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Probabilistic assessment of annual maximum precipitation in Almaty, Kazakhstan Cover

Probabilistic assessment of annual maximum precipitation in Almaty, Kazakhstan

Scientific Review Engineering and Environmental Sciences
Volume 35 (2026): Issue 1 (March 2026)
By: Yerlan Mukhanbet,  Jarosław Chormański,  Dana Tungatar,  Mariusz Paweł Barszcz and  Ainura Aldiyarova  
Open Access
|Mar 2026
Figures & tables

Figures & Tables

FIGURE 1.

Location map of the meteorological stations in Almaty city, with a general map of KazakhstanSource: own work.

FIGURE 2.

Map of annual precipitation distribution in Almaty citySource: own work.

FIGURE 3.

Annual maximum precipitation distribution fitted with the generalized extreme value distribution using the maximum likelihood estimation method for the Almaty meteorological stationSource: own work.

FIGURE 4.

Annual maximum precipitation distribution fitted with the lognormal distribution using the maximum likelihood estimation method for the Almaty meteorological stationSource: own work.

Values of parameter α for estimating the empirical exceedance probability

αCommentReference
0.4Recommended for hydrometeorological data; provides a balanced estimation for both small and large sample sizesCunnane (1978)
0.5Universal, but performs poorly at the tails (rare events). Frequently used in engineering calculations, but may overestimate probabilities at the distribution tailsHazen (1914)
0Simple and widely used, but may introduce biasWeibull (1939)
0.3Based on Soviet tradition, but often inadequate in mountainous regionsChegodayev (1955)
0.375Used in statistics; closer to the normal distributionBlom (1958)
≈0.333Sometimes applied in scientific publicationsTukey (1962)

Statistical characteristics of annual precipitation sum data series observed in the period of 2000–2023 by stations

Statistical test valueMeteorological station
AlmatyShymbulakMynzhylkyBALKemen
Independence|U| = 1.5|U| = 0.83|U| = 0.38|U| = 0.24|U| = 1.23
Stationarity|K| = 1.76|K| = 1.32|K| = 0.74|K| = 0.86|K| = 0.82
Homogeneity|W| = 1.01|W| = 0.22|W| = 1.01|W| = 0.72|W| = 0.43

Probability density functions, equations, and descriptions

DistributionProbability formulaTypical useAdvantagesLimitations
Gumbel fx1βexp−x−μβ−exp−x−μβ \[ f\left( x \right)\frac{1} {\beta }\exp \left[ { - \frac{{x - \mu }} {\beta } - \exp \left( { - \frac{{x - \mu }} {\beta }} \right)} \right] \] Maxima (annual peak precipitation)Simple, widely used in hydrologyMay underestimate tail behavior (underpredict extremes)
Generalized extreme value (GEV) fx=1σ1+ξx−μσ−1ξ−1exp−1+ξx−μσ−1ξ \[ f\left( x \right) = \frac{1} {\sigma }\left( {1 + \xi \frac{{x - \mu }} {\sigma }} \right)^{ - \frac{1} {\xi } - 1} \exp \left[ { - \left( {1 + \xi \left( {\frac{{x - \mu }} {\sigma }} \right)^{ - \frac{1} {\xi }} } \right.} \right] \] Flexible for maxima with different tail behaviorAccounts for skewness and heavy tailsMore complex parameter estimation
Log--Pearson Type III logX~Pearson Type III \[ \log \left( X \right)\~\textit{Pearson}\,\textit{Type}\,III \] Flood and rainfall extremesRecommended by the U.S. Water Resources CouncilSensitive to sample size and outliers
Pearson Type III fx=βαx−x0α−1e−βx−x0Γα \[ f\left( x \right) = \frac{{\beta ^\alpha \left( {x - x_0 } \right)^{\alpha - 1} e^{ - \beta \left( {x - x_0 } \right)} }} {{\Gamma \left( \alpha \right)}} \ , where x > x0General hydrologic useGood fit for varied data with skewnessAssumes continuous positive data
Lognormal fx=1xσ2πexp−lnx−μ22σ2,x>0 $f\left( x \right) = \frac{1} {{x\sigma \sqrt {2\pi } }}\exp \left( { - \frac{{\left( {\ln x - \mu } \right)^2 }} {{2\sigma ^2 }}} \right),x > 0$] Precipitation intensitiesSuitable for moderately skewed dataSensitive to zero and near-zero values
Normal fx=12πσ2exp−x−μ22σ2 $f\left( x \right) = \frac{1} {{\sqrt {2\pi \sigma ^2 } }}\exp \left( { - \frac{{\left( {x - \mu } \right)^2 }} {{2\sigma ^2 }}} \right)$ Not recommended for extremesEasy to interpretPoor fit for skewed and extreme event data

