# Seamless Integration of Rainfall Spatial Variability and a Conceptual Hydrological Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Rainfall Spatial Variability Description

#### 2.2. Statistical Runoff Calculation

_{i}) is greater than the infiltration rate (I

_{i}) according to the infiltration excess (Horton) runoff generation mechanism [39]. Therefore, the statistical distribution of surface runoff (RS) in the SRR model is obtained by combining the pdfs of rainfall and infiltration capacity:

_{c}is the infiltration capacity, F(I

_{c}) is the distribution of the infiltration capacity, I

_{cmax}is the maximum infiltration capacity within the catchment, BF is a parameter that reflects the unevenly distributed infiltration capacity, and I

_{cm}represents the catchment-average infiltration capacity. Thus, the pdf of the infiltration capacity is derived:

_{cm}[51,70]:

_{s}is the stable infiltration rate, KF is the osmotic coefficient, showing the sensitivity coefficient of the influence of soil moisture on the infiltration rate, S

_{cm}is the catchment-average soil moisture capacity, and W refers to the soil moisture content.

_{i}and P

_{i}are independent, integration in math is then applied to Equation (6) to derive the distribution of the surface runoff:

_{min}is the minimum rainfall within the catchment at a given time and r, r

_{1}, and r

_{2}are coefficients independent of RS that can be solved by numerical integration methods using MATLAB (Matrix Laboratory, USA) or R (The R Programming Language, New Zealand).

_{max}is the maximum rainfall within the catchment at a given time.

_{c}is the soil moisture capacity, F(S

_{c}) is the distribution of soil moisture capacity, S

_{cmax}is the maximum soil moisture capacity within the catchment, and $\beta $ is a parameter describing the degree of the spatial variability of soil moisture capacity. The smaller the value of $\beta \left(\beta \ge 0\right)$, the more uniform the distribution of the soil moisture capacity. Under this condition, the mean groundwater runoff (RG) is estimated [41,67]:

_{max}is the maximum catchment-average soil moisture content, W

_{0}is the initial soil moisture content, and S

_{c}

_{0}is the soil moisture capacity corresponding to W

_{0}.

#### 2.3. Stochastic Flow Routing Calculation

_{I}(t) is the stochastic inflow, m

^{3}/s; Q

_{o}(t) is the stochastic outflow, m

^{3}/s; S(t) is the water storage of a calculation cell, m

^{3}; and K is the channel time lag (i.e., the hydrograph movement time), h. The stochastic inflow Q

_{I}(t) (m

^{3}/s) is transformed from the TR (mm) of the catchment:

^{2}.

_{I}(t) can be expressed as Gaussian white noise superimposed on a deterministic mean [54]:

## 3. Study Area and Data

^{2}. The mean annual temperature is 11°–16° and the mean annual precipitation is 1077 mm. However, storms usually occur between June and September (summer in China) and account for 50–80% of the total annual rainfall amount [73,74]. Most floods in this region are caused by rainstorms with characteristics of high intensity, short duration, small rainstorm center range, and high peak discharge.

_{n}(P) is the empirical distribution function, F′(P) equals 1 minus the exceedance distribution function F(P) shown in Equation (3) and D (also called the KS statistic) is the maximum difference between the empirical distribution and the EDD.

_{obs}is the observed flow, m

^{3}/s; Q

_{sim}is the simulated flow, m

^{3}/s; and $\overline{{Q}_{obs}}$ is the mean of the observations, m

^{3}/s. The range of the NSE is between −∞ and 1, and the closer the NSE value approaches 1, the better the model simulation [83].

## 4. Results and Discussion

#### 4.1. Fitness of the Rainfall Distribution

#### 4.2. Model Calibration and Validation

#### 4.3. Comparison between the SRR Model and the XAJ Model

^{3}/s, while ${Q}_{M2}$ is the simulations from Model 2, m

^{3}/s. A positive CEF indicates a closer match between the results of Model 1 and the observations, whereas a negative CEF indicates a closer match between the results of Model 2 and the observations.

