Probabilistic assessment of annual maximum precipitation in Almaty, Kazakhstan

Extreme hydrometeorological events have become a growing global concern due to their direct socio-economic and environmental impacts. Numerous studies have shown that intense rainfall events and flash floods are increasing in both frequency and severity due to climate change. These phenomena not only damage infrastructure and property but also pose significant risks to human safety, particularly in rapidly urbanizing areas with insufficient drainage systems (D’iya et al., 2014).

Globally, urban flooding has become one of the most critical consequences of climate-induced hydrological extremes. For example, severe rainfall events in Europe, such as the 2021 floods in Germany, Belgium, and Poland, resulted in extensive infrastructural damage and loss of life (Kreienkamp et al., 2021; Pietras & Pyrc, 2025). Similar patterns are observed in South Asia, where countries like India, Pakistan, and Bangladesh experience increasingly frequent monsoon-driven floods (Latif et al., 2017; Muhammad Iskandar et al., 2025). In China and Japan, rapid urbanization combined with changing rainfall regimes has intensified flood hazards, prompting the development of advanced stormwater management systems and statistical models for rainfall prediction (Jayawardane et al., 2024; Ling et al., 2025). These international examples highlight that the challenge of managing extreme precipitation and urban flooding is not unique to a single region but represents a widespread, global issue requiring localized yet methodologically comparable approaches.

In Kazakhstan, Almaty city has recently witnessed a marked increase in the frequency and intensity of extreme precipitation events. Several episodes of intense rainfall and sudden surface runoff have caused localized flooding, especially in low-lying districts such as Almalinskiy and Bostandyk. The city’s mountainous surroundings, coupled with rapid urban expansion and the loss of permeable green spaces, accelerate surface runoff and exacerbate the overload of the existing drainage infrastructure (BES Media, 2025). Furthermore, the absence of a comprehensive stormwater management system across many parts of the city leaves Almaty particularly vulnerable to the adverse effects of hydrological extremes.

These challenges are not unique to Almaty city. Similar situations have been observed in other urban centers of Kazakhstan, such as Astana city, where episodes of heavy rainfall and rapid snowmelt have overwhelmed drainage systems, resulting in urban flooding (Nakispekova, 2024). Almaty city has typical characteristics of the Central Asia urbanization processes and, in the authors’ opinion, can explain similar processes in this region, where rapidly developing cities are dominant. Almaty city faces the dual challenge of adapting to intensifying climate extremes while managing the consequences of rapid urbanization. Traditional stormwater infrastructure and planning approaches, based on outdated climatic assumptions, are increasingly inadequate for modern conditions (Dong et al., 2020). This underscores the urgent need for more adaptive and data-driven methods in urban flood risk assessment and management.

International experience demonstrates that statistical modeling of extreme rainfall using probability distribution functions has become a widely accepted approach to understanding and predicting hydrological extremes. The generalized extreme value (GEV) distribution remains the most commonly applied model in Europe, China, and Australia for estimating design rainfall and return periods of floods due to its flexibility in capturing both short-term and long-term extremes (Coles, 2001; Katz et al., 2002; Gentilucci et al., 2023).

Recent studies have shown that lognormal and gamma distributions continue to be preferred in regions where rainfall intensity exhibits moderate variability and observational records are limited, owing to their computational simplicity and stable fitting performance (Stedinger et al., 1993; Cho et al., 2004; Montes--Pajuelo et al., 2024). However, each model presents specific limitations: for instance, the GEV may overestimate rare events when sample sizes are small, while lognormal and gamma distributions tend to underestimate the upper tail of extreme rainfall (Rima et al., 2025).

Therefore, the combined application and intercomparison of multiple statistical models provide a more robust and reliable basis for assessing precipitation extremes under non-stationary climate conditions, as emphasized in recent hydrological analyses (Gentilucci et al., 2023; Rima et al., 2025).

