Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions
<p>Duration curves for Prigor river.</p> "> Figure 2
<p>Determining drought volumes and durations for the Q<sub>80%</sub> threshold.</p> "> Figure 3
<p>Methodological approach.</p> "> Figure 4
<p>The Prigor River location—Prigor hydrometric station.</p> "> Figure 5
<p>Water deficit graph for the Prigor River for the WMO reference period.</p> "> Figure 6
<p>The probability distribution curves for minimum 1-day low flows.</p> "> Figure 6 Cont.
<p>The probability distribution curves for minimum 1-day low flows.</p> "> Figure 7
<p>The probability distribution curves for minimum 7-day low flows.</p> "> Figure 8
<p>The probability distribution curves for minimum 30-day low flows.</p> "> Figure 9
<p>The probability distribution curves for annual deficit volume.</p> "> Figure 10
<p>The probability distribution curves for annual deficit volume.</p> "> Figure A1
<p>The variation diagram of <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>C</mi> <mi>s</mi> </msub> <mo>−</mo> <mi>L</mi> <msub> <mi>C</mi> <mi>k</mi> </msub> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Methodology
- -
- 3 for rivers with strong supply from the aquifer;
- -
- 2 for rivers that never dry up and with a significant supply from the aquifer in periods of low water;
- -
- 1.5 for rivers with a deficient supply from the aquifer and with a strong trend of decreasing flows;
- -
- 0–1 in the case of rivers that dry up (small rivers, without water supply).
2.1. Probability Distributions
2.1.1. Log-Normal (LN3)
2.1.2. Pearson III (PE3)
2.1.3. Pearson V (IPV)
2.1.4. Wilson–Hilferty (WH3)
2.1.5. Fatigue Lifetime (FL3)
2.1.6. Weibull (W3)
2.1.7. The Five-Parameter Wakeby Distribution (WK5)
2.1.8. The Five-Parameter Lambda Distribution (L5)
2.2. Parameter Estimation
2.2.1. Log-Normal (LN3)
2.2.2. Pearson III (PE3)
2.2.3. Pearson V (IPV)
2.2.4. Wilson–Hilferty (WH3)
2.2.5. Fatigue Lifetime (FL3)
2.2.6. Weibull (W3)
2.2.7. The Five-Parameter Wakeby Distribution (WK5)
2.2.8. The Five-Parameter Lambda Distribution (L5)
3. Case Study
4. Results
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
MOM | the method of ordinary moments |
L-moments | the method of linear moments |
expected value; arithmetic mean | |
standard deviation | |
coefficient of variation | |
coefficient of skewness; skewness | |
coefficient of kurtosis; kurtosis | |
linear moments | |
coefficient of variation based on the L-moments method | |
coefficient of skewness based on the L-moments method | |
coefficient of kurtosis based on the L-moments method | |
multiplication factor | |
Distr. | Distributions |
LN3 | three parameters log-normal distribution |
FL3 | fatigue lifetime distribution |
PE3 | Pearson III distribution |
IPV | three-parameter Pearson V distribution |
WH3 | three-parameter Wilson–Hilferty distribution |
W3 | three-parameter Weibull distribution |
WK5 | five-parameter Wakeby distribution |
L5 | five-parameter lambda distribution |
xi, Q | observed values |
Qp% | flow with exceedance probability of p% |
Qm | multiannual average flow |
Appendix A. The Frequency Factors for the Analyzed Distributions
Distr. | Frequency factor, | |
Quantile function (inverse function) | ||
Method of ordinary moments (MOM) | L-moments | |
LN3 | ||
PE3 | ||
IPV | where | |
WH3 | ||
W3 | ||
WK5 | ||
L5 |
Appendix B. The Formula for Sample L-Moments
- —is the L-CV;
- —is the L-Skewness;
- —is the L-Kurtosis
Appendix C. Distribution Equivalence Relations LCs-Cs
Distribution | Skewness |
---|---|
Generalized Extreme Value | |
Log-Normal | if |
Frechet | |
Weibull | |
Log-Logistic | |
Pareto | |
Pearson V | |
Wilson–Hilferty | if |
Pseudo-Weibull | |
K3— Generalized Gumbel | if |
K3—Park |
Appendix D. The Central Moments of Wakeby and Lambda Distributions
Appendix E. The First Raw and Central Moments of IPV Distribution
Appendix F. Built-in Function in Mathcad
- —returns the value of the Euler gamma function of x;
- —returns the value of the incomplete gamma function of x with parameter a;
- —returns the probability density for value x, for gamma distribution;
- —returns the cumulative probability distribution for value x, for gamma distribution;
- —returns the inverse cumulative probability distribution for probability p, for gamma distribution. This can also be found in other dedicated programs (the GAMMA.INV function in Excel).
- —returns the inverse standard cumulative probability distribution for probability p, for Normal distribution,(NORM.INV function in Excel).
