Bjerhammar sphere(R _E −R _B ), and a scale factor associated with the error degree-variances. Three different combinations of the empirical covariance values were used. The scale factor was not well determined from the gravity anomaly covariance values, and the depth to the Bjerhammar sphere was not well determined from geoid height covariance values only. A combination of the two types of auto-covariance values resulted in a well determined model.     

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.     

Azimuth-dependent statistics for interpolating geodetic data

Most authors using statistical interpolation techniques on geodetic data have assumed isotropy for the undulation autocorrelation. Tests of actual data,414 deflections of the vertical, indicate this assumption is not valid. The results of interpolation, however, are not very sensitive to the parameters in the covariance function. A special limiting case for which statistical interpolation degenerates into a completely deterministic process is given in the spherical domain. In this case the covariance function has absolutely no effect on the results, so that the covariance of the output of a prediction need not be that assumed for the interpolation. This provides a self-correcting process whereby the information in the data corrects for a poor choice of covariance function. Estimates of the precision of the interpolation, on the other hand, are very sensitive to the covariance function, particularly to the modeling of azimuth dependence. A simple procedure for generalizing isotropic functions to azimuth dependence is given, which provides sufficiently accurate estimates of precision. The advisability of trend removal is illustrated by some numerical examples.     

Gibbs sampler for computing and propagating large covariance matrices

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model. AcknowledgementsThe authors are indebted to Pieter Visser and Pavel Ditmar for providing simulation output that was used in the GOCE error generation experiments. Furthermore, the NASA/NIMA/OSU team is acknowledged for providing public ftp access to the EGM96 error covariance matrix. The two anonymous reviewers are thanked for their valuable comments.     

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     

利用MLSC法求解高程异常

运用常规的拟合模型求解大范围高程异常必然存在较大的模型误差,鉴于模型误差的不确定性,可以将模型误差看作信号采用最小二乘配置法来处理,最小二乘配置中协方差函数的确定是一个比较棘手的问题,介于此,本文提出了移动最小二乘配置法,对不同的待定点建立不同的配置模型,并通过定权的方式解决复杂的协方差函数计算问题,对一个大测区的部分GPS水准数据进行解算,获得了满意的结果。     

变差函数权值估算的移动最小二乘匹配点云粗差剔除

现有密集匹配点云数据已实现了地表3维信息的精细化表达,然而由于误匹配,此类点云往往包含一定数量粗差点并影响后续应用的处理效果。针对此类数据中误匹配所产生粗差点的剔除问题,将变差甬数引入移动最小二乘(MLS)粗差剔除算法。变差函数对MLS拟合区域内点对间的相关性进行估算,以此为依据设置权值对最小二乘的结果进行优化;然后利用MLS局部、分区域地对点云进行曲面拟合;最终根据拟合结果剔除粗差。利用A3数字测图系统生成的城区、山区密集点云数据进行实验,并将处理结果、等权MLS处理结果与人工剔除结果进行对比。实验结果表明该算法可有效对点云中的粗差进行剔除,相较于等权MLS粗差剔除算法,陔算法在城区、山区的误判率分别降低了5.16%和1.31%。     

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.     

大地测量相关观测抗差估计理论

相关观测异常诊断、质量控制是测量数据处理领域亟待解决的难题之一。分别从方差膨胀模型和相关权元素压缩模型入手研究了相关观测的质量控制理论和方法；给出了误差影响函数；构造了方差膨胀函数和权因子收缩函数；利用观测量的等价协方差阵和等价权矩阵讨论了相关观测质量控制的计算方法。该等价协方差矩阵和等价权矩阵不仅保持了原有协方差矩阵和权矩阵的对称性，而且保持了原有协方差矩阵的相关性不变。计算结果表明异常观测的方差膨胀法和等价权法能有效地控制异常观测对参数估值的影响。     

