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NASA NTRS1985

Determination of Mineral Type, Abundance, and Grain Size with Remotely Obtained Reflectance Spectra

To derive the abundances of minerals in a surface composed of more then one mineral in an intimate mixture, the bidirectional reflectance spectrum must be measured at several angles of incidence and emission. From these measurements, the scattering parameters may be derived, and the effective single scattering albedo can be computed from the models. If the scattering parameters are not derived, and the particles scatter anisotropically, then the computed effective single scattering albedo has been computed, the solution to the model is a nonlinear least squares problem with three unknowns per particle type. The solution is feasible when the complex indices of refraction and the densities of each mineral component are known. The unknowns are: the grain size of each particle type, the mass fraction of each particle type, and a grain internal scattering coefficient that describes grain imperfections.

Clark, R. N.↗

NASA NTRS1992

Development of an efficient analysis for high Reynolds number inviscid/viscid interactions in cascades

An efficient steady analysis for predicting strong inviscid/viscid interaction phenomena such as viscous-layer separation, shock/boundary-layer interaction, and trailing-edge/near-wake interaction in turbomachinery blade passages is described. It uses an inviscid/viscid interaction approach, wherein the flow in the outer inviscid region is assumed to be potential, and that in the inner or viscous-layer region is governed by Prandtl's equations. The inviscid solution is determined using an implicit, least-squares, finite-difference approximation. The viscous-layer solution is obtained using an inverse, finite-difference, space-marching method which is applied along the blade surfaces and the wake streamline. A semiinverse global iteration procedure permits the prediction of boundary-layer separation and other strong-interaction phenomena. Results are presented for two cascades where a range of inlet flow conditions was considered for one of them including conditions leading to large-scale flow separation.

Barnett, M.↗

NASA NTRS1993

An analysis for high Reynolds number inviscid/viscid interactions in cascades

An efficient steady analysis for predicting strong inviscid/viscid interaction phenomena such as viscous-layer separation, shock/boundary-layer interaction, and trailing-edge/near-wake interaction in turbomachinery blade passages is needed as part of a comprehensive analytical blade design prediction system. Such an analysis is described. It uses an inviscid/viscid interaction approach, in which the flow in the outer inviscid region is assumed to be potential, and that in the inner or viscous-layer region is governed by Prandtl's equations. The inviscid solution is determined using an implicit, least-squares, finite-difference approximation, the viscous-layer solution using an inverse, finite-difference, space-marching method which is applied along the blade surfaces and wake streamlines. The inviscid and viscid solutions are coupled using a semi-inverse global iteration procedure, which permits the prediction of boundary-layer separation and other strong-interaction phenomena. Results are presented for three cascades, with a range of inlet flow conditions considered for one of them, including conditions leading to large-scale flow separations. Comparisons with Navier-Stokes solutions and experimental data are also given.

Barnett, Mark↗

NASA NTRS1972

Discrete square root smoothing.

The basic techniques applied in the square root least squares and square root filtering solutions are applied to the smoothing problem. Both conventional and square root solutions are obtained by computing the filtered solutions, then modifying the results to include the effect of all measurements. A comparison of computation requirements indicates that the square root information smoother (SRIS) is more efficient than conventional solutions in a large class of fixed interval smoothing problems.

Kaminski, P. G.↗

NASA NTRS1979

Astrometric observations of the satellites of the outer planets. I - The Galilean satellites in 1977

Astrometric observations of the Galilean satellites of Jupiter performed around the opposition of 1977 with a 67-cm photovisual refractor are reported. The measurements are reduced to absolute and intersatellite positions by two different methods: a linear least-squares plate-constant solution to the AGK3 reference-star frame and the trail-scale method of Pascu (1977). Results of a plate-constant reduction are also presented for two observations of JV (Amalthea). Comparison of the data with Liske's (1978) theoretical predictions for the intersatellite positions indicates no systematic bias in the data; a probable scatter of about 15 arcsec in both right ascension and declination is estimated.

