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The class of all linear predictors of minimal order for a stationary vector-valued process is specified in terms of linear transformations on the associated Hankel covariance matrix. Two particular transformations, yielding computationally efficient construction schemes, are proposed.
This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.
The difference equations of Kalman filtering and smoothing recursively factor and invert the covariance of the output of a linear state-space system driven by a white-noise process. Here it is shown that similar recursive techniques factor and invert the inertia matrix of a multibody robot system. The random field models are based on the assumption that all of the inertial (D'Alembert) forces in the system are represented by a spatially distributed white-noise model. They are easier to describe than the models based on classical mechanics, which typically require extensive derivation and manipulation of equations of motion for complex mechanical systems. With the spatially random models, more primitive locally specified computations result in a global collective system behavior equivalent to that obtained with deterministic models. The primary goal of applying random field estimation is to provide a concise analytical foundation for solving robot control and motion planning problems.
Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting nonmodal transient growth. Failure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.
The current and next generation satellite infrared sensors such as AIRS, IASI, CrIS, and CLARREO all have one thing in common: large number of spectral channels. In order to fully utilize the information content of these sensors, a large number of Radiative Transfer (RT) calculations through the inhomogeneous atmosphere are needed. It is also computationally intensive to invert atmospheric and surface properties using all the channels. Usually, only subsets of channels are used to perform physical inversions for atmospheric profiles. We will explore ways to speed up RT calculations and inversions and test the algorithms using data from the Infrared Atmospheric Sounding Interferometer (IASI) satellite instrument. We will describe a retrieval algorithm using a Principal Component-based Radiative Transfer Model (PCRTM) for generating atmospheric temperature/moisture/ozone profiles and surface properties. The retrieval algorithm performs both the radiative transfer calculations and inversions in the Principal Component (PC) domain. The inversion algorithm is based on a non-linear Levenberg- Marquardt method with climatology covariance matrices and a priori information as constraints. One advantage of this approach is that it uses all information content from the ultraspectral data so that the retrieval is less sensitive to instrument noise and eliminates the need for selecting a sub-set of the channels. We will also use data collected during the Joint Airborne IASI Validation Experiment (JAIVEx) field campaign to validate the algorithm and IASI retrievals.
Using the Stokes waves as a model of nonlinear waves and considering the linear component as a narrow-band Gaussian process, the covariances and spectra of velocity and acceleration components and pressure for points in the vicinity of still water level were derived taking into consideration the effects of free surface fluctuations. The results are compared with those obtained earlier using linear Gaussian waves.
The accuracy with which the GEOS-C altitude may be estimated over long (7 day) and short (40 minute) orbital arcs is investigated. Over the long are excellent agreement was attained between a simulation of the orbit determination process and a covariance analysis. Both approaches yielded RMS altitude errors of about 1.5 meters over the Caribbean calibration area and approximately 7.5 meters overall. The geopotential was identified as the largest error source. For the short arc, the covariance analysis revealed that the propagated altitude error is linearly dependent upon station survey component errors which are also the largest source of altitude errors. An Appendix contains the mathematics of covariance analysis as applied to orbit determination.
This is an old issue in computational fluid dynamics (CFD). What is the so-called contravariant velocity or contravariant velocity component? In the article, we review the basics of tensor analysis and give the contravariant velocity component a rigorous explanation. For a given coordinate system, there exist two uniquely determined sets of base vector systems - one is the covariant and another is the contravariant base vector system. The two base vector systems are reciprocal. The so-called contravariant velocity component is really the contravariant component of a velocity vector for a time-independent coordinate system, or the contravariant component of a relative velocity between fluid and coordinates, for a time-dependent coordinate system. The contravariant velocity components are not physical quantities of the velocity vector. Their magnitudes, dimensions, and associated directions are controlled by their corresponding covariant base vectors. Several 2-D (two-dimensional) linear examples and 2-D mass-conservation equation are used to illustrate the details of expressing a vector with respect to the covariant and contravariant base vector systems, respectively.
A modern control theory design package (ORACLS) for constructing controllers and optimal filters for systems modeled by linear time-invariant differential or difference equations is described. Numerical linear-algebra procedures are used to implement the linear-quadratic-Gaussian (LQG) methodology of modern control theory. Algorithms are included for computing eigensystems of real matrices, the relative stability of a matrix, factored forms for nonnegative definite matrices, the solutions and least squares approximations to the solutions of certain linear matrix algebraic equations, the controllability properties of a linear time-invariant system, and the steady state covariance matrix of an open-loop stable system forced by white noise. Subroutines are provided for solving both the continuous and discrete optimal linear regulator problems with noise free measurements and the sampled-data optimal linear regulator problem. For measurement noise, duality theory and the optimal regulator algorithms are used to solve the continuous and discrete Kalman-Bucy filter problems. Subroutines are also included which give control laws causing the output of a system to track the output of a prescribed model.
