Effective and Median Effective Earth Radius Factor Distribution Characterization for Microwave Link Design Applications

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Effective and Median Effective Earth Radius Factor Distribution Characterization for Microwave Link Design Applications

A. M. Nyete

Department of Electrical and Information Engineering University of Nairobi

Abstract:- In this paper, radiosonde data is processed to obtain the point median and effective values of the effective earth radius factor (k-factor). These point values are the interpolated

is referred to as the effective earth radius factor (k-factor) and is given by [3]:

to obtain the effective earth distribution for the entire country by producing the contour maps from the values obtained. The accuracy of the results obtained is validated using the RMSE

=

= 1 + (5)

and MAE. The error results show that the interpolation methods used have a high degree of accuracy in predicting the k-factor values of the entire country.

  1. INTRODUCTION

    The refractive properties of the lower atmosphere is defined in terms of the vertical profile of the radio refractivity, given

    where is the effective earth radius factor and is the effective earth radius and is the true earth radius. The effective earth radius factor ( k-factor) is a function of the vertical refractivity gradient and can also be determined by the equation [1-3]:

    1

    by [1-6]:

    = 77.6 + 3.3 × 105

    (1)

    = [1 + ]

    157

    (6)

    2

    where is the atmospheric pressure (hPa), is the water vapour pressure (hPa), and is the absolute temperature (K). The above Equation is valid when the radio frequency is below 100GHz with errors of less than 0.5% [5,6]. From the equation above, the radio refractivity has wet and dry components, given by [6-10]:

    77.6

    Due to the tedious work of collecting and processing radiosonde data to obtain the radio refractivity and ultimately the k-factor statistics for different location, interpolation techniques will be applied on sample data for chosen locations in South Africa to generate the contour maps necessary for link design in the whole country.

  2. INTERPOLATION TECHNIQUES Interpolation refers to the process of predicting data values

    and,

    =

    (2)

    at locations where samples are not available using data from surrounding locations within a particular geographical range (area) [14, 15]. There are two broad categories of interpolation techniques namely: global and local methods.

    = 3.3 × 105

    2

    (3)

    Global methods consider all known samples to estimate the value at a given location. Local methods consider only a fixed number of samples within a certain search radius to the

    The vertical refractivity gradient is then given by [9-15]:

    = 77.6 (1 + 4810) (4)

    point being estimated. Global methods are known to produce smooth estimates but are very sensitive to outliers. On the other hand, local methods produce less smooth surfaces but

    2

    are less sensitive to outliers. Global methods include trend

    In Equation (4) above, it is assumed that the radio refractivity does not vary horizontally, that is, it only varies vertically.

    Due to the radio refractivity, electromagnetic waves travelling through the atmosphere suffer a degree of bending. For easier of geometrical analysis, the electromagnetic waves are normally represented as straight lines then compensation is done by assuming an imaginary earth radius, otherwise referred to as effective earth radius,

    [3]. The ratio between the effective and true earth radius

    surfaces, Fourier series and many others while local methods include Radial Basis Functions, Kriging and Inverse Distance Weighting (IDW) [14, 16], among others. The most commonly used interpolation techniques are spatial and they include: Radial Basis Functions (RBFs), Kriging and Inverse Distance Weighting (IDW). Spatial interpolation techniques are based on the proposition that things which are closer to each other are more alike than those further apart.

    Due to the tedious nature of processing clear-air meteorological data, and the cost involved in launching radiosondes, the three spatial interpolation techniques listed above have been employed here in extending both the median and effective k-factor statistics obtained in Chapter 4 to cover the whole of South Africa. A comparison of the interpolation methods is drawn by using the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) criteria and then the interpolated and gridded values are presented in contour maps. The RMSE is sensitive to outliers while the MAE is not; hence both errors complement each other. Both error criteria provide an excellent averaged summary of the variation between measured and predicted values.

