Selection of Hydrological Probability Distributions for Extreme Rainfall Events in the Regions of Colombia
Abstract
:1. Introduction
2. Case Study
3. Methodology
3.1. Selection of Rainfall Stations
3.2. Frequency Analysis
- Gumbel distribution
- GEV distribution
- Pearson type III distribution
- Log-Pearson type III distribution
- Normal distribution
3.3. Goodness of Fit Test and Methods of Estimation of Parameters
3.4. Selection of Hydrological Distribution
- For each rainfall stations the mean, maximum and minimum values, and standard deviation of the chi-squared test were computed for the Gumbel-ML, Gumbel-MV, Log-Pearson Type III-SAM, Pearson Type III-ML, Pearson Type III-WM, Normal-ML, GEV-ML and GEV-WM. These eight methods were used because they have adequately fitted the trend of maximum daily precipitation in various publications [22,23]. Based on this analysis, a regional mean value of the chi-squared test for Colombia was calculated based on the number of stations using a weighted mean.
- Estimation of percentage that establishes times where a hydrological distribution reaches the best fits of the trend of maximum daily precipitation records considering the minimum value of the chi-squared test.
4. Analysis of Results
- In all regions of Colombia, the best fits of the chi-squared test were obtained with the GEV probability distribution. The weighted moment method best fits the parameters for this distribution and has an average regional value for Colombia of 5.04. There are other probability distributions that also fit the trend of the data similarly well: GEV with the maximum likelihood method, Gumbel with the weighted moment and maximum likelihood methods and Pearson’s with the method of weighted moments. The Gumbel distribution using the WM method brings a better estimation of maximum daily precipitation for several return periods in comparison with the ML, obtaining a similar result reported in the literature [22].
- In Colombia, the poorest fits were obtained when employing the Pearson type III probability distribution with the maximum likelihood method, where an average value of the chi-square test of 56.57 was obtained, and the log-Pearson type III distribution with the SAM method which had a value of 10.31. This finding is also verified by analyzing the maximum and minimum values and the standard deviation in these probability functions.
- In the Amazonas region, the best fit in the chi-squared test was obtained with the GEV probability distribution and the weighted moment method, with a value of 4.36. This value may have been obtained because few stations were used in the analyses.
5. Conclusions and Recommendations
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A
1107013 | 2102002 | 2120112 | 2120637 | 2312009 | 2602025 | 2618019 |
1506001 | 2103003 | 2120113 | 2120639 | 2312012 | 2602503 | 2619010 |
