Nelspruit is located 330 km east of Pretoria in Crocodile River catchment (Figure 1). Crocodile River catchment is part of Incomati catchment (Figure 2). Its geographical coordinates are 25°27′S and 30°58′E. Its altitude is 667 m a.s.l. Nelspruit normally receives about 667 mm of rain per year with most rainfall occurring in months of December, January, February and March. On average, it receives the lowest rainfall (2 mm) in June and the highest rainfall (119 mm) in December. The monthly distribution of average daily maximum temperatures shows that the average midday temperatures for Nelspruit range from 21.4 °C in June to 27.9 °C in January. The region is the coldest during July when the temperature drops to 6.2 °C on average during the night. Maps of Crocodile River catchment and Incomati catchment are shown in Figures 1 and 2.

The daily rainfall (mm) data for the years 1961–2015 for Nelspruit were obtained from WR 90 and WR 2005 studies, which were carried out by Water Research Council (WRC) (Middleton & Bailey 2005). In these studies, the methods of ensuring and improving quality of daily rainfall data including data used in this study were outlined (Middleton & Bailey 2005). From the daily rainfall data of each year, the largest annual daily rainfall event was extracted. The extracted largest annual daily rainfall events for the years 1961–2015 formed the AMDR events. The AMDR events for the period 1961–2015 are presented in Table 1.

Empirical continuous probability distribution function (ECPDF) was applied to determine the return periods of the AMDR events for Nelspruit and its environs for the period 1961–2015. The probability of exceedances (p) of the AMDR events was determined by the rank-order method. This method involved ordering the events from the largest event to the smallest event. Rank 1 was assigned to the largest event and rank 55 to the smallest event. The data sample size was 55 events.

To obtain p of each event, Weibull formula (Weibull 1939) was applied:

p is the exceedance probability for an event with rank i, i is the rank of the event, n is the sample size that was equal to 55 in this case. The return period (T) of each event is defined as the inverse of its exceedance probability (Weibull 1939):

The empirical return period of each AMDR event was determined by applying equation 2.

The AMDR events ≥ 100 mm were identified as the events that caused flood hazards resulting in flood disasters in Nelspruit and its environs (Kelbe 1984). AMDR event of magnitude 100 mm was adopted as threshold. AMDR events of magnitude 100 mm or ≥ 100 mm were taken as flood causing events, therefore flood disaster events. Based on this adoption, the model for flood disaster risk analysis (exceedance probability of event X being equal to or greater than threshold x_T at least once in N years) was developed as demonstrated below.

Letting x_T be the threshold of AMDR variable. The exceedance probability (p) of X being equal to or greater than x_T at least once in N years of record was formulated as:

Equation 6 was applied to determine the occurrence probability of flood disaster risk associated with AMDR events of magnitude ≥ 100 mm at least once in 1, 2, 3, 4 and 5 years.

Theoretical continuous probability distribution functions (TCPDFs) were applied to estimate quantiles of AMDR events of return periods up to 50 years. The estimation of the quantiles was carried out by first identifying the TCPDFs which adequately fitted the AMDR events. Masereka et al. (2015) have developed a methodology to identify candidate and best-fit TCPDFs for frequency analysis of hydrometeorological events. This methodology was adopted to identify candidate and best-fit TCPDFs for frequency analysis of the AMDR events. The identified candidate TCPDFs were subjected to three goodness-of-fit tests, namely CS, Kolmogorov–Smirnov (KS) and Anderson–Darling (AD), to identify the best-fit TCPDF.

Three goodness-of-fit tests which were applied to identify the best-fit TCPDF from the candidate TCPDFs are described below.

From the AMDR data, probability–probability(P–P) plots were constructed and applied to visually identify the candidate TCPDFs for analysis of magnitude and frequency of AMDR events. The application of P–P plots involved plotting ECPDF values against TCPDF values of 50 continuous and discrete PDFs supported by EasyFit software (Mathwave 2015). The three TCPDF curves closest to the diagonal of the plots were selected as the candidate TCPDFs (Mathwave 2015).

Quantile–quantile (Q–Q) plots were applied to confirm the identified candidate TCPDFs. The application of Q–Q plot involved plotting the AMDR data: X_i (i = 1 … . .55) against the X-axis, and the corresponding values against the Y- axis: F⁻¹(F_n (X_i) – 0.5/n), where F⁻¹(X) is the inverse cumulative TCPDF and F_n (X) is the ECPDF. The proximity of the TCPDF plots to the displayed diagonal in EasyFit graph indicated the confirmation of the candidate TCPDFs (Mathwave 2015).

Based on the test statistics: X², KS and A² of each candidate TCPDF, the candidate TCPDFs were ranked. The candidate TCPDF with the lowest test statistic was ranked 1 and the candidate TCPDF with the largest test statistic was ranked 3. The total number of identified candidate TCPDF was 3. The candidate TCPDF with the least sum of ranks from the three goodness-of-fit was considered to be the best-fit TCPDF for the analysis of magnitude and frequency of the AMDR events.

