CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
DETERMINATION OF TROPOSPHERIC
II
REFRACTION UTILIZING SATELLITE DOPPLER DATA
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
David John Reed
/
January, 1977
The thesis of David John Reed is approved:
California State University, Northridge
,June, 1976
TABLE OF CONTENTS
Chapter
I.
II.
III.
IV.
II~TRODUCTION
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ATl\'10SPHERIC REFRA.CrriON ........•. , . . . . . . . . . . . . .
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Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Refraction Theory and its Effects...........
Measuring the Meteorological Parameters .....
3
3
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THE IVIATHEI/IATICAL CONCEPT AND COMPUTEH PROGRAM.
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Ir1troduc·tion . ................ · ........... · . ·
Mathematical Theory ...........•....•........
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Cornpu tel""' Prograrn. . . . . . . . . . . . . . . . . . . . . . . . . . . .
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THE TOTAL SYSTEM. . • . • . . . . . . . . . . • . . . . . • . . . . . . . .
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Navy Navigational Satellites ....•...........
Receiving and Recording the Doppler .........
Obtaining the Satellite Orbital Parameters ..
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THE RECEIVING AND MEJI.SURING EQUIPMENT. . . . . . . . .
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Introduction . ......
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Receiving Equipment ...•............•......... 54
IVIeasur ing Equipment. . . • . . . . . . • . . . . . . . • . . . . . . 58
VI.
ACCURACY ANALYSIS. . . . . . . • . . . . . . . . . . . . . . • . . . . . .
In-troduction., ..........
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Navy Navigational Satellite Transmitter
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AN/BRN-3 Receiver and Atmospheric Errors ....
HP5360A Computing/Counter ........•.....•....
Orbi tF.ll Da·ta . ........
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Accuracy as Determined by NAG .•......•......
Accuracy as Determined by Using a Regression
Anctlys~s"."
VII.
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Derivation of the 687.445IviHz Signal .........
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Surnmai"Y . .•
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Summary . . . . .
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RESTJLTS.
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CONCLUSIONS AND RECOMMENDATIONS ...•........... 111
VIII.
Satellite Oscillator Stability ............. ~
Gaussian Noise Error .............•..........
Tracking Loop Errors ........................
Residual Ionospheric Errors .................
Equipment Calibration Errors ................
Multipath Errors .......•....••........•.....
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Orbital Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Summary and Recommendations ..............•.. 121
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REFERENCES . . . . . . .
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APPENDIX . ....... ~ ...•
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LIST OF ILLUSTRATIONS
2.1.
2.2.
2·3·
2.4.
R~dio
Bending of
Wave Due to Refraction.......
Snells Lav1 . . . . . . . . . . . . . . . . .
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2. 5·
Types of Refractive Bending...................
Refractivity Profile ...... ~ ..........
Effect of Trapping Layer on Radar Ray Prop-
2.6.
Difference Between-Trapping Layer and Ducting
2.7.
Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Errors in Tracking Due to Refraction ..........
The Effect of a Radar Hole on Tracking ........
Refractive Index Profile ......................
14
2.8.
2.9.
2.10.
2.11.
2.1.2.
2.13·
2.14.
2.15·
2.16.
2.1?·
2.18.
2.19·
2.20.
ag8.tion.
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Radar Ray Trace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Refractive Index Profile. . • . . . . . . . . . . . . . . . . . . .
Radar Ray Trace .•....................•........
Refraction Index Profil~ ......................
19
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Radar Ray T1. . . ace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Refraction Index Profile ......................
Radar Ray Trace ...............................
"Ghosts" Caused by Super-Refractive Layers ....
Overcoming the Problem of Radar Holes .........
Typical Atmospheric Conditions Over Subtropical-Vlaters ............................. , ....
Ducting Extends Communication Range Between
23
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Two Airplanes . ..........._...........
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2.21.
The Difference as a Function of Height of Two
Radiosondes Compared Against a Reference
2.22.
Ways of Measuring the Refraction Index of
Ra~diosond.e
•
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the Atmosphere . . . . . . . . . . . . . . . . . . : ..
3·1·
3.2.
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Geometry of Ray Path ......................... .
Refraction Index Profile Using Simule.ted
Satellite.
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3·3·
Refraction Index Profile Using Simulated
3.4.
Refraction Index Profile Using Simulated
3·5·
Refraction Index Profile Using Simulated
3.6.
Refraction Index Profile Using Simulated
3·7·
Refraction Index Profile Using Simulated
Sa.telli te . ........
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Satellite. " . . . . . . . . . . . . . . . . . . . . , ........... .
Satellite ..............
Satellite .. .....
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Sa·telli te .
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High Frequency Receiver ...................... .
Refraction Doppler Correction Unit ........... .
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6.1.
Method Used by the HP5J60A Computing/
Counter to Measure Frequency ............ \... .
Accuracy of Signal Leaving Satellite Transh
6,2.
Accuracy of Doppler Signal Leaving First
6.'3·
6.4.
6.5.
Accuracy of Doppler Signal Leaving VCO .... : .. .
Accuracy of Doppler Signal Leaving Receiver .. .
Relationshio Between Interpolator Error and
Measurement Time .........•..•..... ~ ..... ~ ...
Obtaining Input 'l rigger Using the Input
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mi tter . ... • .............................. ·... .
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7.4.
7· 5·
7.6.
7·7·
7.8.
7·9·
7·10.
7·11.
7.12.
7·13·
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7·15·
7.16.
7·17.
7.18.
7·19.
7.20.
7·21.
8.1.
. 8.2.
8.3.
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S i gna 1 S lop e . . . . . . . . . • . . • . . . . . . . . .
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6.6.
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Regre~sion
Analysis of Doppler Data .......
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R~gre~s1on
Analys1s of Doppler Data ........
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(Ti -TilJl) .... ~ ... : .......................... ~~ ..
(Tic-'~Pi
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Rfigre~slfon Analysis of 512KHz Signal ........ ·..
( T ; -T iiJl) ..•••.•. : ••. • •••• • . • • · • • • · · · · · • • • • • ·• •
R~~Be~s1on Analys1s of Vacuum Doppler ........ .
( T. -Tim) .. ...... : ..................... " . e • • • • , •
Re~f·ess 10n Analys lS of Vacuum Doppler .......•.
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Regression Analysis of 532KHz Signal ......... . 98
Regressi?n Analysis o~ ~imu1~ted Doppler
Data W1th a One Par~ 1n 10 Accuracy ....... . 99
R~gre~sion Analysis of Vacuum Doppler ........ . 101
(T• - T j m ) • • • • • • • • • • • • • • • • • • • • • • • 7 • • • • • • • • t P • • • 102
R~gfe9sion Aria1ysis of V?J.cuum Doppler ......•.. 103
( T . -T . ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . :, .... 104
RegtesS~on
Analysis of Vacuum Doppler .•....... 105
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(Tic-Tim) ' ' ' ' ' ' ' ' : ' ' ' . ' ' ' ' ' ' ' ' ' ' ' ' . ' .. ' ' ' ' ' ' ' ' 106
R~gre~s1on Ana1ys1s of Vacuum Doppler .•. ·~ .... 107
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( T . -11 . ) .•••••.•••••.•.•. •..••.•
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R~gtes~Ton Analysis of Vacuum Doppler ......... . 109
( T . -T .. ) .......
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Ion8spi\Wric Refraction Error ......... .: ....... . 117
Ionospheric Refraction Error.~·· .•..•......... 118
AGC Data Taken on J...,aguna Peak Shows Multipathing on the 400MHz and l50Iv1Hz Signals .... 120
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LIST OF TABLES
6.1.
6.2.
6. J.
6.4.
6.5.
8 .1,
Doppler Link Analysis for 150MHz ............. . 66
Doppler Link Analysis for 400MHz .•..•.....•... 67
Error With Measurement Time of .lsec(Hz) ..... . 72
Error With Measurement Time of .ssec(Hz) ..... . 73
Error Due to Noise on the Input Signal ....•... 77
Major Sources of Error ....................... . 111
vii
ABSTRACT
DETERMINATION OF TROPOSPHERIC ·REFRACTION
UTILIZING SATELLITE DOPPIJER DATA
by
David John Reed
Master of Science in Engineering
January, 1977
This project report describes a different method for
calculating the refraction index of the troposphere than
used now and results of attempting to use this method are
given.
Theoretically, if the doppler shift of a satellite
signal were known along with the satellites orbit, the refraction index profile could be determined using a least
squares method.
Attempts at measuring the doppler have yielded results which are not accurate enough to allow the computer
program to converge on a solution.
sons have been discovered.
Several possible rea-
They are satellite oscillator
stability, gaussian noise error, tracking loop errors, residual ionospheric errors, equipment calibration errors,
multipath errors and orbital data errors.
Analysis of the
doppler measurements has yielded accuracies of around one
viii
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part in 10
7
10 .
5
.
and the needed accuracy is about one part in
If a refraction profile is to be generated then it
is recommended :that new mathematical techniques be implemented in order to decrease the required accuracy and the
above error sources 'should be investigated.
ix
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CHAPTER I
INTRODUCTION
The objective of this project is to determine the
refraction index profile of the atmosphere by making use
of the effect that tropospheric refraction has on the doppler signal received from a satellite.
-
divided into eight chapters.
This report is
In order to better under-
stand the project some background material on refraction
in the atmosphere and on the mathematical concept used
to calculate the refraction index have been included.
These are included in chapters II and III.
Following this
is a chapter describing the whole system to show what is
involved in reaching the objective.
After this is a
chapter describing, in more detail, some of the equipment
that is involved in getting the needed doppler data.
Be-
cause of the importance on accuracy of the data, a chapter
on accuracy analysis of some of the equipment is necessary.
Finally results, conclusions and recommendations are presented.
The basic concept of this project is as follows.
A
satellite whi6h is orbiting the earth transmits two signals
down to earth.
Since the satellite is moving toward or
away from the receiver located on the earth, the signals
are doppler shifted.
When the signals travel through the
atmosphere they are bent due to the change in refraction
1
2
index in the atmosphere.
effects the doppler(l).
atmosphere were
~nown
This bending of the signals also
If the refraction index of the
then the effect of the bending of
the signal on the doppler could be calculated.
Converse-
ly, if the doppler were known and if it were known what it
should be without refraction than the refraction index of
the atmosphere could be calculated from the difference in
the two.
In order to obtain the doppler one needs to
track the satellite and in order to obtain what the doppler should be without refraction effects 6ne needs to
know the orbital data of the satellite and the position of
the receiver.
the results.
Finally a computer is needed to calculate
CHAP1'ER II
ATMOSPHERIC REFRACTION
Introduction
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It is well known(2) that many properties of the
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atmosphere(e.g., temperature, pressure, humidity, density,
electron content, etc.) change significantly with altitude.
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The temperature, pressure and humidity will generally
\
decrease with height above the earth but, this is not always the case.
Sometimes warm air can overlay cooler air
and cause temperature to increase with height.
The humid-
ity can also increase if for instance moist air is overlaying dry air.
These three variables, temperature, pres
sure and humidity, are what determine the refraction index,
n, of the atmosphere.
Refraction Theory and its Effects
When an object is put into water such ad a pencil
it appears to bend.
On a hot day what looks like heat
waves are seen coming off the road or, as the sun goes
down its color appears to change from a bright yellow to
a deep __ ()range.
All of these are exam_p1es of refraction.
If radio waves could be seen transmitted from a satellite
down to, earth the bending due to refraction would be apparent.
This is illustrated in Figure 2 .1.
The degree of this bending phenomenon is found in
the index of refraction, n.
The quantity, N, called re-
3
4
n
',}
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-
--
-...
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,.
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REFRACrrED RAY
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EARTH
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Fig. 2.1.
........
/
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.......
--
----
-
.......
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Bending of radio wave due to refraction.
5
fractivi ty is used instead of n so that the number~·s will
be more convenient to work with.
\
The relationship between
refractivity, N, and the index of refraction, n, is;
N
= (n-l)xl0 6
(1)
Also, the relationship between refractivity and the three
parameters, temperature, pressure and humidity has been
shown to be(2);
N
= (77.6/T)(P+4810e/T)
(2)
dN = -1· 27dT+4. 5de+. 27dP
where N is given in parts in 10 6 , T is the temperature in
degrees kelvin, e is the water vapor partial pressure(mb)
and pis the atmospheric pressure(mb)(2).
The equation
for dN shows that m is more dependent on humidity(e) than
temperature or pressure.
The humidity profile for the
atmosphere illustrates the
undergo.
degre~
of change that N can
However, all the information necess~ry to obtain
a refraction profile-cannot be found from a humidity pro
file alone.
The index of refraction and refractivity is related
to the velocity of electromagnetic wave propagation through
the medium by;
n
=
(c/v), N
=
(c/v-l)xlO 6
= (c-v)/vxlO 6
(3)
where c is the velocity of light propagation in a vacuum
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and v is the velocity of propagation in a medium with an
index of refraction, n.
The velocity of propagation of
electromagnetic energy through a medium can therefore be
expressed as a function of temperature, pressure and
humidity by combining equations (1), (2) and (3), i.e.;
v
=
6
c/(1+(7?.6/T)(P+4810e/T)xl0- )
(4)
A definition of refraction is often given by the
following: Refraction is the bending of electromagnetic
energy which occurs in passing through a medium due to
changes in the velocity of propagation through that medium(2).
Figure 2.2 illustrates how electromagnetic energy
is bent when passing from one medium to another.
The angle
of incidence is related to the angle of refraction by(22);
---
-
( 5)
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This equation is commonly referred to in optics as Snells
law.
Now, suppose instead of a sudden change in v, the
change was gradual as the wave moved through a medium
such as the atmosphere.
Figure 2.1.
ly.
The result of this is shown in
The wave is bent gradually instead of abrupt-
Given this, consider what would happen if a signal
was transmitted from the earth into the troposphere.
"standard atmosphere" conditions exist then the signal
If
7
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Incident
Rays
_
-
-- - -..-- -...- -;,.-Sh.---.
~
Bound~'lry
.._
Surface
Transmitted
Rays
Fig. 2.2. Snells Law
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~--)
will be bent in such a way that the earths' radiu\f appears
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to be four-thirds its actual radius.
Standard atmosphere
is defined as a decrease of twelve N units every one-thou'
sand feet of altitude(-12/lOOOft).
this bending.
Figure 2.3 illustrates
The wave still propagates away from the
earth but not as much as it would have if it was travelling through a vacuum.
If the decrease inN units with
altitude is greater than twelve the signal will be bent
more towards the earth.
If N changes by 48 per 1000 feet
the wave will be bent around the earth.
Figure 2.3 and is known as ducting.
This is shown in
Any decrease greater
than twelve is called super-refraction.
When N increases
with altitude a condition known as sub-refraction occurs.
In this case the signal will be bent away from the earth.
Figure 2.3 also shows this.
The profile of the refrac-
tivity, N, for these cases is shown in Figure 2.4 where
N is plotted against height above the earths surface.
The
degree of change of N with height is what effects the
propagation of electromagnetic energy.
When the decrease in refractivity becomes greater
than forty-eight per one-thousand feet, a condition known
as ducting can occur.
This is a condition such that when
a signal is transmitted and is incident on the inside boundary of the layer where N decreases at a rate of fortyeight or more at a very small angle(less than five degrees)
the wave would reflect and be propagated through this
layer
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STANDARD A MOSPHERE
SUB-REFRACTION
SUPER-REFRACTION
EARTH
Fig. 2.3.
Types of refractive bending.
10
HEIGHT
RE:F'RAC IVITY
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SUB-REFRACTION
SUPER REFRACTION
EARTH
Fig. 2.4.
Ref~activity
profile.
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as shown in Figure 2. 5.
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The layer that includes ·t:he de-
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crease inN greater than forty-eight units per one-thousand feet is called an inversion or trapping layer,-
The
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layer that corresponds to the envelope of the signal is
called the ducting layer.
Figure 2.6 illustrates
~he
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reason why these two may not be the same.
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The duct is de\
termined by extending a line with a slope of -48/ldOOft
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from the top of the inversion layer down to a vertical extended from the bottom of the inversion layer.
If the ac-
tual slope was -48/lOOOft than the ducting and inversion
layer would be the same.
Refraction has several effects on the propagation of
electromagnetic energy through the atmosphere.
Figure 2.7.
Consider
When trying to track an object the radar
nal is bent due to refraction.
sig~
Because of the bending of
the ray the true object looks as if it is located at a
ferent location.
di~
The resulting ~rrors are th~t of range,
altitude and elevation angle.
The errors could be cor-
rected by using a computer program if the refraction profile around the atmosphere was known.
of a problem is that of ducting.
What is even more
When a radar is inside a
duct and is attempting to track an object, a phenomenon
known as a "radar hole" occurs.
