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NASA NTRS2012

3D Drop Size Distribution Extrapolation Algorithm Using a Single Disdrometer

Determining the Z-R relationship (where Z is the radar reflectivity factor and R is rainfall rate) from disdrometer data has been and is a common goal of cloud physicists and radar meteorology researchers. The usefulness of this quantity has traditionally been limited since radar represents a volume measurement, while a disdrometer corresponds to a point measurement. To solve that problem, a 3D-DSD (drop-size distribution) method of determining an equivalent 3D Z-R was developed at the University of Central Florida and tested at the Kennedy Space Center, FL. Unfortunately, that method required a minimum of three disdrometers clustered together within a microscale network (.1-km separation). Since most commercial disdrometers used by the radar meteorology/cloud physics community are high-cost instruments, three disdrometers located within a microscale area is generally not a practical strategy due to the limitations of these kinds of research budgets. A relatively simple modification to the 3D-DSD algorithm provides an estimate of the 3D-DSD and therefore, a 3D Z-R measurement using a single disdrometer. The basis of the horizontal extrapolation is mass conservation of a drop size increment, employing the mass conservation equation. For vertical extrapolation, convolution of a drop size increment using raindrop terminal velocity is used. Together, these two independent extrapolation techniques provide a complete 3DDSD estimate in a volume around and above a single disdrometer. The estimation error is lowest along a vertical plane intersecting the disdrometer position in the direction of wind advection. This work demonstrates that multiple sensors are not required for successful implementation of the 3D interpolation/extrapolation algorithm. This is a great benefit since it is seldom that multiple sensors in the required spatial arrangement are available for this type of analysis. The original software (developed at the University of Central Florida, 1998.- 2000) has also been modified to read standardized disdrometer data format (Joss-Waldvogel format). Other modifications to the software involve accounting for vertical ambient wind motion, as well as evaporation of the raindrop during its flight time.

Lane, John↗
NASA NTRS1973

Specification of mesospheric density, pressure, and temperature by extrapolation

A procedure is presented which employs an extrapolation technique to obtain estimates of density, pressure, and temperature up to 90 km from 52 km data. The resulting errors are investigated. The procedure is combined with a special temperature interpolation method around the stratopause to produce such estimates at eight levels between 36 km and 90 km from North American sectional chart data at 5, 2, and 0.4 mb. Fifth charts were processed to obtain mean values and standard deviations at grid points for midseasonal months from 1964 to 1966. The mean values were compared with Groves' model, and internal consistency tests were performed upon the statistics. Through application of the extrapolation procedure, the atmospheric structure of a stratospheric warming event is studied.

Graves, M. E.↗
NASA NTRS1972

Precision state and filter weighting matrix extrapolation, revision 3

The matrix extrapolation routine presented is a coded algorithm for the numerical solution of modified forms of the basic differential equations which are satisfied by the geocentric state vector of a spacecraft's center of mass and by the filter-weighting matrix. The equations are described in detail and the various input and output variables designated. The capability is thus established to extrapolate any spacecraft's geocentric state vector, either backwards or forwards in time, through a force field consisting of the earth's primary central-force gravitational attraction, and a superimposed perturbing acceleration.

Robertson, W. M.↗
NASA NTRS1976

Comparison of the San Marco 3 Nace neutral composition data with the extrapolated Ogo 6 empirical model

An investigation is conducted concerning the feasibility to extrapolate the data of the Ogo 6 empirical composition model to altitudes which are lower than 450 km. Extrapolated Ogo 6 model densities are, therefore, compared with data obtained in the Neutral Atmospheric Composition Experiment (Nace) carried out during the time from April to November 1971. The results of the investigation support the conclusions of an earlier comparison of Ogo 6 and Nace data conducted by Newton et al. (1973).

