Weather & environmental changes affect RF signal severely. Ducting is one of the environmental phenomena that heavily deteriorate the radio performance. This document will give some ideas on root cause, impact & solutions of ducting on Radio Performance.

1. Ducting-Cause and Impact on Radio
Performance
Created by: JakirAhmed
Md. MustafizurRahman

2. Contents
1 HISTORICAL BACKGROUND.............................................................................................................................3
2 WHAT IS TROPOSPHERIC DUCTING ..............................................................................................................4
3 REASONS OF ENVIRONMENTAL IMPACT ....................................................................................................4
3.1 Theoretical Consideration: .............................................................................................................4
3.2 Temperature Inversion:...................................................................................................................6
4 DUCTING AREA & PERIOD:...............................................................................................................................8
4.1 Ducting Area.....................................................................................................................................8
4.2 Ducting Period..................................................................................................................................8
5 INVESTIGATION: DUCTING IN BRITISH CHANNEL...................................................................................9
6 FREQUENCY RANGE..........................................................................................................................................10
7 DUCTING FORECAST & INDEX.......................................................................................................................11
7.1 Geographical Prediction View: ....................................................................................................12
8 IMPACT ON NETWORK.....................................................................................................................................14
8.1 Pattern of Ducting Effect: .............................................................................................................14
8.2 Ducting Impact on Radio Network Performance: .....................................................................17
9 CONTROL STRATEGIES....................................................................................................................................20
9.1 Hardware Modification ..................................................................................................................20
9.2 Soft Modification: ...........................................................................................................................23
10 APPENDIX: ..........................................................................................................................................................26
10.1 Conditions for Ducting: ...............................................................................................................26
10.2 Direction of Enhanced Interference Fields: .............................................................................27
10.3 Width of Duct:...............................................................................................................................28

3. 1 Historical Background
Fig: 01 First Proof of Ducting
This is a picture received by Australian TV channel from around 1340 miles away. It was
a great surprise at that time. New Zealand TV-4 channel 4 (175.25 MHz) video received
on December 3rd, 1975 by Robert Copeman, Sydney, Australia. It was summer time and
later the phenomenon was rightly explained by tropospheric ducting.
Fig: 02
Geographical view – Signal was received in
Sydney from Auckland
Radio waves traveling through free space have little or no external influence on them. But
when such signals are traversed through the atmosphere, their effective propagation will
be determined by atmospheric factors. Some of these atmospheric factors are variations
in geographic height, differences in geographic locations, and changes in time (day, night,

4. season, etc.). Knowledge about these factors is very important for telecommunications
operators in making necessary adjustments, where possible, for optimum result.
2 What is Tropospheric ducting
Tropospheric ducting is a type of radio propagation that tends to happen during periods
of stable, anticyclonic weather. In this propagation method, when the signal encounters
a rise in temperature in the atmosphere instead of the normal decrease (known as a
temperature inversion), the higher refractive index of the atmosphere there will cause the
signal to be bent. Tropospheric ducting affects all frequencies, and signals enhanced this
way tend to travel up to 800 miles (1,300 km) (though some people have received "tropo"
beyond 1,000 miles / 1,600 km), while with tropospheric-bending, stable signals with good
signal strength from 500+ miles (800+ km) away are not uncommon when the refractive
index of the atmosphere is fairly high.
3 Reasons of Environmental Impact
When radio wave signals are propagated through the atmosphere, their transmission
capacity is greatly influenced by atmospheric conditions at that point. Within the
atmosphere, radio waves can refract, reflect, diffract, and even be ducted. For example,
in our recent work, it was demonstrated that when signals are propagated through the
atmosphere, they may be refracted. This refraction tends to further lengthen the ray path
thereby increasing the propagation time, which in turn lead to late arrival time of the signal
at the receiver circuit. This phenomenon is called propagation delay.
Also, wind, air temperature, and water content of the atmosphere can combine to either
extend radio communications or to greatly attenuate wave propagation making normal
communications extremely difficult. So, due to the non homogeneous nature of the
atmosphere, little changes in atmospheric constituents can produce dramatic changes
in ability to communicate.
3.1 Theoretical Consideration:
When radio waves are propagated, it is customary to experience problems caused by
certain atmospheric conditions. These problem-causing conditions results from a lack of
uniformity in the earth’s atmosphere. Many factors can affect atmospheric conditions
either positively or negatively. Before we proceed further here, it is pertinent to clearly
examine the composition of earth’s atmosphere.

