# Extreme Runoff Estimation for Ungauged Watersheds Using a New Multisite Multivariate Stochastic Model MASVC

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multisite Multivariate Stochastic Model MASVC

#### 2.2. Probability Density Functions (PDF)

#### 2.3. Curves IDT

#### 2.4. Soil Conservation Service Curve Number Method (SCS-CN)

#### 2.5. Case Study

^{2}. The region’s predominant climate is subhumid, with an average annual rainfall of 815 mm. The main stream of the Rio Grande de Morelia and its main tributaries (1 Itzicuaros, 2 Alberca, 4 Barajas, 5 Arroyo de Tierras, 6 Rio Chiquito, 8 Atapaneo, 12 Quinceo, 13 Mora Tovar, 14 Calabocito, 15 Calabozo, and 16 Carlos Salazar) will be considered for the subbasins. The weather stations were obtained from the national meteorological service belonging to the National Water Commission (CONAGUA), which are available at https://smn.conagua.gob.mx (accessed on 20 January 2023). Four weather stations with influence in the study area were identified: 16022, 16247, 16055, and 16081. The available data for these stations are from 1980 to 2009 (Table 1).

## 3. Results

#### 3.1. Multisite Multivariate Stochastic Results

_{01}and p

_{11}(4).

#### 3.2. PDFs

#### 3.3. SCS-CN

^{2}), height difference (m), length of the main channel (m), slope (%), and concentration time (h).

#### 3.4. Determination of Surface Runoff for All Subbasins

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Temporal variation of maximum precipitation for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081, blue line is the maxima precipitation (1980 to 2009) and red line is the linear regression.

**Figure A2.**QQ plots normal distribution for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081. Blue plus sings are the normal adjusted precipitation and dotted lines theorical normal quartiles.

**Figure A3.**QQ plots gamma distribution for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055 and (

**d**) 16081. Blue plus sings are the gamma adjusted precipitation and dotted lines theorical gamma quartiles.

**Figure A4.**QQ plots generalized pareto distribution for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081. Blue plus sings are the generalized pareto adjusted precipitation and dotted lines theorical generalized pareto quartiles.

**Figure A5.**QQ plots Gumbel distribution for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081. Blue plus sings are the gumbel adjusted precipitation and dotted lines theorical gumbel quartiles.

**Figure A6.**QQ plots log-normal distribution for the four stations; (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081. Blue plus sings are the log-normal adjusted precipitation and dotted lines theorical log-normal quartiles.

