Journal of Geodesy (1997) 71: 749±767

Theory of ground-track crossovers M. C. Kim Center for Space Research, The University of Texas at Austin, Austin, Texas, 78712-1085, USA Tel: +1 512 471 8121; fax: +1 512 471 3570; e-mail: [email protected]

Received: 19 November 1996 / Accepted: 12 May 1997

Abstract. The fundamental geometry of satellite ground The points where the ground track of a satellite in- tracks and their crossover problem are investigated. For tersects itself on the surface of the earth are called (single- idealized nominal ground tracks, the geometry is gov- satellite) crossover points. When the intersections are erned by a few constant parameters whose variations from ground tracks of two distinct , these are lead to qualitative changes in the crossover solutions. usually called dual-satellite crossover points. As a means On the basis that the theory to locate crossovers has not of obtaining relevant measurements at the same geo- been studied in suf®cient detail, such changes are graphic location separated in time, crossover points hold described in regard to the number of crossover solutions a signi®cant usage in satellite geodesy, especially in the in conjunction with their bifurcations. Employing the realm of satellite oceanography associated with the ap- spinor algebra as a tool for establishing the ground- plications of satellite altimetry. The satellite-altimeter track crossing condition, numerical methodologies to crossover di€erence, which is de®ned as the di€erence in locate crossovers appearing in general dual-satellite two satellite altimeter ranges interpolated at a crossover ground-track con®gurations are also presented. The point along their respective ground tracks, provides pe- methodologies are applied to precisely determined culiar time-di€erenced oceanographics. While the altim- orbital ephemerides of the , ERS-1, and eter crossover di€erences have been also used in the TOPEX/POSEIDON altimeter satellites. evaluation of an altimeter time-tag bias (Schutz et al. 1982), in the re®nements of satellite (Cloutier 1983; Sandwell et al. 1986; Born et al. 1986; Moore and Roth- Key words. Satellite á Ground track á Crossover well 1990; Hsu and Geli 1995; Kozel 1995), in the cali- Bifurcation á Spinor bration of a gravity ®eld model (Klokocnik and Wagner 1994), in the estimation of the altimeter inferred sea state bias (Gaspar et al. 1994), in the improvement of a plan- etary topography model (Stottlemyer 1996), etc., the computational eciency of locating crossovers cannot be overemphasized as one of routine tasks for such a wide 1 Introduction range of applications. Techniques for the determination of crossover loca- In the geodetic applications of an arti®cial satellite as a tions have been introduced in several studies. For single- platform for active remote sensors, associated measure- satellite crossovers, Hagar (1977) presented a computa- ments are frequently sampled along the ground track of tional ¯owchart to obtain circular- ground-track the satellite's nadir point, which is also referred to as the crossovers taking into account the secular subsatellite point. For instance, a satellite-borne altim- e€ects of the earth's oblateness. Hereinafter such an eter measures the distance between the satellite and its idealized , its geocentric projection onto the nadir point so that the measurement, called the satellite surface of a uniformly rotating spherical earth model, and altimeter range, can be used in the mapping of the the associated crossover points will be referred to as the physical shape of the mean sea surface (Kim 1993) and nominal orbit, nominal ground track, and nominal in the monitoring of time-varying oceanographics, such crossover points, respectively. As the nominal orbit and as the oceanic tides (Ma et al. 1994), the oceanic eddy its ground track can be e€ectively described by a few variabilities (Shum et al. 1990b), the global mean sea constant parameters, a set of fundamental mathematical level rise phenomena (Nerem 1995), etc., when the relations between the , longitude, and ground- satellite's orbital trajectory is determined by indepen- track crossing times can be easily established. Therefore dent tracking to the satellite (Tapley et al. 1994). Rowlands (1981) and Shum (1982) presented iterative 750 positioning of actual single-satellite crossovers starting mapping of the satellite's orbital trajectory onto a from nominal crossovers to utilize altimeter crossover prescribed earth-surface model such as a reference di€erences in their orbit adjustment and determination ellipsoid of revolution (Torge 1991) or a spherical studies. As also summarized in Schrama (1989) and harmonics geoid model (Shum 1982). While de®nitions Spoecker (1991), the general procedure of single-satellite of the nadir mapping and the earth surface can be crossover locating has been based on the search for in- dependent on a speci®c problem and earth satellites are tersections between piecewise ground tracks segmented under the in¯uence of a variety of perturbations, the by ascending and descending passes (i.e., by semirevolu- nominal ground track is appropriate analytically to tions), and mainly consists of two steps of the prediction investigate the fundamental geometry of ground tracks. of nominal crossovers and the determination of actual In the rotating earth-®xed geocentric equatorial crossovers by interpolating discrete subsatellite points Greenwich-meridional reference frame (hereinafter, the projected from satellite trajectories. As well as the actual earth-®xed frame), the orbital angular motion of a sat- determination step, the analytical prediction step itself ellite can be represented by the geocentric unit vector needs numerical iterations due to the inherent transcen- dental relation between the latitude and the di€erence of x cos / cos k r 1 crossing timings as discussed by Colombo (1984). Such a r^ 8 y 9 8 cos / sin k 9 two-step procedure has also been adopted in ®nding dual- ˆ r ˆ r ˆ <> z => <> sin / => satellite crossovers. For a particular combination of two nominal orbits, Santee (1985) employed spherical trian- cos s cos # cos i sin s sin # :> ;> :> ;> gular models to make an initial guess of simulated dual- cos s sin # cos i sin s cos # 1 ˆ 8 ‡ 9 † satellite nominal crossovers followed by iterative re®ne- > > ments by updating inclinations. More or less general < sin i sin s = dual-satellite crossover-®nding techniques, all based on where:> x; y; z denotes the Cartesian;> components of the the semirevolutions search, have been also discussed by satellitef positiong vector r associated with its geocentric Shum et al. (1990a), and by Moore and Ehlers (1993). radius r, latitude /, and longitude k, s is the argument of Given orbital ephemerides of actual satellites and a latitude and represents the satellite's orbital rotation, # proper de®nition of ground tracks, the crossover locating is the longitude of the orbital ascending node measured problem is to determine the ground-track crossing time- from the Greenwich meridian and represents the tag pairs. Strictly speaking, the solutions are intersections rotation of the satellite's orbital plane with respect to of two unrelated curves. Depending on the relevant orbit the rotating earth, and i is the inclination of the orbital geometries, the number of dual-satellite crossovers can be plane. For a nominal ground track, i becomes constant a priori unknown. There can exist more than one inter- and the two rotations s and # become uniform variables section (or no intersection) in a semirevolution ground- resulting in their linear dependence, track pair and therefore one may lose some crossovers at the prediction step. Even for the counting of crossovers in # ke qs 2 the nominal single-satellite con®guration, we found some ˆ † aspects overlooked in the previous studies. On the basis where ke represents the longitudinal origin of the that the theory to locate crossovers has not been surveyed nominal ground track, and q denotes the angular in sucient detail and that we need a more systematic velocity ratio between the two uniform rotations. With approach to the problem, this paper presents both ana- the linear dependence we can consider s as a time-like lytical and numerical aspects of locating crossovers in the continuously increasing independent variable (the cases of the general dual-satellite scenario which include along-track location in the nominal ground track) and the single-satellite cases as a subset. We ®rst investigate # as a dependent variable which is continuously the fundamental geometric structure of ground tracks. decreasing, as the rotation of any orbital plane with With some relevant mathematical tools described in the respect to the earth is always westward (i.e., q > 0). Appendix, we establish the ground-track crossing con- Nominal ground tracks are thus on the three-di- dition, followed by numerical strategies for ®nding them. mensional parametric space of 0 i 180;   We will provide an answer to the predictability on the 0 < q < ; 0 k < 360 . Apart from the longitudi- 1  e † number of crossover solutions. With some treatments on nal origin ke, the manner in which a nominal ground the issue of the nadir mapping of satellite ephemerides track is laid down on the spherical earth is primarily onto the reference ellipsoid of revolution, actual appli- dependent on i and q. While i determines the latitudinal cations of the developed techniques to the precisely de- bounds of a nominal orbit and its ground track, it is also termined orbital ephemerides of the historic GEOSAT the parameter to classify conveniently the longitudinal and the currently orbiting ERS-1 and TOPEX/POSEI- direction of orbital motions. We say that an orbit is DON altimeter satellites will be also presented. posigrade when 0 i < 90 , is retrograde when    90 < i 180, and is circumpolar when i 90. How- ever, such a simple classi®cation is valid onlyˆ when or- 2 Nominal ground tracks bits are viewed from the inertial frame. When orbits are viewed from the earth-®xed frame, in which ground The ground track of an earth-orbiting arti®cial satellite tracks are actually depicted, we need careful accounting is the time-trace of its subsatellite point: a nadir for the relative rotational motions between the orbital 751 planes and the earth. For a nominal orbit the relative rotation can be e€ectively quanti®ed by the parameter q. To investigate the behavior of a nominal ground track, using Eqs. (1) and (2), we ®rst formulate the spin- axis component velocity of the unit vector r^,

z 0 2 2 /0cos / sin i cos s sgn cos s cos / cos i r ˆ ˆ ˆ †  p3 † and the spin-axis component angular momentum,

