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NASA NTRS1992

Dominant modes via model error

Obtaining a reduced model of a stable mechanical system with proportional damping is considered. Such systems can be conveniently represented in modal coordinates. Two popular schemes, the modal cost analysis and the balancing method, offer simple means of identifying dominant modes for retention in the reduced model. The dominance is measured via the modal costs in the case of modal cost analysis and via the singular values of the Gramian-product in the case of balancing. Though these measures do not exactly reflect the more appropriate model error, which is the H2 norm of the output-error between the full and the reduced models, they do lead to simple computations. Normally, the model error is computed after the reduced model is obtained, since it is believed that, in general, the model error cannot be easily computed a priori. The authors point out that the model error can also be calculated a priori, just as easily as the above measures. Hence, the model error itself can be used to determine the dominant modes. Moreover, the simplicity of the computations does not presume any special properties of the system, such as small damping, orthogonal symmetry, etc.

Yousuff, A.↗

NASA NTRS1993

Attitude control with realization of linear error dynamics

An attitude control law is derived to realize linear unforced error dynamics with the attitude error defined in terms of rotation group algebra (rather than vector algebra). Euler parameters are used in the rotational dynamics model because they are globally nonsingular, but only the minimal three Euler parameters are used in the error dynamics model because they have no nonlinear mathematical constraints to prevent the realization of linear error dynamics. The control law is singular only when the attitude error angle is exactly pi rad about any eigenaxis, and a simple intuitive modification at the singularity allows the control law to be used globally. The forced error dynamics are nonlinear but stable. Numerical simulation tests show that the control law performs robustly for both initial attitude acquisition and attitude control.

Paielli, Russell A.↗

NASA NTRS1991

Parallel computers - Estimate errors caused by imprecise data

A new approach to the problem of estimating errors caused by imprecise data is proposed in the context of software engineering. A software device is used to produce an ideal solution to the problem, when the computer is capable of computing errors of arbitrary programs. The software engineering aspect of this problem is to describe a device for computing the error estimates in software terms and then to provide precise numbers with error estimates to the user. The feasibility of the program capable of computing both some quantity and its error estimate in the range of possible measurement errors is demonstrated.

Kreinovich, Vladik↗

NASA NTRS1993

Analyzing human errors in flight mission operations

A long-term program is in progress at JPL to reduce cost and risk of flight mission operations through a defect prevention/error management program. The main thrust of this program is to create an environment in which the performance of the total system, both the human operator and the computer system, is optimized. To this end, 1580 Incident Surprise Anomaly reports (ISA's) from 1977-1991 were analyzed from the Voyager and Magellan projects. A Pareto analysis revealed that 38 percent of the errors were classified as human errors. A preliminary cluster analysis based on the Magellan human errors (204 ISA's) is presented here. The resulting clusters described the underlying relationships among the ISA's. Initial models of human error in flight mission operations are presented. Next, the Voyager ISA's will be scored and included in the analysis. Eventually, these relationships will be used to derive a theoretically motivated and empirically validated model of human error in flight mission operations. Ultimately, this analysis will be used to make continuous process improvements continuous process improvements to end-user applications and training requirements. This Total Quality Management approach will enable the management and prevention of errors in the future.

Bruno, Kristin J.↗

NASA NTRS1994

Realtime mitigation of GPS SA errors using Loran-C

The hybrid use of Loran-C with the Global Positioning System (GPS) was shown capable of providing a sole-means of enroute air radionavigation. By allowing pilots to fly direct to their destinations, use of this system is resulting in significant time savings and therefore fuel savings as well. However, a major error source limiting the accuracy of GPS is the intentional degradation of the GPS signal known as Selective Availability (SA). SA-induced position errors are highly correlated and far exceed all other error sources (horizontal position error: 100 meters, 95 percent). Realtime mitigation of SA errors from the position solution is highly desirable. How that can be achieved is discussed. The stability of Loran-C signals is exploited to reduce SA errors. The theory behind this technique is discussed and results using bench and flight data are given.