## References

- Beneyto, C.; Aranda, J.Á.; Benito, G.; Francés, F. New Approach to Estimate Extreme Flooding Using Continuous Synthetic Simulation Supported by Regional Precipitation and Non-Systematic Flood Data. Water
**2020**, 12, 3174. [Google Scholar] [CrossRef] - Abreu, M.C.; Cecílio, R.A.; Pruski, F.F.; dos Santos, G.R.; de Almeida, L.T.; Zanetti, S.S. Criteria for Choosing Probability Distributions in Studies of Extreme Precipitation Events. Rev. Bras. Meteorol.
**2018**, 33, 601–613. [Google Scholar] [CrossRef] - Segura-Beltrán, F.; Sanchis-Ibor, C.; Morales-Hernández, M.; González-Sanchis, M.; Bussi, G.; Ortiz, E. Using Post-Flood Surveys and Geomorphologic Mapping to Evaluate Hydrological and Hydraulic Models: The Flash Flood of the Girona River (Spain) in 2007. J. Hydrol.
**2016**, 541, 310–329. [Google Scholar] [CrossRef] - Kastridis, A.; Theodosiou, G.; Fotiadis, G. Investigation of Flood Management and Mitigation Measures in Ungauged Natura Protected Watersheds. Hydrology
**2021**, 8, 170. [Google Scholar] [CrossRef] - Coronado-Hernández, Ó.E.; Merlano-Sabalza, E.; Díaz-Vergara, Z.; Coronado-Hernández, J.R. Selection of Hydrological Probability Distributions for Extreme Rainfall Events in the Regions of Colombia. Water
**2020**, 12, 1397. [Google Scholar] [CrossRef] - Flowers-Cano, R.S.; Ortiz-Gómez, R. Comparison of Four Methods to Select the Best Probability Distribution for Frequency Analysis of Annual Maximum Precipitation Using Monte Carlo Simulations. Theor. Appl. Climatol.
**2021**, 145, 1177–1192. [Google Scholar] [CrossRef] - Moon, Y.-I.; Lall, U. Kernel Quantite Function Estimator for Flood Frequency Analysis. Water Resour. Res.
**1994**, 30, 3095–3103. [Google Scholar] [CrossRef] - Petroselli, A.; De Luca, D.L.; Młyński, D.; Wałęga, A. Modelling Annual Maximum Daily Rainfall with the STORAGE (STOchastic RAinfall GEnerator) Model. Hydrol. Res.
**2022**, 53, 547–561. [Google Scholar] [CrossRef] - Ciupak, M.; Ozga-Zieliński, B.; Tokarczyk, T.; Adamowski, J. A Probabilistic Model for Maximum Rainfall Frequency Analysis. Water
**2021**, 13, 2688. [Google Scholar] [CrossRef] - Tarpanelli, A.; Franchini, M.; Brocca, L.; Camici, S.; Melone, F.; Moramarco, T. A Simple Approach for Stochastic Generation of Spatial Rainfall Patterns. J. Hydrol.
**2012**, 472–473, 63–76. [Google Scholar] [CrossRef] - Alodah, A.; Seidou, O. Assessment of Climate Change Impacts on Extreme High and Low Flows: An Improved Bottom-Up Approach. Water
**2019**, 11, 1236. [Google Scholar] [CrossRef] - Lele, S.; Keim, J.L. Weighted Distributions and Estimation of Resource Selection Probability Functions. Ecology
**2006**, 87, 3021–3028. [Google Scholar] [CrossRef] [PubMed] - Venkata Rao, G.; Venkata Reddy, K.; Srinivasan, R.; Sridhar, V.; Umamahesh, N.V.; Pratap, D. Spatio-Temporal Analysis of Rainfall Extremes in the Flood-Prone Nagavali and Vamsadhara Basins in Eastern India. Weather Clim. Extrem.
**2020**, 29, 100265. [Google Scholar] [CrossRef] - Dasallas, L.; An, H.; Lee, S. Developing an Integrated Multiscale Rainfall-Runoff and Inundation Model: Application to an Extreme Rainfall Event in Marikina-Pasig River Basin, Philippines. J. Hydrol. Reg. Stud.
**2022**, 39, 100995. [Google Scholar] [CrossRef] - Gabriel, K.R.; Neumann, J. A Markov Chain Model for Daily Rainfall Occurrence at Tel Aviv. Q. J. R. Meteorol. Soc.
**1962**, 88, 90–95. [Google Scholar] [CrossRef] - Hernández-Bedolla, J.; Solera, A.; Paredes-Arquiola, J.; Sanchez-Quispe, S.T.; Domínguez-Sánchez, C. A Continuous Multisite Multivariate Generator for Daily Temperature Conditioned by Precipitation Occurrence. Water
**2022**, 14, 3494. [Google Scholar] [CrossRef] - Hayhoe, H.N. Improvements of Stochastic Weather Data Generators for Diverse Climates. Clim. Res.
**2000**, 14, 75–87. [Google Scholar] [CrossRef] - Richardson, C.W.; Wright, D.A.; Nofziger, D.L.; Hornsby, A.G. WGEN: A Model for Generating Daily Weather Variables; U.S. Department of Agriculture: Washington, DC, USA, 1984.
- Marcello, D.; Gianni, B.; Ephrem, H.; Simone, B.; Roberto, C.; Bettina, B. CLIMA: A Weather Generator Framework. In Proceedings of the 18th World IMACS/MODSIM Congress, Cairns, Australia, 13–17 July 2009. [Google Scholar]