## 5. Conclusions

- (1)
- A transformation form of the generalized exponential function, i.e., the EDD, is proposed to describe the spatial variability of rainfall. The unimodal, skewed, and right-tailed EDD indicates that small rainfall has a relatively high probability over the catchment and vice versa. The exponential type provides the ability to incorporate the EDD into a conceptual rainfall-runoff model. The IDW approach is then applied to obtain more rainfall samples, based on which, parameters of the rainfall distribution (the EDD) are estimated using the least squares method.
- (2)
- The VMR model is a lumped conceptual rainfall-runoff model considering both the infiltration excess and saturation excess runoff generation mechanisms. Additionally, the spatial variability of soil infiltration capacity and soil moisture capacity is described using empirical distributions in the VMR model. The EDD is then coupled with the VMR model to estimate the statistical runoff component. Specifically, the distribution of surface runoff is deduced by the joint distribution of rainfall and soil infiltration capacity according to the infiltration excess mechanism, while the expected value of groundwater runoff is estimated based on the saturation excess mechanism. Considering the complexity of the distribution, the total runoff can be described by the mean and variance where uncertainties in the spatial variability of rainfall, infiltration capacity, and soil moisture capacity are included.
- (3)
- To address the flow routing under the condition of stochastic inflow, the stochastic differential equation is adopted. The study catchment is assumed to be a linear reservoir whose input is the estimated total runoff, while the output is the simulated streamflow. The first two moments of the simulated streamflow can be given while the mean is the focus in this study.
- (4)
- The SRR model is calibrated and verified by 16 flood events that occurred in the Huangnizhuang catchment of China. The fitness of the EDD and observed rainfall is assessed by the KS test. The gamma distribution is also adopted for comparison. The KS test results indicate the good performance of the EDD, which is comparable to that of the gamma distribution. In addition, the SRR model is compared with the XAJ model based on metrics of peak flow and NSE. The results show the advantage of the SRR model when its expected values are considered. Further study will elaborate the probability forecast of the SRR framework when its variance is considered. Moreover, catchments with various hydro-climatic characteristics will be investigated.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Rezaie-Balf, M.; Zahmatkesh, Z.; Kim, S. Soft Computing Techniques for Rainfall-Runoff Simulation: Local Non–Parametric Paradigm vs. Model Classification Methods. Water Resour. Manag.
**2017**, 31, 3843–3865. [Google Scholar] [CrossRef] - Feng, Z.; Niu, W.; Tang, Z.; Jiang, Z.; Xu, Y.; Liu, Y.; Zhang, H. Monthly runoff time series prediction by variational mode decomposition and support vector machine based on quantum-behaved particle swarm optimization. J. Hydrol.
**2020**, 583, 124627. [Google Scholar] [CrossRef] - Nayak, P.C.; Sudheer, K.P.; Ramasastri, K.S. Fuzzy computing based rainfall-runoff model for real time flood forecasting. Hydrol. Process.
**2005**, 19, 955–968. [Google Scholar] [CrossRef] - Khazaei, M.R.; Zahabiyoun, B.; Saghafian, B.; Ahmadi, S. Development of an Automatic Calibration Tool Using Genetic Algorithm for the ARNO Conceptual Rainfall-Runoff Model. Arab. J. Sci. Eng.
**2013**, 39, 2535–2549. [Google Scholar] [CrossRef] - Kunnath-Poovakka, A.; Eldho, T.I. A comparative study of conceptual rainfall-runoff models GR4J, AWBM and Sacramento at catchments in the upper Godavari river basin, India. J. Earth Syst. Sci.
**2019**, 128, 33. [Google Scholar] [CrossRef] [Green Version] - Buzacott, A.J.V.; Tran, B.; Van Ogtrop, F.F.; Vervoort, R.W. Conceptual Models and Calibration Performance—Investigating Catchment Bias. Water
**2019**, 11, 2424. [Google Scholar] [CrossRef] [Green Version] - Kim, S.; Paik, K.; Johnson, F.M.; Sharma, A. Building a flood-warning framework for ungauged locations using low resolution, open-access remotely sensed surface soil moisture, precipitation, soil, and topographic information. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2018**, 11, 375–387. [Google Scholar] [CrossRef] - Knoben, W.J.M.; Freer, J.E.; Fowler, K.J.A.; Peel, M.C.; Woods, R.A. Modular Assessment of Rainfall–Runoff Models Toolbox (MARRMoT) v1.2: An open-source, extendable framework providing implementations of 46 conceptual hydrologic models as continuous state-space formulations. Geosci. Model Dev.
**2019**, 12, 2463–2480. [Google Scholar] [CrossRef] [Green Version] - Qi, P.; Xu, Y.J.; Wang, G. Quantifying the Individual Contributions of Climate Change, Dam Construction, and Land Use/Land Cover Change to Hydrological Drought in a Marshy River. Sustainability
**2020**, 12, 3777. [Google Scholar] [CrossRef] - Kurtzman, D.; Navon, S.; Morin, E. Improving interpolation of daily precipitation for hydrologic modelling: Spatial patterns of preferred interpolators. Hydrol. Process.
**2009**, 23, 3281–3291. [Google Scholar] [CrossRef] - Bárdossy, A.; Das, T. Influence of rainfall observation network on model calibration and application. Hydrol. Earth Syst. Sci.
**2008**, 12, 77–89. [Google Scholar] [CrossRef] [Green Version] - Wilson, C.B.; Valdés, J.B.; Rodriguez-Iturbe, I. On the influence of the spatial distribution of rainfall on storm runoff. Water Resour. Res.
**1979**, 15, 321–328. [Google Scholar] [CrossRef] - Li, D.; Liang, Z.; Zhou, Y.; Li, B.; Fu, Y. Multicriteria assessment framework of flood events simulated with vertically mixed runoff model in semiarid catchments in the middle Yellow River. Nat. Hazards Earth Syst. Sci.
**2019**, 19, 2027–2037. [Google Scholar] [CrossRef] [Green Version] - Mertens, J.; Raes, D.; Feyen, J. Incorporating rainfall intensity into daily rainfall records for simulating runoff and infiltration into soil profiles. Hydrol. Process.
**2002**, 16, 731–739. [Google Scholar] [CrossRef] - Le, M.-H.; Lakshmi, V.; Bolten, J.; Du Bui, D. Adequacy of Satellite-derived Precipitation Estimate for Hydrological Modeling in Vietnam Basins. J. Hydrol.
**2020**, 586, 124820. [Google Scholar] [CrossRef] - Wheater, H.S.; Chandler, R.E.; Onof, C.J.; Isham, V.S.; Bellone, E.; Yang, C.; Lekkas, D.; Lourmas, G.; Segond, M.-L. Spatial-temporal rainfall modelling for flood risk estimation. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 403–416. [Google Scholar] [CrossRef] - Abu Shoaib, S.; Marshall, L.; Sharma, A. Attributing uncertainty in streamflow simulations due to variable inputs via the Quantile Flow Deviation metric. Adv. Water Resour.
**2018**, 116, 40–55. [Google Scholar] [CrossRef] - Anagnostou, E.N.; Maggioni, V.; Nikolopoulos, E.I.; Meskele, T.; Hossain, F.; Papadopoulos, A. Benchmarking High-Resolution Global Satellite Rainfall Products to Radar and Rain-Gauge Rainfall Estimates. IEEE Trans. Geosci. Remote Sens.
**2010**, 48, 1667–1683. [Google Scholar] [CrossRef] - Zhu, D.; Wang, G.; Ren, Q.; Ilyas, A.M. Hydrological evaluation of hourly merged satellite–station precipitation product in the mountainous basin of China using a distributed hydrological model. Meteorol. Appl.
**2020**, 27, e1909. [Google Scholar] [CrossRef] - Niu, J.; Chen, J. Terrestrial hydrological responses to precipitation variability in Southwest China with emphasis on drought. Hydrol. Sci. J.
**2014**, 59, 325–335. [Google Scholar] [CrossRef] [Green Version] - Li, B.; Liang, Z.; Chang, Q.; Zhou, W.; Wang, H.; Wang, J.; Hu, Y. On the Operational Flood Forecasting Practices Using Low-Quality Data Input of a Distributed Hydrological Model. Sustainability
**2020**, 12, 8268. [Google Scholar] [CrossRef] - Deal, E.; Favre, A.-C.; Braun, J. Rainfall variability in the Himalayan orogen and its relevance to erosion processes. Water Resour. Res.
**2017**, 53, 4004–4021. [Google Scholar] [CrossRef] [Green Version] - Negri, D.H.; Gollehon, N.R.; Aillery, M.P. The Effects of Climatic Variability on US Irrigation Adoption. Clim. Chang.
**2005**, 69, 299–323. [Google Scholar] [CrossRef] - Husak, G.J.; Michaelsen, J.; Funk, C. Use of the gamma distribution to represent monthly rainfall in Africa for drought monitoring applications. Int. J. Climatol.
**2007**, 27, 935–944. [Google Scholar] [CrossRef] - Kaźmierczak, B.; Kotowski, A. The suitability assessment of a generalized exponential distribution for the description of maximum precipitation amounts. J. Hydrol.
