UNITED STATES EPARTMENT OF IOMMERCE

UBLICATION NBS TECHNICAL NOTE 610

U.S.

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COMMERCE

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Application of VLF

Theory to Time

Dissemination
NATIONAL BUREAU OF STANDARDS

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IAT10NAL BUREAU OF STANDARDS FEB ?1972

T$U>V occlq«7)

UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

Ui.NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 610

ISSUED NOVEMBER 1971

Nat. Bur. Stand. (U.S.), Tech. Note 610, 170 pages (Nov. 1971) CODEN: NBTNA

Application of YLF Theory to Time Dissemination

W. F. Hamilton and J. L. Jespersen

Frequency-Time Dissemination Research Section

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Final Report Prepared for National Aeronautics and Space Administration

under Contract No. 51740A-G

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

a means for making available scientific data that are

of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (Order by SD Catalog No. C13.46:610), Price $1.50 20402

Contents

Page

List of Tables iv

List of Figures v

1. Introduction 1

2. Time Dissemination Systems 2

3. Path Delay 3

4. Important Parameters for VLF Calculations 6

5. Theory of Calculations 10

6. The Calculations 15

7. Discussion and Conclusions 17

8. References 21

Appendix: Simplified Mathematics of the Theory 23
List of Tables

Page

Table 1. List of Calculations 29

Table 2. Comparison of Experimental and

Theoretical Diurnal Phase 32

Table 3. Comparison of Experimental and

Theoretical Group Delay 35

Table 4. Predicted Diurnal and Phase Delay 36

Table 5. Predicted Group Delay 38
List of Figures

Page

Figure 1. Normal Day Profile 41

Figure 2. Normal Night Profile 41

Figure 3. Deeks 1200 Profile 41

Figure 4. Bain and May Profile 41

Figure 5. Diurnal Phase 42

Figure 6. Relative Group Delay, Normal Day 42

Figure 7. Normal Night 43

Figures 8 - 40: Relative Phase, Normal Day 44

Figures 41 - 6 9: Relative Phase, Normal Night 59

Figures 70 - 102: Amplitude, Normal Day 74

Figures 103 - 12 8: Amplitude, Normal Night 91

Figures 129 - 167: Relative Group Delay, Normal Day 104

Figures 168 - 204: Relative Group Delay, Normal Night .... 124

Figures 205 - 233: Diurnal Phase 143

Figures 234 - 251: Relative Phase 158

Figures 252 - 254: Relative Group Delay 165

APPLICATION OF VLF THEORY TO TIME DISSEMINATION

W. F. Hamilton and J. L. Jespersen

The characteristics of time dissemination systems are

discussed. Particular emphasis is placed on the advantages

and problems of very low frequency (VLF) timing systems.

The parameters affecting VLF propagation along with causes

of variations in these parameters, are discussed. Three

methods of computing VLF propagation delays --mode theory,

geometric-optics, and zonal harmonics - -are compared,,

The method of zonal harmonics, used for the calculations

in this paper, is discussed in more detail. A method of

calculating reflection coefficients for a continuously varying

ionosphere is described. The theoretical values are compared with experimental measurements. Graphical results

of the calculations are included.

Key Words: Diurnal phase; Omega; time; time dissemination; VLF propagation; zonal harmonics.

1. Introduction

There are a number of existing or proposed time dissemination

systems which operate in the VLF region of the radio spectrum.

OMEGA is one of the most widely known such systems operating in

the 10. to 14. kHz range. Although the system is conceived

primarily for worldwide navigation, it has considerable potential for

time dissemination. It is this potential which motivates the present

report. Briefly, OMEGA is attractive for time dissemination for the

following reasons:

a. OMEGA could provide the basis for a unified worldwide time

dissemination system which does not presently exist.

b. A timing signal would be available reliably and continuously

at any location on the earth's surface.
c. The time accuracy available to the user would be commensurate with his equipment investment.

In the following sections we discuss:

a. Characteristics of time dissemination systems,

b. Details of those characteristics which are particularly important for the calculations presented in this report,

C Descriptions of the theory and method of calculation,

d. Graphical presentation of the results of the calculations,

e. Brief summary of the conclusions which may be drawn from

calculations.

2. Time Dissemination Systems

There are a number of common elements which characterize most

time dissemination systems. Among the more important are: accuracy,

repeatability, coverage, and ambiguity. The accuracy of a system is usually limited by ones knowledge of the propagation delay, whereas

repeatability is usually related to short-term variations in the characteristics of the propagation path. We can illustrate ambiguity by considering the following simple time dissemination system. Suppose the system

consists of a sine wave which is a repetitive waveform. We can't tell by looking at the signal which one of the zero crossings is the one that

is on time so the system is ambiguous. If we want to use it to improve

our knowledge of the time we must somehow pick out the correct cycle

and then measure its arrival time with respect to our local clock.

To pick the correct cycle, however, we must already know the time

from some other source to an accuracy equal to the period of our sine

wave (100 microseconds in the case of a 10 kHz sine wave). Many

users won't know the time to 100 microseconds and in addition, they

would prefer to use the same system from beginning to end. To reduce
the ambiguity it is usual to transmit another coherent frequency say at

1 0„ 1 kHz and then inspect the 100 Hz difference frequency. To pick

out the on-time cycle of the 100 Hz difference frequency, we need

only know the time to 10 milliseconds (the period of a 100 Hz sine wave).

We see that by adding more signals at different frequencies, we can reduce the ambiguity of the system to any desired level. However, as we reduce the ambiguity, we must at the same time increase the amount

of frequency spectrum which the system occupies. For very low frequency systems, we can't very well put on modulations which require

large bandwidths. That is, you can't put a signal which occupies 50 kHz

on a 10 kHz carrier. In many cases, the usefulness of the system may

be increased by averaging many measurements.

3. Path Delay

Path delay predictions, and thus, the accuracy of the system are

limited by knowledge of the geometry of the transmitter and user locations and by the availability of information concerning the propagation

medium. At VLF, at great distances from the transmitter, the signal

velocity has a nearly constant value, whereas near the transmitter the

average velocity between the user and the transmitter changes in an

irregular way as a function of distance. In general, the transition from

regular to irregular occurs at a distance of about 10, 000 kilometers

from the transmitter. In this report, the irregular region is of most

interest. It is important to consider this near region because here the

signal-to-noise ratio is greatest and receivers' costs, in principle,

should be at a minimum. To consider in more detail the problem of

path delay, we must introduce the ideas of phase velocity, V , and

group velocity, V , as they apply to VLF.
Let us consider a simple two-frequency timing system. Suppose

we have two closely spaced cw signals at frequencies fl and f2 where

fZ - fl - ^f. In this system, the beat frequency /\f is used as a coarse time marker to identify a specific cycle of either carrier frequency.

This removes ambiguities and a more precise time of arrival measure- ment is made at the higher carrier frequency. The difference frequency signal will travel from transmitter to receiver with a group

velocity which depends upon the phase velocity versus frequency characteristic of the propagation medium. Specifically, the group velocity

is related to phase velocity V by the following expression:

V = 1/

g X~(w /V ) (1) d(D P where U) is the angular frequency. We see that the concept of group

velocity breaks down if the derivative -— ( ^ /V ) is not well behaved.

dm p

In a dispersive medium, such as we are considering at VLF, Vp is a function of oi so that t following expression:

function of uu so that the group delay, t over a path r is given by the

t d

= hHfi-^-x-M (2)

At very low frequencies the phase velocity versus frequency dependence

cannot be expressed as a simple function of the properties of the propagation medium, except perhaps at great distances from the transmitter.

The most accurate results are obtained from numerical solutions of

Maxwell's equations, i.e., full wave solutions, with some realistic

model of the ionosphere, the approach taken in this paper. Since the

difference frequency may travel with a group velocity that is different

than the phase velocity of either one of the two carrier frequencies, the
difference frequency signal may slip in phase with respect to either one

of the carrier frequencies as a function of distance from the transmitter.

This could lead to incorrect cycle identification of the carrier frequency

unless the difference in velocities is known. At great distances, one can correctly identify a cycle by using different group and phase

velocities. Unfortunately, the concept of group velocity applies only

at great distances from the transmitter, where the phase velocity versus frequency characteristics are well behaved, i.e.,

-— ( : i/V ) is well behaved,, Near the transmitter, a different approach

drj) ' p

must be used. To help illustrate what is happening near the transmitter

we will consider first how a VLF pulse would propagate in this region.