Kasprzak, W. T.↗
NASA NTRS1982

Radar prediction of absolute rain fade distributions for earth-satellite paths and general methods for extrapolation of fade statistics to other locations

The first absolute rain fade distribution method described establishes absolute fade statistics at a given site by means of a sampled radar data base. The second method extrapolates absolute fade statistics from one location to another, given simultaneously measured fade and rain rate statistics at the former. Both methods employ similar conditional fade statistic concepts and long term rain rate distributions. Probability deviations in the 2-19% range, with an 11% average, were obtained upon comparison of measured and predicted levels at given attenuations. The extrapolation of fade distributions to other locations at 28 GHz showed very good agreement with measured data at three sites located in the continental temperate region.

Goldhirsh, J.↗
NASA NTRS1983

Extrapolation of optimum design based on sensitivity derivatives

In many applications, it is useful to have information regarding the effect of the perturbations of the design problem constants (parameters) on the optimum solution of the problem. Sobieszczanski-Sobieski et al. (1981) have, therefore, proposed to utilize an optimum sensitivity analysis as a structural design tool to assess the parameter perturbation effects by extrapolations based on optimum sensitivity derivatives. The present investigation is concerned with an example of optimum sensitivity analysis involving a composite panel. The case was selected in connection with the nonlinearity of the constraints and, the tendency for a set of active constraints to change when the parameters are disturbed. Under these adverse conditions, sensitivity derivatives are used in linear extrapolations to extend optimization information over a limited range of variation for a typical parameter.

Barthelemy, J.-F. M.↗
NASA NTRS1987

Extrapolation of velocity for inviscid solid boundary conditions

The present examination of an extrapolation of velocity along an arbitrary direction in the case of inviscid solid boundary conditions demonstrates the error associated with the arbitrary direction and establishes that the most appropriate procedure lies in the extrapolation of the velocity components that are tangential to the body surface. It is noted that, in a typical calculation, mesh lines generally cluster together and are therefore parallel or nearly parallel to the body surface.

Hung, Ching-Mao↗
NASA NTRS1990

Efficient Implementation of Minimal Polynomial and Reduced Rank Extrapolation Methods

The minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) are two effective techniques that have been used in accelerating the convergence of vector sequences, such as those that are obtained from iterative solution of linear and nonlinear systems of equation. Their definitions involve some linear least squares problems, and this causes difficulties in their numerical implementation. Timewise efficient and numerically stable implementations for MPE and RRE are developed. A computer program written in FORTRAN 77 is also appended and applied to some model problems.

Sidi, Avram↗
NASA NTRS1991

J-85 jet engine noise measured in the ONERA S1 wind tunnel and extrapolated to far field

Noise from a J-85 turbojet with a conical, convergent nozzle was measured in simulated flight in the ONERA S1 Wind Tunnel. Data are presented for several flight speeds up to 130 m/sec and for radiation angles of 40 to 160 degrees relative to the upstream direction. The jet was operated with subsonic and sonic exhaust speeds. A moving microphone on a 2 m sideline was used to survey the radiated sound field in the acoustically treated, closed test section. The data were extrapolated to a 122 m sideline by means of a multiple-sideline source-location method, which was used to identify the acoustic source regions, directivity patterns, and near field effects. The source-location method is described along with its advantages and disadvantages. Results indicate that the effects of simulated flight on J-85 noise are significant. At the maximum forward speed of 130 m/sec, the peak overall sound levels in the aft quadrant were attentuated approximately 10 dB relative to sound levels of the engine operated statically. As expected, the simulated flight and static data tended to merge in the forward quadrant as the radiation angle approached 40 degrees. There is evidence that internal engine or shock noise was important in the forward quadrant. The data are compared with published predictions for flight effects on pure jet noise and internal engine noise. A new empirical prediction is presented that relates the variation of internally generated engine noise or broadband shock noise to forward speed. Measured near field noise extrapolated to far field agrees reasonably well with data from similar engines tested statically outdoors, in flyover, in a wind tunnel, and on the Bertin Aerotrain. Anomalies in the results for the forward quadrant and for angles above 140 degrees are discussed. The multiple-sideline method proved to be cumbersome in this application, and it did not resolve all of the uncertainties associated with measurements of jet noise close to the jet. The simulation was complicated by wind-tunnel background noise and the propagation of low frequency sound around the circuit.

