The modified parabolic equation (MPE) is a standard parabolic equation that has been slightly modified to include some clear-air radioclimatic propagation parameters. The two basic parameters that were included in the modifications are the geoclimatic factor and the effective earth radius factor (popularly known as k-factor). Geoclimatic factor is the basic parameter needed to implement the effect of multipath propagation in a clear-air environment [54, 58]. On the other hand, k-factor variable is needed for diffraction fading modeling in clear-air radioclimatic studies [42, 44, 60].
Durban (K) 2003/2004 Botswana (K) 1996 Month
ITU-R approach
AF approach
% deviation
ITU-R approach
AF approach
% deviation January 0.001684 0.001670 0.8314
February 0.170679 0.159465 6.5702 0.010935 0.010091 7.7183 March 0.003329 0.003173 4.6864 0.001010 0.001082 7.1287
April 0.002140 0.002141 0.0467
May 0.002085 0.002003 3.9328 0.008151 0.007338 9.9742
June 0.001047 0.000982 6.2082 0.000712 0.000686 3.6517
July
0.001055 0.001010 4.2654
August
0.021970 0.022697 3.3091 0.000172 0.000172 0
September 0.000809 0.000731 9.6415
October 0.011306 0.010335 8.5884
November 0.000521 0.000480 7.8695 0.002060 0.001959 4.9029 December 0.000737 0.000736 0.1357 0.013906 0.014177 1.9488
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 173
The geoclimatic factor as determined in equation (4.31) of section 4.7 is incorporated into equation (3.93) in chapter three in order to come up with the first modified parabolic equation MPE1. Equation (3.93) is restated as equation (4.31) below:
u = iko
(
1 + 1+ X u)
x
∂ −
∂ (4.32)
where k is the wave number in free space, 0 X = 1 k
(
20)(
∂ ∂2 z2) (
+ n2−1)
and n is the refractive index. It will be noted in equation (4.32) that the variable X is a function of refractive index. So in MPE1 the refractive index portion of variable X is replaced with the geoclimatic factor GF , equation (4.31) which is also a function of refractive index. Then equation (4.32) now becomes:∂u∂xGF = iko
(
−1 + 1+ XGF)
u (4.33) WhereGF
u is the reduced field component of the first modified parabolic equation (MPE1),
GF
X is
( )( )
F
2 2 2
G 0 F
X = 1 k ∂ ∂z + G . The solution of the MPE1 is very similar to that of SPE detailed in previous section. Since GF can be determined numerically, from the radio propagation data available for the research; the remaining task of the solution is resolved by the M-File MATLAB software code developed to implement the solution.
The features of this modification can better be observed by plotting path loss against height and path loss against range separately. This is done in the next set of plots for the two stations and for different seasonal months. Figures 4.35 (a) and (b) below shows the path loss against height for Durban and Botswana in the summer months of February while figures 4.36 (a) and (b) shows path loss against height plots for Durban and Botswana in the winter months of August.
In figures 4.35 and 4.36 where path loss versus height is plotted at two coverage ranges (i.e. 10km and 20km), it can be observed that for all the plots in this section at a range of 10km from the transmitting source, path loss increases initially slowly as the height increases for the first 0 –10m above ground level (a.g.l.) and exponentially later above 10m above ground level (a.g.l). On the other hand, at a longer range of 20km, the behaviour of the plot is quite different from what is explained above as one observes from figures 4.35 – 4.36. In this case, path loss decreases initially slowly from
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 174
a maximum value of about 5.5 dB to a value ranging from 2.85dB – 4dB depending on the location and season. It then starts to increase slowly initially from this minimum value for the first 4m height and increase exponentially beyond 10 m a.g.l.
This behaviour of the plot at 20km range shows duct occurrence probability, ducting is a very significant phenomena in clear-air study because it has the effect of trapping signal as if propagating in a wave guide. The duct depth and thickness varies from season to season and for different locations as seen from figures 4.35 – 4.36.
