Weibull Research Articles

A study of time spectrum identification of induced polarization field based on Weibull distribution function

Identification of time spectrum is one of the core issues of the time-domain induced polarization (IP) method, which can be considered as one of the bases for distinguishing various polarized rocks. Fitting the IP data based on the spectrum forward model and obtaining optimal solution of the model parameters is a key step in spectrum identification. However, the suitable forward model should consider the observation conditions, such as the charging time and the time window. The time spectrum identification may be difficult to implement stably due to the lack of objective reference for the optimal solution. Therefore, our purpose is to improve the forward model and implement the spectrum identification for IP data. First, using the Weibull (WB) distribution function as the basis, a time spectrum forward model considering the charging time and observation time window is provided according to the typical measurement mode. Then, based on the WB spectrum model and Barzilai-Borwein gradient optimization, a method for solution of apparent spectral model parameters for spectrum identification is developed. Finally, this method is used to process the IP data from a mine in which the anomalies related to ore-bearing beds are identified based on the processing results. Results obtained demonstrate that the spectrum forward model based on the WB function is feasible in describing the IP data. The limited charging time and a wide observed time window should be considered to realize accurate simulations and description of the time-domain IP data. The essence of the time spectrum identification is to comprehensively reflect the time-varying state of the whole time channel through the spectral model parameters, whereas the decay field ratio of the adjacent time channel can be used as an objective reference. The parameters of the spectrum model characterizing the time-varying state are independent of polarization and resistivity and thus can be directly used for identification of IP anomalies.

Non-linear Tree Height (h-d) Model Development and Forest Resource Productivity Assessment of Diguyo Limestone Forest within Northern Sierra Madre Natural Park

Forest resource monitoring of different forest types is of great importance in sustainable forest management and climate change mitigation. Monitoring the productivity of forest resources could be achieved by modeling the basic tree parameters necessary for forest growth and yield. This study was conducted to develop a height-diameter at breast height (h-d) model necessary for tree height (h) estimation since h measurement is difficult in the field, especially in dense forests, and to estimate the forest productivity of the Diguyo limestone forest within the Northern Sierra Madre Natural Park (NSMNP). The diameter at breast height (d) and h of 124 trees were measured in seven 400-m2 plots as the basis for the model development. The h-d model was developed using different non-linear models such as the Chapman-Richards (CR), exponential (EX), Korf/Lundqvist (KL), modified logistic (ML), Schnute (SC), and Weibull (WE) models. The models were evaluated using the adjusted coefficient of determination (R2 adjusted), Akaike information criterion (AIC), Bayesian information criterion (BIC), mean absolute error (MAE), root mean square error (RMSE), percentage root mean squared error (PRMSE), and root mean squared percentage error (RMSPE). The performance of the species-specific allometric models and the generic models were compared for the biomass productivity of the limestone forest. Results showed that the CR h-d model performed best with MAE, RMSE, PRMSE, RMSPE, R2 adjusted, AIC, and BIC values of 1.47 m, 1.74 m, 19.31%, 28.71%, 0.79, 32.46, and 36.00, respectively. The highest average predicted tree biomass and carbon stock of the Diguyo limestone forest was 112.52 ± 97.65 t/ha and 50.64 ± 43.94 tC/ha, respectively, which is lower than other karst forests in Asia. The low forest resource productivity is due to the physical condition of the forest aggravated by natural and anthropogenic disturbances, thereby needing immediate attention to achieve forest sustainability.

Best-fit probability distribution models for monthly rainfall of Northeastern Brazil.

The analysis of precipitation data is extremely important for strategic planning and decision-making in various natural systems, as well as in planning and preparing for a drought period. The drought is responsible for several impacts on the economy of Northeast Brazil (NEB), mainly in the agricultural and livestock sectors. This study analyzed the fit of 2-parameter distributions gamma (GAM), log-normal (LNORM), Weibull (WEI), generalized Pareto (GP), Gumbel (GUM) and normal (NORM) to monthly precipitation data from 293 rainfall stations across NEB, in the period 1988-2017. The maximum likelihood (ML) method was used to estimate the parameters to fit the models and the selection of the model was based on a modification of the Shapiro-Wilk statistic. The results showed the chosen 2-parameter distributions to be flexible enough to describe the studied monthly precipitation data. The GAM and WEI models showed the overall best fits, but the LNORM and GP models gave the best fits in certain months of the year and regions that differed from the others in terms of their average precipitation.

