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A system which can generate multiple signals such that each signal can have different quality states is modeled. A measure of the effectiveness of the system as a function of the status of its components is developed. Assuming only that the probability distribution of the status of each component is known, bounds on the probability distribution of the system's measure of effectiveness are developed.
A polar nephelometer has been developed at NCAR to measure the angular variation of the intensity of light scattered by air molecules and particles. The system has been designed for airborne measurements using outside air ducted through a 5-cm diameter airflow tube; the sample volume is that which is common to the intersection of a collimated source beam and the detector field of view within the airflow tube. The source is a linearly polarized helium-neon laser beam. The optical system defines a collimated field-of-view (0.5deg half-angle) through a series of diaphragms located behind a I72-mm focal length objective lens. A photomultiplier tube is located immediately behind an aperture in the focal plane of the objective lens. The laser beam is mechanically chopped (on-off) at a rate of 5 Hz; a two-channel pulse counter, synchronized to the laser output, measures the photomultiplier pulse rate with the light beam both on and off. The difference in these measured pulse rates is directly proportional to the intensity of the scattered light from the volume common to the intersection of the laser beam and the detector field-of-view. Measurements can be made at scattering angles from 15deg to 165deg with reference to the direction of propagation of the light beam. Intermediate angles are obtained by selecting the angular increments desired between these extreme angles (any multiple of 0.1deg can be selected for the angular increment; 5deg is used in normal operation). Pulses provided by digital circuits control a stepping motor which sequentially rotates the detector by pre-selected angular increments. The synchronous photon-counting system automatically begins measurement of the scattered-light intensity immediately after the rotation to a new angle has been completed. The instrument has been flown on the NASA Convair 990 airborne laboratory to obtain data on the complex index of refraction of atmospheric aerosols. A particle impaction device is operated simultaneously to collect particles from the same airflow tube used to make the scattered-light measurements. A size distribution function is obtained by analysis of the particles collected by the impaction device. Calculated values of the angular variation of the scattered-light intensity are obtained by applying Mie scattering theory to the observed size distribution function and assuming different values of the complex index of refraction of the particles. The calculated values are then compared with data on the actual variation of the scattered-light intensity obtained with the polar nephelometer. The most probable value of the complex refractive index is that which provides the best fit between the experimental light scattering data and the parameters calculated from the observed size distribution function.
This paper presents a characterization of performance of 32-GHz Ka-band link in an operational environment. The data come from tracking of Kepler spacecraft by the NASA Deep Space Network (DSN) in the past few years. Ka-band link offers a significant signal-to-noise advantage compared to the more common X- or S-band; however, it is subject to more signal fluctuation caused by the weather. Kepler is the first mission supported by the DSN that solely rely on the higher deep space Ka-band communications link to return its high-rate science data. A well characterization of the operational performance of Kepler would benefit future Ka-band missions, especially for those operating with smaller link margin. The study examines how weather conditions at the DSN facilities (e.g., winds, clouds, rains) affect the received signal, particularly on telemetry data. It addresses questions such as how often the weather affects the link and how much degradation the link could suffer. Among the 22 Ka-band passes in 2012, heavy clouds affected one pass and two passes were impacted by high winds. The adverse weather caused at time as much as 3-dB instantaneous change in the signal to noise ratio (SNR). Estimates of degradation to the SNR as a function of wind speeds are captured based on observations. This study also quantifies the probability distribution of the variation of received signal power. With such information, future missions can better plan their link design. An optimal link design aims for just having enough margin to realize the data return with a targeted probability, with neither having too much margin that it reduces the downlink data rate nor insufficient reserve that causes frequent data outages. The study tries to provide some answers to the questions such as how much does SNR vary from one tracking pass to the next, and what is the cumulative distribution of signal fluctuation. Regarding the signal variation over many tracking passes, we found that the averaged symbol SNR (SSNR) for each pass changed by as much as 4 dB. Some variations were due to geometry such as the changing distance between spacecraft and Earth and different antenna pointing elevations. Other variations seem to be random in nature, reflecting the randomness of operational environments. Some passes were found to be quite stable, with a standard deviation of symbol SNR around 0.25 dB while others had greater variation, up to 1.4 dB. Overall, the cumulative distribution of the symbol SNR reflects that 50% of the fluctuations was less than 0.6 dB, 90% of fluctuation was within 1.6 dB and 95% within 2.2 dB. Some of the challenges in data processing, validation and modeling were captured in this paper. We observed some inconsistent measurements where operational data significantly deviate from a normal expectation. These inconsistent measurements made it hard to develop an accurate and consistent performance model, such as the degradation of signal SNR as a function of high wind. The difficulty was compounded by the fact that there were only a few Ka-band passes observed in high winds.
The dependence of counts in cells on the shape of the cell for the large scale galaxy distribution is studied. A very concrete prediction can be done concerning the void distribution for scale invariant models. The prediction is tested on a sample of the CfA catalog, and good agreement is found. It is observed that the probability of a cell to be occupied is bigger for some elongated cells. A phenomenological scale invariant model for the observed distribution of the counts in cells, an extension of the negative binomial distribution, is presented in order to illustrate how this dependence can be quantitatively determined. An original, intuitive derivation of this model is presented.