      1. Inverse Distance Weighting (IDW)

        IDW is a deterministic interpolation technique. In this method, data values that are closer to each other are assumed to be more similar than those farther apart. IDW uses a weighting scheme to evaluate the impact individual data points have on the predicted value with data points that are closer to the prediction point being assigned more weight, hence having more impact on the predicted value than those farther from the predicted value . The weighting policy in IDW is based only on the Euclidian distance between the known data points and the prediction point. The predicted estimate is then a linear combination of the weighted measured values. The weight assignment in IDW is controlled by a power parameter, that determines how much influence data points have on the predicted value as the Euclidian distance varies. The higher the power

        = 1 (10)

        =1

        Since IDW uses a simple weighting scheme that is only based on the Euclidian distance between the known data point and the prediction point alone, the level of computational subjectivity is low hence computation speeds are faster compared to other sophisticated methods like kriging.

      2. Kriging

    Kriging is a stochastic interpolator but similar to IDW in that it also employs weighting to predict unknown values. Just as in IDW, the predicted value is a linear combination of the known weighted samples [14, 15]. Points closer to the prediction point are also assigned more weight compared to those farther apart. However, the weighting used in kriging is more complex and involves spatial correlation between the predicted point and known data points. This spatial correlation and interdependence in kriging is modeled using the semivariogram. The semivariogram is a plot of the semivariances against the separation distances (lag distances) of the known data points. The mathematical relationship that best describes the semivariogram is then used to build covariance matrices necessary for the determination of the kriging weights. The semivariance is a measure of the dissimilarity of a measured variable and is given by the average squared difference between the data values and is given by [14, 17]:

    parameter, the more the influence closer points have on the 1 2

    predicted value and vice versa. Typically, the power parameter is assigned value of 2 or 3 but the choice can also be made based on error measurement resulting in optimal

    () = ( ) 2()

    (, | = )

    (11)

    IDW. Mathematically, the IDW predictor is given by [14]:

    where () is the number of data values separated by a distance .There are several empirical sevariogram models

    (, ) = (, )

    =1

    (7)

    <>that can be used to model the semivariance. They include; spherical semivariogram, exponential semivariogram, cubic semivariogram, Bessel semivariogram, j-Bessel

    where (, ) is the IDW predictor, is the number of known data points, (, ) is the known data value at the point (, ) and is the weight assigned to (, ). The weight , is given by [14]:

    semivariogram and Gaussian semivariogram, among others [14, 16].

    The semivariogram is characterized by three main coefficients namely; the range, the sill and the nugget. The range is defined as the maximum lag distance between two

    =

    (8)

    points for which the semivariance between them can be

    =1

    where is the power parameter, is the Euclidian distance between the prediction point and the known data point and is given by:

    = ( )2 + ( )2 (9)

    where all parameters are as described in (7) and (8). The weighting in IDW is such that the sum of all the weights sum to unity:

    determined. This is to say that, if the lag distance between two data values is greater than the range, then the points are not spatially correlated or interdependent. The sill is the value of the semivariance between two data values at the lag distance equal to the range. Ideally, when the lag distance is zero, the semivariance is supposed to be zero also. This is not always the case and at times a non-zero value of the semivariance is obtained when the lag distance is zero, and this value of the semivariance is what is referred to as the nugget effect, or simply the nugget. The nugget is usually an indicator of the interpolation error and can also be attributed to spatial sources of variation at distances smaller than the sampling interval. There are different kriging techniques, namely: ordinary kriging, simple kriging, cokriging, kriging

    with trend and universal kriging [15]. Of these, ordinary kriging is the one used in the current study and is the only one discussed further.

    2.2.1 Ordinary Kriging

    In ordinary kriging, the mean is assumed constant and unknown in the local neighbourhood of the prediction point. The kriging weights sum to unity and are computed from [14, 17, 18]:

    = 1 (12)

    where is the kriging weights vector matrix, 1 is the covariance matrix for the known data points and is the covariance vector matrix between the prediction data point and known data points. The kriging predictor is then given by [14, 18, 19]:

    functions (RBFs) are an efficient way of making predictions when one encounters scattered data problems [22]. Scattered data may not be of much use since one will be interested in knowing or estimating the values where data is not available, for continuity purposes. Interpolation and 1-D derivative approximations are well performed using RBFs especially where the data does not form a regular grid [23]. In [23], Carr, Fright and Beatson emphasize that RBFs do not impose any restrictions on the geometry of the measured data and therefore are very well suited for scattered data. RBFs are also known to produce sufficiently accurate interpolated values from relatively few data points and are able to retain small features of the data geometry as well as spatial distribution. Also, in [24], Franke reports that multiquadrics and thin plate splines produced the best outcome for the interpolation of scattered data. RBFs can interpolate values that fall outside the range of the measured data values. They are also known to be very sensitive to

    (, ) =

    ( , )

    (13)

    outliers. The most basic form of the RBF interpolant is given

    =1

    by [25]:

    where

    (, )

    is the data value at point

    (, ),

    is

    () =

    1 ( )

    (17)

    =

    =

    the weight associated with it and is as defined in (7).

