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NASA NTRS1993

Quons, an interpolation between Bose and Fermi oscillators

After a brief mention of Bose and Fermi oscillators and of particles which obey other types of statistics, including intermediate statistics, parastatistics, paronic statistics, anyon statistics, and infinite statistics, I discuss the statistics of 'quons' (pronounced to rhyme with muons), particles whose annihilation and creation operators obey the q-deformed commutation relation (the quon algebra or q-mutator) which interpolates between fermions and bosons. I emphasize that the operator for interaction with an external source must be an effective Bose operator in all cases. To accomplish this for parabose, parafermi and quon operators, I introduce parabose, parafermi, and quon Grassmann numbers, respectively. I also discuss interactions of non-relativistic quons, quantization of quon fields with antiparticles, calculation of vacuum matrix elements of relativistic quon fields, demonstration of the TCP theorem, cluster decomposition, and Wick's theorem for relativistic quon fields, and the failure of local commutativity of observables for relativistic quon fields. I conclude with the bound on the parameter q for electrons due to the Ramberg-Snow experiment.

Greenberg, O. W.↗

NASA NTRS1996

Interpolation Errors in Spectrum Analyzers

To obtain the proper measurement amplitude with a spectrum analyzer, the correct frequency-dependent transducer factor must be added to the voltage measured by the transducer. This report examines how entering transducer factors into a spectrum analyzer can cause significant errors in field amplitude due to the misunderstanding of the analyzer's interpolation methods. It also discusses how to reduce these errors to obtain a more accurate field amplitude reading.

Martin, J. L.↗

NASA NTRS1995

A Global Interpolation Function (GIF) boundary element code for viscous flows

Using global interpolation functions (GIF's), boundary element solutions are obtained for two- and three-dimensional viscous flows. The solution is obtained in the form of a boundary integral plus a series of global basis functions. The unknown coefficients of the GIF's are determined to ensure the satisfaction of the governing equations at selected collocation points. The values of the coefficients involved in the boundary integral equations are determined by enforcing the boundary conditions. Both primitive variable and vorticity-velocity formulations are examined.

Reddy, D. R.↗

NASA NTRS1996

Geostatistical approaches to interpolation and classification of remote-sensing data from ice surfaces

Geostatistical methods for interpolation and extrapolation techniques are used in glaciological data analysis. The results of a program involving the mapping of the Antarctica from satellite radar altimeter data are discussed. A combination of high and low resolution techniques was applied in the analysis of the Bering Glacier (Alaska). The global positioning system (GPS) located video data collected from small aircraft and the ERS-1 synthetic aperture radar (SAR) images were used. From the perspective of SAR data analysis, the Bering Glacier surge was the opportunity to characterize the surface of fast flowing ice and the rapid changes in the surface roughness.

Herzfeld, Ute Christina↗

NASA NTRS1994

Kriging in the Shadows: Geostatistical Interpolation for Remote Sensing

It is often useful to estimate obscured or missing remotely sensed data. Traditional interpolation methods, such as nearest-neighbor or bilinear resampling, do not take full advantage of the spatial information in the image. An alternative method, a geostatistical technique known as indicator kriging, is described and demonstrated using a Landsat Thematic Mapper image in southern Chiapas, Mexico. The image was first classified into pasture and nonpasture land cover. For each pixel that was obscured by cloud or cloud shadow, the probability that it was pasture was assigned by the algorithm. An exponential omnidirectional variogram model was used to characterize the spatial continuity of the image for use in the kriging algorithm. Assuming a cutoff probability level of 50%, the error was shown to be 17% with no obvious spatial bias but with some tendency to categorize nonpasture as pasture (overestimation). While this is a promising result, the method's practical application in other missing data problems for remotely sensed images will depend on the amount and spatial pattern of the unobscured pixels and missing pixels and the success of the spatial continuity model used.