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.
The ways an Independent Landing Monitor (ILM) may be used to complement the automatic landing function were studied. In particular, a systematic procedure was devised to establish the information and display requirements of an ILM during the landing phase of the flight. Functionally, the ILM system is designed to aid the crew in assessing whether the total system (e.g., avionics, aircraft, ground navigation aids, external disturbances) performance is acceptable, and, in case of anomaly, to provide adequate information to the crew to select the least unsafe of the available alternatives. Economically, this concept raises the possibility of reducing the primary autoland system redundancy and associated equipment and maintenance costs. The required level of safety for the overall system would in these cases be maintained by upgrading the backup manual system capability via the ILM. A safety budget analysis was used to establish the reliability requirements for the ILM. These requirements were used as constraints in devising the fault detection scheme. Covariance propagation methods were used with a linearized system model to establish the time required to correct manually perturbed states due to the fault. Time-to-detect and time-to-correct requirements were combined to devise appropriate altitudes and strategies for fault recovery.
Papers are presented on fiber optic links for airborne satellite applications, optoelectronic techniques for broadband switching, and GaAs circuits for a monolithic optical controller. Other topics include the optical processing of covariance matrices for adaptive processors, an optical linear heterodyne matrix-vector processor, and an EHF fiber optic-based array. An adaptive optical signal processing architecture using a signed-digit number system is considered along with microwave fiber optic links for phased arrays.
The Infrared Atmospheric Sounding Interferometer (IASI) is a hyperspectral sensor with 8461 spectral channels and a nominal spectral resolution of 0.25 cm(sup -1). It is computationally intensive to perform radiative transfer calculations and inversions using all these channels. We will present a Principal Component-based Radiative Transfer Model (PCRTM) and a retrieval algorithm which perform all the necessary calculations in EOF domain. Since the EOFs are orthogonal to each other, only about 100 principal components are needed to represent the information content of the 8461 channels. The PCRTM provides the EOF coefficients and associated derivatives with respect to atmospheric and surface parameters needed by the inversion algorithm. The inversion algorithm is based on a non-linear Levenberg-Marquardt method with climatology covariance and a priori information as constraints. The retrieved parameters include atmospheric temperature, moisture and ozone profiles, cloud parameters, surface skin temperature, and surface emissivities. To make the retrieval system even more compact and stable. The atmospheric vertical profiles are compressed into the EOF space as well. The surface emissivities are also compressed into EOF space.
Atmospheric Infrared Sounder (AIRS), Infrared Atmospheric Sounding Interferometer (IASI), and Cross-track Infrared Sounder (CrIS) are all hyper-spectral satellite sensors with thousands of spectral channels. Top of atmospheric radiance spectra measured by these sensors contain high information content on atmospheric, cloud, and surface properties. Exploring high information content contained in these high spectral resolution spectra is a challenging task due to computation e ort involved in modeling thousands of spectral channels. Usually, only very small fractions (4{10 percent) of the available channels are included in physical retrieval systems or numerical weather forecast (NWP) satellite data assimilations. We will describe a method of simultaneously retrieving atmospheric temperature, moisture, cloud, and surface properties using all available spectral channels without sacrificing computational speed. The essence of the method is to convert channel radiance spectra into super-channels by an Empirical Orthogonal Function (EOF) transformation. Because the EOFs are orthogonal to each other, about 100 super-channels are adequate to capture the information content of the radiance spectra. A Principal Component-based Radiative Transfer Model (PCRTM) developed at NASA Langley Research Center is used to calculate both the super-channel magnitudes and derivatives with respect to atmospheric profiles and other properties. There is no need to perform EOF transformations to convert super channels back to spectral space at each iteration step for a one-dimensional variational retrieval or a NWP data assimilation system. The PCRTM forward model is also capable of calculating radiative contributions due to multiple-layer clouds. The multiple scattering effects of the clouds are efficiently parameterized. A physical retrieval algorithm then performs an inversion of atmospheric, cloud, and surface properties in super channel domain directly therefore both reducing the computational need and preserving the information content of the IASI measurements. The inversion algorithm is based on a non-linear Levenberg-Marquardt method with climatology covariance matrices and a priori information as constraints. One advantage of this approach is that it uses all information content from the hyper-spectral data so that the retrieval is less sensitive to instrument noise and eliminates the need for selecting a sub-set of the channels.
