BT plc \[United Kingdom\]

Testing the distance corrections to the bullington method proposed by
switzerland

# Introduction

A recent paper \[1\] submitted into the CG 3K-1 by Switzerland has
proposed a modification to the distance correction used within the
Bullington diffraction method to overcome the objections \[2\]\[3\] that
Bullington underestimates the path loss at long path length. The
correction was based upon performing a polynomial fit to Monte Carlo
data plotted in a UK paper comparing the competing diffraction methods
to the behaviour of ITU-R P.1546.

Various degrees of polynomial were tried and a final choice was made to
use a ninth order polynomial. It was not made clear how this was to be
incorporated into the method. It is assumed here that the correction is
a substitute for and not an addition to the current *10.0+0.02d* terms
in the following equation, which has been widely published by the UK and
not contradicted:

$L_{d} = L_{s} + \left( \text{10}\text{.}0 + 0\text{.}2d \right)\left( 1 - \text{exp}\left( \frac{- L_{s}}{6\text{.}0} \right) \right)$
(1)

Hence the new equations tested were:

$L_{\text{poly}} = \left( 5\text{.}\text{145} + 2\text{.}\text{276}d - 4\text{.}\text{338}d^{2} + 3\text{.}\text{7494}d^{3} + 5\text{.}\text{992}d^{4} - 6\text{.}\text{557}d^{5} - 1\text{.}\text{766}d^{6} + 4\text{.}\text{152}d^{7} - 1\text{.}\text{664}d^{8} + 0\text{.}\text{214}d^{9} \right)$
(2)

$L_{d} = L_{s} + L_{\text{poly}}\left( 1 - \text{exp}\left( \frac{- L_{s}}{6\text{.}0} \right) \right)$
(3)

Since the data points used for the fit were conveniently given in the
paper, a further examination was made of the fitting function since the
degree of ripple present with the polynomial did not seem to be
justified by the data points. A much simpler expression was found that
fitted the data points without any ripple. Both these new distance
corrections were compared to the original Bullington and 3-edge Deygout
methods for two 1000 km smooth earth paths. One of the paths was assumed
to be entirely over water and the other entirely over land. Using a 200
m step size, the path loss for the P.1812 diffraction loss mechanism
together with the overall loss was calculated from the transmitter to a
receiver that was progressively stepped along the path profile to the
full extent of the path. Further plots were made showing the variation
of each model mechanism as functions of time percentage for different
distance paths.

# The Distance Correction Functions

Using the data points provided within the Swiss paper, the following
function was developed to give a plot with far less ripple, as it was
uncertain at an early stage in the study if the ripple would have other
adverse effects at the shorter path lengths where the transition from
the diffraction to troposcatter mechanisms was expected to occur:

$L_{\text{pow}} = 5\text{.}\text{947}^{\left( 2\text{.}\text{586}d - 5\text{.}\text{863} \right)} + \text{10}\text{.}\text{442}\left\lbrack 1\text{.}0 - \left( \frac{d - 3}{4} \right)^{2} \right\rbrack$
(4)

$L_{d} = L_{s} + L_{\text{pow}}\left( 1 - \text{exp}\left( \frac{- L_{s}}{6\text{.}0} \right) \right)$
(5)

Figure 1 The Bullington distance correction functions

![](media/image1.wmf){width="6.736805555555556in"
height="5.971527777777778in"}

# Testing on Smooth Earth Paths

Two smooth earth paths of 1000 km length were generated using a 200 m
step size, one over sea and one over land. A version of P.1812 was
created in which the current 3-edge Deygout diffraction model was
replaced with the Bullington method with the ability to select between
the existing linear and the two new non-linear distance correction
terms. Details for incorporating the original Bullington implementation
into P.1812 were given in an earlier CG paper. \[4\]. This paper
described how the method could be made to work at other than median time
percentages. The two earlier non-linear distance corrections were also
coded into the new version with the ability to software switch between
them at model run time.

A frequency of 100 MHz, terminal heights of 20 m and time percentage of
10% was selected to allow all the model mechanisms to become dominant
under certain conditions.

## An Inland Path

Figure 2 shows the variation in the diffraction loss for an inland path
using each of the Bullington variants in comparison to the current
3-edge Deygout method. It is apparent that all of the Bullington
variants diverge from the 3-edge Deygout method increasingly as the
receiver moves further from the transmitter. There are only marginal
differences between the two non-linear distance correction
implementations and either could be used if a worthwhile effect was
possible although the stability of a ninth order polynomial is likely to
be worse. The non-linear correction crosses over the original Bullington
linear term at approximately 700 km as predicted by Figure 1.

Figure 2 The diffraction loss for the different methods on an inland
path

![](media/image2.wmf){width="7.324305555555555in"
height="5.218055555555556in"}

However, the overall effect of using this type of distance correction is
shown to be minimal on long smooth paths if the intention is to ensure
that the diffraction loss approaches that of Deygout. Figure 3 shows the
overall loss from using the full P.1812 procedure with the different
diffraction distance corrections. The slight change of slope on the
3-edge Deygout line occurs at the point where the model mechanism
changes. This is not evident with any of the Bullington variants.

Figure 3 The overall loss for the different methods on an inland path

![](media/image3.wmf){width="7.317361111111111in"
height="5.218055555555556in"}

The next group of six figures looks in detail at the individual model
mechanisms as functions of time percentage for two path distances, 400
and 100 km using Deygout and Bullington with a linear or non-linear
correction.

