Abstract
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.
HIGHLIGHTS
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
INTRODUCTION
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.
MATERIAL AND METHODS
Study area
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 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.
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.
Mille and Akaki catchments’ climatological stations
Mille catchment . | Akaki catchment . | ||||||
---|---|---|---|---|---|---|---|
Stations . | Altitude (m.a.s.l) . | Longitude . | Latitude . | Stations . | Altitude (m.a.s.l) . | Longitude . | Latitude . |
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 . | ||||||
---|---|---|---|---|---|---|---|
Stations . | Altitude (m.a.s.l) . | Longitude . | Latitude . | Stations . | Altitude (m.a.s.l) . | Longitude . | Latitude . |
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 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.
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 catchment . | Akaki catchment . | |
1 | 5.153 | 1.274 |
2 | 4.353 | 1.221 |
3 | 3.940 | 1.206 |
4 | 3.861 | 1.210 |
5 | 3.900 | 1.219 |
6 | 3.960 | 1.226 |
Idp = n, where n = 1, 2, 3, 4, 5, 6 . | RMSE . | |
---|---|---|
Mille catchment . | Akaki catchment . | |
1 | 5.153 | 1.274 |
2 | 4.353 | 1.221 |
3 | 3.940 | 1.206 |
4 | 3.861 | 1.210 |
5 | 3.900 | 1.219 |
6 | 3.960 | 1.226 |
Ordinary kriging



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 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

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.