Rossi, Richard E.↗

NASA NTRS1997

Adaptation of a Fast Optimal Interpolation Algorithm to the Mapping of Oceangraphic Data

A fast, recently developed, multiscale optimal interpolation algorithm has been adapted to the mapping of hydrographic and other oceanographic data. This algorithm produces solution and error estimates which are consistent with those obtained from exact least squares methods, but at a small fraction of the computational cost. Problems whose solution would be completely impractical using exact least squares, that is, problems with tens or hundreds of thousands of measurements and estimation grid points, can easily be solved on a small workstation using the multiscale algorithm. In contrast to methods previously proposed for solving large least squares problems, our approach provides estimation error statistics while permitting long-range correlations, using all measurements, and permitting arbitrary measurement locations. The multiscale algorithm itself, published elsewhere, is not the focus of this paper. However, the algorithm requires statistical models having a very particular multiscale structure; it is the development of a class of multiscale statistical models, appropriate for oceanographic mapping problems, with which we concern ourselves in this paper. The approach is illustrated by mapping temperature in the northeastern Pacific. The number of hydrographic stations is kept deliberately small to show that multiscale and exact least squares results are comparable. A portion of the data were not used in the analysis; these data serve to test the multiscale estimates. A major advantage of the present approach is the ability to repeat the estimation procedure a large number of times for sensitivity studies, parameter estimation, and model testing. We have made available by anonymous Ftp a set of MATLAB-callable routines which implement the multiscale algorithm and the statistical models developed in this paper.

Menemenlis, Dimitris↗

NASA NTRS1997

Quadrature, Interpolation and Observability

Methods of interpolation and quadrature have been used for over 300 years. Improvements in the techniques have been made by many, most notably by Gauss, whose technique applied to polynomials is referred to as Gaussian Quadrature. Stieltjes extended Gauss's method to certain non-polynomial functions as early as 1884. Conditions that guarantee the existence of quadrature formulas for certain collections of functions were studied by Tchebycheff, and his work was extended by others. Today, a class of functions which satisfies these conditions is called a Tchebycheff System. This thesis contains the definition of a Tchebycheff System, along with the theorems, proofs, and definitions necessary to guarantee the existence of quadrature formulas for such systems. Solutions of discretely observable linear control systems are of particular interest, and observability with respect to a given output function is defined. The output function is written as a linear combination of a collection of orthonormal functions. Orthonormal functions are defined, and their properties are discussed. The technique for evaluating the coefficients in the output function involves evaluating the definite integral of functions which can be shown to form a Tchebycheff system. Therefore, quadrature formulas for these integrals exist, and in many cases are known. The technique given is useful in cases where the method of direct calculation is unstable. The condition number of a matrix is defined and shown to be an indication of the the degree to which perturbations in data affect the accuracy of the solution. In special cases, the number of data points required for direct calculation is the same as the number required by the method presented in this thesis. But the method is shown to require more data points in other cases. A lower bound for the number of data points required is given.

Hodges, Lucille McDaniel↗

NASA NTRS2000

Wavelet-Based Interpolation and Representation of Non-Uniformly Sampled Spacecraft Mission Data

A well-documented problem in the analysis of data collected by spacecraft instruments is the need for an accurate, efficient representation of the data set. The data may suffer from several problems, including additive noise, data dropouts, an irregularly-spaced sampling grid, and time-delayed sampling. These data irregularities render most traditional signal processing techniques unusable, and thus the data must be interpolated onto an even grid before scientific analysis techniques can be applied. In addition, the extremely large volume of data collected by scientific instrumentation presents many challenging problems in the area of compression, visualization, and analysis. Therefore, a representation of the data is needed which provides a structure which is conducive to these applications. Wavelet representations of data have already been shown to possess excellent characteristics for compression, data analysis, and imaging. The main goal of this project is to develop a new adaptive filtering algorithm for image restoration and compression. The algorithm should have low computational complexity and a fast convergence rate. This will make the algorithm suitable for real-time applications. The algorithm should be able to remove additive noise and reconstruct lost data samples from images.