**2015**, 525, 345–351. [Google Scholar] [CrossRef] - Wang, Z.; Zeng, Z.; Lai, C.; Lin, W.; Wu, X.; Chen, X. A regional frequency analysis of precipitation extremes in Mainland China with fuzzy c-means and L-moments approaches. Int. J. Climatol.
**2017**, 37, 429–444. [Google Scholar] [CrossRef] - Raziei, T. Performance evaluation of different probability distribution functions for computing Standardized Precipitation Index over diverse climates of Iran. Int. J. Climatol.
**2021**, 2021, 1–22. [Google Scholar] [CrossRef] - Faurès, J.-M.; Goodrich, D.C.; Woolhiser, D.A.; Sorooshian, S. Impact of small-scale spatial rainfall variability on runoff modeling. J. Hydrol.
**1995**, 173, 309–326. [Google Scholar] [CrossRef] - Vaze, J.; Post, D.A.; Chiew, F.H.S.; Perraud, J.-M.; Teng, J.; Viney, N.R. Conceptual Rainfall–Runoff Model Performance with Different Spatial Rainfall Inputs. J. Hydrometeorol.
**2011**, 12, 1100–1112. [Google Scholar] [CrossRef] - Emmanuel, I.; Payrastre, O.; Andrieu, H.; Zuber, F. A method for assessing the influence of rainfall spatial variability on hydrograph modeling. First case study in the Cevennes Region, southern France. J. Hydrol.
**2017**, 555, 314–322. [Google Scholar] [CrossRef] [Green Version] - Aksoy, H. Use of gamma distribution in hydrological analysis. Turk. J. Eng. Environ. Sci.
**2000**, 24, 419–428. [Google Scholar] - Kaźmierczak, B.; Wartalska, K. Changes in Maximum Rainfall Amounts in Wroclaw (Poland). Int. J. Environ. Sci. Dev.
**2019**, 10, 368–372. [Google Scholar] [CrossRef] - Gupta, R.D.; Kundu, D. Generalized exponential distributions. Aust. N. Zeal. J. Stat.
**1999**, 41, 173–188. [Google Scholar] [CrossRef] - Gupta, R.D.; Kundu, D. Generalized exponential distribution: Different method of estimations. J. Stat. Comput. Simul.
**2001**, 69, 315–337. [Google Scholar] [CrossRef] - Gupta, R.D.; Kundu, D. Generalized exponential distribution: Existing results and some recent developments. J. Stat. Plan. Inference
**2007**, 137, 3537–3547. [Google Scholar] [CrossRef] [Green Version] - Lambert, M.; Kuczera, G. Seasonal generalized exponential probability models with application to interstorm and storm durations. Water Resour. Res.
**1998**, 34, 143–148. [Google Scholar] [CrossRef] - Nadarajah, S.; Kotz, S. The beta exponential distribution. Reliab. Eng. Syst. Saf.
**2006**, 91, 689–697. [Google Scholar] [CrossRef] - Lin, P.; Shi, P.; Yang, T.; Xu, C.-Y.; Li, Z.; Wang, X. A Statistical Vertically Mixed Runoff Model for Regions Featured by Complex Runoff Generation Process. Water
**2020**, 12, 2324. [Google Scholar] [CrossRef] - Horton, R.E. The Rôle of infiltration in the hydrologic cycle. Trans. Am. Geophys. Union
**1933**, 14, 446–460. [Google Scholar] [CrossRef] - Dunne, T.; Black, R.D. An Experimental Investigation of Runoff Production in Permeable Soils. Water Resour. Res.
**1970**, 6, 478–490. [Google Scholar] [CrossRef] - Zhao, R. The Xinanjiang model applied in China. J. Hydrol.
**1992**, 135, 371–381. [Google Scholar] - Sugawara, M. Tank model. In Computer Models of Watershed Hydrology; Singh, V.P., Ed.; Water Resources Publications: Littleton, CO, USA, 1995; pp. 165–214. [Google Scholar]
- Wagener, T.; Boyle, D.P.; Lees, M.J.; Wheater, H.S.; Gupta, H.V.; Sorooshian, S. A framework for development and application of hydrological models. Hydrol. Earth Syst. Sci.
**2001**, 5, 13–26. [Google Scholar] [CrossRef] - Burnash, R.J.C. The NWS River Forecast System-catchment modeling. In Computer Models of Watershed Hydrology; Singh, V.P., Ed.; Water Resources Publications: Littleton, CO, USA, 1995; pp. 311–366. [Google Scholar]
- Zhao, R. Water Hydrological Modeling-Xinanjiang Model and Shanbei Model; China Water Resources Hydropower Publications: Beijing, China, 1984; pp. 125–129. [Google Scholar]
- Liu, Y.; Zhang, K.; Li, Z.; Liu, Z.; Wang, J.; Huang, P. A hybrid runoff generation modelling framework based on spatial combination of three runoff generation schemes for semi-humid and semi-arid watersheds. J. Hydrol.
