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The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
The maximum likelihood estimation is a general estimation procedure. It is often compared to estimation procedures like the ordinary least squares regression or generalized method of moments, to name a few. We discuss some basics about the maximum likelihood estimation, its advantages and disadvantages, and provide an example application to a gamma distribution function.
This report documents the quantitative results of the current industry-average performance for components and initiating events (IEs) at U.S. commercial nuclear power plants (NPPs). It represents the third update of the original analysis in NUREG/CR-6928 with data through 2020. Continuous characterization and updating of current industry-average performance with the latest industry data available are important steps in maintaining up-to-date risk models. Typically, data from 1998–2002 were used in NUREG/CR-6928, data from 1998–2010 in the first update, data from 1998–2015 in the second update, and data from 2006–2020 in this update, although many IEs required longer periods for adequate characterization of frequencies in all these analyses. As with NUREG/CR-6928 and previous updates, four types of events are covered in this report: component unreliability (e.g., a pump that fails to start or fails to run), component or train unavailability resulting from test or maintenance outages, special event probabilities covering operational issues (e.g., pump restarts and injection valve re-openings during unplanned demands), and IE frequencies. Results (in the form of beta distributions for failure probabilities upon demand and gamma distributions for rates) are used as inputs to the U.S. Nuclear Regulatory Commission standardized plant analysis risk models covering U.S. commercial NPPs.
Columnar aerosol size distributions have been inferred by numerically inverting particulate optical depth measurements as a function of wavelength. An inversion formula which explicitly includes the magnitude of the measurement variances is derived and applied to optical depth measurements obtained in Tucson with a solar radiometer. It is found that the individual size distributions of the aerosol particles (assumed spherical), at least for radii greater than or approximately equal to 0.1 micron, fall into one of three distinctly different categories. Approximately 50% of all distributions examined thus far can best be represented as a composite of a Junge distribution plus a distribution of relatively monodispersed larger particles centered at a radius of about 0.5 micron. Scarcely 20% of the distributions yielded Junge size distributions, while 30% yielded relatively monodispersed distributions of the log-normal or gamma distribution types. A representative selection of each of these types will be presented and discussed. The sensitivity of spectral attenuation measurements to the radii limits and refractive index assumed in the numerical inversion will also be addressed.
The modeling assumptions and governing equations required to estimate the heatshield surface recession due to dust particle impacts are presented. The dust particle size distribution can be estimated using a modified gamma distribution. The vertical distribution of dust particles in the atmosphere in terms of the ratio of dust particle to atmospheric density can be related to measured or computed values of the dust opacity. The different approaches to coupling the particle and fluid dynamics are discussed. The equations and modeling assumptions presented in this paper are applied to computing the surface recession due to dust particle impacts on the ExoMars Schiaparelli capsule entering the Martian atmosphere during a representative dust storm. The dust surface recession at the stagnation point of about 1 millimeter is about 40 percent of the recession value due to charring ablation.
In this study, a framework is discussed to apply air/space-borne dual-wavelength radar for the estimation of characteristic parameters of hydrometeors. The focus of our study is on the Global Precipitation Measurements (GPM) precipitation radar, a dual-wavelength radar that operates at Ku (13.8 GHz) and Ka (35 GHz) bands. As the droplet size distributions (DSD) of rain are expressed as the Gamma function, a procedure is described to derive the median volume diameter (D(sub 0)) and particle number concentration (N(sub T)) of rain. The correspondences of an important quantity of dual-wavelength radar, defined as deferential frequency ratio (DFR), to the D(sub 0) in the melting region are given as a function of the distance from the 0 C isotherm. A self-consistent iterative algorithm that shows a promising to account for rain attenuation of radar and infer the DSD without use of surface reference technique (SRT) is examined by applying it to the apparent radar reflectivity profiles simulated from the DSD model and then comparing the estimates with the model (true) results. For light to moderate rain the self-consistent rain profiling approach converges to unique and correct solutions only if the same shape factors of Gamma functions are used both to generate and retrieve the rain profiles, but does not converges to the true solutions if the DSD form is not chosen correctly. To further examine the dual-wavelength techniques, the self-consistent algorithm, along with forward and backward rain profiling algorithms, is then applied to the measurements taken from the 2nd generation Precipitation Radar (PR-2) built by Jet Propulsion Laboratory. It is found that rain profiles estimated from the forward and backward approaches are not sensitive to shape factor of DSD Gamma distribution, but the self-consistent method is.
