Thesis (M.Sc. (Statistics)) -- University of Limpopo, 2020 / Several studies indicated a growing trend in terms of frequency and severity of extreme events. Extreme rainfall could cause disasters that lead to loss of property and life. The aim of the study was to model the monthly rainfall variability in selected provinces of South Africa using extreme value distributions. This study investigated the best-fit probability distributions in the five provinces of South Africa. Five probability distributions: gamma, Gumbel, log-normal, Pareto and Weibull, were fitted and the best was selected from the five distributions for each province. Parameters of these distributions were estimated by the method of maximum likelihood estimators. Based on the Akaike information criteria (AIC) and Bayesian information criteria (BIC), the Weibull distribution was found to be the best-fit probability distribution for Eastern Cape, KwaZulu-Natal, Limpopo and Mpumalanga, while in Gauteng the best-fit probability distribution was found to be the gamma distribution. Monthly rainfall trends detected using the Mann–Kendall test revealed significant monotonic decreasing long-term trend for Eastern Cape, Gauteng and KwaZulu-Natal, and insignificant monotonic decreasing longterm trends for Limpopo and Mpumalanga. Non-stationary generalised extreme value distribution (GEVD) and non-stationary generalized Pareto distribution (GPD) were applied to model monthly rainfall data. The deviance statistic and likelihood ratio test (LRT) were used to select the most appropriate model. Model fitting supported stationary GEVD model for Eastern Cape, Gauteng and KwaZulu-Natal. On the other hand, model fitting supported non-stationary GEVD models for maximum monthly rainfall with nonlinear quadratic trend in the location parameter and a linear trend in the scale parameter for Limpopo, while in Mpumalanga the non-stationary GEVD model, which has a nonlinear quadratic trend in the scale parameter and no variation in the location parameter fitted well to the maximum monthly rainfall data. Results from the non-stationary GPD models showed that inclusion of the time covariate in our models was not significant for Eastern Cape, hence the bestfit model was the stationary GPD model. Furthermore, the non-stationary GPD model with a linear trend in the scale parameter provided the best-fit for KwaZulu-Natal and Mpumalanga, while in Gauteng and Limpopo the nonstationary GPD model with a nonlinear quadratic trend in the scale parameter fitted well to the monthly rainfall data. Lastly, GPD with time-varying thresholds was applied to model monthly rainfall excesses, where a penalised regression cubic smoothing spline was used as a time-varying threshold and the GPD model was fitted to cluster maxima. The estimate of the shape parameter showed that the Weibull family of distributions is appropriate in modelling the upper tail of the distribution for Limpopo and Mpumalanga, while for Eastern Cape, Gauteng and KwaZulu-Natal, the exponential family of distributions was found to be appropriate in modelling the upper tail of the distribution. The dissertation contributes positively to the body of knowledge in extreme value theory application to rainfall data and makes recommendations to the government agencies on the long-term rainfall variability and their negative impact on the economy.

