**4. Results**

#### *4.1. Estimating the Design Rainfall*

The important assumption of RainyDay for estimating design rainfall is that the storms in the transition domain are likely to occur in the study area. In order to illustrate the rationality of the selected transition domain, this study analyzes the spatial distribution and storm occurrence probability of 200 maximum storms under different durations (2 h, 6 h, 12 h, and 24 h) (Figure 3). The spatial distribution of storms with different durations is basically similar to each other. Generally speaking, the frequency of storms in coastal areas is relatively higher, but its spatial distribution is still relatively random, that is, heavy storms may occur everywhere in the selected transition domain (Figure 3). Similar to the spatial distribution, the spatial probability distribution of storms in the transition domain is relatively uniform, but there are still some differences. The storm occurrence probability decreases from south to north (Figure 3), which is in line with the actual distribution of storms (see Wang et al. [60] for evaluation of rainfall distribution in different precipitation products). The selected transition domain is reasonable since the probability of storm occurrence of 200 maximum storms varies from around 0.0002 to 0.0014 in the transition domain, i.e., the storms in the transition domain are likely to occur in the study area or other regions.

**Figure 3.** The probability of storm occurrence and spatial distribution of 200 maximum storms under 2h(**a**), 6 h (**b**), 12 h (**c**), and 24 h (**d**) durations over the transition domain. Shading denotes spatial probability of storm occurrence calculated based on 200 maximum storms. Black dots represent the rainfall centroids of 200 maximum storms, and its size means the relative rainfall depth of each storm.

Figure 4 shows the relationships between IDF formula-based and RainyDay-based estimates for different durations at different return periods. The ensemble spread of 20 realizations is shown as shaded area. Comparing results indicates that RainyDay is generally able to estimate urban extreme rainfall for different durations, but it may relatively underestimate or overestimate the rainfall accumulation. The results show that RainyDay usually underestimates the rainfall accumulation at low return periods or short rainfall durations; the RainyDay-based estimates are usually larger than the IDF formulabased estimates when the rainfall duration is long or the return period is high. Specifically, RainyDay overall underestimates the rainfall accumulation when the rainfall duration is 2 h at different return periods. The degree of underestimation, which varies from 0.4% (100-yr) to 57% (5-yr), decreases with increasing the return period (Table 1). When the duration reaches 6 h, the underestimation is improved. The IDF formula-based estimates, overall, fall within the ensemble spread of RainyDay-based estimates with the increase of duration. At each return period for 6 h or longer durations, the absolute value of the ratio of at least one RainyDay-based estimate (maximum, minimum, or average estimates) to IDF formula-based estimate is less than 10% (Table 1).

**Figure 4.** The relationships between IDF formula-based and RainyDay-based estimates for different durations at different return periods. The shaded areas mean the ensemble spread of RainyDay-based estimates. The solid lines denote the ensemble mean for 20 realizations. The symbols of different shapes represent the IDF formula-based estimates.

**Table 1.** Relative deviations between IDF formula-based and RainyDay-based estimates (%).


The IDF formula-based estimates gradually approach to the lower boundary of the shaded area with increasing return period. It indicates that the RainyDay-based estimates basically can reflect the observed design rainfall for long (6 h or longer) durations. To be consistent with the design specification for outdoor drainage in China, the time distributions of the RainyDay-based and IDF formula-based estimates for urban flood simulation are determined by the Chicago rainfall pattern. The time distribution results show that the main difference comes from the rainfall peak. The rainfall peak is underestimated

