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1.
Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups 总被引:7,自引:1,他引:6
General rigorous and simplified formulae are reported for the best invariant quadratic unbiased estimates of the variance–covariance
components, which can be applied to all least-squares adjustments with the general linear stochastic model. Simplified procedures
are given for two cases frequently recurring in geodetic applications: uncorrelated groups of correlated or uncorrelated observations,
with more than one variance component in each group.
Received: 19 November 1998 / Accepted: 21 March 2000 相似文献
2.
Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation 总被引:5,自引:2,他引:3
Frank Neitzel 《Journal of Geodesy》2010,84(12):751-762
In this contribution it is shown that the so-called “total least-squares estimate” (TLS) within an errors-in-variables (EIV)
model can be identified as a special case of the method of least-squares within the nonlinear Gauss–Helmert model. In contrast
to the EIV-model, the nonlinear GH-model does not impose any restrictions on the form of functional relationship between the
quantities involved in the model. Even more complex EIV-models, which require specific approaches like “generalized total
least-squares” (GTLS) or “structured total least-squares” (STLS), can be treated as nonlinear GH-models without any serious
problems. The example of a similarity transformation of planar coordinates shows that the “total least-squares solution” can
be obtained easily from a rigorous evaluation of the Gauss–Helmert model. In contrast to weighted TLS, weights can then be
introduced without further limitations. Using two numerical examples taken from the literature, these solutions are compared
with those obtained from certain specialized TLS approaches. 相似文献
3.
J. Mäkinen 《Journal of Geodesy》2002,76(6-7):317-322
A bound is established for the Euclidean norm of the difference between the best linear unbiased estimator and any linear
unbiased estimator in the general linear model. The bound involves the spectral norm of the difference between the dispersion
matrices of the two estimators, and the residual sum of squares, all evaluated at the assumed model, but is independent of
the provenance of the observation vector at hand. The bound, a straightforward consequence of first principles in Gauss–Markov
theory, generalizes previous results on the difference between the best linear unbiased estimator and the ordinary least-squares
estimator. In a numerical example from repeated precise levelling, the bound is used to analyse the sensitivity of estimates
of vertical motion to the choice of estimator.
Received: 9 September 1999 / Accepted: 15 March 2002 相似文献
4.
An iterative solution of weighted total least-squares adjustment 总被引:9,自引:0,他引:9
Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. However, its
exact solution is rather complicated, and the accuracies of estimated parameters are too difficult to analytically compute.
Since the EIV model is essentially a non-linear model, it can be solved according to the theory of non-linear least-squares
adjustment. In this contribution, we will propose an iterative method of weighted TLS (WTLS) adjustment to solve EIV model
based on Newton–Gauss approach of non-linear weighted least-squares (WLS) adjustment. Then the WLS solution to linearly approximated
EIV model is derived and its discrepancy is investigated by comparing with WTLS solution. In addition, a numerical method
is developed to compute the unbiased variance component estimate and the covariance matrix of the WTLS estimates. Finally,
the real and simulation experiments are implemented to demonstrate the performance and efficiency of the presented iterative
method and its linearly approximated version as well as the numerical method. The results show that the proposed iterative
method can obtain such good solution as WTLS solution of Schaffrin and Wieser (J Geod 82:415–421, 2008) and the presented numerical method can be reasonably applied to evaluate the accuracy of WTLS solution. 相似文献
5.
Stochastic assessment of GPS carrier phase measurements for precise static relative positioning 总被引:17,自引:11,他引:17
Global positioning system (GPS) carrier phase measurements are used in all precise static relative positioning applications.
The GPS carrier phase measurements are generally processed using the least-squares method, for which both functional and stochastic
models need to be carefully defined. Whilst the functional model for precise GPS positioning is well documented in the literature,
realistic stochastic modelling for the GPS carrier phase measurements is still both a controversial topic and a difficult
task to accomplish in practice. The common practice of assuming that the raw GPS measurements are statistically independent
in space and time, and have the same accuracy, is certainly not realistic. Any mis-specification in the stochastic model will
inevitably lead to unreliable positioning results. A stochastic assessment procedure has been developed to take into account
the heteroscedastic, space- and time-correlated error structure of the GPS measurements. Test results indicate that the reliability
of the estimated positioning results is improved by applying the developed stochastic assessment procedure. In addition, the
quality of ambiguity resolution can be more realistically evaluated.
Received: 13 February 2001 / Accepted: 3 September 2001 相似文献
6.
