Propagation Characteristics of Very Low Latitude Whistlers by

Three-Dimensional Ray Tracing

Kenji Ohta

Faculty of Engineering, Chubu University, Kasugai, Japan 487

Yasuhiro Nishimura, Tomomi Kitagawa, and Masashi Hayakawa

Faculty of Electro-Communications, University of Electro-Communications, Chofu, Japan 182

SUMMARY

In magnetosphere propagation of very low latitude

whistlers occurs at geomagnetic latitudes around 10 de-

grees, both duct propagation and nonduct propagation are

conjectured as propagation mechanisms. The conjecture

for nonduct propagation is based on the fact that the en-

hancement factor that is needed in duct propagation,

reaches several hundred percent at very low latitudes and

thus duct propagation is hardly conceivable. The conjecture

for the duct propagation is mostly based on observation of

echo train whistlers, that must be generated by reverberat-

ing propagation along the same path. From this perspective,

the possibility of echo train whistler generation by nonduct

propagation is investigated using three-dimensional ray

tracing based on the night electron density model and the

IGRF magnetic field model for very low latitudes. We show

that echo train whistlers can be generated by considering

not only vertical emission but also conical emission angle

within a 3° emission angle to the ionosphere. It is also

shown that nonduct propagation can be transmitted down-

ward through the ionosphere over a wide frequency range

and that the point of downward transmission through the

ionosphere depends less on the frequency. © 1998 Scripta

Technica. Electron Comm Jpn Pt 1, 81(6): 42�50, 1998

Key words: Whistler; duct propagation; nonduct

propagation; ray tracing.

1. Introduction

Whistlers are electromagnetic waves in the VLF band

in which a part of the electromagnetic wave generated by

a lightning discharge reaches the other hemisphere, propa-

gating mainly along a geomagnetic force line [1]. Let the

frequency of a whistler propagating in the ionosphere and

the geomagnetosphere be f [Hz], then the propagation time

t [s] is given by:

where D [s1/2] is called the dispersion factor of the whistler.

Let fp be the plasma frequency of the propagation path, fHthe electron gyro-frequency, and let c be the velocity of the

light, then D can be written:

Two propagation configurations may be considered

for propagation of the whistler [1]. One is duct propagation,

CCC8756-6621/98/060042-09

© 1998 Scripta Technica

Electronics and Communications in Japan, Part 1, Vol. 81, No. 6, 1998Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J80-B-II, No. 4, April 1997, pp. 314�321

(1)

(2)

42

where the wave propagates through a path called a duct

formed along a geomagnetic line with a higher electron

density than its surroundings [2]. The other is nonduct

propagation, where the wave propagates through layers

with varying electron densities, following Snell�s law [3].

Since echo train whistlers, which are interpreted as

the result of reverberating propagation along the same path,

are detected in whistler observations, duct propagation is

in general assumed to be the propagation configuration for

whistlers at both high and low latitudes [4, 5]. This conjec-

ture however, is not realistic at very low latitudes for the

following reason. At very low latitudes the angle between

a magnetic line and the ionosphere is very small. The

enhancement factor that represents the electron density in

the duct with respect to the surrounding electron density in

the downward transmission of the whistler through the

ionosphere, reaches several hundred percent [6, 7], which

makes duct propagation unrealistic. Because of this, there

have been debates concerning the propagation configura-

tion of very low latitude whistlers.

A pioneering approach to very low latitude propaga-

tion at geomagnetic latitudes of 20 degrees or less was

presented in 1979 by Ondoh and others [8]. They concluded

that duct propagation occurred, based on observations of

echo train whistlers and on the fact that the distribution of

downward transmission points through the ionosphere con-

centrates in azimuth. In January 1988, the authors also

conducted multi-point simultaneous observations at three

very low latitude points in China, i.e., Wuchang (19.4°N),

Guilin (14.1°N), and Zhanjiang (10.0°N). They inferred the

occurrence of duct propagation based on the fact that there

were stable downward transmission points of whistlers

through the ionosphere and that there was a lower fre-

quency dependence of the downward transmission points

[9]. Another viewpoint is that there were many echo train

whistlers.

