Gamma Probability Density Function Research Articles

The dynamics of concentration fluctuations within passive scalar plumes in a turbulent neutral boundary layer

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

A new mathematical solution of convection‐dispersion equation to describe solute transport in heterogeneous soils

AbstractThere was a long‐time debate about the validation of the convection‐dispersion equation (CDE) and its replacement with the well‐known convective lognormal transfer function model (CLT) to describe solute transport in a heterogeneous soil with uniformity at the longitudinal water flow direction and nonuniformity at the transverse direction. The objective of this study is to prove that the CDE is valid and almost identical to the CLT. Gamma probability density function (pdf) was initially assumed in this study to describe the distribution of pore‐water velocity across capillary tubes in a heterogeneous soil. The capillary bundle model was used to describe solute transport without transverse solute mixing between adjacent tubes. The inverse‐gamma function, a new mathematical solution of the CDE differential equation with scale‐dependent dispersivity, was initially derived from the capillary bundle model and the gamma pdf. The only difference between the inverse‐gamma function and the CLT is that lognormal pdf of pore‐water velocity is assumed in the CLT while the two pdfs are close to each other. The inverse‐gamma function and the CLT were tested with the published data from the miscible displacement experiments on the two repacked soils with different aggregate sizes. Results show that both the inverse‐gamma function and the CLT fit the measured breakthrough curves in the miscible displacement experiments. The estimates of the squared coefficient of variation of the pore‐water velocity in the gamma pdf were 0.314 and 0.0582 for the two soils, and they were consistent with the lognormal pdf in the CLT.

Performance Evaluation of UOWC Systems from an Empirical Channel Model Approach for Air Bubble-Induced Scattering.

Underwater optical wireless communication (UOWC) systems provide the potential to establish secure high-data-rate communication links in underwater environments. The uniqueness of oceanic impairments, such as absorption, scattering, oceanic turbulence, and air bubbles demands accurate statistical channel models based on empirical measurements for the development of UOWC systems adapted to different types of water and link conditions. Recently, generalized Gamma and a mixture of two generalized Gamma probability density functions (PDF) were proposed to describe the statistical behavior of small and large air bubbles, respectively, when considering several levels of particle-induced scattering. In this paper, we derive novel closed-form analytic expressions to compute the bit error rate (BER) and outage performance using both proposed PDFs for various scattering conditions. Furthermore, simple asymptotic expressions are obtained to determine the diversity order of each scenario. Monte Carlo simulation results verify the obtained theoretical expressions. Our results also reveal that UOWC systems present lower BER and outage performance under more turbid water cases with respect to the tap water case due to the higher diversity order and despite the significant increases in pathloss at short link distances. Particle-induced scattering provides an inherent mechanism of turbid waters to mitigate air bubble-induced fluctuations and light blockages.

Long-term trends and probability distribution functions of air pollutant concentrations in the megacity of São Paulo

Air quality conditions in many urban areas have improved in the last decades as a consequence of air pollution control policies and regulations. Mitigation strategies have been successful in reducing the concentration of primary pollutants like inhalable particulate matter (PM10), but the control of secondary pollutants like tropospheric ozone (O3) is still challenging in megacities like the Metropolitan Area of São Paulo (MASP) in Brazil. To support the development of effective mitigation strategies, it is crucial to characterize the statistical behavior of air pollutant concentrations and its long-term evolution. Probability Density Functions (PDF) can be useful to model site-specific air quality conditions, providing estimates for the frequency of extreme pollution events and exceedance of air quality standards. The current study aims to characterize which PDF model better fits and characterizes the variability of PM10 and O3 concentrations in the MASP. For that, daily maximum moving average concentrations were analyzed between 2000 and 2023, characterizing the long-term trends and the frequency of exceedance of air quality standards. PM10 concentrations followed a lognormal PDF, with an expected value of 31 ± 15 µg.m-3. O3 followed a Gamma PDF, with an expected value of 68 ± 25 µg.m-3. A consistent long-term decrease was observed for PM10 (-1.04 ± 0.09 µg.m-3.yr-1), while O3 showed an increasing trend of 0.51 ± 0.04 µg.m-3.yr-1 in the summer. In recent years (2021-2023), the probability of exceedance of the World Health Organization guideline was 17.4 and 11.0%, respectively, for PM10 and O3. In 2020, a statistically significant increase in O3 expected values was observed, possibly associated with changes in the emission patterns of precursors due to the mobility restrictions imposed by the COVID-19 pandemic.

