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Results for “least-squares method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1978

On the determination and investigation of the terrestrial ionospheric refractive indices using GEOS-3/ATS-6 satellite-to-satellite tracking data

When the radio link between two satellites (GEOS-3/ATS-6) is intercepted by the earth's ionosphere and neutral atmosphere, a change in the Doppler frequency results. Travel through the atmosphere causes the Doppler phase to be advanced in the ionosphere's portion and retarded in the neutral portion of the atmosphere. Analysis of the shortening and lengthening of the phase of the Satellite-to-Satellite Tracking (SST) data that passed within 40-700 km above the earth's surface during its ATS-6 to GEOS-3 to ATS-6 path, caused by the atmosphere, results in refractivity versus height profiles. The SST Doppler data were used directly to adjust the GEOS-3 orbit. Perturbation from the Moon, Sun and a 15th order/degree earth gravity field were included in the orbit solution. This orbit was continued through the occultation period and a model ionosphere was estimated by a least-square adjustment of the Chapman ionosphere parameters from the SST data residuals. The refractivity profile obtained by this model ionosphere was compared to a refractivity profile obtained by a direct integral inversion of the SST data residuals. Systematic differences between the 2 methods were caused by orbital errors, which propagated into the solution. The SST data yielded refractive index profiles in a novel economical manner because no additional or special on-board equipment were required.

Liu, A. S.↗
NASA NTRS1979

Photographic stellar photometry with the PDS microdensitometer

A technique for photographic stellar photometry with the PDS microdensitometer is described. It employs a least-squares fit to a model density profile to derive an instrumental magnitude index, an image-abnormality index, and the local value of the background density for each image. The instrumental magnitude index is calibrated in terms of true magnitude by the same methods as for iris photometry. A preliminary test of the method using plates of the open cluster NGC 188 indicates that a precision comparable to or slightly better than that of conventional iris photometry or other methods of PDS reduction may easily be attained. Possibilities for the future elaboration of the technique are mentioned.

Stetson, P. B.↗
NASA NTRS1975

Surface fitting three-dimensional bodies

The geometry of general three-dimensional bodies was generated from coordinates of points in several cross sections. Since these points may not be on smooth curves, they are divided into groups forming segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction through longitudinal curves. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines or specifying slopes at selected points. This method was used to surface fit a 70 deg slab delta wing and the HL-10 Lifting Body. The results for the delta wing were very close to the exact geometry. Although there is no exact solution for the lifting body, the surface fit generated a smooth surface with cross-sectional planes very close to prescribed coordinate points.

Dejarnette, F. R.↗
NASA NTRS1978

Simplified contact analysis

A simple and straightforward method for calculating the elastic deformation at the center of contact between two solids with different radii of curvature is proposed instead of the conventional contact deformation analysis for ball bearings, gears, and cams, which usually involves tedious iterative procedures or the use of design charts. Ellipticity of the contact is approximated from a least-squares power fit. Simplified expressions are derived which allow rapid calculation of deformation from the material properties and geometry of the contacting elements.

Brewe, D. E.↗
NASA NTRS1974

The two-dimensional tunnel structures of K3Sb5O14 and K2Sb4O11

The structures of K3Sb5O14 and K2Sb4O11 have been solved by the single-crystal X-ray direct method and the heavy-atom method, respectively. The structure of K3Sb5O14 is orthorhombic, with space group Pbam and cell parameters a = 24.247 (4), b = 7.157 (2), c = 7.334 (2) A, Z = 4. The structure of K2Sb4O11 is monoclinic, with space group C2/m and cell parameters a = 19.473 (4), b = 7.542 (1), c = 7.198 (1) A, beta = 94.82 (2) deg, Z = 4. A full-matrix least-squares refinement gave R = 0.072 and R = 0.067, respectively. In both structures, oxygen atoms form an octahedron around each Sb atom and an irregular polyhedron around each K atom. By sharing corners and edges, the octahedra form a skeleton network having intersecting b-axis and c-axis tunnels. The K(+) ions, which have more than ten oxygen near neighbors, are located in these tunnels. Evidence for K(+)-ion transport within and between tunnels comes from ion exchange of the alkali ions in molten salts and anisotropic temperature factors that are anomalously large in the direction of the tunnels.

