**I**ONOSPHERIC **C**OMMUNICATIONS

**E**NHANCED **P**ROFILE **A**NALYSIS & **C**IRCUIT

(**ICEPAC**)

####### 

####### PREDICTION PROGRAM

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######## TECHNICAL MANUAL

**TABLE OF CONTENTS**

**Page**

**1. INTRODUCTION 1**

1.1 HF Radio Propagation History 1

1.2 General Description 3

**2. PREDICTABLE IONOSPHERIC PARAMETERS 5**

2.1 D Region 5

2.2 E Region 6

2.3 F Region 8

2.4 Propagation by Way of sporadic E and Other Anomalous Ionization 10

2.5 Electron density profile model 12

**3. CALCULATION OF CIRCUIT PARAMETERS FROM PATH GEOMETRY 14**

3.1 Path Length and Bearings 14

3.2 Reflection Area Coordinates 14

3.3 Sun\'s Zenith Angle 16

3.4 Types of Paths Considered 16

3.5 Ionospheric Parameters 17

3.6 Electron Density Profile 23

3.7 Raypath and Area Coverage Propagation 26

3.8 Probability of Sky-Wave Propagation 36

3.9 Probability of Sporadic-E Propagation 40

3.10 Calculation of Mixed Modes 41

**4. NOISE PARAMETERS 42**

4.1 Galactic Noise 42

4.2 Atmospheric Noise 42

4.3 Man-Made Noise 43

4.4 Combination of Noise 44

**5. HIGH-FREQUENCY TRANSMISSION LOSS CALCULATIONS 46**

5.1 Free Space Loss L~bf~ 47

5.2 Ionospheric Loss L~i~ 48

5.3 Frequency Dependence Effects on Absorption 52

5.4 Loss For Propagation Above the Standard MUF 55

5.5 Sporadic E Loss 56

5.6 System Loss L~s~ 57

5.7 Conclusions 59

**TABLE OF CONTENTS (CONT\'D)**

Page

**6. LONG DISTANCE MODEL 60**

6.1 Qualitative Aspects of Long Distance Communications 60

6.2 The Long Distance Model 64

6.3 Summary 69

**7. HIGH-FREQUENCY SYSTEM PERFORMANCE 71**

7.1 Circuit Reliability 71

7.2 Service Probability 74

7.3 Multipath Evaluation 82

**8. MAXIMUM USABLE FREQUENCY (MUF) MODEL 85**

8.1 Geometry of the MUF 86

**9. ACKNOWLEDGEMENTS 89**

# APPENDIX A. INDIVIDUAL ANTENNA DESCRIPTIONS AND POWER GAIN EQUATIONS 90

**REFERENCES 133**

**TECHNICAL DESCRIPTION OF ICEPAC PROPAGATION PREDICTION PROGRAM**

**FRANK G. STEWART**

Simulation models have been developed for predicting

and analyzing the performance of HF systems that depend

on ionospheric propagation. These models are documented.

**1. INTRODUCTION**

This report describes a propagation predictions model (ICEPAC) that is
an extension of the IONCAP program. It differs in the polar region
structure of the ionosphere and the low and mid latitude ionospheric
structure. The ICED (ionospheric conductivity and electron density)
profile model is a statistical model of the large-scale features of the
northern hemisphere ionosphere. The model recognizes the different
physical processes that exist in the different regions of the
ionosphere. It contains distinct algorithms for the subauroral trough,
the equator-ward portion of the auroral zone, the polward region of the
auroral zone, and the polar cap. This report will be a complete
description of the ICEPAC propagation prediction program.

The predictions are used primarily for long-term (month-to-month,
year-to-year, etc.) frequency management and circuit planning, but are
often used for hour-to-hour and day-to-day operations as well. Most
important, propagation considerations are basic to studies of
electromagnetic compatibility, and analytical computer prediction
methods such as the one described in this report are essential to a
practical solution.

It should be emphasized that a computer program is a tool for
convenience in calculation; the user must exercise his own engineering
judgment in determining the applicability and limitation of the results
to specific problems.

1.1 HF RADIO PROPAGATION HISTORY

For many years, numerous organizations have been employing the High
Frequency (HF) spectrum to communicate over long distances. It was
recognized in the late 30\'s that these communication systems were
subject to marked variations in performance, and it was hypothesized
that most of these variations were directly related to changes in the
ionosphere. Considerable effort was made in the United States, as well
as in other countries, to investigate ionospheric parameters and
determine their effect on radio waves and the associated reliability of
HF circuits. A worldwide network of vertical incidence sounders was
established to measure values of parameters such as foE, foF1, foEs,
foF2, and h\'F. Worldwide noise measurement records were started and
steps were taken to record observed variations in signal amplitudes over
various HF paths. The results of this research established that ionized
regions ranging from approximately 70 to 1000 km above the earth\'s
surface provide the medium of transmission for electromagnetic energy in
the HF spectrum (2 to 30 MHz) and that most variations in HF system
performance are directly related to changes in these ionized regions.
The ionization is produced in a complex manner by the photoionization of
the earth\'s high altitude atmosphere by solar radiation. Within the
ionosphere, the recombination of the ions and electrons proceeds slowly
enough (due to low gas densities) so that some free electrons persist
even throughout the night. In practice, the ionosphere has a lower limit
of 50 to 70 km and no distinct upper limit, although 1000 km is somewhat
arbitrarily set as the upper limit for most application purposes.

The vertical structure of the ionosphere is changing continuously. It
varies from day to night, with the seasons of the year, and with
latitude. Furthermore, it is sensitive to enhanced periods of
short-wavelength solar radiation accompanying solar activity. In spite
of all this, the essential features of the ionosphere are usually
identifiable, except during periods of unusually intense geomagnetic
disturbances.

The Radio Propagation Unit of the U.S. Army Signal Corps provided a
great deal of information and guidance on the phenomena of HF
propagation in 1945. By 1948, a treatise of ionospheric radio
propagation was published by the Central Radio Propagation Laboratory
(CRPL) of the national Bureau of Standards. This document (NBS, Circular
462, 1948) outlined the state of the art in HF propagation. Techniques
were included for: predicting the maximum usable frequencies (MUF);
determining the MUF for any path at any time taking into account the
various possible modes of propagation by combining theory and
operational experiences; and estimating skywave field strength.

Laitinen and Haydon (1962) of the U.S. Army signal Radio Propagation
Agency furthered the science of predicting HF system performance by
developing empirical ionospheric absorption equations and combining them
with the theoretical ground loss, free-space loss, and antenna gain
factors so that expected field strengths could be estimated for radio
signals reflected from the E- and F2-regions, considering the effect of
solar activity and seasonal and diurnal variations.

In the United States, the first automated HF path prediction computer
program was developed in 1957, for the U.S. Army Signal Corps, Radio
Propagation Agency (Contract DA 360039 SC-66438), now part of the U.S.
Army Strategic Communications Command (see Stanford Research Institute
(SRI), 1957). A later version was published in 1961 (Radio Corporation
of America, 1961). The first fully automated program, in which the
oblique transmission equations for parabolic layers were used, was
developed in 1966 (Lucas and Haydon, 1966) by ESSA\'s Institute for
Telecommunication Sciences and Aeronomy (ITSA), which preceded the
Institute for Telecommunication Sciences (ITS).

This work was continued in two separate paths, one for communications
analysis and predictions, reported in ITS-78 (Barghausen et al., 1969)
and another for analysis and prediction of OTH radar systems reported in
NRL Tech. Reports 2226 and 2500 (Headrick et al., 1971; Lucas et al.,
1972).

The culmination of this work was the IONCAP program which uses the above
described development for the shorter paths and other techniques for the
long path predictions (Whale, 1969).

Fundamental to all efficient HF computer prediction programs are the
synoptic numerical coefficient representations of the ionospheric
characteristics. These were first developed by ITSA (formerly the
Central Radio Propagation laboratory, National Bureau of Standards) and
first published in 1960 (Jones and Gallet, 1960). Subsequent
modification led to the technique now used (Jones et al., 1966), which
will be discussed later.

1.2 GENERAL DESCRIPTION

The techniques used in the computer program described in this report are
procedurally similar to the earlier ITS programs (ITSA-1, ITS-78,
HFMUFES, IONCAP), but there have been sufficient significant changes to
warrant further documentation.

The literature on the ionosphere and its role in HF sky-wave radio
communications is very extensive. Theories concerning ionospheric
propagation will not be repeated here in detail, but some background
material will be given where necessary for an understanding of the
prediction processes and the philosophy of the program.

In the basic model, it is assumed that the ionosphere can be represented
by one or more Chapman layers (Dudney, 1983), given sufficient
information concerning the height of maximum ionization, semi-thickness,
and electron density. Sufficient data must be available to predict an
average electron density distribution with height for any possible
transmission path. The model retains the equivalent path theorem (Breit
and Tuve, 1926; Martyn, 1935) and its transmission curve solution
(Smith, 1939), since this is the method for scaling and predicting
ionospheric characteristics.

The program predicts the long-term operational parameters, such as
maximum usable frequency (MUF), optimum traffic frequency (FOT), and
lowest useful frequency (LUF), in terms of the probability of successful
transmission for a particular circuit. The probability of successful
transmission depends on the probability that the transmission frequency
is below the critical frequency (i.e., the maximum frequency for
reflection) of the F2 layer and the probability that the available
signal-to-noise ratio is above a specified level.

Throughout the report, attempts have been made to clarify duplication of
nomenclature and symbols commonly accepted in wave propagation and
antenna studies.

**2. PREDICTABLE IONOSPHERIC PARAMETERS**

The presence of free electrons in the ionosphere produces the reflecting
regions important to High Frequency (HF) radio-wave propagation. In the
principal regions, between the approximate heights of 75 km and 500 km,
the electrons are produced by the ionizing effect of ultraviolet light
and soft x-rays from the sun. for convenience in studies of radio-wave
propagation, the ionosphere is divided into three regions defined
according to height and ion distribution: the D,E, and F regions. Each
region is subdivided into layers called the D,E, Es, F1, and F2 layers,
also according to height and ion distribution. These are not distinctly
separated layers, but rather overlapping regions of ionization that vary
in thickness from a few kilometers to hundreds of kilometers. The number
of layers, their heights, and their ionization (electron) density vary
both geographically and with time. At HF, all the regions are important
and must be considered in predicting the operational parameters of radio
communication circuits.

2.1 The D Region

The D region lies between the approximate limits of 75 and 90 km above
the earth\'s surface. The electron density is relatively small compared
with that of the other regions, but, because of collisions between the
molecules of the atmosphere and free electrons excited by the presence
of an electromagnetic wave, pronounced energy loss occurs. This energy
loss, dissipated in the form of thermal energy of the electrons or
thermal (electromagnetic) noise, is termed absorption. Absorption in the
D region is called non-deviative, since it occurs below the level of
reflection and predominates when the real part of the refractive index
is near unity (μ≈1); i.e., little or no bending takes place. Higher in
the E and F regions, electron collisions with atmosphere molecules can
also affect the condition for reflection that occurs wherever there is a
marked bending of the wave. This is explained by the fact that as the
wave nears its reflecting level, there is a slowing down or retardation
effect, which allows additional time for collisions to occur and thus
for absorption to take place. Absorption of this type is called
deviative absorption.

Because of the low electron density, the D region does not reflect
useful transmissions in the frequency range above 1 MHz. However,
D-region absorption is important at all frequencies and, because its
ionization is produced by ultraviolet solar radiation, it is primarily a
daytime phenomenon. The degree of absorption, expressed by the
absorption factor, is proportional to the product of the collision
frequency and electron density, and approximately inversely proportional
to the square of the wave frequency. The absorption factor variation is
adequately represented by cos χ, where χ is the zenith angle of the sun.
After sunset in the D region, ionization decreases rapidly and
non-deviative absorption becomes negligible 2 to 3 hours later.

Non-deviative D-region absorption is the principal cause of the
attenuation of HF sky waves, particularly at the lower frequencies
during daylight hours. It is accounted for in the program by an
analytical, semi-empirical expressions, which is explained in detail in
section 5. Deviative absorption losses are estimated and included in the
loss calculations as an uncertainty factor (see sec.7).

Another important property of the upper D and lower E regions is the
differential absorption between the ordinary and extraordinary waves
produced by the earth\'s magnetic field. These differential absorption
properties of the characteristic waves and their down-coming state of
polarization are especially important at frequencies below about 3 MHz
and in low-latitude regions. In these areas, when vertically polarized
antennas are used, most of the radiated power is coupled into the
extraordinary wave (Berghausen, 1966). The extraordinary wave is
reflected at a lower level, its critical frequency is somewhat higher,
and it suffers greater absorption. The idealized situation is when a
vertically polarized wave is transmitted in an easterly or westerly
direction along the location of the earth\'s magnetic dip equator and
the wave frequency is exactly equal to the gyromagnetic frequency (\~0.8
MHz). Then, [all]{.underline} the radiated power is coupled into the
extraordinary wave and the ordinary wave does not exist. In this
program, only the ordinary wave critical frequency and absorption
properties are considered.

2.2 THE E REGION

For communication, the most important characteristic feature of the E
region is the temporal and geographic variation of its critical
frequency. In almost all other respects, the features of the E layer are
very predictable compared with those of the F2 layer.

2.2.1 Measurements

A large volume of vertical-incidence ionosonde data has been collected
over about three solar cycles, and many features of the E region are
therefore well known. The minimum virtual height of the E region and the
variation of maximum electron density within this region as a function
of time and geographic location are readily obtained from the ionograms.
The phenomenology of sporadic-E has been investigated, but
classification of sporadic-E types remain unresolved. The effects of
different types of sporadic-E on oblique-incidence radio propagation are
not established; as a result, the compilation of meaningful statistics
to form the basis of predictions is difficult.

The E-region characteristics that have been systematically scaled from
the vertical-incidence ionosonde records include:

foE The critical frequency of the ordinary component of the E layer;
i.e., that frequency at which the signal from the ionosonde just
penetrates the E layer.

h\'E The minimum virtual height of the E layer, measured at the point
where the trace becomes horizontal.

foEs The highest observed frequency of the ordinary component of
sporadic-E (Es).

h\'Es The minimum virtual height of the sporadic-E layer, measured at
the point where the trace becomes horizontal.

fbEs The blanketing frequency; i.e., the lowest ordinary wave frequency
at which the Es layer begins to become transparent, usually determined
from the minimum frequency at which ordinary wave reflections of the
first order are observed from a higher layer.

2.2.2 Predictions

The regular E layer is predicted using three parameters: the monthly
median value of critical frequency (foE), height of maximum ionization
of the layer (hmE), and ratio of hmE to semi-thickness (ymE). In the
past, the E-layer critical frequency has been determined by a
semi-empirical equation involving the sunspot number and the zenith
angle of the sun. Obviously, such a relationship would be inadequate to
estimate foE values at sunrise or sunset and during nighttime. Worldwide
numerical coefficients of monthly median foE are available for computer
applications in terms of geographic latitude, longitude, and universal
time. The numerical coefficients (Leftin, 1976) representing foE were
derived from measurements taken during 1958 and 1964. These years were
selected for analysis because the data are representative of the high
(1958) and low (1964) phases of the sunspot cycle. Linear interpolation
is used between the representative data for the high (SSN = 150) and low
(SSN = 10) sunspot periods to obtain foE estimates at all other phases
of the solar cycle.

Little information is available concerning the statistical distribution
of the monthly median foE. In daytime, the E layer is sufficiently
regular that the distribution spread of the monthly median foE may be
considered negligible. Nighttime data are insufficient, but it appears
justified (Elling, 1961; Wakai, 1966; Wakai, 1967) to assume a similar
regularity for the foE monthly median of the nighttime E layer.

Therefore, we believe that the E-layer characteristics most important
for communication purposes are adequately represented by the available
foE monthly median numerical coefficients. The approximate true height
range of the regular E layer is well established at 90 to 130 km and it
is assumed that the maximum electron density occurs at 110 km and the
semi-thickness is 20 km (Kneckt, 1963; Frihagen, 1965). With the above
assumption, the ratio of the height to the semi-thickness (hmE/ymE) is
assumed to be 5.5.

2.3 THE F REGION

The vertical-incidence ionosonde network with its long series of
measurements over much of the world, provides the basis for F-region
predictions (Martyn, 1959). The following parameters have been
systematically scaled from the vertical ionosonde records (Piggott and
Rawer, 1961), although some stations do not report all of them:

foF2 The critical frequency of the ordinary component of the F2 layer;
i.e., that frequency at which the signal from the ionosonde just
penetrates the F2 layer.

M(3000)F2 The factor for converting vertical-incidence critical
frequencies to oblique incidence for a distance of 3000 km via the F2
layer.

foF1 The critical frequency of the ordinary component of the F1 layer;
i.e., that frequency at which the signal from the ionosonde just
penetrates the F1 layer.

H\'F2 The minimum virtual height of the F layer; i.e., the minimum
virtual height of the night F layer and the day F1 layer. It is measured
at the point where the F trace becomes horizontal. (In earlier years the
minimum virtual height of the night F layer was often combined with that
of the day F2 layer, the combined tabulation being designated h\'F2. In
these cases, the minimum virtual height of the F1 layer, h\'F1, was
tabulated separately.)

hpF2 The virtual height of the F2 layer corresponding to a frequency f,
where f = 0.834 foF2. This is based on the assumption of a parabolic
ionization distribution, which is usually considered justified as an
approximation to the height of maximum ionization of the F2 layer.

For HF radio communications, the F region is the most important part of
the ionosphere. It is not regular and because of its variability, short
time scale estimates of the important F-region characteristics are
required if predictions of the operational parameters of HF radio
systems are to be meaningful.

There are many characteristic features of the F region important to HF
radio communications. It is beyond the scope of this report to describe
all of them, nor is this necessary, since there are many publications
and excellent textbooks on the subject. We will briefly describe only
those F-region characteristics that are relevant to the program.

The F1 layer has not been as well defined as the F2 layer in terms of
its predictable characteristic features. the F1 layer is of importance
to communication only during daylight hours or during ionospheric storms
(Kelso, 1964; Wright et al., 1960-63); it lies in the height range of
about 200 to 250 km and undergoes both seasonal and solar cycle
variations, which are more pronounced during the summer and in high
sunspot periods.

2.3.2 Predictions

The F2 layer is described by three parameters: monthly median value of
critical frequency (fof2), height of maximum ionization (hmF2), and a
ratio of hmF2 to semi-thickness (ymF2). Monthly median values of foF2
and the M(3000)F2 for two solar activity levels are available as
numerical coefficients in terms of a modified magnetic-dip angle and
longitude, and universal time (CCIR, 1966). There is also available a
more recent model of the F2 region of the ionosphere. This model is
based on a combination of observed and theoretical data. The theoretical
data provided stability in large regions where no observed data existed,
such as ocean areas and non-industrialized areas. (CCIR 1989) This F2
region model showed minor improvements in populated regions and
significant improvements over sea area and unpopulated regions when
compared to observed ionosound and satellite measurements. For this
document the old CCIR model will be described. The analytical structure
of the more recent coefficients and the documentation would be
consistent. The solar activity dependence is accounted for by linear
interpolation.

The model divides the F2 region into four distinct zones: (1) normal
low-latitude and mid-latitude ionosphere as described by the numerical
coefficients, (2) the trough, (3) the zone of aurorally enhanced foF2\'s
and (4) the polar cap. The key boundary for the model is the equatorward
edge of the auroral oval. The resulting boundary location is
parameterized by comparing it with standard Feldstein oval boundaries
computed as a function of Kp or Q (Whalen, 1972). The resulting magnetic
index (Kp~eff~ or Q~E~) is an effective auroral energy index because it
is based on the \"current\" state of the high latitude ionosphere.

