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1.
Compactly supported radial covariance functions 总被引：1，自引：0，他引：1
G. Moreaux 《Journal of Geodesy》2008,82(7):431-443
The Least-squares collocation (LSC) method is commonly used in geodesy, but generally associated with globally supported covariance
functions, i.e. with dense covariance matrices. We consider locally supported radial covariance functions, which yield sparse
covariance matrices. Having many zero entries in the covariance matrice can both greatly reduce computer storage requirements
and the number of floating point operations needed in computation. This paper reviews some of the most well-known compactly
supported radial covariance functions (CSRCFs) that can be easily substituted to the usually used covariance functions. Numerical
experiments reveals that these finite covariance functions can give good approximations of the Gaussian, second- and third-order
Markov models. Then, interpolation of KMS02 free-air gravity anomalies in Azores Islands shows that dense covariance matrices
associated with Gaussian model can be replaced by sparse matrices from CSRCFs resulting in memory savings of one-fortieth
and with 90% of the solution error less than 0.5 mGal.
This article is dedicated to Cerbère. 相似文献
2.
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. 相似文献
3.
Least-squares collocation with covariance-matching constraints 总被引：1，自引：0，他引：1
Christopher Kotsakis 《Journal of Geodesy》2007,81(10):661-677
Most geostatistical methods for spatial random field (SRF) prediction using discrete data, including least-squares collocation
(LSC) and the various forms of kriging, rely on the use of prior models describing the spatial correlation of the unknown
field at hand over its domain. Based upon an optimal criterion of maximum local accuracy, LSC provides an unbiased field estimate
that has the smallest mean squared prediction error, at every computation point, among any other linear prediction method
that uses the same data. However, LSC field estimates do not reproduce the spatial variability which is implied by the adopted
covariance (CV) functions of the corresponding unknown signals. This smoothing effect can be considered as a critical drawback
in the sense that the spatio-statistical structure of the unknown SRF (e.g., the disturbing potential in the case of gravity
field modeling) is not preserved during its optimal estimation process. If the objective for estimating a SRF from its observed
functionals requires spatial variability to be represented in a pragmatic way then the results obtained through LSC may pose
limitations for further inference and modeling in Earth-related physical processes, despite their local optimality in terms
of minimum mean squared prediction error. The aim of this paper is to present an approach that enhances LSC-based field estimates
by eliminating their inherent smoothing effect, while preserving most of their local prediction accuracy. Our methodology
consists of correcting a posteriori the optimal result obtained from LSC in such a way that the new field estimate matches
the spatial correlation structure implied by the signal CV function. Furthermore, an optimal criterion is imposed on the CV-matching
field estimator that minimizes the loss in local prediction accuracy (in the mean squared sense) which occurs when we transform
the LSC solution to fit the spatial correlation of the underlying SRF. 相似文献
4.
利用最小二乘配置对非平稳空间随机场进行推估时,趋势项数学模型的选择通常无法完整体现非平稳空间随机场的系统性,这将导致经验协方差函数估计出现偏差,最终推估结果可能错误。提出了一种基于多面函数的改进最小二乘配置方法来解决上述问题。该方法引入多面函数拟合区域内的趋势项,通过多次迭代计算得到稳定的待定系数值与协方差函数的参数值,最后综合趋势项与信号项得到最终估值。分别采用了模拟地震垂直形变数据和2009年意大利L’Aquila地震的合成孔径雷达干涉测量（Interferometric SAR,InSAR）与GPS同震位移数据来对该方法进行验证,并将其结果与常规方法进行比较。结果表明,改进方法在外部检核点估值的均方残差要小于多面函数法与常规的最小二乘配置法,且受采样点位的影响最小。 相似文献
5.
K. H. Neumayer 《Journal of Geodesy》1998,72(12):698-704
An attempt is made to bridge the gap between closed-form harmonic upward continuation (HUC) of analytic covariance functions
of the disturbing potential of the anomalous local gravity field and the numerical shaping filter construction when the local
gravity vector is modelled in the framework of Kalman filtering. Some fundamental concepts of the local gravity field, interpreted
as a stochastic process that is stationary in the plane and harmonic in the upper half space, are reviewed. The shaping-filter
modelling technique for the local gravity vector is introduced. To determine the relation between the disturbing potential
covariance function and the gravity vector covariance matrix, the role of the so-called admissible pair is established. It
is shown that rescaling an admissible pair leads to an analogue rescaling of the shaping filter matrices derived hereof; no
cumbersome numerical recalculations are necessary. The class of covariance functions whose corresponding shaping filters possess
a closed-form HUC are identified as models whose HUC can be interpreted as a rescaling.
