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NASA NTRS1994

Determining the accuracy of maximum likelihood parameter estimates with colored residuals

An important part of building high fidelity mathematical models based on measured data is calculating the accuracy associated with statistical estimates of the model parameters. Indeed, without some idea of the accuracy of parameter estimates, the estimates themselves have limited value. In this work, an expression based on theoretical analysis was developed to properly compute parameter accuracy measures for maximum likelihood estimates with colored residuals. This result is important because experience from the analysis of measured data reveals that the residuals from maximum likelihood estimation are almost always colored. The calculations involved can be appended to conventional maximum likelihood estimation algorithms. Simulated data runs were used to show that the parameter accuracy measures computed with this technique accurately reflect the quality of the parameter estimates from maximum likelihood estimation without the need for analysis of the output residuals in the frequency domain or heuristically determined multiplication factors. The result is general, although the application studied here is maximum likelihood estimation of aerodynamic model parameters from flight test data.

Morelli, Eugene A.↗

NASA NTRS1994

The Accuracy of Parameter Estimation in System Identification of Noisy Aircraft Load Measurement

This thesis focuses on the subject of the accuracy of parameter estimation and system identification techniques. Motivated by a complicated load measurement from NASA Dryden Flight Research Center, advanced system identification techniques are needed. The objective of this problem is to accurately predict the load experienced by the aircraft wing structure during flight determined from a set of calibrated load and gage response relationship. We can then model the problem as a black box input-output system identification from which the system parameter has to be estimated. Traditional LS (Least Square) techniques and the issues of noisy data and model accuracy are addressed. A statistical bound reflecting the change in residual is derived in order to understand the effects of the perturbations on the data. Due to the intrinsic nature of the LS problem, LS solution faces the dilemma of the trade off between model accuracy and noise sensitivity. A method of conflicting performance indices is presented, thus allowing us to improve the noise sensitivity while at the same time configuring the degredation of the model accuracy. SVD techniques for data reduction are studied and the equivalence of the Correspondence Analysis (CA) and Total Least Squares Criteria are proved. We also looked at nonlinear LS problems with NASA F-111 data set as an example. Conventional methods are neither easily applicable nor suitable for the specific load problem since the exact model of the system is unknown. Neural Network (NN) does not require prior information on the model of the system. This robustness motivated us to apply the NN techniques on our load problem. Simulation results for the NN methods used in both the single load and the 'warning signal' problems are both useful and encouraging. The performance of the NN (for single load estimate) is better than the LS approach, whereas no conventional approach was tried for the 'warning signals' problems. The NN design methodology is also presented. The use of SVD, CA and Collinearity Index methods are used to reduce the number of neurons in a layer.

Kong, Jeffrey↗

NASA NTRS1995

The construction of high-accuracy schemes for acoustic equations

An accuracy analysis of various high order schemes is performed from an interpolation point of view. The analysis indicates that classical high order finite difference schemes, which use polynomial interpolation, hold high accuracy only at nodes and are therefore not suitable for time-dependent problems. Thus, some schemes improve their numerical accuracy within grid cells by the near-minimax approximation method, but their practical significance is degraded by maintaining the same stencil as classical schemes. One-step methods in space discretization, which use piecewise polynomial interpolation and involve data at only two points, can generate a uniform accuracy over the whole grid cell and avoid spurious roots. As a result, they are more accurate and efficient than multistep methods. In particular, the Cubic-Interpolated Psuedoparticle (CIP) scheme is recommended for computational acoustics.

Tang, Lei↗

NASA NTRS1995

High Accuracy In-Flight Wavelength Calibration of Imaging Spectrometry Data

Accurate wavelength calibration of imaging spectrometer data is essential if proper atmospheric transmission corrections are to be made to obtain apparent surface reflectance. Accuracies of 0.1 nm are necessary for a 10 nm-sampling instrument in order to match the slopes of the deep atmospheric water vapor features that dominate the 0.7-2.3 micrometer regions. The Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) is calibrated in the laboratory to determine the wavelength position and full-width-half-maximum (FWHM) response for each of the 224 channels. The accuracies are limited by the quality of the monochromator used as a source. The accuracies vary from plus or minus to plus or minus 1.5 nm depending on the wavelength region, in general decreasing with increasing wavelength. Green et al. make corrections to the wavelength calibrations by using the known positions of 14 atmospheric absorption features throughout the 0.4-2.5 micrometer wavelength region. These features, having varying width and intensity, were matched to the MODTRAN model with a non-linear least squares fitting algorithm. A complete calibration was developed for all 224 channels by interpolation. Instrument calibration cannot be assumed to be stable to 0.1 nm over a flight season given the rigors of thermal cycling and launch and landing loads. The upcoming sensor HYDICE will require a means for in-flight spectral calibration of each scene because the calibration is both temperature and pressure sensitive. In addition, any sensor using a two-dimensional array has the potential for systematic wavelength shifts as a function of cross-track position, commonly called 'smile'. Therefore, a rapid means for calibrating complete images will be required. The following describes a method for determining instrument wavelength calibration using atmospheric absorption features that is efficient enough to be used for entire images on workstations. This study shows the effect of the surface reflectance on the calibration accuracy and the calibration history for the AVIRIS B spectrometer over the 1992 flight season.

