Temporal Analysis of Rainfall and Temperature of Ranchi, Jharkhand for Period 1975-2050

The rainfall and temperature pattern of the study area Ranchi is analyzed for the period 1975-2023, and it is observed that the rainfall has an increasing trend for Annual, Monsoon, June, July, August, September, and October while temperature has an increasing trend for August only. The correlation between rainfall and average temperature emphasizes a negative correlation for annual, summer, March, May, and June at a 99 % confidence level while for July, September, and December, the correlation is negative at a 95 % confidence level. ARIMA(0,1,1) is chosen to be the most accurate model for the forecasting of rainfall and temperature for the period 2024-2050. Trend analysis of forecasted rainfall emphasizes an increasing trend for summer, monsoon, post-monsoon, and annual rainfall, whereas a negative trend is observed for winter rainfall. Trend analysis of forecasted temperature emphasizes an increasing trend for Monsoon while Annual and the rest of other seasons have negative trends.


Introduction
The Rainfall of India is heavily dependent upon the southwest monsoon, which runs from mid-July to mid-September.Monsoon rainfall in India contributes nearly 75 % rainfall to the annual rainfall.However, there is a large variation in the temporal rainfall.According to the fifth climate assessment report of the Intergovernmental Panel on Climate Change, the average worldwide combined land and ocean surface temperature rose by 0.85 °C between 1880 and 2012 [1].A warmer climate is probably going to result in more heavy rainfall, which will probably come from fewer but stronger occurrences.The trend of seasonal rainfall of all sub-divisions throughout India was analyzed, and a substantial decline in tendency was discovered over three sub-divisions: Jharkhand, Chhattisgarh, and Kerala [2].A significant downward trend was observed in annual, monsoon, and winter rainfall for the period 1901-2002 [3].Pre-monsoonal rainfall distribution over Ranchi district was disturbed in the period 2000-2020 [4].The seasonal Rainfall over Jharkhand has a decreasing trend for the period 1986-2018, and the average annual maximum and minimum temperatures are also increasing at a rate of 0.58 ºC and 0.46 ºC per decade, respectively [5].Mann Kendall-Sneyers test can be applied to data sets for change point detection [6].The changing pattern of rainfall and temperature has gathered eyes from around the world, and researchers around the globe are working on forecasting these events to understand the upcoming patterns and variability.Several methods, such as interpolation, Holt-Winter exponential smoothing, ARIMA, etc., can forecast rainfall and temperature.Ranchi's rainfall was forecasted for 2023-2030 using the Holt-Winter exponential smoothing method [7].The trend of future rainfall in Chattogram, Bangladesh, projected by ARIMA using the historical rainfall data of the period 1970-2021, was analyzed, and it was reported that rainfall during the monsoon and December months would rise [8].The trend of future weekly rainfall predicted by ARIMA modeling indicated a declining tendency in Iraq's semi-arid Sinjar area [9].
In this paper, historical rainfall and the average temperature have been analyzed for trend, variability, and correlation for 1975-2023, and then, the ARIMA model is used to forecast the future rainfall and average temperature for 2024-2050.The forecasted rainfall and average temperature are further analyzed to determine trends.For regional planning, studying temperature and precipitation at the global or continental scale is not particularly helpful [10,11]; hence, this temporal study of historical and future rainfall and temperature will help understand the changing pattern of rainfall and temperature at a regional scale.These findings can be utilized by government or private agencies to make plans to mitigate the impact of any extreme hydrological events, enhance crop yield, and plan water management.

Description of Study Area
Ranchi is the capital of the state of Jharkhand in India.It is located on the southern part of the Chota Nagpur plateau of eastern India, at 21°58` to 25º18` N latitude and 83°22` to 87°57` E longitude.Ranchi's climate is humid subtropical and receives unique convectional rainfall during summer (March-June).Ranchi receives good annual rainfall; approximately 80 % of rainfall is received during the southwest monsoon season from June to September [12].The location map of Ranchi is displayed in Fig. 1.

Material and Methods
The daily rainfall and average temperature data of study area Ranchi for the period 1975-2023 are obtained from the website of the National Centre for Environmental Information (https://www.ncdc.noaa.gov/cdo-web/datatools/findstation).Monthly, seasonal, and annual data are calculated from daily data series.The statistical descriptions such as mean, standard deviation, and coefficient of variation are calculated for monthly, seasonal, and annual data.The rainfall data are further categorized into deficit and excess rainfall according to rainfall deviation from its long-term mean.Let 'm' be the long-term mean and 'd' be the long-term standard deviation of rainfall.Then rainfall in year 'YY' is categorized as  Trend rainfall and average temperature analysis are done using Mann-Kendall [13] and Sen's slope method [14].The shift point for the trend is analyzed using the Mann-Kendall-Sneyers sequential test [15].The correlation analysis of rainfall and average temperature is done using Pearson's correlation test.Rainfall and average temperature forecasting from 2024 to 2050 are done using the ARIMA model.ARIMA model requires data to be stationary, and hence Augmented Dicky Fuller unit root tests are applied to the data set to know whether data is stationary or non-stationary.

