NTIA Report 99-368

Medium Frequency Propagation

Prediction Techniques and Antenna

Modeling for Intelligent Transportation

Systems (ITS) Broadcast Applications

Nicholas DeMinco

U.S. DEPARTMENT OF COMMERCE

William M. Daley, Secretary

Larry Irving, Assistant Secretary

for Communications and Information

August 1999
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iii

CONTENTS

Page

FIGURES .................................................................... v

ABSTRACT .................................................................. 1

1. INTRODUCTION ........................................................... 1

2. BASIC CONCEPTS IN RADIO-WAVE PROPAGATION .......................... 3

3. GROUND-WAVE PROPAGATION ............................................ 4

 3.1 SPECIFIC GROUND-WAVE PROPAGATION MODELS ..................... 8

4. SKY-WAVE PROPAGATION ............................................... 20

5. ANTENNA MODELING .................................................... 29

6. SYSTEM PERFORMANCE CALCULATIONS AND PREDICTIONS ................ 42

7. CONCLUSION ............................................................ 57

8. REFERENCES ............................................................ 61
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v

FIGURES

Page

Figure 1. Spherical smooth-Earth geometry. ...................................... 10

Figure 2. Spherical smooth-Earth mixed-path geometry. ............................. 15

Figure 3. Geometry for integral equation irregular-Earth model along the great circle path. . . 19

Figure 4. Elevation plane pattern for a quarter-wave monopole on a ground screen. ........ 22

Figure 5. FCC/Region 2 sky-wave curve of median field strength versus distance .......... 24

Figure 6. ITU sky-wave curves of median field strength for several values of geomagnetic

 latitiude at 1000 kHz. ................................................. 27

Figure 7. Wang sky-wave curves of median field strength for several values of geomagnetic

 latitude. ........................................................... 28

Figure 8. Space wave and unattenuated surface wave components of the ground wave for a

 vertical dipole at the Earth’s surface for average ground conditions at 760 kHz. ... 31

Figure 9. Elevation plane pattern for the space wave antenna power gain for the DGPS 

 transmitter site antenna for average ground conditions at 300 kHz. ............. 34

Figure 10. Electric field strength antenna patterns of the space wave and the surface wave

 for the DGPS transmitter site for average ground conditions at 300 kHz. ....... 35

Figure 11. Power gain antenna pattern in the elevation plane for the space wave for the

 DGPS one meter whip mounted in the center of the van roof for average 

 ground conditions at 300 kHz. ......................................... 36

Figure 12. Power gain antenna pattern in the elevation plane for the space wave for the

 one meter whip antenna mounted in the center of the van roof for average ground

 conditions at 760 kHz. ............................................... 37

Figure 13. Power gain antenna pattern in the elevation plane for a quarter-wave monopole 

 antenna on a ground screen for average ground conditions at 760 kHz. ......... 38
vi

Figure 14. Equivalent gain antenna pattern for station KTLK nighttime antenna configuration

 versus azimuth angle for average ground conditions at 760 kHz. .............. 39

Figure 15. Equivalent gain antenna pattern for station KTLK daytime antenna versus

 azimuth angle for average ground conditions at 760 kHz. .................... 40

Figure 16. Power gain antenna pattern in the elevation plane on azimuth beam maximum

 For the KTLK nighttime antenna for average ground conditions at 760 kHz. .... 41

Figure 17. Power gain antenna pattern in the elevation plane on azimuth beam maximum 

 for the KTLK daytime antenna for average ground conditions at 760 kHz. ...... 41

Figure 18. Predicted KTLK day coverage contours (dBuV/m) using the smooth-Earth

 model. ............................................................ 44

Figure 19. Predicted KTLK day coverage contours (dBuV/m) using the smooth-Earth

 mixed-path model. .................................................. 45

Figure 20. Predicted KTLK night coverage contours (dBuV/m) using the smooth-Earth

 model. ............................................................ 46

Figure 21. Predicted KTLK night coverage contours (dBuV/m) using the smooth-Earth

 mixed-path model. .................................................. 47

Figure 22. Electric field strength vs distance from station KTLK at nighttime on 330 degree

 radial using the smooth-Earth model. ................................... 49

Figure 23. Electric field strength vs distance from station KTLK at nightttime on 330 degree

 radial using the smooth-Earth mixed-path model. .......................... 50

Figure 24. Comparison of measured and predicted DGPS field strength for the Appleton to

 Pendleton path. ..................................................... 51

Figure 25. Terrain contour for path from Appleton to Pendleton, Washington. ............ 52

Figure 26. Comparison of measured and predicted data for Canyonlands, Utah path at 

 520 kHz. ......................................................... 53

Figure 27. Comparison of measured and predicted data for Canyonlands, Utah path at 

 1618 kHz. ........................................................ 54
vii

Figure 28. Terrain contour for Canyonlands, Utah measurement site. ................... 55

Figure 29. Contours of electric-field strength in dBuV/m from the Appleton, WA DGPS

 site at nighttime (f=300 kHz). ......................................... 56

Figure 30. Contours of electric-field strength in dBuV/m from the Appleton, WA DGPS

 site at daytime (f=300 kHz). .......................................... 57
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The author is with the Institute for Telecommunication Sciences, National Telecommunication *

and Information Administration, U.S. Department of Commerce, Boulder, CO 80303.

MEDIUM FREQUENCY PROPAGATION PREDICTION TECHNIQUES AND 

ANTENNA MODELING FOR INTELLIGENT TRANSPORTATION SYSTEMS (ITS)

BROADCAST APPLICATIONS

Nicholas DeMinco*

This paper discusses the basic aspects of radio-wave propagation and antenna

modeling in the medium frequency (MF) band. This band covers the frequencies of

300 to 3000 kHz. More specifically, we are concerned with the ground wave and the

sky wave in the 300 kHz to 1705 kHz band. The sky wave models described in this

paper are valid from 150 kHz to 1705 kHz. The ground wave models described in

this paper are valid from 10 kHz to 30 MHz. The AM Broadcast band of 535 to 1605

kHz is in this band and is planned to be used in the Advanced Traveler Information

Systems (ATIS) of Intelligent Transportation Systems (ITS) for rural travelers. This

system would provide information such as road conditions, road hazards, weather,

and incident reporting. The 285 kHz to 325 kHz band is presently being used for a

differential correction signal in another application of ITS called the Differential

Global Positioning System (DGPS) that will be used for precision location of

vehicles. The propagation of radio waves in this band depends on both a ground

wave and a sky wave and is quite different from propagation at any other frequency.

Antenna modeling in this band is also quite unlike that in other bands. This paper

describes radio wave propagation together with antenna modeling in this frequency

band so that a better understanding of the phenomena can be obtained for use in

design and application of ITS subsystems. The models described here can be used

for designing systems and making performance predictions for both of these ITS

applications and any other systems that operate in this band. The paper contains

descriptions of both sky-wave and ground-wave propagation models in addition to

the methodology used to analyze antennas that operate in this band. A method of

calculating and normalizing antenna gain for MF systems computations is also

discussed. Some comparisons of measured and predicted data are also contained in

the descriptions.

Keywords: radio-wave propagation, antennas, communications, intelligent transportation systems

1. INTRODUCTION

This paper contains discussions about radio-wave propagation and antenna analysis in the medium

frequency (MF) band and how analysis methods differ from those used in other frequency bands.

The MF band covers the frequencies of 300 to 3000 kHz. Actually the sky-wave propagation models

are also valid down to 150 kHz, and the ground-wave model is valid down to 10 kHz. The paper

specifically addresses two applications of Intelligent Transportation Systems, but this material is
2

useful for other systems that operate in this band. Radio-wave propagation prediction models, that

can be used for engineering analyses of communication systems, determine the basic transmission

loss between a communications transmitter and receiver. This knowledge is necessary for the

prediction of the behavior of separate functional communication subsystems of the Intelligent

Transportation System (ITS) and other systems operating in this band. This knowledge can be used

for planning, architectural evaluation, system performance prediction, system design and testing,

interference resolution, and standards development. The AM Broadcast band of 535 to 1605 kHz

is in this band and is planned to be used in the Advanced Traveler Information Systems (ATIS) of

ITS for rural travelers. This system would provide information like road conditions, road hazards,

weather, and incident reporting. The subsystems of ITS use computer and telecommunications

technology to provide information to travelers about road and transit travel conditions and also

provide information to monitor, guide, and control the operation of vehicles. ITS can improve

safety, reduce congestion, enhance mobility, minimize environmental impact, save energy, and

promote economic productivity on the Nation's transportation system. The 285 kHz to 325 kHz

band is presently being used for a differential correction signal in another application of ITS called

the Differential Global Positioning System (DGPS) that will be used for precision location of

vehicles. The DGPS signals in this band are used to differentially correct the location determined

by the Global Positioning System (GPS) navigation system. Each DGPS site processes information

from the GPS satellite signal to calculate the location. The DGPS site then compares this location

to its exact fixed location and sends out a correction signal at 300 kHz for others to obtain a more

precise location calculation than originally available from the L-band GPS satellite signals.

Propagation models that are used to analyze radio communication systems require sound engineering

judgement in their use for a particular analysis. For the MF band of 300 kHz to 3000 kHz both the

ground wave and the sky wave must be considered. The propagation of radio waves in this band

depends on both a ground wave and a sky wave and is quite different from propagation at any other

frequency. Antenna modeling in this band is also quite unlike that in other bands. The expected

sky-wave signal combined with the ground-wave signal may be compared with the expected radio

noise environment (consisting of atmospheric, galactic, and man-made noise components) to predict

the likelihood that the communication link will operate satisfactorily. The presence of the sky wave

at night could create potential interference problems between distant stations on the same frequency

or frequencies that are near each other. The sky-wave models provide some means of estimating the

expected field strengths of signals to assist in frequency allocation and to avoid potential interference

problems. Free space loss cannot be used to characterize radio-wave propagation at these

frequencies. Other propagation models must be used to predict radio-wave propagation correctly.

The ground-wave prediction methods presented here are valid from 10 kHz to 30 MHz. 

The sky-wave propagation loss prediction methods discussed in this paper are valid from 150 kHz

to 1705 kHz. Above 1705 kHz and below 150 kHz there are different sky-wave propagation

phenomena that take place and other methods must be used which are not included in this paper,

since our current attention focuses on frequencies around 300 kHz and the AM broadcast band. 

This paper will describe analysis techniques and propagation models used in this frequency band and

will give examples of comparisons of predictions and measurements. Use of the appropriate antenna
T  V V T  HUD

 

3

(1)

(2)

gains with the propagation models is also essential to accurate performance prediction. This paper

will also explain how antenna gains are determined at these MF frequencies to predict system

performance correctly. A method of calculating and normalizing antenna gain for systems

computations has been proposed. Antenna modeling examples will be presented for the average

frequency of 300 kHz used for DGPS, and the 760 kHz test frequency used for the AM subcarrier

Advanced Traveler Information Systems (ATIS) test program. 

