^{1}

^{*}

^{1}

Most planned developments in a catchment for control of excess water using a culvert, bridge or dam spillway are located at a site in a stream where there are no discharge measurements. Even though, for gauged catchments a number of established flood frequency models and rainfall-runoff models do exist, for ungauged catchments mostly regional flood frequency and event-based rainfall-runoff models are used, which depend on regional parameters. In this paper, a regional approach for design floods is presented and risk implication for design of drainage structures assessed. A case study in light of the above has been considered at four ungauged sites in the Limpopo Drainage Basin in north-eastern Botswana.

In the design of drainage structures, hydrological modelling, flood risk assessment and flood plain management design flood estimation are essential elements. Estimation of design flood for a catchment ideally requires sufficiently long period of flood data. Because of the limitations of at-site flow data, rainfall-based flood estimation methods are often preferred by the hydrologists [

The input in the application of rainfall-based flood estimation methods is the expected rainfall intensity for given storm duration and return period, usually known as design storms. Design storm values are needed in many hydrological studies, especially for providing an indirect estimation of the design flood. To this end, intensity-dura- tion-frequency (IDF) curves are often employed. These curves allow one to estimate the design storm, provided that historical rainfall extremes are available using scaling and stochastic models [

Even though, there is a network of autographic rain gauges in Botswana, their spatial distribution as well as at-site records is so short that they merit low credit for extrapolation to any ungauged sites. Most rain gauge records in Botswana have records of daily time interval that one may apply stochastic modeling of extreme daily rainfall series only. There is a limitation with the use of such data in the estimation of flush floods, which would last less than 24 hours. A limited study is carried out that indicated that much of rainfall in Botswana occurs in short spells of high intensity and occasionally very high intensities in short time periods [

Discharge gauging stations for a site of interest may be absent or usually sited far from locations where drainage facilities are needed. Rainfall runoff models are employed to determine design discharges from rainfall intensities of given duration and return period. Developing a unified model for derivation of IDF curves at a national scale remains vital. Otherwise, it is likely to introduce inconsistencies and significant bias in flood estimates for a given location, due to subjective treatment of the rainfall intensity and depth which is a major flood producing variable, leading to under- or over-design of engineering structures that include culverts, bridges, spillways or any flood control structures.

In this study we employ a regional flood estimation approach which relies on a regional parametric Intensity-Duration-Frequency (IDF) model developed for Botswana. The latter is a regional model for intensity-duration-frequency relationship for the estimation of rainfall extremes. It is derived by employing existing data, which is in use in Botswana by the Roads Department. The proposed formulation allows one to estimate the expected rainfall depth for a duration ranging from 5 minutes to 2 hours and beyond, and for low and medium return periods (up to 50 years) in any location in the country.

The proposed storm IDF model uses only annual rainfall data as an input. The use of the annual rainfall data, which is a measure of aridity in Botswana, is fairly reliable available information and its long-term distribution spatially in most regions over Botswana is fairly known, as documented in [

In the determination of design discharge, the transformation of rainfall to runoff is undertaken through hydrological models which use rainfall as a short time event storm inputs (from seconds to hours) and/or rainfall magnitudes that occur in days, weeks, months or beyond. The former commonly known as Design Event Approach is considered to determine design discharges. Design Event Approach for estimation of design floods is the currently practiced technique in different parts of the world [

Event-based simulations are applied to transform storm rainfalls to design discharges by using rainfall-runoff models. IDF curves/equations are used in a number of deterministic models to compute flood peaks, flood volumes, timing, etc. Examples of the currently widely used and available event-based and continuous rainfall runoff models are: i) the Rational method [

The first rainfall-runoff model we considered for evaluation of the risk associated with the regional design storm and flood modeling is the Soil Conservation Society (SCS) Curve Number (CN) method [

The second regional approach we used was the empirical deterministic method developed by Midgley and Pitman [^{2} is based on the following formula:

in which Q_{T} is the T-year flood peak of catchment exit at the bridge site (in m^{3}/s) from a drainage area, A (in km^{2}). The parameter K_{T} is a constant for T-year return period for which flood probability charts are available.