- Stöckle, C.O.; Nelson, R.; Donatelli, M.; Castellvì, F. ClimGen: A Flexible Weather Generation Program. In Proceedings of the 2nd International Symposium Modelling Cropping Systems, Florence, Italy, 16–18 July 2001; pp. 16–18. [Google Scholar]
- Semenov, M.A.; Barrow, E.M. User´s guide: LARS-WG A Stochastic Weather Generator for Use in Climate Impact Studies LARS-WG: Stochastic Weather Generator Contents; Rothamsted: Harpended, Hertfordshire, UK, 2002. [Google Scholar]
- Chen, J.; Brissette, F.P.; Leconte, R. WeaGETS—A Matlab-Based Daily Scale Weather Generator for Generating Precipitation and Temperature. Procedia Environ. Sci.
**2012**, 13, 2222–2235. [Google Scholar] [CrossRef] - Mehrotra, R.; Li, J.; Westra, S.; Sharma, A. A Programming Tool to Generate Multi-Site Daily Rainfall Using a Two-Stage Semi Parametric Model. Environ. Model. Softw.
**2015**, 63, 230–239. [Google Scholar] [CrossRef] - Carter, T.; Posch, M.; Tuomenvirta, H. SILMUSCEN and CLIGEN User’s Guide: Guidelines for the Construction of Climatic Scenarios and Use of a Stochastic Weather Generator in the Finnish; Academy of Finland: Helsinki, Finland, 1995. [Google Scholar]
- Richardson, C.W. Stochastic Simulation of Daily Precipitation, Temperature, and Solar Radiation. Water Resour. Res.
**1981**, 17, 182–190. [Google Scholar] [CrossRef] - Rayner, D.; Achberger, C.; Chen, D. A Multi-State Weather Generator for Daily Precipitation for the Torne River Basin, Northern Sweden/Western Finland. Adv. Clim. Change Res.
**2016**, 7, 70–81. [Google Scholar] [CrossRef] - Humphrey, G.B.; Gibbs, M.S.; Dandy, G.C.; Maier, H.R. A Hybrid Approach to Monthly Streamflow Forecasting: Integrating Hydrological Model Outputs into a Bayesian Artificial Neural Network. J. Hydrol.
**2016**, 540, 623–640. [Google Scholar] [CrossRef] - Portoghese, I.; Bruno, E.; Guyennon, N.; Iacobellis, V. Stochastic Bias-Correction of Daily Rainfall Scenarios for Hydrological Applications. Nat. Hazards Earth Syst. Sci.
**2011**, 11, 2497–2509. [Google Scholar] [CrossRef] - Wang, L.; Onof, C. Analysis of sub-daily rainfall sequences based upon a semi-deterministic multiplicative cascade method. In Proceedings of the International Workshop on Advances in Statistical Hydrology, Taormina, Italy, 23–25 May 2010; pp. 1–9. [Google Scholar]
- Vandenberghe, S.; Verhoest, N.E.C.; Buyse, E.; De Baets, B. A Stochastic Design Rainfall Generator Based on Copulas and Mass Curves. Hydrol. Earth Syst. Sci.
**2010**, 14, 2429–2442. [Google Scholar] [CrossRef] - Katz, R.W.; Parlange, M.B. Generalizations of Chain-Dependent Processes: Application to Hourly Precipitation. Water Resour. Res.
**1995**, 31, 1331–1341. [Google Scholar] [CrossRef] - Koch, E.; Naveau, P. A Frailty-Contagion Model for Multi-Site Hourly Precipitation Driven by Atmospheric Covariates. Adv. Water Resour.
**2015**, 78, 145–154. [Google Scholar] [CrossRef] - Ailliot, P.; Allard, D.; Monbet, V.; Naveau, P. Stochastic Weather Generators: An Overview of Weather Type Models. J. Société Française Stat. Rev. Stat. Appliquée
**2015**, 156, 101–113. [Google Scholar] - Wilks, D.S. Multisite Generalization of a Daily Stochastic Precipitation Generation Model. J. Hydrol.
**1998**, 210, 178–191. [Google Scholar] [CrossRef] - Anderson, R.L. Distribution of the Serial Correlation Coefficient. Ann. Math. Stat.
