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All sourcesNASA NTRSDOE OSTI
NASA NTRS1985

Shape Preserving Spline Interpolation

A rational spline solution to the problem of shape preserving interpolation is discussed. The rational spline is represented in terms of first derivative values at the knots and provides an alternative to the spline-under-tension. The idea of making the shape control parameters dependent on the first derivative unknowns is then explored. The monotonic or convex shape of the interpolation data can then be preserved automatically through the solution of the resulting non-linear consistency equations of the spline.

Gregory, J. A.↗
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 NTRS2019

Updated MISR Over-Water Research Aerosol Retrieval Algorithm - Part 2: A Multi-Angle Aerosol Retrieval Algorithm for Shallow, Turbid, Oligotrophic, and Eutrophic Waters

Coastal waters serve as transport pathways to the ocean for all agricultural and other runoff from terrestrial sources, and many are the sites for upwelling of nutrient rich, deep water; they are also some of the most biologically productive on Earth. Estimating the impact coastal waters have on the global carbon budget requires relating satellite-based remote-sensing retrievals of biological productivity (e.g., chlorophyll alpha concentration) to in situ measurements taken in near-surface waters. The Multi-angle Imaging SpectroRadiometer (MISR) can uniquely constrain the "atmospheric correction" needed to derive ocean color from remote-sensing imagers. Here, we retrieve aerosol amount and type from MISR over all types of water. The primary limitation is an upper bound on aerosol optical depth (AOD), as the algorithm must be able to distinguish the surface. This updated MISR research aerosol retrieval algorithm (RA) also assumes that light reflection by the underlying ocean surface is Lambertian. The RA computes the ocean surface reflectance (R (sub rs)) analytically for a given AOD, aerosol optical model, and wind speed. We provide retrieval examples over shallow, turbid, and eutrophic waters and introduce a productivity and turbidity index (PTI), calculated from retrieved spectral R (sub rs), that distinguished water types (similar to the the normalized difference vegetation index, NDVI, over land). We also validate the new algorithm by comparing spectral AOD and Angstrom exponent (ANG) results with 2419 collocated AErosol RObotic NETwork (AERONET) observations. For AERONET 558 nanometer-interpolated AOD less than 1.0, the root-mean square error (RMSE) is 0.04 and linear correlation coefficient is 0.95. For the 502 cloud-free MISR and AERONET collocations with an AERONET AOD greater than 0.20, the ANG RMSE is 0.25 and r is 0.89. Although MISR RA AOD retrieval quality does not appear to be substantially impacted by the presence of turbid water, the MISR-RA-retrieved Angstrom exponent seems to suffer from increased uncertainty under such conditions. MISR supplements current ocean color sources in regions where sunglint precludes retrievals from single-view-angle instruments. MISR atmospheric correction should also be more robust than that derived from single-view instruments such as the Moderate Resolution Imaging Spectroradiometer (MODIS). This is especially true in regions of shallow, turbid, and eutrophic waters, locations where biological productivity can be high, and single-view-angle retrieval algorithms struggle to separate atmospheric from oceanic features.

Satellite Remote Sensing↗
NASA NTRS1994

Response of a two-dimensional dual-Doppler radar wind synthesis

The wind field resulting from a two-dimensional dual-Doppler synthesis algorithms is spectrally modified from the true wind field. The effects of spatial filtering on wind fields from the processes of interpolation, the averaging of pulses, and the effect of the finite radar pulse dimension were assessed. The effect resulting from the use of different interpolation techniques was also evaluated. Of those techniques tested, the best are the Cressman distance-weighted averaging and linear distance-weighted averaging, with the closest neighbor and uniform weighting having more undesirable characteristics. The optimum influence radius is defined as the influence radius at which the ratio of the rms difference between the Fourier and least-squares responses (a measure of the aliasing) and the variance of the filtered wind field is minimized. This seeks to minimize the effect of energy aliased into scales other than the input wavelength. For the Cressman interpolation technique, the optimum influence radius is between 1.85 and 2.25 times the maximum data spacing. The range of acceptable influence radii includes consideration of the filtering by the radar of the data as it is collected, as well as the resolution of the final dataset. The optimum influence radius is dependent upon the largest data separation in the analysis domain. The absolute optimum influence radius is not significantly affected by inclusion of the radar-beam filtering effects.