Time-domain induced polarization forward modeling and an apparent spectral parameter solution based on the Weibull growth model

The application of the Cole-Cole model within time-domain induced polarization (TDIP) forward field modeling shows that the model parameters can characterize time-varying states of the TDIP field and support observed data analysis. The Cole-Cole model contains real and imaginary parts, and it requires a frequency-to-time conversion for TDIP forward modeling. However, the TDIP field is usually expressed by a real number, and its intuitive time-varying states field intensity increases with charging time. Therefore, the forward model should be constructed in a simpler form. We have aimed to develop a forward model using mathematical functions not based on physical principles. The Weibull (WB) growth model, which is primarily used to describe the time-varying curve features in regression analysis, is introduced into the basic algorithm of the TDIP forward model. Subsequently, a forward expression of the TDIP effect is established. Based on the time-varying shape and scale parameters, this expression describes the time-varying rate and relaxation states of the TDIP fields. Furthermore, based on the extensively used conjugate gradient optimization, an apparent WB parameter scheme is initiated to calculate the spectral parameters that represent the relaxation and time-varying rate obtained from the multi-time-channel TDIP data. Finally, this scheme is applied to interpret the different simulated and actual TDIP data. The results demonstrate that the WB growth model can be used for the TDIP forward model without involving physical principles, the model parameters without specific physical significance can be used to represent the time-varying states of TDIP fields, and apparent WB parameters can be used to discern different TDIP observed data. The setting of the TDIP forward model and model parameters can actually be more flexible and diverse, so as to obtain simpler forward expressions and ensure a highly efficient inverse solution.

A comparison between the performance of Weibull and Log-logistic Aging Models on Saccharomyces cerevisiae lifespan data

Empirical lifespan datasets are often studied with the best-fitted mathematical model for aging. In this study, we focus our attention to the budding yeast S. cerevisiae lifespan and the determination of the best-fitted model of aging. We investigate the influence of model selection in yeast lifespan datasets and the fitting outcomes of the two-parameter Weibull (WE) and Log-logistic (LL) models of aging. Both of these models are commonly studied and implemented in aging research. They show similar tendency as a survival function that they correspond to mortality rates that increase, and then decrease, with time. Studies so far has been usually done with medflies, Drosophila, house flies, flour beetles, and humans with these models. Different than previous research, we focus our attention on the influence of fitting results and calibrations on empirical lifespan data samples. As expected both of the models could be used as a substitute of each other. However, we also find WE model fits the yeast lifespan data significantly better than LL model with an R 2 = 0.86. This finding is especially important in yeast aging study because of typically survival models are applied and therefore one can see which model fits the yeast data best. In this article, comparisons are done and developed and the potential of the approach is demonstrated with a model comparison of yeast replicative lifespan datasets of the laboratory BY4741 and BY4742 wildtype reference strains. Our study highlights that interpreting model fitting results of experimental lifespans should take model selection and resulted variation into account.

Comparative analysis of probability distributions for the Standardized Precipitation Index and drought evolution in China during 1961–2015

As a representative index for monitoring and assessing drought, the Standardized Precipitation Index (SPI) relies on a suitable probability distribution function (PDF) to describe a precipitation series, which allows interregional comparisons following normalization. In this study, we considered nine PDFs (exponential (EXP), extreme value (EV), gamma (GAM), generalized Pareto (GP), logistic (LO), log-logistic (LOL), log-normal (LON), normal (NOR), and Weibull (WEI)) as candidates for use in SPI calculations. Based on monthly precipitation time series data (1961–2015) from 582 stations across China, together with the Kolmogorov–Smirnov (K–S) and Akaike Information Criterion (AIC) methods, differences in optimal PDFs for SPI calculations were compared comprehensively from the perspectives of timescale, record length, and index value. Based on the SPI calculated using the optimal PDF at the 6-month timescale (SPI6-opt), we analyzed the spatiotemporal characteristics of drought trends in China using the Mann–Kendall method. Results indicated both the timescale and the record length would affect the selection of the optimal PDF. The performance of the WEI and GAM distributions was superior to other distributions in describing monthly precipitation (especially for long precipitation records) at short and multiple timescales, respectively. During the entire study period, areas of China with high frequency of drought have transferred from the northwest (1960s), to the northeast (2000s), and to the southwest (most recent 5 years). Trend analysis revealed a noticeable wetting tendency confined mainly to Northwest, Northeast, and Southeast China and a significant trend toward drought in Southwest China and on the North China Plain.