Hong, H. Y.-P.↗
NASA NTRS1982

Crustal interpretation of the MAGSAT data in the continental United States

The processing of MAGSAT scalar data to construct a crustal magnetic anomaly map over the continental U.S. involves removal of the reference field model, a path-by-path subtraction of a low order polynomial through a least-squares fit to reduce orbital offset errors, and a two dimensional spectral filtering to mitigate the spectral bias induced by the path-by-path orbital correction scheme. The resultant anomaly map shows reasonably good correlations with an aeromagnetic map derived from the project MAGNET. Prominent satellite magnetic anomalies are identified in terms of geological provinces and age boundaries. An inversion method was applied to MAGSAT data which produces both the Curie depth topography and laterally varying magnetic susceptibility of the crust. A contoured Curie depth map thus derived shows general agreements with a crustal thickness map based on seismic data.

Won, I. J.↗
NASA NTRS1981

Simulation of the magnetic structure of the inner heliosphere by means of non-spherical source surface

A new method for mapping the Sun's magnetic field B from the photosphere through the corona and interplanetary space is presented. The method entails the derivation of B from a scalar potential within a current-free annular volume bounded inside by photosphere and outside by a prescribed nonspherical source surface to which B is made (as nearly as possible) perpendicular. As usual we obtain the potential for the part of B that arises from currents inside the Sun by fitting an expansion to the observed line-of-sight component of B at the photosphere. A second least-squares fit is introduced to obtain the part of B that arises from currents outside the source surface. Comparisons are made between this model and observed coronal and interplanetary structures. There is evidence that observation data underestimate the strength of photospheric polar magnetic fields.

Levine, R. H.↗
NASA NTRS1982

X-ray observations of Abell 2218 and implications for the Sunyaev-Zel'dovich effect

The Einstein Observatory was used to obtain a high-resolution X-ray image and an X-ray flux measurement of the galaxy cluster Abell 2218. The combined system of intracluster medium and galaxies is modeled as a gas in hydrostatic equilibrium in a common gravitational potential with an isothermal galaxy component. This model is then fitted in the least-squares sense to the X-ray surface brightness distribution and to the optically determined galaxy surface density profile to determine model parameters. The gas is found to be nominally isothermal, moderately dense, and very hot (10-30 keV); these conditions are consistent with the Sunyaev-Zel'dovich effect, i.e., the diminution in the cosmic microwave background radiation intensity in the direction of this cluster. It is noted that the considerable imprecision inherent in the method of distance measurement based on the combination of microwave measurements with X-ray and optical data makes it impossible to make a definitive statement about the distance to Abell 2218.

Boynton, P. E.↗
NASA NTRS1982

Errors in fixed and moving frame of references - Applications for conventional and Doppler radar analysis

Techniques for the correction of errors in Doppler radar scans due to advection effects are presented. A moving frame of reference is shown to be useful in least-squares estimates with stationary observations expressed in scalars or Cartesian coordinates. For non-Cartesian coordinates, such as the deduction of radial velocities from triple Doppler radar data, an integral is defined for accounting for advection effects and consequent coordinate transformations. The multiple radars are necessary for unambiguous characterization of the horizontal wind velocity. A scale analysis is employed to estimate errors with and without the error correction procedure. Improvements in correlations between scans are demonstrated when the error correction method is used.

Tzvi, G.-C.↗
NASA NTRS1974

Curve fits of predicted inviscid stagnation-point radiative heating rates, cooling factors, and shock standoff distances for hyperbolic earth entry

Curve-fit formulas are presented for the stagnation-point radiative heating rate, cooling factor, and shock standoff distance for inviscid flow over blunt bodies at conditions corresponding to high-speed earth entry. The data which were curve fitted were calculated by using a technique which utilizes a one-strip integral method and a detailed nongray radiation model to generate a radiatively coupled flow-field solution for air in chemical and local thermodynamic equilibrium. The range of free-stream parameters considered were altitudes from about 55 to 70 km and velocities from about 11 to 16 km.sec. Spherical bodies with nose radii from 30 to 450 cm and elliptical bodies with major-to-minor axis ratios of 2, 4, and 6 were treated. Powerlaw formulas are proposed and a least-squares logarithmic fit is used to evaluate the constants. It is shown that the data can be described in this manner with an average deviation of about 3 percent (or less) and a maximum deviation of about 10 percent (or less). The curve-fit formulas provide an effective and economic means for making preliminary design studies for situations involving high-speed earth entry.

