Three-parameter Gamma Distribution Research Articles

Forecasting the design maxima water discharges of floods on the Latorica river according to the data of the Mukachevo gauging station using plotting position formulas

This article presents the results of forecasting design maxima discharges on the Latorica River within Mukachevo town based on hydrological observation data at the “Mukachevo” gauging station using plotting position formulas. While solving the task, a novel non-parametric method of forecasting using observation data is applied. The method includes extrapolating the discrepancy (divergence, disagreement) between the estimates of the statistical annual probabilities of exceedance obtained by different plotting position formulas. The task is considered in the frame of the stationarity hypothesis of the maximum river flow employing a time series of maximal discharges of the Latorica River observed at the “Mukachevo” gauging station from 1947 to 1999.We involved the thirteen plotting position formulas. There was no specific criterion for choosing them to solve the task. All applied formulas were considered admissible options, and results obtained after using them – expert judgments reflecting decision-makers’ predisposition to more cautious or less expensive decision options in flood management strategies.The epistemic uncertainty of the different plotting positions was reduced by employing the Fishburn rule. According to this rule, the significance of various plotting positions was given by arranging their estimates in descending order of importance of their values under decision-making. Depending on the selected significance option assignment of the different plotting position formulas, such rank-weighted estimates of the design peak discharges (each of them for annual exceedance probability 1%, 0.5%, and 0.2%) were computed: (1) the rank-weighted upper bound estimate (sup-estimate) corresponding to the predisposition to more cautious decision options; (2) the rank-weighted lower bound estimate (inf-estimate) corresponding to the predisposition to less expensive decision options. As possible control theoretical alternatives for forecasting design maximal discharges considered were five parametric probability distributions: 1) the Kritskyi-Menkel three-parameter gamma distribution; 2) Pearson’s type III distribution; 3) the Extreme value type I distribution (Gumbell’s type I distribution); 4) the Logarithmic Pearson type III distribution; and 5) the Two-parameters logarithmic-normal distribution. The population statistical parameters for these parametric probability distributions were estimated from the sample statistics by the method of moments.

A new reduced contagious zero-inflated model: An application to count data

In this paper, a reduced one-parameter contagious distribution was developed from the joint distribution of three-parameter gamma and Poisson distributions, on which Lakshmi’s three-parameter gamma distribution is based to model a count data. The distribution properties and some common descriptive measures relating to this contagious distribution are derived. The behavior of the probability mass function with changes in parameters was also studied. The parameter estimation by the maximum likelihood and moment-generating function methods is discussed. A simulation study was carried out with the proposed model to check for consistency and bias. The new model show consistency as the sample size increases. The model was applied to a real-life dataset and was seen to be more flexible in capturing excess zero, under, and over-dispersion in count data and proved to be a useful alternative to some existing zero-inflated models.

Newly Developed Multiparameter Bulk Cloud Schemes. Part I: A New Triple-Moment Condensation Scheme and Tests

Abstract Double-moment schemes cannot accurately describe the evolution of the cloud droplet spectrum during condensation. Hence, a new triple-moment condensation scheme is developed to describe the evolution of cloud droplet spectra. In this scheme, a three-parameter gamma distribution function of the cloud droplet mass is adopted, and the prognostic equations of the spectral shape parameter and slope parameter are derived by means of the number concentration, cloud water content, and reflectivity factor of cloud droplets. The new parameterization scheme is compared with high-resolution Lagrangian and Eulerian bin schemes, double-moment schemes, and existing triple-moment schemes by performing simulations under different supersaturation values. The new scheme can reduce the cloud spectral error in the cloud water content and reflectivity factor caused by the fixed shape parameter in some bulk schemes. The spectra simulated with the new scheme match the Lagrangian analytical solutions well, with errors within approximately 1% in the cloud water content and reflectivity factor. The effects of curvature and solution on condensation growth are also tested using the new scheme, and a method of using multiple gamma distribution functions to characterize the multimodal spectrum of cloud droplets is proposed in the new condensation scheme. Ultimately, the formation of rain embryos from giant aerosols can be simulated via the new scheme.

