Skip to main content

Full text of "NASA Technical Reports Server (NTRS) 19730012797: Triangulation Error Analysis for the Barium Ion Cloud Experiment. M.S. Thesis - North Carolina State Univ."

See other formats


ABSTRACT 


Long, Sheila Ann Thibeault. Triangulation Error Analysis for the Barium 
Ion Cloud Experiment. (Under the direction of Dr. Edward R. Manring.) 

The triangulation method developed at the NASA, Langley Research 
Center specifically for the Barium Ion Cloud Project is discussed. 
Expressions for the four displacement errors, the three slope errors, 
and the curvature error in the triangulation solution due to a probable 
error in the lines-of-sight from the observation stations to points on 
the cloud are derived. The triangulation method is then used to deter- 
mine the effect of the following on these different errors in the 
solution: the number and location of the stations, the observation 

duration, east-west cloud drift, the number of input data points, and 
the addition of extra cameras to one of the stations. The pointing 
displacement errors are compared, and the pointing slope errors are 
compared. The displacement errors in the solution due to a probable 
error in the position of a moving station plus the weighting factors 
for the data from the moving station are also determined. 


(NASA-TM^X-69222) TRI ANGULATION ERROR N73-2T524 

ANALYSIS FOR THE BARIUM ION CLOUD 
EXPERIMENT M.S. Thesis - North Carolina 

State Univ. (NASA) 156 p HC $10.00 Unclas 

CSC! 04A G3/20 68628 J 



TRIANGULATION ERROR ANALYSIS FOR THE BARIUM ION CLOUD 

EXPERIMENT 


ty 

SHEILA ANN THIBEAULT LONG 


A thesis submitted, to the Graduate Faculty of 
North Carolina State University at Raleigh 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 

DEPARTMENT OF PHYSICS 


RALEIGH 

19 7 3 

APPROVED BY: 





Chairman of Advisory Committ 





I 


BIOGRAPHY 

Sheila Ann Thibeault Long was born 

She received her elementary and secondary education in 
Richmond, Virginia, graduating from John Marshall High School in June 
1962. 

She entered the College of William and Mary, Williamsburg, 

Virginia, in September 1962. At William and Mary she received a four- 
year Virginia State Teacher's Scholarship plus three National Science 
Foundation Undergraduate Research Grants. She was made a member of the 
following national collegiate fraternities — Phi Beta Kappa, Sigma Pi 
Sigma, Kappa Delta Pi, and Delta Delta Delta. She was the president 
and soloist of Orchesis, the modern dance group at the college. She 
student-taught the P.S.S.C. physics courses at York High School in 
Yorktown, Virginia. She received the Bachelor of Science degree with a 
major in physics and a Virginia State Teacher's Certificate for Second- 
ary Education from William and Mary in June 19 66. 

Having first worked at the National Aeronautics and Space Adminis- 
tration, Langley Research Center (NASA, LRC ) during the summer between 
her junior and senior years in college, she returned there in August 
1966 to become a permanent member of their research staff. At Langley 
she has helped develop their holographic capability and has also done 
studies in vision research. Presently she is in their Environmental and 
Space Sciences Division, Space Physics Branch, Magnetospheric Physics 
Section doing research on the Barium Ion Cloud Project. She is the 






ii 


first and present chairman of the Langley Colloquium Series. She is a 
member of the Optical Society of America and the American Institute of 
Aeronautics and Astronautics. In the November 1971 issue of "New 
Woman" magazine, she was saluted as one of the "2 6 Young Women Who Made 
It Big In Their Twenties"; and she was chosen to appear in the "1972 
Edition of Outstanding Young Women of America." 

She began her graduate study in physics on a part-time basis at 
the College of William and Mary in September 1966. From January 
1968 to June 1969, she was on graduate study leave from NASA, LHC 
to do full-time graduate study in physics at North Carolina State 
University (NCSU), Raleigh, North Carolina. At NCSU she was made a 
member of Phi Kappa Phi . 

She was married to Edward R. Long, Jr., also a physicist, on June 
8, 1968. She and her husband reside in Hampton, Virginia, where they 
both work at the Langley Research Center. She is a member of the Back 
River Area Civic League; the Hampton Roads Civic Ballet Company; the 
Junior Woman's Club of Hampton, Inc.; and the Women's Auxiliary of 
the Hampton Roads Power Squadron. 



iii 


ACKNOWLEDGMENTS 

The author sincerely thanks Mr. David Adamson and Dr. Clifford L. 
Fricke - both of the National Aeronautics and Space Administration, 
Langley Research Center - who formulated the idea for this study and 
who provided invaluable assistance throughout its entire undertaking. 

Sincere appreciation is extended to Dr. Edward R. Manring, chairman 
of her advisory committee, for all of his guidance and advice and to 
the other members of the committee. Dr. Alvin W. Jenkins and Dr. 

Ernest E. Burniston, for their good criticism and helpful suggestions . 

This work was funded entirely by the National Aeronautics and 
Space Administration. 

Lastly, many thanks are given to her parents Mr. and Mrs. William 
R. Thibeault for their faith and to her husband Edward R. Long, Jr., 
for his patience and inspiration. 



iv 


TABLE OF CONTENTS 

Page 

LIST OF TABLES vi 

LIST OF FIGURES vii 

INTRODUCTION 1 

OBJECTIVES • • 7 

REVIEW OF LITERATURE 8 

TRIANGULATION FOR THE BIC PROJECT 9 

The Single-Point Two-Station Tri angulation Problem 10 

Line and Multistation Triangulation Considerations 23 

The LaRC Triangulation Method 26 

TRIANGULATION ERRORS FOR THE BIC EXPERIMENT 48 

Observation Stations for the BIC Project 49 

Pointing Displacement Errors 49 

Pointing Displacement Errors as a Function of the Number and 

Location of the Observation Stations 60 

Pointing Slope Errors 6l 

Pointing Slope Errors as a Function of the Number and 

Location of the Observation Stations T9 

Pointing Curvature Error . 83 

Pointing Curvature Error as a Function of the Number and 

Location of the Observation Stations 87 

Comparison of Pointing Displacement Errors 89 

Comparison of Pointing Slope Errors 89 

Pointing Errors as a Function of Observation Duration 92 

Pointing Errors as a Function of East-West Cloud Drift 94 

Pointing Errors as a Function of the Number of Input 

Data Points 104 

Effects of Additional Cameras at a Particular Observation 

Station. 113 

Station Displacement Errors for the Aircraft Station 11 8 

Resultant Displacement Errors for the Aircraft Station 127 

Aircraft Data Weighting Factors 127 

SUMMARY AND CONCLUSIONS 135 


LIST OF REFERENCES 


139 



APPENDIX . Transformation from Latitude, Longitude, Altitude to 
Azimuth, Elevation, Range Coordinates 



vi 

LIST OF TABLES 

Page 

1 The BIC observation stations and their respective 

coordinates 50 

2 The nine cases of different station combinations 62 

3 Case 1 pointing displacement errors 11^ 

4 Case 1' pointing displacement errors 116 

5 Case l" pointing displacement errors 117 

6 Case 1 resultant displacement errors 128 

7 Case 3 resultant displacement errors 129 

8 Case 4 resultant displacement errors ..... 130 

9 Case 6 resultant displacement errors 131 

10 Aircraft data weighting factors for the displacement errors 

for different station cases 133 



Vll 


LIST OF FIGURES 


1 

2 

3 

4 

5 

6 

T 

8 

9 

10 

11 

12 

13 

14 

15 

16 


Page 


Neutral barium cloud as seen from Mt. Hopkins Baker Nunn 

Site, Arizona, on September 21, 1971 > at 03 05 13 2 

Ionized barium cloud as seen from Mt. Hopkins Baker Nunn 

Site, Arizona, on September 21, 1971, at 03 11 I** 3 


The relative station coordinate system 

The line d between the two non-intersecting lines-of- 
r 

sight • • • • • 

The residual angle <5 

The point t on the azimuth-elevation curve closest to 
the trial solution (paz, pel) 


11 

16 

20 

31 


The local coordinate system centered at the point t plus 
the three residuals dl^, ani ^ 

The increment da + 

The longitude plon which gives the minimum residual sum 

E a 

The vector B 

The vectors A, B, and C • 


. 33 

. 40 

. 42 
. 54 
. 56 


East-west pointing displacement error as a function of 
latitude for the different station cases 


63 


North-south pointing displacement error as a function of 
latitude for the different station cases. ...... 

Vertical pointing displacement error as a function of 
latitude for the different station cases ... . . . . 

Total pointing displacement error as a function of 
latitude for the different station cases. . 

The latitude, longitude, and range of the points 

P K-1 “ d P » 



Vlll 


Page 


IT 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 


28 


29 


30 


31 


A line solution through Pjj_, and- an <i a line 

solution through P N _ 1 and N fl 

The vectors dP and d?' ajid the total output pointing 

slope error aST^ jj ?4 

The total input pointing slope error TT 


Latitude pointing slope error as a function of latitude 
for the different station cases . 


Longitude pointing slope error as a function of latitude 
for the different station cases 

Total pointing slope error as a function of latitude for the 
different station cases 

The radius of curvature R 

Pointing curvature error as a function of latitude for the 
different station cases 

Pointing displacement errors as a function of latitude for 
the five-station case 

Pointing slope errors as a function of latitude for the 
five-station case 


81 

82 

84 

88 

90 

91 


East-west pointing displacement error as a function of 
latitude for the five-station case for different release 
points varying in longitude 96 

North-south pointing displacement error as a function of 
latitude for the five-station case for different release 
points varying in longitude 97 

Vertical pointing displacement error as a function of 
latitude for the five-station case for different release 
points varying in longitude 98 

Total pointing displacement error as a function of latitude 
for the five-station case for different release points 
varying in longitude 99 


Latitude pointing slope error as a function of latitude 
for the five-station case for different release points 
varying in longitude 



ix 


Page 


32 Longitude pointing slope error as a function of 

latitude for the five-station case for different 

release points varying in longitude 101 

33 Total pointing slope error as a function of latitude 

for the five-station case for different release 

points varying in longitude 102 

3U Curvature pointing error as a function of latitude 
for the five-station case for different release 
points varying in longitude 103 

35 East-west pointing displacement error as a function 

of latitude for the five-station case for different 
values of BN 105 

36 North-south pointing displacement error as a function 

of latitude for the five-station case for different 

values of BN 106 

37 Vertical pointing displacement error as a function of 

latitude for the five-station case for different 

values of BN. • 1°7 

38 Total pointing displacement error as a function of 

latitude for the five-station case for different 

values of BN 108 

39 Latitude pointing slope error as a function of latitude 

for the five-station case for different values of BN. . .109 

1+0 Longitude pointing slope error as a function of latitude 

for the five-station case for different values of BN. . .110 

1+1 Total pointing slope error as a function of latitude 

for the five-station case for different values of BN. . .111 

1+2 Pointing curvature error as a function of latitude for 

the five-station case for different values of BN 112 

1*3 East-west station displacement error as a function of 

latitude for the station cases containing the aircraft. .122 

1+1+ North-south station displacement error as a function of 

latitude for the station cases containing the aircraft. 

1*5 Vertical station displacement error as a function of 


.123 



X 


Page 

latitude for the station cases containing the aircraft. . .124 

46 Total station displacement error as a function of latitude 

for the station cases containing the aircraft . . 125 

47 The azimuth, elevation, and range coordinates of the line-of- 

sight from an observation station S to a point in 
space 

48 The geocentric latitude, longitude, and range coordinates 

of a point N Q llt2 



/ 


INTRODUCTION 

Charged particles radiated from the sun flow toward and enshroud 
the earth’s insulating atmosphere, enclosing the earth's magnetic 
field within, thus forming the earth's magnetosphere. Up to the 
present, much data has been collected on the earth's magnetic field 
within the magnetosphere from satellite-borne magnetometers. However, 
data on the earth's electric field within the magnetosphere is lacking. 
In order to understand such dynamic phenomena as geomagnetic storms 
(which greatly interfer with communications), aurorae, etc., this void 
must be filled. For this purpose the National Aeronautics and Space 
Administration (NASA) and the Max Planck Institute, Institute for 
Physics and Astrophysics, Institute for Extraterrestrial Physics (MPE) 
jointly formulated the "NASA/MPE Barium Ion Cloud (BIC) Project. 

A payload of 1.6 kilograms of neutral barium is released from a 
four-stage Scout rocket at an altitude of about 31,633 kilometers, or 
approximately five earth radii. Figure 1 is a photograph of a neutral 
(spherical) barium cloud. The neutral cloud reaches its peak bright- 
ness, which is somewhat brighter than a third magnitude star, in about 
fifteen to twenty seconds after the release. 

The neutral barium rapidly becomes ionized by the incoming solar 
radiation. The charged particles attach themselves to a magnetic 
field line and spiral along it, forming an elongated cloud along the 
direction of the magnetic field. Figure 2 is a photograph of an 
ionized (elongated) barium cloud. After about four minutes the 




Figure 1.- Neutral barium cloud as seen from Mt. Hopkins Baker Nunn Site, Arizona, 
on September 21, 1971, at 03 05 13. 


ro 



Figure 2.- Ionized barium cloud as seen from Mt. Hopkins Baker Nunn 
Site, Arizona, on September 21, 1971, at 03 11 1**. 




4 


elongated cloud subtends an angle of about one half degree, vhich is 
approximately the angle subtended by the diameter of the moon. After 
about thirty minutes the elongated cloud extends to a length of 
10,000 kilometers or greater. The observation time is about one hour 
and fifteen minutes duration. 

Initially, the relatively dense cloud of charged particles 
introduced into the magnetosphere perturbs the weak magnetic field. 
Also, the initial velocity of the ions, the same as that of the Scout 
fourth-stage, is greater than that of the ambient plasma. 

After an extended time the cloud adopts the velocity of the 
drifting ambient plasma. The drift velocity v for all charged 
particles is 

r = if! , for |1| <c III . (1) 

e 

where E is the electric field vector, £ is the magnetic 

induction vector, and c is the speed of light. 

Upon vector multiplication of each side of equation 1 by $ , it is 

seen that 

E = - (v x g) (2) 

Using equation 2 the electric field vector can be computed once the 
drift velocity and the magnetic induction vector are known. 



5 


Since the barium is fluorescent against a dark-sky background, it 
can be photographed from ground-based observation sites. The cameras 
at the different sites are all synchronized in time. The photographs 
of the barium cloud are projected onto appropriate star charts, 
matching the respective star conf igurations , thus obtaining the 
coordinates of particular points along the length of the cloud in 
whatever coordinates were used in constructing the star charts, eg. 
azimuth and elevation. 

By triangulating on the two-dimensional data thus obtained from 
the various observation sites, the position of the cloud in three- 
dimensional space as a function of time and, hence, its velocity is 
determined. From the elongation of the cloud, the geometry of the 
magnetic field line is deduced. The magnetic field line thus deter- 
mined can be compared to the magnetic field line resulting from the 
earth's internal magnetic sources, and any perturbation suffered by 
the magnetic field line as delineated by the cloud can be deduced. 

And, from the drift of the cloud, since the magnetic induction is 
measured by the magnetometer aboard the Scout, the strength and 
direction of the electric field is computed using equation 2. 

It is necessary to know how accurately the position of the cloud 
in space is determined in order to evaluate the final results of the 
magnetic field and electric field determinations. It is therefore 
important to know what errors are introduced into the data through the 
acquisition and reduction of the data and how these errors are mani- 
fested by the triangulation in the solution of locating the barium 



6 


cloud in space. This thesis deals with the problem of triangulation 
and the errors which result from the triangulation for the BIC 


Experiment . 



7 


OBJECTIVES 


I. To define what triangulation errors are meaningful to the BIC 
Experiment. 

II. To exercise the triangulation method developed for the BIC 
Project to obtain data for use in designing the remainder 
of the experiment. 



8 


REVIEW OF LITERATURE 

An extensive literature search using the NASA Library facilities 
yielded no information on triangulation errors which was meaningful 
to and useful in designing the BIC Experiment. 



TRIANGULATION 
FOR THE 
BIC PROJECT 



10 


The Single-Point Two-Station Triangulation Problem 
To begin a paper on triangulation errors, it might be illustrative 
to first consider the simplified triangulation problem of a single 
point in space observed by two observation stations . Most of what 
will be stated here concerning this simplified problem is taken from 
reference 1 . 

