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U.S. DEPARTMENT OF COMMERCENational Teciwicai Information Service
AD-A031 343
Stereographic ProIectionin the Joint.Surveillance System
Mitre Corp Bedford Mass
Sep 76
BestAvailable
Copy
307055
ESD-TR-76-169 MTR-3225
STEREOGRAPHIC PROJECTIONIN THE
J JOINT SURVEILLANCE SYSTEM
SEPTEMBlER 1976
Prepared fo~r
DEPUTY FOR SUIRVE'!Al ' AND tNAVIGATION -1's InWIWELECTRONIC SYSTEN q D11,11IONAIR FORCE SYSTEM. COMW AND
UNITED STATXS AM FORCE-A Hasmcoi Air Poe Due, Bedford, Massachw~uset
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VOCToS9 a
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Project No. 6620__________________ Irepr&J by
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S. W,,II MCI1 A.Sl) WE* . WEI~~orl, LTC, Us"?Cief)5 HioM Eng & To-at Divioion Chief, Setwor Enig DlvisionJoint Surveillance Sjya h-og oI'c Jaint Surveillance SY4 Prog Ofc
FOR TH~ OOAD~
JILYSELL If. WOESME, GS-14Deputy Syritem Progran DirectorJoint Surveillance Syv Prog Oft
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Hanscorn Air Force B~ase, Bledford, MA 01731 07a_____________147UONFiyQkiNG AGECY NAME 4ADORE$Sta 1f..ee* ~~Ol~ Offtc)- -r3S e-CURITV CLASS. (GD tt. off
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4 17 01yo TtolSACMEN? 101 #A* ebtioeI e"E..d too Mae* 20. It W#*"*#~t~ PN .V4.$IO
IS* &84J~PtMMVARY sioves
is 4(ty %allot rC*R~fftWq """o At,-~ it ?104ebprk* i14d ien"for kv blhcib "V4410)
.CK)RDlNAItE CONVERISION AND TflANSPOIA'IIONNEFTTED RADAR SYSTEMRADAR DATASTEREO)GRA PIIC IPROJIEC1ION
ZOx iSYPIACT (Coo,~. onto.049 of6 a" 06Ifi ord block~t ""*IW&*0
The stereographic projeiction process consists of using slant range, azimiuth and heightInformation to obtain radar coordinates, aid transformation of thie radar coordinatisto obtain coordinates on a common plane.
* This report describe-s analysis that was performed on the stereographic projectionprocess. The results indicate that the SAGE / BIC equations for 9tereographicground range produce unacceptably large registration errors when extended to a
00 '0" 1413 EDITION or I -aves, Is oOsoLeICVCIASII
SECUAITV CL ASSIOICATION OP~ THIS PAGE (Mfes' Date Eiftfeso
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE(N'Ten Dat(a Entetd)
20. A13TRACT (Concluded)
large region. The results further Indicate that the error can be corrected by asimple modification of the SAGE/ BUIC equations.
INLS$FEti i' k10M4MPW I-.
ACKNOWLEDGMENTS
This report has been prepared by The MITRE Corporation underProject No. 6620. The contract is sponsored by the ElectronicSystems Division, Air Force Systems Command, Hanscomi Air Force Base,Massachusetts.
TABLE OF CGiiEWTS
PageLIST OF ILLUSTRATIONS 5
LIST OF TABLES 6
SECTION I INTRODUCTION 9
SECTION II EFFECT OF THE CONFORMAL SPHERE ON STEREO-GRAPHIC PROJECTION 11
STEREOGRAPHIC PROJECTION 11
THE RADAR PLANE 13
DETFR4INATION OF STEREOGRAPHIC GROUND RANGE 13
APPROXIMATIOPS TO GROUND RANGE 16
The Series Approximation 17
The First Order Approximation 17
The 35. Approximation 18
T"he Current SAGE/!MTC Approximation 18
EARTH 40DEL 19
Mapping from Ellipsoid to Sphero 19
Radius of the EArth to a Radar Site 21
TIE C(?OO; COORI)INAI'E PLANE 22
Radar Coordinatea on tih Comon Plane 23
Angular Rotation 24
Transformarttn of Radar Coordinateg onkite Coon Plane 24
3 e g p Mu
TABLE OF CONTENTS (Concluded)
Page
THE CONFORMAL SPHERE 25
Scale Factor - Ellipsoid to ConformalSphere 25
Scale Factor - Conformal Sphere toPlane 26
Calculation of the Radius of theConformal Sphere 28