Subsequently, the authors applied two-dimensional

ray tracing (latitude and altitude) to nonduct propagation.

Duct propagation was again supported from a theoretical

point of view, since there were many discrepancies with

nonduct theory from the observations, e.g., that whistlers

in the frequency range below 2.5 kHz cannot be transmitted

through the ionosphere and that there was a clear frequency

dependence at the downward transmission points in the

ionosphere [10].

A problem, however, with the above ray tracing

approach is that a dipole magnetic field model was used.

There is a significant difference at very low altitudes be-

tween the dipole magnetic field model and the actual geo-

magnetic field (the international geomagnetic reference

force line, IGRF), and thus ray tracing should be based on

more realistic magnetic force lines and electron density

profiles. Another viewpoint is that an enhancement factor

as large as several hundred percent has never been observed

up to the present, and there is hesitation in concluding that

all of the frequently observed very low latitude whistlers

are due to duct propagation [11�14]. However, the exist-

ence of echo train whistlers, that are observed at very low

latitudes, continues to be the major basis for inferring duct

propagation.

With this background, we present a propagation

analysis for very low latitude whistlers by three-dimen-

sional ray tracing (latitude, longitude and altitude) based on

the 1990 IGRF magnetic field model and the electron

density profile at very low latitudes. We show that echo

train whistlers can also exist in nonduct propagation.

2. Night Ionosphere Model at Very Low

Latitudes

The electron density distribution in the ionosphere

and the magnetosphere at very low latitudes is in general

as follows (15]:

where r [km] is the distance from the center of the earth,

q[°] is the geomagnetic colatitude [90°-geomagnetic lati-

tude], and f[°] is the geographic longitude. NDE(r) is the

diffusion equilibrium model of Angerami and Thomas [16],

NI(r) is the correction factor for the lower ionosphere given

by Kimura [17], Nq(r, q) is the slope of the electron density

in the latitude direction given by Singh [18], and NL(f) is

the slope of the electron density in the longitudinal direc-

tion given by Zhou.

The generation of whistlers at very low latitudes is

usually observed only at night. Based on the winter night

parameters obtained by ISS-b from 1978 to 1980, the

electron temperature Te [K], the electron density

NRbase [cm-3] and the proportions of H+, He+, and O+ are

determined for a reference altitude of Rbase = 500 km. Table

1 shows these parameters for the electron density profile.

In the calculation of the correction factor for the lower

ionosphere, the altitude of the lower limit of the ionosphere

is set as 90 km.

The slope of the electron density in the latitudinal

direction is given by [15, 18]:

where C(r) is a parameter that determines the distribution

of the electron density at altitude r and a, sin q0 are

parameters that represent the slope of the electron density

in the latitude direction as modeled by Singh [18]. We set

a = 15, as in the study by Zhou and coworkers [15]. Set sin

(3)

(4)

43

q0 = 0.985, since the geomagnetic latitude of Zhanjiang is

10°. C(r) is given as follows by setting C0 = 0.5:

The slope of the electron design in the longitudinal

direction is [15]:

Here LG is a parameter that represents the slope of the

electron density in the longitudinal direction and f0 is the

reference geographic longitude. This study is based on

night observations from 00 hr. to 04 hr. local time in China,

where the observation points are located near longitude

110° E. Consequently, f0 = 110° in this study. LG is set to

�0.2 based on NSSDC (National Space Science Data Cen-

ter) figures. Since whistlers in the northern hemisphere are

considered, the parameters are based on the northern hemi-

sphere model. The wave is sent from a source in the

southern hemisphere.

Using the parameters of Table 1, the electron density

is calculated for the ionosphere in the equatorial plane,

considering the lower end of the ionosphere and the slope

of the electron density in the latitudinal direction. Figure 1

shows the profile.

3. Ray Tracing Calculation

In most previous ray tracings, the electromagnetic

wave from lightning discharge acting as the whistler source

is emitted in the vertically upward direction at a specific

altitude, since the diffractivity of the ionosphere is large.