Braasch, Soo Y.↗

NASA NTRS1993

The importance of robust error control in data compression applications

Data compression has become an increasingly popular option as advances in information technology have placed further demands on data storage capabilities. With compression ratios as high as 100:1 the benefits are clear; however, the inherent intolerance of many compression formats to error events should be given careful consideration. If we consider that efficiently compressed data will ideally contain no redundancy, then the introduction of a channel error must result in a change of understanding from that of the original source. While the prefix property of codes such as Huffman enables resynchronisation, this is not sufficient to arrest propagating errors in an adaptive environment. Arithmetic, Lempel-Ziv, discrete cosine transform (DCT) and fractal methods are similarly prone to error propagating behaviors. It is, therefore, essential that compression implementations provide sufficient combatant error control in order to maintain data integrity. Ideally, this control should be derived from a full understanding of the prevailing error mechanisms and their interaction with both the system configuration and the compression schemes in use.

Woolley, S. I.↗

NASA NTRS1994

Minimizing systematic errors from atmospheric multiple scattering and satellite viewing geometry in coastal zone color scanner level IIA imagery

Water-leaving radiances and phytoplankton pigment concentrations are calculated from coastal zone color scanner (CZCS) radiance measurements by removing atmospheric Rayleigh and aerosol radiances from the total radiance signal measured at the satellite. The single greatest source of error in CZCS atmospheric correction algorithms in the assumption that these Rayleigh and aerosol radiances are separable. Multiple-scattering interactions between Rayleigh and aerosol components cause systematic errors in calculated aerosol radiances, and the magnitude of these errors is dependent on aerosol type and optical depth and on satellite viewing geometry. A technique was developed which extends the results of previous radiative transfer modeling by Gordon and Castano to predict the magnitude of these systematic errors for simulated CZCS orbital passes in which the ocean is viewed through a modeled, physically realistic atmosphere. The simulated image mathematically duplicates the exact satellite, Sun, and pixel locations of an actual CZCS image. Errors in the aerosol radiance at 443 nm are calculated for a range of aerosol optical depths. When pixels in the simulated image exceed an error threshhold, the corresponding pixels in the actual CZCS image are flagged and excluded from further analysis or from use in image compositing or compilation of pigment concentration databases. Studies based on time series analyses or compositing of CZCS imagery which do not address Rayleigh-aerosol multiple scattering should be interpreted cautiously, since the fundamental assumption used in their atmospheric correction algorithm is flawed.

Martin, D. L.↗

NASA NTRS1994

Space charge enhanced, plasma gradient induced error in satellite electric field measurements

In magnetospheric plasmas it is possible for plasma gradients to casue error in electric field measurements made by satellite double probes. The space charge emhanced plasma gradient induced error is discussed in general terms, the results of a laboratory experiment designed to illustrate this error are presented, and a simple expression that quantifies this error in a form that is readily applicable to satellite data is derived. The simple expression indicates that for a given probe bias current there is less error for cylindrical probes than for spherical probes. The expression also suggests that for Viking data the error is negligible.

Diebold, D. A.↗

NASA NTRS1995

Model studies of the beam-filling error for rain-rate retrieval with microwave radiometers

Low-frequency (less than 20 GHz) single-channel microwave retrievals of rain rate encounter the problem of beam-filling error. This error stems from the fact that the relationship between microwave brightness temperature and rain rate is nonlinear, coupled with the fact that the field of view is large or comparable to important scales of variability of the rain field. This means that one may not simply insert the area average of the brightness temperature into the formula for rain rate without incurring both bias and random error. The statistical heterogeneity of the rain-rate field in the footprint of the instrument is key to determining the nature of these errors. This paper makes use of a series of random rain-rate fields to study the size of the bias and random error associated with beam filling. A number of examples are analyzed in detail: the binomially distributed field, the gamma, the Gaussian, the mixed gamma, the lognormal, and the mixed lognormal ('mixed' here means there is a finite probability of no rain rate at a point of space-time). Of particular interest are the applicability of a simple error formula due to Chiu and collaborators and a formula that might hold in the large field of view limit. It is found that the simple formula holds for Gaussian rain-rate fields but begins to fail for highly skewed fields such as the mixed lognormal. While not conclusively demonstrated here, it is suggested that the notionof climatologically adjusting the retrievals to remove the beam-filling bias is a reasonable proposition.

