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1.
Fast spherical collocation: theory and examples   总被引：2，自引：4，他引：2  
 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  相似文献   

2.
苏勇  范东明  游为 《测绘学报》2015,44(2):142-149
欧空局早期公布的时域法和空域法解算的GOCE模型均采用能量守恒法处理轨道数据, 但恢复的长波重力场信号精度较低, 而且GOCE卫星在两极存在数据空白, 利用其观测数据恢复重力场模型是一个不适定问题, 导致解算的模型带谐项精度较低, 需进行正则化处理。本文分析了基于轨道数据恢复重力场模型的方法用于处理GOCE数据的精度, 对最优正则化方法和参数的选择进行研究。利用GOCE卫星2009-11-01—2010-01-31共92 d的精密轨道数据, 采用不依赖先验信息的能量守恒法、短弧积分法和平均加速度法恢复GOCE重力场模型, 利用Tikhonov正则化技术处理病态问题。结果表明, 平均加速度法恢复模型的精度最高, 能量守恒法的精度最低, 短弧积分法的精度稍差于平均加速度法。未来联合处理轨道和梯度数据时, 建议采用平均加速度法或短弧积分法处理轨道数据, 并且轨道数据可有效恢复120阶次左右的模型。Kaula正则化和SOT处理GOCE病态问题的效果最好, 并且两者对应的最优正则化参数基本一致, 但利用正则化技术不能完全抑制极空白问题的影响, 需要联合GRACE等其他数据才能获得理想的结果。  相似文献   

3.
Gravity gradients acquired by the Gravity field and steady-state Ocean Circulation Explorer(GOCE) do not cover the entire earth because of its sun-synchronous orbit leaving data gaps with a radius of about 6.5° in the polar regions.Previous studies showed that the loss of data in the polar regions deteriorates the accuracy of the low order(or near zonal) coefficients of the earth gravity model,which is the so-called polar gap problem in geodesy.In order to find a stable solution for the earth gravity model from the GOCE gravity gradients,three models,i.e.the Gauss-Markov model,light constraint model and the mixed model,are compared and evaluated numerically with the gravity gradient simulated with the EGM2008.The comparison shows that the Best Linear Uniformly Unbiased Estimation(BLUUE) estimator of the mixed model can solve the polar gap problem as effectively as the light constraint model;furthermore,the mixed model is more rigorous in dealing with the supplementary information and leads to a better accuracy in determining the global geoid.  相似文献   

4.
Considering present attempts to develop a gradiometer with an accuracy between 10−3 E and 10−4 E, two applications for such a device have been studied: (a) mapping the gravitational field of the Earth, and (b) estimating the geocentric distance of a satellite carrying the instrument. Given a certain power spectrum for the signal and 10−4 E (rms) of white measurement noise, the results of an error analysis indicate that a six-month mission in polar orbit at a height of 200 km, with samples taken every three seconds, should provide data for estimating the spherical harmonic potential coefficients up to degree and order 300 with less than 50% error, and improve the coefficients through degree 30 by up to four orders of magnitude compared to existing models. A simulation study based on numerical orbit integrations suggests that a simple adjustment of the initial conditions based on gradiometer data could produce orbits where the geocentric distance is accurate to 10 cm or better, provided the orbits are 2000 km high and some improvement in the gravity field up to degree 30 is first achieved. In this sense, the gravity-mapping capability of the gradiometer complements its use in orbit refinement. This idea can be of use in determining orbits for satellite altimetry. Furthermore, by tracking the gradiometer-carrying spacecraft when it passes nearly above a terrestrial station, the geocentric distance of this station can also be estimated to about one decimeter accuracy. This principle could be used in combination with VLBI and other modern methods to set up a world-wide 3-D network of high accuracy.  相似文献   