Ianna, P. A.↗

NASA NTRS1981

Report of MAGSAT project by Survey of India

Magsat data fo quiet days for the period 4.11.79, 17.11.79 to 24.11.79, which pertains to 44 passes, was screened to exclude the the period of exclude the period of maximum diurnal variation. The results of this screening process is a data set that has the effect, (or at least the major part of the effect), of the external field removed from it. Such a data set is the basis for further analysis which is in hand. The data set is now being reduced to horizontal plane. The mathematical techniques uses a finite harmonic series representation of the three dimensional data, combined with least squares approach for the solution of coefficients. A computer programme was developed to reduce satellite data to a common elevation.

Sawhney, J. L.↗

NASA NTRS1986

A recursive solution for a fading memory filter derived from Kalman filter theory

A simple recursive solution for a class of fading memory tracking filters is presented. A fading memory filter provides estimates of filter states based on past measurements, similar to a traditional Kalman filter. Unlike a Kalman filter, an exponentially decaying weight is applied to older measurements, discounting their effect on present state estimates. It is shown that Kalman filters and fading memory filters are closely related solutions to a general least squares estimator problem. Closed form filter transfer functions are derived for a time invariant, steady state, fading memory filter. These can be applied in loop filter implementation of the Deep Space Network (DSN) Advanced Receiver carrier phase locked loop (PLL).

Statman, J. I.↗

NASA NTRS1989

Regional quasigeoid determination in Northern Germany and comparison with GPS

For the northern part of the Federal Republic of Germany, new quasigeoid solutions were computed by least squares collocation and fast Fourier transformation (FFT) techniques using point and mean gravity data, a digital terrain model, and a global geopotential model. As severe accuracy limitations for precise regional quasigeoid determination come from global model uncertainties, different geopotential models were investigated by combining them with gravimetric data and comparing the quasigeoid heights with GPS and leveling. Optimum results were obtained by a global model tailored to gravity data in Europe. Collocation and FFT results based on this model agree well. The comparison with GPS and leveling yields rms discrepancies of + or - 2 cm over approximately 400 km range.

Denker, Heiner↗

NASA NTRS1992

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

Wathen, A. J.↗

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↗

NASA NTRSDate not supplied

Using Distributed Fiber-optic Strain Sensing to Estimate Generalized Modal Coordinates from Flight-test Data

A method for estimating the generalized modal coordinates of an aircraft during flight has been developed. The Fiber-optic Sensing System (FOSS) offers an efficient and cost-effective method of measuring the strain at thousands of points along the wings. The estimation of modal coordinates was implemented as a two-step process. First, a maximum likelihood method is used to estimate the statistical properties of the sensors and generalized modal coordinates. Second, the strain mode shapes from the finite element model are used along with the statistical properties from the first step to estimate the generalized modal coordinates over time. Using simulated data from the X-56A Multi-Utility Technology Testbed (MUTT), different methods of modal coordinate estimation were compared to demonstrate the benefits and weaknesses of each. These were compared against the exact solution and an ordinary least squares (a more traditional method) estimate. Modal coordinate estimation methods were then applied to flight-test data from the X-56A aircraft to show that the method continues to work well with actual test data. The new estimation method provides insights unavailable from more classical approaches.

Jeffrey Ouellette↗

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 NTRS1992

Essentially nonoscillatory postprocessing filtering methods

High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

Lafon, F.↗

NASA NTRS1991

Least-squares/parabolized Navier-Stokes procedure for optimizing hypersonic wind tunnel nozzles

A new procedure is demonstrated for optimizing hypersonic wind-tunnel-nozzle contours. The procedure couples a CFD computer code to an optimization algorithm, and is applied to both conical and contoured hypersonic nozzles for the purpose of determining an optimal set of parameters to describe the surface geometry. A design-objective function is specified based on the deviation from the desired test-section flow-field conditions. The objective function is minimized by optimizing the parameters used to describe the nozzle contour based on the solution to a nonlinear least-squares problem. The effect of the changes in the nozzle wall parameters are evaluated by computing the nozzle flow using the parabolized Navier-Stokes equations. The advantage of the new procedure is that it directly takes into account the displacement effect of the boundary layer on the wall contour. The new procedure provides a method for optimizing hypersonic nozzles of high Mach numbers which have been designed by classical procedures, but are shown to produce poor flow quality due to the large boundary layers present in the test section. The procedure is demonstrated by finding the optimum design parameters for a Mach 10 conical nozzle and a Mach 6 and a Mach 15 contoured nozzle.