Correcting Satellite Precipitation Data and Assimilating Satellite-Derived Soil Moisture Data to Generate Ensemble Hydrological Forecasts within the HBV Rainfall-Runoff Model

An implementation of bias correction and data assimilation using the ensemble Kalman filter (EnKF) as a procedure, dynamically coupled with the conceptual rainfall-runoff Hydrologiska Byråns Vattenbalansavdelning (HBV) model, was assessed for the hydrological modeling of seasonal hydrographs. The enhanced HBV model generated ensemble hydrographs and an average stream-flow simulation. The proposed approach was developed to examine the possibility of using data (e.g., precipitation and soil moisture) from the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) Satellite Application Facility for Support to Operational Hydrology and Water Management (H-SAF), and to explore its usefulness in improving model updating and forecasting. Data from the Sola mountain catchment in southern Poland between 1 January 2008 and 31 July 2014 were used to calibrate the HBV model, while data from 1 August 2014 to 30 April 2015 were used for validation. A bias correction algorithm for a distribution-derived transformation method was developed by exploring generalized exponential (GE) theoretical distributions, along with gamma (GA) and Weibull (WE) distributions for the different data used in this study. When using the ensemble Kalman filter, the stochastically-generated ensemble of the model states generally induced bias in the estimation of non-linear hydrologic processes, thus influencing the accuracy of the Kalman analysis. In order to reduce the bias produced by the assimilation procedure, a post-processing bias correction (BC) procedure was coupled with the ensemble Kalman filter (EnKF), resulting in an ensemble Kalman filter with bias correction (EnKF-BC). The EnKF-BC, dynamically coupled with the HBV model for the assimilation of the satellite soil moisture observations, improved the accuracy of the simulated hydrographs significantly in the summer season, whereas, a positive effect from bias corrected (BC) satellite precipitation, as forcing data, was observed in the winter. Ensemble forecasts generated from the assimilation procedure are shown to be less uncertain. In future studies, the EnKF-BC algorithm proposed in the current study could be applied to a diverse array of practical forecasting problems (e.g., an operational assimilation of snowpack and snow water equivalent in forecasting models).

Multiparameter probability distributions for at-site frequency analysis of annual maximum wind speed with L-Moments for parameter estimation

Estimation of quantiles of Annual Maximum Wind Speed (AMWS) is needed in different environmental fields, engineering risk analysis, design of structures, renewable energy sources, agricultural operations, and climatology. Therefore, wind speed frequency analysis (WSFA) was carried out at nine stations from Pakistan. Multiparameter Probability Distributions (PDs), such as Generalized logistic (GLO), Generalized Extreme Value (GEV), Generalized Normal (GNO), Generalized Pareto (GPA), Weibull (WEI), Pearson type 3 (P3), Log Pearson type 3 (LP3); and two parameter PDs, such as Logistic (LOG), Normal (NOR), Gumbel (GUM), Exponential (EXP), and Uniform (UNI) were used to determine the most suitable distributions for the nine stations. The method of L-moments was used for estimating parameters of the distributions. The Kolmogorov-Smirnov (KS) test, Anderson-Darling (AD) test, Minimum L-Kurtosis (ML-K) Difference Criterion, and L-moment ratio diagram (L-ratio diagram) showed that four distributions, namely GEV, GNO, GPA, and GLO were the most suitable distributions for different stations and were superior to the two-parameter distributions. The quantile estimates (design estimates) from multiparameter PDs provide information on how fast the maximum wind will pass through a certain place and hence are important for policy makers and planners in the design and construction of different structures. The Multivariate Diebold–Mariano (DM) test was applied to check the accuracy of design estimates from the best fitted PDs and results indicated that they were significantly different.

