In the topographic complex catchments, landscape features have a significant impact on the spatial prediction of rainfall and temperature. In this study, performance assessments were made of various interpolation techniques for the prediction of the spatial distribution of rainfall and temperature in the Mille and Akaki River catchments, Ethiopia, through an improved approach on selecting the auxiliary variables as a covariate. Two geostatistical interpolation techniques, ordinary kriging (OK) and kriging with external drift (KED), and one deterministic interpolation technique, inverse distance weighting (IDW), were tested through a leave-one-out cross-validation (LOOCV) procedure. The results indicated that using the multivariate geostatistical interpolation technique (KED) with the auxiliary variables as a covsariate outperformed the univariate geostatistical (OK) and deterministic (IDW) techniques for the spatial interpolation of sampled rainfall–temperature data in both contrasting catchments, Akaki and Mille, with the lowest estimation errors (e.g., for Mille annual mean rainfall: root mean square error=75.32, 77.34, 245.72, mean bias error=3.70, −33.18, −15.61, mean absolute error=67.99, 69.51, 192.64) using KED with the combination of elevation and easting as a covariate, IDW and OK, respectively. Thus, the study confirmed that the use of elevation and easting/northing coordinates as predictors in geostatistical interpolation techniques could significantly improve the spatial prediction of climatic variables.

  • Globally, there is no suitable interpolation technique for the spatial prediction of climatic variables like rainfall and temperature.

  • In the mountainous catchment, geostatistical interpolation outperforms deterministic interpolation techniques.

  • The combination of elevation and easting as a covariate significantly improves the performance of the spatial prediction of climatic variables.

Graphical Abstract

Graphical Abstract
Graphical Abstract

In the topographic complex catchments, optimum spatial predictions of climatic variables, specifically rainfall and temperature, are essential as a principal input for downstream applications, namely hydrological and/or hydraulic modeling (Lebel et al. 1987; Grimes et al. 1999), flood early warning, forecasting, and drought management (Bertini et al. 2020; Lu et al. 2020). However, in developing countries, the spatial array of the weather stations of the aforementioned input rainfall and temperature is irregular and highly sparse, and the low network density (Washington et al. 2006; Parker et al. 2011; Dinku 2019) in Ethiopia is not exceptional.

In Ethiopia, specifically in the lowlands, the spatial coverage of weather stations is highly sparse, and they are below the standard of the World Meteorological Organization (WMO) (Washington et al. 2006; Dinku et al. 2017).

Although historically the ground-based stations have been the key source of rainfall and temperature for catchments’ spatial pattern prediction and areal mean estimation (Taesombat & Sriwongsitanon 2009; Ly et al. 2011; Di Piazza et al. 2015; Adhikary et al. 2017), the satellite-based climate data have been taking a leading role in predicting areal mean products, especially rainfall and temperature using different prediction algorithms based on the satellite imagery, mainly geostationary satellites, i.e. Meteosat Second Generation (MSG-2), which can produce high imagery both at spatial and temporal resolutions (Gebremichael & Hossain 2010; Gebere et al. 2015; Chen & Li 2016).

Although satellite-based weather data take advantage of ground-based highly sparse gauged weather data in many aspects, for instance covering a large area at different spatial and temporal scales, there are still some limitations to using satellite-based climatic data alone. For example, since the precipitation measurement is indirect, the accuracy is less, which requires calibrations using ground-based rainfall data, specifically over mountainous regions (Dinku et al. 2008b), and tends to underestimate high rainfall values in mountainous regions such as Ethiopia (Le Coz & Van De Giesen 2020) and overestimate low rainfall events (Toté et al. 2015). In addition to the aforementioned limitations, the spectral resolution of sensors, such as thermal infrared (TIR) and passive micro-wave (PMW), varies with wavelength and fails to capture more accurate images and predictions of climate variability. For instance, PMW sensors cannot properly capture and identify very cold cloud-based rainfall from ice, especially at the top of mountainous regions, and background emissions from the land surface, which vary significantly depending on landscape characteristics (Toté et al. 2015; Petković & Kummerow 2017). Distinguishing raining clouds from the non-raining cloud, like Cirrus clouds from top cloud temperature, and being unable to detect warm orographic rainfall are some limitations of TIR sensors (Dinku et al. 2008a).

A novel approach, which is blending satellite and ground-based climatic data for optimum areal mean climatic variable estimation, has emerged for three decades to solve the aforementioned limitations (Grimes et al. 1999; Yang et al. 2017; Dinku et al. 2017; Gebremedhin et al. 2021).

There are two types of interpolation methods: deterministic interpolation methods, for example, radial basis function (RBF) (Yang et al. 2017) and inverse distance weighting (IDW) (Goovaerts 2000), and geostatistical interpolation methods such as simple kriging (SK), ordinary kriging (OK), ordinary cokriging (CK), universal kriging (UK), and kriging with external drift (KED) (Phillips et al. 1992; Goovaerts 2000; Haberlandt 2007; Taesombat & Sriwongsitanon 2009; Ly et al. 2011; Mukhopadhaya 2016).

Novikov (1981) investigated the impact of elevation on the prediction of the spatial pattern of precipitation and temperature for the New Hampshire and Vermont mountainous catchment via simple linear regression, and the results indicated that the mean monthly precipitation increases strongly with elevation, whereas the mean monthly temperature decreases with elevation. Cantet (2017) compared several spatial interpolation techniques to map the mean annual and monthly precipitation of a small island, which has a complex topography, and the results indicated that the KED seems to outperform regression methods. Similarly, another study in which the dependency of monthly precipitation on elevation was analyzed by Lloyd (2005), focusing on Great Britain through the comparison of different interpolation techniques, concluded that KED with an elevation as a covariate provides the most accurate estimates of precipitation for most months. Hudson & Wackernagel (1994) noted that the integration of information about elevation as a covariate into the mapping of temperature by kriging improves the performance of prediction. Numerous scholars have used the comparison approach of different interpolation methods to predict the spatial disparity of rainfall and groundwater depth (Kisaka et al. 2016; Adhikary & Dash 2017; Amini et al. 2019; Jalili Pirani & Modarres 2020), and their results indicated that geostatistical interpolation techniques yield more accurate predictions than deterministic techniques.

The aforementioned literature review indicated that there was no globally suitable interpolation technique, and thus while scholars used a comparative approach to assess the performance and select the suitable method for a specific site and a specific objective, and as a knowledge gap, none of them considered the effect of the combination of elevation and easting or northing as a covariate on the spatial prediction of climatic variables. Therefore, this research aimed (i) to assess the performance of the deterministic model (IDW) and two geostatistical models, OK and KED interpolation techniques, and (ii) to select and use the suitable technique for contrasting catchments, Mille and Akaki's climatic variable spatial pattern prediction, based on statistical and graphical evaluation methods.

Study area

This research paper considers two contrasting catchments (in terms of physiographic features like climatic condition and land cover) in the Awash River Basin as the case study area, namely the Mille River catchment and the Akaki River catchment, which are located in the Western escarpment and the upper part of the Awash River Basin, respectively (Figure 1).
Figure 1

Location of the study area.

Figure 1

Location of the study area.

Close modal

The Mille River catchment is situated between 39 °5′ and 40 °9′ longitude and 11 °2′ and 11 °8′ latitude, and covers an area of 5,598.74 km2. Water resource management is an essential issue in the Mille River catchment because of its wide range of water uses in its upper part as well as its lower part user requirements and environmental flow provisions (Ministry of Water Resources 2009). The catchment significantly contributes to the water supply for different purposes like irrigation to communities that reside within it, specifically communities living in the upper catchment (Ministry of Water Resources 2009), and it contributes a considerable share to the Tendaho multipurpose reservoir inflows. Consequently, a more accurate spatial distribution of rainfall in the whole catchment, particularly the upper part, would be essential for downstream water resource management and developments, including Tendaho Reservoir operation.

The topography of the catchment is characterized by steep slopes of ridges and mountains in the upper part to a gentle slope in the low-lying part.

The annual rainfall average in the Mille catchment ranges from 374.2 to 1,032 mm, with the highest and lowest monthly average in August and February, respectively (Figure 2). The maximum and minimum mean temperatures range from 28.820 to 43.210 °C and 10.270 to 18.980 °C, respectively.
Figure 2

Mille catchment's mean monthly climatological variables.

Figure 2

Mille catchment's mean monthly climatological variables.

Close modal

The Akaki River catchment is located in the northwestern escarpment of the Awash River Basin, Ethiopia, and covers an area of about 1,425 km2. It is located between 38 °6′ and 39 °1′ longitude and 8 °8′ and 9 °2′ latitude (see Figure 1). The Akaki catchment is circumscribed by the Intoto Mountains to the north, Mount Menagesha and Wechecha volcanic mountains to the west, and Yerer Mountain to the east. In the Akaki catchment, there are three surface water reservoirs, Legedadi, Dire, and Gefersa Reservoir, which are used for water supply for Addis Ababa city and its surrounding towns, and one hydropower reservoir, Abasamuel. The catchment is very important from a water supply point of view, specifically for Addis Ababa and its surrounding communities and for agricultural production.

The climate of the Akaki catchment has two distinct wet seasonal weather patterns. The main rainy season starts from late June to mid-September, which contributes almost 70% of the total annual rainfall, and the pre-rainy season starts from March to mid-May (Molla et al. 2005). Based on historic climatic data (2000–2016), the mean minimum and maximum rainfall was 935.2 and 1,011.6 mm, respectively. The mean minimum and maximum annual temperatures of the catchment vary from 8.640 to 10.330 °C and 21.610 to 23.460 °C, respectively (Figure 3).
Figure 3

Akaki catchment's mean monthly climatological variables.

Figure 3

Akaki catchment's mean monthly climatological variables.

Close modal

Datasets

Both historic ground-based point data and blended gridded climatic variables, specifically rainfall and temperature data, were obtained from the National Meteorological Agency (NMA), Ethiopia, from 1 January 1983 to 31 December 2016 (Table 1 and Figure 1). The blended pixel climatic dataset was merged from the European Meteorological Satellites (METEOSAT) and ground-based observations at the national level for some African nations, including Ethiopia (Dinku et al. 2017). However, the ground-based climatic datasets were missing climatic variable values on some consecutive days, months, and years (>10% missing data) for most ground stations. As a solution, we took a blended pixel value of the grid in which the gauging station was laid within it, and then the authors performed the correlation and regression analysis (not shown here) with ground-based historic climate datasets to check the similarity between two neighboring sample climatic datasets. We obtained that a pixel value of a grid was much more strongly correlated with ground-based historic climate data because of the close sample distance, and the same result was confirmed by Wilson et al. (1998). Therefore, based on their correlation, we selected and used pixel values as point data instead of gauged station data. As a consequence, we selected and used nine pixel values for the Mille catchment and 10 pixel values for the Akaki catchment as the point station dataset, which were used as principal variables for spatial pattern prediction using different interpolation techniques. Based on collected daily data, monthly and annual data for climatological variables were developed to predict the spatial pattern of both the rainfall and temperature of the interesting study areas, the Mille and Akaki catchments.

Table 1

Mille and Akaki catchments’ climatological stations

Mille catchment
Akaki catchment
StationsAltitude (m.a.s.l)LongitudeLatitudeStationsAltitude (m.a.s.l)LongitudeLatitude
X055 2,089 39.75 11.32 X027 2,202 38.85 8.88 
X058 1,854 40.77 11.43 X029 2,279 38.68 8.93 
X059 1,573 39.62 11.54 X030 2,282 38.67 8.94 
Weranso 643 39.67 11.66 X031 2,197 38.76 8.95 
Waama 1,020 39.61 11.75 X036 2,385 38.75 9.02 
Mille (AVA) 491 40.48 11.35 X038 2,440 38.73 9.03 
Haik 2,003 40.08 11.45 X040 2,606 38.73 9.06 
Chifra 928 40.00 11.75 X041 2,741 38.84 9.06 
Bokeksa 1,771 40.75 11.42 X043 2,771 38.73 9.08 
    X044 2,543 39.02 9.15 
Mille catchment
Akaki catchment
StationsAltitude (m.a.s.l)LongitudeLatitudeStationsAltitude (m.a.s.l)LongitudeLatitude
X055 2,089 39.75 11.32 X027 2,202 38.85 8.88 
X058 1,854 40.77 11.43 X029 2,279 38.68 8.93 
X059 1,573 39.62 11.54 X030 2,282 38.67 8.94 
Weranso 643 39.67 11.66 X031 2,197 38.76 8.95 
Waama 1,020 39.61 11.75 X036 2,385 38.75 9.02 
Mille (AVA) 491 40.48 11.35 X038 2,440 38.73 9.03 
Haik 2,003 40.08 11.45 X040 2,606 38.73 9.06 
Chifra 928 40.00 11.75 X041 2,741 38.84 9.06 
Bokeksa 1,771 40.75 11.42 X043 2,771 38.73 9.08 
    X044 2,543 39.02 9.15 

Unlike deterministic and univariate geostatistical interpolation techniques, multivariate geostatistical interpolation methods, for instance KED, account for secondary information in the prediction of spatial climatological variables. The 90 m spatial resolution digital elevation model (DEM) elevation, longitudinal, and latitudinal positions, and their combination were considered as auxiliary variables (covariates) in this study.

In this research, the integration of R-programming with the gstat package (Pebesma 2004, 2012) and GIS tools were applied for spatiotemporal interpolation techniques, preprocessing raster layers containing the predictive variables, and preparing shapefiles.

Methods

The three stages followed in the study plan were (1) the preparation and export of monthly and annual climatic variable data in ‘.csv’ format, exporting raster of DEM, easting, and northing, and shapefiles for catchments into R-programming, (2) various interpolation technique applications to generate a spatial pattern of rainfall and temperature map and to estimate areal mean rainfall and temperature (minimum and maximum), and (3) the assessment of the performance of various interpolation methods based on statistical evaluation criteria. The details of each step are described as follows.

Collection and preprocessing of sampled historical climatological datasets

The sampled historic climatic data were collected and processed as Excel spreadsheets, and they were prepared and exported in ‘.csv’ format for monthly and annual rainfall and temperature. Consequently, the prepared sampled data were exported into R-programming for the various types of spatial interpolation techniques. Simultaneously, the sampled data location shapefiles, elevation, latitudinal, longitudinal raster, and shapefile of the study area were imported into R-programming. In R-programming, point data were converted to a spatial points data frame (SPDF) using a longitude and latitude coordinate system. Then, for better work in R, the longitudinal–latitudinal coordinate system was transformed to the Universal Transverse Mercator (UTM) coordinate system. Finally, the sampled climatic data were combined with location data to form a new spatial point data frame.