Randomness of optical turbulence generated by Rayleigh–Bénard convection using intensity statistics

The experimental study of optical turbulence proves difficult due to challenges in generating controllable conditions in a laboratory environment. Confined water tanks that produce Rayleigh–Bénard (RB) convection are one method to generate optical turbulence using a controllable temperature gradient. It is of utmost concern to quantify the properties of the optical turbulence generated for characterization of other optical applications such as imaging, sensing, or communications. In this experimental study a Gaussian beam is propagated through a RB water tank where two intensity measurements are made at the receiver’s pupil and focal plane. The pupil and focal plane results include quantification of the intensity fluctuation distribution, scintillation distribution, and refractive index structure constant at various values of the temperature gradient. The angle of arrival fluctuations is also calculated at the focal plane to obtain a second estimate of C n 2. The pupil plane estimate for C n 2 using scintillation index and focal plane angle of arrival fluctuations is compared to preliminary predictions of C n 2 as a function of RB temperature gradient showing C n 2∼ΔT4/3. The outcomes of the study confirm that the RB process produces intensity fluctuations that follow gamma–gamma and log-normal probability density functions. Estimates of the refractive index structure constant C n 2 produce the same trends with different magnitudes when measured from the pupil and focal plane.

Static and dynamic responses to hyperoxia of normal placenta across gestation with T2*-weighted MRI sequences.

T2*-weighted magnetic resonance imaging (MRI) sequences have been identified as non-invasive tools with which to study placental oxygenation in vivo. This study aimed to use these to investigate both static and dynamic responses to hyperoxia of the normal placenta across gestation. We conducted a single-center prospective study including 52 uncomplicated pregnancies. Two T2*-weighted sequences (T2* relaxometry) were performed, one before and one after maternal hyperoxia. The distribution of placental T2* values was modeled by fitting a gamma probability density function (T2* ), describing the structure of the histogram using the mean T2* value, the shape parameter (α) and the rate (β). A dynamic acquisition (blood-oxygen-level-dependent (BOLD) MRI) was also performed before and during maternal oxygen supply, until placental oxygen saturation had been achieved. The signal change over time was modeled using a sigmoid function, to determine the intensity of enhancement (ΔBOLD (% with respect to baseline)), a temporal variation coefficient (λ (min-1), controlling the slope of the curve) and the maximum steepness (Vmax (% of placental enhancement/min)). The histogram analysis of the T2* values in normoxia showed a whole-placenta variation, with a decreasing linear trend in the mean T2* value (Pearson's correlation coefficient (R) = -0.83 (95% CI, -0.9 to -0.71), P < 0.001), along with an increasingly peaked and narrower distribution of T2* values with advancing gestation. After maternal hyperoxia, the mean T2* ratios (mean T2*hyperoxia/mean T2*baseline) were positively correlated with gestational age, while the other histogram parameters remained stable, suggesting a translation of the histogram towards higher values with a similar appearance after maternal hyperoxia. ΔBOLD showed a non-linear increase across gestation. Conversely, λ showed an inverted trend across gestation, with a weaker correlation (R = -0.33 (95% CI, -0.58 to -0.02), P = 0.04, R2 = 0.1). As a combination of ΔBOLD and λ, the changes in Vmax throughout gestation were influenced mainly by the changes in ΔBOLD and showed a positive non-linear correlation with gestational age. Our results suggest that the decrease in the T2* placental signal as gestation progresses does not reflect placental dysfunction. The BOLD dynamic signal change is representative of a free-diffusion model of oxygenation and highlights the increasing differences in oxygen saturation between mother and fetus as gestation progresses (ΔBOLD) and in the placental permeability to oxygen (λ). © 2024 International Society of Ultrasound in Obstetrics and Gynecology.