Estimation of wind speed using regional frequency analysis based on linear‐moments

The quantiles of annual maximum wind speed (AMWS) can be estimated for different meteorological stations of interest by using at‐site frequency analysis and extreme value theory. These estimates are of immense importance for the codification of wind speed. However, the historical data of wind speed at the number of meteorological stations are sometimes unavailable and often insufficient due to the shorter length, especially in developing countries like Pakistan. The scarcity of the data increases the uncertainty of the quantiles estimates regarding policy implications. To cope with the problem, an approach of Regional Frequency Analysis (RFA) is opted here. In this study, RFA of AMWS using linear‐moments (L‐moments) is carried out by considering wind speed data of nine meteorological stations of province Punjab, Pakistan. No station is found to be discordant. A single homogenous region is constituted from these nine stations using a subjective approach based on their geographical locations. Heterogeneity measures justify that these nine stations of Punjab form a single homogeneous region. Regional quantiles estimates are found through the most appropriate probability distribution among generalized normal (GNO), generalized logistic (GLO), Pearson Type 3 (P3), generalized Pareto (GPA), Weibull (WEI), log Pearson Type 3 (LP3) and generalized extreme value (GEV) distributions. Z‐statistic and L‐moment ratio diagram suggest that GLO and GNO distributions are better choices than others. Robustness of both distributions is evaluated through relative bias (RB) and relative root mean square error (RRMSE). Findings indicate that overall, GLO distribution is better than GNO. Further, we also find at‐site quantiles from dimensionless quantities (regional quantiles) using the sample mean and median as scaling factors. Quantiles' estimates calculated from this study can be used in codified structural designs for policy implications.

Evaluation of Energy in Wind Turbine System Using Probability Distribution

In this work, annual energy output of a variable speed wind turbine is analyzed using annual Weibull wind speed probability distribution function. The power coefficient variety with tip speed proportion in torque control district and pitch point variety for most extreme power yield from wind turbine are examined for distinguishing control framework parameters. The wind turbine power output and variation of power coefficient with tip speed ratio as well as pitch angle are examined / reported using annual Wei bull distribution function. Finally the variation of the estimated annual energy output of the given wind turbine with the mean wind speed is presented.

A comparative study to determine the optimal copula model for the wind speed and precipitation of typhoons

ABSTRACTTyphoon is usually one of main reasons to damage the buildings, infrastructures, water operations, and human life in summer in East Asia, which includes China, Japan, and Korea. Typhoons can be considered multivariate extremes that consists of climate variables that affect wind and precipitation. Copula models can be used to investigate the dependence between random variables. In this study, we used copula models to investigate the optimal marginal distribution for the relations between the wind speed and the precipitation of typhoons at the Jeju weather station in South Korea. We used the 10‐min average wind speed (W10) and the total precipitation (PT) from 65 typhoon events that had affected the Jeju station. Two case studies were considered using the typhoon data. Case one included all typhoon events, and case two included 39 typhoon events with the following thresholds:W10 ≥ 10 m s−1andPT ≥ 50 mm. The marginal distributions for the copula models included the generalized extreme value (GEV), generalized logistic (GLO), generalized Pareto (GPA), and Weibull (WBU) distributions as well as three copula models: Clayton, Frank, and Gumbel copulas. Each step was checked by the robust diagnostic, the probability plot correlation coefficient (PPCC) test, and the goodness‐of‐fit test (Snstatistic). The Frank copula model withGLO(W10) – GPA(PT) had the best performance in both cases. The results of this study present the key steps needed to identify an optimum copula for a bivariate distribution in applications related to atmospheric sciences and provides an example of how this applies to wind velocity and precipitation from typhoons.

