Search NASA⌕ Search

SEARCH · Search NASA

Results for “Legendre coefficients”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

Coefficients of associated Legendre functions

The exact coefficients for the explicit forms of the associated Legendre functions Pm for integer values of m,n=0,1,2,...25 are presented in tabular form together with two cross-referenced listings of the zeroes for these functions rounded to five decimal places. The unfactored coefficients and the interger coefficients are presented in adjacent columns for each function. The greatest common factor and divisor have been removed and listed separately for the integer coefficients.

Peasley, Q. D.↗
NASA NTRS2019

Principal Component Analysis of Up-The-Ramp Sampled Infrared Array Data

We describe the results of principal component analysis (PCA) of up-the-ramp sampled infrared (IR) array data from the Hubble Space Telescope wide field camera 3 (WFC3 IR), James Webb Space Telescope NIRSpec, and prototype Wide Field Infrared Survey Telescope’s wide field instrument detectors. These systems use, respectively, Teledyne H1R, H2RG, and H4RG-10 near-IR detector arrays with a variety of IR array controllers. The PCA shows that the Legendre polynomials approximate the principal components of these systems (i.e., they roughly diagonalize the covariance matrix). In contrast to the monomial basis that is widely used for polynomial fitting and linearization today, the Legendre polynomials are an orthonormal basis. They provide a quantifiable, compact, and (nearly) linearly uncorrelated representation of the information content of the data. By fitting a few Legendre polynomials, nearly all of the meaningful information in representative WFC3 astronomical datacubes can be condensed from 15 up-the-ramp samples down to 6 compressible Legendre coefficients per pixel. The higher order coefficients contain time domain information that is lost when one projects up-the-ramp sampled datacubes onto two-dimensional images by fitting a straight line, even if the data are linearized before fitting the line. Going forward, we believe that this time domain information is potentially important for disentangling the various nonlinearities that can affect IR array observations, i.e., inherent pixel nonlinearity, persistence, burn in, brighter-fatter effect, (potentially) nonlinear interpixel capacitance, and perhaps others.

Rauscher, Bernard J.↗
NASA NTRS1985

Phase function, backscatter, extinction, and absorption for standard radiation atmosphere and El Chichon aerosol models at visible and near-infrared wavelengths

Tabular values of phase function, Legendre polynominal coefficients, 180 deg backscatter, and extinction cross section are given for eight wavelengths in the atmospheric windows between 0.4 and 2.2 microns. Also included are single scattering albedo, asymmetry factor, and refractive indices. These values are based on Mie theory calculations for the standard rediation atmospheres (continental, maritime, urban, unperturbed stratospheric, volcanic, upper atmospheric, soot, oceanic, dust, and water-soluble) assest measured volcanic aerosols at several time intervals following the El Chichon eruption. Comparisons of extinction to 180 deg backscatter for different aerosol models are presented and related to lidar data.

Whitlock, C. H.↗
NASA NTRS1988

Helioseismology from the South Pole: Comparison of 1987 and 1981 Results

Full disk images with 10 arc sec pixels and filtered to a 7 A pass band centered on the Ca II K line were obtained from the geographic South Pole in 1981 and 1987. In 1981, 50hr of essentially uninterrupted data were obtained. In 1987, three such runs were obtained over a period of 325 hours for a duty cycle of 47 percent. The 1987 observations are characterized by a much lower level of solar activity than 1981, a much improved CCD camera, considerably better image stability and a varying amount of instrumental scatter. The 1987 data have a substantially better signal-to-noise ratio than the 1981 data so that oscillations with degrees from 0 to 150 and frequencies from 2 to 7 mHz are well observed. The observations were reduced to spectra in l, m, and v. A comparison of p-mode frequencies measured in 1981 and 1987, and coefficients of Legendre polynomial expansions of frequency shifts caused by solar rotation are presented. The time behavior of systematic frequency shifts which depend upon m but which do not arise from rotation is described.

Jefferies, S. M.↗
NASA NTRS1995

Considerations for modeling thin cirrus effects via brightness temperature differences

Brightness temperature difference (BTD) values are calculated for selected Geostationary Operational Environmental Satellite (GOES-6) channels (3.9, 12.7 micrometer) and Advanced Very High Resolution Radiometer channels (3.7, 12.0 micrometer). Daytime and nighttime discrimination of particle size information is possible given the infrared cloud extinction optical depth and the BTD value. BTD values are presented and compared for cirrus clouds composed of equivalent ice spheres (volume, surface area) versus randomly oriented hexagonal ice crystals. The effect of the hexagonal ice crystals is to increase the magnitude of the BTD values calculated relative to equivalent ice sphere (volume, surface area) BTDs. Equivalent spheres (volume or surface area) do not do a very good job of modeling hexagonal ice crystal effects on BTDs; however, the use of composite spheres improves the simulation and offers interesting prospects. Careful consideration of the number of Legendre polynomial coefficients used to fit the scattering phase functions is crucial to realistic modeling of cirrus BTDs. Surface and view-angle effects are incorporated to provide more realistic simulation.