There are two distinct effects which one must consider. First,

the average propagation velocity of the pulse (group velocity) may be

equal to or less than the velocity of light. Second, the pulse may be

changing its shape as it propagates through the medium. If a particular

point on the pulse has been tagged as the time reference point, at the

transmitter, it is necessary to determine how this "tag" point is mapped

into its new position as the pulse distorts during its propagation away

from the transmitter. If this is not known, it will not be clear to the user which point on the received pulse represents the reference point as transmitted. In addition, having correctly identified the tagged

point, one must also consider the fact that the average pulse velocity

may be less than the velocity of light. A multifr equency VLF system

is somewhat similar to the propagation of a pulse in the following sense: A short pulse of length T consists of a spectrum of frequencies, i. e. , a number of Fourier components which occupy a spectral region approximately equal to 1/t„ In a dispersive medium, such as the ionosphere,

each one of these Fourier components will travel with a different phase

velocity. In a sense then, when we use a VLF tone system, we are
sending one Fourier component at a time and the user, if he desired,

could by storing the received tones add them together to produce something looking like a pulse. This composite waveform would have some

well defined shape as it left the transmitter. Because of the dispersive

properties of the propagation medium this composite waveform will

change its shape as a function of distance from the transmitter until

it has propagated out to the region where the phase velocity is well

behaved. Beyond this point, the composite waveform would maintain

essentially the same shape as a function of distance although it would

not be the same as transmitted. To use the time system for accurate

transferral, it is necessary to know in detail the structure of this

composite waveform as a function of distance from the transmitter.

Similarly, use of the VLF multitone system for accurate time transferral requires detailed knowledge of the phase versus frequency and

distance characteristics of the signals. We will turn to the details of

theoretical calculations of VLF phase velocity in Sections 4 and 5.

4. Important Parameters for VLF Calculations

The phase velocity versus frequency characteristics at VLF are

particularly sensitive to the propagation medium. The propagation

medium is characterized by the conductivity of the ground, the electrical

properties of the ionosphere, the direction of propagation with respect to the earth's magnetic field,and the strength of the magnetic field.

The most important electrical properties of the ionosphere for

VLF calculations are:

a. The electron density vs height.

b. The proton density vs height.

c. The collision frequency between charged and neutral particles vs height.
The composition of the ionosphere is not static. Many different

types of events, for example solar flares, can cause short-term

variations in the ionization. The theoretical calculations in this report

do not take these short-term variations into account but are based on

the average properties of the ionosphere as it might appear at noon and

at midnight of the summer solstice at midlatitudes.

There is considerable experimental evidence that the propagation

delay of a VLF signal from one day to the next at the same time of day

is repetitive to a few microseconds, so that consideration of the average

properties of the ionosphere should be sufficient for many applications.

This is an attractive feature of VLF signals for time dissemination.

We list here briefly some physical processes responsible for these

average properties. The most important factors are as follows:

a. Solar X-rays (=10 A) above 85 km at sunspot minimum and

above 7 km at sunspot maximum;

b. The ionization of NO by solar Lyman -a in the 70 to 90 km

range;

c. Primary cosmic rays below 70 km with the flux increasing by

a factor of about 4 from sunspot maximum to minimum. However, this

subject is very volatile and recent work indicates that water vapor

and metastable states of O are important.

d. Various reactions between free electrons, ions and neutral

particles; and

e. Mechanical and electrodynamical forces which redistribute

the ionization.

Although, as stated previously, the calculations in this report are based upon the average properties of the ionosphere, it is important to realize that there are short-term variations in the medium. The shortterm phase variations are one of the most noticeable features of a VLF
record. These fluctuations are much shorter (of the order of 30 minutes) than one would expect from solar controlled phenomena and are

undoubtedly at times related to the motions in the neutral atmosphere.

There is not much information on the turbulent structure of the

mesosphere (the region of the neutral atmosphere which overlaps the

D Region) because this is a region that is too high for balloon soundings

and too low for satellites. However, there is some information on the

irregular structure in this region based upon rocket and meteor trail

investigations. For example, sodium cloud measurements by Blamont

and Jager indicated the existence of irregularities below 100 km which

they explained by a combination of turbulent mixing and molecular

2 diffusion. About 10 years ago, Greenhow and Neufeld suggested the

presence of large scale irregularities in the 80-90 km region based

upon radar observations of meteor trails. However, there is some

controversy about these observations since later measurements by

3 Manring, et al. , via rocket sodium cloud measurements suggested

that the turbulence explanation may be spurious and in fact may be due

to the rocket. An alternative explanation has been made by Charney

4 and Drazin who suggest that the irregularities are produced by the

propagation of gravity waves into the upper atmosphere.

In any case, phase fluctuations of VLF signals are in some fashion

related to the irregular structure of the mesosphere.

As discussed earlier, realistic calculations of VLF radio fields are

obtained from full wave solutions and for the present these solutions

have only been obtained for plane stratified ionospheric models. Thus,

the possibility of using this approach to obtain wavefields for a more

complicated irregular ionosphere does not seem particularly practical

at the present time and it appears necessary to resort to simpler more

approximate methods.
As an example of one such possible approach, suppose that one is

several thousand kilometers from a VLF transmitter. At such a distance,

the phase velocity of the radio signal is nearly constant (this corresponds

to the situation in a waveguide where mode 1 is dominant) and is given

approximately by

V = c/7lT) 2 /l6h

2

(3) p o

where \ is the radio wavelength, h the height of the ionospheric reflecting

layer, and c the velocity of light. To a first approximation, we might

consider that the effect of a large irregularity would be to change the

height of the reflecting layer by an amount n h, which would in turn

alter the phase velocity. Thus from the equation above, we obtain

2 /VV^_2 Ah

( 4 ) V I6h

2 h

p o o since > < < h . o

This change in phase velocity would produce a change in the phase

of the received signal which could be interpreted in a straight forward

manner for this situation. However, the situation becomes more

complicated when the irregularities are small compared to the path

5 length or to the radio wave length. Crombie has attempted this

problem theoretically. He has taken the work by Rice which was

developed specifically for VHF forward scatter problems and has

attempted to adapt it for the interpretation of VLF phase measurements.

Although this work appears to be far from exact, it does give some

interesting results. Specifically, Crombie considers the two cases when VLF phase measurements are made on a distant transmitter using

two VLF receiving sites. In the first case (longitudinal case) the two

receiving sites are in line with the transmitter, and in the second case

(transverse case) occurs when the line joining the two receivers is
perpendicular to the line between one of the receivers and the transmitter,

7 He then takes some experimental data of Hargreaves which apply to the

transverse case and concludes that the average size of the irregularities

is about 65 km and that the horizontal drift velocity is about 35 m/sec.

Q Crombie compares these values with some results from Hines , (which

appear to apply to the region in question) which predict a horizontal

velocity of about 100 m/sec. for internal gravity waves with horizontal

components cf 65 km, Although the predicted value is higher than that

deduced by Crombie, the values agree within an order of magnitude.

5. Theory of Calculations

Three approaches to theoretical VLF calculations have seen wide

9 usage: (1) Wave guide mode theory, (2) geometric optics, and (3) zonal

harmonics. Mode theory has the advantage that at large distances only one mode need normally be considered. As the distance between the

transmitter and receiver decreases, more modes are necessary. Some

disadvantages are that physical interpretation of modes is difficult and

the introduction of varying propagation parameters along the path

becomes very complicated. It is possible that significant modes may

be overlooked. In addition, unless a Fourier transform into the time

domain is employed, mode theory does not account for the observed

difference in arrival time between the ground wave and the various

sky waves. Geometric optics, on the other hand, lends itself well to clear

physical interpretation. Various "hops" arrive later than the ground

wave. The accuracies of geometric optics are greatest near the transmitter, just the opposite of mode theory. The approximations made by

this theory become more valid with increasing frequency, but are in

general not valid at VLF except very near the transmitter. Geometric

10
optics is not a rigorous theory for electromagnetic wave propagation in

the VLF region.

