The gaps in the time series were filled using a suitable criterion to complete the missing data and provide a continuous monthly base. This is the coefficient of variation (C_V). It is defined as the ratio of the standard deviation of a variable to its mean, and is given by the relationship:

C V = σ X X ¯ (1)

where σ_X is the standard deviation and X ¯ is the arithmetic mean of the series X = (x_i).

When the coefficient of variation is less than 20%, the missing data for each month of the year are supplemented by the average of the available values. This assignment is made because of the low dispersion between the observed data [15] . On the other hand, for variables with a coefficient of variation greater than 20%, reflecting high variability, missing values are supplemented by the median, which is more robust than the mean. Indeed, when dispersion is very wide, the mean is an indicator biased by extreme values.

A change-point or shift in the central tendency of a time series can be defined by a change in the probability distribution of the random variables whose successive realizations define the time series under study [16] .

XLSTAT software was used to detect shifts in the flow rate time series. Its various statistical tests enable us to determine whether there has been a change in the behavior of a variable over time. Changes are highlighted by a shift isolating two homogeneous flow periods. These tests are Pettitt test and Buishand test.

1) Pettitt test

Pettitt’s approach is non-parametric and derives from the Mann-Whitney test. A non-parametric test focuses on the overall distribution of data, without any hypothesis about their distribution (free distribution). The absence of a shift in the series (x_i) of size N is the Null hypothesis H₀ (no shift or no change), it is tested against the Alternative hypothesis H₁ (presence of shift or change) [17] .

The test assumes that for any time between 1 and N the time series (x_i), i = 1 at t; j = t + 1 at N belong to the same population. The variable to be tested is the maximum absolute value of the variable U_t, defined by Equation (2):

U t , N = ∑ i = 1 t ∑ j = t + 1 N D i j (2)

where D_ij = sgn(x_i − x_j), with sgn(x) = 1 if x > 0; sgn(x) = 0 if x = 0; sgn(x) = −1 if x < 0.

If the H₀-hypothesis is rejected, an estimate of the shift date is given at this point, defining the absolute maximum of the variable U_t.

2) Buishand test

The Buishand test (1982) can be used on variables with any distributions (case of the U statistic). However, its properties have been particularly studied for the case of normal distribution (case of the V statistics).

In the Buishand approach, the U statistic is defined by [18] [19] :

U = 1 N ( N + 1 ) ∑ k = 1 n − 1 ( S k D x ) 2 (3)

with

S k = ∑ i = 1 k ( X i − X ¯ ) (4)

where k = 1 , ⋯ , n and D_x is the standard deviation of the time series.

The Buishand statistic U assumes a series of random variables X 1 , X 2 , ⋯ , X n based on the following model:

X i = { μ + E i , i = 1 , 2 , ⋯ , τ μ + δ + E i , i = τ + 1 , ⋯ , n (5)

The E i is independent and has normal distribution, with mean value of zero and variance σ². The variables τ, δ and σ correspond to unknown parameters verifying:

1 ≤ τ ≤ n − 1 ,   − ∞ < μ < + ∞ ,   − ∞ < δ < + ∞ ,   σ > 0 (6)

The possible change (in position and amplitude) corresponds to the mode of the a posteriori distribution of τ (position of the shift in time) and δ (the amplitude of the change in comparison with the mean).

The non-parametric Mann-Kendall test [20] [21] is recommended for identifying series trends. It is used to study the presence or absence of trend in a given time series. The Mann-Kendall test is based on the S statistic defined as follows [22] :

S = ∑ i = 1 n − 1 ∑ j = i + 1 n sgn ( x j − x i ) (7)

where S is the relationship between the number of pairs of observations, n is the total number of samples and x_j represents the sequential data values:

sgn ( x j − x i ) = { 1 ,         si   ( x j − x i ) > 0 0 ,         si   ( x j − x i ) = 0 − 1 ,     si   ( x j − x i ) < 0 (8)

Mann (1945) and Kendall (1975) showed that when n ≥ 8, the statistic S has approximately a normal distribution with mean E(S) and variance Var(S) as follows:

E ( S ) = 0 (9)

and

V a r ( S ) = 1 18 [ n ( n + 1 ) ( 2 n + 5 ) − ∑ i = 1 m t i ( t i − 1 ) ( 2 t i + 5 ) ] (10)

where n is the amount of data in the series, m is the number of linked groups and t_i is the amount of data in group of order i. The standard multivariate normal Z_MK is calculated as follows:

Z M K = { S − 1 V a r ( S ) ,     si   S > 0 0 ,                                 si   S = 0 S + 1 V a r ( S ) ,     si   S < 0 (11)

The Null hypothesis H₀ (absence of trend) is rejected when the significance level or p-value is greater than 5%. When H₀ is accepted, the slope of the trend or Sen’s slope is estimated by Sen’s method, where the slope is the median of all slopes calculated between each pair of points. The robustness of the test has been validated by several comparison tests [22] .