    Normally, the weights are determined by minimizing the variance of the interpolation error, otherwise referred to as the estimation error variance, 2. This is usually achieved by the introduction of a function which contains the Lagrange parameter, given by [15]:

    Model

    Model Equation

    Cubic

    () = { + (72 8.753 + 3.755 0.757) 1

    + 1

    Gaussian

    () = + (1 (2))

    Exponent

    () = + (1 (3))

    Spherical

    3 1 3

    () = { + ( ) 1

    2 2

    + 1

    Model

    Model Equation

    Cubic

    () = { + (72 8.753 + 3.755 0.757) 1

    + 1

    Gaussian

    () = + (1 (2))

    Exponent

    () = + (1 (3))

    Spherical

    3 1 3

    () = { + ( ) 1

    2 2

    + 1

    = 2 + 2[1 1 ] (14)

    where denotes the Euclidian distance, , and () is some RBF, is the number of RBFs and is the real- valued weighting coefficient associated with each RBF.

    Table 1: Semivariogram models used [14, 18]

    =

    The minimization of the estimation error variance is then done by taking the derivative of the above function and then equating it to zero, that is, [15]:

    1 = 1

    = 0 (15)

    2

    =1

    Then, the ordinary kriging weights are computed from the following matrix equation [19]:

    Thus, RBF methods employ a linear combination of

    (1,1) (1,2) (1, ) 1 1

    (2,1) (2,2) (2, ) 1 2

    (0,1)

    (0,2)

    translates for one function () of a single variable. In some cases, it is pertinent to add a polynomial of first degree,

    =

    (16)

    (), that describes the constant as well as the linear

    (, 1) (, 2) (, ) 1 (0, ) [1 1 1 0] [ ] [ 1 ]

    where (i,j) is the covariance between measured locations and , (0, ) is the covariance between the prediction location 0 and measured location , and is the simple kriging weight for measured location . The four different kriging semivariogram models shown in Table 1 are used in

    the interpolation of the k-factor for South Africa in this study. In Table 1, is the nugget, is the sill and is the value of the lag distance divided by the range.

    2.3 Radial Basis Functions (RBFs)

    RBFs are real-valued, continuous univariate functions that are radialized by composition with the Euclidian distance as the norm, [20]. Radial basis functions can be thought of as fitting rubber-sheeted surface through measured data values using a mathematical function [21]. Radial basis

    portions of RBF interpolant in Equation (17), and in turn maintains positive-definiteness and stability of the solution; resulting in the following new expression for the RBF interpolant [26]:

    =1

    =1

    () = ( ) + () (18)

    Radial basis functions fall in two main categories: the piece- wise smooth and the infinitely smooth. Piece-wise smooth RBFs include powers (linear, cubic and quantic among others) and thin plate splines (TPS). The power RBFs are given by [27]:

    () = 21 , = 1,2,3, . . (19)

    The TPS is defined by [20, 25]:

    () = 2 (20)

    Infinitely smooth RBFs include inverse multiquadrics (IQ), multiquadric (MQ) and the Gaussian (GA) and are defined by the following expressions respectively [25, 28]:

    performing cross validation and this is the technique used here.

    The RMSE and MAE are used to ompare the performance

    () = 1

    1+()2

    (21)

    of the different interpolation methods used. These errors are given by [18, 31]:

    () = 1 + ()2 (22)

    () = ()2 (23)

    where is the shape parameter.