Figure 4.35 (a) Path Loss against Height at 19.5 GHz for MPE1 February 2004 Durban
Figure 4.35(b) Path Loss against Height at 19.5 GHz for MPE1 February 1996 Botswana
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 175 Figure 4.36(a) Path Loss against Height at 19.5 GHz for
MPE1 August 2004 Durban
Figure 4.36(b) Path Loss against Height at 19.5 GHz for MPE1 August 1996 Botswana
We are just considering MPE1 modification of parabolic equation in these plots, and MPE1 modification is arrived at by incorporating only the geoclimatic factor variable into the parabolic equation. We now look at the effect of geoclimatic factor as captured by our plot, geoclimatic factor is the clear-air variable which is used for multipath fading modeling, it is therefore expected that multipath fading effect should be more prominent in a flat terrain such as Botswana as compared to an hilly terrain like Durban where diffraction fading should be more prominent.
Further analysis of the plots reveals the following: at a height of 10 m a.g.l and range 10 km, a path loss value of 2.5 dB is observed in Durban while at the same height and range, a value of 2.6 dB is observed in Botswana during the summer months of February (see Figure 4.35(a) and (b)). While still at the same height level of 10m a.g.l but now moving to a longer range of 20 km, a path loss value of 5 dB is observed for Durban while that of Botswana is 5.6dB during the summer months of February (see Figure 4.35 (a) and (b)).
Moving to a higher height level of 30m a.g.l for the same months (i.e. February) and at a range of 10 km shows that Durban has 2.8dB path loss while Botswana has a higher path loss value of 3 dB. If we now increase the coverage range to a distance of 20 km at the same height level of 30 m a.g.l for this
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 176
same summer month of February, our result will be as follows: Durban has a path loss value of 5.6 dB while Botswana has a higher value of 6 dB.
Doing the same analysis for the winter months of August in the two stations shows similar trends to our discussion above with slight differences in values. At the height level of 10 m a.g.l and coverage range of 10 km, Durban has a path loss value of 2 dB while Botswana has a higher value of 2.6 dB.
Taking a longer range of 20 km at the same height level of 10 m a.g.l gives Durban a path loss value of 4.3 dB while that of Botswana is 5.5 dB (see Figure 4.36 (a) and (b)).
At a higher height level of 30 m a.g.l in the winter month of August, the result is as follows: for a coverage range of 10 km, Durban has a path loss value of 2.5 dB while Botswana records a path loss value of 2.9 dB. If a longer range of 20 km is considered, then the results will be as follows: Durban has a path loss value of 5.1 dB while Botswana has a path loss value of 6 dB at this range and height level (see Figure 4.36(a) and (b)).
All the discussions above show that for the two seasonal months considered, Botswana always has a higher path loss value at the selected ranges and height levels. This is not surprising since we are at this stage dealing with the first modification of parabolic equation which incorporates multipath fading effect and Botswana being a flatter terrain than Durban is expected to have more fading due to multipath than Durban as explained earlier.
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 177 Figure 4.37 (a) Path Loss against Range at 19.5 GHz for
MPE1 February 2004 Durban Figure 4.37(b) Path Loss against Range at 19.5 GHz for MPE1 February 1996 Botswana
Figure 4.38(a) Path Loss against Range at 19.5 GHz for
MPE1 August 2004 Durban Figure 4.38(b) Path Loss against Range at 19.5 GHz for MPE1 August 1996 Botswana
Clear-Air Radioclimatological Modeling for Terrestrial Line of Sight Links in Southern Africa Odedina P.K. Aug , 2010 178
In figures 4.37 (a) and (b), the path loss is plotted against the coverage range or distance for the summer months of February in both Durban and Botswana. On the other hand in figures 4.38 (a) and (b), path loss is plotted against range for the winter months of August in both Durban and Botswana.
Several deductions can be made from these figures. It will be noticed that all the plot figures 4.35 – 4.38 show a progressive increase of path loss as the coverage range or distance increases, this is expected since theoretically more signal is lost as the propagation distance increases. The way each plot increases in path loss with distance is very unique to each location and season as can be seen from figures 4.35 – 4.38.
The remaining plots (Figure 4.37 – 4.38) corroborate most of our discussions in the preceding paragraphs. This is a plot of path loss versus range graphs and as could be seen from the plots, all the plots conspicuously show a gradual increase of the path loss with increasing range as expected.
However, it should be noted that in all the plots, the two heights (10 m and 20 m a.g.l) selected in these plots coincide for some range distance before the difference becomes obvious after this range.
The reason for this will be study further in future refinement of the model.