最小二乘配置法中局部协方差函数的计算

随着 GPS日益广泛的应用及精度的不断提高 ,在有些实际应用中利用 GPS来代替传统的水准测量进行高程控制已成为可能 ,这也进一步提出了对高精度大地水准面的需求。快速傅立叶变换 (FFT)是目前计算大地水准面比较常用的方法之一 ,但需要将重力观测量进行内插得到规则格网上的平均重力异常。利用最小二乘配置法计算大地水准面可直接利用已有的观测值进行计算 ,同时可综合利用不同类型的数据 ,如重力异常和垂线偏差等计算大地水准面 ,因此最小二乘配置法仍有广泛的应用 ,但制约最小二乘配置应用的关键问题是局部协方差函数的计算。将主要讨论最小二乘配置法中局部协方差函数的计算 ,使所用的协方差函数能更好地反映已知的数据 ,从而获得更精确的结果。     

联合重力异常和GPS水准数据的最小二乘配置方法

本文对最小二乘配置的基本方法进行了简要介绍,讨论了局部协方差函数模型的确定方法,并利用GPS水准和重力数据,根据移去恢复法,运用最小二乘配置方法进行重力异常和GPS水准的联合配置计算,确定了某市2′30″×2′30″区域似大地水准面模型,并将最终结果与GPS水准数据进行比较分析,通过检核,精度达到±1.6cm。     

利用自适应最小二乘配置的GPS水准与重力似大地水准面的拟合

针对GPS水准与重力似大地水准面之差中存在系统误差的问题,使用最小二乘配置估计信号大小,来提高似大地水准面的拟合精度.对于最小二乘配置的噪声与信号的协方差之间的关系不合理,采用自适应因子纠正两者之间的关系,并首次将自适应最小二乘配置算法应用于似大地水准面精化.最后使用我国东部面积将近2万km2的城市A的数据进行验证,计算结果表明,最小二乘配置及自适应最小二乘配置在一定程度上能够提高拟合效果,使似大地水准面更接近于真实值,实现了国内较大城市面积的1.0 cm检核精度的区域似大地水准面成果.     

组合法精化胶州市大地水准面

本文利用全球重力位模型、胶州市地面重力观测数据、胶州市GPS水准数据和数字地面模型(DTM),采用组合法应用移去-恢复技术计算剩余大地水准面,并与地球位模型计算的高程异常进行拟合,得到该地区重力似大地水准面,再和布测、计算得到的GPS/水准所构成的几何大地水准面拟合,利用多项式拟合完成系统改正,获得最终的大地水准面结果及相关的精度信息。     

组合法精化胶州市大地水准面

本文利用全球重力位模型、胶州市地面重力观测数据、胶州市GPS水准数据和数字地面模型(DTM),采用组合法应用移去-恢复技术计算剩余大地水准面,并与地球位模型计算的高程异常进行拟合,得到该地区重力似大地水准面,再和布测、计算得到的GPS/水准所构成的几何大地水准面拟合,利用多项式拟合完成系统改正,获得最终的大地水准面结果及相关的精度信息。     

Kriging方法结合最小二乘配置在GPS高程拟合中的应用

本文利用Kriging方法结合最小二乘配置将GPS高程转换成正常高.研究了将Kriging方法中的变异函数用于计算最小二乘配置中的协方差的方法,并对一局部GPS水准网的高程作了拟合计算.通过将最小二乘配置法与平面拟合模型和多面函数拟合模型等进行比较,其外符合精度从最大的±0.0277m提高到±0.0162m.     

最小二乘配置方法确定局部大地水准面的研究

从最小二乘配置方法的原理出发 ,描述了最小二乘配置法中经验协方差函数的确定方法。在此基础上 ,以我国西部高山地区为例 ,利用实测点重力数据、30″× 30″数值地面模型和EGM96地球重力场模型 ,确定了该地区 2 .5′× 2 .5′大地水准面     

区域似大地水准面确定的最小二乘支持向量机方法

支持向量机(SVM)是近年来发展起来的机器学习的新方法,它较好地解决小样本、非线性、高维数、局部极小点等实际问题.文中研究支持向量机的拓展算法--最小二乘支持向量机(LSSVM),并将其应用于确定大面积复杂似大地水准面.通过工程实例并与神经网络模型和二次曲面多项式拟合模型相比较,验证确定区域似大地水准面的LSSVM方法的有效性.     