GPS signal diffraction modelling: the stochastic SIGMA-δ model 总被引:1,自引:0,他引:1
The SIGMA-Δ model has been developed for stochastic modelling of global positioning system (GPS) signal diffraction errors
in high precision GPS surveys. The basic information used in the SIGMA-Δ model is the measured carrier-to-noise power-density
ratio (C/N0). Using the C/N0 data and a template technique, the proper variances are derived for all phase observations. Thus the quality of the measured
phase is automatically assessed and if phase observations are suspected to be contaminated by diffraction effects they are
weighted down in the least-squares adjustment. The ability of the SIGMA-Δ model to reduce signal diffraction effects is demonstrated
on two static GPS surveys as well as on a kinematic high-precision GPS railway survey. In cases of severe signal diffraction
the accuracy of the GPS positions is improved by more than 50% compared to standard GPS processing techniques.
Received: 27 July 1998 / Accepted: 1 March 1999 相似文献
7.
Success probability of integer GPS ambiguity rounding and bootstrapping 总被引:26,自引:7,他引:19
P. J. G. Teunissen 《Journal of Geodesy》1998,72(10):606-612
Global Positioning System ambiguity resolution is usually based on the integer least-squares principle (Teunissen 1993).
Solution of the integer least-squares problem requires both the execution of a search process and an ambiguity decorrelation
step to enhance the efficiency of this search. Instead of opting for the integer least-squares principle, one might also want
to consider less optimal integer solutions, such as those obtained through rounding or sequential rounding. Although these
solutions are less optimal, they do have one advantage over the integer least-squares solution: they do not require a search
and can therefore be computed directly. However, in order to be confident that these less optimal solutions are still good
enough for the application at hand, one requires diagnostic measures to predict their rate of success. These measures of confidence
are presented and it is shown how they can be computed and evaluated.
Received: 24 March 1998 / Accepted: 8 June 1998 相似文献
8.
W. Prószyński 《Journal of Geodesy》1997,71(10):652-659
For a linear least-squares parametric model analysis is carried out of the structure of the projection operator transforming
the vector of standardised observations into the vector of standardised residuals. On this basis the properties of the model
responses to observational disturbances (i.e. gross errors or blunders) are derived. A final outcome of the research can be
summarised as: (1) proposing the robustness characteristics of a model and linking them with the local measures of internal
reliability, being the diagonal elements in the projection operator; (2) determining the internal reliability levels satisfying
specified robustness requirements, i.e. the possibility of detecting at least one of the k observational disturbances (k=1,2,…) having most disadvantageous locations in the system. The theory and a numerical example show that for the systems
which have been designed to a proper level of internal reliability, the least-squares estimation can demonstrate an accordingly
high level of robustness.
Received: 11 June 1996 / Accepted: 28 April 1997 相似文献
9.
Robust estimation of systematic errors of satellite laser range 总被引:13,自引:0,他引:13
Methods for analyzing laser-ranging residuals to estimate station-dependent systematic errors and to eliminate outliers in
satellite laser ranges are discussed. A robust estimator based on an M-estimation principle is introduced. A practical calculation
procedure which provides a robust criterion with high breakdown point and produces robust initial residuals for following
iterative robust estimation is presented. Comparison of the results from the least-squares method with those of the robust
method shows that the results of the station systematic errors from the robust estimator are more reliable.
Received: 18 March 1997 / Accepted: 17 March 1999 相似文献
10.
On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms 总被引:2,自引:1,他引:1
The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E
Y
= (X−E
X
) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E
Y
and E
X
. Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation
matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix
X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem
by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative
algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition.
For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert
cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing
the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment
of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift
to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares
and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion
matrix of the estimated parameters. 相似文献
11.