Ha, Eunho↗

NASA NTRS1994

Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres

Although neglecting polarization and replacing the rigorous vector radiative transfer equation by its approximate scalar counterpart has no physical background, it is a widely used simplification when the incident light is unpolarized and only the intensity of the reflected light is to be computed. We employ accurate vector and scalar multiple-scattering calculations to perform a systematic study of the errors induced by the neglect of polarization in radiance calculations for a homogeneous, plane-parallel Rayleigh-scattering atmosphere (with and without depolarization) above a Lambertian surface. Specifically, we calculate percent errors in the reflected intensity for various directions of light incidence and reflection, optical thicknesses of the atmosphere, single-scattering albedos, depolarization factors, and surface albedos. The numerical data displayed can be used to decide whether or not the scalar approximation may be employed depending on the parameters of the problem. We show that the errors decrease with increasing depolarization factor and/or increasing surface albedo. For conservative or nearly conservative scattering and small surface albedos, the errors are maximum at optical thicknesses of about 1. The calculated errors may be too large for some practical applications, and, therefore, rigorous vector calculations should be employed whenever possible. However, if approximate scalar calculations are used, we recommend to avoid geometries involving phase angles equal or close to 0 deg and 90 deg, where the errors are especially significant. We propose a theoretical explanation of the large vector/scalar differences in the case of Rayleigh scattering. According to this explanation, the differences are caused by the particular structure of the Rayleigh scattering matrix and come from lower-order (except first-order) light scattering paths involving right scattering angles and right-angle rotations of the scattering plane.

Mishchenko, M. I.↗

NASA NTRS1995

Analysis of discretization errors in LES

All numerical simulations of turbulence (DNS or LES) involve some discretization errors. The integrity of such simulations therefore depend on our ability to quantify and control such errors. In the classical literature on analysis of errors in partial differential equations, one typically studies simple linear equations (such as the wave equation or Laplace's equation). The qualitative insight gained from studying such simple situations is then used to design numerical methods for more complex problems such as the Navier-Stokes equations. Though such an approach may seem reasonable as a first approximation, it should be recognized that strongly nonlinear problems, such as turbulence, have a feature that is absent in linear problems. This feature is the simultaneous presence of a continuum of space and time scales. Thus, in an analysis of errors in the one dimensional wave equation, one may, without loss of generality, rescale the equations so that the dependent variable is always of order unity. This is not possible in the turbulence problem since the amplitudes of the Fourier modes of the velocity field have a continuous distribution. The objective of the present research is to provide some quantitative measures of numerical errors in such situations. Though the focus of this work is LES, the methods introduced here can be just as easily applied to DNS. Errors due to discretization of the time-variable are neglected for the purpose of this analysis.

Ghosal, Sandip↗

NASA NTRS1996

New double-byte error-correcting codes for memory systems

Error-correcting or error-detecting codes have been used in the computer industry to increase reliability, reduce service costs, and maintain data integrity. The single-byte error-correcting and double-byte error-detecting (SbEC-DbED) codes have been successfully used in computer memory subsystems. There are many methods to construct double-byte error-correcting (DBEC) codes. In the present paper we construct a class of double-byte error-correcting codes, which are more efficient than those known to be optimum, and a decoding procedure for our codes is also considered.