The dependence of counts in cells on the shape of the cell for the large scale galaxy distribution is studied. A very concrete prediction can be done concerning the void distribution for scale invariant models. The prediction is tested on a sample of the CfA catalog, and good agreement is found. It is observed that the probability of a cell to be occupied is bigger for some elongated cells. A phenomenological scale invariant model for the observed distribution of the counts in cells, an extension of the negative binomial distribution, is presented in order to illustrate how this dependence can be quantitatively determined. An original, intuitive derivation of this model is presented.
Straggling functions for protons in thick silicon radiation detectors are computed by Monte Carlo simulation. Mean energy loss is constrained by the silicon stopping power, providing higher straggling at low energy and probabilities for stopping within the detector volume. By matching the first four moments of simulated energy-loss distributions, straggling functions are approximated by a log-normal distribution that is accurate for Vavilov k is greater than or equal to 0:3. They are verified by comparison to experimental proton data from a charged particle telescope.
Models that predict the failure probability of monolithic glass and ceramic components under multiaxial loading have been developed by authors such as Batdorf, Evans, and Matsuo. These "unit-sphere" failure models assume that the strength-controlling flaws are randomly oriented, noninteracting planar microcracks of specified geometry but of variable size. This report develops a formulation to describe the probability density distribution of the orientation of critical strength-controlling flaws that results from an applied load. This distribution is a function of the multiaxial stress state, the shear sensitivity of the flaws, the Weibull modulus, and the strength anisotropy. Examples are provided showing the predicted response on the unit sphere for various stress states for isotropic and transversely isotropic (anisotropic) materials--including the most probable orientation of critical flaws for offset uniaxial loads with strength anisotropy. The author anticipates that this information could be used to determine anisotropic stiffness degradation or anisotropic damage evolution for individual brittle (or quasi-brittle) composite material constituents within finite element or micromechanics-based software
A detailed study is presented of some statistical properties of a stochastic process that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf).
A numerical algorithm and computer program are presented for solving the laminar, transitional, or turbulent two dimensional or axisymmetric compressible boundary-layer equations for perfect-gas flows. The governing equations are solved by an iterative three-point implicit finite-difference procedure. The software, program VGBLP, is a modification of the approach presented in NASA TR R-368 and NASA TM X-2458, respectively. The major modifications are: (1) replacement of the fourth-order Runge-Kutta integration technique with a finite-difference procedure for numerically solving the equations required to initiate the parabolic marching procedure; (2) introduction of the Blottner variable-grid scheme; (3) implementation of an iteration scheme allowing the coupled system of equations to be converged to a specified accuracy level; and (4) inclusion of an iteration scheme for variable-entropy calculations. These modifications to the approach presented in NASA TR R-368 and NASA TM X-2458 yield a software package with high computational efficiency and flexibility. Turbulence-closure options include either two-layer eddy-viscosity or mixing-length models. Eddy conductivity is modeled as a function of eddy viscosity through a static turbulent Prandtl number formulation. Several options are provided for specifying the static turbulent Prandtl number. The transitional boundary layer is treated through a streamwise intermittency function which modifies the turbulence-closure model. This model is based on the probability distribution of turbulent spots and ranges from zero to unity for laminar and turbulent flow, respectively. Several test cases are presented as guides for potential users of the software.
The atmospheric effect on the upward radiance emerging from the atmosphere above a nonuniform surface results in a reduction of the separability between the surface classes by broadening the radiance probability distribution of each class, while narrowing the total radiance range. The atmospheric modulation transfer function (MTF) is used in Fourier transform analyses to simulate the atmospheric effect on the imagery of a nonuniform surface and to demonstrate the atmospheric effect on separability of field classes.
Karst vadose zone heterogeneity creates complex transmission and storage dynamics that affect the timing and magnitude of aquifer recharge. Young, high-matrix permeability eogenetic karst aquifers may have significantly higher matrix storage than older, lower matrix permeability counterparts. In vulnerable and water-limited karst regions, the timescales of storage may be important for seasonal and sub-seasonal water resource management. We create a framework to quantify storage dynamics in karst aquifers using high-resolution precipitation and groundwater levels from the Northern Guam Lens Aquifer in the US territory of Guam. We estimate recharge using the Water Table Fluctuation method, and then develop transfer functions between precipitation and recharge to quantify storage and release of water from the vadose zone. The transfer functions are partitioned into different flow pathways including conduit, conduit/matrix, and slowly draining matrix. Probability distributions are fit to each pathway to determine the average travel times of infiltrated waters. The results show that aquifer recharge through secondary porosity features typically occurs within a few hours of a rainfall event, but this rapid recharge accounts for only 12%–28% of total recharge. The majority of aquifer recharge (>70%) occurs within a month of a contributing storm event. An additional 10% of recharge, on average, took longer than a month to reach the water table. The framework established can help improve the hydrological modeling and freshwater management for karst aquifers.
