BT plc \[United Kingdom\]

A Proposal for a hybrid diffraction method

# Introduction

The recently adopted deterministic propagation model, ITU-R P.1812,
contains a modified 3-edge Deygout diffraction model first proposed for
use in ITU-R P.452. The Swiss Administration has expressed concern that
this diffraction model is inaccurate when compared to measurements in
mountainous regions. In documents 3K/171, 3K/113 and others \[5\] they
show that better agreement to measurements can be obtained by using the
Bullington diffraction method. However, several other groups, notably
workers in the United Kingdom \[1\]\[2\]\[3\]\[4\], have shown that the
Bullington method gives totally unacceptable results on smooth earth
paths over land or water, particularly at longer distances.
Correspondence Group 3K-1 was established by SG3 to gain a better
understanding of the issues involved and, if possible, formulate a
revised diffraction method that would meet the concerns of all parties

Much of the effort to date has concentrated upon a linear distance
correction term contained within both diffraction methods with a view to
introducing non-linear terms to make the Bullington method more
acceptable at the longer distances. A recent proposal by Switzerland to
change to a ninth order polynomial distance correction term \[7\] does
not change the large underestimation of diffraction loss on long smooth
earth paths. A separate document is in preparation in which these
findings are demonstrated.

The comparison between the competing diffraction methods has been made
using a database of measured results provided by a number of
Administrations. As part of its remit, the CG adopted a common format
for the data sets and the Polish Administration has worked hard to
collate the datasets into the new format. Unfortunately, extensive
analysis of the profiles contained within the datasets by workers in the
UK has highlighted a number of errors within the profiles themselves. A
reduced dataset has removed the worst of these obvious errors but there
are still some large unresolved mean offsets between datasets from
differing sources (possibly due to clutter) that make calculating a
meaningful overall standard deviation difficult. The datasets also have
a paucity of sea paths and small time percentage data. It is, perhaps,
unsurprising that no unambiguous conclusion over which method performs
best overall over the entire input parameter space has been forthcoming.
Indeed, it is difficult to improve on the statistics from a simple
*(FSL + x) dB* model.

This document is an attempt to examine some of the fundamental reasons,
which have given the Swiss Administration their concerns over using the
3-edge Deygout diffraction method in mountainous terrain. A Monte Carlo
approach using synthetic path profiles has been adopted in an attempt to
define the boundaries of the problem. No comparisons to actual
measurements have been made but it has been assumed that the two
diffraction methods under consideration can safely represent the lower
and upper bounds of diffraction loss for any particular path profile.
Justification for this statement rests with two sets of facts:

-   The Swiss measurements have been very carefully taken using
    height-gain measurements to remove ground reflections and show good
    agreement to the Bullington method for highly mountainous paths
    considered irrespective of the number of obstacles on the path. It
    is difficult to comprehend how a lower total diffraction loss can be
    achieved on a transhorizon path than by representing all the terrain
    features using a single virtual knife-edge.

-   At the other extreme, a number of studies by David Bacon in the UK
    \[3K/149 and \[6\]\] have shown that the 3-edge Deygout method is a
    good approximation to the smooth earth diffraction methods in
    ITU-P.526 and gives a large enough diffraction loss at longer
    distances not to dominate over the troposcatter loss mechanism.

To date, it has been assumed that it is the roughness of mountainous
terrain that is the key distinguishing factor that should be used to
determine which diffraction method to use. This paper focuses upon
another of the factors that is presented in the Swiss papers but not
highlighted to any great degree: the obstacle separation distance. The
obstacle separation distance is shown to be a very good differentiator
between the smooth earth paths and shorter more mountainous paths as the
earth curvature increasingly dominates the path profile.

The approach adopted is to use the obstacle separation distance to drive
an interpolation factor, which smoothly switches between the Deygout and
Bullington methods. Since the losses due to the primary edge in the
Deygout method and the virtual edge in the Bullington method are almost
equal at small obstacle separation distances, it is not necessary to
compute the full Bullington method as well as the Deygout method. It is
only necessary to blend out the two secondary Deygout edges to achieve
an almost identical result.

A suitable interpolation factor has been developed together with a pair
of tuneable parameters to alter the position and range at which the
switch occurs. It is not possible to give precise values for these
parameters without a comparison to actual measurements and it is hoped
that other members of the CG can test the method using the refined
datasets to make a suitable recommendation. Since the method involves a
minimal change to the current 3-edge Deygout method, it is hoped that
some preliminary testing of the method can be undertaken in time for the
3K-1 CG meeting scheduled for March in London.

# The Background

To date, it has been assumed that the terrain roughness is a key factor
in differentiating Swiss terrain from the gentler hills in NW Europe.
Indeed, although this does play a part, it is the ratio of the mountain
height to the maximum extent of earth curvature and the consequent
knock-on effect that this has on the probability of detecting closely
spaced obstacles that is of much more importance. Figure 1shows that two
different profiles can have widely different roughness figures, however
calculated, but still have the same set of three Deygout knife-edges.

It is instructive at this stage to reproduce Figure 2 contained with
document 3K/171 drafted by the Swiss Administration (Figure 2 in this
document also). The thrust of the document is to show that the 3-edge
Deygout method becomes increasingly inaccurate compared to measurements
as the number of obstacles increases in the path. However, the figure
makes it clear that a combination of using a very detailed terrain
database together with jagged mountain peaks is responsible for
attributing three knife edges, separated by only 0.3km, to what, in
essence, is a single diffracting obstacle. The other salient feature of
the terrain is that the higher the main mountain ridge, the less the
probability of earth curvature blocking the stretched string joining
transmitter to receiver. This is illustrated in Figure 3 using a random
placement of 3 synthetic knife edges, no more than 7 km apart on a 100km
smooth earth path profile. In the first picture the maximum obstacle
height is restricted to be less than 25m whilst in the second it is less
than 500m. The principal knife-edges used in the Deygout construction
are shown.