**2020**, 590, 125440. [Google Scholar] [CrossRef] - Hu, C.; Zhang, L.; Wu, Q.; Soomro, S.-E.-H.; Jian, S. Response of LUCC on Runoff Generation Process in Middle Yellow River Basin: The Gushanchuan Basin. Water
**2020**, 12, 1237. [Google Scholar] [CrossRef] - Bao, W.; Wang, C. Application of the vertically mixed runoff model. J. China Hydrol.
**1997**, 03, 19–22. [Google Scholar] - Sivapalan, M.; Wood, E.F.; Beven, K.J. On hydrologic similarity: 3. A dimensionless flood frequency model using a generalized geomorphologic unit hydrograph and partial area runoff generation. Water Resour. Res.
**1990**, 26, 43–58. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Huang, P.; Zhang, J.; Yao, C.; Yao, Y. Construction and application of Xin’anjiang-Haihe model. J. Hohai Univ. Nat. Sci.
**2013**, 41, 189–195. [Google Scholar] - Bao, W.; Zhao, L. Application of Linearized Calibration Method for Vertically Mixed Runoff Model Parameters. J. Hydrol. Eng.
**2014**, 19, 04014007. [Google Scholar] [CrossRef] - Hu, C.; Guo, S.; Xiong, L.; Peng, D. A modified Xinanjiang model and its application in northern China. Nord. Hydrol.
**2005**, 36, 175–192. [Google Scholar] [CrossRef] - Jiang, S. Application of stochastic differential equations in risk assessment for flood releases. Hydrol. Sci. J.
**1998**, 43, 349–360. [Google Scholar] [CrossRef] - Sun, Y.; Rui, X.; Fu, Q.; Xing, Z. Basin flow concentration model based on stochastic differential equation. Shuili Xuebao J. Hydraul. Eng.
**2011**, 42, 187–191. [Google Scholar] - Liang, Z.; Hu, Y.; Wang, J. Application of stochastic differential equation to reservoir routing with probabilistic inflow forecasting and flood control risk analysis. In Proceedings of the EGU General Assembly Conference Abstracts, Vienna, Austria, 22–27 April 2012; p. 401. [Google Scholar]
- Yan, B.; Guo, S.; Chen, L. Estimation of reservoir flood control operation risks with considering inflow forecasting errors. Stoch. Environ. Res. Risk Assess.
**2013**, 28, 359–368. [Google Scholar] [CrossRef] - Cao, L.G.; Zhong, J.; Su, B.-D.; Zhai, J.Q.; Gemmer, M. Probability Distribution and Projected Trends of Daily Precipitation in China. Adv. Clim. Chang Res.
**2013**, 4, 153–159. [Google Scholar] [CrossRef] - Kumari, M.; Singh, C.K.; Bakimchandra, O.; Basistha, A. Geographically weighted regression based quantification of rainfall-topography relationship and rainfall gradient in Central Himalayas. Int. J. Climatol.
**2017**, 37, 1299–1309. [Google Scholar] [CrossRef] - Arvind, G.; Kumar, P.A.; Karthi, S.G.; Suribabu, C.R. Statistical Analysis of 30 Years Rainfall Data: A Case Study. In Proceedings of the IOP Conference Series: Earth and Environmental Science, Tirumalaisamudram, India, 17–18 March 2017; IOP Publishing: Bristol, UK, 2017. [Google Scholar]
- Hirsch, R.M.; Helsel, D.R.; Cohn, T.A.; Gilroy, E.J. Statistical analysis of hydrologic data. In Handbook of Hydrology; McGraw-Hill Inc.: New York, NY, USA, 1992; pp. 17–55. [Google Scholar]
- Miller, S. Handbook for Agrohydrology; Natural Resources Institute (NRI): Chatham, UK, 1994. [Google Scholar]
- Cicek, I. The Statistical analysis of precipitation in Ankara, Turkey. Firat Üniv. Sos. Bilim. Derg.
**2003**, 13, 1–20. [Google Scholar] - Sardeshmukh, P.D.; Compo, G.P.; Penland, C. Need for Caution in Interpreting Extreme Weather Statistics. J. Clim.
**2015**, 28, 9166–9187. [Google Scholar] [CrossRef] - Rashid, M.M.; Beecham, S.; Chowdhury, R.K. Statistical characteristics of rainfall in the Onkaparinga catchment in South Australia. J. Water Clim. Chang.
**2015**, 6, 352–373. [Google Scholar] [CrossRef] - Sage, A.P.; Masters, G.W. Least-Squares Curve Fitting and Discrete Optimum Fitting. IEEE Trans. Educ.
**1967**, 10, 29–36. [Google Scholar] [CrossRef] - Cryer, S.A.; Rolston, L.J.; Havens, P.L. Utilizing simulated weather patterns to predict runoff exceedence probabilities for highly sorbed pesticides. Environ. Pollut.
**1998**, 103, 211–218. [Google Scholar] [CrossRef] - Xiao, Z.; Liang, Z.; Li, B.; Hou, B.; Hu, Y.; Wang, J. New Flood Early Warning and Forecasting Method Based on Similarity Theory. J. Hydrol. Eng.
**2019**, 24, 04019023. [Google Scholar] [CrossRef] [Green Version] - Xie, Z.; Su, F.; Liang, X.; Zeng, Q.; Hao, Z.; Guo, Y. Applications of a surface runoff model with Horton and Dunne runoff for VIC. Adv. Atmos. Sci.
**2003**, 20, 165–172. [Google Scholar] - Liang, Z.; Wang, J.; Li, B.; Yu, Z. A statistically based runoff-yield model coupling infiltration excess and saturation excess mechanisms. Hydrol. Process.
