On a Satellite Coverage

I. INTRODUCTION On a Satellite Coverage Problem DANNY T. CHI Kodak Berkeley Researc Yu T. su National Ciao Tbng University Te eart coverage area for a satellite in an Eart syncronous orbit wit a nonzero inclination angle is a function of te time of day. Locating te area were an Eart terminal can establis a link wit a geosyncronous satellite at a certain period of te day is of great concern to system designers. A special case of muc interest is wen te time period of satellite coverage is 24. 'o simple and rapid algoritlls for locating 24 satellite coverage areas are presented. One of te proposed algoritns can also be applied to determine oter airborne antenna coverage problem. Manuscript received September 9, 1992, revised January 3, 1993. IEEE Log NO. TAES/31/3ll2730. Tis work was presented in part at te 1991 IEEE Military Communications Conference, McLean, VA, Nov. 1991. Autors' addresses: D. T. Ci, Kodak Berkeley Researc, 2120 Haste St., Berkeley, CA 94704; Y. T Su, MIRC Researc Center and Department of Communication Engineering, National Ciao 'ng University, Hsincu, Tiwan. 00189251/95/$4.00 @ 1995 IEEE Because of lunar and solar gravitational attraction, geosyncronous satellite orbits do not remain perfectly circular. Tey can also be intentionally inclined to provide greater coverage to iger altitudes, even as ig as te nort and sout poles [l]. Of special concern to communication engineers is not only weter a ground terminal as a direct view of te desired satellite, or te oter way around, but also weter te communication link can be establised, i.e., weter te link power budget renders enoug signal power margin at te receiver site. A ground terminal is said to be covered by a geosyncronous satellite if bot conditions are met. In te case of inclined orbits, tis status of coverage, given te terminal location, is a function of te time of day because te terminal elevation angle, te atmosperic conditions, and te terminal to satellite distance all vary wit time. A 24 or allday coverage area (ADCA) is defined as te area witin wic it is possible, at any time and under specified weater conditions and a fixed terminal altitude, to establis a link wit a given geosyncronous satellite. For convenience, we assume tat te reference altitude is sea level. Te assumptions of a fixed altitude and uniform climate and atmosperic conditions in te ADCA definition are necessary to eliminate te time variation, altoug a coverage area is very likely to cross different climatic regions and will definitely ave altitude variations. Suc an ADCA can tus be regarded as te best case (maximum coverage area) or te worst case (minimum coverage area) result, depending on te specific atmosperic parameters and link availability requirement used in te calculation. Two efficient algoritms for locating ADCAs in te latitudelongitude (L L) plane are presented. In te next section a relationsip between te link budget computation and te satellite coverage is establised troug a single parameter. Te problem of coverage status cecking is ten simplified to examine an inequality. A few useful properties of ADCAs are derived and ten applied to develop an efficient algoritm in Section 111. Anoter algoritm and numerical examples are presented in te following section. Tis second algoritm uses an observation derived from a simple geometric viewpoint. It can also be extended to evaluate te coverage area in any given time period and for any circular satellite orbit. Section V gives a brief summary of our results and suggests possible extensions. II. LINK BUDGET AND EARTH COVERAGE AREA Te received carriertonoise power ratio (GIN),, for a oneway ground terminal to satellite link is given IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 3 JULY 1995 891