Interpolation techniques

Various spatial interpolation techniques used in this study were briefly introduced and compared in R-programming (https://cran.r-project.org) through gstat (Pebesma & Wesseling 1998; Pebesma 2003) and related packages. For this study, based on their best performance (e.g., Chen & Liu 2012; Rata et al. 2020), one deterministic method, IDW, and two geostatistical methods, OK and KED, were selected among the various spatial interpolation techniques. For details of the description of geostatistical and other interpolation techniques, the reader can refer to geostatistical books (Wackernagel 1998, 2003; Webster & Oliver 2007).

Inverse distance weighting
The IDW method estimates values at ungauged/unsampled points by the weighted average of measured data that found surrounding points. It is based on the assumption that the sampled observations that are close to the estimated points have more weight control on the estimated value than the sampled observations far apart (Goovaerts 2000). According to Shepard (1968), the equation of IDW is given by:
(1)
where is the climatic variable (in our case, either rainfall or temperature) at the unsampled point (U), is the climatic variable at the sampled location , , and are the undetermined weights to be estimated as a function of distance.

The inverse distance power (p) by default is 2 (Shepard 1968; Goovaerts 2000; Otieno et al. 2014). However, the authors took a certain value of inverse distance power (Table 2) in an interval of one unit to test the performance of each power and selected the best power using the LOOCV method. Accordingly, power 4 was selected for the Mille catchment, and power 3 was selected for the Akaki catchment.

Table 2

Inverse distance power selection using the LOOCV method (for the month of January)

Idp = n,
where n = 1, 2, 3, 4, 5, 6
RMSE
Mille catchmentAkaki catchment
5.153 1.274 
4.353 1.221 
3.940 1.206 
3.861 1.210 
3.900 1.219 
3.960 1.226 
Idp = n,
where n = 1, 2, 3, 4, 5, 6
RMSE
Mille catchmentAkaki catchment
5.153 1.274 
4.353 1.221 
3.940 1.206 
3.861 1.210 
3.900 1.219 
3.960 1.226 

Ordinary kriging
Ordinary kriging (OK) is one of the geostatistical interpolation techniques by which local variation is considered by limiting the domain of stationarity of the unknown local mean (m(X)) to the local neighborhood, and the kriging weights sum to one (Goovaerts 1997, 2000; Webster & Oliver 2007). Its equation is given by:
(2)
where ZOK(X) is the climatic variable (in our case, rainfall and temperature) predicted at the unknown location (X) using the OK method, are the sampled values at X’s n data locations, and are the OK weights determined to minimize the estimation of variance while confirming no biasedness of the OK estimator (Goovaerts 1997; Wackernagel 1998). It is mathematically expressed as follows:
(3)
According to Goovaerts (1997, 2000), the weights are acquired by solving a system of linear equations known as the ‘OK system’, which is given by the following equation:
(4)
where μ(X) is the Lagrange parameter.
Unlike IDW, any geostatistical technique such as OK uses a variogram but not Euclidean distance to measure the degree of dissimilarity between sampled data Z(Uα) and unsampled value Z(UO) (Goovaerts 2000; Webster & Oliver 2007). Let two distinct climatic data values Z (Uα) and Z (Uα + h) at two different locations be given, and if we assume isotropy, which is the direction independence of the semi-variance, then the more distant sample value should receive less weight in the estimation of Z (Uo). The experimental variogram is calculated as half the average squared difference between the components of value pairs, as presented in Equation (5) and described in Goovaerts (2000):
(5)
where is the experimental variogram, which is a function of both the direction and the distance (anisotropic–spatial pattern), and N (h) is the number of pairs of climatic variable data locations separated by lag distance/vector h.

In this study, the automap package automatically selects some models, namely spherical, exponential, and Ste Mat (Matern, M. Stein's parameterization) models, which are widely applied (Goovaerts 2000; Webster & Oliver 2007; Stein 2010; Frazier et al. 2016) to model the theoretical variogram.

The equations for the spherical, exponential, and Ste models are as follows:
(6)
(7)
(8)
where is a nugget effect, c is the sill variance, h is the lag distance (m), a is the actual range, is the practical range, which is three times ( = 3a) the actual range (Stein 2010), is the smoothness parameter varying from 0 to ∞, is the gamma function, and is the modified Bessel function.

The most common technique of fitting variogram models to compute experimental variograms is performed using manual fitting procedures (Nalder & Wein 1998; Haberlandt 2007). However, this is not an appropriate approach because it depends on the expertise and the sample size in the data (Ly et al. 2011). In this research paper, an automatic fitting procedure was applied using the ‘autofitVariogram’ function from the package ‘automap’ to choose the appropriate model for fitting a variogram model to an experimental variogram and also calibrated its parameters such as range, nugget, and sill.

Kriging with external drift
KED is a kind of universal kriging that replaces one or more variables as a predictor by the coordinates, and the trend mean, m(u), is modeled as a linear function of smoothly varying covariates y(u) instead of the spatial coordinates as a function (Goovaerts 1997; Webster & Oliver 2007):
(9)
where m(u) is a local constant mean at location u, the trend coefficients ao and a1 are implicitly estimated through the kriging system within each search neighborhood Z, and y(u) is the independent variable used as an influence in the prediction of the dependent variable.
Let y1, y2, y3,…,yn be several independent (external) variables linearly related to Z(u),
(10)
and we might be able to calculate the KED estimator as follows:
(11)
The expectation is:
(12)
where , k = 1, 2, 3,…,K, are unknown coefficients to be determined, are known external variables/covariates at location , ε(u) is the Lagrange multiplier, and are kriging weights calculated by:
(13)
And the estimator is unbiased if:
(14)
where is the semi-variance between the data points and , is the semi-variance between the target point and the data points surrounding the estimated point, and , k = 1, 2,…,K is a Lagrange multiplier.

Among spatial interpolation methods, the geostatistical technique assumes that the variable is normally distributed (Isaaks & Srivastava 1989). However, point data is/are often not symmetrical (skewness either to the right or to the left), which affects spatial reduction of input data in which the few values will overcome all the others. According to Goovaerts (1997), nonsymmetrical distributions are often transformed to conditions of normality using one of the three transformations such as natural logarithmic function, square root transformation to reduce the skewness of input data, and the influence of extreme values. But, for small sampled data such as our case, we have chosen not to use transformation to conditions of normality for the sake of highly sparsely distributed gridded sampled climate data (Rossiter 2014; Bati 2022), and we have also intended to ignore the possibility of anisotropy for this research work for the sake of not missing the remaining sample data and for simplicity of modeling.

Performance evaluation

The performances of three selected interpolation techniques, IDW, OK, and KED, were accomplished via evaluations and comparisons of estimated climatic variable values and observed data. In this study, the available climatic variable data were split into two parts: training and test/validation data. The training data were used to fit the model, while the test data were used to calculate prediction accuracy, and the procedure is called regular-validation.

The cross-validation procedure was applied to compare the spatial interpolation performance of KED with univariate interpolation methods. The basic idea behind cross-validation is that we split our test/validation dataset into k-folds. For this study, the commonly used type of cross-validation, the so-called leave-one-out cross-validation (LOOCV), was applied, where the climatic data consecutively took the role of test data, and the remaining data took the role of training data. We train our model on k − 1 folds and use the resulting model to predict the values of the left-out fold. In our case, sampled climatic data for two contrasting catchments, Mille (number of observations (k = 9)) and Akaki (number of observations (k = 10)), from 2000 to 2016 were used for modeling. Accordingly, the LOOCV technique involves using only one observation data as the test set and the k − 1 remaining observations as the training set.

To assess and select a suitable interpolation technique, this study evaluated and compared different interpolation techniques via four statistical indicators, such as root mean square error (RMSE), mean bias error (MBE), mean absolute error (MAE), and coefficient of correlation (r), between the predicted and observed climatic variable values, which are given as follows (Vicente-Serrano et al. 2003; Li & Heap 2008):
(15)
(16)
(17)
(18)
where and are the mean of observed and estimated climatic variable values, respectively, and n is the number of paired climatic data.

Then, the model showing the lowest error on the test sample (i.e., the lowest test error) is identified as the best one in this study area. This was the reason why the normality condition was not checked.

Variogram parameter estimation and modeling

For the Mille catchment, in the KED interpolation method with elevation and easting as a covariate, an experimental variogram and two variogram models (i.e., exponential and spherical) automatically fit the theoretical variogram with the experimental variogram. Both monthly and annual variogram parameters, namely range, nugget, and sill, were generated using historic climatic variable data (Table 3).

Table 3

Variogram parameters and variogram models were developed for the Mille catchment using the KED interpolation technique with the combination of elevation and easting as covariates

MonthRange (m)Nugget (mm2)Sill (mm2)ModelSelected covariate/predictor
Jan 12,925 18 22 Spherical Elevation + easting 
Feb 12,925 6.8 Spherical Elevation + easting 
Mar 12,925 11 21 Spherical Elevation + easting 
Apr 12,925 19 24 Spherical Elevation + easting 
May 12,925 24 30 Spherical Elevation + easting 
Jun 12,925 10 15 Spherical Elevation + easting 
Jul 38,775 211 888 Exponential Elevation + easting 
Aug 12,925 218 313 Spherical Elevation + easting 
Sep 12,925 26 38 Spherical Elevation + easting 
Oct 38,775 28 43 Exponential Elevation + easting 
Nov 38,775 0.55 0.73 Exponential Elevation + easting 
Dec 12,925 1.6 3.6 Spherical Elevation + easting 
Annual 38,775 2,608 4,029 Exponential Elevation + easting 
MonthRange (m)Nugget (mm2)Sill (mm2)ModelSelected covariate/predictor
Jan 12,925 18 22 Spherical Elevation + easting 
Feb 12,925 6.8 Spherical Elevation + easting 
Mar 12,925 11 21 Spherical Elevation + easting 
Apr 12,925 19 24 Spherical Elevation + easting 
May 12,925 24 30 Spherical Elevation + easting 
Jun 12,925 10 15 Spherical Elevation + easting 
Jul 38,775 211 888 Exponential Elevation + easting 
Aug 12,925 218 313 Spherical Elevation + easting 
Sep 12,925 26 38 Spherical Elevation + easting 
Oct 38,775 28 43 Exponential Elevation + easting 
Nov 38,775 0.55 0.73 Exponential Elevation + easting 
Dec 12,925 1.6 3.6 Spherical Elevation + easting 
Annual 38,775 2,608 4,029 Exponential Elevation + easting 

Dissimilarities expressed by semi-variance were increased following the separation distance (lag) increase and resulted in both sampled climatic data close to each other being more similar; hence, their squared difference was less significant than those that were farther apart. The theoretical variogram model rises to a certain distance and then the model levels off, and the distance at which the model first levels off is termed the range parameter. The semi-variance value that reaches the range parameter is known as the sill. For instance, year long-based mean annual and August mean rainfall were used to simulate the experimental variograms using nine sampled stations, which were then fitted with theoretical variograms using spherical and exponential variogram models, respectively (Figure 4). Accordingly, the exponential and spherical values of the two variogram models provided ranges of approximately 38.8 and 12.9 km, nugget effects of 2,608 and 218 mm2, and sills of approximately 4,029 and 313 mm2 for the annual and August months, respectively.
Figure 4

Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and August monthly mean rainfall, respectively.

Figure 4

Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and August monthly mean rainfall, respectively.

Close modal

As seen from Table 3, among the two models used, the spherical model was the most frequently applied to fit the monthly experimental semivariogram with the theoretical variogram. For most months (9 out of 12), the theoretical variogram was fitted with an experimental variogram by using the same model, and the same covariates resulted in the same range, but the sill and nugget varied, which may be in connection with the spatial pattern and smaller sample size of the sampled rainfall data over the fixed domain (Kaufman & Shaby 2013). Moreover, unlike the spherical model, since the exponential model asymptotically approaches the sill, the effective range was three times the actual range parameter (a) (a* = 3a) (Goovaerts 1997; Nalder & Wein 1998), which means that the proximity by which the spatial dependency decay was longer than the spherical model's effective range (see Table 3).

For the mean minimum and maximum temperatures, both the spherical and exponential variogram models were fitted to the mean annual and monthly (e.g., April month) experimental variograms (Figure 5(a)–5(d)). For the mean minimum annual temperature, the theoretical variogram was fitted to the experimental variogram using an exponential model with a range of 38.8 km, a sill of 0.16°C2, and a nugget effect of 0.06°C2. For April, the theoretical variogram was fitted to the experimental variogram using a spherical model with a range of 12.93 km, a sill of 0.2°C2, and a nugget effect of 0.16°C2. The same procedure was applied for the mean maximum temperature; as a result, unlike the minimum temperature, both mean monthly (e.g., April) and mean annual maximum temperature experimental variograms were fitted using a spherical model with the same range of 12.93 km but with nugget effects of 0.69 and 0.48°C2 and sills of approximately 0.83 and 0.9°C2, respectively.
Figure 5

Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and April mean minimum temperatures and spherical (c and d) for annual and April mean maximum temperatures.

Figure 5

Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and April mean minimum temperatures and spherical (c and d) for annual and April mean maximum temperatures.

Close modal
As depicted in Figure 5(a)–5(d), the number of bins (each possesses 11 and 8 paired climatic datasets) was highly scattered, and these were the results of the sample size and/or the density of sampled climatic data (see Figure 1), which may affect the reliability of the experimental variogram complements to the statistical distribution of the sampled data (Webster & Oliver 2007). According to Webster & Oliver (2007), the reliability of the experimental variogram is affected by factors, namely the size/density of the sampled data (e.g., Ly et al. 2011), the statistical distribution of the sampled data, and the configuration or design of the sample. As the sample data size increases, such scatter decreases, and the plotted paired points (bins) tend to be closer to a theoretical variogram (e.g., Figure 6).
Figure 6

Experimental (bins) and fitted theoretical (curve) variograms of January mean monthly ((a) and (c)) and annual ((b) and (d)) rainfall and maximum temperature, respectively.

Figure 6

Experimental (bins) and fitted theoretical (curve) variograms of January mean monthly ((a) and (c)) and annual ((b) and (d)) rainfall and maximum temperature, respectively.

Close modal

In the case of the Akaki catchment, satisfactory computation results were obtained in response to the success in producing an experimental variogram (Table 4 and Figure 6) and have resulted in successful attempts at fitting theoretical variograms with experimental variograms for both mean monthly and annual rainfall and the mean maximum temperature.