2.8.
This is shown in Figure
As can be seen, if a target is approaching the radar
through this hole it will not be seen on the radar scope,
even though it can be physically seen.
Examples of radar
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RADAR RAY
TRAPPING LAYER
..
-~'
~
..
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''
'
:1
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'
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LAND
SEA
Fig. 2.5. Effect of trapping layer on radar ray propagation.
13
INVERSION LAYER
~
-48N
________
L
___
L
__
·. ____ *, __ _
-48N/1000 ft.
- - --·
/' ---------T- ---r--.
DUCTING LAYER
HEIGHT(ft.)
REFRACTIVITY(N)
Fig. 2.6.
ducting layer.
Differenc~
between trapping .layer and
14.
dY
;
/
dX
where
de' = Elevation angle error
dX = Down range error
dY =Altitude error
Fig. 2.7.
Errors in tracking due to refraction.
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holes and refractivity profiles are shown in Figures 2.9
through 2.16.
The radar ray plots have associated with
them the refractivity profiles so it can be seen how the
refractivity profile effects the radar rays.
Another problem is that of "ghosts'', as shown in Figure 2.17.
They are an image on the radar screen that ac-
tually is not where it appears to be or what it appears to
be.
In tropical and warm water regions the absolute mois-
ture content of the air is generally so high that relatively subtle changes in humidity are often sufficient to
cause large changes in the N values over very short distances.
These may occur throughout the lower and midlevels
of the atmosphere or very close to the surface, thus posing problems to both shipboard and airborne radar systems.
It may be that such subtle and randomly spaced humidity
and temperature gradients near the-ocean surface are partially responsible for the "ghosts''.
The returns from the
rays that impinge the sea look as if there is an object at
that distance from the radar.
When ducts occur and if the general profile of the
atmosphere is known, there are ways to work around the
problem associated with refraction.
For example, if the
location of the duct was known pickett ships and planes
could be placed to eliminate the problem.
trated in Figure 2.18.
This is illus-
Since ducting is dependent on the
angle at which the radar ray impinges the boundary of the
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27
inversion layer the radar height could be adjusted to
change this angle.
In this way the radar ray would im-
pinge the boundary at a large enough angle such that it
will not be trapped.
According to Sherar and Rosenthat(2), in order for
N to decrease with height at the rate required for trapping, the humidity must decrease very sharply with height.
These conditions are most common over subtropical bodies
of water in the summer.
This is the dry season when very
little turbulence of the atmosphere occurs, a fair weather
season.
Warm dry air overlays a shallow, cool and rela-
tively moist layer just above the surface.
Advancing up
in height, the humidity and temperature both change abruptly producing a very strong super-refractive gradient.
This
is illustrated in Figure 2.19.
These refractive layers do not remain fixed, however.
They move around depending on seasonal changes and storms.
These elevated refractive layers are lowest in height and
are stronger due to a large change in N in fair weather 1
such that off the Southern California coast during the
summer, and highest and weakest near storms(2).
The evaporative duct usually lies just above the surface of a body of water.
It is caused by a rapid decrease
of moisture in approximately the first fifty feet above
the water and is directly related to the difference between the temperature of the water and air directly above
28
WARM-DRY AIR
INVERSION AND/OR SUPER-REFRACTIVE LAYER
CLOUDS
COOL-MOIST AIR
.• ...... ~ ... -
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---
EVAPORATIVE DUCT
WATER
Fig. 2.19.
Typical atmospheric conditions over
subtropical waters.
-
-
29
it and the humidity(2).
This duct can benefit shipboard
surface search radars by extending their range or if the
duct does not extend high enough above the ocean to inelude the ships radar a submarine located in the ductcould locate the ship and the ship would not see the submarine.
The duct could also extend the range of two air-
planes trying to communicate with each other provided they
had knowledge of where and how strong the duct was.
This
is shown in Figure 2.20.
A study was done at Saint Nicholes Island(SNI) and
it was shown that, from 570 feet to 5000 feet, trapping
occurred 70% of the time in the summer and 50% of the time
in the winter.
Over subtropical and tropical oceans,
evaporative ducts may occur 85% to 90% of the time(2).
According to the Applied Physics Laboratory(APL), in colder climates, such as the North Sea, X-band ducting by evaporative ducts has been estimated to occur from 60% to 90%
of the time in summer.
In polar regions only are ducts
apparently infrequent.
The hotter, more humid, and fair
regions seem to influence ducting the most.
Measuring the Meteorological Parameters
Measurements of temperature, pressure, and humidity
of the atmosphere are done most often by devices called
radiosondes or refractometers.
These radiosondes are at-
tached onto balloons and released from the earth.
The way
in which radiosondes measure these three parameters vary.
30
REFLECTED SIGNAL
-DUCTING LAYER
EARTH
Fig. 2.20.
Ducting extends communication range between
two airplanes.
31
The most inaccurate measurement is the humidity because
the device is slow to respond.
Even though there have been about ten different radiosondes in use over the globe by 1972(3), none were sufficiently accurate.
Several radiosondes were used at Pay-
ers, Switzerland and the results were compared only to
find considerable disagreement in the measurements(J).
They also tried a technique called dual soundings, where
two identical devices were sent on the same balloon and
their results were compared against a reference radiosonde.
The difference in temperature as a function of height is
shown in Figure 2.21.
Another problem in the earlier radiosondes, due to
the ascent rate of the balloon and the switching time between sensing mechanisms, is that the data points were at
least one hundred feet apart and the lowest data points
were at two hundred feet(2).
Also, the rate at which the
humidity measurement is made, which is very important in
calculating the refractivity profile, was very poor.
Re-
cently radiosondes have been changed so that the switching rate is faster to give closer data points, and more
accurate sensing mechanisms with faster sampling times
have been developed.
However, there is still much room
for improvement in the accuracy of the profiles computed
from this data.
Another problem is control of the balloon.
Once it
32
- ·--·--·---·--···----·-·-.-
-~
~-----~-- ~
----
---
--~-
·-·-·---·----··-
.-zoo
150
100
50
30
Bur s~10
--~~~~~~--~~--~7
..••
I
10
20
30
_Q
E
40
'<lJ
50
"'0
::J
J_ _ _c_~--~~L--L--~-J--~--~~-
-1
0
1 -. -.2 . 3
L.
.5 6
.6tC
7
8
9
Fig. 2.21. The dif'ferenc~ as a function of height of two
radiosondes compared against a reference radiosonde.
JJ
is let go it goes wherever the winds take it.
This causes
problems of obtaining a strict vertical scale and also one
of retrieval of the instrument.
A more accurate instrument for measurements of data
of this type is called the refractometer.
Refractometers
contain a resonate cavity, which when the wind blows
through, measures the refraction index.
are flown aboard planes.
These devices
While the plane spirals over the
area where a profile is needed the information is recorded.
Even though refractometers detect much more detail
than the radiosonde, the frequently large space-time variations of the detailed structure make much of it transient in nature(2).
Therefore, neither of these two methods for determining the refraction index profile of the atmosphere are
very good.
detects.
One does not detect enough, the other over
Also, neither will obtain the parameters below
two hundred feet very good where evaporative ducts lie
although recently progress has been made in this direction.
Other methods are being tested and designed to meet
the needs of all those who require accurate refractivity
data.
These include vertical pointing radars, acoustic
sounders, tethered and de.llayed-release balloon soundings,
instrumented portable and stationary towers and dropsondes(2).
Also, the weather satellite can be used by
looking at the satellite data and determining cloud tops
34
which indicate approximate refractive structures over large:
areas.
Figure 2.22 shows some of the methods stated above.
One other method, which is the concern of the rest
of this paper, is that of using the doppler effect of satellite transmissions.
As will be shown later, if proper
accuracies of the receiving equipment can be met, very
good refraction index profiles can be obtained.
The advan-
tage of this method is that all that is needed is receiving, recording, reducing and computer equipment.
it can be done on a near real time basis.
Secondly,
Once refraction
index data is obtained using this method, it will be compared against that of other methods and the accuracy can
be obtained.
One way in which it will be compared to other
refraction profiles is through the use of it to correct
for the range, altitude, and elevation angle data from
radars.
A plane will fly over SNI where use of a laser
system will give almost its exact position.
Then, all the
refractivity profiles including the one from the satellite
data, will be input into the program to correct the data
that a radar at Point Mugu receives while the plane is
flying over SNI.
Whichever is closer to the laser position
is the most accurate.
In conclusion, to meet the needs of those needing
accurate refractive index data and on a real time basis,
new ways of obtaining this data must be developed, including that of using satellite transmissions.
With this
.
SCHEMATIC VIEW OF MEASUREMENTS
SATELLITE.DOPPLER METHOD
TETtiE~En
Fig. 2.22. Ways of measuring the refractio~ index of the atmosphere.
/
\..U
\n
36
basic understanding of refraction and its effects, the
method of using the doppler data from satellite transmissions becomes more meaningful.
CHAPTER III
THE MATHEMATICAL CONCEPT AND COMPUTER PROGRAM
Introduction
The basic idea of obtaining atmospheric refraction
data from measuring the doppler shift is as follows.
If
the characteristics of the atmosphere were fully known at
a given time, the amount of distortion of the signal caused
by refraction at that time would become quite predictable.
If the atmospheric refraction profile was known along with
the satellite position and velocity and receiver position
and velocity, the doppler shift could be calculated.
Con-
versely, if the distortion as a function of the satellite
position can be determined, then some amount of information
relative to atmospheric refraction characteristics can be
obtained(4).
Mathematical Theory
The following theory was developed by Ralph Claassen
of PMTC(4).
The doppler frequency is related to the rate
of change of the propagation time of the satellite signal,
1' by;
•
T = -df/(fe+df)
( 1)
where df is the doppler frequency and fe is the frequency
emitted from the satellite.
The method utilized the least-squares solution.
37
Val-
38
ues for the refraction index versus height which will minimize the following equation need to be obtained.
i
=1
(2)
to n
where Tim are the measured values and Tic are the calculated values obtained by assuming a refraction profile and
knowing the satellite and receiver position and velocity
for the set of doppler measurements.
If the assumed pro-
file is the same·as the real profile then the difference
T will
between the actual and theoretical
iteration stops.
be zero and the
Therefore, to minimize this equation
with respect to either propagation velocity, v, or height
above the earth, r, a set of J-simultaneous equations of
the form;
k = 1,2, ••.• J, i = 1 ton
( 3)
must be solved for the v's and r's.
Using the truncated
taylor series, the normal equations becomer
2_(v j-v jO) <r(dTic/dvk) J0x(dTic/dv j)
lo=( dTic/dvk j0x(Tim-Tic)
( 4)
In order to solve this equation for the v's we must obtain
values for Tic and its derivatives with respect to the v's.
Using Snells law and Figure 3·1 it can be shown that
the following equation is
f.})_\; ol !\
J'-A /tJ _P ·"'fl'l..- lj"'..
-..)..
=
( 5)
39
Figure 3.1 - Geometry of Ray Path.
40
. where (rr' ~r' ~r) are the polar coordinates of the receiver, (ri, ~i• ~i) are the polar coordinates of the satellite at the time of the ith doppler measurement, v and
r are the corresponding propagation velocities and heights
respectively, andJOi is the ray path constant for the ray
when the ith doppler measurement is taken.
This equation
is used when the satellite and receiver positions and the
refraction index profile are known to solve for..,Pi•
Using
this value for..,..oi and the corresponding atmospheric refraction profile and satellite and receiver positions, the
values for Tic and its derivatives can be found(4).
There-
fore, as each set of doppler frequencies are measured and
the corresponding satellite and receiver positions and velocities are known, there is one iteration and a new atmospheric refraction profile is calculated from the normal.
equations.
The procedure goes as follows.
At each step, the
previously approximated model of the atmosphere is used
to determine.,..Oi from equation ( 5).
The solution of pi
and the approximation to the atmosphere are used in the
equations for Tic and its derivatives, which in turn are
substituted into the normal equations to yield new solutions for altitudes and velocities.
It must be pointed out here that three approximations have been made in order to use this method.
they are as follows.
And
41
Approximation 1- Propagation velocity is a function,
of a single variable, that being the distance from the
center of the earth.
In this way we are assuming that
the propagation velocity is time invariant, and that contour surfaces are spheres with centers at the center of
the earth.
Approximation 2 - Propagation velocity as a function
of altitude can be represented by a sequence of connected
straight line segments.
Therefore, the goal becomes to
find the best values for the end points of the line segments; i.e., the optimal altitude and corresponding propagation velocities at which the changes in slope occur.
Approximation 3 - The position and velocity vectors
of a satellite and receiver corresponding in time to each
doppler shift measurement are accurately known.
necessary in order to
determine~i
This is
from equation (5) ac-
curately.
Therefore, for each set of doppler shift measurements a new atmospheric refraction profile is obtained
which is closer to the actual one than the one that was
obtained before that.
Since the accuracy to which the
doppler shift can be determined is limited, as many data
points as possible should be obtained.
Some typical results of the simulation are shown
in Figures J.2 through 3·7•
In each of these figures the
dashed line represents the boundary between ducting and
42
.. ····------·-------· -----·---·---------------------·------------------~------------------------------------·-----~
5,00~
/
I
I
l
I
.I
J
I
4,000
I
I 1/
v
3,000
~
I /
/l
!:l
0
;:I
1-
i=
l
I
7
_,/"
II
~
!:l
!:l
r
I
•
•I
I l/
..J
<
.; 2,000
IJX
~1
llj
i/ II
1,000
/
l[JI
X
•
ll: ...... -
0
'
"-
X
X
--
0.999660
_ ....
-I
1-7
~[7
0.999680
0.999700
0.999720
0.999740
r
y
2
6
= 0.999663, Acc"racy :::: 1 in 10 •
Fig. ).2. Refraction index profile using simulated satellite.
-<- - -
-~----~
--
----- -- ---
----~--
·-----------
5,00 0
1/
!] /
-----------~-~----------
.•
.I
I
?'
4,00 0
/
il
1/.l •
1='
lj
3,000
I
.f I
w
X
X
w
!!;.
/
w
0
:::>
1-
I
X'/
X
i=
..l
IJ:x/
<
.. 2,000
X
/
/
;7
/
I
·:/
,x
1,000
.f~'/
".;
.!V
0
v
£
bY
L_:-..,.
0.999660
v
..,. ..,.
~..,.
0.999680
0.999700
y
0.999720
0.999740
7
}' = 0. 979663, Accuracy :::: 1 in 10 •
2
Fig. 3·3· Refraction index profile using simulated satellite.
44
--- -----··---------------~--------
---~-----------·----------
----- -------------- - -
5,000
I
/
t
I
l
•
I
I
,-
!
v
4,000
------------- ------
l
/
~t
I
l
X
p
I
3,000
lj
w
w
~
w
0
I
i=
/
<
j_ I
::J
1..J
,.; 2,000
Lll
/
/
j /
l
.f
I--
1_].·"
-- ----
~~
1,000
I .I
!
)
,. ,..
.)1
...
/
\
I
..
\
I
J(
.)1
0
-"
0.999660
0.999680
-
0.999700
y
0.999720
0.999740
y = 0.999675, Accuracy ~ 1 in 10 6•
2
Fig. ).4. Refraction index profile using simulated satellite.
4:5
.
---~------------------------------- ------~-------------------------------
-··-
·~----------------
.•••
I .I
5,000
II
I
I
4,000
.I I
I vv
i
1/
---;'
Il
!='
w
/.i
3,000
l!j
w
,_i
ew
/;
0
;:)
I-
i=
//
...l
<
,; 2,000
v
lj
/V
~
~"-/
;v
1,000
//
...
J> ,..
0
~
v
0.99%60
VI
v
~
J
0.999680
0.999700
0.999720
0.999740
y
y 2 = 0.?99675, Accuracy=::: 1 in 107 •
Fig.
3·5·
Refraction index profile using simulated satellite.
46
-·----~------~--·~-~------~----------·----------------- ----~------------------- ---~---
~.ooo
•
!
I!
I/;
l
4,000
l
/J
1/
.~
,//
~ 3,000
Al
1>.1
1>.1
e
/
1>.1
c
;::>
f-
l
i=
/
..J
.
<
2,000
l
X
I
}7
v
/
,;
)
I
if
1,000
A
0
~
~
0.999660
/l
v·
.s
... ...
p-
0.999680
0.999700
0.999720
0.999740
r
· y 2 = 0 . 999681, Accuracy ~ 1 in 10 •
6
Fig.
J.6. Refraction index
prof~la·using simulated satellite.