Feng, Gui-Liang↗

NASA NTRS1997

Error Analysis System for Spacecraft Navigation Using the Global Positioning System (GPS)

The Flight Dynamics Division (FDD) at the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC) is currently developing improved space-navigation filtering algorithms to use the Global Positioning System (GPS) for autonomous real-time onboard orbit determination. In connection with a GPS technology demonstration on the Small Satellite Technology Initiative (SSTI)/Lewis spacecraft, FDD analysts and programmers have teamed with the GSFC Guidance, Navigation, and Control Branch to develop the GPS Enhanced Orbit Determination Experiment (GEODE) system. The GEODE system consists of a Kalman filter operating as a navigation tool for estimating the position, velocity, and additional states required to accurately navigate the orbiting Lewis spacecraft by using astrodynamic modeling and GPS measurements from the receiver. A parallel effort at the FDD is the development of a GPS Error Analysis System (GEAS) that will be used to analyze and improve navigation filtering algorithms during development phases and during in-flight calibration. For GEAS, the Kalman filter theory is extended to estimate the errors in position, velocity, and other error states of interest. The estimation of errors in physical variables at regular intervals will allow the time, cause, and effect of navigation system weaknesses to be identified. In addition, by modeling a sufficient set of navigation system errors, a system failure that causes an observed error anomaly can be traced and accounted for. The GEAS software is formulated using Object Oriented Design (OOD) techniques implemented in the C++ programming language on a Sun SPARC workstation. The Phase 1 of this effort is the development of a basic system to be used to evaluate navigation algorithms implemented in the GEODE system. This paper presents the GEAS mathematical methodology, systems and operations concepts, and software design and implementation. Results from the use of the basic system to evaluate navigation algorithms implemented on GEODE are also discussed. In addition, recommendations for generalization of GEAS functions and for new techniques to optimize the accuracy and control of the GPS autonomous onboard navigation are presented.

Truong, S. H.↗

NASA NTRS1961

Repeatability of the Over-All Errors of an Airplane Altimeter Installation in Landing-Approach Operations

Flight tests have been conducted to determine the repeatability of the over-all altimetry errors in the landing-approach condition of two sensitive altimeters (Air Force type C-12) installed in the cockpit of a transport airplane and of four precision altimeters (Air Force type MA-1) installed in a photo-observer. Data were obtained through a speed range of 62 to 100 knots during 42 landing-approach operations conducted on four different days. The results of the tests show that the repeatability errors of the two sensitive altimeters are +/- 35 feet and +/- 39 feet. These errors are of the same order as the maximum repeatability error measured in previous tests of eleven airplanes of the same type. For each of the four flights of the present tests the mean values of the data obtained with the two sensitive altimeters shifted by relatively large amounts, apparently because of the interaction of the stability and aftereffect- recovery characteristics of the instruments. For concurrent measurements of the over-all errors of the four precision altimeters, it is concluded that for comparable installations, the repeatability errors measured with these altimeters would be smaller than those measured with the sensitive altimeters.

Gracey, William↗

NASA NTRS2000

Stress Recovery and Error Estimation for 3-D Shell Structures

The C1-continuous stress fields obtained from finite element analyses are in general lower- order accurate than are the corresponding displacement fields. Much effort has focussed on increasing their accuracy and/or their continuity, both for improved stress prediction and especially error estimation. A previous project developed a penalized, discrete least squares variational procedure that increases the accuracy and continuity of the stress field. The variational problem is solved by a post-processing, 'finite-element-type' analysis to recover a smooth, more accurate, C1-continuous stress field given the 'raw' finite element stresses. This analysis has been named the SEA/PDLS. The recovered stress field can be used in a posteriori error estimators, such as the Zienkiewicz-Zhu error estimator or equilibrium error estimators. The procedure was well-developed for the two-dimensional (plane) case involving low-order finite elements. It has been demonstrated that, if optimal finite element stresses are used for the post-processing, the recovered stress field is globally superconvergent. Extension of this work to three dimensional solids is straightforward. Attachment: Stress recovery and error estimation for shell structure (abstract only). A 4-node, shear-deformable flat shell element developed via explicit Kirchhoff constraints (abstract only). A novel four-node quadrilateral smoothing element for stress enhancement and error estimation (abstract only).

