#### 5.1. Preparation of the Rainfallâ€“Runoff Model

In this part of the study, the sensitivity of the runoff peak to the rainfall temporal distribution was evaluated. The sensitivity analysis was repeated two times; once on the models of the rainfall temporal distribution, and once on the randomness of the rainfall temporal distribution. Here, the randomness of the rainfall distribution indicates the transformation of the rainfall temporal distribution from Case 3 to Case 2 or Case 1. In contrast to the very well-organized design for rainfall based on the rainfall temporal distribution model, the observed rainfall event shows very random organization of the rainfall intensity values over the rainfall duration.

This study used the 1-h unit hydrograph (UH) for the rainfallâ€“runoff analysis. The UH was derived using the Clark instantaneous unit hydrograph (IUH), whose two parametersâ€”the concentration time

T_{c} and storage coefficient

Kâ€”were assumed to be the same. Three different UHs were derived with the concentration time 1, 3, and 5 h. The basin area was also assumed to be linearly proportional to the concentration time.

Figure 5 compares the three UHs. In particular, it shows that the peak flow is not proportional to the basin area.

The infiltration loss was considered in the rainfallâ€“runoff analysis by the NRCS-CN (National Resources Conservation Service-Curve Number) method [

38]. Three different CN values of 60, 80 and 100 were considered, to consider the different land cover and soil characteristics. The higher the CN value, the larger the effective rainfall amount. The CN 100 indicates no infiltration, while the CN value 0 indicates no effective rainfall.

#### 5.2. Sensitivity to the Rainfall Temporal Distribution Models

Each model considered in this study derives a different rainfall temporal distribution, which is also related to the shape of the runoff hydrograph. Additionally, the infiltration characteristics (i.e., CN), as well as the runoff characteristics (i.e., T_{c} and K), will also change the shape of the runoff hydrograph. To evaluate this difference caused by applying a different model of rainfall temporal distribution, this study performed the rainfallâ€“runoff analysis with different basin and infiltration characteristics. The annual maximum rainfall event that occurred in 1997 was considered as an example case. This rainfall event was continued for 13 h, and the mean rainfall intensity was 9.7 mm/h. The return period of this rainfall event was estimated to be two years.

With the mean rainfall intensity and the rainfall duration of the rainfall event in 1997, five different rainfall temporal distributions were derived. Also, by considering three different CN values and three different UHs, a total of nine combinations could be prepared for each model of rainfall temporal distribution. That is, for each UH, 15 combinations of rainfall distributions and CNs were prepared.

Table 4 summarizes these combinations for the rainfallâ€“runoff analysis.

First,

Figure 6 compares the 15 histograms of effective rainfall with the different rainfall temporal distributions and CNs. As can be expected, the effect of the CNs is concentrated in the early part of the rainfall, but the effective rainfall peak was also decreased a bit with higher CN. Second,

Figure 7 compares the runoff hydrographs derived by applying the 1-h UH. This figure shows that the runoff hydrographs were considerably affected by the CN value and the UH. The runoff hydrographs were all derived as would be expected.

The key result of

Figure 7 lies in the effect of the model of rainfall temporal distribution. First, the Yen and Chow model and the Huff model produced a far smaller peak flow than the other models. In particular, the sensitivity of the peak flow to the model of rainfall temporal distribution was higher for the case with small

T_{c} and

K. For the case with larger

T_{c} and

K, the sensitivity to the model was found to be much smaller. The peak flow derived by applying the Mononobe model was the highest, but was also similar to that of the Keifer and Chu model.

The above results can also be evaluated quantitatively from the point of the sensitivity of the peak flow to the model of rainfall temporal distribution. For this purpose, the following measure, a ratio of the peak flow of a model to the mean of all five models (

Ratio_{peak}), was introduced:

where

O_{peak} is the peak flow of a model, and

${\stackrel{\xc2\xaf}{O}}_{peak}$ is the mean of the peak flows of the five models of rainfall temporal distribution.

Figure 8 summarizes the results in box plots.

This figure shows that the sensitivity of the peak flow to the model of rainfall temporal distribution became much larger in a small basin (where the concentration time is short). The inter-quantile range estimated for the case of T_{c} = 1 h was estimated to be more than three times of that for the case of T_{c} = 3 or 5 h. On the other hand, the difference between the two cases of T_{c} = 3 h and T_{c} = 5 h was very small. As the rainfall distribution was made at hourly intervals, the effect of the rainfall peak on the peak flow seems to be direct for the case of T_{c} = 1 h. On the other hand, in the case of longer T_{c}, the storage effect significantly dampened the effect of the rainfall peak.