2.4 Propagation by Way of Sporadic E and Other Anomalous Ionization

In the preceding discussion of the important regions of the ionosphere,
we concentrated on the first order characteristics of the various
layers. There are many other characteristic phenomena, e.g., sporadic E,
spread F, F scatter, multiple traces, and other transients, often
observed on ionosonde records (Piggott and Rawer, 1961), that are
important in radio communications; however, present prediction schemes
demand that the general ionospheric structure be statistically
representative and in a continuous sequence. Of these phenomena, the
only one we have been able to partly represent for prediction purposes
is the sporadic-E layer.

2.4.1. Sporadic E

Sporadic E (Smith and Matsushita, 1962; Bowhill, 1966; Whitehead, 1969)
is seen on vertical and oblique ionograms near the height of maximum
ionization of the regular E layer. Sporadic E(Es) is characterized by
little or no retardation at its critical frequency and may be either
blanketing (totally reflecting) or semitransparent (partly reflecting),
or both, up to very high frequencies( \>75 MHz). These characteristics
can be helpful or harmful to radio communications. For example,
blanketing Es may block propagation via a more favorable regular layer
mode in a certain frequency range or cause additional attenuation at
other frequencies. Partially reflecting Es can cause serious multipath
and mode interference, especially detrimental to data transmission
systems. However, Es may extend the useful frequency range and its
presence can be effectively used in system design and operations.

The physical processes that produce sporadic-E ionization are not fully
known, but it is generally accepted that the mechanisms may be quite
different in auroral, temperate, and equatorial geographic areas
(Bowhill, 1966; Whitehead, 1969). In auroral areas, energetic particles
appear to play a vital role in the production of sporadic E (Baily,
1968). Temperate-area Es is best explained by the behavior of upper
atmosphere winds (Matsushita and Reddy, 1968) and a related wind-shear
theory (Axford and Cunnold, 1968). In equatorial areas, i.e., in a
narrow ±6^o^ belt centered on the magnetic dip equator, the production
of sporadic-E is explained by theories on plasma instability phenomena
(Farley, 1963; Waldteufel, 1965). Methods of forecasting sporadic E are
influenced by the physical processes involved and should be considered
in all prediction schemes. In this report, we are not directly concerned
with forecasting techniques, but with predicting operational parameters
when sporadic E is the dominant propagation mode. Therefore, the
numerical coefficients representing the monthly statistical distribution
of foEs for any location are empirically derived estimates of sporadic E
during periods of solar cycle minimum and maximum, and they are used
only when propagation via the regular E layer is not possible (see sec.
3).

It may be helpful to review the general occurrence characteristics of
sporadic E for the three geographic areas mentioned above (CCIR, 1969):

[Auroral Es]{.underline} - Occurs mainly at night at geomagnetic
latitudes greater than about 60^o^, with a maximum near 69^o^. Its
seasonal, diurnal, and solar cycle patterns are not clear. It occurs
more frequently during periods of high magnetic activity and follows the
sudden commencement associated with a solar flare (Hunsucker and Bates,
(1969).

[Temperate Es]{.underline} - Characterized by a pronounced maximum
during the summer solstices (June-July in the Northern Hemisphere and
December-January in the Southern Hemisphere). A seasonal minimum occurs
during the vernal equinox; this minimum changes abruptly at 60^o^
geomagnetic latitude. The diurnal pattern exhibits peaks during
mid-morning hours and near sunset. It is primarily observed during the
daylight hours and shows a complicated dependence on the sunspot cycle.

[Equatorial Es]{.underline} - A regular daytime occurrence without
seasonal dependence. It is highly transparent (partly reflecting) and
reaches high ( ≈ 50 MHz) frequencies. Values of foEs around 10 MHz are
regularly observed by ionosondes near the geomagnetic dip equator. The
reflection properties depend on the direction of propagation; higher
reflection coefficients are to be expected for north-south paths.

2.4.2 PREDICTIONS

Numerical coefficients are available for each month representing the
median and decile values of foEs in terms of a modified magnetic-dip
angle and longitude, and universal time (Leftin et al., 1968). These
numerical maps are from data taken during periods of solar activity
minimum (1954) and solar activity maximum (1958). Linear interpolation
is used for other levels of solar activity. Unless other information is
available, the virtual height of the sporadic-E layer is assumed to be
110 km.

2.5 ELECTRON DENSITY PROFILE MODEL

Frequency versus virtual height traces of the ordinary wave as available
on vertical incidence ionograms can be converted into electron density
profiles by a standard reduction program. These profiles, including
geographic, diurnal, seasonal, and solar cycle variations, are generated
between heights of 70 km and the height of maximum of the F2 layer,
hmF2. The electron density is given by the relationship

N = 1.24 x 10^10^ f~N~^2^ (4)

N = electrons per cubic meter

f~N~ = plasma frequency MHz

A fixed reflection height for the E and F2 layers was used in the
original computer program (Lucas and Haydon, 1961). Then parabolic
layers for both the E and F2 layer were used (Lucas and Haydon, 1966;
Barghausen et al., 1969). The F1 layer was added and the profile was
generated by taking the maximums of these intersecting layers. The
current method of profile generations replaces the parabolic layer
structure with a Chapman layer structure. The parabolic layer is
analytically more tractable but the Chapman layer has the advantage that
a layer whose process is dominated by electromagnetic ionization and
chemical losses is closely described by the Chapman layer. In addition,
the Chapman layer decreases exponentially with altitude above the layer
peak \-- this again more closely describes the ionospheric situation.
(Dudney 1983).

**3. CALCULATION OF CIRCUIT PARAMETERS FROM PATH GEOMETRY**

To determine the operational parameters for an HF ionospheric radio
communication circuit, it is necessary to calculate several parameters
that are based on the geometry of the path, such as path length, path
bearings, and zenith angle of the sun.

3.1 Path Length and Bearings

The first parameter to be calculated, given the geographic latitude and
longitude of the transmitting and receiving locations, is the path
length, which is taken to be the shorter of the great-circle distances
between the two points, and which is computed as follows:

cos d = sin x~1~ sin x~2~ + cos x~1~ cos x~2~ cos(y~1~ - y~2~), (3.1)

where

x~1~ = geographic latitude of transmitter,

y~1~ = geographic longitude of transmitter,

x~2~ = geographic latitude of receiver,

y~2~ = geographic longitude of receiver,

d = path length in radians.

Having obtained the path length, we calculate the bearing of transmitter
to receiver and receiver to transmitter along the great circle path:

cos b~1~ = (sin x~2~ - sin x~1~ cos d)/ (cos x~1~ sin d) (3.2)

cos b~2~ = (sin x~1~ - sin x~2~ cos d)/ (cos x~2~ sin d) (3.3)

where

b~1~ = bearing transmitter to receiver in radians,

b~2~ = bearing receiver to transmitter in radians.

3.2 Reflection Area Coordinates

In the development of a profile of electron density along the path, the
ionospheric parameters at from one to five reflection areas along the
path are evaluated depending on the path length. These five areas are:

1\. The midpoint of the path.

2\. The E-region reflection area nearest the transmitter for the
estimated least possible number of hops.

3\. The E-region reflection area nearest the receiver for the same
number of hops.

4\. The F-region reflection area nearest the transmitter for the
estimated least possible number of hops.

5\. the F-region reflection area nearest the receiver for the same
number of hops.

The estimated least possible number of E-layer and F-layer hops is
determined from the following relationship to path length:

1E, 1F - 0000 km ≤ path length \< 2000 km.

2E, 1F - 2000 km ≤ path length \< 4000 km.

4E, 2F - 4000 km ≤ path length \< 8000 km.

6E, 3F - 8000 km ≤ path length \< 12,000 km.

for paths less than 2000 km, only the path midpoint is considered. This
establishes the reflection areas for determining the ionospheric
characteristics for the entire path.

To evaluate the ionospheric parameters of these five reflection areas,
their geographic coordinates and geomagnetic latitude have to be
computed as follows:

x~n~ = 90^o^ - arccos(cos d~n~ sin x~1~ + sin d~n~ cos x~1~ cos b~1~)
(3.4)

y~n~ = y~1~ - arccos**(**(cos d~n~ - sin x~n~ sin x~1~)/(cos x~n~ cos
x~1~**))** (3.5)

g~n~ = 90^o^ - arccos**(**sin 78.5^o^ sin x~n~ + cos 78.5^o^ cos x~n~
cos

(y~n~ - 69.0^o^)**)**, (3.6)

where

d~n~ = angular distance of reflection area from transmitter,

x~n~ = geographic latitude of reflection area,

y~n~ = geographic longitude of reflection area,

g~n~ = geomagnetic latitude of reflection area.

The modified magnetic dip latitude is required at each control point for
the evaluation of the ionospheric parameters. The magnetic dip is
calculated from the 1963 Jensen and Cain model of the earth\'s magnetic
field. (Jensen, D.C. and Cain, T.C.- 1962) The corrected geomagnetic
latitude and longitude is required to define the location and structure
of the polar ionosphere.

Computation of the corrected geomagnetic coordinates begins by starting
in the equatorial plane at the same point with a dipole field line and a
spherical analysis field line, and then calculating the distance between
the \"landing points\" of the two field lines on the earth. In its
simplest form, the method consists of labeling the spherical analysis
field lines (sometimes called the real field lines) with the coordinates
of the coincident equatorial, dipolar field lines. That is, the
spherical analysis field has numerous irregularities due to regional
anomalies and so it is difficult to assign a meaningful symmetric grid
pattern to such a system. However, superimposing the symmetric dipolar
grid system on the \"realistic\" spherical analysis produces a useful
coordinate system for modeling purposes.

3.3 Sun\'s Zenith Angle

For the first three reflection areas, the zenith angle of the sun is
needed for each hour of the day, to be used later in calculating the
absorption factor, and is computed from the following equation:

cos χ = sin x~n~ sin s~x~ + cos x~n~ cos s~x~ cos(s~y~ - y~n~), (3.7)

where

t~g~ = universal time,

s~y~ = 15 t~g~ - 180 = subsolar longitude,

s~x~ = subsolar latitude for the middle of the month,

χ = sun\'s zenith angle.

3.4 Types of Paths Considered

Up to six ray paths are evaluated for each hour and each designated
frequency. These ray paths are interpolated from the reflectrix table
for the frequency with the distance calculated using a corrected version
of Martyn\'s Theorem for the equivalence of oblique and vertical heights
of reflectivity.

3.5 Ionospheric Parameters

Once the propagation path has been located geographically, the
parameters of the ionosphere at the reflection areas along the path are
needed for further computation. These parameters are the critical
frequency of the layer, height of the maximum electron density of the
layer, height of the bottom of the layer, and semi-thickness of the
layer.

3.5.1 Low and Mid Latitude Model

The critical frequency of the E layer (foE) is obtained from world maps
(Leftin, 1976) and is the median value of that parameter. The height of
the maximum foE (hmE) is set at 110 km for the low and mid latitude with
the semi-thickness (ymE) set at

(3.8)

The maximum solar zenith angle χ~max~ for the occurrence of the F1 layer
is used as the cutoff for the prediction of the monthly median foF1.
This is necessary since the F1 layer is normally observed on vertical
incidence ionograms during daylight hours only. The χ~max~ and critical
frequency of the F1 layer (foF1) are calculated as follows (Rosich et
al., 1973)

χ~max~ = AC~1~ + BC~1~ \* SSN + (AC~2~ + BC~2~ \* SSN) \* cos (GMDIP)
(3.9)

where SSN = 12 month smoothed mean sunspot number

GMDIP = Rawer\'s modified magnetic dip latitude

AC~1~, BC~2~, AC~2~, BC~2~, are coefficients based on a two dimensional
representation of χ~max~ using sunspot numbers and Rawer\'s modified
magnetic dip. A set of coefficients exists for each month.

When χ at the control point is greater than χ~max~ then foF1 = 0.2

When χ at the control point is less than χ~max~ , the following equation
is used:

foF1 = A~1~ + B~1~ \* SSN + (A~2~ + B~2~ \* SSN) cos χ + (A~3~ + B~3~ \*
SSN) cos^2^χ (3.10)

where

SSN is the 12 month smooth mean sunspot number and

χ is the solar zenith angle of the sun at the control point ( χ ≤ 90.0).

A~1~, B~1~, A~2~, B~2~, A~3~, B~3~ are coefficients from the numerical
representations. (Rosich et. al. 1973). The height of the maximum
ionization for the F~1~ layer (hmF1) is calculated as follows:

hmF1 = 165 + 0.6428 \* χ or 200 which ever is less

The semi-thickness of the foF1 layer (ymF1) is calculated as

where χ is the solar zenith angle of the sun and hmF1 is restricted to a
maximum value of 200 km.

The critical frequency of the F2 layer is obtained from world maps
(Jones et al., 1966) and is the median value of that parameter. The true
height of the maximum electron density of the F layer is developed in
two steps. First, the M(3000)F2 factor is obtained from world maps, and
then the true height of the maximum ionization hmF2 in the layer is
calculated on the basis of the following relationship (Dudney, 1983):

XE = foF2/foE if XE \< 1.7, XE = 1.7 (3.11)

(3.12)

(3.13)

(3.14)

The semi-thickness (ymF2 of the F2 layer is calculated on the basis of
coefficients that represent the ratio of the height of maximum
ionization of the F2 region to its semi-thickness ymF2. (Lucas et al.,
1966).

3.5.2 Auroral Zone

The Auroral foE is made up of both solar ionization and auroral zone
precipitating particals. The maximum auroral critical frequency
(foEa~max~) due to precipitating particals (Vondrak et.al. 1978) is
calculated by the following equation:

foEa~max~ = 2.5 + Qe / 9.0 0 ≤ Qe ≤ 2.7

= -1.0 + 7/5 Qe 2.7 \< Qe ≤ 4.2

= 3.2 + 2/5 Qe Qe \> 4.2 (3.15)

where Qe = effective geomagnetic activity index. foEa~max~ is then
adjusted for local time magnetic variations (Maximum ionizations at 0300
magnetic local time and minimum ionization at 1500 magnetic local time).

(3.16)

where TCGM = time corrected geomagnetic time.

A linear interpolation is done between the oval boundaries and the point
of maximum ionization in the oval. The FoEa values of the polarized and
equatorial boundaries of the auroral zone are set at 60% of the
foEa~max~. The model uses linear interpolation to get values between the
equatorial or polarized auroral boundaries and the geomagnetic latitude
of the foEa~max~. FoEa, the interpolated value, is the auroral value of
foE when no solar component is present (Auroral night line, Auroral
sunrise, sunset). The height of maximum ionization for the auroral E
layer is calculated as follows:

hmEa = 130.0 foEa ≤ 1.0

hmEa = 90.0 foEa ≥ 7.0

(3.17)

When there is an Auroral daytime point, the low and mid latitude value
of the foE is used.

foEs = foE obtained from world maps

hmEs = hmE from low and mid latitude = 110 km

When both solar and auroral E components are present, the following
rules are used to select the E layer critical frequency:

foE = (foEs^4^ + 1.5 foEa^4^)^1/4^ (hmEs - hmEa) ≤ 10.0

hmE = corresponding height (hmEa or hmEs) (hmEs - hmEa)\>10.0

Auroral foF1

The auroral zone night time value of the F1 layer are set as follows:

foF1 = 0.2 MHz

hmF1 = 200.0 km

ymF1 = hmF1/4.0

The low and mid latitude values are used otherwise.

Auroral foF2 is calculated from the following equations (Tascione et.al.
1988)

(3.18)

λm = corrected geomagnetic latitude of interest

Φp = polward boundary of the oval

Φa = equatorward boundary of the oval

Φm = middle of the auroral oval

X~1~ = 2 \* X~1~ if λm \> φm

HmF2 for the auroral oval is calculated the same as the low and mid
latitude value.

3.5.3 Polar Cap

The polar E layer has a default set of parameters as follows:

foE = 0.6 Mhz

HmE = 120.0 km

ymE = 120.0/5.5 = 22 km

The polar F1 layer has a default set of parameters as follows:

foF1 = 0.2 Mhz

fmF1 = 200.00 km

ymF1 = 200.0/4.0 = 50.0 km

The polar F2 layer is calculated with the same formulas as the auroral
zone.

3.5.4 Sub Auroral Trough

foF2 = foF2c

foF2 = (1 + ∆N)

where foF2c is the foF2 from coefficients and where ∆N is the depletion
region adjustment factor which is a function of both magnetic local time
and geomagnetic activity. The following definition of ∆N is based upon
Miller and Gibbs (1975):

where D is the Julian day and T is a complicated weighting factor
dependent on local geomagnetic time, magnetic activity, and solar zenith
angle.

T = T~1~ exp (\[χ~1~ - χ~1~^2^\] / 2.0) exp \[-1.0 TPC^2^ / 12.0\]

where

χ~1~ = (λ~m~ - Φ~A~) / (3.7 + 1.3 KP~eff~

TPC = abs (TCGM - 3.0)

and

T~1~ = 0.0 χ ≤ 90 or 6.0 \< TCGM \< 18.0

T~1~ = -0.2 χ \> 94.6 and 18.0 ≤ TCGM ≤ 6.0

T~1~ = \[-0.2(χ - 90)\] / 4.6

90 ≤ χ ≤ 94.6 and 18.0 ≤ TCGM ≤ 6.0

In the above expressions, χ is the solar zenith angle, TCGM is the local
geomagnetic time, λ~m~ is the corrected geomagnetic latitude of the
point being considered, and φ~A~ is the geomagnetic latitude of the
equatorward auroral boundary, Kp~eff~ is the effective Kp geomagnetic
index.

The trough height values are calculated as follows. The low and mid
latitude calculations are made at 4 points, centered on 0300 magnetic
local time and the trough maximum. Calculation of hmF2 for the
subauroral trough is made for each point on the following grid:

Equatorial ┌────────────────┬─────────────────┐ Trough polward

boundary of the│ │ │ boundary

auroral zone │ │ │

│ │ │

├────────────────┼─────────────────┤ Trough maximum

│ │ │

│ │ │

│ │ │ Trough equatorial

└────────────────┴─────────────────┘ boundary

2000 MLT 0300 MLT 0700 MLT

At the trough maximum and 0300 magnetic local time (0300 MLT), the
maximum height of the trough is set at 450 km. The value of hmF2 is
calculated by interpolation in the above grid.

3.6 Electron Density Profile

For the generation of the electron density profile a single set of
critical frequencies and associated parameters are chosen. Depending on
the path length, a process of elimination is used to reduce up to 5
control points to 1 to 3 control points that best represent the
ionosphere for that particular communications circuit. The ionospheric
parameters (foE, hmE, ymE, foF1, hmF1\.....etc) are then used to build
an electron density profile at the control points based on the Chapman
layer structure at the critical frequency and below, with exponential
extensions on the topsides of the layers.

3.6.1 Control Point Selection

For each communication circuit, up to five geographic points are
calculated along the circuit path. From these geographic locations, up
to 3 control points are selected with ionospheric parameters assigned to
each. When the ionospheric profile is generated, the layer parameters
are selected from the values at the three control points.

Distance ≤ 2000 km 1 Control point

foF2 from path midpoint

fof1 from path midpoint

foE from path midpoint

2000 km \< Distance ≤ 4000 km 2 Control points

Control point 1

foF2 from path midpoint

foF1 1000 km from transmitter

foE 1000 from transmitter

Control point 2

foF2 path midpoint

foF1 1000 km from receiver

foE 1000 km from receiver

4000 \< Distance ≤ 8000 km 3 Control points

Control point 1

foF2 2000 km from transmitter

foF1 1000 km from transmitter

foE 1000 km from transmitter

Control point 2

foF2 path midpoint

foF1 path midpoint

foE path midpoint

Control point 3

foF2 2000 km from receiver

foF1 1000 km from receiver

foF2 1000 km from receiver

When circuit parameters that describe characteristics of the path are
calculated, the profile is made up of the most pessimistic layer values,
minimum foF2 or minimum foE when the FoF2 values are approximately
equal.