5. The earth’s atmosphere is divided into three major separate regions:
(1) Troposphere
(2) Stratosphere, and
(3) Ionosphere.
The troposphere is the lowest part of the atmosphere extending from the surface up to
10km (6 miles) see below Figure. Almost all weather phenomena takes place in the
troposphere.
Fig: 03
Layers within the
atmosphere
Temperature in this region decreases rapidly with altitude. There is formation of cloud
and also a lot of turbulence because of variations in temperature, pressure, and density.
All these conditions have profound effects on radio waves.
The stratosphere is located between troposphere and ionosphere. Temperature
throughout this region is almost constant and there is little presence of water vapor.
Because of its relatively calm nature with little or no temperature change, stratosphere
has almost no effect on radio wave signals.
The ionosphere refers to the upper region of the atmosphere where charged gas
molecules have been produced by energy of the sun. The degree of ionization varies with
intensity of the solar radiation. It is the most important region of the atmosphere used for
long distance and point-to-point communication.

6. The ionosphere is not a homogeneous region but consists of rather distinct layers which
have their own individual effects on radio propagation. The layers of distinct interests are
D, E, F1 and F2.
3.2 Temperature Inversion:
Tropospheric ducting is a phenomenon that is brought about as a result of a dramatic
increase in temperature at higher altitudes. Under normal atmospheric conditions, the
warmest air is found near the surface of the earth. The air gradually becomes cooler as
altitude increases. At times, unusual situations may develop in which layers of warm air
are formed above layers of cool air. This condition is known as “temperature inversion”.
This temperature inversion causes ducts of cool air to be sandwiched between the
surface of earth and a layer of warm air. Now, if a radio wave enters the duct at a very
low angle of incidence, VHF and UHF transmission may be propagated beyond normal
line-of-sight. When ducts are present as a result of temperature inversion, propagation
ranges of VHF and UHF are extended.
This extension is as a result of different densities and refractive qualities of warm and
cool air. The duct acts like an open–ended waveguide. Signals trapped in it can travel
thousands of miles. Indeed, there is no theoretical limit to the distance a signal can travel
via tropospheric ducting.
As a result, a distant station may be heard in favor of a closer station on the same
frequency band (a situation similar to “far away condition” in E.M. theory). Sometimes,
conditions are such that multiple ducts are formed, bringing in distant stations from many
different areas at the same time.

7. Fig: 04, Duct Effect Caused by Temperature Inversion
Fig: 05, Duct Effect Caused by Temperature Inversion
Typical conditions required for a good duct to occur are:
1. An increase in temperature by 3C or more per 100ft.
2. A rapid decrease of RH (dew-point) with height.
The depth of the duct required for varying wave-lengths is:
1. 50ft for wavelengths around 3cm (approx. 1000MHz)
2. 600ft for wavelengths around 1m (approx. 300MHz)
Typical meteorological conditions which can be favorable for ducting are:
1. Warm dry air over a cooler surface, especially a cool sea
2. Surface cooling under clear skies overland
3. Anticyclone (high pressure) or developing high pressure ridges with a cold
surface
4. Sea breezes undercutting warm air overland
5. At fronts with a strong thermal contrast
6. In cold downdraughts associated with cumulonimbus clouds (indicated by heavy
showers or thunderstorms)