Table 4

Variogram parameters and models developed for Akaki's catchment mean monthly and annual rainfall and maximum temperature using KED with various covariates

Parameters and variogram model for mean rainfall
Parameters and variogram model for mean Tmax
MonthRangeNuggetSillModelCovariatesRangeNuggetSillModelCovariates
Jan 4,869 0.3 0.65 Spherical Northing 9,128 0.4 Ste Elevation + northing 
Feb 12,471 12 Ste Easting 8,915 0.45 Ste Elevation + northing 
Mar 8,295 24 Ste Easting 9,224 0.5 Ste Elevation + northing 
Apr 5,725 14 Ste Northing 9,954 0.56 Ste Elevation 
May 11,838 24 Ste Easting 10,962 0.72 Ste Elevation 
Jun 4,743 17 Ste Elevation 9,057 0.62 Ste Elevation 
Jul 6,015 231 Ste Northing 9,666 0.71 Ste Elevation 
Aug 8,954 504 Ste Northing 9,447 0.58 Ste Elevation 
Sep 5,240 106 Ste Elevation 10,489 0.51 Ste Elevation 
Oct 2,904 10 Ste Easting 8,931 0.6 Ste Elevation + northing 
Nov 9,547 2.2 Ste Elevation 8,526 0.43 Ste Elevation + northing 
Dec 9,773 1.5 Ste Easting 8,384 0.42 Ste Elevation + northing 
Annual 7,527 1,943 Ste Northing 10,658 2,496 Ste Elevation 
Parameters and variogram model for mean rainfall
Parameters and variogram model for mean Tmax
MonthRangeNuggetSillModelCovariatesRangeNuggetSillModelCovariates
Jan 4,869 0.3 0.65 Spherical Northing 9,128 0.4 Ste Elevation + northing 
Feb 12,471 12 Ste Easting 8,915 0.45 Ste Elevation + northing 
Mar 8,295 24 Ste Easting 9,224 0.5 Ste Elevation + northing 
Apr 5,725 14 Ste Northing 9,954 0.56 Ste Elevation 
May 11,838 24 Ste Easting 10,962 0.72 Ste Elevation 
Jun 4,743 17 Ste Elevation 9,057 0.62 Ste Elevation 
Jul 6,015 231 Ste Northing 9,666 0.71 Ste Elevation 
Aug 8,954 504 Ste Northing 9,447 0.58 Ste Elevation 
Sep 5,240 106 Ste Elevation 10,489 0.51 Ste Elevation 
Oct 2,904 10 Ste Easting 8,931 0.6 Ste Elevation + northing 
Nov 9,547 2.2 Ste Elevation 8,526 0.43 Ste Elevation + northing 
Dec 9,773 1.5 Ste Easting 8,384 0.42 Ste Elevation + northing 
Annual 7,527 1,943 Ste Northing 10,658 2,496 Ste Elevation 

For instance, in KED with northing as a covariate, the theoretical variogram was fitted with an experimental variogram model using the spherical model with a range of 4.9 km, a sill of 0.65 mm2, and a nugget effect of 0.3 mm2 for January mean monthly rainfall, and the Ste model was fitted to the theoretical variogram and the experimental variogram using a range of 7.5 km, a sill variance of 1,943 mm2, and a nugget variance of 0 mm2 for the mean annual rainfall.

Overall, the variogram value is often zero at a lag distance equal to zero in theory. Nevertheless, within the shortest distance, which is less than lag, the variogram often exhibits the phenomenon called the ‘nugget effect’, which is a value greater than zero (Webster & Oliver 2007). The nugget effect can be attributed to measurement errors, which occur because of the error inherent in measuring devices, or spatial sources of variation at microscale distances smaller than the lag distance (or both).

Verifying the performance of interpolation methods via the LOOCV procedure

All the methods described in subsection 2.3.2. were performed and evaluated through a cross-validation technique, specifically the LOOCV method, which allows us to compare estimated and actual values using sampled data (Isaaks 1990) (e.g., Mille catchment rainfall and maximum temperature; Tables 5 and 6). The results in Table 5 show that the KED using the combination of longitude and elevation gives overall the best spatial estimation results with the smallest statistical evaluation parameters, followed by KED with longitude alone as a covariate and IDW and KED with elevation as the covariates.

Table 5

Mille catchment's mean monthly and annual rainfall spatial prediction using various interpolation techniques

MonthEst. rainfallaObr. rainfallaRMSEMBEMAErV.modelMonthEst. rainfallObr. rainfallRMSEMBEMAErV.model
KED with 90 m DEM elevation KED with the combination of 90 m DEM elevation and easting 
Jan 11.79 11.42 5.71 −0.36 5.05 −0.01 Spherical Jan 11.62 11.42 5.41 −0.20 4.56 0.30 Spherical 
Feb 6.75 6.80 3.17 0.05 2.62 0.57 Spherical Feb 6.42 6.80 3.89 0.38 3.21 0.5 Spherical 
Mar 38.33 37.92 9.07 −0.42 8.09 0.86 Exponential Mar 37.17 37.92 7.31 0.75 6.38 0.92 Spherical 
Apr 58.14 57.64 9.92 −0.50 8.37 0.86 Exponential Apr 57.04 57.64 7.69 0.6 6.49 0.93 Spherical 
May 43.95 43.40 9.27 −0.56 8.46 0.84 Exponential May 43.18 43.40 6.68 0.22 5.61 0.92 Spherical 
Jun 16.62 16.49 4.36 −0.13 3.61 0.88 Exponential Jun 16.59 16.49 4.41 −0.1 3.34 0.88 Spherical 
Jul 194.87 192.76 36.00 −2.12 30.40 0.89 Exponential Jul 193.10 192.76 26.85 −0.36 19.22 0.94 Exponential 
Aug 218.10 216.06 34.90 −2.05 32.46 0.88 Exponential Aug 214.80 216.06 20.66 1.26 18.5 0.96 Spherical 
Sep 64.69 64.03 13.45 −0.67 11.27 0.87 Exponential Sep 63.46 64.03 9.3 0.56 7.6 0.95 Spherical 
Oct 24.64 24.16 7.39 −0.48 6.35 0.75 Exponential Oct 24.45 24.16 7.07 −0.3 6.21 0.77 Exponential 
Nov 16.25 16.08 2.77 −0.17 2.50 0.85 Exponential Nov 16.01 16.08 1.28 0.07 1.04 0.97 Exponential 
Dec 12.23 12.00 4.71 −0.24 3.88 0.50 Exponential Dec 11.72 12.00 2.82 0.28 2.41 0.88 Spherical 
Annual 706.40 698.80 124.17 −7.67 110.66 0.88 Exponential Annual 695.1 698.8 75.32 3.70 67.99 0.96 Exponential 
KED with easting alone as a covariate IDW 
Jan 11.81 11.42 4.90 −0.39 4.42 0.40 Exponential Jan 10.88 11.42 3.86 0.54 2.55 0.69 – 
Feb 6.35 6.80 3.63 0.45 2.90 0.52 Spherical Feb 7.30 6.80 2.92 −0.50 2.66 0.67 – 
Mar 37.25 37.92 7.10 0.66 5.71 0.93 Spherical Mar 40.09 37.92 6.96 −2.17 6.23 0.93 – 
Apr 57.24 57.64 7.66 0.40 6.06 0.93 Spherical Apr 59.50 57.64 6.87 −1.86 5.23 0.94 – 
May 43.47 43.40 6.42 −0.07 5.18 0.93 Spherical May 45.02 43.40 6.16 −1.62 5.78 0.94 – 
Jun 16.43 16.49 5.47 0.06 4.58 0.81 Spherical Jun 17.86 16.49 3.81 −1.37 3.33 0.92 – 
Jul 193.78 192.76 25.38 −1.02 19.88 0.95 Exponential Jul 204.34 192.76 29.32 −11.58 26.31 0.94 – 
Aug 215.25 216.06 22.27 0.82 18.45 0.95 Spherical Aug 227.10 216.06 23.15 −11.04 18.97 0.96 – 
Sep 63.44 64.03 8.92 0.58 7.14 0.95 Spherical Sep 66.31 64.03 9.67 −2.28 8.21 0.94 – 
Oct 24.76 24.16 6.37 −0.60 5.46 0.82 Exponential Oct 24.98 24.16 3.40 −0.83 2.72 0.97 – 
Nov 16.02 16.08 1.12 0.06 0.90 0.98 Exponential Nov 16.65 16.08 1.12 −0.56 0.92 0.99 – 
Dec 11.87 12.00 2.81 0.12 2.31 0.86 Exponential Dec 11.91 12.00 2.68 0.08 2.52 0.87 – 
Annual 697.40 698.80 78.39 1.37 66.75 0.96 Exponential Annual 731.90 698.80 77.34 −33.18 69.51 0.97 – 
KED with northing alone as a covariate OK 
Jan 11.45 11.42 5.95 −0.02 4.62 −0.15 Spherical Jan 11.45 11.42 5.85 −0.03 4.77 −0.99 Exponential 
Feb 6.54 6.80 4.65 0.26 4.11 −0.75 Exponential Feb 6.85 6.80 3.93 −0.05 3.55 −0.59 Exponential 
Mar 37.98 37.92 18.01 −0.06 15.36 −0.10 Exponential Mar 38.73 37.92 16.71 −0.81 13.29 0.40 Exponential 
Apr 58.06 57.64 19.88 −0.42 16.58 −0.02 Exponential Apr 58.67 57.64 18.67 −1.03 14.90 0.40 Exponential 
May 43.07 43.40 18.51 0.33 15.13 −0.56 Exponential May 44.00 43.40 17.01 −0.60 13.78 −0.01 Exponential 
Jun 16.19 16.49 10.18 0.30 8.16 −0.26 Exponential Jun 16.71 16.49 9.25 −0.22 7.38 −0.24 Exponential 
Jul 197.60 192.76 75.24 −4.79 56.99 0.41 Exponential Jul 198.00 192.76 71.71 −5.25 54.10 0.60 Exponential 
Aug 217.50 216.06 72.55 −1.47 57.69 0.03 Exponential Aug 267.50 269.30 68.54 −3.67 52.78 0.48 Exponential 
Sep 64.55 64.03 27.91 −0.52 23.91 −0.03 Exponential Sep 65.26 64.03 26.61 −1.23 21.37 0.29 Exponential 
Oct 24.23 24.16 11.88 −0.07 9.26 −0.73 Exponential Oct 24.53 24.16 11.26 −0.37 8.70 −0.15 Exponential 
Nov 16.38 16.08 4.87 −0.29 4.18 0.38 Exponential Nov 16.44 16.08 4.77 −0.35 3.62 0.63 Exponential 
Dec 11.82 12.00 5.02 0.18 4.20 0.43 Exponential Dec 12.13 12.00 5.57 −0.14 4.71 −0.25 Exponential 
Annual 708.80 698.80 258.35 −10.04 204.18 0.25 Exponential Annual 714.40 698.80 245.72 −15.61 192.64 0.53 Exponential 
MonthEst. rainfallaObr. rainfallaRMSEMBEMAErV.modelMonthEst. rainfallObr. rainfallRMSEMBEMAErV.model
KED with 90 m DEM elevation KED with the combination of 90 m DEM elevation and easting 
Jan 11.79 11.42 5.71 −0.36 5.05 −0.01 Spherical Jan 11.62 11.42 5.41 −0.20 4.56 0.30 Spherical 
Feb 6.75 6.80 3.17 0.05 2.62 0.57 Spherical Feb 6.42 6.80 3.89 0.38 3.21 0.5 Spherical 
Mar 38.33 37.92 9.07 −0.42 8.09 0.86 Exponential Mar 37.17 37.92 7.31 0.75 6.38 0.92 Spherical 
Apr 58.14 57.64 9.92 −0.50 8.37 0.86 Exponential Apr 57.04 57.64 7.69 0.6 6.49 0.93 Spherical 
May 43.95 43.40 9.27 −0.56 8.46 0.84 Exponential May 43.18 43.40 6.68 0.22 5.61 0.92 Spherical 
Jun 16.62 16.49 4.36 −0.13 3.61 0.88 Exponential Jun 16.59 16.49 4.41 −0.1 3.34 0.88 Spherical 
Jul 194.87 192.76 36.00 −2.12 30.40 0.89 Exponential Jul 193.10 192.76 26.85 −0.36 19.22 0.94 Exponential 
Aug 218.10 216.06 34.90 −2.05 32.46 0.88 Exponential Aug 214.80 216.06 20.66 1.26 18.5 0.96 Spherical 
Sep 64.69 64.03 13.45 −0.67 11.27 0.87 Exponential Sep 63.46 64.03 9.3 0.56 7.6 0.95 Spherical 
Oct 24.64 24.16 7.39 −0.48 6.35 0.75 Exponential Oct 24.45 24.16 7.07 −0.3 6.21 0.77 Exponential 
Nov 16.25 16.08 2.77 −0.17 2.50 0.85 Exponential Nov 16.01 16.08 1.28 0.07 1.04 0.97 Exponential 
Dec 12.23 12.00 4.71 −0.24 3.88 0.50 Exponential Dec 11.72 12.00 2.82 0.28 2.41 0.88 Spherical 
Annual 706.40 698.80 124.17 −7.67 110.66 0.88 Exponential Annual 695.1 698.8 75.32 3.70 67.99 0.96 Exponential 
KED with easting alone as a covariate IDW 
Jan 11.81 11.42 4.90 −0.39 4.42 0.40 Exponential Jan 10.88 11.42 3.86 0.54 2.55 0.69 – 
Feb 6.35 6.80 3.63 0.45 2.90 0.52 Spherical Feb 7.30 6.80 2.92 −0.50 2.66 0.67 – 
Mar 37.25 37.92 7.10 0.66 5.71 0.93 Spherical Mar 40.09 37.92 6.96 −2.17 6.23 0.93 – 
Apr 57.24 57.64 7.66 0.40 6.06 0.93 Spherical Apr 59.50 57.64 6.87 −1.86 5.23 0.94 – 
May 43.47 43.40 6.42 −0.07 5.18 0.93 Spherical May 45.02 43.40 6.16 −1.62 5.78 0.94 – 
Jun 16.43 16.49 5.47 0.06 4.58 0.81 Spherical Jun 17.86 16.49 3.81 −1.37 3.33 0.92 – 
Jul 193.78 192.76 25.38 −1.02 19.88 0.95 Exponential Jul 204.34 192.76 29.32 −11.58 26.31 0.94 – 
Aug 215.25 216.06 22.27 0.82 18.45 0.95 Spherical Aug 227.10 216.06 23.15 −11.04 18.97 0.96 – 
Sep 63.44 64.03 8.92 0.58 7.14 0.95 Spherical Sep 66.31 64.03 9.67 −2.28 8.21 0.94 – 
Oct 24.76 24.16 6.37 −0.60 5.46 0.82 Exponential Oct 24.98 24.16 3.40 −0.83 2.72 0.97 – 
Nov 16.02 16.08 1.12 0.06 0.90 0.98 Exponential Nov 16.65 16.08 1.12 −0.56 0.92 0.99 – 
Dec 11.87 12.00 2.81 0.12 2.31 0.86 Exponential Dec 11.91 12.00 2.68 0.08 2.52 0.87 – 
Annual 697.40 698.80 78.39 1.37 66.75 0.96 Exponential Annual 731.90 698.80 77.34 −33.18 69.51 0.97 – 
KED with northing alone as a covariate OK 
Jan 11.45 11.42 5.95 −0.02 4.62 −0.15 Spherical Jan 11.45 11.42 5.85 −0.03 4.77 −0.99 Exponential 
Feb 6.54 6.80 4.65 0.26 4.11 −0.75 Exponential Feb 6.85 6.80 3.93 −0.05 3.55 −0.59 Exponential 
Mar 37.98 37.92 18.01 −0.06 15.36 −0.10 Exponential Mar 38.73 37.92 16.71 −0.81 13.29 0.40 Exponential 
Apr 58.06 57.64 19.88 −0.42 16.58 −0.02 Exponential Apr 58.67 57.64 18.67 −1.03 14.90 0.40 Exponential 
May 43.07 43.40 18.51 0.33 15.13 −0.56 Exponential May 44.00 43.40 17.01 −0.60 13.78 −0.01 Exponential 
Jun 16.19 16.49 10.18 0.30 8.16 −0.26 Exponential Jun 16.71 16.49 9.25 −0.22 7.38 −0.24 Exponential 
Jul 197.60 192.76 75.24 −4.79 56.99 0.41 Exponential Jul 198.00 192.76 71.71 −5.25 54.10 0.60 Exponential 
Aug 217.50 216.06 72.55 −1.47 57.69 0.03 Exponential Aug 267.50 269.30 68.54 −3.67 52.78 0.48 Exponential 
Sep 64.55 64.03 27.91 −0.52 23.91 −0.03 Exponential Sep 65.26 64.03 26.61 −1.23 21.37 0.29 Exponential 
Oct 24.23 24.16 11.88 −0.07 9.26 −0.73 Exponential Oct 24.53 24.16 11.26 −0.37 8.70 −0.15 Exponential 
Nov 16.38 16.08 4.87 −0.29 4.18 0.38 Exponential Nov 16.44 16.08 4.77 −0.35 3.62 0.63 Exponential 
Dec 11.82 12.00 5.02 0.18 4.20 0.43 Exponential Dec 12.13 12.00 5.57 −0.14 4.71 −0.25 Exponential 
Annual 708.80 698.80 258.35 −10.04 204.18 0.25 Exponential Annual 714.40 698.80 245.72 −15.61 192.64 0.53 Exponential 