x y 0 y x 0 2 k0cos / r r r r ˆ   cos i q q sin2 i 1 cos2 s 4 ˆ ‡ † cos i q cos2 / ˆ Á where the prime indicates di€erentiation with respect to s, and the sgn function takes 1, 0, or 1 depending on the sign of its argument. Equations (3) and (4) can be combined to yield Fig. 1. Longitudinal direction of nominal ground tracks classi®ed in q; cos i space 1 d/ cos2 / cos2 i † Æ 2 5 cos / dk ˆ cospi q cos / † when cos i < 0, f when cos i > 0), q which represents the instantaneous slope of the nominal 1f cot2 i . In the regionˆ1q < cos i 1, the associat-ˆ1 ground track on the unit sphere. Note that for given ed groundˆ tracks† are always eastward f > 1 . In the values of and , the slope is a double-valued function 1 † i q two regions q < cos i 1and 1 cos i < 0, the as- of the latitude /. sociated ground tracks are always  westward f < 0 . The latitudinal motion of the nominal ground track Hereinafter these two regions are respectively referred to† is bounded in cos / cos i and is j j as the posigrade westward region and the retrograde westward region, where the ``posigrade'' and ``retro- northward when cos s > 0 grade'' indicate the longitudinal direction with respect to stationary when cos s 0 6 the inertial frame (the ``eastward'' and ``westward'' are 8 † > ˆ used for the earth-®xed frame). In the remaining region < southward when cos s < 0 1 of 0 < cos i < min q; q (where 0 < f < 1), the asso- as:> the sign of /0 in Eq. (3) will indicate. Meanwhile the ciated ground tracks change the longitudinal direction; Á 2 1 longitudinal motion can be somewhat complex. As the eastward at higher of cos / < q cos i and 2 1 sign of k0 in Eq. (4) will indicate, the nominal ground westward at lower latitudes of cos / > q cos i.We track is will refer to this region as the eastward/westward mixed region or simply the mixed region. It can be said that 2 2 1 eastward when cos s < f cos / < q cos i ground tracks of all circumpolar orbits cos i 0 are ˆ † 2 2 1 westward k0 q<0. The point q cos i 1 8stationary when cos s f Ácos / q cos i †ˆ ˆ ˆ > ˆ ˆ ( f 1=2 as a limit) corresponds to the geostationary > 2 2 1 ˆ f Ácos / > q cos i orbit whose ground track is motionless over a point on the equator 0 . > 7 / ; k ke :> Á †A nominal ground track † will eventually be parallel to where its preceding trace with an o€set less than some preset limit. To this extent all nominal ground tracks are re- q cos i 1 q cos i cos i peating, although the repeat interval may be as long as f 1 † 8 possible. Hence we restrict our interest to the cases ˆ q sin2 i ˆ q 1 cos i 1 cos i † † ‡ † where q is a rational number, i.e., is a constant which can be considered as a parametric q N=M 9 function of i and q. ˆ † Based on Eqs. (7) and (8), in Fig. 1 we classify the where M and N are positive coprime integers. With Eq. longitudinal direction of nominal ground tracks in the (9), which is a resonance condition between s and #, the two-dimensional parametric space of 0 < q < ; nominal ground track becomes periodic and repeats its 1 cos i 1 . There are four regions separated 1 by trace after M orbital revolutions of the satellite, or q cos i f †1 , q cos i 1 f 0 , cos i 0 f 0 , equivalently, N nodal days. In order to visualize the andˆ surrounded ˆ † by cosˆi 1ˆ f† ˆ, cos iˆ 1† topological evolution of nominal ground tracks, in Fig. 2 f when q < 1; f ˆwhen ˆq > 1†1 , q 0(ˆf we geographically display 21 nominal ground tracks ˆ1 ˆ1 † ˆ ˆ 752

Fig. 2. Nominal ground tracks 753

(A to U) with the same value of k 180 but with generate no crossovers, its neighboring track e ˆ  varying q and i as indicated by the corresponding 21 I q 11=20; i 55 has 160 crossovers showing the points (A to U) in the q; cos i space shown in Fig. 1. e€ect ˆ of aˆ small† change in the value of q. † The two arrows in each of the nominal ground tracks J q 1=2; i 95 is a highly inclined westward track indicate the subsatellite point at the s 0 and which ˆ showsˆ four† crossover points. The following three after a quarter orbital revolution s p= 2 .ˆ As† the tracks (K, L, M) are for the same value of q 2=3 but ˆ † ˆ tracks are periodic, we can limit our interest in a repeat with varying i. While both K i 70:5 and L i 85 period 0 s < 2Mp . are located in the mixed region, ˆ they are† quite di€erentˆ † The ®rst seven† tracks (A±G) correspond to as far as the number of crossovers is concerned; none in q N=M 1=3 but with varying i.Ai 0or K and twelve in L. The track M i 120 is another cosˆi 1 ˆis an eastward equatorial track. In theˆ period typical westward track having the ˆ same† number of of theˆ three† M orbital revolutions, the track makes two crossovers as L but with a quite di€erent topological M N eastward † equatorial revolutions with the net structure. e€ects of† one westward N revolution of the orbital The next three tracks (N, O, P) correspond to † node with respect to the earth. As a degenerate case, the q 1=1. N i 60 is a typical geosynchronous satellite track is con®ned to the equator, and one can say that its groundˆ track ˆ which† forms a ®gure-eight curve. The actual repeat period is not three M revolutions but one track librates both latitudinally and longitudinally † and half, i.e., M= M N , revolutions. B i 60 or spending 12 h successively in each of its northern and cos i 1=2 is a typically † inclined eastward ˆ ground southern loops. With q 1=1, this geosynchronous be- track.ˆ One† can say that the two longitudinal revolutions havior is preserved betweenˆ the geostationary case of the track A have been separated with an equatorial i 0 and the circumpolar geosynchronous case ˆ † symmetry. Now the repeat period becomes three M i 90 which is shown as the track O. As the track † ˆ † orbital revolutions. There are three crossover points (at becomes purely retrograde westward i > 90 , the lon- the equator) where the two longitudinal revolutions in- gitude no more librates but undergoes smooth† rotations tersect each other. C i 70:5 or cos i 1=3 is at the as shown in the track P i 120 . Each of the three border q cos i of the ˆ purely eastwardˆ region† and the tracks has only one crossover ˆ point† (as its epoch posi- eastward/westward ˆ † mixed region shown in Fig. 1. The tion). track crosses the equator vertically so that the angle The last ®ve tracks (Q±U) are for q 2=1 but with ˆ between ascending and descending traces becomes 180 varying i. With the track Q i 30 we are now in the ˆ † at the crossover points. With D i 85 or posigrade westward region. The track generates no ˆ cos i 0:087 we are now in the mixed region. The track crossover point. R i 60 , a cuspidal track, is a lim- is eastward † (resp. westward) above (resp. below) iting case located at theˆ border† q cos i 1 of the mixed / 59:2 , at which it becomes instantaneously meri- region and the posigrade westward region.ˆ † We see two j j  dional k0 0 . There exist nine crossovers with the cusps where the ground-track velocity instantaneously †ˆ appearance of six nonequatorial crossovers whose lon- becomes zero. With the track S i 85 , we are again in gitudes are aligned with those of equatorial crossovers. the mixed region showing a looping ˆ ground† track with E i 90 or cos i 0 is a circumpolar ground track an equatorial asymmetry. The cusps of R have been which ˆ consists of inclinedˆ † straight lines as both the lat- evolved into loops with the appearance of two cross- itude and longitude become linear functions of s, i.e., overs. Following another circumpolar track T with two /0 sgn cos s and k0 q. Each of the north and crossover points, the nominal ground track becomes  †  south poles becomes a collocation point of three cross- westward again as shown in the track U i 120 . overs, and we therefore still have nine crossovers. With While all nominal ground tracks have ˆ longitudinal† F i 120 or cos i 1=2 we are now in the retro- cyclic symmetry (i.e., the same pattern repeats M times), grade ˆ westward region.ˆ While† there still exist nine they also have either a symmetry or an asymmetry with crossover points, the longitudes of nonequatorial respect to the equator. The equatorial symmetry occurs crossover points are no more aligned with those of the when M and N have the same parity (both odd) and the equatorial crossover points. As the last example of equatorial asymmetry occurs when M and N have the q 1=3 cases, G i 180 or cos i 1 is a westward opposite parity. While one can count M equatorial equatorialˆ track. Likeˆ the track A,ˆ the† ground-track crossovers in the same parity cases, no equatorial con®guration is again con®ned to the equator. In the crossover point exists in the opposite parity cases due to period of the three M orbital revolutions the track the asymmetry. In its repeat period, a nominal ground makes four M N westward† equatorial revolutions track undergoes M latitudinal librations such that it and one can say‡ that† its actual repeat period is 3/4, i.e., crosses a zonal line, in sin / < sin i,2Mtimes (along M= M N , revolutions. The equatorial tracks each of its M ascendingj andj M descending traces). cos i ‡ 1† actually form degenerate cases, for which Meanwhile the longitudinal behavior can be diverse. In the ground-trackˆÆ † motion becomes one dimensional. case of a retrograde westward track, there are M N The next six tracks (H±M) are examples of equa- longitudinal revolutions which consist of M revolutions‡ torially asymmetric ground tracks characterized by the due to the satellite's orbital motion and N revolutions opposite parity of M and N. While H q 1=2; i 55 due to the net e€ects of the orbital plane's motion with closely corresponds to a ground track ˆ of the Globalˆ † respect to the earth. Such a track crosses a meridional Positioning System (GPS) satellites (Leick 1995) which line M N times. Since the M N longitudinal revo- ‡ ‡ 754 lutions interweave so that there exist M N 1 cross- satellite nominal ground-track con®gurations, one may over latitudes and each of the crossover‡ latitudes con- need to survey a ®ve-dimensional parametric space of tains M crossover points, the number of crossovers is i1; i2; q1 N1=M1; q2 N2=M2; D ke1 ke2 . Here D M M N 1 . In case of a purely eastward track, there represents ˆ the mutual longitudinalˆ originˆ of two† ground exist ‡M N †longitudinal revolutions which produce tracks, and is introduced to reduce the number of pa- M N 1 crossover latitudes and consequently rameters (from six to ®ve) by using the fact that the M M N 1 crossovers. In case of a posigrade west- relative geometry of two nominal ground tracks is in- ward track, there† exist N M longitudinal revolutions variant upon a simultaneous rotation around a ®xed which produce N M 1 crossover latitudes and con- axis. In Sect. 4 we will provide a more systematic sequently M N M 1 crossovers. For the eastward/ treatment for the count of crossovers in both the single- westward mixed tracks, † as will be veri®ed later, the and dual-satellite nominal ground-track con®gurations. number of crossover latitudes varies between M N 1 and M N 1 depending on the values of 3 Ground-track intersection cosjj iandjq, andj consequently‡ the number of crossovers varies from M M N 1 to M M N 1 with the Let r1 t1 and r2 t2 be two arbitrary orbital trajectories increment M. jj j j ‡ † † † referenced to the earth-®xed frame, where t1 and t2 are Some previous studies have also presented formulas time-tags (to be frequently omitted) used as independent to count the number of single-satellite nominal cross- variables of the two trajectories. To the extent that the overs appearing in a repeat period. Tai and Fu (1986) geocentric nadir mapping is concerned, the ground- claimed that the number equals M M N 1 if ‡ † track crossing condition for the two trajectories can be cos i > 0 and equals M M N 1 if cos i < 0 (note simply written as that M and N change their ‡ notational † roles in Tai and 2 Fu). Spoecker (1991) claimed M . Considering the case r^1 t1 r^2t2 10 q < 1, Cui and Lelgemann (1992) claimed †ˆ † † M M rN 1 where r sgn cos i . Along with a sign where r^1 and r^2 are geocentric unit vectors pointing r1 error in Tai and† Fu, andˆ the complete † ignorance of the and r2, respectively. earth rotation e€ects in Spoecker, these studies passed Let F denote the direction cosine between r1 and r2, over the fact that the crossing geometry of a single- i.e., satellite nominal ground track is more signi®cantly Ft1;t2 cos w t1; t2 r^1t1r^2t2 11 variant in the two-dimensional parametric space of †ˆ †ˆ †Á † † q; cos i . The reciprocal of q has been called the where w represents the spherical angular distance ground-track † parameter. Although some of its impor- between the two ground tracks. Locating ground-track tant roles have been recognized in the counting of crossovers can be considered as a two-dimensional root- crossover latitudes (King 1976; Farless 1985; Parke et al. ®nding problem of a single scalar equation Ft1;t2 1, 1987), we claim that all of these previous studies missed which replaces the vector equation in Eq. (10). †ˆ Since F the existence of the eastward/westward mixed tracks and yields its maximal value 1 at crossovers, the problem can the posigrade westward tracks. be solved by using the extremal condition In Fig. 3 we overlay a pair of nominal ground tracks _ _ with i1 60; q1 1=3; k 1 180 and i2 95; Ft1 ^r1 r^2 0; Ft2 r^1 ^r2 0 12 †ˆ ˆ e ˆ ˆ ˆ Á ˆ ˆ Á ˆ † q2 1=2; ke2 155 . In addition to the three cross- oversˆ of the eastwardˆ † track and the four crossovers of where the subscripts t1 and t2 indicate partial di€eren- tiations. Given a t1; t2 point close to a crossover the westward track, one can count ten points where the fg two tracks intersect. These intersections, ordered from 1 solution, the Newton-Raphson iteration scheme to 10 along the eastward track, are nominal examples of 1 t1 t1 ^r1r^2 ^r_1^r_2^r_1r^2 the dual-satellite crossover point. To fully classify dual- Á Á Á13 t2 t2 ^r_1^r_2 r^1^r2 r^1^r_2 †   Á Á Á may converge to the solution. The iteration fails when the determinant of the 2 2 Hessian matrix, i.e., the Gaussian curvature of F , Â