The deficit of the dry period, compared to the wet period, is evaluated by applying the formula represented by Equation (12):

D = X i − X j X i ⋅ 100 (12)

with D: hydrological deficit; X_i mean before shift in the time series; X_j average after shift in the time series.

In this study, deficits were calculated using the Pettitt test. If there is no shift in a time series, the deficit is calculated in relation to the periods before and after 1970.

In the absence of rainfall, a river is fed by the groundwater in its catchment area. As a result, the volume of water released by this regulating reserve during the period of emptying under uninfluenced conditions is referred to as the recession phase. Through this relationship between surface water and groundwater, it has proved important to understand the evolution of groundwater reserves during climatic variations [1] . The variability of the recession coefficient and the volume of water mobilized by aquifers help to highlight this evolution [20] . The mathematical expression of recession is given by Equation (13):

Q t = Q 0 e − k t (13)

Taking the natural logarithm in the right hand side and in the left hand side, one obtains Equation (14):

k = − 1 t ln [ Q t Q 0 ] (14)

with Q_t: flow at time t (m³/s); Q₀: flow at the start of recession (m³/s); k: recession coefficient (s⁻¹) and t: elapsed or release time (s).

This test is a classic test initially designed to compare two series and check whether they belong to the same data. As part of a stationarity study, this test can be used if the date of presumed shift is known. In this case, the series is subdivided into two sub-series (on both sides of this date), which are compared using this test.

· Conditions for applying the test

Data must have normal distribution and be independent. The variances of the two samples are assumed to be equal. This assumption can be verified using a test for equality of two variances.

· Notations

Let m₁ and m₂ be the theoretical averages of the two subgroups and X₁ and X ₂ their arithmetic means. The two subgroups contain n ₁ and n₂ values. σ₁ and σ₂ denote the theoretical standard deviations of the two samples (assumed to be equal) and S₁ and S₂ are their estimators:

{ S 1 = 1 n 1 − 1 ∑ i = 1 t ( x 1 − X ¯ 1 ) 2 S 2 = 1 n 2 − 1 ∑ i = 1 t ( x 1 − X ¯ 2 ) 2 (15)

The common variance of both samples S² is calculated by:

S 2 = ( n 1 − 1 ) S 1 2 + ( n 2 − 1 ) S 2 2 n 1 + n 2 − 2 (16)

· Assumptions

{ H 0 : m 1 = m 2 H 1 : m 1 ≠ m 2 , bilateraltest H 1 : m 1 > m 2 , unilateraltest (17)

· Statistic of test

Under the assumption H₀ equality of averages, the quantity:

t = X ¯ 1 − X ¯ 2 S ⋅ [ 1 n 1 + 1 n 2 ] − 1 / 2 (18)

is a t student variable with (n₁ + n₂ −2) degrees of freedom.

The results of the various detection tests of shifts (Pettitt and Buishand) applied to the annual flow series (floods, yearly average flows or modules, and low-water) are presented in Table 1 and Table 2.

Here, mu1 is the average before shift; mu2 the average after shift and D the hydrological deficit.

1) Results of Pettitt test

The p-values calculated on the annual flows (module and low flow) of the Sangha River at Ouesso station using the Pettitt test are below the significance level (α = 0.05), so the null hypothesis H₀ (absence of shift) is rejected. The

alternative hypothesis H₁ (presence of a shift date) is retained. The shift date is 1970 for floods, 1971 for modules and 1976 for low-water (Table 1).