2. BASIC CONCEPTS IN RADIO-WAVE PROPAGATION

In all of the radio-wave propagation concepts that are discussed, it is assumed that the propagation

is taking place in the far field of the transmitter and receiver antennas. If a point source radiates

power with a transmitter antenna that has a directional gain, then the received power for a receiver

antenna is given by the following [1-4]:

where P and P are the received and transmitted powers respectively in dBm, and G and G are the r t r t

receiver and transmitter antenna gains respectively in dB. Lfsb is the free-space basic transmission

loss in dB and is given by the following [5]:

where f is the frequency of the radio wave in GHz and r is the distance between the transmitter GHz

and receiver in meters. Free-space loss is a theoretical reference or limit, against which the actual

basic transmission loss L can be compared or expressed as a ratio. b

Equation (1) can be used to calculate the received power for a transmitter and receiver located in free

space. Free space is a highly idealized environment and is not the situation for ITS propagation

channels. In the ITS environment there is at least a conducting ground to consider and in many

situations the wave may have to propagate through a very complicated environment. For example,

the energy may have to propagate: through or be reflected from precipitation, through an atmosphere

with oxygen and water vapor, through buildings, around corners, down side streets, into and around

wooded areas, and over rough terrain. For radio-wave propagation at MF the loss due to

precipitation or atmospheric absorption by water vapor or oxygen is negligible, but the signal must

still diffract around obstacles in the environment. Radio-wave propagation at frequencies between

150 kHz and 1705 kHz depends on both a ground wave and a sky wave. Neither of these two wavepropagation phenomena behave like free-space loss. In this type of environment the received signal

can be expressed by the simple equation given in (1) if the free-space basic transmission loss Lfsb
4

were replaced by the basic transmission loss L computed by propagation models that take all of b

these propagation effects into account. Basic transmission loss is the transmission loss that would

occur if the transmitter and receiver antennas were replaced by ideal isotropic loss-free antennas with

the same polarization as the real antennas [3,4]. 

This paper describes the different types of radio-wave propagation phenomena and antenna behavior

that must be considered for general radio-wave propagation models in the roadway environment in

the MF band. Radio-wave propagation prediction models generally compute basic transmission loss

L which is combined with antenna gains and transmitter power to perform engineering analyses of b

communication systems.

3. GROUND-WAVE PROPAGATION

The ground-wave signal can be determined using one of several models [6,7] that specifically

address the propagation phenomena at these frequencies. One general ground-wave model [6]

computes propagation loss, electric field strength, received power, noise, received signal-to-noise

power ratio, and antenna factors over lossy Earth. The smooth-Earth and irregular-Earth (terrain

dependent) propagation loss prediction methods within this model can be used over either

homogeneous or mixed paths. This model combines three propagation loss prediction methods for

both smooth and irregular Earth, and an antenna algorithm into a single analysis tool. The

propagation loss prediction methods for the ground-wave model compute basic transmission loss

and are valid from 10 kHz to 30 MHz. A model that incorporates the sky wave with the ground

wave [7] specifically addresses the 150 kHz to 1705 kHz frequency range. The frequency limits of

this model have been set by the valid frequency range of the sky-wave model. This model was

previously available on a mainframe computer but now operates in a Windows environment on a PC.

The ground wave includes the direct line-of-sight space wave, the ground-reflected wave, and the

Norton surface wave that diffracts around the curved Earth. The Norton surface wave will hereafter

be referred to as a surface wave in this paper. Propagation of the ground wave depends on the

relative geometry of the transmitter and receiver location and antenna heights. The radio wave

propagates primarily as a surface wave when both the transmitter and receiver are near the Earth in

wavelengths, because the direct and ground-reflected waves in the space wave cancel each other and

as a result the surface wave is the only wave that is left. This cancellation is a result of the fact that

the elevation angle is zero and the two waves (direct and reflected) are equal amplitude and opposite

in phase. This is the condition that exists for the MF band. The surface wave is predominantly

vertically polarized, since the ground conductivity effectively shorts out most of the horizontal

electric field component. What is left of the horizontal component is attenuated at a rate many times

that for the vertical component of the field. When one or both antennas are elevated above the

ground to a significant height with respect to a wavelength, the space wave predominates.

The ground-wave propagation phenomena at these frequencies are basically deterministic processes.

The noise, however, is a stochastic process. The surface wave propagates along and is guided by

the Earth's surface. This is similar to the way that an electromagnetic wave is guided along a


 	  	

 	

5

(3)

(4)

(5)

transmission line. Charges are induced in the Earth by the surface wave. These charges travel with

the surface wave and create a current in the Earth. The Earth carrying this current can be represented

by a leaky capacitor (a resistance shunted by a capacitive reactance). The characteristics of the Earth

as a conductor can therefore be represented by this equivalent parallel RC circuit, where the Earth's

conductivity can be simulated with a resistor and the Earth's dielectric constant by a capacitor. As

the surface wave passes over the surface of the Earth, it is attenuated as a result of the energy

absorbed by the Earth due to the power loss resulting from the current flowing through the Earth's

resistance. Energy is taken from the surface wave to supply the losses in the ground, and the

attenuation of this wave is directly affected by the ground constants of the Earth along which it

travels [8]. 

The attenuation function is the ratio of the electric field from a short dipole over the lossy Earth's

surface to that field from the same short dipole located on a flat perfectly conducting surface. At

distances within the line of sight the surface wave field strength, E, is given by [8]: 

where E is the electric field strength of the surface wave at the surface of the Earth at a small o 

distance on the order of a few wavelengths away from the transmitting antenna (the Earth's losses

are negligible at this small distance, so this could also be considered the field from the short dipole

located on a flat perfectly conducting surface), d is the distance between the transmitting antenna and

reception point, and A is the flat-Earth attenuation function that takes into account the ground losses.

The exact equation derived by Norton [10] for the flat-Earth attenuation function is described later

in this paper. Norton further simplified this exact equation into a form that is more amenable to

calculation. These simplified expressions for the Norton approximations to the flat-Earth attenuation

[9,10] function can be easily implemented on a programmable calculator. They are reasonably

accurate for line-of-sight propagation [11]. 

For p <4.5 and all b: o 

For p >4.5 and all b: o 
 



 	  



  

U

U

 	  



6

(6)

(7)

where for vertical polarization:

and for horizontal polarization:

where is the conductivity of the Earth in siemens per meter, and  is the relative permittivity of r

the Earth. The field strength at this small distance is directly proportional to the square root of

power radiated by the transmitter and the directivity of the antenna in the horizontal and vertical

planes. If the antenna is non-directional in the horizontal plane and has a vertical directional pattern

that is proportional to the cosine of the elevation angle (this corresponds to a short vertical antenna),

then the electric field at one kilometer for an effective radiated power of one kilowatt is 300 mV/m

[8]. The flat-Earth attenuation function A is dependent upon frequency, distance, and the ground

constants of the Earth along which the wave is traveling. A numerical distance, p (equations (6) o

and (7) above) can be computed that is a function of frequency, ground constants, and distance in

wavelengths. If the numerical distance is less than one, then the attenuation function is very close

to one, and as a result for distances close to the transmitting antenna, the losses in the Earth have

very little effect on the electric-field strength of the surface wave. In this region, the electric field

strength is inversely proportional to distance. For situations where the numerical distance becomes

greater than unity, the attenuation function decreases in magnitude rapidly. When the numerical

distance becomes greater than 10, the attenuation factor is also inversely proportional to distance.

The combination of the attenuation factor and the unattenuated electric field both being inversely

proportional to distance results in the electric field strength of the surface wave being inversely

proportional to the square of the distance when the numerical distance is greater than 10.

At lower HF frequencies, AM broadcast (medium frequencies), and lower frequencies in the LF band

(below 300 kHz), the Earth can be regarded as being purely resistive in nature. The equivalent

circuit of the Earth is still a resistor of resistance R and capacitor of capacitance C in parallel.

However, the Earth is predominantly resistive at these frequencies because of the fact that more

current flows through the resistance, because R<<1/C, so under these conditions the resistor has

the major effect. The attenuation factor is then primarily dependent on the conductivity of the Earth

and the frequency. For frequencies above about 10 MHz, the impedance represented by the ground
 	

    



7

(8)

(9)

(10)

is primarily capacitive, so the attenuation factor for the surface wave at a given physical distance is

determined by the dielectric constant of the Earth and the frequency. 

The attenuation of the surface wave is determined by the average values of the Earth conductivity

and dielectric constant down to a depth to which the ground currents penetrate and still maintain an

appreciable magnitude. This is similar to a skin depth phenomenon in a good conductor. The depth

of penetration of the surface wave currents depends upon frequency, dielectric constant, and

conductivity. This ranges from a fraction of a meter at the highest frequencies for HF

communications to tens of meters at AM broadcast and lower frequencies. For this reason

propagation at the lower frequencies is not particularly dependent on conditions at the actual surface

of the ground. Therefore, a recent rainfall would not significantly affect propagation at MF and LF

frequencies. 

The depth of penetration of an electromagnetic wave into the surface of the Earth depends upon the

frequency and ground constants of the Earth. The electric field strength at a distance z below the

surface of the Earth is given by [12]:

where E is the electric field intensity at the surface of the Earth, z is the depth below the surface of o

the Earth, and  is the attenuation per meter of the electric field intensity.

The attenuation per meter  is given by:

where  is the angular frequency and is equal to 2 f, f is radio frequency in Hertz, μ is the

magnetic permeability of the Earth μ = μ x 4 x 10 henries/meter, μ is the relative permeability, r r

-7 

 = the permittivity of the Earth =  (8.85 x 10 ) farads per meter,  is the relative permittivity of r r 

-12

the Earth, and is the conductivity of the Earth in siemens per meter.

The distance the wave must travel in a lossy medium to reduce its amplitude to e =.368 of its value -1

at the surface is  = 1/ meters and is called the skin depth of the lossy medium. For other values

of attenuation of the electric field, r=e , one can use  to determine the distance z below the surface -z

where the electric field is attenuated to that ratio r. The ratio r is always less than or equal to 1. The

distance z is given by:
8

where lnr is the natural logarithm of r. An example is where f=300 kHz, μ = 1 for a nonmagnetic r

Earth,  = 15 for average ground, = .005 for average ground. The attenuation  is calculated as r

.0751 per meter and  is calculated as 1/ as 13.32 meters.

This is the distance at which the electric field is e or .368 (36.8 percent) of its value at the surface -1

of the Earth. If the distance z at which the electric field is .1 (10 percent) of its value at the surface

is desired, then ln r is ln (.1)= -2.3026, and  = .0751, so z= -1/ (-2.3026) = 30.66 meters. Table

1 gives the depth of penetration for f= 300 kHz.

Table 1. Depth of Penetration in the Earth for Different Ground Conditions

Conductivity Permittivity Media Type Alpha () Skin Depth () for r=e =.368 -1

(siemens/m) (no units) (per meter) (meters)

0.001 4.0 poor ground 0.0333 30.04

0.005 15.0 avg. ground 0.0751 13.32

0.020 25.0 good ground 0.1523 06.57

0.010 81.0 fresh water 0.1017 09.83

5.000 81.0 sea water 2.4330 00.41

3.1 Specific Ground-wave Propagation Models

3.1.1 The Smooth-Earth Model

The simplified expressions for field strength of a ground wave described previously in equations (3)

through (7) are only valid for line-of-sight radio-wave propagation. The antennas must also be

located on the ground or very near it in distance with respect to a wavelength. They are the

simplified expressions for the Norton [10] approximation to the Sommerfeld flat-Earth attenuation

function. Sommerfeld originally solved the problem of radiation over a flat lossy Earth for a short

current element in 1909 [13]. Later work by Sommerfeld and many others resulted in an extensive

numerical evaluation of the flat-Earth attenuation function. One of the more prominent evaluations

of the flat-Earth attenuation function was that due to Norton [9,10]. When the geometry of the

receiver antenna, transmitter antenna, and the Earth are such that a flat-Earth attenuation function

[14] is no longer valid, then other methods must be used. In this paper a smooth-Earth model is

described that uses a variety of algorithms to account for this geometry. The smooth-Earth model

described in this paper is that developed by L.A. Berry [15]. It is also described in more detail in

[6,7,14,16,17,18]. The following six computation techniques are used in this smooth-Earth model

to account for all possible propagation geometries: flat-Earth attenuation function, flat-Earth

attenuation function with a curvature correction, the power series expansion, the residue series

calculation, geometric optics, and numerical integration of the full-wave theory. The appropriate

technique is selected according to the relative geometry of the transmitter and receiver antenna

heights and locations with respect to the Earth. Diffraction is included where necessary in the last

five of these techniques. 
E(d)

9.487 PE A()

d

  ' 

 % 

  

 %  



  



  

9

(11)

(12)

(13)

(14)

(15)

(16)

The details of the smooth-Earth model equations are given in Berry [15] and Stewart, et al. [16].