Statistical Characteristics of Warm Season Raindrop Size Distribution in the Beibu Gulf, South China

Raindrop size distribution (DSD) can be used to improve the accuracy of radar quantitative precipitation estimation (QPE) and further understand the microphysical process of precipitation; however, its spatio-temporal characteristics vary with different climates, rain types, and geographical locations. Due to the lack of observations, the DSD characteristics in the Beibu Gulf, especially at the rainfall center of Guangxi in South China, is poorly understood. In this paper, these regional DSD characteristics were analyzed during the warm season with an upgraded version of the OTT Particle Size Velocity (Parsivel) (OTT2) disdrometer. The DSD datasets from June to October 2020 and March to May 2021 were grouped into convective and stratiform precipitation by rain rate (R). The rainfall parameters were calculated from DSDs to further understand the rain characteristics. The results showed that: (1) the regional DSDs feature the lowest concentration of largest-sized drops when compared with the statistical results for other areas such as Zhuhai in South China, Nanjing in East China, Hubei province in Central China and Beijing in North China; (2) the raindrop spectra have an excellent fit with the three-parameter gamma distribution, particularly in regard to the medium-size raindrops; (3) the μ–Λ relation is closer to the coastal regions than the inland area of South China; (4) the localized Z−R relations differ greatly for convective rainfall (Z = 202.542 R1.553) and stratiform rainfall (Z = 328.793 R1.363). This study is the first study on DSDs in the Beibu Gulf region. The above findings will provide a better understanding of the microphysical nature of surface precipitation for different rain types along the Beibu Gulf in southern China, which may improve precipitation retrievals from remote sensing observations.

VALIDATION OF REMOTE MEASUREMENT OF RAIN CHARACTERISTICS BY THE METHOD OF DOUBLE FREQUENCY RADAR SENSING

Subject and Purpose. Th e development of radar methods for measuring liquid-drop precipitation characteristics is of great importance for studying physical processes in the atmosphere and for lots of applied problems to be solved. At the same time, an experimental research of these methods in situ is essential to determine their frames and scope and estimate retrieval errors of the rain parameters. The purpose of the work is to evaluate eff ectiveness of the previously proposed method that is based on the parametrization of the distribution function of drops by size, uses averaged touch-probing data of the three-parameter gamma distribution of the parameters versus the rain intensity, and employs the double-frequency method of precipitation remote sensing. For its validation, the rain intensity results obtained by the radar sensing are compared with the data taken from the ground raingauge. Methods and Methodology. Th e measurement results gained by the proposed method are compared with the measurement results obtained by the standard technique. Results. An experimental study using the previously developed algorithm and with a refi ned double-frequency weather radar MRL-1 has been given to the double-frequency sensing of liquid-drop precipitation. Th e obtained results show that the proposed data processing algorithm for double-frequency sensing of liquid-drop precipitation is good at the rain intensity retrieval and makes it possible to estimate liquid precipitation amounts over long periods of time. Conclusion. Th e rain intensity measured with the use of meteo radar almost completely coincides, both qualitatively and quantitatively, with the data from the ground raingauge. Th e algorithm proposed by the authors for processing double-frequency radar sensing data on liquid-drop precipitation retrieves the rain intensity and, also, evaluates liquid-drop precipitation amounts.

Phosphorus transport in the Three Gorges Reservoir over the past two decades

Phosphorus (P) is an essential nutrient in aquatic ecosystems and its dynamics in river systems are greatly affected by human activities. Based on the in situ measurements of water discharge (Q), suspended sediment concentrations (SSC), and concentrations (C) of total P (TP), particulate P (PP), and total dissolved P (TDP) from 1997 to 2017, P transport in the Three Gorges Reservoir (TGR) and the combined effects of cascade reservoirs operation and anthropogenic emissions have been explored. The whole period was divided into three sequential periods of January 1997 to May 2003 (period 1), June 2003 to September 2012 (period 2), and October 2012 to December 2017 (period 3), corresponding to the period prior to impoundment of the TGR, during the operation of the TGR, and after the impoundment of upstream cascade reservoirs, respectively. Results showed that the P concentrations were positively skewed and generally followed the three-parameter gamma and log-normal distributions. There was significant difference for SSC, TP, PP, and TDP between adjacent periods but not for Q. The SSC, TP, and PP presented downward trends during the periods affected by the impoundment of cascade reservoirs, while the TDP increased in period 2 especially after 2007 and decreased in period 3 due to the upstream reservoirs regulation and the pollution management in the tributaries. For the monthly variation, the SSC, TP, and PP became more evenly distributed throughout the year after the impoundment of cascade reservoirs, while the TDP tended to show a pattern of “higher in dry season and lower in flood season”. The C-Q relationships indicated that the TP in the TGR was positively correlated with Q (transport limited), while negatively with Q for TDP (source limited). Generally, the TP, PP, and TDP were positively correlated with both SSC and anthropogenic P emissions, thus, the P concentrations declined with sediment retention in the reservoir but increased with time due to the increasing pollutant emissions. Correspondingly, the P dynamics were better depicted after considering the effects of sediment and pollutant emissions than using a simple C-Q relationship.