Schmidt, E. O.↗
NASA NTRS1983

The Gravitational Potential Due to Uniform Disks and Rings

The gravitational potential of bodies possessing axial symmetry can be expressed as a power series in distance, with the Legendre polynomials as coefficients. Such series, however, converge so slowly in the neighborhood of thin, uniform disks and rings that too many series terms must be summed in order to obtain an accurate field measure. A gravitational potential expression is presently obtained in closed form, in terms of complete elliptic integrals.

Lass, H.↗
NASA NTRS1988

Direct solution for thermal stresses in a nose cap under an arbitrary axisymmetric temperature distribution

The design of a nose cap for a hypersonic vehicle is an iterative process requiring a rapid, easy to use and accurate stress analysis. The objective of this paper is to develop such a stress analysis technique from a direct solution of the thermal stress equations for a spherical shell. The nose cap structure is treated as a thin spherical shell with an axisymmetric temperature distribution. The governing differential equations are solved by expressing the stress solution to the thermoelastic equations in terms of a series of derivatives of the Legendre polynomials. The process of finding the coefficients for the series solution in terms of the temperature distribution is generalized by expressing the temperature along the shell and through the thickness as a polynomial in the spherical angle coordinate. Under this generalization the orthogonality property of the Legendre polynomials leads to a sequence of integrals involving powers of the spherical shell coordinate times the derivative of the Legendre polynomials. The coefficients of the temperature polynomial appear outside of these integrals. Thus, the integrals are evaluated only once and their values tabulated for use with any arbitrary polynomial temperature distribution.

Davis, Randall C.↗
NASA NTRS1990

Numerical experiments in geomagnetic modeling

Numerical tests were made, using least squares fitting of a spherical harmonic model, to a selection of Magsat data to determine the practical limits of this technique with modern computers. The resulting (M102189) model, whose coefficients were adjusted up to n = 50, was compared with M07AV6, a previous model which used least squares (on vector data) for coefficients up to n = 29, and Gauss-Legendre quadrature (on Z residuals) to adjust the coefficients up to n = 63. For the new least squares adjustment to n = 50 a condition number of 115 was obtained for the solution matrix, with a resulting precision of 11 significant figures. The M102189 model shows a lower and more Gaussian residual distribution than did M07AV6, though the Gaussian envelope fits to the residual distributions, even for the scalar field, gives "standard deviations' never lower than 6 nT, a factor of three higher than the estimated Magsat observational errors. Ionospheric currents are noted to have a significant effect on the coefficients of the internal potential functions.

Cain, Joseph C.↗
NASA NTRS1983

Legendre-Tau approximations for functional differential equations

The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

Ito, K.↗
NASA NTRS1986

Legendre-tau approximations for functional differential equations

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Ito, K.↗
NASA NTRS1993

Solar activity and oscillation frequency splittings

Solar p-mode frequency splittings, parameterized by the coefficients through order N = 12 of a Legendre polynomial expansion of the mode frequencies as a function of m/L, were obtained from an analysis of helioseismology data taken at Big Bear Solar Observatory during the 4 years 1986 and 1988-1990 (approximately solar minimum to maximum). Inversion of the even-index splitting coefficients confirms that there is a significant contribution to the frequency splittings originating near the solar poles. The strength of the polar contribution is anti correlated with the overall level or solar activity in the active latitudes, suggesting a relation to polar faculae. From an analysis of the odd-index splitting coefficients we infer an uppor limit to changes in the solar equatorial near-surface rotatinal velocity of less than 1.9 m/s (3 sigma limit) between solar minimum and maximum.