Effect of E /E 30c s
SECTION III EFFECT OF THE SAGE/BUIC AND JSS STEREOGRAPHIC
GROUND RANGE EQUATIONS ON REGISTRATION 33
- -•INTRODUCTION 33
SIMfULATED RADAR DATA 33
SECTION IV CONCLUSIONS 37
REFERENCES 9
APPENDIX I: GENERATION OF SLANT RANGE AND AZIMUTH DATA 41
APPENDIX I1: SIMULATED DATA FOR THE SEVEN JSS REGIONS 43
4
LIST OF ILLUSTRATIONS
Figure Number Page
I STEREOGRAPHIC PROJECTION 12
II COORDINATE AXES ON RADAR PLANE 14
III RADAR OBSERVATION GEOMETRY 15
IV A CROSS SECTION OF THE ADOPTED FARTH MODEL 20
BT Region Center, Radar Site, and Aircraft LocationsNortheast JSS Region 47
BII Region Center, Radar Site, and Aircraft Locations -
Northwest JSS Region 51
Bill Region Center, Radar Site, and Aircraft LocationsSoutheast JSS Region 55
BIV Region Center, Radar Site, and Aircraft Locations -
Southwest JSS Region 59
IBV Region Center, Radar Site, and Aircraft Locations -
Eastern Canadian JSS Region 63
BVI Region Center, Radar Site, and Aircraft Locations -
Western Canadian JSS Region 67
BVII Region Center, Radar Site, and Aircraft LocationsAlaskan JSS Region 71
4
5
LIST OF TABLES
Table Number Page
I Earth Radius Versus Latitude 22
II Scale Factor- Ellipsoid to Sphere Versus Latitude 26
III Scale Factor-Conformal Sphere to Plane VersusAngular Separation 27
IV E Versus Region Size 30c
V Difference Between Ground Range Equations 8 and 9Versus Region Size 31
VI WORST CASE REGISTRATION ERRORS IN EACH JSS RECTON 35
BI SITE DATA - NORTHEAST JSS REGION 45
Bi SIMULATED RADAR DATA - NORTHEAST JSS RECTON 46
DIII REGISTRATION ERRORS - NORTHEAST JSS RkEON 48
8IV SITE DATA - NORTHWEST JSS REMION 49
BV SIMULATED RADAR DATA -. NORTHWEST JSS REGrION 50
BVI RE;ISTRATION ERRORS - NORTHIJEST JSS REGION 52
BVII SITE DATA - SOUT11FAST JSS REWION 53
UVIII SIMULATEI) RADAR DATA - SOUTHEAST JSS REGION 54
DXx REGISTRATION ERRORS - SOUT1EAST J,1S RE~ZtON 56
DX SITE DATA - SOUTHWEST JSS RErION 57
BXI SIMULATED RADAR DATA - SOITOW4ST JSS REGION 58
1LXII REGISTRATTION ERRORS - SOUTHWEST JSS REG:ION 60
BXIII SITE DATA - EASTERN CANADIAN JSS RE;ON 61
UXIV SIMULATED RADAR DATA - EASTERN CANAI)IAN .JSS RWTION 62
6
LIST OF TABLES (Concluded)
Table Number Pag
BXV REGISTRATION ERRORS - EASTERN CANADIAN JSS REGION 64
BXVI SITE DATA - WESTERN CANADIAN JSS REGION 65
BXVII SIMULATED RADAR DATA - WSTERN CANADTAN JSS REGION 66
BXVIII REGISTRATION ERRORS WESTERN CANADIAN jSS REGTON 68
BXIX SITE DATA - ALASKAN JSS REGION 69
XX SIMULATED RADAR DATA - ALASKAN JS REGION 70
DxXI REGISTRATION ERRORS - ALASKAN JSS REGION 72
I7
SECTION I
INTRODUCTION
In air defense and air traffic control systems, data from the
system radars are stereographically projected onto a common coordi-
nate plane for presentation to system operators. The stereographicprojection of radar data involves two steps; stereographic projection
using slant range, azimuth and height information to obtain polar
coordinates in a plane of projection centered at the radar site, and
transformation of the radar coordinates into cartesian coordinates
on a common coordinate plane.
The steroopraphic projection avd transformation process is
thevaticallv complex. Because of the comoutationalt coplexity,
several assumptions and approximations have beetn made to expedite
processing time ulthout unduly sacrificiu accuracy. The presentSA MG=C Lequatloou for comptting radar vordtnattg, a lthoughsattsf-uctorv for their intended usmpe, Itntrodu~e r!!aceptable rogi,'-
:, !- ga.ion errors when extended to large, regttms aa wil~l be ecuntered4o. in the Joint Survellance S::tem (JSS).