Four different implementations of Stokes' formula are employed for the estimation of geoid heights over Sweden: the Vincent
and Marsh (1974) model with the high-degree reference gravity field but no kernel modifications; modified Wong and Gore (1969)
and Molodenskii et al. (1962) models, which use a high-degree reference gravity field and modification of Stokes' kernel;
and a least-squares (LS) spectral weighting proposed by Sj?berg (1991). Classical topographic correction formulae are improved
to consider long-wavelength contributions. The effect of a Bouguer shell is also included in the formulae, which is neglected
in classical formulae due to planar approximation. The gravimetric geoid is compared with global positioning system (GPS)-levelling-derived
geoid heights at 23 Swedish Permanent GPS Network SWEPOS stations distributed over Sweden. The LS method is in best agreement,
with a 10.1-cm mean and ±5.5-cm standard deviation in the differences between gravimetric and GPS geoid heights. The gravimetric
geoid was also fitted to the GPS-levelling-derived geoid using a four-parameter transformation model. The results after fitting
also show the best consistency for the LS method, with the standard deviation of differences reduced to ±1.1 cm. For comparison,
the NKG96 geoid yields a 17-cm mean and ±8-cm standard deviation of agreement with the same SWEPOS stations. After four-parameter
fitting to the GPS stations, the standard deviation reduces to ±6.1 cm for the NKG96 geoid. It is concluded that the new corrections
in this study improve the accuracy of the geoid. The final geoid heights range from 17.22 to 43.62 m with a mean value of
29.01 m. The standard errors of the computed geoid heights, through a simple error propagation of standard errors of mean
anomalies, are also computed. They range from ±7.02 to ±13.05 cm. The global root-mean-square error of the LS model is the
other estimation of the accuracy of the final geoid, and is computed to be ±28.6 cm.
Received: 15 September 1999 / Accepted: 6 November 2000 相似文献
12.
A general scheme is given for the solution in a least-squares sense of the geodetic boundary value problem in a spherical,
constant-radius approximation, both uniquely and overdetermined, for a large class of observations. The only conditions are
that the relation of the observations to the disturbing potential is such that a diagonalization in the spectrum can be found
and that the error-covariance function of the observations is isotropic and homogeneous. Most types of observations used in
physical geodesy can be adjusted to fit into this approach. Examples are gravity anomalies, deflections of the vertical and
the second derivatives of the gravity potential.
Received: 3 November 1999 / Accepted: 25 September 2000 相似文献
13.
Procedures to calculate mean sea surface heights and gravity anomalies from altimeter-derived sea surface heights and along-track
sea surface slopes using the least-squares collocation procedure are derived. The slope data is used when repeat track averaging
is not possible to reduce ocean variability effects. Tests were carried out using Topex, Geosat, ERS-1 [35-day and 168-day
(2 cycle)] data. Calculations of gravity anomalies in the Gulf Stream region were made using the sea surface height and slope
data. Tests were also made correcting the sea surface heights for dynamic ocean topography calculated from a degree 360 expansion
of data from the POCM-4B global ocean circulation model. Comparisons of the anomaly predictions were carried out with ship
data using anomalies calculated for this paper as well as others.
Received: 19 August 1996 / Accepted: 14 April 1997 相似文献
14.
Possible improvement of Earth orientation forecast using autocovariance prediction procedures 总被引:3,自引:2,他引:1
Autocovariance prediction has been applied to attempt to improve polar motion and UT1-UTC predictions. The predicted polar
motion is the sum of the least-squares extrapolation model based on the Chandler circle, annual and semiannual ellipses, and
a bias fit to the past 3 years of observations and the autocovariance prediction of these extrapolation residuals computed
after subtraction of this model from pole coordinate data. This prediction method has been applied also to the UT1-UTC data,
from which all known predictable effects were removed, but the prediction error has not been reduced with respect to the error
of the current prediction model. However, the results show the possibility of decreasing polar motion prediction errors by
about 50 for different prediction lengths from 50 to 200 days with respect to the errors of the current prediction model.
Because of irregular variations in polar motion and UT1-UTC, the accuracy of the autocovariance prediction does depend on
the epoch of the prediction. To explain irregular variations in x, y pole coordinate data, time-variable spectra of the equatorial components of the effective atmospheric angular momentum, determined
by the National Center for Environmental Prediction, were computed. These time-variable spectra maxima for oscillations with
periods of 100–140 days, which occurred in 1985, 1988, and 1990 could be responsible for excitation of the irregular short-period
variations in pole coordinate data. Additionally, time-variable coherence between geodetic and atmospheric excitation function
was computed, and the coherence maxima coincide also with the greatest irregular variations in polar motion extrapolation
residuals.
Received: 22 October 1996 / Accepted: 16 September 1997 相似文献
15.
XU Xinyu LI Jiancheng ZOU Xiancai CHU Yonghai 《地球空间信息科学学报》2007,10(3):168-172
The principle and method for solving three types of satellite gravity gradient boundary value problems by least-squares are discussed in detail. Also, kernel function expressions of the least-squares solution of three geodetic boundary value problems with the observations {Γ zz },{Γ xz , Γ yz} and {Γ xx -Γ yy ,2 Γxy}are presented. From the results of recovering gravity field using simulated gravity gradient tensor data, we can draw a conclusion that satellite gravity gradient integral formulas derived from least-squares are valid and rigorous for recovering the gravity field. 相似文献
16.
