A number of drought indices have been developed and applied in met-drought assessment over the years. Some of these indices include Aggregated Drought Index (ADI) [3] , Standardized Precipitation Index (SPI) [4] [5] , Palmer Drought Severity Index (PDSI) and Z-Index [6] , Effective Drought Index (EDI) [7] , Keetch-Byram Drought Index (KBDI) [8] , Hybrid Drought Index (HDI) [9] , Vegetation Drought Response Index (VegDRI) [10] , Recconnaissance Drought Index (RDI) [11] , Rainfall Anormally Index (RAI) [12] , Drought Severity Index (DSI) [13] , National Rainfall Index (NRI) [14] and Drought frequency index (DFI) [15] . Among the mereorological drought indices, the SPI and EDI have generally been used more than most of the other drought indices because they require precipitation as a single input variable.

The Standard Precipitation Index (SPI) was developed by [4] to quantify the precipitation deficit and monitor drought conditions within Colorado, USA. The SPI is used to categorize the different drought classes as described in [4] . For calculation of SPI, long-term historical precipitation record of at least 30 years is integrated into a probability distribution function which is then transformed into a normal distribution function. The SPI requires less input data compared to most other drought indices and this makes it flexible for wide applications [16] [17] . The SPI has several advantages which make it more applicable in many river basins. First, it requires only the precipitation as the input data. This makes it ideal for river basins that do not have extensive hydrological data records. Secondly, its evaluation is relatively easy since it uses precipitation data set only. Thirdly, it is a standardized index and this makes it independent of geographical location as it is based on average precipitation values derived from the area of interest. In addition, the SPI exhibits statistical consistency, and has the ability to present both short-term and long-term droughts over time scales of precipitation variation [18] . However, the SPI has some disadvantages in its use as a drought assessment tool. First, it is not always easy to find a probability distribution function to fit and model the raw precipitation data. Secondly, most river basins do not have reliable time-series data to generate the best estimate of the distribution parameters. In addition, application of SPI in arid and semi-arid lands of time-series of less than three months may give inaccurate values.

To overcome the challenge of simulating and modelling the data for SPI outputs, application of different probability distribution functions may be employed. These include the Gamma, Pearson type III, Lognormal, Extreme Value and Exponential distribution functions [19] . However, the Gamma probability distribution function is preferred in hydrological studies. In hydrology, it has an advantage of fitting only positive and zero values since hydrological variables such as precipitation, and runoff are always positive or equal to zero as lower limit values [20] [21] . The Gumbel and Weibull distribution functions are used for study of extreme hydrological variables. The Gumbel distribution function is used for frequency analysis of floods, while the Weibull distribution function is used to analyze low flow values observed in rivers. SPI has been found to perform differently for various time scales. For time scales shorter than 6 months, there is insignificant autocorrelation while for time scales greater than 6 months, the autocorrelation increases significantly [22] .

The effective drought index (EDI) uses effective precipitation which is the accumulation of selected portions of the days before the estimated time period [19] . It estimates droughts more accurately than many other indices in terms of on-set, detection, spatial and temporal analysis. When compared with seven other drought indices in Iran, [23] found that EDI is more accurate and consistent in the study of drought.

The study of drought characteristics such as spatial, temporal, trend is attracting great attention in river basins due to the adverse effects whenever they occur. There is need to understand drought spatial, temporal and trend characteristics for its prioritized integration in planning for timely mitigation measures. In this paper therefore, meteorological drought on spatial, temporal and trend domains was detected using Standardized Precipitation Index (SPI) and Effective Drought Index (EDI) for the upper Tana River basin with a view for its incorporation in drought early warning systems.

The upper Tana River basin has an area of 17,420 km² as presented in Figure 1 and is the focus of this study. The basin lies between latitudes 00˚05' and 01˚30' south and longitudes 36˚20' and 37˚60' east. The upper Tana River basin lies between latitudes 00˚05' and 01˚30' south and longitudes 36˚20' and 37˚60' east.