Обоснование развития гидрологических процессов с использованием динамико-стохастического подхода

A technique has been developed for the probabilistic assessment (forecast) of the hydrological characteristic of the annual water inflow W (t) of the river basin Volga in the formation zone near the city of Volgograd. The proposed method makes it possible to build a predictive supply function that characterizes changes in the future regime of river flow in a given river section. The forecast of river water with a given availability makes it possible to effectively vary the operating modes of the water management systems of the river basin depending on the expected water content. To study the development of hydrological processes in a river basin, it is recommended to use the dynamic-stochastic concept, based on the condition that, along with the stochastic component, river runoff fluctuations also have a deterministic component, which manifests itself in the form of monotonically directed or periodic trends. As the analysis showed, long-term changes in the water content of the Volga River near the city of Volgograd do not fully comply with the laws of fluctuations of random variables. Some parameters of this process (average long-term values) change quite signifi cantly over time, other characteristics (dispersion, asymmetric distribution of members), on the contrary, change very little over time and can practically be considered constant. To construct predictive supply curves, two interrelated tasks are solved, including, firstly, the construction of functional equations of the average annual value (norm) of infl ow over time and, based on the obtained dependencies, extrapolations of the inflow rate for the first half of the 21st century are carried out and, secondly, using three-parameter gamma distribution of S.N. Kritsky and M.F. Menkel and with known predictive norms obtained in the corresponding samples and ratios CS = 2CV, ensembles of predictive probability curves for the Volga River tributary in the formation zone near the city of Volgograd are constructed.

МАКСИМАЛЬНЫЕ ГОДОВЫЕ РАСХОДЫ ВОДЫ МАЛЫХ РЕК СЛАВСКОГО РАЙОНА КАЛИНИНГРАДСКОЙ ОБЛАСТИ

Purpose: to restore the series of maximum annual water discharges (MAWD) of small rivers in Slavsky district Kaliningrad region and determine the calculated values of MAWD with the necessary availability for designers. Materials and methods. The basic material – observational data of Roshydromet for water discharges on the Zlaya, Osa and Nemoninka (Nemonin river basin) rivers and on the Instruch river. The Instruch river was used as an analogue for the restoration of the MAWD on the Zlaya river for a few missed years of observation. The river Osa has the shortest series of observations, for its restoration, as well as for the Nemonin river, the Zlaya river served as an analogue. A hypothesis that the MAWD of the rivers of the Slavsky district obey a three-parameter gamma distribution (Kritsky-Menkel) was put forward. To solve the system of integral equations, a previously developed numerical method was used. Results. A close stochastic relationship between the MAWD of the tributaries of the Nemonin river has been determined for years of joint observations. This made it possible to obtain the restored MAWD series for 1955–2019. The probability curves of three small rivers were constructed, and the hypothesis on possibility of using the Kritsky-Menkel distribution was confirmed. According to the analytical expressions of the specified distribution, the calculated values of the MAWD of the given exceedance probabilities were found. Conclusions. The hydrological characteristics of the studied rivers of the Nemonin basin are strongly influenced by the functions of water intakes of draining reclamation systems. Due to the impossibility of dividing the peak annual discharges into the maxima of rain and melt water for each river, a single series of MAWD was restored, regardless of their origin. At the tributaries of the Nemonin river, the ratio of maximum discharges to average annual ones is noticeably lower than that of the Instruch river, which is under the similar hydrological conditions, but is not subject to such a strong anthropogenic impact. The calculated MAWD modules of the given exceedance probabilities can be used in the design of drainage systems and hydraulic structures.