The Logistic-Exponential Weibull Model as a Tool to Predict Natural Microflora Inactivation of Agave Mapsiaga Aguamiel (Agave Sap) by High Pressure Treatments

Agave sap (aguamiel) is a major source of nutraceutics that may be affected by natural microflora fermentation and thermal processing. High pressure processing (HPP) at 100–400 MPa/25C/1–20 min was investigated as an alternative non-thermal pasteurization by analyzing aerobic mesophiles (AM), coliforms (CL) and yeasts and molds (YM) inactivation. Standard statistics and information theory criteria determined the goodness of fit of first-order kinetics (LKM), Weibull (WBLL) and Gompertz (GMPZ) models. Non-detectable CL were found for 300 MPa/1 min, whereas 400 MPa/4 min HPP treatments reduced AM and YM by 3.86–4.72 log10 units. Statistical analysis showed WBLL best described microbial inactivation. A secondary WBLL model consisting of logistic-exponential b′(P) and exponential decay n(P) functions predicted AM (R2 = 0.835), YM (R2 = 0.867) kinetics for different pressure–time combinations. The secondary WBLL model predicted commercial HPP pasteurization conditions (5-log10 reductions of AM and YM) could be achieved at 400 MPa/5 min or 500 MPa/2 min. Practical Applications High pressure processing (HPP) is a consumer preferred alternative for food pasteurization. Aside from guacamole and some fruit and vegetable juices, there no other significant HPP foods available in México. Our research group believes that the commercialization of food products based on endemic plants such as agave sap (aguamiel), will draw the attention of national and international consumers, researchers and industrial HPP food processors to increase the number of HPP applications. Furthermore, the development of reliable pressure–time predictive models for microbial inactivation would help HPP food processors to ensure food safety and generate better budget estimates for novel products, or redefine current operating conditions since processing costs depend on the applied pressure level and pressure holding time.

Uncertainty in drought monitoring by the Standardized Precipitation Index: the case study of the Abruzzo region (central Italy)

As shown by several authors, drought monitoring by the Standardized Precipitation Index (SPI) presents some uncertainties, mainly dependent on the choice of the probability distribution used to describe the cumulative precipitation and on the characteristics (e.g., length and variability) of the dataset. In this paper, the uncertainty related to SPI estimates has been quantified and analyzed with regards to the case study of the Abruzzo region (Central Italy), by using monthly precipitation recorded at 75 stations during the period 1951–2009. First, a set of distributions suitable to describe the cumulative precipitation at the 3-, 6-, and 12-month time scales was identified by using L-moments ratio diagrams. The goodness-of-fit was evaluated by applying the Kolmogorov–Smirnov test, and the Normality test on the derived SPI series. Then the confidence intervals of SPI have been calculated by applying a bootstrap procedure. The size of the confidence intervals has been considered as a measure of uncertainty, and its dependence on several factors such as the distribution type, the time scale, the record length, and the season has been examined. Results show that the distributions Pearson type III (PE3), Weibull (WEI), Generalized Normal (GNO), Generalized Extreme Value (GEV), and Gamma (GA2) are all suitable to describe the cumulative precipitation, with a slightly better performance of the PE3 and GNO distributions. As expected, the uncertainty increases as the record length and time scale decrease. The leading source of uncertainty is the record length while the effects due to seasonality and time scale are negligible. Two-parameter distributions make it possible to obtain confidence intervals of SPI (particularly for extreme values) narrower than those obtained by three-parameter distributions. Nevertheless, due to a poorer goodness of fit, two-parameter distributions can provide less reliable estimates of the precipitation probability. In any event, independently of the type of distribution, the SPI estimates corresponding to extreme precipitation values are always characterized by a relevant uncertainty. This is due to the explosion of the probability variability that occurs when precipitation values approach the tails of the supposed distribution.

Evaluation of High Pressure Processing Kinetic Models for Microbial Inactivation Using Standard Statistical Tools and Information Theory Criteria, and the Development of Generic Time-Pressure Functions for Process Design