Goetz, Alexander F. H.↗

NASA NTRS1996

Some Considerations in the Determination of the Accuracy of a Measurement in Space of the Newtonian Gravitational Constant (G)

A commonly suggested method for determining the Newtonian constant of universal gravitation (G) is to observe the motion of two bodies of known mass moving about each other in an orbiting laboratory. In low Earth orbit (LEO), bodies constructed of even the densest material available experience a gravitational attraction that is several times smaller than the 'tidal' forces (due to their proximity to the Earth), which tend to pull them apart. While the tidal forces do not preclude stable orbits of the two objects about each other, they and the Coriolis force (in the rotating laboratory) dominate the motion, and the gravitational attraction of the two bodies may be considered a weak (but significant) contribution to the motion. As a result, compared to an experiment that would be performed in a laboratory far from the Earth, greater accuracy of measuring the motion of the two bodies may be required for a given accuracy in the determination of G. We find that the accuracy with which positions must be determined is not much different in an experiment in LEO than in one performed far from the Earth, but that rotational periods must be determined more accurately. Using a curvature matrix analysis, we also find that a value of G may be extracted (with some loss in accuracy, but probably some practical gain) from an analysis of the time dependence of the distance between the bodies rather than of a full specification (distance and direction) of their relative positions. A measurement of the gravitational constant to one part in 10(exp 4) continues to be thinkable, but one part in 10(exp 5) will be very difficult.

Baker, Stephen D.↗

NASA NTRS1997

Accuracy Study of a 2-Component Point Doppler Velocimeter (PDV)

A two-component Point Doppler Velocimeter (PDV) which has recently been developed is described, and a series of velocity measurements which have been obtained to quantify the accuracy of the PDV system are summarized. This PDV system uses molecular iodine vapor cells as frequency discriminating filters to determine the Doppler shift of laser light which is scattered off of seed particles in a flow. The majority of results which have been obtained to date are for the mean velocity of a rotating wheel, although preliminary data are described for fully-developed turbulent pipe flow. Accuracy of the present wheel velocity data is approximately +/- 1 % of full scale, while linearity of a single channel is on the order of +/- 0.5 % (i.e., +/- 0.6 m/sec and +/- 0.3 m/sec, out of 57 m/sec, respectively). The observed linearity of these results is on the order of the accuracy to which the speed of the rotating wheel has been set for individual data readings. The absolute accuracy of the rotating wheel data is shown to be consistent with the level of repeatability of the cell calibrations. The preliminary turbulent pipe flow data show consistent turbulence intensity values, and mean axial velocity profiles generally agree with pitot probe data. However, there is at present an offset error in the radial velocity which is on the order of 5-10 % of the mean axial velocity.

Kuhlman, John↗

NASA NTRS1998

Effects of magnetometer calibration and maneuvers on accuracies of magnetometer-only attitude-and-rate determination

Two different algorithms - a deterministic magnetic-field-only algorithm and a Kalman filter for gyroless spacecraft - are used to estimate the attitude and rates of the Rossi X-Ray Timing Explorer (RXTE) using only measurements from a three-axis magnetometer. The performance of these algorithms is examined using in-flight data from various scenarios. In particular, significant enhancements in accuracies are observed when' the telemetered magnetometer data are accurately calibrated using a recently developed calibration algorithm. Interesting features observed in these studies of the inertial-pointing RXTE include a remarkable sensitivity of the filter to the numerical values of the noise parameters and relatively long convergence time spans. By analogy, the accuracy of the deterministic scheme is noticeably lower as a result of reduced rates of change of the body-fixed geomagnetic field. Preliminary results show the filter-per-axis attitude accuracies ranging between 0.1 and 0.5 deg and rate accuracies between 0.001 deg/sec and 0.005 deg./sec, whereas the deterministic method needs a more sophisticated techniques for smoothing time derivatives of the measured geomagnetic field to clearly distinguish both attitude and rate solutions from the numerical noise. Also included is a new theoretical development in the deterministic algorithm: the transformation of a transcendental equation in the original theory into an 8th-order polynomial equation. It is shown that this 8th-order polynomial reduces to quadratic equations in the two limiting cases-infinitely high wheel momentum, and constant rates-discussed in previous publications.