Bose, Tamal↗

NASA NTRS2004

An Interpolation Method for Obtaining Thermodynamic Properties Near Saturated Liquid and Saturated Vapor Lines

The two most common approaches used to formulate thermodynamic properties of pure substances are fundamental (or characteristic) equations of state (Helmholtz and Gibbs functions) and a piecemeal approach that is described in Adebiyi and Russell (1992). This paper neither presents a different method to formulate thermodynamic properties of pure substances nor validates the aforementioned approaches. Rather its purpose is to present a method to generate property tables from existing property packages and a method to facilitate the accurate interpretation of fluid thermodynamic property data from those tables. There are two parts to this paper. The first part of the paper shows how efficient and usable property tables were generated, with the minimum number of data points, using an aerospace industry standard property package. The second part describes an innovative interpolation technique that has been developed to properly obtain thermodynamic properties near the saturated liquid and saturated vapor lines.

Nguyen, Huy H.↗

NASA NTRS2003

An Interpolation Method for Obtaining Thermodynamic Properties Near Saturated Liquid and Saturated Vapor Lines

The availability and proper utilization of fluid properties is of fundamental importance in the process of mathematical modeling of propulsion systems. Real fluid properties provide the bridge between the realm of pure analytiis and empirical reality. The two most common approaches used to formulate thermodynamic properties of pure substances are fundamental (or characteristic) equations of state (Helmholtz and Gibbs functions) and a piecemeal approach that is described, for example, in Adebiyi and Russell (1992). This paper neither presents a different method to formulate thermodynamic properties of pure substances nor validates the aforementioned approaches. Rather its purpose is to present a method to be used to facilitate the accurate interpretation of fluid thermodynamic property data generated by existing property packages. There are two parts to this paper. The first part of the paper shows how efficient and usable property tables were generated, with the minimum number of data points, using an aerospace industry standard property package (based on fundamental equations of state approach). The second part describes an innovative interpolation technique that has been developed to properly obtain thermodynamic properties near the saturated liquid and saturated vapor lines.

Nguyen, Huy H.↗

NASA NTRS2013

Elastic-Plastic J-Integral Solutions or Surface Cracks in Tension Using an Interpolation Methodology

No closed form solutions exist for the elastic-plastic J-integral for surface cracks due to the nonlinear, three-dimensional nature of the problem. Traditionally, each surface crack must be analyzed with a unique and time-consuming nonlinear finite element analysis. To overcome this shortcoming, the authors have developed and analyzed an array of 600 3D nonlinear finite element models for surface cracks in flat plates under tension loading. The solution space covers a wide range of crack shapes and depths (shape: 0.2 less than or equal to a/c less than or equal to 1, depth: 0.2 less than or equal to a/B less than or equal to 0.8) and material flow properties (elastic modulus-to-yield ratio: 100 less than or equal to E/ys less than or equal to 1,000, and hardening: 3 less than or equal to n less than or equal to 20). The authors have developed a methodology for interpolating between the goemetric and material property variables that allows the user to reliably evaluate the full elastic-plastic J-integral and force versus crack mouth opening displacement solution; thus, a solution can be obtained very rapidly by users without elastic-plastic fracture mechanics modeling experience. Complete solutions for the 600 models and 25 additional benchmark models are provided in tabular format.

Allen, P. A.↗

NASA NTRS2013

Elastic-Plastic J-Integral Solutions or Surface Cracks in Tension Using an Interpolation Methodology. Appendix C -- Finite Element Models Solution Database File, Appendix D -- Benchmark Finite Element Models Solution Database File

Allen, Phillip A.↗

NASA NTRS2012

LU60645GT and MA132843GT Catalogues of Lunar and Martian Impact Craters Developed Using a Crater Shape-based Interpolation Crater Detection Algorithm for Topography Data