1506002 | 2103005 | 2120115 | 2120640 | 2312014 | 2602507 | 2618502 |
1506004 | 2103006 | 2120133 | 2120641 | 2312019 | 2603003 | 2618504 |
1506005 | 2103008 | 2120134 | 2120644 | 2312024 | 2603005 | 2619009 |
1506006 | 2103009 | 2120136 | 2120646 | 2314502 | 2603007 | 2619502 |
1506007 | 2103011 | 2120138 | 2120647 | 2319070 | 2603503 | 2620012 |
1506008 | 2104001 | 2120141 | 2120652 | 2319511 | 2604026 | 2620507 |
1506009 | 2104002 | 2120156 | 2120659 | 2401002 | 2604031 | 2621007 |
1506010 | 2104003 | 2120159 | 2123502 | 2401011 | 2604501 | 2621008 |
1506011 | 2104004 | 2120166 | 2303502 | 2401015 | 2605006 | 2621009 |
1506013 | 2104005 | 2120167 | 2120046 | 2401018 | 2605027 | 2623013 |
1506014 | 2104006 | 2120168 | 2120049 | 2401020 | 2605507 | 2701077 |
1506015 | 2104007 | 2120169 | 2120139 | 2401021 | 2606003 | 2801020 |
1506016 | 2105006 | 2120170 | 2120151 | 2401024 | 2606020 | 2801028 |
1506018 | 2105007 | 2120172 | 2120189 | 2401026 | 2606502 | 2801029 |
1506020 | 2105014 | 2120173 | 2120691 | 2401027 | 2607011 | 3705001 |
4401503 | 2105027 | 2120174 | 2120611 | 2401028 | 2607076 | 3802002 |
3509510 | 2105029 | 2120176 | 2305504 | 2401029 | 2607501 | 3212001 |
2101005 | 2105502 | 2120177 | 2306014 | 2401030 | 2608007 | 3306001 |
2101006 | 2106004 | 2120178 | 2306019 | 2401031 | 2608501 | 4208001 |
2101010 | 2106007 | 2120179 | 2306033 | 2401033 | 2609523 | 4704003 |
2101011 | 2106008 | 2120180 | 2306034 | 2401035 | 2610030 | 3501006 |
2101004 | 2113006 | 2120181 | 2306507 | 2401036 | 2610069 | 3801003 |
2101013 | 2116501 | 2120182 | 2306516 | 2401037 | 2610077 | 3705005 |
2701507 | 2119022 | 2120183 | 2306517 | 2401038 | 2610079 | 3521001 |
2801013 | 2119046 | 2120184 | 2903037 | 2401039 | 2610511 | 3509004 |
2621502 | 2103010 | 2120185 | 1401502 | 2401042 | 2610516 | 4701003 |
2617026 | 2119026 | 2120186 | 2320503 | 2401043 | 2611004 | 3204002 |
2618020 | 2119047 | 2120187 | 2904023 | 2401044 | 2611006 | 4604001 |
1506027 | 2119514 | 2120188 | 2904502 | 2401046 | 2611007 | 3207001 |
1506504 | 2119515 | 2120190 | 2502516 | 2401049 | 2611011 | 3502006 |
1506505 | 2120026 | 2120193 | 2803504 | 2401051 | 2611012 | |
1506510 | 2120027 | 2120194 | 2904511 | 2401052 | 2611015 | |
1506511 | 2120033 | 2120195 | 1308504 | 2401053 | 2611504 | |
1506512 | 2120043 | 2120213 | 1204502 | 2401054 | 2612015 | |
1506513 | 2120044 | 2120214 | 2502519 | 2401055 | 2612017 | |
1507506 | 2120051 | 2120516 | 2321013 | 2401056 | 2612506 | |
1508011 | 2120055 | 2120525 | 2502508 | 2401057 | 2613018 | |
1508503 | 2120060 | 2120540 | 1309005 | 2401058 | 2613020 | |
2101002 | 2120069 | 2120541 | 1702502 | 2401059 | 2613514 | |
2101008 | 2120071 | 2120548 | 1506501 | 2401068 | 2614009 | |
2101012 | 2120073 | 2120557 | 1501505 | 2401110 | 2614012 | |
2101014 | 2120074 | 2120559 | 2906024 | 2401511 | 2614502 | |
