Exploration for Disorder

The assumption of lognormal (parent) field size distributions has for a long time been applied to resource appraisal and evaluation of exploration strategy by the petroleum industry. However, frequency distributions estimated with observed data and used to justify this hypotheses are conditional. Examination of various observed field size distributions across basins and over time shows that such distributions should be regarded as the end result of an economic filtering process. Commercial discoveries depend on oil and gas prices and field development costs. Some new fields are eliminated due to location, depths, or water depths. This filtering process is called economic truncation. Economic truncation may occur when predictions of a discovery process are passed through an economic appraisal model. We demonstrate that (1) economic resource appraisals, (2) forecasts of levels of petroleum industry activity, and (3) expected benefits of developing and implementing cost reducing technology are sensitive to assumptions made about the nature of that portion of (parent) field size distribution subject to economic truncation.

The problem of estimating the number of summands of random variables for a total normal distribution law or a sample average with a normal distribution is investigated. The Central limit theorem allows us to solve many complex applied problems using the developed mathematical apparatus of the normal probability distribution. Otherwise, we would have to operate with convolutions of distributions that are explicitly calculated in rare cases. The purpose of this paper is to theoretically estimate the number of terms of the Central limit theorem necessary for the sum or sample average to have a normal probability distribution law. The article proves two theorems and two consequences of them. The method of characteristic functions is used to prove theorems. The first theorem States the conditions under which the average sample of independent terms will have a normal distribution law with a given accuracy. The corollary of the first theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 1. The second theorem defines the normal distribution conditions for the average sample of independent random variables whose mathematical expectations fall in the same interval, and whose variances also fall in the same interval. The corollary of the second theorem determines the normal distribution for the sum of independent random variables under the conditions of theorem 2. According to the formula relations proved in theorem 1, a table of the required number of terms in the Central limit theorem is calculated to ensure the specified accuracy of approximation of the distribution of the values of the sample average to the normal distribution law. A graph of this dependence is constructed. The dependence is well approximated by a polynomial of the sixth degree. The relations and proved theorems obtained in the article are simple, from the point of view of calculations, and allow controlling the testing process for evaluating students ' knowledge. They make it possible to determine the number of experts when making collective decisions in the economy and organizational management systems, to conduct optimal selective quality control of products, to carry out the necessary number of observations and reasonable diagnostics in medicine.

Spatially and temporally extensive observations of the stratospheric aerosol cloud produced by the eruption of el Chichón in 1982 were made by the Solar Mesosphere Explorer satellite. Measurements of thermal emission at 6.8 μm are inverted to give aerosol extinction coefficients. At this wavelength the extinction coefficient is proportional to the cube of the particle radius, so the measured radiance is proportional to the slant column mass density of aerosols, independent of size distribution. Vertical column mass densities of aerosols and total aerosol mass are found from the 6.8‐μm data. The evolution of the aerosol cloud in time and space is discussed. A peak column mass density above 22 km of 0.43 g m−2 occurred near 20°N latitude 8 weeks after the eruption. A maximum total global aerosol burden above 22 km of 1.3×1013 g occurred a week later. The aerosol mass determinations are used in conjunction with observations of scattered sunlight from the aerosols at 1.27 and 1.87 μm to derive single‐mode log normal columnar size distributions for the aerosol cloud. The results are presented and the time evolution of the particle sizes is discussed.

A comprehensive sensitivity study has been made using Mie theory to determine the effect of realistic variations in values of real and imaginary parts of the complex index of refraction on volume extinction and absorption coefficients for a wide range of log normal particle size distributions (defined by geometric mean radius r(g) and geometric standard deviation sigma(g)). Wavelengths lambda from the visible (0.55 microm) through the middle ir (10.6 microm) were considered. Extinction is independent of the complex index to within 20% for the majority of realistic particle size distributions, providing lambda < 2 microm. However, changes in extinction by up to an order of magnitude are caused by realistic variations in refractive indexes for 2 microm </= lambda </= 10.6 microm, with the real index being more important in affecting extinction than the imaginary index. Similar changes are caused by variations in particle size distribution for values of refractive indexes typical of atmospheric constituents. For bimodal size distributions representative of desert aerosols, values of the complex refractive index that result in minimum and maximum extinction coefficients are given. Absorption is generally less dependent on size distribution than is extinction and is not, in general, linear with the imaginary index, especially for broad particle distributions.

Computations of Mie scattering by marine particles indicate that the ratios between the backward scattering coefficient and the scatterance at 120 degrees are ~6.9 +/- 0.4, 7.2 +/- 0.6, and 7.1 +/- 0.5 for Junge, normal and log-normal particle size distributions, respectively. In situ observations of the scattering confirm this result.

The available coefficient of friction (ACOF) for human locomotion is the maximum coefficient of friction that can be supported without a slip at the shoe and floor interface. A statistical model was introduced to estimate the probability of slip by comparing the ACOF with the required coefficient of friction, assuming that both coefficients have stochastic distributions. This paper presents an investigation of the stochastic distributions of the ACOF of quarry tiles under dry, water and glycerol conditions. One hundred friction measurements were performed on a walkway under the surface conditions of dry, water and 45% glycerol concentration. The Kolmogorov-Smirnov goodness-of-fit test was used to determine if the distribution of the ACOF was a good fit with the normal, log-normal and Weibull distributions. The results indicated that the ACOF appears to fit the normal and log-normal distributions better than the Weibull distribution for the water and glycerol conditions. However, no match was found between the distribution of ACOF under the dry condition and any of the three continuous distributions evaluated. Based on limited data, a normal distribution might be more appropriate due to its simplicity, practicality and familiarity among the three distributions evaluated.