    The TPS and the MQ are the ones used for the interpolation

    1

    = (

    ( )2) (26)

    =1

    of the k-factor for South Africa and are discussed briefly next. TPSs are inspired by the concept of bending an

    =

    1

    | |

    (27)

    infinitely thin metal plate. They belong to a class of RBFs commonly referred to as polyharmonic or simply surface splines. Polyharmonic RBFs do not contain a shape parameter. Thin plate splines mathematically realize the idea that, when a metal plate is fixed at certain heights vertical to the plane, it tends towards a form of minimal energy, which is the form of minimal curvature [29]. They are more attractive method since they provide continuity and minimize the energy function below [23]:

    =1

    where is the measured value and is the cross value.

  3. MATERIALS AND METHODS

    The computed median and effective values of the k-factor values shown in Tables 2 and 3 below. These values are then interpolated to cover the rest of South Africa using the interpolation techniques discussed above, that is, kringing,

    2

    2 2

    2 2

    inverse distance weighting and radial basis function. The

    () = 2 2 + 2 ()

    + ( )

    2

    (24)

    seasonal and annual median and effective k-factor values are entered into a grid together with the locations coordinates;

    over all interpolants for which the energy functional is well

    defined. Thus in this particular sense, the TPS is the smoothest RBF interpolant.

    The function , which minimizes the following factor, below, (25), is an example of an exact RBF technique and is specifically referred to as the exact spline method [30, 31]:

    =1

    =1

    () + 2 [() ()]2 (25)

    where () = () + () is the source of random error,

    is the measured value at point and is the random error associated with it. The term () represents the smoothness

    of the function and the second term represents the proximity to the measured values. The smoothing function

    () determines the level of smoothness with respect to the accuracy of the interpolation. When the function is taken to zero, one performs interpolation but without any smoothing. Alternatively, pushing the value of the function to infinity results in least square fit of the data. Thus, it is important that one maintains a good smoothing trade-off for optimal interpolation. One of the best ways to achieve this is by performing generalized cross validation [17], and this is the approach adopted for this study. Cross validation statistics are the ones used for the error computations. Cross validation involves the removal of one data point at a time and then interpolating its value. This is done in turns for all data points and then the overall error statistics are calculated.

    The MQ, on the other hand, is popular for many applications and has some good approximation and interpolation properties. It represents a class of RBFs that are global in nature, containing a shape parameter and are infinitely differentiable [28, 31]. The MQ can also be optimized by

    the longitude and the latitude. For the case of kriging interpolation, the semivariogram model is then chosen. For RBF, the particular choice of the RBF is done. Else the method chosen is the IDW. The values are then interpolated to cover the rest of the country between the coordinates; latitude (15 S, 35 S) and longitude (22 E, 35.2 E). Cross validation of the interpolation statistics is then performed and then the associated RMSE and MAE errors computed. This process is repeated for the four semivariogram models in Table 1, the TPS and MQ RBFs, and for IDW where the minimum error criteria are used as the optimal value power parameter. The interpolated values for the method that produces the least error (best interpolation method) are then copied and saved in either Microsoft Excel or notepad files.

    The latitude, longitude and the k-factor matrices are then exported to MATLAB. A MATLAB code is then used for the mapping of the k-factor and development of the contour maps of the seasonal and annual k-factor for South Africa.

    Table 2: Annual median (50%) and effective (99.9%) values of the k-factor, 200m a.g.l

    Location

    Median(50%)

    (99.9%)