The hydrologic risk involved in a design flood magnitude corresponding to return period T is usually defied as the probability that the estimated flood will be exceeded at least once during the design life (N) in years of the structure and it is given by:

Equation (2) can be easily used to determine the risk, P for any design if the recurrence interval (T) associated with the design flood is known, along with the proposed design life of the structure. Due to the difficulty to determine the recurrence interval (T) from the flood formulas, we determined it indirectly from the return period of the design storm as a function of Extreme Value Type 1 Distribution (EV1):

where u and α are the parameters of EV1 distribution and y is EV1 reduced variate and x is the design storm intensity (i) for given duration and return period.

For this purpose x was determined by employing an empirical storm IDF model [

in which R is the mean annual rainfall (mm/a); i is average rainfall intensity (mm/hr); t_{d} is storm duration, time of concentration (minutes); T is return period (years); and a, b, c are constants that depend on the units employed.

In the scientific literature, the one used by the Texas Department of Transportation which is based on an exponential type formula for approximating the intensity-dura- tion-frequency curve is commonly used. This is a formula for synthetic hyetograph known as the Chicago method. It is based on the parameters of an assumed relationship of Intensity-Duration-Frequency (IDF), for a given return period, having the following form:

where i is average rainfall intensity (mm/hr or inch/hr); t_{d} = storm duration (minutes); and a, b, c are constants dependent on the units employed and the return period of the storm event. When Equation (6) was applied to storm characteristics in Botswana, it was found to be inferior to the storm IDF model of form given Equation (5) [

Furthermore, one difficulty of the model based on Equation (6) is that is applicable for a given return period which requires three parameter sets derived separately for each return period. Whereas, the former is a parsimonious model in that it is used for any return period, n which is a parameter in built with the model, which requires to

Duration (minutes) | Return period (years) | ||||
---|---|---|---|---|---|

5 | 10 | 25 | 50 | 100 | |

5 | 123 | 155 | 180 | 213 | 269 |

10 | 102 | 130 | 150 | 177 | 224 |

20 | 77 | 97 | 113 | 133 | 168 |

30 | 61 | 78 | 90 | 106 | 134 |

60 | 38 | 49 | 56 | 66 | 84 |

90 | 28 | 35 | 41 | 48 | 61 |

100 | 26 | 32 | 38 | 44 | 56 |

optimize only the 3 parameters (a, b, c) for any n. Whereas, the advantage of the first model proposed in this study, Equation (5), is versatile and robust and parsimonious. The only optimized model parameters are a, b and c, (

The methodology adopted in this study is based on parameter optimization of the above empirical formulation (Equation (5)) using the observed record of rainfall IDF. Finally a test of the models was performed to examine their applicability in reproducing the observed intensities qualitatively and quantitatively for the various return periods and durations. The design storms are inputs used in the event-based rainfall-runoff modeling approach to estimate design flood quantities, and risks associated with the use of the flood estimates in the design of drainage structures.

The observed record of rainfall IDF data is available as charts from the Roads Design Manual [

For Francistown station, which is close to the study catchments, the data digitized and used in the study to illustrate the applicability of the design storm model, for selected durations and return periods is summarized in

We have considered four ungauged catchments which are tributaries to the Limpopo drainage basin (

Nyamambisi River Crossing Marobela-Sebina Road (Nyamambisi).

Gatswane River located at a road crossing along Tshesebe-Butale road (Gatswane 1).

Gatswane River in the downstream side along Butale-Senyawe (Gatswane 2).

Shashe River at Kalakamate along the Msiwizi-Kalakamate road (Shashe).

The characteristics of the four catchments considered including their catchment area (A), length (L), slope (S), time of concentration were determined based on the Kirpich’s formula (Tc) and mean annual precipitation (R) are summarized in

Duration (minutes) | Location | A (km^{2}) | L (km) | S (%) | Tc (hour) | R (mm/a) | |
---|---|---|---|---|---|---|---|

Longitude | Latitude | ||||||

Nyamambisi | 27˚14.52' | 20˚53.25' | 155 | 19 | 0.33 | 5.5 | 450 |

Gatswane 1 | 27˚40.76' | 20˚41.28' | 101 | 25 | 0.75 | 5.5 | 440 |

Gatswane 2 | 27˚40.89' | 20˚39.01' | 111 | 31 | 0.79 | 6.0 | 440 |

Shashe | 27˚17.60' | 20˚37.40' | 117 | 19 | 0.70 | 4.0 | 447 |

Before the results discussion of the flood modeling results and associated risks in the design of drainage structures, we first illustrate the performance of the regional storm model.