Figure 1 Example of terrain profiles with different roughness criteria
but similar Deygout edges

![](media/image1.wmf){width="4.9222222222222225in" height="3.01875in"}

Figure 2 Path profile reproduced from document 3K/171

![](media/image2.wmf){width="7.014583333333333in"
height="2.6222222222222222in"}

It is interesting to note that, with the 3-edge Deygout construction,
the presence of three well-defined obstacles does not guarantee that
both secondary edges will contribute to the total loss. This will depend
upon which of the three edges is deemed to have the largest *ν* value;
both secondary edges will only be present if the centre obstacle is the
main edge. This high sensitivity to small changes in the terrain is one
of the contributory factors leading to the high variability in the total
loss when using the Deygout method. It is also a reason as to why it is
difficult to make meaningful comparisons between the Deygout and
Bullington methods unless the terrain features can be well controlled.
Later sections will show this in more detail.

Figure 3 Three randomly place knife edges whose heights are not allowed
to exceed a) 25m and b) 500m

![](media/image3.wmf){width="4.729166666666667in" height="3.24375in"}

![](media/image4.wmf){width="4.729166666666667in" height="3.24375in"}

Figure 3 also shows that it is the ratio of the height of the three
close-spaced obstacles to the maximum height of the earth curvature,
which is, in turn, governed by the path length, that sets the
probability of determining three Deygout edges in close proximity to one
another. These observations can be exploited to create a hybrid
diffraction model that should address both the Swiss concerns in
modelling mountainous terrain and other nation's disquiet that the
Bullington method is not well suited to the longer smoother paths.
Before developing a new model, it is useful to conduct two studies using
a Monte Carlo technique with some synthetic path profiles to gauge the
magnitudes of the discrepancies between the two competing diffraction
methods.

# The Study Methodologies

Two studies were conducted to gauge the boundary conditions to the
discrepancies between the Deygout and Bullington diffraction methods:

-   ***Study A***

In this method, three obstacles that are within 80% of the same height
as each other were randomly placed on a smooth earth profile of a fixed
length. The maximum allowable height of the obstacles was then allowed
to increment in a series of discrete steps. One hundred samples were
taken at each maximum height value. This was then repeated for a small
number of increasingly longer path lengths. In order to prevent unwanted
short range effects from blurring the results, the obstacles were
prevented from being too close to either terminal. This is in line with
the recommendations in the Swiss papers where receivers were only placed
on open ground with no obstructions in the immediate vicinity in the
direction of the transmitter. Also in line with the Swiss papers, a
relatively high effective transmitter height was selected to mimic the
use of the TV broadcast transmitters employed for the measurements.
Figure 4 and Table 1 show the parameters and values used.

Figure 4 The parameter definitions used in Study A

![](media/image5.wmf){width="5.7652777777777775in"
height="4.032638888888889in"}

-   ***Study B***

In the second method three obstacles that are within 90% of the same
height as each other were randomly placed within a defined width window
on a smooth earth profile of a fixed length. The width of the window was
allowed to vary in a set of discrete steps and its centre position was
randomly variable. The lower and upper bound of the window was not
allowed to fall below or increase beyond a fixed distance from either
terminal respectively. An explanation for this is given in *Study A*.
The maximum height of the obstacle group was set to a fixed value. One
hundred samples were taken at each window width value. This was then
repeated for a small number of increasingly longer path lengths. Figure
4 and Table 2 show the parameters and values used.

For both studies, the complete P.1812 recommendation with either the
Deygout or Bullington diffraction method was calculated for every random
profile generated. A version of the recommendation using the Bullington
method was provided to the CG earlier in the year \[4\]. Both individual
and mean data values were calculated for a wide range of the internal
model parameters. The random number generator was re-seeded between the
Deygout and Bullington calculation runs to ensure that meaningful
diffraction loss and geometric comparisons could be made. Results are
given in the next section.

Table 1

  -------------------- --------- --------- ------------------------ ----------
  **Parameter**        **Min**   **Max**   **Type**                 **Unit**

  Freq                 0.6       \-        Fixed                    GHz

  p%                   50        \-        Fixed                    \%

  step size            100       \-        Fixed                    m

  dt~min~              5         \-        Fixed                    km

  dt~max~              3         \-        Fixed                    km

  d~obs~               dt~min~   dt~max~   Continuously Variable    km

  d                    30        1000      Discrete Variable        km

  Txh                  50 m      \-        Fixed                    m

  Rxh                  10 m      \-        Fixed                    m

  dN                   39        \-        Fixed                    N-unit

  N0                   325       \-        Fixed                    N-unit

  hmax                 0         1000      Discrete Variable        m

  V~fact~              0.8       \-        Fixed                    \-

  \# samples           100       \-        Fixed                    \-
  -------------------- --------- --------- ------------------------ ----------

Figure 5 The parameter definitions used in Study B

![](media/image6.wmf){width="5.764583333333333in"
height="4.075694444444444in"}

Table 2

  -------------------- --------- --------- ------------------------ ----------
  **Parameter**        **Min**   **Max**   **Type**                 **Unit**

  Freq                 0.6       \-        Fixed                    GHz

  p%                   50        \-        Fixed                    \%

  step size            100       \-        Fixed                    m

  dt~min~              5         \-        Fixed                    km

  dt~max~              3         \-        Fixed                    km

  d~obs~               dt~min~   dt~max~   Continuously Variable    km

  w~obs~               0.2       20        Discrete Variable        km

  d                    30        1000      Discrete Variable        km

  Txh                  50 m      \-        Fixed                    m

  Rxh                  10 m      \-        Fixed                    m

  dN                   39        \-        Fixed                    N-unit

  N0                   325       \-        Fixed                    N-unit

  h~max~               500       \-        Fixed                    m

  V~fact~              0.9       \-        Fixed                    \-

  \# samples           100       \-        Fixed                    \-
  -------------------- --------- --------- ------------------------ ----------

# The Study Results

Display of the result of the studies presents a challenge with so many
parameters varying. In general, the easiest format is to plot any
particular model parameter against one of the discrete variable
parameter inputs. In the case of *Study A*, the maximum obstacle height
was used for the abscissa. For *Study B*, the maximum window width was
used for the abscissa. These were plotted for a number of different
total path lengths to best illustrate the features being revealed. Mean
values were calculated at each of the discrete abscissa values. A number
of derived parameters from the model intermediate values were also
calculated for each sample taken and treated in the same manner as one
of the basic model parameters.