The parameters of the identified best-fit TCPDF were estimated using maximum likelihood method using EasyFit 5.5 software (Mathwave 2015:7). Based on the PDF of the identified best-fit TCPDF, the estimated parameters were applied to develop quantile function (QF) (Mathwave 2015). The developed QF was applied to estimate quantiles of AMDR of return periods of 2, 3, 4, 5, 10, 15, 20, 25 and 50 years.

To determine how well the developed QF fitted the AMDR events, a probability difference (P–D) plot was constructed. P–D plot is a plot of the difference between the ECPDF and the TCPDF (Mathwave 2015):

where F_n (X) was the ECPDF and F (X) was the identified best-fit TCPDF which was generalised logistic (GL).

The model (equation 13) proposed by Stedinger, Vogel and Foufoula-Georgiou (1993) to estimate 95% confidence intervals of quantiles estimated by the application of GL distribution function was applied to determine the confidence intervals of estimated quantiles of return periods of 2, 3, 4, 10, 15, 20, 25 and 50 years:

where ZI−∞2 is the upper 100 (∝/2)% percentile of the normal distribution. x^{^} _T is the estimated quantile of return period (T).

The descriptive statistics of AMDR events of Nelspruit for the period 1961–2015 are presented in Table 2.

The mean of AMDR events was 62.74 mm. The median was 58.00 mm. The Skewness of the AMDR events was positive (0.09), indicating that the TCPDF of AMDR events had the tail that was on the right. This tail was longer or taller than on the left. These figures indicated that the AMDR events were not normally distributed.

Equation 2 expresses the relationship between plotting positions of AMDR events and the corresponding empirical return periods that are presented in Table 2. The empirical return period of 100 mm AMDR event that has been adopted in this study as a threshold is 9.17 years (Table 3). Results of several studies have also shown that the return period of flood hazards that result in flood disasters in Nelspruit and its environs is 10 years (Kelbe 1984).

In this study, the AMDR event of magnitude 100 mm was taken as threshold (x_T) as indicated in sections above. AMDR events equal to or greater than x_T were considered as flood disaster events. Taking x_T = 100 mm, from the sample data in this study (Table 3), the number of occurrence of events x_T ≥ 100 mm was 6, and the number of intervals was 5. Therefore, the empirical return period of event X_T was 10 years (55/5 = 10 years).

The exceedance probability of flood disaster event (AMDR event: X ≥ 100 mm) at least once in 1, 2, 3, 4 and 5 years calculated by applying equation 6 is presented in Table 4.

The exceedance probability of flood disaster event (≥ 100 mm) occurring at least once in 5 years is 0.41. The results are important for decision-making on the acceptable level of risk because of flooding that should be associated with designing specific to infrastructure for the reduction of the risk of flood in Nelspruit and environs.

Applying P–P and Q–Q plots, GL, GEV and Gumbel maximum (GM) were identified as the candidate TCPDFs for frequency analysis of the AMDR events by visual identification method. The developed plots are presented in Figures 3 and 4.

The P–P plot (Figure 3) showed that GL, GEV and GM distribution functions were closest to the plot diagonal. This indicated that the GL, GEV and GM were the candidate functions for frequency analysis of the AMDR events (Mathwave 2015). However, it was not possible to identify the best-fit TCPDF for describing the AMDR events. The Q–Q plot (Figure 4) showed that GL, GEV and GM distribution functions underestimated AMDR quantiles of magnitudes < 40 mm. The three functions accurately estimated AMDR quantiles of magnitudes of the range 40 mm – 95 mm. Between 95 mm and 110 mm, all the three models overestimated the quantiles. Quantiles > 110 mm were underestimated by GL and GEV but overestimated by GM. The diagonal line of the Q–Q plot was the reference. If a distribution function curve lays on the diagonal, this indicates that the function accurately estimates the quantiles (Mathwave 2015).

The results of identifying the best-fit PDF for frequency analysis of the AMDR events for Nelspruit and environs are presented in Table 5.

The rankings of the candidate TCPDFs as the best-fit TCPDFs for the frequency analysis of AMDR events based on statistics of the three goodness-of-fit tests are presented in Table 5. The candidate TCPDF with the least sum of the rankings was GL distribution. GL distribution was therefore identified as the best-fit TCPDF because it had the lowest sum of rankings.

Generalised logistic PDF is defined by the QF as (Shin et al. 2011):

α is the scale parameter, β is the shape parameter, Σ is the location parameter.

The parameters, Σ, α and β, were estimated by applying the method of maximum likelihood (ML) in Mathwave software (Mathwave 2015). These parameters are shown in Table 6.