aEst. rainfall, estimated rainfall; Obr. rainfall, observed rainfall.

Table 6

Mille catchment's mean monthly and annual maximum temperature estimated and actual values, and descriptive statistics using various interpolation techniques

MonthEst. TmaxaObr. TmaxaRMSEMBEMAErV.modelMonthEst. TmaxObr. TmaxRMSEMBEMAErV.model
KED with 90 m DEM elevation KED with the combination of 90 m DEM elevation and easting as a covariate 
Jan 31.71 31.62 1.72 −0.09 1.41 0.90 Exponential Jan 31.62 31.62 1.401 0.006 1.29 0.93 Exponential 
Feb 33.45 33.47 0.67 0.02 0.61 0.98 Spherical Feb 33.45 33.47 0.706 0.024 0.625 0.9824 Spherical 
Mar 36.11 36.06 1.68 −0.05 1.50 0.94 Spherical Mar 35.92 36.06 1.734 0.143 1.542 0.9345 Spherical 
Apr 36.77 36.69 1.44 −0.08 1.13 0.97 Spherical Apr 36.77 36.69 1.269 −0.082 1.116 0.9752 Spherical 
May 37.38 37.36 0.90 −0.02 0.80 0.99 Spherical May 37.38 37.36 0.972 −0.014 0.859 0.9828 Spherical 
Jun 39.08 39.01 1.88 −0.07 1.72 0.94 Exponential Jun 39.07 39.01 1.860 −0.056 1.738 0.9426 Exponential 
Jul 38.99 38.86 2.20 −0.14 1.86 0.93 Exponential Jul 38.92 38.86 2.066 −0.059 1.763 0.9404 Exponential 
Aug 35.85 35.77 1.42 −0.08 1.13 0.97 Spherical Aug 35.89 35.77 1.452 −0.115 1.174 0.9747 Spherical 
Sep 34.32 34.38 1.64 0.06 1.24 0.96 Exponential Sep 34.55 34.38 1.779 −0.177 1.498 0.956 Exponential 
Oct 34.36 34.33 1.15 −0.03 0.99 0.98 Spherical Oct 34.43 34.33 1.348 −0.100 1.2 0.97  Spherical 
Nov 33.18 33.04 2.14 −0.14 1.82 0.92 Exponential Nov 33.06 33.04 1.762 −0.022 1.43 0.94 Spherical 
Dec 30.43 30.42 0.62 −0.02 0.52 0.99 Spherical Dec 30.44 30.42 0.595 −0.021 0.531 0.99 Spherical 
Annual 35.13 35.08 1.16 −0.05 1.03 0.97 Spherical Annual 35.11 35.08 1.150 −0.029 1.06 0.97 Spherical 
KED with easting alone as a covariate IDW 
Jan 31.84 31.62 2.92 −0.22 2.41 0.68 Exponential Jan 31.17 31.62 1.54 0.46 1.37 0.94 – 
Feb 33.55 33.47 1.76 −0.08 1.48 0.89 Exponential Feb 32.98 33.47 1.47 0.49 1.26 0.94 – 
Mar 36.25 36.06 3.18 −0.19 2.74 0.77 Exponential Mar 35.44 36.06 2.49 0.62 2.09 0.87 – 
Apr 36.99 36.69 3.76 −0.30 3.31 0.77 Exponential Apr 35.96 36.69 2.12 0.73 1.76 0.94 – 
May 37.56 37.36 2.83 −0.20 2.37 0.85 Exponential May 36.64 37.36 1.70 0.72 1.39 0.96 – 
Jun 39.34 39.01 3.78 −0.33 2.98 0.76 Exponential Jun 38.36 39.01 1.83 0.66 1.55 0.96 – 
Jul 39.18 38.86 4.04 −0.32 3.36 0.76 Exponential Jul 38.04 38.86 2.07 0.82 1.67 0.96 – 
Aug 36.10 35.77 3.98 −0.33 3.55 0.80 Exponential Aug 34.91 35.77 2.44 0.86 1.86 0.93 – 
Sep 34.54 34.38 2.74 −0.16 2.37 0.89 Exponential Sep 33.59 34.38 2.47 0.78 1.98 0.92 – 
Oct 34.61 34.33 3.24 −0.27 2.82 0.83 Exponential Oct 33.61 34.33 2.04 0.72 1.65 0.94 – 
Nov 33.32 33.04 3.82 −0.28 3.41 0.71 Exponential Nov 32.36 33.04 2.65 0.68 2.02 0.87 – 
Dec 30.57 30.42 2.30 −0.15 2.00 0.82 Exponential Dec 29.85 30.42 1.20 0.56 1.08 0.97 – 
Annual 35.32 35.08 3.09 −0.24 2.66 0.81 Exponential Annual 34.41 35.08 1.86 0.68 1.57 0.94 – 
KED with northing alone as a covariate OK 
Jan 31.59 31.62 3.93 0.04 3.82 0.11 Exponential Jan 31.47 31.62 3.55 0.16 3.40 0.61 Exponential 
Feb 33.38 33.47 3.77 0.10 3.54 0.10 Exponential Feb 33.29 33.47 3.50 0.18 3.21 0.60 Exponential 
Mar 35.91 36.06 4.89 0.15 4.64 0.11 Exponential Mar 35.84 36.06 4.47 0.22 4.14 0.55 Exponential 
Apr 36.50 36.69 5.46 0.18 5.26 0.21 Exponential Apr 36.40 36.69 4.97 0.29 4.74 0.72 Exponential 
May 37.21 37.36 5.07 0.16 4.90 0.24 Exponential May 37.09 37.36 4.63 0.28 4.43 0.74 Exponential 
Jun 38.89 39.01 5.60 0.12 5.46 0.07 Exponential Jun 38.76 39.01 5.00 0.25 4.88 0.67 Exponential 
Jul 38.70 38.86 5.91 0.15 5.74 0.23 Exponential Jul 38.55 38.86 5.28 0.31 5.13 0.77 Exponential 
Aug 35.51 35.77 6.02 0.26 5.77 0.27 Exponential Aug 35.43 35.77 5.49 0.35 5.21 0.75 Exponential 
Sep 34.10 34.38 5.62 0.28 4.89 0.42 Exponential Sep 34.04 34.38 5.32 0.33 4.55 0.64 Exponential 
Oct 34.14 34.33 5.48 0.19 5.28 0.20 Exponential Oct 34.04 34.33 5.01 0.29 4.77 0.71 Exponential 
Nov 32.94 33.04 5.16 0.10 5.00 0.22 Exponential Nov 32.79 33.04 4.74 0.25 4.43 0.66 Exponential 
Dec 30.26 30.42 3.79 0.16 3.59 0.25 Exponential Dec 30.20 30.42 3.45 0.22 3.27 0.75 Exponential 
Annual 34.92 35.08 4.96 0.16 4.82 0.21 Exponential Annual 29.85 30.42 1.20 0.56 1.08 0.97 Exponential 
MonthEst. TmaxaObr. TmaxaRMSEMBEMAErV.modelMonthEst. TmaxObr. TmaxRMSEMBEMAErV.model
KED with 90 m DEM elevation KED with the combination of 90 m DEM elevation and easting as a covariate 
Jan 31.71 31.62 1.72 −0.09 1.41 0.90 Exponential Jan 31.62 31.62 1.401 0.006 1.29 0.93 Exponential 
Feb 33.45 33.47 0.67 0.02 0.61 0.98 Spherical Feb 33.45 33.47 0.706 0.024 0.625 0.9824 Spherical 
Mar 36.11 36.06 1.68 −0.05 1.50 0.94 Spherical Mar 35.92 36.06 1.734 0.143 1.542 0.9345 Spherical 
Apr 36.77 36.69 1.44 −0.08 1.13 0.97 Spherical Apr 36.77 36.69 1.269 −0.082 1.116 0.9752 Spherical 
May 37.38 37.36 0.90 −0.02 0.80 0.99 Spherical May 37.38 37.36 0.972 −0.014 0.859 0.9828 Spherical 
Jun 39.08 39.01 1.88 −0.07 1.72 0.94 Exponential Jun 39.07 39.01 1.860 −0.056 1.738 0.9426 Exponential 
Jul 38.99 38.86 2.20 −0.14 1.86 0.93 Exponential Jul 38.92 38.86 2.066 −0.059 1.763 0.9404 Exponential 
Aug 35.85 35.77 1.42 −0.08 1.13 0.97 Spherical Aug 35.89 35.77 1.452 −0.115 1.174 0.9747 Spherical 
Sep 34.32 34.38 1.64 0.06 1.24 0.96 Exponential Sep 34.55 34.38 1.779 −0.177 1.498 0.956 Exponential 
Oct 34.36 34.33 1.15 −0.03 0.99 0.98 Spherical Oct 34.43 34.33 1.348 −0.100 1.2 0.97  Spherical 
Nov 33.18 33.04 2.14 −0.14 1.82 0.92 Exponential Nov 33.06 33.04 1.762 −0.022 1.43 0.94 Spherical 
Dec 30.43 30.42 0.62 −0.02 0.52 0.99 Spherical Dec 30.44 30.42 0.595 −0.021 0.531 0.99 Spherical 
Annual 35.13 35.08 1.16 −0.05 1.03 0.97 Spherical Annual 35.11 35.08 1.150 −0.029 1.06 0.97 Spherical 
KED with easting alone as a covariate IDW 
Jan 31.84 31.62 2.92 −0.22 2.41 0.68 Exponential Jan 31.17 31.62 1.54 0.46 1.37 0.94 – 
Feb 33.55 33.47 1.76 −0.08 1.48 0.89 Exponential Feb 32.98 33.47 1.47 0.49 1.26 0.94 – 
Mar 36.25 36.06 3.18 −0.19 2.74 0.77 Exponential Mar 35.44 36.06 2.49 0.62 2.09 0.87 – 
Apr 36.99 36.69 3.76 −0.30 3.31 0.77 Exponential Apr 35.96 36.69 2.12 0.73 1.76 0.94 – 
May 37.56 37.36 2.83 −0.20 2.37 0.85 Exponential May 36.64 37.36 1.70 0.72 1.39 0.96 – 
Jun 39.34 39.01 3.78 −0.33 2.98 0.76 Exponential Jun 38.36 39.01 1.83 0.66 1.55 0.96 – 
Jul 39.18 38.86 4.04 −0.32 3.36 0.76 Exponential Jul 38.04 38.86 2.07 0.82 1.67 0.96 – 
Aug 36.10 35.77 3.98 −0.33 3.55 0.80 Exponential Aug 34.91 35.77 2.44 0.86 1.86 0.93 – 
Sep 34.54 34.38 2.74 −0.16 2.37 0.89 Exponential Sep 33.59 34.38 2.47 0.78 1.98 0.92 – 
Oct 34.61 34.33 3.24 −0.27 2.82 0.83 Exponential Oct 33.61 34.33 2.04 0.72 1.65 0.94 – 
Nov 33.32 33.04 3.82 −0.28 3.41 0.71 Exponential Nov 32.36 33.04 2.65 0.68 2.02 0.87 – 
Dec 30.57 30.42 2.30 −0.15 2.00 0.82 Exponential Dec 29.85 30.42 1.20 0.56 1.08 0.97 – 
Annual 35.32 35.08 3.09 −0.24 2.66 0.81 Exponential Annual 34.41 35.08 1.86 0.68 1.57 0.94 – 
KED with northing alone as a covariate OK 
Jan 31.59 31.62 3.93 0.04 3.82 0.11 Exponential Jan 31.47 31.62 3.55 0.16 3.40 0.61 Exponential 
Feb 33.38 33.47 3.77 0.10 3.54 0.10 Exponential Feb 33.29 33.47 3.50 0.18 3.21 0.60 Exponential 
Mar 35.91 36.06 4.89 0.15 4.64 0.11 Exponential Mar 35.84 36.06 4.47 0.22 4.14 0.55 Exponential 
Apr 36.50 36.69 5.46 0.18 5.26 0.21 Exponential Apr 36.40 36.69 4.97 0.29 4.74 0.72 Exponential 
May 37.21 37.36 5.07 0.16 4.90 0.24 Exponential May 37.09 37.36 4.63 0.28 4.43 0.74 Exponential 
Jun 38.89 39.01 5.60 0.12 5.46 0.07 Exponential Jun 38.76 39.01 5.00 0.25 4.88 0.67 Exponential 
Jul 38.70 38.86 5.91 0.15 5.74 0.23 Exponential Jul 38.55 38.86 5.28 0.31 5.13 0.77 Exponential 
Aug 35.51 35.77 6.02 0.26 5.77 0.27 Exponential Aug 35.43 35.77 5.49 0.35 5.21 0.75 Exponential 
Sep 34.10 34.38 5.62 0.28 4.89 0.42 Exponential Sep 34.04 34.38 5.32 0.33 4.55 0.64 Exponential 
Oct 34.14 34.33 5.48 0.19 5.28 0.20 Exponential Oct 34.04 34.33 5.01 0.29 4.77 0.71 Exponential 
Nov 32.94 33.04 5.16 0.10 5.00 0.22 Exponential Nov 32.79 33.04 4.74 0.25 4.43 0.66 Exponential 
Dec 30.26 30.42 3.79 0.16 3.59 0.25 Exponential Dec 30.20 30.42 3.45 0.22 3.27 0.75 Exponential 
Annual 34.92 35.08 4.96 0.16 4.82 0.21 Exponential Annual 29.85 30.42 1.20 0.56 1.08 0.97 Exponential 