Figure 6 Example plots from one sample from study types A and B

![](media/image7.wmf){width="3.15625in" height="2.591666666666667in"}
![](media/image8.wmf){width="3.0819444444444444in"
height="2.613888888888889in"}

Figure 6 shows example plots of a single sample of excess diffraction
loss made at each of the abscissa values. It is immediately apparent
that there is, not unsurprisingly, a much greater level of variability
with the Deygout method even with such simple experiments as these. This
variability makes it extremely difficult to formulate experiments using
real terrain, as the number of controllable parameter permutations is
just too great.

## The Results from Study A

### Geometric Considerations

Figure 7 shows an example of the diffraction path profile analysis using
both the Deygout and Bullington methods for a 50 km path. At 50 km, all
but the smallest maximum height obstacles will dominate the earth
curvature and both Deygout principal and secondary edges will be located
at one of the knife-edges. In the example shown the principal edge is
located at the receiver horizon point. Thus there is no *L~r~* loss and
one of the three knife-edges will not be used; the centre one in this
case.

Figure 7 Typical path profile analysis using the Deygout and Bullington
methods for a 50km path and 200 m maximum obstacle height.

![](media/image9.wmf){width="6.163888888888889in"
height="2.8444444444444446in"}

Figure 8 Typical path profile analysis using the Deygout and Bullington
methods for a 100km path and 200 m maximum obstacle height.

![](media/image10.wmf){width="6.170833333333333in"
height="2.8201388888888888in"}

In the Bullington case, the two outermost knife edges will form the two
horizon points and a variable height virtual edge will always exist, the
magnitude of which will be set by the proximity of these edges to the
terminals. Even the smallest maximum height edges will obey these rules
unless they are grouped close to one or other end of the path.

At a 100 km path length, Figure 8 shows that the probability of a
Deygout knife-edge not falling on one of the obstacles has now increased
particularly at low maximum obstacle heights. Once the obstacles are all
located on one or other side of the path midpoint, there is a chance
that even a 200 m high obstacle will not be visible to one or other
terminal, and a point on the smooth earth will become one of the Deygout
knife-edges. The figure shows that when the obstacles are well spread
out that only the lowest maximum height obstacles will not become
horizon points.

In the Bullington 100km case, similar comments to those made in the
previous case will apply

Figure 9 Typical path profile analysis using the Deygout and Bullington
methods for a 200km path and 200 m maximum obstacle height.

![](media/image11.wmf){width="6.271527777777778in" height="2.8125in"}

Figure 10 Typical path profile analysis using the Deygout and Bullington
methods for a 500km path and 200 m maximum obstacle height.

![](media/image12.wmf){width="6.351388888888889in"
height="2.847916666666667in"}

Once the path length reaches 200km, Figure 9 shows that lower maximum
height obstacles need to be carefully positioned if they are to become
Deygout knife-edges. In the example, 200 m high obstacles only meet this
criterion if there is one on either side of the path centre point.

The single virtual Bullington edge is now becoming very high wherever
the obstacles are located

The final example shown in Figure 10 is for a 500 km long path. At this
path length, the maximum earth curvature is of the order of 4000 m and
only taller obstacles are significant in determining the locations of
the Deygout knife-edges. The probability that at least one smooth earth
point is one of the horizon points is high unless the obstacles are
either very high or spaced well apart.

### The Excess Diffraction Loss

The next group of four images shows the excess diffraction loss for each
of the selected path lengths as functions of the maximum allowable
obstacle height. Both the individual samples are shown and the mean
taken by averaging each vertical column of 100 sample points for each
diffraction method. Excess diffraction loss is here defined to be the
decibel path loss value in excess of the free space value. Both
diffraction methods make this available as the parameter *Ld~50~*.

A number of interesting observations can be made that relate directly
back to the geometry of the two methods.

-   In all cases the Deygout method always gives a larger excess
    diffraction loss than the Bullington method; the higher the path
    length or maximum obstacle height, the larger the difference.

-   Experiment has shown that the only way in which the Bullington
    method gives a higher excess diffraction loss than 3-edge Deygout is
    when there are asymmetric and very small horizon distances such as
    found in the eighth test case, "*misc.csv*" in the RAL set of test
    paths \[2\]. The introduction of the terminal guard distances,
    *dt~min~* and *dr~min~* prevents this from occurring in this study.

-   For the shortest path lengths, with low maximum obstacle heights,
    the losses of the two methods approach each other.

-   For paths in excess of 200 km in length there is always at least a
    35 dB difference in excess diffraction loss even with no obstacles
    present (the lowest maximum height is 0 m).

-   For paths in excess of 500 km, the excess diffraction loss for each
    method is practically independent of maximum obstacle height up to
    at least 1000 m.

-   The spread of values using the Deygout method is always higher than
    the Bullington method reflecting the higher number of degrees of
    freedom.

-   At higher path lengths, the spread of values for both methods
    decreases dramatically for the lower maximum obstacle heights.

Figure 11 The excess diffraction loss using the Deygout and Bullington
methods for a 50 km path.

![](media/image13.wmf){width="4.85625in" height="3.890277777777778in"}

Figure 12 The excess diffraction loss using the Deygout and Bullington
methods for a 100 km path.

![](media/image14.wmf){width="4.85625in" height="3.890277777777778in"}

Figure 13 The excess diffraction loss using the Deygout and Bullington
methods for a 200 km path.

![](media/image15.wmf){width="4.85625in" height="3.886111111111111in"}

Figure 14 The excess diffraction loss using the Deygout and Bullington
methods for a 500 km path.

![](media/image16.wmf){width="4.85625in" height="3.890277777777778in"}

### Actual Obstacle Separation

The next group of four images relate to the actual obstacle separation.
This parameter can be easily derived from two of the intermediate model
parameters present within the path analysis section of ITU-R P.1812,
namely the transmitter and receiver horizon distances, *d~lt~* and
*d~lr~*.

Equation 1 defines a new parameter, *d~os~*, which is used extensively
in the remaining sections of this document.