Geographic characteristics of meteorological stations and descriptive statistics of annual precipitation sum data series observed in the period of 2000–2023

Geographical coordinateMeteorological station
AlmatyShymbulakMynzhylky BALKemen
Latitude (N)43°24′43°12′43°08′ 43°05′43°18′
Longitude (E)76°93′77°08′77°07′ 76°98′76°96′
Descriptive statistics of annual precipitation sum
Minimum [mm]489687668 636615
Maximum [mm]1,0101,4801,240 1,3201,360
Mean (M) [mm]678968875 866904
Standard deviation (SD) [mm]135196154 165179
Coefficient of variation (CV)0.1980.2020.176 0.1900.198
Skewness (γ1)0.9951.340.739 1.120.632
Kurtosis coefficient (γ2)3.244.192.87 3.602.88
Data length [year]2000–2023 (24 years)2001–2023 (23 years)2000–2023 (24 years) 2000–2023 (24 years)2000–2023 (24 years)

Design rainfall intensities for return periods of 1%, 2% and 10% calculated by selected empirical probability functions applied to precipitation annual sum time series in meteorological stations of Almaty city

Meteorological stationDesign rainfall for return periods calculated by selected probability distribution functions [mm]
exp.GEVnormallognormalgamma
10%2%1%10%2%1%10%2%1%10%2%1%10%2%1%
Almaty9361,2501,3908511,0201,09085195599184998310308459641,010
Shymbulak1,3501,8202,0301,2101,4801,5901,2201,3701,4201,2101,4001,4701,2101,3801,440
Mynzhylky1,1601,5001,6501,0701,2601,3301,0701,1901,2301,0701,2301,2801,0701,2001,250
BAL1,1801,5701,7301,0801,3101,4101,0801,2001,2501,0701,2301,2901,0701,2101,270
Kemen1,3001,7801,9901,1401,3301,4001,1301,2701,3201,1401,3301,4001,1301,2901,350

Comparison of the analysis for the meteorological station based on AIC, BIC, P(Mi|x), χ2 test, and p-value criteria

Meteorological stationParameterProbability distribution type
exp.GEVnormallognormalgamma
AlmatyAIC305.7304.2306.4303.2304.0
BIC308.1307.7308.7305.6306.3
P(Mi|x)11.213.48.239.927.1
χ29.22.87.55.15.1
p0.050.410.110.270.27
ShymbulakAIC311.4307.2311.0306.9309.2
BIC313.7310.6313.3309.2310.2
P(Mi|x)4.521.15.543.425.3
χ27.12.84.71.63.4
p0.130.410.320.800.48
MynzhylkyAIC310.0312.2312.8310.8311.2
BIC312.4315.7315.2313.1313.6
P(Mi|x)37.47.19.225.820.4
χ28.64.02.84.54.5
p0.070.260.580.330.33
BALAIC315.1312.8316.0312.5313.4
BIC317.4316.3318.4314.9315.8
P(Mi|x)10.918.86.738.924.5
χ212.75.71.65.15.1
p0.010.120.790.270.27
KemenAIC326.0320.3320.2318.5318.5
BIC328.3323.9322.5320.8320.8
P(Mi|x)0.98.616.939.334.1
χ211.06.34.008.06.3
p0.020.090.400.080.17
DOI: https://doi.org/10.22630/srees.10847 | Journal eISSN: 2543-7496 | Journal ISSN: 1732-9353
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Language: English
Page range: 20 - 39
Submitted on: Oct 3, 2025
Accepted on: Dec 30, 2025
Published on: Mar 31, 2026
Published by: Warsaw University of Life Sciences - SGGW Press
In partnership with: Paradigm Publishing Services
Keywords:
urban flooding,
extreme rainfall,
Gumbel distribution,
generalized extreme value,
GEV,
return period analysis
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© 2026 Yerlan Mukhanbet, Jarosław Chormański, Dana Tungatar, Mariusz Paweł Barszcz, Ainura Aldiyarova, published by Warsaw University of Life Sciences - SGGW Press
This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 License.

Volume 35 (2026): Issue 1 (March 2026)
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