More strictly, however, there is a small angular deviation

following Snell�s law away from the vertical direction, in

the transmission through the ionosphere. In the case of the

electron density profile of Fig. 1 the lower end of the

ionosphere is 90 km and the refractivity at the ionospheric

altitude of 120 km is 30 to 50. In this analysis, a maximum

3° conical angle was considered at an ionospheric altitude

of 120 km and waves propagating in the slant direction

along the latitudinal and the longitudinal directions were

examined.

Figure 2 shows the input latitude and the final latitude

based on the electron density profile of Fig. 1. The wave

source is placed at geographic latitude 110° in the southern

hemisphere, and the emission angle is slanted by 0° in the

longitudinal direction (e) and ±3° in the latitudinal direction

(d positive in the south direction). The frequency is set to 5

kHz. The input altitude and the final altitude are both 120

km.

Figure 3 shows the input and final latitudes when the

emission angle is slanted by 0° in the latitudinal direction

and ±3° in the longitudinal direction (positive in the east

direction) only for model A due to space limitations. The

dot indicates that the wave passes downward through the

ionosphere in the northern hemisphere.

In model A of Fig. 2, there are two points through

which the wave can passes downward through the iono-

sphere in the other hemisphere, around 10° and 20° geo-

magnetic latitude. They correspond to the conjugate points

indicated by the slant dashed-line. This is the same ten-

dency as obtained in two-dimensional ray tracing [10]. The

tendency is also the same in models D and E. In model B,

the passing points at low and high latitudes approach each

other. In model F, downward transmission is possible only

for a certain range of angles. In model C, the transmission

is impossible at any angle.

Table 1. Parameters of ion composition at 500 km

(5)

(6)

Fig. 1. Profiles of background magnetospheric electron

density.

44

The same tendency as in model A is observed for the

electron density parameters used by the authors in two-di-

mensional ray t racing a t al t i tude 1,000 km

(Te = 980 K, NRbase = 3,350 cm-3, H+ = 0.80, He+ = 0.10,

O+ = 0.10) [10], for the electron density parameters used

by Liang and coworkers [11] at an altitude of 500 km

(Te = 1,000 K, NRbase = 20,000 cm-3, H+ = 0.015, He+ =

0.058, O+ = 0.927), and the electron density parameters

used by Zhou and others [15] for an altitude of 500 km

(Te = 1,200 K, NRbase = 100,000 cm-3, H+ = 0.0005, He+ =

0.011, O+ = 0.9885). Consequently, the following discus-

sion considers only model A.

In model A, it is interesting that the passing point in

the latitudinal direction at about 10° is not affected greatly

by the deviation in the latitudinal and longitudinal direc-

tions. The passing point at about 20° is clearly affected,

especially by the deviation in the longitudinal direction. In

this study, however, very low latitude whistlers in China are

considered. Consequently, only the transmission charac-

teristics near 10° are examined. In the discussion of the

incidence angle to the ionosphere as well as the reflection

angle, the slopes of the ionosphere in the latitudinal and

longitudinal directions are considered.

The electromagnetic wave radiated from a lightning

discharge acting as a whistler source passes through the

ionosphere in the southern hemisphere at various angles.

Consider the case shown in Fig. 4. The whistler waves

emitted with deviations d in the latitudinal direction and e

Fig. 2. Relationship between input and final latitude (d

= ±3°, e = 0°).

Fig. 3. Relationship between input and final latitude (d

= 0°, e = ±3°).

Fig. 4. Relationship between input and final locations,

and incident angle at 5 kHz.

45

in the longitudinal direction from the two points in the

southern hemisphere at geographic latitude 0.50°S, longi-

tude 110.0°E, and at latitude 0.50°S, longitude 110.2°E,

which are separated by only 0.2° in the longitudinal direc-

tion at the ionospheric altitude of 120 km, are considered.

