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At least 19 records

The beta distribution: A statistical model for world cloud cover

Much work has been performed in developing empirical global cloud cover models. This investigation was made to determine an underlying theoretical statistical distribution to represent worldwide cloud cover. The beta distribution with probability density function is given to represent the variability of this random variable. It is shown that the beta distribution possesses the versatile statistical characteristics necessary to assume the wide variety of shapes exhibited by cloud cover. A total of 160 representative empirical cloud cover distributions were investigated and the conclusion was reached that this study provides sufficient statical evidence to accept the beta probability distribution as the underlying model for world cloud cover.

Falls, L. W.↗

Beta distributions: A computer program for probabilities and fractile points

A beta distribution is specified by range parameters a b, and two shape parameters alpha and beta 0. The computer program presented calculates any desired probability and/or fractile point for specified values of a, b, alpha, and beta. This program additionally computes gamma function values for integer and noninteger arguments.

Brownlow, J. D.↗

The beta distribution - A statistical model for world cloud cover

A statistical distribution representing worldwide cloud cover is obtained as a basis for an empirical global cloud cover model incorporating some 29 regions of the globe which are of interest to NASA's Skylab Program. A total of 160 representative empirical cloud cover distributions were investigated with the conclusion that the beta probability distribution is an adequate model for world cloud cover.

Falls, L. W.↗

Probability Distribution Estimated From the Minimum, Maximum, and Most Likely Values: Applied to Turbine Inlet Temperature Uncertainty

Modern engineering design practices are tending more toward the treatment of design parameters as random variables as opposed to fixed, or deterministic, values. The probabilistic design approach attempts to account for the uncertainty in design parameters by representing them as a distribution of values rather than as a single value. The motivations for this effort include preventing excessive overdesign as well as assessing and assuring reliability, both of which are important for aerospace applications. However, the determination of the probability distribution is a fundamental problem in reliability analysis. A random variable is often defined by the parameters of the theoretical distribution function that gives the best fit to experimental data. In many cases the distribution must be assumed from very limited information or data. Often the types of information that are available or reasonably estimated are the minimum, maximum, and most likely values of the design parameter. For these situations the beta distribution model is very convenient because the parameters that define the distribution can be easily determined from these three pieces of information. Widely used in the field of operations research, the beta model is very flexible and is also useful for estimating the mean and standard deviation of a random variable given only the aforementioned three values. However, an assumption is required to determine the four parameters of the beta distribution from only these three pieces of information (some of the more common distributions, like the normal, lognormal, gamma, and Weibull distributions, have two or three parameters). The conventional method assumes that the standard deviation is a certain fraction of the range. The beta parameters are then determined by solving a set of equations simultaneously. A new method developed in-house at the NASA Glenn Research Center assumes a value for one of the beta shape parameters based on an analogy with the normal distribution (ref.1). This new approach allows for a very simple and direct algebraic solution without restricting the standard deviation. The beta parameters obtained by the new method are comparable to the conventional method (and identical when the distribution is symmetrical). However, the proposed method generally produces a less peaked distribution with a slightly larger standard deviation (up to 7 percent) than the conventional method in cases where the distribution is asymmetric or skewed. The beta distribution model has now been implemented into the Fast Probability Integration (FPI) module used in the NESSUS computer code for probabilistic analyses of structures (ref. 2).