For example, in the case of the August month rainfall, some interpolation techniques with and without predictors, for example, KED using northing as a predictor and OK, both show relatively higher statistical indicators than the rest of the interpolation methods. In contrast, IDW shows higher performance with the lowest statistical indicators than KED with elevation/northing as the covariate.

The combination of predictors, for instance easting and elevation used by KED, significantly improves the prediction of sampled mean monthly and annual rainfall data. Therefore, KED with the combination of two predictors, elevation and easting (Table 5), seems to be the optimum method to predict mean monthly climatic data (e.g., August: RMSE = 20.66, r = 0.96, and mean annual rainfall: RMSE = 75.62, MBE = 3.61, r = 0.96). However, the KED with predictors, i.e. northing and OK, shows the worst results (RMSE = 258.35, MAE = 204.18, and RMSE = 245.72, MAE = 192.64), respectively.

Accordingly, based on the statistical evaluation results, KED with the combination of elevation and easting as a covariate was selected as the optimum spatial interpolation technique for Mille's catchment area mean rainfall estimation.

Similarly, Table 6 illustrates the performance of the different prediction methods for estimating the monthly and annual maximum temperatures for the Mille catchment in terms of statistical indicators, namely RMSE, MBE, MAE, and r. The smaller the values of statistical parameters (and the higher the r-value), the better the predictor/s with the corresponding interpolation techniques (Adhikary et al. 2017). As a result, Table 6 presents the performance of the various prediction techniques for the mean annual and monthly maximum temperatures for 12 months. Based on the estimated results, there was little difference between KED with the combination of elevation and easting as a covariate and KED with elevation alone as a covariate followed by IDW, and these interpolation techniques performed better than the rest of the methods for spatial prediction and mean monthly and mean annual maximum temperature estimation. Nevertheless, KED with the combination of easting and elevation (see Table 6) seems to be the optimum technique to estimate the mean monthly (e.g., April: RMSE = 1.27, r = 0.98) and mean annual maximum temperature (RMSE = 1.15, r = 0.97). Therefore, it was selected for monthly and annual mean maximum temperature spatial prediction as an optimum interpolation technique. For the maximum temperature and mean rainfall, KED with easting and elevation as covariates seems to be the most suitable technique to predict the mean minimum temperature, since, for most months, the statistical parameters were the lowest with the aforementioned method.

Figures 7 and 8 are a map of different interpolation techniques with and without covariates for August rainfall and the mean annual maximum temperature of the Mille catchment to visualize the spatial pattern of predicted mean rainfall and mean maximum temperature, and the maps show the fundamental differences between the various interpolation approaches.
Figure 7

Spatial maps for the August mean rainfall (mm) pattern using OK (a), IDW (b), KED with elevation (c), KED with easting (d), KED with northing (e), and KED with the combination of elevation and easting (f) by the interpolation of nine sampled observations.

Figure 7

Spatial maps for the August mean rainfall (mm) pattern using OK (a), IDW (b), KED with elevation (c), KED with easting (d), KED with northing (e), and KED with the combination of elevation and easting (f) by the interpolation of nine sampled observations.

Close modal
Figure 8

Spatial maps for the annual mean maximum temperature (°C) pattern using IDW (a), OK (b), KED with elevation (c), KED with northing (d), KED with easting (e), and KED with the combination of elevation and easting (f) by the interpolation of nine sampled observations.

Figure 8

Spatial maps for the annual mean maximum temperature (°C) pattern using IDW (a), OK (b), KED with elevation (c), KED with northing (d), KED with easting (e), and KED with the combination of elevation and easting (f) by the interpolation of nine sampled observations.

Close modal

For the Akaki catchment, Tables 7 and 8 present the performance of different interpolation methods for predicting both mean monthly and mean annual climatic variables. These interpolation techniques were quantitatively compared based on evaluation performance scores to identify a suitable method for the spatial prediction of climatic variables at the catchment scale. As depicted in Table 7, overall, KED with northing as a covariate and KED with easting as a covariate perform relatively better than the rest of the interpolation techniques, specifically OK, IDW, and KED, with elevation as a covariate for mean monthly rainfall estimation. For mean annual rainfall, KED with elevation as a covariate performs better than the rest of the interpolation methods (e.g., RMSE = 40.28 mm, MBE = −10.43 mm, and MAE = 33.54 mm). Nevertheless, KED with easting as a covariate shows a high RMSE during the rainy season (June up to Sep). Therefore, based on the overall results shown in Table 7 and considering the rainy season, KED with northing as a covariate was selected as a suitable method for monthly interpolated rainfall datasets, and KED with elevation as a covariate was proven to be the best interpolation technique (in terms of errors but not r) selected for interpolating annual mean rainfall.

Table 7

Akaki catchment long year-based predicted vs. observed mean monthly and annual rainfall using various interpolation techniques

MonthEst. rainfallObr. rainfallRMSEMBEMAErV.modelMonthEst. rainfallObr. rainfallRMSEMBEMAErV.model
KED with 90 m DEM elevation IDW 
Jan 6.650 6.652 0.99 0.002 0.81 0.59 Exponential Jan 6.85 6.65 1.17 −0.19 0.94 0.31 – 
Feb 9.58 9.14 2.48 −0.44 1.93 0.46 Exponential Feb 9.94 9.14 2.05 −0.80 1.60 0.66 – 
Mar 34.73 34.14 2.95 −0.59 2.78 0.54 Ste Mar 35.65 34.14 4.07 −1.51 2.94 0.54 – 
Apr 55.09 55.78 3.92 0.69 1.94 0.47 Ste Apr 55.35 55.78 2.41 0.43 1.90 0.67 – 
May 66.20 64.89 6.99 −1.31 4.68 0.31 Ste May 67.89 64.89 7.60 −3.00 4.61 0.22 – 
Jun 114.60 113.50 4.47 −1.08 3.74 0.80 Ste Jun 115.40 113.50 5.72 −1.93 4.75 0.65 – 
Jul 249.10 248.60 17.49 −0.49 11.84 0.52 Ste Jul 248.30 248.60 17.55 0.35 14.22 0.36 – 
Aug 267.00 265.40 23.48 −1.57 16.48 0.46 Ste Aug 267.80 265.40 25.60 −2.39 20.26 0.23 – 
Sep 128.50 123.90 10.51 −4.56 8.27 0.74 Ste Sep 130.40 123.90 11.51 −6.45 9.83 0.75 – 
Oct 16.77 15.86 4.01 −0.91 2.73 0.50 Ste Oct 17.96 15.86 4.32 −2.10 3.17 0.62 – 
Nov 6.60 6.08 1.16 −0.52 0.78 0.69 Ste Nov 6.74 6.08 1.24 −0.66 0.98 0.77 – 
Dec 5.73 5.69 1.26 −0.05 1.05 0.15 Ste Dec 5.96 5.69 1.14 −0.27 0.84 0.34 – 
Annual 960.10 949.70 40.28 −10.43 33.54 0.65 Ste Annual 968.20 949.70 48.20 −18.53 41.72 0.50 – 
KED with easting alone as a covariate OK 
Jan 6.84 6.65 1.48 −0.19 1.01 −0.42 Exponential Jan 6.70 6.65 1.29 −0.05 0.95 −0.315 Exponential 
Feb 9.48 9.14 1.70 −0.34 1.45 0.76 Ste Feb 9.78 9.14 2.16 −0.64 1.67 0.606 Ste 
Mar 33.95 34.14 2.28 0.19 1.88 0.86 Ste Mar 34.75 34.14 3.82 −0.61 2.67 0.544 Ste 
Apr 55.96 55.78 3.10 −0.18 2.13 0.53 Ste Apr 55.18 55.78 2.38 0.60 1.58 0.687 Ste 
May 64.84 64.89 5.18 0.05 3.32 0.69 Ste May 66.29 64.89 7.00 −1.40 4.66 0.312 Ste 
Jun 113.50 113.50 5.98 −0.05 5.00 0.56 Exponential Jun 113.80 113.50 6.41 −0.30 5.50 0.518 Exponential 
Jul 244.40 248.60 25.23 4.27 19.11 −0.59 Ste Jul 248.40 248.60 20.16 0.22 16.23 −0.702 Exponential 
Aug 259.90 265.40 49.75 5.49 30.66 −0.56 Ste Aug 266.30 265.40 28.40 −0.91 22.06 −0.509 Exponential 
Sep 121.81 123.90 12.24 2.10 8.79 0.75 Exponential Sep 127.70 123.90 8.41 −3.80 6.51 0.865 Ste 
Oct 15.81 15.86 1.85 0.06 1.56 0.92 Ste Oct 15.87 15.86 4.99 −0.01 4.08 −0.997 Ste 
Nov 5.98 6.08 1.58 0.10 1.07 0.61 Ste Nov 6.58 6.08 1.05 −0.50 0.87 0.85 Ste 
Dec 5.70 5.69 0.96 −0.01 0.71 0.58 Ste Dec 5.77 5.69 0.95 −0.08 0.65 0.60 Ste 
Annual 940.50 949.70 91.14 9.13 61.53 0.07 Ste Annual 952.80 949.70 51.53 −3.181 42.24 0.07 Exponential 
KED with northing alone as a covariate KED with the combination 90 m DEM elevation and northing as a covariate 
Jan 6.71 6.65 0.73 −0.05 0.59 0.79 Spherical Jan 6.85 6.65 1.07 −0.20 0.83 0.52 Ste 
Feb 10.17 9.14 4.59 −1.03 2.79 −0.01 Ste Feb 10.17 9.14 3.43 −1.04 2.33 0.08 Ste 
Mar 35.38 34.14 4.88 −1.24 3.12 0.24 Ste Mar 36.46 34.14 8.22 −2.32 4.66 −0.34 Ste 
Apr 55.31 55.78 2.36 0.48 1.63 0.69 Ste Apr 55.51 55.78 3.68 0.27 2.38 0.35 Ste 
May 67.79 64.89 10.16 −2.90 6.23 −0.53 Ste May 69.39 64.89 13.82 −4.50 8.02 −0.82 Ste 
Jun 114.40 113.50 8.22 −0.93 6.19 0.15 Exponential Jun 114.70 113.50 3.69 −1.23 3.19 0.88 Ste 
Jul 249.10 248.60 14.14 −0.46 12.37 0.69 Ste Jul 246.70 248.60 23.78 1.98 17.88 0.18 Ste 
Aug 268.60 265.40 11.35 −3.25 8.88 0.92 Ste Aug 268.10 265.40 20.61 −2.69 15.95 0.62 Ste 
Sep 129.70 123.90 5.16 −5.81 8.77 0.55 Ste Sep 131.90 123.90 16.51 −7.97 11.28 0.20 Ste 
Oct 17.55 15.86 5.16 −1.69 3.14 0.17 Ste Oct 18.73 15.86 8.31 −2.87 4.47 −0.47 Ste 
Nov 6.82 6.08 1.44 −0.73 1.10 0.50 Ste Nov 6.61 6.08 1.37 −0.53 0.84 0.51 Ste 
Dec 5.95 5.69 1.40 −0.26 1.07 0.08 Ste Dec 6.17 5.69 2.20 −0.48 1.54 −0.13 Ste 
Annual 967.10 949.70 41.78 −17.41 35.59 0.67 Ste Annual 969.80 949.70 47.72 −20.10 40.51 0.57 Ste 
MonthEst. rainfallObr. rainfallRMSEMBEMAErV.modelMonthEst. rainfallObr. rainfallRMSEMBEMAErV.model
KED with 90 m DEM elevation IDW 
Jan 6.650 6.652 0.99 0.002 0.81 0.59 Exponential Jan 6.85 6.65 1.17 −0.19 0.94 0.31 – 
Feb 9.58 9.14 2.48 −0.44 1.93 0.46 Exponential Feb 9.94 9.14 2.05 −0.80 1.60 0.66 – 
Mar 34.73 34.14 2.95 −0.59 2.78 0.54 Ste Mar 35.65 34.14 4.07 −1.51 2.94 0.54 – 
Apr 55.09 55.78 3.92 0.69 1.94 0.47 Ste Apr 55.35 55.78 2.41 0.43 1.90 0.67 – 
May 66.20 64.89 6.99 −1.31 4.68 0.31 Ste May 67.89 64.89 7.60 −3.00 4.61 0.22 – 
Jun 114.60 113.50 4.47 −1.08 3.74 0.80 Ste Jun 115.40 113.50 5.72 −1.93 4.75 0.65 – 
Jul 249.10 248.60 17.49 −0.49 11.84 0.52 Ste Jul 248.30 248.60 17.55 0.35 14.22 0.36 – 
Aug 267.00 265.40 23.48 −1.57 16.48 0.46 Ste Aug 267.80 265.40 25.60 −2.39 20.26 0.23 – 
Sep 128.50 123.90 10.51 −4.56 8.27 0.74 Ste Sep 130.40 123.90 11.51 −6.45 9.83 0.75 – 
Oct 16.77 15.86 4.01 −0.91 2.73 0.50 Ste Oct 17.96 15.86 4.32 −2.10 3.17 0.62 – 
Nov 6.60 6.08 1.16 −0.52 0.78 0.69 Ste Nov 6.74 6.08 1.24 −0.66 0.98 0.77 – 
Dec 5.73 5.69 1.26 −0.05 1.05 0.15 Ste Dec 5.96 5.69 1.14 −0.27 0.84 0.34 – 
Annual 960.10 949.70 40.28 −10.43 33.54 0.65 Ste Annual 968.20 949.70 48.20 −18.53 41.72 0.50 – 
KED with easting alone as a covariate OK 
Jan 6.84 6.65 1.48 −0.19 1.01 −0.42 Exponential Jan 6.70 6.65 1.29 −0.05 0.95 −0.315 Exponential 
Feb 9.48 9.14 1.70 −0.34 1.45 0.76 Ste Feb 9.78 9.14 2.16 −0.64 1.67 0.606 Ste 
Mar 33.95 34.14 2.28 0.19 1.88 0.86 Ste Mar 34.75 34.14 3.82 −0.61 2.67 0.544 Ste 
Apr 55.96 55.78 3.10 −0.18 2.13 0.53 Ste Apr 55.18 55.78 2.38 0.60 1.58 0.687 Ste 
May 64.84 64.89 5.18 0.05 3.32 0.69 Ste May 66.29 64.89 7.00 −1.40 4.66 0.312 Ste 
Jun 113.50 113.50 5.98 −0.05 5.00 0.56 Exponential Jun 113.80 113.50 6.41 −0.30 5.50 0.518 Exponential 
Jul 244.40 248.60 25.23 4.27 19.11 −0.59 Ste Jul 248.40 248.60 20.16 0.22 16.23 −0.702 Exponential 
Aug 259.90 265.40 49.75 5.49 30.66 −0.56 Ste Aug 266.30 265.40 28.40 −0.91 22.06 −0.509 Exponential 
Sep 121.81 123.90 12.24 2.10 8.79 0.75 Exponential Sep 127.70 123.90 8.41 −3.80 6.51 0.865 Ste 
Oct 15.81 15.86 1.85 0.06 1.56 0.92 Ste Oct 15.87 15.86 4.99 −0.01 4.08 −0.997 Ste 
Nov 5.98 6.08 1.58 0.10 1.07 0.61 Ste Nov 6.58 6.08 1.05 −0.50 0.87 0.85 Ste 
Dec 5.70 5.69 0.96 −0.01 0.71 0.58 Ste Dec 5.77 5.69 0.95 −0.08 0.65 0.60 Ste 
Annual 940.50 949.70 91.14 9.13 61.53 0.07 Ste Annual 952.80 949.70 51.53 −3.181 42.24 0.07 Exponential 
KED with northing alone as a covariate KED with the combination 90 m DEM elevation and northing as a covariate 
Jan 6.71 6.65 0.73 −0.05 0.59 0.79 Spherical Jan 6.85 6.65 1.07 −0.20 0.83 0.52 Ste 
Feb 10.17 9.14 4.59 −1.03 2.79 −0.01 Ste Feb 10.17 9.14 3.43 −1.04 2.33 0.08 Ste 
Mar 35.38 34.14 4.88 −1.24 3.12 0.24 Ste Mar 36.46 34.14 8.22 −2.32 4.66 −0.34 Ste 
Apr 55.31 55.78 2.36 0.48 1.63 0.69 Ste Apr 55.51 55.78 3.68 0.27 2.38 0.35 Ste 
May 67.79 64.89 10.16 −2.90 6.23 −0.53 Ste May 69.39 64.89 13.82 −4.50 8.02 −0.82 Ste 
Jun 114.40 113.50 8.22 −0.93 6.19 0.15 Exponential Jun 114.70 113.50 3.69 −1.23 3.19 0.88 Ste 
Jul 249.10 248.60 14.14 −0.46 12.37 0.69 Ste Jul 246.70 248.60 23.78 1.98 17.88 0.18 Ste 
Aug 268.60 265.40 11.35 −3.25 8.88 0.92 Ste Aug 268.10 265.40 20.61 −2.69 15.95 0.62 Ste 
Sep 129.70 123.90 5.16 −5.81 8.77 0.55 Ste Sep 131.90 123.90 16.51 −7.97 11.28 0.20 Ste 
Oct 17.55 15.86 5.16 −1.69 3.14 0.17 Ste Oct 18.73 15.86 8.31 −2.87 4.47 −0.47 Ste 
Nov 6.82 6.08 1.44 −0.73 1.10 0.50 Ste Nov 6.61 6.08 1.37 −0.53 0.84 0.51 Ste 
Dec 5.95 5.69 1.40 −0.26 1.07 0.08 Ste Dec 6.17 5.69 2.20 −0.48 1.54 −0.13 Ste 
Annual 967.10 949.70 41.78 −17.41 35.59 0.67 Ste Annual 969.80 949.70 47.72 −20.10 40.51 0.57 Ste 
Table 8