Received: 17 December 1997 / Accepted: 7 September 1998 相似文献
6.
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 10o×20o 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. 相似文献
7.
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。 相似文献
8.
联合重力异常和GPS水准数据的最小二乘配置方法 总被引：1，自引：1，他引：0
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。 相似文献
9.
本文所采用的基于输入－输出系统论的谱方法在计算结果的精度上与最小二乘配置方法相当,却很容易用于异性场的计算。用该谱方法对卫星测高及海洋重力资料进行组合求解重力场量（大地水准面差距和重力异常）,其误差估计结果表明各向异性场的计算精度优于各向同性场的精度。 相似文献
10.
协方差函数的抗差拟合 总被引：3，自引：2，他引：3
如果最小二乘拟合推估法被应用在重力异常、高程异常等的内插中 ,当观测值中含有粗差时 ,由此拟合的协方差函数就不能精确表征其统计性质。本文先从协方差函数的拟合过程入手 ,通过分析传统的协方差函数拟合法的无抗差性 ,提出了协方差函数的抗差拟合法 相似文献
11.
如果最小二乘拟合推估法被应用在重力异常、高程异常等的内插中 ,当观测值中含有粗差时 ,由此拟合的协方差函数就不能精确表征其统计性质。本文先从协方差函数的拟合过程入手 ,通过分析传统的协方差函数拟合法的无抗差性 ,提出了协方差函数的抗差拟合法 相似文献
12.
It is well known that terrain may vary markedly over small areas and that statistics used to characterise spatial variation in terrain may be valid only over small areas. In geostatistical terminology, a non-stationary approach may be considered more appropriate than a stationary approach. In many applications, local variation is not accounted for sufficiently. This paper assesses potential benefits in using non-stationary geostatistical approaches for interpolation and for the assessment of uncertainty in predictions with implications for sampling design. Two main non-stationary approaches are employed in this paper dealing with (1) change in the mean and (2) change in the variogram across the region of interest. The relevant approaches are (1) kriging with a trend model (KT) using the variogram of residuals from local drift and (2) locally-adaptive variogram KT, both applied to a sampled photogrammetrically derived digital terrain model (DTM). The fractal dimension estimated locally from the double-log variogram is also mapped to illustrate how spatial variation changes across the data set. It is demonstrated that estimation of the variogram of residuals from local drift is worthwhile in this case for the characterisation of spatial variation. In addition, KT is shown to be useful for the assessment of uncertainty in predictions. This is shown to be true even when the sample grid is dense as is usually the case for remotely-sensed data. In addition, both ordinary kriging (OK) and KT are shown to provide more accurate predictions than inverse distance weighted (IDW) interpolation, used for comparative purposes. 相似文献
13.
局部重力场最小二乘配置通用表示技术 总被引：1，自引：1，他引：1
在分析局部重力场最小二乘配置法技术特点的基础上，推导出一种能综合多种类型、不同高度重力场元经验协方差函数的通用表达方法，以期实现局部重力场元的内插、外推、延拓或其他不同高度的重力场元估计一体化。分析了最小二乘配置技术的一些性能以及算法实现中应注意的问题。 相似文献
14.
F. Morrison 《Journal of Geodesy》1977,51(2):105-118
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. 相似文献
15.
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. 相似文献
16.
Bernd Eissfeller 《Journal of Geodesy》1996,70(9):539-545
Linear gravity field state space models are still a useful tool to model the anomalous gravity field in vector gravimetry, airborne gravimetry, inertial geodesy and navigation. This paper deals with an idea ofJordan and Heller (1978) to solve analytically the upward continuation problem of Markov gravity models.In contrary to the standard Markov shaping filter approach the height dependency of the covariance function, i.e. variance factor and correlation length as function of height, is strictly introduced in state space and not neglected. Using some basic integral transforms, a general upward continuation integral is derived for the n-th order Markov process. The upward continuation integral is solved for the special and practically important case of 2nd order Markov process in very detail. This leads to the introduction of the special sine and cosine integral functions into the the mathematical covariance model. The features of the covariance model are analyzed analytically and the height dependency is discussed numerically. 相似文献
17.
Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise 总被引：1，自引：0，他引：1
Satellite data that are used to model the global gravity field of the Earth are typically corrupted by correlated noise, which
can be related to a frequency dependence of the data accuracy. We show an opportunity to take such noise into account by using
a proper noise covariance matrix in the estimation procedure. If the dependence of noise on frequency is not known a priori,
it can be estimated on the basis of a posteriori residuals. The methodology can be applied to data with gaps. Non-stationarity
of noise can also be dealt with, provided that the necessary a priori information exists. The proposed methodology is illustrated
with CHAllenging Mini-satellite Payload (CHAMP) data processing. It is shown, in particular, that the usage of a proper noise
model can make the measurements of non-gravitational satellite accelerations unnecessarily. This opens the door for high-quality
modeling of the Earth’s gravity field on the basis of observed orbits of non-dedicated satellites (i.e., satellites without
an on-board accelerometer). Furthermore, the processing of data from dedicated satellite missions – GRACE (Gravity Recovery
and Climate Experiment) and GOCE (Gravity field and steady-state Ocean Circulation Explorer) – may also benefit from the proposed
methodology. 相似文献
18.
Behzad Behnabian Masoud Mashhadi Hossainali Ahad Malekzadeh 《Journal of Geodesy》2018,92(11):1329-1350
The cross-validation technique is a popular method to assess and improve the quality of prediction by least squares collocation (LSC). We present a formula for direct estimation of the vector of cross-validation errors (CVEs) in LSC which is much faster than element-wise CVE computation. We show that a quadratic form of CVEs follows Chi-squared distribution. Furthermore, a posteriori noise variance factor is derived by the quadratic form of CVEs. In order to detect blunders in the observations, estimated standardized CVE is proposed as the test statistic which can be applied when noise variances are known or unknown. We use LSC together with the methods proposed in this research for interpolation of crustal subsidence in the northern coast of the Gulf of Mexico. The results show that after detection and removing outliers, the root mean square (RMS) of CVEs and estimated noise standard deviation are reduced about 51 and 59%, respectively. In addition, RMS of LSC prediction error at data points and RMS of estimated noise of observations are decreased by 39 and 67%, respectively. However, RMS of LSC prediction error on a regular grid of interpolation points covering the area is only reduced about 4% which is a consequence of sparse distribution of data points for this case study. The influence of gross errors on LSC prediction results is also investigated by lower cutoff CVEs. It is indicated that after elimination of outliers, RMS of this type of errors is also reduced by 19.5% for a 5 km radius of vicinity. We propose a method using standardized CVEs for classification of dataset into three groups with presumed different noise variances. The noise variance components for each of the groups are estimated using restricted maximum-likelihood method via Fisher scoring technique. Finally, LSC assessment measures were computed for the estimated heterogeneous noise variance model and compared with those of the homogeneous model. The advantage of the proposed method is the reduction in estimated noise levels for those groups with the fewer number of noisy data points. 相似文献
19.
流动重力空间插值方法比较 总被引：1，自引：0，他引：1
流动重力对于地震监测具有十分重要的指示作用,对其进行空间插值能获取其空间分布特征、弥补测站数据较少的不足。本文使用球面Kriging、反距离加权、改进的Shepard 3种方法对流动重力进行插值,从数据分布、插值点个数、插值分辨率、搜索半径（搜索点数）分析插值结果,并对结果予以验证。结果表明：①与球面Kriging、反距离加权法相比较,改进的Shepard方法对于流动重力的插值能取得较好的结果,当插值区域内流动重力样本点的变化较小、不存在异常点时,使用球面克里金插值会获得连续性好、平滑度较高的插值结果;②流动重力插值过程中要根据样本点分布特征、插值范围选取适当的分辨率;③对于地球要素插值,要充分考虑其地球物理构造性质及空间相关性和变异性。 相似文献
20.
D. Arabelos 《Journal of Geodesy》1989,63(1):69-84
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. 相似文献