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1.
Large-scale least squares problems require tailored numerical techniques to overcome the computational burden. For these types of problems, iterative strategies are suitable because of their flexibility and effectiveness. The only shortcoming of iterative strategies in least squares estimation is that the inverse of the normal equation matrix as the carrier of the covariance information is either unavailable or very expensive to compute. This paper presents algorithms based on Monte Carlo integration, which can be incorporated very efficiently into iterative solvers and which are demonstrated to close the aforementioned gap. Tailored strategies for different types of solution techniques with respect to normal equations, observation equations, and combined models are treated. Finally, the paper presents new criteria to define confidence regions for the estimated covariance matrix of the parameters, as well as for all additional derived quantities. In a case study these techniques are applied to simulated GOCE data, where satellite gravity gradiometry and satellite-to-satellite tracking information are combined for reconstructing the gravity field. The problem of deriving the covariance matrix of gravity fields with high spatial resolution by combined iterative estimation processes, unsolved until now, is treated.  相似文献   

2.
Based on tensor theory, three invariants of the gravitational gradient tensor (IGGT) are independent of the gradiometer reference frame (GRF). Compared to traditional methods for calculation of gravity field models based on the gravity field and steady-state ocean circulation explorer (GOCE) data, which are affected by errors in the attitude indicator, using IGGT and least squares method avoids the problem of inaccurate rotation matrices. The IGGT approach as studied in this paper is a quadratic function of the gravity field model’s spherical harmonic coefficients. The linearized observation equations for the least squares method are obtained using a Taylor expansion, and the weighting equation is derived using the law of error propagation. We also investigate the linearization errors using existing gravity field models and find that this error can be ignored since the used a-priori model EIGEN-5C is sufficiently accurate. One problem when using this approach is that it needs all six independent gravitational gradients (GGs), but the components \(V_{xy}\) and \(V_{yz}\) of GOCE are worse due to the non-sensitive axes of the GOCE gradiometer. Therefore, we use synthetic GGs for both inaccurate gravitational gradient components derived from the a-priori gravity field model EIGEN-5C. Another problem is that the GOCE GGs are measured in a band-limited manner. Therefore, a forward and backward finite impulse response band-pass filter is applied to the data, which can also eliminate filter caused phase change. The spherical cap regularization approach (SCRA) and the Kaula rule are then applied to solve the polar gap problem caused by GOCE’s inclination of \(96.7^{\circ }\). With the techniques described above, a degree/order 240 gravity field model called IGGT_R1 is computed. Since the synthetic components of \(V_{xy}\) and \(V_{yz}\) are not band-pass filtered, the signals outside the measurement bandwidth are replaced by the a-priori model EIGEN-5C. Therefore, this model is practically a combined gravity field model which contains GOCE GGs signals and long wavelength signals from the a-priori model EIGEN-5C. Finally, IGGT_R1’s accuracy is evaluated by comparison with other gravity field models in terms of difference degree amplitudes, the geostrophic velocity in the Agulhas current area, gravity anomaly differences as well as by comparison to GNSS/leveling data.  相似文献   

3.
Most time series of geophysical phenomena have temporally correlated errors. From these measurements, various parameters are estimated. For instance, from geodetic measurements of positions, the rates and changes in rates are often estimated and are used to model tectonic processes. Along with the estimates of the size of the parameters, the error in these parameters needs to be assessed. If temporal correlations are not taken into account, or each observation is assumed to be independent, it is likely that any estimate of the error of these parameters will be too low and the estimated value of the parameter will be biased. Inclusion of better estimates of uncertainties is limited by several factors, including selection of the correct model for the background noise and the computational requirements to estimate the parameters of the selected noise model for cases where there are numerous observations. Here, I address the second problem of computational efficiency using maximum likelihood estimates (MLE). Most geophysical time series have background noise processes that can be represented as a combination of white and power-law noise, \(1/f^{\alpha }\) with frequency, f. With missing data, standard spectral techniques involving FFTs are not appropriate. Instead, time domain techniques involving construction and inversion of large data covariance matrices are employed. Bos et al. (J Geod, 2013. doi: 10.1007/s00190-012-0605-0) demonstrate one technique that substantially increases the efficiency of the MLE methods, yet is only an approximate solution for power-law indices >1.0 since they require the data covariance matrix to be Toeplitz. That restriction can be removed by simply forming a data filter that adds noise processes rather than combining them in quadrature. Consequently, the inversion of the data covariance matrix is simplified yet provides robust results for a wider range of power-law indices.  相似文献   