For this two-station triangulation problem, it is convenient to 
use a relative station coordinate system as shown in Figure 3- The 
two stations are denoted by A and B. The 1,2, 3 Cartesian 

coordinate axes are the geocentric coordinate axes defined by: the 

origin is at the center of the earth; 1 is directed toward the 
intersection of the Greenwich Meridian with the equator; 2 is 
directed toward 90° east longitude, 0° latitude; 3 is directed to- 
ward the geographic north pole. The radial distance of station A 
from the earth's center is denoted by r^; and its 1,2, 3 

components, by r, , r , r , respectively. The radial distance of 
A 1 A 2 A 3 

station B from the earth's center is denoted by r B ; and its 1 , 

2 , 3 components, by , r , r , respectively. The base line 

1 2 3 

AB, projected onto the 1,2, 3 axes, has the projections AB^, 

AB 2 , AB^j respectively, given by 



( 5 ) 




12 


The length of the line AB is then 



( 6 ) 


The radial line from the center of the earth which is perpendicular to 
the base line AB is denoted by r^ This perpendicular intersects 
AB at the point M. The angle m is the angle between r A and AB 
and can be found, using the law of cosines, to be 


m 


cos 


-1 


2 . .2 

r A + d AB 


- r. 


B 


2 r A d AB 


(T) 


The point M on AB is located a distance from station A given 

by ■ ' 


Si = r A C ° S m 


( 8 ) 


And, the length of the radial line r^ is given by 

r M = r A sin m 


(9) 


The relative station coordinate system with axes denoted by 7» 8, 
and 9 is defined by: the origin is at the point M; 7 is directed 
radially outward along the line r M ; 8 is directed toward station B 
along the base line AB; 9 is such as to form an orthogonal system. 
The direction cosines for transforming from the relative station 



13 


coordinates to the geocentric coordinates are 

*1 


r, + 




*1 d 


AB 


'71 


'M 


<^2 


r + 

A 2 d AB 


'72 


M 




'73 


r M 


AB, 


'81 d 


AB 

AB„ 


'82 “ d 


AB 




'83 d 


AB 


Y 91 Y 72 Y 83 ” Y 82 y T3 


Y 92 Y 81 Y 73 " Y 71 Y 83 


Y 93 ' Y 71 Y 82 ' Y 8l Y 72 


( 10 ) 

( 11 ) 

( 12 ) 

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 


The direction cosines for transforming from the relative station 
coordinate system to the topocentric coordinate system with axes 
denoted by 4, 5, and 6 can also be found. The topocentric 



l4 


coordinate system is defined by: the origin is located at a particular 

location on the surface of the earth, eg. an observation station; 4 is 


directed along the radial line from the center of the earth to the 
station, ie vertical from the station; 5 is directed east from the 
station at 0° elevation perpendicular to 4; 6 is directed north from 
the station at 0° elevation perpendicular to 4 and 5* The direction 
cosines for transforming from the relative station coordinates to the 
topocentric coordinates are then 

^ ■ k i v ik* (w) 

where i = 7, 8, 9 and j = 4, 5, 6. The Y jk are the direction 
cosines for transforming from the topocentric coordinates to the geo- 


centric coordinates and are given in reference 1. 

Now, the line-of-sight defined by the azimuth and elevation of 
the point in space from station A may not intersect precisely with 
the line-of-sight defined by the azimuth and elevation of the point in 
space from station B. Such could be the result of human error and/or 
equipment error while acquiring and/or reducing the photographic 
data. 

The lines-of-sight from the two observation stations when pro- 
jected onto the plane formed by the 7 and 9 axes of the relative 
station coordinate system form two lines extending from the relative 
station coordinate system origin M and two angles 0 A and 0 g 
measured from the 7 axis. To circumvent this problem caused by the 
non-intersecting lines-of-sight, it is convenient to speak of an 
angle* say 0, between the two angles © A and 0 fi . The solution to 



15 


the problem of locating the point in space will then lie on the plane 

which passes through the 8 axis making an angle 0 with the 7 axis. 

The shortest distance between the two lines-of-sight and perpendicular 

to both is denoted by d . The distance from station A to the inter- 

r 

section of d and the line-of-sight from station A is denoted by d A ; 
r 

the distance from station B to the intersection of d r and the line- 

of-rsight from station B, by dg. The angles $ A and $ B are measured 

between the base line AB and the lines-of-sight from station A and 

station B, respectively. The quantities 0 A » 6 g, 6 , d r> d A , d^, <1> A , 

and <L, are all shown in Figure 4. 

B 

The 4, 5, and 6 components of the unit lines-of-sight from stations 
A and B - denoted by i^, i A5 > i A g» ^ » * 35 » and 1 g 6 ’ res P ectlvel y ~ 
are given by 


* A 4 = Sin el A 


(20) 


i A5 = cos el A sin az A 


(21) 


i A 6 = C ° S el A COS aZ A 


(22) 


^4 = Sin el B 


(23) 


i B5 = cos el B Sin aZ B 


(24) 


i B 6 = C0S el B C0s aZ B 


(25) 



16 



Figure k.- The line dj. between the two 

non-intersecting lines-of-sight . 



17 


where az A> el A , aZg, and el fi are the azimuths and elevations of the 
point in space from stations A and B, respectively, determined from 

the photographic data. 

The angles <J> A and <j> B are found from 


cos = y 8k A 1 Ah + Y 85A i A5 + Y 86A i A6 


(26) 


cos 4 > b * ■Y 8 1 +B 1 B 1 + + 7g 5B 1 B5 + 706^6’ 


(27) 


where Y 81(A . Y 85A> Y 86a , Y 81(B , Y 05B . and Y 86b are the direction 
cosines for transforming from the relative station coordinate system 
to the topocentric coordinate systems centered at stations A and B, 
respectively, and are found from equation 19. 

The angles 8 A and 0 B are found from 

Y 94A i AU + Y 95A 1 A5 * Y 96A*A6 
tan A y 7Ua 1 aU + y 75A 1 A5 + y 76 a 1 a 6 

Y 9UB 1 B^ + Y 95B X B5 * Y 96B 1 B6 
tan B y 7Ub 1 b4 + y 75B 1 B5 + y 76 b 1 b 6 

where the direction cosines Y^^j ^95A’ Y 96a’ Y ji+A’ Y 75A’ Y 76A’ Y 9^B’ 

v v v , y and are also found from equation 19- 

Y 95B’ y 96b’ y 7Ub’ y 75B t 76b 

The distance d r squared is given hy 


(28) 

(29) 



18 


d r = (d AB * d A COS ' d B COS *B )2 

2 

+ (d A sin <{> A cos d0 - dg sin <j>g) 

+ (d^ sin <|) A sin d0)^ , (30) 


where d0 = 0. - 8^. Minimizing this expression with respect to d. 

A B A 

and d fi by first differentiating it with respect to d A and dg and 
then equating each of these derivatives to zero leads to the following 
solutions for d A and dg 




The location P of the point in space is assumed to lie on the 

line d . If it is assumed that the angular errors from the two 
r 

observation sites are equal, then it is reasonable to assume that the 
location P of the point in space on d^ is at a distance 1 A from 
the line-of-sight from station A and at a distance lg from the 
line-of-sight from station B such that 



19 


1 A d 

.A = _B _ r (33) 

d A S d A + d B 


Now, the residual angle 6 is defined to be the angular deviation 


between the actual and measured lines-of-sight from an observation 


station. The location P of the point in space, the distances 1 A 

and 3 , and the residual angle 6 are shown in Figure 5 • 

B 11 

Since the quantities and ~~ are the tangents of the angular 

a A B 


deviations between the actual and measured lines-of-sight from 
stations A and B, respectively, then 6 is determined from 


tan 6 


d A + d B 


(3M 


The location P of the point in space is at a distance d^ 

from station A and at a distance d^ from station B; the distances 

d' and di are the actual lines-of-sight from stations A and B, 

A B 

respectively, to the location P of the point in space. From Figure 
5 it is seen that 


d! = t 

A cos o 


^ cos 6 


(35) 


( 36 ) 


The components of d^ along the 7 and 9 axes — denoted by 
and &pg , respectively - are 



20 



Figure 5*- The residual angle 6. 


21 


where 

by 


d A7 = d A7 + (d B7 ” d A7 } ( d A + dg ) 
d A9 = d A9 + (d B9 ' d A9 } (d. + d B ) • 


(37) 


(38) 


and d. ^ are the 7 and 9 components of d. and are given 
Ay A 


d A7 = d A Sin ^A COS 6 A 
d A9 = d A Sin ^A Sin 6 A ’ 

and where 0 ^ and are the 7 and 9 components of d B 

given by 


(39) 

(HO) 


and are 


d B7 = <13 sin <|> B cos 6 B 


(Hi) 


*39 = ^ sin Sin e B 


(H2) 


The component of d^ along the 8 axis, denoted by d^g, 
given by 


d A 8 = d i C0S *A 


(1*3) 


where <J>^ is the angle between the actual line-of-sight d^ from 
station A to the location P and the base line AB and, as seen in 



22 ' 


Figure 5 , is determined from 


sin <f>^ 


d^ sin p 


AB 


(WO 


where p is the angle between the actual line-of-sight from station A 
to the location P and the actual line-of-sight from station B to the 
location P and, applying the law of cosines to the geometry of 
Figure 5> is found to be 


p = cos 




<*A >' 


<*B>' 


- d 


2 I 


AB 


d A 




(45) 


The angle 0, lying between the angles 0^ and 0g, which is 
the angle that the plane through the 8 axis and containing the location 
P of the point in space makes with the 7 axis is then given by 


0 = tan 



(46) 


where d' and d' are given by equations 37 and 38, respectively. 

A f Ay 

The geocentric components P^, Pg» P^ °i’ location P of the 
point in space are given by 


P 1 = r A 1 + Y 71A d A7 + Y 8lA d A8 + Y 91A d A9 
P 2 = r A 2 + Y 72A d A7 + Y 82A d A8 + Y 92A d A9 
P 3 = r A 3 + Y 73A d A7 + Y 83A d A8 + Y 93A d A9 


(47) 

(48) 


9 


(49) 



23 


where r , r , and r. are the geocentric components of station A 
A 3_ A 2 A 3 

and where Y ?1A . Yj2A* Y T3A’ Y 8lA’ Y 82A’ Y 83A’ Y 91A’ Y 92A’ and Y 93A 
are the direction cosines for transforming from the relative station 

coordinate system from station A to the geocentric coordinate system. 

Finally, the geocentric latitude, the longitude, and the radial 

distance of the location P of the point in space observed from the 

two stations - denoted by <j>£, 6 p , and r p , respectively - are given 

by 


3 

(50) 

_(P 2 + P 2 + P^) 172 . 



tan (pf) 

(51) 

1 


> o O 1/2 


: + p " * P 3 

(52) 


Line and Multistation Triangulation Considerations 
Now, in this above simplified problem of locating a single point 
in space using the photographic data from only two observation sites, 
the solution is not unique unless the lines-of-sight from the two 
stations to the point in space intersect precisely. In most instances, 
due to h uman error and/or equipment error while acquiring and/or 
reducing the data, the lines-of-sight will not intersect precisely and, 
hence, the solution of locating the point in space will not be unique. 
All that can be done is to work in terms of a most probable angle 0 
lying between the angles 0 A and e fi , which are the angles measured 



2k 


from the 7 axis that are made by the projections of the lines-of-sight 
from stations A and B, respectively, onto the plane formed by the 7 and 
9 axes of the relative station coordinate system, as was just done in 
the above. 

In the problem of locating a line, as opposed to a single point, 
in space using the photographic data from only two observation sites, 
the surface formed by the lines— of— sight from station A to various 
points on the line in space will in general intersect with the surface 
formed by the lines-of-sight from station B to various points on the 
line in space. If not, then extrapolations of these two surfaces will 
intersect. Hence, the two surfaces, or their extrapolations, will 
always intersect in the form of a linej and, therefore, a unique 
solution to the problem of locating a line in space from two observation 
stations can always be found. 

In the BIC Experiment the barium cloud forms a line, as opposed 
to a point, in space as the neutral barium becomes ionized and 
elongates along the magnetic field line. A study of the exist- 
ing triangulation methods revealed that the existing method most 
applicable to the BIC Experiment was the one developed by Fred L. 
Whipple and Luigi G. Jacchia of the Smithsonian Institution Astrophysi- 
cal Observatory (reference 2). This method, herein referred to as 
the SAO method, was originally developed for triangulating on photo- 
graphic meteor trails. It was later used by John E. Hogge of the 

Research Center for reentry experiments (reference 3). The 


Langley 



25 


SAO method is only for two observation stations and approximates each 
photographic trail image by a straight line, which as was pointed out 
above, renders a unique solution always. 

However, this author in a separate paper on an analytical study 
to minimize the triangulation error for an idealized observation site 
arrangement (reference 4) has shown, for the particular case of 2 or 
more observation sites symmetrically located on the circumference of a 
circle with the cloud released on the perpendicular whose foot lies at 
the center of the circle, that the triangulation error is inversely 
proportional to the square root of the number of observation sites em- 
ployed. On the basis of these findings, it was not desired to use only 

two observation stations for the BIC Project. 

Also with regard to the BIC Experiment, the magnetic field lines 
are not straight, but are curved; hence, the barium cloud images are 
also curved. It was, hence, not desired to use a straight-line 
approximation to the curved barium cloud images. 

Therefore , it was felt that , although the SAO method has been 
used for some time with great success for the triangulation of 
photographic meteor data and of reentry experiment data, it was perhaps 
not necessarily the best triangulation method for the BIC Project. 

It was decided to develop an entirely new triangulation method for the 
BIC Project that would incorporate any number of observation stations 
and that would accommodate the curved images. The results of the new 
method could always be compared to the results of the already 
successful SAO method as a check. 



26 


The new triangulation method developed for the BIC Project at the 
Langley Research Center will be herein referred to as the LaRC method. 
This author in a separate paper has explained in complete detail for 
use in computer programs both the SAO and LaRC triangulation methods 
and has compared the two for their applicability to the BIC Experiment 
(reference 5). That paper concluded that the LaRC method was the 
best method for the BIC Experiment; and, hence, it was the method 
adopted for the project. 

It was decided to develop the LaRC method to use the azimuth - 
elevation coordinates. The azimuth-elevation curve of the cloud image 
from each observation station can be obtained from the photographic 
data. This curve from each station defines a conical— like surface in 
space. For two observation stations, as was pointed out earlier, the 
intersection of the two cones is unique. For more than two observation 
stations, however, the intersection of the cones is not unique. Hence, 
an averaging procedure had to be developed in the LaRC method in order 
to incorporate the data from more than two observation stations. 

The LaRC Triangulation Method 

For the reader's convenience a summary of the LaRC method is 
first given here before it is explained in detail. First, the total 
arc length of the azimuth - elevation curve from each station is 
found; and then new azimuth and elevation data points equally spaced 
along the azimuth - elevation curves are calculated. An initial trial 
solution in terms of the latitude, longitude, and altitude coordinates 



27 


which are transformed to the azimuth, elevation, and range 
coordinates - is estimated. The arc lengths between the trial 
solution and each of the new points on the azimuth - elevation curve 
from each station are calculated and compared to find the point on each 
azimuth - elevation curve closest to the initial trial solution. 

Using the closest point as the origin of a local coordinate system, 
three residuals, one being the perpendicular distance from the 
trial solution to the azimuth - elevation curve and the other two 
being the respective distances from the trial solution to the points 
lying on either side of the closest point, are found for each station. 

A residual is the distance from the trial solution to the endpoint of 
a ray on the surface defined by the azimuth - elevation curve from that 
station. The minimum residual for each station is found; and the 
residual sum for all the stations combined, calculated. The initial 
trial solution is then varied in altitude and in longitude, the incre- 
menting of the longitude being nested within the incrementing of the 
altitude and each time calculating the residual sum for all the 
stations combined, until the minimum residual sum from varying both 
the longitude and altitude is found. Then, the trial latitude is 
incremented, the altitude and longitude increments decreased by half, 
and the entire process iterated. Finally, after two iterations the 
unique solution to the problem of locating the barium cloud in space 
which minimizes the triangulation error is found. 

The LaRC method is discussed in more detail in the following. 

(The reader interested in using the method might also see reference 6, 



28 


a computer program.) 

From the photographic data the azimuth and elevation for each point 
on the image from each observation station is obtained. For N points 
and L stations, the original azimuths and elevations from the photo- 
graphic data are denoted by az'i and el') , where n = 1,2,...,N 

X/ j n X/ jii 

and H = 1,2,..., L. The arc length ds. in degrees between points 

X, |il 

n and n-1 is 




el " + el £, 


•O 2 cos V*’V"^ ) 


1/2 

J 


(53) 


where ds £jl = 0, el^ Q = azJ >Q = 0. 