by [2, c. 41 were EIRP transmitter effective isotropic radiated power, x carrier wavelengt, d slant range from satellite to Eart terminal, L, atmosperic and weater loss, L, rain attenuation, k Boltzmann constant, B noise bandwidt of satellite cannel (Hz), G/T receiver antenna gaintonoise temperature ratio. Te atmosperic and rain losses depend on te link availability requirement, te eigt of te terminal, te carrier frequency, te weater conditions, e.g., rain rate, eigt of 0' C isoterm, mean local surface absolute umidity, mean local surface temperature (see [3] for details), and $, te terminal elevation angle. For fixed altitude, frequency and weater conditions, te total propagation pat loss is ten a function of te elevation angle only. For example, if 4 is greater tan 6O, ten te atmosperic loss can be approximated by [3, Fig. 6.231 L,(4) = Krsq4) (2) were K depends on carrier frequency, ground altitude, mean surface relative umidity, and mean local surface temperature. Under te same assumptions, te rain attenuation statistics is also a function of te elevation angle only; see [3, Fig. 6.3.11 for an analytic estimation procedure for L,. An implicit assumption of (1) is tat te terminal antenna boresite always points to te satellite antenna boresite. Tis and te constant terminal altitude assumption imply tat tere is a onetoone correspondence between $ and 8, te angle between te Eart terminal and te satellite, bot viewed from te center of te Eart.' Terefore te product L,L, will be denoted by L(8) encefort. Note tat te carriertonoise power ratio calculated by (1) sould be interpreted statistically, i.e., on te average, te actual (C/N)rcc will be less tan te value so obtained for at most p% of te year, were p is te designed link outage used in estimating L,. If a perfect symmetric satellite antenna pattern is assumed ten it can be sown tat te received carriertonoise ratio of te satellite sould satisfy te inequality were (GIN),, is te required C/N, Re is te radius of te Eart, is te eigt of te satellite above te Eart, R, is te distance between te satellite and te Eart center, and ko = G /TFIRP/(~TA)~](~/~B). Terefore if te maximum angle 6JC wic satisfies (3) is known, te condition under wic an Eart terminal located at (Q,$) can close a link (for at least p% of te year) wit a satellite wose subsatellite point at time t is (\Ef, $f) can be expressed as cos $ cos Q cos y!+ cos \Et + sin $ cos \E sin?,bl cos \El + sinqsinqf > cosoc. (4) In oter words, equating bot sides of (3) and solving te associated equation using establised atmosperic loss and rain attenuation models, we ten obtain te maximum angle 8, tat accounts for te combined effect of all te linkage parameters, te required link availability, bit error probability, te assumed weater conditions and terminal altitude on te coverage problem under investigation. For Eart terminal locations wit a 8 less tan e,, te associated pat loss will be smaller tan tat for locations wose 8 is equal to 8, and ence te requirement (3) will be satisfied. Te angle 8, is encefort referred to as te coverage angle and te corresponding sperical circle radius R as te coverage radius; see Fig. 1. For a perfect geostationary satellite at (\Ef, qf) = (0,O) for all t, (4) becomes cos+ cos q > case,. (5) It is well known tat an inclination of te geostationary orbit will cause te subsatellite point to move in a figure 8 pattern [l, 21. To decide weter a given (+,$) belongs to te ADCA, it is necessary to ceck weter (4) is satisfied for all subsatellite points on te figure 8 pattern. 111. PROPERTIES OF FIGURE 8 PATTERNS AND THE ADCA Te equations governing relative latitude and longitude beavior of te figure 8 pattern for an inclination angle i are given by2 A = sin'(sin i sin@) (6) a = tan'(cosisina,cosa) a (7) were = 2 ~t/t, T = 24, t = te normalized time of te day, and te function tan'(x,y) is a modified version of te conventional arctangent defined by tan'(x,y) = 8 'Tis angle is often referred to as te central angle [Z]. 2A similar set of equations was given in [l, (64)], but its second equation indicates an incorrect moving direction. We tus rederive te figure 8 equations in Appendix A. 892 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 3 JULY 1995