The mathematical derivation of the zonal harmonic and geometric

series representations for electromagnetic wave propagation is well

, , T ,, 10, 11, 12, 13, 14, 15 documented by Johler „ The approach used here will

be primarily nonmathematical in an attempt to give physical insight to this method.

The use of zonal harmonics in propagation formulae was introduced

early in the 20th century by Debye (for electromagnetic waves) and

17 Lord Rayleigh (for sound). It was not, however, until recently that

practical application of this theory could be accomplished. The high

speed digital computer allows summation of the large number of terms

that are required for convergence of the series. (Here convergence is used to mean that additional terms do not significantly effect the result. )

In addition, new theoretical advances have greatly reduced the number

of terms required for convergence.

Zonal harmonics allow the introduction of rigorous analysis while

maintaining the clear physical interpretation of geometric optics. The

term in the classical zonal harmonic series containing both the ground

and ionospheric reflection coefficients can be represented as a geometric series. Inverting the order of summation- -summing the zonal

harmonic series before the geometric series--yields a series, each

term of which is analogous to the geometric optic series. The zeroth

order term of the geometric series is the ground wave and the jth term

represents a wave which has suffered j reflections from the ionosphere.

The large number of terms ordinarily required for convergence of

the zonal harmonic series is found to be necessary only because the

ground wave is included in the calculations. The higher order terms

converge with approximately one- fifteenth the number of terms. Since

1 8 the ground wave may be computed more effectively by other means

the remaining terms of the geometric series can be rapidly computed.

11
Very few terms are required for convergence of the geometric

series at VLF, Under normal conditions, 10 terms (i. e. , hops) have

been found sufficient for convergence to distances of 8, 000 to 10, 000

kilometers. Greater distances may require a few more terms.

Each term of the geometric series may be thought of as a set of

"n-waves" (the n terms of the zonal harmonic series). Each of the n- waves is incident on the ionosphere and the ground at an angle which is an approximate function of n. (The rigorous theory uses complex angles

of incidence which are given exactly by functions of n. Little difference

has been noticed between using real and complex angles of incidence since

normally the imaginary portion is quite small. ) This function of n is the

determining factor for the number of terms required for convergence.

The series converges abruptly as the angle of incidence approaches

90 degrees. The contribution of terms where the angle of incidence

is less than approximately 30 degrees is negligible. Consequently, even fewer terms are required.

Because of lateral displacement at both the ionosphere and the

ground, no specific angle of incidence is required for a wave to reach

a particular distance. Lateral displacement as used here means that

a wave may be trapped and propagate radially within the ionosphere or ground for some distance before emerging again to continue normal

propagation (see figs. 12 through 15, pp. 45-48, Johler and Mellecker,

1970--Ref. 15). Most of the reflection does, however, take place near

the angle predicted by geometric optics.

Each n-wave contains a factor which may be called the effective

reflection coefficient. In essence this factor is the sum of all possible

reflection coefficients for ordinary and extraordinary reflection from

the ground and ionosphere. For the case of vertical electric polarization of both the transmitter and receiver antennae, for example, only

12
ordinary reflection can effect the field for the first hop so the effective

reflection coefficient is T „ Both ordinary and extraordinary reflecee

tion can affect the second hop field and the effective coefficient is T R T + T R T Higher order coefficients become very em m me ee e ee complicated. Here T stands for the ionospheric reflection coefficient

with the first subscript denoting the incident polarization and the second

the reflected polarization (e is vertical electric and m is vertical magnetic). The ground reflection coefficient, R, needs only one subscript

since no abnormal component is generated during ground reflections.

Since most of the reflection for a given "hop" takes place within

a region, local reflection coefficients may be introduced. For example,

a land sea boundary changes the reflection coefficient of the ground and

may be incorporated. Similarly, if part of the path is in sunlight while

the rest is in darkness (i. e. , the "terminator" occurs along the propagation path) this fact may be used in the analysis.

The ionospheric reflection coefficients which are most readily

available are plane Fresnel coefficients. These may be used as an approximation to the true coefficients. The approximation becomes

quite good if the coefficients are multiplied by a convergence-divergence

factor. The best method, however, is to allow complex angles of

incidence. It has been stated earlier that the contribution of the

imaginary term will normally be small, but it may not be mathematically

negligible.

Since the only distance related portion of each n-wave is the

Legendre function, the field at many distances can be calculated simultaneously. Upwards of ZOO distances may be calculated with only

minimal increase in computation time.

It is necessary to compute the reflection coefficients for each n. This is no difficulty for the ground reflections since this calculation

13
is relatively straightforward and does not consume much computer

time. The ionospheric reflection coefficients, however, are computationally quite complicated and can consume as much as 2 minutes

of computer time for each set of coefficients required. Because the

reflection coefficients are a smooth function of angle of incidence

(of n), it is practical to set up a table of reflection coefficients and to

interpolate on this table to obtain the desired coefficient. The table

should be made in such a way that no extrapolation is necessary.

The electrical qualities of the ionosphere may be characterized

by electron and ion density profiles, and a collision frequency model.

To facilitate calculation of the reflection coefficients, the ionosphere

is divided into layers. Each layer is assumed to have uniform densities.

By increasing the number of layers, this representation can be made

as close as desired to the continuous distribution.

Maxwell's equations and the equation of motion of the electron are

solved yielding transmitted and reflected portions of the incident wave

for each of the layers. By keeping track of these upgoing and downgoing waves and solving for continuity at the boundaries between each of the

layers, the reflection coefficients are obtained.

Coupling of the vertical electric and vertical magnetic polarizations within the ionosphere causes the generation of an abnormal component and necessitates calculation of two reflection coefficients for

the vertical electric incident polarization. Also, since ground reflections do not change the polarization of the incident wave (i. e. , there is no coupling at the ground), it is necessary to compute reflection

coefficients for normal and abnormal ionospheric reflection of vertical

magnetic incident polarization. To summarize, four reflection coefficients must be computed for each angle of incidence; (1) vertical electric

polarization of the incident and reflected wave (T ), (2) vertical electric ee

14
polarization of the incident wave with abnormal or vertical magnetic

polarization of the reflected wave (T ), (3) vertical magnetic polarization of the incident and reflected wave (T ), and (4) vertical magnetic mm

polarization of the incident wave and vertical electric polarization of the

reflected wave (T ). me

NOTE: Vertical electric polarization is in quadrature (at 90 degrees)

to vertical magnetic polarization and may also be called horizontal magnetic polarization^ Similarly, vertical magnetic polarization may be

called horizontal electric polarization.

Each reflection coefficient gives the phase and amplitude of the

reflected wave. The phase is normally referenced to the bottom of the

ionosphere. This causes the phase to vary rapidly with angle of incidence.

Interpolation on this wildly varying phase can be very inaccurate. The

problem is solved by re-referencing the phase to a height where the

phase is slowly varying with angle of incidence. This height is sometimes referred to as the effective reflection height.

6. The Calculations

Table 1 lists the calculations that are reported in this paper.

Throughout these calculations a magnetic dip of 70 degrees and a magnetic field strength of 40 ampere-turns /meter have been employed.

These values are a good approximation of the magnetic field vector

experienced in the United States. All of the phase and group delay

graphs are relative to a medium where the index of refraction is 1. 0001.

Two ionospheric profiles (electron density as a function of height

above the surface of the earth) designated normal day (fig. 1) and

normal night (fig. 2), have been used for the majority of the calculations

with six other profiles used to demonstrate the types of fluctuations one might expect under certain conditions. The normal day and normal

night profiles were raised and lowered 2. 5 km. While this is admittedly

15
an artificial alteration of the ionosphere, it was felt that calculations

made by using such profiles would indicate the variations to be expected

19

during ionospherically active periods. The Deeks 1200 profile (fig. 3)

20

and the Bain and May profile (fig. 4) represent sunspot maximum and

sunspot minimum conditions as experienced over Great Britain,

The variable most comprehensively treated herein is magnetic

azimuth. The majority of the calculations were repeated with at least

three changes in azimuth. For purposes of interpolation, propagation

towards the magnetic north (0 degrees) has been assumed to be reciprocal to propagation towards the magnetic south (180 degrees).

