1 Introduction

Precipitation and temperature extremes are the most important meteorological, hydrological, and climatological parameters and have been extensively analysed in the past few decades (Seneviratne et al. 2012). Some of the most significant impacts of climate change on mountain systems include rapid melting of glaciers, changes in vegetation cover, species migration and loss, erratic weather patterns, loss of snow cover, water scarcity, and an increase in the frequency and magnitude of natural disasters. Localised rainfall and temperature extremes can have a profound impact on human activities. Specifically, for water resource management, it is essential to know the probability of exceedance of rainfall from the known amount. Heavy rainfall is one of the most frequent and widespread severe weather hazards which affects Uttarakhand. From past studies, it has been observed that high temperatures with no atmospheric moisture are one of the main reasons for forest fires in Uttarakhand, which cause significant loss to the forest ecosystem, flora, and fauna as well as economic wealth (Negi and Kumar 2016). The main reason behind using statistical modelling techniques for weather extremes is its ability to cope with the weather conditions that may occur in the future. For any civil engineering work, the knowledge of extreme weather conditions at the site of interest is essentially required to design engineering structures such as dams that can withstand extreme weather stresses. For this purpose, a risk model is developed by selecting the best-fit probability distribution method. Statistical models for extreme value analysis are based on extreme value theory (EVT). Extreme value theory is an advanced statistical analysis tool to evaluate and simulate the extremes of an independent identical distribution (Reiss and Thomas 2001). Extreme value theory deals with the stochasticity of natural variability by describing extreme events with respect to their probability of occurrence at large or small levels (Hasan et al. 2012). This statistical modelling method has already been successfully applied in various climatological time series to model the extremities by various researchers worldwide, viz., Uppsala, Sweden (Rydén 2011); Botswana (Wilson Moseki Thupeng 2019); Zimbabwe (Chikobvu and Chifurira 2015); middle Ebro Valley, Spain (Beguería and Vicente-Serrano 2006); Penang, Malaysia (Hasan et al. 2012); Nigeria (Sunday et al. 2020); Mbonge, Cameroon (Arreyndip and Joseph 2015); Ghana (Sampson and Kwadwo 2018); western Black Sea subregion, Turkey (Yozgatlıgil and Türkeş 2018); southwest western Australia (Li et al. 2005); Debuncha, Cameroon (Ayuketang et al. 2016); Maharashtra (Gandhre 2020); Tiruchirappalli, India (Sasireka et al. 2019); etc. Extreme rainfall records from different field stations in Uttarakhand during the last century (1900–2013) were analysed (Nandargi et al. 2016). The decreasing trend in extreme rainfall is nearly 42% of recording stations, especially after 1970. Kumar et al. (2017) analysed the statistical distribution of rainfall over all the districts in the state of Uttarakhand. It was observed that in the earlier statistical investigations of meteorological time series in this region, there had been no significant attempt towards understanding the characteristics of statistical (probabilistic) methods for forecasting extreme events in the Himalayan region. Climate affects humans directly through the weather experienced (physically and psychologically) on a day-to-day basis and the impacts of weather on daily living conditions, and indirectly through its impacts on economic, social, and natural environments. Extreme rainfall may result in landslides, flash floods, and crop damage that might have a major impact on society, the economy, and the environment as a whole. In recent years, a greater percentage of precipitation in Uttarakhand has occurred as extreme one day episodes. In June 2013, a cloudburst centred on the north Indian state of Uttarakhand caused devastating floods and landslides and was reported as the worst natural disaster in our country since the Tsunami (the year 2004). The rainfall was observed to be 200% above the average at many sites, including Rudraprayag, Uttarkashi, Chamoli, Pauri, Tehri, and neighbouring districts. When the event actually occurred, the Weather Research and Forecasting System (WRF) model predicted rainfall to fall between 320 and 400 mm above Kedarnath, which is in a fair amount of agreement with the actual amount of rainfall (325 mm) measured by the Wadia Institute of Himalayan Geology (Shekhar et al. 2015).

Because of the topography of Uttarakhand, it would be fair to say that small changes in temperature melt glaciers and turn ice and snow into water. Where extreme slopes lead to rapid changes in climatic zones over small distances, these will show marked impacts in terms of biodiversity, water availability, agriculture and hazards, and this will have an impact on general human well-being. On February 07, 2021, a landslide at the glacier's terminus at an altitude of 5600 m triggered a snow avalanche, covering approximately 14 km2 area and caused a flash flood downstream of Rishi Ganga (Chamoli), where at least 70 people at Raini are believed to have been buried or washed away in the sludge that the floods carried, and around 35 were reported stuck in a tunnel that was choked by the sludge (Rana et al. 2021; Sati 2022). Although the prediction of such extreme weather events is still fraught with uncertainties, a proper assessment of likely future trends would help in setting up infrastructure for disaster preparedness.