The total arc length sarc^ in degrees of the azimuth-elevation curve 
from the H— station is 


sarc^ 


N 

I 

n=l 



(5^) 


Since it is desirable to have the points on the azimuth-elevation curves 
equally spaced, BN is defined as the spacing desired in degrees be- 
tween consecutive points on the azimuth-elevation curves. For the 
present, BN is set equal to 0.28 degrees. The total number of points 
NB 0 along the azimuth-elevation curve from the i — station is then 

X/ 


NB, 


SarC * 


BN 


(55) 



29 


New azimuth and elevation data points spaced BN = 0.28° apart along 
the azimuth-elevation curves are then calculated using FTLUP , a 
Langley Research Center systems computer subroutine which calculates 
y a F(x) from a table of values using second-order interpolation. A 
second-degree least squares curve is then fitted through these new 
azimuth and elevation data points from each observation station, based 
on three points and centered symmetrically about the point n where 
n = 1 ,2 , . . . ,NB., obtaining the coefficients bc^ ^ r where m = 1,2,3. 
This is accomplished using the following equations and procedure. 


p =c (az„ - az. ) cos ( 
i,J C i,j-l V £,nv fc.n' V 


el Jl.nv + el l,n \ . 
2 / ’ 


(56) 


i = 1,2,3; j = 2,3; nv = nm-2+i; nm = 2, if n < 2, and nm = NB^ - 1, 


if n > HB £ - 1; c 1 ^ = c 2>1 - c 3jl - 1 


a. 

l 


,m 


3 

Z c. . c, 
k=1 k,i k,m 


b. 

l 


3 

J; C k,i (el L,nm-2+k “ el A,n* 


(57) 

(58) 


The problem is to solve the matrix equation a -_ m bc m = b i» where 

a is a square coefficient matrix and b. is a matrix of constant 
i ,m 1 

vectors. The solution is found using SIMEQ, a Langley Research Center 
systems computer subroutine which solves a set of simultaneous equa- 
tions and obtains the determinant. The bc m values found are the 

coefficients be „ , which are stored for later use. 

m,Jo,n 



30 


An initial trial solution in terms of latitude, longitude, and 
altitude coordinates — denoted by plat, plon, and pr, respectively! — is 
estimated. These coordinates are then transformed to the azimuth, 
elevation, and range coordinates — denoted by paz, pel, and pra, 
respectively — ac cording to the transformation in the appendix. Then 
the arc lengths dsn 0 in degrees between the trial solution (paz, 
pel, pra) and each of the n equally-spaced points on the azimuth- 
elevation curve (az„ , el„ ) from each station are calculated. 


dsn 


Jl,n 


■r 


( P el " el £,n )< 


+ (paz - 




COS 


( pei v - H l 


1/2 
( 59 ) 


For a given station these arc lengths to the n points are compared 
to find the one which is the shortest or, in other words, to find 
the point on each azimuth-elevation curve which is the closest to 
the initial trial solution. The value of n for this closest point 
is denoted by t, as indicated in Figure 6. Suppressing the subscript 
t in the following, as it is for a particular point, the horizontal 
and vertical components - —denoted by x^ and y^» respectively of 
dsn^ are 


(paz - az^) cos 



(60) 


H - J> el - el i 


(61) 




32 


The coordinate system of Figure 7 is a local coordinate system 
with the point t as the origin. The coordinates of the point p in 
this system are x^ and y^ as given by equations 60 and 6l, 
respectively. The coordinates of the point r in this system are 
xl^ and yl^, where xl^ and yl^ are given by 

*1 i = *i (62) 

" bo l,Jt + bc 2,it xl i + b0 3.e. xl ? > <63) 

using the be coefficients found from the least squares curve fit 
that correspond to n = t. 

The slope of the tangent to the curve at the point t is given 
by the change in yl { with respect to xl.. 


tin.) 

rsr " bc 2,* + 2 


(6k) 


The angle 0^ as seen in Figure ^ is just 


-i/ d(y V 


9 l = tan l d xl. 


= tan -1 (bc 2 ^ + 2 bc 3 ^ xl^) , 


(65) 


using equation 6k. 




Figure T'~ The local coordinate system centered at the point t plus the 
three residuals dl„ , d2„ , and d3„. 




3 *+ 


The three residuals dl^, d2^, and d3^ as shown in Figure 7 
are computed according to the following. From Figure 7 it is seen that 


dl £ = t(y A - yl^) + (x & - xl £ )^] 


1/2 


2 1/2 

- [ (y £ - yl^) ] , using equation 62 


= " yl * 


( 66 ) 


From Figure 7 it is also seen that the triangle pqr is similar to 
the triangle tsr; hence, the angle rpq is equal to 0 . The 
distance qr is then 

v = (y z - yi z ) sin e z ( 67 ) 

The distance dx^ is then 

dx^ = qr cos 0^ 

= (y^ - yl^) Sin 0£ cos 0^ , (68) 

using equation 67 for qr. 

The coordinates of the point q, which is very near to or the same as 
the closest point t, are x2„ and y2 p and are given by 


/ 



35 


And, 


x2 i = xl £, + (69) 

y2 J l = bc l,£ + bc 2 ,£ x2 Z + hc 3,l x2 i ( ' 7 °^ 

2 o 1/2 

d2 il = [(y £ ‘ y2 £ } + (x £ ~ x2 i y ] (T1) 


The coordinates of the point u are x3 p and y3 0 and are given by 


x h = x2 n + **1 


(72) 


- y h m bc i,i + bc 2,i x h + bc 3 ,Ji x h <T3) 


And, 


2 2 1/2 

d3 £ = ^ y £ " y3 ^ + ( x l ~ x3 z'l ( 7*0 


The origin of the coordinate system is then shifted from the point t 
to the point q to make the calculation of the minimum residual d„ 

X / 

from the £ — station simpler. This shift does not affect the final 
expression for d^. 

The minimum residual d^ can be written in the following 
general form 

+ c i + c s *4 


9 


(75) 



3b 


where the coefficients C q ,C , and C £ need to be determined. Taking 
the first derivative of d^ with respect to dx^ , 


d(d ) 

TST “ C 1 + 2C 2 dx * 


( 76 ) 


The condition for a minimum d 0 with respect to dx„ is then 

% z 


0 * C 1 + 2C 2 dx t 


( 77 ) 


Hence , 


dx„ 


mm 



( 78 ) 


The three residuals dl^, d2 £ , and d3^ can be written in the 
following forms, respectively, using this new coordinate system 


61 1 = C o " C 1 + C 2 *4 
= C o 

d3 £ " °o + C 1 ^ + C 2 


( 79 ) 

( 80 ) 
(81) 


Solving these three equations simultaneously, the coefficients C , 

o 

C 1 , and Cg can be found. The coefficient C q is just equal to 



3T 


d2 because of the origin of the new coordinate system. Subtracting 
£ 

equation 79 from equation 8l, 


d3 Jl 41 Jl = 2 C 1 


(82) 


Hence , 


r d3 & - 
1 " 2 


(83) 


Adding equations 79 and 8l, 


d3 i + “i - 2 


C o + 2 c 2 


= 2 d2^ + 2 C^dx 2 , using equation 80 

(84) 


Hence, 


c 2 = 


d3 * + - 2 d2 * 
2 dx? 


(85) 


Then, equation 78 for dx^ becomes, using equations 83 and 85 for 

min 


and C 2 , respectively, 



38 


dx„ 


mm 


2 C, 


(78) 


- (d3 £ - dl & ) /2 dx £ 

2(d3 £ + dl £ - 2 d2 £ )/2 dx| 

- (d3 £ - dl £ ) dx £ 

2(d3 £ + dl £ - 2 d2 £ ) 


( 86 ) 


Substituting equations 80, 83, 85, and 86 for C q , C^, C^, and dx £ , 
respectively, into equation 75 for d £ , it becomes 


d l - C o + C 1 + C 2 


(75) 


~ di ^ r ~ ~ dx & 

= d2 n + „ ^ 2(d3 £ + dl £ - 2 d2 £ )~ 


'£ 2 dx 


(d3 £ + dl £ - 2 d2 £ ) f - (d3„ - dl p ) dx p 


2 dx" 


• - (d3, - di t ) ax t - 

2(a3 t + dl t - 2 a2 t ) 

A/ 

(d3 £ - dl £ ) 2 (d3 £ - dl £ ) 2 


_ 62 1 " U(d3 £ + dl £ - 2 d2 £ ) + 8(d3 £ + dl £ - 2 62 J 

(87) 


' d2 Jl “ 8(d3 £ + dl £ - 2 d2 £ T 


Therefore, the minimum residual d £ between the trial solution 
and the azimuth-elevation curve from station H is 


(d3 £ - dl £ ) 2 

d £ = d2 £ ' 8(d3 £ + dl £ - 2 d2 £ ) 


( 87 ) 



39 


Then the square root of the sum of the squares of the residuals, 
called the residual sum and denoted by E, is calculated. 


E = 


" L 
E 

£=1 



( 88 ) 


This first value of E is denoted by El . 

3 . 

The initial trial longitude plon is incremented by da + . From 
Figure 8 it is seen that 


da 


+ 


dr 

pr cos (plat) ’ 


( 89 ) 


where pr and plat are the trial altitude and latitude, respectively. 

Initially, dr is set equal to 80 km. The incremented longitude is 

plon_|_ = plon + da + (90) 

Using this new value plon| for the longitude, E is again calculated. 

This value of E is denoted by E2 . The residual sums El and 

a a 

E2 & are compared. 

Al. If E2 & < El & , plon_j_ is incremented by da + (plon^ = plon| 

+ da + ), E is calculated, and this value of E is denoted 
by E3 a . 

Bl. If E2 > El , the values of El and E2 are interchanged, 
the sign of da + is changed and this new increment is 
denoted by da , plon is incremented by da (plon' = 




hi 


plon + da_ ) , E is calculated, and this value of E is 

denoted by E3 . 

8 . 

The residual sums E2 & and E3 a are compared. 

A2 . If E2 < E3 , the comparing terminates. 

B2 . If E2 > E3 , the old value of E2 is given to El , the 

a a a a 

old value of E3 a is given to E2 & , plon” is incremented 

by da + (plon^" = plon+ + da + ) or plon' is incremented by 

da_ (plon" = plon' + da_) depending on whether route A1 or 

route B1 was used, E is calculated, this value of E is 

denoted by E3 , E3 is again compared to E2 , and this 
a. a a 

is continued until an E3 is found such that E2 < EB . 

a a “a. 

This procedure is carried out until an E2 is found such that 

a 

E2 a < E1 a aJld E2 a < E3 a’ where E1 a » E2 a’ and E3 a are three 
consecutive residual sums. Then, using these three residual sums 

the approximate value of the longitude plon which gives the minimum 

residual sum E & from varying the longitude, as shown in Figure 9, 

is calculated. For the purpose of simplifying this calculation of 

Pl° n a , the origin is shifted to the point plon 2 , shown in Figure 9; 

this shift does not affect the final expression for plon . The 

a 

points plon^ , plon 2 , and plon^ are the longitudes which correspond 

to the three residual sums El , E2 , and E3 , respectively. 

a a a. * 

The shift in origin requires a change in the longitudinal 
variable to dlon, where 

dlon = plon - plon 2 , 


(91) 



plon 1 


pl on g Pl° n a plon 3 


longitude 


Figure 9-- The longitude plon which gives the minimum 
residual sum E . 



k3 


where plon is the longitude of the trial solution and plon 0 

longitude which gives the residual sum E2 . The residual sum 

8 . 

be written in the following general form 


E = A + A., dlon + A_ dlon^ , 
o 1 2 


where the coefficients A q , A^, and A^ need to be determined, 
first derivative of E with respect to dlon is 


d E 

d dlon 



+ 2 A, 


dlon 


The condition for a minimum E with respect to dlon is 


0 = A^ + 2 Ag dlon 


Hence , 


- A, 


dlon . = 


'min 2 A^ 


The three residual sums El , E2 , and E3 can be written in 

a’ a’ a 

following respective forms using this new coordinate system. 


is the 
E can 

(92) 

The 

(93) 


( 9h ) 


(95) 

the 



hh 


El = A - A da + A 0 da"" (96) 

a o 1 d 

E2 = A (97) 

cl O 

E3 a = A + A da + A 0 da 2 (98) 

a o 1 d 

Solving these three equations simultaneously gives the coefficients 
A , A , and A 0 . From equation 97 it is seen that A = E2 already 
because of the choice of origin of the new coordinate system. 
Subtracting equation 96 from equation 98 gives 


Therefore , 


E3 - El = 2 A. da 
a a 1 


E3 - El 

a a 

2 da 


(99) 


( 100 ) 


Adding equations 96 and 98 gives 

E3 + El = 2 A + 2 A 0 da 2 
a a o 2 

= 2 E2 + 2 A, da 2 , using equation 97 

8 . c. 

(101) 


Therefore 



45 


= 


E3 + El - 

a a 

2 

2 da 


( 102 ) 


Hence, equation 95 for dlon^ becomes, using equations 100 and 102 
for A and A^, respectively, 


dlon . 


min 



- (E3 - El )/2 da 

d ct 

2(E3 + El - 2 E2 )/2 da 2 

a a a 

- (E3 - El ) da 

a a 

2(E3 + El - 2 E2 ) 

a a a 


(95) 


(103) 


From equation 91 it is seen that 


dlon . = plon - plon 0 , (104) 

min r a r 2 ’ 


where plon is the longitude that gives the minimum residual sum E . 

8 . 0 . 

Hence, 


plon = dlon . + plon„ 

e a min 2 


(104) 


- (E3 & - El ) da 

= 2(E3 + El - 2 E2 ) + plon 2 ’ 

a a a 


(105) 


using equation 103 for dlon ^ . 



46 


Therefore, 

(E3 - El ) da 

Pl ° n a ~ pl ° n 2 “ 2(E3 + El - 2 E2 ) * (105) 

a a a 

where da is da + or da_ depending on whether routes A1 or B1 

were taken. The longitude plon is used to again calculate E. 

This value of E is denoted by E and is the minimum residual sum 

8 . 

from varying the longitude. 

This value of E & is denoted by El . Then the initial trial 
altitude pr is incremented by dr = 80 km, and the entire procedure 
of incrementing the longitude by da and finding a second value for 
E & using this new value of the altitude is carried out. This new 
value of E & is denoted by E2 y . The procedure for incrementing the 
altitude by dr and finding E r> the minimum residual sum from vary- 
ing the altitude, is identical to the procedure for incrementing the 
longitude by da and finding E & , the minimum residual sum from vary- 
ing the longitude. The incrementing of the longitude, after finding 
the first value of E , is nested within the incrementing of the alti- 
tude. The overall procedure of incrementing the altitude is carried 

out until an E2 is found such that E2 < El and E2 < E3 

r r r r r ’ 

where El r> E2^, and E3 r are three consecutive residual sums. 

Using these three residual sums the approximate value of the altitude 

pr^ which gives the minimum residual sum E^ from varying the 

altitude is calculated, using an equation completely analogous to 

equation 105 for plon . The altitude pr is then used to calculate 

a r 



47 


E . The value of E is denoted by E , since it is actually the 
r r m 

minimum residual sum from varying both the longitude and altitude. 

Then using the values of plon and pr which gave the 

Q. i 

minimum residual sum E and incrementing the initial trial latitude 

m 

plat by dna = 1° and decreasing the value of dr by half (dr = 

40 km), the entire procedure of varying the longitude and altitude 

to find the minimum residual sum E ffi is repeated. Then using the 

new values of plon and pr found from the first iteration and 

incrementing the trial latitude again by dna = 1° and decreasing 

the value of dr again by half (dr = 20 km), the entire procedure of 

varying the longitude and altitude to find the minimum residual sum 

E is again repeated. With dr = 20 km this procedure is repeated 
m 

over the range of latitude, in increments of dna = 1°, desired. The 
final values of the longitude and altitude, corresponding to the given 
values of latitude, which give the final minimum residual sums provide 
the unique solution to the problem of locating a barium cloud in 
space which minimizes the error. For the LaRC triangulation output 
this geodetic coordinate solution is finally transformed to geocentric 
coordinates, according to equations derived by this author in a 
separate paper on the transformation from geocentric to geodetic 
coordinates and vice versa in powers of the earth's flattening 
(reference 7)- 



Triangulation Errors 
For The 


BIC Experiment 



Observation Stations for the BIC Project 
For the BIC Project, use had to be made, as much as possible, of 
existing observation stations. Table 1 is a list of the observation 
stations chosen for the BIC Project and their respective coordinates. 
These stations were chosen on the basis of their availability, 
facilities, relative location, and weather conditions during the launch- 
window periods for the experiment . The two prime sites , ones that have 
to be clear for the "go" launch condition, are Mt. Hopkins and Cerro 
Morado. The Wallops station is actually the NASA CV-990 High Altitude 
Research Aircraft (NASA-711), which is equipped as an airborne optical 
observatory and which flies between Bermuda and W&llops Station at 
an altitude of 35,000 feet or higher for the experiment. It was 
decided to have a north-eastern station to improve the triangulation 
accuracy; and since the east coast is frequently plagued by cloud 
cover, it was decided to use an aircraft to fly above the clouds. The 
Baker-Nunn sites are extra sites included to improve the triangulation 
accuracy. 

The observatories at Byrd Station, Antarctica, and Great Whale, 
Canada, were included to obtain data on the geophysical condition at 
the time of the release and to monitor any induced changes that might 
occur, but not to obtain data for triangulation purposes. 