Soderman, Paul T.↗
NASA NTRS1992

Upper bounds for convergence rates of vector extrapolation methods on linear systems with initial iterations

The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.

Sidi, Avram↗
NASA NTRS1992

A regularization method for extrapolation of solar potential magnetic fields

The mathematical basis of a Tikhonov regularization method for extrapolating the chromospheric-coronal magnetic field using photospheric vector magnetograms is discussed. The basic techniques show that the Cauchy initial value problem can be formulated for potential magnetic fields. The potential field analysis considers a set of linear, elliptic partial differential equations. It is found that, by introducing an appropriate smoothing of the initial data of the Cauchy potential problem, an approximate Fourier integral solution is found, and an upper bound to the error in the solution is derived. This specific regularization technique, which is a function of magnetograph measurement sensitivities, provides a method to extrapolate the potential magnetic field above an active region into the chromosphere and low corona.

Gary, G. A.↗
NASA NTRS1993

Efficient Minimum-Polynomial And Reduced-Rank Extrapolation

MPERRE computer program accelerates convergence of sequence of vectors by use of minimum-polynomial extrapolation (MPE) and reduced-rank extrapolation (RRE). Effective in accelerating convergences of such sequences of vectors as those obtained from iterative solution of systems of linear and nonlinear equations. In conjunction with various iterative techniques, successfully employed in finite-difference solution of large-scale elliptic boundary-value problems and problems in computational-fluid-dynamics. Only input required is sequence of vectors, convergence of which is accelerated. Program economical and easy to use. Written in FORTRAN 77.

Sidi, Avram↗
NASA NTRS1956

Extrapolation techniques applied to matrix methods in neutron diffusion problems

A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.

Mccready, Robert R↗
NASA NTRS1942

New method of extrapolation of the resistance of a model planing boat to full size

The previously employed method of extrapolating the total resistance to full size with lambda(exp 3) (model scale) and thereby foregoing a separate appraisal of the frictional resistance, was permissible for large models and floats of normal size. But faced with the ever increasing size of aircraft a reexamination of the problem of extrapolation to full size is called for. A method is described by means of which, on the basis of an analysis of tests on planing surfaces, the variation of the wetted surface over the take-off range is analytically obtained. The friction coefficients are read from Prandtl's curve for turbulent boundary layer with laminar approach. With these two values a correction for friction is obtainable.

Sottorf, W↗
NASA NTRS1995

Essentially nonoscillatory (ENO) reconstructions via extrapolation

In this paper, the algorithm for determining the stencil of a one-dimensional Essentially Nonoscillatory (ENO) reconstruction scheme on a uniform grid is reinterpreted as being based on extrapolation. This view leads to another extension of ENO reconstruction schemes to two-dimensional unstructured triangular meshes. The key idea here is to select several cells of the stencil in one step based on extrapolation rather than one cell at a time. Numerical experiments confirm that the new scheme yields sharp nonoscillatory reconstructions and that it is about five times faster than previous schemes.