The basin has forest land resources located along the eastern slopes of Mount Kenya and Aberdares range which are crucial in controlling hydrological processes of the basin [24] . This basin is located in a fragile ecosystem with all agro-ecological zones of Kenya. The Tana River tributaries originate from the slopes of Mount Kenya and Aberdares range. The basin constitutes a very important resource in Kenya such as being a water supply source, hydro-power generation and agricultural production.

The Standardized Precipitation Index (SPI) was used to quantify precipitation deficit within the basin as a representation of drought condition as defined by [4] . The first step involved fitting the precipitation data into a probability distribution function and then computation of the SPI values. The computed SPI values were used in drought assessment and classification. In the first step, the gamma distribution function was adapted since it fits well in time series precipitation data [25] . The gamma distribution is expressed in terms of its probability density function as:

f ( x , α , β ) = 1 β α Γ ( α ) x α − 1 e − x / β       for   x , α , β > 0 (1)

where; α = the shape parameter, β = scale parameter, x = the precipitation amount (mm), Γ(α) = the value taken by gamma function and x ¯ = mean rainfall (mm).

The Γ(α) is the value defined by the Gamma function which is determined by applying an integral function according to [11] expressed as:

Γ ( α ) = ∫ 0 α x α − 1 e − y d x (2)

where; Γ(α) = the value taken by gamma function, x = the precipitation amount (mm) and α = the shape parameter.

The Gamma function in Equation (2) was evaluated both by the numerical method and use of tabulated values using the selected shape parameter α. A

maximum probability was then used to estimate the optimal values of α and β using Equations (3) and (4):

α = 1 4 A ( 1 + 1 + 4 A 3 ) (3)

β = x ¯ α (4)

where; α = the shape parameter, β = scale parameters, x ¯ = mean precipitation (mm) and A = sample statistic.

The sample statistic is defined as:

A = ln ( x ¯ ) − ln x n (5)

where; x ¯ = the precipitation average (mm) and n = the number of observations

The calculated values were in turn used to compute the cumulative probability for non-zero rainfall using Equations (6) and (7) respectively:

f ( x , α , β ) = ∫ 0 x f ( x , α , β ) d x = 1 β α Γ ( α ) ∫ 0 x x α − 1 e − x / β d x (6)

where; α= the shape parameter, β = scale parameter and x = the precipitation amount (mm)

The Equation (6) above was reduced to:

f ( x , α , β ) = 1 Γ ( α ) ∫ 0 x t α − 1 e t d t       for   t = x β (7)

where; Γ(α) = the value taken by gamma function, x = the precipitation amount (mm), β = scale parameter and t = the time period

The Gamma function was applied for values of precipitation x > 0 for the precipitation time series of the upper Tana River basin. In case of non-zero values, cumulative probability of both zero and non-zero values were computed. This probability is represented by a function H(x) defined as:

H ( x ) = q + ( 1 − q ) F ( x , α , β ) (8)

where; H(x) = Cumulative probability and q = probability of zero precipitation

When m was taken as the number of zero entries in the time series precipitation data, then the q value was estimated by the ratio m/n. The cumulative probability was then transformed into a standard normal distribution function. This gave values of the mean and variance of the SPI as zero and one respectively. This step was carried out using approximate transformation functions adapted from [26] . These functions given in Equations (9) and (10) are expressed as:

S P I = − ( k − c 0 + c 1 k + c 2 k 2 1 + d 1 k + d 2 k 2 + d 3 k 3 )         for   0 < H ( x ) ≤ 0.5 (9)

S P I = + ( k − c 0 + c 1 k + c 2 k 2 1 + d 1 k + d 2 k 2 + d 3 k 3 )         for   0.5 < H ( x ) < 1.0 (10)

where; c₀ = 2.515517, c₁ = 0.802853, c₂ = 0.010328, d₁ =1.432788, d₂ = 0.189269, d₃ = 0.001308

The parameters were used to compute the SPI and were adapted from [19] . The value of k in Equations (9) and (10) was determined from the functions given as:

k = ln   ( 1 H ( x ) 2 )       for   0 < H ( x ) ≤ 0.5 (11)

k = ln ( 1 1 − H ( x ) 2 )       for   0.5 < H ( x ) < 1.0 (12)

In this study, the SPI values were calculated using a monthly time step and the threshold ranges adapted from [25] ranging from extreme drought to extremely wet conditions.