New two-phase and three-phase thermodynamic models for predicting wax precipitation in hydrocarbon mixtures

One of the major issues in flow assurance is the precipitation and subsequent deposition of paraffin wax in petroleum production systems. A new two-phase multicomponent thermodynamic model based on a continuous thermodynamic phase equilibria employing a three-parameter gamma distribution function has been developed to predict the wax appearance temperature (WAT), the amount of precipitated wax, and the wax content of crude oils. Also, a generalized three-phase multicomponent thermodynamic model based on the combined polymer and regular solution theories of mixtures as well as on equations of state (EOS) is presented. The model utilizes the solid-liquid-vapour phase equilibrium relations with some constraint functions. Furthermore, hinged on the regression analysis of published experimental data, a new set of molecular-weight-dependent thermodynamic properties’ correlations were developed and used to calculate the solid-liquid-equilibrium-ratios ( K i S-L ) for the individual component. The models were validated with data obtained from experiments conducted on 32 different hydrocarbon fluids from the Niger Delta, North Sea, Gulf of Mexico, East Asia, and Middle East fields. Results show that the gamma distribution parameters (α, β, η) can be used to classify hydrocarbon mixtures into three categories: light oils or condensate, waxy oil or asphaltenic oils, and biodegraded oils. Analysis shows that when compared with experimental data, the two-phase thermodynamic model is reliable with the average absolute deviations (AAD%) of 0.80% for paraffinic and waxy oils, 0.42% for paraffinic and waxy condensates, and 4.33% for paraffinic-asphaltenic and biodegraded oils. For the three-phase model, results show that the AAD% for the paraffinic and waxy oils, paraffinic and waxy condensates, and biodegraded oils are 0.93%, 1.03%, and 4.88% respectively. However, the two models slightly overestimate the WAT for biodegraded crude oils. Nevertheless, the generalized thermodynamic models developed have proved to be effective for predicting the onset conditions for wax precipitation during petroleum production and processing. It will prove a helpful tool for the fast and accurate assessment of wax deposition tendencies of hydrocarbon mixtures. • A new two-phase multicomponent thermodynamic model based on a continuous thermodynamic phase equilibria employing a three-parameter gamma distribution function has been developed to predict the wax appearance temperature (WAT), the amount of precipitated wax, and the wax content of crude oils. • A generalized three-phase multicomponent thermodynamic model based on the combined polymer and regular solution theories of mixtures as well as on equations of state (EOS) is also presented. • A new set of molecular-weight-dependent thermodynamic properties' correlations were developed and used to calculate the solid-liquid-equilibrium-ratios (KiS-L) for the individual component. • The gamma distribution parameters (α, β, η) can be used to classify hydrocarbon mixtures into three categories: light oils or condensate, waxy oil or asphaltenic oils, and biodegraded oils. • The models were reliably and successfully validated with extensive data obtained from experiments conducted on 32 different hydrocarbon fluids from the Niger Delta, North Sea, Gulf of Mexico, East Asia, and Middle East fields.

Toward an efficient wax precipitation model: Application of multi-solid framework and PC-SAFT with focus on heavy end characterization for different crude types

Several studies have investigated the wax precipitation process, focusing on the equations of state, correlations for fusion properties, and calibration of parameters. In most of them, the impact of the fluid characterization on the wax properties has not been addressed widely. In this study, we show that a single characterization scheme is not applicable to all crudes with wax precipitation. However, it is possible to categorize the crudes based on the compositional data and wax parameters to have a primary understanding of the fluid description for modeling. We employed the PC-SAFT equation of state for modeling of liquid phase behavior and the multi-solid framework to describe solid phase. The heptane plus fraction of the studied fluids is defined by the three-parameter gamma distribution of Whitson, et al. [1], and the continuous description of the plus fraction is discretized by the pseudoization method of Whitson, et al. [2]. We also show the significance of binary interaction parameters on calculating wax appearance temperature and wax weights. The critical volumes of the precipitating hydrocarbons are adjusted to find the best binary interaction parameters values for the WAT and wax weight determination. The WAT and wax weight of eleven crude oil samples are calculated and compared to experimental and modeling results of previous studies. Our model shows considerable improvements in estimating fluid description, WAT, and wax weight compared to other available models. Moreover, we demonstrated that the number of pseudo-components and the heaviest molecular weight in the description of plus fraction highly depend on the compositional data of the heavy end and wax properties. According to our results, 15 pseudo-components with the highest molecular weight of 1200 g/mol would result in the accurate presentation of WAT and wax weight for crudes with WAT higher than 310 K and high wax weights.