Characteristics of the raindrop size distribution in seven tropical cyclones have been studied through impact-type disdrometer measurements at three different sites during the 2004-06 Atlantic hurricane seasons. One of the cyclones has been observed at two different sites. High concentrations of small and/or midsize drops were observed in the presence or absence of large drops. Even in the presence of large drops, the maximum drop diameter rarely exceeded 4 mm. These characteristics of raindrop size distribution were observed in all stages of tropical cyclones, unless the storm was in the extratropical stage where the tropical cyclone and a midlatitude frontal system had merged. The presence of relatively high concentrations of large drops in extratropical cyclones resembled the size distribution in continental thunderstorms. The integral rain parameters of drop concentration, liquid water content, and rain rate at fixed reflectivity were therefore lower in extratropical cyclones than in tropical cyclones. In tropical cyclones, at a disdrometercalculated reflectivity of 40 dBZ, the number concentration was 700 plus or minus 100 drops m(sup -3), while the liquid water content and rain rate were 0.90 plus or minus 0.05 g m(sup -3) and 18.5 plus or minus 0.5 mm h(sup -1), respectively. The mean mass diameter, on the other hand, was 1.67 plus or minus 0.3 mm. The comparison of raindrop size distributions between Atlantic tropical cyclones and storms that occurred in the central tropical Pacific island of Roi-Namur revealed that the number density is slightly shifted toward smaller drops, resulting in higher-integral rain parameters and lower mean mass and maximum drop diameters at the latter site. Considering parameterization of the raindrop size distribution in tropical cyclones, characteristics of the normalized gamma distribution parameters were examined with respect to reflectivity. The mean mass diameter increased rapidly with reflectivity, while the normalized intercept parameter had an increasing trend with reflectivity. The shape parameter, on the other hand, decreased in a reflectivity range from 10 to 20 dBZ and remained steady at higher reflectivities. Considering the repeatability of the characteristics of the raindrop size distribution, a second impact disdrometer that was located 5.3 km away from the primary site in Wallops Island, Virginia, had similar size spectra in selected tropical cyclones.
Spectral energy and angular distributions of gamma radiation through single and double layered shielding barriers
While the total cover of broken cloud fields can in principle be obtained from one-dimensional measurements, the cloud size distribution normally differs between two-dimensional (area) and one-dimensional retrieval (chord length) methods. In this study, we use output from high-resolution Large Eddy Simulations to generate a transfer function between the two. We retrieve chord lengths and areas for many clouds, and plot the one as a function of the other, and vice versa. We find that the cloud area distribution conditional on the chord length behaves like a gamma distribution with well-behaved parameters, with a mean μ=1.1L and a shape parameter β=L−0.645. Using this information, we are able to generate a transfer function that can adjust the chord length distribution so that it comes much closer to the cloud area distribution. Our transfer function improves the error in predicting the mean cloud size, and is performs without strong biases for smaller sample sizes. However, we find that the method is still has difficulties in accurately predicting the frequency of occurrence of the largest cloud sizes.
Large solar particle events (SPEs) present significant acute radiation risks to the crew members during extra-vehicular activities (EVAs) or in lightly shielded space vehicles for space missions beyond the protection of the Earth's magnetic field. Acute radiation sickness (ARS) can impair performance and result in failure of the mission. Improved forecasting capability and/or early-warning systems and proper shielding solutions are required to stay within NASA's short-term dose limits. Exactly how to make use of observations of SPEs for predicting occurrence and size is a great challenge, because SPE occurrences themselves are random in nature even though the expected frequency of SPEs is strongly influenced by the time position within the solar activity cycle. Therefore, we developed a probabilistic model approach, where a cumulative expected occurrence curve of SPEs for a typical solar cycle was formed from a non-homogeneous Poisson process model fitted to a database of proton fluence measurements of SPEs that occurred during the past 5 solar cycles (19 - 23) and those of large SPEs identified from impulsive nitrate enhancements in polar ice. From the fitted model, the expected frequency of SPEs was estimated at any given proton fluence threshold (Phi(sub E)) with energy (E) >30 MeV during a defined space mission period. Corresponding Phi(sub E) (E=30, 60, and 100 MeV) fluence distributions were simulated with a random draw from a gamma distribution, and applied for SPE ARS risk analysis for a specific mission period. It has been found that the accurate prediction of deep-seated organ doses was more precisely predicted at high energies, Phi(sub 100), than at lower energies such as Phi(sub 30) or Phi(sub 60), because of the high penetration depth of high energy protons. Estimates of ARS are then described for 90th and 95th percentile events for several mission lengths and for several likely organ dose-rates. The ability to accurately measure high energy protons (50-300 MeV) in real-time is shown to be a crucial issue for crew protection.