2 2 G Ft t Ft t F ^r1 r^2 r^1^r2 ^r_1^r_2 14 ˆ 1 1 2 2 t1t2 ˆ Á ÁÁ † vanishes. ÁÁÁ Let K be the angle between ^r_1 and ^r_2, i.e.,

^r_1 ^r_2 ^r_1 ^r_2 cos K 15 Á ˆ k † At a crossover point, , using the relation r^1 r^2 ^r r^ ^r_ 2, Eq. (14) yields ˆ Á ˆ 2 2 2 2 2 2 G ^r_1 ^r_2 ^r_1 ^r_2 ^r_1 ^r_2 sin K 16 Fig. 3. Overlay of two nominal ground tracks v ˆ Á † ˆ †

755 where the subscript v, to be frequently omitted on the Now we present an alternative approach by repre- right-hand sides of equations, indicates evaluation at the senting the orbital states of satellites in terms of spinors, crossover point. We note that the iteration failure occurs which enable us to rewrite the ground-track crossing when the ground tracks become collinear condition in a more convenient form. At this point, the K 0 or 180 . author encourages the reader to refer to the Appendix, Aˆ converged† solution of Eq. (13) does not always which presents a summary on spinors, their algebra, and yield a crossover solution (where F 1, a maximum of the two-way transformations between the vector state F , or equivalently w 0, a minimumˆ of w) since other and the spinor state. Some equations in the Appendix, types of extrema alsoˆ satisfy Eq. (12). In other words, e.g., the spinor multiplication laws in Eq. (A1), will be Eq. (12) forms only a necessary condition for a cross- frequently used in the following development. over. Figure 4 shows the variation of w in the two-di- Let e^ ; e^ ; e^ be a right-handed orthogonal triad of f x y zg mensional domain 0 s1=2p < M1 3; 0 s2=2p < unit vectors and suppose it forms the earth-®xed frame.  ˆ  M2 2 associated with the two nominal ground tracks We consider the spinor decomposition of the two orbital shownˆ † in Fig. 3. Among the 14 minimal points of w in position vectors the white area, 10 points correspond to crossovers which r1 Ry e^ R1; r2 Ry e^ R2 18 are numbered from 1 to 10, consistent with Fig. 3. The ˆ 1 x ˆ 2 x † remaining four minima labeled by L are local minima. with One may also count 14 maxima in the gray area. There actually exist 56 extremal points including 14 saddle R1 a1 jb1e^x c1e^y d1e^z ˆ ‡ ‡ ‡ 19 points in each of the white and gray areas. The extrema R2a2jbÁ2e^x c2e^y d2e^z † of F can be classi®ed as ˆ‡‡ ‡ Á

Crossovers F 1 Maxima H < 0 ˆ † Centers G > 0 †hLocal Maxima F < 1 17 Extrema F 0 †hMinima H > 0 † † r ˆ †hSaddles G < 0 † †

where H denotes the mean curvature of F given by where a1; b1; c1; d1 and a2; b2; c2; d2 denote the real f g f g H Ft1t1 Ft2t2 =2. With the iteration scheme, Eq. (13), components of the spinors R1 and R2 respectively, j is weˆ need to‡ check† whether a converged solution is a the unit imaginary number, and the superscript crossover or not. With the symmetry in the numbers of denotes the spinor conjugate. y centers and saddles, the symmetry in the numbers of With Eq. (18) the ground-track crossing condition maxima and minima, and the existence of local maxima, Eq. (10) can be rewritten as less than a quarter of converged solutions will be crossovers. Ry e^ R Ry e^ R 1 x 1 2 x 2 20 r1 ˆ r2 † where, for the magnitudes of the two vectors, we have

r1 Ry R1 R1Ry ; r2 Ry R2 R2Ry 21 ˆ 1 ˆ 1 ˆ 2 ˆ 2 † Premultiplying R1 and postmultiplying R2y , Eq. (20) yields

e^ R1Ry R1Ry e^ 22 x 2 ˆ 2 x † on using Eq. (21). We introduce a new spinor

Q R1Ry A jBe^ Ce^ De^ 23 ˆ 2 ˆ ‡ x‡ y‡ z † Á Substituting Eq. (19) into the product in R1R2y in Eq. (23), which is also given in Eq. (A5), one obtains

Fig. 4. Spherical angular distance (in degree) between two nominal ground tracks 756