    Bloemfontein

    1.22

    0.51

    Cape Town

    1.29

    0.49

    Durban

    1.27

    0.53

    Polokwane

    1.26

    0.63

    Pretoria

    1.20

    0.66

    Upington

    1.18

    0.49

    Bethlehem

    1.16

    0.73

    May

    1.20

    0.38

    Aug

    1.19

    0.44

    Nov

    1.16

    0.37

    Bethlehem

    Feb

    1.16

    0.72

    May

    1.21

    0.78

    Aug

    1.17

    0.58

    Nov

    1.14

    0.64

    May

    1.20

    0.38

    Aug

    1.19

    0.44

    Nov

    1.16

    0.37

    Bethlehem

    Feb

    1.16

    0.72

    May

    1.21

    0.78

    Aug

    1.17

    0.58

    Nov

    1.14

    0.64

    Location

    season

    50%

    99.9%

    Bloemfontein

    Feb

    1.24

    0.49

    May

    1.21

    0.53

    Aug

    1.21

    0.54

    Nov

    1.22

    0.50

    Cape Town

    Feb

    1.26

    0.41

    May

    1.29

    0.48

    Aug

    1.3

    0.52

    Nov

    1.27

    0.46

    Durban

    Feb

    1.28

    0.54

    May

    1.26

    0.58

    Aug

    1.26

    0.55

    Nov

    1.26

    0.63

    Polokwane

    Feb

    1.27

    0.64

    May

    1.27

    0.57

    Aug

    1.23

    0.66

    Nov

    1.23

    0.64

    Pretoria

    Feb

    1.22

    0.59

    May

    1.19

    0.58

    Aug

    1.19

    0.61

    Nov

    1.20

    0.62

    Upington

    eb

    1.14

    0.43

    Location

    season

    50%

    99.9%

    Bloemfontein

    Feb

    1.24

    0.49

    May

    1.21

    0.53

    Aug

    1.21

    0.54

    Nov

    1.22

    0.50

    Cape Town

    Feb

    1.26

    0.41

    May

    1.29

    0.48

    Aug

    1.3

    0.52

    Nov

    1.27

    0.46

    Durban

    Feb

    1.28

    0.54

    May

    1.26

    0.58

    Aug

    1.26

    0.55

    Nov

    1.26

    0.63

    Polokwane

    Feb

    1.27

    0.64

    May

    1.27

    0.57

    Aug

    1.23

    0.66

    Nov

    1.23

    0.64

    Pretoria

    Feb

    1.22

    0.59

    May

    1.19

    0.58

    Aug

    1.19

    0.61

    Nov

    1.20

    0.62

    Upington

    Feb

    1.14

    0.43

    Table 3: Seasonal median and effective k-factor values

    while for August the largest errors are produced by the MQ. For November, the TPS produces the largest RMSE errors while the MQ produces the largest MAE.

    Thus, overall, we note that the IDW method posts the best error performance in terms of either the RMSE or the MAE or both. The error performance variations can be attributed to differences in k-factor values for both seasonal and annual cases as well as median and effective values as well. Also the errors variations are subject to the different parameters in each individual interpolation function as well as the spatial distribution of the data. It is worthy to note there is

  4. INTERPOLATION RESULTS AND DISCUSSION

    The interpolation errors are tabulated in Tables 4-7. Table 4 shows the seasonal median k-factor interpolation errors. From this table, we see that for the months of February, August and November, the Gaussian semivariogram model produces the worst error performance while IDW produces the least error. For the month of May, IDW produces the least RMSE error while the cubic semivariogram model produces the least MAE error. Table 5 shows the annual median k-factor interpolation errors. From this table, we see that IDW method produces the least RMSE and MAE errors while the Gaussian semivariogram model produces the largest interpolation errors. Table 6 shows the annual effective k-factor interpolation errors. From this table, IDW produces the least errors while Gaussian semivariogram model produces the largest RMSE error and TPS produces the largest MAE error. Table 7 shows the seasonal effective k-factor interpolation errors. From this table, we see that IDW performs best in terms of the two error criteria used for the months of February, May and November. For the month of August, the best RMSE performance is produced by IDW while the best MAE is produced by the Gaussian semivariogram model. Also, from the same table, the Gaussian semivariogram model posts the largest errors for the months of February and May

    always no preferred interpolation method over another and that is why the two different measures of fit are used to determine the best method in different application scenarios. In our case, the IDW comes out as the best method (in terms of error) and is the one chosen for the mapping of the k- factor. The strong IDW error performance could be attributed to the fact that all its interpolated values fall within the range of the measured values. All the other methods used can interpolate values outside the range of the measured data. The dismal performance exhibited by the RBFs could be attributed to the fact that they do well with gently varying data and this is not the case with the values measured. Figures 1-5 show the contour maps for the annual and seasonal median k-factor. Figures 6-10 show the contour maps for the annual and seasonal effective k-factor.