一种遥感土壤湿度误差空间协方差的估算方法

针对遥感反演土壤湿度空间相关的误差协方差难以估计的问题,提出了一种遥感反演数据误差空间协方差估算方法——3类数据集成分析误差协方差（triple collocation covariance,TC_Cov）,将土壤湿度场的每个单元（像元）看作一个空间随机变量,用两个随机变量表示的土壤湿度值的时间序列作为样本进行空间协方差估计,由任何两个随机变量的协方差形成多个随机变量（随机场）的协方差矩阵。利用先进散射计（ad-vanced scatterometer,ASCAT）和热带降雨测量卫星（tropical rainfall measuring mission,TRMM）的遥感土壤湿度数据以及ERA-Interim土壤湿度再分析数据作为TC_Cov方法的输入数据,分别估算了ERA-Interim、AS-CAT和TRMM在澳大利亚Murrumbidgee流域的土壤湿度误差协方差矩阵,验证了估算方法的合理性和可行性。     

Fast spherical collocation: theory and examples

 It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This idea has been generalized for the method of least-squares collocation, when using an isotropic covariance function or reproducing kernel. The data only need to be at the same altitude and of the same kind for each latitude. This permits, for example, the combination of gravity data at the surface of the Earth and data at satellite altitude, when the orbit is circular. Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose that it is required to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of only N × N. Results of simulation studies using the method are described. Received: 18 October 2001 / Accepted: 4 October 2002 Correspondence to: F. Sansò     

平面控制网相对点位误差的合理评定与分析

时平面控制网的相对点位精度评定问题做了一些理论分析,探讨了在不同基准条件下协因数阵的转换关系,及依据转换后的协因数阵来评定相对点位精度的问题.在有关数学模型推导的基础上,利用MATLAB语言编制了控制网平差及相对点位精度的评定程序,并结合具体的工程进行了算例分析.     

海洋重力场特征统计模型计算与分析

海洋重力场特征参数在地球重力场逼近计算和海上测量优化设计中具有重要的应用价值。基于卫星测高重力在海域具有覆盖范围广且分布均匀的独特优势,提出了利用最新卫星测高重力数据集开展海洋重力场特征统计模型计算和分析的研究方案,给出了代表误差和协方差函数模型参数的计算公式,定义并研究了海洋广义布格重力异常的变化特征,提出了等精度和非等精度拟合经验协方差函数的计算模型。利用中国近海及西太平洋海区超过50万个5'×5'方块的1'×1'网格卫星测高重力异常数据,首次计算得到一组有代表性的中国周边海域重力场特征统计模型参数,较好地揭示了海洋重力场有别于陆地重力场的变化特征,利用海面船测重力数据对计算结果进行了可靠性检核,提出了相应的模型参数修正方案和使用建议。     

航空重力向下延拓的最小二乘配置Tikhonov正则化法

基于最小二乘配置法向下延拓航空重力的过程中,由于协方差矩阵严重病态,影响延拓结果的稳定性和精度。针对这一问题,提出了航空重力向下延拓的最小二乘配置Tikhonov正则化法。基于全球协方差函数模型建立航空重力数据与地面重力数据的协方差关系,引入基于广义交叉验证法,选择正则化参数的Tikhonov正则化法改善协方差矩阵的病态性,抑制观测噪声对延拓结果的放大影响。基于EGM2008重力场模型,设计了山区、丘陵和海域3种不同地形区域的航空重力数据向下延拓的仿真实验,实验结果验证了该方法的有效性。     

Spectral accuracy of Jgm3 from satellite crossover altimetry

A detailed accuracy assessment of the geopotential model Jgm3 is made based on independent single- and dual-satellite sea-height differences at crossovers from altimetry with Jgm3-based orbits. These differences, averaged over long time spans and in latitude bands, are converted to spectra (latitude-lumped coefficients) by least-squares estimation. The observed error spectra so obtained are then compared directly to error projections for them from the Jgm3 variance–covariance matrix. It is found from these comparisons that Jgm3 is generally well calibrated with respect to the crossover altimetry of and between Geosat, TOPEX/Poseidon (T/P), and Ers 1. Some significant discrepancies at a few lower orders (namely m=1 and 3) indicate a need for further improvement of Jgm3. A companion calibration (by order) of the geopotential model Jgm2 shows its variance–covariance matrix also to be generally well calibrated for the same single- and dual-satellite altimeter data sets (but based on Jgm2 orbits), except that the error projections for Geosat are too pessimistic. The analysis of the dual-satellite crossovers reveals possible relative coordinate system offsets (particularly for Geosat with respect to T/P) which have been discussed previously. The long-term detailed seasonally averaged Geosat sea level with respect to T/P (covering 1985–1996) should be useful in gauging the relative change in sea level between different parts of the ocean over the single 4-year gap between these missions (1988–1992). Received: 16 February 1998 / Accepted: 25 November 1998     

