Path Clearance 
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Refractivity in the Atmosphere
Propagation in Standard Atmosphere
©
20012016, Apus Cloud Project e Luigi Moreno
_______________________________________________________________
In this
Session the effect of the atmosphere on radio ray trajectories is first
considered, by introducing the kfactor concept; possible deviations from
standard conditions are identified, as well as the minimum kfactor value. Then
the Fresnel ellipsoid is defined; the partial obstruction of the ellipsoid
leads to the estimate of the resultant loss. Finally, the previous concepts are
used to set clearance criteria and to discuss their application to path profile
analysis.
The Refractive Index n in a given medium
is defined as the ratio of the speed of radio waves in vacuum to the speed in
that medium. Since the speed of radio waves in the atmosphere is just slightly
lower than in vacuum, then the Refractive Index in the atmosphere is greater
than, but very close to, 1.
However,
also small variations in the atmosphere Refractive Index have significant
effects on radio wave propagation. For this reason, instead of using the
Refractive Index n (close to 1), it is convenient to define the Refractivity N
as :
_{
}
So, N
is the number of parts per million that the Refractive Index exceeds unity; it
is a dimensionless parameter, measured in Nunits.
The
atmosphere Refractivity is a function of Temperature, Pressure, and Humidity.
The ITUR Rec. 453 gives the formula :
_{
}
where :
T = absolute temperature (Kelvin deg);
P = atmospheric pressure (hPa, numerically
equal to millibar);
e = water vapour pressure (hPa).
At sea level,
the average value of N is about No = 315 Nunits. The ITUR gives world maps
with the mean values of No in the months of February and August.
Temperature, atmospheric pressure, and water vapour pressure are not
constant with height. This produces a Vertical Refractivity Gradient G
(measured in Nunits per km, N/km), defined as:
_{
}
where N_{1}
and N_{2} are the refractivity values at elevations H_{1} and H_{2},
respectively.
Under
normal (standard) atmospheric conditions,
Refractivity decreases at a constant rate, moving from ground level up
to about 1 km height. This means that
the Refractivity Gradient G is constant, the typical value being about
40 N/km.
Deviation
from the Standard Atmosphere condition is usually associated with particular
weather events, like temperature inversion, very high evaporation and humidity,
passage of cold air over warm surfaces or vice versa. In these conditions, the
Vertical Refractivity Gradient is no longer constant. A number of different
profiles have been observed and measured. It is worth noting that, at greater
altitude, the Refractive Index is, in any case, closer and closer to 1; so the
Refractivity N decreases to zero.
A
Radio Wave propagates in the
direction normal to the isophase plane (the plane where all the points are
phase synchronous, with respect to the sinusoidal pattern of electric and
magnetic fields).
In a homogeneous
medium, the isophase planes are parallel to each other and the propagation
direction is a straight line normal to them.
As seen above, the Atmosphere is not a
homogeneous medium and the Vertical Refractivity Gradient gives a measure of
that. Different Refractivity at different heights means different propagation
speeds. The wavefront moves faster or slower, depending on the height: this
causes a rotation of the wavefront itself.
Wavefront and ray rotation
caused by a
vertical refractivity gradient in the atmosphere
So, the
propagation trajectory (normal to the wavefront) is not a straight line, but
it is rotated, as shown in the above figure. Taking into account that the
propagation speed is inversely proportional to the refractive index, it is
possible to derive that the radio trajectory curvature
1/r is related to the Vertical Refractivity Gradient G, as :
_{
}
In
Standard Atmosphere, with a typical value of the
Refractivity Gradient G =
40 N/km, the curvature of the radio ray trajectory is :
_{
}
This
means that the radio ray is bent downward, with a curvature 1/r, somewhat lower
(less curved) than the Earth curvature 1/R :
_{
}
Ray bending in standard atmosphere
(CL = clearance, vertical distance from ground to
ray trajectory)
A
convenient artifice is used to account, at
the same time, for both the ray and the earth curvatures. An
"equivalent" representation of
the
above figure
can be plotted by altering both curvatures by an amount equal
to the ray curvature 1/r.
In the new figure (see below) the radio ray trajectory
becomes a straight line, while the modified ("equivalent") earth
curvature 1/R_{E} is :
_{
}
Equivalent representation of the previous figure,
with a modified earth radius R_{E} and a
straight ray trajectory.
Note
that, at any point of the radio path, the vertical distance (CL = clearance)
from the earth surface to the ray trajectory is the same in the real and in the
equivalent representations.