The effective drought index (EDI) was computed using monthly time step data for the weather stations within the study area according to [27] . The computation of the EDI was done through four steps. The first step involved the calculation of the effective precipitation parameter EP_p of the current month using the relation:

E D P = ∑ m = 1 N ( ∑ i = 1 m P E m m ) (13)

where; EP_p = effective precipitation parameter (mm), m = total period before the current month, PE_m = the precipitation in m − 1 months before the current month (mm) and N = duration of summation of the precipitation.

The mean EP is computed annually to represent the climatological characteristics of water resources. For practical application of MEP a 5-months running mean is applied in this computation [28] . Then the deviation time series EP from the mean EP was computed using the relation:

D E P = E P − M E P (14)

where; DEP = deviation of time series EP_p from mean effective precipitation parameter (mm) and MEP = mean effective precipitation parameter (mm)

From the EP_p, both the mean and the standard deviations of the monthly values were determined. The resulting time-series EP was used as inputs to calculate its deviation from the mean. Then the return to normal precipitation (RNP) values was determined using the relation adopted from [29] :

R N P = D E P ∑ ( 1 N ) (15)

where; RNP = return to normal precipitation (mm)

N = previous period (months)

From the calculated RNP, the EDI was derived from the relation:

E D I = R N P S t d ( R N P ) (16)

where; Std (RNP) = Standard deviation of a particular months RNP values

Using the computed EDI values, the severity of the drought was categorized based on the thresholds and classification (ranging from extreme drought to extreme wet conditions) adopted from [30] .

A test-statistic can be used to detect a shift in the mean of values [31] . [32] Identified trend in time series water quality while [33] used a non-parametric statistic detect change point of a temporal data. To test for the trend in drought severity, a non-parametric Mann-Kendall trend test was applied. The method was selected for this study because the capacity to test for increasing, decreasing or no trend [34] as required by this study. The data for the upper Tana River basin was evaluated using ordered time series in Figure 2. The data sets were organized in form of x 1 , x 2 , x 3 , ⋯ , x j n-data points where x_i represent data point at time j. Then the Mann-Kendall statistical trend S was determined using the relation:

S = ∑ k = 1 n = 1 [ ∑ j = k + 1 n s i g n ( x i − x k ) ] (17)

The right hand side of the Equation (17) was simplified using Equation (18) given as:

s i g n ( x j − x k ) = { 1             if   ( x j − x k ) > 0 0             if   ( x j − x k ) = 0 − 1         if   ( x j − x k ) < 0 (18)

The probability linked to the Mann-Kendall statistic S and the selected n-data were determined to quantify the level of significance of the trend. The VAR(S) was calculated and then the normalized test statistic Z was computed using the following equations:

V A R ( S ) = n ( n − 1 ( 2 n + 5 ) − ∑ t t ( t − 1 ) ( 2 t + 5 ) 18 ) (19)

Z = { S − 1 V A R ( S )         if   S > 0 0                                               if   S = 0 S + 1 V A R ( S )         if   S < 0 (20)

where; VAR(S) = the variance of the data set and n = the number of data points

Equation (20) which was adapted from [35] was used to qualify the drought trend in the basin as: no trend, increasing trend and decreasing trend when S = 0, S > 0 and S < 0 respectively. In order to determine whether or not the drought trend in the upper Tana River basin was significant or insignificant, significance levels at 90% and 95% were used. At these significance levels, the null hypothesis of no trend was rejected when | Z | > 1.645 and | Z | > 1.96 respectively where the values of Z were adapted from [36] .

The results for monthly time series SPI and the spatial characteristics of droughts in the upper Tana River basin are presented. The results spatial maps are based on the partitioned basin into four elevations bands; low, lower-middle, middle and high elevations. The results of plotted drought conditions on monthly time series graphs are illustrated using the graphs for meteorological stations Sagana FCF (ID 9037096), Kerugoya DWO (ID 9037031), Nyeri (ID 9036288) and Naro-moru (ID 9037064) as presented in Figures 3-6.