$d_{\text{os}} = d - d_{\text{lt}} - d_{\text{lr}}$ (1)

where:

*d*: path length

*d~lt~*: transmitter horizon distance

*d~lr~*: receiver horizon distance

Obstacle separation distance is shown as a function of maximum allowable
obstacle height for the four chosen path lengths. These plots are shown
with a log ordinate to allow the smaller values to be displayed. The
distribution of 100 sample points at any particular abscissa value is
however roughly normally distributed. This enables a 5-percentile level
to be displayed as the value 1.65 times the standard deviation below the
mean. It is not plausible compute a smaller value as the sample noise
becomes too great. Since obstacle separation distance is purely a
geometric parameter and the random number generator was re-seeded
between the Deygout and Bullington runs, both give the same graph and
only one is shown.

Again a number of observations can be made:

-   At small path lengths, the obstacle separation distance is
    independent of maximum allowable obstacle height for both the mean
    and 5 percentile.

-   The upper bound of the obstacle separation distance is roughly
    constant at the path distance minus the two distance guard intervals
    at each terminal.

-   The lower bound of the obstacle separation distance has a high
    dependency on both the path length and the maximum allowable
    obstacle height becoming almost linear on a log ordinate plot.

-   The probability of the obstacle separation distance being less than
    10 km for paths in excess of 200 km length is very small indeed even
    for a maximum allowable obstacle height of 1000 m.

Figure 15 The actual obstacle separation using the Deygout and
Bullington methods for a 50 km path.

![](media/image17.wmf){width="4.85625in" height="3.886111111111111in"}

Figure 16 The actual obstacle separation using the Deygout and
Bullington methods for a 100 km path.

![](media/image18.wmf){width="4.85625in" height="3.890277777777778in"}

Figure 17 The actual obstacle separation using the Deygout and
Bullington methods for a 200 km path.

![](media/image19.wmf){width="4.925694444444445in"
height="3.886111111111111in"}

Figure 18 The actual obstacle separation using the Deygout and
Bullington methods for a 500 km path.

![](media/image20.wmf){width="4.93125in" height="3.890277777777778in"}

## The Results from Study B

### Geometric Considerations

Example plots showing the geometric construction of both diffraction
methods are displayed as before. For the plots and the simulation runs
the maximum allowable obstacle height is set to 500 m. The maximum
allowable window size was set to be 7 km in the plots to allow
visibility of the three obstacles. The window size was varied from 0.2
to 20 km for the simulation runs.

Again a number of observations can be made:

-   For a 50 km or 100 km path with a 500m high obstacle, the earth
    curvature is insignificant. One of the three knife-edges will always
    be selected as the main edge. If the Deygout principal edge is
    anything other than the centre one then only one secondary edge is
    found as the sub path ν value on the opposite side is below --0.78
    (see Figure 19).

-   For a 50 km or 100 km path, the Deygout principal edge and
    Bullington virtual edge are extremely similar in both position and
    height.

-   For path lengths longer than 200 km, there is a high probability
    that one of the secondary Deygout edges is located at a point on the
    smooth earth profile with a consequent very large value for the
    obstacle separation distance, *d~os~*. (see Figure 22)

-   At intermediate distances between 100 and 200 km, a Deygout
    secondary edge located at a smooth earth profile point may be either
    at a real string point giving a ν greater than 0 or be a sub path
    obstacle with a loss below 6dB.

Figure 19 Typical path profile analysis using the Deygout and Bullington
methods for a 50km path and 500 m maximum obstacle height.

![](media/image21.wmf){width="6.163888888888889in"
height="2.8402777777777777in"}

Figure 20 Typical path profile analysis using the Deygout and Bullington
methods for a 100km path and 200 m maximum obstacle height.

![](media/image22.wmf){width="6.170833333333333in"
height="2.8243055555555556in"}

Figure 21 Typical path profile analysis using the Deygout and Bullington
methods for a 200km path and 200 m maximum obstacle height.

![](media/image23.wmf){width="6.351388888888889in"
height="2.8444444444444446in"}

Figure 22 Typical path profile analysis using the Deygout and Bullington
methods for a 300km path and 200 m maximum obstacle height.

![](media/image24.wmf){width="6.351388888888889in"
height="2.8444444444444446in"}

### Excess Diffraction Loss

Figure 23 through to Figure 26 show the excess diffraction loss as a
function of maximum allowed obstacle separation. Again these figures
contain both the raw sample points and the means averaged at each column
position. The following points can be made:

-   The Bullington method gives an excess diffraction loss that is
    virtually independent of both maximum obstacle separation and path
    distance.

-   The 3-edge Deygout loss increases with maximum obstacle separation
    but the trend is for lower increases at longer path lengths

-   The Deygout and Bullington loss only become equal when the maximum
    obstacle separation is zero and the path length is less than 50 km.

-   The spread of values for the Bullington method increases with path
    length up to a certain distance between 200 and 300 km and then
    reduces again.

-   The spread of values with the Deygout method has a complex
    dependency on both path length and maximum obstacle separation.
    Below 200 km, the range of values is consistently high but the
    standard deviation is smaller at lower path lengths (inferred from
    the density of the red coloration).

-   The lower limit of the Bullington spread is abrupt whereas the that
    for the Deygout method is not except when the path length reaches
    300 km

At first glance, these complex relationships are difficult to explain
until looked at in the light of the next set of four graphs showing the
*actual* obstacle separation as a function of *maximum allowable*
obstacle separation.

### Actual Obstacle Separation

Figure 27 through to Figure 30 show the actual obstacle separation as
defined in equation 1 as a function of the maximum obstacle separation
allowed. The following observations are clearly displayed:

-   Two distinct populations of sample points exist for path lengths
    above 50 km.

-   The distribution of points within each group alters as the path
    length increases as the movement in the mean line indicates. At
    small path lengths, only the lower population exists whereas at 300
    km only the upper population exists.

-   At 50 km with only a single population of points, the actual
    obstacle separation mean is approximately half the maximum allowable
    mean. This reflects the fact that, at any maximum allowable
    separation, the actual knife-edges were selected from a uniform
    distribution within the window. This relationship holds even down to
    values as small as 2 km.

-   The above relationship starts to break down below this as the
    quantisation effect of the profile spacing starts to dominate and
    the probability of the three edges being randomly placed at the same
    profile point increases. This will increase the probability that one
    or other of the secondary Deygout edges will be not be present since
    with synthetic knife-edges the intervening terrain points lie on a
    smooth earth surface and with have too low a *ν* value to give other
    than a 0 dB loss.

-   The upper population of points is independent of the maximum
    allowable separation distance.