11ila report describes an-alyglfA that was verfo0mtd oti the stere-.
graphic pro oproces. Equations for obtai ing stereographic
ground range are derived, Ili# derivation tdicates that the sArCd/-' fUIC ground range etquation lacks a scale factor vital to proper
regittraton in large regions. The scale factor Is a function of
* the radius of the earth at a radar site and the radius of the con-
formal sphere.
9ecedg page
SECTI0N II
EFFECT OF THE CONFORMAL SPHERE ON STEREOGRAPHIC PROJECTION
STEREOGRAPHIC PROJECTION
Stereographic projection is a method for mapping points in
space onto a plane tangent to a sphere. This sphere is termed the
conformal sphere and its radius is arbitrary. Figure I depicts the
projection geometry for a cross section of the sphere. The cross
section is obtained by passing a plane through the center of the
sphere, the aircraft, and the point of tangencv. The intersection
of this plane and the sphere is a great circle. Mapping of point A
in space onto the tangent plane BB' results in the stereographic
ground range DT. DT is obtained by passing a line from point 0,
opposite the point of tangency D, through point P, the point of
projection, and intersecting the tangent plane. Point P is the
intersection of the line containing point A and the center of the
earth, point C, and the great circle. If angle DCP is designated as
V1, then angle DOP equals i/2 since DC=CO=CP. The stereographic
ground range -DT is determined as follows.
DT DO tan (1)
If DT is defined as R and DC, CO, CP are defined as Ec, then
equation (1) takes the form:
R 2E tan
= 2E 1 - . • . -C o j (2)
The stereographic ground rang, R is therefore directly proportional
to the radius of the conformal sphere E.1 c
~Preceding page blank
F 1ur . STEREOGRAPHIC PROJECTION
12
THE RADAR PLANE
Radar data are stereographically projected onto a plane centered
at the reporting radar site and tangent to the conformal sphere. The
plane will be termed the radar plane. The coordinate axes of the
radar plane are oriented such that the positive Y axis is directed
toward true north and the positive X axis towards east. Figure II
shows the orientation of the coordinate axes on the radar plane. An
aircraft's location on the plane is expressed in polar coordinates.
The range R is the stereographic ground range, and the azimuth angle
0 is the azimuth of the radar return. Azimuth angles are measured
clockwise from the positive Y axis.
DETERMINATION OF STEREOGRAPHIC GROUND RANGE
Radar slant range, height of the aircraft above sea level and
elevation of the site above sea level are used to determine the
stereographic ground range o: an aircraft on the radar plane. The
angle 0 between the radar site, the center of the earth and the
aircraft are used in equation (2) to calcqlate the stereographic
ground range. The angle 0 can be calculated if the earth is assumed
to be spherical. Figure III illustrates the radar geometry for a
cross section of a spherical earth where E is the radius of thes
earth, h is the site elevation, S is the measured slant range and H
is the aircraft height. From Figure III the angle 0 mav be calculated
from the law of cosines.
2 2 2S - (E + h) + (E + H) -2(E + h)(E + H) cos
2coiP- +2(E 5 + h)(E + H)
F 2 (3)
13
RADA~TR SIT £AAD
'4G OFTNGN
F [guts U COORDINATE AXES ON RADAR PLANE
14
Ss TARGET
Figure flRADAR OBSERVATION GEOMETRY
15
where:
F2 S2 - (H -h) 2
Substituting the results of equation (3) into equation (2),
the ground range R may be obtained as follows.
F2 11/2R=22
4(E + h)(E + H) -F
E F E 2 j1/-I+ h +2]It/F (4)
E 2 2]
Reference 1 page 7 presents the SAGE/BUIC formulation of the ground
range wherein it is assumed that Ec and Es are equal and thus cancel. It
will be shown that for the large regions that will be encountered in the
JSS system, this assumption results in unacceptable registration errors.
APPROXIMATIONS TO GROUND RANGE
The accuracy of typical common digitizer search radar out-
puts is 0.25 nmi in range and 0.18* in azimuth. Registration
errors are a combination of data errors, radar site location errors,
and errors due to approximations in the stereographic projection
process. Since equation (4) is computationally complex, an approxi-
mation which does not unduly sacrifice accuracy is used to expedite
processing time. A maximum error, induced by approximation,of 0.18
nmi provides a reasonable compromise between processing requirements
and registration accuracy. Four approximations are presented in this
section; the series approximation, the first order approximation,
the JSS approximation and the current SAGE/BUIC approximation.
16
The Series Approximation
Equation (4) may be written in the following form.
R h c F2
(5)E + - +E2 -" /
aE 2 2a E 45
EEs s 4E
The term within brackets in the denominator of equation (5) may then
be expressed by the following series expansion.
( +" 2(1+ X),l i+nx + n~ -UL+2!
where:
-+ a2
x . Q+sh E and
IIThe maximum value of x encountered in a JSS region is 0.005.