Akaki catchment long year-based predicted vs. observed mean monthly and annual maximum temperature using various interpolation techniques

MonthEst. TmaxObr. TmaxRMSEMBEMAErV.modelMonthEst. TmaxObr. TmaxRMSEMBEMAErV.model
KED with 90 m DEM elevation IDW 
Jan 23.56 23.68 0.83 0.1250 0.69 0.80 Ste Jan 23.56 23.68 1.04 0.126 0.89 0.68 – 
Feb 24.83 24.96 0.83 0.13 0.69 0.81 Ste Feb 24.82 24.96 1.08 0.14 0.92 0.68 – 
Mar 25.03 25.17 0.94 0.1415 0.77 0.76 Ste Mar 25.03 25.17 1.09 0.1453 0.93 0.67 – 
Apr 24.53 24.70 0.95 0.16 0.76 0.75 Ste Apr 24.49 24.70 1.15 0.20 0.95 0.64 – 
May 24.56 24.77 1.10 0.22 0.84 0.70 Ste May 24.50 24.77 1.20 0.27 0.95 0.65 – 
Jun 22.91 23.17 0.89 0.26 0.67 0.82 Ste Jun 22.79 23.17 1.23 0.38 0.98 0.67 – 
Jul 20.75 20.99 0.89 0.233 0.67 0.80 Ste Jul 20.66 20.99 1.21 0.32 0.98 0.63 – 
Aug 20.30 20.51 0.79 0.21 0.61 0.83 Ste Aug 20.23 20.51 1.12 0.28 0.92 0.66 – 
Sep 21.49 21.69 0.84 0.19 0.63 0.83 Ste Sep 21.46 21.69 1.13 0.22 0.89 0.70 – 
Oct 22.74 22.89 0.93 0.15 0.78 0.80 Ste Oct 22.74 22.89 1.16 0.15 1.00 0.67 – 
Nov 23.02 23.15 0.86 0.13 0.71 0.81 Ste Nov 23.05 23.15 1.06 0.10 0.10 0.69 – 
Dec 22.77 22.86 0.78 0.10 0.64 0.79 Ste Dec 22.77 22.86 0.99 0.10 0.86 0.67 – 
Annual 23.03 23.20 0.87 0.17 0.69 0.80 Ste Annual 22.75 22.81 0.92 0.06 0.73 0.67 – 
KED with easting as a covariate OK 
Jan 23.84 23.68 1.56 −0.15 1.29 −0.25 Exponential Jan 23.677 23.684 1.31 0.01 1.11 0.120 Exponential 
Feb 25.13 24.96 1.62 −0.17 1.34 −0.26 Exponential Feb 24.96 24.97 1.36 0.01 1.14 0.101 Exponential 
Mar 25.35 25.17 1.66 −0.18 1.36 −0.29 Exponential Mar 25.16 25.17 1.37 0.01 1.15 0.044 Exponential 
Apr 24.89 24.70 1.73 −0.19 1.41 −0.25 Exponential Apr 24.68 24.70 1.41 0.02 1.16 −0.035 Exponential 
May 24.98 24.77 1.83 −0.21 1.46 −0.15 Exponential May 24.74 24.77 1.47 0.03 1.17 −0.062 Exponential 
Jun 23.34 23.17 1.71 −0.17 1.38 0.08 Exponential Jun 23.12 23.17 1.46 0.05 1.09 0.094 Exponential 
Jul 21.16 20.99 1.70 −0.17 1.40 −0.004 Exponential Jul 20.95 20.99 1.45 0.04 1.12 −0.052 Exponential 
Aug 20.64 20.51 1.54 −0.13 1.30 0.004 Exponential Aug 20.47 20.51 1.37 0.04 1.08 −0.0002 Exponential 
Sep 21.81 21.69 1.60 −0.12 1.34 −0.06 Exponential Sep 21.66 21.69 1.42 0.03 1.14 0.112 Exponential 
Oct 23.08 22.89 1.77 −0.19 1.47 −0.28 Exponential Oct 22.88 22.89 1.47 0.01 1.25 0.051 Exponential 
Nov 23.33 23.15 1.64 −0.17 1.34 −0.29 Spherical Nov 23.15 23.15 1.36 0.001 1.14 0.14 Exponential 
Dec 23.04 22.86 1.54 −0.18 1.25 −0.35 Exponential Dec 22.86 22.86 1.25 0.001 1.06 0.05 Exponential 
Annual 23.37 23.20 1.65 −0.17 1.36 −0.18 Exponential Annual 23.18 23.20 1.38 0.022 1.13 0.05 Exponential 
KED with northing as a covariate KED with the combination of northing and 90 m DEM elevation as a covariate 
Jan 23.60 23.68 0.95 0.08 0.88 0.71 Exponential Jan 23.37 23.68 0.85 0.32 0.61 0.86 Ste 
Feb 24.88 24.96 1.01 0.08 0.94 0.69 Exponential Feb 24.63 24.96 0.88 0.34 0.64 0.85 Ste 
Mar 25.07 25.17 0.99 0.10 0.90 0.70 Exponential Mar 24.8 25.17 0.997 0.37 0.71 0.82 Ste 
Apr 24.54 24.70 1.08 0.15 0.93 0.66 Exponential Apr 24.24 24.7 1.22 0.46 0.82 0.75 Ste 
May 24.60 24.77 1.27 0.17 1.07 0.54 Exponential May 24.18 24.77 1.56 0.60 1.02 0.64 Ste 
Jun 23.03 23.17 1.59 0.14 1.34 0.27 Exponential Jun 22.52 23.17 1.62 0.65 1.03 0.57 Ste 
Jul 20.71 20.99 1.39 0.28 1.07 0.44 Exponential Jul 20.36 20.99 1.62 0.63 1.03 0.580 Ste 
Aug 20.21 20.51 1.29 0.30 0.97 0.49 Ste Aug 19.93 20.51 1.52 0.58 0.98 0.59 Ste 
Sep 21.58 21.69 1.36 0.10 1.24 0.47 Ste Sep 21.12 21.69 1.44 0.57 0.96 0.68 Ste 
Oct 22.75 22.89 1.03 0.13 0.93 0.72 Exponential Oct 22.52 22.89 0.97 0.37 0.72 0.85 Ste 
Nov 23.01 23.15 0.79 0.14 0.68 0.82 Ste Nov 22.91 23.15 0.62 0.244 0.50 0.92 Ste 
Dec 22.81 22.86 0.84 0.06 0.79 0.75 Exponential Dec 22.62 22.86 0.62 0.24 0.46 0.91 Ste 
Annual 23.09 23.20 1.16 0.11 1.04 0.59 Exponential Annual 22.76 23.2 1.16 0.45 0.79 0.76 Ste 
MonthEst. TmaxObr. TmaxRMSEMBEMAErV.modelMonthEst. TmaxObr. TmaxRMSEMBEMAErV.model
KED with 90 m DEM elevation IDW 
Jan 23.56 23.68 0.83 0.1250 0.69 0.80 Ste Jan 23.56 23.68 1.04 0.126 0.89 0.68 – 
Feb 24.83 24.96 0.83 0.13 0.69 0.81 Ste Feb 24.82 24.96 1.08 0.14 0.92 0.68 – 
Mar 25.03 25.17 0.94 0.1415 0.77 0.76 Ste Mar 25.03 25.17 1.09 0.1453 0.93 0.67 – 
Apr 24.53 24.70 0.95 0.16 0.76 0.75 Ste Apr 24.49 24.70 1.15 0.20 0.95 0.64 – 
May 24.56 24.77 1.10 0.22 0.84 0.70 Ste May 24.50 24.77 1.20 0.27 0.95 0.65 – 
Jun 22.91 23.17 0.89 0.26 0.67 0.82 Ste Jun 22.79 23.17 1.23 0.38 0.98 0.67 – 
Jul 20.75 20.99 0.89 0.233 0.67 0.80 Ste Jul 20.66 20.99 1.21 0.32 0.98 0.63 – 
Aug 20.30 20.51 0.79 0.21 0.61 0.83 Ste Aug 20.23 20.51 1.12 0.28 0.92 0.66 – 
Sep 21.49 21.69 0.84 0.19 0.63 0.83 Ste Sep 21.46 21.69 1.13 0.22 0.89 0.70 – 
Oct 22.74 22.89 0.93 0.15 0.78 0.80 Ste Oct 22.74 22.89 1.16 0.15 1.00 0.67 – 
Nov 23.02 23.15 0.86 0.13 0.71 0.81 Ste Nov 23.05 23.15 1.06 0.10 0.10 0.69 – 
Dec 22.77 22.86 0.78 0.10 0.64 0.79 Ste Dec 22.77 22.86 0.99 0.10 0.86 0.67 – 
Annual 23.03 23.20 0.87 0.17 0.69 0.80 Ste Annual 22.75 22.81 0.92 0.06 0.73 0.67 – 
KED with easting as a covariate OK 
Jan 23.84 23.68 1.56 −0.15 1.29 −0.25 Exponential Jan 23.677 23.684 1.31 0.01 1.11 0.120 Exponential 
Feb 25.13 24.96 1.62 −0.17 1.34 −0.26 Exponential Feb 24.96 24.97 1.36 0.01 1.14 0.101 Exponential 
Mar 25.35 25.17 1.66 −0.18 1.36 −0.29 Exponential Mar 25.16 25.17 1.37 0.01 1.15 0.044 Exponential 
Apr 24.89 24.70 1.73 −0.19 1.41 −0.25 Exponential Apr 24.68 24.70 1.41 0.02 1.16 −0.035 Exponential 
May 24.98 24.77 1.83 −0.21 1.46 −0.15 Exponential May 24.74 24.77 1.47 0.03 1.17 −0.062 Exponential 
Jun 23.34 23.17 1.71 −0.17 1.38 0.08 Exponential Jun 23.12 23.17 1.46 0.05 1.09 0.094 Exponential 
Jul 21.16 20.99 1.70 −0.17 1.40 −0.004 Exponential Jul 20.95 20.99 1.45 0.04 1.12 −0.052 Exponential 
Aug 20.64 20.51 1.54 −0.13 1.30 0.004 Exponential Aug 20.47 20.51 1.37 0.04 1.08 −0.0002 Exponential 
Sep 21.81 21.69 1.60 −0.12 1.34 −0.06 Exponential Sep 21.66 21.69 1.42 0.03 1.14 0.112 Exponential 
Oct 23.08 22.89 1.77 −0.19 1.47 −0.28 Exponential Oct 22.88 22.89 1.47 0.01 1.25 0.051 Exponential 
Nov 23.33 23.15 1.64 −0.17 1.34 −0.29 Spherical Nov 23.15 23.15 1.36 0.001 1.14 0.14 Exponential 
Dec 23.04 22.86 1.54 −0.18 1.25 −0.35 Exponential Dec 22.86 22.86 1.25 0.001 1.06 0.05 Exponential 
Annual 23.37 23.20 1.65 −0.17 1.36 −0.18 Exponential Annual 23.18 23.20 1.38 0.022 1.13 0.05 Exponential 
KED with northing as a covariate KED with the combination of northing and 90 m DEM elevation as a covariate 
Jan 23.60 23.68 0.95 0.08 0.88 0.71 Exponential Jan 23.37 23.68 0.85 0.32 0.61 0.86 Ste 
Feb 24.88 24.96 1.01 0.08 0.94 0.69 Exponential Feb 24.63 24.96 0.88 0.34 0.64 0.85 Ste 
Mar 25.07 25.17 0.99 0.10 0.90 0.70 Exponential Mar 24.8 25.17 0.997 0.37 0.71 0.82 Ste 
Apr 24.54 24.70 1.08 0.15 0.93 0.66 Exponential Apr 24.24 24.7 1.22 0.46 0.82 0.75 Ste 
May 24.60 24.77 1.27 0.17 1.07 0.54 Exponential May 24.18 24.77 1.56 0.60 1.02 0.64 Ste 
Jun 23.03 23.17 1.59 0.14 1.34 0.27 Exponential Jun 22.52 23.17 1.62 0.65 1.03 0.57 Ste 
Jul 20.71 20.99 1.39 0.28 1.07 0.44 Exponential Jul 20.36 20.99 1.62 0.63 1.03 0.580 Ste 
Aug 20.21 20.51 1.29 0.30 0.97 0.49 Ste Aug 19.93 20.51 1.52 0.58 0.98 0.59 Ste 
Sep 21.58 21.69 1.36 0.10 1.24 0.47 Ste Sep 21.12 21.69 1.44 0.57 0.96 0.68 Ste 
Oct 22.75 22.89 1.03 0.13 0.93 0.72 Exponential Oct 22.52 22.89 0.97 0.37 0.72 0.85 Ste 
Nov 23.01 23.15 0.79 0.14 0.68 0.82 Ste Nov 22.91 23.15 0.62 0.244 0.50 0.92 Ste 
Dec 22.81 22.86 0.84 0.06 0.79 0.75 Exponential Dec 22.62 22.86 0.62 0.24 0.46 0.91 Ste 
Annual 23.09 23.20 1.16 0.11 1.04 0.59 Exponential Annual 22.76 23.2 1.16 0.45 0.79 0.76 Ste 