Riggs, H. R.↗

NASA NTRS1994

Impact of Uncertainties and Errors in Converting NWS Radiosonde Hygristor Resistances to Relative Humidity

A set of 2,600 6-second, National Weather Service soundings from NASA's FIRE-II Cirrus field experiment are used to illustrate previously known errors and new potential errors in the VIZ and SDD ) brand relative humidity (RH) sensors and the MicroART processing software. The entire spectrum of RH is potentially affected by at least one of these errors. (These errors occur before being converted to dew point temperature.) Corrections to the errors are discussed. Examples are given of the effect that these errors and biases may have on numerical weather prediction and radiative transfer. The figure shows the OLR calculated for the corrected and uncorrected soundings using an 18-band radiative transfer code. The OLR differences are sufficiently large to warrant consideration when validating line-by-line radiation calculations that use radiosonde data to specify the atmospheric state, or when validating satellite retrievals. in addition, a comparison of observations of RH during FIRE-II derived from GOES satellite, raman lidar, MAPS analyses, NCAR CLASS sondes, and the NWS sondes reveals disagreement in the RH distribution and underlines our lack of an understanding of the climatology of water vapor.

Westphal, Douglas L.↗

NASA NTRS1994

Impact of Uncertainties and Errors in Converting NWS Radiosonde Hygristor Resistances to Relative Humidity

A set of 2,600 6-second, National Weather Service soundings from NASA's FIRE-II Cirrus field experiment are used to illustrate previously known errors and new potential errors in the VIZ and SDD brand relative humidity (RH) sensors and the MicroART processing software. The entire spectrum of RH is potentially affected by at least one of these errors. (These errors occur before being converted to dew point temperature.) Corrections to the errors are discussed. Examples are given of the effect that these errors and biases may have on numerical weather prediction and radiative transfer. The figure shows the OLR calculated for the corrected and uncorrected soundings using an 18-band radiative transfer code. The OLR differences are sufficiently large to warrant consideration when validating line-by-line radiation calculations that use radiosonde data to specify the atmospheric state, or when validating satellite retrievals. In addition, a comparison of observations of RE during FIRE-II derived from GOES satellite, raman lidar, MAPS analyses, NCAR CLASS sondes, and the NWS sondes reveals disagreement in the RH distribution and underlines our lack of an understanding of the climatology of water vapor.

Westphal, Douglas L.↗

NASA NTRS2002

Multilevel Error Estimation and Adaptive h-Refinement for Cartesian Meshes with Embedded Boundaries

This paper presents the development of a mesh adaptation module for a multilevel Cartesian solver. While the module allows mesh refinement to be driven by a variety of different refinement parameters, a central feature in its design is the incorporation of a multilevel error estimator based upon direct estimates of the local truncation error using tau-extrapolation. This error indicator exploits the fact that in regions of uniform Cartesian mesh, the spatial operator is exactly the same on the fine and coarse grids, and local truncation error estimates can be constructed by evaluating the residual on the coarse grid of the restricted solution from the fine grid. A new strategy for adaptive h-refinement is also developed to prevent errors in smooth regions of the flow from being masked by shocks and other discontinuous features. For certain classes of error histograms, this strategy is optimal for achieving equidistribution of the refinement parameters on hierarchical meshes, and therefore ensures grid converged solutions will be achieved for appropriately chosen refinement parameters. The robustness and accuracy of the adaptation module is demonstrated using both simple model problems and complex three dimensional examples using meshes with from 10(exp 6), to 10(exp 7) cells.

Aftosmis, M. J.↗