Challa, M.↗

NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations That Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations that Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS1999

Accuracy Study of the Space-Time CE/SE Method for Computational Aeroacoustics Problems Involving Shock Waves

The space-time conservation element and solution element(CE/SE) method is used to study the sound-shock interaction problem. The order of accuracy of numerical schemes is investigated. The linear model problem.govemed by the 1-D scalar convection equation, sound-shock interaction problem governed by the 1-D Euler equations, and the 1-D shock-tube problem which involves moving shock waves and contact surfaces are solved to investigate the order of accuracy of numerical schemes. It is concluded that the accuracy of the CE/SE numerical scheme with designed 2nd-order accuracy becomes 1st order when a moving shock wave exists. However, the absolute error in the CE/SE solution downstream of the shock wave is on the same order as that obtained using a fourth-order accurate essentially nonoscillatory (ENO) scheme. No special techniques are used for either high-frequency low-amplitude waves or shock waves.

Wang, Xiao Yen↗

NASA NTRS2000

A Probabilistic Technique for Improving the Mid-Swath Wind Direction Accuracy of Scanning Pencil Beam Scatterometers

The SeaWinds on QuikSCAT scatterometer is a device developed by NASA JPL to measure the speed and direction of ocean surface winds. Simulations performed in order to determine the expected performance of the instrument prior to its launch have indicated that the accuracy of the retrieved wind vectors varies across the swath. In particular the mid-swath accuracy is worse than that of the rest of the swath. This behavior is a general characteristic of scanning pencil beam scatterometers. For SeaWinds, the accuracy of the rest of the swath, and the size of the swath are such that the instrument meets its science requirements despite mid-swath shortcomings. However, by understanding the problem at mid-swath, we can improve the performance there as well, yielding even better coverage and accuracy than was originally intended. The mid-swath performance degradation is due to suboptimal viewing geometry. In particular mid-swath measurements contains measurements from two azimuths, roughly 180 degrees apart. With the standard wind retrieval technique, mid-swath root mean square (RMS) wind direction errors are approximately 30 degrees for low wind speeds (3 - 5.5 m/s) and 20 degrees for moderate to high wind speeds (5.5 - 30 m/s). We discuss the underlying causes of this phenomenon in detail and propose a modification to the wind retrieval technique in order to improve mid-swath performance. By estimating the range of likely wind directions for each measurement cell, one can optimally apply information from neighboring cells in order to reduce random wind direction errors without significantly degrading the resolution of the resultant wind field. In this manner we are able to achieve mid-swath RMS wind direction errors as low as 15 degrees for low winds and 10 degrees for moderate to high winds, while at the same time preserving high resolution structures such as cyclones and fronts. Since our results are based on simulated data, we also outline a procedure for making use of real data to optimize the technique after launch.

Stiles, Bryan↗

NASA NTRS2002

Thermocouple Calibration and Accuracy in a Materials Testing Laboratory

A consolidation of information has been provided that can be used to define procedures for enhancing and maintaining accuracy in temperature measurements in materials testing laboratories. These studies were restricted to type R and K thermocouples (TCs) tested in air. Thermocouple accuracies, as influenced by calibration methods, thermocouple stability, and manufacturer's tolerances were all quantified in terms of statistical confidence intervals. By calibrating specific TCs the benefits in accuracy can be as great as 6 C or 5X better compared to relying on manufacturer's tolerances. The results emphasize strict reliance on the defined testing protocol and on the need to establish recalibration frequencies in order to maintain these levels of accuracy.

Lerch, B. A.↗

NASA NTRS2002

On Accuracy of Adaptive Grid Methods for Captured Shocks

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax-Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

Yamaleev, Nail K.↗

NASA NTRS2003

Navigation Accuracy Guidelines for Orbital Formation Flying Missions

Some simple guidelines based on the accuracy in determining a satellite formation's semi-major axis differences are useful in making preliminary assessments of the navigation accuracy needed to support such missions. These guidelines are valid for any elliptical orbit, regardless of eccentricity. Although maneuvers required for formation establishment, reconfiguration, and station-keeping require accurate prediction of the state estimate to the maneuver we, and hence are directly affected by errors in all the orbital elements, experience has shown that determination of orbit plane orientation and orbit shape to acceptable levels is less challenging than the determination of orbital period or semi-major axis. Furthermore, any differences among the member s semi-major axes are undesirable for a satellite formation, since it will lead to differential along-track drift due to period differences. Since inevitable navigation errors prevent these differences from ever being zero, one may use the guidelines this paper presents to determine how much drift will result from a given relative navigation accuracy, or conversely what navigation accuracy is required to limit drift to a given rate. Since the guidelines do not account for non-two-body perturbations, they may be viewed as useful preliminary design tools, rather than as the basis for mission navigation requirements, which should be based on detailed analysis of the mission configuration, including all relevant sources of uncertainty.