A a1a2 b1b2 c1c2 d1d2 r_1 g1 r1 r_2g2r2 F 1 j e^ m1 e^ ;Fyje^m2e^ ˆ ‡ ‡ ‡ ˆ r ‡ r2 z ‡ g x 2ˆrr2z‡gx B b1a2 a1b2 d1c2 c1d2 1 1 1 222 ˆ ‡ 32 C c1a2 d1b2 a1c2 b1d2 ˆ ‡ † D d1a2 c1b2 b1c2 a1d2 24 where g's are the magnitudes of the total angular ˆ ‡ † momenta, With Eq. (23), Eq. (22) yields 2 g g1g^ r1 r_1 r r^1 ^r_1 1 ˆ 1 ˆ  ˆ 1  33 e^xR1R2y R1R2y e^x e^xQ Qe^x 2 De^y Ce^z 0 2 ˆ ˆ ˆ g g2g^ r2 r_2 r r^2 ^r_2 † 2 ˆ 2 ˆ  ˆ 2  Á25 † and m's are the e€ective normal perturbations of the on using Eq. (A1). Note that the ground-track crossing satellites in the earth-®xed frame. condition is now reduced to Substituting Eq. (31) into Eq. (30), we have C t ; t 0; Dt;t 0 26 1 1 1 2 1 2 Q F 1R1Ry F 1Q †ˆ †ˆ † t1 ˆ 2 2 ˆ 2 34 whereas and can be arbitrary. 1 1 † A B Q R1 Ry F y Q F y We will refer to the two scalar equations in Eq. (26) t2 ˆ 2 2 2 ˆ 2 2 as the spinor representation of the ground-track cross- Now, substituting Eq. (32) into Eq. (34), in comparison ing condition. In the t1; t2 domain the crossovers are with Eq. (30), we obtain located at the intersections † of the two nonlinear curves; 1 r_1 g1 r1 i.e., we are again working with a two-dimensional root- A A D m1 B t1 ˆ 2 r r2 g ®nding problem for a set of the unknown crossover time 1 1 1 pairs. The two equations in Eq. (26) can be solved with 1r_1g1r1 the Newton- Raphson iteration scheme Bt1B2Cm1A ˆ2r1‡r‡g1 1 35 1 1r_1g1r1 † t1 t1 Ct1 Ct2 C 27 Ct1C2Bm1D t2 t2 Dt Dt D † ˆ2r1r1‡g1   1 2   1r_1g1r1 with DDAm1C t1ˆ2r‡r2g 111 C c_1a2 d_1b2 a_1c2 b_1d2 and t1 ˆ ‡ _ _ 1 r_2 g2 r2 Ct2 c1a_2 d1b2 a1c_2 b1d2 A A D m B ˆ ‡ 28 t2 2 2 ˆ 2 r2 ‡ r2 ‡ g2 Dt d_1a2 c_1b2 b_1c2 a_1d2 †  1 ˆ ‡ 1r_gr _ _ B2B2Cm2A Dt2 d1a_2 c1b2 b1c_2 a1d2 t2 22 ˆ ‡ ˆ2r2‡rg2 2 36 obtained on di€erentiating the last two equations in 1r_2g2r2 † Eq. (24). It is quite noteworthy that the condition of Eq. Ct2C2Bm2D ˆ2r2r‡g2 (26) is necessary and sucient, and therefore a converged 2 solution of the spinor-version iteration scheme Eq. (27) 1r_2g2r2 Dt2D2Am2C always yields a crossover, in contrast to the vector- ˆ2r2rg2 2 version iteration scheme of Eq. (13). Equation (27) fails when the determinant of the 2 2 Jacobian matrix, as similar to Eq. (A34).  With Eqs. (35) and (36), at crossovers, where J C D D C 29 C D 0, the Jacobian in Eq. (29) becomes ˆ t1 t2 t1 t2 † ˆ ˆ 1 g1 g2 i.e., the Jacobian of the crossing condition Eq. (26), Jv 2 2 AB 37 vanishes. ˆ 2 r1 r2 † To further analyze the Jacobian, we take partial which, as will be veri®ed, vanishes when the ground derivatives of Q in Eq. (23), i.e., tracks become collinear in accordance with Gv in Eq. (16). _ y Q R1R At1 jBt1e^x Ct1e^y Dt1e^z We consider the two rotating satellite-®xed orbital t1 ˆ 2 ˆ ‡ ‡ ‡ 30 _ † frames (relative to the rotating earth-®xed frame); QR1RyAjBÁe^ Ce^ De^ t2ˆ2ˆt2‡t2 x‡ t2 y‡ t2 z _ _ r^1 e^x r^2 e^x For the derivatives ÁR1 and R2y , from Eqs. (A44) and 1 1 (A45), we have g^1 r^1 R1y e^y R1; g^2 r^2 R2y e^y R2 8  9 ˆ r1 8 9 8  9 ˆ r2 8 9 < g^1 = < e^z = < g^2 = < e^z = _ 1 _ 1 R1 F 1R1; Ry Ry F y 31 38 ˆ 2 2 ˆ 2 2 2 † : ; : ; : ; : ; † with The spinor algebra enables us to write 757

1 between the Jacobian (appearing in the spinor-version g^1 g^2 g^1g^2 R1ye^zR1R2ye^zR2 Á ˆ hiˆr1r2 iteration scheme) and the Gaussian curvature (in the 1DE vector-version) at crossovers. ezR1R2ye^zR2R1y 39 ˆr1r2 † 1 DE1 4 Number of crossovers e^ Qe^ Qy A2B2C2D2 ˆ r r z z ˆrr‡ 1 2 12 The ground-track crossing condition derived in the DEÁ on using Eq. (23). In Eq. (39), the bracket denotes the previous section is exact and general as far as the scalar operator in which the product of two spinors is geocentric nadir mapping is concerned. Given orbital commutative as applied to R1y and e^zR1R2y e^zR2. In similar ephemerides of two satellites in their ®nite mission ways, one can obtain the remaining eight direction periods, the crossover locating problem is now reduced cosines between the two orbital frames to write to solve the crossing condition. How many crossovers

2 2 2 2 r^1 A B C D 2 BC AD 2 BD AC r^2 1 ‡ 2 2 ‡ 2 † 2 † g^ r^1 2 BC AD A B C D 2 CD AB g^ r^2 40 8 1  9 ˆ r r 2 38 2  9 † g^ 1 2 2 BD AC† 2 CD‡ AB A2 B2 ‡C2 † D2 g^ < 1 = ‡ † † ‡ < 2 = 4 5 : ; : ; which generalizes Eq. (A18). With the relation exist? And when do they (pairs of time-tags) occur? Strictly speaking, the number of solutions can be a y 2 2 2 2 Q Q R1R2y R2R1y r1r2 A B C D 41 priori unknown as highly perturbed, irregularly ˆ ˆ ˆ ‡ ‡ ‡ † maneuvered satellites can generate signi®cantly com- Eq. (40) yields r^1 r^2, and ˆ plicated ground-track patterns, and there is nothing 22 special about any crossover point from either satel- g^1 r^1 1AB2AB g^2r^2  222 2 Âlite's point of view. Let us consider instead the g^1 ˆAB2AB A B g^2 v‡ vnumber of crossovers in nominal ground tracks. In 42 practice, as most of geodetic satellites have near- † circular orbits, nominal ground tracks often have at crossovers C D 0 . ˆ ˆ † topologically equivalent structure to true ground With the relations tracks. Nevertheless, no satisfactory answer has been g1 g2 given to the question regarding the number of cross- ^r_1 g^ r^1; ^r_2 g^ r^2 43 ˆ r2 1  ˆ r2 2  † over solutions between two nominal ground tracks 1 2 (even for single-satellite nominal ground tracks as

Eq. (42) states that, at crossovers, the angle between g^1 discussed before). _ _ and g^2 is same as that of ^r1 and ^r2. (The ground-track We ®rst consider the single-satellite nominal ground- crossing angle is essentially formed by two osculating track crossing by representing the spinor Q in terms of orbital planes instantaneously passing through the cross- Euler angles which are less useful than spinor compo- over point. When observed from the earth-®xed frame, nents for numerical purposes but superior in providing the orientations of the orbital planes can be e€ectively some qualitative insights. Substituting Eq. (A48) into represented by the unit vectors g^1 and g^2.) We can write Eq. (24), with some trigonometry one can derive A2 B2 2AB cos K ; sin K 44 A pr1r2 sin s cos i sin # cos s cos i cos # v ˆ A2 B2 v ˆ A2 B2 † ˆ † ‡ ‡ Bpr1r2 sin s sin i sin # cos s sin i cos # ‡ ‡  ‡ ‡ ‡ so that the correct quadrant of the crossing angle Kv is ˆ †‡ ‡ Cpr1r2 cos s sin i sin # sin s sin i cos # de®ned in 0; 360 . ˆ †‡ ‡ ‡ ‡ With Eq.‰ (44), Eq.† (37) yields Dpr1r2 cos s cos i sin # sin s cos i cos # ˆ †‡ ‡ ‡ 48 1 g1 g2  † Jv sin Kv 45 ˆ 4 r1 r2 † with

Meanwhile, with Eq. (43), Eq. (16) can be written as 1 1 s s1 s2 ;ii1i2; Æ 2 Æ2 2 ˆ 1 †Æ ˆ †Æ 49 g1 g2 ##1#2 † ˆ2 † Gv 2 2 sin Kv 46 ˆ r1 r2 †  From the last two equalities in Eq. (48), with i1 i2 i, ˆ ˆ We therefore obtain a simple relation q1 q2 q, ke1 ke2 ke,and# qs , the ground- trackˆ crossingˆ conditionˆ ˆ C 0 and ˆD 0 yield 4J 2 ˆ ˆ G v 47 v ˆ r r † cos s sin qs sin i 0 50 1 2 ‡ ˆ † 758 and cos s sin qs cos i sin s cos qs 51 ˆ † Meanwhile we restrict our interests to a repeat period which corresponds to the two-dimensional domain 0 s2 < s1 < 2Mp 0 < s

