Assessment of the Joint Impact of Rainfall Characteristics on Urban Flooding and Resilience Using the Copula Method

Abstract

The performance of urban drainage system (UDS) and green infrastructures (GI) significantly depends on the accurate determination of the rainfall characteristics of an urban area, especially because these rainfall characteristic variables are closely related. A bivariate copula method is used to evaluate and compare the joint impact of rainfall depth and duration on the hydrological performance of UDS and GI by considering the dependent structure of flood drivers. The peak flow, peak flow occurrence time, outflow, runoff coefficient, overflow, and resilience serve as the performance indicators. The estimated joint probability based on the optimal copula is used to define the boundary conditions for the calibrated Storm Water Management Model to simulate the hydrological process and the magnitude of floods caused by various combination of rainfall depth and durations. Results demonstrate that considering the dependence structure of rainfall characteristics, especially for the peak flow, runoff coefficient, and overflow, is critical for the comprehensive evaluation of the performance of the UDS. Neglecting the interaction between flood drivers can lead to overestimating the performance of UDS and GI in mitigating urban flood risk. The GI can effectively reduce urban flood and improve resilience, which varies with rainfall characteristics. Our results suggest that the utilization of accurate data on rainfall characteristics enhances the performance of the GI in flood mitigation and resilience improvement.

Appendix

Fig. 5

Density scatter plot of rainfall depth and duration with marginal histograms

1.1 C1. Gumbel, Clayton, and Frank Copula

$${C}_{Gumbel}\left(u,v\right)=\mathrm{exp}\{{[{\left(-\mathrm{ln}u\right)}^{\theta }+{(-\mathrm{ln}v)}^{\theta }]}^{1/\theta }\}$$

(7)

$${C}_{Clayton}(u,v)={({u}^{-\theta }+{v}^{-\theta }-1)}^{-1/\theta }$$

(8)

$${C}_{Frank}\left(u,v\right)=-\frac{1}{\theta }\mathrm{ln}[1+\frac{({e}^{-\theta u}-1)({e}^{-\theta v}-1)}{{e}^{-\theta }-1}]$$

(9)

where u and v are uniformly distributed random variables, which is the cumulative distribution function of rainfall depth (X) and duration (Y); C $\in$ [0, 1]² is copula function.

1.2 C2. The Definition of RMSE, BIC, and AIC

$$RMSE=\sqrt{\frac{1}{n}{\sum}_{i=1}^{n}{[C({u}_{i},{v}_{i})-{C}_{n}({u}_{i},{v}_{i})]}^{2}}$$

(10)

$$AIC=2k+N\mathrm{ln}(\frac{1}{N}{\sum}_{i=1}^{N}{[C({u}_{i},{v}_{i})-{C}_{n}({u}_{i},{v}_{i})]}^{2})$$

(11)

$$BIC=k\mathrm{ln}N+N\mathrm{ln}(\frac{1}{N}{\sum}_{i=1}^{N}{[C({u}_{i},{v}_{i})-{C}_{n}({u}_{i},{v}_{i})]}^{2})$$

(12)

where k is the number of parameters of the copula model; N represents the number of samples; $C({u}_{i},{v}_{i})$ represents the theoretical frequency; ${C}_{n}({u}_{i},{v}_{i})$ represents the empirical frequency.

1.3 C3. Performance Indicators

Single factor and synthetical index are adopted to study the impact of marginal distribution and correlation of rainfall depth and duration on the performance of drainage system, and to assess the flood control performance of urban drainage systems with or without LID strategies under joint rainfall events considering both rainfall depth and duration. Single factor indicators include peak flow, time of peak flow, total outflow, runoff coefficient, and overflow. Resilience index (Eq. (13)) indicates the synthetical index used for measuring the resilience of the drainage system is employed. Considering the layout of local drainage system, two system functionalities are included in the definition of resilience, including social benefit and technological benefit. The social benefit represents the capacity to prevent urban flooding, and the technological benefit indicates the capacity of the pump to process extra water during rainfall events.

The definition of resilience is shown below:

$$Res={\sum}_{T}\frac{1}{T}(\frac{1}{1+\frac{{k}_{s}{Sev}_{s}}{{Q}_{R}}+\frac{{k}_{t}{Sev}_{t}}{{Q}_{R}}})$$

(13)

$${Sev}_{s}=\frac{{t}_{fi}}{t}{\int }_{{t}_{0}}^{t}{Q}_{fi}dt$$

$${Sev}_{t}=\frac{{t}_{ti}}{t}{\int }_{{t}_{0}}^{t}{(Q}_{pi}-{Ca}_{pumpi})dt$$

where ${k}_{s}$ and ${k}_{t}$ are the weights of urban flooding and pump capacity, ${k}_{s}={k}_{t}=1$; ${Sev}_{s}$ is the overflow amount of manhole in drainage system indicating the urban flooding; ${Sev}_{t}$ is the residual water volume of front pool exceeding the treatment capacity of pump station water; ${Q}_{R}$ is surface runoff generated by rainfall event; ${t}_{fi}$ represents the overflow time at ith time step; $t$ represents the total time of simulation; ${t}_{ti}$ is the duration when Q_pi exceeds capacity of ith pump; Q_pi represents the inflow of ith pump from its front-pool. ${Ca}_{pumpi}$ represents the capacity of ith pump.

1.4 C4. Detailed Information of GI Design

The application of Green infrastructure (GI) is promoted for runoff control, flood mitigation, and resilience improvement, including green roof, porous pavement, and so on (Mao et al. 2017). Seoul city focused on applying green infrastructure in existing areas to make the city sustainable (Shafique and Kim 2018). In this study, green infrastructures, including Green Roof (GR) and Porous Pavement (PP) are arranged and simulated to assess its performance on improving city safety and resilience. The physical parameter of green infrastructure for each type is defined according to the previous papers in South Korea and US EPA design guidelines. For GR, the applied area is based on the public building and residential/commercial area. For PP, the arranged area is based on the pavement and parking plot. It should be noted that the area proportion of green infrastructure in this study indicates the proportion of the specific site area mentioned above. For both GR and PP, different proportions are designed, including 10%, 20%, 30%, and 50%. Finally, GR and PP with identical proportions are combined to display the combined control.

1.5 C5. Detailed Information of Marginal Distributions and Dependence Structure

The parameters with 95% confidence intervals of copulas are concluded in Appendix Table 2. The parameter values based on the relationship between parameter and rank relative factor are 1.585, 1.170, 3.751 for Gumbel, Clayton, and Frank copulas, respectively, which coincide with the parametric estimates and bracketed in 95% confidence interval excepting Clayton copula.

Table 1 Test Statistics of the fitted CDFs for rainfall depth and duration

Table 2 Estimated parameters of copulas and their 95% confidence intervals

Table 3 Goodness-of-fit of copulas for different combinations of marginal distributions

Table 4 The upper tail dependence coefficients for Gumbel copulas

Fig. 6

Three dimensional plots for the theoretically fitted Gumbel copula

Table 5 Probabilities of the two variables X (rainfall depth) and Y (rainfall duration) under different conditions