Table 8 depicts the descriptive statistical performance of the KED with various descriptors as covariates, IDW, and OK in terms of RMSE, MBE, MAE, and r for both the monthly and mean annual maximum temperature datasets. Based on the results shown in Table 8, KED with the combination of elevation and northing as a predictor shows the lowest statistical errors and the highest r-value compared to the rest of the interpolation techniques for most months and annually, followed by KED with elevation as a predictor. KED with easting as a covariate and OK relatively score the highest errors: RMSE, MBE, and MAE on spatial prediction and mean monthly and annual maximum temperature estimation. As a result, KED with the combination of elevation and northing as a predictor was proven to be used as a suitable technique for the spatial prediction of the mean maximum temperature and the mean minimum temperature. The map shows that the spatial pattern of August mean rainfall, and April mean maximum temperature interpolated with various interpolation techniques, are displayed in Figures 9 and 10, respectively.
Figure 9

Spatial map of 17 years August mean rainfall by (a) IDW, (b) KED 90 m DEM elevation, (c) KED with easting, (d) KED with northing, (e) KED with the combination of 90 m DEM and northing, and (f) OK.

Figure 9

Spatial map of 17 years August mean rainfall by (a) IDW, (b) KED 90 m DEM elevation, (c) KED with easting, (d) KED with northing, (e) KED with the combination of 90 m DEM and northing, and (f) OK.

Close modal
Figure 10

Spatial map of 17 years April mean maximum temperature by (a) IDW, (b) KED with 90 m DEM elevation, (c) KED with easting, (d) KED with the combination of 90 m DEM elevation and northing, (e) KED with northing, and (f) OK.

Figure 10

Spatial map of 17 years April mean maximum temperature by (a) IDW, (b) KED with 90 m DEM elevation, (c) KED with easting, (d) KED with the combination of 90 m DEM elevation and northing, (e) KED with northing, and (f) OK.

Close modal
Figures 11 and 12 show the box plots of mean monthly rainfall using KED with northing as a predictor and mean minimum and maximum temperatures using KED with the combination of elevation and northing as a covariate for the Akaki catchment and for the Mille catchment using KED with a combination of elevation and easting as a covariate, respectively.
Figure 11

Box plots for mean rainfall, Tmin, and Tmax for the Akaki catchment, obtained with KED.

Figure 11

Box plots for mean rainfall, Tmin, and Tmax for the Akaki catchment, obtained with KED.

Close modal
Figure 12

Box plots for mean rainfall, Tmin, and Tmax for the Mille catchment, obtained with KED.

Figure 12

Box plots for mean rainfall, Tmin, and Tmax for the Mille catchment, obtained with KED.

Close modal

As observed from Figures 11 and 12 box plots, much higher scales plotting of climatic variable data were observed for the Mille catchment than the Akaki catchment, and the reason behind this was that both the estimated and actual climatic data were generally highly varying in the spatial pattern than Akaki's climatic variables. The bimodal nature of the rainfall pattern was boldly observed in the Mille catchment, which was higher in the months of April and August than in the Akaki catchment. The maximum temperature was picked from February to May for the Akaki catchment, whereas it was picked approximately from March to July for the Mille catchment (see Figures 11 and 12). From careful inspection and prior knowledge of the authors, the KED with the combination of elevation and easting/northing as a covariate was generally acceptable in its predictive accuracy.

Spatial pattern of rainfall

Based on descriptive statistical evaluation parameters that the authors made (see subsection 3.2.), suitable interpolation techniques were selected for each spatial prediction of mean monthly and annual rainfall for Akaki and Mille catchments, respectively. The IDW and OK approaches resulted in the most constant zonal pattern of rainfall covering regions of different spatial elevations. In contrast, the KED method produced a spatial rainfall distribution that was closely related to the topography, and this was a result of the use of some external variables, namely elevation, easting, and northing, as the predictors (see Figures 7 and 9). The complex topographic relief influence on rainfall spatial distribution was more highly observed in the Mille catchment than in the Akaki catchment, and the reason for the latter catchment was that the catchment is comparatively flat but includes some higher elevation locations and areas such as the ‘Intoto’ mountain. Consequently, KED with various descriptors as a covariate (mentioned in Tables 5 and 7) was selected and performed to map the spatial mean monthly and annual rainfall.

The maps in Figures 7 and 9 depict that the spatial rainfall patterns and trends following the catchment's elevation trend were generally acceptable in their prediction reliability and accuracy. For instance, our obtained results showed that the spatial patterns by KED with northing as a covariate interpolated the mean monthly and annual rainfall gradually increase from south to north of the Akaki catchment (see Figure 9(d)) following the elevation increments, and for the Mille catchment from the east to the west (see Figure 7(f)) as catchment's elevation gradually increased using KED with the combination of elevation and easting as a covariate. It is noted that the weakest correlation is between Mille's January rainfall and elevation because of the stronger correlations with the longitudinal location in summer.

The study by Goovaerts (2000) found that rainfall depths generally vary spatiotemporally and tend to increase with increasing elevations because of the orographic effect of mountainous terrain, which causes warm air to ascend vertically, and condensation occurs due to the adiabatic cooling effect. Havesi (1991) also discovered that there is a significant correlation between a natural log of average annual precipitation (AAP) and station elevation using a cokriging method with 62 rainfall stations in Nevada and southeastern California, which is similar to our obtained research findings, specifically for the Mille catchment. However, unlike the mille catchment, the spatial rainfall distribution in the Akaki catchment is less likely affected by orographic effects as well as longitudinal and/or latitudinal effects.

Spatial distribution of temperature

Similar to spatial rainfall but for temperature, the spatial interpolation techniques using KED with various covariates, IDW and OK, resulted in different spatial distributions of temperature (see Figures 8 and 10) for both the Mille and Akaki catchments. The OK and IDW methods resulted in the most gradual and smooth zonal patterns, while the KED with elevation, easting, and northing as covariates or with the combination of elevation and easting/northing as covariates showed an irregular distribution following the elevation trend; the temperature pattern exhibited a remarkable decrease with increasing elevation, and the research findings obtained by Matsuura (1995) exhibited the same results.

The spatial distribution of temperature determined by KED showed that the temperature gradually decreases from the east to the west for the Mille catchment (see Figure 8(c)–8(f)) and the south to the north for the Akaki catchment (see Figure 10(b)–10(e)). Accordingly, the spatial distribution of temperature produced by KED with the combination of elevation and easting (for the Mille catchment) and elevation and northing (for the Akaki catchment) as covariates exhibited better performance than the two unilateral interpolation techniques, IDW and OK, and they produced the most detailed and irregular spatial pattern compared with the results of the remaining techniques.

The principal contribution of this research resides in the covariate selection used in KED, and based on the proposed method, the selected predictors improved the predictive performance of KED. For instance, KED with the combination of either elevation and easting or elevation and northing as a predictor highly improves the prediction performance compared with KED with elevation, easting, or northing independently as a covariate (see Tables 5,678). Overall, from the results explained in sections 3 and 4, there were strong improvements in the spatial climatological variable prediction by using KED with the combination of elevation and easting/northing as a predictor.

As an important landscape descriptor, considering the combination of elevation and longitudinal/latitudinal location as a covariate in the spatial interpolation technique, especially in catchments with complex topography, is a key issue. Three interpolation techniques, i.e., two geostatistical techniques, OK and KED, and one deterministic interpolation technique, IDW, were described, and their performances were evaluated using cross-validation methodology. Among the geostatistical methods used in the estimation of the areal average climatological variables on both contrasting catchments, the KED approach incorporated secondary data, i.e. elevation, easting, and northing. Two univariate approaches, OK and IDW, served as benchmarks to which the multivariate approach, KED, could be compared to improve interpolation methods. The KED approach using auxiliary variables as a covariate exhibited significant improvement in interpolation accuracy and/or in reductions in interpolation error relative to unilateral interpolation methods, for example, OK and IDW. Among the KED approaches with a covariate, the one that combined DEM dataset (elevation) and the catchment's longitudinal location as a predictor performed the best, specifically for the Mille catchment. Therefore, it is clear that the incorporation of secondary data, as in our case study, can significantly minimize both the rainfall and temperature spatial interpolation errors for catchments with complex topography where climate stations are poor in terms of coverage and quality. Thus, it can be concluded that KED is identified as the best interpolation technique for the spatial interpolation of mean monthly and annual rainfall and temperature using the combination of elevation and easting/northing data as secondary information in both contrasting catchments, which is expected to be very useful in various climatological, hydrological, and water resource planning studies.