**2012**, 26, 2856–2865. [Google Scholar] [CrossRef] - Gan, Y.; Liu, H.; Jia, Y.; Zhao, S.; Wei, J.; Xie, H.; Zhaxi, D. Infiltration-runoff model for layered soils considering air resistance and unsteady rainfall. Hydrol. Res.
**2019**, 50, 431–458. [Google Scholar] [CrossRef] - Rosmann, T.; Domínguez, E. A Fokker–Planck–Kolmogorov equation-based inverse modelling approach for hydrological systems applied to extreme value analysis. J. Hydroinform.
**2018**, 20, 1296–1309. [Google Scholar] [CrossRef] - Elfeki, A.M.M.; Ewea, H.A.R.; Bahrawi, J.A.; Al-Amri, N.S. Incorporating transmission losses in flash flood routing in ephemeral streams by using the three-parameter Muskingum method. Arab. J. Geosci.
**2015**, 8, 5153–5165. [Google Scholar] [CrossRef] - Li, X.; Ren, L. Effect of temporal resolution of NDVI on potential evapotranspiration estimation and hydrological model performance. Chin. Geogr. Sci.
**2007**, 17, 357–363. [Google Scholar] [CrossRef] - Xu, Q.; Chen, X.; Bi, J.; Ouyang, R.; Ren, L. Simulating hydrological responses with a physically based model in a mountainous watershed. Proc. Int. Assoc. Hydrol. Sci.
**2015**, 370, 153–159. [Google Scholar] [CrossRef] [Green Version] - Apaydin, H.; Sonmez, F.K.; Yildirim, Y.E. Spatial interpolation techniques for climate data in the GAP region in Turkey. Clim. Res.
**2004**, 28, 31–40. [Google Scholar] [CrossRef] [Green Version] - Haberlandt, U. Geostatistical interpolation of hourly precipitation from rain gauges and radar for a large-scale extreme rainfall event. J. Hydrol.
**2007**, 332, 144–157. [Google Scholar] [CrossRef] - Chen, D.; Ou, T.; Gong, L.; Xu, C.-Y.; Li, W.; Ho, C.-H.; Qian, W. Spatial interpolation of daily precipitation in China: 1951–2005. Adv. Atmos. Sci.
**2010**, 27, 1221–1232. [Google Scholar] [CrossRef] - Cannarozzo, M.; Noto, L.V.; Viola, F. Spatial distribution of rainfall trends in Sicily (1921–2000). Phys. Chem. Earth Parts A/B/C
**2006**, 31, 1201–1211. [Google Scholar] [CrossRef] - Wilks, D.S. Statistical Methods in the Atmospheric Sciences; Academic Press: Cambridge, MA, USA, 2011; Volume 100, ISBN 0123850223. [Google Scholar]
- Massey, F.J. The Kolmogorov-Smirnov Test for Goodness of Fit. J. Am. Stat. Assoc.
**1951**, 46, 68–78. [Google Scholar] [CrossRef] - Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Krause, P.; Boyle, D.P.; Bäse, F. Comparison of different efficiency criteria for hydrological model assessment. Adv. Geosci.
**2005**, 5, 89–97. [Google Scholar] [CrossRef] [Green Version] - Gupta, H.V.; Kling, H. On typical range, sensitivity, and normalization of Mean Squared Error and Nash-Sutcliffe Efficiency type metrics. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Duan, Q.; Sorooshian, S.; Gupta, V. Effective and efficient global optimization for conceptual rainfall-runoff models. Water Resour. Res.
**1992**, 28, 1015–1031. [Google Scholar] [CrossRef] - Duan, Q.; Sorooshian, S.; Gupta, V.K. Optimal use of the SCE-UA global optimization method for calibrating watershed models. J. Hydrol.
**1994**, 158, 265–284. [Google Scholar] [CrossRef] - Jayawardena, A.W.; Zhou, M.C. A modified spatial soil moisture storage capacity distribution curve for the Xinanjiang model. J. Hydrol.
**2000**, 227, 93–113. [Google Scholar] [CrossRef] - Li, H.; Zhang, Y.; Chiew, F.H.S.; Xu, S. Predicting runoff in ungauged catchments by using Xinanjiang model with MODIS leaf area index. J. Hydrol.
**2009**, 370, 155–162. [Google Scholar] [CrossRef] - Ahirwar, A.; Jain, M.K.; Perumal, M. Performance of the Xinanjiang model. In Hydrologic Modeling; Springer: Berlin/Heidelberg, Germany, 2018; pp. 715–731. [Google Scholar]
- Jie, M.; Chen, H.; Xu, C.; Zeng, Q.; Chen, J.; Kim, J.; Guo, S.; Guo, F. Transferability of Conceptual Hydrological Models Across Temporal Resolutions: Approach and Application. Water Resour. Manag.
**2018**, 32, 1367–1381. [Google Scholar] [CrossRef] - Kyi, K.H.; Lu, M. Development of XAJMISO hydrological model for rainfall-runoff analysis. Hydrol. Res. Lett.
**2019**, 13, 34–40. [Google Scholar] [CrossRef] [Green Version] - Bao, W. Hydrological Forecasting, 3rd ed.; China Water Power Press: Beijing, China, 2006. [Google Scholar]