Performance Analysis of Coherent Source SAC OCDMA in Free Space Optical Communication Systems

In this paper, we investigate the performance of spectral amplitude coding optical code division multiple access (SAC OCDMA) systems under the effect of beat noise and turbulence. Three different multi-laser source configurations are considered in this analysis: shared multi-laser, separate multi-laser, and carefully controlled center frequency separate multi-laser. We demonstrate through Monte Carlo simulation that the gamma–gamma probability density function (pdf) cannot adequately approximate the measured intensity of overlapping lasers and that an empirical pdf is required. Results also show it is possible to achieve error-free transmission at a symmetrical data rate of 10 Gbps for all active users when only beat noise is taken into account by precisely controlling the center frequencies. However, only 30% of the active users can be supported when both beat noise and turbulence are considered.

On DCT-based MMSE estimation of short time spectral amplitude for single-channel speech enhancement

This paper proposes Discrete Cosine Transform (DCT) based speech enhancement algorithms. These algorithms utilize minimum mean square error (MMSE) estimator of clean short-time spectral amplitude, which respectively uses Gaussian, Laplace and Gamma probability density functions (PDF) as speech priors. We consider the noise process is additive and Gaussian. The proposed estimators are closed-form solutions, whereas the conventional Discrete Fourier Transform (DFT) based estimators derived under super-Gaussian speech priors have no closed-form solutions. We also examine the estimators with the Speech Presence Uncertainty (SPU) that addresses the speech or silence problem with probability. Compared to the alternative approaches, such as the Ephraim and Malah or the Erkelens et.al MMSE-STSA estimators, the proposed methods demonstrate superior performance in terms of Segmental SNR (SegSNR), Perceptual Evaluation of Speech Quality (PESQ), short-time objective intelligibility measure (STOI), and mean subjective preference score, while exhibiting an equal or lower complexity.

Trade-offs in the performance of a Free Space Optical communication system employing DCO-OFDM transmission with a PAPR-dependent DC-bias

Free Space Optical (FSO) communication technology has gained popularity because of its high bandwidth, which supports high-speed data transmission. Despite its merits over Radio Frequency technology, the FSO system performance is restricted by the combined impact of beam divergence, beam scintillation, and beam-wander effects triggered by atmospheric turbulence. A Gamma–Gamma probability density function (PDF) channel model is employed to model the irradiance fluctuations due to turbulence. In this paper, we evaluate the performance metrics of a FSO link employing the Direct Current-biased Optical Orthogonal Frequency Division Multiplexing (DCO-OFDM) modulation technique for different turbulent conditions. In our paper, we determine the DC-bias, which is proportional to peak-to-average-power ratio (PAPR) and a preclipping factor of nclip (<1) in order to reduce the transmitted optical power of the DCO-OFDM signal. We evaluate the average bit error rate (BER) and outage probability of the FSO link by considering the impact of system design parameters related to the OFDM signal, laser beam, and atmospheric turbulence on the received signal. Numerical simulation results show that, depending on the chosen values of PAPR and preclipping, an FSO link can experience either a power saving or a power penalty, for meeting the BER requirement. It is further demonstrated that trade-offs in performance at various levels exist between data transmission rate (QAM size), system power efficiency (preclipping), signal-PAPR and link-span to ensure target signal quality (BER and Outage). Under turbulence, the link with a longer span not only incurs a power penalty but also suffers from a reduced operating range by 3–4 dB.

Estimation of Wind Speed Probability Distribution Parameters by Using Four Different Metaheuristic Algorithms

The inclusion of energy produced from renewable energy sources (RES) such as solar and wind energy into existing energy systems is important to reduce carbon emissions, air pollution and climate change, and to ensure sustainable development. However, the integration of RES into the energy system is quite difficult due to their highly uncertain and intermittent nature. In this study, considering three different probability density functions in total, the scale and shape parameters of the Weibull probability density function (PDF), the scale parameter of the Rayleigh PDF, and the scale and shape parameters of the Gamma PDF were estimated for the wind speed data obtained from urban stations located in Istanbul by using the four different metaheuristic algorithms, namely Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO) and Grey Wolf Optimization (GWO) algorithms. Calculating the mean absolute error (MAE), root mean squared error (RMSE), and R2 values for each PDF at each station, the PDF that characterizes the wind speed probability distribution the best was identified.