-----------
47
~--
------
---~----
-~----
- - - - - - -----------
- - - - - - -------
~- - - · - - - - - - - - - - - - - - - - - - - - - ~- - - " 1
5,000
I
II
I
I
I
I
I
I
I
IL
4,000
I
j_
I
I
I
I/
1/./
I/
lt
I
I
3,000
!='
tLI
tLI
!=.
tLI
I
c
k
ll
;:I
1-
j::
..J
<
"
2,000
I
--
l
-
}I
1,000
.
;
;
I
X
;
('
0
I=-·- ::.£..
0.999660
~~
1/
; tl
/;
1/
--
I--
0.999680
0.999700
y
0.999720
0.999740
y = 0.999681, Accuracy~ 1 in 107 •
2
Fig. 3·7· Refraction index profile u'sing simulated satellite.
48
nonducting, the solid is the solution and the cross-hatched:
is the assumed atmospheric model.
These figures show that
an accuracy of at least one part in 10 7 should be obtained
for the doppler frequency and orbital.velocities of the
satellite for the method to predict accurately whether or
not ducting occurs.
Computer Program
The computer program that was developed by Claassen
in reference (4) was set up to use data from a simulated
satellite pass.
In order to use real data several minor
changes had to be implemented.
The program is written in
Fortran IV and is listed in Appendix A.
CHAPTER IV
THE TOTAL SYSTEM
Introduction
The system involved in getting data to input into
the computer program which computes the refraction index
might be called the doppler and orbital system.
The basic
components in this system are the navigational satellites,
receiver, frequency counter, digital recorder, computer
and the Orbital Improvement Program(OIP).
Accuracy is of
utmost importance in this project so it is necessary to
understand the entire system.
In this way the areas where
inaccuracies could occur would be known.
Also, as in most
systems, if one part of that system is not working properly then the effect of the whole system is degraded.
What
follows in this chapter is a discussion of the path of the
data as it starts from the satellite and ends in the
computer program.
Navy Navigational Satellites
The satellites that will be used for this project
are the Navy Navigational Satellites called Oscar satellites.
The satellites are in near polar orbits about the
earth with an altitude of about six hundred nautical miles
and a speed of about four nautical miles per second.
They
are used for navigational purposes by transmitting their
exact orbit to the receiver every two minutes.
The trans-
mitted signals are doppler shifted because of the relative
velocity between the satellite and receiver.
As the satel-
lite is coming closer to the receiver the frequency received is higher than that transmitted and as it goes
away it is lower.
Therefore, if frequency is plotted ver-
sus time a curve would be obtained from which, by measuring the slope at the time of closest approach, the position of an object could be determined.
The navigational
satellites transmit two phase coherent signals.
The
reason for this is that when a multiple of these two signals is combined the ionospheric refraction effects are
eliminated(5).
A satellite is overhead about once every
hour as of the present time.
This will enable a refrac-
tion index profile to be calculated before the state of
the atmosphere changes significantly.
There are a total
of six satellites in orbit at this time with the oldest
being about six years old.
The satellite transmitter will
be discussed in greater detail in Chapter
v.
Receiving and Recording the Doppler
Once the signal is transmitted and doppler shifted,
it must be received and tracked while the frequency is
changing.
At the Navy Astronautics Group(NAG) the anten-
na used to receive the signals is a helix.
There is one
helix for each of the signals and they both turn together
on one mounting.
The beamwidth for the antenna is about
thirty degrees and the gain is about lJ.5dB.
When track-
ing a pass the antenna is moved to the correct azimuth
and elevation angle by just following the minute by minute
azimuth and elevation data which is obtained before the
satellite pass.
The receiver used to track the satellite
is an AN/BRN-3 navigational receiver designed especially
for these satellites.
When the low and/or high frequency
signal is received the receiver locks on to it by means of
a phase locked loop and follows the doppler signal.
The
doppler signal is acquired and tracked and is converted
down to a lower frequency.
Then, both the high and low
signals are combined in such a way as to eliminate ionospheric refraction effects and as a result there is one
doppler signal which is called the vacuum doppler.
The frequency of this doppler signal must be put
into a computer.
If the analog signal is recorded and then
digitized there is a possibility of recording errors such
as variations in the tape speed.
So instead, the signal
is first fed into a Hewlett-Packard 5360A Computing/Counter
to be digitized and than to a Hewlett-Packard 5050B digital
recorder.
By adjustments on the counter the frequency can
be counted and recorded from once every 10- 6 seconds up
to once every 100 seconds.
Along with the frequency it is also necessary to
have the time that the frequency is obtained.
This is
done manually by sight because of the availability of a
digital clock.
52
----------
--·-~-----
-~---------
----
----- --- -·
~-----
--~---- -~------
.. -----
----------------------
Once the signal is recorded it can be put on punched
cards along with the time and then it is ready to be put
into the computer to be run in the refraction index program.
Obtaining the Satellite Orbital Parameters
The satellite position and velocity are needed each
time a doppler frequency is recorded.
If the doppler is
recorded every one half second then the satellite orbital
data is needed every one half second.
In other words if
a doppler frequency is recorded in a certain second of the
day then the orbital data for the satellite must coincide
with that second of the same day.
If an error of one tenth
of a second was introduced between the orbital and doppler
data the satellite position could be off by about four
tenths of a nautical mile which greatly effects the data
calculated in the computer program.
Therefore it is very
important to obtain the best orbital data possible at the
exact time that it is needed.
At NAG they are responsible for providing this orbital data.
Their Orbital Improvement Program(OIP) is res-
ponsible for transforming the received doppler data that
if obtained from tracking stations, into orbit improvement
data for injection into a navigational satellite memory.
One function of this program is to provide an improved estimate of the satellite orbit at an epoch just prior to the
data used in the improvement.
In other words, by tracking
53
the satellite and recording doppler they use the predicted
doppler frequency shift for that satellite along with the
·actual doppler frequency shift and least squares fit these
two to improve the predicted orbital data.
The reason
they do this is to obtain from this improved orbit a predicted orbit for the satellite for the next day or two.
This predicted orbit is injected into the satellite so that:
navigators can use it to perform their navigation.
So, the
best possible orbital data is that improved orbit.
It is
this data that is being used in the computer program for
calculating the tropospheric refraction index.
Normally a
position and velocity is given once every two minutes but
by adjusting a parameter it can be obtained every one half
seconds or whatever time interval that is needed.
Once the
orbital data is obtained for the tracking span that is required it can be input into the computer.
,--·--·-
----------------------------------~-------------------------
--·----·----··---·· ---'"·------·-·--·----·----------------·---------
---------,
CHAPTER V
THE RECEIVING AND
~mASURING
EQUIPMENT
Introduction
The equipment that is being used to obtain the doppler frequency are the AN/BRN-3 receiver which is used
aboard nuclear submarines for navigation and a Hewlett
Packard 5360A Computing/Counter.
Both are included here
with an accuracy analysis of the two done in Chapter VI
because both are very important in achieving the desired
accuracy of the doppler signal.
Receiving Equipment
The Navy Navigational Satellites transmit two frequencies, one at 400MHz and the other at
150~ffiz.
The rea-
son for the two is that after multiplication and combining
of the two signals the ionospheric refraction effect on
the doppler signal is removed leaving what is called a va• ·
cuum doppler(5).
This vacuum doppler contains only tropo-
spheric refraction effects and third order ionospheric effects.
These two signals are received by a high frequency
and a low_frequency receiver.
Only the high frequency re-
ceiver will be described here, and shown in Figures 5.1
and 5.2.
The low frequency receiver being the same except
for the signals that are mixed and the doppler signal at
the output.
The high frequency receiver acquires and tracks the
TO OA.TA DETECTOR
400 MHI
-AF
22 MH&
-D.F
1ST
MIXER
4.~
2ND
MH&
+6FI
BPF
3~
~ ±"·~~'~.
MIX
MIXER
l.....
r
378 KHI
'
LIMITER
500 KHI +Aft
L
L.O.
.
l8l.IH1
L.O.
500i<lh
-l f:>.F
MIXER
~COK'h
vc
i32
TRACKING
FILTER
L.O.
PHASE
COMPARATOR
I
!
TO REFRACTION CORRECTION IJNIT
6
F • 32. KH1 OfFSET +DOPPLER
6F1 • 32KHz OFFSET +VCO DOPPLER COMPEI'SATION
VCO OUTPUT VA!l::S ±12KHz ABOUT ~ 32KH1.
Pig.
5.1.
Hjgh frequency receiver.
Vt.
Vt.
56
'
'512KHz
X2
'
10~4KHz
/
'
.....
<
MIXER
680KH'
.
512KHz
~
-'/
XJ/4
7
z
"V
') ')10{7,
/
.184KJ
MIXER
'
/
'-
r I'
Figure 5.2 - Refraction Doppler Correction Unit.
57
---~--------------------------~-----'·-----------~---------
--------------
----~--·--
-----------------------------------.
;
higher of the two satellite carrier signals and converts
this phase modulated signal to a 500KHz signal to use in
the phase modulation functional section for decoding.
With the aid of input signals from the frequency standard
the receiver produces a high frequency VCO tracking signal
for the refraction/doppler correction section.
In addition.
the receiver lock-on signals for the phase modulation decoder section and the data processor are produced along
with a high frequency doppler shift signal for the processor.
These last signals are not involved in obtaining the
doppler so they will not be described further.
The primary function of each receiver is to acquire
and track the associated carrier.
The secondary functions
are to amplify and convert the received carrier(and its
modulation sideband) to a phase modulated 500KHz IF signal
for application to the phase modulation decoding section
and to supply a signal which is phase locked to the carrier, to the refraction/doppler correction section.
The tracking circuits lock the receiver to the RF
carrier to provide low noise doppler signals to the refraction correction unit and to the data processor.
The 500KHz
IF signal and the 500KHz signal from the standard are phase
compared as shown in Figure 5.1.
If they are out of phase
by 90 degrees then there is zero output from the phase
parator.
co~
If there is some output, then the two frequencies
are not the same, the output is applied to the VCO to cor-
58
·--~-----·-------·--·-----------------·- ---~---
-- ---~-------· --·
··---·----~---------·---- ~-----------------~
rect the frequency from the second IF to be exactly 4.5MHz ••
• Therefore, the VCO changes in frequency corresponding to
the doppler frequency.
The VCO signals from both the high
and low frequency receiver are sent to the refraction/correction unit.
The high frequency VCO signal is multiplied
by two and the low one is multiplied by three quarters.
In this way first order ionospheric effects to the doppler
signal are removed leaving only second and third order effects(5).
This is shown in Figure 5.2.
Measuring Equipment
In order to obtain the best possible accuracy for
the doppler frequency it would be necessary to count the
frequency directly from the receiver and record it digitally.
If the signal was recorded directly from the receiver
in analog form then variations in the tape speed could effect the frequency count when it is processed.
By record-
ing it digitally then these errors are not involved.
The
counter being used to digitize the doppler signal is the
HP5360A model.
It utilizes an interpolation technique
which decreases the plus or minus one count inaccuracy
which is present in most counters by one-thousand fold.
An accuracy analysis of this counter is given in Chapter
v.
The doppler frequencies from the counter are being
recorded on an HP5050B digital printer.
For ease in input-
ting the data into the computer it would be appropriate to
record the digitized data on magnetic tape with time in
milliseconds.
59
---~---·---~-------~-------
----------------------------·---------------- ---- ---- -··---- ------
-----------------------'"~---------~-----------
The method used to calculate the frequency by the
counter is shown in Figure 5·3·
When the counter is trig-
gered the start interpolator starts the T1 measurement
and the T0 measurement. The minimum measurement time is
set by front panel measurement time controls. When this
time has elapsed, the stop arm signal occurs and the stop
interpolator begins T2 measurement. The counting of No
pulses stops on the next lOMHz clock pulse. When the stop
interpolator finishes, the result is computed and displayed.
The use of the interpolator reduces the normal plus
or minus one count error by a factor of one thousand.
The
basic measurement is an extension of the time interval as
shown in Figure 5· 3, where time interval T = T0+T 1 -T2 is
the time that is sought.
Times T0 , T1 , and T2 represent three separate measurements. To is the total N0 clock pulses occurring over
the measured time interval. T1 is the time interval between start pulse and first c:J..ock pulse.
T2 is the time
interval between stop pulse and the next clock pulse.
The start interpolator measures T1 by charging a
capacitor with a constant current. The capacitor is then
discharged by a second current one thousand times smaller.
The time taken to discharge the capacitor to its initial
state is then one thousand times longer than charging time
T1·
Real time T1 is thus stretched by a factor of one
thousand and is measured by counting the number of clock
60
- '--- . ----- ---- ---
·---~-~----~
------------~
----
~------------------------------------
.. --- _________________________________ ._ _____!
- - - - T ----1
INPUT
SIGNAL
0
2
3
4 . . . . . H0
IOMHz
CLOCK
INTERPOLATED
TIMES
I 2 3
H2
llll~rllll
COUNTED
IOMHz
CLOCK
·~p me
Interval T
=
T
0
+
T
1
- T
2
Gated Clock Pulses, Start to Stop
= N1
Start lnterpolatio!1 Counts
Stop Interpolation Counts
N
N
N
0
1
2
proportional to T
= N0
= N2
0
proportional to T ~
T
proportional toT~ = T
1
2
x 1000
x 1000
Fig. 5·3· Method used by the HP5360A Computing/
Counter to meas·ure frequency.
61
,----------·--------------------·~----------·---·-····~--
pulses N1 that occur in time T1 '.
works in the same way.
----~-----------
·-----------------------------·
The stop interpolator
By changing the measurement time on the front panel
the time over which the period average measurement is
made can be changed.
The longer the period average meas-
urement the more accurate the frequency count.
So by the
use of the interpolators and the high frequency internal
oscillator as a time base and by measuring the time base
instead of the input frequency, the accuracy of the measurement is much improved over conventional counters.
By
the use of an external time base which is much more stable
the measurements are even better.
Therefore, a cesium
beam frequency standard is being used with a short term
stability of parts in 10 12 . The counter and frequency
standard are about the state of the art at this time.
----·-
-~--~--------------------
-- - - - - -- --
-
------~--
--- --
- -- --
----""--~··-·------ ··-------------~---
CHAPTER VI
ACCURACY ANALYSIS
Introduction
As previously stated in Chapter III, it is necessary
to have an accuracy of at least one part in 107 for the
doppler frequency if the method is going to be able to pre-·
diet ducting conditions in the atmosphere.
A nominal val-
ue for the doppler frequency is 10KHz therefore the maximum permissible error is .OOlHz.
Since this is an extreme-
ly small number it is necessary to analyze all the equipment involved to see if this degree of accuracy can be realized.
The accuracy of the orbital velocities must be the
same therefore it is also necessary to check to see if the
Orbital Improvement Program at NAG, which is the most accurate at this time in computing the orbit, can realize
this degree of accuracy.
Navy Navigational Satellite Transmitter Errors
The Oscar satellites that are used in the Navy Navigational Satellite System transmit two very stable frequencies, one at 149.988MHz and the other at 399.968MHz.
Both of these signals are derived from one oscillator which
is rated to have a short term stability of about one part
in 10 10 (6). In this analysis the accuracy of the signal
will be defined by the standard deviation of the frequency
of the signal,c-, divided by the magnitude or mean freq-
62
6J
uency of the signal, u.
Both frequencies are derived from
·a 4.9996MHz frequency oscillator.
When the signal is mul-
tiplied electronically, both the standard deviation and the
mean are multiplied.
stays the same.
Therefore, the ratio or accuracy
So both frequencies are transmitted at the
same accuracy but with different standard deviations or errors.
This is illustrated in Figure 6.1.
If the oscillator is operating at a stability of one part in 10 10 then
10
the transmitted frequency is accurate to one part in 10
when referenced to their respective frequencies.
Ultimate-
ly the errors must be referenced to 10KHz since this is the
reference used to define the needed accuracy.
AN/BRN-3 Receiver and Atmospheric Errors
The receiver locks on to these two frequencies and
the doppler frequency is derived from them after being
converted to a lower frequency.
conversion of the two signals.
Figure 6.2 shows the first
The incoming signals are
mixed with signals from .a frequency standard with rated
short term stability of about one part in 10 12 • Utilizing
the fact that the standard deviation of two numbers which
are added is the square root of the sum of the squares of
the individual standard deviations the numbers shown in
Figure 6.2 for the standard deviations are obtained.
From here the first intermediate frequencies are
mixed with signals from a voltage controlled oscillator or
VCO as shown in Chapter IV.
Since the VCO is tracking the
64
---------- ------------- - - - -
4.9996MHz
:t • 00049996Hz
...
-------------
----------~---;---~----~--------------·---'"1
J.YIULTIPLIER
NETWORK
149.988MHz
± .Ol4988Hz
)
399·968MHz
)
± .0399968Hz
Fig. 6.1. Accuracy of signal leaving satellite transmitter.