Pointing Displacement Errors 

The two-dimensional input data to the triangulation program 
could have, and probably will have, errors which occurred during the 



Table 1: The BIC observation stations and their respective coordinates. 


Station 

Geodetic 

Latitude 

(deg) 

East 

Longitude 

(deg) 

Altitude 

(km) 

Byrd Station, Antarctica 

-80.0167 

-119.5167 

0 

Cerro Morado, Chile 

-30.1657 

-70.7673 

2.1346 

Edwards Air Force Base Baker-Nunn, California 

34.9641 

-117.9146 

0.0781 

Great Whale, Canada 

55.2700 

-77.7800 

0 

Mt. Hopkins, Arizona 

31.6853 

-110.8774 

2.3640 

Mt. Hopkins Baker-Nunn, Arizona 

31.6853 

-110.8774 

2.3640 

Natal Baker-Nunn, Brazil 

-5.9306 

-35.1617 

0.0421 

Wallops Station, Virginia 

37.9324 

-75.4717 

0.0106 

White Sands, New Mexico 

32.4238 

-106.5528 

1.6500 




51 


data acquisition and data reduction phases of the experiment. It is 

o 

important to know how such errors are manifested by the triangulatiori 
in the three-dimensional solution of locating the barium cloud in 
space. A reasonable error to assume for the total error occurring 
during the acquisition and reduction of the data is a probable error 
ed of 0.01 degrees in the lines-of-sight from the observation stations 
to the points on the cloud. 

The probable error ed = 0.01 degrees is introduced into the 

lines-of-sight from one observation station at a time. The perturbed 

azimuth and elevation coordinates, az.' and elJ , corresponding to 

A jll A 

the probable error ed = 0.01° in the n lines-of-sight from station 
X are 


az 


el 


I 

X ,n 


I 


X ,n 


az 


A,n cos el'. 


ed _ del 

, dae 
A ,n 


-i , . daz 

el. + ed - — 

A,n dae 


(106) 


(107) 


where az^ n is the unperturbed azimuth and el^ ^ is the unperturbed 
elevation of the n — point on the cloud from station X and where 


daz = (az X,n+l - az X,n> cos ( X,ntl a il£ -) (108 > 

del = el X,n+l - el X,n <109 > 

dae = [ ( daz ) ^ + ( del ) ^ ] 


( 110 ) 



52 


First, the solution of locating the barium cloud in space is 
found, using points along the magnetic field line which passes through 
the chosen BIC nominal release point for the input data; the solution 
in latitude, longitude, and range coordinates is denoted by <j>^, 8 , 
and r^. The solution is then found, using the same input data but 
introducing the probable error ed = 0.01° into the lines-of-sight from 
one station, say station X; this perturbed solution is denoted by 
<t>' jj, 0j^ jj, and rj^ ^ . The two solutions are then compared by finding 
the displacement between the two respective curves in space according 
to the following. 

Initially, for the unperturbed solution a least squares poly- 
nominal fit of fourth degree is found for 0^ = 811(1 also for 

r N = r N^N^ us * ng LSQPOLj a Langley Research Center systems computer 
subroutine which determines the M coefficients of the polynomial of 


degree M-l which gives the best fit in the least squares sense. 


The coefficients of the polynomial for 0^ = are denoted by 

b& M jj, where M = 1,2, 3, *+,5; the coefficients of the polynomial for 

r » ' r n ( V* by br M,r 

The polynomial for 8^ = 0^(4^) using the b£ coefficients is 


0 N = b£ l,N 


+ W 2,N^H + bA 3,H*N 


<f>H + bL. + b& c 
T N 4 ,N N 5 ,N y N 


(111) 


The polynomial for r^ = using the br coefficients is. 


similarly. 



53 


r N = br l,N + br 2,N^N + br 3,N^N + br 4,N*N + br 5,N <J> N 


( 112 ) 


The first derivative 0„ of 0„ with respect to d>„ is 

N N ^ t N 


6 N ~ b£ 2,N + 2 b£ 3,N*N + 3 b ^4 ,N^N + k bS, 5,N^N 


(113) 


The first derivative r^ of r^ with respect to <j> is, similarly. 


• o o 

r = br„ + 2 br d> +3 br> <b + U br dr 
N 2,N 3,N^N 3 U,1TN 5 ,N V N 


(114) 


From Figure 10 it is seen that the vector B, which is tangent to 
the unperturbed solution curve at the point C, is given by 


B = Vd) + Ve 


+ B i 
r r 


= r 6(t i , + r cos <b 0 6<j) i„ + r 6<b i 
c c d> c T c c c 0 c c r 



r 

c 


A ^ 

cos 4 8 i„ + r i )6<J> 
T c c 0 c r c 


(115) 


where 6<f> c is the difference in latitude between the point C and the 

. A 

A A 

nearby point C' on the same curve and i^, ig, and i^ are unit 

vectors in the directions of increasing latitude, longitude, and 

• • 

range, respectively, and 0 c> r c , Q^, and r c are given by equations 
111, 112, 113, and 114, respectively, for the point N = C. The 
point A in Figure 10 is the point on the perturbed solution curve. 





55 


due to a probable error ed = 0.01° i n the lines-of-sight from 
station X, having the same latitude <J> c as the point C on the 
unperturbed solution curve. 

As is shown in Figure 11 the vector from the point A to the 
point C is A. The vector from the point A which is perpendicular 
to B is C. It is assumed that the vector distance from the point 

•f t 

A to the unperturbed solution curve is C. From Figure 11 it is seen 
that 


C = A + FB, (ll6) 


where F needs to be determined. Dotting both sides of equation ll6 

■f 

with B gives 


B*C=B'A+FB*B (117) 

But, B • C = 0, as C is perpendicular to B; so 

0 = B • A + FB • B (118) 

Therefore , 

F = - ^4 (119) 

B • B 

Substituting equation 119 for F into equation 116, the expression 




for C becomes 


57 


-* -+ ->■ 
C = A + FB 


(116) 


B • B 


= A - 


— T A A A . 

A • (r i. + r cos 4 0 i. + r i )o6 
v c (t> c cc6 cr y c 

/ 2 . 2 2 , A2 , ’ 2 \ r , 2 

(r + r cos 6 0 + r )o6 

c c c c c c 


x (r i, + r cos 4 6 i. + r i )6(J> 
c <j> c T c c 0 cr c 


4- A A A 

A • (r i, + r cos 4 0 i + r i ) 
-*■ c 6 c T cc0 cr 


= A - 


. ^ 

(r 2 + r 2 cos 2 <J> 0 2 + r 2 ) 
c c c c c 


x (r i. + r cos 60 i Q + r i ) 
c 4> c cc0 cr 


(120) 


Now, as can be seen from Figures 10 and 11, the vector A is given by 


A - r X,A (e c - e x,A> 003 *c i e + <r c - r X,A U r • (l2l) 


where 0, . and r, are the longitude and range, respectively, of 

A jA A jA 

the point A, which is on the perturbed solution curve due to a 
probable error ed = 0.01° in the lines-of-sight from station A. 
Substituting equation 121 for A into equation 120, the expression 


for C becomes 



58 


5 * r X,A (9 c - e x,A> 003 *c*e + (r c - r X,A )J r 


[r X.A (e c- e X.A )cos W’V'x/rH'-.V r c COS »cW r c 1 r 1 

(r 2 + r 2 cos 2 *? + r 2 ) 

' c c c c c 


x (r i, + r cos 40 i Q + r i ) 
c 4> c cco cr 


- r X,A (9 c ' e x,A >C ° 3 *e 1 e * (r c • r X ,A)^r 


[r, a r„(0„ - 6, A )cos 


•V A-*- \ V v ■» . / 

A .A c v c A, A 


c c 


(r* + r^ cos 2 0 6 2 + r 2 ) 
c c c c c 


x (r i + r cos <t> 0 i A + r i ) 
c <p c cco cr 


= [r X.A (9 c- e X.A )cos V r X.A r c ( V 9 X.A ),: ° 3 \ 9 c- r c (r c- r X.A )l: ° i " t c 6 c r c H e 


• • A 


/ 2 , 2 2. 12 . *2. 

(r + r cos 00 + r ; 

c c c c c 


• A 


[-r, . r 2 (0 -0, ) cos 2 (f> 0 - r (r -r, . ) r ] i, 

A, A c c A, A T c c c c A,A c (ft 

/ 2 2 2. A2 . *2x 

(r + r cos ())0 + r / 

c c c c c 


» r c- r X.A ) - r >.A r c ( 9 c- 6 X.A )c ° s 2 |> c e c r c 
+ * *""0 O o r o • o . 


(r^ + r^ cos 2 (ft 0^ + r 2 ) 
c c c c c 


• p ~ 

( r a) r ,J 
c 2 L (122) 


Therefore, the east-west, north-south, and vertical components —denoted 
by dl^ A , d2^ A> and d3^ A » respectively- of the total pointing dis- 
placement error in the solution point A due to a probable error 
ed = 0.01° in the lines-of-sight from station A are, respectively. 



59 


dl 


A, A 


, [r. (0 -0, A )-r (r -r. A )0 r -r, A r^(0 -6, A 

cos4> c A t A c A,A c c A, A cc A, Ac c A, A 


)cos‘ L (}> G" - ] 
c c 


/ 2 2 2 *2 ‘ 2 , 
(r + r cos <p 0 +r ) 
c c T c c c 


(123) 


d2. a ■ -r 
A, A c 


tr X.A r c ( V 6 A.A )cos *c B o * (r c- r X.A> r c ] 

(r 2 + r 2 cos 2 (t 6 2 + r 2 ) 
c c c c c 


(124) 


d3 


A,A 


= [(r c- r A,A ) - r A,A r c (e c- 6 A,A )coS ^c 6 


r -(r -r, .)r^] 
c c c A, A c 


/ 2 2 2. 12 A '2v 

(r + r cos cb 0 + r ) 

c c c c c 


(125) 


It is recalled that in the above expressions for dl^ A , d2^ A , and 
d3^ A the subscript C is for the point on the unperturbed solution 
curve which has the same latitude as the point A on the perturbed 
solution curve. 

Therefore, the east-west, north-south, and vertical components — 

denoted by dSl^ d S2^ A , and dS3 M , respectively - of the total point- 

'til 

ing displacement error in the NA— solution point due to a probable 
error ed = 0.01° in the lines-of-sight from the observation stations 
to the points on the cloud are, respectively. 


dSl 


dS2 


NA 


NA 



(41 x,sa> 


(d2 X,»A> 


1 

■] 


1/2 

1/2 


(126). 

(127) 



60 


as 3. 


NA 


(d3 X, NA ): 


1/2 


, 1 < NA < NT, 


( 128 ) 


where NT is the total number of points on the solution curve and the 
summation over X means that the error in the lines-of-sight is only 
put into the data from one station, station X, at a time. 

The total pointing displacement error dST^ in the NA — 
solution point due to a probable error ed = 0.01° in the lines-of-sight 
from the observation stations to the points on the cloud is then 


, j, P 1/2 

^ T NA * t(aS V + (dS V + < dS %A> 1 (129) 


The dimensions of dSl , dS2 KA> dS3^, and dST^ are kilometers. 

Pointing Displacement Errors as a Function of the 
Number and Location of the Observation Stations 

It was decided to exercise the LaRC triangulation method to its 
fullest in order to extract meaningful information to aid in designing 
the remainder of the experiment. To begin with, the errors in the tri- 
angulation solution as a function of the number and the location of 
the observation stations were desired. These were important to know for 
the formulation of the "go"-"no go" launch criteria for the experiment 
in the event of unfavorable weather or equipment malfunction at one or 
more of the stations. Eight cases - which were composed of all the 
possible combinations of from two to five stations, always keeping the 
two prime sites Mt. Hopkins and Cerro Morado - plus a ninth case - which 



6i 


was composed of just Mt. Hopkins and Cerro Morado, but with two 
cameras at Mt. Hopkins — were investigated. These nine cases of 
different station combinations are listed in Table 2. Figures 12, 13, 
14, and 15 are plots of the pointing displacement errors — east-west, 
north-south, vertical, and total, respectively — as functions of the 
latitude for the nine cases of different station combinations. It is 
seen from the figures that case 9> with just the two prime stations, 
is the worst and that case 1, with all five stations, is the best. 

The cases composed of three and four observation stations give 
intermediate results. Case 8, which denotes the case of one camera 
at Cerro Morado and two cameras at Mt. Hopkins, is considerably 
better than case 9, which denotes the case of just one camera at each 
of these same two stations . 

Pointing Slope Errors 

In addition to determining what pointing displacement errors to 
expect in the triangulation solution of the cloud position, it was 
decided to determine what pointing errors in slope and curvature to 
also expect. First, it is necessary to find the latitude, longitude, 
and range coordinates of the line between two points in geocentric 
coordinates. In Figure l6 the points P N _^ and denote the two 

points in question. 

If the point P^ is the point P in Figure 11, where the vector C 
drawn from the point A on the perturbed solution curve perpendicular- 
ly intersects the vector B which is tangent to the point C on the 



Table 2: The nine cases of different station combinations 


Case 

1: 

Cerro Morado 
Mt. Hopkins 
Natal Baker Nunn 
Wallops 
White Sands 

Case 

2: 

Cerro Morado 
Mt. Hopkins 
Natal Baker Nunn 
White Sands ' 

Case 

3: 

Cerro Morado 
Mt. Hopkins 
Natal Baker Nunn 
Wallops 

Case 

4: 

Cerro Morado 
Mt. Hopkins 
Wallops 
White Sands 

Case 

5: 

Cerro Morado 
Mt . Hopkins 
White Sands 

Case 

6: 

Cerro Morado 
Mt. Hopkins 
Wallops 

Case 

7: 

Cerro Morado 
Mt. Hopkins 
Natal Baker Nunn 

Case 

8: 

Cerro Morado 
Mt . Hopkins 

Mt. Hopkins Baker Nunn 

Case 

9: 

Cerro Morado 
Mt. Hopkins 






ERROR, km 


6 


!! 


!!!! 


III Hill 
!i! iilii 


Hill lillilillililHI II II II 


i! Hill 


t 


linn 

1111! t! 
mini! ii 
lilliUlllO HU 


iiiiiin 


1 


nil IIU 11 
I1IIH tiitl 


n 


n 


||lil!!| 


Hll 


III 


III 


Hi 


llllllllIgBBS lSP^lllgl 

I I mUM 


8 10 
LATITUDE, de« 


Figure 13.- North-south pointing displacement error as a function of 
latitude for the different station cases. 


VERTICAL ERROR, km 


65 



Figure 14.- Vertical pointing displacement error as a function of 
latitude for the different station cases. 


TOTAL 


66 



-a -» 


6 8 10 12 
LATITUDE, *g 


24 26 28 30 


Figure 15.- Total pointing displacement error as a function of latitude 
for the different station cases. 




68 


unperturbed solution curve, then the latitude, longitude, and range of 

the point P„ — denoted by <J> , 6 , and r , respectively — are 

r N r N r K 

given by 


4> p =<{) + F (130) 


6_ =6 + F 6 (131) 

P N C 


r p = r c + F r c * (132) 


where 4> c is the latitude of the point C, which is the same as the 

• • 

latitude of the point A. The quantities 0 , r , 0 , and r are 

C O C 

the longitude, range, change in longitude with respect to latitude, 
and change in range with respect to latitude, respectively, of the 
point C and are given by equations 111-114, respectively, for 


N = C. 

Analogous expressions also exist for the latitude, longitude, and 

range of the point P„ , — which are denoted by 4> , 0 , and 

N-l N-l 

r p , respectively. 

