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NASA NTRS2000

Low-dimensional Representation of Error Covariance

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.

Tippett, Michael K.↗

NASA NTRS2013

Powered Descent Guidance with General Thrust-Pointing Constraints

The Powered Descent Guidance (PDG) algorithm and software for generating Mars pinpoint or precision landing guidance profiles has been enhanced to incorporate thrust-pointing constraints. Pointing constraints would typically be needed for onboard sensor and navigation systems that have specific field-of-view requirements to generate valid ground proximity and terrain-relative state measurements. The original PDG algorithm was designed to enforce both control and state constraints, including maximum and minimum thrust bounds, avoidance of the ground or descent within a glide slope cone, and maximum speed limits. The thrust-bound and thrust-pointing constraints within PDG are non-convex, which in general requires nonlinear optimization methods to generate solutions. The short duration of Mars powered descent requires guaranteed PDG convergence to a solution within a finite time; however, nonlinear optimization methods have no guarantees of convergence to the global optimal or convergence within finite computation time. A lossless convexification developed for the original PDG algorithm relaxed the non-convex thrust bound constraints. This relaxation was theoretically proven to provide valid and optimal solutions for the original, non-convex problem within a convex framework. As with the thrust bound constraint, a relaxation of the thrust-pointing constraint also provides a lossless convexification that ensures the enhanced relaxed PDG algorithm remains convex and retains validity for the original nonconvex problem. The enhanced PDG algorithm provides guidance profiles for pinpoint and precision landing that minimize fuel usage, minimize landing error to the target, and ensure satisfaction of all position and control constraints, including thrust bounds and now thrust-pointing constraints.

Carson, John M., III↗

NASA NTRS2017

Board Level Proton Testing Book of Knowledge for NASA Electronic Parts and Packaging Program

This book of knowledge (BoK) provides a critical review of the benefits and difficulties associated with using proton irradiation as a means of exploring the radiation hardness of commercial-off-the-shelf (COTS) systems. This work was developed for the NASA Electronic Parts and Packaging (NEPP) Board Level Testing for the COTS task. The fundamental findings of this BoK are the following. The board-level test method can reduce the worst case estimate for a board's single-event effect (SEE) sensitivity compared to the case of no test data, but only by a factor of ten. The estimated worst case rate of failure for untested boards is about 0.1 SEE/board-day. By employing the use of protons with energies near or above 200 MeV, this rate can be safely reduced to 0.01 SEE/board-day, with only those SEEs with deep charge collection mechanisms rising this high. For general SEEs, such as static random-access memory (SRAM) upsets, single-event transients (SETs), single-event gate ruptures (SEGRs), and similar cases where the relevant charge collection depth is less than 10 μm, the worst case rate for SEE is below 0.001 SEE/board-day. Note that these bounds assume that no SEEs are observed during testing. When SEEs are observed during testing, the board-level test method can establish a reliable event rate in some orbits, though all established rates will be at or above 0.001 SEE/board-day. The board-level test approach we explore has picked up support as a radiation hardness assurance technique over the last twenty years. The approach originally was used to provide a very limited verification of the suitability of low cost assemblies to be used in the very benign environment of the International Space Station (ISS), in limited reliability applications. Recently the method has been gaining popularity as a way to establish a minimum level of SEE performance of systems that require somewhat higher reliability performance than previous applications. This sort of application of the method suggests a critical analysis of the method is in order. This is also of current consideration because the primary facility used for this type of work, the Indiana University Cyclotron Facility (IUCF) (also known as the Integrated Science and Technology (ISAT) hall), has closed permanently, and the future selection of alternate test facilities is critically important. This document reviews the main theoretical work on proton testing of assemblies over the last twenty years. It augments this with review of reported data generated from the method and other data that applies to the limitations of the proton board-level test approach. When protons are incident on a system for test they can produce spallation reactions. From these reactions, secondary particles with linear energy transfers (LETs) significantly higher than the incident protons can be produced. These secondary particles, together with the protons, can simulate a subset of the space environment for particles capable of inducing single event effects (SEEs). The proton board-level test approach has been used to bound SEE rates, establishing a maximum possible SEE rate that a test article may exhibit in space. This bound is not particularly useful in many cases because the bound is quite loose. We discuss the established limit that the proton board-level test approach leaves us with. The remaining possible SEE rates may be as high as one per ten years for most devices. The situation is actually more problematic for many SEE types with deep charge collection. In cases with these SEEs, the limits set by the proton board-level test can be on the order of one per 100 days. Because of the limited nature of the bounds established by proton testing alone, it is possible that tested devices will have actual SEE sensitivity that is very low (e.g., fewer than one event in 1 × 10(exp 4) years), but the test method will only be able to establish the limits indicated above. This BoK further examines other benefits of proton board-level testing besides hardness assurance. The primary alternate use is the injection of errors. Error injection, or fault injection, is something that is often done in a simulation environment. But the proton beam has the benefit of injecting the majority of actual SEEs without risk of something being missed, and without the risk of simulation artifacts misleading the SEE investigation.