3.6.2 Vertical Profile Generation

Once the layer characteristics are selected, then the profile based on
the layer values (FoF2, FoF1, FoE, \...) is constructed. The initial
step in the profiling routine is to extract the semi-thickness of the F2
and F1 (if χ ≤ 105^o^) regions from the E-R model. The semi-thickness
(ST) are then normalized by use of the empirical Wrobel function (Damon
and Hartranft, 1970) as follows:

ST (F2) = ST (E-R)\*W (hmF2~ICED~)/W (hmF2~E-R~)

ST (F1) = ST (E-R)\*W (hmF1~ICED~)/W (hmF2~E-R~)

where

W(h) = 1n(h)/0.02186 - 203.447

is the expression for the Wrobel scale height and the subscripts
indicate the source of the height of the maximum density value (E-R =
Elkins-Rush 1973).

The profile is then constructed in the following manner:

\(1\) At and above the F2 layer peak, we closely follow the Elkins-Rush
model with the exception of the modified F-region semi-thickness, and a
modified top-side scale height using DMSP in-situ ion-electron
measurements (if available).

\(2\) At and below the E layer peak, a Chapman layer with scale height
of 16 km is used unless the F1 layer contribution at the E layer peak
height exceeds the E layer density computed from either solar or
particle ionizations. In this case, the E layer is disregarded and the
F1 layer is extended downward.

\(3\) Between the F2 layer peak and the E layer height of maximum
electron density, the modeling depends on whether or not the F1 layer is
present.

\(a\) In the absence of the F1 layers, the electron density at any
height is the sum of the density contributions from the E and F2 layers.

\(b\) When the sun is visible at F1 layer heights, the intermediate
region is modeled by subtracting the F1 layer contribution at E and F2
layer heights from the maximum densities produced by the model for these
two layers independently. The reduced E and F2 layer maximum densities
are then used in the Chapman function representing each layer. Finally,
the total electron density at any level is the sum of the density
contributions from the F1 layer and the modified E and F2 layers. In
this procedure, the Chapman scale height of the F2 layer is decreased,
if necessary, to insure that the F1 layer peak falls at least two (F2)
scale heights below the F2 peak. The above technique, then allows the F2
and E layer peak densities to be modeled exactly and generally results
in an F1 density within 5 percent of the target value.

3.7 Virtual Height Ray Path and Area Coverage

Model For Single Hop Propagation

This section describes a simple computation method for obtaining all the
single-hop ray paths through an ionosphere described by an electron
density profile. It uses the classical relationships between the
virtual-height ionograms and the oblique path (however, Martyn\'s
equivalence theorem is used in a \"corrected\" form as described in
Section 3.7.4). First, the ionogram is obtained using numerical
integration techniques. Then reflectrices are obtained as single table
entering all ray-path information. Finally, the correction to Martyn\'s
theorem and a table look up and interpolation procedure are used to find
the ray sets which describe the propagation for a particular operating
frequency.

3.7.1 Virtual Height Ionograms

Virtual heights for the ordinary trace are found from the electron
density profile by numerically integrating the equation.

(3.19)

f~v~ is a selected vertical sounding frequency;

h~o~ is the lowest true height of the profile, i.e., 70 km;

h\' is the virtual height of the profile;

h~r~ is the true height corresponding to f~v~;

h is the true height of reflection;

µ\' is the group index of refraction;

f~N~ is the plasma frequency.

The area is found using a Gaussian integration technique (see Figure 1).
The effect of the cusp at h~r~ can be lessened by using a nonlinear
transformation from

the interval \[h~o~, h~r~\], to \[-1, 1\]. The transformation and
integration equations are:

h~j~ = true height corresponding to (w~j~, X~j~);

X~j~ = Gaussian abscissa;

w~j~ = Gaussian weight;

N = number of Gaussian terms (at least 40) for electron density profiles
described in Section 3.6.2.

A forty-point Gaussian integration was found to be adequate when the
electron density profile was sampled at true height intervals of 4 km
and the vertical sounding frequencies were selected at intervals of 0.2
MHz.

3.7.2 Skywave Propagation

Skywave radio propagation paths may be described by a set of parameters
known as raysets (Croft, 1967). For most HF communication applications,
this consists of operating frequency, takeoff angle, virtual height of
reflection, true height of reflection, and ground distance. The basic
inputs are true and virtual heights as a function of critical-incidence
frequency.

The ray paths are calculated using the following simplifying
assumptions:

1\. Horizontal and azimuthal variations in the ionospheric electron
density profiles are negligible for each hop (on a multi-hop path,
different sample area are used).

2\. The magnetic field may be ignored.

3\. The ionosphere is spherically symmetrical to the earth.

With these simplifications, the equivalence between a given frequency on
an oblique path (f~ob~) and a vertical incidence frequency (f~v~) with
same vertical height specified by Snell\'s law is

f~ob~ = f~v~ secθ~t~ (3.20)

where

θ~t~ is the angle between the apparent ray path and the normal to the
earth at the true height of reflection.

By simple geometry, the virtual height of the oblique path is related to
the takeoff angle ∆ by (see Figure 2).

a cos ∆ = (a + h\'~ob~) sin Φ (3.21)

where

∆ = takeoff angle of the ray,

a = earth\'s radius,

h\'~ob~ = virtual height of the oblique path,

Φ = is the angle between the virtual ray and the normal to the earth at
h\'~ob~.

Martyn\'s theorem for a plane ionosphere specified the ray path by the
equality of the virtual height of the oblique path, with the virtual
height of the ionogram at the equivalent frequency f~v~. For a curved
ionosphere, this leads to a consistent error at higher frequencies for
thicker layers. The Breit and Tuve theorem states that the time taken to
transverse the actual path is the same as that which would be taken to
transverse the equivalent path in free space. Both theorems are
corrected in the following model by an empirically derived correction
factor which depends only on the electron density profile and the
curvature of the ionosphere:

(3.22)

where

foF2 is the F2 critical frequency,

h\'~v~ is the virtual height corresponding to f~v~,

h is the true height of reflection.

This correction has errors of less than one percent as compared with the
distances calculated by a ray-trace program based on Haselgrove\'s
equation method (Haselgrove, 1954; Finney, 1963). This is described in
more detail in the following section.

The model uses equations 3.20 and 3.21 to generate Table 1. The first
row is the vertical-incidence frequency in megahertz. The second row is
the true height of reflection, h; the third row is the virtual height of
reflection, h\'; and the rows following are the equivalent oblique
operating frequencies of the transmitter for ray paths with
corresponding takeoff angles; e.g., the row after 0.0 is for ray paths
with a takeoff angle of O^o^. When the information contained in Table 1
is plotted in constant frequency contours of virtual reflection height
and distances as in Figure 3, the displayed contours are called
reflectrices (Lejay and Lepechinsky, 1950). At each given operating
frequency, the area coverage is found by interpolating in Table 1 for
the desired ray paths. This procedure yields the desired reflectrix for
the given frequency in the form of a table of raysets (Table 2).
Calculations are not necessary at each frequency as all the desired
information is contained in Table 1. For two-hop modes, a table of
reflectrix information (as in Table 1) is generated for the second-hop
sample area, and at the given operating frequency, a table of rayset
information is generated (as in Table 2). The two-hop modes are found by
matching the takeoff angle of the second hop with the arrival angle of
the first hop. Note that there has been no mention of individual layers
(E, F1, or F2) since the electron density profile was generated. The
ionosphere is treated as a single region by the program, and all
possible mode combinations are generated. In order to keep the
traditional layer nomenclature, the ray paths are named according to
where their equivalent vertical frequencies lie (e.g., below foE); thus
the modes may be E-E, F2-F2, or E-F2 according to the label of each hop.
Since the sporadic-E modes are assumed to exist with some degree of
probability with reflection at a constant height, the rayset information
is the same for each operating frequency and there is no need to
generate the equivalent of Table 1 for the Es mode.

Table 1 Showing the Relationship Between Vertical Incidence Data
(Ionogram)

and Oblique Propagation (Reflectrix)

IONOGRAM AND REFLECTRIX

FVERT .01 .91 1.82 2.72 2.83 2.95 3.06 3.17 3.29 3.40 3.40 3.51 3.62
3.73 3.84

HTRUE 70.0 74.2 89.0 98.0 99.0 100.0 101.3 102.9 105.0 110.0 153.8 155.0
156.7 158.5 160.4

HPRIM 70.0 83.9 103.0 114.0 114.6 116.3 119.6 124.4 132.1 175.7 328.4
241.0 231.3 226.7 226.5

ANGLES OBLIQUE FREQUENCIES kHz

0.0 68 6036 10983 15684 16257 16822 17358 17867 18325 18532 15760 16204
16630 17046 17454

.5 67 6026 10968 15665 16238 16802 17337 17847 18304 18512 15748 16192
16617 17035 17440

1.0 67 5997 10924 15608 16179 16741 17276 17784 18242 18452 15711 16154
16579 16997 17402

2.0 66 5885 10753 15385 15950 16507 17039 17542 17998 18216 15567 16007
16430 16846 17249

3.0 64 5711 10484 15034 15590 16138 16661 17160 17613 17844 15336 15772
16191 16603 17003

4.0 61 5492 10141 14582 15125 15661 16174 16666 17115 17359 15029 15459
15873 16280 16676

5.0 58 5245 9746 14057 14585 15106 15608 16089 16532 16791 14661 15083
15491 15892 16283

6.0 55 4984 9321 13487 13998 14504 14991 15461 15897 16168 14246 14660
15060 15454 15839

8.0 49 4466 8452 12305 12779 13249 13706 14148 14565 14856 13335 13729
14111 14489 14858

10.0 44 3991 7030 11167 11604 12039 12461 12875 13268 13567 12391 12762
13125 13483 13825

12.0 39 3578 6895 10136 10538 10937 11328 11712 12081 12378 11478 11826
12168 12506 12840

14.0 35 3227 6257 9230 9599 9966 10327 10683 11028 11319 10626 10955
11276 11595 11909

16.0 32 2930 5709 8444 8784 9123 9457 9787 10107 10390 9855 10164 10465
10765 11061