Suttles, J. T.↗
NASA NTRS1975

Consolidation of fatigue and fatigue-crack-propagation data for design use

Analytical methods developed for consolidation of fatigue and fatigue-crack-propagation data for use in design of metallic aerospace structural components are evaluated. A comprehensive file of data on 2024 and 7075 aluminums, Ti-6Al-4V alloy, and 300M steel was established by obtaining information from both published literature and reports furnished by aerospace companies. Analyses are restricted to information obtained from constant-amplitude load or strain cycling of specimens in air at room temperature. Both fatigue and fatigue-crack-propagation data are analyzed on a statistical basis using a least-squares regression approach. For fatigue, an equivalent strain parameter is used to account for mean stress or stress ratio effects and is treated as the independent variable; cyclic fatigue life is considered to be the dependent variable. An effective stress-intensity factor is used to account for the effect of load ratio on fatigue-crack-propagation and treated as the independent variable. In this latter case, crack-growth rate is considered to be the dependent variable. A two term power function is used to relate equivalent strain to fatigue life, and an arc-hyperbolic-tangent function is used to relate effective stress intensity to crack-growth rate.

Rice, R. C.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal estimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed.

Rodriguez, G.↗
NASA NTRS1983

A function space approach to state and model error estimation for elliptic systems

An approach is advanced for the concurrent estimation of the state and of the model errors of a system described by elliptic equations. The estimates are obtained by a deterministic least-squares approach that seeks to minimize a quadratic functional of the model errors, or equivalently, to find the vector of smallest norm subject to linear constraints in a suitably defined function space. The minimum norm solution can be obtained by solving either a Fredholm integral equation of the second kind for the case with continuously distributed data or a related matrix equation for the problem with discretely located measurements. Solution of either one of these equations is obtained in a batch-processing mode in which all of the data is processed simultaneously or, in certain restricted geometries, in a spatially scanning mode in which the data is processed recursively. After the methods for computation of the optimal esimates are developed, an analysis of the second-order statistics of the estimates and of the corresponding estimation error is conducted. Based on this analysis, explicit expressions for the mean-square estimation error associated with both the state and model error estimates are then developed. While this paper focuses on theoretical developments, applications arising in the area of large structure static shape determination are contained in a closely related paper (Rodriguez and Scheid, 1982).

Rodriguez, G.↗
NASA NTRS1974

Surface fitting three-dimensional bodies

The geometry of general three-dimensional bodies is generated from coordinates of points in several cross sections. Since these points may not be smooth, they are divided into segments and general conic sections are curve fit in a least-squares sense to each segment of a cross section. The conic sections are then blended in the longitudinal direction by fitting parametric cubic-spline curves through coordinate points which define the conic sections in the cross-sectional planes. Both the cross-sectional and longitudinal curves may be modified by specifying particular segments as straight lines and slopes at selected points. Slopes may be continuous or discontinuous and finite or infinite. After a satisfactory surface fit has been obtained, cards may be punched with the data necessary to form a geometry subroutine package for use in other computer programs. At any position on the body, coordinates, slopes and second partial derivatives are calculated. The method is applied to a blunted 70 deg delta wing, and it was found to generate the geometry very well.

Dejarnette, F. R.↗
NASA NTRS1971

Development of techniques for measuring pilot workload

An objective method of assessing information workload based on physiological measurements was developed. Information workload, or reserve capacity, was measured using a visual discrimination secondary task and subjective rating of task difficulty. The primary task was two axis (pitch and roll) tracking, and the independent variables in this study were aircraft pitch dynamics and wind gust disturbances. The study was structured to provide: (1) a sensitive, nonloading measure of reserve capacity, and (2) an unencumbering reliable measurement of the psychophysiological state. From these, a measured workload index (MWI) and physiological workload index (PWI) were extracted. An important measure of the success of this study was the degree to which the MWI and PWI agreed across the 243 randomly-presented, four-minute trials (9 subjects X 9 tasks X 3 replications). The electrophysiological data collected included vectorcardiogaram, respiration, electromyogram, skin impedance, and electroencephalogram. Special computer programs were created for the analysis of each physiological variable. The digital data base then consisted of 82 physiological features for each of the 243 trials. A prediction of workload based on physiological observations was formulated as a simultaneous least-squares prediction problem. A best subset of 10 features was chosen to predict the three measures of reserve capacity. The cannonical correlation coefficient was .754 with a chi squared value of 91.3 which allows rejection of the null hypothesis with p of .995.

Spyker, D. A.↗