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NASA NTRS2021

Error Propagation in Microwave Soil Moisture and Vegetation Optical Depth Retrievals

Satellite soil moisture and vegetation optical depth (VOD; related to the total vegetation water mass per unit area) are increasingly being used to study water relations in the soil-plant continuum across the globe. However, soil moisture and VOD are typically jointly estimated, where errors in the optimization approach can cause compensation between both variables and confound such studies. It is thus critical to quantify how satellite microwave measurement errors propagate into soil moisture and VOD. Such a study is especially important for VOD given limited investigations of whether VOD reflects in-situ plant physiology. Furthermore, despite new approaches that constrain (or regularize) VOD dynamics to reduce soil moisture errors, there is limited study of whether regularization reduces VOD errors without obscuring true vegetation temporal dynamics. Here, we find that, across the globe, VOD is less robust to measurement error (more difficult for optimization methods to find the true solution) than soil moisture in their joint estimation. However, a moderate degree of regularization (via time-constrained VOD) reduces errors in VOD to a greater degree than soil moisture and reduces spurious soil moisture-VOD coupling. Furthermore, despite constraining VOD time dynamics, regularized VOD variations on sub-weekly scales are both closer to simulated true VOD time series and have global VOD post-rainfall responses with reduced error signatures compared to VOD retrievals without regularization. Ultimately, we recommend moderately regularized VOD for use in large scale studies of soil-plant water relations because it suppresses noise and spurious soil moisture-VOD coupling without removing the physical signal.

Vegetation optical depth↗

NASA NTRS1978

A singularity free analytical solution of artificial satellite motion with drag

An analytical satellite theory based on the regular, canonical Poincare-Similar (PS phi) elements is described along with an accurate density model which can be implemented into the drag theory. A computationally efficient manner in which to expand the equations of motion into a fourier series is discussed.

Mueller, A.↗

NASA NTRS1972

Optimal filter design subject to output delobe constraints

The design of filters for detection and estimation in radar and communications systems is considered, with inequality constraints on the maximum output sidelobe levels. A constrained optimization problem in Hilbert space is formulated, incorporating the sidelobe constraints via a partial ordering of continuous functions. Generalized versions (in Hilbert space) of the Kuhn-Tucker and Duality Theorems allow the reduction of this problem to an unconstrained one in the dual space of regular Borel measures. A convergent algorithm is presented for computational solution of the dual problem.

Fortmann, T. E.↗

NASA NTRS1974

Vibration analysis of a completely free elliptical plate

The Ritz method is applied to the variation analysis of a completely free elliptical plate as an efficient alternative to a procedure using a truncated series of regular and modified Mathieu functions. It is shown that a solution as accurate as desired can be obtained with the Ritz method in solving thin plate vibration problems with difficult geometries.

Beres, D. P.↗

NASA NTRS1984

Estimating location parameters in a mixture

The problem of estimating the parameters in a finite mixture is considered. The approach is based on an integral equation formulation of the form h sub t (x) = integral (limits b and a) f(x-y) g sut t (y)dy where h sub t is a smoothed version of h and g sub t is a prior function that tends to be concentrated on the translation values. A solution for g sub t that uses the method of regularization and one based on a posterior operator approach is considered. Numerical simulations are presented to bring out some of the estimation and numerical problems of these approaches.

Heydorn, R. P.↗

NASA NTRS1999

Representation and Integration of Scientific Information

The objective of this Joint Research Interchange with NASA-Ames was to investigate how the Tsimmis technology could be used to represent and integrate scientific information. The funding allowed us to work with researchers within NAS at the NASA Ames Research Center, to understand their information needs, and to work with them on integration strategies. Most organizations have a need to access and integrate information from multiple, disparate information sources that may include both structured as well as semi-structured information. At Stanford we have been working on an information integration project called Tsimmis, supported by DARPA. The main goal of the Tsimmis project is to allow a decision maker to find information of interest from such sources, fuse it, and process it (e.g., summarize it, visualize it, discover trends). Another important goal is the easy incorporation of new sources, as well the ability of deal with sources whose structure or services evolve. During the Interchange we had research meetings approximately every month or two. The participants from NASA included Michael Cox and Peter Vanderbilt. The Stanford PI, and various students and Stanford staff members also participated. NASA researchers also participated in some of our regular Tsimmis meetings. As planned, our meetings discussed problems and solutions to various information integration problems.

Source record↗

NASA NTRS1991

Time-reversibility and particle sedimentation

This paper studies an ODE model, called the Stokeslet model, and describes sedimentation of small clusters of particles in a highly viscous fluid. This model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n-sided polygon. When n = 3, Hocking and Caflisch et al. (1988) proved the existence of periodic motion (in the frame moving with the center of gravity in the cluster) in which the particles form an isosceles triangle. Here, the study of periodic and quasi-periodic solutions of the Stokeslet model is continued, with emphasis on the spatial and time-reversal symmetry of the model. For three particles, the existence of a second family of periodic solutions and a family of quasi-periodic solutions is proved. It is also indicated how the methods generalize to the case of n particles.