2101016 | 2120075 | 2120561 | 2502530 | 2401515 | 2614503 | |
2101017 | 2120077 | 2120562 | 1309003 | 2401518 | 2615006 | |
2101018 | 2120080 | 2120565 | 2502013 | 2401519 | 2615015 | |
2101019 | 2120085 | 2120629 | 2903004 | 2401520 | 2615511 | |
2101020 | 2120088 | 2120630 | 1501502 | 2401521 | 2616010 | |
2101021 | 2120089 | 2120631 | 2904019 | 2401531 | 2616012 | |
2101022 | 2120096 | 2120632 | 2903078 | 2403041 | 2616016 | |
2101023 | 2120103 | 2120633 | 2803503 | 2405007 | 2617015 | |
2101024 | 2120104 | 2120634 | 1701501 | 2406006 | 2617018 | |
2101025 | 2120106 | 2120635 | 2502509 | 2406503 | 2617019 | |
2101028 | 2120111 | 2120636 | 2903508 | 2602002 | 2618018 |
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Region | Number of Rainfall Stations | Percentage of Used Rainfall Stations (%) | Location of Rainfall Stations by Departments of Colombia |
---|---|---|---|
Andean | 250 | 69 | Antioquía, Boyacá, Caldas, Cauca, Cundinamarca, Huila, Quindío, Risaralda, Santander, Tolima |
Caribbean | 59 | 16 | Atlántico, Bolívar, César, Córdoba, Magdalena, San Ándres y Providencia, Sucre |
Pacific | 37 | 10 | Valle, Cauca |
Orinoquía | 11 | 3 | Arauca, Vichada, Meta, Casanare |
Amazonas | 5 | 2 | Vaupés, Putumayo, Guaviare, Amazonas, Caquetá |
Total | 362 | 100 | N/A |
Region | Sta. | Probability Distribution | |||||||
---|---|---|---|---|---|---|---|---|---|
Gum ML | Gum WM | LP SAM | Pea ML | Pea WM | Nor ML | GEV ML | GEV WM | ||
Values of the Chi-Squared Test | |||||||||
Andean | Me | 5.85 | 5.63 | 6.40 | 45.01 | 7.09 | 8.04 | 5.11 | 4.60 |
Mx | 24.60 | 25.78 | 273.0 | 360.0 | 252.0 | 64.9 | 27.6 | 18.2 | |
Mn | 0.26 | 0.26 | 0.36 | 0.26 | 0.29 | 0.29 | 0.29 | 0.29 | |
Sd | 4.30 | 4.13 | 18.55 | 77.43 | 17.89 | 7.63 | 4.05 | 3.52 | |
Caribbean | Me | 7.72 | 5.64 | 16.81 | 91.57 | 6.34 | 7.41 | 5.41 | 5.18 |
Mx | 26.12 | 20.61 | 287.0 | 392.0 | 27.22 | 42.5 | 12.9 | 14.8 | |
Mn | 0.43 | 0.74 | 0.89 | 0.89 | 0.50 | 0.89 | 0.89 | 0.50 | |
Sd | 5.08 | 3.72 | 50.57 | 111.0 | 4.67 | 6.35 | 3.04 | 3.48 | |
Pacific | Me | 9.85 | 8.89 | 9.06 | 48.21 | 8.16 | 9.87 | 8.13 | 7.52 |
Mx | 22.42 | 25.52 | 20.40 | 280.0 | 24.89 | 23.9 | 17.8 | 16.4 | |
Mn | 1.66 | 1.46 | 0.92 | 0.80 | 0.80 | 2.00 | 0.80 | 1.20 | |
Sd | 5.25 | 6.00 | 5.19 | 91.97 | 5.47 | 5.97 | 4.44 | 4.26 | |
Orinoquía | Me | 13.91 | 8.09 | 33.49 | 102.1 | 7.35 | 9.24 | 8.89 | 6.31 |
Mx | 34.48 | 16.62 | 252.0 | 252.0 | 20.97 | 20.9 | 31.6 | 13.7 | |
Mn | 4.15 | 2.42 | 2.64 | 4.11 | 3.00 | 3.68 | 1.50 | 1.50 | |
Sd | 9.06 | 3.99 | 72.82 | 101.1 | 5.31 | 5.60 | 9.03 | 3.83 | |
Amazonas | Me | 6.64 | 6.32 | 87.14 | 183.0 | 4.93 | 6.70 | 4.37 | 4.36 |
Mx | 15.50 | 17.62 | 416.0 | 416.0 | 7.50 | 11.7 | 7.00 | 7.50 | |
Mn | 1.60 | 0.92 | 1.46 | 4.00 | 1.46 | 0.38 | 1.46 | 1.46 | |
Sd | 5.83 | 6.71 | 183.9 | 171.1 | 2.28 | 4.09 | 2.24 | 2.70 | |