4.
Summary Using a data set of 260 000 gravity anomalies it is shown that common characteristics for a local covariance function exist in an area as large as Canada excluding the Rocky Mountains. After eliminating global features by referencing the data to the GEM-10 satellite solution, the shape of the covariance function is remarkably consistent from one sample area to the next. The determination of the essential parameters and the fitting of the covariance function are discussed in detail. To test the reliability of the derived function, deflections of the vertical are estimated at about 230 stations where astrogeodetic data are available. Results show that the standard error obtained from the discrepancies is about1″ for each component and that the error covariance matrix of least-squares collocation reflects this accuracy remarkably well.  相似文献   

5.
One of the products derived from the gravity field and steady-state ocean circulation explorer (GOCE) observations are the gravity gradients. These gravity gradients are provided in the gradiometer reference frame (GRF) and are calibrated in-flight using satellite shaking and star sensor data. To use these gravity gradients for application in Earth scienes and gravity field analysis, additional preprocessing needs to be done, including corrections for temporal gravity field signals to isolate the static gravity field part, screening for outliers, calibration by comparison with existing external gravity field information and error assessment. The temporal gravity gradient corrections consist of tidal and nontidal corrections. These are all generally below the gravity gradient error level, which is predicted to show a 1/f behaviour for low frequencies. In the outlier detection, the 1/f error is compensated for by subtracting a local median from the data, while the data error is assessed using the median absolute deviation. The local median acts as a high-pass filter and it is robust as is the median absolute deviation. Three different methods have been implemented for the calibration of the gravity gradients. All three methods use a high-pass filter to compensate for the 1/f gravity gradient error. The baseline method uses state-of-the-art global gravity field models and the most accurate results are obtained if star sensor misalignments are estimated along with the calibration parameters. A second calibration method uses GOCE GPS data to estimate a low-degree gravity field model as well as gravity gradient scale factors. Both methods allow to estimate gravity gradient scale factors down to the 10−3 level. The third calibration method uses high accurate terrestrial gravity data in selected regions to validate the gravity gradient scale factors, focussing on the measurement band. Gravity gradient scale factors may be estimated down to the 10−2 level with this method.  相似文献   

6.
This is the third of a four-part series on the development of a general framework for error analysis in measurement-based geographic information systems (MBGIS). In this paper, we study the characteristics of error structures in intersections and polygon overlays. When locations of the endpoints of two line segments are in error, we analyze errors of the intersection point and obtain its error covariance matrix through the propagation of the error covariance matrices of the endpoints. An approximate law of error propagation for the intersection point is formulated within the MBGIS framework. From simulation experiments, it appears that both the relative positioning of two line segments and the error characteristics of the endpoints can affect the error characteristics of the intersection. Nevertheless, the approximate law of error propagation captures nicely the error characteristics under various situations. Based on the derived results, error analysis in polygon-on-polygon overlay operation is also performed. The relationship between the error covariance matrices of the original polygons and the overlaid polygons is approximately established.This project was supported by the earmarked grant CUHK 4362/00H of the Hong Kong Research grants Council.  相似文献   

7.
Error analyses of CHAMP data for recovery of the Earth’s gravity field   总被引：1，自引：0，他引：1  
A preliminary commission error analysis whereby orbit perturbation theory and other techniques are used to assess and predict the recovery of the Earths gravity field from the challenging microsatellite payload (CHAMP) mission is developed and implemented. With CHAMP launched in July 2000, accumulated evidence is now available to quantify the errors in the recovery procedure including the orbital precision from GPS, attitude errors, accelerometer noise and thruster mismatch/misalignment. For the latter, numerical integrations using a variable length single-step Runge–Kutta integrator and a fixed length multi-step method are compared to assess the error associated with assuming that the thruster misalignment can be spread uniformly across a step interval. Error degree variances from simulated studies are compared to results from a recently released CHAMP-based gravity field, EIGEN-1S. It is seen that the orbital positioning, as derived from the onboard GPS receiver, is critical, with accelerometer noise contributing at a lower level. Attitude error, at currently quoted accuracy, is not significant as an error source. AcknowledgementsThe authors would like to thank the UK Natural Environment Research Council (Grant No. NER/A/0000/00612) for financing this study and GFZ for supplying the data and technical support.  相似文献   

8.
We describe the computation of the first Australian quasigeoid model to include error estimates as a function of location that have been propagated from uncertainties in the EGM2008 global model, land and altimeter-derived gravity anomalies and terrain corrections. The model has been extended to include Australia’s offshore territories and maritime boundaries using newer datasets comprising an additional \({\sim }\)280,000 land gravity observations, a newer altimeter-derived marine gravity anomaly grid, and terrain corrections at \(1^{\prime \prime }\times 1^{\prime \prime }\) resolution. The error propagation uses a remove–restore approach, where the EGM2008 quasigeoid and gravity anomaly error grids are augmented by errors propagated through a modified Stokes integral from the errors in the altimeter gravity anomalies, land gravity observations and terrain corrections. The gravimetric quasigeoid errors (one sigma) are 50–60 mm across most of the Australian landmass, increasing to \({\sim }100\) mm in regions of steep horizontal gravity gradients or the mountains, and are commensurate with external estimates.  相似文献   