18.0 29 2679 5239 7765 8079 8393 8703 9010 9310 9580 9166 9453 9736
10018 10297

20.0 27 2465 4835 7179 7471 7762 8051 8338 8619 8876 8562 8821 9088 9353
9616

22.0 25 2283 4488 6672 6944 7216 7486 7754 8018 8263 8006 8260 8511 8761
9010

24.0 23 2126 4186 6231 6486 6741 6994 7246 7494 7728 7523 7762 7999 8236
8471

26.0 21 1989 3924 5846 6086 6326 6564 6801 7036 7258 7093 7320 7545 7769
7992

28.0 20 1871 3694 5508 5734 5961 6186 6410 6633 6845 6711 6926 7140 7353
7565

30.0 19 1766 3492 5210 5424 5639 5852 6065 6276 6480 6370 6574 6778 6981
7183

32.0 18 1675 3314 4945 5150 5354 5557 5759 5961 6155 6065 6260 6454 6648
6842

34.0 17 1593 3155 4710 4905 5100 5294 5487 5679 5866 5791 5978 6164 6350
6535

36.0 16 1521 3013 4501 4687 4873 5058 5244 5428 5607 5545 5724 5903 6081
6259

38.0 15 1456 2886 4313 4491 4670 4848 5025 5202 5375 5323 5495 5667 5839
6010

40.0 15 1398 2772 4143 4315 4487 4658 4829 4999 5166 5123 5289 5454 5619
5784

42.0 14 1346 2670 3991 4157 4322 4487 4652 4816 4978 4941 5101 5261 5421
5580

44.0 14 1298 2577 3853 4013 4173 4333 4492 4651 4608 4777 4932 5086 5241
5395

46.0 13 1256 2493 3729 3884 4038 4193 4347 4501 4653 4627 4777 4926 5077
5227

48.0 13 1217 2417 3616 3766 3916 4066 4216 4365 4513 4491 4637 4783 4928
5074

50.0 12 1182 2348 3513 3659 3805 3951 4096 4242 4386 4367 4509 4651 4793
4934

52.0 12 1151 2286 3420 3562 3704 3846 3988 4130 4270 4255 4393 4531 4669
4807

54.0 12 1122 2229 3335 3474 3613 3751 3889 4028 4165 4152 4287 4422 4557
4692

56.0 12 1095 2177 3258 3394 3529 3665 3800 3935 4069 4058 4190 4322 4454
4586

58.0 11 1072 2130 3188 3321 3453 3586 3718 3851 3982 3973 4103 4232 4361
4490

60.0 11 1050 2088 3125 3255 3385 3515 3644 3774 3908 3696 4023 4150 4276
4403

65.0 11 1005 1998 2991 3115 3240 3364 3489 3613 3737 3733 3854 3976 4098
4219

70.0 10 970 1929 2888 3009 3129 3249 3359 3489 3609 3607 3725 3842 3960
4078

75.0 10 944 1878 2812 2930 3047 3164 3281 3398 3515 3514 3629 3743 3858
3973

80.0 10 927 1843 2760 2875 2990 3105 3220 3335 3450 3450 3562 3675 3788
3900

85.0 10 916 1823 2730 2843 2957 3071 3184 3298 3412 3412 3524 3635 3746
3858

90.0 10 9l3 l8l6 2720 2833 2946 3059 3173 3286 3399 3400 3511 3622 3733
3844

> Table 2. Rayset Table for 10 MHz. Great‑Circle Distance (km),
>
> Vertical Angle (Degrees), Virtual and True Reflection
>
> Heights (km), Propagation Mode, and Equivalent Vertical
>
> Incidence Frequency (f~v~ in MHz)
>
> RAYSET FOR FREQUENCY =10.00 MHz
>
> DISTANCE ANGLE VIRTUAL TRUE MODE FV
>
> 2240.69 0.00 99.81 86.03 E 1.64
>
> 2132.85 .50 99.85 86.07 E 1.64
>
> 2031.62 1.00 100.03 86,19 E 1.65
>
> 1848.52 2.00 100.66 86.68 E 1.68
>
> 1689.82 3.00 101.70 87.46 E 1.73
>
> 1553.09 4.00 103.08 88.51 E 1.79
>
> 1431.40 5.00 104.33 89.49 E 1.87
>
> 1323.49 6.00 105.49 90.43 E 1.96
>
> 1147.44 8.00 108.15 92.59 E 2.18
>
> 1012.11 10.00 111.13 95.02 E 2.42
>
> 906.11 12.00 114.34 97.62 E 2.68
>
> 819.51 14.00 117.39 100.15 E 2.96
>
> 1024.54 16.93 181.35 109.96 E 3.40
>
> 1817.87 16.93 359.68 153.64 1 3.40
>
> 1213.47 18.00 234.01 158.34 1 3.73
>
> 1121.66 20.00 237.04 163.87 1 4.01
>
> 1194.88 22.00 280.89 174.09 1 4.31
>
> 1496.14 22.52 374.11 182.41 1 4.40
>
> 2184.89 22.52 595.08 210.45 2 4.40
>
> 1349.37 24.00 354.18 215.75 2 4.68
>
> 1209.20 26.00 340.38 221.60 2 4.96
>
> 1133.73 28.00 343.83 228.35 2 5.23
>
> 1094.45 30.00 358.10 236.32 2 5.50
>
> 1084.19 32.00 383.12 246.06 2 5.77
>
> 1127.25 34.00 432.03 259.43 2 6.05
>
> 1942.44 35.70 874.76 290.49 2 6.30
>
> 3.7.3 Corrected Martyn\'s Theorem
>
> The model derives the rays as described above in the following manner:
> first the virtual height of reflections are derived by numerical
> integration (40 point Gaussian quadrature)
>
> (3.23)
>
> h~r~ is the true height of reflection and µ\'(f~v~ , h) is the group
> index of refraction.
>
> For each radiation (takeoff) angle ∆, Snell\'s law is used to find the
> oblique frequency corresponding to f~v~, the vertical sounding
> frequency:
>
> f~ob~ = f~v~ sec Φ
>
> sin Φ = cos ∆ / (1 + h\'/a).
>
> Then the distance is found using Martyn\'s theorem for equivalence of
> oblique and vertical heights of reflectivity (first source of error as
> it is valid only for a \"flat\" ionosphere). See figure 2.
>
> D = 2a (π - ∆ - Φ\')
>
> where
>
> sin Φ\' = cos ∆ / (1 + h\'/a)
>
> h\' is the virtual height corresponding to h~r~ as derived above.
>
> Finally the group path is found using the law of Breit and Tuve
> (second source of error as it is valid only for a \"flat\"
> ionosphere):
>
> P\' = 2a cos (Φ\'+ ∆)/ sin Φ\'.
>
> To compensate for the above errors, the following correction was used.
>
> (3.24)
>
> 3.8 Probability of Sky-Wave Propagation
>
> Since the MUF calculated is based upon median values of the
> ionospheric characteristics, the probability of a sky-wave path for
> this frequency is assumed to be 50 percent. The frequency that would
> have a 90 percent probability of propagating (FOT) and the frequency
> which has 10 percent probability (HPF) are obtained by multiplying the
> MUF by the factors given in table 3.
>
> The values in table 3 were derived from a study (R.M. Davis and N.L.
> Groome, 1964, private communication) of the distribution of daily
> values of MUF about their monthly median. The data used in this study
> were values of the standard MUF, which is the product of the foF2 and
> M(3000)F2 factors scaled from vertical incidence ionograms. Data from
> 13 stations located at geomagnetic latitudes from 71^o^S to 88^o^N
> were analyzed. The temporal variations of the MUF distribution were
> determined by considering observations at all hours of the day, each
> month of the year, and periods representing low, medium, and high
> solar activity. Values were derived for the ratios of upper and lower
> decile MUFs to median MUF for a given season, a given solar activity,
> 4-hour local time blocks, and each 10^o^ of geographic latitude from
> 10^o^ to 80^o^, north or south.
>
> The values in table 3 reflect some general trends. The MUF
> distribution is wider at night than during the day, and wider at low
> latitudes than high latitudes in daytime. Also, in daytime, the
> distribution is wider in summer than winter, except at high latitudes
> where the reverse is true. Dependence on solar activity is weak, but
> during daytime hours the difference between the decile ratios
> generally seems to increase with solar activity at latitudes above
> 40^o^. The study by Davis and Groome (1964, private communication)
> indicates that the distribution is mostly dependent on foF2, not on
> the M(3000)F2 factor; therefore, the distributions are assumed valid
> for any oblique path.
>
> TABLE 3. FACTORS FOR CALCULATING FOT AND HPF FROM MUF LOCAL TIME
>
> N. Hemisphere (Nov., Dec., Jan., Feb.)
>
> Winter
>
> { S. Hemisphere (May, June, July, Aug.)
>
> Low (0-50) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.44 .60 │ 1.34 .65 │ 1.45 .69 │ 1.32 .72 │ 1.33 .68 │ 1.40
> .67 ║ \> 75│
>
> │65-75 ║ 1.37 .68 │ 1.29 .71 │ 1.38 .75 │ 1.23 .76 │ 1.24 .75 │ 1.35
> .70 ║ 65-75│
>
> │55-65 ║ 1.30 .74 │ 1.24 .76 │ 1.27 .80 │ 1.15 .80 │ 1.17 .82 │ 1.30
> .73 ║ 55-65│
>
> │45-55 ║ 1.25 .79 │ 1.21 .78 │ 1.16 .83 │ 1.12 .85 │ 1.12 .84 │ 1.25
> .76 ║ 45-55│
>
> │35-45 ║ 1.23 .81 │ 1.20 .79 │ 1.13 .85 │ 1.11 .87 │ 1.11 .89 │ 1.23
> .77 ║ 35-45│
>
> │25-35 ║ 1.28 .81 │ 1.30 .74 │ 1.15 .86 │ 1.17 .82 │ 1.15 .85 │ 1.28
> .78 ║ 25-35│
>
> │15-25 ║ 1.34 .78 │ 1.37 .67 │ 1.19 .87 │ 1.20 .75 │ 1.24 .77 │ 1.32
> .79 ║ 15-25│
>
> │ ≤ 15 ║ 1.27 .71 │ 1.38 .70 │ 1.18 .88 │ 1.15 .86 │ 1.14 .87 │ 1.20
> .79 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> Medium (50-100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.45 .76 │ 1.39 .78 │ 1.44 .68 │ 1.40 .67 │ 1.33 .62 │ 1.45
> .70 ║ \> 75│
>
> │65-75 ║ 1.39 .79 │ 1.31 .81 │ 1.37 .74 │ 1.32 .70 │ 1.29 .73 │ 1.41
> .73 ║ 65-75│
>
> │55-65 ║ 1.33 .82 │ 1.24 .83 │ 1.25 .79 │ 1.21 .75 │ 1.22 .80 │ 1.33
> .76 ║ 55-65│
>
> │45-55 ║ 1.30 .84 │ 1.19 .82 │ 1.14 .83 │ 1.15 .81 │ 1.16 .84 │ 1.29
> .78 ║ 45-55│
>
> │35-45 ║ 1.27 .83 │ 1.17 .81 │ 1.12 .85 │ 1.14 .86 │ 1.14 .86 │ 1.28
> .79 ║ 35-45│
>
> │25-35 ║ 1.30 .78 │ 1.31 .76 │ 1.16 .85 │ 1.18 .85 │ 1.18 .85 │ 1.32
> .78 ║ 25-35│
>
> │15-25 ║ 1.33 .74 │ 1.38 .71 │ 1.17 .85 │ 1.22 .83 │ 1.26 .82 │ 1.40
> .76 ║ 15-25│
>
> │ ≤ 15 ║ 1.21 .77 │ 1.26 .69 │ 1.14 .87 │ 1.13 .86 │ 1.15 .85 │ 1.23
> .78 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> High (\>100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo.║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat.║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.36 .62 │ 1.27 .70 │ 1.41 .74 │ 1.42 .67 │ 1.40 .64 │ 1.43
> .73 ║ \> 75│
>
> │65-75 ║ 1.31 .69 │ 1.25 .74 │ 1.34 .77 │ 1.30 .72 │ 1.16 .72 │ 1.34
> .78 ║ 65-75│
>
> │55-65 ║ 1.26 .77 │ 1.23 .78 │ 1.24 .81 │ 1.18 .80 │ 1.11 .79 │ 1.26
> .82 ║ 55-65│
>
> │45-55 ║ 1.19 .83 │ 1.19 .80 │ 1.16 .84 │ 1.11 .87 │ 1.09 .84 │ 1.20
> .86 ║ 45-55│
>
> │35-45 ║ 1.15 .86 │ 1.14 .81 │ 1.13 .87 │ 1.09 .90 │ 1.09 .87 │ 1.14
> .87 ║ 35-45│
>
> │25-35 ║ 1.22 .83 │ 1.26 .76 │ 1.12 .89 │ 1.09 .90 │ 1.11 .88 │ 1.13
> .86 ║ 25-35│
>
> │15-25 ║ 1.32 .78 │ 1.35 .70 │ 1.12 .89 │ 1.12 .89 │ 1.14 .89 │ 1.20
> .83 ║ 15-25│
>
> │ ≤ 15 ║ 1.18 .83 │ 1.25 .76 │ 1.14 .89 │ 1.13 .90 │ 1.15 .89 │ 1.20
> .84 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> TABLE 3. FACTORS FOR CALCULATING FOT AND HPF FROM MUF LOCAL TIME
> (CONT.)
>
> Equinox (Mar., Apr., Sept., Oct.)
>
> Low (0-50) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.42 .67 │ 1.32 .72 │ 1.29 .74 │ 1.26 .73 │ 1.33 .80 │ 1.48
> .65 ║ \> 75│
>
> │65-75 ║ 1.38 .70 │ 1.25 .75 │ 1.25 .76 │ 1.23 .74 │ 1.26 .82 │ 1.40
> .69 ║ 65-75│
>
> │55-65 ║ 1.32 .73 │ 1.21 .78 │ 1.22 .80 │ 1.20 .75 │ 1.20 .81 │ 1.31
> .73 ║ 55-65│
>
> │45-55 ║ 1.26 .75 │ 1.19 .80 │ 1.20 .81 │ 1.18 .76 │ 1.16 .81 │ 1.26
> .76 ║ 45-55│
>
> │35-45 ║ 1.22 .77 │ 1.20 .81 │ 1.19 .81 │ 1.16 .77 │ 1.16 .80 │ 1.25
> .78 ║ 35-45│
>
> │25-35 ║ 1.22 .78 │ 1.26 .80 │ 1.18 .82 │ 1.15 .78 │ 1.16 .81 │ 1.28
> .74 ║ 25-35│
>
> │15-25 ║ 1.30 .77 │ 1.32 .75 │ 1.16 .83 │ 1.14 .81 │ 1.18 .83 │ 1.33
> .69 ║ 15-25│
>
> │ ≤ 15 ║ 1.23 .76 │ 1.40 .66 │ 1.13 .86 │ 1.13 .89 │ 1.19 .86 │ 1.16
> .75 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> Medium (50-100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.45 .64 │ 1.31 .61 │ 1.27 .73 │ 1.28 .74 │ 1.30 .74 │ 1.47
> .67 ║ \> 75│
>
> │65-75 ║ 1.41 .68 │ 1.22 .71 │ 1.23 .77 │ 1.26 .74 │ 1.26 .78 │ 1.38
> .70 ║ 65-75│
>
> │55-65 ║ 1.35 .70 │ 1.17 .75 │ 1.20 .80 │ 1.23 .72 │ 1.18 .78 │ 1.29
> .73 ║ 55-65│
>
> │45-55 ║ 1.28 .73 │ 1.15 .77 │ 1.17 .81 │ 1.21 .74 │ 1.13 .76 │ 1.20
> .75 ║ 45-55│
>
> │35-45 ║ 1.22 .75 │ 1.16 .78 │ 1.16 .82 │ 1.18 .78 │ 1.12 .76 │ 1.17
> .76 ║ 35-45│
>
> │25-35 ║ 1.22 .77 │ 1.22 .76 │ 1.15 .82 │ 1.17 .83 │ 1.14 .78 │ 1.23
> .72 ║ 25-35│
>
> │15-25 ║ 1.32 .75 │ 1.30 .73 │ 1.13 .84 │ 1.15 .87 │ 1.17 .81 │ 1.37
> .69 ║ 15-25│
>
> │ ≤ 15 ║ 1.18 .79 │ 1.39 .68 │ 1.11 .86 │ 1.13 .89 │ 1.20 .84 │ 1.23
> .80 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> High (\>100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo.│
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat.│
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.46 .66 │ 1.37 .67 │ 1.35 .75 │ 1.40 .66 │ 1.38 .70 │ 1.46
> .72 ║ \> 75│
>
> │65-75 ║ 1.42 .67 │ 1.31 .71 │ 1.30 .73 │ 1.31 .70 │ 1.33 .70 │ 1.37
> .72 ║ 65-75│
>
> │55-65 ║ 1.30 .69 │ 1.25 .75 │ 1.27 .71 │ 1.24 .71 │ 1.25 .71 │ 1.24
> .72 ║ 55-65│
>
> │45-55 ║ 1.18 .73 │ 1.20 .78 │ 1.25 .70 │ 1.20 .72 │ 1.16 .74 │ 1.17
> .73 ║ 45-55│
>
> │35-45 ║ 1.15 .79 │ 1.16 .82 │ 1.17 .75 │ 1.16 .78 │ 1.12 .80 │ 1.14
> .82 ║ 35-45│
>
> │25-35 ║ 1.25 .81 │ 1.18 .82 │ 1.10 .87 │ 1.10 .87 │ 1.11 .87 │ 1.15
> .86 ║ 25-35│
>
> │15-25 ║ 1.31 .81 │ 1.32 .77 │ 1.11 .89 │ 1.11 .92 │ 1.12 .90 │ 1.20
> .85 ║ 15-25│
>
> │ ≤ 15 ║ 1.21 .80 │ 1.23 .79 │ 1.09 .86 │ 1.20 .90 │ 1.14 .90 │ 1.23
> .82 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> TABLE 3. FACTORS FOR CALCULATING FOT AND HPF FROM MUF LOCAL TIME
> (CONT.)
>
> N. Hemisphere (May., June., July., Aug.)
>
> Summer
>
> { S. Hemisphere (Nov., Dec., Jan., Feb.)
>
> Low (0-50) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥───────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo. │
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat. │
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫───────┤
>
> │ \> 75 ║ 1.26 .68 │ 1.24 .79 │ 1.15 .84 │ 1.17 .87 │ 1.21 .85 │ 1.22
> .76 ║ \> 75 │
>
> │65-75 ║ 1.22 .70 │ 1.18 .81 │ 1.14 .83 │ 1.15 .86 │ 1.16 .86 │ 1.18
> .77 ║ 65-75 │
>
> │55-65 ║ 1.18 .72 │ 1.17 .84 │ 1.14 .83 │ 1.15 .84 │ 1.14 .86 │ 1.15
> .81 ║ 55-65 │
>
> │45-55 ║ 1.17 .75 │ 1.20 .85 │ 1.15 .82 │ 1.16 .83 │ 1.14 .85 │ 1.15
> .84 ║ 45-55 │
>
> │35-45 ║ 1.17 .79 │ 1.25 .85 │ 1.17 .80 │ 1.17 .82 │ 1.15 .83 │ 1.16
> .85 ║ 35-45 │
>
> │25-35 ║ 1.18 .79 │ 1.30 .82 │ 1.17 .78 │ 1.20 .80 │ 1.19 .81 │ 1.20
> .80 ║ 25-35 │
>
> │15-25 ║ 1.20 .77 │ 1.34 .78 │ 1.14 .77 │ 1.24 .79 │ 1.22 .79 │ 1.23
> .73 ║ 15-25 │
>
> │ ≤ 15 ║ 1.20 .74 │ 1.37 .75 │ 1.12 .80 │ 1.30 .83 │ 1.27 .82 │ 1.20
> .69 ║ ≤ 15 │
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨───────┘
>
> Medium (50-100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥──────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo. │
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat. │
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫──────┤
>
> │ \> 75 ║ 1.27 .82 │ 1.23 .80 │ 1.20 .82 │ 1.18 .85 │ 1.25 .80 │ 1.23
> .79 ║ \> 75│
>
> │65-75 ║ 1.23 .83 │ 1.19 .82 │ 1.19 .79 │ 1.17 .82 │ 1.17 .82 │ 1.19
> .82 ║ 65-75│
>
> │55-65 ║ 1.20 .83 │ 1.18 .82 │ 1.19 .77 │ 1.17 .79 │ 1.14 .82 │ 1.17
> .83 ║ 55-65│
>
> │45-55 ║ 1.17 .81 │ 1.19 .81 │ 1.21 .76 │ 1.17 .77 │ 1.15 .81 │ 1.16
> .82 ║ 45-55│
>
> │35-45 ║ 1.17 .78 │ 1.22 .78 │ 1.23 .75 │ 1.18 .78 │ 1.17 .78 │ 1.17
> .78 ║ 35-45│
>
> │25-35 ║ 1.20 .77 │ 1.30 .73 │ 1.22 .75 │ 1.19 .79 │ 1.19 .77 │ 1.18
> .74 ║ 25-35│
>
> │15-25 ║ 1.26 .77 │ 1.38 .69 │ 1.17 .78 │ 1.23 .82 │ 1.23 .78 │ 1.28
> .73 ║ 15-25│
>
> │≤ 15 ║ 1.26 .79 │ 1.44 .63 │ 1.11 .84 │ 1.28 .85 │ 1.28 .81 │ 1.22
> .77 ║ ≤ 15│
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨──────┘
>
> High (\>100) Sunspot Number
>
> ┌──────╥────────────┬────────────┬────────────┬────────────┬────────────┬────────────╥───────┐
>
> │ Geo. ║ 22-02 │ 02-06 │ 06-10 │ 10-14 │ 14-18 │ 18-22 ║ Geo. │
>
> │
> ╟────────────┼────────────┼────────────┼────────────┼────────────┼────────────╢
> │
>
> │ Lat. ║ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl │ Fu Fl ║ Lat. │
>
> ├──────╫────────────┼────────────┼────────────┼────────────┼────────────┼────────────╫───────┤
>
> │ \> 75 ║ 1.30 .73 │ 1.27 .74 │ 1.17 .82 │ 1.15 .83 │ 1.23 .79 │ 1.24
> .75 ║ \> 75 │
>
> │65-75 ║ 1.22 .75 │ 1.22 .75 │ 1.20 .77 │ 1.18 .80 │ 1.21 .80 │ 1.23
> .77 ║ 65-75 │
>
> │55-65 ║ 1.16 .77 │ 1.18 .76 │ 1.26 .74 │ 1.21 .77 │ 1.19 .80 │ 1.21
> .80 ║ 55-65 │
>
> │45-55 ║ 1.14 .79 │ 1.15 .76 │ 1.30 .73 │ 1.26 .75 │ 1.19 .80 │ 1.18
> .84 ║ 45-55 │
>
> │35-45 ║ 1.14 .80 │ 1.14 .76 │ 1.30 .75 │ 1.27 .75 │ 1.19 .79 │ 1.16
> .84 ║ 35-45 │
>
> │25-35 ║ 1.16 .81 │ 1.15 .76 │ 1.25 .82 │ 1.20 .81 │ 1.17 .79 │ 1.15
> .83 ║ 25-35 │
>
> │15-25 ║ 1.21 .81 │ 1.22 .77 │ 1.18 .85 │ 1.15 .86 │ 1.18 .81 │ 1.19
> .80 ║ 15-25 │
>
> │≤ 15 ║ 1.25 .80 │ 1.21 .79 │ 1.13 .86 │ 1.17 .89 │ 1.22 .85 │ 1.23
> .78 ║ ≤ 15 │
>
> └──────╨────────────┴────────────┴────────────┴────────────┴────────────┴────────────╨───────┘
>
> .1 Based on the values in table 3, the probability of ionospheric
> support at a given frequency is determined by evaluating the Gaussian
> probability distribution function.
>
> The standard deviation is first evaluated as follows:
>
> where F(I,J,K) are the multiplying factors from table 3 and I,J,K are
> indexs that represent latitude, low or high sunspot number, and time
> of day.
>
> Then the probability that the operating frequency (F) is greater than
> the FOT, MUF, or HPF for a given layer using the MUF distribution is
>
> The E layer is stable and very predictable; therefore, if the E layer
> supports propagation, it is assumed to have a probability of 0.99.
>
> 3.9 Probability of Sporadic-E Propagation
>
> The program contains an option for considering sporadic-E propagation
> in the system performance predictions. Sporadic-E propagation is not
> considered when calculating the MUF, because the median value of foEs
> does not represent a true critical frequency and is used simply to
> estimate the probability of occurrence of sporadic E. It is only
> considered when regular E-layer propagation is not possible.
>
> The median and upper and lower decile values of foEs are obtained from
> the numerical coefficients (see sec. 2.4), and converted into values
> for the oblique path by the secant law relationship. The
> multiplicative factor, sec Φ, is computed as follows:
>
> (3.35)
>
> where
>
> d/2 = half the great-circle path length in radians,
>
> β = takeoff angle obtained from (3.17), with h\' = 110 km.
>
> The probability of sporadic-E propagation is then calculated for the
> operating frequency from these median and decile values using the
> normal probability function. As discussed in section 3.4, no
> additional loss is determined for the partial transparency of the E~s~
> layer to radio waves.
>
> 3.10 Calculation of Mixed Modes
>
> For path lengths equal to or greater than 3000 km, two N-type mixed
> modes of propagation are considered. This type consists of one E or
> E~s~ hop, with the remainder of the propagation path via the F layer.
> If E-layer propagation is not possible, the probability of E~s~
> propagation is determined and combined with the probability of
> subsequent F-layer propagation. Normally, a mixed mode is not the most
> reliable mode.
>
> **4. NOISE PARAMETERS**
>
> The probability of successful transmission depends on the probability
> that the available signal-to-noise ratio at the receiving location is
> above a specified level. This level is the one required for a
> specified grade of service (see sec. 7) in the presence of noise, but
> in the absence of any other unwanted signals. The three major types of
> external noise with which the HF signal must compete are galactic,
> atmospheric, and man-made. In general, these noise sources have
> spectral energy distributions that vary more or less uniformly over
> the entire high frequency range. All values of noise are considered
> representative of those that would be expected with a short vertical
> lossless receiving antenna. No allowance is made in the program to
> account for the directional and polarization properties of other types
> of receiving antennas that could alter the available signal-to-noise
> ratio.
>
> 4.1 Galactic Noise
>
> Estimates of galactic (cosmic) noise are obtained from measurements
> (Spaulding and Disney (1974) CCIR Report 258-4(1982)).
>
> The median galactic noise is represented by
>
> N~g~ = 52.0 + 23 log~10~ (f) - 204 dBW (4.1)
>
> where
>
> 10 log~10~ kTo = 204 db
>
> k = (Boltzmann;s constant) 1.38 \* 10^-23^ Joules/degree Kelvin
>
> To = Reference temperature 288^o^ K
>
> N~g~ = expected median value of the galactic noise power in decibels
> relative to 1 W in a 1 Hz bandwidth,
>
> f = frequency in megahertz.
>
> The temporal variation is estimated at ±2 dB (CCIR, 1982), and values
> represent the upper limit of galactic noise. Ionospheric shielding of
> galactic noise is determined by evaluating the F-layer critical
> frequency above the receiving location. Directional properties of the
> receiving antenna are not considered. Estimates of the uncertainty in
> predicting the median and deciles are 0.5 dB and 0.2 dB, respectively.
>
> 4.2 Atmospheric Noise
>
> In the HF band, N~a~, the atmospheric noise, is the most erratic of
> the three major types of noise. It is generally characterized by short
> pulses with random recurrence superimposed upon a background of random
> noise (Spaulding and Disney (1974)). Averaging these short pulses of
> noise power over several minutes yields average values that are nearly
> constant during a given hour. The variations seldom exceed ± 2 dB,
> except during sunrise or sunset periods and when local thunderstorms
> are present. Worldwide maps published in CCIR Report 322-3,
> representing the median of hourly medians of atmospheric noise at 1
> MHz within 4-hour time blocks for the four (3-month) seasons of the
> year, are used as the basis for estimating this noise at any given
> receiving locations. Levels of atmospheric noise for other frequencies
> and its associated distribution about the median are available for
> each time block and season.
>
> The maps in CCIR Report 322-3 were used to generate, by means of a
> least squares fit based on Fourier analysis, numerical coefficients
> representing the worldwide distribution of atmospheric noise
> (Spaulding and Washburn, 1985) as a function of geographic location.
> The curves of frequency dependence and variability of the radio noise
> were generated from a power series of least squares fit. The numerical
> maps and complete listings of the numerical coefficients used in the
> prediction program are given in NTIA Report 85-173 (Spaulding and
> Washburn).
>
> The numerical coefficients in latitude, longitude, and local mean time
> at the receiving location give the median atmospheric noise power in
> decibels above k T~o~ b, at 1 MHz for each time block and season. The
> value of k (Boltzmann\'s constant) is 1.38 x 10 ^-23^ Joules per
> degree Kelvin, the reference temperature is 288^o^K, 10 log~10~kT~o~ =
> 204 dB below one joule, and b is the bandwidth in Hz. Noise values for
> other frequencies and their distributions are evaluated from numerical
> arrays derived from the frequency curves for each time block and
> season.
>
> 4.3 Man-Made Noise
>
> At certain receiving locations, unintended man-made radio noise may be
> the predominant external noise with which the communication system
> must compete. It may arise from any number of sources, such as power
> lines, industrial machinery, ignition systems, etc., and thus have
> wide geographic and short-term variations. Information for deriving
> specific man-made noise levels as a function of geographic location
> and time is insufficient, but from the limited observations available,
> it is possible to express typical levels of unintended radiation
> (JTAC, 1968).
>
> Four models are used to designate the median level of man-made noise,
> all based on the following equation:
>
> N~m~ = c + d log~10~ f - 204 dBW (4.2)
>
> where
>
> N~m~ = man-made noise power in decibels below 1 W in a 1 Hz bandwidth,
>
> f = frequency in megahertz,
>
> c,d = constants derived from measurements and given below.
>
> Values of c,d for each of the four models, are given below.
>
> [Values of Parameters for the Man-Made Noise Models]{.underline}
>
> [Environmental category]{.underline} [c]{.underline} [d]{.underline}
>
> Business (curve A) 76.8 27.7
>
> Residential (curve B) 72.5 27.7
>
> Rural (curve C) 67.2 27.7
>
> Quiet rural (curve D) 53.6 28.6
>
> Galactic noise (curve E) 52.0 23.0
>
> Urban location is defined as one within the industrial-business area
> of large cities, residential is one near a large city or within a
> small town, and rural location is one well removed from all populated
> areas and chosen to be as free as possible of man-made noise. The
> upper and lower decile values of N~m~ are 9.7 dB and 7.0 dB. Estimates
> of the uncertainty in predicting the median and deciles are 5.4 db and
> 1.5 db respectively.
>
> When the user inserts his own value of man-made noise power in
> decibels below 1 watt in a 1 Hz bandwidth at 3 MHz (N~user~), the
> suburban noise curve is used. In the program, the suburban curve is
> shifted vertically and includes the noise value at 3 MHz specified by
> the user.
>
> 4.4 Combination of Noise
>
> The three basic noise sources, man-made, galactic, and atmospheric
> noise, are independent of each other and must be combined by a
> statistically accurate method in order to represent the overall noise
> level at the site of interest. The current noise distributions are
> considered log normal and the method used to combine them is to
> determine the log normal distributions that best approximate the true
> distribution of the sum. The method used is a rather complex
> mathematical process. For specific information refer to an updated
> noise model for use in IONCAP (Spaulding, A.D. and Stewart, F.G.