Woodard, M. F.↗
NASA NTRS1995

Extensional flow convecting a reactant undergoing a first order homogeneous reaction and diffusional mass transfer from a sphere at low to intermediate Peclet and Damkohler numbers

Forced convective diffusion-reaction is considered for viscous axisymmetric extensional convecting velocity in the neighborhood of a sphere. For Peclet numbers in the range 0.1 less than or equal to Pe less than or equal to 500 and for Damkohler numbers increasing with increasing Pe but in the overall range 0.02 less than or equal to Da less than or equal to 10, average and local Sherwood numbers have been computed. By introducing the eigenfunction expansion c(r, Theta) = Sum of c(n)(r)P(n)(cos Theta) into the forced convective diffusion equation for the concentration of a chemical species undergoing a first order homogeneous reaction and by using properties of the Legendre functions Pn(cos Theta), the variable coefficient PDE can be reduced to a system of N + 1 second order ODEs for the radial functions c(sub n)(r), n = 0, 1, 2,..., N. The adaptive grid algorithm of Pereyra and Lentini can be used to solve the corresponding 2(N + 1) first order differential equations as a two-point boundary value problem on 1 less than or equal to r less than or equal to r(sub infinity). Convergence of the expansion for a specific value of N can thus be established and provides 'spectral' behavior as well as the full concentration field c(r, Theta).

Shah, N. Y.↗
NASA NTRS1977

A note of expansions of functions of velocity in the two-body problem

Fourier expansions based on both the true anomaly and the mean anomaly are obtained for the functions of velocity in the two-body problem; the series of coefficients is written from classical formulae involving associated Legendre polynomials, Gegenbauer polynomials, or Bessel functions. The Fourier expansions are compared with the expansions in powers of eccentricity developed by Broucke (1974) through use of computerized Poisson series manipulation.

Kinoshita, H.↗
NASA NTRS2013

RUSHMAPS: Real-Time Uploadable Spherical Harmonic Moment Analysis for Particle Spectrometers

RUSHMAPS is a new onboard data reduction scheme that gives real-time access to key science parameters (e.g. moments) of a class of heliophysics science and/or solar system exploration investigation that includes plasma particle spectrometers (PPS), but requires moments reporting (density, bulk-velocity, temperature, pressure, etc.) of higher-level quality, and tolerates a lowpass (variable quality) spectral representation of the corresponding particle velocity distributions, such that telemetry use is minimized. The proposed methodology trades access to the full-resolution velocity distribution data, saving on telemetry, for real-time access to both the moments and an adjustable-quality (increasing quality increases volume) spectral representation of distribution functions. Traditional onboard data storage and downlink bandwidth constraints severely limit PPS system functionality and drive cost, which, as a consequence, drives a limited data collection and lower angular energy and time resolution. This prototypical system exploit, using high-performance processing technology at GSFC (Goddard Space Flight Center), uses a SpaceCube and/or Maestro-type platform for processing. These processing platforms are currently being used on the International Space Station as a technology demonstration, and work is currently ongoing in a new onboard computation system for the Earth Science missions, but they have never been implemented in heliospheric science or solar system exploration missions. Preliminary analysis confirms that the targeted processor platforms possess the processing resources required for realtime application of these algorithms to the spectrometer data. SpaceCube platforms demonstrate that the target architecture possesses the sort of compact, low-mass/power, radiation-tolerant characteristics needed for flight. These high-performing hybrid systems embed unprecedented amounts of onboard processing power in the CPU (central processing unit), FPGAs (field programmable gate arrays), and DSP (digital signal processing) elements. The fundamental computational algorithm de constructs 3D velocity distributions in terms of spherical harmonic spectral coefficients (which are analogous to a Fourier sine-cosine decomposition), but uses instead spherical harmonics Legendre polynomial orthogonal functions as a basis for the expansion, portraying each 2D angular distribution at every energy or, geometrically, spherical speed-shell swept by the particle spectrometer. Optionally, these spherical harmonic spectral coefficients may be telemetered to the ground. These will provide a smoothed description of the velocity distribution function whose quality will depend on the number of coefficients determined. Successfully implemented on the GSFC-developed processor, the capability to integrate the proposed methodology with both heritage and anticipated future plasma particle spectrometer designs is demonstrated (with sufficiently detailed design analysis to advance TRL) to show specific science relevancy with future HSD (Heliophysics Science Division) solar-interplanetary, planetary missions, sounding rockets and/or CubeSat missions.

Figueroa-Vinas, Adolfo↗
NASA NTRS1980

Orthogonality of spherical harmonic coefficients

Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.

Mcleod, M. G.↗
NASA NTRS1983

Computational methods for estimation of parameters in hyperbolic systems

Approximation techniques for estimating spatially varying coefficients and unknown boundary parameters in second order hyperbolic systems are discussed. Methods for state approximation (cubic splines, tau-Legendre) and approximation of function space parameters (interpolatory splines) are outlined and numerical findings for use of the resulting schemes in model "one dimensional seismic inversion' problems are summarized.

Banks, H. T.↗