The surface reflection coefficients were computed on the basis of a conductivity of . 005 mhos per centimeter and a ground dielectric

constant of 1 5 for land, and a conductivity of 5 mhos per centimeter

and dielectric of 80 for sea. Unless the graphs are otherwise marked,

the values for land were used for the computations.

Table 2 gives a comparison of the theory and experimental data.

These data are in the form of diurnal phase shift measurements.

Diurnal phase shift refers to the phase difference between day and

night, resulting from increased ionization of the upper atmosphere

caused by the sun's rays. This phase difference was computed as the

night phase at a given geodetic distance minus the daytime phase at

that same distance. A few group delay comparisons have been made

in table 3

.

Table 4 lists the predicted daytime phase delay and the diurnal

variation for various sites. Table 5 gives the associated group delays

for these same sites. The values given for Canberra, Australia

represent an extrapolation from the graphs.

16
Figure 5 gives a comparison of theoretical and experimental

diurnal phase shifts measured by NBS on the 20. kHz transmissions

from WWVL. Figure 6 compares theoretical and experimental relative

phase values for the same transmissions.

Figures 8 through 69 give a phase correction in radians for various

frequencies and magnetic azimuths under normal day and normal night

conditions. This phase correction is that value which must be added

to the phase delay of a hypothetical wave traveling at the speed of light

in a medium where the index of refraction is 1. 0001. Consequently,

these figures are labeled "Relative Phase."

Figures 70 through 128 give the amplitude in volts per meter,

referenced to unity dipole current, for the various frequencies and

azimuths under normal day and normal night conditions.

Figures 129 through 204 give the group delay correction in microseconds for various frequencies and azimuths under normal night

conditions. Again, because this correction is obtained as the difference

between the absolute group delay and the delay of a wave traveling at

the speed of light and an index of refraction of 1. 0001, the figures are

labeled "Relative Group Delay. " The actual method of calculation of

21

this group delay has been developed by Fey and Looney and is analogous

to the method presented earlier in the text. Figures 205 through 233, "Diurnal Phase, " give the phase shift to be expected between night and day using the normal day and normal

night profiles.

Figures 234 through 254 are special cases of the previous calculations in that either the profile or ground conditions are changed from

previous figures for the same frequency and azimuth.

7. Discussion and Conclusions

The analysis of the theoretical data presented here is aimed at timing implications. It should be noted during the analysis of these

17
graphs that the calculations have a tendency to deteriorate with distance.

This does not affect many of the graphs, but at the point where the trend

towards linearity with increasing distance breaks down, the graph should

be disregarded or, conversely, the linearity may be extrapolated to obtain the desired results. The cause of the degeneracy is a need for

more terms (hops) in the geometric series summation,

The fluctuations in the amplitude of the sum of the various hops is highly correlated to the points at which the amplitudes of two of the hops

o are equal (see fig. 78 - -Amplitude, 12, 5 kHz, 150 , Normal Day).

These amplitude fluctuations are also well correlated to fluctuations in

phase. Unfortunately, it is precisely these points of equal amplitude

that are most likely to be displaced most by small changes in the

ionosphere. Therefore, there are certain critical distances at which

the phase is likely to be unstable. Since there are more fluctuations

in both phase and amplitude under nighttime conditions, this may account

for the large phase deviations normally experienced at night. An interesting presentation of this idea is given in figure 7. Here phase and

amplitude are combined on one graph for 13. 6 kHz, 90°, Normal Night.

(r is a logarithamic scale giving amplitude in volts/meter and A is relative phase in degrees. ) The points on the curve are at 500 km increments.

It is interesting to note that the gross features of the nighttime phase

and amplitude graphs at the Omega frequencies are very similar to the

daytime features near Z0 kHz. This points out the dependence of the

field upon the ratio of wave length to ionospheric height. One implication of this dependence is that the Omega frequencies may be nearly as

reliable for time synchronization at night as are the WWVL transmissions

during the day. The graphs of the theoretical group delay indicate that,

with a few exceptions, this may indeed be the case. The smoothness of

the night phase from several of the Omega stations monitored at the

National Bureau of Standards again points towards reliable time transfer

1
during the night at these frequencies. Near 20 kHz, however, it is

apparent that synchronization cannot be accomplished with any degree

of reliability during the night.

During the day, synchronization at the Omega frequencies should

be very reliable. The small fluctuations in the group delay graphs

are dissipated in nearly all cases by approximately Z, 000 km. From

this point on the group delay becomes nearly linear and very predictable.

One is able to arrive at an estimate of the maximum timing error

to be expected using a given pair of frequencies, by examining the

group delay plots. In the majority of cases these plots have fluctuations

which are similar to damped sine waves. The amplitude of these waves

is then the maximum expected error. Neglecting the first 500 to 1000

kilometers, the following estimates of maximum timing error are given:

a. for the Omega frequencies during the day the expected error

is approximately ± Z0 fisec;

b. for the frequencies near Z0 kHz during the day, approximately

± 50 Msec;

c. for the Omega frequencies at night, approximately ± 70 Msec;

d. for the frequencies near Z0 kHz during the night, approximately

± 1Z0 Usee.

Because of the similarity of the relative phase graphs with changes

in azimuth it appears that linear interpolation may be used for both

azimuth and distance from the transmitter during the day. The night

phase, though, is not nearly so well behaved. In fact, the night phase

at Z70° is very dissimilar to the 90" and the 180° phases (which are

rather alike). A possible area for further study would be to determine

whether these changes take place linearly with azimuth.

19
The fluctuations (high frequency Fourier component) in group delay- are very much correlated to the frequency separation. Wide frequency

separations have fewer and smaller fluctuations than do narrow frequency separations. At some point then, one must choose between the

high resolution and subsequently large ambiguities of wide separations,

and the low resolution but small ambiguity of small separations,,

The Omega navigation system, with its proposed worldwide coverage, has great potential as a time synchronization tool. The variety

of frequencies and frequency separation, coupled with the possibility of

24 -hour utilization, available at each of the eight stations in the network

offer medium to high resolution time synchronization to almost all locales.

2
8. References

[l] Blamont, J. E. , and de Jager, C. , "Upper -atmospheric turbulence

determined by means of rockets, " J. Geophys. Res., 67, No. 8, pp. 3113-3119, July 1962.

[2] Greenhow, J. S. , and Neufeld, E. L. , "Large scale irregularities

in high altitude winds, " Proc. Phys. Soc. , 75, pp. 228-234, I960.

[3] Manring, E. , Bedinger, J. F. , and Knaflich, H. , "Measurements

of winds in the upper atmosphere during April 1961, " J. Geophys.

Res., 67_, No. 10, pp. 3923-3925, September 1962.

[4] Charney, J. G. , and Drazin, P. G. , "Propagation of planetaryscale disturbances from the lower into the upper atmosphere, "

J. Geophys. Res., 66_, No. 1, pp. 83-109, January 1961.

[5] Crombie, D. D. , "Effects of normal ionospheric fluctuations on propagation of VLF signals to great distances, " Phase and Frequency

Instability in Electromagnetic Wave Propagation (AGARD-EWP Sym. , Ankara, Turkey, October 4-12, 1967), AGARD Conf. Proc. No. 33,

K. Davies (Ed.), pp. 35-43 (Technivision Services, Slough, England,

July 1970).

[6] Rice, S. O. , "Statistical fluctuations of radio field strength far

beyond the horizon, " Proc. IRE, 41_, No. 2, pp. 274-281,

February 1953.

[7] Hargreaves, J. K., "Random fluctuations in VLF signals reflected

obliquely from the ionosphere, " J. Atmos. Terr. Phys., 20,

pp. 155-166, 1961.

[8] Hines, C. O. , "Internal gravity waves at ionospheric heights, " Can. J. Phys., 38, No. 11, pp. 1441-1481, November I960.

[9] Wait, J. R. , and Spies, K. P., "Characteristics of the earthionosphere waveguide for VLF radio waves, " Nat. Bur. Stand. (U.S.)

Tech. Note 300, 1964. IstSuppl., February 1965; 2nd Suppl. , March 1965.

[10] Johler, J. R. , "On the analysis of LF ionospheric radio propagation

phenomena," J. Res., 6 5D, No. 5, pp. 507-529, September -October

1961.