from RainyDay-based estimates at low return periods or short rainfall durations, while it is generally matched or slightly overestimated at high return periods or for long rainfall duration. In order to better explain this fact, the time distributions at different return periods for 6 h duration and at 20-yr return period for different durations are selected as in the below examples (Figures 5 and 6). When the duration is 6 h, the rainfall peak of the RainyDay-based estimates is relatively smaller than the IDF formula-based estimates at 5- and 10-yr return period, but the rainfall peak of IDF formula-based estimates generally falls within the ensemble spread of RainyDay-based estimates, and the average of the ensemble spread is generally matched to the IDF formula-based estimates when the return period reaches 50-yr or higher (Figure 5). On the other hand, when the return period is at 20-yr return period, the time distributions of RainyDay-based and IDF formula-based estimates are essentially coincidental, and the coincidence increases with lengthening rainfall duration (Figure 6). Overall, the RainyDay-based estimates show a good performance for design rainfall analysis. The relationship between the time distributions of RainyDay-based and IDF formula-based estimates at other return periods for other rainfall durations are similar to the above selected rainfall scenarios, so the time distributions of other scenarios are not shown.

**Figure 5.** The time distributions of RainyDay-based and IDF formula-based estimates at 5-yr (**a**), 10-yr (**b**), 20-yr (**c**), 50-yr (**d**), and 100-yr (**e**) return periods for 6 h duration.

**Figure 6.** The time distributions of RainyDay-based and IDF formula-based estimates at 20-yr return period for different durations.

#### *4.2. Simulating the Runoff Process Based on RainyDay-Based Estimates*

The simulated results show that the RaiyDay-based estimates basically can be used for runoff process simulation. The difference between the runoff processes of RainyDay-based and IDF formula-based estimates is similar to the time distributions of design rainfall, but the difference of peak discharge is smaller than the rainfall peak. Similar to the analysis of time distribution of rainfall estimates, we also take the runoff processes at different return periods for 6 h duration and at 20-yr return period for different durations as in the below example (Figures 7 and 8). The runoff processes of RainyDay-based and IDF formula-based estimates indicate that the difference of runoff process decreases as the rainfall duration lengthens. The difference of peak discharge at high return periods (20-yr or higher) or for long durations (6 h or longer) is very small. For example, the difference of the RainyDay-based and IDF formula-based rainfall peaks is relatively significant, but the differences of peak discharges are very small at 5- and 10-yr return periods (Figures 5 and 7). The RainyDay-based peak discharges become closer and closer, and even approximate overlapping IDF formula-based peak discharges with increasing return period. For the same return period (take 20-yr return period for example), the peak discharge is still slightly underestimated for 2 h duration, but the runoff process is predicted pretty well with the lengthening duration (Figure 8). In addition, we use NSE to evaluate the predicted performance of RainyDay-based estimates, i.e., the difference of runoff processes between RainyDay-based and IDF formula-based estimates. Results show that the values of NSE are generally small for short duration or at low return period (e.g., NSE = 0.53 at 5-yr return period for 6 h duration, NSE = 0.77 at 20-yr return period for 2 h duration); however, the values become larger with increasing rainfall duration or rainfall return period (e.g., NSE = 0.98 at 100-yr return period for 6 h duration, NSE = 0.99 at 20-yr return period for 24 h duration). For long duration (6 h or longer) or high return period (10-yr or higher), the values of NSE are generally above 0.5, i.e., the RainyDay-based estimates of long duration or high return period are satisfied to analyze the runoff process.

**Figure 7.** The runoff processes at the outlet of the case-study area at 5-yr (**a**), 10-yr (**b**), 20-yr (**c**), 50-yr (**d**), and 100-yr (**e**) return periods for 6 h duration.

**Figure 8.** The runoff processes at the outlet of the case-study area at 20-yr return period for different durations.

#### *4.3. Analyzing Flood Characteristics Based on RainyDay-Based Estimates*

Results show that the characteristics of urban flooding are generally underestimated based on RainyDay-based estimates at low return periods or short rainfall durations. For short durations or at low return periods, the underestimation of the values of these indicators at each manhole are more significant than runoff processes at the outlet. Specifically, the RainyDay-based estimates underestimate the values of flood time, maximum rate, and total inundation volume when the return period is lower than 10-yr or duration is shorter than 6 h. The underestimation decreases with increasing return period or lengthening rainfall duration. The values of flood time, maximum rate, and total inundation volume simulated based on IDF formula-based estimates generally fall within the ensemble spread of RainyDay-based estimates at high (20-yr or high) return period or long (6 h or longer) duration (Figures 9 and 10). That is to say, the RainyDay-based estimates can be used to assess the flood characteristics at each manhole at relatively high return periods or long rainfall durations.