9.
Since the advent of CHAMP, the first in a series of low-altitude satellites being almost continuously and precisely tracked by GPS, a new generation of long-wavelength gravitational geopotential models can be derived. The accuracy evaluation of these models depends to a large extent on the comparison with external data of comparable quality. Here, two CHAMP-derived models, EIGEN-1S and EIGEN-2, are tested with independent long-term-averaged single satellite crossover (SSC) sea heights from three altimetric satellites (ERS-1, ERS-2 and Geosat). The analyses show that long-term averages of crossover residuals still are powerful data to test CHAMP gravity field models. The new models are tested in the spatial domain with the aid of ERS-1/-2 and Geosat SSCs, and in the spectral domain with latitude-lumped coefficient (LLC) corrections derived from the SSCs. The LLC corrections allow a representation of the satellite-orbit-specific error spectra per order of the models spherical harmonic coefficients. These observed LLC corrections are compared to the LLC projections from the models variance–covariance matrix. The excessively large LLC errors at order 2 found in the case of EIGEN-2 with the ERS data are discussed. The degree-dependent scaling factors for the variance-covariance matrices of EIGEN-1S and –2, applied to obtain more realistic error estimates of the solved-for coefficients, are compatible with the results found here.  相似文献   

10.
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.  相似文献   

11.
Vector gravimetry using a precise inertial navigation system continually updated with external position data, for example using GPS, is studied with respect to two problems. The first concerns the attitude accuracy requirement for horizontal gravity component estimation. With covariance analyses in the space and frequency domains it is argued that with relatively stable uncompensated gyro drift, the short-wavelength gravity vector can be estimated without the aid of external attitude updates. The second problem concerns the state-space estimation of the gravity signal where considerable approximations must be assumed in the gravity model in order to take advantage of the ensemble error estimation afforded by the Kalman filter technique. Gauss-Markov models for the gravity field are specially designed to reflect the attenuation of the signal at a specific altitude and the omission of the long-wavelength components from the estimation. With medium accuracy INS/GPS systems, the horizontal components of gravity with wavelengths shorter than 250 km should be estimable to an accuracy of 4–6 mgal (µg); while high accuracy systems should yield an improvement to 1–2 mgal.  相似文献   

12.
The issue of optimal regularization is investigated in the context of the processing of satellite gravity gradiometry (SGG) data that will be acquired by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite. These data are considered as the input for determination of the Earths gravity field in the form of a series of spherical harmonics. Exploitation of a recently developed fast processing algorithm allowed a very realistic setup of the numerical experiments to be specified, in particular: a non-repeat orbit; 1-s sampling rate; half-year duration of data series; and maximum degree and order set to 300. The first goal of the study is to compare different regularization techniques (regularization matrices). The conclusion is that the first-order Tikhonov regularization matrix (the elements are practically proportional to the degree squared) and the Kaula regularization matrix (the elements are proportional to the fourth power of the degree) are somewhat superior to other regularization techniques. The second goal is to assess the generalized cross-validation method for the selection of the regularization parameter. The inference is that the regularization parameter found this way is very reasonable. The time expenditure required by the generalized cross-validation method remains modest even when a half-year set of SGG data is considered. The numerical study also allows conclusions to be drawn regarding the quality of the Earths gravity field model that can be obtained from the GOCE SGG data. In particular, it is shown that the cumulative geoid height error between degrees 31 and 200 will not exceed 1 cm. AcknowledgmentsThe authors thank Dr. E. Schrama for valuable discussions and for computing the orbit used to generate the long data set. They are also grateful to Prof. Tscherning and two anonymous reviewers for numerous valuable remarks and suggestions. The orbit to generate the short data set was kindly provided by J. van den IJssel. Computing resources were provided by Stichting Nationale Computerfaciliteiten (NCF), grant SG-027.  相似文献   