Mann-Kendall test
It is a non-parametric test used to find trends in the data set.It is very useful as it does not require data to be normally distributed.Let  1 ,  2 ,  3 , … ,   be the data set of length n.The indicator function Sgn(  −   ) is given as follows: The mean S and variance Var(S) of Sgn(  −   )are given as follows: Where t is the extent of any given tie.The Z statistics for the MK test is given by < 0 The value of Z>0 represents a monotonic upward trend in the data series, whereas Z<0 represents a monotonic downward trend.

Sen's Slope method
It is a non-parametric test that is highly efficient in finding linear trends in univariate data series.This can be applied to a data set having missing values and outliers in the data series.The Sen's estimator β of slope is calculated as follows: Where   and   are the data values at time j and i (j>i) respectively.β>0 indicates an upward trend, whereas β<0 indicates a downward trend in the data series.

Mann-Kendall-Sneyers test
It is used in the detection of change points in significant trends.A progressive series and a retrograde series are created for this test.If the two series cross and diverge beyond a certain threshold value, then there is a statistically significant trend.The point at which they intersect indicates the approximate year when the trend begins.Let X={ 1 ,  2 ,  3 , … ,   } be the total number of elements   preceding   (j<i) where   <   .The test statistic   derives the cumulative   for each year The mean of   is given by And the variance of   is given by Now, the forward sequence    ℎ   ′  based on three variables (  , (  ), (  )) is derived as follows: Now reverse the time series sequence X and call it Y.An intermediate sequence.  is then calculated using data sequence Y. Now, in terms of sequence   is reversed, and a negative sign is added to the reversed values.Thus, the newly obtained sequence is called ′  .

Pearson's correlation test
It is a bivariate correlation test that measures the linear correlation between two data sets.It is the ratio between the covariance of two variables and the product of their standard deviation.Pearson correlation between two data points X and Y is given by: Where Cov(X ,Y) is covariance of X and Y and is given by Where ̅ and  ̅ are mean of data set X and Y, respectively and n is the sample size.
And   and   are standard deviations of X and Y, respectively.The Pearson Correlation is the actual correlation value that denotes magnitude and direction, and the Sig.(2-tailed) is the p-value that is interpreted to check the significance of correlation.If the p-value is less than 0.05, then the correlation is statistically significant between the two data sets, and if the p-value is more than 0.05, then the correlation is not a statistically significant association between the two data sets [16].

Augmented Dickey-Fuller Unit Root test
The unit root test is used to check the stationarity of the data set.When a data set has no unit root means that the data set is stationary.
The unit root test in the time series   , the Dickey-Fuller equation is given by Where  represents the intercept and is constant, β represents the coefficient of trend, and p is the order of lag of the autoregressive (AR) process.The unit root test is then carried out under the null hypothesis γ=0 against the alternative hypothesis of  < 0. If γ < 0, then the series is stationary because there is no trend in the time series.

ARIMA model
ARIMA means Autoregressive Integrated Moving Average.It is used in the time series analysis to forecast upcoming series points.ARIMA consists of three components: Autoregressive (AR), Integrated (I), and Moving Average (MA).ARIMA models are denoted by ARIMA (p, d, q) where p is the number of auto-regressive orders, d is the order of differencing applied to the time series, and q denotes the number of moving average orders of the data series.The parameters p, d, and q are non-negative integers.
ARIMA model requires the data set to be stationary.It can also be applied to nonstationary data sets by eliminating the non-stationarity of the mean function by introducing an initial differencing step one or more times.ARIMA models for different p, d, and q values are developed, and the best model is selected using the Bayesian information criterion (BIC).ARIMA (p, d, q) is given as follows: Where ̅  =   −  and   is the shock.Equation ( 1) can be applied after finding the backward shift operator (B) as follows:

Descriptives analysis
The mean, standard deviation, and coefficient of variation are calculated for monthly, seasonal, and yearly rainfall and average temperature (Table 1 2].Excess and deficit rainfall in the monsoon season are observed for 8 years and 12 years, respectively.An interrelation between excess and deficit rainfall is observed for monsoon and annual rainfall.If the monsoon receives deficit rainfall in a year, then annual rainfall is also deficit in that year (except year 1982).If the monsoon receives excess rainfall in a year, then annual rainfall is also excess in that year (except in the year 2022).Post-monsoon and winter rainfall are never deficit.The mean average temperature for the period 1975-2023 is 24 ºC with a standard deviation of 0.45 ºC.The lowest average temperature was observed in year 1978 (23.18 ºC), and the highest average temperature was observed in 2016 (25.49ºC).The yearly average temperature has a variation of 1.90 %.The least seasonal variation is observed for monsoons (2.24 %), whereas the highest variation is observed for winter (4.82 %).The least monthly variation for average temperature is observed for August (1.87 %), whereas the highest variation is observed for February (6.05 %).
La Nina causes excess Rainfall in India, whereas El Nino causes deficit or at times normal rainfall [17].The rainfall pattern of Ranchi does not show any fluctuation because of El Nio or La Nia events.In the period 1975-2023, 18 La Nina and 17 El Nino episodes happened (Table 3).During this 17 El Nino period, the annual rainfall is normal in eleven episodes, excess in two episodes, and deficit only in four episodes (1976,1979,1986,1987).During 18 La Nina episodes, annual rainfall is normal in eleven episodes, deficit in three episodes and excess only in four episodes (2008,2011,2017,2021).

Trend analysis
Seasonal rainfall for summer, monsoon, post-monsoon, winter, and annual rainfall is analyzed for trend (Table 4).Monsoon and annual rainfall have a significant upward trend, and no other seasons have any significant trend.The shift point for the increasing trend of monsoon and annual rainfall is observed to be 1993 and 1996, respectively (Fig. 2).The rainfall pattern started to move significantly in an upward direction for monsoon rainfall from year 1993 and for annual rainfall from year 1996.The monthly rainfall of June, July, August, September, and October have significant positive trends, and no other months have any significant trends.The seasonal and annual average temperatures are analyzed for possible trends, and it is observed that no significant trend is present for seasonal or annual average temperatures.However, the monthly average temperature for August has a significant positive trend.

Correlation analysis
Correlation coefficients are calculated to find out the interrelation of seasonal and annual rainfall with seasonal and annual average temperature (Table 5).Seasonally, summer rainfall is negatively correlated with summer average temperature at a 95 % confidence level.Annual rainfall is also negatively correlated with annual average temperature at a 95 % confidence level.Monthly Rainfall and monthly average temperature have a negative correlation for March, May, and June at a 99 % confidence level and for July, September, and December at a 95 % confidence level.

Conclusion
The historical rainfall and average temperature for the period 1975-2023 have been analyzed, and it is observed that the annual mean rainfall is 863.57mm and the annual mean temperature is 24 ºC.Monsoon (79.68 %) and August (23.98%) contribute the highest rainfall to the annual rainfall, whereas winter (3.46 %) and December (0.98 %) contribute the least rainfall.It is also observed that the El Nio and La Nina have no impact on Ranchi's rainfall pattern.Trend analysis of rainfall suggests a significant positive trend in the Annual Monsoon, June, July, August, September, and October.However, the trend analysis of temperature showed a significant positive trend only for August.Correlation analysis of rainfall and average temperature suggests that a negative correlation exists for annual, summer, March, May, and June at a 99 % confidence level while for July, September, and December, the correlation is negative at a 95 % confidence level.The forecasting of rainfall and average temperature is done using the ARIMA model, and ARIMA (0,1,1) was found to be the most accurate model for the prediction.The best-fit model is selected on the basis of a lower normalized BIC value.The trend analysis of future rainfall suggests that a positive trend is present in summer, monsoon, post-monsoon, and annual, whereas a negative trend is present in winter.Also, the trend analysis of future average temperature shows a positive trend only for monsoon while the rest of other seasons and annual trend is negative.

( 1 )
deficit if Rainfall in YY < m -d (2) Normal if m -d≤ Rainfall in YY ≤ m + d (3) Excess if rainfall in YY> m + d.

Table 1 .
Statistical descriptives of rainfall and temperature for the period 1975-2023.

Table 2 .
Excess and deficit rainfall for the period 1975-2023.

Table 4 .
Trend analysis of rainfall and temperature for the period 1975-2023.

Table 5 .
Correlation analysis of rainfall and temperature.

Table 8 .
Forecasted values of rainfall and temperature.

Table 9 .
Trend analysis of forecasted rainfall for the period 2024-2050.