Modeling and Parameter Estimation of Radar Sea-Clutter with Trimodal Gamma Population

Real radar data often consist of a mixture of Gaussian and non-Gaussian clutter. Such a situation creates one or more inflexion points in the curve of the empirical cumulative distributed function (CDF). In order to obtain an accurate fit with sea reverberation data, we propose, in this paper, a trimodal gamma disturbance model and two parameter estimators. The non-linear least-squares (NLS) fit approach is used to avoid computational issues associated with the maximum likelihood estimator (MLE) and moments-based estimator for parameters of the mixture model. For this purpose, a combination of moment fit and complementary CDF (CCDF) NLS fit methods is proposed. The simplex minimization algorithm is used to simultaneously obtain all parameters of the model. In the case of a single gamma probability density function, a zlog(z) method is derived. Firstly, simulated life tests based on a gamma population with different shape parameter values are worked out. Then, numerical illustrations show that both MLE and zlog(z) methods produce closer results. The proposed trimodal gamma distribution with moments NLS fit and CCDF NLS fit estimators is validated to be in qualitative agreement with different cell resolutions of the available IPIX database.

HONEY BEE CLEANSING FLIGHTS…JUST CLEANSING?

The first major cleansing flight during the winter in a honey bee apiary was witnessed on February 6, 2021. A light snowfall and freezing rain prior to the flight allowed delineation of the distribution of distances from the aggregation of hives of fecal depositions and immobilized honey bees fallen to the ground. Five transects were surveyed radiating from the aggregation of four overwintered hives to collect the data. The frequency distributions of these events were fit to Gamma probability density functions and then a test of the following hypothesis was constructed. Are cleansing flights solely comprised of local defecation by workers and then return to the hive? Or do workers also commence foraging in suboptimal air temperatures for flight? We found evidence that cleansing flights are comprised of two behaviors, local deposition of feces and long-distance foraging. This is based upon a significant difference (P < 0.001) between the mean distance from the hives of fecal deposition (3.01 ± 0.36(se)) and immobilized bees on the ground (9.49 ± 1.21(se)).

Probability density function estimation for filament creation in lossy, turbulent, nonlinear media.

Optical Kerr effects induced by the propagation of high peak-power laser beams through real atmospheres have been a topic of interest to the nonlinear optics community for several decades. Previous work has focused on estimating the Filamentation Onset Distance (FOD) in real atmospheres but not its statistical variance. This paper describes two ad hoc engineering models for predicting the FOD Probability Density Function (PDF) for lossy, turbulent, nonlinear media. Specifically, these models characterize the FOD variation with turbulence. One model uses a log-normal PDF with mean and variance proportional to the Rytov Variance. The other uses a gamma PDF employing the same mean and variance equations. These two PDFs will be compared to previous computer simulation results. Both show reasonable agreement between PDF predictions and computer simulation results for long-range filamentation. In fact, both give similar results, and there is no preference given to the data comparisons presented.

Form of Distribution of Dendro/Morphometric Variables for Brazilian Pine in Southern Brazil

The form of distribution found for the dendro/morphometric variables determines the structure, stability, productivity of forest stands, being a tool to propose silvicultural interventions, management, conservation of species, and dynamics of this environment. Thus, this study evaluates, using probability density functions (pdf), the form of distribution of these variables for araucaria in five sites in southern Brazil, aiming to establish the dynamics and identify the existence of a standard&amp;mdash;or the lack thereof&amp;mdash;to propose the need for silvicultural interventions to conserve the species and the future forest structure. The Normal, Log-Normal, Weibull and Gamma probability density functions were tested. Results show no significant changes in the shape and dimension in the forest structure dynamics, but a period of stability in the pattern of dendro/morphometric values, resulting from the stagnation of the values of the variables, non-intervention in the forest, relationship with the site, density, competition, and position of the tree in the forest stratum, which compromises the future structure of this forest typology. The study proves that the distribution probability of the variables can be used in management for species conservation and future structure development, as this influences the growth dynamics and processes, resource availability, and the stability, diversity, vitality, and productivity of the species.