Holland, Frederic A., Jr.↗

An Updated Secondary Lunar Meteoroid Ejecta Model for Engineering Design

Introduction: The surface of the Moon is constantly being bombarded by a flux of meteoroids of various sizes. Impacts due to these meteoroids produce secondary ejecta material at much lower speeds but with a total mass larger than the original impactor. Details about the secondary ejecta are important for planning missions on the lunar surface. In this work, an updated ejecta model is presented called the Meteoroid Model of Secondary Ejecta (MeMoSeE), to re-place the Apollo-era ejecta model, NASA SP-8013 [1], in the SLS-SPEC 159 Design Specification for Natural Environments (DSNE) [2]. The model pro-duces secondary ejecta flux environments for a user-specified location on the lunar surface, and sorts the incoming secondary flux by angular direction and speed. Methods: MeMoSeE is separated into three parts: the inputs, the conversion step, and the integration of fluxes. Inputs to the model include the primary meteoroid fluxes and the primary near-Earth object (NEO) fluxes. Meteoroid fluxes, both asteroidal and cometary, are calculated using the Meteoroid Engineering Model (MEM3) [3] for different locations on the Moon. For each surface location, an ephemeris is generated using the JPL HORIZONS System [4] that feeds into MEM3. The NEO fluxes are approximated by the high-density population of MEM3 (i.e., only the directionality), where the speed distribution of the NEO fluxes is renormalized to match observations [5]. The regolith properties are used as defined in the DSNE [2]. The conversion step utilizes scaling laws given by Housen & Holsapple 2011 [6] to convert the primary impactor flux to the total mass of secondary ejecta. The ejecta distribution, at the point-of-impact (POI), is separated into a zenith angle and azimuthal angle distribution. The zenith angle distribution follows a beta distribution where the peak depends on the impact altitude angle and the impact azimuth [7]. We employ an ejecta azimuth distribution that is based on Rival & Mandeville 1999 [8] which focuses ejecta in the downstream direction for more oblique impacts. Finally, during the integration step, we sum secondary ejecta number fluxes at a particular region-of-interest (ROI) that originated from many POI locations over the entire surface of the Moon. We keep track of both altitude and azimuth angle bins as well as a range of speed bins, following the igloo gridding as done in MEM3 [9]. The ejecta particle size distribution and density is assumed to be the same as the lunar regolith [10, 2]. Results: The primary fluxes are computed for one Metonic cycle (19 years) for various locations over the lunar surface with a fixed orientation. Both the angular and speed distributions of the primary fluxes are dependent on the latitude and longitude. The over-all primary fluxes show a roughly 13% increase from the eastern limb to the western limb. In general, the speed distribution of the secondary fluxes span from a user-defined minimum speed to the escape speed of the Moon (2.38 km/s), roughly following a power-law relation [6]. Different parts of the speed distribution come from different primary impact locations on the Moon. The secondary ejecta is dominated by the slowest speeds, where these particles originate nearby the ROI. For speeds around 71% of the escape speed, the secondary ejecta originates from locations near the antipodal point. On the other hand, for secondary ejecta speeds that exceed roughly 90% the escape speed, the ejecta particles come from all over the lunar surface to the ROI. Comparing the secondary ejecta fluxes from MeMoSeE with NASA SP-8013 [1], there is a reduction by about 2-3 orders of magnitude for secondary ejecta particles greater than 1 µg. These estimates agree with recent findings from Bjorkman & Christiansen 2019 [11]. The secondary ejecta fluxes are also compared with the primary fluxes, where the ejecta fluxes are roughly an order of magnitude greater than the primary fluxes. References: [1] Cour-Palais, B. G., (1969) NASA SP-8013. [2] NASA SLS-SPEC-159 Rev. H (2020). [3] Moorhead, A. V., et al. (2019) JS&R, 1-17. [4] Giorgini, J. D., (2015) IAUGA, 29, 2256293. [5] Moorhead, A. V., (2020) Memo OSMA/MEO/Lunar-001. [6] Housen, K. R., and Holsapple, K. A., (2011) Icarus, 211(1), 856-875. [7] Gault, D. E., and Wedekind, J. A., (1978) L&PSCP, 9, 3843-3875. [8] Rival, M., and Mandeville, J., (1999) Space Debris, 1(1), 45-57. [9] Moorhead, A. V., (2019) MEM3 User Guide. [10] Carrier, W. D., (1973) The Moon, 6(3-4), 250-263. [11] Bjorkman, M. D. & Christiansen, E. L., (2019) ODC

impacts↗

The Galaxy as the origin of gamma-ray bursts

The gamma-ray burst spectrum N(greater than S) versus S is reviewed and found to be inconsistent with a completely isotropic distribution. Extragalactic distributions are considered but rejected in favor of a galactic origin. Spherical halo models, particularly those with central concentration, are found to be unacceptable. Noncentrally concentrated disk models are acceptable, especially those with larger scale height (beta) distributions. Spiral structure effects prevent the total rejection of small beta distributions but impose space density requirements which exclude many extreme Population I candidates. Integrated burst luminosities range from about 10 to the 38th to 2 x 10 to the 39th ergs while burst rate densities vary between about 10 to the -6th and 10 to the -8th/cu pc yr; both are uniquely determined by the distribution scale height. It is shown that galactic burst sources must be repetitive with repetition rates between about 10 to the -5th/yr and about 10 to the 6th/yr depending on the source space density and the importance of beaming. Bursts are unimportant in the galactic energy balance and, on average, in the energy history of their sources. They may, however, dominate specific events.