t 0 EARTH EQUATOR PLANE sntsllirc orbit plane Fig. 1. Inclined geosyncronous satellite wit coverage angle 0 and coverage radius R. if and only if sine = 2 case = X JW J x2 + y2 # 0. For convenience, it is often assumed tat at t = 0, te subsatellite point is at @ = 0, X = 0 (see Appendix B). Tis normalization is used trougout te subsequent discussion. From te above equations, we can easily derive some useful properties of te figure 8 pattern. Tese properties are summarized below. PROPERTY 1 Te figure 8 pattern is symmetric wit respect to bot latitude and longitude axes. PROPERTY 2 Te maximum latitude deviation from te equator for te figure 8 pattern is equal to i. PROPEW~V 3 Te figure 8 equation can be rewritten as 5(@)=ta11 (cositan@)@~ ifo<t<g i>o. Te maximum longitude deviation from te ascending node is terefore given by A bruteforce approac to locating te 24 coverage area would be to ceck every points in te L L plane weter te condition (4) is met for all t E [0,24). Looking for possible reductions in te number of points to be cecked, we first observe tat Property 1 and (3) lead to te following property. PROPERTY 4 Te ADCA is symmetric wit respect to bot latitude and longitude axes. Terefore, we need only to searc te ADCA in te first quadrant. Te second reduction comes from Property 5. PROPERTY 5 An upper bound for te intersection of an ADCA wit te altitude axis is &(e i)j wit te longitude axis intersection * cos (coso/ cos i). Note tat if i > 8, ten te subsatellite point of te ascending node (A in Fig. 2) can be covered eiter f I s Fig. 2. Subsatellite points of inclined syncronous satellite orbit. at t = 6 or at t = 18 only. Its ADCA is tus equal to zero. Te tird reduction can be accomplised by noting tat te coverage area for a fixed satellite (or subsatellite) position is a sperical circular area wic is a convex set and tat te intersection of convex sets on a spere is still a convex set. Terefore, we ave te following. PROPERTY 6 te Eart. Te area of an ADCA is a convex set on In oter words, for eac longitude (or latitude) circle we need only two points to determine te portion of te ADCA tat belongs to tat circle. Anoter reduction can be derived from te following. PROPERTY 7 Te latitude of an ADCA boundary in te first quadrant is a decreasing function of longitude. Te proofs of te above properties are straigtforward and are omitted. Taking tese properties into account, we can eliminate a large portion of te L L plane from te searc domain and tus an efficient searc algoritm for te ADCA suc as tat described in Fig. 3 can be obtained. However, we still ave to deal wit te problem of cecking te condition imposed by (4) for all t in a certain time period. In oter words, te ADCA evaluated by te above algoritm is only a close approximation since we ave quantized te searc domain (see Fig. 3). In te next section, we develop a CHI & SU: ON A SATELLITE COVERAGE PROBLEM 893

0. dl i. I. JM.n"' * d.mr,,e. Fig. 4. Subsatellite point trajectory and coverage area. Note if L is subauplane curve of airplane wit constant altitude and antenna coverage radius R, ten region Q(A; R) n Q(B; R) is always covered by airplane during period wen its subairplane point travels troug L. fast algoritm wic allows us to ceck only a finite set of points on te figure 8 pattern. IV. FAST ALGORITHM FOR LOCATING COVERAGE AREAS DEFINITION 1 Let A be a point on a spere S. Ten Q(A; R) denotes te sperical circle centered at A wit sperical radius R on S. DEFINITION 2 A curve L joining points A and B S is said to satisfy te Rcondition if and only if L enclosed by Q(C; R) and Q(D; R), were C and D points of intersection of Q(A; R) and Q(B; R). on is are If a curve L (suc as a figure 8 pattern) does not satisfy te Rcondition, ten we can decompose L into a number of subarcs tat do satisfy te Rcondition. As is seen later, te number of subarcs in a decomposition is proportional to te complexity of te given coverage problem. On te oter and, te decomposition of a given curve is quite straigtforward, as te following algoritm demonstrates (Fig. 4). Let us consider te decomposition of curve AB. We can start wit te initial point A, select a nearby point PI on L, and ten ceck weter te arc API satisfies te Rcondition. Tis test can be done analytically or by cecking weter te points A + ja, j = 1,...,n, A = APl/n, along te arc AP 1 are all witin te intersection of Q(A; R) and Q(B; R). If AP1 passes te test, we ten coose anoter point P2 between PI and B on L and perform te test on P1P2. If it doesn't, we replace PI by a point Ql between A and PI