Since x < < 1, all but the first order term of the series may beignored and equation (5) may be expressed as follows.
EFC
R ct I t + h - F2)\ (6)Es + 2E+ + 28E
2Es E ,
The First Order Approximation
The bracketed term in the demoninator of equation (6) contains
a first order term and two second order terms. The maximum value ofH+ hW7 is 0.00248, the maximum value of 2 is 0.000000721, and thes 2E
maximum value of is 0.000424. Since the second order terms are8E
2
• S
smaller than the first order term, they may be neglected and equation
(6) may be expressed as follows.
17
E FRc (7)
TheJSApoximtion
By replacing the aircraft height term (H) in the demoninator of
equation (7) by a constant, equation (7) may be expressed as follows.
E FR a C
(,/2 )+ hi (8)
where:
His the maximum expected aircraft altitude equal to 100,000 ft.
Equation (8) may be expressed tn the following form.
where.
Ibin is a particularly od proximation since the stereographle
ground range to obtained from a stwaple scale multiplication of the
quantity F. This areatly detreses the tiae reured to process
radar returns. Th constant C to adaptatton defined on a site-by-
the E is assume to be te sam as E *equation ()Is expressed
as follow.
+(0/2)+)
18.
" ERTI MODEL
The earth is not a perfect sphere. Therefore, for precise
calculations of stereographic ground range, a model for the geometric
shape of the earth must be adopted. An appropriate first order
representation Is an ellipsoid. The ellipsoid is generated by
.revolving an ellipse about Its sesiminor axis. The earth model can
therefore be specified by its sealmajor axis or equatorial radius
E and the eccentricity e. A cross section of the adopted earthq
model is shown in Figure IV. The eccentricity is defined as follows.
e 2 Zf- f 2 (10)
where:E -E
E
E is the seaminor axis or oolar radius
The International Ellipsoid of 1924 will be used for the purpose of
this r-port- thus, E equals 3444.054 valt ad equals .00672267.
• - IaPpt fra ~Itti~oid. ;to spher~e
The stereographie projection equations have been derived for a
sphere. It is therefore necessary to transform points on or above
the ellipsoid to points on or above the sphere. The transformation
• uAt be conforsal, I.e., angle preserving, if the final stereographic
projection i to be conformal. Reference 2 page 86 derives the re-
lAtionahip between the ellipsoid and the conforiwl sphere. The
sapping of polnts on or above a location on the ellipsoid onto the
coeformal sphere is petfotaed as follows.
19
ILI'i
hi N 0
.a.
-1 -
w
060
owI-
200
+ ta + I 1 -esinLi e/2
In these equations:
L, A are the geographical latitude and longitude of the point
on the ellipsoid
0, A' are the latitude and longitude of the corresponding
point on the conformal sphere. The latitude 0 is
called the conformal latitude.
e is the eccentricity of the earth.
Since the mapping process involves only a transformation of latitudec,
the stereographic projection equations are valid for the ellipsoid
if the conformal latitude (0) is used in place of the geographic
latitude (L).
Radius of the Earth to a Radar Site
In determination of the stereographic ground range, a spherical
earth was used to compute the angle that subtends the radar site
and the target. Since the earth is actually modeled by an ellipsoid,
the use of a spherical earth in calculating stereographic ground
range without introducing corrections to slant range and heigh% to
allow for the conformal projections will introduce a certain auount
of error. To minimize this error, the radius of the spherical earth
E is set equal to the distance from the center of the ellipsoid tosthe surface of the ellipsoid at the radar site as showu in Pigure
IV. The distance E is calculated as follows.8
where:
X N cosL
Y - (1 - e2) N siaL
21
EN =
L is the geographic latitude.
Table I gives the radius of the Earth at latitudes between 0 and
90 degrees.
Table T
Earth Radius Versus Latitude
Geographic Latitude Earth Radius (nnt)
0 3444.054
100 3443.707
20* 3442.;)830 3441.17340' 3439,286500 3437.27360" 3435.37570O 3433.824800 3432.810900 3432.458
Calculations show that the reaultnag error Is greateat for
a target north or south of the radar and does not exceed + 0.03 nami
at a range of 250 nat.
THE CO."O0 COORDINATE PLANE
In a large air surveillance regicn such as vill be encountered Iin the JSS, several radars are linked together in order to display a
composite air surveillance picture. Ir is therefore eceasary to
trauaform coordinates an the itdividual rauar planes to coordinates
on a comon plane. The traasforwtion process requires that the
coordinateS of the radars on the como place and tht angular ro-
tations betueen the radar planes and the common plane be known.
