Geoclimatic Factor K Mapping in Nigeria through Spatial Interpolation.
Spatial interpolation techniques have been applied in a number of fields of study to generate a continuous surface data for informed decisions. It has found applications in climatology [1,2], in electromagnetic field estimation [3,4], in radioactive contamination , aerospace engineering , in optics , and many others. The problem in spatial interpolation technique is always the choice of the best interpolation method to be adopted and the best performance analysis criteria [3,8] to be adopted. There is no single ideal method for data interpolation. Selection of method depends on actual data, required level of accuracy, available time and/or computer resources, quality of secondary information, data variance, grid size or resolution and surface type [3,8]. In this paper, we employed the two mostly used spatial interpolation techniques: Inverse distance weighting (IDW) and Kriging. In the Kringing interpolation, we used the spherical, exponential and Gaussian methods. The performance of the interpolation methods were evaluated using the mean absolute error and the root mean square error to determine the best interpolation method to employ for the mapping.
The rest of the paper is organized as follows. Section 2 described how the geoclimatic factor was determined while section 3 described the spatial interpolation method adopted for the study. In section 4, the methodology deployed for the mapping was described while the results and discussion was presented in section 5 and finally a conclusion was drawn in section 6
In this paper, the two methods were described: kriging and inverse distance weighting. Techniques that produce smooth surfaces include various approaches that may combine regression analyses and distance-based weighting averages. The key difference among the methods is the criteria used in weight values in relation to distance .
2. GEOCLIMATIC FACTOR DETERMINATION
The propagation of radio waves in the troposphere is influential on the variations of climatic conditions (temperature, pressure and humidity) as earlier stated in Section 1. These parameters are related to the atmospheric refractivity and can be expressed as in (1),
N =(n-1)x[10.sup.6]=77.6/T (P + 4810 e/T) (1)
where N is the atmospheric refractivity, n is the refractive index of the atmosphere, T is the temperature (K), P is the atmospheric pressure (hpa) and e is the water vapour pressure (hpa) [11,12]: The water vapour pressure e is given by (2), where H (%) is the relative humidity and t ([degrees]C) is the air temperature :
e = 6.1121H/100 exp (17.502t/t+240.97) (2)
The refractivity gradient can be defined as the rate at which the refractivity is varied with respect to the height of antenna and has a greater interest to designers of LOS links. Signal do experience Multipath fading if the refractivity gradient in the atmosphere varies with height . Equation (3) can be used to estimate Refractivity gradient with [N.sub.1] and [N.sub.2] are the refractivity heights [h.sub.1] and [h.sub.2] h2 respectively :
dN/dh [approximately equal to] [[N.sub.2] - [N.sub.1]]/[[h.sub.2] - [h.sub.1]] (3)
The rate at which the refractivity gradient occur in the first 65 m above the ground level is calculated. Thereafter, cumulative distribution of dN/dh is calculated from the frequency of occurrence. From the cumulative distribution curve, the point refractivity gradient not exceeded for 1% for each months is then determined. Whenever there is no availability of 1% of the refractivity gradient from the distribution curves, then the inverse distance square technique described in the preceding section is adopted for the estimate of value from the nearest values. Equation (4) can be used to determine the geoclimatic factor (for quick planning) where [dN.sub.1] represents the point refractivity gradient in the lowest 65 m of the atmosphere not exceeded for 1% of an average year 
[mathematical expression not reproducible] (4)
3. SPATIAL INTERPOLATION TECHNIQUES
Spatially continuous data or spatially continuous surfaces over a region of interest are needed by scientists to make acceptable analyses. But, such data are usually not available and often very difficult and costly to obtain. Additionally, environmental data collected from the field are usually from point sources. Therefore, the values of an attribute at unsampled points need to be calculated in order to generate spatially continuous data. Spatial interpolation techniques provide tools for valuing the values of unobserved variable at unsampled sites using data from point observables .