In this research, we have developed several novel visualization methods for spatial probability density function data. Our focus has been on 2D spatial datasets, where each pixel is a random variable, and has multiple samples which are the results of experiments on that random variable. We developed novel clustering algorithms as a means to reduce the information contained in these datasets; and investigated different ways of interpreting and clustering the data.
Development of a stochastic model of the probability distribution for the random variable representing the number of microorganisms on a surface as a function of time. The first basic principle associated with bioburden estimation is that viable particles are removed from surfaces. The second notion important to the analysis is that microorganisms in environments and on surfaces occur in clumps. The last basic principle relating to bioburden modeling is that viable particles are deposited on a surface. The bioburden on a spacecraft is determined by the amount and kind of control exercised on the spacecraft assembly location, the shedding characteristics of the individuals in the vicinity of the spacecraft, its orientation, the geographical location in which the assembly takes place, and the steps in the assembly procedure. The model presented has many of the features which are desirable for its use in the spacecraft sterilization programs currently being planned by NASA.
The asymptotic properties of feedback systems are discussed, containing uncertain parameters and subjected to stochastic perturbations. The approach is functional analytic in flavor and thereby avoids the use of Markov techniques and auxiliary Lyapunov functionals characteristic of the existing work in this area. The results are given for the probability distributions of the accessible signals in the system and are proved using the Prohorov theory of the convergence of measures. For general nonlinear systems, a result similar to the small loop-gain theorem of deterministic stability theory is given. Boundedness is a property of the induced distributions of the signals and not the usual notion of boundedness in norm. For the special class of feedback systems formed by the cascade of a white noise, a sector nonlinearity and convolution operator conditions are given to insure the total boundedness of the overall feedback system.
Bistatic radar observations of quasi-specular scatter from the lunar surface at 13 and 116 cm wavelengths have been expressed in terms of probability densities of the underlying surface slope distributions. These show best agreement with a Hagfors scattering function, although other forms having predominantly Gaussian or exponential characteristics also occur. Estimates of root mean square (rms) surface slope derived from these function range from 4 deg rms in maria to at least 8 deg rms in highlands. These are values appropriate to 25 m horizontal scales and represent averages over tens of square kilometers of surface area. The effective scattering roughness varies with wavelength, most strongly in maria, where its behavior is consistent with that expected from an immature, i.e., unsaturated cratered surface. There is very little variation in scattering roughness with wavelength in highlands area.
A straightforward statistical analysis is presented of the problem of discriminating between two distributed targets in a radar image. The probability of making an error is calculated as a function of the target contrast, the number of samples averaged to form each observation, and the system noise. In the absence of system noise, it is observed that a target contrast of 1 dB could not be detected with less than 10 samples averaged; it is also seen that the error probability is sensitive to the number of samples averaged, as expected. Setting the target contrast at the radiometric resolution, a comparison is made of different definitions for this resolution. The comparison shows that each definition yields a different error probability, ranging from less than 0.05 to 0.35. Attention is also given to the effect of system noise, and it is shown that thermal noise does not have a significant effect on the error probability for a fixed target contrast.
We present a uniform X-ray spectral analysis of nine type-1 active galactic nuclei (AGN) that have been previously found to harbor relativistically broadened iron emission lines. We show that the need for relativistic effects in the spectrum is robust even when one includes continuum "reflection" from the accretion disk. We then proceed to model these relativistic effects in order to constrain the spin of the supermassive black holes in these AGN. Our principal assumption, supported by recent simulations of geometrically-thin accretion disks, is that no iron line emission (or any associated Xray reflection features) can originate from the disk within the innermost stable circular orbit. Under this assumption, which tends to lead to constraints in the form of lower limits on the spin parameter, we obtain non-trivial spin constraints on five AGN. The spin parameters of these sources range from moderate (a approximates 0.6) to high (a > 0.96). Our results allow, for the first time, an observational constraint on the spin distribution function of local supermassive black holes. Parameterizing this as a power-law in dimensionless spin parameter (f(a) varies as absolute value of (a) exp zeta), we present the probability distribution for zeta implied by our results. Our results suggest 90% and 95% confidence limits of zeta > -0.09 and zeta > -0.3 respectively.
Year to year variations in the cumulative distributions of rain rate or rain attenuation are evident in any of the published measurements for a single propagation path that span a period of several years of observation. These variations must be described by models for the prediction of rain attenuation statistics. Now that a large measurement data base has been assembled by the International Radio Consultative Committee, the information needed to assess variability is available. On the basis of 252 sample cumulative distribution functions for the occurrence of attenuation by rain, the expected year to year variation in attenuation at a fixed probability level in the 0.1 to 0.001 percent of a year range is estimated to be 27 percent. The expected deviation from an attenuation model prediction for a single year of observations is estimated to exceed 33 percent when any of the available global rain climate model are employed to estimate the rain rate statistics. The probability distribution for the variation in attenuation or rain rate at a fixed fraction of a year is lognormal. The lognormal behavior of the variate was used to compile the statistics for variability.