The current study focuses on determining the effects of excessive precipitation and temperature, employing a regional approach. In this study, we looked at a century-long time-series dataset (1901–2002) for all 13 districts of Uttarakhand, which is part of the Indian western Himalayas. The monthly precipitation and temperature data were analysed at a seasonal level with the primary goal of maximising the information in the data series and providing a more accurate estimation of the distribution parameters. The physiographic and climatological features of the region are described in section 2. The mathematical basis for the EVT approach is presented next, followed by results and discussions. Concluding thoughts on the statistical modelling of meteorological factors are presented in the final section.

2 Physiographic and climatologic description of the region

In the northern region of India, Uttarakhand is found between 28°43′–31°27′N latitudes and 77°34′–81°02′E longitudes (figure 1). This Indian state is located on the southern slope of the majestic Himalayas, with Tibet to the north and Nepal to the east. Himachal Pradesh is located northwest of Uttarakhand and Uttar Pradesh is located south of it. Uttarakhand is primarily a hilly state, with mountains covering 86% of the land and forests covering the remaining 65%. Ice and bare rock cover the high Himalayan highlands, which are glaciated. However, at lower elevations, there is a lush tropical forest. In general, the altitude of the state varies between 180 and 7800 m amsl (meters above mean sea level). The glaciers in this state are the source of the rivers Ganga and Yamuna. The western half of Uttarakhand is known as the Garhwal region, while the eastern half is known as the Kumaun region. The topography and ecology of this state are diverse, and it is rich in natural resources, particularly water and forest.

Figure 1
figure 1

Map of the study site indicating all 13 districts residing in Kumaon and Garhwal regions of the states of Uttarakhand in NW Himalaya India.

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The tropical monsoon in the summer and western disturbances in the winter affect the state's climate, which is defined by variable seasonal temperatures. According to the India Meteorological Department (IMD), seasons of the state are characterised as winter (December–February), pre-monsoon (March–May), monsoon (June–September), and post-monsoon (October–November). The monsoon season receives about 90% of the annual rainfall. Heavy rainfall events in Uttarakhand were statistically analysed in detail (Nandargi et al. 2016). Uttarakhand is eco-sensitive and extremely vulnerable to natural disasters (viz., cloud bursts, floods, landslides, earthquakes, etc.). Table 1 illustrates the characteristics of Uttarakhand state where population and density of the state are taken from census data available at https://www.census2011.co.in/census/state/districtlist/uttarakhand.html. Elevation (metres) represents average elevation of each district from mean sea level; the area of the districts is represented in km square, while JJAS mean rainfall (mm/month) for each district is calculated from a monthly century-long dataset (1901–2002) which we have considered for the study.

Table 1 Characteristics of Uttarakhand state.
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3 Material and methods

3.1 Data source

The present research examines monthly maximum precipitation and temperature time series data for all 13 districts of Uttarakhand during a 102-year span. The dataset used in the present analysis is obtained from the Climate Research Unit (CRU) TS2.02, Tyndall Centre for Climate Change Research, School of Environmental Sciences, and the University of East Anglia in Norwich, UK (Mitchell and Jones 2005) (https://crudata.uea.ac.uk/cru/data/hrg/timm/grid/CRU_TS_2_0_text.html). The empirical analysis is performed using the ‘extRemes’ (Gilleland and Katz 2016) and ‘evd’ (Stephenson 2018) packages within the R-open source software (R Core Team 2020).

3.2 Extreme value theory (EVT)

EVT is a theoretical framework for modelling the extremities and provides a robust statistical tool for the probabilistic prediction of extreme and rear events. The generalized extreme value (GEV) distribution forms a family of continuous probability distributions which are developed between extreme value theory (EVT). EVT is a branch of statistics dealing with extreme deviations from the median of a probability distribution. Let X1, …, Xn be a series of independently identical distributions (iid) of random variables with an unknown distribution. EVT deals with the stochastic behaviour of the maximum and minimum of identically independent random variables. The essential concept of extreme data is about the limiting behaviour of the series as \(n\to \infty\).

The peaks over threshold (POT) method (Leadbetter 1991) and the Block Maxima (BM) method (Huang et al. 2016) are two principal models that exist for extreme values.