Regional mean for Colombia based on the number of stations | Me | 6.82 | 6.05 | 10.31 | 56.57 | 7.06 | 8.14 | 5.57 | 5.04 |
Conventions Sta.: Statistic Me: mean Mx: maximum Mn: minimum Sd: standard deviation | Gum: Gumbel LP: Log-Pearson III Pea: Pearson III Nor: Normal GEV: Generalized extreme value | ML: Maximum likelihood WM: Weighted moments SAM: SAM method |
Station | Code | Region | Gum ML | Gum WM | LP SAM | Pea ML | Pea WM | Nor ML | GEV ML | GEV WM |
---|---|---|---|---|---|---|---|---|---|---|
Doña Juana | 2120630 | Andean | 2.79 | 1.53 | 1.53 | 1.53 | 1.53 | 3.42 | 1.53 | 1.53 |
Apto Rafael Núñez | 1401502 | Carribean | 7.71 | 7.71 | 7.43 | 7.43 | 4.57 | 7.14 | 7.14 | 7.71 |
El Placer | 2610069 | Pacific | 5.51 | 3.87 | 7.56 | 7.97 | 5.92 | 19.87 | 9.21 | 7.56 |
Santa Rita | 3306001 | Orinoquía | 10 | 7.00 | 5.00 | 168.00 | 5.00 | 5.00 | 7.00 | 5.50 |
Puerto Asis | 4701003 | Amazonas | 15.5 | 5.46 | 416 | 416 | 5.46 | 6.15 | 5.46 | 5.46 |
Region | Total Used Rainfall Stations | Reached Percentage of Hydrological Distributions | ||||
---|---|---|---|---|---|---|
GEV | Gum | Pea | LP | Nor | ||
Andean | 250 | 52% | 36% | 31% | 28% | 22% |
Caribbean | 59 | 44% | 42% | 32% | 20% | 27% |
Pacific | 37 | 54% | 30% | 43% | 19% | 22% |
Orinoquía | 11 | 73% | 18% | 36% | 27% | 27% |
Amazonas | 5 | 40% | 60% | 60% | 20% | 20% |
Total | 362 | 52% | 36% | 33% | 25% | 23% |
Region | Total Used Rainfall Stations | GEV and Gum |
---|---|---|
Andean | 250 | 74% |
Caribbean | 59 | 73% |
Pacific | 37 | 73% |
Orinoquía | 11 | 82% |
Amazonas | 5 | 60% |
Total | 362 | 74% |
Region | Extreme Values | Return Period | ||||
---|---|---|---|---|---|---|
5 yr. | 10 yr. | 25 yr. | 50 yr. | 100 yr. | ||
Andean | Min | 37.4 | 39.6 | 41.3 | 42 | 42.6 |
Max | 147 | 173 | 218 | 259 | 242 | |
Caribbean | Min | 64.6 | 84.3 | 97.5 | 99.3 | 100 |
Max | 167 | 199 | 241 | 272 | 306 | |
Pacific | Min | 35.3 | 40.5 | 47.8 | 53.9 | 60.4 |
Max | 121 | 135 | 151 | 162 | 172 | |
Orinoquía | Min | 119 | 131 | 141 | 145 | 149 |
Max | 145 | 152 | 186 | 220 | 262 | |
Amazonas | Min | 124 | 134 | 144 | 150 | 154 |
Max | 139 | 158 | 183 | 200 | 217 |
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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. https://doi.org/10.3390/w12051397
Coronado-Hernández ÓE, Merlano-Sabalza E, Díaz-Vergara Z, Coronado-Hernández JR. Selection of Hydrological Probability Distributions for Extreme Rainfall Events in the Regions of Colombia. Water. 2020; 12(5):1397. https://doi.org/10.3390/w12051397
Chicago/Turabian StyleCoronado-Hernández, Óscar E., Ernesto Merlano-Sabalza, Zaid Díaz-Vergara, and Jairo R. Coronado-Hernández. 2020. "Selection of Hydrological Probability Distributions for Extreme Rainfall Events in the Regions of Colombia" Water 12, no. 5: 1397. https://doi.org/10.3390/w12051397