**1942**, 13, 1–13. [Google Scholar] [CrossRef] - Moors, D.S.; Stubblebine, J.B. Chi-Square Tests for multivariate normality with application to common stock prices. Commun. Stat.-Theory Methods
**1981**, 10, 713–738. [Google Scholar] - Hu, S. Akaike Information Criterion Statistics. Math. Comput. Simul.
**1987**, 29, 452. [Google Scholar] [CrossRef] - Lima, A.O.; Lyra, G.B.; Abreu, M.C.; Oliveira-Júnior, J.F.; Zeri, M.; Cunha-Zeri, G. Extreme Rainfall Events over Rio de Janeiro State, Brazil: Characterization Using Probability Distribution Functions and Clustering Analysis. Atmos. Res.
**2021**, 247, 105221. [Google Scholar] [CrossRef] - Simolo, C.; Brunetti, M.; Maugeri, M.; Nanni, T. Improving Estimation of Missing Values in Daily Precipitation Series by a Probability Density Function-Preserving Approach. Int. J. Climatol.
**2010**, 30. [Google Scholar] [CrossRef] - Li, C.; Singh, V.P.; Mishra, A.K. Simulation of the Entire Range of Daily Precipitation Using a Hybrid Probability Distribution. Water Resour. Res.
**2012**, 48, 3521. [Google Scholar] [CrossRef] - Shin, Y.; Park, J.S. Modeling Climate Extremes Using the Four-Parameter Kappa Distribution for r-Largest Order Statistics. Weather Clim. Extrem.
**2023**, 39, 100533. [Google Scholar] [CrossRef] - Alahmadi, F.S.; Rahman, N.A. Climate Change Impacts on Extreme Rainfall Frequency Prediction. J. Water Clim. Change
**2020**, 11, 935–943. [Google Scholar] [CrossRef] - Nwaogazie, I.L.; Sam, M.G.; Enciso, R.Z.; Gonsalves, E. Probability and Non-Probability Rainfall Intensity-Duration-Frequency Modeling for Port-Harcourt Metropolis, Nigeria. Int. J. Hydrol.
**2019**, 3, 66–75. [Google Scholar] [CrossRef] - Bajirao, T.S. Comparative Performance of Different Probability Distribution Functions for Maximum Rainfall Estimation at Different Time Scales. Arab. J. Geosci.
**2021**, 14, 2138. [Google Scholar] [CrossRef] - Devkota, S.; Shakya, N.M.; Sudmeier-Rieux, K.; Jaboyedoff, M.; Van Westen, C.J.; Mcadoo, B.G.; Adhikari, A. Development of Monsoonal Rainfall Intensity-Duration-Frequency (IDF) Relationship and Empirical Model for Data-Scarce Situations: The Case of the Central-Western Hills (Panchase Region) of Nepal. Hydrology
**2018**, 5, 27. [Google Scholar] [CrossRef] - Pizarro, R.; Valdés, R.; García-Chevesich, P.; Vallejos, C.; Sangüesa, C.; Morales, C.; Balocchi, F.; Abarza, A.; Fuentes, R. Latitudinal Analysis of Rainfall Intensity and Mean Annual Precipitation in Chile. Chil. J. Agric. Res.
**2012**, 72, 252–261. [Google Scholar] [CrossRef] - Villón-Béjar, M. HidroEsta, Software for Hydrological Calculations. Rev. Tecnol. En Marcha
**2016**, 29, 95–108. [Google Scholar] [CrossRef] - Villón Béjar, M. HidroEsta, Software Para Cálculos Hidrológicos. Tecnol. En Marcha
**2005**, 18, 67. [Google Scholar] - Villón Béjar, M. HidroEsta, Software Para Cálculos Hidrológicos y Estadísticos Aplicados a La Hidrología. Rev. Digit. Matemática Educ. E Internet
**2014**, 12, 1–8. [Google Scholar] [CrossRef] - García Castro, E.G. Estimación de caudales máximos en el rio Chira, utilizando métodos estadisticos de Gumbel y de Pearson tipo III; Universidad Nacional de Piura: Castilla, Piura, Peru, 2023. [Google Scholar]
- Mendoza, R.; Zavala, J.; Villa, S. Revisión de Gastos de Diseño de La Presa Huites Mediante Relaciones Lluvia-Escurrimiento. Ing. Hidráulica Y Ambient.