For the Deygout method at a 400 km path length, it is seen in Figure 4
that the dominant model mechanism changes from ducting to troposcatter
at about 10% time as the time percentage increases. Diffraction is never
the dominant mechanism. Contrast this to the two Bullington cases in the
two subsequent figures. Here the dominant mechanism is that of
diffraction until the time percentages reaches about 2-3% when ducting
finally takes over.

For a 100 km path, the situation is slightly changed in that, even with
the Deygout method, diffraction manages to dominate above 7% of time.
With Bullington, the switch to ducting dominance is barely achievable
within the time percentage applicability of the recommendation, the
threshold being only about 2%.

Figure 4 Different model losses as functions of time % on an inland
path.

![](media/image4.wmf){width="5.50625in" height="4.452083333333333in"}

Figure 5 Different model losses as functions of time % on an inland
path.

![](media/image5.wmf){width="5.50625in" height="4.452083333333333in"}

Figure 6 Different model losses as functions of time % on an inland
path.

![](media/image6.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 7 Different model losses as functions of time % on an inland
path.

![](media/image7.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 8 Different model losses as functions of time % on an inland
path.

![](media/image8.wmf){width="5.50625in" height="4.456944444444445in"}

Figure 9 Different model losses as functions of time % on an inland
path.

![](media/image9.wmf){width="5.5125in" height="4.456944444444445in"}

## A Sea Path

The tests made in the previous section were repeated here but with the
zone code set to be sea. The expectation is that the ducting mechanism
will become even more dominant, even for higher percentages of time.
Figure 10 is very little changed from Figure 1. Small changes are caused
by the different *β~0~*% time altering the lower of the two fixed points
from which a 10% time percentage value is interpolated. However there is
a marked change in the overall loss shown in Figure 11 for the sea path.
All of the methods are superimposed. The explanation is that, at 10%
time, the ducting mechanism is always dominant, whichever method is
chosen. This plot does, however, mask the changes at higher time
percentages which Figure 12 to Figure 17 reveal. As before, these
figures show the model mechanism dominance changes with time percentage.

Figure 12 illustrates that, indeed, the ducting mechanism is pre-eminent
for Deygout at 400 km. The switch to troposcatter only occurs at median
time. Interestingly the P.1812 model blending procedures, which prevent
the ducting loss falling lower than the free space value on long
transhorizon path kicks in at around 2% time. Again the two Bullington
methods in Figure 13 and Figure 14 obviate the need for a troposcatter
model mechanism as ducting dominance is replaced by diffraction
dominance at about 25%. Very similar trends are seen in Figure 15
through to Figure 17 at 100 km path lengths.

Figure 10 The diffraction loss for the different methods on a sea path

![](media/image10.wmf){width="6.522916666666666in"
height="4.647222222222222in"}

Figure 11 The overall loss for the different methods on a sea path

![](media/image11.wmf){width="6.522916666666666in"
height="4.647222222222222in"}

Figure 12 Different model losses as functions of time % on a sea path.

![](media/image12.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 13 Different model losses as functions of time % on a sea path.

![](media/image13.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 14 Different model losses as functions of time % on a sea path.

![](media/image14.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 15 Different model losses as functions of time % on a sea path.

![](media/image15.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 16 Different model losses as functions of time % on a sea path.

![](media/image16.wmf){width="5.5125in" height="4.456944444444445in"}

Figure 17 Different model losses as functions of time % on a sea path.

![](media/image17.wmf){width="5.5125in" height="4.456944444444445in"}

# Discussion

The foregoing graphs all point towards a similar conclusion. It is very
difficult to formulate a distance correction that has the path distance
as the only input parameter without introducing a very large dB offset
at the longer distances to overcome the effects of earth curvature,
particularly on very smooth paths where knife edge terrain string points
are largely absent. The single virtual knife-edge that Bullington uses
is incapable of creating the high enough diffraction losses required for
troposcatter to become dominant over diffraction.

It is questionable whether creating ever-more sophisticated distance
corrections is the correct approach to take as it is a tacit admission
that Bullington is little better than *FSL +x(d)* dB on these longer
paths.

# Conclusion

This paper has shown the use of a Bullington distance correction whether
it is linear or non-linear in nature does not achieve the high
diffraction losses required to dominate troposcatter on long smooth
paths without introducing some arbitrarily high dB offsets. Indeed,
these offsets can be a higher dB figure than the loss the virtual edge
itself is creating.

Nevertheless, the detailed and carefully taken measurements by the Swiss
administration show that there are occasions, particularly in
mountainous terrain when the current Deygout method is also in error by
up to 20-30 dB. A critical re-appraisal of the documents produced by
Switzerland is required in an attempt to identify the root causes that
lead Bullington to achieve losses closer to measurements. It is equally
evident from the numerous studies undertaken to date that making
deductions using real terrain profiles is fraught with difficulty as the
number of permutations is so very high.

Both diffraction methods use relatively simple geometric constructs to
determine a few key parameters from the terrain profile points and then
ignore the remainder. An experiment is being devised to bypass this
stage by creating these key features directly using synthetic terrain
profiles in a very controllable manner. The essential differences that
lead to both the Bullington and Deygout method being in error over
different parts of the parameter space can then be identified. It is
hoped that this deeper understanding will lead to an entirely different
approach to creating a hybrid model, which can combine the best parts of
both Bullington and Deygout. It is intended that preliminary results
will be available in time for the CG workshop in London in March.

# References

\[1\] CGD-14 "Proposals for a Distance correction for Bullington",
Markus Liniger, Switzerland

\[2\] CGD-05 Bullington and 3-edge diffraction methods compared with
measured results, UK

\[3\] CGD-13, "Long Path Testing", Richard Rudd, UK

\[4\] CGD-12 "Deygout Bullington Comparison Draft B", Malcolm Hamer, UK

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