Guertin, Steven M.↗

NASA NTRS1971

The complexity of deterministic source encoding with a fidelity criterion

Block encoding of discrete, memoryless sources with equiprobable outputs subject to a fidelity criterion is discussed. The encoder is assumed to be deterministic and a lower bound is derived on the amount of computational work required to encode such sources as a function of the code rate and fidelity of the encoding. An example is given to show that the bound cannot be substantially improved. These bounds apply primarily to sources and distortion measures which are such that the minimum distortion given by the rate-distortion bound is not achievable with a finite block length, deterministic encoder.

Savage, J. E.↗

NASA NTRS1972

Radiative transition probabilities and recombination coefficients of the ion C IV.

Bound-bound and bound-free radiative transition probabilities, as well as radiative recombination coefficients of the ion C IV, are computed with a semi-empirical polarization potential method. The nonhydrogenic probabilities and coefficients are given for all bound states of the ion up to the principal quantum number n = 7.

Leibowitz, E. M.↗

NASA NTRS1986

A methodology for airplane parameter estimation and confidence interval determination in nonlinear estimation problems

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. With the fitted surface, sensitivity information can be updated at each iteration with less computational effort than that required by either a finite-difference method or integration of the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, and thus provides flexibility to use model equations in any convenient format. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. The degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels and to predict the degree of agreement between CR bounds and search estimates.

Murphy, P. C.↗

NASA NTRS1984

Error analysis of analytic solutions for self-excited near-symmetric rigid bodies - A numerical study

Analytic error bounds are presented for the solutions of approximate models for self-excited near-symmetric rigid bodies. The error bounds are developed for analytic solutions to Euler's equations of motion. The results are applied to obtain a simplified analytic solution for Eulerian rates and angles. The results of a sample application of the range and error bound expressions for the case of the Galileo spacecraft experiencing transverse torques demonstrate the use of the bounds in analyses of rigid body spin change maneuvers.

Kia, T.↗

NASA NTRS1987

An analysis of the vertical structure equation for arbitrary thermal profiles

The vertical structure equation is a singular Sturm-Liouville problem whose eigenfunctions describe the vertical dependence of the normal modes of the primitive equations linearized about a given thermal profile. The eigenvalues give the equivalent depths of the modes. The spectrum of the vertical structure equation and the appropriateness of various upper boundary conditions, both for arbitrary thermal profiles were studied. The results depend critically upon whether or not the thermal profile is such that the basic state atmosphere is bounded. In the case of a bounded atmosphere it is shown that the spectrum is always totally discrete, regardless of details of the thermal profile. For the barotropic equivalent depth, which corresponds to the lowest eigen value, upper and lower bounds which depend only on the surface temperature and the atmosphere height were obtained. All eigenfunctions are bounded, but always have unbounded first derivatives. It was proved that the commonly invoked upper boundary condition that vertical velocity must vanish as pressure tends to zero, as well as a number of alternative conditions, is well posed. It was concluded that the vertical structure equation always has a totally discrete spectrum under the assumptions implicit in the primitive equations.

