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Joe Frisbee - One of the best experts on this subject based on the ideXlab platform.
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Position Error Covariance Matrix Validation and Correction
2016Co-Authors: Joe FrisbeeAbstract:In order to calculate operationally accurate collision probabilities, the position Error Covariance matrices predicted at times of closest approach must be sufficiently accurate representations of the position uncertainties. This presentation will discuss why the Gaussian distribution is a reasonable expectation for the position uncertainty and how this assumed distribution type is used in the validation and correction of position Error Covariance matrices.
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An Empirical State Error Covariance Matrix Orbit Determination Example
2015Co-Authors: Joe FrisbeeAbstract:Short Abstract: The concept of the empirical state Error Covariance matrix for batch estimation is investigated. This matrix represents a direct and timely sampled data estimate of state Errors, under unknown or mismodeled Error sources. Of special interest is the problem of orbit determination under modeling Errors. The problem investigated here is that of the empirical Covariance matrix behavior and performance under simple but nontrivial Errors for gravity, drag and tracking uncertainty. Comparisons are made to the usual, theoretical Error Covariance associated with the batch estimate. Results, as compared to ideal modeling, are presented for epoch and a 24 hour propagation interval.
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Empirical State Error Covariance Matrix for Batch Estimation
2015Co-Authors: Joe FrisbeeAbstract:State estimation techniques effectively provide mean state estimates. However, the theoretical state Error Covariance matrices provided as part of these techniques often suffer from a lack of confidence in their ability to describe the uncertainty in the estimated states. By a reinterpretation of the equations involved in the weighted batch least squares algorithm, it is possible to directly arrive at an empirical state Error Covariance matrix. The proposed empirical state Error Covariance matrix will contain the effect of all Error sources, known or not. This empirical Error Covariance matrix may be calculated as a side computation for each unique batch solution. Results based on the proposed technique will be presented for a simple, two observer and measurement Error only problem.
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An Empirical State Error Covariance Matrix for Batch State Estimation
2011Co-Authors: Joe FrisbeeAbstract:State estimation techniques serve effectively to provide mean state estimates. However, the state Error Covariance matrices provided as part of these techniques suffer from some degree of lack of confidence in their ability to adequately describe the uncertainty in the estimated states. A specific problem with the traditional form of state Error Covariance matrices is that they represent only a mapping of the assumed observation Error characteristics into the state space. Any Errors that arise from other sources (environment modeling, precision, etc.) are not directly represented in a traditional, theoretical state Error Covariance matrix. Consider that an actual observation contains only measurement Error and that an estimated observation contains all other Errors, known and unknown. It then follows that a measurement residual (the difference between expected and observed measurements) contains all Errors for that measurement. Therefore, a direct and appropriate inclusion of the actual measurement residuals in the state Error Covariance matrix will result in an empirical state Error Covariance matrix. This empirical state Error Covariance matrix will fully account for the Error in the state estimate. By way of a literal reinterpretation of the equations involved in the weighted least squares estimation algorithm, it is possible to arrive at an appropriate, and formally correct, empirical state Error Covariance matrix. The first specific step of the method is to use the average form of the weighted measurement residual variance performance index rather than its usual total weighted residual form. Next it is helpful to interpret the solution to the normal equations as the average of a collection of sample vectors drawn from a hypothetical parent population. From here, using a standard statistical analysis approach, it directly follows as to how to determine the standard empirical state Error Covariance matrix. This matrix will contain the total uncertainty in the state estimate, regardless as to the source of the uncertainty. Also, in its most straight forward form, the technique only requires supplemental calculations to be added to existing batch algorithms. The generation of this direct, empirical form of the state Error Covariance matrix is independent of the dimensionality of the observations. Mixed degrees of freedom for an observation set are allowed. As is the case with any simple, empirical sample variance problems, the presented approach offers an opportunity (at least in the case of weighted least squares) to investigate confidence interval estimates for the Error Covariance matrix elements. The diagonal or variance terms of the Error Covariance matrix have a particularly simple form to associate with either a multiple degree of freedom chi-square distribution (more approximate) or with a gamma distribution (less approximate). The off diagonal or Covariance terms of the matrix are less clear in their statistical behavior. However, the off diagonal Covariance matrix elements still lend themselves to standard confidence interval Error analysis. The distributional forms associated with the off diagonal terms are more varied and, perhaps, more approximate than those associated with the diagonal terms. Using a simple weighted least squares sample problem, results obtained through use of the proposed technique are presented. The example consists of a simple, two observer, triangulation problem with range only measurements. Variations of this problem reflect an ideal case (perfect knowledge of the range Errors) and a mismodeled case (incorrect knowledge of the range Errors).