N-l 

From Figure l6 it is seen that the x,y,z components of P^ and 


P * denoted by x^ , y , z 

N N N 

respectively — are 


Xp , y p , z p 
N-l N-l N-l 


and 




69 

Xp = r COS <J> cos 0 

N N N N 

(133) 

y p = r cos cf> sin 0 
N N N N 

(134) 

z = r sin <j> 

N N N 

(135) 

x p = r p cos 4>-p cos 0 

N-l N-l N-l N-l 

(136) 

y P = r p cos <t>p sin e P 

N-l N-l N-l N-l 

(137) 

Z P = r P Sin ^P 

N-l N-l N-l 

(138) 

The differences between the respective rectangular components of P R 
and P ■ — denoted by dx I}> dy^, and dz N — are 

= *p ’ X P 
r N N-l 

(139) 

^N ~ ^P ~ ^P 

N N-l 

(1^0) 

^ Z N ^P z p 

N N-l 

(lUl) 

Therefore* the line between P^_^ and Pjj in latitude 

, longitude, 

and range coordinates — denoted by d<{>^, <3-6^ j and dr N > 
given by 

respectively — is 



70 


d<j> N = sin -1 


d *n 

2 ^ , 2 . . 2 , 1/2 
i (dx B + + “V 


d0, T = tan 
N 


-i i ^’n 


dr B = (ax| + dy* ♦ *4> 


1/2 


(1U2) 

(143) 

(144) 


As is seen in Figure 17 there is a line solution through the 

points P„ and P„ and also a line solution through the points 

r N-l N 

P , and P! where PJ „ is the perturbed solution point due to 
N-l X,N A,N 

the probable error ed = 0.01° in the lines-of-sight from station X. 
Just as the latitude, longitude, and range of the line between 
P N and P N were found, the latitude, longitude, and range of the 
line between P N 1 and P^ N can similarly be found. The rectangular 
components of Pj^ N — denoted by xj^ H » y ^ N> and z^ N — are 


■ * =: r» * 

A,N A,N 

cos 

<b * 

*A,N 

cos 

6 a,n 

(145) 

A’N r X,N 

cos 

^X,N 

sin 

CD 

** 

5g 

(146) 

* zz y* * 

'X,N A,N 

sin 

*A,N 

» 


(147) 


where 4>! 8! , and r{ are the latitude, longitude, and range, 
respectively, of the perturbed solution point Pj^ The differences 
between the respective rectangular components of P^ ^ and Pj^ — 




72 


denoted by dx^ N> dyj^ N » and dzj^ N — are 

= X X,N - \ x 
^A.N = y A,N ‘ y P N _ x 
dz A,N = Z A,N ‘ Z P K _ 1 


(1U8) 

( 1 ^ 9 ) 


( 150 ) 


Therefore, the line between P N _ 1 and Pj^ N in latitude, longitude, 
and range coordinates — denoted by d<f>^ d0j^ and drj^ K , 
respectively— -is given by 


d *A,N 


• -1 
sm 


dz A.N 


d6J „ = tan 


[< to x,/ + <*1.* )2 + <^x, H )2 ] 1/2 y 

dy{ 


A,N 




-1 [ “■'X.N 
ld *X,H 


= j^X.N 


> 2 + (dy x,ii )2 + (dz X 


,/] 


1/2 


( 151 ) 


( 152 ) 


( 153 ) 


Since the magnetic field lines of the earth lie approximately in 
planes of constant longitude, the longitude dO^ of the line between 
P and P N is very nearly equal to 0 , the longitude of the 

point P^ Since the slope error is small, the longitude d0j^ ^ 
of the line between P N _ 1 and P^ N is also very nearly equal to 




73 


It is convenient to construct a new coordinate system x' , y', z' — 

with its origin at P^ x' axis in the plane of constant longitude 

0 , z' axis the same as the z axis in Figure 17, and y* axis to 

N-l 

form a right-handed orthogonal set — as shown in Figure 18. The vector 

-4 

from P_ T i to P_ T is denoted by dP; and the vector from P„ , to 

N-l N * N-l 

Pj^ by dP'. The latitude, longitude, range components of dP are 
dffjj, d0^, dr^ as given by equations 142-144, respectively; and the 
latitude, longitude, range components of dP' are d(j)^ ^ , d0j^ dr^ N 
as given by equations 151-153, respectively. The total output pointing 

-4 -4 f 

slope error aST^ ^ is the angle between the vectors dP and dP' . 

The total output pointing slope error can be resolved into two compon- 
ents — one in the plane of constant longitude and the other perpendicular 
to this plane. The parallel component is called the latitude output 
pointing slope error and is denoted by aSl^ N ; the perpendicular 
component is called the longitude output pointing slope error and is 
denoted by aS2^ 

A 

The rectangular coordinates of the unit vector dP in the primed 
coordinate system of Figure 18 — denoted by ddx^, ddy^, ddz^ — 8X6 


ddx^ = cos d<j>^ cos 

K ' 9 P ) 

\ r N-l / 

(154) 

ddy N = cos d<f> H sin 

K - v ) 

' N-l ; 

(155) 

ddz N = sin d<f> N , 


(156) 




Figure 18.- The vectors dP and dP' 

output pointing slope error 


and the total 

aST, «• 

A ,N 



75 


where 0 is the longitude of the point and and d 9jj 

w-l 

are given by equations lk2 and li+3. Similarly, the rectangular coordi- 
nates of the unit vector dP' in the primed coordinate system of Figure 
18 - denoted by ddxj^ ddy^ N’ ddz A N “ are 


ddx i,n = c08 d *M 003 ( d8 ht> ' VJ 

ad3r x,» = 003 d *M 3in ( de x,N - 

ddz X,» = 8i ° d *X,H • 

where d<}>^ N and dQ^ N are given by equations 151 and 152. 

Hence, the latitude and longitude components of the total output 
pointing slope error, denoted by aSl^ jj and aS2^ N’ and i°i a i 
output pointing slope error aST^ ^ in the If- — solution point due to a 
probable error ed = 0.01° in the lines-of-sight from station X are, 
respectively. 


(157) 

(158) 

(159) 



(160) 

(161) 

(162) 


The latitude and longitude components of the total output pointing 
slope error, denoted by aSl^ and aS2^, and the total output 



76 


pointing slope error aST^ A in the NA — solution point due to a 
probable error ed = 0.01° in the lines-of-sight from the observation 
stations to the points on the cloud are, respectively. 


-[* L 

Lx=i 


aS1 NA = (aS1 X,NA J ' 


^A 


1/2 


1/2 


aST 


NA 


"A (aS2 M/] 

2 pi 1/2 

(aSl M ) d + (aS2 NA AJ , 1 < NA < NT , 


(163) 


(16U) 


(165) 


where NT is the total number of points on the solution curve and the 

summation over X means that the error in the lines-of-sight is only 

put into the data from one station, station X, at a time. 

The total pointing slope error AST is defined as the ratio of 

the total output pointing slope error aST to the total input 

pointing slope error sa. In Figure 19, the points P n-1 and P n are 

consecutive points spaced BN degrees apart on the unperturbed 

azimuth-elevation curve. It is recalled that BN = 0.28°. The point 

p^, a line from which perpendicularly intersects the unperturbed 

curve at the point p , is on the perturbed azimuth-elevation curve 

due to a probable error ed = 0.01° in the line-of-sight from the 

station to the point p . The total input pointing slope error sa 

n 

is given by 


tan sa = 


ed 

BN 


(166) 




78 


Since ed = 0.01° and BN = 0.28°, tan sa is small and, hence, is 
approximately equal to sa. Hence, 


sa 


ed 

BN 


(167) 


Therefore, the total pointing slope error AST^ A in the NA — solution 
point due to a probable error ed = 0.01° in the lines-of-sight from 
the observation stations to the points on the cloud is 


AST 


NA 


aST. 


NA 


sa 


(168) 


where aST KA is given by equation 165 and sa is given by equation 

167. 

Now, 


aST 


AST. 


NA 


NA sa 


(168) 


( 


(aSl Mft ) 2 + (aS2 M ) 2 _ 


1/2 


NA' 


sa 


, using equation 165 




1/2 


(169) 


Therefore, from equation 169 it is seen that the latitude pointing 
slope error AS1^ A , the component of the total pointing slope error 
in the plane of constant longitude, and the longitude pointing slope 
error AS2 Mfi , the component of the total pointing slope error 



79 


perpendicular to the plane of constant longitude, in the NA 
solution point due to a probable error ed = 0.01° in the lines-of- 
sight from the observation stations to the points on the cloud are, 
respectively, 


AS1 


NA 


AS2 


NA 


aSl 


NA 


sa 


aS2 


NA 


sa 


(170) 

(171) 


where aSl , aS2„„ , and sa are given by equations 163, l64, and 167 , 
NA NA 

respectively. 

The three slope errors AS1 M , P>S2 ^ t 811(1 AST NA are 821 
dimensionless . 

Pointing Slope Errors as a Function of the 
Number and Location of the Observation Stations 

Figures 20, 21, and 22 are plots of the latitude pointing slope 
error, the longitude pointing slope error, and the total pointing slope 
error, respectively, as functions of the latitude for the nine 
different station combinations. As for the pointing displacement 
errors, the greatest accuracy is for the five-station case, case 1, 
and the least accuracy is for the two-station case, case 9> the 
combinations of four and three stations giving intermediate accuracies. 
Again, case 8, with two cameras at Mt. Hopkins and one at Cerro 
Morado, is considerably more accurate than case 9, with just one 
camera at each of these same two stations. 



80 



LATITUDE, deg 


Figure 20. _ Latitude pointing slope error as a function of 
latitude for the different station cases. 



LONGITUDE SLOPE ERROR 


81 



Figure 21.- Longitude pointing slope error as a function of 
latitude for the different station cases. 



0 10 12 14 16 

LATITUDE, deg 


-UU i i i ' I • i 1 i i i_dll • i 'i If i . j 

JQ 20 22 24 26 28 30 


Figure 22.- Total pointing slope error as a function of 
latitude for the different station cases. 




83 


Pointing Curvature Error 

The pointing curvature error is defined as the ratio of the output 

pointing curvature error to the input pointing curvature error. For 

defining the input pointing curvature error, consider Figure 23, in 

which R is the radius of curvature of the perturbed azimuth-elevation 

curve through the three consecutive points p^ 2* P n-1’ and ^ ue 

a probable error ed = 0.01° in the line-of-sight from the station to 

the point p^ ^ on the unperturbed azimuth-elevation curve through 

the three consecutive points p p , and p spaced BN = 0.28° 

n-2 n-1 n 

apart. The angle a is small, as BN is small. From Figure 23 it is 
seen that 


cos a = 


R - ed 
R 


(172) 


R cos a = R - ed 


ct 

R(l - — +_ ...) = R - ed, expanding cos a 


R - R ~ = R - ed, as a is small 


R f- - ed 


Hence , 


R = 


2 ed 


a 


(173) 




85 


Also from Figure 23 it is seen that 


BN 

sin a = — 


(174) 


Hence , 


a 


BN 

R 


as a is small 


(175) 


Substituting equation 175 for a into equation 173 for R, it becomes 


R = 



(173) 


2 ed 
(BN/R) 2 


using equation 175 


2 ed R 2 
BN 2 


Hence , 


R = 



2 ed 


(176) 


Therefore, the input pointing curvature error defined as the 

reciprocal of the radius of curvature of the perturbed azimuth-elevation 
curve , is given by 



86 


_ l 
C i R 


2 ed 
2 ' 


( 177 ) 


using equation 176 for R. 


Analogously, the output pointing curvature error c in the 


X. I- © * 

N — solution point due to a probable error ed = 0.01 in the lines-of- 


sight from station X is 


2[(a V/ * (d2 A./ * 


( 178 ) 


where dl^ N , d2^ N , and d3^ K are given by equations 123, 124, and 

125 for the point N, respectively, and dr N is given by equation 144. 

"b h 

The pointing curvature error CS^ ^ in the N — solution point 


due to a probable error ed = 0.01 in the lines-of-sight from station 


X, defined as the ratio of c 


to c., is then 
°X,N 1 


X,N c. 

* l 


(179) 


1/2 

2[ (d l / + <d2 ) 2 ♦ (d3 x ) 2 ] /dr 2 


2 ed/BN c 


1/2 

[<di ) 2 ♦ (d2 B ) 2 ♦ <d3 x ) 2 ] BH 2 


, ( 180 ) 


ed dr. 


using equations 177 and 178 for c . and c , respectively . 

1 °X,N 


c-£ 


87 


Therefore, the pointing curvature error in the NA 

solution point due to a probable error ed = 0.01° in the lines-of- 
sight from the observation stations to the points on the cloud is 


CS 


NA 



< 0S X,M> 


2 1/2 




1 < NA < NT, 


(181) 


where NT is the total number of points on the solution curve and the 
summation over X means that the error in the lines-of-sight is only- 
put into the data from one station, station X, at a time. The 
dimensions of the pointing curvature error are degrees/kilometers. 

Pointing Curvature Error as a Function of the 
Number and Location of the Observation Stations 

Figure 2h is a plot of the pointing curvature error as a function 
of the latitude for the nine different station combinations. As for 
the pointing displacement errors and the pointing slope errors, the 
greatest accuracy is for the five-station case and the least accuracy 
is for the two-station case. The cases of four and three stations 
give intermediate accuracies in the following decreasing order - 
2, H, 3> 7, 8, 6, and 5. Also, as for the pointing displacement 
errors and the pointing slope errors, case 8, with two cameras at Mt. 
Hopkins and one at Cerro Morado, is considerably more accurate than 
case 9, with just one camera at each of these same two stations. 

Therefore, from Figures 12, 13, 1^, 15, 20, 21, 22, and 2L, the 
different error components — east— west displacement , north— south 
displacement, vertical displacement, total displacement, latitude slope, 







89 


longitude slope, total slope, and curvature , respectively - in the 
triangulation solution due to a probable error ed = 0.01° in pointing 
can be seen as functions of the number and location of the observation 
stations . In the event of unfavorable weather or equipment malfunction 
at one or more of the stations , these plots can be consulted to see 
if the respective errors can be tolerated or not before deciding whether 
or not to launch. 

Comparison of Pointing Displacement Errors 
It was decided of interest to compare the three pointing displace- 
ment error components to each other to determine which component is the 
greatest. Figure 25 is a plot of the east-west pointing displacement 
error, the north-south pointing displacement error, the vertical point- 
ing displacement error, and the total pointing displacement error as 
functions of the latitude for case 1. Case 1, which is the case composed 
of five stations , was chosen for this comparison because it is the case 
which gives the smallest values for all three displacement error 

components. As can be seen from Figure 25, the vertical component is 

o 0 

the largest component throughout the latitudinal region of -9 to 15 , 
whereas the north-south component is the largest component throughout 
the latitudinal region of 16° to 28°. 

Comparison of Pointing Slope Errors 
Also of interest is a comparison of the two pointing slope 
error components. Figure 26 is a plot of the latitude pointing 
slope error, the longitude pointing slope error, and the total 



urn ‘saoaua 


90 



Figure 25.- Pointing displacement errors as a function of 
latitude for the five-station case. 





SLOPE ERRORS 



Figure 26*- Pointing slope errors as a function of 
latitude for the five-station case. 


92 


pointing slope error as functions of the latitude for case 1, which 
gives the smelliest values for both slope error components. As can be 
seen from Figure 26, the latitude component is much ledger than the 
longitude component and the total pointing slope error very closely 
approximates the latitude component. 

Pointing Errors as a Function of 
Observation Duration 

The barium cloud elongates along the length of the magnetic field 
line. Since the magnetic field lines are approximately constant in 
longitude, the barium cloud is essentially elongating in latitude. 
Looking back to Figures 25, 26, and 24 , the four pointing displacement 
errors, the three pointing slope errors, and the pointing curvature 
error, respectively, as functions of the latitude for the five-station 
case can be seen. Since the position in latitude of the elongating 
cloud is a function of the time after the barium is released, these 
pointing errors are also functions of the time after release. From 
Figures 25, 26, and 2k, then, the pointing errors as functions of the 
observation time after release can be obtained. 

The rate of elongation of the cloud was predicted to be about 
1.2 km/sec in each direction. Observation durations of 10 ,000 sec, 
6,000 sec, and 1,000 sec would then correspond to total cloud lengths 
of 24,000 km, 14,400 km and 2,400 km, respectively. The angle of 
latitude 4>. that the cloud length subtends at the time t after 

"G 


release is approximately 



93 


<J> t = tan" 1 (~) , ( 182 ) 

where L. is the length of the cloud at the time t after release and 
H = 31,633.008 km is the altitude of the nominal release point. Using 
equation 182 the angles of latitude subtended for the cloud lengths of 

24.000 km, 14,400 km, and 2,1+00 knr— which correspond to the observation 
durations of 10,000 sec, 6,000 sec, and 1,000 sec, respectively— are 
37.188°, 24.1+76°, and 4.339°, respectively. Centering these three 
angles of latitude about the nominal release point latitude, which is 
9.229°, gives the regions of latitude covered by the elongating cloud 
during these respective observation durations. The regions of 
latitude corresponding to the observation durations of 10,000 sec, 

6.000 sec, and 1,000 sec are (-9*365° to 27.823°), (-3.009° to 21.467°), 
and (7.060° to 11.398°), respectively. By examining these three 
regions of latitude in Figures 25, 26, and 24, the pointing displacement 
errors, the pointing slope errors, and the pointing curvature error 

can be seen, respectively, as functions of these three observation 
durations . 

From Figure 25 it is seen that the east-west pointing displacement 
error practically remains constant throughout the three observation 
durations, that the north-south pointing displacement error increases 
toward the lower-latitude end and decreases toward the higher-latitude 
end over the 1,000-sec observation duration and first increases said 
then decreases toward the lowerilatitude end and decreases toward the 



higher-latitude end over the 6,000-sec and 10,000-sec observation 
durations , and that the vertical and total pointing displacement 
errors increase toward the lower-latitude end and decrease toward the 
higher-latitude end over all three of the observation durations. From 
Figure 2 6 it is seen that the longitude pointing slope error practically 
remains constant throughout the three observation durations and that 
the latitude and total pointing slope errors increase toward the lower- 
latitude end and decrease toward the higher-latitude end over all 
three of the observation durations. From Figure 24 it is seen that 
the pointing curvature error also increases toward the lower-latitude 
end and decreases toward the higher-latitude end over all three of 
the observation durations . 