The whistler wave emitted with a conical angle of 3°

from the point with geographic latitude 0.50°S and longi-

tude 110.0°E in the southern hemisphere arrives in the

region shown by a circle in the upper left of Fig. 4, at an

ionospheric altitude of 120 km in the northern hemisphere.

The refractivity of the ionosphere at this point is 40. The

angle shown in the figure is the incidence angle to the

ionosphere (called the input angle). The white region in the

circle indicates downward transmission through the iono-

sphere with an input angle of less than 1.4°. Other regions

indicate that the wave is fully reflected by the ionosphere.

The mark M indicates the arrival point when the wave

is emitted with angles d = 0.00° and e = 0.20°. This point is

at geographic latitude 17.32°N and latitude 110.0°E in the

northern hemisphere. The input angle is 0.52°. Since the

maximum angle for which the wave can pass through the

ionosphere is 1.4°, it is obvious that the wave can pass

downward through the ionosphere.

On the other hand, a whistler wave that is emitted

with a conical angle of 3° from the point at geographic

latitude 0.50°S and longitude 110.2°E in the southern hemi-

sphere arrives at the circular region shown in the upper right

of Fig. 4. The mark ´ indicates the arrival point when the

wave is emitted with angles d = 0.00° and e = �1.00°. This

point is the same as the point shown by M at geographic

latitude 17.32°N and longitude 110.0°E in the northern

hemisphere. The input angle is 1.54°, which exceeds the

maximum angle 1.4° at which the wave can pass through

the ionosphere. Consequently, the wave is fully reflected at

an ionospheric altitude of 120 km.

Thus, at the point with geographic latitude 17.32°N

and longitude 110.0°E, there are two waves: the wave

passing downward through the ionosphere at an input angle

less than the maximum transmission angle 1.4°, and the

wave that is fully reflected, at an angle of 1.54°. It should

be noted that these two waves are emitted to the magneto-

sphere from two points in the southern hemisphere sepa-

rated by a very small distance.

In previous two-dimensional ray tracing, only verti-

cal emission of waves at a particular point was considered.

Consequently, in the other hemisphere, there is only a wave

that is transmitted or is fully reflected. In three-dimensional

ray tracing considering the emission angle, on the other

hand, there exist many whistler waves emitted from the

southern hemisphere at various angles, that arrive at various

points in the northern hemisphere at various angles. In other

words, there are a large number of whistler waves at a

particular point. Some pass downward through the iono-

sphere at an angle less than the maximum transmission

angle and other waves are fully reflected by the ionosphere

and propagate again in the magnetosphere.

4. Generation of Echo Train Whistlers

The major reason for conjecturing that very low

latitude whistlers are due to duct propagation is that echo

train whistlers are frequently observed. In two-dimensional

ray tracing in general, with vertical emission the whistler

waves emitted from the 10° and 20° points in the southern

hemisphere, shown in Fig. 2, can pass downward through

the ionosphere in the other hemisphere. The whistler wave

emitted from a point at 10° or 20° in the northern hemi-

sphere generally cannot pass downward through the iono-

sphere in the southern hemisphere, and are fully reflected,

since the magnetic IGRF force lines are not completely

symmetrical with respect to the magnetic equator. Further-

more, the roundtrip paths are not necessarily the same.

Thus, the ratio of the dispersion factors for short whistlers

and echo train whistlers will not be 1:3, causing the actual

observation to differ from theory. From this viewpoint, the

authors used three-dimensional ray tracing and examined

whether or not echo train whistlers can be generated.

4.1. Echo train whistlers caused by earth

reflection

Figure 5 shows the case where a short whistler re-

flected from the earth forms an echo train. Consider the

whistler wave emitted from geographic latitude 0.60°S and

longitude 110.0°E in the southern hemisphere with angles

d = 2.12° and e = �0.60°. The wave passes downward

Fig. 5. Ray path of echo-train whistler reflected from

the Earth.

46

through the ionosphere at input angle 1.16°, at geographic

latitude 17.54°N, and longitude 109.8°E. The dispersion

factor of this short whistler is 12.3 s1/2.