    Table 4: Seasonal median k-factor interpolation errors

    Errors

    Kriging semivariogram models

    Radial Basis Functions

    Inverse Distance weighting

    Spherical

    Exponential

    Gaussian

    Cubic

    TPS

    MQ

    Feb

    RMSE

    0.064

    0.065

    0.205

    0.096

    0.099

    0.090

    0.046

    MAE

    0.059

    0.060

    0.174

    0.081

    0.091

    0.081

    0.037

    May

    RMSE

    0.038

    0.042

    0.080

    0.038

    0.054

    0.065

    0.034

    MAE

    0.030

    0.033

    0.055

    0.029

    0.038

    0.049

    0.031

    Aug

    RMSE

    0.037

    0.040

    0.091

    0.049

    0.056

    0.061

    0.033

    MAE

    0.039

    0.036

    0.074

    0.043

    0.050

    0.051

    0.029

    Nov

    RMSE

    0.056

    0.055

    0.204

    0.088

    0.088

    0.076

    0.040

    MAE

    0.051

    0.052

    0.177

    0.074

    0.078

    0.70

    0.032

    Table 5: Annual median k-factor interpolation errors

    Errors

    Kriging semivariogram models

    Radial basis functions

    Inverse distance weighting

    Cubic

    Exponential

    Spherical

    Gaussian

    TPS

    MQ

    RMSE

    0.071

    0.053

    0.052

    0.134

    0.076

    0.078

    0.042

    MAE

    0.061

    0.049

    0.047

    0.110

    0.069

    0.068

    0.033

    Table 6: Annual effective k-factor interpolation errors

    Errors

    Kriging semivariogram models

    Radial basis functions

    Inverse distance weighting

    Cubic

    Exponential

    Spherical

    Gaussian

    TPS

    MQ

    MAE

    0.126

    0.086

    0.088

    0.023

    0.126

    0.112

    0.07

    RMSE

    0.146

    0.101

    0.105

    0.285

    0.159

    0.134

    0.066

    30

    30

    E

    E

    35

    35

    E

    E

    1.22

    1.22

    1.2

    1.2

    1.18

    1.18

    15

    15

    E

    E

    20

    20

    E

    E

    Table 7: Seasonal effective k-factor interpolation errors

    Errors

    Kriging semivariogram models

    Radial Basis Functions

    Inverse Distance weighting

    Spherical

    Exponential

    Gaussian

    Cubic

    TPS

    MQ

    Feb

    RMSE

    0.106

    0.102

    0.402

    0.169

    0.176

    0.138

    0.067

    MAE

    0.086

    0.085

    0.355

    0.157

    0.147

    0.114

    0.046

    May

    RMSE

    0.134

    0.129

    0.447

    0.20

    0.218

    0.179

    0.080

    MAE

    0.110

    0.105

    0.392

    0.181

    0.173

    0.142

    0.062

    Aug

    RMSE

    0.058

    0.063

    0.056

    0.052

    0.105

    0.107

    0.044

    MAE

    0.042

    0.046

    0.030

    0.038

    0.068

    0.075

    0.036

    Nov

    RMSE

    0.060

    0.065

    0.094

    0.058

    0.106

    0.106

    0.048

    MAE

    0.050

    0.055

    0.075

    0.047

    0.076

    0.081

    0.045

    25 E

    25 E

    1.22

    1.22

    1.24

    1.24

    1.2

    1.2

    1.26

    1.26

    1.28

    1.28

    1.2

    1.2

    1.24

    1.24

    1.22

    1.22

    1.26

    1.26

    1.2

    1.2

    1.22

    1.22

    1.26

    1.26

    1.24

    1.24

    1.18

    1.18

    25

    25

    S

    S

    30

    30

    S

    S

    35

    35

    S

    S

    Figure 1: Three-year annual median k-factor contours for South Africa

    25

    25

    E

    E

    25 E

    25 E

    1.2

    1.2

    25 E

    25 E

    30

    30

    E

    E

    30 E

    30 E

    30 E

    30 E

    35

    35

    E

    E

    35 E

    35 E

    35 E

    35 E

    1.24

    1.24

    1.2

    1.2

    1.22

    1.22

    1.18

    1.18

    1.24

    1.24

    1.16

    1.16

    1.23

    1.23

    1.24

    1.24

    1.21

    1.21