> (1987).
>
> Three major changes have been made with the new noise model: The
> replacement of the world wide atmospheric radio noise estimates with
> the current, much improved estimates of CCIR report 322-3; the
> replacement of the man-made noise estimates with the much more modern
> estimates of CCIR Report 258-4; and the overall distribution and its
> statistical variations have been updated.
>
> **5. IONOSPHERIC LOSS MODEL**
>
> This section describes the equations used to calculate ionospheric
> losses for a single-hop mode. The model is intended to cover the
> frequency band from 3 to 30 MHz. All modes which result from a
> complete electron density profile are covered, as well as sporadic-E
> modes. Lower decile, median, and upper decile values of field strength
> (signal or signal-to-noise ratio) can be determined by the methods
> described. The theoretical background and method of measurements are
> given in the manual on Ionospheric Absorption Measurements (Rawer,
> 1976). The equations described below are intended to be used on a
> worldwide basis, and are limited by the available worldwide prediction
> data base.
>
> The equations are based on the CCIR 252-2 (1970) loss equation using a
> philosophy that modifications are made only when measured values
> demand a change. The CCIR loss equation was primarily derived from F2
> low-angle modes with operating frequencies not greater than the FOT.
> For these conditions, there is no need to modify the equation. For
> E-layer low-angle modes, two modifications were necessary. A
> correction factor to account for less E-Layer bending was added
> \[Equation (5.8)\]. Since an arbitrary electron density profile can be
> used, the true height of reflection may be written into the D-layer
> absorbing region, resulting in lower absorption. This has been
> accounted for by modifying the collision frequency factor in the
> denominator \[Equation (5.10)\]. In addition to the D-region losses
> due to absorption, there may be losses at the area of reflection if
> the ionization of the region is insufficient to reflect all the radio
> energy. This loss increases as frequency increases, starts to be
> significant as the frequency nears the MUF, and increases rapidly
> above the MUF. Since this loss is closely associated with the MUF, the
> day-to-day variation of the MUF within the month can be used to
> estimate the day-to-day variation of these losses within the month.
> This loss due to partial failure of ionospheric support is referred to
> as an \"over-the-MUF\" loss\[Equation (5.12)\]. Since the basic loss
> equations contain the average effect of deviative absorption, an
> additional loss is only added to the high-angle modes (including those
> F modes on the upper side of the E MUF (see Equation 41). Sporadic-E
> mode losses are determined as are the regular E modes, but the
> deviative loss is zero and the over-the-MUF loss is what is usually
> termed reflection loss \[Equation (5.13)\].
>
> The equations referred to provide monthly median losses when used with
> monthly median parameters of the ionosphere. In order to evaluate the
> performance of a communication system, the distribution of these
> losses within the month is needed. These distributions have been given
> as tables of transmission loss distributions (CCIR 252-2, 1970) and
> are uniform for operation frequency and for mode type. For geomagnetic
> latitudes below 40 degrees, these tables are for frequencies less than
> the FOT and are the variation of the F2 mode losses about the CCIR
> 252-2 equation based on 83 circuit years of data (Laitinen and Haydon,
> 1950). The rest of the tables correspond to 10.0 MHz and is based on
> the difference between minimum loss during the month and the
> distribution of losses above this difference within the Arctic
> regions. These tables are used in the following manner; the frequency
> that corresponds to the table of losses is established as either the
> F2 FOT or as 10.0 MHZ. Then the sporadic-E obscuration loss and the F2
> over-the-MUF loss are subtracted from the table value. The residual
> table values are then added to the losses at each frequency for each
> mode. Auroral losses form the main part of the residual table. Decile
> values of losses are determined by calculating the Es obscuration loss
> and the over-the-MUF losses using the proper decile values of the Es
> and E, F1, or F2 mode MUFs. It should be especially noted in this
> model that the over-the-MUF losses limit the successful operation of
> the radio systems at the higher frequencies; i.e., the mode is always
> there but the losses become sufficiently high that the system becomes
> inoperative. The circuit reliability falls because of weak signal
> levels in contrast to earlier prediction methods which use a concept
> of \"fraction-of-days\" where the mode existed to limit the
> reliability at the higher frequencies. both the original table and the
> derived values are decibel with respect to the loss equations given
> here. If other loss equations are used, the distribution tables must
> be adjusted. the derivation of these loss equations is given below.
>
> 5.1 Free-Space Basic Transmission Loss
>
> Free-space losses result from the geometrical spreading of energy as
> the radio wave progresses away from the transmitter. In ionospheric
> propagation, the incremental cross section of the ray bundle at the
> receiver depends upon the physical properties of the ionosphere and
> the geometry of the propagation path. Simplifying asumptions are made
> in the program so that transmission losses can be calculated in a
> practical manner.
>
> In the simplest model of sky-wave propagation, it is assumed that the
> earth and the ionosphere are both flat and that the reflection is
> specular (mirror-like). For this type of propagation, the energy
> density diminishes as the inverse square of the ray-path distance
> (Piggott, 1959). This means that for an isotropic transmitting antenna
> radiating p watts of power, the power flux density at a slant range
> distance D is p/(4 π D^2^). The total area of an isotropic receiving
> antenna in free space is λ^2^ /(4 π), where λ is the wavelength of the
> radio wave. Therefore, the total power received by the antenna is p
> λ^2^ / (4 π D)^2^. The basic free-space transmission loss is the ratio
> between the power radiated and the power received by a loss-free
> receiving antenna, and is given by:
>
> \(26\)
>
> Espressing the wavelength λ in terms of the frequency f of the
> electro-magnetic wave, (26) becomes:
>
> \(27\)
>
> where f is in Megahertz and D is in kilometers.
>
> 5.2 CCIR 252-2 Ionospheric Loss
>
> The absorption in the D-E regions of the ionosphere is usually the
> major loss (after free space) in radio wave propagation via the
> ionosphere. To describe this effect, Martyn\'s theorem relating
> vertical and oblique path and the quasi-longitudinal approximation to
> the absorption loss (e.g., Budden, 1966) are used:
>
> (5.1)
>
> (5.2)
>
> where
>
> C = constant,
>
> h~o~ = height at bottom of ionosphere,
>
> h(f~v~)= height of reflection for frequency f~v~,
>
> N = N(h) electron density profile,
>
> ν = ν(h) collision frequency,
>
> µ = refractive index,
>
> f~ob~ = oblique sounding frequency,
>
> f~v~ = vertical sounding frequency,
>
> f~H~ = gryofrequency, (f~L~, the longitudinal component of
>
> gyrofrequency, is usually put into equation)
>
> φ~o~ = angle of earth\'s normal to ray path at 100 km.
>
> Equation (5.2) can be put into the form (Budden, 1966)
>
> L(f~v~)= (ν/c)(h\' - hp) (5.3)
>
> where
>
> ν = effective collision frequency
>
> h\' = virtual height of reflection,
>
> hp = phase height of reflection,
>
> c = velocity of light.
>
> Since hp is bounded and h\' is not, there is a strong dependence on
> frequency which cannot be explained simply by an
> inverse-frequency-squared law. The usual method of analysis has been
> to write equation (5.2) in the form (definition of the global
> absorption parameter A)
>
> (5.4)
>
> Most analyses of absorption are based on estimated A(f~v~). This has
> sometimes been done by assuming non-deviative absorption only, and
> thereby trying to ignore the frequency dependence. When this is done
> with a small data base and no adjustment is made for the frequency
> dependence, the results can be misleading (and possibly inaccurate).
>
> The CCIR absorption equation is based upon the U.S. Army signal Radio
> Propagation Agency study (Laitinen and Haydon, 1950), with a
> modification for lower frequencies by Lucas and Haydon (1966).
>
> The exponent of the frequency term (f~v~ + f~H~) and values for the
> terms
>
> (ν/2π)^2^ and A(f~v~) were determined by least square curve fitting.
> The oblique
>
> loss measurements were normalized to virtual loss measurements to give
> a standard comparison method, as well as to expand the data base, by
> the equation
>
> (5.5)
>
> The data used were for F-layer modes only. The fitted equation is
>
> (see Figure 4).
>
> (5.6)
>
> That is, the averaged value of A(f~v~) is
>
> \_ + 677.2I
>
> where
>
> I = -.04 + exp (-2.937 + 0.8445foE). (5.7)
>
> The formula for absorption index I is in terms of foE which includes
> the variation in zenith angle and solar activity. This formula is an
> inversion of that formerly used to obtain foE from I. There have been
> attempts to modify and replace equation (5.6) by independent
> researchers. Schultz and Gallet(1970) (used in Barghausen et al.,
> 1969) used the methods described by Piggott (1953) to describe \_
> without frequency dependence (non-deviative absorption) on a smaller
> data base than used to derive Equation (5.6), but did not add the
> frequency dependence suggested by Piggott (1953) \[essentially the
> same as Equation (5.3), but for a parabolic E layer\]. George (1971)
> has developed an absorption equation using an A(f~v~) which has an
> implicit (h\' - h~p~) curve.
>
> The method of supplementing equation (5.6) described below was based
> upon area coverage and radar backscatter data and was developed in two
> steps, first to correct equation (5.6) for E-layer modes and then for
> frequency dependence (deviative absorption). It is essentially based
> upon the suggestion in Piggott and in George, but with the advantage
> of having the electron density profiles available on a worldwide
> basis. (See also Rawer,1976; Bibl et al., 1961; and Fejer, 1961 for
> further discussion.)
>
> 5.3 Frequency Dependence Effects On Absorption
>
> The absorption equation (5.6) contains an averaged value of A(f~v~)
> for F2-layer modes, i.e., the effects of E-region electron density on
> non-deviative absorption collision frequency (and magnetic field
> effects, etc.) were averaged in the curve fitting process. Therefore,
> a correction for E-layer modes was derived:
>
> L~c~ = A + B log~e~ (X~e~), (5.8)
>
> where
>
> L~c~ is the loss correction factor for E modes:
>
> A = 1.359 for f~o~E \> 2 MHz,
>
> = 1.359 (f~o~E-0.5) / 1.5 for 0.5 ≤ f~o~E ≤ 2 MHz,
>
> = 0.0 for f~o~E , 0.5 MHz;
>
> B = 8.617 for f~o~E \> 2 MHz,
>
> = 8.617 (f~o~E - 0.5) / 1.5 for 0.5 ≤ f~o~E ≤ 2 MHz,
>
> = 0.0 for f~o~E , 0.5 MHz;
>
> X~E~ = f~v~/ f~o~E for h \> 90 km,
>
> = f~v~(90) / f~o~E for h ≤ 90 km;
>
> h = true height of reflection;
>
> f~v~ = equivalent vertical sounding frequency.
>
> The equations for E-mode and F-mode losses assume that the mode goes
> through the absorbing region (true height of reflection above 95 to
> 100 km). Equation (38) represents the effect of E-layer losses which
> were included in Equation (36). When the true height of reflection is
> below 90 km, which will happen with the complete electron density
> profiles, the use of Equations (36) and (38) will give losses much
> higher than those observed. The absorption should reach a maximum as
> the operating frequency is lowered and then decreased. Theoretically
> the absorption reaches a maximum at the height where ν/2π = f + f~H~.
> However the electron density profile available does not represent the
> lower ionosphere adequately, so this behavior has been introduced into
> the model in an ad hoc manner by replacing the constant value of
> (ν/2π)^2^ = 10.2 by a calculated value.
>
> When the true height of reflection, h,varies between 70 and 88 km the
> value of (ν/2π)^2^ varies between 40, the value it would have at a
> height of 61 km, and 10.2, the value it would have at a height of 64
> km. The equation for the height h~v~ is (h~v~ does not correspond to
> any physical parameter)
>
> h~v~ = 61 + 3(h-70)/18. (5.9)
>
> The equation for effective collision frequency is
>
> (ν(h~v~)/2π)^2^ = (ν~o~/2π)^2^ exp(-2(h~v~-60)/H) (5.10)
>
> where
>
> (ν~o~ / 2π)^2^ = 63.07
>
> H = 4.39
>
> The equation (5.10) gives realistic values for ν. The transformation
> from h to h~v~ was derived using measured values of field strength on
> the Kogani to Akito path (Japan) at operating frequencies of 2.5 and
> 5.0 MHz, for the year 1970.
>
> Since a complete electron density profile is used, any mode, high- or
> low- angle, will be considered. To account for the deviative (and
> spreading) losses of the high-angle modes, the following procedure is
> used. Deviative losses are considered to be averaged into Equation
> (5.6) for those modes with reflection heights less than that at the
> layer MUF. For modes with reflection height greater than that at the
> layer MUF and for modes just past the E-F cusp, a deviative loss term
> is added.
>
> The equation is based on the relationship that the loss is
> proportional to the product of collision frequency with the difference
> between group path and phase path, i.e.,
>
> A~p~(f~v~) = B(f~v~)(h\'-h)~N~sec (φ~o~), (5.11)
>
> where A~p~(f~v~) is the deviative absorption loss in dB and
>
> B(f~v~) = C~E~ exp \[ -2(h-70)/ymE\] for the E layer,
>
> = C~F~ exp \[-2(h-hmF2 + ymF2)/ymF2\] for the F2 layer,
>
> B(f~v~) = C~1~ exp \[-2(h-hmF1 + ymF1)/ymF1\] for the F1 layer and
>
> B(f~v~) = C~2~ exp \[-2(h-hmF2 + ymF2)/ymF2\] for the F2 layer
>
> when the F1 layer is present.
>
> The constants have not been mapped on a worldwide basis as yet.
> Interim values used are
>
> C~E~= 0.25,
>
> C~F~= 0.40,
>
> C~1~= 0.60,
>
> C~2~= B(foF1).
>
> In order to preserve continuity at the F1-layer to F2-layer
> transition, C~2~ is the value of B(f~v~) with f~v~ taken as foF1. This
> assures a smooth calculation of loss for all possible electron density
> profiles. Note that C~F~ and C~2~ are forced to have minimum values of
> 0.05.
>
> When sporadic E is present, the loss is supplemented by an Es
> obscuration factor as described in Section 5.4.
>
> 5.4 Loss For Propagation Above The Standard MUF
>
> The model does not assume that propagation ceases at the junction
> frequency (or skip distance). It is known that the geometric optics
> approximation does not suffice in the neighborhood of the skip zone.
> By using wave methods, it is shown that energy difracts into the skip
> region (Bremmer, 1949). However, the observed signals are too strong
> to be explained in terms of wave propagation at the skip distance. The
> physical mechanism for the propagation of significant signals within
> the skip zone (or above the MUF when the distance is fixed) is not
> completely understood. One possibility is that the ray-path
> approximation for the calculation of skywave signal paths is probably
> inadequate since the skywave signal will normally return from a large
> region of the ionosphere (at least one Fresnel zone). As the frequency
> approaches the MUF, the energy arriving at the sharper angles will
> tend to penetrate and a portion of the Fresnel zone becomes
> ineffective as a reflector. As the frequency is further increased,
> less and less of energy is reflected and losses will sharply increase.
> On the longer paths where the fresnel zone is larger, losses increase
> slowly as the frequency exceeds the MUF compared to the sharp increase
> on shorter paths where the Fresnel zone is smaller. Other mechanisms
> may also contribute to the signal propagation and explain the
> significant signals observed above the MUF. For example, propagation
> may possibly be explained by scattering from horizontal irregularities
> in the ionosphere (Budden, 1966). The method used here is based on the
> empirical statistical method of Philips-Abel (Wheeler, 1966) rather
> than wave theory. The basic assumption is that a reflecting layer of
> the ionosphere may be considered as composed by quasi-random
> (quasi-independent) elemental patches of ionization. These elemental
> patches are considered to be composed of sub-patches, each with a
> classical MUF (JF). The median value of received power is proportional
> to the probability that there are reflecting areas with elemental
> classical MUF values that are equal to or greater than the
> transmission frequency. Experience indicates that the normal
> distribution is a fair estimate of the spatial probability of the
> elemental classical MUF values and that the mean value of this
> distribution is a fair estimate of the standard MUF. Further, at any
> given operating frequency there is some probability that the frequency
> will exceed the median monthly predicted MUF on some days. Thus this
> loss term includes both the effect of not being below the MUF on some
> days of the month and the loss on those days when the frequency is
> above the MUF. So this term is added at all operating frequencies.
>
> The equation is
>
> L = 10 log~10~ P sec (φ) (dB) (5.12)
>
> where
>
> and
>
> f~m~ is the MUF for the circuit elevation angle and distance (note
> that f~m~ is higher for a two-hop mode than for a one-hop mode on the
> same circuit). For vertical incidence, the loss is less than 0.5 dB
> for frequencies less than the FOT, and less than 3 dB for frequencies
> less than the MUF. Note that all modes from all layers (E, F~1~, F~2~,
> and E~s~) now look the same (absorption losses plus over-the-MUF
> loss). The difference is in the distribution of the MUF\'s, being
> narrow for the E and F~1~, somewhat wider for the F~2~, and the widest
> for the E~s~.
>
> Since signal losses are associated with the ratio of the operating
> frequency to the MUF at a given time, the day-to-day distribution of
> expected losses near the monthly median MUF\> As the frequency is well
> above the monthly median MUF; e.g., above the HPF only the scatter
> components remain and the signal is very weak, but the day-to-day
> distribution is not great.
>
> 5.5 Sporadic-E Losses
>
> This section discusses the loss models associated with modes of
> propagation for which ray theory is not valid. The sporadic-E layer
> was not included in the electron density profile. It is modeled as a
> thin layer occurring at the height h\'Es (usually 100 to 110 km). Its
> effect upon modes of propagation passing through it is given by Es
> obscuration loss. It is calculated by a method proposed by Phillips
> (1962) and modified to use the now available maps of foEs:
>
> L~o~ = 10 log~10~ (1 - P) (dB) (5.13)
>
> where
>
> For modes which have reflected from the sporadic-E layer, the loss is
> the absorption losses supplemented by a reflection loss (corresponding
> to the over-the-MUF loss) defined by
>
> L~R~ = 8.91 p^-0.7^ (5.14)
>
> Note that this is effectively the same as L~r~ = 8.91 - 10 log(P)
> (dB).
>
> 5.6 SYSTEM LOSS
>
> The system loss of a radio circuit is defined as the ratio of the
> signal power available at the receiving antenna terminals relative to
> that available at the transmitter antenna terminals, in decibels. This
> excludes any transmitting or receiving antenna transmission line
> losses, since such losses are considered readily measurable. The
> system loss does include all the losses in the transmitting and
> receiving antenna circuits - not only the transmission loss caused by
> radiation from the transmitting antenna and re-radiation from the
> receiving antenna, but also any ground losses, dielectric losses,
> antenna loading coil losses, and terminating resistor losses. Antenna
> gain is taken as antenna power gain which is the product of antenna
> directive gain in the direction aligned with the propagation path in
> both elevation and azimuth and of antenna efficiency.
>
> The system loss is summarized as (Rice, et al., 1965)
>
> L~b~ = L~bf~ + L~i~ + L~g~ - (G~t~ + G~r~) (dB) (5.15)
>
> where
>
> L~bf~ = the basic free-space transmission loss expected
>
> between ideal, loss-free, isotropic, transmitting
>
> and receiving antennae in free space;
>
> L~i~ = losses caused by ionospheric absorption, this term
>
> includes sporadic E obscuration loss, L~o~ , for F-layer modes.
>
> For all modes, this factor includes the
>
> deviative losses at high reflection heights, over-
>
> the-MUF loss as well as the tables of measured
>
> deviations which include auroral losses.
>
> L~g~ = ground reflection losses.
>
> G~t~ = transmitting antenna power gain relative to an
>
> isotropic antenna in free space;
>
> G~r~ = receiving antenna power gain relative to an isotropic
>
> antenna in free space;
>
> In this report, G~t~ and G~r~ are in the direction of the propagation
> path and include all antenna losses, so that G~t~ + G~r~ is an
> approximation of the gain G~p~ . The values G~t~ and G~r~ are required
> for any elevation angle, azimuth direction, and frequency.
>
> The skywave field strength is directly related to the transmission
> loss, L~b~. This is the loss that would be observed if the actual
> antennas were replaced by ideal, loss-free, isotropic transmitting and
> receiving antennas. the field strength is:
>
> E = 107.2 + 20 log f~ob~ + G~t~ + P - L~b~ (dB) (5.16)
>
> where
>
> E = rem field strength in dB referred to one microvolt
>
> per meter,
>
> G~t~ = transmitting antenna gain in the direction of the ray
>
> path used to determine L~s~ (decibels referred to an
>
> isotropic antenna),
>
> P = transmitter power delivered to the transmitter antenna
>
> in decibels referred to one watt,
>
> f~ob~ = operating frequency in megahertz.
>
> In cases when the reference field strength is 300 millivolts per
>
> meter at one kilometer (rms field produced by 1 kw input to a short
>
> dipole over perfect earth), the skywave field strength, E, is
>
> E = 142 + 20 log f~ob~ - L~b~ (dB) (5.17)
>
> Likewise, when the reference field strength is 222 millivolts per
>
> meter at one kilometer, the skywave field strength, E, is
>
> E = 139.4 + 20 log f~ob~ - L~b~ (dB) (5.18)
>
> 5.7 Conclusions
>
> A semi-empirical method to include recent measurements, especially
> back-scatter measurement, in the calculation of ionospheric losses has
> been developed. It includes deviative and non-deviative effects as
> well as sporadic-E effects. The average effects of magnetic field and
> of polarization changes are included implicitly as the constants in
> the equations are taken from measurements which included these
> effects. The median excess system loss, Table 4 (Lucas and Haydon,
> 1966) is no longer used as it was mainly the average effect of
> sporadic-E and deviative losses.
>
> **6.0 LONG DISTANCE MODEL**
>
> A model to describe the virtual-height ray path and area coverage with
> single ionosphere reflections has been described (Section 5). This
> method was a combination of classical geometric-ray theoretical
> methods and empirical statistical methods. It has long been recognized
> that a continuation of such methods have severe limitations when
> applied to multi-hop propagation (Pedersen, 1927; Harnischmacher,
> 1960; Whale, 1969). As the radio path length increases, individual ray
> paths become increasing difficult to define, and the composite signal
> from all modes becomes increasingly important relative to the signal
> from an individual mode (Bremmer, 1949).
>
> Even for short paths there may be significant contribution from
> multi-hop paths, especially when the antenna pattern does not favor
> the vertical angle associated with the single-hop mode. This is not
> necessarily an indication that there is anything basically wrong with
> the prediction techniques, but rather that there are fundamentally
> unpredictable (or random) factors present. The following section will
> describe some qualitative aspects of multi-hop propagation and a
> quantitative method for estimating the effect of the ionosphere upon
> multi-hops paths.
>
> 6.1 Qualitative Aspects Of Long Distance Propagation