If the earth were flat, this wave would pass upward

through the ionosphere at geographic latitude 16.91°N and

longitude 112.6°E and at ionospheric altitude 120 km with

angles d = 0.40° and e = 1.05°. Then, the wave is incident

on the ionosphere at geographic latitude 0.89°S and longi-

tude 112.6°E in the southern hemisphere with input angle

9.94°, and is fully reflected. The wave again arrives at

geographic latitude 18.55°N and longitude 112.7°E in the

northern hemisphere, and passes downward through the

ionosphere input angle 1.06°.

The passing points of the short whistler and the echo

train whistler that pass downward through the ionosphere

in the northern hemisphere are separated by approximately

350 km, but two whistlers separated by such a short dis-

tance may be observed at a single point. In fact, for the

multi-point observations in China, most of the whistlers

that pass downward through the ionosphere near Zhanjiang

were also observed at Wuchang, approximately 1,000 km

away. In this observation, the dispersion factors of the short

whistler and the echo train whistler are 12.3 s1/2 and 36.9

s1/2, respectively by ray tracing, or in the 1:3 proportion.

4.2. Echo train whistler caused by ionosphere

reflection

In the previous case, the short whistler passes down-

ward through the ionosphere in the northern hemisphere

and is partly or totally reflected by the earth to generate an

echo train whistler. In the following, we consider the case

where the whistler cannot enter the transmission cone in the

northern hemisphere, and is fully reflected to generate an

echo train whistler.

In Fig. 6, consider a whistler that is emitted from

geographic latitude 0.50°S and longitude 110.0°E of the

southern hemisphere with angles d = 0.53° and e = �0.60°.

The wave passes downward through the ionosphere at

geographic latitude 17.38°N and longitude 109.9°E with

input angle 1.20°, generating a short whistler. The disper-

sion factor of this whistler is 12.3 s1/2.

On the other hand, a whistler emitted from geo-

graphic latitude 1.00°S and longitude 110.0°E of the south-

ern hemisphere, approximately 50 km distant from the

above point, with the angles d = 0.89° and e = �0.80°,

arrives at geographic latitude 17.51°N and longitude

109.8°E of the ionosphere in the northern hemisphere, with

input angle 4.68°. The wave is fully reflected and arrives at

geographic latitude 0.54°S and longitude 109.9°E of the

ionosphere in the southern hemisphere with input angle

1.91°. It is again fully reflected and propagates to the

northern hemisphere. It then arrives at geographic latitude

17.51°N and longitude 109.8°E of the ionosphere in the

northern hemisphere with input angle 0.72°. The wave

passes downward and produces an echo train whistler with

dispersion factor 36.9 s1/2.

The points of downward transmissions through the

ionosphere of the above short whistler and the echo train

whistler in the northern hemisphere are only approximately

15 km apart, and the two waves can be observed at the same

observation point. The dispersion factors are in 1:3 propor-

tion. Thus, it is seen that echo train whistlers can be gener-

ated by nonduct propagation, even if the whistler wave is

reflected at the ionosphere.

4.3. Echo train whistlers

Up to the present, the generation of echo train whis-

tlers has been discussed for the cases where the short

whistler is reflected from the earth, and where it is reflected

from the ionosphere. In practice, however, the losses in

reflection and transmission at the earth and the ionosphere

must be considered. In the following discussion, only the

loss in the northern hemisphere is considered.

When echo train whistlers are generated by short

whistlers that can be observed on the earth the wave must

pass downward through the ionosphere, be reflected from

the earth, and pass upward through the ionosphere. When

echo train whistlers are generated by short whistlers reflec-

ted from the ionosphere they experience only reflection loss

at the ionosphere. In reflection at the ionosphere, any angle

is possible, while in the case of the reflection from the earth,

only whistlers wave coming into the transmission cone are

reflected from the earth and pass upward through the iono-

sphere, i.e., the emission angle at an altitude of 120 km is

limited to angles within the transmission cone. From the

Fig. 6. Ray path of echo-train whistler reflected from

the ionosphere.