and ceck weter AQl satisfies te Rcondition. Tis process can be repeated until we reac end point B. To reduce te number of subarcs tat satisfy te Rcondition, we can replace te Rcondition by te R'condition, were R' is sligtly larger tan R. For example, if R = 6378 km (te radius of te Eart), we can replace R by R(1+ E), E = 0.0001. Te error introduced by suc a substitution is negligible, but te number of subarcs reduced may be significant. Our second ADCA algoritm is based on te following teorem. THEOREM. Let L be a curve joiningpoints A and B on a spere S. Suppose L lies witin te area G(C, D; R) Ten G(A, B; R), te area on S enclosed by Q(A; R) and Q(B; R), is equal to te area G(L; R) enclosed by all sperical circles Q@; R), were p E L. Te proof of tis teorem is given in Appendix B. As a consequence of tis teorem we immediately Obtain te COROLLARY. Let L be te curve of all te subsatellite points of a satellite in an inclined circular syncronous orbit aving a coverage radius R Suppose L as a decomposition POPI U... U P,, 1 P,, suc tat eac subarc satisfies te Rcondition. Ten an Eart terminal can establis a communication link wit te satellite all te time if and only if te terminal is covered by all Q(P,,P), 0 < j < n + 1. All te points PO,.., P,, can be precalculated and terefore te satellite coverage problem is reduced to te task of cecking only a finite number of points in a figure 8 pattern. To demonstrate te utility of te fast algoritm, let us consider te case were te angle of inclination i is 5 deg and d,, te coverage angles, are 61.8', 52.5', 25.7', and 12.8' (teir corresponding terminal antenna elevation angles are 20', 30', 60, and 75O, respectively). Te number of points on te figure 8 pattern tat need to be cecked for eac ADCA is listed in Table I. Here we assume tat A = 0.05 = 3 min and E = (see Section 111). Te resulting Eart ADCA contours are illustrated in Fig. 5. As mentioned before, we need to searc only te first quadrant; terefore, for te case 8, = 61.8' only 11 points on te figure 8 pattern (Table I) ave to be cecked. Te ratio of te reduction in te number of points cecked is at least (24/A)/ll = 44.6. Evidently, tis improvement is an increasing function of te required resolution A in te fast algoritm. 894 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 3 JULY 1995

100 120 140 160 180 200 220 240 260 Degrees Longitude Fig. 5. ADCAs or syncronous satellite wit inclined angle 5". V. CONCLUSION In tis paper we ave derived a general condition for determining te coverage area of a satellite. Two algoritms for determining te continuous coverage area of an inclined circular geosyncronous satellite ave been presented. Altoug we ave restricted our discussion to one way (up link) coverage problems only, extensions to simplex or duplex Eart station links are straigtforward. Te second algoritm can also be applied to similar airborne coverage problems if 1) te coverage area is a sperical circle, i.e., te antenna pattern is symmetric wit respect to its boresite and 2) te aircraft is moving wit constant eigt above te Eart. Tere is no need to restrict te time period to 24, as te first algoritm does. Wit only minor modifications, it may also be used to predict te coverage area were an Eart station can simultaneously establis links wit several satellites in te same orbit. Anoter application example is illustrated in Fig. 3. Suppose an airplane wit coverage radius R wants to monitor te region Q(A; R) n Q(B; R). Ten te airplane could coose any pat as long as its subairplane pat is witin te region Q(C;R) n Q(D; R). APPENDIX A. POINTS FIGURE 8 PATTERN OF SUBSATELLITE Let T be te period of a satellite orbit and point A be te subsatellite point on te equator; see Fig. 2. Suppose te subsatellite point passes A at time zero. After t ours, te subsatellite point is at point S and A is rotated along te equator to a point P wose coordinate is given by (x,y,z) = (cos@,sin@,o), were @ = LAOP = LAOS = 2wt/T and t is te normalized time of day. Te coordinates of point S TABLE I e, 61.8 52.5 25.7 12.8 No. of points 11 15 19 23 can be obtained from P by rotating te xaxis troug te angle of te inclination of te satellite i. If te xaxis is rotated