Given, Terence↗
NASA NTRS2016

Simple Test Functions in Meshless Local Petrov-Galerkin Methods

Two meshless local Petrov-Galerkin (MLPG) methods based on two different trial functions but that use a simple linear test function were developed for beam and column problems. These methods used generalized moving least squares (GMLS) and radial basis (RB) interpolation functions as trial functions. These two methods were tested on various patch test problems. Both methods passed the patch tests successfully. Then the methods were applied to various beam vibration problems and problems involving Euler and Beck's columns. Both methods yielded accurate solutions for all problems studied. The simple linear test function offers considerable savings in computing efforts as the domain integrals involved in the weak form are avoided. The two methods based on this simple linear test function method produced accurate results for frequencies and buckling loads. Of the two methods studied, the method with radial basis trial functions is very attractive as the method is simple, accurate, and robust.

Raju, Ivatury S.↗
NASA NTRS1974

A simplified package for calculating with splines

This package is designed to solve some of the elementary problems of spline interpretation and least squares fit. The subroutines fall into three basic categories. The first category involves computations with a given spline and/or knot sequence; the second category involves routines which calculate the coefficients of splines which perform a certain task (such as interpolation or least squares fit); and the last category is a banded equation solver for specific linear equations.

Smith, P. W.↗
NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗
NASA NTRS2006

Rapid Aeroelastic Analysis of Blade Flutter in Turbomachines

The LINFLUX-AE computer code predicts flutter and forced responses of blades and vanes in turbomachines under subsonic, transonic, and supersonic flow conditions. The code solves the Euler equations of unsteady flow in a blade passage under the assumption that the blades vibrate harmonically at small amplitudes. The steady-state nonlinear Euler equations are solved by a separate program, then equations for unsteady flow components are obtained through linearization around the steady-state solution. A structural-dynamics analysis (see figure) is performed to determine the frequencies and mode shapes of blade vibrations, a preprocessor interpolates mode shapes from the structural-dynamics mesh onto the LINFLUX computational-fluid-dynamics mesh, and an interface code is used to convert the steady-state flow solution to a form required by LINFLUX. Then LINFLUX solves the linearized equations in the frequency domain to calculate the unsteady aerodynamic pressure distribution for a given vibration mode, frequency, and interblade phase angle. A post-processor uses the unsteady pressures to calculate generalized aerodynamic forces, response amplitudes, and eigenvalues (which determine the flutter frequency and damping). In comparison with the TURBO-AE aeroelastic-analysis code, which solves the equations in the time domain, LINFLUX-AE is 6 to 7 times faster.

Trudell, J. J.↗
NASA NTRS1966

Simple scale interpolator facilitates reading of graphs

Set of cards with scale divisions and a scale finder permits accurate reading of the coordinates of points on linear or logarithmic graphs plotted on rectangular grids. The set contains 34 different scales for linear plotting and 28 single cycle scales for log plots.

Fazio, A.↗
NASA NTRS1989

Bilinear modeling and nonlinear estimation

New methods are illustrated for online nonlinear estimation applied to the lateral deflection of an elastic beam on board measurements of angular rates and angular accelerations. The development of the filter equations, together with practical issues of their numerical solution as developed from global linearization by nonlinear output injection are contrasted with the usual method of the extended Kalman filter (EKF). It is shown how nonlinear estimation due to gyroscopic coupling can be implemented as an adaptive covariance filter using off-the-shelf Kalman filter algorithms. The effect of the global linearization by nonlinear output injection is to introduce a change of coordinates in which only the process noise covariance is to be updated in online implementation. This is in contrast to the computational approach which arises in EKF methods arising by local linearization with respect to the current conditional mean. Processing refinements for nonlinear estimation based on optimal, nonlinear interpolation between observations are also highlighted. In these methods the extrapolation of the process dynamics between measurement updates is obtained by replacing a transition matrix with an operator spline that is optimized off-line from responses to selected test inputs.