Statistical Characteristics of Raindrop Size Distribution in Monsoon Season over South China Sea

The South China Sea (SCS) is the largest and southernmost sea in China. Water vapor from the SCS is the primary source of precipitation over coastal areas during the summer monsoon season and may cause the uneven distribution of rainfall in southern China. Deep insight into the spatial variability of raindrop size distribution (DSD) is essential for understanding precipitation microphysics, since DSD contains abundant information about rainfall microphysics processes. However, compared to the studies of DSDs over mainland China, very little is known about DSDs over Chinese ocean areas, especially over the South China Sea (SCS). This study investigated the statistical characteristics of the DSD in summer monsoon seasons using the second-generation Particle Size and Velocity (Parsivel2) installed on the scientific research vessel that measured the size and velocity of raindrops over the SCS. In this study, the characteristics of precipitation over the SCS for daytime and nighttime rains were analyzed for different precipitation systems and upon different rain rates. It was found that: (1) rain events were more frequent during the late evening to early morning; (2) more than 78.2% of the raindrops’ diameters were less than 2 mm, and the average value of mass-weighted mean diameter Dm (1.46 mm) of the SCS is similar to that over land in the southern China; (3) the stratiform precipitation features a relatively high concentration of medium to large-sized rain drops compared to other regions; (4) the DSD in the SCS agreed with a three-parameter gamma distribution for the small raindrop diameter. Furthermore, a possible factor for significant DSD variability in the ocean compared with the coast and large islands is also discussed.

Discovering spatiotemporal characteristics of passenger travel with mobile trajectory big data

Mobile trajectory big data, such as global positioning system (GPS) trajectories of taxicabs, have enormous social and economic values in intelligent transportation systems, and efficient mining and in-depth analysis of these data can provide beneficial decision-making for traffic operation and especially for passenger travel. To this end, in this paper, we discover the temporal, spatial, and spatiotemporal characteristics of passenger travel by analyzing the travel duration and the travel distance with large-scale taxi trajectory data, which can not only effectively reveal the resident travel patterns but also dynamically perceive the traffic conditions. Specifically, we propose an information framework to preprocess the taxi GPS trajectory data that can thoroughly identify passenger travel trajectories. Moreover, we develop a novel distribution model to explore the rules of citizen travel accurately. Finally, we put forward a parallel clustering approach on Spark, which can discover the spatiotemporal characteristics of passenger travel in a fine-grained manner, to obtain the characteristic changes of inhabitant travel at different periods of the day. In particular, the experimental results from an empirical study show that the travel duration and the travel distance follow a log-normal distribution, and the tail of the travel distance is highly fitting to a three-parameter gamma distribution and the long-distance travel of the day is mainly to the airport between 5:00 and 6:00 AM.

Bayesian and Classical Inference for the Generalized Log-Logistic Distribution with Applications to Survival Data.

The generalized log-logistic distribution is especially useful for modelling survival data with variable hazard rate shapes because it extends the log-logistic distribution by adding an extra parameter to the classical distribution, resulting in greater flexibility in analyzing and modelling various data types. We derive the fundamental mathematical and statistical properties of the proposed distribution in this paper. Many well-known lifetime special submodels are included in the proposed distribution, including the Weibull, log-logistic, exponential, and Burr XII distributions. The maximum likelihood method was used to estimate the unknown parameters of the proposed distribution, and a Monte Carlo simulation study was run to assess the estimators' performance. This distribution is significant because it can model both monotone and nonmonotone hazard rate functions, which are quite common in survival and reliability data analysis. Furthermore, the proposed distribution's flexibility and usefulness are demonstrated in a real-world data set and compared to its submodels, the Weibull, log-logistic, and Burr XII distributions, as well as other three-parameter parametric survival distributions, such as the exponentiated Weibull distribution, the three-parameter log-normal distribution, the three-parameter (or the shifted) log-logistic distribution, the three-parameter gamma distribution, and an exponentiated Weibull distribution. The proposed distribution is plausible, according to the goodness-of-fit, log-likelihood, and information criterion values. Finally, for the data set, Bayesian inference and Gibb's sampling performance are used to compute the approximate Bayes estimates as well as the highest posterior density credible intervals, and the convergence diagnostic techniques based on Markov chain Monte Carlo techniques were used.