22
The origin of the common plane or region center is defined as
the center of the smallest circle that will circumscribe all radars
tied into the air surveillance region. The common plane is tangent
to the conformal sphere. The coordinates axes are oriented suchthat the positive Y axis is directed towards true n~orth and the
positive X axis is directed towards east. Coordinates on the common
plane are expressed in cartesion coordinates.
Radar Coordinates on the Cormon Plane
A radar site or other known location can be stereographicallv
projected onto the comon coordinate plane. Reference 3 page 53derives the equations necessary to project a point or. the conformawlsphere onto the co=-on coordinate plone. The rectangular coordinativs
of a radAr Site or other ktown location X Yr an the comon plane
are obtained as follow.
..... r Sio o + co sinoso COSA0 02
ZE~ + (14)+.sinosn o CSO44CO5AX
where.,
0, Z are the conforl latitude and longitude of OW
point to t-O Projected
i region center
AA A A0 If longitudes are %easured posittivev est of the-
prime meridian
S1 0 if longitude# are measured positive east of the
S"prime meridian
23
Angular Rotation
As indicated in reference 3 page 6, an angular rotation of the
radar plane with respect to the common coordinate plane is necessary
for the transformation process. The effect of the rotation is to
make the axes of the radar plane more nearly parallel to the axes of
the common plane. The angle of rotation is shown i- reference 3 to
be:
n[ - (sins + sino)sn3(r COSCOSO + (I + sin -sino)COSAX
Transformation of Radar Coordinates an the Common-Plane
The equations for transformation of radar coordinates to
p coordinates in the coson coordinate plane are derived tn reference
3 pages 3 - 15. The exact transformation equations involve an in-
I .finite series. To lessen the processing requirement without unduly
sacrificin accuracy a second order approximation is used. Rectangular
coordinates X., Y are obtained as follows.
SX X tn(O + 0) + ARt(2(8 +16) -y]) (16)
SV V + K(ICoSO + 0) + A.cos[2(0 + 0) -j)
where: . 2
+
rf A4g2
c
2 2 1/2• " (Xr + Yrr r r
24L1
L- R is the stereographic ground range0 is the azimuth angle measured clockwise from north at the
radar site
Xr, Y are the coordinates of the reporting radar on therr
common plane
Since the coordinates of an aircreft on the common plane are a
: - : function of the stereographic ground range R, any error in the
stereographic ground range will appear as a misregistration on the
common coordi;iate plane.
THE CONFORMAL SPHERE
The common coordinate plane and all radar planes are tangent to
the conformal sphere. The radius of the conformal sphere is arbi-
trary, but is chosen to minimize the scale errors that will be en-
countered in the air surveillance region. Scale errors result in
r. .ing from the ellipsoid to the conformal sphere and in mapping
from the co-formal sphere onto th tangent plane.
Scale Factor - Ellipsoid to Conformal Sphere
The scale factor associated with mapping from the ellcpsuid to
the conformal sphere is the ratio of a linear element on the conformal
sphere to a corresponding linear element on the ellipsoid. From
reference 2 page 86, the :,ale factor associated with a point of
projection is:E cos
kl=ck (18)NcosL
where:
is the conformal latitude of the point to be projected
L is the geographic latitude of the point to be projected
25
The scale factor is a function of the radius of the conformal sphere
and the conformal and geographic latitude of the point to be pro-
jected. The scale factor can be. expressed as follows.
Ec (1 - e2sin 2)l/2cosok 1 E cosL
q
E= ..kf (19)
q
Table 11 gives values of k' for several different latitudes.1
Table II
Scale Factor - Ellipsoid to Sphere versus latitude
Geographic Latitude Scale Factor k'
00 1.00000000100 1.0001007120* 1.00039099300 1.0008366740* 1.00138493500 1.0019702460* 1.00252203706 1.0029731480* 1.00326821900 1.00337838
Scale Factor - Conformal SpLere to Plane
The scale factor associated with mapping from the conformal
sphere to the plane of projection is the ratio of a linear element
on the plane of projection to a corresponding linear element on the
conformal sphere, The scale factor is therefore a function of the
angular separation between the origin of the plane and the point of
projection. The distance on the sphere between the origin of the
plane and the point of projection is calculated as follows.
D a E c (20)
26
The stereographic ground range R is given by equation (1) as
follows.
R - 2Ectan(M) (21)
The scale factor associated with mapping from the conformal sphere
to the plane of projection is calculated as follows.
-dR/ d
2 dD/d,
sec2(J)
2 (22)
1 + COS*~
Table III presents the scale factor k2 as a function of angular
separation.