- —returns the cumulative probability distribution for value x, for log-normal distribution;
- —returns the inverse cumulative probability distribution for probability p, for log-normal distribution, (LOGNORM.INV function in Excel).
- —returns the error function;
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A. Distribution | New Elements |
---|---|
Fatigue Lifetime | inverse function; approximate relation for the shape parameter estimation with the L-moments method |
Wakeby | the first four central moments for parameter estimation with the method of ordinary moments |
Lambda | the first four central moments for parameter estimation with the method of ordinary moments |
B. Frequency factors | the frequency factors for expressing the inverse functions of the distributions with MOM and L-moments The international materials do not contain references regarding the expression of the inverse function of statistical distributions with the L-moments method. |
C. Distribution equivalence relations LCs-Cs | The equivalence relations between the two methods are presented for an easy transition from MOM to L-moments. |
D.relations | The variation is presented for a number of 15 distributions. In the case of the existing relations, presented in other materials, for PE3, GEV, W3, LN3, the relations have been improved, and in the case of the Pearson V, WH3, Chi, Inverse Chi, Pseudo-Weibull, Fréchet, Paralogistic, Inverse Paralogistic distributions, the relations are new. |
E. Duration curve | Approximation of duration curves with polynomials with logarithmic argument |
Length [km] | Average Stream Slope [‰] | Sinuosity Coefficient [-] | Average Altitude, [m] | Drainage Area, [km2] |
---|---|---|---|---|
33 | 22 | 1.83 | 713 | 153 |
Minimum 1-Day Low Flows | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | ||
Flow | [m3/s] | 0.257 | 0.335 | 0.250 | 0.249 | 0.170 | 0.310 | 0.235 | 0.796 | 0.400 | 0.500 | 0.258 |
2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||
Flow | [m3/s] | 0.268 | 0.330 | 0.294 | 0.300 | 0.450 | 0.610 | 0.450 | 0.241 | 0.350 | 0.520 | 0.193 |
2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | ||||
Flow | [m3/s] | 0.216 | 0.173 | 0.687 | 0.180 | 0.388 | 0.207 | 0.182 | 0.185 | 0.390 |
Low Flows | Statistical Indicators | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
[m3/s] | [m3/s] | [-] | [-] | [-] | [m3/s] | [m3/s] | [m3/s] | [m3/s] | [-] | [-] | [-] | |
minimum 1-day | 0.335 | 0.157 | 0.469 | 1.359 | 4.593 | 0.335 | 0.084 | 0.026 | 0.013 | 0.251 | 0.307 | 0.152 |
95% Threshold | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | ||
W | [1000 m3] | 28 | 0.0 | 49.6 | 0.4 | 164.4 | 0.0 | 4.4 | 0.0 | 0.0 | 0.0 | 1.4 |
2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||
W | [1000 m3] | 1.5 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 131.7 | 0.0 | 0.0 | 100.8 |
2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | ||||
W | [1000 m3] | 82 | 133.8 | 0.0 | 179.7 | 0.0 | 233 | 141.1 | 122.8 | 0.0 | ||
W | [1000 m3] | 3511.8 | 1907.7 | 0.0 | 1276.2 | 359.1 | 2402.3 | 1642.2 | 1338.4 | 366.9 |
95% Threshold | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1990 | 1991 | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | ||
D | [days/year] | 23 | 0.0 | 23 | 2.0 | 54 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 2.0 |
2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | ||
D | [days/year] | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 59 | 0.0 | 0.0 | 34 |
2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | ||||
D | [days/year] | 59 | 39 | 0.0 | 35 | 0.0 | 67 | 33 | 27 | 0.0 |
P [%] | Probability Distributions | |||||||
---|---|---|---|---|---|---|---|---|
LN3 | PE3 | IPV | WH3 | FL3 | W3 | WK5 | L5 | |
[m3/s] | ||||||||
50 | 0.304 | 0.299 | 0.306 | 0.272 | 0.301 | 0.296 | 0.290 | 0.297 |
70 | 0.240 | 0.235 | 0.241 | 0.218 | 0.238 | 0.232 | 0.228 | 0.237 |