Some of the basic equations will be summarized in this section. Figure 1 describes the geometry for

a spherical smooth Earth. The flat-Earth attenuation function is used for line-of-sight propagation

when the path is short and the Earth can be assumed to be flat [19]. If in addition, the transmitter and

receiver antennas are near the Earth with respect to a wavelength, then the flat-Earth attenuation

function [19] is used and the electric-field strength E(d) in V/m is given by [15,16,19,20]:

where PE is the effective radiated power in watts, d is the distance between the transmitter and

receiver in kilometers, and A() is the flat-Earth attenuation function. The equations (3) through

(7) given earlier for the flat-Earth attenuation function A are an approximation to A(). The

following equations are the more precise expressions used to predict ground-wave loss [13,19]:

where d=a (Figure 1) is the great circle path distance between the transmitter and the receiver base

locations, h and h are the heights of the transmitter and receiver antennas respectively, D is the line 1 2

of sight distance between the transmitter and the receiver, and k=2/, and erfc() is the

complementary error function [21].

The surface impedance of the ground is given by . If the polarization is vertical, then is:

If the polarization is horizontal, then is: 
   



 	 	 	  	

10

Figure 1. Spherical smooth-Earth geometry.

(17)

(18)

where  is the relative dielectric constant of ground,  is the conductivity of the ground in siemens g g

per meter,  is the permittivity of free space, 8.85 x 10 farads per meter,  = 2 f, and f is the o

-12

operating frequency in Hertz. The surface impedance, , is a function of the ground constants of

the Earth's surface. 

When the transmitter and receiver antenna locations are high enough such that an observer

positioned at either the receiver or transmitter location is well above the radio horizon when viewed

from the other, the field strength computation involves the use of geometrical optics [15,16]. The

electric field E in V/m is given by:

where R is the appropriate ground reflection coefficient for vertical or horizontal polarization. The g

direct line-of-sight distance D between the transmitter and receiver antennas is given by [15,16]:
    

 

%




	

  

 

       

 






	






	

11

(19)

(20)

(21)

(22)

(23)

(24)

The radius of the Earth is denoted as a. Closed form solutions for D and D are not possible, but 1 2

Stewart et al. [16] gives details on calculating these distances using an iterative method.

When the receiving antenna is near the radio horizon of the transmitting antenna, but not beyond it,

then the field depends on diffraction effects in addition to the direct wave, and in this case the

computation is performed by numerical integration of the full-wave theory integral [15]. The

expression for calculating the electric field using the full-wave theory integral is [15,16,19]:

where v=(ka/2) , x=v, and q=-jv.  is a contour enclosing the poles of F (q,t). 1/3

I

H ( h ) and H ( h ) are height gain functions given by: 1 1 2 2

where y=kh /v, and W (t) denotes the differential of W (t) with respect to t. The functions W (t) i1 1 n

are Airy functions [19] that satisfy the differential equation: 

where the contour  is taken as the straight line segment from e to the origin and out along the 1

j2%/3
  

 



	 % 	



 




 %

12

(25)

(26)

(27)

(28)

 real axis to  , and contour  is taken as the straight line segment from e to the origin and out 2

-j2%/3

along the real axis to .

The poles of the function F (q,t) occur at points t and they satisfy the differential equation [19,22]: I i 

For long paths, the Earth cannot be considered flat. If, in addition, the geometry is such that a

straight line connecting the transmitter and receiver antennas intersects the curved Earth so that the

transmitter and receiver antennas are beyond line of sight of each other and propagation is beyond

the horizon, then the full-wave theory integral must be evaluated using the residue series [19,22].

It is necessary to calculate the series by summing the residues at the poles t of i F (q,t) I in equation

(21) above [19,22]. The electric field is given by [19]:

For cases where the antennas are close to the Earth and the path lengths are long enough such that

the Earth cannot be considered flat, the field-strength computation is performed using either a flatEarth attenuation function with a small-Earth curvature correction or a power series expansion.

These two techniques reduce the need to use the numerical integration of the full-wave theory

integral (equation 20), since it is a very time consuming computation [23]. These two techniques

bridge the gap for loss computation between the case where the Earth is flat (flat-Earth attenuation

function) and that where the receiving antenna is near the radio horizon of the transmitting antenna.

The computation technique (either flat-Earth attenuation function with curvature correction or power

series expansion) is selected depending on whether the magnitude of the factor q (already defined)

is small or large [24]. Recall that q is given by:

where k=2/,  is the wavelength of the radio wave in meters, a is the radius of the Earth in meters,

and is the normalized surface impedance of the ground below the antenna in question. The

expressions for are the same as given in equations (15) through (17). The surface impedance 

is a function of the ground constants of the Earth's surface. 

If the magnitude of q is small (<0.1), then a power series expansion is used for the attenuation

function f(x). The electric field E at a great circle distance d on the Earth's surface is E f(x). E is 0 0

the electric field of the same dipole source located on a flat perfect conductor. The power series

expansion for f(x) for small q (<0.1) is given by [24-30]:








  

 

 

 

   

  

   

  

      

  

  	

 

13

(29)

(30)

(31)

(32)

where

Higher order coefficients are not available in the literature, but the accuracy is adequate with the

coefficients given above [24].

If the magnitude of q is large (>0.1), then a small curvature expansion is more appropriate for the

attenuation function [25,27,28]. The expression for the small curvature expansion is given by:

The erfc is the complementary error function defined previously [21]. The implementation of these

two techniques reduces the need for the numerical integration technique and reduces computation

time considerably. 

If the terrain contour is “smooth” or the terrain irregularities are much smaller than a wavelength,
      

      

14

(33)

(34)

then the smooth-Earth method is mathematically and numerically accurate for ground-wave

predictions in a frequency range from 10 kHz to 100 MHz; however above 30 MHz the irregularities

of the atmosphere make statistical methods more appropriate. All of the models discussed so far are

part of the smooth-Earth method. 

When the terrain contour is smooth, the smooth-Earth model is valid for all combinations of antenna

heights, frequency, and dielectric constants by virtue of the computation techniques contained within

its structure. It should be used only out to the maximum distances considered useful for groundwave propagation at each frequency, since the sky wave will become significant from those distances

to points beyond. 

3.1.2 The Smooth-Earth Mixed-Path Model

A mixed path is one where the ground constants change along the propagation path between the

transmitter and receiver. The path can be described by a series of finite length segments, each with

different ground constants. The smooth-Earth mixed-path model is a specific sequence of smoothEarth model runs over each of the segments that are then combined in a particular order as

determined by the Millington algorithm [31]. The Millington algorithm is based on reciprocity

considerations. This smooth-Earth, mixed-path model is valid for the same frequency and distance

ranges as the smooth-Earth method. The antenna heights are set to zero for each smooth-Earth run

over each of the segments and combination of segments required by the algorithm. A height-gain

function is then applied to the transmitter and receiver antennas using the ground constants under

each antenna and the antenna heights. The result is the propagation loss over a mixed path with

compensation for antenna heights. The Millington algorithm implemented in the smooth-earth,

mixed-path model will be discussed for the three section mixed path of Figure 2. Expansion to more

sections is a straightforward process.

The first step involves the calculation of losses using the smooth-Earth model over single sections

and combinations of sections using the different ground constants. With a transmitter at T as a

source, (Figure 2), compute the loss L in decibels (dB). tr

where L ( d ) is the loss in dB over distance d using  and  , L ( d ) is the loss in dB over 1 1 1 1 1 2 1 

distance d using  and  , L ( d +d ) is the loss in dB over distance d +d using  and  L( 1 2 2 2 1 2 1 2 2 2 , 3 

d +d ) 1 2 1 2 3 3 3 1 2 3 is the loss in dB over distance d +d using  and  , and L ( d +d +d ) is the loss in dB

over distance d +d +d using  and  . 123 3 3

With a transmitter at R as a source (Figure 2), compute the loss L in decibels. rt

where L (d ) is the loss in dB over distance d using  and  , L ( d ) is the loss in dB over 3 3 3 3 3 2 3 

 
T

R

    

15

Figure 2. Spherical smooth-Earth mixed-path geometry.

(35)

 

distance d using  and , L ( d +d ) is the loss in dB over distance d +d using  and , 3 2 2 2 3 2 3 2 2 2

L ( d +d ) is the loss in dB over distance d +d using  and , and L ( d +d +d ) is the loss 1 3 2 3 2 1 1 1 3 2 1 

in dB over distance d +d +d using  and . 321 1 1

The total loss L (dB) is then computed: t

G(z) and G(z ) denote the magnitude of the height-gain functions for the transmitter and receiver t r

antennas respectively.

The height-gain function is given approximately by the first two terms of the Taylor series expansion

for the exact height-gain function arising from the smooth-Earth diffraction series [19,22,28,32]:
  



  

16

(36)

(37)

where z is the transmitter height, z , or receiver antenna height, z , in meters. The factor k = 2/, t r 

and  is the wavelength in meters of the radio wave. The normalized surface impedance of the 

ground below the antenna in question is as defined previously. The imaginary part of a complex

number is denoted by j. 

The function is valid up to about the first maximum in the height-gain pattern [19,29]. This height

gain function is used in the smooth-Earth mixed-path model and the irregular-Earth mixed-path

model discussed in the next section.

3.1.3 The Irregular-Earth Mixed-Path Model

Irregularities in the terrain have a greater effect at higher frequencies, so an irregular terrain model

is more appropriate when terrain irregularities become appreciable in size with respect to a

wavelength. The smooth-Earth model is much more computationally efficient and many orders of

magnitude faster than the irregular-Earth model, so in cases where the terrain is smooth enough, the

smooth-Earth model can be used with minimal sacrifice in accuracy. Specific comparisons of

smooth-Earth and irregular-Earth predictions with actual measurements have been made for different

terrain irregularities [6]. 

For the MF band it was demonstrated that for terrain variations along the path of less than a

wavelength, the smooth-Earth and irregular-Earth models were in close agreement. The irregularEarth, mixed-path model in this ground-wave model uses an integral equation [33,34] to compute

the propagation loss of a vertically polarized electromagnetic wave over irregular terrain. This

approach is a point-to-point prediction method valid for frequencies between 10 kHz and 30 MHz.

Later versions [33,34] of this model simulate terrain that is covered with forests, buildings, or snow,

where the terrain cover is modeled as a slab of user-specified thickness, length, conductivity, and

dielectric constant. Antenna heights of the transmitter and receiver antennas without a slab are

included in the irregular-Earth propagation loss computations using the same height-gain functions

as the smooth-Earth, mixed-path model. When a slab is included, a special height-gain function [33]

is used for the antennas within or above the slab for the irregular-Earth mixed-path model. 