Golubitsky, Martin↗

NASA NTRS1990

Fully elliptic incompressible flow calculations on regular grid by a new pressure substitution method

A new method is presented for the solution of incompressible flow in generalized coordinates. This method is based on the substitution of the pressure weighted form of the momentum equations into the continuity equation. The algorithm is rigorously derived and a Fourier analysis is used to assess its suitability to act as an error smoother. Linear stability analysis results indicate that the performance of the new pressure substitution method (PSM) and the pressure correction method (PCM) is about the same at low Reynolds numbers, with no significant pressure gradient. At high Reynolds numbers the PSM shows much faster convergence. Likewise prediction of various flows indicate that the PSM has better accuracy for high Reynolds number flows with significant pressure gradients. Since most practical aerodynamic flows have significant pressure gradients, the PSM seems to be attractive for such flows. Solutions for both laminar and turbulent flow are compared with the experimental data. A two-equation low Reynolds number turbulence model is used to resolve the turbulent flowfield.

Hobson, G. V.↗

NASA NTRS1980

SPAR VI Technical Report for Experiment 76-22: Directional Solidification of Magnetic Composites

Samples of eutectic Bi/MnBi were directionally solidified during a low-g interval aboard the SPAR 6 flight and in a l-g environment under identical furnace velocity and thermal conditions. The Bi/MnBi eutectic is characterized by a regular rod eutectic whose morphology may be sensitive to thermo-solutal convection and by its components, MnBi, which is ferromagnetic. Morphological analyses on samples show statistically smaller interrod spacings and rod diameters with respect to samples grown under identical solidification furnace conditions in l-g. An adjustment between the interrod spacing, growth velocity, and total undercooling at the solidification interface is proposed. Morphological analyses on samples grown in l-g indicate little difference between results for different growth orientations with respect to the gravity vector. The magnetic properties are significantly affected, however, by the presence of a nonequilibrium magnetic phase and the nonequilibrium phase transforms to the equilibrium ferromagnetic phase during isothermal heat treatment.

Pirich, R. G.↗

NASA NTRS1993

A hybrid Pade-Galerkin technique for differential equations

A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.

Geer, James F.↗

NASA NTRS1993

Construction and parameterization of all static and dynamic H2-optimal state feedback solutions, optimal fixed modes, and fixed decoupling zeros

This paper considers an H2 optimization problem via state feedback. The class of problems dealt with here are general singular type which have a left invertible transfer matrix function from the control input to the controlled output. This class subsumes the regular H2 optimization problems. The paper constructs and parameterizes all the static and dynamic H2 optimal state feedback solutions. Moreover, all the eigenvalues of an optimal closed-loop system are characterized. All optimal closed-loop systems share a set of eigenvalues which are termed here as the optimal fixed modes. Every H2 optimal controller must assign among the closed-loop eigenvalues the set of optimal fixed modes. This set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations of pole-zero cancellations present in any H2 optimal design. It is shown that both the sets of optimal fixed modes and optimal fixed decoupling zeros do not vary depending upon whether the static or the dynamic controllers are used.

Chen, Ben M.↗

NASA NTRS2011

Sparse Solutions for Single Class SVMs: A Bi-Criterion Approach

In this paper we propose an innovative learning algorithm - a variation of One-class nu Support Vector Machines (SVMs) learning algorithm to produce sparser solutions with much reduced computational complexities. The proposed technique returns an approximate solution, nearly as good as the solution set obtained by the classical approach, by minimizing the original risk function along with a regularization term. We introduce a bi-criterion optimization that helps guide the search towards the optimal set in much reduced time. The outcome of the proposed learning technique was compared with the benchmark one-class Support Vector machines algorithm which more often leads to solutions with redundant support vectors. Through out the analysis, the problem size for both optimization routines was kept consistent. We have tested the proposed algorithm on a variety of data sources under different conditions to demonstrate the effectiveness. In all cases the proposed algorithm closely preserves the accuracy of standard one-class nu SVMs while reducing both training time and test time by several factors.

Das, Santanu↗

NASA NTRS1987

A hybrid perturbation Galerkin technique with applications to slender body theory

A two step hybrid perturbation-Galerkin method to solve a variety of applied mathematics problems which involve a small parameter is presented. The method consists of: (1) the use of a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter; (2) construction of an approximate solution in the form of a sum of the perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions); and (3) the use of the classical Bubnov-Galerkin method to determine these amplitudes. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the degree of applicability of the hybrid method to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1989

A hybrid perturbation Galerkin technique with applications to slender body theory

A two-step hybrid perturbation-Galerkin method to solve a variety of applied mathematics problems which involve a small parameter is presented. The method consists of: (1) the use of a regular or singular perturbation method to determine the asymptotic expansion of the solution in terms of the small parameter; (2) construction of an approximate solution in the form of a sum of the perturbation coefficient functions multiplied by (unknown) amplitudes (gauge functions); and (3) the use of the classical Bubnov-Galerkin method to determine these amplitudes. This hybrid method has the potential of overcoming some of the drawbacks of the perturbation method and the Bubnov-Galerkin method when they are applied by themselves, while combining some of the good features of both. The proposed method is applied to some singular perturbation problems in slender body theory. The results obtained from the hybrid method are compared with approximate solutions obtained by other methods, and the degree of applicability of the hybrid method to broader problem areas is discussed.