90 | 0.169 | 0.172 | 0.167 | 0.199 | 0.171 | 0.174 | 0.179 | 0.179 |
95 | 0.142 | 0.152 | 0.138 | 0.198 | 0.147 | 0.158 | 0.169 | 0.159 |
97 | 0.126 | 0.142 | 0.121 | 0.198 | 0.134 | 0.151 | 0.165 | 0.147 |
P [%] | Probability Distributions | |||||||
---|---|---|---|---|---|---|---|---|
LN3 | PE3 | IPV | WH3 | FL3 | W3 | WK5 | L5 | |
[m3/s] | ||||||||
50 | 0.290 | 0.287 | 0.291 | 0.279 | 0.284 | 0.287 | 0.287 | 0.286 |
70 | 0.234 | 0.228 | 0.236 | 0.215 | 0.230 | 0.227 | 0.227 | 0.227 |
90 | 0.180 | 0.180 | 0.180 | 0.186 | 0.185 | 0.180 | 0.180 | 0.181 |
95 | 0.163 | 0.169 | 0.160 | 0.184 | 0.172 | 0.169 | 0.170 | 0.170 |
97 | 0.153 | 0.164 | 0.149 | 0.183 | 0.165 | 0.165 | 0.166 | 0.166 |
Parameter | Distributions | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
LN3 | PE3 | IPV | WH3 | FL3 | W3 | WK5 | L5 | LN3 | PE3 | IPV | WH3 | FL3 | W3 | WK5 | L5 | |
MOM | L-moments | |||||||||||||||
−1.121 | 2.024 | 13.0 | 0.129 | 0.482 | 1.264 | 2.7 × 10−4 | −3.487 | −1.64 | 1.176 | 6.142 | 0.153 | 0.801 | 1.07 | 0.027 | −4.1 | |
0.421 | 0.110 | 5.451 | 0.519 | 3.490 | 0.212 | 3.984 | 0.045 | 0.644 | 0.152 | 1.296 | 0.507 | 6.312 | 0.181 | 3.984 | 0.043 | |
−0.022 | 0.112 | −0.161 | 0.198 | 14.77 | 0.138 | 0.199 | 2.338 | 0.095 | 0.155 | 0.022 | 0.183 | 125.6 | 0.158 | 0.177 | 2.77 | |
- | - | - | - | - | - | - | −1.353 | - | - | - | - | - | - | - | −1.49 | |
- | - | - | - | - | - | −0.130 | 0.013 | - | - | - | - | - | - | −0.041 | 0.001 | |
- | - | - | - | - | - | 0.159 | - | - | - | - | - | - | - | 0.160 | - |
Distributions | Probability [%] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | |
95% threshold—[1000 m3] | ||||||||||
LN3 | 193.01 | 154.51 | 106.01 | 66.71 | 23.51 | 0 | 0 | 0 | 0 | 0 |
PE3 | 196.16 | 154.69 | 106.06 | 66.63 | 23.35 | 0 | 0 | 0 | 0 | 0 |
W3 | 194.24 | 156.61 | 107.05 | 65.83 | 21.01 | 0 | 0 | 0 | 0 | 0 |
Distributions | Probability [%] | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
5 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | |
95% threshold—[days/year] | ||||||||||
LN3 | 60.84 | 49.81 | 35.53 | 23.62 | 10.14 | 0 | 0 | 0 | 0 | 0 |
PE3 | 60.85 | 49.83 | 35.54 | 23.61 | 10.13 | 0 | 0 | 0 | 0 | 0 |
W3 | 61.11 | 50.33 | 35.83 | 23.42 | 9.45 | 0 | 0 | 0 | 0 | 0 |
Distribution | Methods of Parameter Estimation | Observed Data | ||||||
---|---|---|---|---|---|---|---|---|
MOM | L-moments | |||||||
RME | RAE | RME | RAE | |||||
[-] | [-] | [-] | [-] | |||||
LN3 | 0.0169 | 0.0577 | 0.0087 | 0.0342 | 0.307 | 0.197 | 0.307 | 0.152 |
PE3 | 0.0114 | 0.0431 | 0.0057 | 0.0257 | 0.159 | |||
IPV | 0.0187 | 0.0632 | 0.0102 | 0.0392 | 0.212 | |||
WH3 | 0.0124 | 0.0573 | 0.0096 | 0.0444 | 0.083 | |||
FL3 | 0.0141 | 0.0495 | 0.0066 | 0.0306 | 0.186 | |||
W3 | 0.0087 | 0.0367 | 0.0056 | 0.0254 | 0.155 | |||
WK5 | 0.0058 | 0.0273 | 0.0055 | 0.0253 | 0.152 | |||
L5 | 0.0110 | 0.0401 | 0.0056 | 0.0256 | 0.152 |
Distribution | Methods of Parameter Estimation | |
---|---|---|
L-moments | ||
RME | RAE | |
95% threshold | ||
LN3 | 0.2118 | 0.7498 |
PE3 | 0.2119 | 0.7498 |
W3 | 0.2121 | 0.7497 |
Distribution | Methods of Parameter Estimation | |
---|---|---|
L-moments | ||
RME | RAE | |
95% threshold | ||
LN3 | 0.2087 | 0.7343 |
PE3 | 0.2087 | 0.7343 |
W3 | 0.2093 | 0.7357 |
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Anghel, C.G.; Ilinca, C. Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions. Appl. Sci. 2023, 13, 3055. https://doi.org/10.3390/app13053055
Anghel CG, Ilinca C. Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions. Applied Sciences. 2023; 13(5):3055. https://doi.org/10.3390/app13053055
Chicago/Turabian StyleAnghel, Cristian Gabriel, and Cornel Ilinca. 2023. "Hydrological Drought Frequency Analysis in Water Management Using Univariate Distributions" Applied Sciences 13, no. 5: 3055. https://doi.org/10.3390/app13053055