Jennings, M. C.↗

Two-dimensional leaf orientation distributions

Combined inclination/azimuth leaf angle distributions are important for accurate models of vegetation canopy reflectance. It is shown that appropriate mathematical representations can be constructed from beta distributions under most circumstances. This is illustrated by analyzing observational data on soybean leaves and balsam fir needles. There are some problems when the data is imprecise and when correlations between inclination and azimuth angle are induced by heliotropism. Otherwise, the two-dimensional beta-type distribution appears to be a versatile tool for describing complete inclination/azimuth leaf angle distributions.

Strebel, D. E.↗

The Area Coverage of Geophysical Fields as a Function of Sensor Field-of View

In many remote sensing studies of geophysical fields such as clouds, land cover, or sea ice characteristics, the fractional area coverage of the field in an image is estimated as the proportion of pixels that have the characteristic of interest (i.e., are part of the field) as determined by some thresholding operation. The effect of sensor field-of-view on this estimate is examined by modeling the unknown distribution of subpixel area fraction with the beta distribution, whose two parameters depend upon the true fractional area coverage, the pixel size, and the spatial structure of the geophysical field. Since it is often not possible to relate digital number, reflectance, or temperature to subpixel area fraction, the statistical models described are used to determine the effect of pixel size and thresholding operations on the estimate of area fraction for hypothetical geophysical fields. Examples are given for simulated cumuliform clouds and linear openings in sea ice, whose spatial structures are described by an exponential autocovariance function. It is shown that the rate and direction of change in total area fraction with changing pixel size depends on the true area fraction, the spatial structure, and the thresholding operation used.

Key, Jeffrey R.↗

Using an Informative Missing Data Model to Predict the Ability to Assess Recovery of Balance Control after Spaceflight

Astronauts show degraded balance control immediately after spaceflight. To assess this change, astronauts' ability to maintain a fixed stance under several challenging stimuli on a movable platform is quantified by "equilibrium" scores (EQs) on a scale of 0 to 100, where 100 represents perfect control (sway angle of 0) and 0 represents data loss where no sway angle is observed because the subject has to be restrained from falling. By comparing post- to pre-flight EQs for actual astronauts vs. controls, we built a classifier for deciding when an astronaut has recovered. Future diagnostic performance depends both on the sampling distribution of the classifier as well as the distribution of its input data. Taking this into consideration, we constructed a predictive ROC by simulation after modeling P(EQ = 0) in terms of a latent EQ-like beta-distributed random variable with random effects.

Feiveson, Alan H.↗

Backscatter Modeling at 2.1 Micron Wavelength for Space-Based and Airborne Lidars Using Aerosol Physico-Chemical and Lidar Datasets

Space-based and airborne coherent Doppler lidars designed for measuring global tropospheric wind profiles in cloud-free air rely on backscatter, beta from aerosols acting as passive wind tracers. Aerosol beta distribution in the vertical can vary over as much as 5-6 orders of magnitude. Thus, the design of a wave length-specific, space-borne or airborne lidar must account for the magnitude of 8 in the region or features of interest. The SPAce Readiness Coherent Lidar Experiment under development by the National Aeronautics and Space Administration (NASA) and scheduled for launch on the Space Shuttle in 2001, will demonstrate wind measurements from space using a solid-state 2 micrometer coherent Doppler lidar. Consequently, there is a critical need to understand variability of aerosol beta at 2.1 micrometers, to evaluate signal detection under varying aerosol loading conditions. Although few direct measurements of beta at 2.1 micrometers exist, extensive datasets, including climatologies in widely-separated locations, do exist for other wavelengths based on CO2 and Nd:YAG lidars. Datasets also exist for the associated microphysical and chemical properties. An example of a multi-parametric dataset is that of the NASA GLObal Backscatter Experiment (GLOBE) in 1990 in which aerosol chemistry and size distributions were measured concurrently with multi-wavelength lidar backscatter observations. More recently, continuous-wave (CW) lidar backscatter measurements at mid-infrared wavelengths have been made during the Multicenter Airborne Coherent Atmospheric Wind Sensor (MACAWS) experiment in 1995. Using Lorenz-Mie theory, these datasets have been used to develop a method to convert lidar backscatter to the 2.1 micrometer wavelength. This paper presents comparison of modeled backscatter at wavelengths for which backscatter measurements exist including converted beta (sub 2.1).

