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Stereographic ProIectionin the Joint.Surveillance System

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Sep 76

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STEREOGRAPHIC PROJECTIONIN THE

J JOINT SURVEILLANCE SYSTEM

SEPTEMBlER 1976

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DEPUTY FOR SUIRVE'!Al ' AND tNAVIGATION -1's InWIWELECTRONIC SYSTEN q D11,11IONAIR FORCE SYSTEM. COMW AND

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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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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

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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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