The ratio
between the equivalent and the real earth radius is defined as the
"effective earthradius factor k" (briefly, the kfactor).Taking account of previous formulas, giving 1/R_{E},1/R,and 1/r
, the kfactor is given by :
_{
}
In Standard
Atmosphere (G = 40 N units/km), this gives :
_{
}
The
kfactor gives an indication about the atmosphere state at a given time and
about the bending effect on the radio ray trajectory. So, the statement
"propagation at k = 4/3" is a synonymous of "propagation in
Standard Atmosphere".
On the other hand, k < 4/3 corresponds to
"Subrefractive" conditions,
in which the ray curvature is less than normal
or even is an upward curvature ( k
< 1, G > 0 ),
thus
reducing the clearance over ground.
With k
> 4/3 we are in a "Superrefractive" atmosphere; in particular,
with k =
¥
, the ray trajectory is parallel
to the earth surface and
the
signal can propagate over large distances, beyond the normal horizon
.
The figure
below compares the ray trajectories with different kfactors, using a
"real earth" representation.
Ray bending in different atmospheric conditions
(different kfactor values)
A
further alternative in plotting radio ray trajectories over the earth surface,
is called "flat earth" representation
Again,
both the earth and the ray curvature are altered, but in this case the earth
profile is forced to be flat, while the ray curvature is modified accordingly.
The "real earth" and the "flat earth" diagrams are
equivalent in the sense that,
at any point of the radio path, the vertical distance (CL = clearance)
from the earth surface to the ray trajectory is the same in both
representations.
Equivalent representation of the previous figure,
over flat earth
Using
the "flat earth" representation, we can plot on the same diagram the
path profile and multiple rays, corresponding to different values of the
kfactor. This is the most usual
diagram shown in computer applications for radio hop design.
kFactor variability
We have seen that the kfactor is
related to the atmosphere state and is a function of the refractivity vertical
gradient. So, it is a variable parameter,
depending on daily and seasonal cycles and on
current meteorological
conditions.
In a
"standard atmosphere" state the kfactor value is 4/3; this is close
to the median value in most climates (particularly, temperate climates). Around
this median value,
the
range of variations is rather wide in tropical regions, with increasing temperature
and/or humidity, while it is more limited in cold and temperate climates.
Experimental
observations show for example that the probability of k<0.6 in temperate
climates is generally well below 1%. In tropical climates the same event is
observed with probability in the range 5%  10%. This means that, in tropical
regions, there is the highest probability of propagation anomalies due to
extreme kfactor values. The ITUR gives world maps of the time percentage with
G < 100 Nunits / km (k > 2.75), in different months.
In
discussing kfactor variability, as applied to radio hop design and to
clearance criteria, we have to consider that:
·
In subrefractive conditions
(minimum kfactor) the clearance
over ground is reduced and the
probability of obstruction is maximum.
·
We are not interested in the minimum "local" kfactor,
but in the overall effect through the whole radio path. So an "equivalent
kfactor" (k_{ea}) is defined, whose minimum value depends (f
or given climatic conditions
) on the path length.On long hops k_{ea}
is likely to be not far from
standard values, because extreme atmosphere conditions are probably not present
at a time on the whole path, while in shorter hops it is more likely that
particular events affect almost the whole path andproduce lower
k_{ea}values.
The ITUR (Rec. P530) gives a curve of minimumk_{ea}
values as a function of hop length
(temperate climate).
Minimum equivalent kfactor vs. path length
(from ITUR Rec. P530, by IT permission).
From a geometrical point of view,
the Fresnel ellipsoid is defined as the set of points (P) in the space which satisfy
the equation :
_{
}
where Tx and Rx are the two
antennas (radio path terminal points), representing the two focuses of the
ellipsoid.
The Fresnel ellipsoid, F1 = ellipsoid radius;
CL = clearance, measured from earth surface to the
ray trajectory (that is the ellipsoid longitudinal axis)
The radioelectrical interpretation
of the Fresnel ellipsoid is that two rays, following the paths TxRx and
TxPRx, arrive at the Rx antenna in phase opposition (halfwavelength path
difference, then 180 deg phase shift).
The Fresnel ellipsoid radius F1 (in meters),
at a distance D1 from one of the radio sites, is given by :
_{
}
where D
(km) is the path length, F (GHz) is the frequency and
l
(m) is the wavelength.Some examples are given in the figure below;
note that the Fresnel ellipsoid radius reduces as frequency increases.