Thesis (M. Sc. (Statistics)) -- University of Limpopo, 2020 / The main purpose of modelling rare events such as heavy rainfall, heat waves, wind speed, interest rate and many other rare events is to try and mitigate the risk that might arise from these events. Heavy rainfall and floods are still troubling many countries. Almost every incident of heavy rainfall or floods might result in loss of lives, damages to infrastructure and roads, and also financial losses. In this dissertation, the interest was in modelling average monthly rainfall for South Africa using extreme value theory (EVT). EVT is made up mainly of two approaches: the block maxima and peaks-over thresh old (POT). This leads to the generalised extreme value and the generalised Pareto distributions, respectively. The unknown parameters of these distri butions were estimated using the method of maximum likelihood estimators in this dissertation. According to goodness-of-fit test, the distribution in the Weibull domain of attraction, Gumbel domain and generalised Pareto distri butions were appropriate distributions to model the average monthly rainfall for South Africa. When modelling using the POT approach, the point process model suggested that some areas within South Africa might experience high rainfall in the coming years, whereas the GPD model suggested otherwise. The block maxima approach using the GEVD and GEVD for r-largest order statistics also revealed similar findings to that of the GPD. The study recommend that for future research on average monthly rainfall for South Africa the findings might be improved if we can invite the Bayesian approach and multivariate extremes. Furthermore, on the POT approach, time-varying covariates and thresholds are also recommended. / National Research Foundation (NRF) and South African Weather Service (SAWS)

The objectives of this research were to derive a design 24-hour duration rainfall distribution for use in southwest Florida, and peak rate factors for use in the Soil Conservation Service (SCS) unit hydrograph method for two watersheds, also in the southwest Florida area. The rainfall distribution is derived by applying a least squares polynomial curve fitting technique to National Weather Service hourly rainfall data collected in the study area. The screening criteria for data included in the curve fitting procedure are: storm duration of 18 to 26 hours, at least three inches of rainfall volume, and peak intensity period falling near the center of the storm. The analysis technique includes converting the raw data to dimensionless form which allows the flexibility of applying the 24-hour storm. Peak rate or attenuation factors are determined for the Hickory Creak (2400 acres) and the Gallagher Ditch (300 acres) the U.S. Geological Survey is used for the analysis. The screening criteria for the hydrographs produced from this data include a stable baseflow condition and a single hydrograph peak. The resulting hydrographs provide input to the Soil Conservation Service triangular unit hydrograph. An average of the peak rate factors calculated from the screened hydrographs is taken as the suitable factor for each watershed. The project results are then compared to the currently used rainfall distributions and the SCS peak rate default value of 484. The comparisons are accomplished by modeling a hypothetical watershed using both the SCS unit hydrograph and the Santa Barbara methods. The model Is run on a microcomputer using seven distributions, three return frequency volumes of rainfall, differing watershed sizes, times of concentrations and antecedent moisture conditions. The conclusion recommends a rainfall distribution and peak rate factor best suited to estimate hydrographs.

The concept of Intensity Duration Frequency (IDF) relationship curve presents crucial design contribution for several decades under the assumption of a stationary climate, the frequency and intensity of extreme rainfall nonetheless seemingly increase worldwide. Based on the research conducted in recent years, the greatest increases are likely to occur in short-duration storms lasting less than a day, potentially leading to an increase in the magnitude and frequency of flash floods. The trend analysis of the precipitation influencing the climate variability and extreme rainfall in the state of Florida is conducted in this study. Since these local changes are potentially or directly related to the surrounding oceanic-atmospheric oscillations, the following oscillations are analyzed or highlighted in this study: Atlantic Multi-Decadal Oscillation (AMO), El Niño Southern Oscillation (ENSO), and Pacific Decadal Oscillations (PDO). Collected throughout the state of Florida, the precipitation data from rainfall gages are grouped and analyzed based on type of duration such as short-term duration or minute, in hourly and in daily period. To assess statistical associations based on the ranks of the data, the non-parametric tests Kendall’s tau and Spearman’s rho correlation coefficient are used to determine the orientation of the trend and ultimately utilize the testing results to determine the statistical significance of the analyzed data. The outcome of the latter confirms with confidence whether there is an increasing or decreasing trend in precipitation depth in the State of Florida. The main emphasis is on the influence of rainfall extremes of short-term duration over a period of about 50 years. Results from both Spearman and Mann-Kendall tests show that the greatest percentage of increase occurs during the short rainfall duration period. The result highlights a tendency of increasing trends in three different regions, two of which are more into the central and peninsula region of Florida and one in the continental region. Given its topography and the nature of its water surface such as the everglades and the Lake Okeechobee, Florida experience a wide range of weather patterns resulting in frequent flooding during wet season and drought in the dry season. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Several previous studies have found that various characteristics of the Indian monsoon rainfall have shown secular changes over the past century. In this study, using a gridded (1degree) daily rainfall dataset, we analyse the spatio-temporal characteristics of the intensity and duration of monsoon (June through September) rainfall for secular changes over the last 50 years. The characteristics of the duration of rain events are described by wet and dry spells. A wet/dry spell is defined as a period of consecutive days with rainfall above/below a particular threshold. We choose to use a threshold that is a function of the local climatological mean, given the spatial heterogeneity of mean monsoon rainfall. The wet and dry spells are then divided into three categories: short [1 to 7 days], moderate [8 to 10 days], long [11 and more days] and analysed for changes over the past 50 years [19512000]. We find that while the number of short duration wet spells show a significant increase over the last 50 years (~15% change), the number of long duration wet spells show a significant decrease (~25%). Furthermore, while the numbers of short duration dry periods have shown a significant increase, the moderate and long duration dry spells do not shown an appreciable change. This increase and decrease in the short and long duration wet spells offset each other and consequently the total number of rainy days during the season has not shown any significant change over the past 50 years. In addition to the duration of wet and dry spells, we also analysed for changes in the accumulated rainfall of the short, medium and long duration wet spells. Our analysis suggests that while the depth of accumulated rainfall in short duration wet spells has shown a significant increase (~20%), the depth of rain in the long duration spells has shown a significant decrease (~30%) in the past fifty years.

