| 1. |
- Bombrun, A., et al.
(författare)
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A conjugate gradient algorithm for the astrometric core solution of Gaia
- 2012
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Ingår i: Astronomy & Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 538
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Tidskriftsartikel (refereegranskat)abstract
- Context. The ESA space astrometry mission Gaia, planned to be launched in 2013, has been designed to make angular measurements on a global scale with micro-arcsecond accuracy. A key component of the data processing for Gaia is the astrometric core solution, which must implement an efficient and accurate numerical algorithm to solve the resulting, extremely large least-squares problem. The Astrometric Global Iterative Solution (AGIS) is a framework that allows to implement a range of different iterative solution schemes suitable for a scanning astrometric satellite. Aims. Our aim is to find a computationally efficient and numerically accurate iteration scheme for the astrometric solution, compatible with the AGIS framework, and a convergence criterion for deciding when to stop the iterations. Methods. We study an adaptation of the classical conjugate gradient (CG) algorithm, and compare it to the so-called simple iteration (SI) scheme that was previously known to converge for this problem, although very slowly. The different schemes are implemented within a software test bed for AGIS known as AGISLab. This allows to define, simulate and study scaled astrometric core solutions with a much smaller number of unknowns than in AGIS, and therefore to perform a large number of numerical experiments in a reasonable time. After successful testing in AGISLab, the CG scheme has been implemented also in AGIS. Results. The two algorithms CG and SI eventually converge to identical solutions, to within the numerical noise (of the order of 0.00001 micro-arcsec). These solutions are moreover independent of the starting values (initial star catalogue), and we conclude that they are equivalent to a rigorous least-squares estimation of the astrometric parameters. The CG scheme converges up to a factor four faster than SI in the tested cases, and in particular spatially correlated truncation errors are much more efficiently damped out with the CG scheme. While it appears to be difficult to define a strict and robust convergence criterion, we have found that the sizes of the updates, and possibly the correlations between the updates in successive iterations, provide useful clues.
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| 2. |
- Bombrun, A., et al.
(författare)
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Complexity of the Gaia astrometric least-squares problem and the (non-)feasibility of a direct solution method
- 2010
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Ingår i: Astronomy & Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 516
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Tidskriftsartikel (refereegranskat)abstract
- The Gaia space astrometry mission (to be launched in 2012) will use a continuously spinning spacecraft to construct a global system of positions, proper motions and absolute parallaxes from relative position measurements made in an astrometric focal plane. This astrometric reduction can be cast as a classical least-squares problem, and the adopted baseline method for its solution uses a simple iteration algorithm. A potential weakness of this approach, as opposed to a direct solution, is that any finite number of iterations results in truncation errors that are difficult to quantify. Thus it is of interest to investigate alternative approaches, in particular the feasibility of a direct (non-iterative) solution. A simplified version of the astrometric reduction problem is studied in which the only unknowns are the astrometric parameters for a subset of the stars and the continuous three-axis attitude, thus neglecting further calibration issues. The specific design of the Gaia spacecraft and scanning law leads to an extremely large and sparse normal equations matrix. Elimination of the star parameters leads to a much smaller but less sparse system. We try different reordering schemes and perform symbolic Cholesky decomposition of this reduced normal matrix to study the fill-in for successively longer time span of simulated observations. Extrapolating to the full mission length, we conclude that a direct solution is not feasible with today's computational capabilities. Other schemes, e. g., eliminating the attitude parameters or orthogonalizing the observation equations, lead to similar or even worse problems. This negative result appears to be a consequence of the strong spatial and temporal connectivity among the unknowns achieved by two superposed fields of view and the scanning law, features that are in fact desirable and essential for minimizing large-scale systematic errors in the Gaia reference frame. We briefly consider also an approximate decomposition method a la Hipparcos, but conclude that it is either sub-optimal or effectively leads to an iterative solution.
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| 3. |
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| 4. |
- Hobbs, David, et al.