5.
Efficient gravity field recovery from GOCE gravity gradient observations   总被引：4，自引：2，他引：2  
 An efficient algorithm is proposed for gravity field recovery from Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite gravity gradient observations. The mathematical model is formulated in the time domain, which allows the inclusion of realistic observational noise models. The algorithm combines the iterative solution of the normal equations, using a Richardson-type iteration scheme, with the fast computation of the right-hand side of the normal equations in each iteration step by a suitable approximation of the design matrix. The convergence of the iteration is investigated, error estimates are provided, and the unbiasedness of the method is proved. It is also shown that the method does not converge to the solution of the normal equations. The performance of the approach for white noise and coloured noise is demonstrated along a simulated GOCE orbit up to spherical harmonic degree and order 180. The results also indicate that the approximation error may be neglected. Received: 30 November 1999 / Accepted: 31 May 2000  相似文献   

6.
In this contribution, we describe the global GOCE-only gravity field model ITG-Goce02 derived from 7.5 months of gradiometer and orbit data. This model represents an alternative to the official ESA products as it is computed completely independently, using a different processing strategy and a separate software package. Our model is derived using the short arc approach, which allows a very effective decorrelation of the highly correlated GOCE gradiometer and orbit data noise by introducing a full empirical covariance matrix for each arc, and gives the possibility to downweight ‘bad’ arcs. For the processing of the orbit data we rely on the integral equation approach instead of the energy integral method, which has been applied in several other GOCE models. An evaluation against high-resolution global gravity field models shows very similar differences of our model compared to the official GOCE results published by ESA (release 2), especially to the model derived by the time-wise approach. This conclusion is confirmed by comparison of the GOCE models to GPS/levelling and altimetry data.  相似文献   

7.
Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.  相似文献   

8.
Improvements in height datum transfer expected from the GOCE mission   总被引：1，自引：1，他引：1  
 One of the aims of the Earth Explorer Gravity Field and Steady-State Ocean Circulation (GOCE) mission is to provide global and regional models of the Earth's gravity field and of the geoid with high spatial resolution and accuracy. Using the GOCE error model, simulation studies were performed in order to estimate the accuracy of datum transfer in different areas of the Earth. The results showed that with the GOCE error model, the standard deviation of the height anomaly differences is about one order of magnitude better than the corresponding value with the EGM96 error model. As an example, the accuracy of the vertical datum transfer from the tide gauge of Amsterdam to New York was estimated equal to 57 cm when the EGM96 error model was used, while in the case of GOCE error model this accuracy was increased to 6 cm. The geoid undulation difference between the two places is about 76.5 m. Scaling the GOCE errors to the local gravity variance, the estimated accuracy varied between 3 and 7 cm, depending on the scaling model. Received: 1 March 2000 / Accepted: 21 February 2001  相似文献   

9.
利用GOCE模拟观测反演重力场的Torus法   总被引：1，自引：1，他引：0  
在介绍Torus方法反演地球重力场模型的基本原理和方法的基础上,基于圆环面上均匀分布的卫星引力梯度模拟观测值解算了200阶次的地球重力场模型,在无误差情况下,Torus方法解算模型的阶误差RMS小于10-16,验证了该方法的严密性。利用61dGOCE卫星轨道上无误差的模拟引力梯度观测值解算了200阶次的地球重力场模型,分析了格网化误差、极空白对解算精度的影响,迭代3次后,在不考虑低次系数情况下,模型的大地水准面阶误差和累积误差均较小,最大值仅为0.022mm和0.099mm。在沿轨卫星引力梯度模拟数据中加入5mE/Hz1/2的白噪声,基于Torus方法和空域最小二乘法解算了200阶次的地球重力场模型,Torus方法的精度略低于空域最小二乘法的精度,在不考虑低次项的情况下,两种方法解算模型的大地水准面阶误差最大值分别为1.58cm和1.45cm,累积误差最大值分别为6.37cm和5.55cm。但由于采用了二维快速傅里叶技术和块对角最小二乘法,极大地提高了计算效率。本文数值结果说明Torus方法是一种独立有效的方法,可用于GOCE任务海量卫星引力梯度观测值反演重力场的快速解算。  相似文献   