The approach used by Ott [34] was to solve the irregular-Earth mixed-path model integral equation

numerically using a technique based on Wagner's method [35]. The integral equation is a solution

to a parabolic differential equation [34]. The complete derivation of the integral equation as well

as the actual numerical evaluation of the integral equation can be found in the literature [36,37].

The parabolic wave equation can be derived from the general wave equation [34,36]. The general

wave equation is given by:
0Q

0 
 Q

 U


 U


0 5

0  0 50 	 05

0 
	 %-

0 5

0 	 05

0 
	 %-


 	 P ! 	 1 ! &

W ! ! 	 	 !

	!   ! 	 ! & ! 	! !

17

(38)

(39)

(40)

(39)

The solution to this equation satisfies an impedance boundary condition [36]:

where Q is the vertical component of the electric field for vertical polarization or the vertical

component of the magnetic field for horizontal polarization with time dependence e [36]. The j7t

function -(x,y) is the source distribution. 

If we let Q(x,y)=e 5(x,y), and substitute this into equation (37), then equation (37) is transformed -jkx

into:

If it is then assumed that the term 0 5/0 x is much smaller than the other terms in the equation, then 2 2

the equation becomes:

This is a parabolic differential equation. The parabolic equation may be used to approximate the

scalar wave equation if the polarization of the wave remains constant along the path and if the

direction of propagation is at low angles and nearly horizontal [38]. The solution to this parabolic

differential equation is an integral equation, where the path of integration is along the irregular

ground on a line between the transmitter and the receiver. The integral equation has the form [34]:

where f(x) is the field normalized to twice the free-space field, and  is the normalized surface r

impedance at the transmitter antenna. The normalized surface impedance can be computed from ),J,

and the frequency f using the expressions given earlier in this paper in the smooth-Earth discussion.

(!) is a function of these parameters at the integration point on the path !, and (!)- is zero for r

a homogeneous path. The surface impedance  (! ) along the path can be continuous or have abrupt

changes along the path. 
  

  

   

 

  

   

   

 	

 



	

	



18

(42)

(43)

(44)

(45)

(46)

(47)

(48)

This integral definition of the complementary error function (equation 48) is defined in [21]. Figure

3 shows the geometry for the irregular-Earth mixed-path model. The symbol x denotes the distance

from the transmitter antenna at which the receiver antenna is located, y(x) denotes the height of the

receiver antenna with respect to the transmitter antenna height,  is the distance of the integration

point measured from the transmitter, y(  ) denotes the height of the integration point with respect

to the transmitter height, and g(x,y) is the antenna pattern factor for the transmitter antenna. The

transmitter terrain is at zero height and serves as a reference height since all heights are used as

program input only after the terrain height at the transmitter is subtracted from each terrain height.
19

Figure 3. Geometry for integral equation irregular-Earth model along the great circle path. 

W( x, ) is the flat-Earth attenuation function. The terms within the bracket under the integral sign

represent the surface impedance, height, and slope of the ground at the integration point . The

integral equation computes the relative field f(x) at a point x along the path in terms of f (  ), its

value at all previous points along the path.

The numerical solution of the integral equation is obtained by dividing the path up into discrete

intervals. The field f() must be known initially at a series of discrete points. The field at the very

next point is determined by fitting a second order polynomial in each of the intervals and then

performing a numerical integration. The initial points are determined using the Sommerfeld

attenuation function, and then the integral term of the equation is evaluated. The computer program

implementation calculates the electric field at all desired points between the transmitter and receiver.

The integral is evaluated over the irregular ground between the terminals, and the ground is

considered inhomogeneous in the direction of propagation along the path between the transmitter

and receiver.
20

It has also been assumed that the electrical properties of the ground may be described by its surface

impedance which is a function of the ground constants ) and J. The integral equation method can

be used with any variations of terrain represented as heights along the propagation path. The terrain

heights can change abruptly or continuously. The terrain does not need to be described by standard

geometrical features or canonical shapes. The terrain can be represented by a completely arbitrary

profile. The program assumes that the terrain varies linearly between points input by the terrain file

or user. If the program decides that it needs additional terrain points between those given, then linear

interpolation is used to determine terrain heights between available terrain heights. The program

automatically chooses the spacing so that the terrain is sampled frequently enough for an accurate

representation of the terrain variation, and so that the numerical integration of the integral equation

is sufficiently accurate, but a compromise is also made so as to prevent excessive computation time.

The distance between points should be long so as to minimize computation time, since the

computation time is proportional to the square of the number of computation points, but it should

be short to accurately represent the terrain and provide an accurate numerical integration. 

The analytical details of the integral equation and its derivation are described in the references

[33,34]. In both of these references, good agreement is found between this method and other

analytical computation methods. Comparisons of calculations with measurements have also been

made [6,39,40]. It has been found that for terrain variations smaller than a wavelength, the smoothEarth and smooth-Earth mixed-path models result in comparable accuracy to the irregular-Earth

mixed-path model, so it may be more efficient to use one of the smooth-Earth models.

4. SKY-WAVE PROPAGATION

A medium frequency sky wave will be returned back to Earth by the ionosphere if the degree of

ionization in the appropriate regions is sufficient to refract and reflect the incident electromagnetic

wave. Ionospheric propagation models for medium frequencies can predict this degree of ionization

in the different layers to determine the amount of signal that is refracted and reflected and hence the

system performance. The two regions that are responsible for the refraction and reflection of medium

frequencies are the D region and the E region. The first region encountered by the sky wave is the

D region which extends in a layer that is 50 to 90 km above the Earth's surface [41,42]. It is a region

of low electron density whose degree of ionization is determined primarily by solar photoionization.

This region usually exists during the daytime. This region has a low electron density and the

electrons collide with predominantly neutral gases, so this region absorbs the energy in the MF radio

waves that pass through it during the daytime hours [41,42]. The MF sky wave is therefore highly

attenuated as it passes through the D layer during the daytime. 

At night in the absence of the photoionization created by the sunlight, the ionization in the D region

is at a much lower level or is nonexistent, so the D region no longer absorbs the energy from the MF

sky wave passing through it. The MF sky wave proceeds to the E region above this D region where

it is reflected and refracted. The E-region ionization is from multiple sources that exist all of the

time, so it is active during both the daytime and the nighttime. E-region ionization in the daytime
21

is predominantly caused by solar ultraviolet and x-rays, while E-region ionization at night is caused

predominantly by cosmic rays and meteors. The E region occurs at heights of 90 to 140 km, and it

attains its maximum electron density near 100 km [41,42]. This is the height within the E region that

is the predominant reflecting medium for MF propagation at night. Since the sky wave at medium

frequencies is strongly attenuated by the electron density in the daytime D region, long distance

radio-wave propagation during the daytime is limited by how far the surface wave component of the

ground wave can diffract around the Earth and its terrain features. Consequently, ionospheric radiowave propagation at MF is practical only at night [41,42]. 

The sky wave can contribute to a desired signal and also generate undesirable interference to the

desired signal depending on the distance between the transmitter and receiver. At night the

undesirable interference from the sky wave can manifest itself as adjacent and co-channel

interference to stations that it would not normally reach in the daytime. System performance for

close and far distances between the transmitter and the receiver depend on the frequency and ground

conductivity, since the magnitude of the ground wave is a function of these two parameters. The sky

wave and the ground wave will add vectorially. At 0.5 MHz over average ground, the ground wave

predominates over the sky wave from the transmitter site out to distances of about 150 km, where

the two signals are equal. The signals add as vectors, and destructive and constructive interference

can occur. At distances beyond 150 km the sky wave is the predominant signal. At a signal

frequency of 1.5 MHz, the distance where the two signals are equal reduces to 45 km, because of the

increased loss at the higher frequency. At even higher frequencies the attenuation losses of both the

ground-wave and the sky-wave signal are greater, so the distance where the signal levels are equal

in amplitude becomes smaller. If the transmitter and receiver are close to each other, the groundwave signal is usually predominant over the sky-wave signal, and the ground-wave signal is

normally considered the desired signal. For large separations, the sky-wave signal predominates and

is the desired signal. In the region between close and far transmitter to receiver separations, either

the sky-wave or ground-wave signal can interfere with the dominant signal and cause possible signal

cancellation. The desired signal in many practical cases is the ground-wave signal. 

The sky wave at AM broadcast frequencies can actually become stronger with increasing distance

from the transmitter, because the sky wave that reaches the receiver represents energy radiated from

increasingly lower elevation angles, and the characteristics of MF broadcast antennas in the elevation

plane are such that the gain that launches or receives the sky wave increases in magnitude over the

lower elevation angles as it passes through and goes beyond the main beam. Beyond this point the

gain that launches the sky wave decreases as shown in Figure 4. Figure 4 is an elevation pattern for

a quarter-wave monopole antenna that is typically used at a transmitter site. 

The MF sky-wave model [7] predicts sky-wave propagation parameters using empirical formulas

derived from measured data and assumes an undisturbed ionosphere. The propagation of the sky

wave is latitude (actually geomagnetic latitude) dependent, but not all models include this effect.

The field strength decreases with increasing latitude [47]. The amount of the decrease is

proportional to the path length along which propagation takes place. The field strength can decrease

as much as 10 dB with a 10 degree increase in latitude at large distances on the order of 4000 km
-30

-20

-10

0

-30 -20 -10 0

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Elevation Angle (Degrees)

Power Gain (dB)

22

Figure 4. Elevation plane pattern for a quarter-wave monopole on a ground 

 screen.

[43,47]. At 1000 km the decrease in field strength for a 10 degree increase in latitude is only 5 dB.

Curves are available for field strengths at other distances and latitudes [47]. The sky-wave models

that have latitude and path length dependence can predict the amount of decrease [43]. During

fading the field strength amplitude follows a Rayleigh distribution [44] for latitudes less than 40

degrees. The propagation of the sky wave at MF is also dependent on distance, time of day, and

frequency. The effects due to frequency are minor. 

Semi-empirical algorithms have been produced for ionospheric radio-wave propagation. One such

algorithm is that developed by the International Telecommunications Union (ITU) [43,46]. The ITU

method is the only one that accounts for the variation of sky-wave propagation with frequency, but

it has a frequency-dependent term that reduces the predicted field strength with increasing frequency.

One reference, [47], notes that measurements in the United States show a frequency dependence that

is exactly the opposite of what the ITU method predicts. The measurements show that signals

measured at 1530 kHz at sunset or sunrise are about 15 dB greater than those at 700 kHz. This

difference becomes smaller and smaller as the time of day approaches midnight and reduces to 3 to

5 dB at two hours after sunset or two hours before sunrise. The difference at midnight is negligible.

Rather than adding the error that is incurred when using the ITU model [47], the effects of frequency

for the sky wave are generally neglected. The frequency dependence over the range of 500 kHz to

1600 kHz is small, so it is often ignored and the field strength at 1000 kHz is taken as representative

of the entire band [44]. 





	

23

(49)

Seasonal dependence has been demonstrated by measurements. The field strength received from a

transmitter is at a minimum during the summer and reaches a maximum during spring and fall. The

field strength decreases in the winter, but is not as low as the minimum achieved during the summer.