8. 4 Ducting Area & Period:
4.1 Ducting Area
Temperature inversions occur most frequently along coastal areas bordering large bodies
of water. This is the result of natural onshore movement of cool, humid air shortly after
sunset when the ground air cools more quickly than the upper air layers. The same action
may take place in the morning when the rising sun warms the upper layers.
High mountainous areas and undulating terrain between the transmitter and receiver
can form an effective barrier to tropospheric signals. Ideally, a relatively flat land path
between the transmitter and receiver is ideal for tropospheric ducting. Sea paths also
tend to produce superior results.
In certain parts of the world, notably the Mediterranean Sea and the Persian Gulf,
tropospheric ducting conditions can become established for many months of the year to
the extent that viewers regularly receive quality reception of signals over distances of
1,000 miles (1,600 km). Such conditions are normally optimum during very hot settled
summer weather.
Tropospheric ducting over water, particularly between California and Hawaii, Brazil and
Africa, Australia and New Zealand, Australia and Indonesia, Strait of Florida, and Bahrain
and Pakistan, has produced VHF/UHF reception ranging from 1000 to 3,000 miles (1,600
– 4,800 km).
4.2 Ducting Period
Overland paths are usually the strongest at sunrise and weakest at mid afternoon.
Generally ducting happens at the beginning & ending of winter.
Water paths are usually the strongest at mid afternoon and weakest at sunrise.
Combination land-water paths may peak at various times depending on the local weather
conditions.

9. 5 Investigation: Ducting in British Channel
British Channel:
Time Period: August 2003 to September 2004
Three transhorizon, 2 GHz radio paths have been established in the British Channel
Islands in order to investigate the characteristics of long-range UHF propagation over the
sea. During the measurement period (August 2003 to September 2004), signal strength
enhancements occurred in the late afternoons and evenings and in the summer months.
Experiment Paths:
Three transhorizon 2 GHz radio paths were established in the British Channel Islands:
(1) Jersey to Alderney (48.5 km)
(2) Jersey to Guemsey (33.5 km &
(3) Jersey to Sark (21 km)
Fig: 06, Map depicting
transmitter and receiver
locations in the British Channel
Islands
It was observed that enhanced signal strengths occur predominantly during the late
afternoon and evening periods and, on a seasonal scale, occur more often during the
summer months. This strong diurnal variation may be evidence of a sea breeze/land
breeze oscillation. During the measurement period, approximately 8% of signals received
at the Alderney high antenna show enhancements, however the monthly values peak at
38% in May 2004 and have a minimum of 0% in October 2003, December 2003, January
2004 and February 2004.

10. Fig: 07, Diurnal variation in the
frequency of occurrence of enhanced
signals at the Alderney high antenna
Fig: 08, Monthly statistics of enhanced
signal strength data at the Alderney
high antenna (August 2003 to
September 2004)
6 Frequency Range

11. Fig: 09, Frequency Spectrum Distribution
Ultra High Frequency range (UHF) : 300–3000 MHz ----> used in microwave
devices/communications, radio astronomy,
mobile phones, Land Mobile
Very high frequency (VHF) : 30–300 MHz -----> used in FM, television
broadcasts, weather radio
As both UHF & VHF are suffered by signal inversion and GSM spectrum lies in UHF range
mobile operators face ducting problem with GSM spectrum.
7 Ducting Forecast & Index

12. First ducting forecast was introduced by William R. Hepburn and this forecast is called
Hepburn Ducting Index. Hepburn Tropo Index is the degree of tropospheric bending
forecast to occur over a particular area which is an indication of the overall strength of
tropospheric radio signal strengths (and hence interference) on a linear scale from 0 to
10.
Ducting index is based on the prediction for VHF (Very High Frequency- 30 MHz to 300
MHz) UHF (Ultra High Frequency - 300 MHz to 3 GHz) and/or microwave radio bands
for signal
 Scatter
 Reflection &
 Refraction
Hence it indicates interference to occur.
Some general rules of thumb:
A higher index = stronger signals.
A larger area = longer distances.
7.1 Geographical Prediction View:

13. Fig: 10, Prediction by Hepburn Tropo Index
Nov 22, 2011
Fig: 11, Prediction by Hepburn Tropo Index
Nov 23, 2011

14. 8 Impact on Network
8.1 Pattern of Ducting Effect:
 Network often faces severe performance degradation without any visual reason.
 Environmental variation might be the possible reason in this sharp degradation of
performance which is called ducting.
 2 times/year Bangladeshi Cellular networks face ducting, October-December &
March-April
 Ducting Effect is observed in morning & in evening
 Span of Ducting Effect is about 4 hours in morning & 6~7 hours in evening
 In the day time, Ducting free/minimal time span was 9AM~4PM in October
 Dhaka City is almost excluded or minimally affected by ducting effect. Other big
cities like Chittagong & Sylhet are also less affected.
 Eastern Bangladesh is not affected. In general, the part which is closer to the hills
are not/less affected by the ducting effect.
 Zone closest to Sundarbans are less affected

15. Fig: Ducting Impact on MPD & Ducting Time
Fig: Ducting Presence Identification from TA sample

16. Ducting Effect- March ’07
Fig: 13, Ducting impact on Bangladesh
Ducting Effect- April ’08
Fig: 14, Ducting impact on Bangladesh

17. 8.2 Ducting Impact on Radio Network Performance:
 Ducting Effect is found in morning & in evening.
 Span of Ducting Effect is about 3 hours in morning & 6~7 hours in evening.
 In the day time, Ducting free/minimal time span was 9AM~4PM in October,
10AM~3PM in March, 11AM~3PM in April & 8AM~6PM in May.
 In late April/May, little ducting effect was observed all through the day time
 93%~94% of total drop increment is contributed by the morning slab & evening
slab. Evening slab is dominating because of its high traffic generation
characteristics. But ducting severity is no less in morning slab
Fig: TA Sample vs Ducting Intensity

18. Fig: Call Drop Comparison between Ducting & Non-ducting Period
Fig: 15, Ducting impact - Time
Observations:
 3 types of Bad Quality Drops (Uplink, downlink & both link) are contributing more
than 50% of total drop increment. Out of them Bad Quality- downlink is the most
dominant, contributing about 35% of total.
 Though, ‘bad quality drop’ is the dominant drop in case of ducting effect; other
types of drops are affected as well, like- Signal Strength drop, Sudden drop & Other
Drop Increment (27-Apr-08 Vs 11-May-08)
5AM~10AM
37.3%
12AM~5AM
0.0%
10AM~4PM
6.6%
4PM~12AM
56.0%

19. drop. This can be explained as- quality degradation (higher interference) causes
Signal Strength Drop, Sudden Drop & ‘Other’ Drops to increase as well.
 The Signal Strength what is sufficient in case of low interference scenario to retain
a call, suddenly become insufficient when interference increases.
 High TA sample also increases during the ducting period. By checking TA sample
during the morning & evening along with bad quality drop trend checking, ducting
can be easily identified.
Drops Distribution:
Fig: 16, Ducting impact - Drops
Distribution
Drop cause Distribution-Total-27th Apr'08 Vs 11th May'08
0.00
0.05
0.10
0.15
0.20
BQ DR-
DL
BQ DR-
UL
BQ DR-
BL
Sudden
Loss DR
SSDR-
DL
SSDR-
UL
SSDR-
BL
Other DR
%
27-Apr-08 11-May-08

20. Fig: 17, Ducting impact - Drops
Distribution
Percentage of Bad quality drops is around 46% during ducting period which is the main
contribution of performance degradation.
9 Control Strategies
There is no complete solution to address ducting effect. But some measures may be
taken to reduce its impact:
9.1 Hardware Modification
Probable Steps to Reduce Ducting Impact:
Here are some possible ways for trial in the selected cluster –
(1) HW Modification:
(a) Use Null fill antenna:
It is used to prevent too much of the signal from overshooting the nearest
part of intended coverage area. Phasing is used between antenna elements to take
power away from the upper lobe & redirected it to lower lobe. As upper lobe is
suppressed ducting is supposed to be reduced by using this
Drop Increment Contribution (27-Apr-08 Vs 11-May08)
SS-BL
4.9%
Sudden
22.9%
TA
0.1%
SS-UL
9.2%
BQ-BL
5.0%
SS-DL
7.6%
BQ-UL
12.7%
BQ-DL
29.7%
Other
7.9%

