Correspondence Group 3K-1: Workshop 11-13 March 2008, London

**Investigation into adjusting the mean loss predicted by the Bullington
model**

# Introduction

The Bullington diffraction model is a candidate to replace the 3-edge
model in P.1812. It has demonstrated lower standard deviation of
discrepancies when compared to measured results for paths generally less
than 100 km. But for paths longer than about 100 km it increasingly
predicts lower losses than the 3-edge method. Unless corrected, this
would lead to seriously under-estimated interference levels when P.1812
is used for coordination.

This document reports an investigation into an empirical correction to
the Bullington model to avoid this effect. At present the Bullington
model has an empirical correction which depends only on path length, as
does the 3-edge model. The new correction depends also on frequency and
antenna heights. It is described here as additional to the current
empirical correction.

The new correction modifies the mean of predicted losses. However, the
new correction makes least change at short distances, and does not alter
the way the Bullington responds to path-to-path differences in terrain
shape. It is thus expected that similar standard deviations when
compared to measurements will be retained.

To obtained the required correction, both diffraction models were run
for a large number of paths selected randomly in the UK grid rectangle E
0 to 700 km and N 0 to 1100 km for different combinations of frequency
and antenna heights. The path lengths were arranged such that log(*d*)
was uniformly distributed between 0 and 3 where *d* is path length in
km. Both diffraction models were used as currently implemented for tests
conducted in the UK, including their present empirical corrections, and
with taper applied to the Bullington method.

The differences between the two diffraction losses for each path were
used to derived an empirical correction which, when added to the loss
predicted by the Bullington method, tends to reduce the difference to
zero.

# Definition of Bullington model

The model to which the correction described here is applied is as
follows. The Bullington construction is applied to a NLOS path to obtain
the knife-edge diffraction loss *L~de~* produced by the virtual point
defined by the rays from each terminal which just graze the terrain, and
for a LOS path by the profile point with the highest knife-edge
diffraction parameter ν. The final diffraction loss is then given by:

$L_{d} = L_{\text{de}} + \left( \text{10} + 0\text{.}2d \right) \cdot \left( 1 - \text{exp}\left( \frac{L_{\text{de}}}{6\text{.}\text{02}} \right) \right)$
dB (2.1)

This is the form in which the Bullington model has been tested in the
UK. It gives a diffraction loss which rises smoothly from zero as ν
increases from -0.67.

# Description of the correction

'Delta-loss, defined as 3-edge loss minus Bullington loss, was
calculated for 10,000 random paths around the British Isles for all
combinations of 30, 100, 300, 1 000 and 3 000 MHz, and for the following
pairs of antenna metres above ground: 10/10, 100/10, 100/100 and 200/10.
When plotted against path length delta-loss shows a concentration of
points forming a line. Except for short distances this line tends to be
central in the scatter. Systematic features were found in this line
which could be represented by three characteristic points as illustrated
in Figure 1, which is for 1 GHz and both antenna heights 10 m agl.

![](media/image1.png){width="5.072916666666667in" height="2.375in"}

**Figure 1. Delta-losses losses: 1 GHz antennas both 10 m**

a)  Point 1 is zero delta-loss at the maximum distance for which a
    smooth path has 0.6 Fresnel clearance. This distance can be
    estimated from the frequency and antenna heights.

b)  Point 2 occurs where there is a changed in slope. This is the least
    definite point, but its distance and delta-loss change
    systematically with frequency.

c)  Point 3 is the delta-loss at 1 000 km which again varies
    systematically with frequency.

Equations were derived by inspection to match these 3 points for the
ranges of frequency and antenna-height combinations given above. The
correction was then defined as a function of path-length, frequency and
antenna heights by straight-line segments between the points, with an
auxiliary correction to smooth the junction at Point 2. The complete
correction is defined in §4 below. For the purposes of the correction
the antenna heights are taken relative to the smooth-earth surface used
defined in the anomalous-propagation model.

The correction function is not limited to any minimum or maximum
distance, although it has not been tested beyond 1 000 km. It not
expected that diffraction would dominate for any path longer than
1 000 km.

The correction should be added to the Bullington diffraction model as
described in §2 above. This brings the mean delta-loss between the two
models close to zero at all distances. Since the correction is not a
function of any terrain features, the Bullington performance with
respect to standard deviation should not be impaired.

# Definition of the correction

The correction, *C*, is calculated as follows.