Cohn, Stephen E.↗

NASA NTRS1987

Efficient computation of parameter confidence intervals

An important step in system identification of aircraft is the estimation of stability and control derivatives from flight data along with an assessment of parameter accuracy. When the maximum likelihood estimation technique is used, parameter accuracy is commonly assessed by the Cramer-Rao lower bound. It is known, however, that in some cases the lower bound can be substantially different from the parameter variance. Under these circumstances the Cramer-Rao bounds may be misleading as an accuracy measure. This paper discusses the confidence interval estimation problem based on likelihood ratios, which offers a more general estimate of the error bounds. Four approaches are considered for computing confidence intervals of maximum likelihood parameter estimates. Each approach is applied to real flight data and compared.

Murphy, Patrick C.↗

NASA NTRS1989

An analysis of the vertical structure equation for arbitrary thermal profiles

Cohn, Stephen E.↗

NASA NTRS1990

Reliability analysis of continuous fiber composite laminates

A composite lamina may be viewed as a homogeneous solid whose directional strengths are random variables. Calculation of the lamina reliability under a multi-axial stress state can be approached by either assuming that the strengths act separately (modal or independent action), or that they interact through a quadratic interaction criterion. The independent action reliability may be calculated in closed form, while interactive criteria require simulations; there is currently insufficient data to make a final determination of preference between them. Using independent action for illustration purposes, the lamina reliability may be plotted in either stress space or in a non-dimensional representation. For the typical laminated plate structure, the individual lamina reliabilities may be combined in order to produce formal upper and lower bounds of reliability for the laminate, similar in nature to the bounds on properties produced from variational elastic methods. These bounds are illustrated for a (0/plus or minus 15)sub s Graphite/Epoxy (GR/EP) laminate. And addition, simple physically plausible phenomenological rules are proposed for redistribution of load after a lamina has failed. These rules are illustrated by application to (0/plus or minus 15)sub s and (90/plus or minus 45/0)sub s GR/EP laminates and results are compared with respect to the proposed bounds.

Thomas, David J.↗

NASA NTRS1990

Computational efficiency of parallel combinatorial OR-tree searches

The performance of parallel combinatorial OR-tree searches is analytically evaluated. This performance depends on the complexity of the problem to be solved, the error allowance function, the dominance relation, and the search strategies. The exact performance may be difficult to predict due to the nondeterminism and anomalies of parallelism. The authors derive the performance bounds of parallel OR-tree searches with respect to the best-first, depth-first, and breadth-first strategies, and verify these bounds by simulation. They show that a near-linear speedup can be achieved with respect to a large number of processors for parallel OR-tree searches. Using the bounds developed, the authors derive sufficient conditions for assuring that parallelism will not degrade performance and necessary conditions for allowing parallelism to have a speedup greater than the ratio of the numbers of processors. These bounds and conditions provide the theoretical foundation for determining the number of processors required to assure a near-linear speedup.

Li, Guo-Jie↗

NASA NTRS1992

Photoexcited escape probability, optical gain, and noise in quantum well infrared photodetectors

We present a detailed and thorough study of a wide variety of quantum well infrared photodetectors (QWIPs), which were chosen to have large differences in their optical and transport properties. Both n- and p-doped QWIPs, as well as intersubband transitions based on photoexcitation from bound-to-bound, bound-to-quasi-continuum, and bound-to-continuum quantum well states were investigated. The measurements and theoretical analysis included optical absorption, responsivity, dark current, current noise, optical gain, hot carrier mean free path; net quantum efficiency, quantum well escape probability, quantum well escape time, as well as detectivity. These results allow a better understanding of the optical and transport physics and thus a better optimization of the QWIP performance.

Levine, B. F.↗

NASA NTRS1993

Structure of the dayside reconnection layer in resistive MHD and hybrid models

Numerical simulations were performed to investigate the structure of the reconnection layer at the dayside magnetopause. Two typical cases are examined in detail; both are asymmetric in magnetic field and plasma density. In case 1, the guide fields in the magnetosheath and in the magnetosphere are set at zero and thus the tangential magnetic fields on the two sides of the initial current sheet are exactly antiparallel. In case 2, the angle between the tangential magnetic fields on the two sides of the initial current sheet is 145 deg. The results obtained from a resistive MHD model and from a hybrid model are found to be different. In the MHD simulation of case 1, a 2-4 intermediate shock is found to bound the reconnection layer on the magnetosheath side, while an Alfven wave pulse bounds the reconnection layer on the magnetospheric side. In case 2, it is found that a time-dependent intermediate shock (TDIS) bounds the reconnection layer on the magnetosheath side, with a slow expansion wave propagating behind. With the MHD simulations, in the general case in which the tangential magnetic fields on the two sides of the initial current sheet are not exactly antiparallel, a rotational discontinuity across which the tangential magnetic field rotates, a large angle is found to bound the reconnection layer on the magnetosheath side.