The 2 arc-minute × 2 arc-minute geoid model (GEOID96) for the United States supports the conversion between North American
Datum 1983 (NAD 83) ellipsoid heights and North American Vertical Datum 1988 (NAVD 88) Helmert heights. GEOID96 includes information
from global positioning system (GPS) height measurements at optically leveled benchmarks. A separate geocentric gravimetric
geoid, G96SSS, was first calculated, then datum transformations and least-squares collocation were used to convert from G96SSS
to GEOID96.
Fits of 2951 GPS/level (ITRF94/NAVD 88) benchmarks to G96SSS show a 15.1-cm root mean square (RMS) around a tilted plane (0.06 ppm,
178∘ azimuth), with a mean value of −31.4 cm (15.6-cm RMS without plane). This mean represents a bias in NAVD 88 from global mean
sea level, remaining nearly constant when computed from subsets of benchmarks. Fits of 2951 GPS/level (NAD 83/NAVD 88) benchmarks
to GEOID96 show a 5.5-cm RMS (no tilts, zero average), due primarily to GPS error. The correlated error was 2.5 cm, decorrelating
at 40 km, and is due to gravity, geoid and GPS errors. Differences between GEOID96 and GEOID93 range from −122 to +374 cm
due primarily to the non-geocentricity of NAD 83.
Received: 28 July 1997 / Accepted: 2 September 1998 相似文献
17.
Robust estimation of geodetic datum transformation 总被引:18,自引:1,他引:17
Y. Yang 《Journal of Geodesy》1999,73(5):268-274
The robust estimation of geodetic datum transformation is discussed. The basic principle of robust estimation is introduced.
The error influence functions of the robust estimators, together with those of least-squares estimators, are given. Particular
attention is given to the robust initial estimates of the transformation parameters, which should have a high breakdown point
in order to provide reliable residuals for the following estimation. The median method is applied to solve for robust initial
estimates of transformation parameters since it has the highest breakdown point. A smooth weight function is then used to
improve the efficiency of the parameter estimates in successive iterative computations. A numerical example is given on a
datum transformation between a global positioning system network and the corresponding geodetic network in China. The results
show that when the coordinates are contaminated by outliers, the proposed method can still give reasonable results.
Received: 25 September 1997 / Accepted: 1 March 1999 相似文献
18.
Spherical cap harmonic analysis: a comment on its proper use for local gravity field representation 总被引:1,自引:0,他引:1
Spherical cap harmonic analysis is the appropriate analytical technique for modelling Laplacian potential and the corresponding
field components over a spherical cap. This paper describes the use of this method by means of a least-squares approach for
local gravity field representation. Formulations for the geoid undulation and the components ξ, η of the deflection of the
vertical are derived, together with some warnings in the application of the technique. Although most of the formulations have
been given by another paper, these were confusing or even incorrect, mainly because of an improper application of the spherical
cap harmonic analysis.
Received: 16 January 1996 / Accepted: 17 March 1997 相似文献
19.
In the linear estimation problem associated with an experiment that is exactly repeated a number of times, the estimation
parameters may naturally be partitioned into two groups, those that are common to all repetitions, and those that are particular
to each repeat experiment. We derive least-squares solutions that minimise in norm either group of parameters, as also the
trace of the corresponding covariance matrix. These solutions are applied to the station adjustment of triangulation surveying,
and to the estimation problem of satellite radar altimetry: to estimate simultaneously mean sea surface heights and residual
radial orbit errors, while minimising the norm of either group of parameters. This altimetry problem is considered in the
cases of collinear, local crossover and global crossover data.
Received: 6 January 1997 / Accepted: 21 December 1998 相似文献
20.
B. Kasietczuk 《Journal of Geodesy》1997,71(3):131-136
This paper presents an extension of the geodetic network adjustment model. The proposed extension makes possible the estimation
of the 3-rd and 4-th central moments for the vector of measurement errors in the process of network adjustment by the least-squares
method with application of orthogonal matrices. It allows to estimate the asymmetry and kurtosis of the measurement errors
distribution.
Received 13 April 1993; Accepted 8 July 1996 相似文献