Published Article / The objective of this study was to characterize meteorological droughts in the Central Region of South Africa, Modder River Basin, C52A quaternary catchment using two popular drought indices: Standardized Precipitation Index (SPI) and Standardized Precipitation-Evapotranspiration Index (SPEI) and to compare the two indices. Drought events were characterized based on their frequency, duration, magnitude and intensity. The indices were computed for the time-scales that are important for planning and management of water resources, i.e. 3-, 6- and 12-month time-scales. The basic meteorological input data used in the computation of these indices were 57 years (1950-2007) of monthly precipitation and monthly temperature data which were recorded at The Cliff weather station in the quaternary catchment. It was found that both SPI and SPEI responded to drought events in similar fashion in all time-scales. During the analysis period, a total of 37, 26 and 17 drought events were identified in the area based on 3-, 6-, and 12-month times-scales, respectively. Considering event magnitude as severity parameter, results from both indices identified the periods 1984-1985, 1992-1993 and 2003-2005 as the severest drought periods in the area. However, when the effects of both drought duration and magnitude are considered (drought intensity), the most severest drought events were identified during the years 1982/83, 1966 and 1973 based on 3-, 6- and 12-month timescales, respectively. It was concluded that although the SPEI generally exhibits veracity over SPI by including, apart from precipitation, additional meteorological parameter, mean temperature, SPI should be adopted as an appropriate drought monitoring tool in an area, like Africa, where meteorological data are scarce.

The dissertation focuses on the application of entropy theory in hydrologic analysis and simulation, namely, rainfall analysis, streamflow simulation and drought analysis. The extreme value distribution has been employed for modeling extreme rainfall values. Based on the analysis of changes in the frequency distribution of annual rainfall maxima in Texas with the changes in duration, climate zone and distance from the sea, an entropy-based distribution is proposed as an alternative distribution for modeling extreme rainfall values. The performance of the entropy based distribution is validated by comparing with the commonly used generalized extreme value (GEV) distribution based on synthetic and observed data and is shown to be preferable for extreme rainfall values with high skewness. An entropy based method is proposed for single-site monthly streamflow simulation. An entropy-copula method is also proposed to simplify the entropy based method and preserve the inter-annual dependence of monthly streamflow. Both methods are shown to preserve statistics, such as mean, standard deviation, skenwess and lag-one correlation, well for monthly streamflow in the Colorado River basin. The entropy and entropy-copula methods are also extended for multi-site annual streamflow simulation at four stations in the Colorado River basin. Simulation results show that both methods preserve the mean, standard deviation and skewness equally well but differ in preserving the dependence structure (e.g., Pearson linear correlation). An entropy based method is proposed for constructing the joint distribution of drought variables with different marginal distributions and is applied for drought analysis based on monthly streamflow of Brazos River at Waco, Texas. Coupling the entropy theory and copula theory, an entropy-copula method is also proposed for constructing the joint distribution for drought analysis, which is illustrated with a case study based on the Parmer drought severity index (PDSI) data in Climate Division 5 in Texas.