Pointing Errors as a Function of 
East-West Cloud Drift 

The barium cloud is expected to drift eastward or westward. It 
was decided to investigate how the triangulation results are affected 
if the cloud drifts in such a fashion into other areas of the sky. 

An east-west drift corresponds to a longitudinal drift; hence, it is 
convenient for this investigation to look at, actually, the effect of 
using different release points — release points having the same 
latitude and altitude, but different longitudes. 

It is reasonable to assume that in order to observe the barium 
cloud at an altitude of 32,000 kilometers from a particular observation 
station the elevation angle from that station to the cloud should not 



95 


be less than 20 degrees. From the equations derived in the appendix, 
the elevations were calculated for all integer values of the longitude 
between 0 ° and 180 ° and between 0 ° and - 180 ° for the five observation 
stations — Mt. Hopkins, Cerro Morado, Wallops, Arequipa, and White Sands. 
It wan found for the range of longitudes between -49° and - 119° that 
the elevation angles were approximately greater than or equal to 20 
degrees for all five stations simultaneously. Six different release 
points — each having latitude equal to 9-229 degrees and altitude equal 
to 31 , 633.008 kilometers, with longitudes equal to - 49 °, - 63 °, - 77 °, 

- 91°, - 105°, and - 119°, respectively — were chosen for the investiga- 
tion. Points along the respective magnetic field lines through these 
particular release points were used for the input data to the LaRC 
method . 

Figures 27, 28, 29, 30, 31, 32, 33, and 34, respectively, are 
plots of the pointing errors — east-west displacement, north-south 
displacement, vertical displacement, total displacement, latitude 
slope, longitude slope, total slope, and curvature— ^as functions of 
the latitude for the five-station case for different release points 
varying in longitude. Now, the coordinates of the nominal release 
point chosen for the BIC Experiment are latitude = 9.229°, longitude = 

- 75.000°, and altitude = 31,633-008 km. Hence, these plots can be 
examined to see if the triangulation errors increase or decrease as 
the cloud drifts eastward or westward from this chosen nominal 
release point. From Figures 27-34 it is seen that all of the pointing 
errors increase as the cloud drifts eastward into the longitudinal 



EAST -WEST ERROR, km 


96 



-12 -10 -8 -6 


LATITUDE, dsg 


Figure 27.- East-west pointing displacement error as a function of 

latitude for the five-station case for different release 
points varying in longitude. 



ERROR, km 


9 ' 


: ~ | : 1 1 - ! : | 


1 

ill 

If 

HI 

III 1 III 

| !i|j i j: |i| 

||| 

[jiff 


RELEASE POINT 
LONGITUDE. dt« 

ip 


i i i;i||| 

| | 

1 

jtj i| ||i 

|p !lj| 

1 lit! isji 

• iiij |'!l 

o 

n 

- 49 

- 63 

II 

I j 

j jlll { 



| 


! lip iji 1 

tffliiR iiii 

<> 

- 77 

il 

it 

1 iiijlllll 

tl It! 


tfltti ttl 

m il 1 i 

1 1 

j Ii;: iiii 


-103 


ii 

i iittltti! 

tt lit 


ft) [tt| (tl 

1 

1 j 

uflili 

C> 

-1 19 

iiii i:i 

fHr+Fr 

... 

I 

1 I I 

it ttt 


}|P | 

tilt It IP 

I j 1 

Mil if 


4,:ilj t !ij Uj Jijif 

HIM 

Rip 

1 ! P 

It ft) 


ItP III 

m 1) |i 

1 

ill r 



!;|! iiii 

liiii 

piff 

! ffi 

j 

1 ml {A 

ffllJPp 

lift 

Will 


|i J jif-tJ 


FI 


tt ttt 

| 

II If pl 

1 'M Pi 

i iiii i 



j 

in 

: ::i :’;i 

ij ;i:i 

1 m 


pp | 

1 ll ti In! 

4 nil ft] 




hit aii 


ipll 

ijj} (It 

1 ! 

III ill I 

I j[| ]| | !j| j!j| 

Jllliili! 

i ii ill ! ll i 



if 



w 

II 

jl 

i 1!|| j) 

r t|| }p iiij 

1¥P 

; jiilpTff 



I 



■ A 


i! ill 

1111 

|j iK Ii! :l : ; 



Al 

i 



-• 

2 

A 

11 

If 

m 

11 

Ilf 

111 

ii™ 

; : ,i: :: : 

i ' ]TV: 

. i ; : i •: ■ 

ii 

Hi mi iiii i 
iMPf ii 

pi| 



! i 

4- 


T 


i ! ii! 

If h; 

■ ! ! i;i' 

■ ■ 



‘ 

j i i i 

1 A. 


\:f~. ; ;V l' : I W | ! !j III, ^ 41 I . ii I'H- 

■ i il\ ‘;ii i ! - 1 I jl } i i ii iili. 

;ii_iiiLffpjt4 |pm§ 
= i^iifiili||fciL 4 if 

fKIuti Ilf Li;;: 


iimypi 




A ■ Ur -I 


wmmr. 


mm 

# H i 4 i]!: i 1 ■ T ’ r ; ' 


FlFlJi 


III! 


' / / r':' : xR:^ N -,\ i 

17 7^ I f f “ I ~ 4 “ T'" “■■■ ' F f v“ - kt; k: 4 -T- 

Tt^FnrijjBisij^ 

-^FifeipiiiiiHiFlrfln 


UMlliU ilUiilu i 


MMilllll 


T^bT -I o ; 4 6 8 lo ia 14 is w So tX St w “ So 

LATITUDE, deg 


Figure 28.- North-south pointing displacement error as a function of 
latitude for the five-station case for different release 
points varying in longitude. 


VERTICAL ERROR, km 


98 



Figure 29.- Vertical pointing displacement error as a function of 
latitude for the five-station case for different 
release points varying in longitude. 



ERROR, km 



Figure 30.- Total pointing displacement error as a function 

of latitude for the five-station case for different 
release points varying in longitude. 





Figure 31.- Latitude pointing slope error as a function 

of latitude for the five-station case for different 
release points varying in longitude. 








102 



8 10 (2 14 

LATITUDE, deg 


Figure 33.- Total pointing slope error as a function 
of latitude for the five-station case for 
different release points varying in longitude. 


CURVATURE ERROR, O^An 


103 



Figure 34. - Curvature pointing error as a function 

of latitude for the five— station case for 
different release points varying in longitude. 



104 


regions of — 63^ and. - 49 and that all of the pointing errors decrease 
as the cloud drifts westward into the longitudinal region of - 91° • As 
the cloud drifts farther westward into the longitudinal regions of - 105° 
and - 11 9°, the pointing errors increase or decrease depending on the 

cloud's position in latitude. 

Pointing Errors as a Function of the 
Number of Input Data Points 

Since the LaRC triangulation method requires for input data a 
number of points from the original azimuth-elevation curves from the 
observation stations, it was decided to determine just how many such 

raw data points are required to be read. It is recalled from the 
discussion of the LaRC method that the number of points NB^ along the 
azimuth-elevation curve of arc length sarc^ in degrees from station 
£, is given by equation 55 


= BN ’ 

where BN is the spacing in degrees between consecutive points along 
the curve. A convenient parameter for this determination is the 
spacing BN required between points instead of the actual number of 
points required to be read, as the arc lengths of the azimuth- 
elevation curves vary. 

The LaRC method was tested for the following six different values 
of the spacing in degrees — BN = 0.28, 0.56, 1.12, 2.24, 4.48, and 8.96. 
Figures 35, 36, 37, 38, 39, 40, 4l, and 42 are plots of the pointing 
errors — east-west displacement, north-south displacement, vertical 



105 



LATITUDE, deg 


Figure 35-- East-vest pointing displacement error as a 
function of latitude for the five-station 
case for different values of BN. 




NORTH-SOUTH ERROR, km 


106 



LATITUDE, deg 


Figure 36, _ North-south pointing displacement error as a 

function of latitude for the five-station case 
for different values of BN. 



VERTICAL ERROR, km 


107 



LATITUDE, deg 


Figure 37.- Vertical pointing displacement error as a 
function of latitude for the five-station 
case for different values of BN. 



TOTAL DISPLACEMENT ERROR, km 


108 



LATITUDE, dej 


Figure 38.- Total pointing displacement error as a 

function of latitude for the five-station 
case for different values of BN. 



LATITUDE SLOPE ERROR 


109 



Figure 39 Latitude pointing slope error as a function 
of latitude for the five-station case for 
different values of BN. 




LONGITUDE SLOPE ERROR 


110 



Figure U0 Longitude pointing slope error as a function 
of latitude for the five-station case for 
different values of BN. 



TOTAL SLOPE ERROR 


111 



LATITUDE, deg 


Figure Ul.- Total pointing slope error as a function 
of latitude for the five-station case for 
different values of BN. 


CURVATURE ERROR, deg/km 


112 



LATITUDE, deg 


Figure 42.- Pointing curvature error as a function 
of latitude for the five-station case 
for different values of BN. 




113 


displacement, total displacement, latitude slope, longitude slope, 
total slope, and curvature, respectively — as functions of the latitude 
for the five-station case for these six different values of BN. As 
can be seen from the figures, the curves are nice and smooth for the 
shorter spacings of BN = 0.28, O. 56 , and 1.12 degrees, whereas they 
are highly erratic for the longer spacings of BN = U.48 and 8.96 
degrees. The intermediate spacing of BN = 2.2k degrees produces only 
slightly erratic behavior. It is, therefore, recommended that the 
maximum spacing allowed between the input data points along the 
azimuth-elevation curves be BN = 1.12 degrees. For example, a 
reasonable arc length for a long barium cloud azimuth-elevation 
curve would be 37 degrees; using this maximum spacing of 1.12 degrees 
between points would require that at least 3 1 * input data points be 
read from the curve. 

Effects of Additional Cameras at a Particular 
Observation Station 

From Figures 12, 13, lH, 15, 20, 21, 22, and 2k for the pointing 
errors, it was seen that case 8, which is composed of one camera at 
Cerro Morddo and two cameras at Mt. Hopkins, gave significantly less 
errors than case 9, which is composed of just one camera at each of 
these same two stations. Therefore, adding a second camera to 
Mt. Hopkins in the two-station case improves the triangulation results. 

It was decided to see if adding a second camera to Mt. Hopkins in 
the five-station case would also increase the accuracy. Table 3 gives the 
values of the pointing displacement errors as functions of the latitude 



Ill 


Table 3: Case 1 pointing displacement errors. 


Latitude 

(deg) 

E-W . 
Error 
(km) 

N-S 

Error 

(km) 

Vertical 

Error 

(km) 

Total 

Error 

(km) 

-9 

3.81 

3.48 

36.77 

37-14 

-8 

3.87 

4.74 

36.52 

37-02 

-7 

3-91 

5-99 

36.31 

37.01 

-6 

3.91 

7.20 

35-99 

36.91 

-5 

3-97 

8.36 

35.54 

36.73 

-1+ 

3.97 

9-38 

34.70 

36.17 

-3 

4.00 

10.42 

34.08 

35-86 

-2 

4.02 

11.37 

33.33 

35.45 

-1 

4.04 

12.25 

32.51 

34.98 

0 

4.04 

12.96 

31.41 

34.22 

1 

4.05 

13.66 

30.45 

33.62 

2 

4.04 

14.23 

29.35 

32.86 

3 

4.04 

14.73 

28.26 

32.12 

4 

4.03 

15.13 

27.10 

31.29 

5 

4.02 

15.44 

25.94 

30.46 

6 

4.00 

15.63 

24.70 

29.50 

7 

3.97 

15.71 

23.41 

28.46 

8 

3-93 

15.73 

22.17 

27.47 

9 

3.89 

15.65 

20.91 

26.41 

10 

3.86 

15.55 

19.74 

25.43 

■ 11 

3.83 

15-39 

18.60 

24.44 

12 

3-79 

15.18 

17.48 

23.46 

13 

3.74 

14.85 

16.32 

22.38 

l4 

3.70 

14.54 

15.28 

21.42 

15 

3.65 

l4.l4 

14.22 

20.38 

l6 

3.59 

13.71 

13.22 

19.38 

17 

3.55 

13.31 

12.31 

18.48 

18 

3.50 

12.85 

11.41 

17.54 

19 

3.45 

12.38 

IO.56 

16.64 

20 

3.4l 

11.92 

9.78 

15.80 

21 

3-37 

11.45 

9.04 

14.98 

22 

3.32 

10.98 

8.35 

14.19 

23 

3.28 

10.52 

7-71 

13.45 

24 

3.25 

10.05 

7.10 

12.73 

25 

3.21 

9-59 

6.54 

. 12.04 

26 

3.17 

9.13 

6.00 

11.37 

27 

3.13 

8.65 

5-50 

10.72 

28 

3.14 

8.40 

5.16 

10.34 












115 


for case 1, the five-station case. Then a second camera was added to 
Mt. Hopkins, and this new case was denoted by case 1'. Table H gives the 
values of the pointing displacement errors as functions of the latitude 
for case 1'. The east-west component has remained the same or has de- 
creased very slightly; but the north-south, vertical, and total pointing 
displacement errors have increased slightly. Then a third camera was 
added to Mt. Hopkins, and this case was denoted by case l". Table 5 gives 
the values of the pointing displacement errors as functions of the lati- 
tude for this case. Once again the east-west component has remained the 
same or has decreased very slightly, while the north-south, vertical, and 
total pointing displacement errors have increased slightly. This process 
of adding an extra camera to Mt. Hopkins was continued up to the final 
case of twelve cameras at Mt . Hopkins ; and each time the east-west 
component remained the same or decreased very slightly, while the 
north-south, vertical, and total pointing displacement errors increased 
slightly. 

At first these increased errors might be a little alarming in view 
of the results found earlier for the two-station case, in which adding 
a second camera to Mt. Hopkins improved the accuracy. However, for the 
two-station case the intersection of the two conical-like surfaces in 
space defined by the respective azimuth-elevation curves from the two 
stations is unique; whereas, with three or more observation stations, 
especially with errors in pointing, the intersection of these surfaces 
is no longer unique. What happens when more and more cameras are 
added to one station in a case of three or more stations is that, in 



116 


Table 4: Case 1' pointing displacement errors. 