Lin, Y.↗

NASA NTRS1992

An algorithm for the estimation of water temperatures from thermal multispectral airborne remotely sensed data

A method for water temperature estimation on the basis of thermal data is presented and tested against NASA's Thermal IR Multispectral Scanner. Using realistic bounds on emissivities, temperature bounds are calculated and refined to estimate a tighter bound on the emissivity of the source. The method is useful only when a realistic set of bounds can be obtained for the emissivities of the data.

Jaggi, S.↗

NASA NTRS1992

Information models of software productivity - Limits on productivity growth

Research into generalized information-metric models of software process productivity establishes quantifiable behavior and theoretical bounds. The models establish a fundamental mathematical relationship between software productivity and the human capacity for information traffic, the software product yield (system size), information efficiency, and tool and process efficiencies. An upper bound is derived that quantifies average software productivity and the maximum rate at which it may grow. This bound reveals that ultimately, when tools, methodologies, and automated assistants have reached their maximum effective state, further improvement in productivity can only be achieved through increasing software reuse. The reuse advantage is shown not to increase faster than logarithmically in the number of reusable features available. The reuse bound is further shown to be somewhat dependent on the reuse policy: a general 'reuse everything' policy can lead to a somewhat slower productivity growth than a specialized reuse policy.

Tausworthe, Robert C.↗

NASA NTRS1993

The importance of environmental conditions in reflectance spectroscopy of laboratory analogs for Mars surface materials

Reflectance spectra are presented here for a variety of particulate, ferric-containing analogs to Martian soil (Fe(3+)-doped smectites and palagonites) to facilitate interpretation of remotely acquired spectra. The analog spectra were measured under differing environmental conditions to evaluate the influence of exposure history on water content and absorption features due to H2O in these samples. Each of these materials contains structural OH bonded to metal cations, adsorbed H2O, and bound H2O (either in a glass, structural site, or bound to a cation). Previous experiments involving a variety of Mars analogs have shown that the 3 micron H2O band in spectra of palagonites is more resistant to drying than the 3 micron H2O band in spectra of montmorillonites. Other experiments have shown that spectra of ferrihydrite and montmorillonites doped with ferric sulfate also contain sufficient bound H2O to retain a strong 3 micron band under dry conditions. Once the effects of the environment on bound water in clays, oxides, and salts are better understood, the hydration bands measured via reflectance spectroscopy can be used to gain information about the chemical composition and moisture content of real soil systems. Such information would be especially useful in interpreting observations of Mars where subtle spatial variations in the strengths of metal-OH and H2O absorptions have been observed in telescopic and ISM spectra. We measured bidirectional reflectance spectra of several Mars soil analogs under controlled environmental conditions to assess the effects of moisture content on the metal-OH and H2O absorptions. The samples analyzed include chemically altered montmorillonites, ferrihydrite. and palagonites from Hawaii and Iceland. Procedures for preparation of the cation-exchanged montmorillonites, ferric-salt doped montmorillonites, and ferric oxyhydroxides are described in detail elsewhere.

Bishop, J.↗

NASA NTRS1994

Achieving unequal error protection with convolutional codes

This paper examines the unequal error protection capabilities of convolutional codes. Both time-invariant and periodically time-varying convolutional encoders are examined. The effective free distance vector is defined and is shown to be useful in determining the unequal error protection (UEP) capabilities of convolutional codes. A modified transfer function is used to determine an upper bound on the bit error probabilities for individual input bit positions in a convolutional encoder. The bound is heavily dependent on the individual effective free distance of the input bit position. A bound relating two individual effective free distances is presented. The bound is a useful tool in determining the maximum possible disparity in individual effective free distances of encoders of specified rate and memory distribution. The unequal error protection capabilities of convolutional encoders of several rates and memory distributions are determined and discussed.

Mills, D. G.↗