Table III
Scale Factor - Conformal Sphere to Plane versus Angular Separation
Angular Separation Scale ractor k
00 1.0000000020 1.0003046840 1.00121946
60 1.0027465880 1.00488976
100 1.00765427120 1.01104690140 1.01507605160 1.01975173180 1.02508563200 1.03109120
Aircraft and radar locations in the air surveillance region are
presented on the common plane. The scale factor k2 , associated with
the common plane is a function of the angular separation between the
region center and the point of interest. From reference 3 page 23,
the angular separation between the region center and a point of
27
interest can be expressed .in terms of their locations on the conformal
sphere as follows.
cos* - sin~sin 0 + coscos0COSAX (23)
Combining equations (22) and (23) the scale factor k2 is expressed
as follows.2 (4
k2 - 1 + sinosinO + coscos 0cosAX (24)
Calculation of the Radius of the Conformal Sphere
The total scale factor in mapping a point from the ellipsoid to
the common coordinate plane is the product of kI and k The total
scale factor is expressed as follows.
2E cost (25)NcosL(I + sin~sin 0 + cosocoS 0COSAX) (5
The scale fpctor expresses the ratio of a linear element on the
common coordinate plane to the corresponding element on the ellipsoid.
The scale factor therefore represents the ratio of the velocity on
the common plane to the corresponding velocity on the ellipsoid. A
unity scale factor is highly desirable since velocity on the common
plane will represent actual ground speed. The scale error is defined
as the difference between the scale factor at a point in the region
and a unity scale factor. The scale error e is expressed as follows.
2t coso
NcosL(1 + sinosin 0 + cosocos*oCos&X)
=E A- 1 (26)c
The extent of an air surveillance region is defined by the location
of the radars that are tied into it. Since the scale error varies
as a function of the location and separation of a point from the
region center, it is highly desirable to minimize the maximum scale
28
errors that will be encountered. From equation (26) the value of
the scale error at a point can be varied by varying the radius of
the conformal sphere E . It is therefore possible to obtain bothc
positive and negative scale errors. To minimize the magnitude of
the largest scale error, E is chosen such that the magnitude of theC
maximum negative scale error is equal to the maximum positive scale
error. The radius of the conformal sphere is obtained as follows.
Cmin + ax - 0
E Ain -1 + E cA-ma 1 -0
E(c A i n + A a - 2 -0
2E A i A(27)
where Am n and Amax are the smallest and largest A calculated for
the region center and all radars tied to the region. Substituting
equations (19) and (24) into equation (21) the value of A may be
calculated as follows.k.k
A 12 (28)q
The values of Amax and Amin for a given region can therefore be
determined from Tables IT and ITI. The following general conclusions
can be drawn from examination of the tables.
1. Am n usually corresoonds to the region center since its
value of k2 is unity.2. A usually corresponds to the most distant radar since it
maxhas the largest angular separation. Tf two radars have the
same angular separation, Amax will correspond to the more
northerly since it will have the larger k1 value.
29
Table IV shows values of E calculated for a region center at a
geographic latitude of 450 and radar sites directly north and south
of the region center. Table IV shows that the radius of the conformal
sphere decreases markedly as the region size increases.
Table IV
E Versus Region SizeC
Geographic Latitude Distance From Eof Most Distant Radar Region Center c
470 120.012 3437.561490 240-10 3435.787
510 359.997 3435.963
530 479.970 3429.088570 719.881 3418.17061° 959.744 3403.004650 1199.563 3383.553
430 120.026 3437.96441 240.064 3436.596390 360.116 3434.179370 480.179 3430.71433* 720.342 3420.62829* 960.553 3406.3i425° 1200.808 3387.749
Effect of c/E5
The JSS stereographic ground range equation, equation (8), and
the current SAGE/BUIC stereographic ground range equation, equation
(9), differ by the scale factor E c/E . The effect of E c/E can be
expressed as the difference between the two equations as follows.
6 p=(l + !- 2 _ "c
I S - (29)
30
Since stereographic ground range is transformed into coordinates on
the common plane, a difference in the ground ranges will result in a
corresponding misregistration on the common plane. The quantity R
in equation (29) represents an approximation. Since the design
registration error budget in JSS is .1i nmi, R must differ from its
actual value by no more than .18 nmi. (A detailed review of the
errors induced by an error in R is beyond the scope of this report.)
Therefore a difference calculated by equation (29) greater rhan .36
nmi will guarantee an unacceptable registration error if the SAGE/
BUIC stereographic ground r --e is used. To show the effect ofEc/E as a function of region size, values of 5 are shown in Table
V for the radar locations, region center and values of E indicatedc
in Table IV. The value of R was arbitrarily chosen to be 100 and
200 nmi. Region size is defined as the distance of the most distant
radar from the region center.