47

above reasoning, it is conjectured that there is less possibil-

ity that short whistler reflected from the earth generate echo

train whistlers. Thus, the echo train whistlers observed at

very low latitudes are whistlers produced by reflection at

the ionosphere.

It is expected that the ratio of the dispersion factors

for short whistlers and echo train whistlers is 1:3, since the

propagation paths are the same [1]. In Figs. 5 and 6,

however, the relation of 1:3 is maintained even though the

propagation paths are not the same. The reason for this is

probably as follows.

Consider the whistler dispersion given by Eq. (2). In

the case of middle and high latitude whistlers where the

highest point in the propagation path is high, fH is small; it

is inversely proportional to the cube of the distance. Con-

sequently, the dispersion has the largest effect on the result

of integration at the highest point of the propagation path

[1]. Thus, the dispersion factor is determined by the elec-

tron density at the highest point of the propagation path. In

other words, the dispersion factor is greatly affected if the

altitude of the highest point of the propagation path differs.

In order to realize the same dispersion factor, the

altitude must be the same, i.e., the propagation path must

be the same. In the case of very low latitude whistlers, on

the other hand, the altitude of whistler propagation is less

than 1,000 km and fH is at least half the value at the earth

surface. Furthermore, most of the entire propagation path

is spent in the F2 layer, where the electron density is the

highest. Thus, the dispersion factor is affected not only by

the electron density at the highest point, but also by the

entire propagation path.

Consider the case where the wave is emitted verti-

cally from geographic longitude 110°E and the latitude is

varied. Figure 7 shows the relation between the dispersion

and the highest altitude of the whistler propagation path. As

can be seen from Figs. 5 and 6, when the wave is emitted

from around geographic latitude 10° the altitude of the

highest point is 600 to 700 km. The dispersion factor at this

altitude is almost constant at 12.3 s1/2. From this argument,

it is conjectured that for very low latitude of around 10°

there are points for which the dispersion is almost constant

even if the propagation path is different. The dispersion

ratio between short whistlers and echo train whistlers is 1:3.

Thus, it is expected that the whistlers at very low latitudes

have propagation characteristics that differ from those at

other latitudes.

5. Ionospheric Transmission

Characteristics for Whistler Frequency

In our observation of the whistlers at very low latitude

in China, frequency components from 1 kHz to 8 kHz are

present by the sonogram despite the BPF (band-pass filter,

2 to 6 kHz) inserted into the observation system [19]. In our

previous two-dimensional ray tracing, on the other hand,

there is a frequency band in which the wave cannot pass

downward [10].

From this viewpoint, the ionospheric downward

transmission characteristics for the whistler frequency are

examined as in two-dimensional ray tracing. It is assumed

that the wave is emitted from geographic latitude 0.44°S

and longitude 110.0°E of the southern hemisphere, by

varying the whistler frequency in steps of 0.2 kHz. Then

three-dimensional ray tracing is applied. As a result we find

that downward transmission in the ionosphere cannot be

realized for the frequency band from 1.0 kHz to 2.2 kHz.

The tendency is almost the same as for the two-dimensional

ray tracing.

When the emission angle is considered, however, the

situation is different. When a wave is emitted from the same

point with angles d = �1.00° and e = 0.50° downward

transmission is realized for the frequency band of 1 to 10

kHz. The point of downward transmission moves to higher

latitudes with increasing frequency, as for two-dimensional

ray tracing. For 1 kHz and 8 kHz, the shift in the latitude

direction is as small as 4 km. Consequently, it seems

difficult to discriminate whether duct propagation or non-

duct propagation occurs from directional measurements of

the arriving wave based on electromagnetic field analysis

[19]. According to the results of ray tracing, the dispersions

at 1 and 8 kHz are 12.13 s1/2 and 12.46 s1/2, respectively. If

a more detailed dispersion analysis is applied in the future

[20, 21], it may be possible to discriminate between duct

propagation and nonduct propagation.

Fig. 7. Relationship between altitude of the path and

the dispersion of whistler.