Calculate the distance to point 1 given by: (see P.1546-2 Annex 5 §15)

$d_{1} = \frac{d_{f}d_{h}}{d_{f} + d_{h}}$ km (4.1)

where

$d_{f} = 0\text{.}\text{0000389} \cdot f \cdot h_{1} \cdot h_{2}$ km
(4.1a)

$d_{h} = 4\text{.}1\left( h_{1^{0\text{.}5}} + h_{2^{0\text{.}5}} \right)$
km (4.1b)

*d* = path length, km

*f* = frequency, MHz

*h*~1~ = antenna height above the smooth-earth surface at start of path,
m

*h*~2~ = antenna height above the smooth-earth surface at end of path, m

If *d* ≤ *d*~1~ , the correction *C* is given by:

$C = 0$ dB (4.2)

and no further calculation is required.

If *d* \> *d*~1~ , continue as follows.

Calculate the distance to point 2 given by:

$d_{2} = \text{200} \cdot \left\lbrack 0\text{.}\text{7568} + 0\text{.}\text{00046111} \cdot \left( h_{1} + h_{2} \right) \right\rbrack^{2 \cdot \text{log}\left( \frac{f}{\text{30}} \right)}$
km (4.3)

Calculate the correction at point 2 given by:

$C_{2} = 6\text{.}3 + 2\text{.}\text{45} \cdot \text{log}(f)$ dB (4.4)

Calculate an auxiliary correction used to smooth results in the vicinity
of point 2:

$C_{\text{aux}} = 4\text{exp}\left\lbrack 5\text{log}\left( R_{d} \right) \right\rbrack$
dB (4.5)

where:

$R_{d} = \frac{d}{d_{2}}$ if *d* \< *d*~2~ (4.5a)

$R_{d} = \frac{d_{2}}{d}$ otherwise (4.5b)

If *d* ≤ *d*~2~ , the correction *C* is given by:

$C = C_{2} \cdot \left( \text{log}\left( \frac{d}{d_{1}} \right)\text{log}\left( \frac{d_{2}}{d_{1}} \right) \right) + C_{\text{aux}}$
dB (6)

and no further calculation is required.

If *d* \> *d*~2~ , continue as follows.

Calculate the correction at point 3 given by:

$C_{3} = \text{34} + \text{19}\text{.}8 \cdot \text{log}(f)$ dB (6)

The correction *C* is given by:

$C = C_{2} + \left( C_{3} - C_{2} \right) \cdot \left( \text{log}\left( \frac{d}{d_{2}} \right)\text{log}\left( \frac{\text{1000}}{d_{2}} \right) \right) + C_{\text{aux}}$
dB (7)

# Testing the correction

Figures 2 to 6 show plots corresponding to Figure 1 with a blue trace
added to show the correction function, for 30, 100 and 300 MHz, and 1
and 3 GHz, respectively. These indicate how the shape of the correction
must change with frequency.

![](media/image2.png){width="5.020833333333333in"
height="2.3645833333333335in"}

**Figure 2. Delta-losses plus fitted function: 30 MHz, antennas both 10
m**

![](media/image3.png){width="5.010416666666667in"
height="2.3541666666666665in"}

**Figure 3. Delta-losses plus fitted function: 100 MHz, antennas both 10
m**

![](media/image4.png){width="5.010416666666667in"
height="2.3541666666666665in"}

**Figure 4. Delta-losses plus fitted function: 300 MHz, antennas both 10
m**

![](media/image5.png){width="5.010416666666667in" height="2.34375in"}

**Figure 5. Delta-losses plus fitted function: 1 GHz, antennas both 10
m**

![](media/image6.png){width="5.010416666666667in" height="2.34375in"}

**Figure 6. Delta-losses plus fitted function: 3 GHz, antennas both 10
m**

Results for other antenna heights appear in Annex A. In these cases,
delta-loss scatter diagrams were plotted with the correction added to
the Bullington method. This produces delta-losses closer to zero over
the range of path lengths, which is convenient when plotting many
graphs.

# Comparison with measurements

Figures 7 and 8 show the effect of the correction on comparisons with
measurements. In viewing these graphs it should be remembered that few
measured paths exceed 100 km in length. Thus the effect of the
correction at longer distances is not being exercised in these cases.

As indicated by the legends, the traces give results for:

Solid red: 3-edge method

Dashed blue: Bullington plus taper as described in §2

Solid blue: Bullington plus taper + long-path correction defined in §4

![](media/image7.png){width="5.708333333333333in"
height="2.4479166666666665in"}

**Figure 7. Effect of correction on mean discrepancies with
measurements**

![](media/image8.png){width="5.708333333333333in" height="2.46875in"}

**Figure 8. Effect of correction on SD of discrepancies with
measurements**

Figure 7 shows that the long-path correction tends to adjust the
Bullington means upwards towards the 3-edge means. Since most Bullington
means are negative, this improves results in most cases.