Error characteristics associated with satellite-derived precipitation products are important for atmospheric and hydrological model data assimilation, forecasting, and climate diagnostic applications. This information also aids in the refinement of physical assumptions within algorithms by identifying geographical regions and seasons where existing algorithm physics may be incorrect or incomplete. Examination of relative errors between independent estimates derived from satellite microwave data is particularly important over regions with limited surface-based equipments for measuring rain rate such as the global oceans and tropical continents. In this context, analysis of microwave based satellite datasets from the Tropical Rainfall Measuring Mission (TRMM) enables to not only provide information regarding the inherent uncertainty within the current TRMM products, but also serves as an opportunity to prototype error characterization methodologies for the TRMM follow-on program, the Global Precipitation Measurement (GPM) . Most of the TRMM uncertainty evaluation studies focus on the accuracy of rainfall accumulated over time (e.g., season/year). Evaluation of instantaneous rainfall intensities from TRMM orbital data products is relatively rare. These instantaneous products are known to potentially cause large uncertainties during real time flood forecasting studies at the watershed scale. This is more so over land regions, where the highly varying land surface emissivity offers a myriad of complications, hindering accurate rainfall estimation. The error components of orbital data products also tend to interact nonlinearly with hydrologic modeling uncertainty. Keeping these in mind, the present thesis fosters the development of uncertainty analysis using instantaneous satellite orbital data products (latest version 7 of 1B11, 2A25, 2A23, 2B31, 2A12) derived from the passive and active microwave sensors onboard TRMM satellite, namely TRMM Microwave Imager (TMI) and precipitation radar (PR). The study utilizes 11 years of orbital data from 2002 to 2012 over the Indian subcontinent and examines the influence of various error sources on the convective and stratiform precipitation types. Two approaches are taken up to examine uncertainty. While the first approach analyses independent contribution of error from these orbital data products, the second approach examines their combined effect. Based on the first approach, analysis conducted over the land regions of Mahanadi basin, India investigates three sources of uncertainty in detail. These include 1) errors due to improper delineation of rainfall signature within microwave footprint (rain/no rain classification), 2) uncertainty offered by the transfer function linking rainfall with TMI low frequency channels and 3) sampling errors owing to the narrow swath and infrequent visits of TRMM sensors. The second approach is hinged on evaluating the performance of rainfall estimates from each of these orbital data products by accumulating them within a spatial domain and using error decomposition methodologies. Microwave radiometers have taken unprecedented satellite images of earth’s weather, proving to be a valuable tool for quantitative estimation of precipitation from space. However, as mentioned earlier, with the widespread acceptance of microwave based precipitation products, it has also been recognized that they contain large uncertainties. One such source of uncertainty is contributed by improper detection of rainfall signature within radiometer footprints. To date, the most-advanced passive microwave retrieval algorithms make use of databases constructed by cloud or numerical weather model simulations that associate calculated microwave brightness temperature to physically plausible sample rain events. Delineation of rainfall signature from microwave footprints, also known as rain/norain classification (RNC) is an essential step without which the succeeding retrieval technique (using the database) gets corrupted easily. Although tremendous advances have been made to catapult RNC algorithms from simple empirical relations formulated for computational expedience to elaborate computer intensive schemes which effectively discriminate rainfall, a number of challenges remain to be addressed. Most of the algorithms that are globally developed for land, ocean and coastal regions may not perform well for regional catchments of small areal extent. Motivated by this fact, the present work develops a regional rainfall detection algorithm based on scattering index methodology for the land regions of study area. Performance evaluation of this algorithm, developed