МОДЕЛИРОВАНИЕ ОТКАЗОВ ЭЛЕМЕНТОВ ГОРОДСКОЙ ЭЛЕКТРИЧЕСКОЙ СЕТИ (10 КВ) НАПРИМЕРЕПРАВОБЕРЕЖНОГООКРУГАГОРОДАИРКУТСКА

The paper presents mathematical models for probabil-istic evaluation and forecasting of emergency shutdowns in electric networks by the example of the Pravoberezhniy District of the Irkutsk City from 2008 through 2017. At the first stage, the autocorrelation function of the parameter series is determined to estimate its randomness. According to the calculated statistical parameters and the consent criteria, a number of equipmentfailures may be described by a three-parameter gamma distribution. A method of two-level identification of extreme (maximum and minimum) values of the parameter under study is proposed; accord-ing to this method, a significant polynomial trend is ob-tained for predicting the largest number of failures. The evaluation of the presence of trends based on monthly data showed that polynomial and power trends may be used to predict failures on electric networks. At the same time, sig-nificant trends were identifiedonly for January, February, May and December. At the next stage, trend-seasonal models are constructed; the least squares method is used to calculate their components. According to the obtained seasonality indices, the greatest increase in emergency shutdowns takes place in April and July, and a decrease -in February and March. On the basis of the correlation-regression model, factor models of failures of electrical network elements and the accumulated average daily tem-peratures for months and time are constructed. Linear and nonlinear models with and without trends are obtained. To evaluate the accuracy of the forecasts of the obtained models, the results of the retrospective forecast for 2017 were compared with the actual values. According to the resultsobtained for predicting failures on electric networks in February, June, July and September, the best result is shown by a trend-seasonal model, in May -a polynomial trend, in November -a factorial one taking into account time, in March, May and October-a nonlinear regression equation, and in December -a power trend. There are no qualitative forecast models for the months of January, April, and August. In this regard, the values of emergency shut-downs on these months may be estimated using a proba-bilistic model.

Bayesian inference for the simple step-stress accelerated life tests under order-restriction

In this work, we investigate the order-restricted Bayesian estimation for a simple step-stress accelerated life tests. Based on the three-parameter gamma distribution as a conditional prior, we ensure that the failure rates increase as the stress level increases. In addition, its conjugate-like structure enables us to derive the exact joint posterior distribution of the parameters without a need to perform an expensive MCMC sampling. Upon these distributional results, several Bayesian estimators for the model parameters are suggested along with their individual/joint credible intervals. Through Monte Carlo simulations, the performance of our proposed inferential methods are assessed and compared. Finally, a real engineering case study for analyzing the reliability of a solar lighting device is presented to illustrate the methods developed in this work.

A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres

Three-parameter (3-p) Gamma distribution is widely used to model for skewed data in the reliability field. Thus, the problem of parameter estimation for the Gamma distribution has remained significant and interesting in all times. The maximum likelihood (ML) and the least square (LS) are the most popular methods in the parameter estimation. In this study, a novel Differential Evolution (DE) algorithm is proposed for the ML and LS estimation of the parameters of the 3-p Gamma distribution. This approach overcomes the problem of how to determine the search space of the DE by utilizing a new search space based on the confidence interval. The modified maximum likelihood and the profile likelihood methods are considered to constitute the confidence interval. In order to examine the performance of the proposed approach, an extensive Monte Carlo simulation study and a real data application are performed. The results show that this proposed approach is effective for estimating the parameters of the 3-p Gamma distribution with respect to mean square error and deficiency criteria.

Comparison of parametric and nonparametric standardized precipitation index for detecting meteorological drought over the Indian region

In this study, Standardized Precipitation Index (SPI) derived from parametric and nonparametric methods using 0.25∘ gridded rainfall data from 1901 to 2013 (113 years) generated by India Meteorological Department (IMD) was compared for understanding drought conditions over the Indian region. The parametric SPI was computed using a three-parameter Gamma distribution function, whereas nonparametric SPI was computed using Gringorten, Weibull, and Hazen plotting positions, on a 4-month cumulative rainfall data of June–September (SPI-4) representing the southwest monsoon season. Nonnormality is a major concern if equal-sized intervals are drawn for interpretation, and SPI being a normalized index wherein classes are standard deviations from normal, its impact on drought assessment needs to be understood. Accordingly, in our study, normality tests were performed using the Shapiro-Wilk method on SPI derived from both parametric and nonparametric methods. The SPI showed 100% of grid cells conforming to normality in the case of nonparametric methods, whereas in the case of parametric approach it was only 80%. The remaining 20% of nonnormality in parametric SPI is spread over montane, tropical wet, and semi-arid regions of India. Furthermore, differences in the estimation of dryness are observed in the range of 1.0 to 2.5% between nonparametric and parametric SPI for the drought years considered this study. The quantile analysis on all grid cells for the drought year 2002 showed an important fact that at 0.025 quantile only 2.6% of grid cells are in the extremely dry condition as per parametric SPI, whereas in the case of nonparametric SPI it is 6.9%. For the drought year 1939 in grid cells where normality is not followed in parametric SPI, Cohen’s kappa (κ = 0.15) under extreme dryness category indicates large disagreements between parametric and nonparametric SPI. The temporal analysis of Cohen’s kappa computed for each grid cell over drought years shows that in 22.5% of cases the drought category between nonparametric and parametric SPI is not in perfect agreement. Hence, the nonparametric SPI can better categorize the drought classes, representing well the extent of dryness and normality conditions, it is highly recommended for drought assessment over India.