Latitude 

(deg) 

E-W 

Error 

(km) 

— 

N-S 

Error 

(km) 

Vertical 

Error 

(km) 

Total 

Error 

(km) 

-9 


3-76 

39-72 

kO. 08 

-8 


5.11 

39.37 

39.89 

-T 

3-90 

6. k5 

39.07 

39-79 

-6 

3-93 

7-73 

38.6k 

39.60 

-5 

3.97 

8.96 

38.08 

39-32 

-k 

3.97 

10.0k 

37-lk 

38.68 

-3 

k.00 

11.13 

36. kO 

38.27 

-2 

k .02 

12.13 

35.5k 

37-77 

-1 

k.ok 

13.0k 

3k. 60 

37.20 

0 

k.03 

13-77 

33.38 

36.3k 

1 

k.ok 

lk.k9 

32.31 

35.6k 

2 

k.ok 

15.07 

31.08 

3k. 77 

3 

k.ok 

15-57 

29.87 

33-93 

k 

It. 03 

15-97 

28.60 

33.01 

5 

It. 02 

16.27 

27.33 

32.06 

6 

It. 00 

l6.k5 

25.98 

31.01 

7 

3.96 

16.50 

2k. 58 

29.87 

8 

3.93 

16. k9 

23.25 

28.77 

9 

3.89 

16.38 

21.89 

27.61 

10 

3.86 

16.25 

20.63 

26.5k 

11 

3-83 

16.06 

19. k0 

25. k7 

12 

3.79 

15.81 

18.20 

2k. kO 

13 

3.7k 

15. k3 

16.97 

23.2k 

lk 

3.70 

15.09 

15.85 

22.20 

15 

3.6k 

lk.6k 

lk.73 

21.08 

16 

3.59 

lk. 17 

13.66 

20.01 

17 

3-55 

13.7k 

12.70 

19.0k 

18 

3-50 

13.2k 

11.75 

18. 0k 

19 

3.k5 

12.73 

10.86 

17.09 

20 

3-kl 

12.2k 

10.0k 

16.19 

21 

3.36 

11.73 

9.26 

15-32 

22 

3.32 

11.23 

8.5k 

lk.50 

23 

3.28 

10.7k 

7.87 

13.71 

2 k 

3.25 

10.25 

7.2k 

12.96 

25 

3.21 

9-76 

6.65 

12.2k 

26 

3.17 

9-27 

6.10 

11.5k 

27 

3.12 

8.77 

5-57 

IO.85 

28 

3-lk 

8.50 

5.22 

10. k6 











Table 5: Case l" pointing displacement errors 


Latitude 
( deg ) - 

E-W 

Error 

(km) 

N-S 

Error 

(km) 

Vertical 

Error 

(km) 

Total 

Error 

(km) 

-9 



1+1.13 

41.49 

-8 


5.29 

40.74 

41.26 

-7 

3-89 

6.67 

1+0.39 

41.12 

-6 

3-93 

7-99 

39.91 

40.89 

-5 

3-96 

9-25 

39-30 

40.57 

-4 

3.97 

10.36 

38.30 

39-88 

-3 

3-99 

11. 47 

37.50 

39-42 

-2 

4.02 

12.1+9 

36.60 

38.88 

-1 

4.04 

13.1+1 

35.60 

38.26 

0 

4.03 

11+.16 

3U.33 

37-35 

1 

U.OU 

lU.89 

33.19 

36.60 

2 

4.04 

15.1+7 

31.90 

35.69 

3 

4.04 

15.98 

30.61+ 

34.79 

1* 

U.03 

16.37 

29.32 

33.82 

5 

4.02 

16.67 

28.00 

32.83 

6 

4.00 

16.83 

26.59 

31.73 

7 

3.96 

16.87 

25.15 

30.54 

8 

3:93 

16.86 

23.76 

29.39 

9 

3.89 

16.73 

22.35 

28.19 

10 

3.86 

16.58 

21.05 

27.07 

11 

3.83 

16.38 

19.78 

25.97 

12 

3.79 

16.11 

18.55 

24.85 

13 

3.7U 

15.71 

17-27 

23.65 

lit 

3.70 

15.35 

16.13 

22.57 

15 

3.64 

14.88 

14.97 

21.42 

l6 

3-59 

ll+.l+O 

13.88 

20.31 

17 

3.55 

13. 9U 

12.89 

19.31 

18 

3.50 

13.1+2 

11.92 

18.29 

19 

3.U5 

12.90 

11.01 

17.30 

20 

3.U1 

12.39 

10.16 

16.38 

21 

3.36 

11.87 

9-37 

15.49 

22 

3.32 

- 11.35 

8.63 

14.64 

23 

3.28 

10.84 

7.94 

13-84 

24 

3. 24 

10. 31* 

7.30 

13.07 

25 

3.21 

9.81+ 

6.71 

12.33 

26 

3.17 

9-33 

6.14 

11.61 

27 

3.12 

8.83 

5.61 

10.92 

28 

3.1U 

8.55 

5.25 

10.51 











118 


effect, the data from this one station is receiving more and more 
weighting and the solution is becoming more and more constrained to 
lie on the surface originating from this one station. Hence, the 
triangulation errors should be expected to increase, as was seen in 
Tables 3-5. 

Station Displacement Errors for the 
Aircraft Station 

As was mentioned earlier the Wallops station is really an air- 
craft. The position of the moving aircraft contains some error, as 
the aircraft is tracked by radar and because of its particular flight 
path for the experiment there are times when it is outside of the range 
of the existing radar stations. This error in the Wallops station 
position produces errors in the triangulation results, called station 
errors, just as an error in the lines-of-sight from the different 
stations produces errors in the triangulation results , called pointing 
errors. A reasonable probable error to assume in the aircraft 
position is a three-kilometer sphere of uncertainty (a sphere of 
radius equalling three kilometers) about its position. This total 
uncertainty is denoted by a^. It is then reasonable to assume that 
this sphere of uncertainty is due to equal uncertainty in the three 
rectangular components east— west , north— south, and vertical of the 

aircraft. These three uncertainties are denoted by °EW’ a NS’ 

1/2 

Oy, respectively, and are each equal to +_(3) km., as 


and 



119 


«» - < 4 ♦ 4 + 4 > 


1/2 


(183) 


a. = [(+ S3) 2 + (+ Si) 2 + (+ S3) 2 ] 


1/2 


= (3 + 3 + 3) 


1/2 


= (9) 


1/2 


= 3 


(184) 


If T denotes the range of the aircraft, slat denotes itB latitude, 

and a , . , a . , and a_ denote the uncertainties in its latitude, 

slat’ slon’ T 

longitude, and range, respectively, then 


= T cos (slat) a 
EW slon 


(185) 


a NS r °slat 


(186) 


°v = a r 


(187) 


For computing the numerical values for O g ^ a ^ and a s i on » ‘'"I 16 latitude 
and range used were those for the aircraft over Wallops . The latitude 
of Wallops is 37-9324 degrees. The aircraft flies at an altitude of 
35,000 feet or higher. If 35>000 feet, which equals 10.6680 kilometers 



120 


is assumed for the altitude of the aircraft, then the range of the 

aircraft is this altitude plus the earth's radius of 6371*2 kilometers; 

hence, T = 6381.9 kilometers. Solving for o in equation 185, 

slon 

a - °EW 

slon F cos (slat) 

G = ± (3) 1/2 km 

slon (6381.9 km) cos (37*9324o) 

= +0.0003^3 radians 


(188) 


(189) 


Solving for o .. . in equation 186, 
slat 


o 


slat 



_ ± ( 3 ) 1 ^ 2 km 
: slat ~ (6381.9 km) 


= + 0 . 000271 radians 


(190) 


(191) 


And since the uncertainty in range is just equal to the uncertainty in 
the vertical component , as given by equation 187 , 

= + (3) 1/2 km 

= + 1.732051 km (192) 

The triangulation results were again computed using the magnetic 



121 


field line data through the BIC nominal release point for the input data - 
only this time instead of introducing the probable error ed = 0.01° in- 
to the lines-of-sight from each station, one at a time, the uncertainties 

a , , , a , , and O-n were introduced into the aircraft coordinates over 

slat’ slon’ r 

Wallops, one at a time. The four station displacement errors due to the 
uncertainties a n . , a n , and C r - given by equations 191, 189, and 
192, respectively - were calculated, using an analogous procedure to 
that used earlier for calculating the four pointing displacement errors 
due to the probable error ed. Figures 13, 11, 15, and 16 are plots of 
the station displacement errors - east-west, north-south, vertical, and 
total, respectively — as functions of the latitude due to the uncertain- 
ties 0,^,0, , and O-r in the coordinates of the aircraft over Wallops . 

slat slon i 

Only the cases 1, 3, 4, and 6, which are defined in Table 2, were con- 
sidered, as these are the only cases which include the Wallops station. 

It is seen from Figures 11, 15, and 16 for the north-Bouth, 
vertical, and total station displacement errors, respectively, that 
the errors for case 3, composed of four stations, are less than the 
errors for case 1, composed of five stations, and also that the 
errors for case 6, composed of three stations, are less than the 
errors for case 1, composed of four stations. At first this might 
be a little alarming in view of the fact that reference 1, even 
though it is for an idealized situation, indicates that the tri- 
angulation error decreases as the number of observation stations 
increases. However, it is noticed when looking back to Table 2 that 



122 



Figure 43 .- East-west station displacement error as a function 
of latitude for the station cases containing the 
aircraft . 


40 



LATITUDE, deg 


Figure 44.- North-south station displacement error as a function 
of latitude for the station cases containing the 
aircraft . 



VERTICAL ERROR, km 



Figure 45.- Vertical station displacement error as a function 
of latitude for the station cases containing the 
aircraft . 


TOTAL DISPLACEMENT ERROR, km 


125 



Figure 46.- Total station displacement error as a function 

of latitude for the station cases containing the 
aircraft . 


126 


case 1 is equal to case 3 with White Sands added and that also case 4 
is equal to case 6 with White Sands added. From Table 1 it is noticed 
that the coordinates of White Sands are almost the same as the coordi- 
nates of Mt. Hopkins. Hence, case 1 is approximately the same as case 
3 with a second camera added to Mt. Hopkins and also case 1+ is approxi- 
mately the same as case 6 with a second camera added to Mt. Hopkins. 

It is recalled from the previous discussion on the effects of additional 
cameras at a particular observation station that adding a second camera 
to Mt. Hopkins in case 1 causes the north-south, vertical, and total 
pointing displacement errors to increase slightly. As was explained 
just previously, for three or more observation stations, especially with 
errors present, the intersection of the conical-like surfaces in space 
defined by the respective azimuth-elevation curves from the stations is 
not unique and that when a second camera is added to one station that 
station's data essentially receives a higher weighting and the solution 
is more constrained to lie on the surface from that one station. Hence, 
the results of Figures 44, 45, and 46 for the north-south, vertical, 
and total station displacement errors, respectively, are not as alarming 
as they first might seem. 

In the situation of the pointing displacement errors, however, 
when White Sands was added to a given station case not originally 
containing it these results did not occur. Hence, the north-south, 
vertical, and total station displacement errors are more sensitive to 
this phenomenon than their corresponding pointing displacement errors. 



127 


Resultant Displacement Errors for the 
Aircraft Station 

The resultant displacement errors , defined as the square root of 
the sum of the squares of the corresponding components of the pointing 
displacement errors and the station displacement errors, were calcu- 
lated. Then, the per cent differences between the resultant displace- 
ment errors and their corresponding pointing displacement errors were 
calculated. 

These resultant displacement errors and these per cent differences 
between the resultant displacement errors and their corresponding 
pointing displacement errors are shown in Tables 6, 7, 8, and 9 for 
the cases 1, 3, U , and 6, respectively. From these tables it is 
seen that the per cent differences between the resultant displacement 
errors and their corresponding pointing displacement errors are 
indeed significant. 

Aircraft Data Weighting Factors 

Since the coordinates of the moving aircraft contain uncertainties 
which are manifested in the triangulation as station displacement errors 
in addition to any pointing displacement errors, rendering resultant 
displacement errors whose per cent differences with the corresponding 
pointing displacement errors are significant - it was decided that the 
data from the moving aircraft should not be weighted as heavily as the 
data from the fixed, ground-based stations. Some scheme of weighting 
the data from the different observation stations needed to be devised. 



128 



Latitude 

(deg) 


E-W 

Error 

(km) 

% differ- 
ence with 
E-W 

Pointing 

Error 

only 

1 

32 


32 

5-31- 

32 

5-3** 

33 

5-35 ' 

33 

5-35 

33 

5-3? 

33 

5-37 

3 *t 

5.38 

3*t 

5.36 

3*t 

5-35 

35 

5-33 

35 

5-31 

35 

5-27 

36 

5.2l* 

36 

5.20 

36 

5-15 

36 

5-09 

36 

5.02 

37 

**•97 

37 

U .91 

37 

It . 85 

38 

it . 78 

38 

It. 71 

38 

it . 63 

38 ' 

It. 55 

39 

It. 1+8 

39 

U.UO 

39 

It. 33 

*t0 

It. 26 

*t0 

! It. 19 

*tl 

It. ll 

*tl 

lt.Olt 

*t2 

3.98 

*t2 

3.91 

**3 

3- 8U 

*t *t 

3-77 

It *t 

3-7 1 ! 

*t6 


■7 

7*t 

10 

*t3 

13 

00 

15 

**3 

17 

70 

19 

75 

21 

65 

23 

35 


% differ- 
ence with 
Vertical Vertical 
Error Pointing 
(km) Error 
only 


Total 

Error 

(km) 

% differ- 
ence with 
Total 
Pointing 
Error 
only 

81.1*9 

80.60 

79.63 

78.1*8 

■ 

77-18 

12 

75-59 

12 

7 *t. 01 

13 

72.32 

13 

70.52 

13 

68.50 

13 

66.51 

ll* 

6 it.i*o 

lit 

62.26 

l*t 

60.06 

15 

57.83 

15 

55 . 51 * 

15 

53.20 

' 16 

50.91 

16 

1 * 8.58 

16 

1 * 6.33 

16 

1 * 1*. 10 

17 

1 * 1.91 

17 

39.70 

17 

37.60 

18 

35.50 

18 

33.1*7 

18 

31.56 

19 

29.67 

19 

27.86 

20 

26 . ll* 

20 

2*t . 1*9 

21 

22.90 

. 21 

21.1*2 

22 

20.00 

23 

18.65 

2l* 

17.37 

21+ 

16.15 

25 

15. IT 

27 


















129 



Latitude 

(deg) 


E-W 

Error 

(km) 

% differ- 
ence with 
E-W 

Pointing 

Error 

only 

4.96 

27 

5-37 

33 

5-39 

33 

5-Ul. 

33 

5-43 

3>* 

5-U3 

31+ 

5-43 

31* 

5.1* It 

31* 

5.1*1* 

35 

5.1*2 

35 

5-1*0 

35 

5-38 

35 

5.36 

36 

5-33 

36 

5-30 

36 

5.2l* 

36 

5-19 

37 

5.11* 

37 

5-08 

37 

5.02 

37 

U . 96 

38 

It. 90 

38 

U. 82 

38 

1* . 76 

38 

1* . 68 

39 

1* . 60 

39 

1*. 53 

39 

1* . 1*5 

1*0 

1*. 38 

1*0 

1*. 30 

1*0 

1* . 23 

1*1 

l*.l6 

1*1 

l*. 09 

1*2 

1*. 02 

1*3 

3.95 

1*3 

3.88 

1*1* 

3.81 

1*1* ' 

3.78 

1*6 



% differ- 
ence with 

N-S 

N-S 

Error 

Pointing 

(km) 

Error 

only 


Vertical 

Error 

(km) 

% differ- 
ence with 
Vertical 
Pointing 
Error 
only 

91.22 

1* 

89.80 

1* 

88.15 

5 

86.22 

5 

81*. 05 

5 

81.57 

5 

79.00 

5 

76.25 

6 

73.39 

6 

70.36 

6 

67.31 

6 

61*. 19 

6 

61.06 

7 

57.90 

7 

51* . 78 

7 

51.67 

7 

1*8.60 

8 

1*5-60 

8 

1*2.68 

8 

39-88 

8 

37.16 

9 

34.56 

9 

32.04 

9 

29-67 

9 

27.38 

10 

25-22 

10 

23-21 

10 

21.29 

11 

19.50 

11 

17-83 

11 

16.28 

12 

lit. 84 

12 

13.50 

13 

12.26 

ll* 

ll.ll 

ll* 

10.06 

15 

9.08 

15 

8.25 

16 



% differ- 


ence with 

Total 

Total 

Error 

Pointing 

(km) 

Error 


only 






















Latitude 

(deg) 


-It 


5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 
21 
22 
23 
2 k 

25 

26 

27 

28 


E-W 

Error 

(km) 

% differ- 
ence with 
E-W 

Point ing 
Error 
only 

N-S 

Error 

(km) 

% differ- 
ence with 
N-S 

Pointing 

Error 

only 

Vertical 

Error 

(km) 

% differ- 
ence with 
Vertical 
Pointing 
Error 
only 

Total 

Error 

(km) 

% differ- 
ence with 
Total 
Pointing 
Error 
only 

7.62 

1*1 

JO. 03 

23 

105.18 

23 

105.93 

23 

1 . 6 k 

1*1 

13.51 

23 

103.56 

23 

10lt.72 

23 

7-69 

ll 

16.87 

23 

101.91 

2l* 

103.59 

2l* 

7-73 

111 

20.05 

2l 

99-95 

2l* 

102.23 

2 k 

7-77 

1*2 

23-03 

2l 

97.70 

2l* 

100.68 

2 k 

7-79 

1*2 

25.80 

2 k 

95.18 

25 

98.91 

25 

7.81 

1*2 

28.31 

25 

92.1*1 

25 

96.96 

25 

7.81 

1*2 

30.53 

25 

89.31* 

25 

9 k. Ik 

25 

7.80 

1*3 

32.53 

25 

86.21 

26 

92.1*8 

26 

7.80 

1*3 

31*. 25 

26 

82.92 

26 

90.05 

26 

7-77 

1*3 

35-71 

26 

79-51 

26 

87-51 

26 

7.72 

1*3 

36.83 

26 

75-88 

27 

8U. 70 

27 

7.69 

1*1* 

37.71* 

27 

72.32 

27 

81.95 

27 

1.66 

1*1* 

38. 1*6 

27 

68.81* 

27 . 

79-23 

27 

7.60 

1*1* 

38.88 

28 

65.21* 

28 

76.33 

28 

7-52 

1*1* 

39-01 

28 

61.57 

28 . 