Srivastava, V.↗

The interaction between hot and cold gas in early-type galaxies

SO and Sa galaxies have approximately equal masses of H I and X-ray emitting gas and are ideal sites for studying the interaction between hot and cold gas. An X-ray observation of the Sa galaxy NGC 1291 with the ROSAT position sensitive proportional counter (PSPC) shows a striking spatial anticorrelation between hot and cold gas where X-ray emitting material fills the large central black hole in the H I disk. This supports a previous suggestion that hot gas is a bulge phenomenon and neutral hydrogen is a disk phenomenon. The X-ray luminosity (1.5 x 10(exp 40) ergs/s) and radial surface brightness distribution (beta = 0.51) is the same as for elliptical galaxies with optical luminosities and velocity dispersions like that of the bulge of NGC 1291. Modeling of the X-ray spectrum requires a component with a temperature of 0.15 keV, similar to that expected from the velocity dispersion of the stars, and with a hotter component where kT = 1.07 keV. This hotter component is not due to emission from stars and its origin remains unclear. PSPC observations are reported for the SO NGC 4203, where a nuclear point source dominates the emission, preventing a study of the radial distribution of the hot gas relative to the H I.

Bregman, Joel N.↗

Cloud cover models derived from satellite radiation measurements

Using daily measurement of day and night infrared and incoming and absorbed solar radiation obtained from a TIROS satellite over a period of approximately 45 months, and integrated over 2.5 degree latitude-longitude grids, the proportion of cloud cover over each grid each day was derived for the entire period. For each of four three-month periods, estimates a and b of the two parameters of the best-fit beta distribution were obtained for each grid location. The (a,b) plane was divided into a number of regions. All the geographical locations whose (a,b) estimates were in the same region in the (a,b) plane were said to have the same cloud cover type for that season. For each season, the world was thus divided into separate cloud cover types. Using estimates of mean cloud cover for each season, the world was again divided into separate cloud cover types. The process was repeated for standard deviations. Thus for each season, three separate cloud cover models were obtained using the criteria of shape of frequency distribution, mean cloud cover, and variability of cloud cover. The cloud cover statistics were derived from once-a-day, near-local-noon satellite radiation measurements.

Bean, S. J.↗

Some new worldwide cloud-cover models

Using daily measurements of day and night infrared, and incoming and absorbed solar radiation obtained from a Tiros satellite over a period of approximately 45 months, and integrated over 2.5 deg latitude-longitude grids, the proportion of cloud cover over each grid each day was derived for the entire period. For each of four 3-month periods, for each grid location, estimates a and b of the two parameters of the best-fit beta distribution were obtained. The (a, b) plane was divided into a number of regions. All the geographical locations whose (a, b) estimates were in the same region in the (a, b) plane were said to have the same cloud cover type for that season. For each season, the world is thus divided into separate cloud-cover types.

Bean, S. J.↗

Cloud cover archiving on a global scale - A discussion of principles

Monitoring of climatic variability and climate modeling both require a reliable global cloud data set. Examination is made of the temporal and spatial variability of cloudiness in light of recommendations made by GARP in 1975 (and updated by JOC in 1978 and 1980) for cloud data archiving. An examination of the methods of comparing cloud cover frequency curves suggests that the use of the beta distribution not only facilitates objective comparison, but also reduces overall storage requirements. A specific study of the only current global cloud climatology (the U.S. Air Force's 3-dimensional nephanalysis) over the United Kingdom indicates that discussion of methods of validating satellite-based data sets is urgently required.

Henderson-Sellers, A.↗

How Well Does the Latest Anthropomorphic Test Device Mimic Human Impact Responses?