The intensity distributions of gamma-ray bursts observed by BATSE from 19 April 1991 to 19 September 1994 are presented. For this data set, (V/V(sub max)) is 0.329 +/- 0.011, which is 15.5 sigma away from the value of 0.5 expected for a homogeneous distribution. Standard cosmological model parameters are obtained by fitting the differentially binned peak flux distribution expressed in units of photons cm(exp -2) s(exp -1) in the energy range 50-300 keV. The value of z calculated for a peak flux of 1 photon cm(exp -2) s(exp -1) is 0.8 +/- 0.33. The procedures used to produce the peak flux data and C(sub p)/C(sub lim) data are presented. The differences between the two representations of burst intensity are emphasized so that researchers can determine which type of data is most appropriate for their studies. The sky sensitivity correction as a function of intensity for the peak flux data is also described.
Expressions are derived for higher-order skewness and excess coefficients using central moments and cumulants up to 8th order. These coefficients are then calculated for three probability distributions: (1) Log-normal, (2) Rice-Nakagami, and (3) Gamma distributions. Curves are given to shown the variation of skewness with excess coefficients for these distributions. These curves are independent of the particular distribution parameters. This method is useful for studying fluctuating phenomena, which obey non-Gaussian statistics.
The first problem addressed concerns passive-microwave rain retrievals. Most current approaches start by building off-line a cloud-model-derived database. Given data, the retrieval algorithms search the database for the microwave temperatures "closest" to the observed data, then after some fine-tuning (performed in different ways by different implementations) the rain is estimated to be that which corresponds to the selected (and fine-tuned) set of database temperatures. These approaches have three drawbacks: they cannot properly take into account the ambiguities which arise from the fact that several rain scenarios can produce the same observed temperatures; they are quite inefficient since they require manipulating a large database along with often complex "fine-tuning" procedures; and they cannot refine their estimates if additional data is available. This past year we have derived closed formulae relating observed microwave brightness temperatures, T(sub b), and the underlying rain rates, R: average T(sub b) =f (rain) and average rain = g (T(sub b)), along with the corresponding covariance matrices. These results are sufficient to describe the conditional probabilities p(R/T(sub b)) and p(T(sub b)/R) to second order. Progress has also been made towards deriving a robust description of the rain drop size distribution (DSD). The widespread approach consisting in parameterizing the DSD as a gamma-distribution in terms of the drop diameter D suffers from the facts that, in reality, the DSD is not a smooth function of D and that the largely arbitrary Gamma model imposes unintended behavior, which has implications on any quantities derived from the DSD model. We have therefore developed a non-parametric yet practical description of the DSD, which is particularly well-suited for use in remote-sensing applications. The diagram on the left shows a comparison between an actual DSD sample and the truncated non-parametric representation. One figure shows the relation between radar reflectivity and rain rate derived using this representation. Validation of the Tropical Rainfall Measuring Mission (TRMM) radar-radiometer combined R and DSD algorithm is underway. This algorithm was designed to make optimal use of the instantaneous reflectivity profiles measured by the TRMM radar and the microwave brightness temperatures measured by the TRMM passive radiometer. So far, it appears to be the most reliable TRMM rain algorithm.
The homogeneity of the burst source population in space with the log N(greater than P sub c)-log P sub c and log (dN/dP sub c)-log P sub c relations is studied on the basis of a sample of 225 observed gamma-ray bursts between September 1978 and July 1988 from the Pioneer Venus Orbiter. The relations N(greater than P sub c) is approximately equal to P sub c exp -1.5 and dN/dP sub c is approximately equal to P sub c exp -2.5 are expected if the gamma bursters are uniformly distributed in space. The Kolmogorov-Smirnov statistic gives a probability of 0.93 that the observed integral distribution N(greater than P sub c) is drawn from the expected uniform distribution P sub c exp -1.5. A least-squares 1 standard deviation fit to the observed differential distribution yields dN/dP sub c of about P sub c -2.42 +/- 0.07. Both are consistent with a uniform distribution in space of the parent population of burst sources.
The Burst and Transient Source Experiment (BATSE) on NASA's Compton Gamma-Ray Observatory has shown that the sources of gamma-ray bursts have an angular distribution consistent with isotropy, yet the bursts are spatially inhomogeneous. Detection of a statistically significant deviation from isotropy in the burst distribution would provide a clue to the distance of the burst sources. BATSE's ability to detect slight deviations of the angular distribution from isotropy is, in part, dependent on the accuracy of the computed burst locations. Using bursts whose locations have been determined accurately and independently, we estimate BATSE's location uncertainties and incorporate them into 3D Monte Carlo simulations of the burst distributions to show that, despite individual burst location uncertainties of about 5 deg, the ability of BATSE to place stringent limits on the anisotropy present in the angular distribution of the gamma-ray bursts is not significantly degraded. Observational effects of possible galactic halo distributions of the burst sources are noted.