The short-distance propagation model rigorously extended to long paths
would lead to the expectation that failure of propagation at any of the
ionospheric reflection areas would cause propagation to fail altogether.
Empirically, however, it has been observed that propagation usually does
not fail until ionospheric \"control areas,\' along the great circle
path near each end of the path, fail to support transmission (CRPL,
1948). This is apparently true, within limits, no matter now long the
great circle path is between the control points, and no matter what the
condition of the ionosphere is in between.

Figure 10 illustrates some of the additional modes of propagation
available over great distances along the great circle path. Actual
propagation is not of course limited to a geometric plane, but travels
by waves within some volume of space \[ hence we speak here of control
\'areas\" and reflection \'areas,\" not points, (see Figure 5)\].

The modes depicted include chordal modes (i.e. those which because of
ionospheric tilts are deflected over the absorbing regions) and M-modes,
which reflect above a regular E- or sporadic-E layer on some part of the
path. Any wave which leaves the earth and strikes the F2 layer excites
scatter sources on its way up through the E region (short scatter), and
on its way down, after being reflected from the F2 layer (long scatter).
Re-radiation takes place from both regions. The short scatter source
provides fairly intermittent energy because of the intermittent nature
of the ionization causing the re-radiation. However, the wave reflected
from the F2 layer is spread over a wide area so that there are enough
patches of irregularities to provide an appreciable amount of
re-radiation.

The irregularities consist of rather small departures of the electron
density from the average of background value so the \"blobs\" in the
ionosphere are not opaque, but are regions in which the refractive index
is only slightly different from the average value. Such a medium can be
referred to as a diaphanous screen (Whale, 1969). The power spectrum
from such a medium contains two components: a pseudo-specular component
and a scattered component. On short-distance paths, the specular
component, which follows the geometric ray path, predominates. On
long-distance paths, the scattered component becomes increasingly
important. The situation is further complicated by horizontal and
lateral inhomogeneities, presence of the magnetic field, and ground and
ionospheric scatter, both along and from off the great circle path.

The proposed model for multi-hop propagation loss considers a wave which
travels on a great circle path by considering three parts of this path:

\- losses associated with the launching of a skywave from the
transmitter,

\- losses associated with the propagation of the wave between

the launch and intercept area,

\- losses associated with the intercept of a skywave at the

receiver.

It is assumed that the effective launch area of the ionosphere may be
identified by examining earth-ionosphere propagation paths and selecting
the most efficient one as determined by combining the short-distance
propagation model with appropriate transmitter antenna gains. The
effective intercept area is determined in a similar manner using the
appropriate receiving antenna gains.

It is assumed that between the launch and intercept areas the energy
travels in a complex combination of specular reflection and scatter
components (see figure 10), always seeking a path with the least loss.

In order to keep the models (single-hop and multi-hop) internally
consistent, the single-hop models are used to estimate the launch and
inherent areas of the multi-hop model and to estimate the losses between
these areas. The following section gives a more detailed description

6.2 THE LONG-DISTANCE MODEL

Sampling areas are selected the same for both models; i.e., two at each
end, and one in the middle. No new set of equations nor new computer
code arises this way.

The basic transmission loss equation used in the single-hop model
assumes continual divergence as if the medium were free space. A
convergence factor is added to this equation for the multi-hop model.
The theoretical angular (horizontal) convergence factor (Davis, 1965,
1969) is used and the resultant functional form is shown in figure 6.

C~H~ = 15 dB at the antipodal (Ψ = π)

(6.1)

P\' is the group path

where

C~H~ is the convergence factor, dB, and

∆ is the radiation angle,

a is the earth\'s radius,

Ψ is the distance angle.

Ionospheric losses for the multi-hop model are in three parts as
follows:

\(1\) The loss associated with the launch of the skywave. The
characteristics of the dominate launch path are taken from the samples
of ionospheric parameters at the transmitter and of the circuit. The
radiation angle of the transmitted ray is taken from that path which
will have a maximum field strength at a ground distance of 4000 km. The
ionospheric loss associated with the selected ray is then taken as
one-half the loss associated with a one-hop mode using this ray. The
distance corresponding to the launch of the skywave is one-half the
ground distance associated with the one-hop mode.

\(2\) The loss associated with the intercept of the skywave. The
dominant intercept path is assumed to be at the angle at which the
ionospheric loss rate is the lowest; i.e., the least loss per kilometer
of ground distance. As in the launch case, this angle and associated
distance may be determined directly from the \"ray set\' table.

\(3\) Loss between the launch and intercept area. The distance between
the launch and intercept areas is the great circle distance minus the
distance to the launch area from the transmitter and the distance from
the intercept area to the receiver. The loss rate along this distance is
assumed to be approximated by the average of the loss rates at the three
sample areas.

The model therefore assumes that absorption on long paths is related to
the signal power that reaches the ionosphere at the transmitter end less
the power that is attenuated at the receiver and less that power lost
along the path. The expected behavior of the long-path model is depicted
in Figure 7. For long-path propagation, the mode structure is lost,
being a composite of ground and ionospheric reflection and scatter.

The total system loss for multi-hop propagation is then (figure 8)

L~T~ = L~bf~ - C~H~ + L~T~ + L~i~ + L~R~ + L~G~ + Y~P~ (6.2)

where

L~T~ is total system loss for long sky-wave paths.

L~bf~ is free space loss based on ray path distance.

C~H~ is a horizontal convergence gain.

L~T~ is loss in the launch of the radio wave at the

transmitter. A combination of transmitter antenna gain

and propagation losses for the most efficient mode at the

sample area near the transmitter.

L~i~ is loss over the portion of the great circle path between

the launch and intercept areas. Average of the loss

rates at the sample areas times the distance between the

launch and intercept areas.

L~R~ is loss in the intercept of the radio wave at the receiver.

Calculated in the same way at L~T~ using receiver antenna gain.

L~G~ is an average ground loss rate times distance between

launch and intercept areas.

Y~P~ is the prediction system loss correction factor which

includes effect of auroral zones and sporadic E.

It is possible to designate the factor A(f~v~) in Equation (5.4) as a

frequency term by defining a special lowest usable high frequency (LUF).
Take the LUF as that value of frequency, f~LUF~, where the field
strength of 1000 kW of effective radiated power is 0 dBu (Beckmann,
1958). Combining Equations (30), (5.4), (5.15), (5.16), and (6.2), the
field strength is

E = (E~0~ + 34.8) \[1 - L~T~ - L~i~ - L~R~\] + L~G~ + L~M~ + Y~P~ - 34.8
(6.3)

The reference field is E~o~

E~0~ = 104.8 - 20 log~10~D

L~M~ is the over-the-MUF losses from each end.

Rearranging the algebraic terms, the other factors are:

(6.4)

(6.5)

(6.6)

I~T~ , I~R~ are absorption indices at each end of the path.

f~o~ = f~ob~ + f~H~

L~i~ = A~M~ D~M~ + A~F~ D~F~

D~M~ is the distance along the path supporting an M mode

(Fig.14)

D~F~ is the total distance less D~M~ and the one-half hop

distances at each end.

Using the night minimum absorption index (I = .1); an average

distance factor (α) to replace the number of hops, and the loss

per kilometer function, A~H~ is obtained from the rearrangement

of the ionospheric loss Equation (5.6).

b = .1047 10^7^

(6.7)

I is average absorption index.

6.3 SUMMARY

The short-distance model based on a complete electron density

profile of the D, E, F~1~ , and F~2~ regions with a separate sporadic-E
layer predicts the ray path at any vertical angle of departure.
Propagation losses for multipath propagation, which is essential for
long paths, is considered to be approximated through a multiple
application of the single-hop model. The steps are:

1\. Sample the ionosphere at control areas near the transmitter and
receiver and at the path midpoint.

2\. Use the short-distance propagation model to develop a ray-set table
at each sample location. This table includes propagation distance for
each angle of departure including absorption and over-the-MUF losses.

3\. Combine the transmitting antenna gain pattern with the propagation
ray set for the sample area near the transmitter to determine the most
efficient launch path for a distance of 4000 km. (Path with the lowest
loss rate including transmitter antenna gain.) Note the launch loss
(one-half the losses from the ray-set table plus transmitter antenna
gain) and the corresponding launch distance (one-half the ground wave
range for a single hop).

4\. Repeat #3 above to determine the most efficient intercept path using
the sample area near the receiver and the receiving antenna vertical
gain pattern.

5\. Determine the distance between the ionospheric launch and intercept
areas. (Ground great-circle distance minus the launch and intercept
distance is an adequate approximation.)

6\. Estimate a loss for the distance obtained in #5 above by averaging
the loss rates at the three sample areas and multiply by the distance.

7\. Determine the basic free-space loss and modify by a horizontal
convergence factor.

8\. Estimate ground losses (loss rate for type of intervening terrain
times distance in #5 above).

9\. Use the system loss adjustment factor table.

10\. Combine 3, 4, 6, 7, 8, 9 to estimate the system loss.

**7. HIGH-FREQUENCY SYSTEM PERFORMANCE**

Two methods are used in the program to obtain estimates of the expected
system performance. The first gives performance estimates in terms of
the circuit reliability, which is defined as the fraction of days that
successful communication may be expected at a given hour within the
month at a specific operating frequency. The circuit reliability is
based upon monthly median estimates of all operational parameters and
their distributions and represents only the expected fraction of days in
the month at the given hour that successful communication is expected.
The second method gives estimates of the service probability, which is
defined as the probability of obtaining a predetermined grade of service
or better during a specified percentage of time (Barsis et al., 1961).
It includes estimates of the prediction uncertainties associated with
the operational parameters as a function of the percentage of time the
specified grade of service is required, and it is therefore a measure of
the chance of successful communication with the system employed.

7.1 Circuit Reliability

The primary factor in determining the circuit reliability is the
signal-to-noise ratio, which is directly associated with a grade of
service. The grade of service defines the quality of communication
desired as, for example, the percentage of satisfied observers of
facsimile transmissions. A minimum required signal-to-noise ratio is
associated with the desired grade of service. This ratio is determined
in terms of the class of emissions, modulation index, modulation rate,
and signaling codes, and includes the effects of fading,
error-correcting schemes, noise reducers, optimum modulation and
detection techniques, and diversity schemes. The required
signal-to-noise ratio for a minimum acceptable grade of service must be
specified by the user.

The method for estimating the circuit reliability for a particular path
consists of (1) calculating the maximum frequency that will be supported
by the ionosphere; (2) determining the probability that a specified
frequency will be propagated at a given hour within the month; (3) based
on relationships between the specified system components and the
propagation medium, calculating the distribution of signal-to-noise
ratios from the medians and standard deviations of both signal power and
noise power; (4) determining the probability that the signal-to-noise
ratio will exceed the required signal-to-noise ratio; and (5) combining
the two probabilities to give an estimate of the circuit reliability.

The median value of received signal power P~e~ required for minimum
acceptable grade of service during the month at a given hour is (CCIR,
1988)

P~e~ = F~am~ + R~md~ + B - 204 (dBW) , (7.1)

where

F~am~ = median of the hourly values of radio noise power density F~a~
within a seasonal time block system,

R~md~ = required system signal-to-noise power density ratio,

B = effective receiver bandwidth (10 log~10~b; b in Hz).

The values of F~am~ are given in terms of the effective antenna noise
factor that results from the external noise power available from a
loss-free antenna. Note that the values of F~am~ are representative of
those that would be obtained with a short vertical lossless antenna over
a perfectly conducting plane earth. Use of other antennas requires some
correction because of directivity and polarization; it is not now
accounted for in the program. To account for this, atmospheric noise
source maps are now under study. The available signal-to-noise ratio
R~S/N~ at the receiving antenna terminals is calculated by combining the
received signal power with the received noise power:

R~S/N~ = S - N, (7.2)

where

S = monthly median signal in decibels,

N = monthly median of the hourly median radio noise power in decibels.

The probability q~S/N~ that R~S/N~ exceeds a given level is evaluated by
the chi-square probability functions given by (5.20), where the value of
χ^2^ is related to R~mo~. The median, upper, and lower decile values of
the required signal-to-noise ratio, R~mo~, R~.90~, R~.10~, respectively,
are substituted in (5.21), (5.23), (5.24), and (5.26) for the MUF
values, and the calculation proceed as before. The ratios R~.90~ and
R~.10~ are defined as follows:

where

R~m0~ = median value D~50~ of the required signal-to-noise power ratio,

σ~Nu,l~ = departures of the standard deviation of the radio noise from
the median,

S~u,l~ = departures of the standard deviation of the received signal
from the median.

Values of N and 1.28 σ~N~ are obtained from numerical representations of
the atmospheric, galactic, and man-made noise as described in section 4.
Values of S(med.), S~u~, and S~l~ are obtained by the methods described
in section 7 in terms of the system loss and are shown in tables 7.3 and
7.4.

The probability q~f~ that at least one sky-wave mode will be present to
produce the median available signal-to-noise is described in section 5,
and the combined probabilities gives the circuit reliability

REL = q~f~ q~S/N~ . (7.5)

Therefore, the reliability is based on the available signal or
signal-to-noise ratio and the fraction of days a sky-wave path is
expected to be present. These two factors do not need to be associated
with the same propagation mode, however. The program selects the lowest
loss of all likely modes and combines this with the highest likelihood
of ionospheric support to calculate the circuit reliability.

\(a\) the reliability computation is based (among other things) upon an
estimate of the day-to-day variation in signal level;

\(b\) the semi-empirical determination of these day-to-day variations as
used in this prediction method includes all modes of propagation (see
sec. 7.4), i.e., signal levels associated with specific modes of
propagation ate not separated out;

\(c\) the minimum loss (corresponding essentially to the highest hourly
median signal level observed during the month) is more predictable than
the monthly median of hourly medians, especially when predictions must
be applicable worldwide;

\(d\) the minimum loss for any path is assumed to be associated with any
mode that has a q~f~ of 0.01 or more; and

\(e\) the fraction of days for any propagation mode is assumed to be
approximated by the fraction of days for the most probable mode.

The program lists the individual mode for any frequency and hour that
has the highest reliability, but this particular mode may not be the
mode with the lowest loss or the highest likelihood of ionospheric
support q~f~ used to calculate the circuit reliability. The lowest
useful frequency (LUF) is defined as the minimum frequency at which the
calculated circuit reliability is above a specified level. It has been
common practice to designate the LUF as the minimum frequency with a
circuit reliability of 0.90. An option is available in the program to
specify any reliability level to designate the LUF.

7.2 Service Probability

The service probability Q~t~ is the probability that the grade of
service, g~r~, defined by the required signal-to-noise ratio will be
achieved or exceeded for a specified time availability, q~r~. In other
words, the problem is to assess the probability that the specified grade
of service for a given percentage of the days within the month at some
specified hour will be provided by the calculated available
signal-to-noise ratio. Suppose, for example, that the fraction of days,
q~r~, within the month that the grade of service, g~r~, is desired is
0.95, and that the calculated service probability is 0.70. This means
that, for 0.95 of the days within the month, there is a 0.70 probability
at the particular hour and frequency that the grade of service will be
achieved.