-   The spread of values in the upper population is remarkable constant
    with both maximum allowable obstacle separation and path length.

-   There are a wide range of values from 20 km to 90 km in which no
    *actual* obstacle separations can exist

When all of these observations are taken as a whole it is clear that the
actual obstacle separation distance defined in equation (1) embodies
both the combination of the ratio of obstacle height and maximum earth
curvature into a single parameter that can be exploited. At small actual
obstacle separations, the knife-edges should really be treated as single
object for which the Bullington method clearly gives a more consistent
value, which the Swiss measurements indicate is closer to being correct.
As either the obstacle separation increases due *either* to increasing
separation of real edges *or* the sudden intervention of earth curvature
as path length increases, then the 3-edge Deygout gives the necessary
extra excess diffraction loss to ensure that diffraction does not
dominate troposcatter.

A method is now required to switch between the Bullington and Deygout
methods using the actual obstacle separation distance as the controlling
parameter. The next section examines some of the ways in which this
might be possible.

Figure 23 The excess diffraction loss using the Deygout and Bullington
methods for a 50 km path.

![](media/image25.wmf){width="4.63125in" height="3.886111111111111in"}

Figure 24 The excess diffraction loss using the Deygout and Bullington
methods for a 100 km path.

![](media/image26.wmf){width="4.63125in" height="3.890277777777778in"}

Figure 25 The excess diffraction loss using the Deygout and Bullington
methods for a 200 km path.

![](media/image27.wmf){width="4.63125in" height="3.886111111111111in"}

Figure 26 The excess diffraction loss using the Deygout and Bullington
methods for a 300 km path.

![](media/image28.wmf){width="4.70625in" height="3.890277777777778in"}

Figure 27 The actual obstacle separation using the Deygout and
Bullington methods for a 50 km path.

![](media/image29.wmf){width="4.78125in" height="3.886111111111111in"}

Figure 28 The actual obstacle separation using the Deygout and
Bullington methods for a 150 km path.

![](media/image30.wmf){width="4.85625in" height="3.890277777777778in"}

Figure 29 The actual obstacle separation using the Deygout and
Bullington methods for a 200 km path.

![](media/image31.wmf){width="4.85625in" height="3.886111111111111in"}

Figure 30 The actual obstacle separation using the Deygout and
Bullington methods for a 500 km path.

![](media/image32.wmf){width="4.85625in" height="3.890277777777778in"}

# The Development of a New Hybrid Model

Before addressing the issues of the model itself, it is useful to
examine a few more plots taken from *Study B*. These will be used to
show how it is possible to take the best part from the Bullington method
and incorporate it into the current Deygout method without recourse to
calculating both, which would be undesirable.

## The Principal Knife-edge

Figure 31 compares the Deygout principal or main edge loss to that of
the Bullington virtual edge for the four different path lengths. As seen
in the previous section, two populations of sample points exist, albeit
not as distinct or well separated as before. However the same general
trend is seen for the points to split into two populations at middle
distance path lengths before coalescing again at 300 km. The important
point to derive is that, at shorter path lengths, where the actual
obstacle separation is small, the Deygout principal loss, *L~m50~*,
appears to be within a few dBs of the Bullington virtual edge loss,
*L~s50~*. Figure 32 through to Figure 35 shows this in more detail for a
50km path length.

Figure 31 Comparison between the Deygout Lm~50~ and Bullington Ls~50~
diffraction losses for different length paths.

![](media/image33.wmf){width="3.2569444444444446in"
height="2.890277777777778in"}![](media/image34.wmf){width="3.2569444444444446in"
height="2.890277777777778in"}

![](media/image35.wmf){width="3.2569444444444446in"
height="2.8979166666666667in"}![](media/image36.wmf){width="3.2569444444444446in"
height="2.890277777777778in"}

Figure 32 shows the difference between the Deygout principal and
Bullington virtual edge as a function of maximum allowable obstacle
separation. Again the mean value is approximately half the maximum as
uniform distributions of edges were used within the window. In the
region where individual knife-edges should really be treated as a single
obstacle, the difference is less than 1 dB. This observation is even
more apparent, if the differential loss is plotted as function of
*actual* obstacle separation in Figure 33.

## The Deygout Secondary Knife-edges

All of the considerations to date have tended to ignore any Deygout sub
path losses since the edge will no longer be located at a string point
and hence detectable by the obstacle separation distance parameter. The
results to date have been able to ignore the effect of sub path edges as
the intervening terrain between obstacles is the smooth earth surface
and, for the most part, any sub path loss will be so small that it is
set to zero (ν \< -0.78). Figure 34 and Figure 35 illustrate this. The
density of the *L~t50~* or *L~r50~* sample points below a 6 dB
threshold, indicative of a finite sub path loss is very low compared to
either the points above the line or those on the abscissa. Points on the
abscissa indicate no secondary edge was found (ν \< -0.78).

Figure 35 shows the combined *L~t50~* plus *L~r50~* loss as a function
of actual obstacle separation and is has an unusual distribution of
points on the ordinate between 0 and 6 dB. The only feasible explanation
for the group is as follows: For there to be a combined secondary loss
below 6 dB, at zero actual obstacle separation implies that all three
knife-edges are located at the same profile point. In this case, the
single edge becomes the principal one and then either one or both
secondary edges have small sub path losses that sum to less than 6 dB.
Also the population of values above 6 dB splits into two groups. The
larger group peaking at a value of about 25 dB derives from there being
only a single secondary edge (the principal edge is not the centre one)
whereas the group above implies a centre principal edge with a secondary
contribution from the edge on either side.

Figure 32 Difference between the Bullington *L~s50~* and Deygout
*L~m50~* diffraction losses vs. maximum obstacle separation for a 50 km
path.

![](media/image37.wmf){width="4.420833333333333in"
height="3.902083333333333in"}

At longer path distances or much lower obstacle heights, it would be
unwise to ignore the sub path losses and a later section will
re-consider how to treat them.

Figure 33 Difference between the Bullington *L~s50~* and Deygout
*L~m50~* diffraction losses vs. actual obstacle separation for a 50 km
path.

![](media/image38.wmf){width="4.3125in" height="3.902083333333333in"}

Figure 34 Secondary edge losses for the Deygout method vs. actual
obstacle separation for a 50 km path.