2 2 2 1 cos sà cos qsà Kà q KÃq 058 † ˆcos s sin qs  † à à Á which indicate the range of the extremun KÃ. In other words, the function K is monotone when K < 0 and 1 1 when min q; q

Fig. 5. The s1; s2 domain for single-satellite nominal crossover 2M solutions of s and the L solutions of s ) in the two- solutions † dimensional domain‡ of 0 < s < Mp; 0 s < 2Mp . †  ‡ 759

Since we are working in its semidomain 0 < s when both ground tracks are circumpolar, it is essen- < Mp; s s < 2Mp s , we say that the number tially a quadratic equation and may yield 0, 1, or 2 real of single-satellite  ‡ nominal crossovers† is ML. solutions depending on the parametric values. The As we have seen in the two-dimensional parametric existence of a solution (or solutions) implies that the space of single-satellite cases (which forms a subspace of corresponding parametric point is in bifurcating regions the ®ve-dimensional parametric space of dual-satellite where the number of crossovers can be changed as the cases), changes may occur in the number of crossover parameter values vary. 1 solutions in the mixed region 0 < cos i < min q; q . In Fig. 7 we display eleven i1; i2 sections of the † These changes correspond to ``bifurcations'' i1; i2; q1; q2 space by varying the values of q1; q2 . The Á † † (Guckenheimer and Holmes 1986, p. 117): the term eleven points (A±K) in the q1; q2 section at the right † originally used to describe the splitting of equilibrium lower corner depict the approximate values of q1; q2 † solutions in a family of di€erential equations at para- corresponding to the eleven i1; i2 sections. In each of † metric points or regions where the Jacobian derivative of the i1; i2 sections, bifurcation regions are striped the di€erential equations has a zero eigenvalue. In our (horizontally, † vertically, or in both ways). The hori- case, the splitting of crossover solutions occurs where zontally striped regions correspond to the existence of the Jacobian of the ground-track crossing condition the 0 crossing angle solution, and the vertically striped Eq. (26) is singular. We know that the Jacobian vanishes regions correspond to the existence of the 180 crossing when the ground-track crossing angle becomes 0 or angle solution. While both roots exist in the regions 180. In other words, as it can be quite naturally un- where the horizontal stripes and vertical stripes cross, no derstood, births and deaths of crossover solutions can real root exists in the white areas, which will be called occur at parametric regions where the two ground tracks regular regions. The lines labeled a±g are can be collinear. Such regions will be referred to as bi- a. sin i2 i1 q1sin i2 q2 sin i1 furcating regions. †ˆ We now return to the ground-track slope equation b. sin i2 i1 q1sin i2 q2 sin i1 (5). One can write the ground-track collinear condition †ˆ‡ ‡ c. cos i1 cos i2 by ˆ‡ d. cos i1 cos i2 64 1 d/ 1d/ ˆ † 60 e. q2 cos i2 1 cos / dk 1ˆcos / dk 2 † ˆ  f. q1 cos i1 1 ˆ which yields g. Y 2 4XZ 0 ˆ 2 2 2 cos / cos i2 cos i1 q1 cos / 61 at which the characteristics of the parametric space can 2 2 2 be signi®cantly changed. pcos / cosÁi1 cos i2 q2 cos / † ˆÆ In each of the eleven sections, there essentially exist where,p in the dual sign,Á ``plus'' is for the 0 crossing four regular regions (left, right, lower, and upper white angle cases (i.e., both tracks are north-going or south- areas) which are separated by the diagonal and antidi- going) and ``minus'' is for the 180 crossing angle cases agonal bifurcating regions. We focus our discussion on (i.e., one track is north-going and the other south- the section A q1 N1=M1 1=3; q2 N2=M2 1=2 †ˆ ˆ ˆ ˆ going). It is noteworthy that D is absent in Eq. (61), and which is typical for two satellites below the geosyn- we can therefore work with a four-dimensional para- chronous altitude (i.e., q < 1). In each of the four reg- metric space of i1; i2; q1; q2 to search the bifurcating ular regions, the number of crossovers is invariant. Since † regions. As the roles of two tracks can be exchanged, we one can consider the four limiting points i1 0; ˆ can set q2 q1. Taking squares of both sides, Eq. (61) i2 90 , i1 180; i2 90 ,i190; i2 0 ,i1  ˆ † †ˆ ˆ †ˆˆ ˆ can be written as 90; i2 180 as representative points of the four reg- ular regions,ˆ the† corresponding numbers of crossovers 6 4 2 X cos / Y cos / Z cos / 0 are readily summarized as 2M2 M1 N1 ;2M2M1N1, ‡ ‡ ˆ jj †‡ 2 2 2 2M1M2N2;2M1M2N2, respectively. For in- with max cos i1; cos i2 cos / 1 62 jj †‡  † stance, with i1 0; i2 90 each of the M2 revolu- †ˆ ˆ and ÈÉ tions of the circumpolar ground track crosses with each of the M1 N1 revolutions of the eastward equatorial X = q2 q2 track twicejj (one along its ascending trace and one along 1 2 its descending trace). In the bifurcating regions, the 8Y =2q2cos i2 q1 cos i1 †22 2 2 number of crossovers cannot be easily guessed and may >qcos i1 q cos i2 63 >‡2 1 † depend on the value of the remaining parameter D.In >2 2 2cosi1 cos i2 q1 cos i2 q2 cos i1 >‡ † ground tracks, such a small change can be easily intro- > Equation:> (62) is a cubic equation in cos2 /. Excluding duced as the parameters experience perturbed varia- the peculiar triple solution cos2 / 0, which can occur tions. ˆ 760

