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A procedure to aid in the deciding between four and five parameters in a Jensen's type bivariate gamma distribution is presented. It is based upon the CDF of the ratio of correlated gamma distributed variates. The criteria are posed in a test of hypothesis setting and results are presented.
Analytical and numerical computational methods to determine the location of the mode as a function of the parameters of a class of the bivariate gamma distribution are presented.
The radial distribution of gamma ray emissivity in the Galaxy was derived from flux longitude profiles, using both the final SAS-2 results and the recently corrected COS-B results and analyzing the northern and southern galactic regions separately. The recent CO surveys of the Southern Hemisphere, were used in conjunction with the Northern Hemisphere data, to derive the radial distribution of cosmic rays on both sides of the galactic plane. In addition to the 5 kpc ring, there is evidence from the radial asymmetry for spiral features which are consistent with those derived from the distribution of bright HII regions. Positive evidence was also found for a strong increase in the cosmic ray flux in the inner Galaxy, particularly in the 5 kpc region in both halves of the plane.
The radial distribution of gamma ray emissivity in the Galaxy was derived from flux longitude profiles, using both the final SAS-2 results and the recently corrected COS-B results and analyzing the northern and southern galactic regions separately. The recent CO surveys of the Southern Hemisphere were used in conjunction with the Northern Hemisphere data, to derive the radial distribution of cosmic rays on both sides of the galactic plane. In addition to the 5 kpc ring, there is evidence from the radial asymmetry for spiral features which are consistent with those derived from the distribution of bright H II regions. Positive evidence was also found for a strong increase in the cosmic ray flux in the inner Galaxy, particularly in the 5 kpc region in both halves of the plane.
The statistical characteristics of radar return from vegetation targets are investigated through a simulation study based upon the first-order scattered field. For simulation purposes, the vegetation targets are modeled as a layer of randomly oriented and spaced finite cylinders, needles, or discs, or a combination of them. The finite cylinder is used to represent a branch or a trunk, the needle for a stem or a coniferous leaf, and the disc for a decidous leaf. For a plane wave illuminating a vegetation canopy, simulation results show that the signal returned from a layer of disc- or needle-shaped leaves follows the Gamma distribution, and that the signal returned from a layer of branches resembles the log normal distribution. The Gamma distribution also represents the signal returned from a layer of a mixture of branches and leaves regardless of the leaf shapes. Results also indicate that the polarization state does not have a significant impact on signal distribution.
A five-parameter bivariate gamma distribution having two shape parameters, two location parameters and a correlation parameter was developed. This more general bivariate gamma distribution reduces to the known four-parameter distribution. The five-parameter distribution gives a better fit to the gust data. The statistical properties of this general bivariate gamma distribution and a hypothesis test were investigated. Although these developments have come too late in the Shuttle program to be used directly as design criteria for ascent wind gust loads, the new wind gust model has helped to explain the wind profile conditions which cause large dynamic loads. Other potential applications of the newly developed five-parameter bivariate gamma distribution are in the areas of reliability theory, signal noise, and vibration mechanics.
A five-parameter gamma distribution (BGD) having two shape parameters, two location parameters, and a correlation parameter is investigated. This general BGD is expressed as a double series and as a single series of the modified Bessel function. It reduces to the known special case for equal shape parameters. Practical functions for computer evaluations for the general BGD and for special cases are presented. Applications to wind gust modeling for the ascent flight of the space shuttle are illustrated.
Gamma-ray bursts are transient astronomical phenomena which have no quiescent counterparts in any region of the electromagnetic spectrum. Although temporal and spectral properties indicate that these events are likely energetic, their unknown spatial distribution complicates astrophysical interpretation. Monte Carlo samples of gamma-ray burst sources are created which belong to Galactic disk populations. Spatial analysis techniques are used to compare these samples to the observed distribution. From this, both quantitative and qualitative conclusions are drawn concerning allowed luminosity and spatial distributions of the actual sample. Although the Burst and Transient Source Experiment (BATSE) experiment on Gamma Ray Observatory (GRO) will significantly improve knowledge of the gamma-ray burst source spatial characteristics within only a few months of launch, the analysis techniques described herein will not be superceded. Rather, they may be used with BATSE results to obtain detailed information about both the luminosity and spatial distributions of the sources.