The objective of the present study is to forecast annual rainfall taking into account the periodicities and structure of the stochastic component. This study has six Chapters. Chapter 1 presents introduction to the problem and objectives of the study. Chapter 2 consists of review of literature. Chapter 3 deals with the model formulation and development. Chapter 4 gives an account of the application of the model. Chapter 5 presents results and discussions. Chapter 6 gives the conclusions drawn from the study. In this thesis the following model formulations are presented in order to achieve the objective. Fourier analysis model is used to identify periodicities that are present in the rainfall series.1 These periodic components are used to obtain discrotized ranges which is an essential input for the Fourier series model. Auto power regression model is developed for estimation of rainfall and hence to compute the first order residuals errlt The parameters of the model are estimated using genetic algorithm. The auto power regression model is of the form, ( Refer the PDF File for Formula) where αi and βi are parameters and M indicates modular value. Fourier series model is formulated and solved through genetic algorithm to estimate the parameters amplitude R, phase Φ and periodic frequency wj for the residual series errlt. The ranges for the parameters R, Φ and wj were obtained from Fourier analysis model. errl't= /µerrlt+ Σj Rcos(wjt+ Φ) Further, an integrated auto power regression and Fourier series model developed (with parameters of the model being known from the above analysis) to estimate new rainfall series Zesťt=Zµ Σ t αi(ZMi-t ) βi+µerrl+ Σj Rcos(wjt+ Φ) and the second order residuals, err2t is computed using, err2t = (zt –Zesťt) Thus, the periodicities are removed in the errlt series and the second order residuals err'2f obtained represents the stochastic component of the actual rainfall series. Auto regressive model is formulated to study the structure of the stochastic component err2t. The auto regressive model of order two AR(2) is found to fit well. The parameters of the AR(2) model were estimated using method of least squares. An exponential weighting function is developed to compute the weight considering weight as a function of AR{2) parameters. The product of weight and Gaussian white noise N(0, óerr2) is termed as weighted stochastic component. Also, drought analysis is performed considering annual (January to December) and summer monsoon (June to September) rainfall totals, to determine average drought interval (idrt) which is used in assigning signs to the random component of the forecasting model. In the final form of the forecasting model. Zest”t = Z µ Σ t αi(ZMi-t ) βi+µerrl+ Σj Rcos(wjt+ Φ) ± WT(Φ1, Φ2)N(0, óerr2) The weighted stochastic component is added or subtracted considering two criteria. Criterion I is used for all rainfall series except all-India series for which criterion II is used. The criteria also consider average drought interval Further, it can be seen that a ± sign is introduced to add or subtract the weighted stochastic component, albeit the stochastic component itself can either be positive or negative. The introduction of ± sign on the already signed value (instead of absolute value) is found to improve the forecast in the sense of obtaining more number of point rainfall estimates within 20 percent error. Incorporating significant periodicities, and weighted stochastic component along with average drought interval into the forecasting formulation is the main feature of the model. Thus, in the process of rainfall prediction, the genetic algorithm is used as an efficient tool in estimating optimal parameters of the auto power regression and the Fourier series models, without the use of an expensive nonlinear least square algorithm. The model application is demonstrated considering different annual rainfall series relating to IMD-Regions (RI...R5), all-india (AI), IMD-Subdivisions (S1...S29), Zones (Z1...Z10) and all-Karnataka (AK). The results of the proposed model are encouraging in providing improved forecasts. The model considers periodicity, average critical drought frequency and weighted stochastic component in forecasting the rainfall series. The model performed well in achieving success-rate of 70 percent with percentage error less than 20 percent in 4 out of 5 IMD Regions (R2 to R5), all-India, 17 out of 29 IMD Subdivisions (S1 to S5, S7 to S9, S18, S19, S21, S24 to S29) and all-Karnataka rainfall series. The model performance for Zones was not that-satisfactory as only 2 out of 10 Zones [Z1 and Z2) met the criterion. In a separate study, an effort was made to forecast annual rainfall using IMSL subroutine SPWF -which estimates Wiener forecast parameters. Monthly data is considered for the study. The Wiener parameters obtained were used to estimate monthly rainfall. The annual estimates obtained by simple aggregation of the monthly estimates compared extremely well with the actual annual rainfall values. A success rate of more than 80 percent with percentage error less than 10 percent is achieved in 4 out of 5 IMD Regions (R2 to R5), all-India, 18 out of 29 IMD Subdivisions (S1 to S8, S14, S18, S19, S22 to S24, S26 to S29) and all-Karnataka rainfall series. Whereas a success rate of 80 percent within 20 percent error is achieved in 4 out of 5 IMD Regions (except R1), all-India, 25 outof 29 IMD Subdivisions (except S10, S11, S12 and S17), all- Karnataka and 8 out of 10 Zones (except Z6 and Z8)(Please refer PDF File for Formulas)

The thesis is concerned with development of efficient regional frequency analysis (RFA) approaches to estimate quantiles of hydrometeorological events. The estimates are necessary for various applications in water resources engineering. The classical approach to estimate quantiles involves fitting frequency distribution to at-site data. However, this approach cannot be used when data at target site are inadequate or unavailable to compute parameters of the frequency distribution. This impediment can be overcome through RFA, in which sites having similar attributes are identified to form a region, and information is pooled from all the sites in the region to estimate the quantiles at target site. The thesis proposes new approaches to RFA of precipitation, meteorological droughts and floods, and demonstrates their effectiveness. The approach proposed for RFA of precipitation overcomes shortcomings of conventional approaches with regard to delineation and validation of homogeneous precipitation regions, and estimation of precipitation quantiles in ungauged and data sparse areas. For the first time in literature, distinction is made between attributes/variables useful to form homogeneous rainfall regions and to validate the regions. Another important issue is that some of the attributes considered for regionalization vary dynamically with time. In conventional approaches, there is no provision to consider dynamic aspects of time varying attributes. This may lead to delineation of ineffective regions. To address this issue, a dynamic fuzzy clustering model (DFCM) is developed. The results obtained from application to Indian summer monsoon and annual rainfall indicated that RFA based on DFCM is more effective than that based on hard and fuzzy clustering models in arriving at rainfall quantile estimates. Errors in quantile estimates for the hard, fuzzy and dynamic fuzzy models based on the proposed approach are shown to be significantly less than those computed for Indian summer monsoon rainfall regions delineated in three previous studies. Overall, RFA based on DFCM and large scale atmospheric variables appeared promising. The performance of DFCM is followed by that of fuzzy and hard clustering models. Next, a new approach is proposed for RFA of meteorological droughts. It is suggested that homogeneous precipitation regions have to be delineated before proceeding to develop drought severity - areal extent - frequency (SAF) curves. Drought SAF curves are constructed at annual and summer monsoon time scales for each of the homogeneous rainfall regions that are newly delineated in India based on the proposed approach. They find use in assessing spatial characteristics and frequency of meteorological droughts. It overcomes shortcomings associated with classical approaches that construct SAF curves for political (e.g., state, country) and physiographic regions (e.g., river basin), based on spatial patterns of at-site values of drought indices in the study area, without testing homogeneity in rainfall. Advantage of the new approach can be noted especially in areas that have significant variations in temporal and spatial distribution of precipitation (possibly due to variations in topography, landscape and climate). The DFCM is extended to RFA of floods, and its effectiveness in prediction of flood quantiles is demonstrated by application to Godavari basin in India, considering precipitation as time varying attribute. Six new homogeneous regions are formed in Godavari basin and errors in quantile estimates based on those regions are shown to be significantly less than those computed based on sub-zones delineated in Godavari basin by Central Water Commission in a previous study.