    1.26

    1.26

    1.2

    1.2

    1.2

    1.2

    1.2

    1.2

    1.2

    1.2

    1.22

    1.22

    1.16

    1.16

    1.22

    1.22

    1.22

    1.22

    1.18

    1.18

    1.24

    1.24

    1.18

    1.18

    1.2

    1.2

    1.21

    1.21

    1.18

    1.18

    1.22

    1.22

    1.18

    1.18

    1.24

    1.24

    1.16

    1.16

    1.2

    1.2

    1.22

    1.22

    1.14

    1.14

    1.28

    1.28

    1.2

    1.2

    1.26

    1.26

    1.2

    1.2

    1.22

    1.22

    1.22

    1.22

    1.26

    1.26

    1.18

    1.18

    1.21

    1.21

    1.24

    1.24

    1.16

    1.16

    1.2

    1.2

    1.24

    1.24

    1.22

    1.22

    1.23

    1.23

    1.26

    1.26

    1.14

    1.14

    1.16

    1.16

    1.22

    1.22

    1.21

    1.21

    1.2

    1.2

    1.19

    1.19

    1.24

    1.24

    1.22

    1.22

    1.26

    1.26

    1.22

    1.22

    1.18

    1.18

    1.22

    1.22

    1.24

    1.24

    15 E

    15 E

    15

    15

    E

    E

    25 S

    25 S

    25

    25

    S

    S

    30 S

    30 S

    30

    30

    S

    S

    35 S

    35 S

    35

    35

    S

    S

    15 E

    15 E

    25 S

    25 S

    20 E

    20 E

    20

    20

    E

    E

    20 E

    20 E

    Figure 2: Three-year seasonal median k-factor contours for South Africa for the month of February

    25 E

    25 E

    1.18

    1.18

    30 E

    30 E

    35 E

    35 E

    1.22

    1.22

    1.24

    1.24

    1.22

    1.22

    1.23

    1.23

    1.25

    1.25

    1.2

    1.2

    1.26

    1.26

    1.27

    1.27

    1.2

    1.2

    1.28

    1.28

    1.2

    1.2

    1.25

    1.25

    1.18

    1.18

    1.2

    1.2

    1.22

    1.22

    1.24

    1.24

    1.26

    1.26

    1.27

    1.27

    1.25

    1.25

    1.18

    1.18

    1.16

    1.16

    1.26

    1.26

    30 S

    30 S

    35 S

    35 S

    15 E

    15 E

    25 S

    25 S

    1.2

    1.2

    1.23

    1.23

    20 E

    20 E

    1.24

    1.24

    Figure 3: Three-year seasonal median k-factor contours for South Africa for the month of May

    1.26

    1.26

    1.28

    1.28

    1.24

    1.24

    1.26

    1.26

    1.28

    1.28

    1.24

    1.24

    1.26

    1.26

    1.28

    1.28

    30 S

    30 S

    35 S

    35 S

    1.24

    1.24

    Figure 4: Three-year seasonal median k-factor contours for South Africa for the month of August

    Figure 5: Three-year seasonal median k-factor contours for South Africa for the month of November

    25

    25

    E

    E

    30 E

    30 E

    30

    30

    E

    E

    35 E

    35 E

    35

    35

    E

    E

    0.5

    0.5

    0.55

    0.55

    0.65

    0.65

    0.6

    0.6

    0.55

    0.55

    0.45

    0.45

    0.5

    0.5

    0.55

    0.55

    0.5

    0.5

    0.6

    0.6

    0.5

    0.5

    0.6

    0.6

    0.55

    0.55

    0.5

    0.5

    0.55

    0.55

    0.65

    0.65

    0.55

    0.55

    0.6

    0.6

    0.65

    0.65

    0.65

    0.65

    0.6

    0.6

    0.65

    0.65

    0.5

    0.5

    0.65

    0.65

    0.6

    0.6

    15

    15

    E

    E

    25

    25

    S

    S

    30

    30

    S

    S

    35

    35

    S

    S

    15 E

    15 E

    25 S

    25 S

    20

    20

    E

    E

    20 E

    20 E

    Figure 6: Three-year annual effective k-factor contours for South Africa

    25 E

    25 E

    0.55

    0.55

    0.5

    0.5

    0.45

    0.45

    0.55

    0.55

    0.5

    0.5

    0.45

    0.45

    0.4

    0.4

    30 S

    30 S

    35 S

    35 S

    Figure 7: Three-year seasonal effective k-factor contours for South Africa for the month of February