![](media/image39.wmf){width="5.99375in" height="2.734027777777778in"}

Figure 35 Sum of the secondary edge losses for the Deygout method vs.
actual obstacle separation for a 50 km path.

![](media/image40.wmf){width="4.395833333333333in"
height="3.902083333333333in"}

## The New Model

Having determined a suitable parameter to use as a switch, an
implementation method without any discontinuities is needed. It is also
desirable that any function employed has tuneable constants whose
effects are easily understood and optimised. This is important, as the
model ultimately needs to be tuned against real terrain as opposed to
the synthetic terrain used here. Figure 36 is such a function, similar
to one that has been employed within the model blending sections of both
ITU-R P.452 and P.1812. Equation 2 gives a suitable function that
implements Figure 36.

$\Theta(x) = \left\{ \begin{matrix}
1\text{.}0\ \text{                       if  }d_{\text{osd}} = 0 \\
1\text{.}0 - 0\text{.}5\left\lbrack 1\text{.}0 + \text{tanh}\left\lbrack \frac{r_{\text{os}}\left( \text{log}(x) - \text{log}(d_{\text{osd}}) \right)}{\text{log}(d_{\text{osd}})} \right\rbrack \right\rbrack\ \text{otherwise }
\end{matrix} \right.\ $ (2)

where:

*r~os~* = \[3\]: constant to set the midpoint of the transition

*d~osd~* = \[1.5\]: constant to set the range of the transition

The two constants can be used to alter both the midpoint and range of
the transition as indicated. This will need to be tuned against
measurements. The suggested values given have been used in the remainder
of the paper.

### Methods of Use

The new interpolation function can be used in more than one manner and
the advantages and drawbacks for each are examined:

#### Method A

In method A, the interpolation factor is used to simply switch between
the full 3-edge Deygout and Bullington losses using equation 3:

$L_{d\text{50}} = L_{d\text{50}D} + \left( L_{d\text{50}B} - L_{d\text{50}D} \right)\Theta\left( d_{\text{os}} \right)$
(dB) (3)

where:

*L~d50~*: new diffraction loss for 50% time

*L~d50D~*: original Deygout diffraction loss

*L~d50B~*: original Bullington diffraction loss

Figure 36. New interpolation function for the hybrid diffraction method.

![](media/image41.wmf){width="4.81875in" height="3.9611111111111112in"}

#### Method B

As has been shown, the principal 3-edge Deygout loss, *L~m50~*, is
highly correlated to the Bullington virtual edge loss, *L~s50~*,
particularly at small obstacle separation distances, *d~os~*. The next
method exploits this relationship to switch out the secondary edges of
the Deygout method at small obstacle separation distances as shown
below:

$\begin{array}{r}
L_{d\text{50}} = L_{m\text{50}} + \left( 1 - e^{- \frac{L_{m\text{50}}}{6}} \right)\ \left( \left( L_{t\text{50}} + L_{r\text{50}} \right)\left( 1 - \Theta\left( d_{\text{os}} \right) \right) + \text{10} + 0\text{.}\text{04}d \right)\ \text{if }\nu_{m\text{50}} - 0\text{.}\text{78} \\
\text{        } = 0\text{                                                                  }\ \text{   }\ \text{otherwise}
\end{array}$ (4)

which is a straight replacement for equation (23) in ITU-R P.1812.

## Testing the New Model

The effects of these methods are shown in Figure 37. This figure is a
reproduction of Figure 23 with reduced intensity sample points to
highlight the effects of the blending methods. Note that the abscissa
shows the maximum allowable obstacle separation distance, which, on
average, is approximately twice the actual obstacle separation achieved.
It is also possible to see a difference between the two methods near to
0 km separation due to a different the distance correction term in the
Deygout and Bullington methods. The former uses *0.4d* and the latter
*0.2d*. There would appear to be no particularly strong reason why these
two could not be aligned.

Figure 37 The excess diffraction loss after applying the interpolation
factor on a 50 km path.

![](media/image42.wmf){width="5.645833333333333in"
height="4.8180555555555555in"}

Figure 38 shows the mean losses for new model as a function of the mean
actual obstacle separation distance. It is clear from this plot that the
Swiss example profile given in Figure 2 would return an excess
diffraction loss very close to the Bullington value with either of the
two methods but any path where the obstacle separation distance exceeded
about 10 km the full Deygout value would be reached.

### Susceptibility of the New Model to Profile Step Size

In document 3K/171 from Switzerland, reference was made to the fact that
'*only profiles which fulfil a minimum number of points and a good
resolution'* should be used. It is interesting to test the validity of
this argument using the new method.

A separate study was established in which the window containing three
randomly placed knife-edges in *Study B* was replaced by a single
randomly placed Gaussian distribution. To simulate the effect of the
terrain ruggedness the height of each profile point was randomly varied
by 5% of the height at that point. The standard deviation of the
Gaussian was fixed but its position was allowed to randomly vary. Only
the 50 km path length was examined in detail. The major parameters are
given in Table 3. All other relevant parameters are identical to those
in Table 2.

Figure 38 The mean excess diffraction loss versus actual obstacle
separation after applying the interpolation factor on a 50 km path.

![](media/image43.wmf){width="4.783333333333333in"
height="4.058333333333334in"}

Table 3

  --------------------- --------- --------- ------------------------ ----------
  **Parameter**         **Min**   **Max**   **Type**                 **Unit**

  step size             10        5000      Discrete Variable        m

  dt~min~               5         \-        Fixed                    km

  dt~max~               5         \-        Fixed                    km

  σ                     3         \-        Fixed                    km

  h~max~                500       \-        Fixed                    km

  d                     50        \-        Fixed                    km
  --------------------- --------- --------- ------------------------ ----------

These basic synthetic profiles were generated at a profile spacing of 10
m giving a 5001-point path profile. The variable step size was obtained
by down-sampling the points to give step sizes from 10 m to 5000 m.
Again, 100 samples were taken at each profile step spacing. Figure 39
shows an example of the geometric analysis for each method at the full
profile resolution of 10 m.

Figure 39 Example profile analyses for Deygout and Bullington used in
the step size study for a 50 km path. The peak of each profile is
magnified to show the edges more clearly.