Figure 8 shows that in most cases the long-path correction reduces the
Bullington SDs.

It can thus be concluded, on the basis of this comparison, that the
long-path correction does not reduce the accuracy of the Bullington
model for shorter paths, and in most cases improves it.

# Alternative correction

An alternative correction has been published ("Distance correction for
Bullington.pdf") in the form of a 9^th^-order polynomial of distance.
This rises from about zero at 100 m, is stated to be matched to
measurements in Switzerland for middle distances, and to be adjusted
according to Fig 7.4 of a report from the UK on the under-prediction of
loss by Bullington for long paths. It is not entirely clear whether the
polynomial is intended as a replacement to the 2^nd^ term in eq (2.1) or
as an additional correction.

The results in Figures 2 to 6 above show that the necessary correction
must adapt for frequency and antenna heights. Since the alternative
correction is a function of path length only, it clearly cannot do this.

Figure 9 corresponds to Figure 5 with the polynomial plotted in blue.
The point is not that this is not a good fit, but that it cannot adapt
as necessary for frequency and antenna heights.

![](media/image9.png){width="5.010416666666667in"
height="2.3541666666666665in"}

**Figure 9. Delta-losses plus alternative correction: 3 GHz, antennas
both 10 m**

# Further work

The long-path correction performs least well between points 2 and 3, as
shown in Figure 1. This is modelled as a straight line, for the
logarithm of distance (apart from the smooth correction in the vicinity
of Point 2), whereas small degrees of overall curvature are evident in
Figures 2 to 6. The result on the overall model over a wide range of
frequencies and antenna heights is visible in Annex A, where the
concentration of points in the approximate distance range from 100 km to
1 000 km shows the most deviation from zero.

The curvature can be in either direction, but appears to behave fairly
systematically. It is thus intended to do further work in this area to
improve the fit to the difference between the 3-edge and Bullington
means.

It is also intended to test the correction for a wider range of path
types. The correction presented in this paper were derived for 10 000
paths around the British Isles, using in effect national terrain-height
data. It is proposed to test the correction using SRTM height data for
different regions of the world.

# Conclusions

The correction defined in §4 was derived from the differences found
between the 3-edge and Bullington diffraction models. The effect of the
correction, when added to diffraction loss predicted by the Bullington
model, is to minimise the mean differences between the two methods. The
corrected Bullington model thus retains its performance in terms of
standard deviation, but with mean losses closer to the 3-edge model,
thus avoiding the under-prediction of loss for long paths characteristic
of the uncorrected Bullington model.

The 3-edge model was used in this way because it is known to match at
least approximately the existing evidence from P.1546 curves and results
from P.452 at long distances. Using the two diffraction models in this
way allows delta-loss data to be generated for any frequency and pair of
antenna heights, allowing a wide parameter range to be covered.
Comparing the Bullington model to P.1546 curves is convenient for
demonstration purposes, but would provide less information for defining
the correction.

As reported in §7, comparisons with measurements shows that the
correction generally improves the Bullington prediction accuracy.

As indicated in §8, the correction is not perfect and further work can
usefully be done. Nevertheless the basic correction method appears to be
sound. If the 3-edge diffraction model were to be replaced by the
Bullington method, without further correction, there would be
unacceptable errors at long distances, with serious consequences for
coordination studies. The method described here would seem to be a
suitable means to preserve the standard deviation performance of the
Bullington model, whilst adjusting the mean prediction of diffraction
loss upwards, particularly at long distances.

**ANNEX A:\
Delta-losses between the 3-edge and corrected Bullingont models**

This annex presents 3-edge minus *corrected*-Bullington delta-loss
scatter diagrams, in each case for 10,000 random UK paths. Each diagram
is for frequencies from 30 MHz to 3 GHz for a given pair of antenna
heights. The measure of success is that characteristic deltas should be
close to zero.

![](media/image10.png){width="4.864583333333333in"
height="6.958333333333333in"}

**Figure A.1\
Corrected delta-losses for antenna heights of 10 and 10 m above smooth
surface**

![](media/image11.png){width="4.84375in" height="6.958333333333333in"}

**Figure A.2\
Corrected delta-losses for antenna heights of 100 and 10 m above smooth
surface**

![](media/image12.png){width="4.84375in" height="6.989583333333333in"}

**Figure A.3\
Corrected delta-losses for antenna heights of 100 and 100 m above smooth
surface**

![](media/image13.png){width="4.84375in" height="6.958333333333333in"}

**Figure A.4\
Corrected delta-losses for antenna heights of 200 and 10 m above smooth
surface**