Generic kinetic models for microbial inactivation by high pressure processing (HPP) would accelerate the development of commercial applications. The aim of this work was to develop a generic model obtained by fitting peer-reviewed microbial inactivation data (124 kinetic curves) to first-order kinetics (LKM), Weibull (WBLL), and Gompertz (GMPZ) primary and secondary models. Standard statistics (coefficient of determination (R 2), variance, residuals plots, experimental vs. predicted plots) and information theory criteria (Akaike Information Criteria, AIC; Akaike differences, ∆AIC i , Bayesian Information Criteria) determined their goodness of fit. Standard statistics showed no differences between WBLL and GMPZ, whereas information theory criteria identified WBLL as the best model (lowest AIC i value, 61.3 % of cases). LKM performed poorly according to all statistics (e.g., ∆AIC i > 10, 58.1 % of cases). The dispersion of model parameters prevented the derivation of a secondary model for the whole dataset, but clear trends and sufficient data (56 kinetic curves) were found to develop one for milk. A secondary WBLL model (b′ = 0.056–2.230, n = 0.758 − 0.403; 150–600 MPa) was the best alternative (AIC i = 183.8). A GMPZ model yielded similar predictions, but registered ∆AIC i = 19.3 reflecting its larger number of parameters (p = 8). Selecting datasets with pressure holding times of commercial interest (t ≤ 10 min) yielded different parameter estimates for the generic WBLL model (b′ = 0.079–1.859, n = 1.340–0.557; 300–600 MPa). In conclusion, information theory criteria complemented standard statistics, and the simpler WBLL secondary model (p = 4) provided a product-specific time-pressure function of industrial relevance.

Mechanical annealing model of damage in arrays of nanopillars

The mechanical damage accumulation occurring in an array of axially loaded nanopillars is studied within the stochastic approa ch. To each nanopillar, an initial strength- -threshold is assigned which is drawn from the Weib ull probability distribution. Under the influence of load exceeding the strength-thresh old the pillar height is reduced. Then the reduced pillar gets new higher strength-thresho ld and its load is transferred to other working pillars. We analyse slip loads causing the system height reductions and avalanches of the pillar height reductions.

Quantitative genetic variation of resistance against Piscirickettsia salmonis in Atlantic salmon (Salmo salar)

Piscirickettsiosis (Piscirickettsia salmonis) is one of the diseases that cause large economic losses in Chilean salmon industry. Genetic improvement of disease resistance represents one strategy for controlling infectious diseases in farmed fish. However, knowledge of whether genetic variation exists for piscirickettsiosis resistance is needed in order to determine the feasibility of including this trait into the breeding goal. Using data from a challenge test performed on 2601 Atlantic salmon (Salmo salar) from 118 full-sib groups (40 half-sib groups) we found significant genetic variation for resistance to piscirickettsiosis. We used a cross-sectional linear model (CSL) and a binary threshold (probit) model (THR) to analyze the test-period survival, a linear model (LIN), Cox (COX) and Weibull (WB) frailty proportional hazard models to analyse the day at death, and a survival score (SS) model with a logit link to analyze the test-day survival. The estimated heritabilities for the different models ranged from 0.11 (SS) to 0.41 (COX). The Pearson and Spearman correlation coefficients between full-sib families estimated breeding values (EBVs) from the six statistical models were above 0.96 and 0.97, respectively. We used different data subsets, splitting the entire dataset both at random and by tank, in order to predict the accuracy of selection for each model. In both cases COX (0.8 and 0.79) and CSL (0.76 and 0.71) models showed the highest and the lowest accuracy of selection, respectively. These results indicate that resistance against P. salmonis in Atlantic salmon might be genetically improved more efficiently by means of using models which take both time to death and data censoring into account in the genetic evaluations.

Discriminating between the generalized Rayleigh and Weibull distributions

Generalized Rayleigh (GR) and Weibull (WE) distributions are used quite effectively for analysing skewed lifetime data. In this paper, we consider the problem of selecting either GR or WE distribution as a more appropriate fitting model for a given data set. We use the ratio of maximized likelihoods (RML) for discriminating between the two distributions. The asymptotic and simulated distributions of the logarithm of the RML are applied to determine the probability of correctly selecting between these two families of distributions. It is examined numerically that the asymptotic results work quite well even for small sample sizes. A real data set involving the annual rainfall recorded at Los Angeles Civic Center during 25 years is analysed to illustrate the procedures developed here.

