Analytical techniques

The procedures used in this study for the GCMs selection, downscaling of the GCMs simulations, preparation of MME and projection of drought are outlined below:

The 20 GCMs were re-gridded using bilinear interpolation to a common resolution of 2° × 2° and the CRU/GPCC data were aggregated to the same resolution for comparison;

The GCM simulations were compared with GPCC/CRU data using entropy-based methods to select a subset of 20 GCMs, based on their ability to replicate historical rainfall/temperature;

The simulations of the selected GCMs were re-gridded to a resolution of 0.5° × 0.5° using the bilinear interpolator and the bias in GCM simulated rainfall and temperature were corrected based on the GPCC rainfall and CRU temperature, respectively. Four bias correction methods, namely power transformation (PT), linear scaling (SCL), general quantile mapping (GEQM) and gamma quantile mapping (GAQM) were used to correct for systematic bias, inherent in the products;

The bias correction method was selected based on a set of statistical indices and the coefficients of the bias correction methods were used to correct the bias in the GCM simulations for RCP 2.6, 4.5 and 8.5 for the period of 2010–2100;

Random Forest (RF) was used for the generation of the MME of downscaled GCM projections;

The SPEI was estimated from the MME precipitation/temperature for all the three RCPs at each grid point (323 grid points), covering Nigeria, for different 50-year period with a time interval of 10-year during 2010–2100;

The SPEI for each growing season was fitted with the best probability distribution function (PDF) for the estimation of the return periods for different drought categories and different growing seasons;

Trends in projected rainfall, temperature, and SPEI were estimated using modified Mann-Kendall test (MMK) and their associations were examined using Kendall-tau correlation coefficient for different seasons for different 50 year periods.

Three entropy-based methods, namely symmetrical uncertainty (SU), gain ratio (GR), and entropy gain (EG), were used to assess similarity of GCM historical simulated rainfall with GPCC rainfall and temperature with CRU temperature for the period 1961 to 2005. The EG is an information-based concept, which is a measure of uncertainty of a random variable. It can be used for the estimation of information of one variable compared to another⁶³. This capability of entropy is used for assessing the ability of GCM to simulate observed climate. The GR and SU are the modified versions of EG, which can overcome the bias in GR estimation to higher values. Details about the entropy methods used in this study can be found in Khan et al.⁶⁴, Salman et al.¹⁵, and Shiru et al.⁶⁵.

The MCDA was used for ranking of the GCMs according to EG, GR and SU values, estimated at different grid points over Nigeria. Higher weight was given to the GCMs that was found to attain a particular rank at most of the 323 grid points. For example, if a GCM was ranked top by an entropy measure at most of the grid point, it was given the highest weight and vice versa. The GCMs were ranked separately for EG, GR and SU and then averaged to get the final rank of the GCMs.

Four bias correction methods namely PT, SCL, GEQM and GAQM were used in this work for downscaling of selected GCM simulations. Biases of the selected models were corrected by comparing the GCM simulated rainfall/temperature with the GPCC rainfall/CRU temperature for historical period. Bias correction parameters were derived by comparing data for the period 1961–1992, and the parameters were then used to correct bias of the GCM simulations for 1993–2005. This was done to assess performance of the bias correction methods. Standard statistical indices namely, percentage of bias (Pbias), normalized root mean square error (NRMSE), Nash-Sutcliff efficiency (NSE), modified index of agreement (MD) and relative standard deviation (RSD) were considered. The best performing method was then selected for downscaling future rainfall and temperature of Nigeria.

The regression-based MME has the ability to preserve variance in the average of the MME, and is widely utilized in recent times. In this study, RF regression was used for the estimation of GCM ensemble. In RF, nonlinear relationship between the observed rainfall/temperature and the GCM simulated rainfall/temperature was generated. The outcome was then used for the preparation of ensemble of GCM projections. The RF is an effective and robust algorithm for generating the MME because: (1) it can avoid over-fitting; (2) many different types of input variables can be implemented without deleting and regularizing variable; and (3) it has analytical and operational flexibility.

In computing the SPEI, water surplus or deficit for different time scales (D) was estimated from the difference of precipitation and PET⁵³. The SPEI values were then estimated from the best-fitted distribution parameters of D. The values, ranging from −1.0 to −1.5, are considered moderate droughts, −1.5 to −2.0 are severe, while values > −2.0 are defined as extreme droughts⁵¹.

The calculation of PET for the estimation of the SPEI can be conducted using different methods including temperature, radiation, and mass transfer methods⁶⁶. The SPEI is found more sensitive to radiation-based PET estimation and less to temperature-based proxy methods⁶⁷. Begueria et al.⁶⁸ observed that the use of different PET methods in semi-arid and arid regions results in significant differences in the SPEI series. They recommended Penman-Monteith method to be chosen for PET estimation, followed by the Hargreaves and the Thornthwaite methods. The Thorntwaite method, in comparison with other techniques, requires less number of meteorological variables for the estimation of PET, which makes it more suitable for PET calculations in data sparse locations such as Nigeria. For the determination of droughts of a season, last month SPEI value of the season was used.

The return periods (RP) of seasonal droughts were estimated from the SPEI values of different growing seasons. Droughts were defined if SPEI values were lower than −1.0. Hence, zero values were assigned to years with no drought. Droughts frequency analyses were conducted only on the non-zero values and corrections were made using non-exceedance probability (F′) in order to account for the zero values.

where, F is the non-exceedance probability value derived by the use of frequency analysis on the non-zero values, and q is estimable as the ratio of the number of years without drought events to the total number of years^25,69–71.

The rate of change in the SPEI, temperature, and precipitation was assessed using Sen’s slope estimator⁷², where change (Q_med) is computed as the median of N slopes estimated from the consecutive two points of the series as follows:

Trends significance in precipitation, temperature, and the SPEI changes was carried out using the MMK test. This test was used considering its ability to separate natural variability of climate from unidirectional climate change due to global warming^57,73. The classical Mann-Kendall test statistic (S) for a time series, x with n number of data points is calculated as⁷⁴:

where,

The significance of the trend is calculated using Z statistics as:

where, Var(S) is the variance of S.

The significance of trend estimated by the use of MK test is first removed from the time series in MMK test⁷⁵, then equivalent normal variants of rank (R_i) of the de-trended series is estimated as:

where, ϕ−1 is the inverse standard normal distribution function. The self-similarity correlation matrix of the time series or the Hurst coefficient (H) can be derived using the equation that follows⁷⁶:

where, ρl is the autocorrelation function of lag l for a given H. The value of H is obtained by the use of maximum log likelihood function. The significance level of H is determined by the use of mean and standard deviation for H = 0.5. If H is found significant, the biased estimate of the variance of S is calculated for a given H as:

The bias in estimation of V(S)H is removed using a bias correction factor, B as below:

where, B is a function of H. The significance of MMK test is computed using Eq. (⁵) by replacing V(S) with V(S)H.