**2014**, XXXV, 77–89. [Google Scholar] - Yu, B. Theoretical Justification of SCS Method for Runoff Estimation. J. Irrig. Drain. Eng.
**1998**, 124, 306–310. [Google Scholar] [CrossRef] - Hawkins, R.H.; Hjelmfelt, A.T.; Zevenbergen, A.W. Runoff Probability, Storm Depth, and Curve Numbers. J. Irrig. Drain. Eng.
**1985**, 111, 330–340. [Google Scholar] [CrossRef] - Yu, B. Validation of SCS Method for Runoff Estimation. J. Hydrol. Eng.
**2012**, 17, 1158–1163. [Google Scholar] [CrossRef] - Boughton, W.C. A Review of the USDA SCS Curve Number Method. Aust. J. Soil Res.
**1989**, 27, 511–523. [Google Scholar] [CrossRef] - Hooshyar, M.; Wang, D. An Analytical Solution of Richards’ Equation Providing the Physical Basis of SCS Curve Number Method and Its Proportionality Relationship. Water Resour. Res.
**2016**, 52, 6611–6620. [Google Scholar] [CrossRef] - Kirkby, M.; Cerdà, A. Following the Curve? Reviewing the Physical Basis of the SCS Curve Number Method for Estimating Storm Runoff. Hydrol. Process.
**2021**, 35, e14404. [Google Scholar] [CrossRef] - Stathi, E.; Kastridis, A.; Myronidis, D. Analysis of Hydrometeorological Characteristics and Water Demand in Semi-Arid Mediterranean Catchments under Water Deficit Conditions. Climate
**2023**, 11, 137. [Google Scholar] [CrossRef] - Verma, S.; Verma, R.K.; Mishra, S.K.; Singh, A.; Jayaraj, G.K. A Revisit of NRCS-CN Inspired Models Coupled with RS and GIS for Runoff Estimation. Hydrol. Sci. J.
**2017**, 62, 1891–1930. [Google Scholar] [CrossRef] - Satheeshkumar, S.; Venkateswaran, S.; Kannan, R. Rainfall–Runoff Estimation Using SCS–CN and GIS Approach in the Pappiredipatti Watershed of the Vaniyar Sub Basin, South India. Model. Earth Syst. Environ.
**2017**, 3, 24. [Google Scholar] [CrossRef] - Halwatura, D.; Najim, M.M.M. Application of the HEC-HMS Model for Runoff Simulation in a Tropical Catchment. Environ. Model. Softw.
**2013**, 46, 155–162. [Google Scholar] [CrossRef] - Gimeno, L.; Sorí, R.; Vázquez, M.; Stojanovic, M.; Algarra, I.; Eiras-Barca, J.; Gimeno-Sotelo, L.; Nieto, R. Extreme Precipitation Events. Wiley Interdiscip. Rev. Water
**2022**, 9, e1611. [Google Scholar] [CrossRef] - Chen, X.; Hossain, F. Understanding Future Safety of DAMs in a Changing Climate. Bull. Am. Meteorol. Soc.
**2019**, 100, 1395–1404. [Google Scholar] [CrossRef] - Yin, C.; Wang, J.; Yu, X.; Li, Y.; Yan, D.; Jian, S. Definition of Extreme Rainfall Events and Design of Rainfall Based on the Copula Function. Water Resour. Manag.
**2022**, 36, 3759–3778. [Google Scholar] [CrossRef] - Zhu, B.; Chen, J.; Chen, H. Performance of Multiple Probability Distributions in Generating Daily Precipitation for the Simulation of Hydrological Extremes. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 1581–1592. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Zielinski, P.A. Constraining Frequency Distributions with the Probable Maximum Precipitation for the Stochastic Generation of Realistic Extreme Events. J. Extrem. Events
**2015**, 2, 1550009. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Chaumont, D.; Braun, M. Performance and Uncertainty Evaluation of Empirical Downscaling Methods in Quantifying the Climate Change Impacts on Hydrology over Two North American River Basins. J. Hydrol.
**2013**, 479, 200–214. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Zhang, X.J. Hydrological Modeling Using a Multisite Stochastic Weather Generator. J. Hydrol. Eng.
**2016**, 21, 04015060. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. Downscaling of Weather Generator Parameters to Quantify Hydrological Impacts of Climate Change. Clim. Res.
**2012**, 51, 185–200. [Google Scholar] [CrossRef] - Hernández-Bedolla, J. Análisis de Datos Climáticos Como Predictor Para La Gestión Anticipada de Sequias. Ph.D. Thesis, Universidad Politecnica de Valencia, Valence, Spain, 2022. [Google Scholar]
- Sparks, N.J.; Hardwick, S.R.; Schmid, M.; Toumi, R. IMAGE: A Multivariate Multi-Site Stochastic Weather Generator for European Weather and Climate. Stoch. Environ. Res. Risk Assess.