21
[ll] Johler, J. R. , and Berry, L. A., "Propagation of terrestrial

radio waves of long wavelength --theory of zonal harmonics with

improved summation techniques, " J. Res., 66D, No. 6, pp. 7 37-

773, November -December 1962.

[12] Johler, J. R. , "Concerning limitations and further corrections

to geometric -optical theory for LF, VLF propagation between the

ionosphere and the ground, " Radio Sci. J. Res., NBS/USNCURSI, 6 8D, No. 1, pp. 67-78, January 1964.

[13] Johler, J. R., "Zonal harmonics in low frequency terrestrial

radio wave propagation, " Nat. Bur. Stand. (U.S.) Tech. Note

335, April 13, 1966.

[14] Johler, J. R., "Theory of propagation of low frequency terrestrial

radio waves --mathematical interpretation of D-region propagation

studies, " ESSA Tech. Report IER48-ITSA 47, August 1967.

[15] Johler, J. R., and Mellecker, C, "Theoretical LF, VLF field

calculations with spherical wave functions of integer order, " ESSA Tech. Report, ERL165-ITS 106, April 1970.

[l6] Debye, P., "Der lichtdruck auf kugeln von beleibigem material, " (English translation: "The pressure of light on spheres made of any material. "), Ann. Physik (Leipzig), 30_, pp. 57-136, 1909.

[17] Lord Rayleigh, "On the acoustic shadow of a sphere, " with

Appendix giving the values of Legendre's functions from P to P at intervals of 5 degrees by Prof. A. Lodge, Phil. Trans.

Roy. Soc, London, A203 , pp. 87-110, 1904.

[18] Johler, J. R., "Ground wave propagation in a normal and an

ionized atmosphere, " ESSA Tech. Report, ERL12 1 -ITS 85, 1969a.

[19] Deeks, D. G. , "D-region electron distributions in middle latitudes

deduced from the reflexion of long radio waves, " Proc. Roy. Soc,

A291, pp. 413-437, 1966.

[20] Bain, W. C. , and May, B. R. , "D-region electron-density

distributions from propagation data, " Proc. IEE, 114, No. 11,

pp. 1593-1597, November 1967.

[2l] Fey, L. , and Looney, C. H. , Jr., "A dual frequency VLF timing

system," IEEE Trans. Instrum. andMeas,, IM-1

, No. 4, pp. 190-

195, December 1966.

22
APPENDIX

In a series of papers, Johler has combined the zonal harmonic

series with the geometric series and by extracting the ground wave

from the total field, greatly reduced the number of terms required

for convergence of the series. The material presented in this Appendix is an attempt to condense this theory.

To gain insight into this application of zonal harmonics we will

first consider the case of a sharply bounded isotropic ionosphere and

then generalize to an anisotropic ionosphere with a continuously varying electrical makeup.

For the isotropic case, and particularly for vertical electric

polarization, the zonal harmonic representation is

E =— — > n(n+l) (2n+l) P (cosfi)^ ' >Jr r s 4 /^j n la 1 V Ar! n=0

( 2 ) \ i

1 + R -r-— 1 + T —j-^ 111- RnTn [

' n ^ I ) n „ 2 )

J \ ia

I I is the intensity of the electrical source, a point dipole, which for

convenience may be taken to be 1. \d is the permeability of space

— 7 8

(4tt x 10 henry/meter) and c is the speed of light (2. 997925x10 m/s),

a is the radius of the earth and k is the wave number of air. P (z) is

l n

the solution of Lesendre's differential equation. R and T are the e n n ground and ionospheric reflection coefficients respectively (the subscript denotes that the reflection coefficients must be recomputed for

each n). £, is the spherical wave function and ifr = — t, , + t,

.

i a iaZL 1 a 1 a_l<

23
The geometric series may be introduced by noting that the denominator of eq (1) can be expressed as a geometric series.

[i-v„r= i+jKT

„]

j <2 »

Substituting into (1) and interchanging the order of summation yields

e = e +y E

r r ; o £_4 r ,j (3)

j = l where

e = ^44f F ( n )^

(2) ^

(1)

r

< i+r ) ^ , o 8n 2 4 j[_4 i a l a n k^a n=o

and

U c I t co I ^i ) ^(a ) E . = -£- -^-V F(n) C(2)

^(1) (1+R ) 3 U^ ^4 [R ] J_1 [T ] J (5) r,j 8TT s 4 ^

'ia la v n' ) „(s ) (i ) [ L n

J n

J

i n=0 bia ig

and F(n) = n(n+l)(2n+l) P (cosB). n At this point an analogy may be drawn with geometric optics.

E is the ground wave and E . is a wave which has been reflected

r,° b r,j

from the ionosphere j times. Because of lateral displacement at the

ionosphere, and also at the ground, the wave is not reflected from a

point but rather from a region. The largest portion of the reflection

does occur at or near the point predicted by geometric optics. The

actual angle of incidence at the ground, t., or at the ionosphere, cp.,

24
for each of the n waves is given by

Vn(n+1) . VnTn+T) ... sin t. —,——

and sin en. ~ J—! - (t> i k x a n k lg

with the convention that the sine - 1 when or k x a k lg

greater than one. The spherical wave functions may easily be evaluated from the

(IS) [1 2) recurrence relation given (z = k a) and £,

' - v ' ' ( z ) i ia n

n+i z n n-l v '

where.

and

C

( 1,2)

(z) =T i exp [± iz] (8)

^i'

2)

(z) = eXp ^ iz] °

( 9 )

Similarly the Legendre function can be found for integral n from

the recurrence relation

(2n-l)z

p (z) „n^ p

n n n-i n n-2 v

25
where

Po(z) = 1

and

Px (z) = z.

Note: With z = cosG, 9 is the central angle of the sphere subtended by

a given distance,,

For the case of an anisotropic ionosphere, eq (5) must be modified

to read,

.c I n £

r >J 8tt kx a n=0

V"> ( 3

)

( i ) 3 t > F(n) V U (1+R ) * X-* l ' s

i a la ein

A 1

) ,( 2) ia Kg

ia i g

c. (n;

j

where C is the "effective reflection coefficient. " C. can be graphiJ J cally evaluated by analogy with geometric optics,

C = T

i ee C=RT2 +RT T

2 e ee m em me

C = 2 R R T T T + R T~ + R^ T T T (12) 3 em ee em me e ee m mm em me

or from

C. 1 d

J

J j!R e dxJ

l-fA,x "I _1-A x-A x

2 J x = (13)

26
where

A = R T

1 m mm

A„ = R T + R T

Z e ee m mm

A = R R - T T +T T

3 em ee mm me em

The ground reflection coefficients are defined by,

R =

; a x ^ a d4) m' n

-c

(2) k * "i a 3 a

ia i s a

and,

where

ia i

t

(1) k. *.

e ' n -rb

' k $

ia _2 2 a £ ' k ^

ia 2 2 a

(! , 2 ) 4 ( z ) T~71

= ±i° £ ' ' (z)

27
Note that

^(1)

(z) * (z) n n C'< z

> Vz >

Since most of the reflection takes place near the points predicted

by geometric optics the concept of local reflection coefficients may be

introduced in the expression for C.. That is, changes in ionospheric or ground parameters along the propagation path may be included in the

field equation.

The convergence of the zonal harmonic series (1) is notoriously

slow primarily because of the inclusion of the groundwave in this

expression. Equation (3), however, separates the groundwave, E

r , u, from the skywaves. By using other methods of calculation for the

groundwave, the number of terms required for convergence may be

reduced by a factor of approximately 15. That is, convergence of the

skywave series occurs rapidly as n approaches k a (sin (p. — 1),

whereas for the groundwave 15 k a terms are required. Furthermore,

o the contribution of terms with small angles of incidence, cp. < 30 , are

nearly negligible so that for frequencies up to 20 kHz, only a few hundred terms may be required for convergence.

