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The statistics are considered of the Monte Carlo method relative to the interpretation of the NUGAM2 and NUGAM3 computer code results. A numerical experiment using the NUGAM2 code is presented and the results are statistically interpreted.
The present analysis uses the Monte Carlo method to solve the problem of rarefied diffusion between parallel walls. The diffusing molecules are evaporated or emitted from one of two parallel walls and diffused through another molecular species. The analysis treats the diffusing molecule as undergoing a Markov random walk and the local macroscopic properties are found as the expected value of the random variable, the random walk payoff. By biasing the transition probabilities and changing the collision payoffs the expected Markov walk payoff is retained but its variance is reduced so that the M. C. result has a much smaller error.
A Monte Carlo technique involving Stokes vectors is used to obtain the state of polarization and intensity of solar Lyman-alpha photons as they diffuse through a plane-parallel homogeneous model of earth's hydrogen envelope. Fine structure of Lyman-alpha and Doppler redistribution of frequencies are taken into account. Comparison of the results with Heath's observed upper limit for polarization of 1.5 per cent implies an optical thickness tau greater than 7 and intensities of 8-10 kilorayleighs for a solar Lyman-alpha flux of 5.8 ergs per sq cm per sec.
A Monte Carlo method is given for analyzing the transfer of resonance radiation in a rapidly and differentially expanding nebula. Coherence in the frame of the atom is shown to lead to interesting modifications of the emergent radiation field, as compared with the other more usual assumptions of coherence or redistribution in the fluid frame. Results are presented in tabular form suitable for further use. Sample line profiles produced by pure scattering out of the continuum are generated, and it is shown that an arbitrarily high peak can be obtained. This conclusion is independent of the coherence assumption used. The distribution of the number of scatterings is found in all cases. An analytic solution is presented for the case of coherence in the fluid frame.
A program is presented which uses a Monte Carlo technique to compute the geometrical factor of a detector, and the spatial and angular distributions of simulated particles at designated layers within the detector. As the program is written, each layer is assumed to be rectangular, parallel to every other layer with its edges parallel to the edges of every other layer, and centered about a common axis. This program is written in Fortran 4 for the IBM 360 computers and is currently being run under the Fortran 4 (h) compiler. Same features such as the subroutine ERRSET, RANDU, and REMTIM and the type declarations REAL *8 and LOGICAL *1 may need to be changed for use with a different computer system. The use of input data cards, and a sample data set are presented. Lists of the symbol definition, card images of the FORTRAN object deck and subroutines are included along with a flow diagram.
It is pointed out that none of the existing models of lunar regolith evolution take into account the relationship between regolith thickness, crater shape, and volume of debris ejected. The results of a Monte Carlo computer simulation of regolith evolution are presented. The simulation was designed to consider the full effect of the buffering regolith through calculation of the amount of debris produced by any given crater as a function of the amount of debris present at the site of the crater at the time of crater formation. The method is essentially an improved version of the Oberbeck and Quaide (1968) model.
The HD 220 program was created as part of the space shuttle solid rocket booster recovery system definition. The model was generated to investigate the damage to SRB components under water impact loads. The random nature of environmental parameters, such as ocean waves and wind conditions, necessitates estimation of the relative frequency of occurrence for these parameters. The nondeterministic nature of component strengths also lends itself to probabilistic simulation. The Monte Carlo technique allows the simultaneous perturbation of multiple independent parameters and provides outputs describing the probability distribution functions of the dependent parameters. This allows the user to determine the required statistics for each output parameter.
The results are reported of two unrelated studies. The first was an investigation of the formulation of the equations for non-uniform unsteady flows, by perturbation of an irrotational flow to obtain the linear Green's equation. The resulting integral equation was found to contain a kernel which could be expressed as the solution of the adjoint flow equation, a linear equation for small perturbations, but with non-constant coefficients determined by the steady flow conditions. It is believed that the non-uniform flow effects may prove important in transonic flutter, and that in such cases, the use of doublet type solutions of the wave equation would then prove to be erroneous. The second task covered an initial investigation into the use of the Monte Carlo method for solution of acoustical field problems. Computed results are given for a rectangular room problem, and for a problem involving a circular duct with a source located at the closed end.
This report examines the extent to which sample size affects the accuracy of a low order polynomial approximation of an experimentally observed quantity and establishes a trend toward improvement in the accuracy of the approximation as a function of sample size. The task is made possible through a simulated analysis carried out by the Monte Carlo method, in which data are generated by using several transcendental or algebraic functions as models. Contaminated data of varying amounts are fitted to linear quadratic or cubic polynomials, and the behavior of the mean-squared error of the residual variance is determined as a function of sample size. Results indicate that the effect of the size of the sample is significant only for relatively small sample sizes and diminishes drastically for moderate and large amounts of experimental data.