**Figure 1.**Structure of the vertically mixed runoff (VMR) model. The diagram of (

**a**) the infiltration capacity distribution curve and the generation of surface runoff (RS); (

**b**) the soil moisture capacity distribution curve and the generation of groundwater runoff (RG). P is the net rainfall (or effective rainfall), I is the infiltration, I

_{c}is the infiltration capacity, I

_{cmax}is the maximum infiltration capacity within the catchment, S

_{c}is the soil moisture capacity, S

_{cmax}is the maximum soil moisture capacity within the catchment, W

_{0}is the initial soil moisture content, ΔW is the changes in the soil moisture content, and S

_{c}

_{0}is the soil moisture capacity corresponding to W

_{0}.

**Figure 2.**Locations of the Huangnizhuang catchment, rain gauging stations, and a hydrological station.

**Figure 3.**Comparisons of the fitness of the distributions of observed rainfall and the EDD for the rainfall event that occurred on 02/09/2005. The left panels in the subplots represent the probability density function (pdf) of observed rainfall (black bars) and EDD (red lines); the right panels of each subplot represent the cumulative distribution function (CDF) of observations (black dotted lines) and the EDD (red lines). The subplots represent the cases of hourly rainfall that occurred from 11:00 (Figure 3a) to 16:00 (Figure 3f) on 02/09/2005.

**Figure 4.**Comparisons between the observed flow and simulated mean flow (

**a**) for the flood event that occurred on 10/05/1989 in the calibration period and (

**b**) for the flood event that occurred on 10/07/2005 in the validation period.

**Table 1.**The description and ranges of the parameters calibrated in the statistical rainfall-runoff (SRR) model.

Parameter | Description | Ranges |
---|---|---|

KC | The ratio of potential evapotranspiration to pan evaporation | 0.5–1.5 |

ImA | The impermeable area ratio | 0–1 |

BF | The exponent of the infiltration capacity distribution curve | 0–0.5 |

I_{s} | Stable infiltration capacity, mm/h | 2–60 |

KF | Osmotic coefficient of soil | 0.5–2 |

S_{cmax} | Mean areal soil moisture capacity, mm | 50–200 |

$\beta $ | The exponent of the soil moisture capacity curve | 0–2 |

K | Channel time lag or the hydrograph movement time, h | 1–24 |

**Table 2.**Results of the KS test for the 16 rainfall events described by the exponential difference distribution (EDD) and gamma distribution.

Rainfall Event (Day/Month/Year) | The EDD | The Gamma Distribution | ||||
---|---|---|---|---|---|---|

Mean p-Value | H = 0 (%) | H = 1 (%) | Mean p-Value | H = 0 (%) | H = 1 (%) | |

25/06/1983 | 0.35 | 68.8 | 31.2 | 0.38 | 68.8 | 31.2 |

01/05/1987 | 0.30 | 63.2 | 36.8 | 0.31 | 57.9 | 42.1 |

09/08/1988 | 0.14 | 61.5 | 38.5 | 0.31 | 76.9 | 23.1 |

26/08/1988 | 0.18 | 52.6 | 47.4 | 0.17 | 42.1 | 57.9 |

10/05/1989 | 0.35 | 90.9 | 9.1 | 0.51 | 93.4 | 6.6 |

01/07/1990 | 0.31 | 72.2 | 27.8 | 0.38 | 77.8 | 22.2 |

20/09/1993 | 0.12 | 48.0 | 52.0 | 0.13 | 48.0 | 52.0 |

24/06/1995 | 0.27 | 69.2 | 30.8 | 0.36 | 84.6 | 15.4 |

08/07/1995 | 0.43 | 83.3 | 16.7 | 0.55 | 88.9 | 11.1 |

23/06/2002 | 0.45 | 94.7 | 5.3 | 0.65 | 98.9 | 1.1 |

18/07/2004 | 0.57 | 91.4 | 8.6 | 0.71 | 94.3 | 5.7 |

13/08/2004 | 0.42 | 85.1 | 14.9 | 0.50 | 87.2 | 12.8 |

10/07/2005 | 0.18 | 64.3 | 35.7 | 0.44 | 78.6 | 21.4 |

02/09/2005 | 0.29 | 67.2 | 32.8 | 0.39 | 67.2 | 32.8 |

22/07/2006 | 0.33 | 66.7 | 33.3 | 0.41 | 66.7 | 33.3 |

16/08/2008 | 0.45 | 71.1 | 28.9 | 0.53 | 73.7 | 26.3 |

Flood Event (Day/Month/Year) | Observed Peak Flow (m^{3}/s) | Simulated Peak Flow (m^{3}/s) | Absolute Value of the Flood Peak Relative Error (%) | Absolute Value of the Flood Volume Relative Error (%) | NSE | |
---|---|---|---|---|---|---|