28
TABLE 1 LIST OF CALCULATIONS

Frequency Magnetic Azimuth

(kHz) (Degrees)

10. 20 90

10. 20 90

10. 20 180

10. 20 180

10. 20 270

10. 20 270

12. 20 90

12. 20 90

12. 25 90

12. 25 90

12. 30 90

12. 30 90

12. 50 90

12. 50 90

12. 50 135

12. 50 135

12. 50 150

12. 50 180

12. 50 180

12. 50 225

12. 50 270

12. 50 270

12. 70 90

12. 70 90

12. 70 135

12.70 135

12. 70 150

12. 70 180

12. 70 180

12.70 225

12. 70 270

12. 70 270

Profile Identification

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Day

Normal Night

Normal Day

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Day

Normal Night

Normal Day

Normal Day

Normal Night

29
TABLE 1 (Continued)

Frequency Magnetic Azimuth

(kHz) (Degrees)

13. 60 90

13. 60 90

13. 60 180

13. 60 180

13. 60 270

13. 60 270

19.90 90

19.90 90

19.90 180

19.90 180

19.90 270

19.90 270

20. 00 90

20. 00 90

20. 00 180

20. 00 180

20. 00 270

20. .00 270

20. 50 90

20. 50 90

20. 50 180

20. 50 180

20. 50 270

20. 50 270

20. 90 90

20. 90 90

20. 90 180

20. 90 180

20.90 270

20.90 270

10. 20 90

12. 50 90

12. 70 90

20. 00 90

Profile Identification

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Normal Day

Normal Night

Deeks 1200

Deeks 1200

Deeks 1200

Deeks 1200

30
Frequen cy M agnetic Azimuth

(kHz) (Degrees)

10. 20 90

12. 50 90

12. 70 90

20. 00 90

TABLE 1 (Continued)

Profile Identification

Bain & May

Bain & May

Bain & May

Bain & May

10. 20 90 Normal Day

(Sea Water)

12. 50 90 Normal Day

(Sea Water)

12. 70 90 Normal Day

(Sea Water)

20. 00 90 Normal Day

(Sea Water)

12. 20 90 Normal Day

(Raised 2. 5 km)

12. 20 90 Normal Day

(Lowered 2. 5 km)

12.20 90 Normal Night

(Raised 2. 5 km)

12. 20 90 Normal Night

(Lowered 2. 5 km)

12. 30 90 Normal Day

(Raised 2. 5 km)

12. 30 90 Normal Day

(Lowered 2. 5 km)

12. 30 90 Normal Night

(Raised 2. 5 km)

12.30 90 Normal Night

(Lowered 2. 5 km)

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33
References for

Table 2

1. W. D. Westfall, "Diurnal Changes of Phase and Group Velocity

of VLF Radio Waves, " Radio Science, Vol. 2 (New Series), No. 1, January 1967„

2. B. E. Blair and A „ H. Morgan, "Control of WWV and WWVH

Standard Frequency Broadcasts by VLF and LF Signals, " Radio

Science J„ of Res. NBS/USNC-URSI, Vol 69D, No. 7, July 1965.

3. Yoshiyuki Yasuda, private communication.

4. P. B. Rao and W. A. Teso, "An Investigation of Propagation Phase

Changes at VLF, " The Radio and Electronic Engineer, November

1966„

5. A. L. el Sayed, "Comparison of WWVL and the Local Standard, " Proc. of the Mathematical and Physical Society, U.A.R. , 1969.

6. Measurements made by NBSBL.

7. J. H. Roeder, unpublished report.

34
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103
ys

us

280

240

200

160

120 _

!0 -

40 -

280 i240 -

200 -

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 10.2 & 12.5 kHz

90° Az

2000 4000 6000

Figure 129.

BETWEEN 10.2 & 12.5 kHz

180° Az

2000 4000 6000

DISTANCE (km)

Figure 130.

104

10000

10000
us

us

280

240

200

160

120 _

280

240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 10.2 & 12.5 kHz

270° Az

2000 4000 6000

Figure 131.

BETWEEN 10.2 & 12.7 kHz

90° Az

2000 4000 6000

DISTANCE (km)

Figure 132„

105

S000 10000

10000
us

ys

280

240

2 00

160 |-

120

80 -

40 -

280 r

240

200

160

120

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 10.2 & 12.7 kHz

180° Az

2000 4000 6000

Figure 133.

BETWEEN 10.2 & 12.7 kHz

270° Az

2000 4000 6000

DISTANCE (km)

Figure 134.

106

10000

10000
us

MS

280 i240 -

200 -

160 -

120 -

280 i240 -

200 -

160 -

120 -

80 -

40 -

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 10.2 & 13.6 kHz

90° Az

2000 4000 6000

Figure 135.

BETWEEN 10.2 & 13.6 kHz

180° Az

2000 4000 6000

DISTANCE (km)

Figure 136

107

8000

10000

10000
us

us

280

240

200

160

120

10 -

40 -

280 rRELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 10.2 & 13.6 kHz

270° Az

2000 4000 6000

Figure 137.

BETWEEN 12.5 & 12.7 kHz

2000 4000 6000

DISTANCE (km)

Figure 138'. 108

1000 10000

0000
ps

us

280 |-

240 -

200 -

160 -

120

80 k

40

280 p240

200 -

160

120

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.5 & 12.7 kHz

135° Az

2000 4000 6000

Figure 139.

BETWEEN 12.5 & 12.7 kHz

150° Az

;000

2000 4000 6000

DISTANCE (km)

Figure 140.

109

8000

10000

10000
us

ys

280

240

200

160

120

80 -

40 -

280 r240 -

200

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.5

180° Az

12.7 kHz

2000 4000 6000

Figure 141.

BETWEEN 12.5 & 12.7 kHz

225° Az

2000 4000 6000

DISTANCE (km)

Figure 14Z.

110

8000

10000

10000
us

us

280

240

200

160

120

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.5 & 12.7 kHz

270° Az

2000 4000 6000

Figure 143.

IETWEEN 12.5 & 13.6 kHz

2000 4000 6000

DISTANCE (km)

Figure 144„

111

10000

10000
us

us

280

240

200

160

120

iO -

40

280 i240 -

200 -

160 -

120 -

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.5 & 13.6 kHz

180° Az

2000 4000 6000

Figure 145.

BETWEEN 12.5 & 13.6 kHz

270° Az

10000

2000 4000 6000

DISTANCE (km)

Figure 146.

112

8000 10000
us

us

280

p

240 -

200

160 -

120 -

80 -

40

280 r

240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.7 & 13.6 kHz

90° Az

2000 4000 6000

Figure 147.

BETWEEN 12.7 & 13.6 kHz

180° Az

2000 4000 6000

DISTANCE (km)

Figure 148.

113

10000

10000
us

JS

280

240

200

160

120

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 12.7 & 13.6 kHz

270° Az

iO -

40 -

280 p2000 4000 6000

Figure 149.

BETWEEN 19.9 & 20.0 kHz

8000 0000

2000 4000 6000

DISTANCE (km)

Figure 1 50. 114

8000 10000
PS

MS

280

240

200

160

120

80

40

280

240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 19.9 & 20.0 kHz

180° Az

2000 4000 6000

Figure 151.

BETWEEN 19.9 & 20.0 kHz

270° Az

;ooo 10000

2000 4000 6000

DISTANCE (km)

Figure 15Z.

115

8000 10000
MS

us

280 i240 -

200

160

120

40 I280

240 h

200

160 |-

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 19.9 & 20.5 kHz

90° Az

2000 4000 6000

Figure 1 53.

BETWEEN 19.9 & 20.5 kHz

183° Az

2000 4000 6000

DISTANCE (km)

Figure 1 54.

116

8000

10000

10000
us

ys

280 .-

240 -

200 -

160

120 -

280 iRELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 19.9 & 20.5 kHz

270° Az

2000 4000 6000

Figure 155.

BETWEEN 19.9 & 20.9 kHz

2000 4000 6000

DISTANCE (km)

Figure 156,

117

10000

10000
MS

US

280

240

200

160

120 _

80

40 -

280

240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 19.9 & 20.9 kHz

180° Az

2000 4000 6000

Figure 1 57.

BETWEEN 19.9 & 20.9 kHz

270° Az

2000 4000 6000

DISTANCE (km)

Figure 1 58. 118

1000

0000

10000
280

240

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 20.0 & 20.5 kHz

90° Az

ps

ys

200

160

20

iO -

40 -

10 r240

200

160

120

80

40

2000 4000 6000

Figure 159.

BETWEEN 20.0 & 20.5 kHz

180° Az

;ooo 0000

2000 4000 6000

DISTANCE (km)

Figure 160.