The Monte Carlo technique is proposed for the determination of the acoustical pressure-time history at chosen points in a partial enclosure, the central idea of this technique being the tracing of acoustical rays. A statistical model is formulated and an algorithm for pressure is developed, the conformity of which is examined by two approaches and is shown to give the known results. The concepts that are developed are applied to the determination of the transient field due to a sound source in a homogeneous medium in a rectangular enclosure with perfect reflecting walls, and the results are compared with those presented by Mintzer based on the Laplace transform approach, as well as with a normal mode solution.
The vibration of clamped and simply supported elastic panels due to subsonic and supersonic turbulent boundary-layer flows is investigated by a Monte Carlo technique. The resulting generalized random forces are simulated numerically from boundary-layer turbulence spectra, and the response analysis is performed in the time domain. The mutual interaction between panel motion and external and/or internal airflow is included. Response studies are performed with respect to rms response, probability structure, peak distribution, threshold crossing and spectral density. The effect on the response statistics of in-plane loading, static pressure differential, and cavity pressure is investigated.
The Monte Carlo (MC) and continuous slowdown approximation (CSDA) approaches to the spatial deposition of energy by electrons are compared using the same detailed atomic cross section (DACS). It is found that the CSDA method overestimates the amount of energy that is deposited near the end of the path for electrons above a few hundred electron volts. The MC results are in approximate agreement with experimental data in such a way as to be relatively independent of the actual gas used. Our MC results are extended to obtain the three-dimensional deposition of energy by sub-keV electrons in molecular hydrogen.
The Monte Carlo method is used for two types of problems. First, there are interpretation problems of optical observations from meteorological satellites in the short wave part of the spectrum. The sphericity of the atmosphere, the propagation function, and light polarization are considered. Second, problems dealt with the theory of spreading narrow light beams. Direct simulation of light scattering and the mathematical form of medium radiation model representation are discussed, and general integral transfer equations are calculated. The dependent tests method, derivative estimates, and solution to the inverse problem are also considered.
An output error estimation algorithm was used to evaluate the effects of both static and dynamic instrumentation errors on the estimation of aircraft stability and control parameters. A Monte Carlo error analysis, using simulated cruise flight data, was performed for a high-performance military aircraft, a large commercial transport, and a small general aviation aircraft. The results indicate that unmodeled instrumentation errors can cause inaccuracies in the estimated parameters which are comparable to their nominal values. However, the corresponding perturbations to the estimated output response trajectories and characteristics equation pole locations appear to be relatively small. Control input errors and dynamic lags were found to be in the most significant of the error sources evaluated.
An output error estimation algorithm was used to evaluate the effects of both static and dynamic instrumentation errors on the estimation of aircraft stability and control parameters. A Monte Carlo analysis, using simulated cruise flight data, was performed for a high performance military aircraft, a large commercial transport, and a small general-aviation aircraft. The effects of variations in the information content of the flight data, resulting from two different choices of control input maneuvers, were also determined. The results indicate that unmodeled instrumentation errors can cause inaccuracies in the estimated parameters which are comparable to their nominal values. Control input errors and angular accelerometer lags were found to be most significant of the instrumentation errors evaluated, and the perturbations they produce are much larger than those arising from the combined effects of static errors and white noise in the output response measurements.
The performance of the Viking lander has been evaluated by using a Monte Carlo simulation, and all results are presented in statistical form. The primary objectives of this analysis were as follows: (1) to determine the three sigma design values of maximum rigid body accelerations and the minimum clearance of the lander body during landing; (2) to determine the probability of an unstable landing; and (3) to determine the probability of the lander body striking a rock. Two configurations were analyzed with the only difference being in the ability of the primary landing gear struts to carry tension loads.
A previous (Moss and Giuli, 1971) Monte Carlo analysis of the solar-wind modulation of galactic cosmic-ray rigidity spectra is extended to include particles in the C-N-O group. A solar cavity 'radius' of 7 AU is found to be consistent with observations of the C-N-O component at solar maximum. The variation in modulation between solar maximum and solar minimum may be explained solely by variation of the solar cavity size, with no change in the scattering mean free path.
A set of detailed atomic cross sections has been used to obtain the spatial deposition of energy by 1-20-eV electrons in molecular hydrogen by a Monte Carlo simulation of the actual trajectories. The energy deposition curve (energy per distance traversed) is quite peaked in the forward direction about the entry point for electrons with energies above the threshold of the electronic states, but the peak decreases and broadens noticeably as the electron energy decreases below 10 eV (threshold for the lowest excitable electronic state of H2). The curve also assumes a very symmetrical shape for energies below 10 eV, indicating the increasing importance of elastic collisions in determining the shape of the curve, although not the mode of energy deposition.