10.
天问一号是中国首次独立开展的行星际探测任务,实现了对火星的环绕、着陆、巡视探测。天问一号正常科学任务阶段环绕以极轨设计为主,与历史火星任务类似,对当前火星重力场模型精度改进有限,因此其拓展任务轨道选取至关重要。通过对极轨圆轨道和近赤道大偏心率轨道进行仿真模拟,分析两种典型轨道构型对现有火星重力场模型改进的可能性,基于不同误差考量仿真解算了对应6个重力场模型。借助重力场功率谱分析,发现在测量噪声为0.1 mm/s的情况下,不论采用极轨还是近赤道轨道,一个月的跟踪数据均可较好地反演出42阶次的火星重力场模型;考虑综合误差影响之后,发现两种轨道对于重力场解算精度类似,其中实施近赤道大偏心率轨道对35阶次以上约束略强。  相似文献   

11.
We have applied efficient methods for computing variances and covariances of functions of a global gravity field model expanded in spherical harmonics, using the full variance–covariance matrix of the coefficients. Examples are given with recent models derived from GRACE (up to degree and order 150), and with simulated GOCE derived solutions (up to degree and order 200).  相似文献   

12.
GOCE gravitational gradients along the orbit   总被引：6，自引：3，他引：3  
GOCE is ESA’s gravity field mission and the first satellite ever that measures gravitational gradients in space, that is, the second spatial derivatives of the Earth’s gravitational potential. The goal is to determine the Earth’s mean gravitational field with unprecedented accuracy at spatial resolutions down to 100 km. GOCE carries a gravity gradiometer that allows deriving the gravitational gradients with very high precision to achieve this goal. There are two types of GOCE Level 2 gravitational gradients (GGs) along the orbit: the gravitational gradients in the gradiometer reference frame (GRF) and the gravitational gradients in the local north oriented frame (LNOF) derived from the GGs in the GRF by point-wise rotation. Because the V XX , V YY , V ZZ and V XZ are much more accurate than V XY and V YZ , and because the error of the accurate GGs increases for low frequencies, the rotation requires that part of the measured GG signal is replaced by model signal. However, the actual quality of the gradients in GRF and LNOF needs to be assessed. We analysed the outliers in the GGs, validated the GGs in the GRF using independent gravity field information and compared their assessed error with the requirements. In addition, we compared the GGs in the LNOF with state-of-the-art global gravity field models and determined the model contribution to the rotated GGs. We found that the percentage of detected outliers is below 0.1% for all GGs, and external gravity data confirm that the GG scale factors do not differ from one down to the 10−3 level. Furthermore, we found that the error of V XX and V YY is approximately at the level of the requirement on the gravitational gradient trace, whereas the V ZZ error is a factor of 2–3 above the requirement for higher frequencies. We show that the model contribution in the rotated GGs is 2–35% dependent on the gravitational gradient. Finally, we found that GOCE gravitational gradients and gradients derived from EIGEN-5C and EGM2008 are consistent over the oceans, but that over the continents the consistency may be less, especially in areas with poor terrestrial gravity data. All in all, our analyses show that the quality of the GOCE gravitational gradients is good and that with this type of data valuable new gravity field information is obtained.  相似文献   

13.
利用GOCE卫星轨道反演地球重力场模型   总被引：1，自引：1，他引：0  
根据积分方程法反演地球重力场的数学模型,利用GOCE卫星2009-11-02～2010-01-02共61d的精密轨道数据反演了几组地球重力场模型。结果表明,GOCE卫星轨道能有效提取地球重力场的长波信息,弥补了GOCE卫星重力梯度带宽的限制,在106阶次的大地水准面误差为±9.6cm,该阶次精度优于EIGEN-CHAMP03S及GRACE卫星两个月轨道反演地球重力场的精度,但由于两极空白,反演的带谐位系数精度偏低。联合GOCE及GRACE卫星轨道反演的模型在106阶次的大地水准面误差为±6.9cm,弥补了GOCE卫星轨道的缺陷。  相似文献   