119

8000 10000
us

280

240

200

160

120

80

40

280 r240 -

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 20.0 & 20.5 kHz

270° Az

2000 4000 6000

Figure 161.

BETWEEN 20.0 & 20.9 kHz

90° Az

10000

ys

200

160

120

40

2000 4000 6000

DISTANCE (km)

Figure 162.

120

10000
ys

ys

280

240

200

280 r240

200 -

160

120

80

40

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 20.0 & 20.9 kHz

180° Az

2000 4000 6000

Figure 163

BETWEEN 20.0 & '20.9 kHz

270° Az

10000

2000 4000 6000

DISTANCE (km)

Figure 164.

121

8000 10000
us

us

280 iRELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 20.5 & 20.9 kHz

90° Az

280 i2000 4000 6000

Figure 165.

BETWEEN 20.5 & 20.9 kHz

2000 4000 6000

DISTANCE (km)

Figure 166.

122

1000

10000

10000
280

240

RELATIVE GROUP DELAY

NORMAL DAY

BETWEEN 20.5 & 20.9 kHz

270° Az

us

200 -

160

120

SO -

40 -

2000 4000 6000

Figure 167.

10000

123
PS

MS

280 r

240

200

160

120

80

40

280

240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 10.2 & 12.5 kHz

90° Az

2000 4000 6000

Figure 168.

BETWEEN 10.2 & 12.5 kHz

180° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 169.

124

8000

10000

10000
us

MS

280 r280 r240

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 10.2 & 12.5 kHz

270° Az

2000 4000 6000

Figure 170.

8000

BETWEEN 10.2 & 12.7 kHz

90° Az

t 1 1 2000 4000 6000

DISTANCE (km)

Figure 171.

125

000

10000

10000
us

MS

280

RELATIVE GROUP DELAY

NORMAL NIGHT

2000 4000 6000

Figure 172.

BETWEEN 10.2 & 12.7 kHz

270° Az

2000 4000 6000

DISTANCE (km)

Figure 173.

126

8000

10000

10000
RELATIVE GROUP DELAY

NORMAL NIGHT

4000 6000

DISTANCE (km)

Figure 175.

10000

10000

127
MS

us

280

240 -

280

240 -

200

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 10.2 & 13.6 kHz

270° Az

2000 4000 6000

Figure 176.

BETWEEN 12.5 & 12.7 kHz

90° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 177,, 128

10000

10000
280 r240 -

200 -

160

120

80

40

280

240 h

200

160

120

80

40

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 12.5 & 12.7 kHz

135° Az

2000 4000 6000

Figure 178.

BETWEEN 12.5 & 12.7 kHz

180° Az

\

I

000

2000 4000 6000

DISTANCE (km)

Figure 179.

129

8000

10000

J L J 10000
280 r240 -

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 12.5 & 12.7 kHz

270° Az

ys

us

280

240

200

160

1 20

80

40

2000

BETWEEN

90° Az

4000 6000

Figure 180.

2.5 & 13.6 kHz

2000 4000 6000

DISTANCE (km)

Figure 181.

000 10000

10000

130
ys

MS

280

240

200

160

1 20

40

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 12.5 & 13.6 kHz

180° Az

280

240 -

200 -

160 -

120 -

80 -

40

2000 4000 6000

Figure 18Z.

BETWEEN 12.5 & 13.6 kHz

270° Az

8000 10000

2000 4000 6000

DISTANCE (km)

Figure 183,

131

000 10000
us

MS

280 r240 -

200 -

160 -

120 -

80

40

280 i240

200

160

120

80

40

I—

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 12.7

90° Az

13.6 kHz

2000

5ETWEEN 12.7

80° Az

40-00 6000

Figure 184.

& 13.6 kHz

8000

2000 4000 6000

DISTANCE (km)

Figure 185.

132

10000

10000
us

ps

280 r240 -

200 -

160 -

120 -

280

240 -

200 -

160 -

120 -

80

40 -

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 12.7

270° Az

13.6 kHz

2000 4000 6000

Figure 186.

BETWEEN 19.9 & 20.0 kHz

90° Az

2000 4000 6000

DISTANCE (km)

Figure 187.

000 0000

10000

133
us

ys

280 r240 -

200

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 19.9 & 20.0 kHz

180° Az

160

120

280

240

200

160

120

80

40

2000 4000 6000

Figure 188

BETWEEN 19.9 & 20.0 kHz

270° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 189.

134

000

10000

10000
us

MS

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 19.9 & 20.5 kHz

90° Az

4000 6000

DISTANCE (km)

Figure 191.

135

10000

10000
ys

ys

280 r240 -

200 -

160

120

80 -

40 -

-750i-

-790

-830

-870

-910

-950

-990

1030

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 19.9 & 20.5 kHz

270° Az

2000 4000 6000

Figure 192.

BETWEEN 19.9 & 20.9 kHz

90° Az

8000 10000

2000 4000 6000

DISTANCE (km)

Figure 193.

136

8000

J 10000
1620

1660

us

us

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 19.9 & 20.9 kHz

180° Az

2000 4000 6000

Figure 194

8000

4000 6000

DISTANCE (km)

Figure 195.

137

10000

10000
MS

MS

280 i240 -

200 -

160 -

120

80

40

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 20.0 kHz & 20.5 kHz

90° Az

2000 4000 6000

280

240

200 -

Figure 196.

BETWEEN 20.0 & 20.5 kHz

180° Az

2000 4000 6000

DISTANCE (km)

Figure 197.

138

10000

10000
us

us

280

240

200 -

160 -

120 -

80 -

40 -

-830

-870-

-910-

-950

-990

1030

1070

1110

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 20.0 & 20.5 kHz

270° Az

2000 4000 6000

Figure 198.

BETWEEN 20.0 & 20.9 kHz

90° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 199.

139

10000

10000
us

PS

-1820,-

-1860-

-1900-

-1940-

-1980

-2020

-2060

-2100

280

RELATIVE GROUP DELAY

NORMAL NIGHT

BETWEEN 20.0 & 20.9 kHz

180° Az

J I U I J L 2000 4000 6000

Figure 200.

BETWEEN 20.0 & 20.9 kHz

270° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 201.

140

8000

10000

10000
RELATIVE GROUP DELAY

NORMAL NIGHT

US

PS

2500

-4720

r

-4760-

-4800-

-4840

-4880

-4920

-4960

-5000

4000 6000

Figure 202.

BETWEEN 20.5 & 20.9 kHz

180° Az

J L 2000 4000 6000

DISTANCE (km)

Figure 203.

141

8000

10000

J 1 J 10000
us

RELATIVE GROUP DELAY

NORMAL NIGHT

2000 4000 6000

Figure 204.

8000 10000

142
DIURNAL PHASE

90° Az 10.2 kHz

u s

us

-80 -

-100

2000 4000 6000

Figure 205.

180° Az 10.2 kHz

8000

2000 4000 6000

DISTANCE (km)

Figure 206.

143

8000

10000

10000
DIURNAL PHASE

40 r- 270° Az 10.2 kHz

us

us

2000

90° Az

4000 6000

Figure 207.

12.2 kHz

2000 4000 6000

DISTANCE (km)

Figure 208.

144

000

sooo
DIURNAL PHASE

90° Az 12.25 kHz

us

ys

-100

100

2000 4000 6000

Figure 209.

90° Az 12.3 kHz

2000 4000 6000

DISTANCE (km)

Figure Zl o 145

8000 10000

8000 10000
DIURNAL PHASE

90°. Az 12.5 kHz

us

us

40 i2000

135° Az

4000 6001

Figure 211.

12.5 kHz

2000 4000 6000

DISTANCE (km)

Figure 212.

146

10000

8000 10000
DIURNAL PHASE

180° Az 12.5 kHz

us

us

-100

40

20

2000 4000 6000

Figure Z13.

270° Az 12.5 kHz

2000 4000 6000

DISTANCE (km)

Figure 214.

147

8000 10000

10000
DIURNAL PHASE

90° Az 12.7 kHz

us

us

2000

135° Az

4000 6000

Figure 21 5.

12.7 kHz

8000

2000 4000 6000

DISTANCE (km)

Figure Zl 6. 148

8000

10000

10000
DIURNAL PHASE

180° Az 12.7 kHz

us

us

-100

2000

270° Az

4000 6000

Figure 217.