In the present analysis, the BM method is applied. The BM method is widely suitable for applying to the GEV distribution, which is a single three-parametric model with the cumulative distribution function as follows:

$$F(x) ={\mathrm{ exp }}\left\{-{\left[1+\xi \left(\frac{x-\mu }{\sigma }\right)\right]}^{-1/\xi }\right\},$$
(1)

where μ is a location parameter that represents the centre of the GEV distribution, σ is a scale parameter that determines the size of deviations from μ, and finally, the maximum distribution's tail behaviour (i.e., how quickly the upper tail decays) is represented by ξ, which is the shape parameter.

The GEV distribution incorporates three limiting extreme value distributions depending on the value contained by the shape parameter ξ. The shape parameter ξ is also referred to as the extreme value index, and ξ–1 here represents the rate of tail decay.

$${\mathrm{GEV}} \left(x,\xi ,\sigma ,\mu \right)={\mathrm{exp}}\left\{-{\left[1+\xi \left(\frac{x-\mu }{\sigma }\right)\right]}^{-1/\xi }\right\}, \xi \ne 0,$$
(2)
$${\mathrm{GEV}}\left(x,\xi ,\sigma ,\mu \right)={\mathrm{exp}}\left\{-{\mathrm{exp}}{\left[-\left(\frac{x-\mu }{\sigma }\right)\right] }\right\}, \xi =0.$$
(3)

The Gumbel, Frechet, and Weibull distributions are the three members of the GEV distribution family. These three cases are referred to be bounded, light-tailed, and heavy-tailed and can be used to model maxima statistically. When \(\xi =0\), the GEV takes on a lightly-tailed Gumbel distribution. For \(\xi >0\), the GEV takes on a heavy-tailed Frechet distribution, whereas, for \(\xi <0\), the GEV takes on a bounded Weibull distribution. GEV is a more generic or convenient technique for modelling block maxima since it can represent all three types of tail behaviour.

3.3 Stationary/non-stationary test

If a time series has no trend, constant variance, and a consistent autocorrelation structure over time, it is considered to be ‘stationary’.  The ADF test (Augmented Dickey-Fuller test) is a typical statistical technique for determining if a time series is stationary. When it comes to analysing the stationary of a series, it is one of the most widely employed statistical tests. The null and alternative hypotheses are as follows:

  • H0: The time series is non-stationary. In other words, it has a time-dependent structure and does not have constant variance over time.

  • HA: The time series is stationary.

If the tested p-value is less than a certain level of significance (=0.05), we can reject the null hypothesis and conclude that the time series is stationary.

3.4 Parameter estimation

The generalised extreme value distribution was used to investigate the extreme behaviour of precipitation and temperature. Maximum likelihood estimation (MLE) (Naveau et al. 2005), Generalised maximum likelihood estimation (GMLE), L-moments, and Bayesian approaches are used to estimate the three model parameters (location, scale, and shape). The MLE fitting method is the most common way to fit data, even though it can overestimate the parameters for small samples. GMLE, L-moments, or a Bayesian technique may be used to partially solve this problem. Eventually, for parameter estimate and return value estimation, the best-fitted model was chosen. The return level is defined as a value that is expected to be equalled or exceeded on average once every interval of time (T) (with a probability of 1/T). Therefore, we obtain the equation: CDF of the GEV distribution (XT) = 1–1/T (where XT is the level exceeded on an average only once in T years).

4 Results and discussion

This present study utilised precipitation and temperature records from 13 districts in the Indian state of Uttarakhand for statistical modelling and forecasting. All datasets are subjected to the ADF test, which yields a p-value of <0.05. As a consequence, the null hypothesis must be rejected. This indicates that the time series is stationary. In other words, it has a structure that seems to be time-independent and has constant variance over time. The analysis is based on seasonal maxima of monthly mean precipitation and temperature. The precipitation and temperature time series are depicted in figure 2(a–d).

Figure 2
figure 2

Annual maximum precipitation and temperature trends in Uttarakhand from 1901 to 2002. (a and b) representing the Kumaun region and (c and d) representing the Garhwal region, respectively.

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4.1 Precipitation

The rainfall in the state varies from place to place due to its rugged topography. The region of study is strongly influenced by the monsoon, which results in a significant contribution of annual precipitation during June–September. Extreme precipitation and flooding are most common in Uttarakhand during the monsoon season. Because of this, the predictability of seasonal precipitation characteristics, particularly extremes, is important. Because of this fact, there is an interest in the predictability of seasonal precipitation characteristics, including extremes. Understanding how heavy precipitation works enables us to forecast the likelihood of a flood, as well as plan for water supplies and do hydrological modelling.