![](media/image44.wmf){width="6.59375in" height="3.0319444444444446in"}

![](media/image45.wmf){width="6.59375in" height="3.035416666666667in"}

Figure 40 shows the excess diffraction loss for the Deygout, Bullington
and the two hybrid diffraction methods. Points to note are as follows:

-   The excess diffraction loss using the Deygout method is very
    susceptible to the step size as suggested in the Swiss papers for
    paths where the principal and secondary edges are very close to one
    another.

-   The excess diffraction loss when using the Bullington method, by
    contrast, is almost unaffected by the step size.

-   A large spread in the samples is seen in the Deygout case at all
    step sizes. This only reduces at the 5000 m step size where there
    are only 11 points left in the profile.

-   Both the new hybrid methods perform well showing almost the same
    independency from the step size.

-   Small offsets between the hybrid methods and Bullington relate to
    the different distance dependency terms being assumed in the models.

Figure 40 The excess diffraction loss using the Deygout and Bullington
methods as a function of profile step size for a 50 km path.

![](media/image46.wmf){width="6.010416666666667in"
height="4.916666666666667in"}

As a check that the diffraction edges were being correctly identified at
the very coarse spacing, Figure 41 to Figure 44 were produced. In Figure
41, the actual obstacle separation distance is plotted as a function of
the profile step size. The mean obstacle spacing was being maintained
even at the 2000 and 5000 step sizes although this has little real
meaning with only two or three quantised values from which to choose. A
0 km obstacle spacing was possible for profile step sizes down to 50 m.
For this to occur the path must have a common terminal horizon point and
the possibility that sub path losses are being ignored, although in this
case they are likely to be close to the principal edge.

Figure 42 illustrates the Deygout principal and Bullington virtual edges
are very well correlated although the small group of points near to the
bottom of the line occur when the path becomes highly asymmetric with a
very small horizon distance. The Bullington virtual edge loss will then
increase more rapidly than the Deygout principal edge. Figure 43 shows
that these high differential losses are attributable to points that come
from the 2000 and 5000 m step sizes.

Finally, Figure 44 shows a plot of the combined loss of the Deygout
secondary edges as a function of actual object separation. The points
congregate around an obstacle separation of 1 km, which is a critical
region within which obstacles in mountainous terrain should probably be
considered as one object. The plot also shows that no profile step size
produced a single knife-edge with a combined sub path loss of less than
6 dB.

Figure 41 The actual obstacle separation as a function of profile step
size for a 50 km path.

![](media/image47.wmf){width="4.916666666666667in" height="4.125in"}

Figure 42 The Deygout principal and Bullington virtual edges as a
function of profile step size for a 50 km path.

![](media/image48.wmf){width="4.40625in" height="3.9166666666666665in"}

Figure 43 The differential loss between the Deygout and Bullington edges
as a function of profile step size for a 50 km path.

![](media/image49.wmf){width="4.802083333333333in" height="4.0in"}

Figure 44 The combined loss of the Deygout secondary edges as a function
of profile step size for a 50 km path.

![](media/image50.wmf){width="4.552083333333333in"
height="3.9166666666666665in"}

### Sub Path obstacles

In many of the previous sections, the issue of any sub path losses has
been put to one side. This was valid for the types of profiles under
examination where it was demonstrated that the numbers of sample points
that included Deygout sub path losses was very low in comparison to the
overall number of samples. There are, however, a number of types of path
where this blending out of sub path losses should not be done. The basic
problem that needs to be solved is how to identify when a sub path edge
loss is significant in a Deygout context. Sub path edges by definition
do not produce *stretched string* points or terminal horizon distances
in their own right.

Figure 45 illustrates a case where it would not be prudent to ignore the
sub path edges. However, by having a common terminal horizon distance,
the obstacle separation distance becomes 0 km and the full switch to the
Bullington method occurs. This will eliminate both sub path losses. A
solution is required and the relative position of the secondary Deygout
edge and the terminal horizon distance provides the answer. Figure 46
shows a situation in which the secondary Deygout edge on the transmitter
side is further from the transmitter than the horizon point.

Figure 45 An example of significant sub path losses.

![](media/image51.wmf){width="5.999305555555556in"
height="4.470833333333333in"}

Normally the terminal horizon point and corresponds to one or other of
the 3 Deygout edges, but as the plot shows, there are certain
combinations of obstacle heights and separations in which this is not
the case. The crucial difference between these two figures is:

-   when a Deygout edge represents a sub path loss, then *d~lt~* \>
    *d~it50~* or *d~lr~* \< *d~ir50~*

-   when a Deygout edge represents a fully obstructed loss , then
    *d~lt~* \<= *d~it50~* or *d~lr~* \>= *d~ir50~*

Figure 46 An example where the position of the secondary Deygout edge on
the transmitter side is further from the terminal than the terminal
horizon point.

![](media/image52.wmf){width="5.992361111111111in"
height="4.102777777777778in"}

### A refinement to the new model

The preceding observation provides the means to refine the hybrid model
to take account of genuine sub path obstacles that would otherwise be
blended out. Exactly the same considerations with regard to the
proximity of edges apply to sub path obstacles as to fully obstructed
edges. If a sub path edge becomes too close to a fully obstructed edge,
then the whole should really be treated as a single obstacle. The same
interpolation function as described earlier can also be used in this
case, with the crucial difference that, instead of being controlled by
the obstacle separation distance, *d~os~*, the function is controlled by
the difference between the horizon distance and the secondary Deygout
edge, *d~sost~* or *d~sosr~*:

$d_{\text{sost}} = d_{\text{lt}} - d_{\text{it}\text{50}}$ (5)

or:

$d_{\text{sosr}} = d_{\text{ir}\text{50}} + d_{\text{lr}} - d$ (6)

where:

*d*: path length

*d~lt~*: transmitter horizon distance

*d~lr~*: receiver horizon distance

*d~it50~*: Deygout secondary edge on the transmitter side

*d~ir50~*: Deygout secondary edge on the receiver side

Since the two potential secondary edges may now need to be blended out
differently from one another, the new interpolation function has to be
applied at an earlier stage in the method. Hence the original equation
(23) in P.1812 is retained with a change of parameter name and two new
equations are inserted following equations (19) and (22) respectively:

A new equation (19a) is defined:

$\begin{array}{r}
L_{t\text{50}b} = L_{t\text{50}}\left( 1 - \Theta\left( d_{\text{os}} \right) \right)\ \text{if }i_{\text{lt}} \leq i_{t\text{50}} \\
\text{        } = L_{t\text{50}}\left( 1 - \Theta\left( d_{\text{sost}} \right) \right)\ \text{if }i_{\text{lt}} > i_{t\text{50}}
\end{array}$ (7)

A new equation (22a) is defined:

$\begin{array}{r}
L_{r\text{50}b} = L_{r\text{50}}\left( 1 - \Theta\left( d_{\text{os}} \right) \right)\ \text{if }i_{\text{lr}} \geq i_{r\text{50}} \\
\text{        } = L_{r\text{50}}\left( 1 - \Theta\left( d_{\text{sosr}} \right) \right)\ \text{if }i_{\text{lr}} < i_{r\text{50}}
\end{array}$ (8)

Equation (23) becomes

$\begin{array}{r}
L_{d\text{50}} = L_{m\text{50}} + \left( 1 - e^{- \frac{L_{m\text{50}}}{6}} \right)\ \left( L_{t\text{50}b} + L_{r\text{50}b} + \text{10} + 0\text{.}\text{04}d \right)\ \text{if }\nu_{m\text{50}} - 0\text{.}\text{78} \\
\text{        } = 0\text{                                                                  }\ \text{   }\ \text{otherwise}
\end{array}$ (9)

Further testing on the effect of these changes using the synthetic path
profiles has been deferred until after the CG Meeting, in the interests
of making the method available to the group for independent testing in
time for the March CG 3K-1 Workshop in London.

# Conclusion

A detailed attempt has been made to understand the fundamental issues
causing the 3-edge Deygout diffraction method to over predict the loss
in mountainous terrain compared to the simpler Bullington method. The
proximity of the three Deygout edges in rugged terrain was shown to be a
major factor in determining this loss. A series of controlled Monte
Carlo simulations using synthetic terrain profiles has been used to
identify the complex relationships that exist between the height of the
terrain and the maximum extent of the earth curvature as the path length
increases. A single parameter, the separation between the horizon
distances, was shown to neatly encapsulate the transition from smooth
earth profiles to more mountainous conditions. A technique to exploit
the parameter to control an interpolation function to switch between the
Bullington and Deygout methods was shown. This new approach removes the
need to find an ever more complicated distance correction for the
Bullington method which may be an unrealistic goal given the variety of
terrain types around the globe.

The relationship between the principal Deygout edge and the Bullington
virtual edge was examined. Because these were shown to be almost equal
in the region of operation of the interpolation function, a procedure to
switch out the Deygout secondary edges was shown to achieve virtually
identical results. This obviates the need to calculate both the Deygout
and Bullington approach. Whilst this basic approach was shown to work
well for Deygout edges that coincide with terrain profile '*string
points'*, losses caused by Deygout sub path edges were switched out. A
modified technique using the relative positions of the horizon distances
and secondary Deygout edges was shown to predict cases where the sub
path losses should be retained.

The susceptibility of the Deygout, Bullington and the two new methods to
changes in resolution of the profile spacing in mountainous terrain was
made. The high but unwanted sensitivity of the current Deygout method to
the profile spacing was demonstrated. The new methods were shown to
reduce this susceptibility to the profile step size to the level seen
when using the Bullington method whilst retaining the more desirable
aspects of the Deygout method.

The gross insensitivity of the Bullington method to reflect changes to
the diffraction loss in response to large variations in terrain profiles
provides an elegant reason for why the Bullington and *(FSL+x) dB*
methods appear to give very similar results when averaged over many
measured paths. However, it is clear from the results that to switch
completely to the Bullington method would be, at best, a major
simplification and, at worst, equivalent to using a very coarse terrain
profile spacing.

Although the basic approach has been described there is still further
work required before the model can replace the current method. No
consideration has been given to time percentages other than median. The
current method fixes the Deygout edges at 50% time and reuses these same
edges at β~0~% time before interpolating a result for *p%* time. Given
the reliance upon these edge positions in the new method, it is open to
question as to whether the edges may need to be re-calculated at β~0~%
time. Further work is in progress to assess this issue.

Once the assessment of how best to treat the β~0~% time case has been
made, testing of the method within the full context of the full ITU-R
P.1812 can be undertaken. It is too early to take this decision at this
stage.

The new method is readily tuneable with only a small number of
constants. These need to be tuned against measurements. It is
conceivable that these constants may themselves need to be made
functions of other specific terrain related features. The distance over
which edges need to be treated as a single obstacle could be dependent
upon whether the obstacles are well worn rounded hills or jagged
mountain peaks. Since the model development has been approached from a
geometric standpoint, no frequency dependency has been studied. Again,
the constants may need some frequency tuning.

The method described has to be viewed as a pragmatic engineering
response to a real world problem of attempting to model complex
diffraction losses over a wide range of terrain types using less than
perfect models. It has the advantage of being simple to implement using
intermediate parameters that are already required within the overall
P.1812 model. A unified diffraction model incorporating all of the many
methods described within ITU-R P.526 may eventually be required but the
new method may prove useful as a stopgap measure.

Although the salient features of real terrain profiles were simulated,
there is no substitute for testing the model against real terrain
profiles for which measured data exists. It is hoped that other members
of the CG can implement the method and undertake a much more rigorous
test than has been possible to date. As such, the document should be
viewed as a draft input to the CG, which will need further work prior to
any submission to a 3K Block Meeting.

# References

\[1\] CGD-05 Bullington and 3-edge diffraction methods compared with
measured results, UK

\[2\] "Using the Test Results", text file + supporting .csv files
submitted to the CG in Oct 2007, UK

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

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

\[5\] 3K/TEMP/91-E, "Proposed alternate methods of calculating
diffraction losses for path specific propagation prediction", Markus
Liniger, Switzerland

\[6\] CGD-03 "Comparison between diffraction methods under consideration
for the path specific point-to-area DNR for smooth/sea paths", David
Bacon, UK

\[7\] CGD-14 "Proposals for a Distance correction for Bullington",
Markus Liniger, Switzerland

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