73.27 

28 

7.1*5 

1*5 

38.97 

28 

58.01* 

28 

70.30 

28 

7.37 

1*5 

38.71* 

28 

5 1 * - 57 

29 

67.32 

29 

7.28 

1*5 

38.26 

29 

51.10 

29 • 

61*. 25 

29 

7.18 

1*5 

37.68 

29 

1*7-79 

29 

61.28 

29 

7.09 

1*6 

36.96 

29 

1*1*. 62 

29 

58.37 

30 

7.00 

1*6 

36.09 

30 

1*1.51* 

30 - 

55.1*7 

30 

6.88 

1*6 

35.07 

30 

38.53 

30 

52.55 

30 

6.77 

1*6 

33.99 

30 

35-70 

30 

1*9-76 

30 

6.65 

1*6 

32.79 

30 

32.98 

31 

1*6.98 

31 

6.53 

1*7 

31. 5l* 

31 

30.39 

31 

1*1*. 28 

31 

6.1*1 

1*7 

30.26 

31 

27-98 

31 

1*1.71 

31 

6.28 

1*7 

28.93 

31 

25.68 

31 

39.18 

32 

6.16 

1*7 

27.57 

32 

23.52 

32 

36.76 

32 

6.03 

1*8 

26.23 

32 

21.52 

32 

3!+. 1*6 

33 

5.92 

. 1+8 

21.86 

32 

19-63 

33 

32.23 

33 

5-79 

1*8 

23.51 

33 

17. 88 

33 

30.10 

33 

5.66 

1*9 

22.17 

33 

16.25 

33 

28.07 

3l* 

5.5l* 

1*9 

20.86 

3l 

ll* • 75 

3 k 

26. ll* 

3l* 

5.1*1 

1*9 

19-57 

3l* 

13-35 

31* 

2I+.32 

35 

5-29 

50 

18.32 

35 

12.07 

35 

22.56 

35 

5.16 

50 

17-11 

35 

10.87 

35 

20.92 

36 

5.08 

51 

16.12 

36 

9.90 

36 

19.58 

37 















131 


Latitude 

(deg) 


-3 

-2 

-1 

0 


5 

6 

7 

8 
9 

10 

11 

12 

13 

lit 

15 

16 

17 

18 

19 

20 
21 
22 

23 

24 

25 

26 

27 

28 


Table 9 - Case 6 resultant displacement errors. 


E-W 

Error 

(km) 

% differ- 
ence with 
E-W 

Point ing 
Error 
only 

N-S 

Error 

(km) 

% differ- 
ence with 
N-S 

Pointing 

Error 

only 

Vertical 

Error 

(km) 

% differ- 
ence with 
Vertical 
Pointing 
Error 
only 

Total 

Error 

(km) 

% differ- 
ence with 
Total 
Pointing 
Error 
only 

7-70 

1+1 

10.57 

14 

110.53“ 

14 

111.31 

l4 

7-73 

111 

14.23 

14 

108.84 

l4 

110.03 

15 

7-77 

111 

17.75 

15 

107.01 

15 

108.75 

15 

7-81 

1+2 

21.07 

15 

104.86 

15 

107.24 

15 

7.84 

1+2 

24.18 

15 

102.39 

15 

105.50 

16 

7.86 

1+2 

27.05 

16 

99-66 

16 

103.57 

16 

7.87 

1+2 

29.66 

16 

96.68 

16 

101.43 

16 

7.86 

1+3 

31.96 

16 

93.40 

16 

99-02 

16 

7.87 

1+3 • 

34.01 

17 

90.03 

17 

96.56 

17 

7.85 

*13 

35-79 ' 

17 

86.52 

17 

93.96 

17 

7-83 

1+U 

37.28 

17 

82.91 

17 

91.24 

18 

7-77 

lilt 

38.42 

17 

79-08 

18 

88.26 

18 

7 - 7l+ 

uu 

39.35 

18 

75-32 

18 

85.34 

18 

7-70 

l+U 

40.05 

18 

71.62 

18 

82.42 

19 

7.65 

1+5 

40.46 

19 

67.83 

19 

79.35 

19 

7-57 

1*5 

40.58 

19 

64.00 

19 

76.16 

19 

7.50 

1+5 

40.52 

' 19 

60.29 

19 

73.03 

19 

7-1+1 

1+5 

40.21 

19 

56.58 

19 

69.81 

20 

7.33 

1*5 

39-75 

20 

53.00 

20 

66.69 

20 

7.23 

1+6 

39-12 

20 

49.59 

20 

63.59 

20 

7. lit 

46 

38.35 

20 

46.26 

20 

60.53 

21 

7-0*1 

46 

37.45 

21 

43.07 

21 

57.50 

21 

6.93 

46 

36.37 

21 

39.94 

21 

54.46 

21 

6.82 

46 

35-24 

21 

37-00 

21 

51.55 

22 

6.70 

47 

33.99 

22 

34.17 

22 

48.66 

22 

6.58 

47 

32.69 

22 

31.47 

22 

45.85 

22 

6.1+6 

47 

31.35 

22 

28.97 

22 

43.17 

23 

6.33 

• 47 

29.96 

23 

26.58 

23 

40.55 

23 

6.21 

48 

28.55 

23 

24.34 

23 

38.02 

23 

6.09 

48 

27-14 

23 

22.26 

23 

35.63 

24 

5-95 

48 

25.72 

24 

20.30 

24 

33-31 

24 

5.81+ 

48 

24.31 

24 

18.48 

24 

31.09 

25 

5-71 

49 

23.16 

25 

16.79 

25 

28.98 

25 

5.58 

49 

21.55 

25 

15.23 

25 

26.97 

26 

5.U6 

49 

20.21 

25 

13.78 

25 

25.06 

26 

5-32 

50 

18.91 

26 

12.45 

26 

23.26 

27 

5.19 

50 

17-64 

26 

' 11.22 

26 

21.55 

27 

5.12 

51 

16.59 

27 

10.19 

27 

20.12 

29 





















132 


The per cent difference for the i — case, j — component, and 

NA — point is denoted by PD. , where i = 1, 3, 4, 6 and denotes 

1 ,J ,1'iA 

the particular combination of observation stations employed and j = 
east-west, north-south, vertical, or total and denotes the particular 
displacement error of interest and NA = 1, 2,..., NT, where NT is the 
total number of solution points. The average per cent difference 


(PD. ) for the i — case and the j — component is then 
1,0 AV 




NT 
E PD 
NA=1 


i , i ,NA 


AV 


NT 


(193) 


If the weighting factors for the data from all of the fixed, ground- 
based observation stations are chosen as unity, then it seems appropri- 
ate to choose for the aircraft data the weighting factors w. given by 

1 * J 


\ , - 1 - iro .) > 

x » <J A d AV 


(194) 


where i = 1, 3» 4, 6 and j = east-west, north-south, vertical, or total 

and (PD. ) is given by equation 193. Table 10 lists the aircraft 
1,J AV 

data weighting factors for these displacement errors and these station 

cases containing the aircraft, as calculated using equation 194. 

The median per cent difference (PD. ) , the per cent difference 

1,J MED 

for the solution point with latitude equal to nine degrees, is identical 
to the average per cent difference in nine instances and differs only 
by one per cent from the average per cent difference in the remaining 
seven instances (the four cases times the four components give the 



Table 10: Aircraft data weighting factors 

for the displacement errors for 
different station cases. 


Case # 

E - W 

N - S 

Vertical 

Total 

1 

.63 

.85 

.83 

.83 

3 

.62 

• 91 

■ 91 

• 91 

k 

• 55 

.71 

.71 

■ 71 

6 

• 5k 

.80 

.80 

.80 


oo 

00 












13h 


sixteen instances). Hence, the median per cent difference can be used 
instead of the average per cent difference, thereby omitting the calcu- 
lation of the per cent differences for (NA-l) of the solution points 
and the calculation of the average per cent difference. Hence, a second 

equation for the aircraft data weighting factors w is then 

1 » J 


w i,j ■ 1 - (PD i.j> 


MED 


(195) 


Also, for any given case and any given component, the weighting 
factor as calculated by equation 19^ is numerically equal to the 
average of the ratios of the pointing displacement errors to their 

"bll 

corresponding resultant displacement errors. If PE. denotes the j — 

b/ll 

component of the pointing displacement error for the i — case and RE. 

1 9 J 

I.T. 

denotes the j — component of the resultant displacement error for the 

"til 

i— case, then a third equation for the aircraft data weighting factors 
is 

r PE. 


w 


i > J 


JoJ. 

RE. , 

i.J, 


(196) 


AV 


where the average is again taken over the NT solution points. 

Just as the median per cent difference is almost identical to the 
average per cent difference, the median of the ratios of the pointing 
displacement errors to their corresponding resultant displacement 
errors is almost identical to the average of the ratios of the pointing 
displacement errors to their corresponding resultant displacement errors. 
Hence, a final equation for the weighting factors for the aircraft data 



135 


is just 

where PE. is the j— component of the pointing displacement error for 
i » j 

the i— case, RE. is the j— component of the resultant displacement 

1 > J 

error for the i— — case, i = 1, 3, ^ 6 and denotes the particular com- 
bination of observation stations employed, j = east-west, north-south, 
vertical, or total and denotes the particular displacement error of 
interest , and the subscript MED means that the ratio is for the solution 
point with latitude equal to nine degrees. 

Summary and Conclusions 

The single-point two-station triangulation problem, line and 
multistation triangulation considerations, and the LaRC triangulation 
method - developed specifically for the Barium Ion Cloud (BIC) Project 
at the NASA, Langley Research Center - were discussed. 

Expressions for the four pointing displacement errors, the three 
pointing slope errors, and the pointing curvature error in the trian— 
gulation solution due to a probable error in the lines-of-sight from 
the observation stations to the points on the cloud were derived. For 
a probable error of 0.01 degrees in the lines-of-sight, the pointing 
displacement, slope, and curvature errors were plotted as functions of 
the latitude for the nine different combinations of the observation 
stations chosen for the BIC Project to determine the effect of the 
number and location of the observation stations on these pointing 



MED 


( 197 ) 



136 


errors. It was concluded that the pointing errors are the smallest for 
the five— station case and are the largest for the two— station case, with 
the combinations of four and three stations giving intermediate values. 

The four pointing displacement errors were plotted for comparison 
on a single plot and also the three pointing slope errors were plotted 
for comparison on a single plot as functions of the latitude for the 
five-station case. It was concluded that the vertical component is 
the dominant component of the total pointing displacement error through- 
out the latitudinal region of - 9° to 15°, whereas the north-south 
component is the dominant component of the total pointing displacement 
error over the latitudinal region of 16° to 28°. It was concluded 
that the latitude component is the dominant component of the total 
pointing slope error. 

The pointing errors were examined for the observation durations 
of 10,000 sec, 6,000 sec, and 1,000 sec. It was concluded that the 
east-west pointing displacement error and the longitude pointing slope 
error practically remain constant throughout the three observation 
durations; that the north-south pointing displacement error increases 
toward the lower-latitude and and decreases toward the higher-latitude 
end over the 1,000-sec observation duration and first increases and 
then decreases toward the lower-latitude end and decreases toward the 
higher-latitude end over the 6,000-sec and 10,000-sec observation 
durations; and that the vertical and total pointing displacement errors, 
the latitude and total pointing slope errors, and the pointing curvature 
error increase toward the lower-latitude end and decrease toward the 



137 


higher-latitude end over all three of the observation durations. 

The pointing errors were plotted as functions of the latitude 
for different release points varying in longitude to determine the 
effect of east-west cloud drift on the pointing errors. It was con- 
cluded that for the chosen BIC nominal release point - latitude = 
9.229°, longitude = - 75-000°, and altitude = 31,633.008 km - the point 
ing errors increase as the cloud drifts eastward into the longitudinal 
regions of - 63° and - ^9° and decrease as the cloud drifts westward 
into the longitudinal region of - 91°; as the cloud drifts farther 
westward into the longitudinal, regions of - 105° and - 119°, the 
pointing errors increase or decrease depending on the cloud's position 
in latitude. 

The pointing errors were plotted as functions of the latitude for 
different spacings between the points along the azimuth-elevation 
curves from the observation stations to determine the effect of the 
number of input data points on the pointing errors. It was concluded 
that the spacing between the input data points along the azimuth- 
elevation curves from the observation stations should be no greater 
than 1.12 degrees. 

The pointing errors were plotted as functions of the latitude for 
the two-station case with an extra camera added to the Mt. Hopkins 
station. It was concluded that for the two-station case adding an 
extra camera to Mt. Hopkins decreases the pointing errors. The 
pointing displacement errors were calculated for the five-station case 
with from one to eleven extra cameras added to Mt. Hopkins. It was 



138 


concluded that for the five-station case adding extra cameras to Mt. 
Hopkins increases the north-south, vertical, and total pointing dis- 
placement errors. 

The four station displacement errors in the triangulation solution 
due to a probable error of three kilometers in the position of the 
moving Wallops aircraft station were calculated, using an analogous 
procedure to that used for calculating the four pointing displacement 
errors. The station displacement errors were plotted as functions of 
the latitude for the station cases containing the Wallops aircraft. It 
was concluded that the north-south, vertical, and total station dis- 
placement errors increase when an extra fixed station whose coordinates 
are close to those of Mt. Hopkins is added. The resultant displacement 
errors, which are the resulting errors of the corresponding components 
of the pointing and station displacement errors, and the per cent 
differences between the corresponding resultant and pointing displace- 
ment errors were calculated f<!>r the aircraft. It was concluded that 
the station displacement errors were significant enough to warrant 
reduced weighting of the aircraft data. Expressions for the weighting 
factors were derived, and the weighting factors for the Wallops air- 


craft data were calculated. 



139 


LIST OF REFERENCES 


1. Justus, C. G. ; Edwards, H. D. ; and Fuller, R. N. : Analysis 

Techniques for Determining Mass Motions in the Upper Atmosphere 
from Chemical Releases. AFCRL-6U-I87 > 196^, pp. 1-16. 

2. Whipple, Fred L. ; and Jacchia, Luigi G. : Reduction Methods for 

Photographic Meteor Trails. Smithsonian Contrib. Astrophys., 
vol. 1, no. 2, 1957 5 PP* 183-206. 

3. Hogge, John E. : Three Ballistic Camera Data Reduction Methods 

Applicable to Reentry Experiments. NASA TN D-H26O, 1967* 

k. Long, Sheila Ann T.: Analytical Study to Minimize the Triangulation 

Error for an Idealized Observation Site Arrangement. NASA TN in 
preparation. 

5. Long, Sheila Ann T. : Comparison of Three Triangulation Methods 

Applicable to the Barium Ion Cloud Project. NASA TN in 
preparation. 

6. Fricke, Clifford L. : Triangulation of Multistation Camera Data 

to Locate an Elongated Barium Cloud. NASA, Langley Research 
Center computer program, 1971* 

7. Long, Sheila Ann T. : A Transformation from Geocentric to Geodetic 

Coordinates and Vice Versa in Powers of the Earth's Flattening. 
NASA TN in preparation. 



Appendix. Transformation from Latitude, Longitude, Altitude 
to Azimuth, Elevation, Range Coordinates 

If the geodetic coordinates of an observation station and of a 
point in space are known, the line-of-sight from the station to the 
point in space in azimuth, elevation, and range coordinates can be 
computed. 

The coordinate system of Figure kj is a local coordinate system 
centered at the station S. The x axis points east, the y axis points 
north, and the z axis is vertical (or radial). The azimuth az is meas- 
ured in the north-east plane, the elevation el is measured out of the 

north-east plane, and the range r is the distance from the station S 

8 > 

to the point P in space. 

First, the geodetic coordinates of the station and of the point 
in space are transformed to geocentric coordinates according to 
equations derived in reference 7* 

The latitude, longitude, and range of the station are denoted 
"by 4>a> 6111(1 r o> respectively; the latitude, longitude, and 

b b b 

range of the point in space, by $ , 0p, and r p , respectively. 

From Figure U8 it is seen that the rectangular coordinates of 
the station S - denoted by x g , y g , and z g - 


are 





143 


x = r cos d) cos 0 (198) 

S S S S 

y s = r g cos 4> s sin 0 g (199) 


z g = r g sin 4> g (200) 

And, also from Figure 48, the rectangular coordinates of the point P 
in space - denoted by x p , y p , and z p - are 


Xp = r p cos <{> p cos 0 p (201) 
y p = r p cos <j> p sin 0 p (202) 
z p = r p sin <j> p (203) 


The differences - denoted by x„ , y . and z - between the 

as, a 

respective x, y, z components of the point in space and the station 
are just 


X a = *P - X S 


(204) 



(205) 


z 

a 


z„ - 


(206) 



Ihk 


The azimuth az and the elevation el of the line-of-sight from the 
observation station to the point in space are given by 



(207) 



( 208 ) 


where 


x, = - x sin 0 C + y cos 0,, (209) 

t a b a b 

y = - x & sin 4» g cos 0 g - y & sin <|>g sin 0 g + cos $ g (210) 

z = x & cos <|)g cos 0g + y & cos <J>g sin 0g + z & sin <t>g (21l) 

And, the range r of the line-of-sight from the observation station 

8 . 

to the point in space is given by 

„ „ 1/2 
r = (x? + y? + z 2 ) 


( 212 )