Fresnel ellipsoid radius vs. path length and
frequency
(max radius,
computed at mid path length).
From a practical
point of view, the Fresnel ellipsoid gives a rough measure of the space volume
involved in the propagation of a radiowave from a source (Tx) to a sensor (Rx).
About half of the Rx signal energy travels through the Fresnel ellipsoid. So,
any obstruction within the Fresnel ellipsoid has some impact on the Rx power
level.
This
leads to consider radio visibility in terms of clearance of the Fresnel
ellipsoid, as discussed below.
A
note on radio propagation and visual analogies
We
are familiar with our visual experience and this can be of help in describing
some aspects of radio propagation.
However,
the Fresnel ellipsoid shows that radio propagation (like any EM propagation
effect) cannot be explained only in terms of geometric optics, that
is adequate so long as any
discontinuities encountered through the propagation path are very large
compared with the wavelength.
The
ellipsoid radius is proportional to
the wavelength square root. In our visual experience, the light wavelength is
so small (about 5 10^{4} mm) that the radius of the Fresnel ellipsoid
is negligible, at least as a first approximation. Diffraction effects can be
observed only with accurate experiments, showing the role of Fresnel ellipsoid
also in the optical field.
On
the other hand, in radio communications the wavelength is in the range from 1 m
(frequency 300 MHz) to about 1 cm (frequency 30 GHz), that is almost one million
times larger then in visible waves.
In
conclusion, much care must be paid in establishing an analogy between radio
propagation and visual experience. Even if in both cases we deal with EM waves,
the large difference in wavelength makes practical results quite different in
most conditions. For example, the concept of Visibility is quite different in
Radio Engineering and in our visual experience.
At any point
of the path profile, the Clearance (CL)
is defined as the vertical distance form the ray trajectory to the ground.Since for different kfactor values a
different ray trajectory is observed, then the Clearance at a given point
depends on the kfactor (atmosphere state).
A
negative Clearance means that an obstacle is higher than the ray trajectory
(note that this is the sign convention used in ITUR Rec. P530, while the
opposite is adopted in ITUR Rec. P526).
Single
obstacle loss
The
effect of a single obstacle, that in some measure impedes the propagation of a
radio signal, is analyzed in terms of Fresnel ellipsoid obstruction. So, a
Normalized Clearance is defined as C_{NORM} = Cl / F1, where F1 is the
Fresnel ellipsoid radius.
A theoretical evaluation of diffraction loss is usually made with reference to two idealized obstacle models :
·
the knifeedge obstruction, that
is an obstacle with negligible thickness along the path profile;
·
the smooth spherical earth, that
is the obstruction produced by the earth surface for transmission beyond the
horizon.
The two
models represent extreme and opposite conditions and most practical cases can
be assumed as intermediate between them.
The ITUR Rec. P530 gives obstruction loss
curves (see below) for the two models mentioned above and for an intermediate
case (the smooth earth result is for k = 1.33
and frequency 6.5 GHz).
Diffraction Loss vs. Normalized Clearance, with different
obstacles: A) knife edge; B) smooth
spherical earth; C) intermediate
(from ITUR Rec. P530, by ITU permission)
More
on obstruction loss computation
A
more detailed analysis of obstruction loss is reported in ITUR Rec. P526,
where general formulas are given. The knifeedge model is also extended to
rounded obstacles and to the case of multiple obstructions.
Knifeedge obstacle
 A good approximation of the
obstruction loss produced by a knifeedge obstacle is given by :
_{
}
where_{
}
and the
approximation holds for C_{NORM} < 0.5.
Single rounded obstacle

The obstacle geometry is shown in
the figure below, where also the relevant parameters are graphically
defined.
Geometrical parameters in a
rounded obstacle
(from ITUR Rec. P526, by ITU permission).
A n approximate formula for the obstruction loss is :
_{ }
where L_{knife}is given above and D L is the additional loss, compared with a sharp (knifeedge) obstacle, given by:
_{ }
The normalized parameters nandrare computed as:
_{ }
_{ }
where l is the signal wavelength and the geometrical parameters (d, d_{a}, d_{b}, R, q ) are defined in the figure above.
The approximation holds for :
n > 0 that is for negative clearance (obstacle above the ray trajectory);
r < 1 that, for frequency above 1 GHz, means, in practical terms, that the obstacle should not be very close to one hop terminal.