Between 2015 and 2025, Almaty city experienced several instances of localized flooding and mudflows, primarily triggered by heavy rainfall, rapid snowmelt, and natural runoff from the surrounding mountainous terrain. Due to its geographic location at the foothills of the Trans-Ili Alatau (a subrange of the Tian Shan mountain belt), Almaty is particularly vulnerable to sudden water surges from upstream rivers and glacial sources. Each spring and summer, rising temperatures accelerate glacial melt, leading to swollen rivers and, occasionally, the formation and bursting of moraine--dammed glacial lakes (Kyrgyzbay et al., 2023; Choudhary & Kumar, 2025).

Although none of these incidents matched the scale of the 2015 glacial lake outburst flood, they collectively posed a significant challenge to the urban infrastructure. The city’s peripheral districts – such as Alatau, Turksib, and Bostandyk – were the most frequently affected, often facing obstructed roads, flooded basements, and temporary evacuations (Duskayev et al., 2023; Mirlas et al., 2024). Rapid urban expansion has increased vulnerability: changes to the hydrographic network due to infrastructural development, land use patterns, and inadequate drainage have amplified local flood risks. Human activity, particularly the construction of new neighborhoods and modification of watercourses, has further complicated natural water flow, leading to more severe and recurrent urban flood events (Duskayev et al., 2023).

Despite ongoing efforts by the city and national authorities to improve early warning systems and build protective infrastructures such as mudflow barriers and improved river channels, these events highlight the central role of sustainable urban planning and adaptation to climate change. The continued increase in average temperatures and more frequent episodes of intense precipitation intensify the risk of flash floods and mudflows in Almaty’s metropolitan area (Duskayev et al., 2023; Kyrgyzbay et al., 2023; Choudhary & Kumar, 2025).

This study provides a preliminary analysis of rainfall data collected from meteorological stations in Almaty city prior to their application in urban flood modeling.

The aim of this study is to analyze trends and spatial variability of extreme precipitation and runoff potential in Almaty using ground-based meteorological data (2000–2023) through statistical analysis.

This section presents the methodology used to identify the most suitable probability distribution function for estimating design values of annual maximum precipitation. Such a model can be applied by engineers and water resource specialists for a priori assessments in frequency analysis.

During the first stage, five different probability distribution functions (PDFs) were tested on the series of annual maximum precipitation values to determine which best fit the data. These were selected due to their widespread use in contemporary hydrometeorological studies (Kundzewicz et al., 2013; Dong et al., 2020). The annual maximum precipitation series, obtained from various meteorological stations, were fitted using the HYFRAN software package (Duskayev et al., 2023), a practical tool for evaluating probability distributions. The list of applied distribution functions found in the software is presented in Table 3.

TABLE 3.

Distribution	Probability formula	Typical use	Advantages	Limitations
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 hydrology	May 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 behavior	Accounts for skewness and heavy tails	More complex parameter estimation
Log--Pearson Type III	logX~PearsonTypeIII \[ \log \left( X \right)\~\textit{Pearson}\,\textit{Type}\,III \]	Flood and rainfall extremes	Recommended by the U.S. Water Resources Council	Sensitive 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 > x0	General hydrologic use	Good fit for varied data with skewness	Assumes 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 intensities	Suitable for moderately skewed data	Sensitive 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 extremes	Easy to interpret	Poor fit for skewed and extreme event data

Source: own work.

To fit the parameters of each PDF, maximum likelihood estimation (MLE) and method of moments (MoM) approaches were used. These methods enabled the estimation of distribution parameters based on a series of annual maximum precipitation values.

In addition, it was necessary to determine the empirical exceedance probability of the observed annual maximum precipitation values. For this purpose, the following formula was used:

F(xk)=k−αn−2α+1,0≤α≤0.5, \[ F\left( {x_k } \right) = \frac{{k - \alpha }} {{n - 2\alpha + 1}},0 \leqslant \alpha \leqslant 0.5, \]

where: F(x_k) is empirical probability corresponding to the value x_k, k is rank of the value in the ascending ordered series, n is total number of observations, α is offset parameter depending on the selected formula (cf. Table 4).