(författare)
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Determining PPN gamma with Gaia's astrometric core solution
- 2010
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Ingår i: Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis (IAU Symposium). - 1743-9213 .- 1743-9221. ; 261, s. 315-319
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Konferensbidrag (refereegranskat)abstract
- The ESA space astrometry mission Gala, due for launch in early 2012, will in addition to its huge output of fundamental astrometric and astrophysical data also provide stringent tests of general relativity. In this paper we present an updated analysis of Gaia's capacity to measure the PPN parameter gamma as part of its core astrometric solution. The analysis is based on small-scale astrometric solutions taking into account the simultaneous determination of stellar astrometric parameters and the satellite attitude. In particular, the statistical correlation between PPN gamma and the stellar parallaxes is considered. Extrapolating the results to a full-scale solution using some 100 million stars, we find that PPN gamma could be obtained to about 10(-6), which is significantly better than today's best estimate from the Cassini mission of 2 x 10(-5).
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| 5. |
- Hobbs, David, et al.
(författare)
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Gaia and the Astrometric Global Iterative Solution
- 2008
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Ingår i: Proceedings of IInternational Astronomical Union, IAU Symposium. ; 248, s. 119-120
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Konferensbidrag (refereegranskat)abstract
- Gaia is an ESA space astrometry mission due for launch in 2011-12. We describe part of the work carried out in the Gaia Data Processing and Analysis Consortium, namely the Astrometric Global Iterative Solution (AGIS) currently being implemented at the European Space Astronomy Center (ESAC) in Spain and largely based on algorithms developed at Lund Observatory. Some provisional results based on simulated observations of one million stars are presented, demonstrating convergence at microarcsec level independent of starting conditions.
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| 6. |
- Holl, Berry
(författare)
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Characterization and analysis of the astrometric errors in the global astrometric solution for Gaia
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The space astrometry mission Gaia, planned for launch in 2013 by the European Space Agency (ESA), will provide the most comprehensive and accurate catalogue of astrometric data for galactic and astrophysical research in the coming decades. It will observe roughly one billion stars, quasars and other point like objects for which the five astrometric parameters (position, parallax and proper motion) will be determined. The resulting catalogue will become available to the scientific community around 2020. The self-calibrating nature of Gaia requires that both the ~5 billion astrometric and ~50 million additional 'nuisance' parameters are estimated from 1000 billion observations. The interconnectivity of the parameters requires them to be estimated together using a global astrometric solution. The high connectivity together with the sheer number of parameters makes a direct solution computationally infeasible and therefore an iterative approach is adopted using the Astrometric Global Iterative Solution (AGIS). The main part of this thesis discusses the estimation and characterization of the astrometric errors that result from observations containing random errors. Because the observations will be dominated by photon noise this is a good approximation of reality. Using Monte-Carlo experiments we find that the astrometric parameters of sources with angular separation within roughly the field of view size of Gaia will be correlated due to observations that are affected by common (random) attitude errors, and that this correlation scales inversely with the number of sources per attitude parameter. We derive a covariance series expansion model that allows the efficient and accurate estimation of the covariance between any pair of astrometric parameters using only a limited amount of input data. This estimation was not possible before, but is now proposed as a tool in the Gaia catalogue. Additionally the identification and calibration of systematic errors due to radiation damage is studied. We use electron-level Monte-Carlo simulations of the observation process to characterize the biases and standard errors that result from radiation induced traps in the CCDs. Subsequently these standard errors and biases are rigorously propagated through the astrometric solution in numerical experiments. We find that the resulting biases in the astrometric parameters can easily be identified in the data, and that it is likely that they can be calibrated by the methods foreseen in the Gaia data processing. The resulting standard errors in the astrometric parameters are expected to increase by about 10% due to radiation damage, in which case Gaia can still reach its required scientific performance.
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| 7. |
- Holl, Berry, et al.