Detail applicability of the storm IDF model written in computer spreadsheet program developed to optimize the three parameters of the proposed IDF model and to evaluate model performance and undertake error and necessary sensitivity analyses is provided in [

Using the observed rainfall data of Francistown which is close to the ungauged catchments we considered in this study, with mean annual rainfall R = 458 mm, the optimized parameter sets of the IDF curve using model form of Equation (5) are shown in

As suggested by [

Index of Volumetric Fit (IVF)-This is as a criterion representing the ratio of the observed to simulated volume of the storm under the observed and simulated IDF curves.

Parameter | Range | Typical value |
---|---|---|

a | 1960 - 1980 | 1970 |

b | 2900 - 3200 | 3100 |

c | 18 - 21 | 19 |

The model efficiency criteria R^{2} [

The correlation coefficient, r, between the observed and simulated series of rainfall intensity for the durations considered (5, 10, 20, …, 100 minutes).

Summary of results of application of the proposed storm model for Francistown in terms of the above model efficiency criteria is presented in ^{2} range from 92.9% to 99.9%, underlying the high percentage of the initial variance being accounted for by the proposed storm IDF model.

Extensive model calibration and validation tests with sensitivity analyses of the model are discussed in [

For the Nyamambisi catchment, rainfall intensities for selected durations and return periods simulated by the proposed regional storm design model, using a mean annual rainfall of 450 mm for the Nyamambisi watershed is shown in

In the calculation of risk associated with the above design flood estimates, we adopted the General Extreme Value (GEV) as a regional frequency model. GEV was found to be a robust model for the Limpopo basin system through regional analysis [

As it can be seen from

Duration (minutes) | Return period (years) | |||
---|---|---|---|---|

5 | 10 | 25 | 50 | |

IVF | 1.03 | 0.95 | 0.87 | 0.83 |

R^{2} (%) | 92.9 | 98.1 | 99.9 | 99.6 |

r | 0.995 | 0.996 | 0.997 | 0.992 |

Catchment | CN method | Empirical method | ||
---|---|---|---|---|

50 years | 100 years | 50 years | 100 years | |

Nyamambisi | 167 | 186 | 230 | 279 |

Gatswane 1 | 224 | 248 | 210 | 289 |

Gatswane 2 | 286 | 318 | 200 | 280 |

Shashe | 350 | 388 | 222 | 302 |

Catchment | Hydrological risk level | ||
---|---|---|---|

0.1% | 1.0% | 5.0% | |

Nyamambisi | 167 | 186 | 230 |

Gatswane 1 | 224 | 248 | 210 |

Gatswane 2 | 286 | 318 | 200 |

Shashe | 350 | 388 | 222 |

We have undertaken a risk assessment in design flood estimation using regional storm modeling approach. A generalized IDF model is proposed to produce the IDF relationships in Botswana. The performance of the model was judged against a number of model performance indices, which gave very high model efficiency (R^{2}) that was well above 90%, and a correlation coefficient (r) and IVF, both close to unity. In similar studies [

In this study, the consistent agreement in quality and magnitude of rainfall intensity curves shows the acceptability for the use of the proposed design storm Intensity- Duration-Frequency (IDF) model for the calculation of design flood estimates using an event based design flood modeling approach. This approach was also used to subsequently calculate the risk associated with the use of this design flood magnitudes for the design of drainage structures like culverts, bridges and other flood protection works, for any design period.

The authors wish to appreciate the support of the University of Botswana and Department of Metrological Services of the Botswana Government for encouraging this research directly or indirectly by providing the necessary information and support.

Alemaw, B.F. and Chaoka, R.T. (2016) Regional Design Storm and Flood Modelling―Risk Implications in Ungauged Catchments. Journal of Water Re- source and Protection, 8, 1211-1221. http://dx.doi.org/10.4236/jwarp.2016.813093