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An Empirical State Error Covariance Matrix for the Weighted Least Squares Estimation Method
2011Co-Authors: Joe FrisbeeAbstract:State estimation techniques effectively provide mean state estimates. However, the theoretical state Error Covariance matrices provided as part of these techniques often suffer from a lack of confidence in their ability to describe the un-certainty in the estimated states. By a reinterpretation of the equations involved in the weighted least squares algorithm, it is possible to directly arrive at an empirical state Error Covariance matrix. This proposed empirical state Error Covariance matrix will contain the effect of all Error sources, known or not. Results based on the proposed technique will be presented for a simple, two observer, measurement Error only problem.
Zhao Yuhong - One of the best experts on this subject based on the ideXlab platform.
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A new robust direct method for measurement Error Covariance estimation
International Conference on Neural Networks and Signal Processing 2003. Proceedings of the 2003, 2003Co-Authors: Zhao YuhongAbstract:Estimation of the measurement Error Covariance matrix is an essential requirement in data reconciliation methods. It is common practice to assume that the measurement Errors are normal and have a known Covariance matrix. A new robust direct algorithm for measurement Error Covariance estimation is proposed in this paper. Hampel's three-part redescending M-estimators are used to nullifies the effect of large outliers. A direct scheme treating the measured process variables is adopted to make it be used in the cases of nonlinear constraints. Implementation results show that credible results can be achieved either with or without the presence of external causes.
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Robust Estimation of Measurement Error Covariance
Control theory & applications, 2001Co-Authors: Zhao YuhongAbstract:Conventional indirect estimations of measurement Error Covariance are very sensitive to gross Errors. A new robust indirect algorithm based on Hampel's three part redescending M estimators is developed. Credible results can be achieved either with or without the presence of external causes. Two examples are provided to demonstrate the robustness of the proposed method.
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Robust estimation of measurement Error Covariance
Proceedings of the 3rd World Congress on Intelligent Control and Automation (Cat. No.00EX393), 1Co-Authors: Zhao Yuhong, Gu ZhongwenAbstract:Conventional indirect estimations of measurement Error Covariance are very sensitive to gross Errors. A robust indirect algorithm based on Hampel's three-part redescending M-estimators is developed. Credible results can be achieved either with or without the presence of external causes. Two examples are provided to demonstrate the robustness of the proposed method.
François-xavier Le Dimet - One of the best experts on this subject based on the ideXlab platform.
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Posterior Covariance vs. Analysis Error Covariance in Data Assimilation
2013Co-Authors: François-xavier Le Dimet, Victor Shutyaev, Igor GejadzeAbstract:The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain Errors (observation and background Errors), hence there is an Error in the analysis. For mildly nonlinear dynamics, the analysis Error Covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis Error Covariance is not the posterior Covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior Covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term.
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Analysis Error Covariance versus posterior Covariance in variational data assimilation
Quarterly Journal of the Royal Meteorological Society, 2013Co-Authors: Igor Gejadze, Victor Shutyaev, François-xavier Le DimetAbstract:The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain Errors (observation and background Errors), hence there is an Error in the analysis. For mildly nonlinear dynamics, the analysis Error Covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis Error Covariance is not the posterior Covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior Covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term.
Dacian N. Daescu - One of the best experts on this subject based on the ideXlab platform.
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The Adjoint Sensitivity Guidance to Diagnosis and Tuning of Error Covariance Parameters
Data Assimilation for Atmospheric Oceanic and Hydrologic Applications (Vol. II), 2013Co-Authors: Dacian N. Daescu, Rolf H LanglandAbstract:Adjoint techniques are effective tools for the analysis and optimization of the observation performance on reducing the Errors in the forecasts produced by atmospheric data assimilation systems (DASs). This chapter provides a detailed exposure of the equations that allow the extension of the adjoint-DAS applications from observation sensitivity and forecast impact assessment to diagnosis and tuning of parameters in the observation and background Error Covariance representation. The Error Covariance sensitivity analysis allows the identification of those parameters of potentially large impact on the forecast Error reduction and provides a first-order diagnostic to parameter specification. A proof-of-concept is presented together with comparative results of observation impact assessment and sensitivity analysis obtained with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System – Accelerated Representer (NAVDAS-AR) and the Navy Operational Global Atmospheric Prediction System (NOGAPS).