**Figure 9.** The flood characteristics of each manhole at different return periods for 6 h duration. The flood time at 5-yr, 10-yr, 20-yr, 50-yr, and 100-yr return periods is shown in (**a**), (**d**), (**g**), (**j**), and ( **m**), respectively. The maximum rate at 5-yr, 10-yr, 20-yr, 50-yr, and 100-yr return periods is shown in (**b**), (**e**), (**h**), (**k**), and (**n**), respectively. The total inundation volume at 5-yr, 10-yr, 20-yr, 50-yr, and 100-yr return periods is shown in (**c**), (**f**), (**i**), (**l**), and (**o**), respectively. The grey boxes indicate the spread of RainyDay-based, and points represent IDF formula-based, values.

**Figure 10.** The flood characteristics of each manhole at 20-yr return period for different durations. The flood time for 2 h, 6 h, 12 h, and 24 h durations is shown in (**<sup>a</sup>**–**d**), respectively. The maximum rate for 2 h, 6 h, 12 h, and 24 h durations is shown in (**<sup>e</sup>**–**h**), respectively. The total inundation volume for 2 h, 6 h, 12 h, and 24 h durations is shown in (**i**–**l**), respectively. The grey boxes indicate the spread of RainyDay-based, and points represent IDF formula-based, values.

In order to better clarify the changing characteristics of urban flooding at each manhole with rainfall return period or rainfall duration, we also take the flood characteristics of each manhole at 20-yr return period for different durations and at different return periods for 6 h duration as an example. For 6 h rainfall duration, the RainyDay-based estimates significantly underestimate the values of the selected indicators at 5-yr return period; when the return period increases to 10-yr, the RainyDay-based estimates can reflect the flood characteristics at each manhole to a certain extent, but it is still relatively underestimated; while the rainfall return period reaches 20-yr or more, the values of indicators simulated by IDF formula-based estimates basically fall within the ensemble spread of RainyDay-based estimates. On the other hand, when the rainfall return period is 20-yr, the RainyDay-based estimates can basically reflect the flood characteristics of each manhole under different rainfall duration scenarios, especially for long (6 h or longer) rainfall duration. The flood characteristics of some manholes will be slightly overestimated with the increasing rainfall duration.

The results shown in Figures 9 and 10 cannot comprehensively assess the flood characteristics of each manhole, therefore, the projection pursuit algorithm is used to reduce three dimensions (i.e., three indicators) to one dimension. The one-dimension values (i.e., the projection values) indicate the comprehensive characteristics of urban flooding for each manhole. Results show that the flood hotspot manholes are J3, J7, and J13, but they are significant underestimated based on RainyDay-based estimates at low return periods or short rainfall durations (Figures 9 and 10). The changing characteristics of projection values with return periods or duration are similar to the values of the three indicators, but the degree of underestimation for the projection values is larger than the values of indicators (Figures 11 and 12). However, the degree of underestimation decreases with increasing return period or duration. Similar to the values of three indicators, the projection values estimated based on IDF formula-based estimates fall within the RainyDaybased ensemble spread at high (20-yr or higher) return periods or long (6 h or longer) durations. The comprehensive analysis results for urban flooding demonstrates that the RainyDay-based estimates can be used for urban flood analysis, especially for high (20-yr or high) return periods or long (6 h or longer) durations.

**Figure 11.** The projection values of each manhole at 5-yr (**a**), 10-yr (**b**), 20-yr (**c**), 50-yr (**d**), and 100-yr (**e**) return periods for 6 h duration.

**Figure 12.** The projection values of each manhole at 20-yr return period for2h(**a**), 6 h (**b**), 12 h (**c**), and 24 h (**d**) durations.