Dwyer, Thomas A. W., III↗
NASA NTRS1989

Thin-plate spline quadrature of geodetic integrals

Thin-plate spline functions (known for their flexibility and fidelity in representing experimental data) are especially well-suited for the numerical integration of geodetic integrals in the area where the integration is most sensitive to the data, i.e., in the immediate vicinity of the evaluation point. Spline quadrature rules are derived for the contribution of a circular innermost zone to Stoke's formula, to the formulae of Vening Meinesz, and to the recursively evaluated operator L(n) in the analytical continuation solution of Molodensky's problem. These rules are exact for interpolating thin-plate splines. In cases where the integration data are distributed irregularly, a system of linear equations needs to be solved for the quadrature coefficients. Formulae are given for the terms appearing in these equations. In case the data are regularly distributed, the coefficients may be determined once-and-for-all. Examples are given of some fixed-point rules. With such rules successive evaluation, within a circular disk, of the terms in Molodensky's series becomes relatively easy. The spline quadrature technique presented complements other techniques such as ring integration for intermediate integration zones.

Vangysen, Herman↗
NASA NTRS2009

The GMAO ADVAR System: Preliminary Results

The Goddard Earth Observing System (GEOS-S) Data Assimilation System is now configurable to run strong constraint 4DVAR. The system combines the 4DVAR-capable Grid-point Statistical Interpolation analysis with the GEOS-S general circulation model and an early version of the tangent linear and adjoint models of the finite-volume hydrodynamics with simplified physics. The GMAO 4DVAR uses a Lanczos-based conjugate gradient algorithm in a nested resolution inner-loop setting. A range of tests and experiments will have been carried out by the time of the Symposium. Results on balance-related issues, resolution configuration, and choice of assimilation time window will be shown. A comparison with the Global Modeling and Assimilation Office (GMAO) 3DVAR and a discussion of future directions of development will also be presented.

Todling, Ricardo↗
NASA NTRS1987

Image reconstruction from multiple 1-D scans using filtered localized projection

The spatial resolution that can be attained using scanning linear arrays (consisting of discrete IR solid-state detectors) for image acquisition is considered, and a filtered local projection (FLP) method is described which efficiently combines all available scan information into one rectangular grid without the need for explicit interpolation. Mathematically, the FLP algorithm consists of a localized summation followed by an inverse-filter operation, and it has application to nonlinear restoring techniques. The present method is applied, using a linear array, to simulated data for staggered parallel scans and to multiple scan directions. Noise effects and limitations of the technique are also considered.

Frieden, B. Roy↗
NASA NTRS1977

Comment of fitting ab initio intermolecular potentials for scattering calculations

The angular dependence of the interaction between two rigid linear rotors is considered, and one of the two expansions in current use is shown to be inappropriate. Interpolation of tabulated radial interactions is also discussed, and numerical tests are presented which demonstrate the accuracy of Lagrange and spline algorithms as a function of the spacing between points.

Green, S.↗
NASA NTRS1981

An analysis of LANDSAT MSS scene-to-scene registration accuracy

Measurements were made for 12 registrations done by ERL for 8 registrations done by SRS. The results indicate that the ERL method is significantly more accurate in five of the eight comparison. The difference between the two methods are not significant in the other three cases. There are two possible reasons for the differences. First, the ERL model is a piecewise linear model and the EDITOR model is a cubic polynomial model. Second, the ERL program resamples using bilinear interpolation while the EDITOR software uses a nearest neighbor resampling. This study did not indicate how much of the difference is attributable to each factor. The average of all merged scene error values for ERL was 31.6 meters and the average for the eight common areas was 32.6 meters. The average of the eight merged scene error values for SRS was 40.1 meters.

