# Relationship between rainfall intensity duration and frequency

Rainfall Intensity-Duration-Frequency (IDF) relationship is one among the plethora of tools used for planning, designing and operating water resource. Intensity-Duration-Frequency (IDF) curves describe the relationship between rainfall duration, and return period (or its inverse, probability of exceedance). This paper presents a rainfall intensity-duration-frequency curve for 2, 5, 10, In addition to this, the relationship between the maximum rainfall.

Planning and protection of these basins, requires estimates of expected discharge from rainfall events of different magnitudes. Traditional estimation techniques are not feasible due to the lack of available runoff data. The practice regarding the hydrologic design in Jordan has been using the IDF curves developed by Ibbitt[2].

An independent study analyzing the rainfall data for 40 major stations in the Kingdom was published by the WAJ[3]. The accuracy of the recording rainfall charts mass curves was limited to 20 min the results for shorter duration were estimated by extrapolation.

The curves corresponding to 2 and 5 years return period were extended visually, which can be considered as an insubstantial point in this technique. Hydrologic phenomena are characterized by great variability, randomness, and uncertainty[5]. Precipitation and streamflow must be treated as random variables, with associated measures of frequency that represent percentage of time, or probability[6,7].

Rainfall IDF relationships can be expressed in various formats. An isohytal map prepared by the U. Geological Survey for Illinois[8], NWS atlases consist of sets of maps covering particular regions, with a map for each combination of recurrence interval and duration. The Texas Department of Transportation[9] hydraulic design Manual, related rainfall intensity in inch per hour to rainfall duration in minutes as a function of recurrence interval for each of the countries of the state.

In evaluation the frequency of intense rainfall, mention must be made of some classic studies. The later one of these is by Dillon[10] there are deferent methods such as: Chen[14] utilized a method similar to Bell method[12].

The previously developed formula was based on the average depth-duration ratio and the mean depth-frequency ratio. The recent Population of Rwanda is growing fast and counts around 12 million according to the National Institute of Statistics of Rwanda. Agriculture being the mainstay of the majority of the rural population, erratic equatorial rainfall pattern endangered the agricultural production.

Even though Rwanda is situated in the equatorial rain-forest belt, it perceives a modified humid climate characterized by both equatorial rainforest and savannah type. The rainfall pattern is dominated by the subtropical anticyclone as a consequence of the Inter Tropical Convergence Zone positions permitting bimodal rainfall pattern to the region.

Majority of the eastern belts of the country receive low seasonal rainfall and are characterized as drought prone areas. Materials and Methods Short to long period 14 - 83 years daily observed rainfall records have been collected from Rwanda Meteorological Agency under Ministry of Natural Resources.

The concise information of rainfall stations considered for present analysis are presented in Table 1. The rainfall data at each stations has undergone through preliminary data scrutiny for consistency. Using stations spatial proximity principle missing daily rainfall records are accounted. Maximum daily rainfall magnitudes are disaggregated into sub-daily values of 0.

Multiples of probability distributions are used to fit the sample data for selected rainfall durations so as to reinforce the statistical argument. Moment ratio diagram MRD and L-moment ratio diagram LMRD techniques are used to estimate parameters of the distribution and test the goodness of fit of probability distributions. The best fitted probability distribution is utilized to estimate the quantile estimates for different return period.

Based on regional homogeneity analysis, stations having similar rainfall pattern are identified and the entire country is divided into five homogenous daily maximum rainfall zones. IDF Curve Parameter Estimation The intensity-duration-frequency relationship is established for each station and parameters of IDF curves are identified using the following relationship. The typical generalized IDF parameters can be estimated using the following relationship [16].

Converting Equation 1 into logarithmic form and reducing the sum of the squared deviation to minimum, we have, 2 3 Equation 2 and 3 are utilized to compute the required intensities for respective stations and durations. Quantile Estimation The relationship between return period and probability of non-exceedence is expressed as: Typical characteristics of rainfall stations under consideration.

The point estimate of certain quantile corresponding to a return period may be insignificant unless there is a proof of estimate of accuracy. The validity of estimated quantile checked by the standard error of estimate, ST. The most efficient method of parameter estimation is the one which gives the least standard error of estimate.

Regionalization Rainfall Frequency Analysis The observed at-site hydrologic time series data are very short in length and hence substituting space for time is deployed to obtain representative average information about the region.

The regional frequency analysis based on index flood method [17] [18]L-moments [19] - [22]region of influence [23]canonical correlation analysis [24] and others have been in use in literature. In the present study, the method suggested by Hosking and Wallis [25] is applied to identify candidate homogenous regions for maximum daily rainfall magnitudes. Invariant stations are identified by discordance measure.

Regionalization was made on statistical values Cs, Ck, LCs, LCk of maximum rainfall of the selected duration for each station based on the concept that stations in the same region are assumed to be drawn from similar parent distribution.

Thus, similarity of the stations Cs, Ck and LCs, LCk plots to the theoretical probability distributions is accounted to classify the stations and determine the best fitting probability distribution helpful for subsequent quantile estimation.

Results and Discussion 4. Estimation of IDF Parameters As available data in majority of the rainfall stations is daily record, reducing the available data in manageable sub-daily scale has been carried out using the uniform random disaggregation model.

The disaggregated sub- daily data is further statistically checked against the historical records for corresponding duration. It has been found that there is no statistically significant variability between the desegregated and historical observations for selected stations.

The IDF parameters are computed for all 26 stations for return period of 2, 5, 10, 25, 50 and years. The computed IDF parameters are presented in Table 2.

### Analysis of Rainfall Intensity-Duration-Frequency Relationship for Rwanda

The parameters exhibit similarity over return period, however, there is no well-defined relationship with respect to station location.

The IDF curves have been developed for all station for different return period Figure 2. To aid water resources planners and decision Table 2.

Computed IDF parameters for selected 15 stations. These maps will assign particular rainfall intensity magnitude to particular points over the study area through spatial interpolation.

But L-moment ratios plot well separated and allows identifying of distribution.

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The L moment ratios LCs and LCk for each station based on specific duration rainfall is plotted against its regional averages on L-moment ratio diagrams. A suitable parent distribution is that which averages the scattered data and around which the data spread consistently.

The delineation result indicated that five 5 homogeneous regions were established. The transect starts with region-1 in the North-west part of the country and extends progressively to region-5 in the South-east parts in the transverse direction.

This region covers a very limited North-west part of the country. Region-2 covers the Busasamana, Gisenyi, gishyita and Kabaya stations residing to the south of region Region-4 accounts for Gakoma, Gitega, Kigali and Nyamiyaga stations. All other stations are grouped into region 5, the south western region Figure 5. The rainfall stations grouped into particular regions and corresponding best fitting distributions are listed in Table 3. The regional IDF parameters are estimated for five homogeneous regions.