21. Fig: 18
Limitations:
 only portion of upper lobe is converted to lower lobe – hence ducting will not be
reduced so effectively
 needs high costing to purchase new antenna
 needs heavy manpower
(b) Use shade over antenna:
Shading over upper lobe may be used to absorb signal which causes ducting after
traveling long distance.
Upper
lobe is
suppresse
d & power
is used to

22. Fig: 19
Limitations:
 difficult to mechanical setup
 needs high expenditure
 needs heavy manpower
(c) RET Antenna:
if antenna have remote electric tilting facility, it will be possible to put high tilt
during ducting & then withdraw during normal condition. This method will restrict
signal propagation cause less ducting.
Limitations:
 High tilting will certainly lessen cell coverage
 RET antenna are expensive
(d) Antenna Down-tilting:
To be such that- there is no or negligible signal remain above horizontal plane from
the antenna, Specially antenna of 900band BTS. This is required to prevent signal
to enter into high temperature ‘ducting layer’ at first place. More tilting needed for
antenna with higher vertical opening angle. We also need to look into restricted
use of higher vertical opening angle antennas. Back lobe of antennas also need to
be restricted by all possible means. Persistent and relatively longer term effort is
needed in this area to gradually get rid of signal above horizontal plane where we
Shade may
be used to
suppress
upper lobe

23. don’t need it (other than dense urban areas). We can prioritize the areas for this
activity according to their contribution on ducting effect.
9.2 Soft Modification:
SW Modification / Feature Implementation:
As mentioned earlier, ducting impact cannot be minimize completely. Features &
optimization initiatives can slightly improve the performance. Optimum Frequency
Planning, Site Design & Neighbor Planning are the best solutions to combat the
Ducting.
 TA Controlling in inside Country Border:
Inside the country border, maximum TA is defined to retain network coverage from the
neighbors’ sites during any outage. During the Ducting period, signal can travel through
long distance & this TA settings will allow the long distance cell to consider as the
neighbor cell.
By limiting TA threshold inside country boarder can be an effective solution to restrict
the long distance cell as neighbor. This will disallow cells taking traffic from long
distance which will stop unnecessary camping, Handover failure and eventual call
drops. However, it is not possible to avoid the interference comes from distant site due
to ducting by this Optimization. This optimization will only help during campaign &
Handover state. 5-7% quality gain can be achieved by this optimization.
TA limit settings need to be done carefully, excessive and unnecessary TA control can
reduce the coverage and accessibility.
Listed three criterions can be checked for inside country TA control cell selection.
Cells which meets all 3 criteria 1) Having traffic from TA>X in ducting but not without
ducting 2) During ducting HO success rate fall by >Y% 3) During ducting period
Quality drop increases by >Z%; can be selected for tuning TA control.
After optimization Traffic & Accessibility KPIs need to be checked for coverage &
accessibility checking. Optimization rollback or TA threshold relaxing might be needed
to maintain coverage areas unchanged.
Gains:

24. Fig: Minute per Drop Gain
Fig: Handover SR Gain

25. Fig: Handover SR Gain
 BTS Power Control:
Feature adjustment. In ‘ducting effect’ affected areas- Signal strength from long
distance cells are coming to interfere heavily the serving cells at that area. So, the
serving cells have to be that much stronger to ‘beat’ the interferer. To help the
serving cells to be strong enough compared to enhanced interferer (when quality
is low)- BTS Power Control feature has to be more quality driven and less signal
strength driven. We mean- the power regulation parameters need to adjusted so
that it become more responsive to quality and less responsive to signal strength.
 DTX in downlink:
DTX in downlink will limit signal transmission both for serving cell & interfering cell
and ultimately will reduce collision probability between them thus reducing
interference. At present, DTX-D is used mainly in urban areas but this will really
help the scenario in Rural areas specially in the high ducting season (January to
May).
 Frequency planning:
Taking care of ‘ducting effect’. Taking the real interference scenario (interference
matrix) of the ‘field’ while planning the frequencies for cells (in 900 band). The
theme is that- when propagation pattern changes, frequency plan should follow
that. NCS for Ericsson system or similar tool in Huawei System can be used to
measure the real life interference relationship of cells, when ducting is ‘ON’. NCS
recording can be done for example for whole 3 days (3*24) so that it takes an
aggregated impact of both ducting & non-ducting hours (to ensure a better whole
day weighted average scenario). This special frequency set can be in place for 5

26. months- January to May, every year, until we get ducting effect minimized by
higher density of BTS in Rural areas or through antenna tilting/modification
(propagation control) mechanism.
 Interference Rejection Combining:
IRC may be implemented to reduce ducting where UL signals contribute. But as
UL quality drops have much less contribution to ducting compared to DL will be an
expensive feature.
10 Appendix:
10.1 Conditions for Ducting:
The index of refraction is defined as
n = √εr = c/v (2)
where
εr is the dielectric constant of the troposphere,
c is the speed of light and
v is the phase velocity of the electromagnetic wave in the medium.
Since n near the earth’s surface is slightly greater than unity (1.00025–1.00040),it seems
more practical to use the scaled index of refraction N, which is called refractivity and is
defined as:
N = (𝑛 − 1) ・ 10 6
=
77.6𝑝
𝑇
−
5.6𝑒
𝑇
+
3.75 ・ 105𝑒
𝑇2
where
p is the total pressure in mbar,
e is the water vapor pressure and
T is the temperature in ◦Kelvin.

27. In order to examine the N gradients, the modified refractivity index is used. It is defined
as:
𝑀 = (𝑛 − 1 +
ℎ
𝑎
) ∗ 106
= 𝑁 + 0.157ℎ
The computation of the refractive conditions, characterized as Sub refraction, Standard,
Surer refraction and Trapping is achieved by its gradient 𝑑𝑀/𝑑ℎ. Tropospheric ducting
phenomena occur when either:
𝑑𝑀/𝑑ℎ < 0
Or
𝑑𝑁/𝑑ℎ < −157 is met.
The tropospheric ducting effects to radio wave propagation, are similar to that of the metal
waveguides; therefore, only modes with a wavelength shorter than the cut-off wavelength
can propagate (the cut-off frequency being a function of the duct’s width). In
transmitter/receiver systems, the height of the antennas, together with the vertical
refractivity profile, can lead to a variety of phenomena. Usually, the radio waves are
trapped inside the duct, leading to an over the-horizon propagation. On the other hand,
radar with the antenna positioned below the ducting layer can miss a target flying inside
or above the duct. If both receiving and transmitting antennas are located inside the duct,
the field in the receiver is stronger, compared to the field received in the absence of the
waveguide.
10.2 Direction of Enhanced Interference Fields:
For the avoidance of interference as required by radio regulations [ITU, 1976; NTIA,
1979J, the terminals of co-channel systems are normally positioned to avoid an efficient
inter-system (interference) propagation path. For the usual trans-horizon potential
interference path that then results, the available modes of propagation are volume
scattering (troposcatter), diffraction, and (in the presence of tropospheric layers)
turbulence-layer scatter, reflection, scatter in the presence of rain, and the strongly
refracted (ducted) modes of propagation. Their order in the preceding sentence is that of
generally increasing efficiency, but of decreasing availability.