The lateral distribution of atmospheric Cerenkov photons emitted by gamma ray showers of energy 100 GeV is calculated. The lateral distribution shows a characteristic hump at a distance of approx. 135 meter from the core. The hump is shown to be due to electrons of threshold energy 1 GeV, above which the mean scattering angle becomes smaller than the Cerenkov angle.
The development of a vector wind gust model that is suitable for orbital flight test operations and trade studies was studied. Verification of the hypothesis that gust component variables are gamma distributed, gust modulus is approximately Weibull distributed, and zonal and meridional gust components are bivariate gamma distributed is emphasized. A method of testing for bivariate gamma distributed variables, and two distributions for gust modulus are described. The results of extensive hypothesis testing of one of the distributions are presented, and the validity of the gamma distribution for representation of gust component variables is established.
A model of the largest gust amplitude and gust length is presented which uses the properties of the bivariate gamma distribution. The gust amplitude and length are strongly dependent on the filter function; the amplitude increases with altitude and is larger in winter than in summer.
To enable development of a vector wind gust model suitable for orbital flight test operations and trade studies, hypotheses concerning the distributions of gust component variables were verified. Methods for verification of hypotheses that observed gust variables, including gust component magnitude, gust length, u range, and L range, are gamma distributed and presented. Observed gust modulus has been drawn from a bivariate gamma distribution that can be approximated with a Weibull distribution. Zonal and meridional gust components are bivariate gamma distributed. An analytical method for testing for bivariate gamma distributed variables is presented. Two distributions for gust modulus are described and the results of extensive hypothesis testing of one of the distributions are presented. The validity of the gamma distribution for representation of gust component variables is established.
The SAS-2 gamma ray experiment and its detection of celestial gamma rays are described. Data also cover intensity of high energy gamma rays, gamma ray distribution, gamma ray origin, and diffuse radiation.
The radial distribution is derived of gamma ray emissivity and the cosmic ray intensity in the Galaxy using the final SAS-2 results, the recently corrected COS-B results, and recent CO surveys of the Northern and Southern Galactic Hemispheres. In addition to the 5 kpc ring of enhanced emission, there is evidence for spiral features. A strong increase in the cosmic ray flux is found in the inner Galaxy, particularly in the 5 kpc region, in both halves of the plane.
Supercooled liquid water (SLW) clouds, where liquid droplets exist at temperatures below 0 C present a well known aviation hazard through aircraft icing, in which SLW accretes on the airframe. SLW clouds are common over the Southern Ocean, and climate-induced changes in their occurrence is thought to constitute a strong cloud feedback on global climate. The two recent NASA field campaigns POlarimeter Definition EXperiment (PODEX, based in Palmdale, California, January-February 2013) and Studies of Emissions and Atmospheric Composition, Clouds and Climate Coupling by Regional Surveys (SEAC4RS, based in Houston, Texas in August- September 2013) provided a unique opportunity to observe SLW clouds from the high-altitude airborne platform of NASA's ER-2 aircraft. We present an analysis of measurements made by the Research Scanning Polarimeter (RSP) during these experiments accompanied by correlative retrievals from other sensors. The RSP measures both polarized and total reflectance in 9 spectral channels with wavelengths ranging from 410 to 2250 nm. It is a scanning sensor taking samples at 0.8deg intervals within 60deg from nadir in both forward and backward directions. This unique angular resolution allows for characterization of liquid water droplet size using the rainbow structure observed in the polarized reflectances in the scattering angle range between 135deg and 165deg. Simple parametric fitting algorithms applied to the polarized reflectance provide retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, Rainbow Fourier Transform (RFT),which allows retrieval of the droplet size distribution without assuming a size distribution shape. We present an overview of the RSP campaign datasets available from the NASA GISS website, as well as two detailed examples of the retrievals. In these case studies we focus on cloud fields with spatial features varying between glaciated and liquid phases at altitudes as high as 10 km, which correspond to temperatures close to the homogeneous freezing temperature of pure water drops (about -35 C or colder). The multimodal droplet size distributions retrieved from RSP data in these cases are consistent with the multi-layer cloud structure observed by correlative Cloud Physics Lidar (CPL) measurements.