48

6. Conclusions

There are many debates as to whether the whistlers

observed at very low latitudes (about 10°) are due to duct

propagation or nonduct propagation. The basis for conjec-

turing nonduct propagation is that the enhancement factor

needed for duct propagation increases to several hundred

percent at very low latitudes, in contrast to high latitudes,

and thus the assumption of nonduct propagation is unreal-

istic. The major reason for conjecturing duct propagation

is that echo train whistlers are observed owing to reverber-

ating propagation of the wave along the same propagation

path.

From this perspective, this paper has investigated

whether or not echo train whistlers can be generated by

nonduct propagation, based on the electron density model

for the very low latitudes and three-dimensional ray tracing

using the IGRF magnetic field model. We have shown that

echo train whistler generation is not confined to the vertical

direction and a conical angle of 3° or less is considered as

the input angle to the ionosphere for the cases of reflection

of a short whistler from the earth or from the ionosphere.

The frequency dependence of the point of downward

transmission through the ionosphere between 1 kHz and 8

kHz is only 4 km at an ionosphere altitude of 120 km. It is

thus seen that discrimination of the propagation configura-

tion by determination of the direction of arrival using

electromagnetic field analysis is still impossible. It is noted,

on the other hand, that there is a difference of approxi-

mately 0.33 s1/2 in the whistler dispersion factor, and there

remains a possibility that the propagation configuration can

be discriminated by a more detailed analysis of the disper-

sion.

The characteristics of downward transmission of

whistler waves through the ionosphere greatly depends on

the electron density profile, as well as on the parameters

specifying the slopes of the electron density in the longitu-

dinal and latitudinal directions. Although an analysis using

only certain parameters is given in this paper, it will be

necessary to continue the examination for a wider realistic

range of parameters. We intend in future to study the use of

more realistic electron density profiles, which are different

in the northern and southern hemispheres, and to analyze

downward transmission characteristics in the ionosphere.

Acknowledgments. The authors are most grateful

for the provision of electron density profiles at very low

latitudes by Prof. B.X. Liang, Prof. Z. T. Bao, and Prof. J.S.

Xu, Space Physics Institute, Wuhan University. As the

parameters of the latitude dependence of the electron den-

sity, the data of NSSDC are used in this study, for which

the authors are likewise grateful.

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AUTHORS (from left to right)

Kenji Ohta (member) graduated in 1966 from Department of Communications, Shinshu University, and became research

associate, Chubu University. Engaged in development of atmospheric noise observation equipment, especially automatic

whistler observation equipment, as well as signal analysis. Now Professor at Chubu University. Dr. Eng. Japan Society Atm.

Elect. Award 1991. Member, Japan Society Atm. Elect.

Yasuhiro Nishimura (member) graduated in 1994 from Department of Electronic Engineering, Chubu University.

Completed 1st half of doctor�s program 1996. Presently, with Nissei Electric Works. In graduate school, engaged in research

on propagation characteristics of very low latitude whistlers.

Tomomi Kitagawa (student member) graduated in 1995 from Department of Electronic Engineering, Chubu University,

and entered 1st half of doctoral program. Previously, engaged in research propagation characteristics of middle latitude

whistlers.

Masashi Hayakawa (member) graduated in 1966 from Department of Electrical Engineering, Nagoya University.

Completed master�s program in 1968. While in doctoral program became research associate and subsequently Assoc. Prof.

(1979) in Atmos. Elect. Research Laboratory, Nagoya University. Presently, Professor, University Electrocomm. Visiting

Lecturer 1975�76 Sheffield University, U.K. Researcher 1980�81 Nat. Space Research Laboratory, France. Engaged in

observational and theoretical studies of plasma wave in very high layers, examination of wave-particle and wave-wave

interactions based on direction measurement, earthquake prediction by electromagnetic waves, and environmental electromag-

netic engineering. Dr. Eng. Tanakadate Award, 1983, Japan Society Geoelectromagnetics. Award, 1989, Japan Society

Atmospheric Electrical. Member, Geoelectromagnetics/Earth-planet Society, Japan Society Atom. Elect., and AGU.

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