MIXER
149. 988MHz
... MIXER
~±~·~0-0~0-1-2~8-H-z------~,
128MHz
± .000128Hz
21.968MHz
± .0)999858 Hz
21.988MHz
± .Ol49993Hz
T
Fig. 6.2. Accuracy of doppler signal leaving first
mixer.
-
65
first IF, its error can be no better than the first IF error.
Therefore, for this analysis it is assumed that the
VCO frequencies have the same error, due to oscillator instabilities, as the first IF.
Phase noise on the signal out of the VCO perturbs the
zero-crossings of the doppler frequency, so an error will
occur when the number of zero-crossings per time interval
are counted.
The rms phase jitter on the VCO output, which
occurs due to the amplitude jitter on the input signal, is
given by(7);
rk
= ~N/2C
'Yrms
( 1)
where C is the carrier power at the receiver input and N
is the effective noise input power due to the atmosphere
and receiver noise.
Tables 6-1 and 6-2 show the doppler
link analysis for both the
pectively(?).
150~ffiz
and 400MHz signals res-
Typical C/N values are shown and using these
values the following Orms value are calculated.to be;
~rms(400) = .0056Hz
~rms(l50) = .00169Hz
In the doppler cycle counting process a zero-crossing
of a doppler cycle opens a gate to instrumentation which
counts cycles from a stable reference oscillator.
ceeding doppler zero-crossing closes the gate.
A sue-
The gate
operation, and herice the cycle count, can be in error due
-
--·-
PART B
c
-+-
(Elevation
In Degrees)
c
Max
-dbm -dbw
Pt :Minimum
c
C/N
db
--·
I
Pt Nominal
C/N
c
c
IC/N
c
c
-dbm
Min
-dbw
db
-dbm
!vlax
-dbw
I
c
C/N
db
-dbm
1!in
-dbw
db
I
5
122.1 152.1 39.5
131.3
161.3 30.2
120.3 150.3
41.3
129.5
159.5
32.4
10
121.0 151.0 4.0.6
130.5
160.5 31.1
119.2 149.2
42.4
128.7
158.7
32.9
20
118.9 148.9 42.8
129.0
159.0 32.6
117.0 147.0
44.6
127.2
157.2
34.4
30
117.4 147.4 44.2
131.8
161.8 29.8
115.6 145.6
46.0
130.0
160.0
31.6
40
116.7 146.7 44.9
140.7
170.7 20.9
114.9 144.9
46.7
138,9
168.9
22.7
50
117.7 147.7 43.9
148.7
17().7 12.9
115.9 145.9
45.7
146.9
176.9
14.7
60
118.4 148.4 43.2
149.6
179.6 12.0
116.6 146.6
45.0
147.8
177.8
13.8
70
118.7 148.7 42.9
147.1
177.1 14.5
116.9 146.9
44.7
145.3
175.3
16.3
80
120.6 150,6 41.0
-
118.8 148.8
42.8
-
-
-
I
-
-
Factors Used in Calc'llations
(~~)
c
=
2
Jo (a)
c
=
0.
C(dbw)
= ( Pr
Pt ) db - 5 . 6 dbw
h
5G( ~~ ) 0. 50w
Noise Bandwidth = 3Hz
NF = 6 db. System Temperature = 1670 •K (95% radio sky)
'c
=
0. 55
C(dbw) =
N
C/N (db)
=
=
(~~) o. 75w
~!;)
db- 3.8 dbw
-191.6 dbw
C dbw + 191.6 dbw
Table 6.1. Doppler link analysis for lSOMHz.
0'-
o--
PArtT B
"'
(Elevation in
Degrees)
c
~1.\!llil.m
c
c
c
J\Iax
-dbm
db
c
C;N
-dbm
-dbw
Pt_tlrun ina!
c
C!N
C
c
C/N
db
-dbm
-dbw
db
-dbm
Min
-dbw
.i'l'lill
I
-dbw
c
J:..lax
db
5
132.3 162.3 31.8
144.5
174,5 19.6
129.9
159.9
34.2
142.1
172.1
22,0
10
123,9 159.9 34.2
142.3
172.3 21.8
127.5
157.5
36.6
139.9
169.9
24.2
20
127.0 157,0 37.1
139.0
169.0 25.1
124.6
154.6
39.5
136.6
166.6
27,5
30
124.6 154.6 39.5
l:JG.G
166.6 27,5
122.5
152.2
41.9
134.2
164.2
29.9
40
122.5 152.5 41.6
135.0
165.0 29.1
120.1
150. 1
44.0
132.6
162.6
31.5
!50
122.0 152.0 42.1
134.0
164.0 30,1
119.6
149.6
44.5
13~.6
161. 6
3Z.5
60
122.0 152.0 42.1
134.7
1G4.7 29.4
119.6
149.6
44.5
132.3
162.3
31.8
70
123.3 153.3 40.8
136.7
166.7 27.4
120.9
150,9
43.2
134;3
164.3
29.8
80
127.4 157.-1 36.7
-
125,0
155.0
39.1
-
-
-
-
-
Factors Used in Calculations
~pP~
c
= Jo 2 (a)
c
= 0. 55 ( :; ) 0. 75w
)
Pt
c
= 0,55w
C(dbw)
=
'
C(dbw)
= (;;)
db - 3, 8 dbw
Noise Bandwidth = 3Hz
NF = 6 db. System Temperature
I'
=940 °K (95%
radio sh-y)
N
C/N (db)
=
(~~)1.3 w
~ ~~ J db
= 1. 4 dbw
-194.1 dbw
= C dbw + 194. 1 dbw
Table 6.2. Doppler link analysis for 400MHz.
L_
0'\
-...:1
68
to gaussian noise by the above values.
In addition to amplitude noise induced phase jitter
errors in the tracking loop there are also tracking loop
errors resulting from 1) phase error due to time delay when
the doppler frequency changes and 2) phase error due to
residual FM as a result of transmitter imperfections.
a. time delay errors
When the doppler frequency is changing there is an
error in the VCO signal due to the time delay of the signal
in the loop.
This error is given by the following equa-
tion(?).
where k1 is given by the change in doppler frequency with
time and is constant and G is the gain. G is approximately
48xl03 and a nominal value for k1 is
k1 =
dl'Iftyp/dt = 5Hz
( 3)
Therefore,
fe = 5/48xl03 = .OOOlHz
If the rate of change of doppler frequency could be
maintained for three minutes, which is a nominal time for
a pass between 0 and 5 degrees elevation, the accumulated
phase error would be;
69
b. residual FM errors
The phase error due to residual FM in the spacecraft
transmitters is given by;
( 4)
In the double-doublet pattern which phase modulates
the carriers, the fundamental frequency of the individual
phase excursions is 200Hz.
For a value of FM residual of
.01 radians the phase error would be;
This is a negligible error and will not be included into
the analysis.
By taking the square root of the sum of the squares
of the errors the values in Figure 6.J are derived.
From here the two signals go into the refraction
correction unit.
The two signals are multiplied and mixed
and the results of this are shown in Figure 6.4.
The over-
all accuracy can be calculated by dividing the standard
deviation of the signal coming out of the last mixer by the
mean of the doppler frequency.
The results of this is an
accuracy of .09Hz/lOKHz which is 9xlo- 6 . Clearly, this is
not as good as needed.
This is the accuracy of a signal through the receiver
on the assumption that the alignment of all the components
are exactly what they should be.
ality.
This, however, is not re-
The mixers and multipliers introduce errors also.
70
21. 968MHz
,
+ · 039998586Hz "
vco
532KHz
+ .0445287Hz
21.988MHz
+ .Ol49993Hz
vco
512KHz
+ .024Hz
'
'./
Fig. 6.3. Accuracy of doppler signal leaving VCO.
532KHz
X2
• .o445287Hz
1064KHz
....
+.0
z
...
512KHz
+ .024Hz
X3/4
\
384
MIXER
z
+. OI8Hz
55KHz
±·09086Hz
MIXER
680KHz
I'+. 09Hz
_,
1
625KHz
Fig. 6.4. Accuracy of doppler signal leaving receiver.
71
.~-----------·-·-------------------·------~-----·-··---·--------~-------------·-----------·---------
:So, in order to measure the overall accuracy of thereceiver, a test signal was input into the receiver.
A
149. 988MHz and 399. 968MHz signal from:, . a frequency standard
were input and the doppler frequency was measured.
The
signal strength of the signals entering the receiver was
varied over the range which sould be normally obtained
when tracking because the accuracy is dependent upon reeeived signal strength.
The accuracy is also dependent
upon the bandwidth the receiver is operating on so this
was varied accordingly.
In order to keep everything con-
sistent measurement times on the counter which would be
used when tracking were used.
The way in which this was done was to measure the
standard deviation of one hundred samples of the doppler
frequency and use this as an indication of the accuracy.
The following formula was used.
whereo-N is the accuracy of the vacuum doppler frequency,
df is the change in frequency between two measurements and
f
0
is 10KHz.
The results obtained are shown in Tables 6.3
and 6. 4.
These tables show that the two larger frequencies
have a slightly smaller error but still. the error is to
large.
This was for a fixed frequency input derived from
a frequency standard therefore tracking loop errors and
72
Table
6.J. Error with measurement time of .lsec(Hz).
BANDWIDTH( Hz;
RECEIVED POWER(dBm)
-110
55KHz
512KHz
532KHZ
-120
1
.038
.039
3
.043
.044
10
.059
.08
30
. 259
.298
1
.021
.022
3
.023
.026
10
.04
.058
30
.221
.234
1
.025
.023
3
.023
.029
10
.046
.054
30
.242
.258
73
-.
------·------------~--------------------------
---
-------~---·-----------------------------------------·-------------
'
Table 6.4. Error with measurement time of .5sec(Hz).
BANDWIDTH( Hz)
55KHz
.512KHz
.532KHz
RECEIVED POWER(dBm)
-110
-120
1
.021
.022
3
.018
.019
10
.014
.015
30
.05
.0)4
1
.017
.017
3
.OJ
.032
10
.01
.016
30
.038
.04
1
.013
.014
3
.008
.009
10
.009
.011
30
.047
. 052
74
errors due to atmospheric gaussian noise is not present.
If the noise from the receiver and standard were as low as
previously assumed then the error should be much less.
However, receiver alignment was not considered and could
be the main cause for such a large error.
Another interesting result from these tables is that
for a measurement time of .1 seconds, a bandwidth of 1Hz
is best, and for a measurement time of .5 seconds, a bandwidth of 10Hz is best.
Theoretically a 1Hz bandwidth
should be more accurate because the smaller the bandwidth
the less frequency variation is allowed in the tracking
loop.
Therefore it is possible that due to a longer aver-
aging time a wider bandwidth appears best.
HP5360A Computing/Counter
Measurement error in the counter is due to the following three items, all of which can be found in the HP
manual for this counter.
a. interpolator error
b. trigger error
c. time base error
a. interpolator error
This is the plus or minus one count error due to the
extra clock period counted because the input frequency
period does not equal the clock period.
The error is given
by equation 6.
E1
= lxl0-9/Measurement
Time
( 6)
75
The measurement time is the actual time and not
necessarily the time set by the front panel controls.
If
the input signal period is longer than the measurement time·
controls then the period determines the measurement time.
The relationship between interpolator error and measurement
time is shown in Figure 6.5.
The interpolator error is
decreased by one-thousand due to charging of a capacitor
over the period of ambiguity at the beginning and end of
the measurement.
b. trigger error
Trigger error consists of input trigger circuit error and error due to noise on the input signal.
The input trigger error may be found from the input
signal slope in "volts :Per second" as shown in Figure 6.6.
After obtaining a value for the trigger error, dT, the accuracy may be calculated by dividing .dT by the measurement
time.
Error due to noise on the input signal may be found
for a sine wave from Table 6.5.
The accuracy may be cal-
culated by dividing the above error by the above error by
the number of input cycles or by multiplying by the period
and dividing by the measurement time.
The total trigger error is the sum of errors due ·to
signal noise and trigger circuit noise.
c. time base error
Errors due to the time base result from aging, short
76
~--,----r-----------------r---r--~11
w
0
NUMBER Of' OIGITS
1-__:.-.:_----l . OISPlAY(O
0
tiC
~
AUTO DISPLAY MODE
~~+---~---_,.--......::..:r---.---1---+---1 t
~
..
-t
0
0
0
~
~
c
w
c
0
u
5
~
c
n
c
c
';: 10-· 1---1----1-~.:-+----.:.i<--x.£-+-_;_-T--:.....,...;__-i 7
~
~
Q
~
<;
0
~·~--~--~-~--~~,-+---~--+--~5 ~
...
a:
0
~
~
z
IO~L--~--~--~--~~--~---L--~--~
.,..
10" 1
10...
&llASUR(UtNT
Tllol[, . . c
Fig. 6.5. Relationship between interpolator error and
measurement time.
10- r
.
:
""~~
'
~
"",""-"\~CHANNEL
_CHANNEL A
'
1'..
I
""
8
I
"" ,"'
~
10•
INPU:J
' ''"''
" ",,
to•
SLOP£,
10.
10.,
V/tec
Fig. 6.6. Obtaining input trigger using the input signal
slope.
77
Table 6.5. Error due to noise on the input signal.
S/N
Error
40dB
J.2xlo- 2
).2xlo- 3
60dB
).2xl0
80dB
).2xl0
lOOdB
).2xl0
20dB
-4
-5
-6
78
term stability, temperature error, line voltage error, and
load error.
Each of these are specified for a given oscil-
lator and all that is necessary is to add them up.
The total accuracy of the counter is the sum of these
individual errors.
Some· approximate figures for the
dop~
pler frequency that have been obtained are as follows.
Measurement Time = .9 seconds
Interpolator error = 6.lxlo-5 Hz
f = 55KHz
-12
Time base errors = 1x10_ x 55KHz = 55xlo-9Hz
Total error = 6.1055xl0 5Hz
Orbital Data
The satellite will have some wobble in its orbit because the earth is not exactly a spheroid but has depressions.
There will be atmospheric drag which is changing
with time because the satellites are not completely free
of the atmosphere.
The data that is obtained is dependent
upon how well the theoretical doppler curve can be matched
to the real. doppler curve.
So, in addition, a fitting er-
ror is introduced.
On the average it is estimated(S) that the position
is accurate to within about ten meters and at about ten
thousand kilometers this is abo~t an accuracy of one part
in 10 6 . It is estimated that the velocity has about the
same accuracy and travelling at four nautical miles per
second the error would be 4xl0
-6
nautical miles per second.
Therefore the accuracy to which the orbit can be obtained
is about an order of magnitude to large.
79
Accuracy as Determined by NAG
At NAG the satellites are tracked and the doppler
data is recorded.
From this doppler data a curve fitting
procedure is done which is used to predict the orbit of the
satellite.
From this curve fitting procedure there is
generated a value for the error of the data which is derived from the residuals which occur in this procedure.
This error is referenced to JOOMHz so that to obtain the
accuracy the error is divided by JOOMHz.
This error term varies from pass to pass but a typical value is about .02Hz. When dividing by JOOMHz the
accuracy is 6 . 67xl0 -11 . It will be shown later that the
received doppler signal is referenced to 687.445MHz.
By
using the above accuracy the true error is .0458Hz. Since
the accuracy of one part in 10 7 which is needed was referenced to 10KHz, this error must be divided by 10KHz.
'
.
nominal value would therefore be approximately 4.58xl0
A
-6
.
Accuracy as Determined by using a Regression Analysis
After the data is digitized and recorded a polynomial regression analysis is performed on it.
A measure of
the accuracy of the data is defined as the standard deviation of the table of residuals which remains.
The assump-
tion which is made here is that the data should be a smooth
curve if it were noiseless.
A nominal value for this standard deviation is about
.135Hz.
This error is 1.96x10-lO when referenced to
80
'----~--~-------·----- -~-~ ·---~--~---------------·--------------·--------
'
·------
---------.---------------~------------------·----------~~
68?.445MHz or 1·35xl0-5 when referenced to 10KHz.
The difference between the error generated by NAG
and that generated by the regression analysis can be ac- ·
counted for by the pre filtering of the NAG data and the
higher gain antenna they used.
Derivation of the 68?.445MHz Signal
The signal starts out at 4.9996MHz in the satellite
transmitted and is converted to 399.968MHz and 149.988MHz.
These two signals are transmitted from the satellite and
received by the ground station.
After conversion to a low-
er frequency and tracking by a phase lock loop the two signals become 532KHz plus or minus doppler and 512KHz plus
or minus doppler.
The noise on these signals is still ref-
erenced to 399.968MHz and 149.988MHz respectively.
The first signal is multiplied by two and the second
by .75.