To calculate the empirical exceedance probability of annual maximum precipitation in Almaty city, the Cunnane formula (Cunnane, 1978) was selected. The choice of this formula is justified by the region’s topographic and climatic characteristics. Almaty is situated in the foothills of the Zailiyskiy Alatau, where orographic influences result in high spatial and temporal variability in precipitation. This is particularly evident in the case of intense convective rainfall events occurring during the warm season, often associated with moisture-laden mountain air and strong atmospheric instability.

Under such conditions, the Cunnane formula provides the most balanced probability estimates, especially when dealing with a limited number of observations (n ≤ 25 years), which is typical for meteorological stations in the city.

The Cunnane formula minimizes systematic errors at the distribution tails, offering a more accurate representation of both frequent and rare (extreme) events. This makes it particularly useful for comparing empirical data with theoretical distributions, such as the Gumbel, GEV, or log-Pearson Type III distributions, which are widely used in hydrological and climate-related studies (Kundzewicz et al., 2013; Dong et al., 2020). Moreover, the World Meteorological Organization recommends the use of the Cunnane formula for analyzing extreme hydrometeorological data in mountainous regions (WMO, 2009, 2017).

Thus, the use of the Cunnane formula is both scientifically and methodologically justified for analyzing extreme precipitation in the mountainous conditions of Almaty city. It provides a reliable foundation for estimating return periods and assessing the risks of pluvial flooding.

In this study, the 24-year annual precipitation records (2000–2023) from five meteorological stations near Almaty city were analyzed. For hydrological planning and design purposes, the best available long-term precipitation data were used. The annual maximum precipitation values (in mm) were statistically processed to estimate design rainfall intensities for return periods of 10 (10%), 50 (2%), and 100 (1%) years using several PDFs, namely the exponential, GEV, normal, lognormal, and gamma distributions.

The goodness of fit of each distribution was evaluated using multiple criteria: the chi-square (χ²) test, Akaike information criterion (AIC), Bayesian information criterion (BIC), and posterior model probability P(M_i|x). Table 5 presents the comparison of empirical and theoretical precipitation quantiles (in mm) for each station.

Figures 3 and 4 show the results of fitting the GEV and lognormal distributions to the observed values of annual precipitation maxima at the Almaty weather station. The red line shows the theoretical model, the black crosses show the empirical data, and the blue lines show the 95% confidence interval. These two graphs are provided as illustrative examples.

FIGURE 3.

Annual maximum precipitation distribution fitted with the generalized extreme value distribution using the maximum likelihood estimation method for the Almaty meteorological station

Source: own work.

FIGURE 4.

Annual maximum precipitation distribution fitted with the lognormal distribution using the maximum likelihood estimation method for the Almaty meteorological station

Source: own work.

The analysis of AIC, BIC, and P(M_i|x) for the Almaty meteorological station, which serves as a representative location for an urban area in this study, shows that the lognormal distribution demonstrates superior performance, yielding the highest posterior probability P(M_i|x) = 39.94 along with the lowest AIC (303.2) and BIC (305.6) values (Table 6).

Although the GEV showed the best agreement with empirical data according to the χ² test, the lognormal distribution proved more favorable when accounting for model simplicity and goodness-of-fit through information-theoretic metrics. The gamma distribution also performed relatively well, with a posterior probability of 27.1 and competitive AIC (304.0) and BIC (306.3) values (Table 6).

The variation in the best-fit distribution across different stations and criteria underscores the complexity of hydrological modeling in a region with complex climatic conditions, such as those in Almaty city. The discrepancy between the χ² test (favoring the GEV) and information criteria/Bayesian probabilities (favoring lognormal/gamma) highlights the importance of using multiple goodness-of-fit measures for robust model selection. The GEV, while showing good agreement with empirical data via the χ² test, might be penalized by AIC and BIC if its additional parameter does not significantly improve the fit relative to its complexity.