using low frequency channels (of 19 GHz, 22 GHz), are statistically tested for individual case study events during 2011 and 2012 Indian summer monsoonal months. Contingency table statistics and performance diagram show superior performance of the algorithm for land regions of the study region with accurate rain detection observed in 95% of the case studies. However, an important limitation of this approach is comparatively poor detection of low intensity stratiform rainfall. The second source of uncertainty which is addressed by the present thesis, involves prediction of overland rainfall using TMI low frequency channels. Land, being a radiometrically warm and highly variable background, offers a myriad of complications for overland rain retrieval using microwave radiometer (like TMI). Hence, land rainfall algorithms of TRMM TMI have traditionally incorporated empirical relations of microwave brightness temperature (Tb) with rain rate, rather than relying on physically based radiative transfer modeling of rainfall (as implemented in TMI ocean algorithm). In the present study, sensitivity analysis is conducted using spearman rank correlation coefficient as the indicator, to estimate the best combination of TMI low frequency channels that are highly sensitive to near surface rainfall rate (NSR) from PR. Results indicate that, the TMI channel combinations not only contain information about rainfall wherein liquid water drops are the dominant hydrometeors, but also aids in surface noise reduction over a predominantly vegetative land surface background. Further, the variations of rainfall signature in these channel combinations were seldom assessed properly due to their inherent uncertainties and highly non linear relationship with rainfall. Copula theory is a powerful tool to characterize dependency between complex hydrological variables as well as aid in uncertainty modeling by ensemble generation. Hence, this work proposes a regional model using Archimedean copulas, to study dependency of TMI channel combinations with respect to precipitation, over the land regions of Mahanadi basin, India, using version 7 orbital data from TMI and PR. Studies conducted for different rainfall regimes over the study area show suitability of Clayton and Gumbel copula for modeling convective and stratiform rainfall types for majority of the intraseasonal months. Further, large ensembles of TMI Tb (from the highly sensitive TMI channel combination) were generated conditional on various quantiles (25th, 50th, 75th, 95th) of both convective and stratiform rainfall types. Comparatively greater ambiguity was observed in modeling extreme values of convective rain type. Finally, the efficiency of the proposed model was tested by comparing the results with traditionally employed linear and quadratic models. Results reveal superior performance of the proposed copula based technique. Another persistent source of uncertainty inherent in low earth orbiting satellites like TRMM arise due to sampling errors of non negligible proportions owing to the narrow swath of satellite sensors coupled with a lack of continuous coverage due to infrequent satellite visits. This study investigates sampling uncertainty of seasonal rainfall estimates from PR, based on 11 years of PR 2A25 data product over the Indian subcontinent. A statistical bootstrap technique is employed to estimate the relative sampling errors using the PR data themselves. Results verify power law scaling characteristics of relative sampling errors with respect to space time scale of measurement. Sampling uncertainty estimates for mean seasonal rainfall was found to exhibit seasonal variations. To give a practical demonstration of the implications of bootstrap technique, PR relative sampling errors over the sub tropical river basin of Mahanadi, India were examined. Results revealed that bootstrap technique incurred relative sampling errors of <30% (for 20 grid), <35% (for 10 grid), <40% (for 0.50 grid) and <50% (for 0.250 grid). With respect to rainfall type, overall sampling uncertainty was found to be dominated by sampling uncertainty due to stratiform rainfall over the basin. In order to study the effect of sampling type on relative sampling uncertainty, the study compares the resulting error estimates with those obtained from latin hypercube sampling. Based on this study, it may be concluded that bootstrap approach can be successfully used for ascertaining relative sampling errors offered by TRMM-like satellites over gauged or ungauged basins lacking in in-situ validation data. One of the important goals of TRMM Ground Validation Program has been to