For Mars, 57,633 craters from the manually assembled catalogues and 72,668 additional craters identified using several crater detection algorithms (CDAs) have been merged into the MA130301GT catalogue. By contrast, for the Moon the most complete previous catalogue contains only 14,923 craters. Two recent missions provided higher-quality digital elevation maps (DEMs): SELENE (in 1/16° resolution) and Lunar Reconnaissance Orbiter (we used up to 1/512°). This was the main motivation for work on the new Crater Shape-based interpolation module, which improves previous CDA as follows: (1) it decreases the number of false-detections for the required number of true detections; (2) it improves detection capabilities for very small craters; and (3) it provides more accurate automated measurements of craters' properties. The results are: (1) LU60645GT, which is currently the most complete (up to ∼D>=8 km) catalogue of Lunar craters; and (2) MA132843GT catalogue of Martian craters complete up to ∼D>=2 km, which is the extension of the previous MA130301GT catalogue. As previously achieved for Mars, LU60645GT provides all properties that were provided by the previous Lunar catalogues, plus: (1) correlation between morphological descriptors from used catalogues; (2) correlation between manually assigned attributes and automated measurements; (3) average errors and their standard deviations for manually and automatically assigned attributes such as position coordinates, diameter, depth/diameter ratio, etc; and (4) a review of positional accuracy of used datasets. Additionally, surface dating could potentially be improved with the exhaustiveness of this new catalogue. The accompanying results are: (1) the possibility of comparing a large number of Lunar and Martian craters, of e.g. depth/diameter ratio and 2D profiles; (2) utilisation of a method for re-projection of datasets and catalogues, which is very useful for craters that are very close to poles; and (3) the extension of the previous framework for evaluation of CDAs with datasets and ground-truth catalogue for the Moon.

Craters↗

NASA NTRS2014

Polyhedral Interpolation for Optimal Reaction Control System Jet Selection

An efficient algorithm is described for interpolating optimal values for spacecraft Reaction Control System jet firing duty cycles. The algorithm uses the symmetrical geometry of the optimal solution to reduce the number of calculations and data storage requirements to a level that enables implementation on the small real time flight control systems used in spacecraft. The process minimizes acceleration direction errors, maximizes control authority, and minimizes fuel consumption.

flight optimization↗

NASA NTRS2015

A Module for Assimilating Hyperspectral Infrared Retrieved Profiles into the Gridpoint Statistical Interpolation System for Unique Forecasting Applications

Hyperspectral infrared sounder radiance data are assimilated into operational modeling systems however the process is computationally expensive and only approximately 1% of available data are assimilated due to data thinning as well as the fact that radiances are restricted to cloud-free fields of view. In contrast, the number of hyperspectral infrared profiles assimilated is much higher since the retrieved profiles can be assimilated in some partly cloudy scenes due to profile coupling other data, such as microwave or neural networks, as first guesses to the retrieval process. As the operational data assimilation community attempts to assimilate cloud-affected radiances, it is possible that the use of retrieved profiles might offer an alternative methodology that is less complex and more computationally efficient to solve this problem. The NASA Short-term Prediction Research and Transition (SPoRT) Center has assimilated hyperspectral infrared retrieved profiles into Weather Research and Forecasting Model (WRF) simulations using the Gridpoint Statistical Interpolation (GSI) System. Early research at SPoRT demonstrated improved initial conditions when assimilating Atmospheric Infrared Sounder (AIRS) thermodynamic profiles into WRF (using WRF-Var and assigning more appropriate error weighting to the profiles) to improve regional analysis and heavy precipitation forecasts. Successful early work has led to more recent research utilizing WRF and GSI for applications including the assimilation of AIRS profiles to improve WRF forecasts of atmospheric rivers and assimilation of AIRS, Cross-track Infrared and Microwave Sounding Suite (CrIMSS), and Infrared Atmospheric Sounding Interferometer (IASI) profiles to improve model representation of tropopause folds and associated non-convective wind events. Although more hyperspectral infrared retrieved profiles can be assimilated into model forecasts, one disadvantage is the retrieved profiles have traditionally been assigned the same error values as the rawinsonde observations when assimilated with GSI. Typically, satellitederived profile errors are larger and more difficult to quantify than traditional rawinsonde observations (especially in the boundary layer), so it is important to appropriately assign observation errors within GSI to eliminate potential spurious innovations and analysis increments that can sometimes arise when using retrieved profiles. The goal of this study is to describe modifications to the GSI source code to more appropriately assimilate hyperspectral infrared retrieved profiles and outline preliminary results that show the differences between a model simulation that assimilated the profiles as rawinsonde observations and one that assimilated the profiles in a module with the appropriate error values.