There are very many types of kriging, but here we discuss the ordinary kriging. Kriging uses linear combination of values at observable points to estimate values at un-observable points. It is prevalent because it is an exact interpolator, i.e., the predictor for an input that has already been observed equals the observed output. The predicted value [Z.sub.po] is given by Eq. (5) where [Z.sub.oi] are the observable values and [k.sub.i] are the Kriging weights [4,5]:
[Z.sub.po] = [[summation].sup.N.sub.i] [k.sub.i*] x [Z.sub.oi] (5)
with the constraints that;
[[summation].sup.N.sub.i] [k.sub.i*] = 1 (6)
Kriging suggests that the closer the input data are, the more positively correlated are their outputs. Kriging is modelled over a covariance process that is second-order stationary, which denotes that the first two statistical moments (the means and the covariance) are constant [4,6]. Kriging is modelled through a semi-variogram. Semi-variogram is a mathematical function that indicates the spatial autocorrelation in observations measured at sample points or locations. It is represented in a graph that shows the semi-variance in measure with distance between the sampled locations. It portrays the spatial continuity or smoothness of a set of data . From the semi-variance cloud graph, we develop a model that describes the variability of the measurement with location. The model developed also acts as a projection tool for estimating the value of a measured value at an un-observed position or location. Some of the commonly used semi-variogram models include; spherical, exponential and Gaussian.
3.2 Inverse Distance Weighting
Inverse Distance Weighting (IDW), like many other spatial interpolation technique, data anal ysis is estimated as a set of observations related with a set of sampled points or places to a set of unsampled points or places where observations are not available . The inverse distance weighting or inverse distance weighted (IDW) method evaluate the values of an attribute at un-sampled points using a linear addition of values at sampled points weighted by an inverse function of the distance from the point of interest to the sampled points :
Z' ([x.sub.0]) = [[summation].sup.n.sub.i=1] [[lambda].sub.i] Z([x.sub.i]) (7)
where Z' is the is the approximated value at the point of interest [x.sub.o], Z is the known value at the sampled point [x.sub.i], [[lambda].sub.i] is the weighting parameter and, n denotes the number of sampled points used for approximation. The weighting bias can be represented by (8), where [d.sub.i] is the distance between [x.sub.0] and [x.sub.i], p is a power parameter, and n is the as defined previously :
[[lambda].sub.i] = [1/[([d.sub.i]).sup.p]]/ [[summation].sup.n.sub.i=1] 1/([d.sub.i]).sup.p] (8)
with the constraint that [[summation].sup.n.sub.i=1] [[lambda].sub.i] = 1 (9)
4.1 Data Collection
Seasonal geoclimatic factor estimated from the procedure given in section 2 is used in interpolation and mapping of geoclimatic factor for the whole of Nigeria. Table 1 shows six stations with their geographical locations expressed in latitudes and longitudes. The four month values were derived from three year radiosonde sounding taken twice per day and represent the four seasons exhibited in Nigeria. The sounding data were reported after every ten seconds which gives a fairly good height resolutions.
4.2 Data Processing
Data processing, analysis and representation was carried out using GS+ Geostatistics for the Environmental Sciences. GS+ is a geostatistical analysis and mapping program that allows you to readily measure and illustrate spatial relationships in geo-referenced data.
It is a data analysis package that includes common statistical, plotting and modelling functions. GS+ analyzes spatial data for autocorrelation and then uses this information to make optimal, statistically rigorous maps of the area sampled. The maps can be created in GS+ or in other mapping programs or geographic information systems.
4.3 Error Analysis
Since there is no preferred interpolation method, the choice of the appropriate method for a certain task is made based on its accuracy [11-16] among other factors. One of the methods used in method assessment is error analysis. There are very many measures of fit for error analysis; in our study, we have applied the root mean square error (RMSE) and the mean absolute error (MAE). RMSE gives measure of inaccuracy though is sensitive to outliers as it emphasizes a lot of weight to large errors while MAE is less sensitive to extreme values. However, they are amongst the best measures of performance of a model because they give a summary of the average difference in units of estimated and observed values .
Control sites where discrete data exist are used in performance analysis by removing these observed values from data to be processed and then using interpolation techniques to predict these observed values (i.e., cross validation). Error calculation is then done using (6) and (7). The method that gives the minimum values of error is assumed to be the best performing method. [Pr.sub.j] is the predicted value and [Ob.sub.j] is the observed value at various control sites, and n is the number of control sites. The control points are chosen randomly for large data points; however, where data points are few, all points become control points.