Geer, James F.↗

NASA NTRS1993

The computation of the post-stall behavior of a circulation controlled airfoil

The physics of the circulation controlled airfoil is complex and poorly understood, particularly with regards to jet stall, which is the eventual breakdown of lift augmentation by the jet at some sufficiently high blowing rate. The present paper describes the numerical simulation of stalled and unstalled flows over a two-dimensional circulation controlled airfoil using a fully implicit Navier-Stokes code, and the comparison with experimental results. Mach numbers of 0.3 and 0.5 and jet total to freestream pressure ratios of 1.4 and 1.8 are investigated. The Baldwin-Lomax and k-epsilon turbulence models are used, each modified to include the effect of strong streamline curvature. The numerical solutions of the post-stall circulation controlled airfoil show a highly regular unsteady periodic flowfield. This is the result of an alternation between adverse pressure gradient and shock induced separation of the boundary layer on the airfoil trailing edge.

Linton, Samuel W.↗

NASA NTRS2010

Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes

Cell-centered and node-centered approaches have been compared for unstructured finite-volume discretization of inviscid fluxes. The grids range from regular grids to irregular grids, including mixed-element grids and grids with random perturbations of nodes. Accuracy, complexity, and convergence rates of defect-correction iterations are studied for eight nominally second-order accurate schemes: two node-centered schemes with weighted and unweighted least-squares (LSQ) methods for gradient reconstruction and six cell-centered schemes two node-averaging with and without clipping and four schemes that employ different stencils for LSQ gradient reconstruction. The cell-centered nearest-neighbor (CC-NN) scheme has the lowest complexity; a version of the scheme that involves smart augmentation of the LSQ stencil (CC-SA) has only marginal complexity increase. All other schemes have larger complexity; complexity of node-centered (NC) schemes are somewhat lower than complexity of cell-centered node-averaging (CC-NA) and full-augmentation (CC-FA) schemes. On highly anisotropic grids typical of those encountered in grid adaptation, discretization errors of five of the six cell-centered schemes converge with second order on all tested grids; the CC-NA scheme with clipping degrades solution accuracy to first order. The NC schemes converge with second order on regular and/or triangular grids and with first order on perturbed quadrilaterals and mixed-element grids. All schemes may produce large relative errors in gradient reconstruction on grids with perturbed nodes. Defect-correction iterations for schemes employing weighted least-square gradient reconstruction diverge on perturbed stretched grids. Overall, the CC-NN and CC-SA schemes offer the best options of the lowest complexity and secondorder discretization errors. On anisotropic grids over a curved body typical of turbulent flow simulations, the discretization errors converge with second order and are small for the CC-NN, CC-SA, and CC-FA schemes on all grids and for NC schemes on triangular grids; the discretization errors of the CC-NA scheme without clipping do not converge on irregular grids. Accurate gradient reconstruction can be achieved by introducing a local approximate mapping; without approximate mapping, only the NC scheme with weighted LSQ method provides accurate gradients. Defect correction iterations for the CC-NA scheme without clipping diverge; for the NC scheme with weighted LSQ method, the iterations either diverge or converge very slowly. The best option in curved geometries is the CC-SA scheme that offers low complexity, second-order discretization errors, and fast convergence.

Diskin, Boris↗

NASA NTRS1990

Fresh look at floating shock fitting

A fast implicit upwind procedure for the two-dimensional Euler equations is described that allows accurate computations of shocked flows on nonadapted meshes. Away from shocks, the second-order accurate upwinding is based on the split-coefficient-matrix (SCM) method. In the presence of shocks, the difference stencils are modified using a floating shock fitting technique. Rapid convergence to steady-state solutions is attained with a diagonalized approximate factorization (AF) algorithm. Results are presented for Riemann's problem, for a regular shock reflection at an inviscid wall, for supersonic flow past a cylinder, and for a transonic airfoil. All computed shocks are ideally sharp and in excellent agreement with other numerical results or 'exact' solutions. Most importantly, this has been accomplished on unusually crude meshes without any attempt to align grid lines with shock fronts or to cluster grid lines around shocks.

Hartwich, PETER-M.↗

NASA NTRS2024

High-Order/Low-Dissipation Chain-Rule Flux Solution Reconstruction Schemes in FUN3D

In this paper, we report progress in the development of economically high-order flux-solution-reconstruction (FSR) schemes, which are second-order accurate on general unstructured grids but achieve high-order accuracy when a grid is regular, i.e., has the same stencil (with the same spacing) throughout the domain. Two variants of the FSR schemes are discussed: chain-rule-flux-solution reconstruction (CFSR) and quadratic-form-flux-solution reconstruction (QFSR), where the former is based on the chain rule and the latter on the flux reconstruction expressed as a function of solution variables. These schemes are tested for flows with shock waves with a limiter incorporated in the flux and solution reconstructions. Improved results, compared with second-order methods are demonstrated for inviscid and viscous flows with smooth grids.

high-order↗