(författare)
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Characterizing the Astrometric Errors in the Gaia Catalogue
- 2011
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Ingår i: EAS Publications Series. - : EDP Sciences. - 1633-4760 .- 1638-1963. ; 45:GAIA: At the Frontiers of Astrometry, s. 117-122
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Tidskriftsartikel (refereegranskat)abstract
- Accurate characterization of the errors in the global astrometric solution for Gaia is essential for making optimal use of the catalogue data. We investigate the structure of the covariance between the estimated astrometric parameters by studying the properties of the astrometric least squares solution. We find that astrometric errors can be separated in a star and an attitude part, due to the estimation of the star and attitude parameters respectively. Hence the covariances can be separated in a star, an attitude and a cross term. This is demonstrated using our scalable simulation tool AGISLab, where the covariances are estimated statistically using Monte Carlo techniques.
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| 8. |
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| 9. |
- Holl, Berry, et al.
(författare)
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Error characterization of the Gaia astrometric solution I. Mathematical basis of the covariance expansion model
- 2012
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Ingår i: Astronomy & Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 543
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Tidskriftsartikel (refereegranskat)abstract
- Context. Accurate characterization of the astrometric errors in the forthcoming Gaia Catalogue will be essential for making optimal use of the data. This includes the correlations among the estimated astrometric parameters of the stars as well as their standard uncertainties, i.e., the complete (variance-)covariance matrix of the relevant astrometric parameters. Aims. Because a direct computation of the covariance matrix is infeasible due to the large number of parameters, approximate methods must be used. The aim of this paper is to provide a mathematical basis for estimating the variance-covariance of any pair of astrometric parameters, and more generally the covariance matrix for multidimensional functions of the astrometric parameters. The validation of this model by means of numerical simulations will be considered in a forthcoming paper. Methods. Based on simplifying assumptions (in particular that calibration errors can be neglected), we derive and analyse a series expansion of the covariance matrix of the least-squares solution. A recursive relation for successive terms is derived and interpreted in terms of the propagation of errors from the stars to the attitude and back. We argue that the expansion should converge rapidly to useful precision. The recursion is vastly simplified by using a kinematographic (step-wise) approximation of the attitude model. Results. Low-order approximations of arbitrary elements from the covariance matrix can be computed efficiently in terms of a limited amount of pre-computed data representing compressed observations and the structural relationships among them. It is proposed that the user interface to the Gaia Catalogue should provide the tools necessary for such computations.
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| 10. |
- Holl, Berry, et al.
(författare)
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Error characterization of the Gaia astrometric solution II. Validating the covariance expansion model
- 2012
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Ingår i: Astronomy & Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 543
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Tidskriftsartikel (refereegranskat)abstract
- Context. To use the data in the future Gaia catalogue it is important to have accurate estimates of the statistical uncertainties and correlations of the errors in the astrometric data given in the catalogue. Aims. In a previous paper we derived a mathematical model for computing the covariances of the astrometric data based on series expansions and a simplified attitude description. The aim of the present paper is to determine to what extent this model provides an accurate representation of the expected random errors in the astrometric solution for Gaia. Methods. We simulate the astrometric core solution by making least-squares solutions of the astrometric parameters for one million stars and the attitude parameters for a five-year mission, using nearly one billion simulated elementary observations for a total of 26 million unknowns. Two cases are considered: one in which all stars have the same magnitude, and another with 30% brighter and 70% fainter stars. The resulting astrometric errors are statistically compared with the model predictions. Results. In all cases considered, and within the statistical uncertainties of the numerical experiments (typically below 0.4%), the theoretically calculated variances and covariances are consistent with the simulations. To achieve this it is however necessary to expand the covariances to at least third or fourth order, and to apply a (theoretically motivated and derived) "fudge factor" in the kinematographic model. Conclusions. The model provides a feasible method to estimate the covariance of arbitrary astrometric data, accurate enough for most applications, and as such it should be available as part of the user's interface to the Gaia catalogue. A main assumption in the current model is that the observational errors are uncorrelated (e.g., photon noise), and further studies are needed on how correlated modelling errors, in particular in the attitude, can be taken into account.
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