Since the noise and reference signal are also mul-
tiplied the references are now 799·936MHz and ll2.491MHz
respectively.
These two signals are then subtracted from
each other so the reference is 68?.445MHz.
Summary
The error of the doppler frequency by both the theoretical and experimental evaluation is too large by about
two orders of magnitude.
be about .09Hz.
Theoretically the error should
A nominal value as determined by NAG is
.045Hz and by a regression analysis is about .135Hz.
One
additional error which has not been considered is the error
.due to misalignment of the _receiyer.
It is
possib_le~_th,a~----
81
____ . ___________________________________________ ·-- --
-------
----
---~--~
------ -------- --------------------------------
this is the reason for such a large fixed frequency error.
The error due to the counter appears to be small as compared to other errors and therefore can be neglected.
The orbital data, which needs to be as accurate as
the·doppler frequency, is estimated to have an accuracy of
6
about one part in 10 .
If this is an accurate estimate
than the orbital data would not be good enough to yield results through the computer program.
CHAPTER VII
RESULTS
Several different satellite passes were tracked using
the equipment described in the preceeding chapters.
The
passes were of different maximum elevation angles,
dif-
ferent azimuths and also, entirely different satellites.
Since there are six possible satellites to track, it is entirely possible to track anyone of them depending on the
time of day that the track takes place.
In each of these passes that were tracked the doppler
frequency taken from the receiver was recorded through the
frequency counter.
While recording the data it would be
time tagged to the nearest second.
Only data below five
degrees elevation angle would be taken.
When the satellite
is rising it is frequently very difficult to lock on to the
doppler signal in the first five degrees therefore only the
last five degrees at satellite set is used to record the
doppler.
Along with the doppler frequency the
AGC(Automa~
tic Gain Control) data would also be recorded to see what
the received signal strength is and also to see whether or
not multipath is effecting the data.
data could be used to determine
data.
th~
In addition the AGC
cutoff point for the
After this the data is analyzed first by doing a
regression analysis on it.
data actually is.
This determines how smooth the
This also identifies "bad" points which
82
83
-----
--~-
-------
--~--------------
--· ~--
-----~----
--~
---
""
------~------·-----
------------------------------.
could result from keypunch or anomalous atmospheric errors.
Then, along with the orbital data for that pass, the data
is input into the refraction profile program and the re..,.
sults obtained.
The first try at this was done on January 21, 1976
at about 7a30 P.M.
The satellite tracked was number 30120.
The maximum elevation angle was eight degrees and the azimuth from rise to set was 240 to 324 degrees.
The last
five degrees in elevation was from 311 to 325 degrees in
azimuth.
From 95 to 315 degrees azimuth the antenna is
looking over water and rest is over land.
Therefore, part
of this pass was over water and part over land.
The measurement time set on the counter was one half
second.
In actuality the measurement time was slightly
longer because of the computational time inherent in the
counter which is about three milliseconds.
After each
measurement is displayed the following measurement begins.
Therefore, the data is time tagged to the nearest half second.
For each measurement the data is slightly longer
then 50 milliseconds away from the previous measurement due
to the three millisecond compute time.
The data was run through a regression analysis and
the results of a third order polynomial fit are shown in
Figure 7.1.
In the top graph in this
~igure
the real dop-
pler data is plotted along with the regression line that
is calculated from this data. _The data looks extremely
smooth in this graph.
The bottom graph is a plot of the
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Fig. 7.1. Regression analysis of vacuum doppler taken on January 21, 1976
at 7:30 P.M.
())
.{::"
85
, residuals and it illustrates the errors in the data.
In
order for the data to be accurate to a part in 107 the residuals should be zero out to the sixth decimal place.
This is assuming, though that the data should fit a smooth
polynomial curve which is r.easonable provided there is not
any subtle changes in the atmosphere and the satellites
orbit.
Both of the signals received for this pass were
greater than -llOdBm.
The high frequency AGC varied more
than the low frequency AGC but it was very little.
If
they both varied together than multipathing could have occured.
This doppler data was put into the computer program
along with the orbital data.
Because of such a large dif-
ference in Tim and 1ic the program could not converge on
a solution.
A graph of this is shown in Figure 7.2.
No-
tice that besides the large difference(.055) between them
there is also an increase in the difference with change in
.
Tim•
In order for the program to plot results which make
any sense the differences have to be no greater then one
part in 10 5 in magnitude as related to Tim" This is the
approximate magnitude of the tropospheric refraction effect.
Therefore, besides being off by at least three orders of
magnitude, there is a slope present which must be corrected
for.
Several things could be the cause for such a large
difference.
The difference expressed in time comes out to
86
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r 3•16 3~17. 3.1a 3.19 3.20 3.21 3.22_ 3.23_ 3'·24 3.25 3 126, 3,2~ 3.28 J-29
Tim
Fig. 7.2. (Tic-Tim) for pass taken on January 21, 1976
at 7:30PM.
87
be about five to six seconds in magnitude in time delay.
That is, if the time was shifted on all the doppler measurements by five or six seconds the difference would be
almost zero.
This large time difference, however, cannot
be accounted for.
Other possible reasons for the large
difference could be an error in receiver position coordinates or an
~rror
in fe(transmitter frequency).
It was
later found out that the receiver position was correct but
in an investigation of the transmitter frequency it was
discovered that the frequency offset in each transmitter
had a long term drift.
Therefore, the normal 80 parts
per million(PPM) offset frequency could be 79 or 81 PPM.
Therefore, for each satellite pass tracked the appropriate
offset was obtained from NAG and applied to the program.
The reason for the slope was entirely unknown for
quite awhile until the time tagging of the doppler frequency was considered.
Each time a measurement was dis-
played it was a little over one half second later in coming
and it was being time tagged to the nearest one half second.
So for each measurement the doppler slowly drifted
away from the true value.
Therefore, what was needed was
an external trigger to the counter to trigger the measurement at a time that was known exactly.
This was achieved
by a one pulse per second signal from a frequency standard
which occured exactly on the second.
The question is now what measurement time should be
used. First of all an accuracy of about one part in 10 7
88
had to be achieved for the doppler signal which is approximately 10KHz.
This means that the signal must be counted
when the doppler changes less than .OOlHz.
For the pre-
vious pass the doppler changed at a rate of about 22Hz per
second.
This means that the measurement' time should not
exceed .045 milliseconds.
But at this measurement time the
counter and receiver combined are only accurate to within
6
about one part in 10 .
This was determined by running a
fixed frequency analysis at this measurement time.
In or-
der to have an accuracy of at least one part in 10 7 a
measurement time of ten milliseconds is needed which gives
an accuracy of about two parts in 10 8 .
With this measurement time and with the use of an
external trigger(offset still uncorrected), a pass was
tracked again.
7.4.
The results are shown in Figure 7·3 and
The satellite that was tracked was number 30140 on
April 9, 1976 at 9:30 A.M.
The maximum elevation angle
was twelve degrees and the azimuth in the last five degrees
of elevation was from 321 to 333 degrees which is entirely
over land.
As can be seen from Figure 7·3 the data is not
quite as good as the first pass but the slope has been
taken out of the differences and the magnitude is slightly
smaller as shown in Figure 7.4.
There are still very
abrupt changes in the differences which is most probably
caused by the accuracy of the data due to the shorter measurement time.
Figure 7.2 and 7.4 show that the changes are
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Fig. 7.3. Regression analysis of vacuum doppler taken on
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Fig. 7. 4. (Tic-Tim) for a pass taken on April 9, 1976 ·
a:t 9t )OAM.
91
?
--------------
-----------------~~----~~----------
--------------------- ----------------,
larger in the second case which, as stated above, is due
to the shorter measurement time of ten milliseconds.
In
this pass the AGC data was approximately the same as the
first pass.
The data from both of these passes was obtained from
the vacuum doppler.
It was shown in the accuracy analysis
in Chapter VI that the two VCO frequencies which enter the
refraction correction unit(512KHz and 532KHz nominal) are
slightly more accurate.
Therefore, a third pass, still
uncorrected for frequency offset, was taken and the 512KHz
signal was recorded.
and 7.6.
The results are shown in Figures 7·5
The satellite was number 30120 and it was track-
ed on April 16, 1976 at 1:00 P.M.
The maximum elevation
angle was 25 degrees and the azimuth in the last five degrees of elevation was from 336 to 342 degrees.
As can
be seen from Figure 7.5 the data was better then the second
pass but not quite as good as the first.
But looking at
Figure 7.6, even though the slope has been taken out, the
difference is larger than the previous two passes(.0575)
and the variation in the points is greater.
Since the
vacuum doppler is ionospheric refraction corrected, it
appears that it is necessary for this data to be used because the theory assumes no refraction in the ionosphere.
This is borne out in the second two passes.
Again the AGC
data for this pass is much the same as for the first pass.
Another pass was recorded with a measurement time of
100 milliseconds to fine out if the measurement time really
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Fig. 7.5. Regression analysis of 512KHz signal taken on April 16, 1976 at l:OOP.M.
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:·---.--:---~
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Fig. 7.6. (Tic-Tim) for a pass taken on April 16, 1976
at 1r00PM
~------~--------·~-----·--~-~----·-----~-~-----------~--------------------------·-·---·----·-----~------·-·-----·-------~
did effect the variation in the difference of
(T.J.m-T· ).
l.C
Satellite 30120 was tracked on May 4, 1976 at approximately
1:30 P.M.
The maximum elevation angle was twelve degrees
and the azimuth in the last five degrees of elevation went
from 321 to 330 degrees.
Therefore, this pass was recorded
when the satellite was entirely over land.
The results of
the regression analysis are shown in Figure 7·7·
Since
the bounds on the residuals were at .0005 in the graph of
the residuals, the doppler was more accurate than that
shown in Figure 7·3·
Figure 7.8 is a graph of (Tim-Tic)
versus Tim for this case.
The variations in the differ-
ences are smaller than Figure 7.4 where the vacuum doppler
was recorded with a measurement time of ten milliseconds.
Therefore the measurement time does effect the variation
in the difference between the calculated and measured
T.
In order to compare the vacuum doppler with the high
frequency signal entering the refraction correction unit
a pass was tracked and both signals were recorded.
sults of this are shown in Figures 7-9 and 7.10.
The reBy look-
ing at the graph of the residuals for both cases the signal
that enters the refraction correction unit is cleaner then
the signal that is leaving it.
As an example of a one part in 10
8
accuracy in the
doppler data Figure 7.11 shows some simulated data.
This
data is extremely accurate and it just exemplifies what
the actual data should look like in order to obtain favorable results.
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4~
1976 at 1:30 P.M.
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STANDARD ERROR OF REGRESSION COEFFIC!ENlS
.2055JE-05
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ANALYSIS OF VARIANCE FOR POLYNOMIAL OF DEGREE
SOURCE OF VARIANCE
DECREE OF
FREEDOM
SU!'I OF
SQUARES
MEAN
SQUARE
1.58029
.52676
DEVIATION ABOUT REGRESSION • •
82
.00000
.ocooo
TOTAL •• , .. ,......
85
1.58029
DUE TO REGRESSION ......... ..
Fig. 7.10. Regression
correction unit.
analys~s
F
VALUE
of 532KHz. signal taken before the refraction
\.0
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-1~.20637
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ANALYSIS OF VARIANCE FOR POLVNOI'IIAL OF DBREE
SOURCE Of' VARIANCE
DEGREE OF
FREEDOI'I
St:l'l OF
SQUARES
I'£ AN
SQUARE
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3
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96
. ocooo
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99
.00~03
F
VALUE
Fig. 7.11. Regression analysis of simulated doppler data with a one part in
10
8
accuracy.
\0
'-.0
100
simulated along with a certain refraction profile.
The re-
siduals were zero down to the fifth decimal place which is
the limit using the current regression program.
After the offset frequency error was discovered
se~.
veral passes were tracked and the offset was removed.
Fi-
gure 7-12 through 7.21 are examples of some of the passes
taken.
By examining each of these it is apparent that the
bias has been virtually eliminated.
However the data is
still too inaccurate to enable the program to converge on
any meaningful results.
Summary
Several problems have been eliminated.
the timing and offset frequency errors.
These include
It appears that
practically all the bias has been taken out but the noise
level is still to high in the doppler frequency.
Also. the
necessity of the elimination of the first order ionospheric
effect has been established.
In addition it was shown that
the high frequency VCO signal is more accurate then the vacuum doppler signal.
At this time and with the present
equipment it seems that the necessary accuracy cannot be
realized.
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----....
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STANDARD ERROR OF REGRESSION COEFFICIENTS
-~'118'1E-05
.1'1727E-04
.1382'1£-08
ANALYSIS OF VARIANCE FOR POLYN0/'11 AL OF DEGREE
SOURCE OF VARIANCE
DEGREE OF
FREE00/'1
SUM OF
SQUARES
MEAN
SQUARE
DUE TO REGRESSION .......... ,
3
.07567
.02522
DEVIATION ABOUT REGRESSION • •
67
• 00000
.00000
TOTAL ............
70
.075&7
F
VALUE
'107'1756.2
Fig. 7.12. Regression analysis of vacuum doppler taken on June 21, 1976 at
3:30P.M.
j-J
0
I-'
-
0
•r-1
•E-t
I
s
·r-1
•E-t
........,
!
I
-.00010
.00000
6.239
6.240
6.241
6.242
6.243
6.244
6.245
6.247
6.249
6.251
. 6.246
6.248
6.250
6.252
Tim
Fig. 7.13. (Tic-Tim) for pass taken on June 21, 1976 at 3:30P.M.
i-'
0
1\.)
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-1~.16
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.........
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66610.
66630.
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TIME
POLYNOMIAL REGRESSION OF DEGREE
INTERCEPT (A VALUEI ...........
REGRESSION COEFFICIENTS
-.22302E-02
.120HE-O~
666~0.
66650.
•XC II= 66585.250
SUBTRACTED OUT FOR
USE AS A BIAS IN COI'I'UTATIONS
-1~.17620
-,25~52E-07
STANDARD ERROR OF REGRESSION COEFFICIENTS
.3~859£-05
.11~5~E-06
.1060lE-08
ANAL VS IS OF VARIANCE FOR POLYNOMIAL OF DEGREE
SOURCE OF VARIANCE
DEGREE OF
FREEOOI'I
SUM OF
SQUARES
MEAN
SQUARE
DUE TO REGRESS I ON .......... ,
3
• 07006
• 02335
DEVIATION ABOUT REGRESSION ..
68
• 00000
• 00000
TOTAL •• ,, ....... ,
71
.07006
F
VALUE
5816~88.1
Fig. 7 .14. Regression analysis .of vacuum doppler taken on June 25, 1976 at 11:10 A.M.
1-'
0
'-..>
..--...
0
•r-i
·E-1
I~
•r-i
•E-1
'-"
-.00010
.00000
6.i82
6.184
6.183
6.186
6.185
6.187
6i188
6.190
6.192
6.194
. 6.189
6.191
6.193
6.195
Tim
. .
Fig.· 7,15. (T.1C -T 1m) for pass taken on June 25, 1976 at 11:10 A.M.
I
'
I-'
0
+:-
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-
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0
...w
0
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0
w
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OP. DATE
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0
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....a:
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•
11~10.
• • •
71~80.
INTERCEPT lA VALUE! ...........
REGRESSION COEFFICIENTS
• •
•• •
77~90.
POLYNOMIAL REGRESSION OF DEGREE
-.6856~£-02
-----....._ .......__
I .. I
•
77500.
•
1· .. I
I
•
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•
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i
•
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17~15.0~9
SUBTRACTED OUT FOR
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-.33822E-07
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STANDARD ERROR OF REGRESSION COEFFICifNTS
.60~6~£-05
.19868E-06
.183881: ·08
ANALYSIS OF VARIANCE FOR POLYNOMIAL OF DEGREE
SOURCE OF VARIANCE
DEGREE OF
FREEDOM
SUM OF
SQUARES
MEAN
SQUARE
DUE TO REGRESSION ...........
3
.9~119
• 31393
DEVIATION ABOUT REGRESSION , •
68
• 00000
.00000
TOTAL ............
71
• 9~119
F
VALUE
25988718 •
Fig. 7.16. Regression analysis of vacuum doppler taken on June 25, 1976
at 2:15 P.M.
1-'
0
""
,......,.
...;.....
•E-1
Is
..-t
·E-1
.__..
-.00010
.00000
+.00010
5.480
5.485
5·490
5·495
5·500
5·505
5·510
0
. 5.)20 . 5·530
5·
5·515
5·525
5·535
Tim .
Fig. 7.17. (Tic-Tim) for pass taken on June 25, 1976 at 2:15P.M.
1-'
0
0'\
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OPERATION NO.
-
-1~.58
0
I~
LLI
....