Hyper-spectral remote sensors, such as Atmospheric Infrared Sounder (AIRS) and Infrared Atmospheric Sounding Interferometer (IASI), provides top of atmospheric radiances with high information content on atmospheric and surface properties. In order to analyze these data in real time, fast, and accurate forward and inverse models are needed. We will describe a method of simultaneously retrieving atmospheric temperature, moisture, cloud, and surface properties using all available spectral channels without sacrificing computational speed. The method has been successfully applied to AIRS, IASI, and NAST-I data. By applying the same method to the current and future hyperspectral sounders, the derived products will have less errors due to biases introduced by different retrieval methods. We have compared the retrieved products to radiosondes, aircraft measurements and other validation dataset. Simulations have done to quantify retrieval errors associated with the retrievals. The essence of the method is to convert channel radiance spectra into super-channels by an Empirical Orthogonal Function (EOF) transformation. A Principal Component-based Radiative Transfer Model (PCRTM) developed at NASA Langley Research Center is used to calculate both the super-channel magnitudes and derivatives with respect to atmospheric profiles and other properties. The inversion algorithm is based on a non-linear Levenberg-Marquardt method with climatology covariance matrices and a priori information as constraints. One advantage of this approach is that it uses all information content from the hyper-spectral data so that the retrieval is less sensitive to instrument noise and eliminates the need for selecting a subset of the channels. The PCRTM forward model has also been used to performance end-to-end sensor performance simulations of the Climate Absolute Radiance and Refractivity Observatory (CLARREO).
The research reported here extends the mathematical formulation of nonlinear, sigma-point estimators to enable consider covariance analysis for dynamical systems. This paper presents a novel sigma-point consider filter algorithm, for consider-parameterized nonlinear estimation, following the unscented Kalman filter (UKF) variation on the sigma-point filter formulation, which requires no partial derivatives of dynamics models or measurement models with respect to the parameter list. It is shown that, consistent with the attributes of sigma-point estimators, a consider-parameterized sigma-point estimator can be developed entirely without requiring the derivation of any partial-derivative matrices related to the dynamical system, the measurements, or the considered parameters, which appears to be an advantage over the formulation of a linear-theory sequential consider estimator. It is also demonstrated that a consider covariance analysis performed with this 'partial-derivative-free' formulation yields equivalent results to the linear-theory consider filter, for purely linear problems.
Nuclear data evaluation methods conventionally make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate for minimization of a cost function (even for non-linear models); and that both the model (of, e.g., neutron cross section) and experimental data (including their covariance data) are perfect. These assumptions are inherent to the well-known generalized linear least squares (GLLS) minimization method commonly used for evaluations of resolved resonance region (RRR) neutron cross sections. However, these assumptions are often not justified due to the presence of non-normal PDFs, non-linear models (e.g. R -matrix formalism), and inherent imperfections in data and models (e.g. discrepant data sets, discrepancies between the previous evaluation and newly measured data, or imperfect covariance data). We remove the said assumptions in a mathematical framework of Bayes’ theorem, and implement it using the Metropolis-Hastings Monte Carlo method. Parameters of a new kind are introduced to parameterize inherent imperfections, e.g. , any discrepancies between the theoretical model and measured data. These new parameters enable evaluators to quantify their expert judgement about any discrepancies or imperfections in a reproducible manner. We demonstrate the framework with an ongoing evaluation of 233 U in the eV region using the ENDF-B/VIII library and transmission data measured by Guber, et al. , and compare the posterior parameters to those obtained by conventional evaluation methods.
Powerful methods are now available for solving linear parametric inverse problems. However, many inverse problems which arise in geohysics are nonlinear. Fortunately, it is possible to treat most of these with the air of linear perturbation theory and liner inversion. But a convenient method is needed for assessing the importance of nonlinearity in these quasi-linear problems. The present paper provides such a method. Matsu'ura and Jackson (1984) have presented a simple algorithm for evaluating the asymptotic covariance matrix fo estimation errors. In the present investigation, aspects of linear inversion are discussed, taking into account linear parametric inverse problems, nonuniqueness, prior information, confidence limits, conditional and marginal statistics, the relative importance of the prior and observational data, and standardized variables. Attention is also given to nonlinear inversion, and the application of the considered approaches to a number of examples.