European dry spell length distributions, years 1951–2000

Daily precipitation records of 267 European rain gauges are considered to obtain dry spell length (DSL) series along the second half of the twentieth century. A dry spell consists of consecutive days with daily rain amount below a given threshold, R0. Four DSL series are obtained for R0 values equal to 0.1, 1.0, 5.0, and 10.0 mm/day, and their empirical distributions are properly fitted to different statistical models: Pearson type III (PE3), Weibull (WEI), generalised Pareto, (GPA) and lognormal distributions. The parameters of every model are estimated by L-moments, and the goodness of fit is assessed by quantifying discrepancies between empirical and theoretical distributions in the L-skewness–kurtosis diagrams. The most common best-fitting model is PE3, especially for 0.1 and 1.0 mm/day. Nevertheless, a few stations in southern Europe are better modelled by the WEI distribution. For 5.0 and 10.0 mm/day, the spatial distribution of the best-fitting model is more heterogeneous than for the lowest thresholds. While PE3 is still the preferred model for Western Europe, some DSL series are better fitted to WEI or GPA models. Maps of DSL average and standard deviation and expected lengths for return periods of 2, 5, 10, 25, and 50 years show some common features. Whereas for thresholds of 0.1 and 1.0 mm/day, a N–S gradient is detected, especially in Mediterranean areas; for 5.0 and 10.0 mm/day, a NW–SE gradient is observed in the Iberian Peninsula and a SW–NE gradient in the Scandinavian Peninsula. Then, the vicinity to Atlantic and Arctic Oceans and the Mediterranean Sea, as well as orographic features, are more determining factors than the latitude in patterns associated with the highest R0 thresholds. Finally, a regional frequency analysis based on a clustering algorithm is attempted for the four thresholds R0, with the PE3 model as the parent distribution for the largest clusters.

DEPENDABLE DISCHARGES OF THE UPPER AND MIDDLE DIYALA BASINS

The Diyala river is a major tributary of the Tigris river that runs through Iran and Iraq which drains an area of 32600 km2. The catchment area consist of four parts which are the area of the basin above Derbendi-Khan, Upper Diyala, Middle Diyala and lower Diyala. Through the last two decade the water coming to the upstream Derbendi-Khan reduced and this reduction reach to about 35%. This research focusing on the water dependable discharges reached to upper and middle Diyala river basin as compared with Russian study(water balance). The water balance mention that the discharge can be supplied the above basins with probability of 95%for the year (1947-1948). In this research consider the mean discharge down stream Derbendi-Khan dam for the period (1989-2007) in order to find the dependable probability for discharge which can be satisfied and equivalent to the water balance study .In this research considered different probabilities (95%,80% and 50%) of each months (Oct-Sep) water year for the period (1989-2007) by using the empirical Wei Bull probability equation, and it was conclude that the probability of 50% can be considered the suitable probability for the water requirement to upper and middle basins.

Pasture height distribution of grazed tropical native grammas in Veracruz State, Mexico

AbstractPasture height measurement and its statistical distribution have been neglected topics in tropical pasture research. Thus, the objective of this study was to fit the single normal (SN), double normal (DN), log normal (LN), gamma (GM) and Weibull (WB) models to relative frequency distributions of pasture height (PHT, cm) of native gramma pastures of the Mexican humid tropics. PHT measurements were done in a study where fixed rotational grazing was used (3‐day grazing/27‐day rest paddock–1) at three stocking rates of two, three and four F1 (Holstein × Zebu) cows ha−1. Two paddocks per stocking rate were used. Approximately 400 PHT measurements were done on each paddock, before (22 times) and after (23 times) grazing. The models were fit by least squares, using the Marquardt–Lavenberg iteration. The Akaike’s information criterion corrected (AICc) was used to select the model with the best fit. Before grazing, the models with the best fit were the DN and the GM, and after grazing, the best fit was shown by the DN and the LN. With respect to stocking rate, the best models were: LN and GM for two cows ha−1; DN and LN for three cows ha−1; and DN for four cows ha−1. However, the differences between the AICc values of the model with the best fit and the second in rank were small; thus, either could fit the data equally well. In conclusion, PHT of native gramma pastures did not distribute normally, and either the double normal or the log normal models can be used as means to describe the PHT frequency distributions of this type of pastures.