RMSE = 1/n [[summation].sup.n.sub.j=1] [([Pr.sub.j] - [Ob.sub.j]).sup.2] (6)
MAE = 1/n [[summation].sup.n.sub.j=1] [absolute value of ([Pr.sub.j] - [Ob.sub.j])] (7)
5. RESULTS AND DISCUSSION
The interpolation process was carried out using Inverse distance weighting, and ordinary Kriging. Three models of variogram were used in Kriging: Spherical, Exponential and Gaussian. It should be noted that the algorithms calculate the raster layer, considering all pixels within the rectangle defined by the latitudes and longitudes and may at times fall out of the intended Nigeria region. It is therefore important for one to know the longitudes and latitudes for the region/place of interest.
Table 2 shows the measures of fit carried out. From the interpolation results, it was found that the lowest MAE and RMSE is recorded by Kriging (Gaussian) method for all the seasons. Hence we can conclude that Kriging (Gaussian) is the best performing method in this study. This is then followed by Kriging (exponential). Kriging performance is influenced by the variogram model used. For our data set, the gaussian model has the lowest error among the three variogram models, followed by exponential model. At some points the error between the exponential model and spherical model is negligible. Figures 1, 2, 3 and 4 shows the contour maps for seasonal geoclimatic factor for February, May, August and November using Kriging (Gaussian) technique. Contour maps for other methods are not shown here but their error analysis is given in Table 2.
In this paper, it can be seen that Inverse distance weighting gives the best performance in interpolating the geoclimatic factor K. Geoclimatic factor depend on the variability of the atmosphere specifically temperature, pressure and relative humidity. If we assume that regions close to each other tend to exhibit the same climatic conditions, then, we can ascertain that IDW is the best method for interpolating the geoclimatic factor. But, it is also known that the geoclimatic factor also depends on the topology of the area in addition to climatic conditions, hence more study could be done on this by integrating fading data of given regions to the climatic conditions for a more decisive conclusion on the distribution of this factor in the region of study. It can be shown that different techniques used in evaluation give different results making it a must to choose the best technique which gives the minimum error of performance. The best technique can further be optimized for better accuracy. The IDW here can be optimized by adjusting the power parameter and the search radius.
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M.O. Olla and I.B. Oluwafemi
Center for Research in Electrical Communication(CRECO)
Department of Electrical and Electronic Engineering, Ekiti State University,
Email: firstname.lastname@example.org and email@example.com
Caption: Figure 1. Contour plot for the Geoclimatic factor K February using Kriging (Gaussian)
Caption: Figure 2. Contour plot for the Geoclimatic factor K for May using Kriging (Gaussian)
Caption: Figure 3. Contour plot for the geoclimatic factor K for August using Kriging (Gaussian)
Caption: Figure 4. Contour plot for the geoclimatic factor K for November using Kriging (Gaussian)
Table 1: Discrete Observable Data used in Mapping and their Latitude/Longitude City Lat Long Feb May Aug Nov Kaduna 7.4 10.20 2.40 1.78 8.92 6.18 50 E-04 E-04 E-03 E-05 Lagos 3.3 6.45 2.98 2.22 2.22 2.26 80 E-04 E-04 E-04 E-04 Abuja 7.3 9.07 6.45 2.02 2.98 8.81 9 E-04 E-02 E-02 E-03 PortHarco 7.0 4.81 5.60 5.14 4.81 1.38 urt 4 E-05 E-04 E-04 E-04 Enugu 5.2 13.00 2.80 7.13 1.39 6.15 4 E-03 E-03 E-02 E-03 Kano 8.5 12.00 7.60 4.02 2.85 9.96 9 E-05 E-03 E-02 E-02 Table 2. Error analysis applied to interpolation techniques Kriging Month Measure IDW of fit Spherical Exponential Gaussian Feb RMSE 2.843E-08 7.414E-08 3.010E-08 11.383E-07 MAE 6.883E-05 1.112E-04 4.250E-04 9.110E-04 May RMSE 2.982E-08 8.004E-08 3.241E-08 1.478E-07 MAE 7.050E-05 1.155E-04 7.350E-05 1.570E-04 Aug RMSE 1.262E-05 9.753E-04 1.666E-13 1.667E-13 MAE 8.702E-03 7.650E-04 1.000E-06 1.000E-06 Nov RMSE 5.508E-07 5.478E-07 5.478E-07 1.667E-11 MAE 1.818E-03 1.813E-03 1.812E-03 1.000E-05
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|Author:||Olla, M.O.; Oluwafemi, I.B.|
|Publication:||International Journal of Digital Information and Wireless Communications|
|Date:||Oct 1, 2018|
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