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0
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~ r--..
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-
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-
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-
_
. .l ··I··· ·I···· ·I···· ·I· ··I· I··· ..1· ···I··. "I·
0.
,0002
78555.
.
78560.
78565.
POLYNOMIAL REGRESSION OF DEGREE
78575.
78580.
TIME
78585.
3
78590.
18595.
78600.
•XI II= 78555. '150
SUBTRACTED OUT FOR
USE AS A BIAS IN COI"pUTATIONS
-1~.58'181
INTERCEPT (A VALUE I ...........
REGRESSION COEFFICIENTS
-.IJ889E-02
.12118E-O~
78570.
I
-.25678E-07
STANDARD ERROR OF REGRESSION COEFFICIEfHS
.5'1Z69E-05
.28201£-06
.'IIITIE-08
ANALYSIS OF VARIANCE FOR
SOURCE OF VARIANCE
PO~YNOMIAL
OF DEGREE
I'£ AN
SQUARE
DEGREE OF
FREEDOM
SUM OF
SQUARES
DUE TO REGRESS I ON ...........
3
.00656
.00219
DEVIATION ABOUT REGRESSION ••
'12
.00000 .
,00000
TOTAL ............
'15
• 00656
Fig. 7.18. Regression
2:30P.M.
an~lysis
F
VALUE
833811.51
of vacuum doppler taken on June 25, 1976 at
1-'
0
'"-.l
.........
0
•.-t
•8
I.
s
•.-t
•8
........
-.00010
.ooooo
+.00002
6. 360
6.361
6. 362
6.363
6 ° 364·
. 6. 366
6. 3"68
6. 370 . . 6. 372
. 6. 374
6.365
. 6.367
6.369
6.371
6.373
Tim
Fig. 7.19, (T.1c -Ti)
m for pass taken on June 25, 1976 at 2:30P.M.
1-'
0
():)
"iMOQ5R OUTPUT
-...
-12."2
Q
UJ
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c
a:
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...
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c
w
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-.0005
--------
-----
------------
[. . ·.·l.·. .:·l :. ··I.· . ' ·I·.·' ·.-l, .......!. ·..,~~~~~.[.I
57~~0.
57~50.
5H60.
57~70.
57~80TIME
smo.
POL VNOMI AL REGRESS I ON OF DEGREE
INTERCEPT I A VALUE I ...........
REGRESSION COEFFICIENTS
-.881q6E-02
.Z803~E-O~
51500.
•XI I I=
-12.2~302
H510.
51520.
SH~O.Z50
SUllTRACTEO OUT FOR
USE AS A BIAS IN COI'PUTATIONS
-.q7q56E-07
STANDARD ERROR OF REGRESSION COEFFICIENTS
.5719~E-05
.15688E-06
.12128E-08
ANALYSIS OF VARIANCE FOR POLYNOMIAL OF DEGREE
SOURCE OF VARIANCE
DEGREE OF
FREEDOM
DUE TO REGRESS I ON ...........
3
DEVIATION ABOUT REGRESSION • ,
82 .
TOTAL ... ,, .......
85
3
SUM OF
SQUARES
MEAN
SQUARE
2.qz1z1
• 80709
• 00000
.00000
F
VALUE
~q239650.
z.qzlz&
Fig, 7,20, Regression analysis of vacuum doppler taken on July 6, 1976 at
8:45 A.M.
1-'
0
'-!)
........
0
•r-1
•E-1
I
s
·r-1
.·E-1
..._..
.00000
-.00010
.5· )30
5. 340
5. 3·50 · 5· 3'6o
.5. 3·70
5. 3·so · 5. 390
. 5·33.5
_5.)4.5
.5·355 . 5·365
5·375
5·38.5
.5·395
Tim
Fig, 7.21.
(T.1c -T.1m)
for pass taken on July 6, 1976 at 8:45A.M.
I
i
J
1-'
1-'
0
-----·---
---·--------·--·-·~---------·---~··--------·------·----------·-~--~--------
------------
--~
-
--·-~------·---- --~---~1
CHAPTER VIII
CONCLUSIONS AND RECOMMENDATIONS
Judging by the results already obtained it is apparent that much is needed in the way of accuracy in the
doppler frequency in order to obtain data that will provide
a refraction index profile through the computer program.
If the accuracy of the data can be increaded to one part in
10 7 then the variations in the (Tim-Tic) measurement can
be significantly reduced.
Some of the major reasons for the error are shown in
Table 8.1 below.
TABLE 8.1
a) Satellite Oscillator Stability
b) Gaussian Noise Error(Atmospheric and Receiver)
c) Tracking Loop Errors
d) Residual Ionospheric Errors
e) Equipment Calibration Errors
f) Multipath Errors
g) Orbital Data Errors
Each of the above errors is described and recommendations are made on each.
Although these errors
compris~
the
majority of the contributing errors it is not a completely
exhaustive list.
These appear to be the major errors but
-thera_could be__ others_that are nat as easily recognizable_. ____
111
112
---------------------------- --·--- -Y----- ---------
------- -- - - - - -
----~-
----~----~~-----
a) Satellite Oscillator Stability
The satellite oscillator stability is estimated to be
about one part in l0- 10
With a 5MHz oscillator this tran-,
slates to an error of .0005Hz.
When this signal is multi-
plied by 30 and 80 two signals are derived and transmitted
(150 and 400MHz).
pectively.
The errors are now .04Hz and .Ol5Hz res-,
Since the maximum error that can be tolerated
is .OOlHz, already the error is much to large.
The oscillator is an oven controlled crystal oscillater.
To obtain an error of only .OOlHz in each trans-
mitted signal the accuracy should be at least 2.5xl0 -12 •
The state-of-the-art accuracies for atomic frequency stan-12
dards are about one part in 10
Therefore, even with
the state-of-the-art the errors are just barely small
enough.
In the Global Positioning System(GPS) the satel-
lites will have atomic standards.
If these oscillators
can be used for a satellite transmitting two signals then
better accuracies could be achieved.
b) Gaussian Noise Error
When there is noise present at the input of the tra
tracking loop of-the receiver, a phase error results at the
output of the loop in the doppler signal.
In Chapter VI
it was determined that the frequency error due to nominal
values for the signal to noise ratio(C/N) were .0056Hz for
the high frequency loop and .00169Hz for the low frequency
loop.
Already these errors are to large.
With proper de-
113
------------~----~-
-~------- --~------~-----~--------
---
---------
-------------~-----~-----------
sign of the receiver system these errors could be reduced.
Possibly a higher gain antenna could be used along with
better filters to increase the signal power and decrease
the noise power.
If the transmitted signal power were in-
creased the received signal power would also increase,
thereby increasing the signal to noise ratio.
c) Tracking Loop Errors
Since the doppler frequency is continuously changing
and since there is a time delay in the tracking loop there
is an error introduced due to this time delay.
In Chapter
VI this time delay error was estimated to be about .01875
Hz.
This error is also much to large therefore new design
techniques must be utilized.
At the present no techniques
are known which will reduce these errors.
d) Residual Ionospheric Errors
The first order effects of the ionosphere
o~.the
doppler are taken out in the receiver but the second and
third order effects still remain.
The third order effects
are larger than the second order effects and it is estimated that the third order effects are on the same order
of magnitude as the tropospheric effects which is what is
needed.
The dependence of the doppler frequency(df) on ionospheric refraction is shown below.
( 1)
...
114
~----------------------------------~-------··-----------~~-------------------·------
The primary satellites being tracked are the Navy
Navigational Satellites of which there are six.
Each of
them transmits two frequencies, one at 150MHz and the other·
at 400MHz.
The follwoing equation shows how these two
signals are combined to eliminate a 1 (t)/f, which is the
largest term, from equation (1).
df
6
150
= a (t)/150xl0 +a (t)/(150xl0
1
2
6 ) 2+
6
a (t)/(150xl0 )3
3
6
df 400 = a 1 (t)/400xl0 +a 2 (t)/(400xl0 6 ) 2+
(2)
6
a (t)/(400xl0 ) 3
3
2
-75xdf 150 -2df
= 2.0833xlo- 1 7a (t)+l.9xl0- 5a (t)
400
2
3
After these two frequencies are combined, all that
is left are the second and third order effects, neglecting
·higher order effects.
It turns out that the third order
effect is larger than the second.
In addition• the third
order effect is on the same order of magnitude as the
tropospheric effect.
Since the latter is the phenomenon
which needs to be measured, it appears that this effect
would be very difficult to analyze due to the third effect.
Indeed, up to the present, obtaining the tropospheric refraction effects has been very difficult.·
There are two methods that can be used to eliminate
the third order effect.
The first method, the three fre-
quency method, is an extension of the two frequency method
of eliminating the first order effect.
With an additional
_fE_eq~~ncy and_~~_ing the resu!:_~~--?~-~11:~- ~~:t'-~~---~-~-~~~~~~~?_n, __
115
the third order effect can be eliminated.
An example of
the method is illustrated below for three frequencies, an
even multiple apart.
2
df -2df 2 =(a1 /f +a 2/f 1 +a /f 1 3)-2(a 1 /f 2 +a 2/f 2 2+a 3/f 2 3)
1
1
3
2
=a 2/2f 1 +Ja 3/4f 1 3
df 2 -2df 3=(a 1/f 2 +a 2/f 2 2+a 3/f 2 3 )-2(a 1/f +a 2/f 2+a /f 3 )
3 3
3
3
2
3
=a 2/8f +Ja 2/J2f
1
1
2
(df -2df )-8(df -2df )=a 2/2f 1
(3)
2
2
3
1
where,
This method would eliminate the first and third orqer
terms exactly, provided the transmitted frequency
An additional benefit is that the
1
~ (t)
could be determined and from these
content of the ionosphere can be
was.know~
13-
and a (t) terms
3
model of the electron
calcul~ted.
At the present, there is a Space Technical Program
(STP) satellite now transmitting ten phase coherent signals.
This satellite is owned by the Defense Nuclear Agency(DNA)
and is used by Stanford Research Institute(SRI) for
spheric scintillation studies.
iono~­
The main disadvantage of
using these ten signals is the inaccuracyof the oscillator
aboard the satellite.
The satellite, however, a'Iso has the
capability of transmitting 150MHz and 4001\lliz from a very
stable oscillator.
Therefore, with the combination of the
two stable frequencies to measure absolute doppler and
three of the ten phase coherent frequencies to measure
116
a 1 (t)/f and a (t)/fJ, the first and third order terms can
3
be eliminated.
The second method, the electron content profile method(ECPM), is to use the two frequency method to eliminate
the first order effect and then with an electron content
profile and using equations for a (t) from reference (5),
3
the·third order effect can be eliminated.
Figures 8.1 and
8.2 illustrate the results of using this method.
The cir-
cles represent the third order effect as calculated using
four frequencies in a similar way as the three frequency
method.
The pluses represent the first order.effect using
this same technique.
The dashed lines for both the third
and first order effect were calculated using two frequencies and an electron content profile.
The solid lines·re-
present the same method using a more accurate model.
Even
with this more accurate model this method is not as precise as the three frequency method described above.
e) Equipment Calibration Errors
In order to insure that the equipment is not introducing any additional errors proper calibration is necessary.
It may be necessary to be much more exact in the
alignment procedure than is done now due to the high degree
of accuracy needed.
One of the major calibration errors
could be in the mixers where the received signal is _heterodyned with a reference signal.
Any error in the reference
signal caused by misalignment translates directly to frequency errors.
117
..
-~-
--·---~---· ··--------------~------- ---------~
------·-
···------~--------------~------------------------
+2
+1
SATELLITE 01165
66
DAY
RISE TIME
1950 UT
CA TIME
72031 UTS
MAX ELEV 33
RISE AZ
205
CA AZ
134
SET AZ
67
TIME - sec
-1
.. -2
_ _ _..;.. CORRECTED ·
- - - - - - UNCORRECTED
:I:
I
0:·
0
0:
0:
w
~
-3
0:
w
:I:
a.
"'
0
§.
-4:
-5
-6
-7
-8
Fig. 8.1. Ionospheric refraction error.
.
---------- -------·
118
---- -
---------~---·--
- - - · - - - ------------- -~--- -- --~-----------~
+6
,,.,.,--..,
/
MAX ELEV 83
RISE AZ
300
CA AZ
30
120
SET AZ
SATELLITE 01165
88
DAY
2009 UT
RISE TIME
CA TIME
73193 UTS
I
I
/
''
'
\
\
\
\
\
\
\
I
CORRECTED
+4
I
I
I
I
I
I
-~-----....,
\
I
- - - - - - UNCORRECTED
\
·II
........
...,. _
/
+++
\\
++.#+
\
l
I
I
I
+3
I
I
I
I
+
I
.
I
+2
I
:c
I
I
I
~
I
-0
~
~
I
I
w+l!::1
I
"
~
w
:c
I
.,
0..
z
+
I
+
...........
.,;/
I+
0
Q
I
72754
T
_
_......
........ ....
-
.... --1+-......
d>
0
0
0
0 0 0
0+
o~~~~~~--~-~--~-~~~~~-~-~·4:~~~+-~~fl~~~~~~~-~-~-~~~~~~~~------------
+ 1cA
. 73074
+I
.1'1
73394
TIME -
sec
+I
++ /·
*I
-1
++ I
+.pI_
+.+t+·
-_rl-+....,+-t-F-'+'
~+~+·~
I
I
I
I
I
-2
I
I
-3
'
.....
__
/
,.,.
/
I
I
I
I
I
I
I
I
/
-4
Fig. 8.2.· Ionospheric refraction error.
73714
119
f) Multipath Errors
The phenomenon known as multipathing is another consideration.
An example of this is shown in Figure 8.J.
Upon analyzing .the AGC data for the satellite passes, however, multipathing did not seem to be present.
It is pos-
sible that there is a small amount of multipath which is
not recognizable but which does introduct a small error.
·The maximum deviation in frequency due to multipathing can
be expressed simply as(9);
( 4)
where f
e
is the lobing frequency and m is the ratio of the
indirect to the direct intensity.
Both of these values
can be measured from the AGC data.
However, in this case
lobing cannot be identified at all.
In order to reduce
any multipathing errors it is best to track the satellite
over land.
g) Orbital Data
The orbital data 'for the satellites which is input
into the refraction program is approximated to be about
one part in 10 6 .
This is an order of magnitude to small
and improvements need to be made.
Errors are primarily
due to the gravitational madel, atmospheric drag and computational accuracies.
All of these areas are being im•
proved and in time the accuracy should improve.
e'
Fig. 8.3. AGC data taken on Laguna Peak shows multipathing on the 400MHz and 150MHz
signals
1-'
N
0
121
Summary and Recommendations
There are seven areas which need to be addressed in
order to obtain the proper accuracy.
The satellite oscil-
lator stability, gaussian noise error, tracking loop errors·
and equipment calibration errors are equipment related
er~
rors therefore work needs to be done in securing the proper equipment or performing R&D to advance the state-of-.
the-art in the equipment.
The residual ionospheric errors
can be eliminated by either additional equipment or using
more sophisticated mathematical techniques.
.1
The orbital
data errors need to be improved by research and larger
computers and the multipath error can be decreased by only
tracking passes over land.
With additional emphasis on
these areas the data could be obtained with an accuracy
which is two orders of magnitude better than that now achieved.
The limitation on the accuracy required is due to
the mathematics used in the program.
It is believed that
techniques can be used to allow a greater error to be present in the data.
These techniques will be initiated by
M. A. Bondelid, Ph.D of PMTC.
In addition, the initial
solution for the refraction profile can be improved thereby
increasing the possibility of convergence.
The motivations for the success of this method are
many.
The largest being perhaps the Integrated Refraction
Effects Prediction System(IREPS) which is to be used aboard ships at sea(lO).
This system is designed to pro-
122
vide an onboard capability to assess the effect of atmospheric refractive anomalies on sensor performance.
A quick
refraction index profile would be very beneficial in this
system for early warning purposes.
123
REFERENCES
( 1). Stansell, Thomas A.. "The Navy Navigation Satellite
System a Description and Status," Journal of
the Institute of Navigation (Fall 1968)z 229.
( 2). Sherar, Captain R. C. , and Rosenthal, Jay "Don't
Fall in _the Radar Hole," U. S. Naval Institute
(197J).
(J). Hinzpeter, Max F. E. "The Reference Radiosonde as a
Tool for Improving Meteorological Data from
Conventional Radiosondes," Source Unknown.
( 4). Claassen, R. W. "A Method for Determination of At-
mospheric Refraction Characteristics Through
Use of Navigational Satellite Data," Technical
Memorandum TM 68-67, Naval Missile Center,
(March 5, 1969).
v. L.,
the u. S.