Energy scale for pulse height distributions from gamma ray spectrometers based on inorganic scintillators
Marine boundary-layer clouds cover large regions of the globe and are known to strongly influence radiative balances. The microphysical properties and persistence of these clouds are tightly coupled with cloud-top entrainment and aerosol properties within both the boundary layer and the overlying free troposphere. This work addressed the microphysical response to entrainment and aerosol properties in marine stratocumulus clouds, using data from the Aerosol and Cloud Experiments in the Eastern North Atlantic (ACE-ENA) project. Specifically, emphasis was placed on 1. extensive in situ measurements taken with the Atmospheric Radiation Measurement (ARM) G1 aircraft using the Holographic Detector for Clouds (Holodec) instrument; 2. airborne measurements obtained with the helicopter-borne Airborne Cloud Turbulence Observation System (ACTOS) operated by the Leibniz Institute for Tropospheric Research (TROPOS) during the first phase of ACE-ENA; and 3. above-boundary-layer, mountain-top measurements taken at the Pico Mountain Observatory (OMP) by the research group of the co-investigator and scientists from TROPOS during the first phase of ACE-ENA. The work is aligned with the topic “Warm Boundary-Layer Atmospheric Processes.” Key results include the following. Data from the Holographic Detector for Clouds (Holodec) were reprocessed, refined, and validated through a careful instrument intercomparison, resulting in a high-quality dataset available on the ARM archive for ACE-ENA. High-resolution ACTOS measurements of thermodynamics, microphysical, and turbulence properties were analyzed to explore the behavior of the entrainment velocity at cloud top. Analysis of the vertical variability of cloud droplet size distribution shape suggests that mixing is more inhomogeneous near cloud top, and more homogeneous deeper into the cloud. Analysis of aerosol measurements from OMP explored the mixing state and cloud condensation nucleus properties, as well as the implications for radiative forcing. Finally, a machine learning algorithm was developed for identifying characteristic cloud droplet size distributions, and was employed to explore microphysical regimes in stratocumulus clouds observed during ACE-ENA. The characteristic size distributions are narrow, and only when spatially averaged do they produce the broad “gamma” distributions typically assumed in models. This implies that precipitation development should account for variability and correlations in the distribution shape as well as the number concentration of cloud droplets.
A fully calibrated low profile 2D video disdrometer (2DVD) has been recording many different rainfall events in Northern Alabama (USA) since June 2007. An earlier publication reported drop shapes and axis ratio distributions determined for some of the events. For one of the cases examined, a noticeable shift in the 3.5 - 3.75 mm drop axis ratio distribution was noted. In this paper, we extend the earlier work by separating the 2DVD measurements into stratiform and convective rain. The separation is made possible by using the minute-by-minute drop size distribution (DSD) measured by the 2DVD. The 1-minute DSDs are fitted to a gamma distribution, and using a simple indexing technique which involves two of the fitted parameters, periods of convective and stratiform rain are separated for a given event. The output of the DSD indexing technique is qualitatively confirmed by comparing with simultaneous time series observations from a co-located UHF profiler which continuously records height profiles of reflectivity, Doppler mean and spectral width, all of which enable the identification of bright-band periods and, furthermore, periods of moderate and deep convection. Excellent consistency is found between the output of the DSD-based separation method and the profiler observations. Next, we utilize the output of DSD index-based separation method to flag the periods of severe convection for a given event. Drop axis ratios during the flagged periods are derived and compared with those during stratiform rain periods. Five cases have been considered. Axis ratio distributions do not show appreciable differences between stratiform and convective periods for four of the cases. The fifth case (the same case as reported earlier) shows a shift in the 3.5 - 3.75 mm drop axis ratios during a prolonged period of convection. The contoured shapes for these drops determined from the 2DVD camera data indicate the possibility of non-axisymmetric oscillations, compared with the contoured images for other events which fit well to our reference drop shapes. For all of above cases, observations from a C-band polarimetric radar - situated 15 km away are examined. The variations between the co-polar radar reflectivity and the differential reflectivity as well as the specific differential phase are compared with the 2DVD data based scattering calculations for the 5 events. The implications will be discussed.