The difference between the summer minimum and the spring/fall maximums is about 15 dB at 500

kHz and reduces to about 3 dB at 1700 kHz [44]. 

There are a number of sky-wave field-strength prediction methods in use. Three methods have been

accepted for use in sky-wave propagation prediction: (1) the FCC/Region 2 method for computing

sky-wave field strengths [45], (2) the ITU-R (formerly CCIR) recommended method [43,46], and

(3) a procedure proposed by Wang [47] for use in the United States and Region 2. All of the

procedures assume a reference field strength at 1 km from the transmitter in the development of their

field strength curves (FCC Region 2 method) or field strength algorithms (ITU and Wang methods).

Using the distance along the path plus other parameters that may be needed, such as frequency and

geomagnetic latitude, the annual median field strength can be predicted. Some of the models do not

use all of the parameters. This power is modified by the model using the actual transmitter power.

The actual antenna gain in the elevation plane for the particular take-off angle needed to reach the

reception area is factored in by the model. The take-off angles are also computed by the model using

the geometry between the transmitter and receiver site. These methods are each described in more

detail in the paragraphs that follow.

The FCC/Region 2 model is actually a combination of the FCC's curve of field strength versus

distance for 50 percent of the time (for a year) for distances up to 4250 km, and the Region 2

expression for distances greater than 4250 km [7]. Measurements performed in 1935 were used by

the FCC to generate the curve of field strength versus distance. The assumptions made during the

development of this curve were that there is no dependency on frequency or latitude for the United

States. The FCC considers this curve accurate for all frequencies and latitudes in the United States,

but there is latitude dependence outside of the United States. The curve for field strength exceeded

for 50 percent of the year is shown in Figure 5. From additional measurements performed from 1939

to 1944, the FCC determined that there is a dependence on latitude [7,48]. The new curves from

those measurements can be used for interference analyses. The field strengths are further modified

for each analysis by adjusting transmitter power and antenna gain. Because the curve only goes out

to about 4250 km, the Region 2 algorithm is used beyond that distance [45]. The Region 2 algorithm

is given by [45]:

where Fc is the characteristic field strength in dBuV/m, and d is the distance in km. This expression

is adapted from the results of long-distance measurements made in the late 1930s across the Atlantic

and from North to South America. They are referred to as the ITU Cairo curves [47]. The field

strength that is exceeded 10 percent (upper decile value) of the time is found by adding 8 dB to the
-20

0

20

40

0 2000 4000 6000 8000 10000

Distance (km)

Ele

ctric

Field

Stre

n

gth (d

B

u

V/m)

FCC/Region 2 Sky-Wave Curve of Median Field StrengthVersus Distance

24

Figure 5. FCC/Region 2 sky-wave curve of median field strength versus distance.

median value of field strength versus distance [47]. The entire curve of field strength exceeded for

50 percent of the time versus distance for the combined FCC and Region 2 methods is shown in

Figure 5.

The development of the ITU sky-wave field strength prediction method is a result of many

modifications made since the 1930s [47,48]. These modifications were a result of knowledge gained

from the measurements. The current ITU method uses the USSR method with certain modifications

(such as the United Kingdom sea-gain correction) [7,48]. The expression [7,43] for field strength of

the sky wave is:

  	  	 1 	 	 	


 


  1

25

(50)

(51)

(52)

where F is the field strength in dBuV/m, V is the transmitter cymomotive force above the reference

300 V in dB, Gs p is the sea-gain correction in dB, L is the excess polarization-coupling loss in dB

(defined graphically in ITU Recommendation 435-7 [43]), 1 is the average of the transmitter and

receiver geomagnetic latitudes, p is the slant-propagation distance in km, L is the hourly loss factor s t

in dB, and Kr is the loss factor in dB including ionospheric absorption, frequency, focusing and

terminal losses, and losses between hops.

where R is the twelve-month smoothed international relative sunspot number, bsa is the solar activity

factor (b =0 for LF band; b =4 for MF band for North American paths, 1 for Europe and Australia, sa sa

and 0 elsewhere). For paths where the terminals are in different regions use the average value of bsa

(ITU Recommendation 435-7) [43].

where f is the frequency in kHz, and 1 is the average geomagnetic latitude defined previously. For

1 > 60 degrees evaluate expression at 1 = 60 degrees. For 1 < -60 degrees evaluate expression at

1 = -60 degrees. For paths shorter than 3,000 km, 1 = 0.5(1 +1 ) where 1 and 1 are the TR T R

latitudes at the transmitter and receiver respectively. The value of L is nearly 0 dB for the time t

period from 4 hours after sunset to 2 hours before sunrise, so this term is ignored for nighttime

propagation predictions. 

During the daytime, 30 dB is subtracted from the nighttime field-strength prediction or added to the

sky-wave loss prediction for the value of L (dB). A transition period occurs immediately after sunset t 

and lasts till approximately four hours after sunset, and another occurs during the period from 2

hours before sunrise until sunrise where the field strength goes through this 30 dB change with a

very steep slope. The shapes of the curves are not symmetrical for the transition from day-to-night

and night-to-day [49].

The D layer of the ionosphere is characterized as having a strong dependence on frequency, but this

is present only during the daytime. The E layer is the dominant contributor to LF and MF

propagation at night and is only mildly dependent on frequency, so the effects of frequency of this

layer can be neglected for most practical purposes [49].

For long distance paths (1000 to 6000 km), when the path is over sea or at least one end of the link

is located on or near the seacoast, the phenomenon of sea gain can add from 3 to 10 dB to the

predicted field strength [7,43]. However, a knowledge of the land-sea boundary information is

necessary to assess the sea-gain phenomena. The sea-gain correction is normally set to 0 dB without

this knowledge. 
E    O

26

(53)

The ITU method makes predictions that depend on both frequency and geomagnetic latitude. The

field strength values are not symmetrical about the geomagnetic latitude equal to 0 degrees. A

family of field strength curves for 1000 kHz is shown in Figure 6. The ITU notes that this field

strength expression predicts lower field strength values as the frequency is increased in the MF band

[43], but measurements performed in the United States show that the field strengths are higher at the

higher frequencies in the MF band when compared to those measurements at the lower frequencies.

Because of this discrepancy, the ITU method has not found wide acceptance as a worldwide

prediction method. Davies [43] also refers to measurements and gives an equation that is a function

of frequency. He recommends the use of predictions at 1000 kHz to represent the entire MF band.

In an effort to create a valid model that would give reasonably accurate predictions in Region 2,

Wang [47] developed an MF sky-wave model after examining all of the available MF methods. The

original FCC curves have a hump at roughly 100 km which Wang concluded was due to groundwave interference present in the 1935 data. The curves become smoother and better behaved after

removal of these data points. The Wang expression for field strength is:

where F is the characteristic field strength in dBuV/m referenced to 100 mV/m at 1 km, d is the c

distance in km,  = geomagnetic latitude of the midpoint of the path in degrees. m

Wang recommends that if d is less than 250 km, then the expression should be evaluated at 250 km.

There are also limits that the latitude, m, be less than or equal to +60 degrees, and greater than -60

degrees. When compared to the ITU expression, Wang's expression is symmetrical about zero

degrees latitude and is not dependent on frequency. Figure 7 shows a family of field strength curves

for several values of m.
-60

-30

0

30

60

90

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

60

50

40

30

0

Geomagnetic Latitude (degrees)

Distance (km)

27

Figure 6. ITU sky-wave curves of median field strength for several values of geomagnetic 

 latitude at 1000 kHz.
28

Figure 7. Wang sky-wave curves of median field strength for several values of geomagnetic 

 latitude.
29

5. ANTENNA MODELING

Accurate antenna modeling is necessary to determine the elevation and azimuth gain variability for

prediction of the actual gain to be used in launching the sky wave at the appropriate take-off angles,

and the ground-wave gain at the horizon angle. The performance of an antenna on or near the

surface of the Earth is very dependent on the interaction with the lossy Earth. This is especially true

for antennas at MF. Currently available techniques for analyzing antennas over ground with

computer algorithms are time consuming and require conversion or normalization for use in system

computations. This process will be described in this section. The gain of the antenna is a function

of the antenna geometry, materials used, ground conductivity, ground dielectric constant, frequency,

elevation angle, and azimuth angle. The gain required for systems performance analysis is usually

calculated with respect to an isotropic radiator in free space or some other reference antenna such

as a dipole. Conventional methods such as free space analysis could not be used due to the close

proximity of the antennas to a lossy Earth. Antenna modeling techniques at medium frequencies

must account for the actual gain that launches the ground wave. The end result for the antenna gain

described in this paper has been transformed to be referenced to an isotropic radiator in free space.

There are several methods for modeling antennas that are in close proximity to a lossy Earth. Some

of these are not valid for an antenna located right on the surface and require that the antenna be 0.2

wavelengths above the surface or the algorithm used in the calculation will not be valid. These

algorithms that assume free-space conditions or negligible ground effects should not be used to

model antennas that are too close to the Earth, because the results are poor. The model selected for

analyzing antennas makes use of extensive method-of moments calculations and is implemented in

a computer program titled the Numerical Electromagnetics Code (NEC) [50]. It is an accurate

method for analyzing antennas at these low and medium frequencies where the antennas are small

or comparable to a wavelength in size. Numerical methods are required to solve this problem and

NEC is a good algorithm to use. The NEC program operates using a computation mode that

implements a Sommerfeld integral computation for the determination of electromagnetic fields for

antenna structures that are buried or penetrate the ground-air interface. This computation technique

includes the reflected field below the interface, the field transmitted across the interface, and the

fields above the ground-air interface. The algorithms used are also valid for antennas very close to

the interface. The NEC program can also model near fields of the antenna very close to the antenna

structure, in addition to being able to model the far field and compute antenna gain. 

A different antenna gain phenomenon occurs with antennas at medium frequencies as compared to

antennas at VHF, UHF and higher frequencies. The antenna principles are the same, but the surface

wave becomes more signficant at MF. This is important because it is the surface wave that accounts

for practically all of the energy transmitted and received at these frequencies. The concepts of gain

and efficiency for antennas close to the surface of the Earth at MF are difficult to comprehend, since

a major portion of the power launched by the antenna is absorbed by the lossy Earth. The electric

field propagating along the surface of the Earth decreases faster than the usual reciprocal of the

distance decay rate. In order to see this phenomenon, it is necessary to examine the Norton

expressions [10] for the electric field resulting from an electric dipole above the surface of a finitely

conducting Earth. Norton [10] in his effort to simplify the expressions developed by Sommerfeld


	



	



  

	

  

30

(54)

(55)

[13] came up with equations that clearly show the surface wave and space wave components. Jordan

[54] deleted the higher order terms that are inversely proportional to the distance terms of Norton's

equations for the vertical and radial directed components of the electric field in cylindrical

coordinates. These higher order terms represent the induction and near field of the antenna and

diminish in amplitude rapidly with distance. Jordan [54] further reduced the equation complexity

by vectorially combining the equations for the vertical and radial directed field components, and then

separating the resulting equation into a total space and total surface wave component. The resulting

equations are:

where k=2/ as defined previously, A() is the flat-Earth attenuation function defined previously,

I is the peak dipole current amplitude in amperes, L is the length of the dipole in meters, R is the 1 

distance between the dipole and the observation point in meters, R is the distance between the dipole 2

image and the observation point in meters, R is the complex reflection coefficient for vertical v

polarization, u =( +j/ ) , and  is the angle representing the direction of the incident wave 2

r 0

-1

measured with respect to the Earth's surface.