(5). Piscane,
and Feen, M. M. "The Potential of
Navy Navigation Satellite System in
the Prediction of Ionospheric Characteristics
for High Frequency Sky Wave Telecommunica- .
tions," Technical Memorandum TG 1182, John
Hopkins University, Applied Physics Lab.,
(August 1971).
(6). Groepler, CDR David "Accuracy Analysis of the Navy
Navigation Satellite System Related to Atmosphere Refraction," Technical Report TR-73-bs-1
Point Mugu: Pacific Missile Test Center,
(January 1973)·
(7). Navy Navigation Satellite Program, Astro-Electronics
Division, RCA, Princeton N.J. 2 (March 30,
1967).
(8). Crawford, Mike. Navy Astronautics Group, Point Mugu,
California.
Interview, 11 March 1976.
(9). Smith, Tom. "Effect of Multipath on Laguna Peak Doppler Data Quality,'' Memorandum for Commanding Officer (April 6, 1973).
( 10). Shlanta, Alexis, and Cornette, William M. "Fleet
Radar Performance Modeling Including Environmental Effects," Technical Note 4073-71
Naval Weapons Center, China, California
(September 1975): 53·
(11). Bondelid, M.A .. Ph.D. "An Heuristic Exposition of
Methods to Obtain Tropospheric and Underwater Refraction Profiles," Working Note
l29A, (April 19, 1967) .
(12). Report on Atmospheric Anomalies Detection Method,
by the Navy Science Assistance Program, TR3062 (January 1974).
(lJ). Orbital Improvement Program,
u.s.
Naval Astronau-
tics Group, Point Mugu, Ca.
(14). Chernoff, Joseph. "Application of Satellite Navigation Techniques to Marine and Air Naviga-
125
;---------~--~~--------------------·--.--------·--·----------------·--------·-------------.--
------------------.,
tion, .. Journal of the Institute of Navigation, (Summer 1969): 133·
( 15). Newton, Robert R. "Everyman's Doppler Satellite Nav-.
igation System," IEEE Transactions on Aerospace and Electronic Systems, AES-3 (May
1967): 527.
(16). Quillin, CDR. James. "The
u.s.
Navy Transit Naviga-
tional Satellite System."
(17). Guier, W.H., and Weiffenbach, G. C. "A Satellite Doppler Navigation System," Proceedings of the
IRE (April 1960).
(18). Anderle, R.J. Memorandum on Tropospheric Refraction
Phenomenon (July 3, 1972).
(19). Commander, Pacific Missile Range.to Commander, Naval
Weather Service Command, (August 9, 1972).
(20). Hopfield, H.S. "The Effect of Tropospheric Refraction on the Doppler Shift of a Satellite
Signal," Journal of Geophysical Research 68
(September 15, 1963): 5157·
(21). Claassen, R.W., and Thorne, C.J. "Refraction Theory
and Applications," Technical Publication TP-·
70-58 Point Mugu: Naval Missile Center,
(November 10, 1970).
(22). Guier, W.H.J Newton, R.R.; and Weiffenbach, G.C.
"Analysis of the Observational Contributions
to the Errors of the Navy Satellite Doppler
126
Geodetic System," Technical Memorandum TG-
653, John Hopkins University, Applied Physics Laboratory (January 1965).
(23). Berbert, John H., and Parker, Horace
c.
"Gees Satel-
lite Tracking Corrections for Refraction in
the Troposphere," International Symposium on
Electromagnetic Distance Measurement and
Atmospheric Refraction Boulder, Colorado:
(June 1969).
(24). Jordan, Edward
c.,
and Balmain, Keith G. Electro-
magnetic Waves and Radiating Systems. Englewood Cliffs, N.J., Prentice Hall, 1960.
127
---·----·--~-----·-·-----···-----·-------------~~-·----·-··--------~---------·-------·------·-·--------·---··--·-------··--;
APPENDIX
Listing of the computer program which calculates the
refraction index profile of the troposphere.
Necessary
inputs are the satellite doppler data and orbital data.
128
SJBFTC PQP>'I'l
n!"~Nq,N SONC 1411. ntv 140), ZJ 1301, VJ-130i~ O~LZ 1231,
1 a ll:JJJl, IIO'll ll:JTJOI, OELF !lOOO), SOJT ll:JOOI, 0 llOi>OI,
2 P IIOOfJI, QD"T ll:l:JOI
Ol~fNSlON
~ATJO
I~JI
nt"'~'lSI':I!>l
T I ;>O'JO I, 1FT 141
CQ~~1'l I
PAT I DATI~
tr'l•n"' SON(, NT, OTV, J, ZJ, VJ,
I ~01T, R, O~LF, DELZ
Cn~~1~ I
A"'OOEZ I l 12001
nn!l"lE P"I'C I S!OIII OH F, SOOT
f"l''1lJ"lE ODE( I SlOIII SJNC
fOUIVIILENCE IT, 01
<lFW!'IO 10
n'1 I 0 I
I , 40
tO RHT':l Ill
O.
<;O'lC I II = O. 5
=
RMAX 1
It
A,
AOOr, 0
1
IDIV II-11 + 1.1
= Cl
10'3 ~A') 15,
R') = QQ
FnR~H
21
RO,
=
I"'IJV Ill = 3.
on 5 I = 2, Z5
SQ'l: IJI = SO'lC II-II
SQNC 11-11 I
5 I'JIV I l l = OIV 11-11 + 2.
'IT
RRHAX,
flEL"
CfJSL
Sl'll
!JEl c
FE,
211
VJ:l;
PHI,
AlAM, RO, FAC,
VJ Ill
21
r Af.
•
17f.lt.OI
ALA"
orns II'JEL!'I
22
f'ST~'
23
IOEl~'l
DI-ll
cnv
ocns
24
SJ'lD
TA'IP
JSJ,N
25
1 nELFl
IIJEI.Fl
ST'lP I f."SP
'1($
PO * CfJSP "' COSL
v<: = ~0 "' ClJ~P * SINL
T<; = P 0 * $I NP
T = 0
I +
22 I
REA') (5, 211. f Ill, (, S
['c l c I l.l = - C
VFL I IF r: + C l
I" IS .EO. 'J.I GO TO 22
J = I
B.C\~ 1)1), 291 TK,
'(1(, YK,
ZK, XDOTK, YDOTK, ZOOTK
~·<A'i '(l:J, 2AI TK, XK, YK, lK, XOOTK 1 YOJTK, ZOOTK
2R FnP~H I \0:1'. F9.3, 3Fl2.h, 3Fll.91
JF (TK .L". T .11'Jil. Tl( .GT. 0.1 GO TO 30
\ill I Tl' f b , 2 0 0 I
200
~"P~AT
(5HTI( ~ I
\IPJTI' (6, 291 Til:
29 Fnq YAf 1 3Flf .81
Oll ll"lLOIIO 1101
CALL FXJT
30 TK ~ TK ~SORT IIXI<- XSl
l + IYK- YSI
2 +
1 I VFL
=
21
*
**
**
WQ!TF
!6, ?50)
FO~~AT I 16HXS TH~OJGH TK
\iRJTI:: (~, 331 XS, YS, lS,
250
33
34
37
38
39
4l
IZK-
lSI** 21
43
44
=I
XK,
31 IF IT~ .L<=. Tl G1 T~ 24
nn 12 K
2, I
T I KI
T I '<-1 l
32 D!'lc ( K - l ) = (lfl F {K I
l = l - f
GO TO 31
23 R<=A'J I 10, 281 TK, XK, YK,
TK
TK ~ SOQT IIXK- XSI
YK,
7Kr
45
TK
=
=
ZIC,
**
=
l
I
• f 0.
-=
snJ T
Al~~1(
C
0
=
=
X I<
=
**
2 ~
VI<
*
*
I$00T *
c
=
= (lSCPT
~>nnTK
**
-
*
2
*
YK
YOOTK
znnTK - ZK
~O'JTKI
IlK- ZSI
**
60
~~
61
qVlT = Yl(
S"nT 121
XK
J\IA.,'(
!J4T AN2
*
Q)
I C I
SOJT
S
66
I RK
67
=
I SO'l.T,
I
65
ISOOTI
= IQ + lK * ZOOTKI
~ pc I S
= OATAN ISOnT)
=_
Z'lOTK
2 +
**
• AW). T K .l T • T I GO TO 2 4
**
srnT =
ST1'1T
D4!K
l
IYK
YOJT~
p-:;')QT rsonr1
~n~T • lK
2
YK
xnnT~ + Y~
PHI1<
PI(
YOJTK,
YFL
p:: I J
s
XDOTK,
2 ~ IYK- YSI
~QOT
__121 I _
68
129
- --- ·---- ----
--~----
---- ---------- -----
2/t II= IT< .GT. T IJII GO T::J"25
xo
XK
yp =
-------------------~-----~-----
VI(
= l1t
7P
Xll1TP
VOCITO
"'
=
lWlTP
TP = Tl(
lCilOTK
YOOTK
lll£lTK
l" IJ .F'l.
. lllA"IlP
PHl'"JP =
=
= Rl(
P01TP =
11 GO Tl) 23
Ali\MilK
f>HinK
RP
P~JD'
RnOTK
= PHIK
Alt.'-'<>=t.LAI'K
r.n
rn 23
25 Jl= I J • GT.
=
S"JT
11 Gfl Hl 26
~~
(XP
XI>
2
**
Y~
•
2
*
=
* Y01TP - YP X~OTP) I SOJT
s OSO~T (SOOT)
r
snoT • ZP ••
0 = XP
xn')TP f. VI>
YDOTP
P~Jn<> = tSnQT * Z'JOTP- ZJ>
Ql I 1: I S
Sf'r"IT = C
Ill>
[I<;ORT ( SOJTJ
ROflT~
IQ + ZP
ZOOTP) I RP
snrn
ZP 1 s
1'!-i I i> = 0 4 TA 'I IS D'J T)
sonr • yp
SOOT 12) : XP
ALA"P = '1AlAN2 ISOJT, SOOT (2)1
2ft C =IT I J I - TPI I ITK- TPI
AlA~'1P
=
=
s •
z
*
*
*
*
=
S + ALA"01( * C
Sf. PH!OK
C
il I J I = ll 0 " S + R'( * C
POOT (J) = POOTP " S + ROOTK
C
snnT
PHIP
S + D~IK
C,
stN~
ns!N 1sonT1
= P~IOP *
~
82
83
*
*
I'C"'
I 5nflT I
I C:>SP
Tn"'l
S I'IP
SI)()T
AlA'11>
SI'It
C'JS I
DSIN ISOOTI
DC'1S
*
S
•
ALAMK
I SC''JT I
COSl - COS!
r. = SPH
T-= TAN!
-
TANP
ST"'l •
ss
s ....
cc •
1. .. c
TT:: TAN(
88
89
*
*
SIN!
*
*
cnso
S
81
c
1. -
Ill"'! = Al-\"OP
DtAT
80
CIJSI
*
*
*
C
90
91
Sl~l
C£lSl
2
0<
T.!.NP"
11. + 2. • TT 1 cc 1
0 = OSIJR T I SD'1TI
Af"lnT IJI-= IS* Ill)~!+ DL.!.T * I T - TANI ICC* SSll ,/ Q
Jl= I Anr")T I Jl .GT. O. I GO, TO 33
O.ll J"lLIJt.D I 101
WRITF !6, 300)
300
H!Q\OAT 112fHtA"no
ETC)
WOJTI= ll,. 291 ALf>'~fl", ALA"'DK, OLQN, PH!OP, PHIOK, OLAT, PHIP,
I p; l -< , T 11 ) , Ala "'P, t.L AMK, S
snnT = T •• 2 + ss •
worn:
350
~n~~AT
1s
, 3 5o 1
17HAOOT-= , 7HryELF =)
l.IPIT\0 !!>, 291 AOOT IJI,
CALL EXIT
33 <;OPT = 0 I ITT + C)
A IJI =OAT/IN IS'lflTJ
J = J ... l
.
T" I J
.u:.
DELF CJI
98
99
100
lOt
102
105
107
II GrJ T1 24
113
CALL U'IL'1Af' ( 101
o'-'A"< -= RO
!C
l
=
.... l
=
0" 34 J
l' l
3 4 n 1 J 1 .. fl"L 1= t J
J .-= l
CALL '"'0'1E~G IZ,
CALL SIO"TShlG tz,
CALL scrsuG ( l ,
C~Ll SETShlG !Z,
35 <;'I
c.
$"(
0.
1
0)
22,
20,
45,
4095.)
l.CI
1.01
12!>
128
no
l32
lJ9
SY
<;l(z
. aa.•
a.
a.
a.
<;XZY = a.
5'1" = a.
on 3b l =
)lC)
SX4
Sl(Y "
I' J
S'l'l = Stl'' + 0 Ill
** 2
S'l = 5'1 + 1 •
$1( = SlC
5~
X = $"1
'l
sxz
SX 2 + )(
X
Sl!3
S'l
sx 1
)(
$1(4
sxr.
'l
sx y + s~
Sl!Y
D Ill
I) ILl
X
SX'lY= sx ?Y+
36 SY = SY + I' Ill
X= Sl!? * ISX2 ** Z- 5X * SX3l- SX * IS~3 * SX2- S~4 * SXI
1 + S'l
15)'3
2 - SX2 * SX'ol,
U
= I)Y * ISX2 ** 2 - SX * ~X3l- SX * ISXY * SX2- SXZY
SU
1 + S '·!
IS X·v
S l! 3 - 5 X2
S X2Y I I I X
PR = ISX2
ISl!Y • 51(2 - <;XZY
SXI - SY
ISX3
SXZ- SX4 * SXJ
1 + S'l * ($X)* SX2V- SXY * SX411 I X
CC c ISY2
ISX2
SXZY- SX3 * S~YI - SX
ISX3
SXZY- SX4 *
1 SXY I + SV * I SX1 ** 2 - $12 * <;JI4 I l I X
X= SU~ + AA
(AA
SX4 + 2.
IRR
SX3 + CC
SXZ- SX2YII + BB
1
1'1" • SX2 + 'l.
ICC * SX - SXYll + CC
ICC * SP-1 - 2.
SYI
WRIT!= 15, <t50l
lt50
Fno~AT ll7HSUM TH~OUGH,CC
l
WI'I!TE lb, 381 SU'I, X, S'l, u,, 8!3, CC
s•~ = o.
s ll"
0.
...
••
•.
*
••
*
•
*
*
*
*
**
*
*
*
*
*
*
*
*
*
*
*
*
*
*
=
f''lT
:
=
1,
*
I Bl'l + SN
b"Hl ITClP, '1 I l l )
h"lNl IRnr, 0 I l l )
SOPT I SIJ" I S'l I
loiDJ·TF (6, 'i'iOI
Fn""AT I 161-'SU", TJP, IIIJT
I
HQ!Tr. 16, 31") SU"', TOP, ROT
TI"IP
37 AnT
$11'1
)p
F'1P."'~T
141>
0.
.. = ;> * ..
IF !"' .LT.
lH
1., R'1T, SN, TOP!
1 , 1 • ; 0. , S "l , a • I
q)
S"',
.
27 ,J
0
Ill,
1,
GO TO 45
K
I( + 1
.. = 1
lt5 J = I I
GO TO 140, 42. lt4).
40 00 Itt L = 1, J
lol (l Ill = OC:l F Ill
Gil 1'1 35
42 11'1 43 L = 1' J
43 n Ill = h!1'1T I l l
GIJ T., 15
n IJ I =
K
= 1•
o.
27 R"4.X = A MAX l IP"'AX, ~ IJI I
P'l"~'IC = l.
I Q"AX
7J = o.
ZJ I 21
zoo. I 32'10.8?.3
ZJ I 'II
!lOCO. I 3 ?BC. AJ'I
1J 141
ZJ (';)
7J I 6 I
7J I 71
J = A
V.l ( 21
VJ I 3 I
AA I
162
lb4
1Sb
=
3'1 l
l, J
S"' = SN + l.
3<; CALL LE:;~JOG IZ,
()"'!
*
16c16.R)
FDA''C:G Ill
Cf•ll SI!RJ~G tz,
C AL l S E G >~T G I ! ,
[''1
145
lbO
C~LL
""
*
15q
=
<.N
*
J
""~
+ l.
n I l J = 0 I L I - CC - S'l
Sll"~
Sll" + 0 I l l ~• 2
550
*
Q.
rnp = o.
0'1 37 L
S'l
*
20000.
100000.
250000.
ROOOOO.
I
I
I
I
3230.833
32i10.833
3 290. 833
32R~.833
VJ
I I . + 3.
*
VJ I .I 4.
1HXI
173
13L
VJ ! ~I = I 1. • VJ I I
f'l"l l fo I(
5. 7
16 VJ I q
I.