Seyfarth, B. R.↗
NASA NTRS2001

Computational Aeroacoustic Analysis System Development

Many industrial and commercial products operate in a dynamic flow environment and the aerodynamically generated noise has become a very important factor in the design of these products. In light of the importance in characterizing this dynamic environment, Rocketdyne has initiated a multiyear effort to develop an advanced general-purpose Computational Aeroacoustic Analysis System (CAAS) to address these issues. This system will provide a high fidelity predictive capability for aeroacoustic design and analysis. The numerical platform is able to provide high temporal and spatial accuracy that is required for aeroacoustic calculations through the development of a high order spectral element numerical algorithm. The analysis system is integrated with well-established CAE tools, such as a graphical user interface (GUI) through PATRAN, to provide cost-effective access to all of the necessary tools. These include preprocessing (geometry import, grid generation and boundary condition specification), code set up (problem specification, user parameter definition, etc.), and postprocessing. The purpose of the present paper is to assess the feasibility of such a system and to demonstrate the efficiency and accuracy of the numerical algorithm through numerical examples. Computations of vortex shedding noise were carried out in the context of a two-dimensional low Mach number turbulent flow past a square cylinder. The computational aeroacoustic approach that is used in CAAS relies on coupling a base flow solver to the acoustic solver throughout a computational cycle. The unsteady fluid motion, which is responsible for both the generation and propagation of acoustic waves, is calculated using a high order flow solver. The results of the flow field are then passed to the acoustic solver through an interpolator to map the field values into the acoustic grid. The acoustic field, which is governed by the linearized Euler equations, is then calculated using the flow results computed from the flow solver.

Hadid, A.↗
NASA NTRS1988

Paradigmatic error calculations for adaptive finite element approximations of convection dominated flows

New error estimation techniques are described for two-dimensional convection-dominated flows. Examples of applications to linear convection problems are given. The methods are based on discontinuous finite-element techniques. Results show that conventional interpolation methods may lead to gross underestimates of actual error and be inadequate for use in adaptive methods.

Oden, J. T.↗
NASA NTRS1996

Atmospheric Pre-Corrected Differential Absorption Techniques to Retrieve Columnar Water Vapor: Theory and Simulations

Two different approaches exist to retrieve columnar water vapor from imaging spectrometer data: (1) Differential absorption techniques based on: (a) Narrow-Wide (N/W) ratio between overlapping spectrally wide and narrow channels; (b) Continuum Interpolated Band Ratio (CIBR) between a measurement channel and the weighted sum of two reference channels. (2) Non-linear fitting techniques which are based on spectral radiative transfer calculations. The advantage of the first approach is computational speed and of the second, improved retrieval accuracy. Our goal was to improve the accuracy of the first technique using physics based on radiative transfer. Using a modified version of the Duntley equation, we derived an "Atmospheric Pre-corrected Differential Absorption" (APDA) technique and described an iterative scheme to retrieve water vapor on a pixel-by-pixel basis. Next we compared both, the CIBR and the APDA using the Duntley equation for MODTRAN3 computed irradiances, transmissions and path radiance (using the DISORT option). This simulation showed that the CIBR is very sensitive to reflectance effects and that the APDA performs much better. An extensive data set was created with the radiative transfer code 6S over 379 different ground reflectance spectra. The calculated relative water vapor error was reduced significantly for the APDA. The APDA technique had about 8% (vs. over 35% for the CIBR) of the 379 spectra with a relative water vapor error of greater than +5%. The APDA has been applied to 1991 and 1995 AVIRIS scenes which visually demonstrate the improvement over the CIBR technique.

Borel, Christoph C.↗