**2018**, 32, 771–784. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. A Daily Stochastic Weather Generator for Preserving Low-Frequency of Climate Variability. J. Hydrol.
**2010**, 388, 480–490. [Google Scholar] [CrossRef] - Gu, L.; Chen, J.; Xu, C.; Kim, J.; Chen, H.; Xia, J.; Zhang, L. The Contribution of Internal Climate Variability to Climate Change Impacts on Droughts. Sci. Total Environ.
**2019**, 684, 229–246. [Google Scholar] [CrossRef] - Li, Z.; Brissette, F.; Chen, J. Finding the Most Appropriate Precipitation Probability Distribution for Stochastic Weather Generation and Hydrological Modelling in Nordic Watersheds. Hydrol. Process.
**2013**, 27, 3718–3729. [Google Scholar] [CrossRef] - Rawat, K.S.; Singh, S.K. Estimation of Surface Runoff from Semi-Arid Ungauged Agricultural Watershed Using SCS-CN Method and Earth Observation Data Sets. Water Conserv. Sci. Eng.
**2017**, 1, 233–247. [Google Scholar] [CrossRef] - Ouaba, M.; Saidi, M.E.; Alam, M.J. Bin Flood Modeling through Remote Sensing Datasets Such as LPRM Soil Moisture and GPM-IMERG Precipitation: A Case Study of Ungauged Basins across Morocco. Earth Sci. Inform.
**2023**, 16, 653–674. [Google Scholar] [CrossRef] - Meresa, H. Modelling of River Flow in Ungauged Catchment Using Remote Sensing Data: Application of the Empirical (SCS-CN), Artificial Neural Network (ANN) and Hydrological Model (HEC-HMS). Model Earth Syst. Environ.
**2019**, 5, 257–273. [Google Scholar] [CrossRef] - Topçuoğlu, M.E.; Karagüzel, R.; Doğan, A. Comparison of the SCS-CN and Hydrograph Separation Method for Runoff Estimation in an Ungauged Basin: The Izmit Basin, Turke. Int. J. Econ. Environ. Geol.
**2022**, 12, 22–31. [Google Scholar] [CrossRef] - Ningaraju, H.J.; Kumar, S.B.G.; Surendra, H.J. Estimation of Runoff Using SCS-CN and GIS Method in Ungauged Watershed: A Case Study of Kharadya Mill Watershed, India. Int. J. Adv. Eng. Res. Sci.
**2016**, 3, 36–42. [Google Scholar] - Hashim, H.Q.; Sayl, K.N. Incorporating GIS Technique and SCS-CN Approach for Runoff Estimation in the Ungauged Watershed: A Case Study West Desert of Iraq. Iraqi J. Civ. Eng.
**2022**, 14, 1–6. [Google Scholar] [CrossRef] - Nageswara Rao, K. Analysis of Surface Runoff Potential in Ungauged Basin Using Basin Parameters and SCS-CN Method. Appl. Water Sci.
**2020**, 10, 47. [Google Scholar] [CrossRef] - Jeon, J.H.; Lim, K.J.; Engel, B.A. Regional Calibration of SCS-CN L-THIA Model: Application for Ungauged Basins. Water
**2014**, 6, 1339–1359. [Google Scholar] [CrossRef] - Moid Mohammed, A.; Lakshmi Thatiparthi, V.; Rao Pyla, K.; Maryada, A. Estimation of Surface Runoff in an Ungauged Basin Using SCS-CN Method, A Case Study of Manair River Basin in Telangana, India. Appl. Ecol. Environ. Sci.
**2020**, 8, 340–350. [Google Scholar] [CrossRef] - Faouzi, E.; Arioua, A.; Hssaisoune, M.; Boudhar, A.; Elaloui, A.; Karaoui, I. Sensitivity Analysis of CN Using SCS-CN Approach, Rain Gauges and TRMM Satellite Data Assessment into HEC-HMS Hydrological Model in the Upper Basin of Oum Er Rbia, Morocco. Model Earth Syst. Environ.
**2022**, 8, 4707–4729. [Google Scholar] [CrossRef] - Juma, B.; Olang, L.O.; Hassan, M.A.; Mulligan, J.; Shiundu, P.M. Simulation of Flood Peak Discharges and Volumes for Flood Risk Management in the Ungauged Urban Informal Settlement of Kibera, Kenya. Phys. Chem. Earth
**2022**, 128, 103236. [Google Scholar] [CrossRef] - Bharali, B.; Misra, U.K. Numerical Approach for Channel Flood Routing in an Ungauged Basin: A Case Study in Kulsi River Basin, India. Water Conserv. Sci. Eng.
**2022**, 7, 389–404. [Google Scholar] [CrossRef] - Ouaba, M.; El Khalki, E.M.; Saidi, M.E.; Alam, M.J. Bin Estimation of Flood Discharge in Ungauged Basin Using GPM-IMERG Satellite-Based Precipitation Dataset in a Moroccan Arid Zone. Earth Syst. Environ.
**2022**, 6, 541–556. [Google Scholar] [CrossRef] - Forootan, E. GIS-Based Slope-Adjusted Curve Number Methods for Runoff Estimation. Environ. Monit. Assess.
**2023**, 195, 489. [Google Scholar] [CrossRef]