Growing space in a 35-year-old Pinus taeda stand

Considering the economic importance of the Pinus species in Brazil, especially in the southern region, this study was carried out to fill some gaps and assist in the production processes of this species. The specific objective was to evaluate the influence of different growing space regimes in experimental plantations of Pinus taeda L., aged 35 years, without thinning, located in the state of Santa Catarina. Diameter at breast height (DBH) and total height data were collected, and the Gamma and Weibull (3P) probability density functions (PDF) were used to describe the structure of the diameter and height variables, respectively. Then multivariate analysis and discriminant functions were applied to identify the correlation of these variables. For qualitative variables, tree health was assessed in three classifications: 1 - Healthy, with no apparent damage; 2 - Initial stage of deterioration by pests and/or diseases; and 3 - Advanced stage of deterioration by pests and/or diseases. The trunk quality variable was also evaluated in three classifications: 1 - Trunk without apparent defects; 2 - Slightly crooked trunk, however cylindrical and devoid of considerable ramifications; and 3 - Trunk with strong tortuosity or bifurcation. The results showed that growing space directly affected the average diameter. Stands with greater space per tree maximized the variables when considering height and DBH together. Stands with less growing space, on the other hand, favored better trunk quality, and the dominant individuals in height had superior health and trunk quality compared to the others.

Spatial distribution of fungi from the analysis of aerobiological data with a gamma function

Fungi are an important component of ecosystems. Some fungi are widely distributed, while others are limited to certain habitats. Studies based on airborne fungal spores can help to know the geographical distribution of fungi in the territory. Our aim was to show that a gamma probability density function (gpdf) based on a database of 20 airborne fungal spore taxa concentrations in eight localities of Catalonia (NE Spain) for a period of 20 years was a useful tool to map the distribution of these taxa in this region, as well as to establish a general classification on their sporulation through the alpha parameter of the validated model. This allows a more efficient study of the atmospheric dynamics of the different taxa, since the number of taxa is reduced to a representative taxon for each of the categories of the generic classification. In general, the results obtained confirmed that the scale parameter of the gamma distribution changes from year to year, depending on the meteorological conditions, while the shape parameter remains fairly stable. At the temporal scale, airborne fungal spores of Agrocybe sp. showed the highest stability; at the spatial scale, Cladosporium sp. showed the highest stability. Regarding localities, Girona was the station with greater interannual variation, while Barcelona and Vielha showed the lowest. In addition, the results obtained allowed a non-subjective classification of these taxa in five groups, based on the gamma (shape) parameter. The taxa Alternaria sp., Cladosporium sp., Ganoderma sp., Pleospora sp., Leptosphaeria sp., Aspergillus sp.-Penicillium sp. were cosmopolitan and showed a similar behavior across the whole study area, with any of them possible candidates for used in predictive models; airborne fungal spores of Agrocybe sp., Arthrinium sp., Epicoccum sp., Drechslera sp.–Helminthosporium sp., Pithomyces sp., Thelephoraceae, Stemphylium sp., Xylariaceae can be used as meteorological indicators and Agaricus sp., Coprinaceae sp., Torula sp. can be used as indicators of anthropogenic activities. The results obtained could be used to reduce the number of spore taxa analyzed and subsequently develop generic predictive models.