Table V
Difference Between Ground Range Equations 8 and 9 versus Region Size
Geographic Latitude Approximate E /E 6(nmi) 6(nmi)of Most Distant Radar Region Size c s R-100nmi R-200nmi
47o 120 .9999 .009 .018490 240 .9995 .049 .098510 360 .9988 .120 .239530 480 .9978 .221 .442
570 720 .9948 .517 1.033610 960 .9906 .937 1.875650 1200 .9852 1,485 2.969
430 120 .9998 .021 .042
410 240 .9993 .072 .145390 360 .9985 .154 .308370 480 .9973 .266 .533330 720 .9942 .581 1.163
29* 960 .9898 1.018 2.03625* 1200 .9842 1.576 3.152
31
Exination of Table V reveals that for R equal to 200 nmi,
values of 8 will exceed .36 nmI for regions somewhere between 360 =mI
and 480 tu. This indicates that the SA*GE/BUIC atereographic ground
range equation ill produce unacceptable results in large regions.
Some of the current SAGEfMJIC regions exceed these limits, and all
JSS regions will exceed these limits by a considerAle margin.
Section III of this report is devoted to, depicting the registration
errors produced by the SAGEfNJIC and JSS stereographic ground range
equations In the seven JSS regions.
32
SECTION III
EFFECT OF THE SAGE/BUIC AND JSS STEREOIrRAPHIC GROUND RANGE
EQUATIONS ON REGISTRATION
INTRODUCTION
Accurate stereographic projection of radar data onto the conmon
coordinate plane is vital to the operation of air defense and air
traffic control systems. Large registration errors seriously down-
grade the performance and stability of the active tracking algorithm.
For each of the seven JSS regions, simulated radar data was used to
devionstrate the effect of the SAGE/BUIC and JSS stereographic ground
range equations on registration.
11~Raa SslantE RD r ange and azimuth data were renerated for four air-
craftloainatattdso30W 4500id6000f toec
JSregion. Apni .niae h loih tdt rdc h
daa or a particular radar-, a slant range-Azimuth pair U&3 getter-.
&ted only it the aircraft vasn vithio 250 nal of the radar And ab~ove
the radar hortion. Aircraft locationa vere chosen to that slant
ranges for the reporting radars would be greater than 170 nwi.
Radar slant ranges were converted into stereographic ground
ranges using the SAr*EICJIC and JSS stereographic ground range e-
Vquations (equations (8) and (9)). ftereographic ground ranges were
transformed Into coordinates on the coumoo plante usitni equations (16)
and (17). Since the location of the aircraft *9r knowatl
coordinates on the coon plan* were ito~uted using equations (13)
and (11.). 'Me registration error +oas obtained by taking the magnitudeI * of the difference betweva the actual coordinates on the cosmon plane
and those obtained by the -RAGE/bUIC and JSS stereographic ground range
equations.I 33
Appendix II presents the data for the seven JSS regions. For
each region, there is a Site Data, a Simulated Radar Data, and a
Registration Error table. In addition there is a figure shoving
the location of the region center, radar sites, and aircraft. The
following information is given in the tables.
Site Data Table:
1. Approximate geographic latitude and longitude of the
region center and radar sites.
2. Radius of the earth to each radar site.
3. Coordinates on the counon plane for each radar site.
4. The radius of the conformal sphere.
Simulated Radar Data Table:
1. The geographic latitude and longitude and altitude of the
aircraft in the region.
2. The designation of all reporting radars.
3. Slant range and azimuth data for all reporting radars.
Registration Error Table:
1. Stereographic ground range calculated using the SAGE/BUIC
and JSS ground range squations.
2. Coordinates on the comon plane obtained from the SAGE/
BUIC and JSS ground range equations.
3. Coordinates on the common plane obtained from the actual
aircraft locations.
4. The registration error induced by the SAGE/BUIC and JSS
ground range equations.
Examination of the tables In Appendix 11 reveals that use of the
SAC/BUIC stereographic ground range equation resulted in registration
errors that exceeded the .18 nmi JSS registration error budget in all
cases tested. Use of the JSS atereographic ground range equation
resulted in acceptable registration errors. Table VI sumarizes the
34
U) c0 0 00 C4 C" - co CO.4w .I LA .4 a .4
.M C.co 0 0 0
0, N M C C4
.4-4 .
U, N s0o. N
It N4 It ' 4
00 0 a %D c
bdOItn 0 g) CA
tA%
35
worst case results for slant ranges of approximately 180 nmi in each
of the seven JSS regions.
The omission of the factor E /EA is responsible for the large
registration errors produced by the SAGE/EUIC stereographic ground
range equation. For a particular region, the value of %Is constant.