Both time series SPI and precipitation were plotted for ease of comparison as given in Figures 3-6 for the four meteorological stations. The area exhibits significant time series and spatial variability in the monthly precipitation. For instance, from Figures 3-6, the maximum monthly precipitation for Sagana FCF, Kerugoya DWO, Nyeri and Naro-moru meteorological stations is 600, 50, 700 and 800 mm respectively. This highly variable precipitation was used to derive the SPI values. The results show that the SPI varies with the monthly precipitation within the study period and across the river basin. For all the stations, extreme drought events based on SPI were detected using SPI for the periods 1972-1974, 1983-1984, 1987-1988, 1999-2000 and 2011 within which the monthly SPI values were consistently below −2.00. The SPI is used to detect the occurrence of drought (negative values of SPI) or the wetness (positive values of SPI) in a river basin. The other drought conditions detected by SPI for the upper Tana River basin as defined in the SPI criterion that includes: severe drought, moderate drought, near normal, moderate wet, very wet and extremely wet conditions. Results of SPI time series within the upper Tana River basin show extreme wetness for 1985-1886, 1992, and 1998 with SPI values being relatively above +2.00.

Drought severities for the upper Tana River basin were computed and mapped using the Kriging approach for the selected years; 1970, 1980, 1990, 2000 and 2010. From Figure 7, it is observed that the spatial drought distribution in the south-eastern areas of the basin exhibit drought severities ranging from 2.044 to 2.835 and from 4.416 to 5.207. In addition, the results show that the north-western parts of the basin experienced drought severity values of 1.822 to 2.463 and 3.745 to 4.384 for 1970 and 2010 respectively. These results indicate that the south-eastern parts of the basin exhibit the highest drought severities while the north-western areas have the lowest. The spatial variation of drought is comparable with the drought distribution generated in other river basins for instance by [37] in the Tel river basin and [6] in the upper Seonath sub-basin.

Based on the SPI, the areal-extend of drought severities increased in both the South-eastern and North-western areas from 4868.7 km² to 6880 km², and 6163.9 km² to 6985.5 km² from 1970 to 2010 respectively. Between 1970 and 1980, the drought areal-extend is almost the same but a significant increase occurred between 1980 and 2010.

From Figure 8(a), the results show that the average drought frequency between 1970 and 2010 for the South-eastern and North-western areas ranged from 12.16 to 14.93 and 3.82 to 6.63 respectively. The drought characteristics were also subjected to Mann-Kendall trend test across the basin. Results of the Mann-Kendall test show that drought trend increased in the South-eastern parts of the basin at 90% and 95% significant levels. However, the results given in Figure 8(b) shows that there was no significant trend that was detected in the North-western areas. This is an indication that the South-eastern parts are drought-prone areas compared to the North-western areas of the upper Tana River basin.

Monthly time series of EDI for meteorological stations Nyeri (ID 9036288), Kerugoya DWO (ID 9037031), Sagana FCF (ID 9037096) and Naro-moru (ID 9037064) are presented in Figures 9-12.

The results of the monthly time series EDI show that this index can be used to detect both the drought and wetness for different years. Typical droughts as presented by this index include the extreme droughts represented by the negative values of −2.5, −2.2, −2.2, −2.5, −2.5, and −2.5 for the years 1972, 1973, 1992, 1994, 2000 and 2010 respectively. At the same time, the index was used to detect the wet conditions of the basin where positive values of +3.0, +3.0 and 4.3 for the years 1986, 1989 and 1998 respectively as illustrated by Figures 9-12 indicate wetness.

From the results of spatial distribution of drought based on EDI shown in Figures 13(a)-(e), it is observed that the drought severity values differ slightly from

those determined using the SPI. It is also noted that the drought severity values in South-eastern areas of the basin range from 3.850 to 4.486 and 4.804 to 5.584 in 1970 and 2010 respectively. Based on the spatially distributed EDI from 1970 to 2010, drought severity has shown some significant increase as per the Figure 13.

For the North-western parts, these values range from 1.822 to 2.463 and 3.745 to 4.384 for the years 1970 and 2010 respectively. Although the drought severity

based on EDI is generally higher than the SPI, both indices exhibit similar trends in terms of spatial distribution, frequency and Mann-Kendall trend test as given in Figure 14(a) and Figure 14(b).