estimate the random and systematic uncertainty associated with TRMM rainfall estimates. Disentangling uncertainty in seasonal rainfall offered by independent observations of TMI and PR enables to identify errors and inconsistencies in the measurements by these instruments. Motivated by this thought, the present work examines the spatial error structure of daily precipitation derived from the version 7 TRMM instantaneous orbital data products through comparison with the APHRODITE data over a subtropical region namely Mahanadi river basin of the Indian subcontinent for the seasonal rainfall of 6 years from June 2002 to September 2007. The instantaneous products examined include TMI and PR data products of 2A12, 2A25 and 2B31 (combined data from PR and TMI). The spatial distribution of uncertainty from these data products was quantified based on the performance metrics derived from the contingency table. For the seasonal daily precipitation over 10x10 grids, the data product of 2A12 showed greater skill in detecting and quantifying the volume of rainfall when compared with 2A25 and 2B31 data products. Error characterization using various error models revealed that random errors from multiplicative error models were homoscedastic and that they better represented rainfall estimates from 2A12 algorithm. Error decomposition technique, performed to disentangle systematic and random errors, testified that the multiplicative error model representing rainfall from 2A12 algorithm, successfully estimated a greater percentage of systematic error than 2A25 or 2B31 algorithms. Results indicate that even though the radiometer derived 2A12 is known to suffer from many sources of uncertainties, spatial and temporal analysis over the case study region testifies that the 2A12 rainfall estimates are in a very good agreement with the reference estimates for the data period considered. These findings clearly document that proper characterization of error structure offered by TMI and PR has wider implications in decision making, prior to incorporating the resulting orbital products for basin scale hydrologic modeling. The current missions of GPM envision a constellation of microwave sensors that can provide instantaneous products with a relatively negligible sampling error at daily or higher time scales. This study due to its simplicity and physical approach offers the ideal basis for future improvements in uncertainty modeling in precipitation.

MESHWR / Department of Hydrology and Water Resources / This study was aimed at investigating the long term seasonal and annual changes in rainfall duration and magnitude at Luvuvhu River Catchment (LRC). Rainfall in this catchment is highly variable and is characterised of extreme events which shift runoff process, affect the timing and magnitude of floods and drought, and alter groundwater recharge. This study was motivated by the year to year changes of rainfall which have some effects on the availability of water resources. Computed long term total seasonal, annual rainfall and total number of seasonal rainy days were used to identify trends for the period of 51 years (1965- 2015), using Mann Kendal (MK), linear regression (LR) and quantile regression methods. The MK, LR and quantile regression methods have indicated dominance of decreasing trends of the annual, seasonal rainfall and duration of seasonal rainfall although they were not statistically significant. However, statistical significant decreasing trends in duration of seasonal rainfall were identified by MK and LR at Matiwa, Palmaryville, Levubu, and Entabeni Bos stations only. Quantile regression identified the statistically significant decreasing trends on 0.2, 0.5 and 0.7 quantiles only in the Palmaryville, Levubu and Entabeni Bos, respectively. Stations with non-statistically significant decreasing trends of annual and seasonal rainfall had magnitude of change ranging from 0.12 to 12.31 and 0.54 to 6.72 mm, respectively. Stations with non-statistically increasing trends of annual and seasonal rainfall magnitude had positive magnitude of change ranging from 1.51 to 6.78 and 2.05 to 6.51 mm, respectively. The Study recommended further studies using other approaches to determine the duration of rainfall to improve, update and compare the results obtained in the current study. Continuous monitoring and installation of rain gauges are recommended on the lower reaches of the catchment for the findings to be of complete picture for the whole catchment and to also minimize the rainfall gaps in the stations. Water resources should be used in a sustainable way to avoid water crisis risk in the next generations. / NRF