The most southerly radar wili have the largest value of E.. There-
fore the value of 6 as calculated by equation (29) will be largest
for the most southerly radar. This Indicates that the worst case
errors will be produced by the most southerly radar, and the beat
case errors by the most uiortherly. Further examination of Appendix
11 supports this conclusion.
36
SECTION IV
CONCLUSIONS
The scale factor Ec A s should be included in the stereographic
ground range equation to avoid large registration errors. This is
especially important in large regions. The JSS stereogranhic ground
range equation for processing returns with height data is:
E (S2 . 2 1/2
E (I + i
The present SAGE/BUIC stereographic ground range equations
should be modified to reflect the scale factor E /E5 The modifi-
cation vould require a change In the adaptation parameters ou as ite--
by-site b"is.
37
-N\\
REFERENCES
1. J.J. Burke, An Improved Stereographic Coordinate ConversionApproximation for Radar Data Processing, MTR-2548, ContractF19628-73-C-OOO1, The MITRE Corporation, Bedford, MA,January 1973.
2. P. Thomas, Conformal Projections in Geodesy and Cartography,Special Publication 251, U.S. Coast and Geodetic Survey, 1952.
3. J.J. Burke, Stereographic Projection of Radar Data in a NettedRadar System, ESfl-TR-73-210, AD 771544, November 1973.
AI
39eed n ag ban
APPEWDIX I
GENERATION OF SLANT RANGE AND AZIMUTH DATA
Radar slant range and azimuth data are calculated for aircraft
locations using vector operations. The vector V from the centpvrof the earth to a point on or above the ellipsoid is calcul~ated as
follows.
X (N + H)cos LcosA
V (N + H)cosLsiuA
Z (Nl-e) + HJsinL
where:
H is the height cf the point above the ellipsoid
A~ is measured positive easit of the prime meridian
The slant range S is compuited as follows.
S T
- jS~(A-2)
where.
Sare vectora f rom the center of the earth to the aircraft
and radar respectively.
The aircraft xuat be above the radar horizon for a radar return Lo
be possible. The aircraft is above the radar horizon 1f:
S 2 0
where:
Z is a unit vector directed along the zenith of the reporting
radar as follows:
41 f~~
.~PP
Z - coaL C05)Lzr r
- Z - sinL(A3z r(A)
L I are the geographic latitude and longitude of the report-
ing radar
The azimuth angle 6 is calculated as follows.
e tart~ i (A-4)
where:
E is a unit vector directed due east of the reporting radar as
follows.
IX E -sinx
E r
- E 0 (A-5)
N is a unit vector directed due north of the reporting radar asI ~~~follows.fX snLCr
-sinL sinX
IZN rOL(4
42
APPENDIX II
SIMULATED DATA FOR THE SEVEN JSS RM*IONS
Radar slant range and azimuth data were generated for four air-
craft locations at altitudes of 30,000, 45,000 and 60,000 feet in
each JSS region. For a particular radar, a slant range azimuth pair
was generated if the aircraft was within 250 nmi of the radar and
above the radar horizon. Radar slant ranges were converted into
stereographic ground ranges using the SAGE/BUIC and JSS ground range
equations. Stereographic ground ranges were transformed into co-
ordinates on the common coordinate plane. The actual coordinates on
the common plane were also obtained. The registration errors induced
by the SAGE/BUIC and JSS ground range equations were calculated.
For each JSS region, there is a Site Data, Simulated Radar Data,
and Registration Errors table. Tn addition, there is a figure
showing radar site, aircraft, and region center locations.
Explanation of Tables
The Site Data Table shows the following.
a) Approximate latitude and longitude of the radar sites and
region center.
b) The earth radius to each radar site calculated using
equation (12).
c) Coordinates on the common plane for the radar sites calcu-
lated using equations (13) and (14).
d) The radius of the conformal sphere calculated using equation
(27).
The Simulated Radar Data Tabl.e shows the following.
a) The latitude, longitude and altitude of the aircraft.
b) The slant range and azimuth for all reporting radars. A
43
- I .
radar is a reporting radar if the target is within 250
inil and above the radar horizon. Slant range and azimuth
data are calculated using the algoritm in Appendix T.
The Registration Error Table shown the following.
a) The stereographic ground range calculated using the SAGE/
WJIC and JSS ground range equations (equations (8) and
(9)).b) Coordinates on the co~n plane obtained from the SAGE/WUIC
and JS5 steroographic ground range equations using
equations (16) and (17).
c) Coordinates on the coinon plane obtained ftrm the actual
location of the aircraft using equations (13) and (14).
d) Tbs registration error induced by the SAGE/MUI.C and JSS
stereographic ground range equations.
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