14.
地球重力场和海洋环流探测（gravity field and steady-state ocean circulation explorer,GOCE）卫星重力梯度数据有色噪声和低频系统误差的滤波处理是反演高精度地球重力场的一个关键问题。针对GOCE卫星重力梯度数据的滤波处理,基于移动平均（moving average,MA）方法和CPR(circle per revolution)经验参数方法设计了两类低频系统误差滤波器,并分别将这两类滤波器与基于自回归移动平均（auto-regressive and moving average,ARMA）模型设计的有色噪声滤波器组合起来形成级联滤波器。为了分析滤波器处理的实际效果,基于空域最小二乘法采用70 d的GOCE观测数据,并联合重力恢复与气候实验（gravity recovery and climate experiment,GRACE）数据分别反演了224阶次的重力场模型GOGR-MA(MA+ARMA级联滤波)和GOGR-CPR(CPR+ARMA级联滤波)。将反演模型与采用同期数据求解的第一代GOCE系列模型及GOCE和GRACE联合模...  相似文献   

15.
First GOCE gravity field models derived by three different approaches   总被引：28，自引：10，他引：18  
Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10?cm in terms of geoid heights, and 3?mGal in terms of gravity anomalies, globally at a resolution of 100?km (degree/order 200).  相似文献   

16.
Knudsen 《Journal of Geodesy》1987,61(2):145-160
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.  相似文献   

17.
Collocation is widely used in physical geodesy. Its application requires to solve systems with a dimension equal to the number of observations, causing numerical problems when many observations are available. To overcome this drawback, tailored step-wise techniques are usually applied. An example of these step-wise techniques is the space-wise approach to the GOCE mission data processing. The original idea of this approach was to implement a two-step procedure, which consists of first predicting gridded values at satellite altitude by collocation and then deriving the geo-potential spherical harmonic coefficients by numerical integration. The idea was generalized to a multi-step iterative procedure by introducing a time-wise Wiener filter to reduce the highly correlated observation noise. Recent studies have shown how to optimize the original two-step procedure, while the theoretical optimization of the full multi-step procedure is investigated in this work. An iterative operator is derived so that the final estimated spherical harmonic coefficients are optimal with respect to the Wiener–Kolmogorov principle, as if they were estimated by a direct collocation. The logical scheme used to derive this optimal operator can be applied not only in the case of the space-wise approach but, in general, for any case of step-wise collocation. Several numerical tests based on simulated realistic GOCE data are performed. The results show that adding a pre-processing time-wise filter to the two-step procedure of data gridding and spherical harmonic analysis is useful, in the sense that the accuracy of the estimated geo-potential coefficients is improved. This happens because, in its practical implementation, the gridding is made by collocation over local patches of data, while the observation noise has a time-correlation so long that it cannot be treated inside the patch size. Therefore, the multi-step operator, which is in theory equivalent to the two-step operator and to the direct collocation, is in practice superior thanks to the time-wise filter that reduces the noise correlation before the gridding. The criteria for the choice of this filter are investigated numerically.  相似文献   

18.
 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  相似文献   

19.
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   

20.
When the orbit of the Landsat I spacecraft was liberated to natural forces, the loss of observations to the remote sensing community was balanced by a modest gain for geodesy. The orbit’s long ground-track repeat period of eighteen days gives rise to a shallow resonance with fourteenth, twenty-eighth and forty-second order terms in the geopotential. A single continuous span of twenty-four days of Unified S-Band tracking data, collected at a single station in 1976, has been analyzed to define constraints on the dominant resonance terms of these orders and of fourteenth-order fringe resonance effects depending on the eccentricitye≈.002. Tracking observations from other stations collected during 1974 and 1975 gave essentially the same results, which provided error estimates for the lumped resonance coefficients. The application of the resonance model can considerably improve the definition and prediction of the Landsat 1 orbit. Direct numerical estimates of the influence coefficients in the resonance constraint equations were made to confirm the accuracy of analytical expressions which allow the equations to be applied to geopotential fields of arbitrarily high degree and order. Several recently derived gravity fields were tested against the Landsat resonance constraints and their comparative agreement is discussed.  相似文献   

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