**Figure 2.**Location of the subbasins in Morelia. 1 Itzicuaros, 2 Alberca, 4 Barajas, 5 Arroyo de Tierras, 6 Rio Chiquito, 8 Atapaneo, 12 Quinceo, 13 Mora Tovar, 14 Calabocito, 15 Calabozo, and 16 Carlos Salazar. Patzcuaro basin 1, Angulo basin 2, Cuitzeo basin 3, Zirahuen basin, Hydrologic Región 12.

**Figure 3.**Transition probabilities for all stations (

**a**) ${p}_{01}$ and (

**b**) ${p}_{11}$, (+) extreme data points considered outliers.

**Figure 4.**Daily average of skewness coefficient (1980−2009) with confidence Anderson limits: (

**a**) historic rainfall and (

**b**) normalized rainfall.

**Figure 5.**Residual series for all subbasins: (

**a**) autocorrelation lag−10 and (

**b**) normal standard distribution (blue) and residual (bars).

**Figure 6.**Scatter plots for rainfall occurrence (mean of 30 observed years and 1000 simulated series) for the four stations: (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081.

**Figure 7.**Scatter plots for observed mean versus daily simulated rainfall (mean of 30 observed years and 1000 simulated series) for stations: (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081.

**Figure 8.**Observed and simulated maximum daily rainfall for 30 years (mean of 30 observed years and 1000 simulated series) for the four stations: (

**a**) 16022, (

**b**) 16247, (

**c**) 16055, and (

**d**) 16081.

**Figure 9.**Urban subbasins in Morelia. 1 Itzicuaros, 2 Alberca, 4 Barajas, 5 Arroyo de Tierras, 6 Rio Chiquito, 8 Atapaneo, 12 Quinceo, 13 Mora Tovar, 14 Calabocito, 15 Calabozo, and 16 Carlos Salazar.

Station | Latitude (°) | Longitude (°) | Elevation (msnm) | Years | Total Annual Precipitation (mm/year) | Pmax * (mm/year) |
---|---|---|---|---|---|---|

16022 | 19.625 | −101.281 | 2096 | 1980−2009 | 811.8 | 78 |

16247 | 19.675 | 101.392 | 2097 | 1980−2009 | 700.7 | 75.3 |

16055 | 19.652 | −101.151 | 2180 | 1980−2009 | 1092.25 | 97 |

16081 | 16.289 | −101.176 | 1913 | 1980−2009 | 772.21 | 80.1 |

Statistical/Station | 16055 | 16081 | 16022 | 16247 |
---|---|---|---|---|

Mean | −0.0084 | −0.0035 | 0.080 | −0.078 |

Standard deviation | 1.0431 | 1.0843 | 1.3315 | 1.0895 |

Skewness coefficient | −0.1212 | 0.3246 | 0.5622 | −0.0939 |

Lag-one autocorrelation | 0.0239 | 0.0420 | 0.0461 | −0.0581 |

AIC | −2145 | −1995 | −4733 | −2652 |

**Table 3.**Smirnov–Kolmogorov test for each of the PDFs fitted to the maximum 24 h precipitation series for each station.

Function/Station | 16055 | 16081 | 16022 | 16247 |
---|---|---|---|---|

Normal | 0.1977 | 0.1138 | 0.1932 | 0.1193 |

Log-Normal 3P | 0.1064 | 0.0592 | 0.0696 * | 0.1044 |

Log-Normal 2P | 0.1187 | 0.0618 | 0.1164 | 0.0907 * |

Gamma 2P | 0.1427 | 0.0798 | 0.1433 | 0.1025 |

Gamma 3P | N/A | 0.05595 | N/A | 0.09575 |

Log Pearson III | 0.09761 | 0.0511 | N/A | N/A |

Gumbel | 0.1347 | 0.045 * | 0.1478 | 0.0994 |

Log Gumbel | 0.0871 * | 0.0596 | 0.079 | 0.0924 |

Model | TR | 16022 | 16247 | 16055 | 16081 |
---|---|---|---|---|---|

MASVC | 2 | 34.99 | 40.05 | 47.73 | 31.65 |

5 | 43.78 | 50.47 | 59.49 | 47.45 | |

10 | 58.26 | 64.33 | 71.87 | 59.36 | |

20 | 69.58 | 79.31 | 84.41 | 72.07 | |

50 | 87.33 | 99.25 | 101.34 | 91.33 | |

100 | 102.25 | 112.2 | 116.51 | 105.23 | |

2 | 40.66 | 41.01 | 31.27 | 42.7 | |

5 | 52.61 | 56.2 | 43.61 | 53.75 | |

10 | 61.63 | 66.27 | 54.35 | 61.07 | |

20 | 71.03 | 75.93 | 67.13 | 68.09 | |

50 | 84.27 | 88.49 | 88.23 | 77.18 | |

100 | 95.02 | 98 | 108.28 | 83.99 |

Subbasin * | Area (km^{2}) | Height Difference (m) | Length of the Main Channel (m) | Slope (%) | Concentration Time (h) | CN |
---|---|---|---|---|---|---|