Berndt, Emily↗

NASA NTRS2013

Geostationary Enhanced Temporal Interpolation for CERES Flux Products

The Clouds and the Earth's Radiant Energy System (CERES) instruments on board the Terra and Aqua spacecraft continue to provide an unprecedented global climate record of the earth's top-of-atmosphere (TOA) energy budget since March 2000. A critical step in determining accurate daily averaged flux involves estimating the flux between CERES Terra or Aqua overpass times. CERES employs the CERES-only (CO) and the CERES geostationary (CG) temporal interpolation methods. The CO method assumes that the cloud properties at the time of the CERES observation remain constant and that it only accounts for changes in albedo with solar zenith angle and diurnal land heating, by assuming a shape for unresolved changes in the diurnal cycle. The CG method enhances the CERES data by explicitly accounting for changes in cloud and radiation between CERES observation times using 3-hourly imager data from five geostationary (GEO) satellites. To maintain calibration traceability, GEO radiances are calibrated against Moderate Resolution Imaging Spectroradiometer (MODIS) and the derived GEO fluxes are normalized to the CERES measurements. While the regional (1 deg latitude x 1 deg longitude) monthly-mean difference between the CG and CO methods can exceed 25 W m(sub -2) over marine stratus and land convection, these regional biases nearly cancel in the global mean. The regional monthly CG shortwave (SW) and longwave (LW) flux uncertainty is reduced by 20%, whereas the daily uncertainty is reduced by 50% and 20%, respectively, over the CO method, based on comparisons with 15-min Geostationary Earth Radiation Budget (GERB) data.

Doelling, David R.↗

NASA NTRS2019

Fusion of Surface Ceilometer Data and Satellite Cloud Retrievals in 2D Mesh Interpolating Model with Clustering

For accurate cloud ceiling information, a data fusion approach is proposed that utilizes satellite data to extend surface station information to much wider areas. Cloud base height (CBH) retrieved from satellite observations provides for much larger spatial coverage and higher resolution. The direct comparison of GOES-16 CBH with surface station ceiling yields a local bias that has to be corrected for in the initial GOES-16 cloud base information. This sparsely sampled bias correction presents an irregular 2D mesh of control points, which is then interpolated by constructing a continuous smooth field using polyharmonic splines. The influence of remote stations is restricted by grouping the control points into clusters depending on an effective distance. This cluster-based approach allows for constructing separate spline surfaces corresponding to physically different clouds. The obtained continuous bias correction function is then applied to the entire GOES-16 pixel level CBH except for areas far away from surface stations in data sparse regions such as offshore. The described method is currently being tested using daytime-only observations over the central and eastern United States. Overall, this approach has potential to provide more accurate, high spatial resolution cloud ceiling information for the aviation community.

Khlopenkov, Konstantin↗

NASA NTRS2021

A Functional Interpolation Approach to Compute Period Orbits in the Circular Restricted Three-body Problem

In this paper, we develop a method to solve for periodic orbits, i.e., Lyapunov and Halo orbits, using a functional interpolation scheme called the Theory of Func- tional Connections (TFC). Using this technique, a periodic constraint is analyti- cally embedded into the TFC constrained expression. By doing this, the system of differential equations governing the three-body problem is transformed into an unconstrained optimization problem where simple numerical schemes can be used to find a solution, e.g., nonlinear least-squares is used. This allows for a simpler numerical implementation with comparable accuracy and speed to the traditional differential corrector method.

Mortari, Daniele↗