Spherical earth  At frequencies above 1 GHz, the spherical earth formulas give :
_{ }
where :
_{ }
_{ }
_{ }
_{ }
and finally F is the frequency (GHz), R_{E} is the equivalent earth radius (8500 km for k = 1.33), D is the path length (km), H is the antenna height (m) over the earth surface; Y_{1}, Y_{2} in the first formula refer to the first and second path terminal, respectively (in the Y formula, use the appropriate antenna height).
Multiple obstacles  Several approximate methods have been suggested to estimate the obstruction loss produced by multiple obstacles in a radio hop.It is to be noted that pointtopoint links should be usually designed in such a way to avoid multiple obstacles along the radio path. However, it is useful to have computational techniques to deal also with this problem.
A reliable solution is the socalled Deygout model. Let us consider, at first, a path with two obstacles, as shown below.
Evaluation of two obstacle
loss with the Deygout model
(from ITUR Rec. P526, by ITU permission).
First, the clearance is estimated at each obstacle, as if that obstacle is the only obstacle in the path. So, the "most significant obstacle" is identified, as the obstacle producing the worst (most obstructing) clearance (in the example above, this corresponds to point M1).
The overall obstruction loss L_{TOT }is then estimated as :
_{ }
where L[XY, YZ, H] is the knifeedge obstruction loss in a radio path from X to Z, where an obstacle is at point Y with height H.
The method can be iteratively extended to more than two obstacles. For the total radio path and then for each "subpath", the most significant obstacle is identified.
ITUR Rec. P526 applies the Deygout model to both knifeedge and rounded obstacles, with introduction of a correction factor (which is negligible when the obstacles are evenly spaced).
We now
have all the elements to establish Clearance Criteria in the design of a radio
hop :
·
the ray trajectory has been discussed and the minimum kfactor value (most critical condition) has been
assessed;
·
the
loss produced by path
obstructions
has been evaluated as a function of the Normalized Clearance and
using the Fresnel ellipsoid concept.
The Clearance Criteria given by ITUR (Rec. P530) are summarized in the figure below. They must be applied both in standard k and in minimum k conditions and take account of different climates and different obstacle shapes.
A chart showing the ITUR (Rec. P 530) criteria for
path clearance.
The red circle is the Fresnel ellipsoid transversal
section,
as seen from one hop terminal, partially obstructed
by the ground.
The more stringent criteria for tropical climate are justified by the wider variability in kfactor values observed in those regions.
According to ITUR, the above rules can be made less tight, in some measure, when frequencies below 2 GHz are used. This means that smaller fractions (by about 30%) of the Fresnel radius can be adopted.
An example of application, with a single isolated obstacle, is given below, in a flat earth representation of the path profile; tropical climate is assumed. First , we check the standardk condition (100% of the Fresnel ellipsoid free of obstacles). The two lines indicates :
·
gray line: ray trajectory (ellipsoid axis) for k = 1.33;
·
blue line: lower margin of the Fresnel ellipsoid (100% of the Fresnel
ellipsoid radius).
Then we check the minimumk condition (60% of the Fresnel ellipsoid free of obstacles). The three lines, in the figure below, indicates :
·
gray line: ray trajectory (ellipsoid axis) for k = k min;
·
red line: lower margin of the Fresnel ellipsoid (100% of the Fresnel
ellipsoid radius).;
·
green line: 60% of the Fresnel ellipsoid radius.
The blue and the green lines, respectively in the two diagrams, are the limiting lines to satisfy the Clearance criteria (the vertical distance from such lines to the ground is usually indicated as the "Margin").
In most cases it is sufficient to indicate those two lines (as derived for k standard and minimum values) on the profile plot and to check that none of them intercepts the path profile (positive Margin).
Further Readings
Doble J., Introductionto Radio Propagation for Fixed and Mobile Communications (Ch. 1), Artech House Inc., 1996.
Schiavone J.A., "Prediction of positive refractivity gradient for lineofsight microwave radio path", BSTJ, vol. 60, n. 6, July 1981, pp. 803822.
Vigants A., "Microwave Radio Obstruction Fading", BSTJ, vol. 60, n.8, August 1981, 785801.
Schiavone J.A., "Microwave radio meteorology: fading by beam focusing", Int. Conf. Communications, Philadelphia, 1982.
Mojoli L.F., "A new approach to visibility problems in lineofsight hops", National Telecomm. Conf., Washington, 1979.
End
of Session #3
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©
20012016, Apus Cloud Project e Luigi Moreno