Carpenter, J. Russell↗

NASA NTRS2004

Navigation Accuracy Guidelines for Orbital Formation Flying

Some simple guidelines based on the accuracy in determining a satellite formation s semi-major axis differences are useful in making preliminary assessments of the navigation accuracy needed to support such missions. These guidelines are valid for any elliptical orbit, regardless of eccentricity. Although maneuvers required for formation establishment, reconfiguration, and station-keeping require accurate prediction of the state estimate to the maneuver time, and hence are directly affected by errors in all the orbital elements, experience has shown that determination of orbit plane orientation and orbit shape to acceptable levels is less challenging than the determination of orbital period or semi-major axis. Furthermore, any differences among the member s semi-major axes are undesirable for a satellite formation, since it will lead to differential along-track drift due to period differences. Since inevitable navigation errors prevent these differences from ever being zero, one may use the guidelines this paper presents to determine how much drift will result from a given relative navigation accuracy, or conversely what navigation accuracy is required to limit drift to a given rate. Since the guidelines do not account for non-two-body perturbations, they may be viewed as useful preliminary design tools, rather than as the basis for mission navigation requirements, which should be based on detailed analysis of the mission configuration, including all relevant sources of uncertainty.

Carpenter, J. Russell↗

NASA NTRS2004

Orbit Determination Accuracy for Comets on Earth-Impacting Trajectories

The results presented show the level of orbit determination accuracy obtainable for long-period comets discovered approximately one year before collision with Earth. Preliminary orbits are determined from simulated observations using Gauss' method. Additional measurements are incorporated to improve the solution through the use of a Kalman filter, and include non-gravitational perturbations due to outgassing. Comparisons between observatories in several different circular heliocentric orbits show that observatories in orbits with radii less than 1 AU result in increased orbit determination accuracy for short tracking durations due to increased parallax per unit time. However, an observatory at 1 AU will perform similarly if the tracking duration is increased, and accuracy is significantly improved if additional observatories are positioned at the Sun-Earth Lagrange points L3, L4, or L5. A single observatory at 1 AU capable of both optical and range measurements yields the highest orbit determination accuracy in the shortest amount of time when compared to other systems of observatories.

Kay-Bunnell, Linda↗

NASA NTRS2004

Geopositional Accuracy Assessment of EarthSat Geocover Landsat Orthorectified Imagery

NASA purchased EarthSat GeoCover orthorectified Landsat imagery of global land areas covering three historical frames: (1) mid-1970's imagery from the Landsat Multispectral Scanner (MSS); (2) late 1980's-early 1990's imagery from the Landsat Thematic Mapper (TM); and (3) year 2000 imagery from the Landsat Enhanced Thematic Mapper Plus (ETM+). Because of the distinct time frames covered by these datasets, this imagery is valuable to land cover change research. Because geopositional accuracy plays a critical role in this area of scientific research. NASA performed an independent assessment of the geopositional accuracy of each EarthSat dataset using an independent set of government-provided ground control points (GCPs). These points were instrumental in the geopositional accuracy assessment of the TM imagery. Because of the orthorectification processes of the MSS imagery and the MSS pixel size, the aforementioned GCPs could not be used, and an alternate relative assessment procedure using the previously validated TM imagery as a "truth" dataset was used for the MSS data. Finally, the ETM data specification was defined in both an absolute sense with respect to ground coordinated and relative to the previously validated TM dataset. Therefore, two separate methods were used in validating the ETM-data. Results of the NASA independent assessments showed that the accuracies of the EarthSat GeoCover datasets met the defined specifications or were within the error and limitations of the verification methods employed.

Smith, Charles M.↗

NASA NTRS2005

Accuracy and stability of fringe measurements in SIM

In order to identify habitable planets beyond our solar system it is necessary to determine their masses. SIM PlanetQuest is the only space mission which can discover and measure planets as small and cool as those in our solar system. This mission, however, requires unprecedented accuracy of angular rneasurements to the level of a microarcsecond, which are three orders of magnitude better than accuracy of the recent space astrometric missions. For microarcsecond accuracy of astrometric measurements the corresponding fringe delay must be determined to the accuracy of picometers, and it is a challenging task to measure fringe delays to this level of precision.

interferometer↗