    25

    25

    E

    E

    30

    30

    E

    E

    35

    35

    E

    E

    0.5

    0.5

    0.46

    0.46

    0.6

    0.6

    0.52

    0.52

    0.54

    0.54

    0.58

    0.58

    0.56

    0.56

    0.52

    0.52

    0.56

    0.56

    0.54

    0.54

    0.58

    0.58

    0.4

    0.4

    0.6

    0.6

    0.62

    0.62

    0.5

    0.5

    0.55

    0.55

    0.64

    0.64

    0.45

    0.45

    0.5

    0.5

    0.4

    0.4

    0.6

    0.6

    0.65

    0.65

    0.35

    0.35

    0.6

    0.6

    0.55

    0.55

    0.55

    0.55

    0.7

    0.7

    0.5

    0.5

    0.5

    0.5

    0.6

    0.6

    0.45

    0.45

    0.4

    0.4

    0.65

    0.65

    0.45

    0.45

    0.65

    0.65

    0.48

    0.48

    15

    15

    E

    E

    25

    25

    S

    S

    15 E

    15 E

    25 S

    25 S

    30 S

    30 S

    35 S

    35 S

    0.7

    0.7

    20 E

    20 E

    20

    20

    E

    E

    Figure 8: Three-year seasonal effective k-factor contours for South Africa for the month of May

    25 E

    25 E

    30 E

    30 E

    35 E

    35 E

    0.48

    0.48

    0.5

    0.5

    0.54

    0.54

    0.52

    0.52

    0.52

    0.52

    0.5

    0.5

    0.54

    0.54

    0.56

    0.56

    30

    30

    S

    S

    35

    35

    S

    S

    15 E

    15 E

    20 E

    20 E

    Figure 9: Three-year seasonal effective k-factor contours for South Africa for the month of August

    0.4

    0.4

    0.65

    0.65

    0.55

    0.55

    0.6

    0.6

    0.5

    0.5

    0.45

    0.45

    0.4

    0.4

    0.6

    0.6

    0.45

    0.45

    0.5

    0.5

    0.65

    0.65

    0.6

    0.6

    0.5

    0.5

    0.55

    0.55

    0.4

    0.4

    25 S

    25 S

    30 S

    30 S

    0.45

    0.45

    0.6

    0.6

    0.65

    0.65

    0.55

    0.55

    0.5

    0.5

    0.45

    0.45

    35 S

    35 S

    Figure 10: Three-year seasonal effective k-factor contours for South Africa for the month of November

  5. CONCLUSION

25 E

25 E

30 E

30 E

35 E

35 E

0.6

0.6

0.5

0.5

0.55

0.55

0.65

0.65

In this paper, three spatial interpolation methods with seven different interpolation options have been considered in an effort to extend discrete k-factor values measured at seven locations in South Africa to cover the rest of the country. In order to determine which method best fits the k-factor values measured, two error criteria have been used to identify the most appropriate method. The performance of any spatial interpolation method is driven by the geometrical distribution of the data, the variation between the maximum and minimum values as well as the interpolation function used and the parameters that control the interpolation. The IDW method has been used to generate the contour maps of the k-factor for South Africa owing to its strong error performance as compared to all the other six options considered. Overall, the median k-factor contours show a decreasing trend of the k-factor as we move from the coastal regions towards the semi-arid regions in the northern and north-western parts of South Africa. Also, the effective k- factor contours show a decreasing trend of the k-factor as we move from east to west. The decreasing trend of the k-factor is consistent with the values obtained by Afullo and Odedina for Maun, Botswana [32]. This would mean that, moving further north towards Botswana, the k-factor values would diminish and a value close to the one obtained in their earlier reports would be obtained. All the interpolated values lie between the respective maximum and the values of the k- factor observed and, thus, the theory of the edge effect associated with IDW interpolation has been proven.

ACKNOWLEDGENTS

The author wishes to acknowledge the South African Weather Service (SAWS) for the data used in this work.

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