We present comparisons of cloud droplet size distributions retrieved from the Research Scanning Polarimeter (RSP) data with correlative in situ measurements made during the North Atlantic Aerosols and Marine Ecosystems Study (NAAMES). This field experiment was based at St. Johns airport, Newfoundland, Canada with the latest deployment in May - June 2016. RSP was onboard the NASA C-130 aircraft together with an array of in situ and other remote sensing instrumentation. The RSP is an along-track scanner measuring polarized and total reflectances in9 spectral channels. Its unique high angular resolution allows for characterization of liquid water droplet size using the rainbow structure observed in the polarized reflectances in the scattering angle range between 135 and 165 degrees. A parametric fitting algorithm applied to the polarized reflectances provides retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, Rainbow Fourier Transform (RFT), which allows us to retrieve the droplet size distribution (DSD) itself. The latter is important in the case of clouds with complex structure, which results in multi-modal DSDs. During NAAMES the aircraft performed a number of flight patterns specifically designed for comparison of remote sensing retrievals and in situ measurements. These patterns consisted of two flight segments above the same straight ground track. One of these segments was flown above clouds allowing for remote sensing measurements, while the other was at the cloud top where cloud droplets were sampled. We compare the DSDs retrieved from the RSP data with in situ measurements made by the Cloud Droplet Probe (CDP). The comparisons show generally good agreement with deviations explainable by the position of the aircraft within cloud and by presence of additional cloud layers in RSP view that do not contribute to the in situ DSDs. In the latter case the distributions retrieved from the RSP data were consistent with the multi-layer cloud structures observed in the correlative High Spectral Resolution Lidar (HSRL) profiles. The comparison results provide a rare validation of polarimetric droplet size retrieval techniques, which can be used for analysis of satellite data on global scale.
We present comparisons of cloud droplet size distributions (DSDs) retrieved from the research scanning polarimeter (RSP) data with correlative in situ measurements made during the North Atlantic Aerosols and Marine Ecosystems Study (NAAMES). The airborne portion of this field experiment was based out of St. John's airport, Newfoundland, Canada with the focus of this paper being on the deployment in May - June 2016. RSP was onboard the NASA C-130 aircraft together with an array of in situ and other remote sensing instrumentation. The RSP is an along-track scanner measuring the polarized and total reflectance in 9 spectral channels. Its uniquely high angular resolution allows for characterization of liquid water droplet sizes using the rainbow structure observed in the polarized reflectance over the scattering angle range from 135 to 165.degrees The rainbow is dominated by single scattering of light by cloud droplets, so its structure is characteristic specifically of the droplet sizes at cloud top (within unit optical depth into the cloud, equivalent to approximately 50m). A parametric fitting algorithm applied to the polarized reflectance provides retrievals of the droplet effective radius and variance assuming a prescribed size distribution shape (gamma distribution). In addition to this, we use a non-parametric method, the Rainbow Fourier Transform (RFT), which allows us to retrieve the droplet size distribution itself. The latter is important in the case of clouds with complex microphysical structure, or multiple layers of cloud, which result in multi-modal DSDs. During NAAMES the aircraft performed a number of flight patterns specifically designed for comparisons between remote sensing retrievals and in situ measurements. These patterns consisted of two flight segments above the same straight ground track. One of these segments was flown above clouds allowing for remote sensing measurements, while the other was near the cloud top where cloud droplets were sampled. We compare the DSDs retrieved from the RSP data with in situ measurements made by the Cloud Droplet Probe (CDP). The comparisons generally show good agreement (better than 1 micron for effective radius and in most cases better than 0.02 for effective variance) with deviations explainable by the position of the aircraft within the cloud, or by the presence of additional cloud layers between the cloud being sampled by the in situ instrumentation and the altitude of the remote sensing segment. In the latter case, the multi-modal DSDs retrieved from the RSP data were consistent with the multi-layer cloud structures observed in the correlative High Spectral Resolution Lidar (HSRL) profiles. The results of these comparisons provide a rare validation of polarimetric droplet size retrieval techniques, demonstrating their accuracy and robustness and the potential of satellite data of this kind on a global scale.
The density function and corresponding moments for the ratio of correlated gamma distributed variates were derived. A class of bivariate gamma distributions was also considered, and additional distributional results which use this class of functions were also derived. Similar results, by using a different class of bivariate gamma distributions, are presented.