Estimates of the service probability combine statistically the
uncertainties of all values included in the prediction process. The most
important of these are related to the prediction of the received signal
and noise. The required signal-to-noise ratio, R~mo~, that defines the
grade of service, g~r~, has an associated uncertainty because of errors
in estimating equipment performance, short-term noise and signal
variations, multipath propagation, etc. In the absence of sufficient
test data, this uncertainty, expressed as a standard deviation, σ~r~,
has arbitrarily been taken to be equal to 2 dB for all grades of
service.

Prediction uncertainties associated with estimates of the received
signal level or, conversely, the excess system loss, are shown in table
4 and 5. Values represent estimates of the uncertainty, expressed as a
standard deviation in predicting the median, σ~P~, the upper decile
σ~Su~, and the lower decile σ~Sl~ of the excess system loss. These
estimates of prediction uncertainties were obtained from comparisons of
predicted values of received signals with those observed on operating
circuits (R.M. Davis, 1969, private communication).

TABLE 4 EXPECTED EXCESS SYSTEM LOSS (dB)

(Paths Less than 2500 km)

WINTER (NOV., DEC., JAN., FEB.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med
Sl Su │ Med Sl Su │ Med Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╫───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.4 4.3 9.0 │ 9.3 4.3 8.3 │ 10.1 4.6 8.6 │ 9.1 4.5 7.1 │ 9.1
4.2 6.2 │ 9.6 4.6 8.1 │ 9.7 4.6 7.9 │ 9.3 4.2 9.1 ║ 40-45 │

│ 45-50 ║ 9.9 4.7 9.1 │ 9.6 4.6 9.0 │ 11.2 5.2 9.6 │ 9.2 5.1 7.8 │ 9.2
4.4 6.5 │ 10.2 5.2 8.6 │ 10.5 5.2 8.3 │ 9.6 4.5 9.2 ║ 45-50 │

│ 50-55 ║ 10.3 5.1 9.2 │ 10.0 5.0 9.7 │ 12.2 5.9 10.7 │ 9.3 5.7 8.7 │
9.4 4.6 6.9 │ 10.8 5.8 9.1 │ 11.3 5.8 8.7 │ 9.9 4.8 9.3 ║ 50-55 │

│ 55-60 ║ 11.0 5.3 10.0 │ 13.5 6.7 9.6 │ 15.6 8.2 14.6 │ 10.4 5.0 10.6 │
9.8 4.8 7.2 │ 11.7 6.5 9.0 │ 12.0 5.4 10.6 │ 11.0 5.1 9.5 ║ 55-60 │

│ 60-65 ║ 16.0 8.0 13.5 │ 25.0 12.7 13.0 │ 23.0 12.3 23.7 │ 12.6 6.8
20.5 │ 11.0 5.8 8.7 │ 15.8 8.3 14.1 │ 18.8 8.9 17.2 │ 15.6 7.2 9.9 ║
60-65 │

│ 65-70 ║ 15.2 7.7 14.6 │ 24.6 13.5 13.2 │ 21.8 11.8 22.5 │ 11.3 6.0
22.0 │ 10.5 5.4 8.2 │ 13.6 7.8 11.3 │ 16.0 7.8 18.6 │ 14.0 6.8 11.5 ║
65-70 │

│ 70-75 ║ 12.4 6.3 9.4 │ 17.3 8.9 15.2 │ 17.6 9.9 14.3 │ 9.9 5.4 13.9 │
9.5 4.8 7.5 │ 11.5 6.5 10.5 │ 12.0 5.8 14.8 │ 12.0 6.0 9.0 ║ 70-75 │

│ 75-80 ║ 11.0 5.6 9.4 │ 15.6 7.7 8.8 │ 15.2 8.4 10.2 │ 10.5 6.3 10.7 │
9.5 4.7 6.7 │ 10.4 5.4 8.6 │ 10.5 4.9 11.5 │ 10.0 5.4 8.5 ║ 75-80 │

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

EQUINOX (MAR., APR., SEPT., OCT.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med
Sl Su │ Med Sl Su │ Med Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────║───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.5 4.5 10.0 │ 9.6 4.4 11.5 │ 10.6 5.3 9.8 │ 10.0 4.7 9.0 │
9.7 4.5 8.9 │ 10.3 5.0 11.3 │ 10.0 4.8 10.0 │ 10.3 4.7 10.0 ║ 40-45 │

│ 45-50 ║ 10.1 5.0 11.1 │ 10.3 4.8 14.1 │ 12.3 6.6 12.0 │ 11.0 5.4 11.6
│ 10.4 5.0 11.4 │ 11.6 6.0 15.0 │ 11.0 5.6 12.5 │ 11.6 5.4 11.0 ║ 45-50
│

│ 50-55 ║ 10.7 5.6 12.2 │ 11.0 5.2 16.6 │ 14.0 8.0 14.3 │ 12.0 6.2 14.2
│ 11.2 5.6 13.9 │ 13.0 7.0 18.7 │ 12.0 6.4 15.0 │ 13.0 6.2 12.0 ║ 50-55
│

│ 55-60 ║ 11.4 5.7 17.6 │ 13.4 6.4 22.0 │ 16.5 8.3 15.3 │ 14.0 7.6 18.3
│ 11.6 5.6 15.5 │ 13.8 7.5 20.2 │ 14.5 7.7 19.5 │ 15.1 7.4 13.3 ║ 55-60
│

│ 60-65 ║ 15.7 7.7 30.3 │ 20.2 9.5 29.3 │ 26.0 14.0 23.4 │ 18.0 10.6
33.0 │ 16.2 8.3 19.2 │ 18.0 10.3 27.0 │ 19.9 11.3 29.0 │ 24.0 13.0 26.7
║ 60-65 │

│ 65-70 ║ 15.5 8.1 28.0 │ 21.0 11.1 31.0 │ 30.4 18.2 26.9 │ 17.5 10.0
27.9 │ 13.8 7.0 18.0 │ 15.0 8.4 24.0 │ 19.0 11.3 28.8 │ 22.7 11.3 17.5 ║
65-70 │

│ 70-75 ║ 12.3 7.0 21.7 │ 20.0 13.8 20.8 │ 20.6 12.8 20.2 │ 14.1 8.8
18.9 │ 11.6 6.2 14.2 │ 13.0 7.2 18.0 │ 15.0 8.6 22.0 │ 16.0 8.0 16.5 ║
70-75 │

│ 75-80 ║ 10.4 6.1 15.5 │ 11.5 7.5 18.7 │ 16.4 9.7 14.4 │ 12.8 7.5 13.6
│ 10.0 5.4 12.0 │ 11.4 6.2 14.1 │ 11.4 6.4 20.6 │ 12.3 6.3 15.7 ║ 75-80
│

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

SUMMER (MAY, JUNE, JULY, AUG.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med Sl Su │ Med
Sl Su │ Med Sl Su │ Med Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╫───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.8 4.4 9.8 │ 9.6 4.3 9.9 │ 9.8 4.7 9.1 │ 9.7 4.5 7.2 │ 9.8
4.5 8.1 │ 10.6 5.1 8.6 │ 10.5 4.9 9.5 │ 10.3 4.7 10.0 ║ 40-45 │

│ 45-50 ║ 10.6 4.8 10.6 │ 10.2 4.7 12.2 │ 10.6 5.5 10.7 │ 10.5 5.0 8.1 │
10.7 5.0 9.8 │ 12.3 6.2 9.7 │ 12.1 5.9 11.4 │ 11.6 5.4 11.0 ║ 45-50 │

│ 50-55 ║ 11.5 5.3 11.4 │ 10.8 5.1 14.6 │ 11.4 6.3 12.3 │ 11.3 5.5 9.0 │
11.6 5.5 11.6 │ 14.0 7.3 10.8 │ 13.6 6.9 13.4 │ 13.0 6.1 12.0 ║ 50-55 │

│ 55-60 ║ 12.2 5.5 17.8 │ 11.7 5.5 16.1 │ 13.0 7.2 15.6 │ 12.0 6.0 10.6
│ 12.0 5.8 13.5 │ 15.5 8.5 11.0 │ 15.7 7.8 14.1 │ 14.0 6.4 15.0 ║ 55-60
│

│ 60-65 ║ 15.5 7.5 24.5 │ 13.8 5.8 22.7 │ 18.0 10.7 26.0 │ 15.0 7.8 18.8
│ 13.8 6.2 19.9 │ 18.1 9.1 14.2 │ 18.5 10.1 20.5 │ 17.9 8.9 21.1 ║ 60-65
│

│ 65-70 ║ 13.9 6.5 22.1 │ 13.2 6.2 21.8 │ 15.2 9.2 26.7 │ 13.5 6.7 19.5
│ 12.8 5.7 16.6 │ 14.4 7.2 13.8 │ 16.7 7.9 21.9 │ 16.2 7.8 19.0 ║ 65-70
│

│ 70-75 ║ 11.4 5.4 15.6 │ 12.0 5.8 15.0 │ 11.8 6.8 18.2 │ 12.0 6.2 12.0
│ 11.7 5.5 13.3 │ 13.2 6.9 11.1 │ 13.8 7.6 17.8 │ 13.5 6.8 16.0 ║ 70-75
│

│ 75-80 ║ 11.2 5.7 12.8 │ 11.7 6.0 11.3 │ 10.2 5.3 16.8 │ 11.3 5.7 9.5 │
11.2 5.3 12.1 │ 12.8 7.0 10.2 │ 13.2 7.2 11.8 │ 12.8 6.8 12.7 ║ 75-80 │

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

TABLE 4 EXPECTED EXCESS SYSTEM LOSS (dB) - Continued

(Paths Greater than 2500 km)

WINTER (NOV., DEC., JAN., FEB.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su
│ Med. Sl Su │ Med. Sl Su │ Med. Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╫───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.3 4.2 9.1 │ 9.3 4.3 7.8 │ 9.4 4.2 9.0 │ 9.0 4.2 7.3 │ 9.0
4.0 6.5 │ 9.3 4.3 7.7 │ 9.5 4.3 7.6 │ 9.4 4.3 9.2 ║ 40-45 │

│ 45-50 ║ 9.6 4.4 9.3 │ 9.6 4.6 8.0 │ 9.8 4.5 10.4 │ 9.1 4.4 8.2 │ 9.1
4.1 6.7 │ 9.6 4.7 7.8 │ 10.0 4.6 7.6 │ 9.5 4.6 9.4 ║ 45-50 │

│ 50-55 ║ 9.9 4.6 9.5 │ 10.0 5.0 8.2 │ 10.3 4.8 11.9 │ 9.1 4.6 9.2 │ 9.1
4.1 6.9 │ 10.0 5.1 7.9 │ 10.5 4.9 7.6 │ 10.4 4.9 9.6 ║ 50-55 │

│ 55-60 ║ 10.6 4.7 9.6 │ 11.4 5.5 8.3 │ 11.7 5.6 12.4 │ 9.6 5.2 10.4 │
9.4 4.6 7.4 │ 10.7 5.5 7.6 │ 10.8 5.0 7.9 │ 12.5 5.3 9.8 ║ 55-60 │

│ 60-65 ║ 14.3 5.7 11.4 │ 16.0 6.8 9.5 │ 14.9 7.3 14.1 │ 11.0 5.2 15.2 │
9.7 4.1 8.3 │ 13.0 6.7 8.0 │ 13.5 5.6 8.9 │ 19.0 7.8 13.8 ║ 60-65 │

│ 65-70 ║ 14.5 6.5 10.2 │ 14.0 6.3 10.9 │ 16.0 8.3 14.2 │ 10.9 4.4 15.8
│ 9.5 5.1 7.7 │ 12.0 6.2 7.3 │ 13.5 6.1 8.5 │ 18.6 8.6 11.1 ║ 65-70 │

│ 70-80 ║ 11.5 3.4 9.7 │ 13.5 6.5 7.8 │ 13.3 6.8 11.2 │ 9.8 4.8 11.2 │
9.3 4.7 7.1 │ 10.6 5.4 7.4 │ 11.0 5.1 8.6 │ 13.8 6.5 9.2 ║ 70-75 │

│ 80-85 ║ 10.5 5.1 9.2 │ 11.7 5.9 8.1 │ 11.5 5.7 10.2 │ 9.8 4.8 9.2 │
8.8 4.4 7.0 │ 9.9 5.0 7.5 │ 9.8 4.8 8.2 │ 10.6 5.0 9.7 ║ 75-80 │

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

EQUINOX (MAR., APR., SEPT., OCT.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su
│ Med. Sl Su │ Med. Sl Su │ Med. Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╫───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.3 4.1 10.0 │ 9.3 4.1 8.5 │ 9.6 4.2 8.3 │ 9.6 4.4 7.9 │ 9.4
4.3 8.8 │ 10.0 4.5 9.0 │ 9.9 4.3 8.9 │ 9.6 4.2 10.6 ║ 40-45 │

│ 45-50 ║ 9.6 4.2 11.0 │ 9.6 4.2 9.4 │ 10.2 4.5 9.0 │ 10.3 4.9 9.4 │ 9.8
4.6 11.2 │ 11.0 4.9 10.4 │ 10.8 4.6 10.2 │ 10.3 4.4 12.3 ║ 45-50 │

│ 50-55 ║ 9.9 4.4 12.1 │ 9.9 4.3 10.3 │ 10.8 4.8 9.7 │ 11.0 5.4 11.0 │
10.3 5.0 13.7 │ 12.1 5.4 11.9 │ 11.7 5.0 11.6 │ 11.0 4.7 14.0 ║ 50-55 │

│ 55-60 ║ 10.4 4.5 13.2 │ 11.0 4.6 10.6 │ 12.2 5.7 9.8 │ 12.8 6.3 11.2 │
10.8 5.1 14.2 │ 13.0 5.7 12.0 │ 14.4 6.2 12.4 │ 12.4 5.1 17.5 ║ 55-60 │

│ 60-65 ║ 12.9 5.7 15.5 │ 14.2 5.9 10.8 │ 16.6 7.9 11.4 │ 18.0 9.4 15.7
│ 14.0 6.5 17.2 │ 16.5 7.5 14.2 │ 19.0 8.5 16.7 │ 16.9 7.5 22.1 ║ 60-65
│

│ 65-70 ║ 12.7 5.7 14.3 │ 14.6 6.6 10.6 │ 16.7 7.7 13.8 │ 17.1 9.0 13.4
│ 12.5 5.7 17.3 │ 14.0 6.5 13.3 │ 18.5 8.5 16.0 │ 16.3 7.1 17.5 ║ 65-70
│

│ 70-75 ║ 10.8 4.9 13.1 │ 11.9 5.3 9.8 │ 13.1 6.1 10.9 │ 14.7 7.7 12.4 │
10.5 4.9 15.9 │ 12.2 5.8 12.5 │ 14.6 6.9 13.8 │ 12.6 5.6 16.3 ║ 70-75 │

│ 75-80 ║ 10.0 4.8 11.0 │ 10.2 4.6 9.0 │ 11.7 5.7 10.6 │ 12.0 6.5 12.2 │
9.6 5.0 14.8 │ 11.0 5.5 11.0 │ 11.3 5.5 13.7 │ 10.6 4.9 15.6 ║ 75-80 │

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

SUMMER (MAY, JUNE, JULY, AUG.)

┌───────╥────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────┬────────────────╥───────┐

│ G.M. ║ 01-04 LMT │ 04-07 LMT │ 07-10 LMT │ 10-13 LMT │ 13-16 LMT │
16-19 LMT │ 19-22 LMT │ 22-01 LMT ║ G.M. │

│
╟────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╢
│

│ LAT. ║ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su │ Med. Sl Su
│ Med. Sl Su │ Med. Sl Su │ Med. Sl Su ║ LAT. │

├───────╫────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────┼────────────────╫───────┤

│ 00-40 ║ 9.0 4.0 9.0 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 6.4 │ 9.0
4.0 6.4 │ 9.0 4.0 7.6 │ 9.0 4.0 7.6 │ 9.0 4.0 9.0 ║ 00-40 │

│ 40-45 ║ 9.7 4.4 9.1 │ 9.8 4.4 9.1 │ 9.7 4.5 8.1 │ 9.3 4.2 6.9 │ 9.7
4.5 7.6 │ 9.4 4.2 8.1 │ 10.4 4.9 8.2 │ 9.8 4.4 9.6 ║ 40-45 │

│ 45-50 ║ 10.4 4.8 9.2 │ 10.6 4.9 10.6 │ 10.4 5.0 8.6 │ 9.6 4.4 7.5 │
10.4 5.0 8.8 │ 9.8 4.5 8.6 │ 11.9 5.8 8.9 │ 10.6 4.9 10.3 ║ 45-50 │

│ 50-55 ║ 11.2 5.2 9.4 │ 11.4 5.4 12.2 │ 11.2 5.6 9.2 │ 10.0 4.7 8.1 │
11.2 5.6 10.1 │ 10.2 4.8 9.1 │ 13.4 6.7 9.6 │ 11.5 5.4 11.0 ║ 50-55 │

│ 55-60 ║ 11.8 5.4 9.6 │ 12.8 6.2 13.0 │ 12.7 6.5 9.7 │ 10.4 4.9 9.2 │
11.7 5.9 12.3 │ 10.6 5.1 9.7 │ 13.5 6.8 11.8 │ 12.2 5.7 13.4 ║ 55-60 │

│ 60-65 ║ 14.5 6.7 9.8 │ 17.5 8.8 16.8 │ 16.6 9.3 13.8 │ 11.6 5.6 13.1 │
13.2 6.8 17.4 │ 12.2 6.1 11.3 │ 14.5 7.1 17.2 │ 14.7 7.2 18.4 ║ 60-65 │

│ 65-70 ║ 13.4 6.1 10.0 │ 15.3 7.4 16.7 │ 14.9 8.2 14.5 │ 10.9 5.4 13.1
│ 12.3 6.3 14.0 │ 10.7 5.4 10.4 │ 13.4 6.4 15.2 │ 13.5 6.4 15.1 ║ 65-70
│

│ 70-75 ║ 10.9 4.8 8.9 │ 13.6 6.5 11.9 │ 11.9 5.9 11.1 │ 10.3 5.0 10.1 │
11.2 5.5 12.2 │ 10.3 5.1 9.0 │ 11.8 5.8 12.3 │ 11.7 5.3 12.1 ║ 70-75 │

│ 75-80 ║ 10.9 5.3 8.2 │ 12.8 6.0 10.0 │ 11.2 5.5 10.1 │ 10.1 4.8 8.6 │
11.1 5.4 11.4 │ 10.2 5.0 8.2 │ 11.3 5.6 9.9 │ 11.4 5.5 10.3 ║ 75-80 │

└───────╨────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────┴────────────────╨───────┘

TABLE 6. PREDICTION ERRORS IN THE ATMOSPHERIC RADIO NOISE LEVELS

Numerical Coefficients for use in Evaluation of the

Standard Deviation of the [Median]{.underline} Value of the Atmospheric
Radio Noise (σ~FAM~)

DEC - JAN - FEB

LMT A1 A2 A3 A4 A5

00-04 .743379E+00 .184805E+01 -.890621E+00 -.199749E+01 .452507E+01

04-08 -.428242E+00 .302502E+00 .742824E+00 -.160900E+01 .417567E+01

08-12 -.342594E+00 .585453E+00 .177053E+01 -.985735E+00 .445041E+01

12-16 -.109963E+01 -.410156E+00 .272162E+01 -.191082E+00 .466347E+01

16-20 -.225828E+00 .262167E+00 .212719E+00 -.114419E+01 .468956E+01

20-24 .735803E+00 .143984E+01 -.177913E+01 -.191275E+01 .529731E+01

MAR - APR - MAY

00-04 .168882E+00 .653291E+00 .919879E-01 -.640683E+00 .367957E+01

04-08 .136798E+01 .256034E+01 -.222544E+01 -.292737E+01 .497452E+01

08-12 .114881E+01 .282530E+01 -.154909E+01 -.383357E+01 .603194E+01

12-16 .557917E+00 .171872E+01 -.111954E+01 -.284310E+01 .663045E+01

16-20 -.719275E-01 .119484E+01 .787773E+00 -.246763E+01 .451286E+01

20-24 -.104108E+00 .258124E+00 .256638E+00 -.575034E+00 .388959E+01

JUN - JUL - AUG

00-04 .441078E+00 .754692E+00 -.149459E+01 -.123047E+01 .552806E+01

04-08 .179997E+01 .285387E+01 -.427602E+01 -.421217E+01 .702824E+01

08-12 .122894E+01 .264191E+01 -.285556E+01 -.389196E+01 .766424E+01

12-16 .975428E+00 .875660E+00 -.440646E+01 -.204496E+01 .902340E+01

16-20 .147046E+01 .320094E+01 -.152498E+01 -.313552E+01 .587587E+01

20-24 .193233E-01 -.494468E+00 -.143067E+01 .118105E+00 .482582E+01

SEP - OCT - NOV

00-04 .507302E+00 .168124E+01 .292958E-01 -.211237E+01 .387806E+01

04-08 -.111661E-01 .110632E+01 .733434E+00 -.185950E+01 .449589E+01

08-12 .279820E+00 .177786E+01 .746844E+00 -.243937E+01 .494175E+01

12-16 .212440E+00 .130495E+01 -.603455E-02 -.221375E+01 .604906E+01

16-20 .820952E-01 .142235E+01 .685795E+00 -.248459E+01 .444219E+01

20-24 .425376E+00 .118228E+01 -.472298E+00 -.142879E+01 .427191E+01

TABLE 6. PREDICTION ERRORS IN THE ATMOSPHERIC RADIO NOISE LEVELS (CONT.)