1 | 308.32 | 898 | 30,833.01 | 2.41 | 3.62 | 76.35 |

2 | 47.62 | 1132 | 16,106.46 | 5.72 | 1.56 | 78.36 |

4 | 24.3 | 459 | 11,451.81 | 5.32 | 1.49 | 84.47 |

5 | 26.67 | 561 | 12,862.88 | 4.58 | 1.58 | 83.14 |

6 | 86.79 | 416 | 21,086.86 | 1.13 | 3.13 | 77.03 |

8 | 18.85 | 230 | 9394.3 | 2.46 | 1.55 | 84.16 |

12 | 40.19 | 792 | 14,307.83 | 4.77 | 1.56 | 85.09 |

13 | 10.01 | 258 | 4614.52 | 5.14 | 0.65 | 86.87 |

14 | 6.11 | 220 | 4114.46 | 3.95 | 0.61 | 86.02 |

15 | 11.3 | 671 | 9878.71 | 3.67 | 1.08 | 84.67 |

16 | 10.71 | 1.93 | 453.54 | 0.43 | 0.29 | 87.21 |

Model | * Subbasin/Tr | 2 | 5 | 10 | 20 | 50 | 100 |
---|---|---|---|---|---|---|---|

MASVC-SCS-CN | 1 | 9.68 | 35.01 | 88.48 | 166.82 | 298.97 | 394.90 |

2 | 0.32 | 1.49 | 9.86 | 20.72 | 44.11 | 68.85 | |

4 | 0.11 | 1.45 | 5.39 | 11.93 | 25.73 | 38.10 | |

5 | 1.45 | 6.11 | 13.08 | 22.26 | 36.62 | 53.63 | |

6 | 4.38 | 13.76 | 28.25 | 47.08 | 78.03 | 110.21 | |

8 | 0.11 | 1.17 | 25.30 | 30.72 | 38.93 | 44.85 | |

12 | 0.93 | 2.51 | 9.08 | 19.89 | 42.50 | 62.65 | |

13 | 0.05 | 0.47 | 2.86 | 7.39 | 17.72 | 27.20 | |

14 | 0.02 | 0.10 | 1.75 | 4.29 | 10.12 | 16.44 | |

15 | 0.01 | 0.12 | 1.93 | 4.64 | 10.78 | 17.52 | |

16 | 0.23 | 0.94 | 2.95 | 6.09 | 12.43 | 17.95 | |

PDF-SCS-CN | 1 | 10.56 | 52.59 | 104.17 | 142.81 | 217.25 | 280.62 |

2 | 0.98 | 7.06 | 16.79 | 25.42 | 44.15 | 62.26 | |

4 | 0.91 | 4.24 | 5.08 | 7.64 | 17.92 | 23.30 | |

5 | 0.33 | 1.11 | 5.81 | 11.47 | 28.00 | 48.51 | |

6 | 2.21 | 3.06 | 13.73 | 25.48 | 59.66 | 101.38 | |

8 | 0.67 | 3.16 | 3.79 | 5.68 | 13.39 | 17.45 | |

12 | 1.40 | 6.67 | 7.99 | 12.02 | 28.33 | 36.87 | |

13 | 0.11 | 1.73 | 2.15 | 3.72 | 10.53 | 14.29 | |

14 | 0.10 | 0.90 | 2.74 | 4.70 | 8.98 | 13.24 | |

15 | 0.06 | 1.34 | 3.74 | 6.01 | 11.05 | 16.06 | |

16 | 0.60 | 2.29 | 2.61 | 3.91 | 8.65 | 11.06 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hernández-Bedolla, J.; García-Romero, L.; Franco-Navarro, C.D.; Sánchez-Quispe, S.T.; Domínguez-Sánchez, C.
Extreme Runoff Estimation for Ungauged Watersheds Using a New Multisite Multivariate Stochastic Model MASVC. *Water* **2023**, *15*, 2994.
https://doi.org/10.3390/w15162994

**AMA Style**

Hernández-Bedolla J, García-Romero L, Franco-Navarro CD, Sánchez-Quispe ST, Domínguez-Sánchez C.
Extreme Runoff Estimation for Ungauged Watersheds Using a New Multisite Multivariate Stochastic Model MASVC. *Water*. 2023; 15(16):2994.
https://doi.org/10.3390/w15162994

**Chicago/Turabian Style**

Hernández-Bedolla, Joel, Liliana García-Romero, Chrystopher Daly Franco-Navarro, Sonia Tatiana Sánchez-Quispe, and Constantino Domínguez-Sánchez.
2023. "Extreme Runoff Estimation for Ungauged Watersheds Using a New Multisite Multivariate Stochastic Model MASVC" *Water* 15, no. 16: 2994.
https://doi.org/10.3390/w15162994