The equations (54 and 55) for the electric-field components of the space wave and the surface wave

are plotted in Figure 8 for conditions representing MF frequencies with average ground.

Examination of an electric-field strength pattern such as that displayed in Figure 8 shows that the

electric field due to the space-wave pattern indicates a very small component at the horizon, but this

is not what launches the surface wave component of the ground wave. The actual gain that launches

the surface wave can be determined from a numerical electromagnetic approach by analyzing the

specific antenna geometry. The normal pattern shown in Figure 8 as a dotted line is the electric-field

antenna pattern responsible for launching the space-wave component of the ground wave consisting

of the direct and reflected ground wave. This is equation (54). It is also responsible for launching

the sky wave. The amplitude of the space-wave electric field decreases with distance and is

proportional to the reciprocal of the distance. The electric-field pattern that launches the surface

wave is shown as a solid line in Figure 8. This is equation (55). It is a plot of the unattenuated

surface wave with A()=1 and represents the surface wave close to the antenna. For large numerical

distances, the amplitude of the surface-wave electric field decreases as the reciprocal of the square

of the distance. Notice that it has a maximum level at the surface and decreases rapidly with

increasing elevation angle. For larger elevation angles greater than a few degrees the space wave
31

Figure 8. Space wave and unattenuated surface wave components of the 

 ground wave for a vertical dipole at the Earth’s surface for 

 average ground conditions at 760 kHz.

contribution becomes more significant and is responsible for the launching of the space wave

component of the ground wave and the sky wave. The relative geometry between most transmitters

and receivers used for propagating ground-wave signals at MF frequencies will be at very low

elevation angles with respect to each other. These elevation angles are on the order of one degree

or less. The ground wave for these conditions will consist predominantly of the surface wave with

little contribution by the space wave. Correct performance predictions will be obtained only when

using this unique electric-field pattern that launches the surface wave to calculate an equivalent gain

that correctly represents the magnitude of electromagnetic energy transmission or reception. Both

the surface wave and the space wave components are included to represent the total electric field of

a dipole or any antenna near a lossy Earth. As one or both antennas are raised in height a distance

comparable to a wavelength above the Earth's surface, the surface wave becomes less significant and

eventually negligible. 

Receiver antennas at MF tend to be small in physical size with respect to a wavelength, and therefore

have very low gains due to radiation efficiency considerations. The transmit antennas are usually

more efficient since they are comparable to a wavelength in size. The antenna algorithm used to

predict the antenna gain for low and medium frequency antennas computes an equivalent gain and

represents the effectiveness of the antenna in launching or receiving the surface wave component of

a ground wave. The gains for these antennas were derived from behavior analysis of extensive

method-of-moments calculations using the Numerical Electromagnetics Code [50,51].
  







32

(56)

(57)

(58)

(59)

A short dipole was selected as a reference antenna due to simplicity of analysis and convenience in

measurement. Many loss calculations in ground-wave analysis are referenced to short dipoles or

monopoles. The computer program NEC is used to determine the equivalent gain of a short dipole

over a lossy Earth with respect to an isotropic radiator in free space for the appropriate ground

conductivities, ground dielectric constants, and frequencies. The relative performance of the subject

antenna with respect to the short dipole is then determined using the NEC program. These two

factors are then combined to obtain the equivalent gain with respect to an isotropic radiator in free

space.

The gain of a short dipole antenna at the surface of a lossy Earth referenced to an isotropic radiator

in free space can be derived from some basic relationships. The power density in free space for an

isotropic radiator is given by Norton [52,53]:

where e(V/m) is the electric field strength in volts per meter,  (ohms) is the impedance of free space

(120ohms), D(m) is the distance from the antenna in meters, and Pr(W) is the radiated power in

watts.

Taking the square root of both sides and rearranging terms, the primary electric field strength from

an isotropic antenna is:

The primary electric field strength from any antenna in free space is:

where g is the equivalent antenna gain ratio referenced to an isotropic radiator in free space and P

is the transmitter input power in watts. If the units are changed the above equation becomes:

where e(mV/m) is the electric field strength in millivolts per meter, P(kW) is the transmitter power

in kilowatts, and d(km) is the distance in kilometers.


  

 

 

33

(60)

(61)

(63)

If the antenna is a short vertical dipole element (g=3) at but not touching the surface of a perfectly

conducting Earth, then:

This is the familiar 300 mV/m at one km for a radiated power of one kW over a perfectly conducting

Earth.

The electric field strength for a lossy Earth and an arbitrary gain g is given by Terman [8] as:

where A is the Norton approximation [9] to the Sommerfeld attenuation function [13]. This

attenuation function was described previously in the discussion on ground-wave propagation. If this

expression is rearranged to solve for g and the logarithm of both sides is taken, the gain of a short

dipole antenna at but not touching the surface of a lossy Earth referenced to an isotropic radiator in

free space is:

:

(62)

The effectiveness of an antenna in launching a surface wave which will be referred to as “equivalent

gain,” is then determined by first calculating a ratio called the relative communication efficiency for

the antenna as a function of the ground constants, frequency, antenna geometry, and azimuthal

direction. The relative communication efficiency in decibels (dB) is then added to the reference

dipole gain. The relative communication efficiency (RCE) [53] of an antenna is defined by:

where E (V/m) is the electric field strength predicted by NEC at reference distance d and input power r

P for a short dipole, P (W) is the dipole antenna input power from NEC used to compute E (V/m), r r r

Et (V/m) is the electric field strength predicted by NEC at reference distance d and input power Pt

for the subject antenna, and P (W) is the subject antenna input power from NEC used to compute t 

E .

t
 

-30

-20

-10

0

-30 -20 -10 0

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Elevation Angle (degees)

Gain (dB)

34

(64)

Figure 9. Elevation plane pattern for the space wave antenna power gain for 

 the DGPS transmitter site antenna for average ground conditions 

 at 300 kHz.

The “equivalent gain” G (dBi) of the subject antenna over lossy Earth referenced to an isotropic a

radiator in free space is given by:

The “equivalent gain” referenced to an isotropic radiator in free space can now be used for

communication system analysis. The result is a term similar to power gain that can be defined in

the presence of a lossy Earth. A number of antennas were modeled using this technique. 

One antenna that was modeled for the DGPS correction signal at 300 kHz consisted of a top-loaded

monopole on a ground screen. This antenna is typical of a DGPS antenna that was installed at a site

in Appleton, Washington. This site is one of the first installation sites for the DGPS system. DGPS

signals radiated from this antenna site were measured extensively [55]. The monopole antenna is

a tapered triangular shaped steel tower consisting of steel tubing 1.52 cm in diameter. The monopole

length is 91.16 meters which is only 0.0912 wavelength at 300 kHz. The cross section of the

monopole is an equilateral triangle measuring 0.61 meters on a side for most of its length and is

tapered to a point on each end. There is a copper radial wire ground screen with 120 radials of 0.32

cm diameter that are each 100 meters in length. There are also twelve copper top-loading elements

approximately 138 meters in length symmetrically located around the monopole extending from the

top of the monopole down to the ground just outside but not touching the edge of the radial wire

ground screen. The top-loading elements are 0.95 cm in diameter. Their purpose is to increase the

efficiency of the antenna. The antenna is omnidirectional in the azimuth plane. The elevation plane
0

0.5

1.0

1.5

2.0

0 0.5 1.0 1.5 2.0

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Surface Wave

Space Wave

Elevation Angle (Deg.)

Electric Field (V/m)

35

Figure 10. Electric field strength antenna patterns of the space wave and the 

 surface wave for the DGPS transmitter site for average ground 

 conditions at 300 kHz.

pattern for the space wave antenna power gain is shown in Figure 9. Notice the amplitude of the

space wave pattern at the horizon. Figure 10 shows antenna patterns in terms of electric field

strength for both the space wave and the surface wave for this antenna. The equivalent gain for

launching the surface wave component was determined to be 3.2 dBi using the technique described

previously, which includes the efficiency factor of 53 percent (-2.75 dB). This antenna gain occurs

in all azimuth directions since the antenna is symmetrical in all azimuth planes. This equivalent

antenna gain occurs at an elevation angle of zero degrees, and rapidly decreases in amplitude with

increasing elevation angle. The loss due to impedance mismatch was assumed to be negligible, since

the voltage standing wave ratio (VSWR) was less than 1.5:1. This antenna gain computation was

verified by measurement of field strength from the actual DGPS site that uses this antenna [55]. 

Reception characteristics of a 300 kHz signal from a one meter antenna mounted on a van test

vehicle were also determined using the equivalent gain technique. A short whip (one meter in

length) on a van test vehicle was modeled to determine the antenna gains to use for system

performance calculations. These gains will be much lower than what may be expected for this type

of antenna due to the inefficient operation of small (with respect to a wavelength) vehicle antennas.

Antennas were modeled at 300 kHz for DGPS and 760 kHz for the AM subcarrier ATIS. The

antennas are much shorter than a wavelength and can also suffer from impedance mismatch loss into

the receiver. Since the van test vehicle was also much smaller than a wavelength, it was found to

have a negligible effect on the antenna as expected. The model for the van consisted of closely

spaced metal wires that represented the geometry of the metal surface of the entire test van. There

is only a slight pattern asymmetry of a few tenths of a dB created by the van asymmetry. Figure 11
-30

-20

-10

0

-30 -20 -10 0

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Elevation Angle (Degrees)

Power Gain (dB)

36

Figure 11. Power gain antenna pattern in the elevation plane for the space 

 wave for the DGPS one meter whip mounted in the center of the 

 van roof for average ground conditions at 300 kHz.

shows the power gain pattern in the elevation plane for the space wave for the one meter whip at 300

kHz mounted in the center of the van roof. Notice the gain amplitude in comparison to Figure 9,

even though the patterns have the same general shape. The equivalent gain for launching the surface

wave was calculated for this antenna to be -21.3 dB. This gain exists in all azimuth directions since

the antenna is symmetrical in this plane. The elevation angle for this equivalent gain is zero degrees

(the horizon). The power gain in the elevation plane of the antenna pattern for the same one meter

antenna on the van roof at 760 kHz is shown in Figure 12. The equivalent gain for launching the

surface wave was calculated to be -13.1 dB. The impedance mismatch loss can add a significant

amount of loss to the antenna gain if the antenna is severely mismatched. If the characteristics of

the receiver impedance and the cabling between the receiver and the antenna are known, then the

mismatch loss can be calculated and included in the calculation of gain. An impedance mismatch

characterized by a VSWR of less than 1.5:1 has been measured, so the impedance mismatch loss for

this case is negligible. 

A single quarter-wavelength monopole on a large ground screen (120 radials each /4 in length) was

also modeled to simulate a broadcast transmitter for an AM radio station at 760 kHz. The monopole

has a symmetrical pattern in azimuth. The resulting power gain pattern in the elevation plane at any

azimuth angle is shown in Figure 13. The equivalent gain that launches the surface wave was

determined to be 4.7 dB for this antenna. 

A more complex antenna was also modeled. This antenna is a model of the actual antenna used at

station KTLK at 760 kHz in Denver, Colorado. Two configurations are used: one for nighttime
37

Figure 12. Power gain antenna pattern in the elevation plane for the space 

 wave for the one meter whip antenna mounted in the center of 

 the van roof for average ground conditions at 760 kHz.

operation, and one for daytime operation. The nighttime configuration consists of four monopole

towers arranged in a physical array with the geometry and electrical phasing given in Table 2. All

antenna tower positions and phasing are relative to tower 1. The azimuth pattern is shown in Figure

14. This pattern is at an elevation angle of zero degrees and represents an equivalent gain that

launches a surface wave. It is not the space wave pattern which would have a very low gain at this

elevation angle of zero degrees. The daytime configuration consists of two monopole towers

arranged in the geometry and phasing given in Table 2. The azimuth pattern is shown in Figure 15.