13 l = 3
?. •
=
=
l 5 C:H l ~>1-1nrnT I Sr.IJT, l I
CALL FXI T
VJ !J-11 = VJ !J-11 + OEL7
loOO
17
=
IF IL .FO. 31 VJ !21
VJ 121 • OFll 121
nn
'<
t, 4
7J (I()
7J !Kl + DF.lZ IKI
7FT {I()
]J {K) * 328'l.A33
Wll IT E I 6, 4 J 0 I
1'(111-.ar !f.,.ZFT = , 5'-lV T
I
wo I T E I(, , o 0 I IZFT IKI, .. K
1' .,,
Wll I T f I h , 9 7 I I VJ I K 1, K = 1' 71_
IF ! l .:::0. 21 GO TO 17 ·
t = 2
en T'l 15
n'1 1 4 u = 1 , 3
CAll QHOOnT IS~IJT, 11
f1'1liK=I,4
ZJ 1'<1
7J !Kl + DEll IKl
ZFT I(}: ZJ IKI
3280.1533
VJ IJ-11 = VJ IJ-ll • OHZ
VJ 141 = VJ 141 • nftZ 151
W"l TF 16, 6'i01
F'JQ-.4T { 747FT2 = , !>HVJ2 =· l
WIIJTt lb, 00) fZI=T {1(), K = lt 41
W~ITE (6, 921
IVJ IKI, K
1, 71
C'"l'IT I'IIJE
=
zn
20
=
=
!.50
Zftl
=
=
11
223
ZZ5
264
*
277
278
283
=
14
(llll
~HJ0'1T
!SOOT, ll
VJ,. VJ 121 + 15000.- lJ (zll
1 1J 1211
WI:{JTE 16, 5001
500
FnR-.AT 18HVJI11 =, SHRATTO
I
WI:( 1 T!' I 6, 921 VJ Ill
*
IVJ 131- VJ
1211
I
291
IZJ
131293
=
><Kilt-
Do
'o'~l
·'"'1'-'
'-''•
"llllll
29lt
!81.
RATl:l 1121, RATIO IZ5lt
l RAT In I 40 l
9 3 FO Q.,. AT f 5 I' l 6. 8 I
Gfl r C1 I 0 3
90 F'JR'IAT I 101' 12.0)
91 FlR"AT 1111
92 FQ't'IAT t 10Fl2.71
fNO
$ T SF TC Dl-f'1JT
SURQ '11.1Tl NE QH'"IOOT I <;Q'lT, t Z l
DIM':'NSTJN SO~:c 1401, nrv 1401, ZJ 1301, VJ 13)), VJZ 130), RJ 1301
1 , RJ2 1301, ~J 1101, AJ2 1301, fJ 1301, FJ 1101, GJ 1301 0 CJ 1301
2 , ".VJ 1301, VIIJ 1301, DELZ 1231, ll 110001, AOOT 110001,
3 nHF 110·)01, ~nor 110001, 0 11000), II 110001, ROfJT 110001
4 , <; IZOI, 0 1201, W 1201, SS 1201, OEP.III 1201, SET 120, 241
C'J'I\41'1 S(''IC, 'lT, 111Vr J, lJ, VJ, RMJI'(, 'tQ_MAJC, ~0, !, A, ADOT, Dr
1 t!.O~T, Q' OELF, DELl
C0'-''10'1 I A"fl0E1 I l I 2001
onu~lE
~RFC!Slf''l
f1~URLE
P~ECISI!'N
D~LF,
SO~T
SO'It, SFRTES
DQI'RLF POF( ISTf1"l F, !12, O, OF, Ut Y, W, X, SS, S,
l , VJZ, <>J, RJ7, t.J, !.J;>, EJ, GJ, CJ, RVJ, VRJ
{)n = ..i/.t!."AX
i.. = J - 1
{'n 20 I(
1' l
=
= PO
oJ IKI
RJ2
DV J
Eo G
+ lJ IK!
**
{Ill
2
f K I "' !:!. J PO
VII.J (I(J
VJ IKI I t!.J IK)
V.l2 (I()
VJ 11<1 •• 2
p: 100 .LT. V0 J l'<ll on
(I<)
I 1<1
IIJ
VJ
295
V"J IKI
ll G'J T, 20
X= 1J IKI - ZJ 1k-11
AJ fl(l
IVJ !Kl - VJ IK-111 I X
AJ21KI
AJ (I()
2
CJ IKI
AJ (KI * ~J !10- VJ (I()
FJ 10
X* IRJ ( I ( ) * VJ IK-11 + RJ IK-11
'IJ IK))
· GJ 110
1( * CJ fKl
* IVRJ (1() + VRJ IK-111
2 0 en NT T "Jllf
C·'ll F<A""G Ill
C~LL SURJFG IZ, DEL"
Ill, 0.05, DELF 11), 0.0~1
Wll{TF 16, 7'i0)
7'i0
!=nil~ AT I19HnELF{ll,
DELFI I l = l
WOJTF (6, Oq) OH" Ill, DELF Ill
IF
(I(
.FQ.
'"*
*
51
54
s;
55
132
------ ------- ---- -·
• on
ftJ IJI = l t . - VJ llll I (AII4AJ(- RJ llll
AJ21Jl = IIJ IJI
2
CJ IJJ = AJ IJI
ll"AX- 1.
IF I IZ .FO. 01 Gn TO q
J - 4
--~ ~-
on ,. t • J OJ 1
~
---------~--------------------
--
--~-
-----··-------- ------·-
**
*
.. =
Ml4 :
..
IF I I l
• 1'0. 1 I M"' :
8 l = 1,
1''1 7 K = l, 1'"1
7 SET I l , K I "' 0.
8 S"T ll,Z4l "' 0.
I""'
on
SU"'
SN
=
5
O.
= o.
sx = o.
o.
sxz
o.
SY =
so
o.
sxr.
sxv
o.
o.
=
SlC2Y
q
na t5 l
0•
=
1,
1
~J
IJI
~ Ill
VJ I J I
AJ I J I
A J I J I - CJ I J I
VJ21 Jl
VJ IJJ ** 2
PJ21 Jl
RJ IJI ** 2
RVJIJI
VJ IJI * ~J IJI
V!!JIJI
VJ IJI I ~J IJI
!' J I J I
R J I J I - RJ I J-11
GJ I J I
I' J I J l * C J I J I * IV R J I J I + VA J
FJ I Jl
!'J IJl * IRJ IJI * VJ IJ-ll + RJ
IF I 0 I L1 • LT • nn I 0 I U
DO
N = 0 ~
E" = 0 Ill
*
10 02 = F
0'1 1 t K
F = 02
I J -1 II
IJ-1 I
*
VJ I Jll
** 2
= \,
J
RJ2 IKI - VJ2 flO
IF IF .r.E. 0.1 GO TJ 11
SIJOT Ill
O.
0 IL I
o.
GO T IJ 15
1 1 0 1'0
DS 0 ll T I F I
F
nF
= 0.
*
=
=
=
139
o.
=
I 2 '( = 2, J
O"
II = AJ2 1'<1 - D2
Y = VJ IKl * 0 IK-ll + VJ IK-11
Q IKI
·W (I() = EJ 110 I 'f I IRJ IKI
0 IK-ll + RJ 1<.-11
0 IKII
X= W 1'<1 * 10 IKJ • Q IK-111
AJ = II
X
2
SS IKl
~ * SEqiFS IAJJ
S 1'<1 = AJ 11'1
SS IKI
I" l'l ."lF. OJ GO TJ 12
V = GJ I K I I Y
II.J = y ... 2
F = F - \ol 11(1 * IV 1U IKI • 0 IKI • VllJ IK-ll
0 IK-lll
1 - S (!'") + Y * '\EDTES IAJI
12 OF= IJ" + E * IW 10 • IRVJ 11<1 I 0 IKI • RVJ IK-11 I Q IK-111
1-S(Kl/IJ)
JF 1'1 .NE. OJ GO 1'1 l3
F
I A ILl + F I I '"IF
r: = E - r=
I r= I~ II c; I r= I I F .t T. I • E-9 I N
1
I"' I Tl .J>Jr;. 0 .nP. 'l .EO. 01 GO TO ·to
13 srmT Ill= IADOT ILl + ::l IJJ I AVJ (JJ *ROOT lUI IE
n Ill = E
U
57.29577Q5 * SOPT I 1. - IVJ I RO I J ILl I ** 21
OIF
SD~'Till- DF.LI=Ill
wcqTE 16, t-.001
00
FnR~AT 17HOELF =, 7HSODT
l
WPIT<= lb, Qg) DELl= Ill, SOOT Ill, AOOT I l l , : , U, OIF
IF 117 .EO. 01 GO TO 15
SU~ = SU" + SOOT Ill
2
S"J
S"J • 1.
~X = <;x + ~N
SY
SY + SnOT I l l
X = S"J
"Z
S"<2
SX7 + X
*
*
=
*
*
*"'
*
*
158
171
=
=
=
=
=
**
=
*"'
Sl(3
SX3+X*<;~
S~4
'\X'+
+ X ** l
199
202
203
*
=
=
<;XY
<:;'J(Y • SN
Sll:H Ill
<:X2Y
SX?V + ~ * SOOT Ill
Sn'JT Ill : OFLF Ill - t;l'OT Ill
Jl' lt;l)nT Ill .GT. 0.05 ,A"'n. SO[lT Ill .LT. 0.061 CAll lEGNDG
1 17, O~'l" Ill, snnT Ill. t, tHXI
p: IU .LT. n. 7 ,'Jq, If ,GT. 3.251 GO Tn 15
U = (AllOT I l l - VA.J IJI 1·0 IJI *ROOT llll I OF
X = S S I 21 I I AJ 2 I 2 I - 07 I I 11 J I 21 - l J I
0'1141<=2,!'1
Y
S~ 1~+11 I IAJ? ll<+ll.- 021 I IZJ 11<+1)- ZJ CKII
E
I!J Ct<l I 0 IK I
G
W IK+11 • Y
E
E*WII<I-X
223
=" *
Of I> l V C< I = - U * CE + :; I
IF (I( .FO. 41 nEA.JV 151. = DEI!TV IKI
p: I'< .GI'O. IZI OEI!IV IKJ = ll • IAJ IKI * E + AJ IK+ll
14 X
=Y
*
Gl
=
OEPJV
tJ *ISS IJ-11 I {AJ2 IJ-11- 021 I llJ I J - l ) - ZJ IJ-211
1 - SS IJJ I IAJ2 I J I - )2) I IRJ I J I - RJ IJ-111- RJ CJ-111
2 Q IJ-11 • IW CJI + W IJ-11))
0'1 1 7 ~ = 1 • """
nn 1 f. I( = N, ""~
16 SFT 1'1, Kl
SFT 1'1, I() • rJERTV l~l) * DERJV (I()
17 S"T 1"1,241 = SFT 1'11,241 .... DfqJV INI *SOOT Ill
1 5 c~·n f'JIII'
{t= 117 ,1'0, 01 A.I'T'J<!N
X
SX2 * I SX2
2 - SX * ';'0 I - SX * I SX3 * SX2 - SX4
SXJ
1 • <;~
!SJl
2 - SX2
SX41
AA = I~Y * ISX2 ** 2 - SX
SX3l - SX * ISXY * SX2- SX2Y * SXI
1 + 5'11 * ISYY * SXl- SX2 * SX2YII I X
~~: !SX2
ISXY * <;X2 - SX2V * SXl - SY * ISX3 * SX2- SX4 * SXI
1 + S'l * lt;Y] * SX2V- SXY * SX4ll I X
CC = !SX2 * (SX2 * SX2Y- SX3 * SXYI
- SX * ISX3 * SX2Y- SX4 *
1 SXYI + SY *IS'O ** 2_;SX2 *SX411 I X
X= t;U'~ + AA * fAA* SX4 • 2. * IBB
SX3 + CC * SX2- SX2YII + 88
1 * ~~~ * SY2 + 2. * ICC * SX- SXYII + CC
ICC * S"J- 2. * SYI
=
*
**
*•
*
*
*
*
*
700
W'l!TO:
(~,
1=n1>~AT
I 61lSU"
4'-IX
16, '?<J) SUM, X,
WOJTE
7001
=,
,
SN,
*
286
=)
5HS'I
AA, fl!l, CC
287
qq I'•Jfi'"AT 16f16.81
= .. =
=
ll
~
l
DO (q N
1, l l
l
N + 1
0'1 18 K
L, "!,.
SFT IK, Nl = SET PI, Kl
CONT!IIlliE
DO 26 L = 1, ll
KK = l + 1
'( = A'3S ISI'T !L, lll
N = l
On 21 K
KK, MM
I F I 4 !l S I S !' T I K, Ll I • l T. X I G0 TO 21
I(
N
X = A8<; I SET I K, Lll
cnp.~T l ••VE
p: I 'I .E 0. l l GO rn 23
Dr' 22 I( = l • .. 'I
X = SI=T I "'• K)
SI:T I !'1, I(} = SFT I L, Kl
SE'T l l . Kl = X
24)
)( = SFT
SET !l. 24)
SlOT l"l, 241
S"T l l . 241 = X
on :t'i I( = I( I(.
X = Sl' T !K' l l I SET IL, l l
on 24 N = KK, ,.. ..
<;ET I K, Nl - X
SET I l , Nl
S"T I K' 'II
S£::T IK,241
SFT ( <' 24) .. X
SET llt24)
CfiNT1'.1UE
DELZ ('I'll = SET I ToO'~, 241 I SH fl'"~•
n1 2B I( I( = 1. l l
KK
L =
l + 1
N
rn 27 1<
N, ""'
SET IL,241
SET ll,24l- DELZ !KI *SET
Ot:ll Ill "' SET IL,2C.I I SET n, Ll
IIET\JR'II
ElllO
=
18
l<J
=
=
21
22
'"'·
2 3.
24
25
26
"'"'
=
=
27
28
=
....
•
•
"""''
-
=
IL, K)
'i"''~
U8Flt
llllUI'll' I'RFC l'dOIII Ftl"lCTION SfiiiES lXI
DI"'~IIISIO'-~
C~~un"l
~ONC 1401, DIV 140), S 140)
PATIO 1~01
I RAT ~ RATIO
Snlllr, I, DtV
D~U~l"'
PRFCISI~N
ntM~'IIISIOIII
tnM~~N
SO~C,
X, S, Y, Z
Y = OARS lXI
IF I 0 , 1 • LT , Y l GO T 0 l 0
I"' IY ,GT. I.F-31 GO TO 13
IF I Y .GT. 1.E-91 GO TO 15
SERIF'S = X I b.
IIJOHII!III
10 tF 10.2 .GT, Yl GO TO 17
IF IX .LT. 0.) GO TJ 30
I" 10.5 .GT. Yl GO TO 19
IF II .NE. 0) GO TJ 22
I
0~
23
22
1q
17
15
l 3
24
25
26
=
1
?1 J
30
31
32
33
34
35
36
IDIV IJ-11 + 1.1
=
=
=
•
-
27
=
26, 40
<;O"lC IJI =·SOIIIC IJ-11 -.SONC (J-ll I
DIV IJI = OIV IJ-ll + 2.
J = 40
Gil TIJ 24
J = 25
Gn TO 24
n
J
Gil T") 24
J
3
Gf1 Trl 2~
'l
J
<; Ill
X
0'1 25 I(
2. J
X
s (I()
s IK-ll
or 7S IC
1' J
SOIIIC lKl
s IKI I OIV IK I
s ('()
<; I Jl
SF~ IF S
'"I = J
1
011 27 I( = 1 • 't
L = J - I(
c; ILl
SEll Tt: S
SF IllES
[)A !IS IS I Jl I SERIES I
s I Jl
p:: IS I J I • GT. PA TlJ (J)) RATIO I J I
R!=TIJ!l'.J
flS:)P"T I 1. + Y)
l
1.
y
y
l
2
IF 10.2 .GT. Yl GO TO 32
I" IO.'l .GT.• VI GO TO .33
l)n 31 J
26, 40
OTV I J I
OJV IJ-11 + 2.
40
J
GO TO 34
J = 12
G!J T'l 34
J = 25
y
s Ill
nll3'>K
2,J
S I'< I
Y * S IK-11
SFRIES
S IJI I !)JV IJl
J - I
00 36 K = 1, "'
l
J - K
SEOI!:S
SFIIIES + S I l l I OIV I l l
SERIFS =
I Z
SERIES - Y
S IJI
DABS 12.
S IJI I OIV IJI I
IF IS IJI .GT, RATIO (Jil RATIO IJI
PHUIIN
HID
*
=
•
=
=
=
=
'
s
(Jl
•
••
82
=
"' =
=
=
=
=
=
z.
=
*
*
SERIES I
S IJI
=
Zl