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Error Covariance sensitivity and impact estimation with adjoint 4d var theoretical aspects and first applications to navdas ar
Quarterly Journal of the Royal Meteorological Society, 2013Co-Authors: Dacian N. Daescu, Rolf H LanglandAbstract:This article presents the adjoint-data assimilation system (adjoint-DAS) approach to evaluate the forecast sensitivity with respect to the specification of the observation-Error Covariance (R-sensitivity) and background-Error Covariance (B-sensitivity) in a four-dimensional variational (4D-Var) DAS with a single outer-loop iteration. Computationally efficient estimates to the forecast impact of adjustments in the Error Covariance models are obtained by exploiting the mathematical properties of the R- and B-sensitivity matrices and their relationship with the observation sensitivity vector. An additional contribution of this work is that it establishes a synergistic link between various methodologies to analyze the DAS performance: observation sensitivity and impact assessment, Error Covariance sensitivity, and a posteriori diagnosis. The practical ability to obtain sensitivity information with respect to R- and B-parameters is presented with the adjoint versions of the Naval Research Laboratory Atmospheric Variational Data Assimilation System–Accelerated Representer (NAVDAS-AR) and the Navy Operational Global Atmospheric Prediction System (NOGAPS). The adjoint approach is used to provide guidance on the forecast impact of weighting the radiance data in the DAS according to observation-Error variance estimates derived from an a posteriori diagnosis. The results indicate that information extracted from both Error Covariance diagnosis and sensitivity analysis is necessary to design parameter tuning procedures that are effective in reducing the forecast Errors. Copyright © 2012 Royal Meteorological Society
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Adjoint sensitivity of the model forecast to data assimilation system Error Covariance parameters
Quarterly Journal of the Royal Meteorological Society, 2010Co-Authors: Dacian N. Daescu, Ricardo TodlingAbstract:The development of the adjoint of the forecast model and of the adjoint of the data assimilation system (adjoint-DAS) makes feasible the evaluation of the local sensitivity of a model forecast aspect with respect to a large number of parameters in the DAS. In this study it is shown that, by exploiting sensitivity properties that are intrinsic to the analyses derived from a minimization principle, the adjoint-DAS software tools developed at numerical weather prediction centres for observation and background sensitivity may be used to estimate the forecast sensitivity to observation- and background-Error Covariance parameters and for forecast impact assessment. All-at-once sensitivity to Error Covariance weighting coefficients and first-order impact estimates are derived as a particular case of the Error Covariance perturbation analysis. The use of the sensitivity information as a DAS diagnostic tool and for implementing gradient-based Error Covariance tuning algorithms is illustrated in idealized data assimilation experiments with the Lorenz 40-variable model. Preliminary results of forecast sensitivity to observation- and background-Error Covariance weight parameters are presented using the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS and its adjoint developed at the Global Modeling and Assimilation Office. Copyright © 2010 Royal Meteorological Society
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ICCS - Forecast sensitivity to the observation Error Covariance in variational data assimilation
Procedia Computer Science, 2010Co-Authors: Dacian N. DaescuAbstract:The development of the adjoint of the forecast model and of the adjoint of the data assimilation system (adjointDAS) make feasible the evaluation of the derivative-based forecast sensitivity to DAS input parameters in numerical weather prediction (NWP). The adjoint estimation of the forecast sensitivity to the observation Error Covariance in the DAS is considered as a practical approach to provide all-at-once first order estimates to the forecast impact as a result of variations in the specification of the observation Error statistics and guidance for tuning of Error Covariance parameters. The proposed methodology extends the capabilities of the adjoint modeling tools currently in place at major NWP centers for observation sensitivity and observation impact analysis. Illustrative numerical results are presented with the fifth-generation NASA Goddard Earth Observing System (GEOS-5) atmospheric DAS and its adjoint.
Jacques Verron - One of the best experts on this subject based on the ideXlab platform.
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efficient parameterization of the observation Error Covariance matrix for square root or ensemble kalman filters application to ocean altimetry
Monthly Weather Review, 2009Co-Authors: Jeanmichel Brankart, Clement Ubelmann, Charlesemmanuel Testut, Emmanuel Cosme, Pierre Brasseur, Jacques VerronAbstract:Abstract In the Kalman filter standard algorithm, the computational complexity of the observational update is proportional to the cube of the number y of observations (leading behavior for large y). In realistic atmospheric or oceanic applications, involving an increasing quantity of available observations, this often leads to a prohibitive cost and to the necessity of simplifying the problem by aggregating or dropping observations. If the filter Error Covariance matrices are in square root form, as in square root or ensemble Kalman filters, the standard algorithm can be transformed to be linear in y, providing that the observation Error Covariance matrix is diagonal. This is a significant drawback of this transformed algorithm and often leads to an assumption of uncorrelated observation Errors for the sake of numerical efficiency. In this paper, it is shown that the linearity of the transformed algorithm in y can be preserved for other forms of the observation Error Covariance matrix. In particular, quit...