The research was financed by the thesis and dissertation support fund through the Africa Center of Excellence for Water Management (ACEWM), Addis Ababa University, Ethiopia. We are also grateful to the Ethiopia National Meteorology Agency for providing the data and information used in this study. Additionally, the authors sincerely thank the Editor-in-Chief and the three anonymous reviewers for their constructive comments and valuable suggestions on the original version of the paper.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

Adhikary
P. P.
&
Dash
C. J.
2017
Comparison of deterministic and stochastic methods to predict spatial variation of groundwater depth
.
Applied Water Science
7
(
1
),
339
348
.
https://doi.org/10.1007/s13201-014-0249-8
.
Adhikary
S. K.
,
Muttil
N.
&
Yilmaz
A. G.
2017
Cokriging for enhanced spatial interpolation of rainfall in two Australian catchments
.
Hydrological Processes
31
(
12
),
2143
2161
.
https://doi.org/10.1002/hyp.11163
.
Amini
M. A.
,
Torkan
G.
,
Eslamian
S.
,
Zareian
M. J.
&
Adamowski
J. F.
2019
Analysis of deterministic and geostatistical interpolation techniques for mapping meteorological variables at large watershed scales
.
Acta Geophysica
67
(
1
),
191
203
.
https://doi.org/10.1007/s11600-018-0226-y
.
Bertini
C.
,
Buonora
L.
,
Ridolfi
E.
,
Russo
F.
&
Napolitano
F.
2020
On the use of satellite rainfall data to design a dam in an ungauged site
.
Water (Switzerland)
12
(
11
),
1
20
.
https://doi.org/10.3390/w12113028
.
Cantet
P.
2017
Mapping the mean monthly precipitation of a small island using kriging with external drifts
.
Theoretical and Applied Climatology
127
(
1–2
),
31
44
.
https://doi.org/10.1007/s00704-015-1610-z
.
Chen
F.
&
Li
X.
2016
Evaluation of IMERG and TRMM 3B43 monthly precipitation products over mainland China
.
Remote Sensing
8
(
6
),
1
18
.
https://doi.org/10.3390/rs8060472
.
Chen
F. W.
&
Liu
C. W.
2012
Estimation of the spatial rainfall distribution using inverse distance weighting (IDW) in the middle of Taiwan
.
Paddy and Water Environment
10
(
3
),
209
222
.
https://doi.org/10.1007/s10333-012-0319-1
.
Dinku
T.
2019
Challenges with availability and quality of climate data in Africa
.
Extreme Hydrology and Climate Variability: Monitoring, Modelling, Adaptation and Mitigation
71
80
.
https://doi.org/TT
.
Dinku
T.
,
Funk
C.
&
Grimes
D.
2008a
The potential of satellite rainfall estimates for index insurance
.
Earth
1
5
.
Dinku
T.
,
Chidzambwa
S.
,
Ceccato
P.
&
Connor
S. J.
2008b
Validation of high-resolution satellite rainfall products over complex terrain
.
International Journal of Remote Sensing
29
(
14
),
4097
4110
.
https://doi.org/10.1080/01431160701772526
.
Dinku
T.
,
Thomson
M. C.
,
Cousin
R.
,
del Corral
J.
,
Ceccato
P.
,
Hansen
J.
&
Stephen
J. C.
2017
Enhancing National Climate Services (ENACTS) for development in Africa
.
Climate and Development
10
(
7
),
664
672
.
https://doi.org/10.1080/17565529.2017.1405784
.
Di Piazza
A.
,
Conti
F. L.
,
Viola
F.
,
Eccel
E.
&
Noto
L. V.
2015
Comparative analysis of spatial interpolation methods in the Mediterranean area: application to temperature in Sicily
.
Water (Switzerland)
7
(
5
),
1866
1888
.
https://doi.org/10.3390/w7051866
.
Frazier
A. G.
,
Giambelluca
T. W.
,
Diaz
H. F.
&
Needham
H. L.
2016
Comparison of geostatistical approaches to spatially interpolate month-year rainfall for the Hawaiian Islands
.
International Journal of Climatology
36
(
3
),
1459
1470
.
https://doi.org/10.1002/joc.4437
.
Gebere
S. B.
,
Alamirew
T.
,
Merkel
B. J.
&
Melesse
A. M.
2015
Performance of high resolution satellite rainfall products over data scarce parts of eastern Ethiopia
.
Remote Sensing
7
(
9
),
11639
11663
.
https://doi.org/10.3390/rs70911639
.
Gebremedhin
M. A.
,
Lubczynski
M. W.
,
Maathuis
B. H. P.
&
Teka
D.
2021
Novel approach to integrate daily satellite rainfall with in-situ rainfall, Upper Tekeze Basin, Ethiopia
.
Atmospheric Research
248
,
105135
.
https://doi.org/10.1016/j.atmosres.2020.105135
.
Gebremichael
M.
&
Hossain
F.
2010
Satellite rainfall applications for surface hydrology
.
Satellite Rainfall Applications for Surface Hydrology
1
327
.
https://doi.org/10.1007/978-90-481-2915-7
.
Goovaerts
P.
1997
Geostatistics for Natural Resources Evaluation
.
Goovaerts
P.
2000
Geostatistical approaches for incorporating elevation into the spatial interpolation of rainfall
.
Journal of Hydrology
228
(
1–2
),
113
129
.
https://doi.org/10.1016/S0022-1694(00)00144-X
.
Grimes
D. I. F.
,
Pardo-Igúzquiza
E.
&
Bonifacio
R.
1999
Optimal areal rainfall estimation using raingauges and satellite data
.
Journal of Hydrology
222
(
1–4
),
93
108
.
https://doi.org/10.1016/S0022-1694(99)00092-X
.
Haberlandt
U.
2007
Geostatistical interpolation of hourly precipitation from rain gauges and radar for a large-scale extreme rainfall event
.
Journal of Hydrology
332
(
1–2
),
144
157
.
https://doi.org/10.1016/j.jhydrol.2006.06.028
.
Havesi
J. A.
1991
Precipitation Estimation in Mountainous Using Multivariate Geostatistics. Part I:
Structural Analysis. Journal of Applied Meteorology
31
(
7
),
661
676
.
Hudson
G.
&
Wackernagel
H.
1994
Mapping temperature using kriging with external drift: theory and an example from Scotland
.
International Journal of Climatology
14
(
1
),
77
91
.
https://doi.org/10.1002/joc.3370140107
.
Isaaks
E. H.
1990
Applied geostatistics
.
Choice Reviews Online
28
(
01
).
https://doi.org/10.5860/choice.28-0304
.
Isaaks
E. H.
&
Srivastava
R. M.
1989
An introduction to applied geostatistics
. In:
Geographical Analysis
, Vol.
26
,
3
, pp.
282
283
.
https://doi.org/10.1111/j.1538-4632.1994.tb00325.x
.
Jalili Pirani
F.
&
Modarres
R.
2020
Geostatistical and deterministic methods for rainfall interpolation in the Zayandeh Rud basin, Iran
.
Hydrological Sciences Journal
65
(
16
),
2678
2692
.
https://doi.org/10.1080/02626667.2020.1833014
.
Kaufman
C. G.
&
Shaby
B. A.
2013
The role of the range parameter for estimation and prediction in geostatistics
.
Biometrika
100
(
2
),
473
484
.
https://doi.org/10.1093/biomet/ass079
.
Kisaka
M. O.
,
Mucheru-Muna
M.
,
Ngetich
F. K.
,
Mugwe
J.
,
Mugendi
D.
,
Mairura
F.
,
Shisanya
C.
&
Makokha
G. L.
2016
Potential of deterministic and geostatistical rainfall interpolation under high rainfall variability and dry spells: case of Kenya's Central Highlands
.
Theoretical and Applied Climatology
124
(
1–2
),
349
364
.
https://doi.org/10.1007/s00704-015-1413-2
.
Lebel
T.
,
Bastin
G.
,
Obled
C.
&
Creutin
J. D.
1987
On the accuracy of areal rainfall estimation: a case study
.
Water Resources Research
23
(
11
),
2123
2134
.
https://doi.org/10.1029/WR023i011p02123
.
Le Coz
C.
&
Van De Giesen
N.
2020
Comparison of rainfall products over sub-saharan Africa
.
Journal of Hydrometeorology
21
(
4
),
553
596
.
https://doi.org/10.1175/JHM-D-18-0256.1
.
Li
J.
&
Heap
A. D.
2008
A review of spatial interpolation methods for environmental scientists
. In:
Australian Geological Survey Organisation, GeoCat# 68(2008/23)
. p.
154
.
https://doi.org/http://www.ga.gov.au/image_cache/GA12526.pdf
.
Lloyd
C. D.
2005
Assessing the effect of integrating elevation data into the estimation of monthly precipitation in Great Britain
.
Journal of Hydrology
308
(
1–4
),
128
150
.
https://doi.org/10.1016/j.jhydrol.2004.10.026
.
Lu
S.
,
Veldhuis
M. C. T.
&
Van De Giesen
N.
2020
A methodology for multiobjective evaluation of precipitation products for extreme weather (In a data-scarce environment)
.
Journal of Hydrometeorology
21
(
6
),
1223
1244
.
https://doi.org/10.1175/JHM-D-19-0157.1
.
Ly
S.
,
Charles
C.
&
Degr
A.
2011
Geostatistical Interpolation of Daily Rainfall at Catchment Scale: the Use of Several Variogram Models in the Ourthe and Ambleve Catchments, Belgium
, pp.
2259
2274
.
https://doi.org/10.5194/hess-15-2259-2011
.
Matsuura
C. J. W. A. K.
1995
Smart Interpolation of Annually Air Temperature in the United States
.
American Meteorological Society
,
Boston, MA
.
Ministry of Water Resources
2009
Federal Democratic Republic of Ethiopia Ministry of Water Resources Mille and Dirma Integrated Sub-Watershed Management Study Annex A: Climate and Water Resources (Final)
.
Molla
D.
,
Stefan
W.
,
Gizaw
B.
&
Stichler
W.
2005
Groundwater recharge in the Akaki catchment, central Ethiopia: evidence from environmental isotopes (d18O, d2H and 3H) and chloride mass balance
.
2274
(
November 2008
),
2267
2274
.
https://doi.org/10.1002/hyp
.
Mukhopadhaya
S.
2016
Rainfall mapping using ordinary kriging technique: case study: Tunisia
.
Journal of Basic and Applied Engineering Research.
3
(
1
),
1
5
.
Nalder
I. A.
&
Wein
R. W.
1998
Spatial interpolation of climatic Normals: test of a new method in the Canadian boreal forest
.
Agricultural and Forest Meteorology
92
(
4
),
211
225
.
https://doi.org/10.1016/S0168-1923(98)00102-6
.
Novikov
S. L.
1981
Elevation: a major influence on the hydrology of
New Hampshire and Vermont
, USA
.
Hydrological Sciences Bulletin
26
(
4
),
399
413
.
https://doi.org/10.1080/02626668109490904
.
Otieno
H.
,
Yang
J.
,
Liu
W.
&
Han
D.
2014
Influence of rain gauge density on interpolation method selection
.
Journal of Hydrologic Engineering
19
(
11
),
04014024
.
https://doi.org/10.1061/(asce)he.1943-5584.0000964
.
Parker
D.
,
Good
E.
&
Chadwick
R.
2011
Reviews of observational data available over Africa for monitoring, attribution and forecast evaluation
.
Hadley Centre Technical Note
86
(
June
),
1
63
.
Pebesma
E. J.
2003
Gstat: multivariable geostatistics for S. Dsc
. In:
Proceedings of DSC
(Hornik, K., Leisch, F. & Zeileis, A., eds.), March 20–22, 2003, Vienna, Austria
.
Pebesma
E. J.
2004
Multivariable geostatistics in S: the gstat package
.
Computers and Geosciences
30
(
7
),
683
691
.
https://doi.org/10.1016/j.cageo.2004.03.012
.
Pebesma
E.
2012
Spacetime: spatio-temporal data in R
.
Journal of statistical software
51
,
1
30
.
Pebesma
E. J.
&
Wesseling
C. G.
1998
GSTAT: a program for geostatistical modelling, prediction and simulation
.
Computers & Geosciences
24
(
1
),
17
31
.
Petković
V.
&
Kummerow
C. D.
2017
Understanding the sources of satellite passive microwave rainfall retrieval systematic errors over land
.
Journal of Applied Meteorology and Climatology
56
(
3
),
597
614
.
https://doi.org/10.1175/JAMC-D-16-0174.1
.
Phillips
D. L.
,
Dolph
J.
&
Marks
D.
1992
A comparison of geostatistical procedures for spatial analysis of precipitation in mountainous terrain
.
Agricultural and Forest Meteorology
58
(
1–2
),
119
141
.
https://doi.org/10.1016/0168-1923(92)90114-J
.
Rata
M.
,
Douaoui
A.
,
Larid
M.
&
Douaik
A.
2020
Comparison of geostatistical interpolation methods to map annual rainfall in the Chéliff watershed, Algeria
.
Theoretical and Applied Climatology
141
(
3–4
),
1009
1024
.
https://doi.org/10.1007/s00704-020-03218-z
.
Rossiter
D. G.
2014
Applied Geostatistics Exercise 5b: Predicting From Point Samples (Part 4) Lognormal Kriging
. Part 4, University of Twente, Faculty of Geo-Information Science & Earth Observation (ITC), Enschede, Netherlands.
Shepard
D.
1968
A Two-Dimensional Interpolation Function for Irregularly-Spaced Data
. Proceedings of the 1968 ACM National Conference, New York, 27-29 August 1968, 517–524. http://dx.doi.org/10.1145/800186.810616.
Stein
M.
2010
Asymptotics for Spatial Processes
.
https://doi.org/10.1201/9781420072884-c6
.
Taesombat
W.
&
Sriwongsitanon
N.
2009
Areal rainfall estimation using spatial interpolation techniques
.
ScienceAsia
35
(
3
),
268
275
.
https://doi.org/10.2306/scienceasia1513-1874.2009.35.268
.
Tas
E.
&
Taş
E.
2017
Comparison of monthly TRMM and ground-based precipitation data in Akarcay Basin, Turkey
. In:
8th Atmospheric Sciences Symposium
,
December
.
Toté
C.
,
Patricio
D.
,
Boogaard
H.
,
van der Wijngaart
R.
,
Tarnavsky
E.
&
Funk
C.
2015
Evaluation of satellite rainfall estimates for drought and flood monitoring in Mozambique
.
Remote Sensing
7
(
2
),
1758
1776
.
https://doi.org/10.3390/rs70201758
.
Vicente-Serrano
S. M.
,
Saz-Sánchez
M. A.
&
Cuadrat
J. M.
2003
Comparative analysis of interpolation methods in the middle Ebro Valley (Spain): application to annual precipitation and temperature
.
Climate Research
24
(
2
),
161
180
.
https://doi.org/10.3354/cr024161
.
Wackernagel
H.
1998
Multivariate Geostatistics: An Introduction with Applications
, Vol.
148
.
Kluwer Academic Publishers
,
the Netherlands
.
Wackernagel
H.
2003
Multivariate Geostatistics: an Introduction with Applications (Third, com, Vol. 148)
.
Kluwer Aeademic Publishers
,
The Netherlands
.
Washington
R.
,
Harrison
M.
,
Conway
D.
,
Black
E.
,
Challinor
A.
,
Grimes
D.
,
Jones
R.
,
Morse
A.
,
Kay
G.
&
Todd
M.
2006
African climate change: taking the shorter route
.
Bulletin of the American Meteorological Society
87
(
10
),
1355
1366
.
https://doi.org/10.1175/BAMS-87-10-1355
.
Webster
R.
&
Oliver
M.
2007
Geostatistics for environmental scientists
.
Vadose Zone Journal
1
(
2
).
https://doi.org/10.2136/vzj2002.3210
.
Wilson
J. P.
,
Spangrud
D. J.
,
Nielsen
G. A.
,
Jacobsen
J. S.
&
Tyler
D. A.
1998
Global positioning system sampling intensity and pattern effects on computed topographic attributes
.
Soil Science Society of America Journal
62
(
5
),
1410
1417
.
Yang
Z.
,
Hsu
K.
,
Sorooshian
S.
,
Xu
X.
,
Braithwaite
D.
,
Zhang
Y.
&
Verbist
K. M. J.
2017
Merging high-resolution satellite-based precipitation fields and point-scale rain gauge measurements – a case study in Chile
.
Journal of Geophysical Research
122
(
10
),
5267
5284
.
https://doi.org/10.1002/2016JD026177
.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/).