troug te angle i, te coordinate of S is ;:I 0 0 cos@ cos @ * [ [%;os; [si;@] = (9) Let @ and X be te relative longitude and altitude of S wit respect to P; ten cos@ = cosxcos(@ + 5) (10) cosisin@ = cosxsin(@ + 5) (11) sinisin@ = sinx (12) and terefore x = sin'(sinisin@) (13) ip = tan'(cosisin@,cos~) ip (14) APPENDIX B. PROOF OF THE MAIN THEOREM Let E and F be two points on a spere S. Ten EF denotes te (great circle) arc joining E and F and lefl is te lengt. LEMMA. Let Q(A; R) and Q(B; R) be two sperical circles on a spere S. Suppose Q(A; R) and Q(B; R) intersect at points C and D on S. Let G(A, B; R) be CHI & SU: ON A SATELLITE COVERAGE PROBLEM 895

~ ~~ te area on S enclosed by Q(A; R) and Q(B; R) and G(C, D; R) be te area on S enclosed by Q(C; R) and Q(D; R). Ten te sperical distance between an arbitrary point in G(A, B; R) and an arbitrary point in G(C, D; R) is less tan or equal to R PROOF. Let E be an arbitrary point in G(A, B; R) and F be an arbitrary point in G(C, D; R). Suppose EF or its extension intersects te convex set Q(Z; R) at a point V, were I = C or D, in te order of A V B on Q(Z; R) and EF or its extension intersects te convex set Q(J; R) at a point U, were J = A or B, in te order of C U D on Q(J; R) (see Fig. 3, were we assume tat J = A and I = C). It is clear tat JU,. always intersects IV and ence JU is always between,. JV and JI. We now assume tat V # J and U # I, oterwise, IEFJ 5 1UVI = R and te Lemma is proved. Notice tat in te sperical triangle AJZV, (ZV I = JZJ/, tus LJVZ = LVJZ and LVJU = LVJZ LUJI 5 LVJI + LIVU = LJVI + LIVU (15) (16) (17) = LJVU. (18) Similarly, we observe tat in te sperical triangle AUJV, LVJU 5 LJVU. Terefore, JU 5 UV and JEFJ 5 JUVJ 5 JJUJ = R, and te Lemma is proved. Wit tis Lemma we now give te proof of te main teorem as follows. PROOF. 1) G(A, B; R) c G(L; R). Let E be an arbitrary point in G(A,B : R) and p be an arbitrary point on te curve L. L is enclosed by G(C, D; R) and tus P E G(C, D; R). Te above Lemma implies tat te sperical distance between P and E is equal to or less tan R and E is inside Q(P; R). Terefore, E E G(L, R) and G(A,B; R) C G(L; R). 2) G(L; R) c G(A,B; R). Let E be an arbitrary point in G(L; R). Ten E is enclosed by all te sperical circles Q(P; R), were P E L. In particular, E is enclosed by Q(A; R) and Q(B; R). Hence, G(L; R) C G(A,B;R). COROLLARY. Let L be a curve joiningpoints A and B on a spere S. Suppose L sati@es te Rcondition. Ten a point T is enclosed by all Q(P; R), P E L, if and on& if T is enclosed by G(A, B; R). REFERENCES [l] Spilker, J. (1977) Digital Communications by Satellite. Englewood Cliffs, NJ: PrenticeHall, 1977. [2] Ha, T. T (1990) Digital Satellite Communications (2nd ed.). New York: McGrawHill, 1990. [3] Ippolito, L. J. (1989) Propagation Effects Handbook for Satellite System Design (4t ed.). NASA, 1989. Danny T. Ci received is P.D. in matematics from University of Soutern California, Los Angeles, 1984. From 1984 to Aug. 1988 e worked at LinCom Corporation, Los Angeles, as a system analyst. In Aug. 1988 e joined Cyclotomics (currently, Kodak Berkeley Researc) as a project engineer. He as done researc on te ReedSolomon encoding and decoding algoritms. He as been responsible for a number of projects including Cinema Digital Sound system, NASA's OMV project, Space Station error detection and correction system, and more recently, Kodak's Poto CD project. His researc interests are in areas of coding teory, digital signal processing, satellite communications and data compression. He as publised numerous papers in tese areas. Yu T. Su received te B.S.E.E. degree from Tatung Institute of Tecnology, Taiwan, in 1974, and te M.S. and P.D. degrees in electrical engineering from te University of Soutern California, Los Angeles, in 1983. From 1983 to 1989 e worked at LinCom Corporation, Los Angeles. Since Sept. 1989, e as been a member of te faculty of National Ciao Tung University, Hsincu, Taiwan, at te MIRC Researc Center and te Department of Communication Engineering. His main researc interests are in te areas of communication teory and statistical signal processing. 896 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 31, NO. 3 JULY 1995