Avoiding numerical stability problems of long duration DGPS/INS Kalman filters

A current pursuit of the geodetic community is the optimal integration of differential GPS (DGPS) and inertial navigation system (INS) data streams for precise and efficient position and gravity vector surveying. Therein a complete INS and multiple-antenna GPS receiver payload, mounted on a moving platform, is used in conjunction with a network of ground-fixed single antenna GPS receivers. This paper presents a complete, GPS-based, external updating measurement model for the applicable Kalman filter. The model utilizes four external observation types for every GPS satellite in-view: DGPS range differences, single phase differences, and single phase-rate differences; as well as the mobile, multipleantenna GPS receiver's measurement of theerrors in the INS's estimate of the phase difference between any two vehicle-borne GPS antennae. Although not widely conveyed in the geodetic world, the inertial navigation community has long known that traditional Kalman filter covariance propagation recurrences are inherently unstable when such highly accurate external updates are repeatedly applied (every 1 second) over long time durations. A hybrid square root covariance/U — D covariance factorization approach is a numerically stable alternative and is reviewed herein. The hybrid makeup of the algorithm is necessitated by the correlated nature of the fourth type of GPS external measurement listed above (each vehicle-borne GPS antenna formstwo baselines). Such measurement correlations require a functional transformation of the overall external updating model to permit the multiple updates (simultaneously available at each updating epoch) to be sequentially (and efficiently) processed. An appropriate transformation is given. Stable covariance propagation relationships are presented and the transformed Kalman gain is also furnished and its use in the determination of the externally updated error states is discussed. Specific DGPS/INS instabilities produced by the traditional recurrences are displayed. The stable alternative method requires about 25% more CPU time than the traditional Kalman recurrences. With the ever-increasing computational speeds of microprocessors, this added CPU time is of no real concern.     

北京城市下垫面大气CO2反演算法

大气温室气体监测仪GMI（Greenhouse gases Monitor Instrument）是高分五号（GF-5）卫星载荷之一,主要用于全球温室气体含量监测和碳循环研究。高精度反演是卫星大气CO₂遥感的基本需求。地表反射率影响卫星遥感辐射量及辐射传输过程中的地气耦合过程,严重制约着CO₂的反演精度,针对GMI开发高精度的大气CO₂反演算法,地表反射是一个需要重点考虑的因素。城市是CO₂重要的发射源,且城市下垫面存在明显的二向反射特性,加上城市大气条件不良,复杂的地气耦合效应存在这都考验反演算法的准确性和鲁棒性。本文针对北京城市地区,利用2011年—2016年共5年的MODIS（MODerate-resolution Imaging Spectroradiometer）地表二向反射分布函数BRDF（Bidirectional Reflectance Distribution Function）数据,构建了适合利用单次观测数据反演的BRDF模型,并提出一种同时反演地表BRDF参数和大气CO₂含量的算法。结果表明在550 nm波长处气溶胶光学厚度AOD（Aerosol Optical Depth）小于0.4时,大部分GMI模拟数据的反演误差控制在0.5%（~2 ppm）内。利用GOSAT （Greenhouse gases Observing SATellite）实测数据的反演结果与修正后的日本国立环境研究所NIES（National Institute for Environmental Studies）反演结果进行对比,其平均误差为1.25 ppm,相关性达到0.85。本算法满足GMI数据在北京城市区域高精度CO₂反演的需求,并使得反演高值气溶胶区域数据成为可能,增加了GMI观测数据的利用率。