12.7 kHz

8000 10000

2000 4000 6000

DISTANCE (km)

Figure 218.

149

S000 10000
DIURNAL PHASE

90° Az 13.6 kHz

us

us

-60 -

100

40 i2000 4000 6000

Figure 219.

180° Az 13.6 kHz

8000 0000

2000 4000 6000

DISTANCE (km)

Figure 220.

150

1000 10000
DIURNAL PHASE

270° Az 13.6 kHz

us

us

-20 -

-40 -

2000 4000 6000

Figure 221.

90° Az 19.9 kHz

2000 4000 600(

DISTANCE (km)

Figure 222„

151

8000 10000

8000 10000
DIURNAL PHASE

100 r

us

100

180° Az 19.9 kHz

2000 4000 6000

Figure 223.

270° Az 19.9 kHz

000 10000

80

60

us

40

20

20

-40 J I I L 2000 4000 6000

DISTANCE (km)

Figure 224„

152

8000
DIURNAL PHASE

90° Az 20.0 kHz

US

us

100

80

60

40

20

2000 4000 6000

Figure 225.

180° Az 20.0 kHz

8000 10000

•20

•40 J I J_ J L 2000 4000 6000

DISTANCE (km)

Figure 226.

153

J L 8000 10000
DIURNAL PHASE

270° Az 20.0 kHz

us

us

2000

90° Az

4000 6000

Figure 227.

20.5 kHz

2000 4000 6000

DISTANCE (km)

Figure 228„

154

8000 10000

1000 10000
DIURNAL PHASE

ys

us

100

80

60

40

20

-20 -

-40

100r80 -

60 -

40 -

20

-20 -

-40

180° Az 20.5 kHz

I _L J_ ±

2000 4000 6000

Figure 229.

270° Az 20.5 kHz

8000 10000

J L J L I 2000 4000 6000

DISTANCE (km)

Figure 230.

155

8000 10000
DIURNAL PHASE

90° Az 20.9 kHz

us

PS

2000 4000 6000

Figure 231.

180° Az 20.9 kHz

2000 4000 6000

DISTANCE (km)

Figure 232.

156

000

1000
DIURNAL PHASE

us

100

80

60

40

20

-20

-40

270° Az 20.9 kHz

J L 2000 4000 6000

Figure 233.

8000 10000

157
RELATIVE PHASE

Normal Day (lowered 2„ 5 km)

Q

2

-2

12.3 kHz 90° Az

J_

2000 4000 6000

Figure 234.

!000 10000

RELATIVE PHASE

Normal Day (raised 2. 5 km)

12.30 kHz 90° Az

4000 6000 8000 10000

DISTANCE (km)

Figure 23 5„ 158
RELATIVE PHASE

Normal Night (raised 2. 5 km)

12.20 kHz 90° Az

2000 4000 6000

Figure 236.

8000 10000

12.30 kHz 90° Az

2000 4000 6000

DISTANCE (km)

Figure 237.

159

8000 10000
RELATIVE PHASE

Normal Day (sea water)

10.2 kHz 90° Az

2000 4000 6000

Figure 238.

RELATIVE PHASE

Normal Day (lowered 2. 5 km)

12.2 kHz 90° Az

8000 10000

2000 4000 6000

Figure 239.

10000

RELATIVE PHASE

Normal Day (raised 2. 5 km)

DISTANCE (km

Figure 240.

160

10000
RELATIVE PHASE

Normal Day (sea water)

12.5 kHz 90° Az

O^

-4

2000 4000

Figure 241.

10.2 kHz 90° Az

6000 ;000 10000

RELATIVE PHASE

Bain and May Profile

10000

10000

DISTANCE (km)

Figure 243.

161
RELATIVE PHASE

Normal Day (sea water]

00

a:

2000 4000 6000

Figure 244.

8000 10000

00<

rr;

12.5 kHz 90° Az RELATIVE PHASE

Bain and May Profile

2000 4000 6000

Figure 245.

8000 10000

o

to 9

12.5 kHz

_L 2000

90° Az RELATIVE PHASE

Deeks 1200 Profile

_L 4000 6000

DISTANCE (km

Figure 246.

162

_l 8000 10000
RELATIVE PHASE

Normal Day (sea water)

2 r- 20.0 kHz 90° Az

2000 4000 6000

Figure 247.

1 u U

in<c

-2

12.7 kHz

_L

90° Az RELATIVE PHASE

Bain and May Profile

2000 4000 6000

Figure 248.

_l 1000 10000

u~> <

Q

12.7 kHz 90° Az RELATIVE PHASE

Deeks 1200 Profile

< -2

ina.

_L ± ±

2000 4000 6000

DISTANCE (km)

Figure 249.

163

sooo 10000
RELATIVE PHASE

Bain and May Profile

10 20.0 kHz 90° Az

<

a: 4 -

00

10 i2000 4000 6000

20.0 kHz

Figure 250.

RELATIVE PHASE

Deeks 1200 Profile

90° Az

DISTANCE (km)

Figure 251.

164

8000 10000

10000
RELATIVE GROUP DELAY

Normal Day (sea water)

US

ys

280 r240 -

200

160

120

10 -

40

280

240 -

200

160 -

120

80

40

12.5 to 12.7 kHz

90° Az

2000 4000 6000

Figure 252.

RELATIVE GROUP DELAY

Deeks 1200 Profile

12.5 to 12.7 kHz

90° Az

8000

2000 4000 6000

DISTANCE (km)

Figure 253.

165

10000

10000
us

280

240

200

160

120

!0 -

RELATIVE GROUP DELAY

Bain and May Profile

12.5 to 12.7 kHz

90° Az

40

2000 4000 6000

Figure 254,

8000 0000

166
form NBS-114A 11-711 U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-610

2. Gov't Accession No.

4. TITLE AND SUBTITLE

"Application of VLF Theory to Time Dissemination"

7. AUTHOR(S)

W. F. Hamilton and J. L. Jespersen

9. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS , DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

Boulder Labs

12. Sponsoring Organization Name and Addtess National Aeronautics and Space Administration

Goddard Space Flight Center

Glenn Dale Road

Greenbelt, Maryland 20771

15. SUPPLEMENTARY NOTES

3. Recipient's Accession No.

5. Publication Date November 1971

6. Performing Organization Code

8. Performing Organization

10. Project/Taslc/Work Unit No. Project #2730416

11. Contract/Grant No. S-51740 A-G

13. Type of Report & Period Covered Final (5-18-70/2-28-711

14. Sponsoring Agency Code

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) The characteristics of time dissemination systems are discussed. Particular

emphasis is placed on the advantages and problems of very low frequency (VLF)

timing systems. The parameters affecting VLF propagation along with causes of variations in these parameters, are discussed. Three methods of computing VLF

propagation delays--mode theory, geometric-optics, and zonal harmonics--ar

compared. The method of zonal harmonics, used for the calculations in this paper,

is discussed in more detail. A method of calculating reflection coefficients for a continuously varying ionosphere is described. The theoretical values are compared

with experimental measurements. Graphical results of the calculations are included.

17. KEY WORDS (Alphabetical order, separated by semicolons) Diurnal phase; Omega; time; time

dissemination; VLF propagation; zonal harmonics.

18. AVAILABILITY STATEMENT

HXl UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

170

22. Price $1.50

GPO 838 - 561 USCOMM-DC 66244-P71

NBS TECHNICAL PUBLICATIONS

PERIODICALS NONPERIODICALS JOURNAL OF RESEARCH reports National Bureau of Standards research and development in physics, mathematics, chemistry, and engineering.

Comprehensive scientific papers give complete details of the work, including laboratory data, experimental

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Published in three sections, available separately:

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Papers of interest primarily to scientists working in these .fields. This section covers a broad range of physical and chemical research, with major emphasis

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Federal Information Processing Standards

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standards adopted and promulgated under the Public Law 89-306, and Bureau of the Budget Circular A-86

entitled, Standardization of Data Elements and Codes

in Data Systems. TECHNICAL NEWS BULLETIN

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NBS Special Publication 305, Supplement 1, Publications of the NBS, 1968-1969. When ordering, include Catalog No. C13.10:305. Price $4.50;

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