大地水准面差距计算方法的研究

本文首先证明了三种大地水准面差距计算方法(迈塞尔方法、文策尔方法、最小二乘配置法)之间的关系。通过对某盆地的大地水准面差距的计算及和多普勒结果的比较,得到了一些对计算我国大地水准面差距有益的结论。     

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。     

Modelling local GPS/levelling geoid undulations using artificial neural networks

The use of GPS for establishing height control in an area where levelling data are available can involve the so-called GPS/levelling technique. Modelling of the GPS/levelling geoid undulations has usually been carried out using polynomial surface fitting, least-squares collocation (LSC) and finite-element methods. Artificial neural networks (ANNs) have recently been used for many investigations, and proven to be effective in solving complex problems represented by noisy and missing data. In this study, a feed-forward ANN structure, learning the characteristics of the training data through the back-propagation algorithm, is employed to model the local GPS/levelling geoid surface. The GPS/levelling geoid undulations for Istanbul, Turkey, were estimated from GPS and precise levelling measurements obtained during a field study in the period 1998–99. The results are compared to those produced by two well-known conventional methods, namely polynomial fitting and LSC, in terms of root mean square error (RMSE) that ranged from 3.97 to 5.73 cm. The results show that ANNs can produce results that are comparable to polynomial fitting and LSC. The main advantage of the ANN-based surfaces seems to be the low deviations from the GPS/levelling data surface, which is particularly important for distorted levelling networks.     

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.     

Local geoid determination combining gravity disturbances and GPS/levelling: a case study in the Lake Nasser area, Aswan, Egypt

 The use of GPS for height control in an area with existing levelling data requires the determination of a local geoid and the bias between the local levelling datum and the one implicitly defined when computing the local geoid. If only scarse gravity data are available, the heights of new data may be collected rapidly by determining the ellipsoidal height by GPS and not using orthometric heights. Hence the geoid determination has to be based on gravity disturbances contingently combined with gravity anomalies. Furthermore, existing GPS/levelling data may also be used in the geoid determination if a suitable general gravity field modelling method (such as least-squares collocation, LSC) is applied. A comparison has been made in the Aswan Dam area between geoids determined using fast Fourier transform (FFT) with gravity disturbances exclusively and LSC using only the gravity disturbances and the disturbances combined with GPS/levelling data. The EGM96 spherical harmonic model was in all cases used in a remove–restore mode. A total of 198 gravity disturbances spaced approximately 3 km apart were used, as well as 35 GPS/levelling points in the vicinity and on the Aswan Dam. No data on the Nasser Lake were available. This gave difficulties when using FFT, which requires the use of gridded data. When using exclusively the gravity disturbances, the agreement between the GPS/levelling data were 0.71 ± 0.17 m for FFT and 0.63 ± 0.15 for LSC. When combining gravity disturbances and GPS/levelling, the LSC error estimate was ±0.10 m. In the latter case two bias parameters had to be introduced to account for a possible levelling datum difference between the levelling on the dam and that on the adjacent roads. Received: 14 August 2000 / Accepted: 28 February 2001     

Tuning a gravimetric quasigeoid to GPS-levelling by non-stationary least-squares collocation

This paper addresses implementation issues in order to apply non-stationary least-squares collocation (LSC) to a practical geodetic problem: fitting a gravimetric quasigeoid to discrete geometric quasigeoid heights at a local scale. This yields a surface that is useful for direct GPS heighting. Non-stationary covariance functions and a non-stationary model of the mean were applied to residual gravimetric quasigeoid determination by planar LSC in the Perth region of Western Australia. The non-stationary model of the mean did not change the LSC results significantly. However, elliptical kernels in non-stationary covariance functions were used successfully to create an iterative optimisation loop to decrease the difference between the gravimetric quasigeoid and geometric quasigeoid at 99 GPS-levelling points to a user-prescribed tolerance.     

确定大地水准面的Tikhonov最小二乘配置法

LSC法（最小二乘配置法）因能融合不同种类重力观测数据确定大地水准面的特性而受到广泛关注,但由于协方差矩阵存在病态性,微小的观测误差将被协方差矩阵的小奇异值放大,导致计算的配置结果不稳定且精度偏低。本文提出Tikhonov_LSC法,即在LSC法中引入Tikhonov正则化算法,基于GCV法选择协方差矩阵的正则化参数,利用正则化参数修正协方差矩阵的小奇异值,以抑制其对观测误差的放大影响。基于Tikhonov_LSC法计算大地水准面,能有效提高其稳定性和精度。通过以EGM2008重力场模型分别计算山区、丘陵和海域重力异常作为基础数据确定相应区域大地水准面的实验,验证了该方法的有效性。     