Suresh, Ambady↗
NASA NTRS1996

Theoretical Basis for Finite Difference Extrapolation of Sonic Boom Signatures

Calculation of sonic boom signatures for aircraft has traditionally followed the methods of Whitham' and Walkden. The wave disturbance generated by the vehicle is obtained by area rule linearized supersonic flow methods, which yield a locally axisymmetric asymptotic solution. This solution is acoustic in nature, i.e., first order in disturbance quantities, and corresponds to ray acoustics. Cumulative nonlinear distortion of the signature is incorporated by using this solution to adjust propagation speed to first order, thus yielding a solution second order in disturbance quantities. The effects of atmospheric gradients are treated by Blokhintzov's method of geometrical acoustics. Both nonlinear signature evolution and ray tracing are applied as if the pressure field very close to the vehicle were actually that given by the source term (the 'F-function') of the asymptotic linearized flow solution. The viewpoint is thus that the flow solution exists at a small radius near the vehicle, and may be treated as an input to an extrapolation procedure consisting of ray tracing and nonlinear aging. The F-function is often regarded as a representation of a near-field pressure signature, and it is common for computational implementations to treat it interchangeably with the pressure signature. There is a 'matching radius' between the source function and the subsequent propagation extrapolation. This viewpoint has been supported by wind tunnel tests of simple models, and very typically yields correct results for actual flight vehicles. The assumption that the F-function and near-field signature are interchangeable is generally not correct. The flowfield of a vehicle which is not axisymmetric contains crossflow components which are very significant at small radii and less so at larger distances. From an acoustical viewpoint, the crossflow is equivalent to source diffraction portions of the wave field. Use of the F-function as a near field signature effectively assumes that the diminution of the crossflow/diffraction component may be applied all at once at the matching radius noted above. This approximation, though not rigorously validated, is responsible for the usual correct far-field results. On the other hand, if an actual near-field signature (either from wind tunnel or CFD data) is used at a starting point rather than one based on th effective source distribution, the predicted far-field signature is generally wrong.

Plotkin, Kenneth J.↗
NASA NTRS1999

Mid-Field Sonic Boom Extrapolation Methodology

In the design cycle of low-boom airplanes, sonic boom prediction must be accurate and efficient. The classical linear method, Whitham's F-function theory, has been widely applied to predict sonic boom signatures. However, linear theory fails to capture the nonlinear effects created by large civil transport. Computational fluid dynamics (CFD) has been used successfully to predict sonic boom signals at the near and mid fields. Nevertheless, it is computationally expansive in airplane design runs. In the present study, the method of characteristics is used to predict sonic boom signals in an efficient fashion. The governing equations are the axisymmetric Euler's equations with constant enthalpy. Since the method solves Euler's equations, it captures more nonlinear effects than the classical Whitham's F-function technique. Furthermore, the method of characteristics is an efficient marching scheme for initial value problems. In this study, we will first review the current CFD extrapolation technique and the work previously done in sonic boom extrapolation. Then, we will introduce the governing equations and the method of characteristics. Finally, we will show that the present method yields the same accurate results as previous CFD techniques, but with higher efficiency.

Cheung, Samson↗
NASA NTRS2017

Extrapolating the Trends of Test Drop Data with Opening Shock Factor Calculations: the Case of the Orion Main and Drogue Parachutes Inflating to 1st Reefed Stage

We describe a new calculation of the opening shock factor C (sub k) characterizing the inflation performance of NASA's Orion spacecraft main and drogue parachutes opening under a reefing constraint (1st stage reefing), as currently tested in the Capsule Parachute Assembly System (CPAS) program. This calculation is based on an application of the Momentum-Impulse Theorem at low mass ratio (R (sub m) is less than 10 (sup -1)) and on an earlier analysis of the opening performance of drogues decelerating point masses and inflating along horizontal trajectories. Herein we extend the reach of the Theorem to include the effects of payload drag and gravitational impulse during near-vertical motion - both important pre-requisites for CPAS parachute analysis. The result is a family of C (sub k) versus R (sub m) curves which can be used for extrapolating beyond the drop-tested envelope. The paper proves this claim in the case of the CPAS Mains and Drogues opening while trailing either a Parachute Compartment Drop Test Vehicle or a Parachute Test Vehicle (an Orion capsule boiler plate). It is seen that in all cases the values of the opening shock factor can be extrapolated over a range in mass ratio that is at least twice that of the test drop data.

Potvin, Jean↗