Calibration | 25/06/1983 | 1350 | 1100 | 18.5 | 8.5 | 0.91 |

01/05/1987 | 651 | 527 | 19.0 | 5.3 | 0.92 | |

09/08/1988 | 447 | 413 | 7.6 | 31.5 | 0.71 | |

26/08/1988 | 933 | 767 | 17.8 | 11.0 | 0.76 | |

10/05/1989 | 460 | 459 | 0.2 | 21.1 | 0.92 | |

01/07/1990 | 315 | 304 | 3.5 | 25.8 | 0.73 | |

20/09/1993 | 614 | 578 | 5.9 | 17.2 | 0.71 | |

24/06/1995 | 833 | 646 | 22.4 | 4.3 | 0.84 | |

08/07/1995 | 375 | 425 | 13.3 | 7.6 | 0.85 | |

23/06/2002 | 586 | 585 | 0.2 | 3.1 | 0.88 | |

18/07/2004 | 669 | 534 | 20.2 | 5.6 | 0.77 | |

13/08/2004 | 521 | 509 | 2.3 | 1.6 | 0.79 | |

Validation | 10/07/2005 | 587 | 664 | 13.1 | 20.3 | 0.85 |

02/09/2005 | 986 | 843 | 14.5 | 12.9 | 0.92 | |

22/07/2006 | 384 | 335 | 12.8 | 4.8 | 0.73 | |

16/08/2008 | 705 | 640 | 9.2 | 7.8 | 0.78 |

Flood Event (Day/Month/Year) | Observed Peak Flow (m^{3}/s) | Simulated Peak Flow (m^{3}/s) | Absolute Value of the Flood Peak Relative Error (%) | Absolute Value of the Flood Volume Relative Error (%) | NSE | |
---|---|---|---|---|---|---|

Calibration | 25/06/1983 | 1350 | 1150 | 14.8 | 21.3 | 0.80 |

01/05/1987 | 651 | 571 | 12.3 | 40.3 | 0.72 | |

09/08/1988 | 447 | 266 | 40.5 | 17.0 | 0.79 | |

26/08/1988 | 933 | 1200 | 28.6 | 24.3 | 0.84 | |

10/05/1989 | 460 | 422 | 8.3 | 17.8 | 0.86 | |

01/07/1990 | 315 | 342 | 8.6 | 19.5 | 0.89 | |

20/09/1993 | 614 | 599 | 2.4 | 4.9 | 0.96 | |

24/06/1995 | 833 | 653 | 21.6 | 32.4 | 0.89 | |

08/07/1995 | 375 | 449 | 19.7 | 27.1 | 0.72 | |

23/06/2002 | 586 | 669 | 14.2 | 21.1 | 0.90 | |

18/07/2004 | 669 | 685 | 2.4 | 12.8 | 0.87 | |

13/08/2004 | 521 | 610 | 17.1 | 0.1 | 0.92 | |

Validation | 10/07/2005 | 587 | 901 | 53.5 | 13.0 | 0.87 |

02/09/2005 | 986 | 1170 | 18.7 | 5.8 | 0.86 | |

22/07/2006 | 384 | 182 | 52.6 | 45.7 | 0.56 | |

16/08/2008 | 705 | 796 | 12.9 | 2.3 | 0.96 |

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**MDPI and ACS Style**

Zhou, Y.; Liang, Z.; Li, B.; Huang, Y.; Wang, K.; Hu, Y.
Seamless Integration of Rainfall Spatial Variability and a Conceptual Hydrological Model. *Sustainability* **2021**, *13*, 3588.
https://doi.org/10.3390/su13063588

**AMA Style**

Zhou Y, Liang Z, Li B, Huang Y, Wang K, Hu Y.
Seamless Integration of Rainfall Spatial Variability and a Conceptual Hydrological Model. *Sustainability*. 2021; 13(6):3588.
https://doi.org/10.3390/su13063588

**Chicago/Turabian Style**

Zhou, Yan, Zhongmin Liang, Binquan Li, Yixin Huang, Kai Wang, and Yiming Hu.
2021. "Seamless Integration of Rainfall Spatial Variability and a Conceptual Hydrological Model" *Sustainability* 13, no. 6: 3588.
https://doi.org/10.3390/su13063588