Korte, John J.↗

NASA NTRS1996

A Full Multi-Grid Method for the Solution of the Cell Vertex Finite Volume Cauchy-Riemann Equations

The system of inhomogeneous Cauchy-Riemann equations defined on a square domain and subject to Dirichlet boundary conditions is considered. This problem is discretised by using the cell vertex finite volume method on quadrilateral meshes. The resulting algebraic problem is overdetermined and the solution is defined in a least squares sense. By this approach a consistent algebraic problem is obtained which differs from the original one by O(h(exp 2)) perturbations of the right-hand side. A suitable cell-based convergent smoothing iteration is presented which is naturally linked to the least squares formulation. Hence, a standard multi-grid algorithm is reported which combines the given smoother and cell-based transfer operators. Some remarkable reduction properties of these operators are shown. A full multi-grid method is discussed which solves the discrete problem to the level of truncation error by employing one multi-grid cycle at each current level of discretisation. Experiments and applications of the full multi-grid scheme are presented.

Borzi, A.↗

NASA NTRS1985

Development and application of a unified balancing approach with multiple constraints

The development of a general analytic approach to constrained balancing that is consistent with past influence coefficient methods is described. The approach uses Lagrange multipliers to impose orbit and/or weight constraints; these constraints are combined with the least squares minimization process to provide a set of coupled equations that result in a single solution form for determining correction weights. Proper selection of constraints results in the capability to: (1) balance higher speeds without disturbing previously balanced modes, thru the use of modal trial weight sets; (2) balance off-critical speeds; and (3) balance decoupled modes by use of a single balance plane. If no constraints are imposed, this solution form reduces to the general weighted least squares influence coefficient method. A test facility used to examine the use of the general constrained balancing procedure and application of modal trial weight ratios is also described.

Zorzi, E. S.↗

NASA NTRS1997

Gaseous abundances and methane supersaturation in Titan's troposphere

Various properties of Titan's troposphere are inferred from an analysis of Voyager I infrared spectrometer (IRIS) data between 200 and 600/ cm. Two homogeneous spectral averages acquired at widely separated emission angles are chosen for the analysis. Both data sets are associated with northern low latitudes very close to that of the radio science ingress occultation point. Solutions require simultaneous nonlinear least-squares fits to the two IRIS data sets, coupled with iteration of the radio occultation refractivity data. Values and associated 1-sigma uncertainties of several parameters are inferred from our analysis. These include mole fractions for molecular hydrogen (approx. 0.0011), argon (small), and methane near the surface ( approx. 0.057). Solutions are also obtained for the hydrogen parafraction (close to equilibrium, with considerable uncertainty), air temperature near the surface ( approx. 93 K), surface surface temperature discontinuity (approx. 1 K), and maximum degree of methane supersaturation in the upper troposphere (approx. 1.5). Actual values for the above-mentioned parameters depend on the amount of ethane cloud near the tropopause. There is no evidence for methane clouds in the upper troposphere, nor is their presence compatible with large degrees of supersaturation. A wave number dependence for the stratospheric haze opacity is inferred similar to that found for a polymeric residue created in laboratory discharge experiments. This haze appears to be uniformly distributed with latitude between altitudes of 40 and 160 km, provided those nighttime data at southern high latitudes that are discounted.

Samuelson, Robert E.↗

NASA NTRS1973

Free geometric adjustment of the SECOR Equatorial Network (Solution SECOR-27)

The basic purpose of this experiment is to compute reduced normal equations from the observational data of the SECOR Equatorial Network obtained from DMA/Topographic Center, D/Geodesy, Geosciences Div. Washington, D.C. These reduced normal equations are to be combined with reduced normal equations of other satellite networks of the National Geodetic Satellite Program to provide station coordinates from a single least square adjustment. An individual SECOR solution was also obtained and is presented in this report, using direction constraints computed from BC-4 optical data from stations collocated with SECOR stations. Due to the critical configuration present in the range observations, weighted height constraints were also applied in order to break the near coplanarity of the observing stations.

Mueller, I. I.↗