13.
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   

14.
Latitude-lumped coefficients (LLC) are defined, representing geopotential-orbit variations for dual-satellite crossovers (DSC). Formulae are derived for their standard errors from the covariances of geopotential field models. Numerical examples are presented for pairs of the altimeter-bearing satellites TOPEX/Poseidon, ERS 1, and Geosat, using the error matrices of recent gravity models. The DSC, connecting separate missions, will play an increasingly important role in oceanography spanning decades only when its nonoceanographic signals are thoroughly understood. In general, the content of even the long-term averaged DSC is more complex then their single satellite crossover (SSC) counterpart. The LLC, as the spatial spectra for the geopotential-caused crossover effects, discriminate these source-differences sharply. Thus, the zero-order LLC in DSC data contains zonal gravity information not present in SSC data. In addition, zero- and first-order LLC of DSC data can reveal a geocenter discrepancy between the orbit tracking of the separate satellite missions. For example, DSC analysis from orbits computed with JGM 2 show that the y-axis of the geocenter for Geosat in 1986–1988 is shifted with respect to T/P by 6–9 cm towards the eastern Pacific. Also, where the time-gap is necessarily large (as between, say, Geosat and T/P missions) oceanographic (sea-level) differences in DSC may corrupt the geopotential interpretation of the data. Most importantly, as we illustrate, media delays for the altimeter (from the ionosphere, wet troposphere and sea-state bias) are more likely sources of contamination across two missions than in SSC analyses. Again, the LLC of zero order best shows this contrast. Using the higher-order LLC of DSC for both Geosat-T/P and ERS 1-T/P as likely representation of geopotential-only error, we show by comparison with the predicted standard errors of JGM 2 that the latter's previously calibrated covariance matrix is generally valid. Received: 14 February 1996 / Accepted: 27 March 1997  相似文献   

15.
Compactly supported radial covariance functions   总被引：1，自引：0，他引：1  
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   

16.
The Altimetric Bathymetry from Surface Slopes (ABYSS), which is the proposed science payload on the International Space Station (ISS), is a Johns Hopkins University Applied Physics Laboratory-developed flight-proved delay-Doppler phase-monopulse radar altimeter capable of measuring ocean surface slope in the 6–200-km half-wavelength frequency band range with an accuracy of 0.5 $muhbox{rad}$ , with autonomous gimbal control to compensate for the ISS structural motions. This measurement allows an improved mapping of the global bathymetry, enabling a wide range of scientific research works and applications. The nonrepeat ISS orbital ground track is ideal for ABYSS. This letter describes a simulation study on the effects of the Earth's gravity field and other errors, including thermal bending of the ISS, on the orbit determination of the altimeter instrument antenna phase center location, fulfilling the science objectives of ABYSS. Our study concluded that the error due to mean gravity field is no longer limiting due primarily to the recent Gravity Recovery and Climate Experiment gravity modeling and that the ABYSS/ISS radial orbit slope error budget in the presence of various force and measurement model errors is estimated at the 0.2-$mu hbox{rad}$ root-sum-squared (RSS) level, which satisfies the ABYSS orbit accuracy science requirement to provide an improved mapping of global bathymetry.   相似文献   

17.
When combining satellite and terrestrial networks, covariance matrices are used which have been estimated from previous data. It can be shown that the least-squares estimator of the unknown parameters using such an estimated covariance matrix is not necessarily the best. There are a number of cases where a more efficient estimator can be obtained in a different way. The problem occurs frequently in geodesy, since in least-squares adjustment of correlated observations estimated covariance matrices are often used. If the general structure of the covariance matrix is known, results can often be improved by a method called covariance adjustment. The statistical model used in least-squares collocation leads to a type of covariance matrix which fits into this framework. It is shown in which way improvements can be made using a modified approach of principal component analysis. As a numerical example the combination of a satellite and a terrestrial network has been computed with varying assumptions on the covariance matrix. It is shown which types of matrices are critical and where the usual least-squares approach can be applied without hesitation. Finally, a simplified representation of covariances for spatial networks by means of a suitable covariance function is suggested. Paper presented at the International Symposium on Computational Methods in Geometrical Geodesy-Oxford, 2–8 September, 1973.  相似文献   

18.
Global gravity field models have been determined based on kinematic orbits covering an observation period of one year beginning from March 2002. Three different models have been derived up to a maximum degree of n=90 of a spherical harmonic expansion of the gravitational potential. One version, ITG-CHAMP01E, has been regularized beginning from degree n=40 upwards, based on the potential coefficients of the gravity field model EGM96. A second model, ITG-CHAMP01K, has been determined based on Kaulas rule of thumb, also beginning from degree n=40. A third version, ITG-CHAMP01S, has been determined without any regularization. The physical model of the gravity field recovery technique is based on Newtons equation of motion, formulated as a boundary value problem in the form of a Fredholm-type integral equation. The observation equations are formulated in the space domain by dividing the one-year orbit into short sections of approximately 30-minute arcs. For every short arc, a variance factor has been determined by an iterative computation procedure. The three gravity field models have been validated based on various criteria, and demonstrate the quality of not only the gravity field recovery technique but also the kinematically determined orbits.  相似文献   

19.
 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   

20.
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   

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