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.
RESULTS
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).
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
Month . | Range (m) . | Nugget (mm2) . | Sill (mm2) . | Model . | Selected covariate/predictor . |
---|---|---|---|---|---|
Jan | 12,925 | 18 | 22 | Spherical | Elevation + easting |
Feb | 12,925 | 6 | 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 |
Month . | Range (m) . | Nugget (mm2) . | Sill (mm2) . | Model . | Selected covariate/predictor . |
---|---|---|---|---|---|
Jan | 12,925 | 18 | 22 | Spherical | Elevation + easting |
Feb | 12,925 | 6 | 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 |
Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and August monthly mean rainfall, respectively.
Empirical and theoretical semivariogram models: exponential (a) and spherical (b) for annual and August monthly mean rainfall, respectively.
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).
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.
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.
Experimental (bins) and fitted theoretical (curve) variograms of January mean monthly ((a) and (c)) and annual ((b) and (d)) rainfall and maximum temperature, respectively.
Experimental (bins) and fitted theoretical (curve) variograms of January mean monthly ((a) and (c)) and annual ((b) and (d)) rainfall and maximum temperature, respectively.
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.
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 . | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Month . | Range . | Nugget . | Sill . | Model . | Covariates . | Range . | Nugget . | Sill . | Model . | Covariates . |
Jan | 4,869 | 0.3 | 0.65 | Spherical | Northing | 9,128 | 0 | 0.4 | Ste | Elevation + northing |
Feb | 12,471 | 0 | 12 | Ste | Easting | 8,915 | 0 | 0.45 | Ste | Elevation + northing |
Mar | 8,295 | 0 | 24 | Ste | Easting | 9,224 | 0 | 0.5 | Ste | Elevation + northing |
Apr | 5,725 | 0 | 14 | Ste | Northing | 9,954 | 0 | 0.56 | Ste | Elevation |
May | 11,838 | 0 | 24 | Ste | Easting | 10,962 | 0 | 0.72 | Ste | Elevation |
Jun | 4,743 | 0 | 17 | Ste | Elevation | 9,057 | 0 | 0.62 | Ste | Elevation |
Jul | 6,015 | 0 | 231 | Ste | Northing | 9,666 | 0 | 0.71 | Ste | Elevation |
Aug | 8,954 | 0 | 504 | Ste | Northing | 9,447 | 0 | 0.58 | Ste | Elevation |
Sep | 5,240 | 0 | 106 | Ste | Elevation | 10,489 | 0 | 0.51 | Ste | Elevation |
Oct | 2,904 | 0 | 10 | Ste | Easting | 8,931 | 0 | 0.6 | Ste | Elevation + northing |
Nov | 9,547 | 0 | 2.2 | Ste | Elevation | 8,526 | 0 | 0.43 | Ste | Elevation + northing |
Dec | 9,773 | 0 | 1.5 | Ste | Easting | 8,384 | 0 | 0.42 | Ste | Elevation + northing |
Annual | 7,527 | 0 | 1,943 | Ste | Northing | 10,658 | 0 | 2,496 | Ste | Elevation |
Parameters and variogram model for mean rainfall . | Parameters and variogram model for mean Tmax . | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Month . | Range . | Nugget . | Sill . | Model . | Covariates . | Range . | Nugget . | Sill . | Model . | Covariates . |
Jan | 4,869 | 0.3 | 0.65 | Spherical | Northing | 9,128 | 0 | 0.4 | Ste | Elevation + northing |
Feb | 12,471 | 0 | 12 | Ste | Easting | 8,915 | 0 | 0.45 | Ste | Elevation + northing |
Mar | 8,295 | 0 | 24 | Ste | Easting | 9,224 | 0 | 0.5 | Ste | Elevation + northing |
Apr | 5,725 | 0 | 14 | Ste | Northing | 9,954 | 0 | 0.56 | Ste | Elevation |
May | 11,838 | 0 | 24 | Ste | Easting | 10,962 | 0 | 0.72 | Ste | Elevation |
Jun | 4,743 | 0 | 17 | Ste | Elevation | 9,057 | 0 | 0.62 | Ste | Elevation |
Jul | 6,015 | 0 | 231 | Ste | Northing | 9,666 | 0 | 0.71 | Ste | Elevation |
Aug | 8,954 | 0 | 504 | Ste | Northing | 9,447 | 0 | 0.58 | Ste | Elevation |
Sep | 5,240 | 0 | 106 | Ste | Elevation | 10,489 | 0 | 0.51 | Ste | Elevation |
Oct | 2,904 | 0 | 10 | Ste | Easting | 8,931 | 0 | 0.6 | Ste | Elevation + northing |
Nov | 9,547 | 0 | 2.2 | Ste | Elevation | 8,526 | 0 | 0.43 | Ste | Elevation + northing |
Dec | 9,773 | 0 | 1.5 | Ste | Easting | 8,384 | 0 | 0.42 | Ste | Elevation + northing |
Annual | 7,527 | 0 | 1,943 | Ste | Northing | 10,658 | 0 | 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.
Mille catchment's mean monthly and annual rainfall spatial prediction using various interpolation techniques
Month . | Est. rainfalla . | Obr. rainfalla . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.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 |
Month . | Est. rainfalla . | Obr. rainfalla . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.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.
Mille catchment's mean monthly and annual maximum temperature estimated and actual values, and descriptive statistics using various interpolation techniques
Month . | Est. Tmaxa . | Obr. Tmaxa . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.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 |
Month . | Est. Tmaxa . | Obr. Tmaxa . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.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.
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.
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.
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.
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.
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.
Akaki catchment long year-based predicted vs. observed mean monthly and annual rainfall using various interpolation techniques
Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.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 |
Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. rainfall . | Obr. rainfall . | RMSE . | MBE . | MAE . | r . | V.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 |
Akaki catchment long year-based predicted vs. observed mean monthly and annual maximum temperature using various interpolation techniques
Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.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 |
Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.model . | Month . | Est. Tmax . | Obr. Tmax . | RMSE . | MBE . | MAE . | r . | V.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 |
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.
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.
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.
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.
Box plots for mean rainfall, Tmin, and Tmax for the Akaki catchment, obtained with KED.
Box plots for mean rainfall, Tmin, and Tmax for the Akaki catchment, obtained with KED.
Box plots for mean rainfall, Tmin, and Tmax for the Mille catchment, obtained with KED.
Box plots for mean rainfall, Tmin, and Tmax for the Mille catchment, obtained with KED.
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.
DISCUSSION
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 5,67–8). 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.
CONCLUSIONS
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.
ACKNOWLEDGMENTS
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.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.