CN also seemed to have a significant effect on the peak flow, even though that effect was not as high as that of the concentration time. As the CN value increased, the inter-quantile range of the box plot increased, which was also obvious for the cases of T_{c} = 3 h and T_{c} = 5 h. For the case of T_{c} = 1 h, the effect of CN seemed to be minimum. This result indicates that as the basin area increases, the sensitivity of the peak flow to the model of rainfall temporal distribution becomes smaller. More generally, the effect of rainfall distribution on the rainfallâ€“runoff model for a small basin with small T_{c} and K can be very limited. On the other hand, in a large basin with larger T_{c} and K, the effect of CN could be seen more clearly.

#### 5.3. Preparation of the Rainfallâ€“Runoff Model

The result in

Section 4 shows that the alternating block method produced the most similar rainfall temporal distribution to the observed. However, this result was limited, as the comparison was carried out with the redistributed observed rainfall. That is, the comparison did not consider the randomness of the observed rainfall distribution. Thus, in this part of the study, the sensitivity of the peak flow to the randomness of the rainfall temporal distribution was evaluated. As an example case, the same rainfall event in the previous section was considered.

First, a total of 10 rainfall temporal distributions were randomly generated using the rainfall distribution derived by applying the alternating block method (

Figure 9). Except for the original distribution (H1), all the others seem more realistic, and similar to the observed. The same rainfallâ€“runoff analysis was also repeated for each combination of

T_{c},

K and CN. That is, nine different combinations of

T_{c},

K, and CN were considered in the rainfallâ€“runoff analysis for one rainfall temporal distribution.

Figure 10 compares the resulting runoff hydrographs.

The hydrographs in

Figure 10 show several important factors to be considered in the analysis of the result. First, when evaluating the sensitivity to the randomness of the rainfall distribution, the effect of

T_{c} (or the size of the basin area) is not that important. This result can be seen more clearly in the case of CN = 100. Second, particularly for the rainfall distribution with its peak at the rainfall beginning, the effect of CN is important. This is obvious, as most of the infiltration occurs at the beginning of a rainfall. However, if we consider only the rainfall events with their peaks on the second, third or fourth quantile, which is believed to result in some severe floods, the effect of CN may be excluded in the analysis of the result. Finally, it should be mentioned that the runoff peak time is directly related to the rainfall peak time. Thus, in this study, only the runoff peak flow was analyzed.

Under the above assumptions regarding considering the rainfall distributions with their peaks in the second, third, or fourth quantile, we derived the following results. First, the highest peak flow was produced for the rainfall distribution H1, i.e., the original distribution. Other random rainfall distributions produced more or less the same peak flow. This result could be seen more clearly for the case of CN = 100. That is, under the condition of saturated soil, a very well-organized rainfall distribution like the model of rainfall temporal distribution can produce higher peak flow than a random distribution.

However, under the condition of smaller CN values, this effect of well-organized rainfall distribution became smaller. For the case of CN = 60, no obvious high peak flow could be found for any case of concentration times. All the small peak flows were generated by the rainfall distribution with their peak in the first quantile (i.e., H5 and H7). This result was also the case for CN = 80. Overall, the sensitivity of the peak flow to the randomness of rainfall temporal distribution seemed to be small. This result can also be confirmed by the box plots (

Figure 11) of the

Ratio_{peak} defined by Equation (11).

The box plots in

Figure 11 show several important features of the randomness of the rainfall distribution and its effect on the runoff peak flow. First, when the CN was high (i.e., CN = 100 in this study), for

T_{c} = 1 h, no sensitivity of the peak flow to the randomness of the rainfall temporal distribution was found. For the case of

T_{c} = 3 h and

T_{c} = 5 h, the result was a bit higher, but still only a small sensitivity was found. On the other hand, when the CN was small, the sensitivity of the peak flow to the randomness of the rainfall temporal distribution was found to be very high. This was basically because the effect of infiltration on the effective runoff became significant. However, as

T_{c} increased, this sensitivity became smaller.

Overall, it could be concluded that when the CN is small, the effect of the randomness of the rainfall temporal distribution on the runoff peak flow was found to be significant. The infiltration, rather than the rainfall temporal distribution, played an important role in controlling the runoff peak flow. Thus, when the effect of the infiltration was small (or the CN value was high), the sensitivity of the runoff peak flow to the rainfall temporal distribution became insignificant. This situation was able to be obtained in urban basins or saturated basins.

Additionally, it is also important to compare the sensitivity to the randomness and the model of rainfall temporal distribution. By comparing

Figure 8 and

Figure 11, it can be easily concluded that the sensitivity of the runoff peak flow to the model of rainfall temporal distribution was much higher. With regard to the flooding condition, i.e., under the condition of high CN values in urban basins and saturated natural basins, the sensitivity of the runoff peak flow to the randomness of the rainfall temporal distribution was insignificant. Even when the CN values were small, the sensitivity of the runoff peak flow to the randomness of the rainfall temporal distribution was minimal, unless the rainfall peak was located in the first quantile.