This pattern for the daytime antenna configuration is also the equivalent gain that launches the

surface wave.

Figure 16 shows an elevation cut on the azimuth beam maximum at an azimuth angle of 225 degrees

for the nighttime KTLK antenna configuration. Figure 17 shows an elevation cut on the beam

maximum at an azimuth angle of 195 degrees for the daytime KTLK antenna. Figures 16 and 17

are elevation cuts of conventional antenna patterns of power gain showing the small magnitude of

the gain at the horizon and for small elevation angles. 
-30

-20

-10

0

-30 -20 -10 0

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Elevation Angle (Degrees)

Power Gain (dB)

38

Figure 13. Power gain antenna pattern in the elevation plane for a quarter-

 wave monopole antenna on a ground screen for average ground 

 conditions at 760 kHz.

Table 2. Geometry and Phasing for Station KTLK Daytime and Nightime Antenna Arrays.

Daytime Antenna Array

Tower Field Ratio Phase (deg) Spacing (deg) Bearing (deg) Height (deg)

1 1.000 360.0 00.0 00.0 89.0

2 1.350 451.0 134.8 116.5 89.0

Nighttime Antenna Array

Tower Field Ratio Phase (deg) Spacing (deg) Bearing (deg) Height (deg)

1 1.000 360.0 00.0 00.0 89.0

2 0.980 462.0 90.0 75.0 89.0

3 0.800 272.0 134.8 116.5 89.0

4 0.820 530.0 90.0 158.0 89.0
0

2

2

4

4

6

6

8

8

8 6 4 2 0 2 4 6 8

90

o

100

o

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o

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o

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170 o

o 180

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o 0

o 10

o 20

o

30

o

40

o

50

o

60

o

70

o

80

o

39

Radial Scale is in dBi

Figure 14. Equivalent gain antenna pattern for station KTLK nighttime antenna 

 configuration versus azimuth angle for average ground conditions at 

 760 kHz.
-10

-5

-5

0

0

5

5

10

10

10 5 0 -5 -10 -5 0 5 10

90o

100o

110o

120o

130o

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160 170 o

o 180o

190o

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220o

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260o

270o

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310o

320o

330o

340o

350o 0

o 10o 20o

30o

40o

50o

60o

70o

80o

40

Radial Scale is in dBi

Figure 15. Equivalent gain antenna pattern for station KTLK daytime antenna 

 versus azimuth angle for average ground conditions at 760 kHz.
-40

-30

-20

-10

0

10

-40 -30 -20 -10 0 10

0

o

10o

20o

30o

40o

50o

60o

70o

80 90 o o

Elevation Angle (degrees)

Power Gain (dB)

41

Figure 17. Power gain antenna pattern in the elevation plane on azimuth beam

 maximum for the KTLK daytime antenna for average ground 

 conditions at 760 kHz.

Figure 16. Power gain antenna pattern in the elevation plane on azimuth beam

 maximum for the KTLK nighttime antenna for average ground 

 conditions at 760 kHz.

   	 

42

(65)

6. SYSTEM PERFORMANCE CALCULATIONS AND PREDICTIONS

System performance calculations can be made using the propagation models described previously

in this paper. Antenna gains for the system performance predictions are factored in to represent the

actual gains of these antennas. The final calculations take into account the antenna gains of the

transmitter antenna at the transmitter site and the receiver antenna at the receiver site or test vehicle.

In some systems analysis applications, it is necessary to calculate electric field strength near a

receiver antenna knowing the received power level into a spectrum analyzer. When using a spectrum

analyzer or receiver, correction must be made to the received power levels to calculate electric field

strength in the vicinity of the receiver antenna, since most coverage calculations are in field strength.

The equation to calculate electric field strength, E(dBuV/m) from received power is [16]:

where P (dBm) is the received signal power which is referenced at the output of the antenna r

terminals but by appropriate adjustment of the losses can be moved to the reference point of the input

to the spectrum analyzer or receiver, f(MHz) is the frequency in MHz, G (dBi) is the gain of the r

receiver antenna, and L (dB) is the loss in the cable. c

The receiver antenna gain in this expression must also be corrected to account for the degradation

in gain as a function of elevation angle for receiving a surface wave, sky wave, or space wave shown

previously in Figure 4. The line-of-sight from the receiver to the transmitter antenna is typically at

(depending on distance from the transmitter and receiver, and transmitter and receiver antenna

heights above mean sea level) an elevation angle of less than a few degrees. Over this range of

elevation angles the gain of the receiver antenna for a sky wave can vary from 10 to 20 dB or more

below the main beam gain. The peak gain of this antenna is small due to the low antenna efficiency

at these frequencies. This additional reduction in antenna gain due to elevation angle increases the

electric field strength required for system operation, which is proportional to the decrease in antenna

gain in decibels. This can be seen from equation (65) above.

The gain for launching and receiving a ground wave is calculated using the NEC code mentioned

earlier and converting the computations in the output to an equivalent gain referenced to an isotropic

radiator. The gain and pattern for launching and receiving a sky wave can also be computed using

NEC directly, since it computes a space wave antenna pattern in addition to the specific information

needed to obtain the ground-wave equivalent gain. The results of these calculations were shown in

the previous section. 

Once confidence is obtained in the prediction algorithms for antenna models, then predictions of area

coverage of a station can be performed. Figures 18 and 19 show KTLK station coverage computed

from the ITS model [7] for the daytime coverages for both smooth-Earth and smooth-Earth mixedpath propagation models. The differences in the two figures are a result of considering a mixed path.
43

The borders for the states of Colorado, Wyoming, and Nebraska are shown. Figures 20 and 21 show

station coverage computed from the ITS model [7] for the nighttime coverages for both smoothEarth and smooth-Earth mixed-path propagation models. The electric-field strength values are given

in units of dB microvolts per meter (dBuV/m). They illustrate the difference in coverages when the

mixed-path data are taken into consideration. The prediction with the smooth-Earth mixed-path

model shows greater coverage near the station where the ground wave propagation predominates.

The ground constants were taken from the data base contained in the LF/MF model [7]. The ground

constants and the mixed path only affect the ground wave propagation, but station KTLK uses

different antenna configurations for daytime and nighttime, which also affects antenna performance

and coverage area. The transmitter power also changes from 1kW at night to 50 kW during the

daytime. The nighttime antenna configuration was specifically designed to mininize the chance for

interference at night to an adjacent or cochannel station northeast of station KTLK, because the sky

wave makes it possible to propagate appreciable level signals at very long distances. The antenna

array configurations for station KTLK were described in Table 2 previously. The patterns in these

coverage plots are different than those in the actual antenna pattern predictions of Figures 14 and 15,

because Figures 14 and 15 are equivalent gain antenna patterns for launching the surface wave

described previously. Figures 20 and 21 were created by the model using a composite of the

equivalent gain antenna patterns for the surface wave and the space wave antenna patterns for the

skywave. The resulting combination will look different than the spatial gain distribution of the

individual antenna patterns. 
44

Figure 18. Predicted KTLK day coverage contours (dBuV/m) using the smooth-Earth model.
45

Figure 19. Predicted KTLK day coverage contours (dBuV/m) using the smooth-Earth mixed-

 path model. 
46

Figure 20. Predicted KTLK night coverage contours (dBuV/m) using the smooth-Earth model.
47

Figure 21. Predicted night coverage contours (dBuV/m) using the smooth-Earth mixed- 

 path model.

The data used to plot Figures 22 and 23 were taken from the ITS model [7] using the point-to-point

computation mode for the nighttime antenna configuration for the smooth-Earth and smooth-Earth

mixed-path propagation models. Both figures are for a particular radial 330 degrees wrt due North

for a long range nighttime prediction that includes both the ground-wave and the sky-wave

components. The propagation model factors in the appropriate gains from the receiver and

transmitter input geometry. The appropriate height-gain functions are also factored in to take into

account nonzero antenna heights at both the transmitter and receiver locations. The combination of
48

the ground-wave and sky-wave propagation models also has an effect on the coverage, so the

contours will look different than the spatial gain distribution of the antenna patterns. Both the

ground wave and sky wave are plotted separately. When the sky-wave and ground-wave magnitudes

become comparable to each other, they can combine constructively or destructively to either

vectorially add together or produce a null at that particular distance from the station. The ground

wave is a deterministic phenomenon, but the sky wave is a stochastic phenomenon, so the

combination of the two is a stochastic process that will vary with time as ionospheric conditions

vary. The smooth-Earth ground-wave model and the sky-wave model are equal at 170 km in Figure

22, but the smooth-Earth mixed-path ground-wave model in Figure 23 predicts that the ground wave

and sky wave are equal at 190 km. The smooth-Earth mixed-path model is generally more accurate,

since it takes into account the presence of a mixed-path ground constant condition if the data base

of ground constants shows that it exists. The smooth-Earth model assumes a homogeneous Earth

with one pair of values for the ground constants over the entire path.

System performance computations were made for several examples where measured data were

available. Data were available from the DGPS site at Appleton [55,56]. Figure 24 compares the

results of performance prediction calculations with the measured data for the Appleton, WA to

Pendleton, WA path. The measured data for the Appleton site were obtained from reference [55].

The equivalent gain technique was used to determine antenna gains for system calibration. Figure

25 is a plot of the terrain contour over the Appleton to Pendleton path. The nulls of Figure 24 appear

to correspond to when the measurement vehicle was located in deep canyons and behind large terrain

peaks. Other extensive data were available for demonstrating agreement between predictions and

measurements at various sites in the continental United States in the DGPS band of 285 to 325 kHz

[56]. The reference [56] describes the results of comparisons of actual measurements to predictions

over many paths. This DGPS signal is used for differential position correction of the GPS signal.

An extensive measurement program was performed at medium frequencies [40]. Figures 26 and 27

compare the measured data from two of the data runs at Canyonlands, Utah along a 45 km path from

this reference with the model predictions. Data [40] were available from two medium frequency

signals both at 520 kHz and 1618 kHz. There is good agreement between measurements and

predictions. The irregular-Earth mixed-path model in the LF/MF model was used to perform the

predictions, since the terrain is considered rough in comparison to a wavelength as shown in Figure

28. A smooth-Earth or smooth-Earth mixed-path prediction would not be accurate for this

computation, since the terrain variations are large with respect to a wavelength. This reasoning was

described earlier in the section on ground-wave propagation.

Propagation modeling and systems analysis can be used to resolve and explain actual measurements.

An example of this can be demonstrated with recent measurements. A measurement was performed

in Boulder, CO to determine the nighttime and daytime electric field strength received from the

Appleton, WA DGPS site at 300 kHz (private communication J.R. Hoffman, February 8, 1999).
0

25

50

75

100

0 100 200 300

Sky Wave

Ground Wave

Distance (km)

49

Figure 22. Electric field strength vs distance from station KTLK at nighttime on 330 degree 

 radial using the smooth-Earth model.