Comparing computing formulas for estimating concentration ratios

Chapter 2 - Probability and Risk

We investigate stochastic microcracks of a heterogeneous microstructure as the primary mechanism determining the non‐shock ignition probability of octahydro‐1,3,5,7‐tetranitro‐1,2,3,5‐tetrazocine‐based polymer‐bonded explosives. To quantify the ignition probability, we modify the viscoelastic cracking constitutive model by considering randomly distributed microcracks and combine this model with the hot‐spot model and the Monte Carlo method. This method is validated by using a standard Steven test simulation, and we quantify how the microcrack stochasticity affects the ignition probability. The high‐temperature region clearly changes compared with the case of single microcrack size. The discrepancy in the mechanical response of each element due to the heterogeneous microcracks causes a shear deformation, which increases the temperature. The difference between a uniform distribution and a normal distribution in microcrack size is also discussed. For a given impact velocity, the ignition probability for the normal microcrack size distribution exceeds that for the uniform microcrack size distribution. Although the two distributions have the same mean and variance, the real range in microcrack size corresponding to the normal distribution exceeds that for the uniform distribution. The results show that the sample radius does not significantly affect the ignition probability, whereas the opposite is true for the sample thickness. The ignition‐probability curve is obtained in these cases. For sample dimensions of φ70 mm×13 mm, φ98 mm×13 mm, φ140 mm×13 mm, and φ98 mm×26 mm, the impact velocity thresholds corresponding to ignition probabilities in the range 0 %–99 % are 41.0–43.7, 41.0–44.3, 40.0–44.6, and 62.0–67.2 m/s, respectively, which is consistent with experimental results.

Simple fixed bed models such as the Bohart–Adams and Thomas equations are often used to describe the breakthrough characteristics of water and air contaminant adsorption in fixed bed adsorbers. However, these popular models are confined to correlating highly symmetric breakthrough curves. The present study explores the feasibility of using two probability distributions (normal and log-normal) to track asymmetric breakthrough curves for water and air contaminant adsorption in fixed bed columns (ciprofloxacin, ammonium, hydrogen chloride, and hydrogen sulfide). The normal and log-normal probability distributions provided accurate fits to the slightly asymmetric ciprofloxacin breakthrough curve. They also provided a good representation of the overall shape of the ammonium breakthrough curve, but failed to describe the leakage of ammonium during the initial period of column operation. The log-normal distribution was able to match the asymmetric HCI and H2S breakthrough curves. The normal distribution, by comparison, failed to describe these two asymmetric breakthrough curves. Because the log-normal distribution has a floating inflection point, it was very effective in describing the asymmetric HCI and H2S breakthrough curves that were sharp initially and then broadened significantly as the column approached saturation. The ability of the normal distribution to track such asymmetric breakthrough data was impeded by its invariant inflection point.

PNS212 Applying a Normal Distribution to Costs in Probabilistic Sensitivity Analysis – NOT Necessarily a Negative Outcome

This study used statistical simulations to investigate the performance of the population bioequivalence test applied to image-based particle size measurements (such as morphologically directed Raman spectroscopy) and methods for designing in vitro bioequivalence trials using prior information. Simulations of in vitro population bioequivalence trials were conducted across a range of representative D50 (number-weighted median particle diameter from a log-normal particle size distribution) and span (which is defined as [Formula: see text] where D90 and D10 are the number-weighted 90th and 10th percentiles in particle diameters sampled from a log-normal particle size distribution) values respectively. The performance of the population bioequivalence test in the simulations was driven by an interplay between overall test variability and the widening or narrowing of the bioequivalence region due to variance terms in the test statistic definition. These findings were dependent upon differences in the variability of D50 and span and may generalise to a wider range of in vitro metrics. Trial design optimisation using power and assurance approaches followed patterns consistent with these findings. As more novel scientific methods are applied to the development of complex generic drug products, the procedures outlined in this study may be used at the inception stage of future in vitro bioequivalence trials to reduce the risk of conducting costly trials with low probabilities of success.

The efficiency of cyclones varies widely with the size of particles and complex mathematic formulations are used for predicting the collection efficiency of a single particle size. Numerical integration of the latter, in conjunction with the appropriate in each case input particle size distribution, or direct use of nomographs if lognormal input particle size distribution can be assumed, yields the overall efficiency. In either case, no information is available regarding the size distribution of the penetrating particles. As shown in this paper, lognormal input particle size distributions yield lognormal output particle size distributions. Based on this and on the analysis of functional relations, a set of nomographs is developed directly yielding the mean diameter and standard deviation of penetrating particles as a function of the input particle size distribution parameters and key cyclone design and operating conditions. The above set the stage for the rigorous computation of the size-specific emissions from cyclones and the overall and size-specific efficiencies of secondary control systems that may follow. This analysis is in line with current trends, which, on the basis of health considerations, place emphasis on the size-specific rather than on the overall particle emissions.

Corrosion behavior of electrodeposited Ni with normal and bimodal grain size distribution

This chapter presents the foundations of two of the most important frequency distributions in environmental monitoring: normal and log-normal distributions. The main characteristics, properties, and parameters are presented, with indications on how to define and interpret them. Normal distributions are widely used and are associated with a symmetric interpretation of data distribution; however, this may not be the case with environmental monitoring data. Therefore, we provide conceptual background to think about data in a nonsymmetrical way, taking the log-normal distribution to represent this typical asymmetrical behaviour of environmental data. Chapters 9 and 10 will make use of the concepts presented here.