The search of changes in the flow series of the Sangha River at Ouesso station shows the existence of two homogeneous flow periods: 1) a surplus period, known as the wet period, running from 1961 to 1070/1971 for floods, 1961 to 1971 for modules and from 1961 to 1976 for low-water, corresponding to an increase in flows; 2) a deficit period, known as the dry period, running from 1971/1972 to 2020 for floods, 1972 to 2020 for modules and from 1977 to 2020 for low-water, corresponding to a decrease in flows. The wet period evolves around the averages 3660 m³/s for floods, 1859 m³/s for modules and 871.938 m³/s for low-water; and the dry period evolves around the averages 3051 m³/s for floods, 1439 m³/s for modules and 560.757 m³/s for low-water (Figure 2). The deficits between the wet and dry periods are 16.64% for floods, 21.70% for modules and 35.67% for low-water, respectively. Annual maximum flows, on the other hand, do not show shift during the study period.

2) Results of Buishand test

The p-values calculated on the annual flows (modules, floods, and low-water) of the Sangha River at Ouesso station using the Buishand test are below the significance level (α = 0.05), so the Null hypothesis H₀ (absence of shift) is rejected. The Alternative hypothesis H₁ (presence of a date showing a shift) is retained. This date is 1971 for modules and floods, and 1976 for low-water (Table 1). The search of changes in the flow series of the Sangha River at Ouesso station shows the existence of two homogeneous flow periods: 1) a surplus period, known as the wet period, running from 1961 to 1971 for modules and floods, and from 1961 to 1976 for low-water, corresponding to an increase in flows; 2) a deficit period, known as the dry period, running from 1972 to 2020 for modules and floods, and from 1977 to 2020 for low-water, corresponding to a decrease in flows. The wet period evolves around the averages 1834 m³/s for modules; 3614 m³/s for floods and 871.938 m³/s for low-water and the dry period evolves around the averages 1436 m³/s for modules, 3049 m³/s for floods and 560.757 m³/s for low-water) (Figure 3). The deficits between the wet and dry periods are 21.70% for modules, 15.63% for floods and 35.67% for low-water, respectively.

The results of the Mann-Kendall trend test are shown in Table 3 and illustrated in Figure 4. These results show a significant downward trend in flow rates for both modules and low-water levels (Figure 4) from the 1970 onwards, more specifically between 1970 and 1971 for modules, and in 1976 for low-water levels. This drop in flow is more seen for low-water levels. Floods, on the other hand, show a no insignificant trend.

To elucidate the impacts of climate change on water resources in the Sangha basin at Ouesso, the recession coefficients are determined at both monthly and annual time scale over the study period (1961-2020). Over this period, one will track changes in the recession coefficient over the 1961-1990 and 1991-2020 sub-periods, and then over the entire study period. The results of the calculations are

represented graphically (Table 4, Figure 5 and Figure 6) to highlight the changes in the discharge of the basin’s aquifers. This recession phenomenon did not occur in the same month throughout the two sub-periods: it occurred in March for the 1961-1990 sub-period and in April for the 1991-2020 sub-period. Over the entire study period (1961-2020), this recession corresponds to that observed during the 1961-1990 sub-period, when recession took place in March (Figure 5 and Table 4).

On the other hand, at the annual time scale, recession occurred in 1985 and 2007 for the 1961-1990 and 1991-2020 sub-periods, respectively. However, over the entire study period (1961-2020), recession occurred in 1985 (Figure 6). The years 1985 and 2007 correspond to severe drought years in the Sangha basin at Ouesso. Similarly, the year 2011, close to recession, corresponds to the period when the basin experienced very severe low-water levels.

The results obtained from calculations of changes in Sangha River flows at Ouesso station from 1961 to 2020 are presented in Table 5 and illustrated graphically in Figure 7. Trends in flood flows on the Sangha River at Ouesso station over the periods 1961-1970 and 1971-2020 show that these have been declining since 1970. This date reflects a change in flood flows. This decrease is most significant during the rainy season (September, October, and November). However, flood flows gradually increase from December to August. Similarly, trends in the modules of the Sangha River at Ouesso station over the periods 1961-1971 and 1972-2020 show that it has been declining since 1971.

These modules show a similar pattern to that of floods. In addition, low-water trends over the periods 1961-1976 and 1977-2020 show that low-water have been declining since 1976, the year in which the hydrological behavior of the Sangha River at Ouesso station changed. The decline is most observed in October. Flows gradually increase from November to September, where they reach their peak.

The changes observed in these flows (flood, modules and low-water) reflect the growing impact of climate variability on water resources in the Sangha watershed at Ouesso.