Marine gravity and geoid determination by optimal combination of satellite altimetry and shipborne gravimetry data

. Satellite altimetry derived geoid heights and marine gravity anomalies can be combined to determine a detailed gravity field over the oceans using the least-squares collocation method and spectral combination techniques. Least-squares collocation, least-squares adjustment in the frequency domain and input-output system theory are employed to determine the gravity field (both geoid and anomalies) and its errors. This paper intercompares these three techniques using simulated data. Simulation studies show that best results are obtained by the input-output system theory among the three prediction methods. The least-squares collocation method gives results which are very close to but a little bit worse than those obtained using input-output system theory. This slightly poorer performance of the least-squares collocation method can be explained by the fact that it uses isotropic structured covariance (thus approximate signal PSD information) while the system theory method uses detailed signal PSD information. The method of least-squares adjustment in the frequency domain gives the poorest results among these three methods because it uses less information than the other two methods (it ignores the signal PSDs). The computations also show that the least-squares collocation and input-output system theory methods are not as sensitive to noise levels as the least-squares adjustment in the frequency domain method is. Received 19 January 1996; Accepted 17 July 1996     

Non-stationary covariance function modelling in 2D least-squares collocation

Standard least-squares collocation (LSC) assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing of, e.g., the gravity field in mountains and under-smoothing in great plains. We introduce the kernel convolution method from spatial statistics for non-stationary covariance structures, and demonstrate its advantage for dealing with non-stationarity in geodetic data. We then compared stationary and non- stationary covariance functions in 2D LSC to the empirical example of gravity anomaly interpolation near the Darling Fault, Western Australia, where the field is anisotropic and non-stationary. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation against data not used in the interpolation, demonstrating that the use of non-stationary covariance functions can improve upon standard (stationary) LSC.     

顾及卫星钟随机特性的抗差最小二乘配置钟差预报算法

为了更好地反映钟差特性并提高其预报精度,采用抗差最小二乘配置方法建立一种能够同时考虑星载原子钟物理特性、钟差周期性变化与随机性变化特点的钟差预报模型。首先使用附有周期项的二次多项式模型进行拟合提取卫星钟差的趋势项与周期项,然后针对剩余的随机项及其可能存在的粗差,采用抗差最小二乘配置的原理进行建模,其中最小二乘配置的协方差函数通过对比协方差拟合的方法并结合试验进行确定。使用IGS精密钟差数据进行预报试验,将本文方法与二次多项式模型、灰色模型进行对比,预报精度分别提高了0.457 ns和0.948 ns,而预报稳定性则分别提高了0.445 ns和1.233 ns,证明了本文方法能够更好地预报卫星钟差,同时说明本文的协方差函数确定方法的有效性。     

Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry,altimetry and GOCE data: evaluation by simulation

The determination of local geoid models has traditionally been carried out on land and at sea using gravity anomaly and satellite altimetry data, while it will be aided by the data expected from satellite missions such as those from the Gravity field and steady-state ocean circulation explorer (GOCE). To assess the performance of heterogeneous data combination to local geoid determination, simulated data for the central Mediterranean Sea are analyzed. These data include marine and land gravity anomalies, altimetric sea surface heights, and GOCE observations processed with the space-wise approach. A spectral analysis of the aforementioned data shows their complementary character. GOCE data cover long wavelengths and account for the lack of such information from gravity anomalies. This is exploited for the estimation of local covariance function models, where it is seen that models computed with GOCE data and gravity anomaly empirical covariance functions perform better than models computed without GOCE data. The geoid is estimated by different data combinations and the results show that GOCE data improve the solutions for areas covered poorly with other data types, while also accounting for any long wavelength errors of the adopted reference model that exist even when the ground gravity data are dense. At sea, the altimetric data provide the dominant geoid information. However, the geoid accuracy is sensitive to orbit calibration errors and unmodeled sea surface topography (SST) effects. If such effects are present, the combination of GOCE and gravity anomaly data can improve the geoid accuracy. The present work also presents results from simulations for the recovery of the stationary SST, which show that the combination of geoid heights obtained from a spherical harmonic geopotential model derived from GOCE with satellite altimetry data can provide SST models with some centimeters of error. However, combining data from GOCE with gravity anomalies in a collocation approach can result in the estimation of a higher resolution geoid, more suitable for high resolution mean dynamic SST modeling. Such simulations can be performed toward the development and evaluation of SST recovery methods.