Fig. 7. Examples of i1; i2 parametric sections †

5 Applications TOPEX/POSEIDON (which has been orbiting since 10 August 1992). For each of the three satellites, after We now aim to ®nd crossovers associated with ephemer- checking their repeatability using the ephemerides to be ides of actual satellites. Those satellites are GEOSAT described, we selected a complete repeat cycle: (which ¯ew from March 1985 to December 1989), ERS-1 17.050458355 days from 02:55:43 10 April 1987 in the (which has been orbiting since 22 April 1991), and GEOSAT Exact Repeat Mission (ERM), 34.999940729 761 days from 03:45:56 22 February 1993 in the ERS-1 Phase- E gg C mission, and 9.915647338 days from 04:29:45 2 March Eg 1994 in the TOPEX/POSEIDON mission. The three Iterate if E isn't small enough repeat missions have, respectively, iG 108; qG j j   t x_ t y_ t z_ NG=MG 17=244 , iE 98:5; qE NE=ME 35=501 ,x ; y ; z  † †  ˆ o 1 o 1 o 1 and iT 66; qT NT=MT 10=127 where the sub- ˆ g ˆ g ˆ kg †   t ‡ t t ‡ t ‡ scripts denote mission initials. We can say that the E 2 xox yoy kzoz ˆ o ‡ o ‡ o GEOSAT and ERS-1 ground tracks are retrograde Et westward, and the TOPEX/POSEIDON ground track is _Á gg posigrade eastward. ˆE t g t g t g Orbital trajectories of actual satellites are often pro- x_o xo g_xo; y_o yo g_yo; z_o zo g_zo ˆ ‡ ˆ ‡ ˆ ‡ vided in terms of tabulated ephemerides. In addition to the TOPEX/POSEIDON ephemerides (Tapley et al. starting from a proper positive value of g. The 1994), the GEOSAT-ERM and ERS-1 Phase-C ephe- superscript t and g appearing in the above equations merides used were consistently generated using the imply partial di€erentiations. Although the ground- UTOPIA orbit determination software system at the track crossing condition derived before was based on the Center for Space Research at the University of Texas at mapping of satellite states to a spherical-earth model, it Austin. Each of the three satellites' orbital trajectories is noteworthy that a generalized surface mapping can be was given in the form of a discretely tabulated time- re¯ected simply by using the nadir-mapped ground- series of earth-®xed geocentric equatorial Cartesian track states instead of the satellite's orbital states in states at 10-s intervals. For usage of such tabulated constructing the crossing condition and solving it to ®nd states in the search of ground-track crossovers, we need crossovers. Here, for the computation of the three an ecient and accurate interpolator to compute the satellites' ground tracks, we use the ellipsoidal nadir state at any instant time in the table. For the state in- mapping of satellite states on the TOPEX/POSEIDON terpolation, we adopted the Hermite interpolation standard reference ellipsoid of revolution with scheme (Ferziger 1981, p. 12). To ®nd the location of a ae 6378136:3 m and be 6356751:6 m (the reciprocal speci®c time in the state tables, the routine ``hunt'' in ¯atteningˆ of 298.257). ˆ Press et al. (1922, pp. 111±113) is used. To ®nd actual crossovers, one can pursue an ap- The spherical earth so far used to describe ground proach speci®c to a particular combination of ground tracks can be replaced by a more realistic earth-surface tracks. However, in general we have to map out the full model. Let the subsatellite point ro xo; yo; zo be a t1; t2 domain. ``There are no good, general methods for normal projection of the satellite's positionˆ fgr x;y;z solving† systems of more than one nonlinear equation'' as ˆf g onto a generalized earth-surface model E ro 0. The stated in Press et al. (1992, p. 372). Here, we pursue a surface normal projection to minimize the †ˆ satellite's al- grid-search scheme by setting initial guesses of solutions titude above the surface can be formulated by on a regular grid of the corresponding t1; t2 domain; the grid scheme can be outlined as † 2 Minimize P ro; g rro gEro 65 †ˆjj‡ † † Loop for t1 t1 ; t1 t1 ; t1 t1 Dt1 k ˆ i k  f k ˆ k ‡ † where g is the Lagrange multiplier. A proper earth- Loop for t2 t2 ; t2 t2 ; t2 t2 Dt2 k ˆ i k  f k ˆ k ‡ † surface model is the geoid, especially for oceanographic 0. Initialize iteration: t1; t2 t1 ;t2 ; f gˆf k kg applications of satellite altimetry, as employed by Shum 1. Compute a1; b1; c1; d1; a_1; b_1; c_1; d_1 at t1 and (1982) by utilizing a spherical harmonics geopotential a2; b2; c2; d2; a_2; b_2; c_2; d_2 at t2; model. In practice, as it is widely adopted in satellite 2. Compute Ct1 ; Dt1 ; Ct2 ; Dt2 using Eq. (28); geodesy, the geoid is well approximated by a reference 3. Compute J using Eq. (29); ellipsoid of revolution (Torge 1991, pp. 44±49) with its 4. If J is tiny, abort iteration; semimajor axis ae and semiminor axis be, such that 5. Update t1; t2 using Eq. (27); f g 6. If t1 t1k > pDt1 or t2 t2k > pDt2, abort 2 j j j j 2 2 2 2 ae iteration; E xo yo kzo ae with k 2 66 ˆ ‡ ‡ ˆ be † 7. Compute C; D using Eq. (24); 8. If C and D are not suciently small, go to 1; Albeit for the ellipsoidal nadir mapping, closed-form 9. If t1i t1 t1f and t2i t2 t2f , store t1; t2 as formulas exist with some complexities (Grafarend and a crossover  time-tag pair;  f g Lohse 1991), we prefer the sequences End Loop x y z End Loop x ; y ; z o 1 o 1 o 1 k ˆ g ˆ g ˆ g Sort the stored crossover time-tag pairs and remove ‡ ‡ 2 ‡ E xoxo yoyo kzozo a duplicated pairs. ˆ ‡ ‡ e g xo g yo g kzo xo ; yo ; zo In this outline, and denote the initial and ®nal ˆ 1 g ˆ 1 g ˆ 1 kg t1i t1f g ‡ g g ‡ g ‡ times for the ®rst satellite, t2i and t2f denote the initial E 2 xox yoy kzoz ˆ o ‡ o ‡ o and ®nal times for the second satellite, t1k and t2k denote Á 762 initial guesses of crossover solutions on the grid knots, 6 Conclusions Dt1 and Dt2 denote the grid-knot steps, and p denotes a parameter to determine the extent of duplicated survey Techniques to locate satellite ground-track crossovers by controlling the size of a box, which is centered at the have been frequently presented as an auxiliary proce- initial grid knot t1k; t2k and brackets the solution dure of other main topics related to the applications of search. For the computationf g of spinor states, we need satellite altimetry. In this paper, we have studied the interpolation of the vector states using ephemerides ta- issue as an independent topic. The principal new aspects bles, their nadir mappings, and vector-to-spinor ground- investigated (or presented) herein are: (1) classi®cation track state transformations. The outline is written for an of idealized nominal ground tracks in the two- illustrative purpose of a general dual-satellite crossover dimensional parametric space of q; cos i (2) classi®ca- locating problem, and may not be an optimal scheme for tion of dual-satellite nominal ground-track † bifurcations a speci®c problem. in the four-dimensional parametric space of i1; i2; q1; q2 There are six ground-track con®gurations for the in conjunction with the number of crossover solutions,† three repeat cycles of GEOSAT-ERM, ERS-1, and and (3) numerical searching schemes of actual cross- TOPEX/POSEIDON; three single-satellite cases and overs in both vector and spinor forms. Indeed, some of three dual-satellite cases. With the Hermite state inter- these interesting aspects seem not to have been surveyed polator, the ellipsoidal nadir mapping, and the two-way before. transformations between the vector and spinor states at In a problem related to satellite orbits, a proper se- our disposal, we applied the grid-search scheme to all lection of variables often leads to simpli®cation of the the six cases. We have deployed the initial guesses of problem, as we have used the spinor representation of crossover solutions on a 1=8 1=8-revolution grid re- the satellite state to establish the ground-track crossing gardless of single- or dual-satellite ground-track sce- condition. In satellite altimetry, a more conventional narios. In other words, the grid-knot step size is 754.69, crossover locating method is based on usage of the 754.49, and 843.22 s, respectively for GEOSAT-ERM, geodetic latitude and longitude, see, e.g., Rowlands ERS-1, and TOPEX/POSEIDON. We used p 1:0so (1981) or Wisse et al. (1995, pp. 72±76). The use of that the search at a grid knot covers a 1=4 ˆ1=4-re- geodetic coordinates originated from the fact that the volution box.  satellite-altimeter geophysical data record (GDR) al- The number of single-satellite crossovers numerically ready contains them along with the altimeter-range found are 63440, 268035, and 14732, respectively for the measurement, its time-tag, and other environmental/ three cycles. These numbers agree exactly with the geophysical corrections. However, when one pursues the number of nominal single-satellite crossovers MG MG direct locating of crossovers from satellite-state tables, ‡ NG 1 ; ME ME NE 1 ; MT MT NT 1 of the the vectors or spinors will be more adequate. Once the three repeat† missions.‡ The† number of dual-satellite† unknown crossover time-tags are obtained, it is a matter crossovers found are 261522, 117234, 57638, respec- of data hunting and interpolation to get the corre- tively, for the three combinations of GEOSAT-ERM sponding altimeter data records. One may also prefer and ERS-1, ERS-1 and TOPEX/POSEIDON, and the vector version of the iteration scheme to the spinor GEOSAT-ERM and TOPEX/POSEIDON. The ®rst version, as it can be more understandable. The main two numbers exactly match with 2ME MG NG and advantage of using spinors is that a converged solution ‡ † 2ME MT NT , indicating that they are located at the of iteration always yields a crossover. The main dis- regular regions. † However the number of crossovers be- advantage of using spinors is that, as the satellite states tween the GEOSAT-ERM and TOPEX/POSEIDON are usually given by Cartesian vectors, one needs con- cannot be easily explained. Returning to Fig. 7 and versions between the vector and spinor state. There is a seeing the section J q1 17=244; q2 10=127 , which trade-o€ unless the satellite states are initially given by is topologically equivalent ˆ to the sectionˆ A, one† may spinors. However, since no trigonometrics are involved approximately locate the parametric point in the state conversions, the increase of computational i1 108; i2 66 corresponding to the dual ground labor produced by using the spinor- language should not tracks of the GEOSAT-ERM† and TOPEX/POSEI- be overestimated. The selection of variables will be DON. The point is located in the vertically striped bi- highly dependent on one's object. furcating region along the diagonal. It is located between the lower regular region and the right regular Acknowledgements. Thanks are due to B. Tapley, B. Schutz, and region and the number 57638 is between R. Eanes for their encouragement and interest on this subject, to J. Ries for providing ephemerides of the three altimeter satellites, 2MG MT NT 57096 and 2MT MG NG 66294. For the combination †ˆ of the GEOSAT-ERM ‡ †ˆ and TO- to S. Bettadpur for reading a part of the paper in its ®rst draft, and to Y. Hahn in Jet Propulsion Laboratory and M. Anzenhofer in PEX/POSEIDON, Eq. (62) yields a real solution GFZ (GeoForschungsZentum, Potsdam) for ®nding some refer- 2 cos / 0:785, indicating that crossovers near 27:5Sor ences. Sincere appreciation is due to anonymous reviewers whose ˆ 27:5N have small acute angle and therefore a slight comments were helpful in improving the paper. perturbation can easily generate or destroy crossovers. These results support the bifurcation analysis presented in the previous section. 763