The specific QF of GL probability function for estimating AMDR events for Nelspruit and environs was developed based on the parameters in Table 6 as indicated in Equation 15:

Equation 15 was applied to estimate quantiles of return periods 2, 3, 4, 5, 10, 15, 20, 25 and 50 years. The estimated quantiles are presented in Table 7.

The estimated quantiles of the AMDR events of return periods between 2 and 50 years which were obtained by applying equation 15 are presented in Table 7. The comparison of results of magnitude analysis of the AMDR events based on ECPDFs and TCPDFs (Table 7) shows that GL PDF fairly accurately estimates the AMDR events of magnitudes 61.61 mm – 101.51 mm. The performance of GL PDF in estimating magnitudes of AMDR events is further demonstrated by the Q–Q plot results. The Q–Q plot based on GL PDF is presented in Figure 5.

The GL PDF was identified as the best-fit TCPDF in this study (Table 5). It should be noted that GL PDF accurately estimated the annual maximum daily rainfall events of range 40 mm – 90 mm (Figure 5). This range corresponds to quantiles of return period range 1.38–7.86 years (Table 1, Figure 5). GL PDF overestimated AMDR events of range 90 mm – 112 mm (Figure 5). This range corresponds to quantiles of return period range 9.17–27.50 years (Table 1, Figure 5). GL PDF under estimated the annual maximum daily rainfall events which were > 112 mm (Figure 5). These results demonstrate the fact that even the magnitude estimates obtained from the identified best-fit TCPDF should be used in infrastructure design with caution.

The results of probability difference (P–D) plot between ECPDF (based on the sample) and GL PDF are presented in Figure 6.

The P–D plot (Figure 6) showed that the probability difference between the AMDR events (the sample) and the quantiles estimated by GL > 100 mm is < 0.03 (3%). This showed that GL estimated the quantiles > 100 mm fairly accurately.

The 95% confidence interval of the estimated quantiles is presented in Table 8.

From the confidence intervals, the relative size of uncertainty at different return periods and the importance of different sources of uncertainty could be determined (Kjeldsen & Jones 2004). However, this was not part of this study.

From the literature, AMDR events of magnitude 100 mm or greater led to flood hazards which resulted in flood disasters in Nelspruit and its environs. Records of flood disasters from 1974 to 2014 cited in this article indicated that flood hazards which resulted in flood disasters in that period were caused by AMDR events of magnitudes ≥ 100 mm. The AMDR event of magnitude 100 mm was adopted as a threshold. The AMDR events of magnitude ≥ 100 mm were assumed to be flood disaster events.

From the empirical distribution function frequency analysis, the return period of 100 mm AMDR event was 10 years. The return period of 100 mm AMDR event was 9.17 years according to plotting point applying Weibull (1939) method (Table 3). The estimate of the return periods from the two methods was in agreement. The results indicated that if the design return period of a hydraulic infrastructure being designed is less or equal to data record period, estimation of quantiles by empirical distribution function or plotting point methods is recommendable.

Generalised logistic probability distribution function was identified as the best-fit function for frequency analysis of AMDR events in Nelspruit and environs. In other studies elsewhere, different TCPDFs have been identified as the best-fit TCPDFs for frequency analysis of AMDR events. In USA, GEV was identified as the best-fit PDF for frequency analysis of AMDR events (Bonnin et al. 2006; Naghavi & Yu 1995). In Malaysia, Daud et al. (2002) identified Log-Gumbel as the best-fit function for frequency analysis of AMDR events. In Cote de Ivoire, Goula Bi et al. (2010) identified Gumbel and Lognormal functions as the best-fit TCPDFs for frequency analysis of AMDR events in Cote de Ivoire, and Smithers and Schulze (2001) proposed Log Pearson 3 function as the best-fit PDF for AMDR events in South Africa.

The performance of GL TCPDF was assessed by applying Q–Q and P–D plots (Figures 4 and 5). Both plots affirmed that the performance of GL was satisfactory. However, there seem to be no universally accepted TCPDF for frequency analysis of AMDR events. Therefore, it is advisable to carry out best-fit TCPDF identification study first for designing any hydro meteorological infrastructure projects in specific regions or sites rather than just adopting any TCPDF.

The results of the study are applicable in flood hazard mapping. The results can also be applied in formulating mitigation of flood hazard in two ways. An engineering approach in designing flood control structures, for example designing and planning storm water and flood management and control infrastructure like urban drainage systems, culverts and bridges. Regulatory approach designed to reduce vulnerability, for example flood plain zoning. The results of the study can also be applied in formulating methods of flood disaster risk reduction which include, among others, among others disaster preparedness.

The main limitation of the study was the short period of record of daily rainfall events data from which the AMDR events were extracted. The record was only 55 years. Length of record data affects the performance of empirical distribution function to estimate return period of magnitudes and therefore the associated risks. The performance is poor with short records. Deciding on the magnitude of AMDR events as a threshold for frequency analysis was another limitation.

Further studies on the topics below are recommended: