Introduction
The issue of climate change impacts on urban drainage system attracted a lot of attention in recent decade, since the densely populated urban areas with a high speed economic development (especially in developing countries) are extremely vulnerable in the face of torrential rainfall. Currently, General Circulation Models (GCMs) and Regional climate models (RCMs) are the main tool for the projections of future precipitation. And to explore the potential climate change impacts on urban drainage systems, temporal downscaling should be applied, since its temporal resolution is too course for the potential impact simulation and assessment. At the same time, intensity-duration-frequency (IDF) relationships are mostly hired to provide the information of rainfall properties to evaluate the design storm for hydrological infrastructures. Therefore, before the application of climate change impact assessment in urban drainage systems, the IDF curves, which derived through the downscaled climate model results, should be validated in the first place. And the performance comparison of different temporal downscaling methods with the aim of selecting an appropriate one is crucial.
Focusing on the two important steps in temporal precipitation downscaling, i.e. the possibility distribution selection and determination of scale invariance options, this paper examined the performances of different ways to convert the daily rainfall intensities into IDF curves. Three possibility distributions and two scale invariance options were tested for the scaling properties, and four statistical indicators were employed for the quantitative evaluation. On the basis of the precipitation data in Kunming city, China, the appropriate combination was identified and recommended to apply the downscaling process for local usage.
Methods
The concepts of scaling provide insights to the apparent complexities of hydrological phenomena, through simple mathematical formulations. Fedder (1988) defined that a scaling distribution function f(x) is scaling if the following equation exists, which provides a fundamental basis of temporal downscaling (by Equation (1), where ŠË¡Ê[0,1]). Burlando and Rosso (1996) demonstrated a power law form of IDF relationship can be derived from the scale invariance concept, shown in Equation (2), where Pd and Psd represent the series of annual maximum rainfall intensity at daily scale and sub-daily scale, respectively.
Since different distribution forms of AMD allow diverse daily rainfall intensities as input for temporal downscaling, and disparate scale invariance assumptions influence how to determine the scaling parameter ŠÂ. So in this paper, three most cited AMD distributions, i.e., Generalized Extreme Value (GEV), Gamma, and Log-Normal, are considered. Meanwhile, two scale invariance options including a simple scaling and a two-stage scaling are tested.
The IDF calculation for Kunming City involves 5 types of return periods (2, 5,10,15, and 25 years) and 8 rainfall durations (1, 2, 3, 6, 10, 12, 18, 24 hours). For the sake of quantitative performance evaluation, four indicators are used: mean bias error (MBE), root mean square error (RMSE), index of agreement (d) and coefficient of determination (R2), showed as Equations (3) -- (6). And the frame of reference for comparison is the local existing IDF relationship obtained by the storm intensity formula, which is shown as the Eq. (7)
Where yi is observed values, yi' is calculated values, yavg is the mean of observations.
Results and Discussion
For fitting the AMD precipitations, all the three distribution forms shown the suitability with relative error controlled within 3%. However, for extreme values, only GEV distribution provided acceptable results while the other two failed in high prediction errors. With regard to the value of 25-year return period, the relatively error was negative 5%, 21% and 28% for GEV, Gamma and Log Normal distributions, separately. As to the calculated IDF curves, First of all, it shown clearly that disparities among IDF results caused by different downscaling options exists, and the extent was increasing when return period increasing. Secondly, as to the IDF curves of the drizzle (2-year return period selected as example), the values of rainfall intensities based on the simple scaling matched the SIF values well, with relative errors controlled within 5%. In contrast, the situation of extreme events, which represented by the case of 25-year return periods, exhibits a much less concise and explicit image, not only the variance between the results computed according to the different scale invariance options still exist, but also the disparities of various fitted distributions shown up. P>Table 1 reports the values of statistical indicators for the performance comparison of different methods. All the four indicators shown that the results calculated by simple scaling is better than multiple scaling. And moreover, the values of MBE indicated that all the results obtained by different methods are smaller than the SIF values to some extent.
Table 1 performance comparison of different downscaling methods
Conclusion
The performance of the six temporal downscaling methods, with the aim of climate change impacts investigations on urban drainage system, were compared based on statistical evaluations. The results indicated that, for fitting the distribution forms, there is no big difference among the three candidates for drizzles but varied a lot for extreme events. As to the scale invariance options, the simple scaling appeared to be more reliable, and multiple scaling mainly underestimated the precipitation intensities with the durations bigger than 6 hours. Overall, the combination of GEV distribution and simple scaling has overwhelming advantage than the others, and is recommend for Kunming city consequently. 1. Fedder, J. (1988) Fractals. Plenum Press, New York.
2. Burlando, P., Rosso, R., (1996) Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation. Journal of Hydrology 187: 45-64.
3. Berggren, K., et al.(2012) Hydraulic Impacts on Urban Drainage Systems due to Changes in Rainfall Caused by Climatic Change. Journal of Hydrologic Engineering 17(1): 92-98.
4. Willems, P., Vrac, M.(2011) Statistical precipitation downscaling for small-scale hydrological impact investigations of climate change. Journal of Hydrology 402(3-4): 193-205.