1=2 1=2 1=2 Appendix R RRy RyR a2 b2 c2 d2 j jˆ † ˆ † ˆ ‡ ‡ ‡ † A7 Satellite state in spinor space † The spinor R can be decomposed into R R^ where R^ is a Let e^x; e^y; e^z and e^u; e^v; e^w be right-handed orthogo- ^ j j f g f g unimodular spinor (i.e., R 1. nal triads of geocentric unit vectors which form the The spinor algebra isj equivalentjˆ † to the quaternion rotating earth-®xed frame and the non- rotating inertial algebra, which has long been known as the ideal tool to frame, respectively. In the spinor algebra, which forms a describe rotational motions of rigid bodies. In practice, subalgebra of the so-called Cli€ord algebra [geometric the four components of a unimodular spinor correspond algebra in Hestenes (1983)], such a triad corresponds to to the Euler parameters (Goldstein 1980, p. 155). For the Pauli spin matrices (Goldstein 1980, p.156) and instance, the earth's rigid body rotation can be e€ec- yields the spinor multiplication laws (Cartan 1981, p.44; tively represented by Landau and Lifshitz 1977, p.202), e.g., e^x e^u e^u e^ e^ e^ e^ e^ e^ 1 x x y y z z e^ E^y e^ E^ x e^ A8 ˆ ˆ ˆ A1 8 y 9 8 v 9 e 8 v 9 e^ e^ je^ e^e^ (in cyclic order) † e^ ˆ e^ ˆ  e^ † x y ˆ z ˆ y x < z = < w = < w = where j is the unit imaginary number. where: ;E^ is a: unimodular; spinor: which; accounts for the Let v1 b1e^ c1e^ d1e^ and v2 b2e^ c2e^ precession, nutation, spin axis rotation and polar ˆ x ‡ y ‡ z ˆ x ‡ y d2e^z be two arbitrary vectors. With the multiplication motion of the earth (Lambeck 1988), and xe is the ‡laws of Eq. (A1), one can readily verify the relation corresponding angular velocity vector. There is a complete analogy between the orbital an- v1v2 v1 v2 j v1 v2 A2 ˆ Á ‡  † gular motion of a satellite and the rotational motion of a rigid body. Let s ue^u ve^v we^w represent the orbital between the spinor product, the scalar product, and the position vector ofˆ a satellite‡ ‡ in the geocentric inertial vector product of the two vectors. Equation (A2) can be frame. Let S a j be^u ce^v de^w be a spinor based also represented by on the triad ˆe^ ;‡e^ ; e^ .‡ As it‡ can† be directly veri®ed f u v wg v v 1 v v v v using the multiplication laws of Eq. (A1), which is also 1 2 2 1 2 2 1 applicable to e^ ; e^ ; e^ , the spinor algebra enables us Á ˆ 1 ‡ † u v w v1 v2 jv1v2 v2v1 A3 f g  ˆ2 † † to write The spinor space is a combination of the scalar and s Sye^uS A9 vector spaces such that a spinor R a j v and its ˆ † ˆ ‡ conjugate Ry a j v, where v be^x ce^y de^z, are with de®ned by theˆ four real-valuedˆ scalar‡ components‡ 2 2 2 2 a; b; c; d . It is equivalent to the two-dimensional u a b c d ; v 2 bc ad ; w 2 bd ac f g ˆ ‡ ˆ ‡ † ˆ † complex vector space appearing in quantum mechanics A10 in conjunction with the wave functions (Greiner 1994, p. † 306). We will frequently denote the scalar part of R with Let s be the magnitude of s. We also have R , i.e., 1=22 h i s s u2 v2 w2 SSSySyS 1 ˆjjˆ ‡ ‡ ˆˆˆA11 R a Ry R Ry A4 2222 † h iˆ ˆh iˆ2 ‡ † † aÁbcd ˆ‡‡‡ where the bracket will be called the scalar operator. hi On di€erentiating Eq. (A9), we have Let R1 a1 j v1 and R2 a2 j v2 be two arbitrary spinors. Withˆ Eq.‡ (A2), oneˆ obtains‡ _ _ s_ Sye^ S Sye^ S A12 ˆ u ‡ u † R1R2y a1 jv1 a2 j v2 which can be expanded to relate the scalar components ˆ ‡ † † _ _ _ a1a2 v1 v2 j a2v1 a1v2 v1 v2 A5 of s u_e^u v_e^v w_ e^w and those of S a_ j be^u c_e^v ˆ ‡ Á ‡ ‡  † † ˆ ‡ ‡ ˆ ‡ ‡ d_e^w . One can actually write which can be further expanded in terms of the real ‡ † components a ; b ; c ; d and a ; b ; c ; d to yield u u_ ab c d a2a_ 1 1 1 1 2 2 2 2 the equationsf given in Eq.g (24) inf the main text.g v v_ dcb a b2b_ 2 3 2 32 3 A13 The products of the two spinors are in general not w w_ ˆ cd ab c2c_ † commutative, e.g., R1R2y R2y R1. But they are commuta- 6 s s_ 7 6abc d76d2d_7 6ˆ 6 7 6 76 7 tive inside the scalar operator, i.e., with Eq. (A5), we have 4 5 4 54 5 on using the equations in (A10), (A11), and their _ R1R2y R2yR1 a1a2 v1 v2 di€erentiations. As the six components of s and s h iˆh iˆ ‡ Á constitute the orbital state in the vector space, the eight a1a2 b1b2 c1c2 d1d2 A6 ˆ ‡ ‡ ‡ † components of S and S_ constitute the orbital state in the For the norm of the spinor R, we use a positive scalar spinor space. While Eq. (A13) stands for the spinor-to- 764 vector state transformation, we need more information where the four-parametric representation of the 3 3 to write its inverse transformation (note that s and s_ are transformation matrix can be also found in classical auxiliaries and the 4 4 matrix is inherently singular). dynamics textbooks, e.g., Goldstein (1980, p. 153). We As the vector s can be decomposed into its dilation s denote the nine elements of the transformation matrix and rotation ^s, i.e., s s^s, where ^s is the unit vector by T11; T12; ...;T33. These can be easily obtained from ˆ pointing s, the spinor S can be also decomposed into its the vector state s; s_ , as they correspond to the f g ^ ^ dilation S and rotation S^, i.e., S S S^. As shown in Cartesian components of s; h s; sh. Note that the ˆ transverse unit vector h^ ^s is pointing towards the Hestenes (1983), the unimodular spinor S^ can be de-  velocity of the radial unit vector composed into the Eulerian rotations, i.e., _ h ^ ^s 2 h ^s A19 ^ 1 1 1 ˆ s  † S exp je^w2s exp je^u2i exp je^w2X A14 ˆ † † † † which itself is not a unit vector. We have where the three Euler angles X; i; s (Goldstein 1980, p. _ h 146) correspond, respectively, to the longitude of the ^s 2 A20 ascending node measured from the equinox, the incli- j jˆs † nation, and the argument of latitude of the orbit. With Based on Shepperd (1978), an ecient means of the relation S s1=2, we can write computing the four components of S from the trans- ˆ formation matrix is to write

4a2 2T T s 2 ˆ ‡ 4ad T12 T21; 4bc T12 T21 4b 2T11 T s ˆ ˆ ‡ 8 2 ˆ ‡ and 4ab T23 T32; 4cd T23 T32 A21 > 4c 2T22 T s 8 ˆ ˆ ‡ † > 2 ˆ ‡ 4ac T31 T13; 4bd T31 T13 < 4d 2T33 T s < ˆ ˆ ‡ ˆ ‡ > : :> 1=2 1 1 1 1 with T T T T : the trace of the transformation S s cos 2s je^w sin 2s cos 2i je^u sin 2i 11 22 33 ˆ ‡ † ‡ † matrix.ˆ Several‡ ways‡ exist to obtain the four unknowns, cos 1X je^ sin 1X A15  2 ‡ w 2 † † e.g., which can be expanded to yield the spinor components T s T23 T32 a s1=2 cos 1 X s cos 1i; b s1=2 cos 1 X s sin 1i a ‡ ; b ˆ 2 ‡ † 2 ˆ 2 † 2 ˆÆ 4 ˆ 4a 1=2 1 1 1=2 1 1 r c s sin X s sin i; d s sin X s cos i T31 T13 T12 T21 ˆ 2 † 2 ˆ 2 ‡ † 2 c ; d A22 A16 ˆ 4a ˆ 4a † † showing their correspondence to the Euler parameters where the sign of a cannot be determined but can be (Goldstein 1980, p. 155) and to the Cayley-Klein selected freely assuming that we are working with a one- parameters (Whittaker 1937, p. 12). to-one correspondence, or isomorphism, between s and Let h be the orbital angular momentum vector of the the pair S. One may say that S is a double-valued satellite, i.e., functionÆ of the transformation matrix. While the solution given in Eq. (A22) is singular when a 0, we h hh^ s s_ s^s s_^s s^s_ s2^s ^s_ A17 ˆ ˆ ˆ  ˆ  ‡ †ˆ  † can write three more solutions by changing the compo- nent to be used as the divisor. At least one nonzero where the scalar h and the unit vector h^ represent the component exists as long as s 0. For numerical magnitude and direction of h, respectively. Now we purposes, the optimal choice among6ˆ the four possibil- introduce a new triad ^s; h^ ^s; h^ , which consists of the ities corresponds to the largest among absolute values of radial, transversal, andf normal g unit vectors of the , whose magnitude order is same as that of rotating satellite-®xed orbital frame. The spinor algebra a; b; c; d . enables us to write the transformation T ; T11; T22; T33

2 2 2 2 ^s 1 e^u 1 a b c d 2 bc ad 2 bd ac e^u ^ ^ ^ ‡ 2 2 ‡ 2 † 2 † h ^s Sy e^v S 2 bc ad a b c d 2 cd ab e^v A18 8 Â^ 9 ˆ s 8 9 ˆ s 2 † ‡ 2 2 ‡ 2 † 2 38 9 † < h = < e^w = 2 bd ac 2 cd ab a b c d < e^w = 4 ‡ † † ‡ 5 : ; : ; : ; 765