Nighttime measurements of electric field strength consistently averaged 30 dBuV/m over several

days with good quality DGPS messages. This is in good agreement with model predictions using

the LF/MF model [7]. These results are shown in Figure 29 where contours of electric field strength

from the Appleton DGPS site are plotted over a map of the Northwestern United States. Daytime

measurements of 14 dBuV/m were also measured over several days, but there were no successful

DGPS messages, since the signal levels from the Appleton DGPS site were too low. Predictions of

electric field strength from the Appleton DGPS site resulted in -10 dBuV/m levels. This result is

shown in Figure 30. 

After searching a database of signals at this frequency, three other sites were found at 300 kHz. The

LF/MF model was used to predict the daytime levels of electric-field strength from these sites. Two

of these sites, one in Spearfish, SD and the other in Auburn, NE, were found to be capable of

generating electric-field strengths at levels of 10 to 15 dBuV/m in the Boulder, CO area. Even

though these two sites were at the same frequency, no successful DGPS messages were decoded

since they were a different type of beacon site and were due to Federal Aviation Administration

(FAA) beacons radiating at 300 kHZ. The propagation model was able to account for the levels of

electric field strength at 300 kHz in Boulder, CO that were measured. 
0

25

50

75

100

0 100 200 300

Sky Wave

Ground Wave

Distance (km)

50

Figure 23. Electric field strength vs distance from station KTLK at nighttime on 330 degree 

 radial using the smooth-Earth mixed-path model.
40

60

80

100

0 50 100 150 200

Predicted Field Strength

Measured Field Strength

Distance from Appleton (km)

51

Figure 24. Comparison of measured and predicted DGPS field strength for the Appleton to 

 Pendleton path.
100

300

500

700

900

0 50 100 150 200

Pendleton

Appleton

Distance From Appleton (km)

52

Figure 25. Terrain contour for path from Appleton to Pendleton, Washington.
50

65

80

95

110

125

0 10 20 30 40 50

LF/MF Model Prediction 

Measured Data

Distance From Transmitter (km)

53

Figure 26. Comparison of measured and predicted data for Canyonlands, Utah path at 520 kHz.
40

60

80

100

120

0 10 20 30 40 50

LF/MF Model Prediction

Measured Data

Distance From Transmitter (km)

54

Figure 27. Comparison of measured and predicted data for Canyonlands, Utah path at 1618 kHz.
1000

1250

1500

1750

2000

0 10 20 30 40 50

Distance From Transmitter Site (km)

55

Figure 28. Terrain contour for Canyonlands, Utah measurement site.
56

Figure 29. Contours of electric-field strength in dBuV/m from the Appleton, WA DGPS site at 

 nighttime (f=300 kHz).
57

Figure 30. Contours of electric-field strength in dBuV/m from the Appleton, WA DGPS site at 

 daytime (f=300 kHz).

7. CONCLUSION

This paper covered the basic aspects of radio-wave propagation and antenna modeling in the MF

band. The MF band covers the frequencies of 300 to 3000 kHz. The sky wave models described

in this paper are valid from 150 kHz to 1705 kHz. The ground wave models described in this paper

are valid from 10 kHz to 30 MHz. In this paper the propagation models and antenna modeling

techniques were shown to be quite different than those at other radio frequency bands. The

propagation of radio waves in this band depends on both a ground wave and a sky wave and their

interaction with the environment. The antennas are strongly influenced by the presence of ground

and require numerical analysis techniques more sophisticated than free space techniques used in

higher frequency bands. This paper describes radio wave propagation together with antenna
58

modeling in this frequency band so that a better understanding of the phenomena can be obtained

for use in design and application of ITS subsystems. The models described here can be used for

designing systems and making performance predictions for both of these ITS applications and any

other systems that operate in this band. 

The radio-wave propagation loss computed using these models is a basic transmission loss and not

free-space transmission loss. Free-space is a highly idealized environment and is not the

environment encountered in actual propagation situations. The ground wave includes the direct lineof-sight space wave, the ground-reflected wave, and the surface wave that diffracts around the

curved Earth. Propagation of the ground wave depends on the relative geometry of the transmitter

and receiver location and antenna heights in addition to the presence of a conductive Earth. The

radio wave propagates as a surface wave when both the transmitter and the receiver are near the

Earth in wavelengths, because the direct and ground-reflected waves in the space wave cancel each

other out, and as a result the surface wave is the only wave that is left. The surface wave propagates

along and is guided by the Earth's surface. As the surface wave passes over the surface of the Earth,

it is attenuated as result of the energy absorbed by the Earth due to the power loss resulting from the

current flowing through the Earth's resistance. Energy is taken from the surface wave to supply the

losses in the ground and the attenuation of this wave is directly affected by the ground constants of

the Earth along which it travels. This cancellation is a result of the fact that the elevation angle is

near zero and the two ground waves (direct and reflected) are equal in amplitude and opposite in

phase. This is the condition that exists for the MF band and lower frequencies. The surface wave

is predominantly vertically polarized, since the ground conductivity effectively shorts out most of

the horizontal electric field component. What is left of the horizontal field component is attenuated

at a rate many times the vertical component of field strength. When one or both of the antennas are

elevated above the ground to a significant height with respect to a wavelength, the space wave

predominates. 

The exact equations derived by Norton [10] can be used for analysis, or some simpler equations also

by Norton [9,10] can be used to describe the flat-Earth attenuation function when the geometry is

such that the flat-Earth attenuation function is appropriate. When the geometry is such that the flatEarth attenuation function is not appropriate, then smooth-Earth, smooth-Earth mixed-path, and

irregular-Earth mixed path models such as those described in this paper should be used to take into

account non-zero antenna heights, mixed-path ground conditions, and terrain. 

A medium frequency sky wave will be returned back to Earth by the ionosphere if the degree of

ionization in the appropriate regions is sufficient to refract and reflect the incident electromagnetic

wave. Ionospheric propagation models for medium frequencies can predict this degree of ionization

in the different layers to determine the amount of signal that is refracted and reflected and hence the

system performance. The two regions that are responsible for the refraction and reflection of medium

frequencies are the D region and the E region. The first region encountered by the sky wave is the

D region and the E region is located above it. The D region is a region of low electron density

whose degree of ionization is determined primarily by solar photoionization. This region usually

exists during the daytime and absorbs the energy in the MF radio waves that pass through it during

the daytime hours. The MF sky wave is therefore highly attenuated as it passes through the D layer
59

during the daytime. Reflection and refraction of the sky wave by the E region does not usually

occur, because of the large amount of attenuation experienced in the D region. At night the solar

photoionization in the D region is reduced and hence the attenuation of the sky wave in this region

is much less, so the sky wave travels on to the E region where it is refracted and reflected back to

Earth. The D region of the ionosphere is characterized as having a strong dependence on frequency,

but this is present only during the daytime. The E layer is the dominant contributor to LF and MF

propagation at night and is only mildly dependent on frequency, so the effects of frequency on radiowave propagation of this E layer can be neglected for most practical purposes. Three sky-wave

propagation models were described in this paper and their relative merits were discussed. The ITU

method is the only one that accounts for the variation of sky-wave propagation with frequency, but

it has a frequency-dependent term that reduces the predicted field strength with increasing frequency.

One reference, [47], notes that measurements in the United States show a frequency dependence that

is exactly the opposite of what the ITU method predicts. Rather than adding the error that is incurred

when using the ITU model [47], the effects of frequency for the sky wave are generally neglected.

The frequency dependence over the range of 500 kHz to 1600 kHz is small, so it is often ignored and

the field strength at 1000 kHz is taken as representive of the entire band [44]. The ITU method

makes predictions that depend on both frequency and geomagnetic latitude. The field strength

values are not symmetrical about the geomagnetic latitude equal to 0 degrees. The ITU notes that

this field strength expression predicts lower field strength values as the frequency is increased in the

MF band [43], but measurements performed in the United States show that the field strengths are

higher at the higher frequencies in the MF band when compared to those measurements at the lower

frequencies. Because of this discrepancy, the ITU method has not found wide acceptance as a

worldwide prediction method. Davies [43] also refers to measurements and gives an equation that

is a function of frequency. He recommends the use of predictions at 1000 kHz to represent the entire

MF band. In an effort to create a valid model that would give reasonably accurate predictions in

Region 2, Wang [47] developed an MF sky-wave model after examining all of the available MF

methods. When compared to the ITU expression, Wang's expression is symmetrical about zero

degrees latitude and is not dependent on frequency. 

Accurate antenna modeling is necessary to determine the elevation and azimuth gain variability for

prediction of the actual gain to be used in launching the sky wave at the appropriate take-off angles,

and the ground-wave gain at the horizon angle. The performance of an antenna on or near the

surface of the Earth is very dependent on the interaction with the lossy Earth. This is especially true

for antennas at MF. Currently available techniques for analyzing antennas over ground with

computer algorithms are time consuming and require conversion or normalization for use in system

computations. This process was described in the propagation section. The gain of the antenna is a

function of the antenna geometry, materials used, ground conductivity, ground dielectric constant,

frequency, elevation angle, and azimuth angle. The gain required for systems performance analysis

is usually calculated with respect to an isotropic radiator in free space or some other reference

antenna such as a dipole. Conventional methods such as free space analysis could not be used due

to the close proximity of the antennas to a lossy Earth. Antenna modeling techniques at medium

frequencies must account for the actual gain that launches the ground wave. The end result for the
60

antenna gain described in this paper has been transformed to be referenced to an isotropic radiator

in free space. 

There are several methods for modeling antennas that are in close proximity to a lossy Earth. Some

of these are not valid for an antenna located right on the surface and require that the antenna be 0.2

wavelengths above the surface or the algorithm used in the calculation will not be valid. These

algorithms that assume free-space conditions or negligible ground effects should not be used to

model antennas that are too close to the Earth, because the results are poor. The model selected for

analyzing antennas makes use of extensive method-of moments calculations and is implemented in

a computer program titled the Numerical Electromagnetics Code (NEC) [50]. It is an accurate

method for analyzing antennas at these low and medium frequencies where the antennas are small

or comparable to a wavelength in size. Numerical methods are required to solve this problem and

NEC is a good algorithm to use. The NEC program operates using a computation mode that

implements a Sommerfeld integral computation for the determination of electromagnetic fields for

antenna structures that are buried or penetrate the ground-air interface. This computation technique

includes the reflected field below the interface, the field transmitted across the interface, and the

fields above the ground-air interface. The algorithms used are also valid for antennas very close to

the interface. The NEC program can also model near fields of the antenna very close to the antenna

structure, in addition to being able to model the far field and compute antenna gain. 

A different antenna gain phenomenon occurs with antennas at medium frequencies as compared to

antennas at VHF, UHF and higher frequencies. The antenna principles are the same, but the surface

wave becomes more signficant at MF. This is important because it is the surface wave that accounts

for practically all of the energy transmitted and received at these frequencies. The concepts of gain

and efficiency for antennas close to the surface of the Earth at MF are difficult to comprehend, since

a major portion of the power launched by the antenna is absorbed by the lossy Earth. A method for

calculating antenna gain of MF antennas including the normalization process to interface with the

propagation models has been discussed in this paper. Agreement with measurements has been

shown.

The propagation and antenna models described here can be used for performance prediction of ITS

applications in the roadway environment to determine design criteria and parameters for building

and testing these systems. The models can also be used to resolve measurement problems. These

models can also be used for performance prediction in an interference environment, and aid in the

resolution of interference problems. Propagation models that are used to analyze radio

communication systems require sound engineering judgement in their use for a particular analysis.
61

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