Numerical Coefficients for use in Evaluation of the

Standard Deviation of the [Upper Decile]{.underline} of the Atmospheric
Radio Noise (σ~DL~)

DEC - JAN - FEB

LMT A1 A2 A3 A4 A5

00-04 .126904E+00 .268785E-01 -.770942E+00 .226285E-01 .248548E+01

04-08 .614280E+00 .571794E+00 -.235541E+01 -.100525E+01 .404069E+01

08-12 -.119745E+00 -.151771E-01 -.408071E+00 -.438922E+00 .347453E+01

12-16 .248158E+00 .202147E+00 -.149445E+01 -.538425E+00 .379969E+01

16-20 .511612E+00 .347771E+00 -.232743E+01 -.632405E+00 .429346E+01

20-24 -.343668E+00 -.414469E+00 .343420E+00 .494973E+00 .241309E+01

MAR - APR - MAY

00-04 -.794447E-01 -.421282E-01 -.106715E+00 -.450397E+00 .275694E+01

04-08 .400025E+00 .417995E+00 -.156035E+01 -.869965E+00 .369667E+01

08-12 .996029E+00 .112322E+01 -.368354E+01 -.221057E+01 .577247E+01

12-16 .795788E+00 .912118E+00 -.388410E+01 -.251610E+01 .719725E+01

16-20 .416649E+00 .821670E+00 -.199945E+01 -.223434E+01 .521205E+01

20-24 .137420E+00 .572644E+00 -.345254E+00 -.150924E+01 .284754E+01

JUN - JUL - AUG

00-04 .232038E+00 .183979E+00 -.116380E+01 -.550627E+00 .294489E+01

04-08 .686566E+00 .435191E+00 -.290795E+01 -.878487E+00 .455870E+01

08-12 .760254E+00 .494537E+00 -.318959E+01 -.969695E+00 .499249E+01

12-16 .956414E+00 .375093E+00 -.436184E+01 -.729336E+00 .636323E+01

16-20 .680902E+00 .331788E+00 -.308260E+01 -.609270E+00 .496306E+01

20-24 .302133E-01 .200021E+00 -.166200E+00 -.521770E+00 .197852E+01

SEP - OCT - NOV

00-04 -.104326E+00 -.796536E-01 -.663268E-01 -.221128E+00 .244773E+01

04-08 .547742E+00 .505893E+00 -.187325E+01 -.747520E+00 .365203E+01

08-12 .927719E+00 .123641E+01 -.299950E+01 -.222166E+01 .514314E+01

12-16 .403634E+00 .695742E+00 -.198732E+01 -.177995E+01 .532871E+01

16-20 .148801E+00 .170572E+00 -.108537E+01 -.747057E+00 .383926E+01

20-24 .208183E-01 .125692E+00 -.290634E+00 -.662367E+00 .244939E+01

TABLE 6. PREDICTION ERRORS IN THE ATMOSPHERIC RADIO NOISE LEVELS (CONT.)

Numerical coefficients for use in Evaluation of the

Standard Deviation of the [Lower Decile]{.underline} of the Atmospheric
Radio Noise (σ~DU~)

DEC - JAN - FEB

LMT A1 A2 A3 A4 A5

00-04 .586943E+00 .910163E+00 -.144586E+01 -.979273E+00 .320932E+01

04-08 .688151E+00 .149756E+01 -.147183E+01 -.217741E+01 .371525E+01

08-12 .667779E+00 .149891E+01 -.188272E+01 -.231266E+01 .518980E+01

12-16 .637561E+00 .154566E+01 -.159770E+01 -.217261E+01 .513591E+01

16-20 .272473E+00 .134363E+01 -.234189E+00 -.202113E+01 .372761E+01

20-24 .201483E+00 .709483E+00 -.493240E+00 -.963046E+00 .327557E+01

MAR - APR - MAY

00-04 .292770E+00 .588747E+00 -.399142E+00 -.827666E+00 .261711E+01

04-08 .121490E+01 .155130E+01 -.327356E+01 -.227474E+01 .483126E+01

08-12 .888091E+00 .104200E+01 -.334305E+01 -.175927E+01 .627552E+01

12-16 .510005E+00 .166035E+00 -.306647E+01 -.646328E+00 .690508E+01

16-20 .783329E+00 .768176E+00 -.290281E+01 -.155714E+01 .553903E+01

20-24 .239284E+00 .681585E+00 -.304964E+00 -.122301E+01 .269532E+01

JUN - JUL - AUG

00-04 .396174E+00 .566412E+00 -.116124E+01 -.792150E+00 .278700E+01

04-08 .787777E+00 .107465E+01 -.281629E+01 -.192217E+01 .480766E+01

08-12 .476864E+00 .300840E+00 -.257711E+01 -.614125E+00 .541123E+01

12-16 .294015E+00 -.609974E+00 -.290586E+01 .986589E+00 .631989E+01

16-20 .785736E+00 .705832E+00 -.332976E+01 -.124018E+01 .563061E+01

20-24 .576982E+00 .110841E+01 -.134597E+01 -.171088E+01 .270130E+01

SEP - OCT - NOV

00-04 .248494E+00 .485690E+00 -.744257E+00 -.847143E+00 .271592E+01

04-08 .846006E+00 .108394E+01 -.260229E+01 -.170798E+01 .439021E+01

08-12 .426888E+00 .161761E+00 -.255403E+01 -.525122E+00 .576006E+01

12-16 .350895E+00 .363107E-02 -.258306E+01 -.531405E+00 .598093E+01

16-20 .537617E+00 .877538E+00 -.189475E+01 -.172580E+01 .452224E+01

20-24 .160974E-01 .437416E+00 -.768397E-02 -.858681E+00 .262708E+01

Uncertainties in predicting the atmospheric radio noise are given in
CCIR Report 322 (1988) as standard deviations representing the errors in
predicting the median, σ~Fam~, upper decile, σ~Du~, and lower decile,
σ~Dl~. The curves representing the above variables were generated by a
least squares fit with the polynomial (Lucas and Haydon, 1966)

Y(x) = A~1~ + A~2~x + A~3~x^2^ + A~4~x^3^ + A~5~x^4^, (7.1)

where

Y(x) = σ~Fam~, σ~Dl~, or σ~Du~,

x = log~e~f,

f = frequency in megahertz.

The coefficients of the polynomial for each of the variables as a
function of season, hemisphere, and time of day are given in table 6.

As already noted, the time availability, q~r~, refers to the expected
percentage of the hours in the month, used in defining the grade of
service, during which the specified grade of service or better is
achieved. The required grade of service is specified by a required
signal-to-noise ratio and when this ratio is equaled or exceeded, the
grade of service will be achieved. However, in calculating the available
signal-to-noise ratio at the receiver, some allowance must be made for
the variations of the hourly median signal levels and hourly median
noise levels during the specified time period. This allowance is termed
a protection factor (CCIR, 1988). The day-to-day variations in the
hourly median signal level S and noise level N are assumed to be
normally distributed and can be described by the median values and the
deviation D~S~ or D~N~ from the median.

The protection factor, C, in decibels, necessary to provide the required
signal-to-noise ratio, R~m~, for the specified time availability, q~r~,
is determined, with no correlation assumed, from

(7.2)

where

D~N~ = the value in decibels of the median noise power exceeded with a
probability (1 - q~r~),

D~S~ = the value in decibels of the median loss exceeded with a
probability (q~r~).

Values of D~S~ and D~N~ are determined as follows. The standard normal
variate ε, corresponding to a given value of q~r~, is first obtained
from a table in the computer program. Then,

D~N~ = εD~u~ / 1.28 (7.3)

D~S~ = εS~u~ / 1.28 (7.4)

for values of q~r~ ≥ .50 and

D~N~ = εD~l~ / 1.28 (7.5)

D~S~ = εS~l~ / 1.28 (7.6)

for values of q~r~ \< .50

The values of S~u~ (the difference between the excess losses exceeded 16
and 50 percent of the time) and the values of S~l~ (the difference
between the excess losses exceeded 50 and 84 percent of the time) are
given in tables 4 and 5. The upper decile D~u~ and the lower decile D~l~
of the median noise power (atmospheric, galactic, or man-made) are
obtained by the methods described in section 5.

With the values of the required signal-to-noise ratio for the specified
grade of service R~mo~, and the required protection factor for the
specified time availability C, the median value of the expected required
signal power is (CCIR, 1988)

(7.7)

and is the equivalent of (8.1).

The prediction uncertainties are combined into a total uncertainty σ~t~,
on the assumption that the errors are un-correlated, from

where

σ~r~ = uncertainty in predicting the required signal-to-noise ratio
R~m~, standard deviation assumed to be 2 dB,

σ~p~ = standard deviation of estimates of the median received signal
level S,

σ~Fam~ = standard deviation of estimates of the median received noise
level N,

σ~C~ = standard deviation of C and is composed of the standard error in
predicting the radio noise level D~N~ and the signal level D~S~
corresponding to the given time availability; σ~C~ is a function of
σ~Su~ (or σ~Sl~) and σ~Du~ (or σ~Dl~); therefore,

The service probability Q~t~ corresponding to the specified time
availability q~r~ and a given required grade of service g~r~ is now
calculated by the normal probability function.

The above service probability calculation is only valid for frequencies
having a probability of ionospheric support q~f~ equal to one. for
frequencies having a probability of support less than one, the total
service probability would be

SPROB = Q~t~ . q~f~. (7.9)

Figure 8 illustrates the calculations described above that the program
evaluates to obtain estimates of circuit performance based on the
service probability concept. Remember that we are concerned with the
system signal levels where the decile values R~.10~ and R~.90~ are the
reverse of those associated with system losses. The two curves
illustrate the distributions and service probability estimates for two
different time availabilities q~r~. The calculated available
signal-to-noise ratio for the frequency and path is 57.5 dB. The
probability of ionospheric support q~f~ for the frequency and path is
assumed to be equal to one. Note that the higher time availability gives
a lower service probability. When service probability is used as a
prediction of the expected systems performance, the lowest useful
frequency (LUF) may be defined as the minimum frequency at which the
service probability is above a specified level.

7.3 Multipath Evaluation

The computer program contains a provision for calculating the likelihood
of multipath on a given transmission circuit (Lucas and Haydon, 1966).
It is an important consideration for high-speed data transmission
systems where interference due to multipath propagation can cause
intolerable bit error rates (Bailey, 1959).

Multipath occurs when signals of comparable amplitude are propagated
with different delay times via two or more paths with a different number
of hops. It can occur a large percent of the time over a wide frequency
range. The number of modes depends on the relationship of the actual
operating frequency to the path MUF. At frequencies near the MUF,
multipath is very low or entirely absent. As the operating frequency is
decreased relative to the MUF, the likelihood of other modes being
received is increased, and it is therefore desirable to have some
indication of the presence of multipath for selecting operating
frequencies.

If the operator has an adequate choice of frequencies to permit
operation where the likelihood of multipath is low, an adequate
signal-to-noise ratio will suffice to determine circuit performance. If
the reliability or service probability of two frequencies is equal, and
one has a greater likelihood of multipath, then the frequency with the
lowest multipath is preferable. It is also possible to reduce multipath
by antenna design or selection.

The probability of multipath is a function of frequency, path length,
location, local time, and season. The program predicts the probability
of multipath based upon some discrete minimum difference in received
power and maximum tolerable delay time between modes. The probability of
multipath is assumed to be the probability that a second sky-wave mode
will be present to produce a minimum difference in received signal
powers.

The minimum tolerable difference in received power and maximum tolerable
delay time difference between modes are input parameters to the program
and depend on the communication system used.

**8. MAXIMUM USABLE FREQUENCY (MUF) MODEL**

The maximum frequency which will support communications between two
points is an important parameter in radio system design and operation.
Because of the stochastic nature of the ionosphere, it is a very
difficult parameter to determine. This communication analysis program
follows the classical philosophy of determining a MUF based on monthly
median ionospheric condition; i.e., to determine the frequency where the
high and low angle rays merge JF (Junction Frequency) using a predicted
ionosphere and to designate this frequency as MUF. Using the day-to-day
distribution of F~2~MUF (3000) (i.e., MUF at a distance of 3000
kilometers for the F~2~ layer ), the percentage of days a given
frequency will be below the JF of the F~2~ region can be estimated.
Frequencies which will be below the JF 90 percent of the days are
classically called optimum traffic frequencies (FOT) and those below the
JF only 10 percent of the days are called highest probable frequencies
(HPF). For the E and F~1~ layers, the distribution is taken as normal
with a standard deviation of 10 percent of the MUF.

The sampling of the ionosphere to determine the MUF is the same as it is
for the rest of the models. For short paths, only the middle of the
great-circle path is sampled. for long paths, the MUF is determined from
the sample area with the lower ionization at the ends of the path. It
should be noted that in this model communication does not fail at the
MUF, only the signal level starts to sharply decrease as frequency is
increased above the daily MUF.

The MUF calculated this way does not correspond to the maximum observed
frequency (MOF) between two radio systems. The MOF may differ for any of
several reasons: the ionosphere (e.g., foF2) may differ sufficiently
from the median, the signal may be caused by an over-the-MUF mode or
some other process such as ground scatter, or the signal may result from
the sporadic-E mode. The 50 percent sporadic-E MUF is available as an
optional output. The predicted MUF corresponds to a junction frequency,
and is not normally measured. Note that no account of ionospheric storms
nor of operational characteristics is taken in the calculation of the
MUF. finally, note that the MUF is not the maximum frequency at which
the program will predict a mode existing between two points on the
earth\'s surfaces. High-angle rays, over-the-MUF modes, and possible
E~s~ modes will predict existing frequencies over the predicted MUF.

8.1 Geometry of the MUF (JF)

A straightforward method to find the MUF would be simply to evaluate the
complete electron density profile, integrate to find the corresponding
virtual heights, and then use some search technique to find the
predicted junction frequency. In order to simplify the calculations, an
alternate procedure has been devised which is basically an extension of
Lucas and Haydon (1966). The major insight is to devise a method of
finding the junction frequency curve as a function of the ionospheric
distance, and frequency. The basic parameter is X~T~, the ratio of the
tangent frequency f~t~ to the critical frequency of the layer (e.g.,
foF1). The term tangent frequency comes from the manual method of
transforming a vertical incidence ionogram into an oblique ionogram for
a given distance using a transmission curve. The JF curve is
approximated in three steps. First the equivalent vertical sounding
frequency corresponding to the JF, F~t~, is estimated using the
equation:

(8.1)

where

X~t~ is the ratio of f~t~ to the layer critical frequency,

f~t~ is to correspond with the tangent point on the ionogram when using
the transmission curves (NBS, 1948),

y~m~ is the semi-thickness of the layer,

h~m~ is the height of maximum ionization of the layer,

α is an adjustment factor (α = 0.5).

For α = 1, Equation (8.1) is the equation for the point on the ionogram
corresponding to minimum time delay for propagation over a spherical
earth and in a flat ionosphere. When the corrected Martyn\'s theorem
(see Section 3.7.4) is used, X~t~ starts at the flat ionosphere value
for zero distance and approaches the JF value for longer distances. A
value of α = 0.5 approximates the JF line somewhat better with the
ionosphere used in the program. For the F~2~ layer, a correction to X~t~
is used to force the MUF to approach the zero MUF as a function of
distance:

(8.2)

where

X~t~ is modified (i.e., as in FORTRAN),

A = -1. + 1./X~t~,

B = 9.5,

D = ground distance,

D~o~ = base distance (2000 km).

The JF curve given by Equation (8.2) with force the curve to rise (or
fall) much more sharply than the actual curve, thus giving a more
conservative estimate of the MUF. Note that the modified X~t~
approaches 1. at zero distance and approaches the original value of X~t~
at D~o~ .

The distinction between long paths and short paths exists for the MUF
calculation also. For long paths, the MUF is that corresponding to an
elevation angle of zero or the minimum angle desired. An electron
density profile is generated, X~t~ is calculated for each layer, table
lookup is used to find the true height, and the secant law is used to
find the MUF (no correction factor as the true height is used). The
relevant equations are:

f~t~ = X~t~ f~c~ , (8.3)

where

f~t~ = vertical tangent frequency,

X~t~ = is given by Equation (57),

f~c~ = zero MUF,

= foE for E layer,

= foF1 for F1 layer,

= foF2 + 1/2 f~H~ for the F2 layer,

f~H~ = gyrofrequency.

For the short path, the mode structure must be determined. This implies
calculation of the elevation angle, and thus use of the corrected form
of Martyn\'s theorem. The latter implies an integration of the electron
density profile and an iteration for the virtual height. For the E
layer, the correction to Martyn\'s theorem is small and may be omitted.
The equations are:

(8.4)

where

N = number of hops. (\[ \] here implies truncation and the first ∆ is
zero or minimum takeoff angle ∆),

D = ground distance,

a = earth radius,

φ\' = angle at virtual height h\',

ψ = hop distance angle,

φ = angle at true height h^T^,

f~t~ = vertical tangent frequency.

For the E layer, using the parameters ymE = 20, hmE = 110 (which can be
changed by user via input), the value of X~t~ is

X~t~ = .957,

which gives, using the electron density profile described in Section 3,

h~T~ = 104 as true height, and

h\' = 130 as virtual height.

For the F1 and F2 layers, Equation (8.3) and (8.4) are used to find an
initial (h\' , ∆) pair. Then the virtual height is corrected (as for
corrected Martyn\'s theorem) and a new ∆ and MUF is found. This
corrective procedure is repeated until the change is MUF is less than a
specified value or until a maximum number of iterations have been
performed (only one repetition normally results).

The sporadic-E MUF is found using the predicted foEs and the secant
factor with the true height and virtual height taken as equal (normally
110 km in the program).

**9. ACKNOWLEDGEMENTS**

This report is the result of the efforts of many groups and individuals.
It is a technical description of ICEPAC with orientation toward the PC
computer use. The wording is sometimes an extraction of reports
describing the scientific basis for this program. In regard to
individuals, the following list may not include everyone, but hopefully
the majority will be mentioned. The author wishes to acknowledge the
contributions of Mr. J. Washburn, Mr. A.F. Barghausen, Dr. Mark T. Ma,
Mr. D.H. Zacharisen, Dr. E.L. Crow, Mr. G.W. Haydon, Mr. D.L. Lucas, Mr.
J.L. Lloyd, Mr. G.R. Hand, Dr. A.D. Spaulding, Dr. R.K. Rosich, Mr. L.R.
Teeters, Mr. R.M. Davis Jr., and many others who have been instrumental
in the evolution of the propagation programs and documentation.