Analysing Outage Probability and Diversity for Matrix Based Shadowed Attenuation in Multipath Fading Environment

In this research outage probability is analysed for a proposed matrix based method using selection combining diversity under shadowed attenuation in a typical macrocell environment. The conventional Lognormal shadowing model approximates the gamma probability density function which is used in proposed matrix based method. The closed form expressions are developed for evaluating the probability of outage in multipath fading environment with the help of shadow matrix, shape, and scale factors. The graphical results using proposed method are validated by comparing them with the Lognormal method. The presented results have been investigated by varying threshold in terms of shadow based attenuation. The results obtained using proposed method is compared with the Lognormal model where shadowing spread is varied. It is clear from the plots that the outage probability using proposed model shows better results the outage probability as compared to Lognormal shadowed model.

Non-conservative kinetic model of wealth exchange with saving of production

In this paper we propose a non-conservative kinetic model of wealth exchange with saving of production as an extension of the Chakraborti–Chakrabarti model of money exchange. Using microeconomic arguments, we achieve rules of interaction between economic agents that depend on two exogenous parameters, the exchange aversion of the agents (λ) and the saving of production (s), such that in the limit s = 0, these rules can be reduced to the ones of the Chakraborti–Chakrabarti model. The non-conservative dynamics are approached analytically through a mean field approximation and the Boltzmann kinetic equation. Both approximations allow us to compute a theoretical rate of exponential growth (g) and to fit the emergent distributions of wealth to a gamma probability density function, in such a way that g, the fit parameters and the Gini index can be expressed analytically in terms of λ and s. In general, the emergent distributions do not reach a stationary state, however it is possible to study the emergence of self-similar distributions that hold the gamma pattern and maximize the Shannon entropy. With the purpose of addressing labor income, we explore additionally the effect of salary income in the model by defining a two-class structure where population is separated into workers and producers. This assumption leads to an emergent rate of economic growth ge. The macroeconomic implications of this model are studied by means of the wealth/income ratio, which can be predicted as s∕ge, in accordance with the Solow model of economic growth. The results in this paper allow to tie some of the important facts of the modern economic speech, as well as the microeconomic theory, with some methods and ideas developed in the context of non-conservative exchange models.

Performance assessment of hybrid PPM–BPSK–SIM based FSO communication system using time and wavelength diversity under variant atmospheric turbulence

In this paper, free space optical communication system employing a hybrid modulation technique called pulse position modulation–binary phase shift keying–subcarrier intensity modulation technique, is investigated using wavelength and time diversity, by considering log-normal and gamma–gamma probability density function as the channel-fading statistics under weak, moderate and strong turbulence. The analytical average bit error rate and closed form of expression of the proposed hybrid modulation scheme are derived. The outcomes of the simulation demonstrate the outstanding consistency of the derived closed form expression with those achieved by the Gauss–Hermite approximation and the Monte Carlo simulations.

Risk of Occurrence of Water Deficit in Soybean Cultivated in Lowland Soils

AbstractThe occurrence of water deficit is intensified in lowland soils. Generating information with regard to its risk of occurrence is essential to avoid seed yield losses. The objective of this study was to determine the probability of water deficit in soybean cultivated in lowlands of the Vacacaí and Piratini River basins in the southern portion of Rio Grande do Sul in Brazil as a function of the sowing date. Soybean development was simulated considering three sets of cultivars of relative maturity groups (RMG) delimited by 5.9–6.8, 6.9–7.3, and 7.4–8.0, with a 10-day interval between the sowing dates making up the period between 21 September and 31 December. Daily meteorological data were used from 1971 to 2017 obtained from the Pelotas meteorological station and from 1968 to 2017 from the Santa Maria meteorological station. Water deficit (mm) in the subperiods and soybean development cycle was obtained from the calculation of evapotranspiration and daily sequential crop water balance. Data of water deficit were subjected to a probability distribution analysis, in which the exponential, gamma, lognormal, normal, and Weibull probability density function (pdf) adjustments were tested using chi-square and Kolmogorov–Smirnov adhesion tests, with a 10% significance level. The water deficit is lower in the Pelotas region than in Santa Maria. Sowings performed from 11 and 1 November present the lowest risk of occurrence of water deficit throughout the soybean cycle in Santa Maria and Pelotas, respectively. Risk of water deficit decreases for the beginning of flowering–beginning of seed (R1–R5) subperiod when soybean sowing occurs from the beginning of November.