One of the goals of the NASA Occupant Protection Group is to understand the human tolerance to dynamic loading. This knowledge has to come through indirect approaches such as existing human response databases, anthropometric test devices (ATD), animal testing, post‐mortem human subjects, and models. This study investigated the biofidelity of the National Highway Traffic Safety Administration's ATD named the THOR (test device for human occupant restraint). If THOR responds comparably to humans, then it could potentially be used as a human surrogate to help validate space vehicle requirements for occupant protection. The THOR responses to frontal and spinal impacts (ranging from 8 to 12 G with rise times of 40, 70, and 100 ms) were measured and compared to human volunteer responses (95 trials in frontal and 58 in spinal) previously collected by the U. S. Air Force on the same horizontal impact accelerator. The impact acceleration profiles tested are within the expected range of multi‐purpose crew vehicle (MPCV) landing dynamics. A correlation score was calculated for each THOR to human comparison using CORA (CORrelation and Analysis) software. A two‐parameter beta distribution model fit was obtained for each dependent variable using maximum likelihood estimation. For frontal impacts, the THOR head x‐acceleration peak response correlated with the human response at 8 and 10‐G 100 ms but not 10‐G 70 ms. The phase lagged the human response. Head z‐acceleration was not correlated. Chest x‐acceleration was in phase, had a higher peak response, and was well correlated with lighter subjects (Cora = 0.8 for 46 kg vs. Cora = 0.4 for 126 kg). Head x‐displacement had a leading phase. Several subjects responded with the same peak displacement but the mean of the group was lower. The shoulder x‐displacement was in phase but had higher peaks than the human response. For spinal impacts, the THOR head x‐acceleration was not well correlated. Head and chest z‐acceleration was in phase but had a higher peak response. Chest z‐acceleration was highly correlated with heavier subjects at lower G pulses (Cora = 0.86 for 125 kg at 8 G). The human response was variable in shoulder z‐displacement but the THOR was in phase and was comparable to the mean peak response. Head xand z‐displacement was in phase but had higher peaks. Seat pan forces were well correlated, were in phase, but had a larger peak response than most subjects. The THOR does not respond to frontal and spinal impacts exactly the same way that a human does. Some responses are well matched and others are not. Understanding the strengths and weaknesses of this ATD is an important first step in determining its usefulness in occupant protection at NASA

Newby, N.↗

How Well Does the Latest Anthropomorphic Test Device Mimic Human Impact Responses?

One of the goals of the NASA Occupant Protection Group is to understand the human tolerance to dynamic loading. This knowledge has to come through indirect approaches such as existing human response databases, anthropometric test devices (ATD), animal testing, post-­‐mortem human subjects, and models. This study investigated the biofidelity of the National Highway Traffic Safety Administration's ATD named the THOR (test device for human occupant restraint). If THOR responds comparably to humans, then it could potentially be used as a human surrogate to help validate space vehicle requirements for occupant protection. The THOR responses to frontal and spinal impacts (ranging from 8 to 12 G with rise times of 40, 70, and 100 ms) were measured and compared to human volunteer responses (95 trials in frontal and 58 in spinal) previously collected by the U. S. Air Force on the same horizontal impact accelerator. The impact acceleration profiles tested are within the expected range of multi-­‐purpose crew vehicle (MPCV) landing dynamics. A correlation score was calculated for each THOR to human comparison using CORA (CORrelation and Analysis) software. A two-­‐parameter beta distribution model fit was obtained for each dependent variable using maximum likelihood estimation. For frontal impacts, the THOR head x-­‐acceleration peak response correlated with the human response at 8 and 10-­‐G 100 ms but not 10-­‐G 70 ms. The phase lagged the human response. Head z-­‐acceleration was not correlated. Chest x-­‐acceleration was in phase, had a higher peak response, and was well correlated with lighter subjects (Cora = 0.8 for 46 kg vs. Cora = 0.4 for 126 kg). Head x-­‐displacement had a leading phase. Several subjects responded with the same peak displacement but the mean of the group was lower. The shoulder x-­‐displacement was in phase but had higher peaks than the human response. For spinal impacts, the THOR head x-­‐acceleration was not well correlated. Head and chest z-­‐acceleration was in phase but had a higher peak response. Chest z-­‐acceleration was highly correlated with heavier subjects at lower G pulses (Cora = 0.86 for 125 kg at 8 G). The human response was variable in shoulder z-­‐displacement but the THOR was in phase and was comparable to the mean peak response. Head x-­‐ and z-­‐displacement was in phase but had higher peaks. Seat pan forces were well correlated, were in phase, but had a larger peak response than most subjects. The THOR does not respond to frontal and spinal impacts exactly the same way that a human does. Some responses are well matched and others are not. Understanding the strengths and weaknesses of this ATD is an important first step in determining its usefulness in occupant protection at NASA

Newby, Nate↗

Trending in Probability of Collision Measurements via a Bayesian Zero-Inflated Beta Mixed Model

We investigate the performance of a generalized linear mixed model in predicting the Probabilities of Collision (Pc) for conjunction events. Specifically, we apply this model to the log(sub 10) transformation of these probabilities and argue that this transformation yields values that can be considered bounded in practice. Additionally, this bounded random variable, after scaling, is zero-inflated. Consequently, we model these values using the zero-inflated Beta distribution, and utilize the Bayesian paradigm and the mixed model framework to borrow information from past and current events. This provides a natural way to model the data and provides a basis for answering questions of interest, such as what is the likelihood of observing a probability of collision equal to the effective value of zero on a subsequent observation.

Conjuction Assessment↗