28. Fig: 20, Ducting Interference Field Direction
the TaRa and TbRb das h-dot trajectories represent LOS service paths, i so1ated from
one another by their antenna patterns and the intervening terrain, such as at O. The
dashed-line path TaORb represents a weak (diffracted or troposcatter) interference path
but the continuous-line path TaLRb represents a potentially strong interference path, via
the ducting layer at L. Of course, additional reflecting layers can provide additional
interference path trajectories. Reflection from terrain (between Ta and 0 or Rb and 0) can
also provide additional components via the layer at L.
10.3 Width of Duct:
The bending of a radio wave's trajectory through the atmosphere results primarily from
the spatial variation of the atmospheric refractive index n(h) with elevation h above the
surface. However, a more sensitive measure of this n(h) structure is given by the
refractivity structure N(h):
𝑁(ℎ) = [𝑛(ℎ) – 𝑙] 106 𝑁 𝑢𝑛𝑖𝑡𝑠,
and its vertical gradient
𝑑𝑁/𝑑ℎ = 106 𝑑𝑛/𝑑ℎ 𝑁 𝑢𝑛𝑖𝑡𝑠/𝑘𝑚

29. Nevertheless, in describing the effects of this structure upon radio propagation, it is
often more convenient to describe the gradient in terms of the modified refractivity
structure M(h),
𝑀(ℎ) = 𝑁(ℎ) + 157ℎ 𝑀 𝑢𝑛𝑖𝑡𝑠,
and its gradient
𝑑𝑀/𝑑ℎ = 𝑑𝑁/𝑑ℎ + 157 𝑀 𝑢𝑛𝑖𝑡𝑠/𝑘𝑚
This modified refractivity results from the geometrical transformation from a spherically
stratified atmosphere above a spherical path to a planar stratification above a flattened
earth.
For those angles of incidence at the layer base that are equal to or less than the critical
value of , the incident wave from below is refracted (bent) by a trapping layer of (dM/dh <
0 M units/km) and may propagate efficiently (be ducted) for long distances. This is
illustrated in the below figures. To the left of the figure, the vertical refractivity profile N(h)
depicts a layer N(h) meters thick with a gradient (dN/dh)o < -157 N units/km and at a base
height of ho meters above ground. Above and below the layer, there are standard
gradients (𝑑𝑁/𝑑ℎ = −39 𝑁 𝑢𝑛𝑖𝑡𝑠/𝑘𝑚). The remainder of Figure depicts the layer
extending over an effective earth curvature Re that is 4/3 that of the true earth, i.e., Re =
4/3 Ro' Ro = 6370 km. In this situation, a radio wave launched at the layer base height
ho at (or less than) the critical angle will follow a curved path within the layer. A radio
wave traveling exterior to the layer will follow a straight path.

30. Fig: 21, Elevated layer with a ducting gradient, its refractivity profile, and two trapped
trajectories. The reference elevation is for an effective earth radius 𝑅𝑒
Fig: 22, Elevated layer with a ducting gradient, its modified refractivity profile, and two
trapped trajectories. The reference elevation is for a flattened earth, D is the duct
thickness; 𝛿ℎ s the layer thickness.

31. Figure 22 is a replotting of Figure 21 for the flattened earth. The modified refractivity profile
𝑀(ℎ) is shown at the left. Note that the top of the duct at ℎ 𝑎 and the bottom of the duct at
ℎ 𝑏 have the same M value,
𝑀 (ℎ 𝑎) = 𝑀(ℎ 𝑏)
The remainder of Figure 6 illustrates the conventional symmetrical trajectories associated
with duct propagation. The trajectories are well approximated as parabolic within a layer
of constant gradient. the duct width is given by:
D=𝜕ℎ [1 − {(
𝑑𝑀
𝑑ℎ
) 𝑜 − (
𝑑𝑀
𝑑ℎ
) 𝑏}]
Where (
𝑑𝑀
𝑑ℎ
)
0
the (negative) is modified refractivity gradient across the ducting layer and
(
𝑑𝑀
𝑑ℎ
)
𝑏
is the (positive) modified refractivity gradient below the ducting layer.