Now, to obtain the components of the time derivative on using Eqs. (A27) and (A30). Now, with Eq. (A33), S_ , we consider the Eulerian angular velocity vector of one can expand Eq. (A32) to yield the four components the orbit expressed as (Hestenes 1983) of S_ , ds di dX 1 s_ h s _1s_hs ^ ^ a_ a 2 d n b ;bb2cna xs h f e^w A23 ˆ 2 s s h ˆ2s‡s‡h ˆ dt ‡ dt ‡ dt † 1s_hs_1s_hs c_2Áscs2bnhd;d2Ásds2anhcA34 with ˆ‡ˆ‡ † ToÁ obtain the right-hand sidesÁ in Eq. (A34) we need h^ sin i sin X e^u sin i cos X e^v cos i e^w A24 to know n. However, having the same instantaneous ˆ ‡ † position and velocity as the true orbit, we can set n 0, and which relates the spinor state to the osculating Keplerianˆ orbit. From Eq. (A34) one can extract f^ cos X e^ sin X e^ A25 ˆ u ‡ v † ^ ca_ db_ ac_ bd_ 0 A35 where f represents the unit vector toward the ascending ‡ ˆ † . For the time derivatives of the Euler and angles, we have the Gauss variation of parameter equations (Battin 1987, p.488) s2 ba_ ab_ dc_ cd_ n A36 dX s sin s di s ds h s sin s cos i ‡ ‡ ˆ h † n ; n cos s; n dt ˆ h sin i dt ˆ h dt ˆ s2 h sin i The relation in Eq. (A35) is called the spinor gauge A26 condition (Hestenes and Lounesto 1983) and also † appears in the Kustaanheimo-Stiefel transformation where n represents the normal component of the satellite (Stiefel and Scheifele 1971; Deprit et al. 1994). The acceleration. two relations of Eqs. (A35) and (A36) compensate the Substituting Eqs. (A24±A26) into Eq.(23), with some two redundancies in the spinor state. manipulation, one can obtain So far we have established the two-way transforma- tion between the vector and spinor states observed in the 1 h s x Sy e^ n e^ S A27 inertial frame. The transformation is general and s ˆ s s2 w ‡ h u †  equivalently applicable to states for the earth-®xed frame. To clarify, let us ®rst denote the orbital position or and velocity vectors of a satellite observed in the earth- h s ®xed frame by r xe^x ye^y ze^z Rye^xR and r_ x_e^x A28 _ ˆ ‡ _ ‡ ˆ ˆ xs 2 n y_e^ z_e^ Rye^ R Rye^ R, where R a jbe^ ce^ ˆ s ‡ h † ‡ y ‡ z ˆ x ‡ x ˆ ‡ x‡ y de^ is a spinor based on the rotating earth-®xed triad on using the ®rst equality in Eq. (A18). Meanwhile, the ‡ z† e^ ; e^ ; e^ . Note that r s but their Cartesian compo- velocity vector s_ can be decomposed into (Hestenes and x y z nents are di€erent, i.e., ˆx; y; z u;v;w , as they are Lounesto 1983) referencedÈÉ to the earth-®xedf g6ˆf frame andg the inertial s_ frame, respectively. Also note that r_ s_ as the time s_ s_^s x s s 1 j x s s x 1 Wys sW 6ˆ ˆ ‡ s  ˆ s 2 s s†ˆ2 ‡ † derivatives are observed in the two di€erent reference frames. The di€erence is the e€ects of the earth rotation, A29 † i.e., with r_ s_ xe s A37 s_ ˆ  † W j xs A30 The total angular velocity (to be denoted by x ) of the s r ˆ ‡ † orbit observed in the earth-®xed frame combines the Comparing Eqs. (A12) and (A29), we obtain the time orbital angular velocity and the earth-rotation e€ect in derivative, the sense of taking their di€erence, i.e.,

S_ 1 S W A31 xr xs xe A38 ˆ 2 † ˆ † _ which can be also written as and therefore the total angular momentum g r r is due to the orbital angular momentum h s ˆs_ and the ˆ  S_ 1G S A32 earth-rotation e€ect, i.e., ˆ 2 † g h r x r h r2x r x r A39 with ˆ  e  ˆ e ‡ †Á e † 1 s_ h s on using Eq. (A37). G S W Sy j e^ n e^ A33 Substituting Eq. (A28) into Eq. (A38) and using ˆ s ˆ s ‡ s2 w ‡ h u †  Eq. (A39), we can write 766

g r where x m A40 r ˆ r2 ‡ g † # X h A49 with ˆ † is the argument of ascending orbital node measured r s m n r^ x A41 from the Greenwich meridian. g ˆ h Á e † where m can be called the normal component acceler- References ation of the satellite e€ective in the rotating earth-®xed frame. Battin RH (1987) An introduction to the mathematical methods of Note that the two two-way transformations astrodynamics. American Institute of Aeronautics and Astro- u; v; w; u_; v_; w_ a;b;c;d;a_;b_;c_;d_and x; y; z; x_; y_; z_ nautics, New York †$a;b;c;d;a_;b_;c_;d_are equivalent under † Eq. (A41) Born GH, Tapley BD, Santee ML (1986) Orbit determination $ Á using dual crossing arc altimetry. Acta Astronaut 13±4:157±163 which is nothing but r^ xr r^ xs r^ xe. We can ÁÁ ˆ Á Á Cartan E (1981) The theory of spinors. Dover, New York write the counterparts of Eqs. (A30)±(A36) simply by Cloutier JR (1983) A technique for reducing low-frequency, time- using new variables. Let dependent errors present in network-type surveys. J Geophys Res 88:659±663 r_ Colombo OC (1984) Altimetry, orbits and tides. NASA Technical V j x A42 ˆ r ‡ r † Memorandum 86180, Goddard Space Flight Center Cui C, Lelgemann D (1992) On cross-over di€erences of the radial then, orbital perturbations as functions of force model parameters. In: Geodesy and physics of the earth, International Union of R_ 1 R V A43 Geodesy and Geophysics. Springer, Berlin Heidelberg New ˆ 2 † York and Deprit A, Elipe A, Ferrer S (1994) Linearization: Laplace vs. Stiefel. Celestial Mech Dyn Astron 58:151±201 Farless DL (1985) The application of periodic orbits to TOPEX R_ 1 F R A44 ˆ 2 † mission design. AAS 85±301:13±36 Ferziger JH (1981) Numerical methods for engineering application. with Wiley, New York Gaspar P, Ogor F, Le Traon P-Y, Zanife O-Z (1994) Estimating 1 r_ g r the sea state bias of the TOPEX and POSEIDON altimeters F R V Ry j e^z m e^x A45 ˆ r ˆ r ‡ r2 ‡ g † from crossover di€erences. J Geophys Res 99:24981±24994  Goldstein H (1980) Classical mechanics (2nd edn) Addison-Wesley, and so on. Reading Mass The vector state r; r_ is independent of the e€ective Grafarend EW, Lohse P (1991) The minimal distance mapping of normal acceleration m which changes R_ (but, not R). the topographic surface onto the (reference) ellipsoid of revo- ÈÉ lution. Manuscr Geod 16:92±110 From Eq. (A41), we note that m is caused by n and the Greiner W (1994) Quantum mechanics, an introduction (3rd edn) radial component of x (i.e., r^ x ). For n one may re- e Á e Springer, Berlin Heidelberg New York ¯ect the e€ects of any perturbation, e.g., the earth's Guckenheimer G, Holmes P (1986) Nonlinear oscillations, dy- oblateness with namical systems, and bifurcations of vector ®elds. Springer, 2 Berlin Heidelberg New York le re z Hagar H (1977) Computation of circular orbit ground trace in- n 3J2 cos i A46 ˆ r2 r r † tersections. Engineering Memorandum 315±29, Jet Propulsion where l is the gravitational parameter of the earth, r is Laboratory e e Hestenes D (1983) Celestial mechanics with geometric algebra. the mean radius of the equator, and J2 is the second Celestial Mech 30:151±170 zonal harmonic of the geopotential. Again, for the Hestenes D, Lounesto P (1983) Geometry of spinor regularization. purpose of state transformation, we can work with an Celestial Mech 30:171±179 osculating Keplerian orbit, i.e., we can set n 0. Like n, Hsu SK, Geli L (1995) The e€ect of introducing continuity con- ˆ ditions in the constrained sinusoidal crossover adjustment r^ xe does not change r; r_ and R. However, the Á _ method to reduce satellite orbit errors. Geophys Res Lett component of xe normal to r changes R and R as well as _ ÈÉ 22:949±952 r (but, not r). We consider a simpli®ed earth-rotation Kim MC (1993) Determination of high-resolution mean sea surface model uniformly rotating along the e^w-axis, i.e., and marine gravity ®eld using satellite altimetry. Ph.D Disser- x x e^ x e^ A47 tation. Dept Aerospace Engineering and Engineering Mechan- e  o w  o z † ics, The University of Texas at Austin where xo represents the constant rate of the Greenwich King JC (1976) Quantization and symmetry in periodic coverage mean to be denoted by h. Then, as the patterns with applications to earth observation. J Astronaut Sci counterpart of Eq. (A16), the components of R can be 24:347±363 Klokocnik J, Wagner CA (1994) A test of GEM-T2 from GEO- expressed as SAT crossovers using latitude lumped coecients. Bull Geod 68:100±108 a r1=2 cos 1 # s cos 1 i; b r1=2 cos 1 # s sin 1 i ˆ 2 ‡ † 2 ˆ 2 † 2 Kozel BJ (1995) Dual-satellite altimeter crossover measurements c r1=2 sin 1 # s sin 1 i; d r1=2 sin 1 # s cos 1 i for precise orbit determination. Ph.D. Dissertation. Dept ˆ 2 † 2 ˆ 2 ‡ † 2 Aerospace Engineering and Engineering Mechanics, The Uni- A48 versity of Texas at Austin † 767

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