Estimating design rainfall with different return periods (e.g., 1%, 2%, 5%, and 10%) is crucial for infrastructure planning and for mitigating risks from extreme precipitation events, such as flash floods and urban inundations, which are intensified by ongoing urbanization and climate variability in Almaty. The selection of the most appropriate PDF directly impacts the accuracy of these design values.

Similar approaches have been widely applied in other countries, where various probability distributions have been tested to model extreme rainfall events. For instance, the GEV is the most commonly used model in Europe, China, and Australia for estimating design rainfall and flood return periods due to its flexibility in capturing both short-term and long-term extremes (Coles, 2001; Burnham & Anderson, 2002; Katz et al., 2002; Gentilucci et al., 2023). In contrast, lognormal and gamma distributions are frequently applied in regions with moderate rainfall variability and limited data availability, such as parts of South Asia and Africa, owing to their computational simplicity and stable fitting performance (Stedinger et al., 1993; Cho et al., 2004; Montes-Pajuelo et al., 2024). However, these models differ in their ability to represent the upper tail of extreme precipitation: the GEV tends to perform better for rare events, whereas the lognormal and gamma distributions may underestimate extremes. Therefore, comparing multiple statistical distributions, as presented in this study, provides a more robust and comprehensive understanding of precipitation extremes under non-stationary climate conditions.

This study presents a comprehensive probabilistic assessment of extreme precipitation in the city of Almaty, based on 24 years of ground-based meteorological observations (2000–2023) from five stations: Almaty, Shymbulak, Mynzhylky, BAL, and Kemen. The analysis used annual maximum precipitation values and applied five theoretical probability distribution functions – exp., GEV, normal, lognormal, and gamma – to model extreme rainfall behavior.

All rainfall series were confirmed to be independent, stationary, and homogeneous, as verified by statistical tests (Wald–Wolfowitz, Kruskal–Wallis, and Pettitt). The results of the MLE showed that the GEV and lognormal distributions provided the best fit across most stations.

Quantitatively, the maximum annual precipitation during the study period ranged from 615 mm (Kemen) to 1,010 mm (Shymbulak), while the mean annual precipitation varied between 678 mm (Almaty) and 866 mm (BAL). The CV ranged from 0.17 to 0.20, indicating moderate interannual variability.

Based on the best-fitting probability models, the estimated rainfall quantiles showed a clear increase with longer return periods. On average across all stations, the 10-year event (10% exceedance probability) corresponded to rainfall amounts of approximately 950–1,050 mm, the 50-year event (2% exceedance) to around 1,100–1,200 mm, and the 100-year event (1% exceedance) to 1,200–1,300 mm. These values represent the expected magnitudes of extreme precipitation for the respective return periods and can serve as reference thresholds for hydrological design and risk assessment in Almaty city. The lognormal distribution demonstrated the lowest AIC and BIC values and the highest posterior probability P(M_i|x) > 0.35 for the Almaty and Shymbulak meteorological stations, while the GEV was slightly better for BAL and Kemen meteorological stations. The gamma model also performed reasonably well (AIC differences < 5), suggesting its potential suitability for areas with complex mountain–urban topography.

These quantitative results confirm that extreme rainfall events have intensified in Almaty city, especially during the warm season (May–September). The observed upward trend in the magnitude of extreme precipitation highlights growing risks of urban flash floods and surface runoff overload.

From a practical standpoint, the derived rainfall quantiles can be directly applied to the design of stormwater drainage networks, flood risk assessments, and climate-resilient infrastructure planning in Almaty. For instance, a 100-year design rainfall of approx. 1,250 mm can serve as a baseline parameter for the dimensioning of retention basins and flood control structures.