Figures 3 and 4 provide probability density graphs for the GEV fit for precipitation in Garhwal and Kumaun regions, respectively. Return period values for 50, 100, 200, 300, and 500 years were investigated further and are shown in figures 5 and 6 accordingly. The return periods indicate how long a specific amount of precipitation is predicted to be exceeded once for a given period.

Figure 3
figure 3

Probability density for seven districts in the Garhwal region. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and seven rows indicate seven districts. The blue colour indicates empirical and the red colour depicts the best model fit using either MLE, GMLE, L-moments, or Bayesian methods. Corresponding location scale and shape parameters are embedded in each plot.

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Figure 4
figure 4

Probability density for six districts in the Kumaun region. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and six rows indicate six districts. The blue colour indicates empirical and the red colour depicts the best model fit using MLE, GMLE, L-moments, and Bayesian methods. Corresponding location scale and shape parameters are embedded in each plot.

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Figure 5
figure 5

Estimated return values for seven districts in the Garhwal region where x-axis represents return period and y-axis represents the return level. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon is presented in four different columns, respectively, and seven rows indicate seven districts (indicated right side of each row).

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Figure 6
figure 6

Return value estimation for six districts in the Kumaun region where x-axis represents the return period and y-axis represents the return level. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon is presented in four different columns, respectively, and six rows indicate six districts (indicated on the right side of each row).

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4.2 Temperature

The state is hilly, except for a few plain areas near its southern border. As a result, temperatures in the state vary widely based on elevation, location, slope, and geography.

Figures 7 and 8 provide probability density graphs for the GEV fit for maximum temperatures in the Garhwal and Kumaun regions, respectively. Return period values for 50, 100, 200, 300, and 500 years were investigated further and are shown in figures 9 and 10 accordingly. The return periods indicate how long a specific temperature is predicted to be exceeded once for a given period.

Figure 7
figure 7

Probability density for seven Garhwal districts. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and seven rows indicate seven districts. The blue colour indicates empirical and the red color depicts the best model fit using either MLE, GMLE, L-moments, or Bayesian methods. Corresponding location scale and shape parameters are embedded in each plot.

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Figure 8
figure 8

Probability density for six districts in the Kumaun region. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and six rows indicate six districts. The blue colour indicates empirical and the red colour depicts the best model fit using either MLE, GMLE, L-moments, or Bayesian methods. Corresponding location scale and shape parameters are embedded in each plot.

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Figure 9
figure 9

Return value estimation for seven districts in the Garhwal region where x-axis represents the return period and y-axis represents the return level. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and seven rows indicate seven districts (indicated right side of each row).

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Figure 10
figure 10

Return value estimation for six districts in the Kumaun region where x-axis represents the return period and y-axis represents the return level. Seasonal analysis, viz., winter, pre-monsoon, monsoon, and post-monsoon, is presented in four different columns, respectively, and six rows indicate six different districts (indicated on the right side of each row).

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From figures 3–10, the model diagnostic plot looks reasonable for precipitation and temperature data for GEV distribution. All of the diagnostic plots for all the districts provide evidence that GEV is a good fit for the block maxima. Each density plot is embedded with location, scale, and shape parameters, where location (μ) denotes the position of the GEV mean, scale (σ) is a multiplier that scales function, and shape (ξ) is the parameter that describes the relative distribution of probabilities.

The density plots indicate that the normality of the residual is obtained. The normality of the residual is an assumption made when running a linear model. Therefore, if our residuals are normal, it means our assumption is valid, and model inference should also be valid (confidence intervals, model predictions).

Next, we fitted a GEV to the monthly maximum precipitation and temperature by applying the maximum likelihood estimation, generalised maximum likelihood estimation, Baysian, and L-moments methods to seasonwise data for all the districts of Uttarakhand and considered the best method for each seasonal dataset in order to estimate the return values and their return level. The parameter estimates and their corresponding ‘standard errors’ at a 95% confidence interval for the maximum precipitation and temperature for the Garhwal and Kumaun regions are represented in tables 2 and 3, respectively.