Two-Dimensional Distribution Law of Random Variables Having S. N. Kritskii and M. F. Menkel Three-Parameter Gamma Distributions: A Symmetrical Case

A method for constructing two-dimensional distribution law in a symmetrical case for three-parameter Kritskii and Menkel distribution is considered, and some results of the application of the model in applied hydrological studies are discussed. A system of orthogonal functions with a weight function in the form of a three-parameter gamma distribution is proposed to obtain linear correlation between three-parameter variables. Orthogonalization method was used to obtain an expression for symmetric two-dimensional density satisfying Markov equation and having a linear regression equation. The results of stochastic simulation of water level variations in the drainless Chany Lake, carried out with the use of the proposed model, showed that the use of linear correlation leads to a value of asymmetry of the level distribution that is in a good agreement with sample estimates of these parameter based on observation data.

Concordance Modeling With a Gold Standard for Variables From the Three-Parameter Gamma Distribution

A way to compare two or more measurements for the same random variable can be achieved by using a negligible error reference measurement, which is called the gold standard, obtained by consolidated measurement methods. This paper presents a new methodology for comparing measurements in the presence of a gold standard with random variables from the multivariate three-parameter (shape, scale, and location) gamma distribution. The errors between gold standard measures and approximate measures have a gamma difference distribution with the same three parameters of the gamma distribution. The concordance measurements were obtained by mean of a coefficient, which measures the degree of agreement as a ratio between the variances of the gold standard and the errors. The developed methodology is illustrated with climatic data which is divided into four ranges. The measurements analyzed are rainfall forecasts of the following four national centers: Canadian Meteorological Center (CMC), European Center for Medium-Range Weather Forecasts (ECMWF), National Centers for Environmental Prediction (NCEP), and Center for Weather Forecasting and Climate Studies (CPTEC). The forecast range was 240 hours for the West mesoregion of Paraná – Brazil, and in the October 1–March 31 period of the 2010/2011 –2015/2016 harvest years. The period was selected because it is related to soybean crop development in the region and because several crop estimation models use rainfall forecast data in this period. The methodology applied spatially indicated the center to be selected in each geographical location according to each rainfall range interval. The gamma model fit well with the data and is an alternative to the normal one for modelling rainfall, in particular to estimate concordances between rainfall forecasts and the gold standard, which are used to improve the selection of rainfall forecast centers.

Formulation of Autoconversion and Drop Spectra Shape in Shallow Cumulus Clouds

Abstract Two-moment autoconversion parameterizations as compared to accretion parameterizations exhibit significant errors suggesting that additional moments are needed to increase their accuracy. We develop a three-moment autoconversion parameterization using output from an LES model with size-resolved microphysics. Adding the third moment decreases the errors of parameterization and improves precipitation prediction. However, the errors are still significantly larger than errors of accretion rate. An analysis of the cloud drop size distributions (DSDs) in the simulated tropical convective cloud system reveals that most DSDs have a significant fraction of cloud liquid water content qc in the midsize droplet range (radii from 20 to 40 μm). Our data indicate that more than 30% of DSDs have over half of qc contained in the midsize range and about 60% of spectra have, at least, one-third of qc in this range. Even when the rain/drizzle mode is small (radar reflectivity Z < −10 dBZ), there is a significant number of spectra in which fraction of qc in the midsize range is as large as 60%. These DSDs are more complex than the frequently used gamma or lognormal distributions, which exhibit a single mode and can be defined by three microphysical moments. The need to define DSDs by more than three moments explains the large errors in the three-moment autoconversion parameterization. The limitation of three-parameter gamma or lognormal distributions should be kept in mind when applying them in precipitating shallow cumulus clouds.