Table 2 Precipitation and temperature (EVA and return value estimation) for Garhwal region.
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Table 3 Precipitation and temperature (EVA and return value estimation) for Kumaun region.
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To study the impact of climate change on a global scale, many studies have come forth showing an increasing trend of extreme events in the Uttarakhand region. If the occurrence of such extreme events can be forecasted based on return periods and their corresponding return levels, then the loss of precious lives and property can be minimised. So far, many studies have been carried out to determine extreme events based on large datasets by using both techniques, namely the generalised extreme value distribution method and the generalised Pareto distribution method. The GEV technique is based on the block maxima and minima approach, while the GP technique is based on the peaks over threshold approach. The present study is dedicated to the statistical modelling of monthly maximum precipitation and temperature for all the 13 districts of Uttarakhand during the last century (1901–2002). The generalised extreme value (GEV) analysis approach is utilised for the probabilistic forecasting of the extremities, as our motive is to determine the occurrence of all the extreme events that may occur in the future. Yearly as well as seasonal analysis is performed for a detailed understanding of these parameters.

The analysis based on the extreme value theory has been carried out for various return levels and return periods and is summarised as follows:

  1. (i)

    Since the dataset is too large, we have divided it into two time periods, i.e., 1901–1950 and 1951–2002, to observe any changes in the last 50 years as compared to the first 50 years. The analysis shows an average increase in temperature of about 2.44°C during the winter season and an average decrease in temperature of about 0.44°C is observed during the monsoon season.

  2. (ii)

    For the Garhwal region, with a 50-year return level, the maximum temperatures of 26.54°, 41.57°, 40.78°, and 33.95°C are expected to exist for Haridwar district (altitude ~314 m amsl) for winter, pre-monsoon, monsoon, and post-monsoon seasons, respectively. Similarly, the maximum precipitation of 135.50, 600.97, and 166.92 mm is expected to exist for Chamoli district (alt. ~1550 m amsl) for winter, monsoon, and post-monsoon seasons, respectively, whereas, for the pre-monsoon season, the maximum precipitation of 149.53 mm is expected to exist for Rudraprayag district (alt. ~895 m amsl).

  3. (iii)

    For the Kumaun region, with a 50-year return level, the maximum temperatures of 28.09°, 42.65°, 41.03°, and 34.33°C are expected to exist for Udham Singh Nagar district (alt. ~550 m amsl) for winter, pre-monsoon, monsoon, and post-monsoon seasons, respectively. Similarly, the maximum precipitation of 131.02, 196.17, and 225.22 mm is expected to exist for Pithoragarh district (alt. ~1630 m amsl) for winter, pre-monsoon, and post-monsoon seasons, respectively, whereas for the monsoon season, the maximum precipitation of 723.42 mm is expected to exist for Champawat district (alt ~1660 m).

  4. (iv)

    From our investigations, we have observed that the Pithoragarh, Champawat, Rudraprayag, and Chamoli districts being at high elevation zones, are more prone to extreme events due to low temperatures and a consistent increase in cloud cover in association with precipitation, whereas Udham Singh Nagar and Haridwar, being at a lower elevation, have high temperatures, low precipitation, and cloud cover. Therefore, having a lower probability of high precipitation events based on the climatic trends. Undoubtedly, this will impact the physical, social, as well as cultural settings of the state.

  5. (v)

    The present analysis shows that the temperature in hilly areas of Uttarakhand is rising more prominently than in the plains. This increment will affect wind patterns, cloud formation, and rainfall. Therefore, more cloud bursts are expected to occur in the future during the monsoon as well as pre- and post-monsoon seasons. The Uttarakhand Himalaya is enriched with a diversity of forests distributed from subtropical to alpine zones. The seasonwise temperature analysis, however, shows that the state is losing its seasonal contrast as winters are becoming warmer and the rainy season is losing its heat. Therefore, phonological behaviour, such as flowering, fruiting, and germination, will be influenced by the seasonal and climatic change. This analysis indicates that the climate of Uttarakhand is changing notably. Although at this stage, the misery of climate change may not be visible, the scenario that is emerging warns that these changes may alter the regional energy balance and, thus may change the pace of ongoing geomorphologic processes.

The significance of this study rests on the fact that Uttarakhand has varied topography, and the climate and vegetation vary greatly with elevation. Conservation strategies for the important temperate medicinal plants need to be developed. Long-term changes in these parameters can directly or indirectly affect many aspects of society in potentially disruptive ways. Due to changes in temperature and rainfall, there has been an increased migration of lowland species towards higher altitudes. The effective use of remote sensing technology can help to improve the current knowledge of vegetation responses to a changing climate. Also, extreme events such as floods, landslides, and cloudbursts are quite common in most of the districts of Uttarakhand. The estimated extreme values and their return level may help the decision-makers plan appropriate risk mitigation measures to reduce the damage caused by extreme weather phenomena. Furthermore, research will be focused on a non-stationary approach.