Theory of satellite ground-track crossovers

M. C. Kim

Center for Space Research, The University of Texas at Austin, Austin, Texas, 78712-1085, USATel: +1 512 471 8121; fax: +1 512 471 3570; e-mail: mkim@csr.utexas.edu

Received: 19 November 1996 /Accepted: 12 May 1997

Abstract. The fundamental geometry of satellite groundtracks and their crossover problem are investigated. Foridealized nominal ground tracks, the geometry is gov-erned by a few constant parameters whose variationslead to qualitative changes in the crossover solutions.On the basis that the theory to locate crossovers has notbeen studied in suf®cient detail, such changes aredescribed in regard to the number of crossover solutionsin conjunction with their bifurcations. Employing thespinor algebra as a tool for establishing the ground-track crossing condition, numerical methodologies tolocate crossovers appearing in general dual-satelliteground-track con®gurations are also presented. Themethodologies are applied to precisely determinedorbital ephemerides of the GEOSAT, ERS-1, andTOPEX/POSEIDON altimeter satellites.

Key words. Satellite á Ground track á CrossoverBifurcation á Spinor

1 Introduction

In the geodetic applications of an arti®cial satellite as aplatform for active remote sensors, associated measure-ments are frequently sampled along the ground track ofthe satellite's nadir point, which is also referred to as thesubsatellite point. For instance, a satellite-borne altim-eter measures the distance between the satellite and itsnadir point so that the measurement, called the satellitealtimeter range, can be used in the mapping of thephysical shape of the mean sea surface (Kim 1993) andin the monitoring of time-varying oceanographics, suchas the oceanic tides (Ma et al. 1994), the oceanic eddyvariabilities (Shum et al. 1990b), the global mean sealevel rise phenomena (Nerem 1995), etc., when thesatellite's orbital trajectory is determined by indepen-dent tracking to the satellite (Tapley et al. 1994).

The points where the ground track of a satellite in-tersects itself on the surface of the earth are called (single-satellite) crossover points. When the intersections arefrom ground tracks of two distinct satellites, these areusually called dual-satellite crossover points. As a meansof obtaining relevant measurements at the same geo-graphic location separated in time, crossover points holda signi®cant usage in satellite geodesy, especially in therealm of satellite oceanography associated with the ap-plications of satellite altimetry. The satellite-altimetercrossover di�erence, which is de®ned as the di�erence intwo satellite altimeter ranges interpolated at a crossoverpoint along their respective ground tracks, provides pe-culiar time-di�erenced oceanographics. While the altim-eter crossover di�erences have been also used in theevaluation of an altimeter time-tag bias (Schutz et al.1982), in the re®nements of satellite orbits (Cloutier 1983;Sandwell et al. 1986; Born et al. 1986; Moore and Roth-well 1990; Hsu and Geli 1995; Kozel 1995), in the cali-bration of a gravity ®eld model (Klokocnik and Wagner1994), in the estimation of the altimeter inferred sea statebias (Gaspar et al. 1994), in the improvement of a plan-etary topography model (Stottlemyer 1996), etc., thecomputational e�ciency of locating crossovers cannot beoveremphasized as one of routine tasks for such a widerange of applications.

Techniques for the determination of crossover loca-tions have been introduced in several studies. For single-satellite crossovers, Hagar (1977) presented a computa-tional ¯owchart to obtain circular-orbit ground-trackcrossovers taking into account the secular perturbatione�ects of the earth's oblateness. Hereinafter such anidealized circular orbit, its geocentric projection onto thesurface of a uniformly rotating spherical earthmodel, andthe associated crossover points will be referred to as thenominal orbit, nominal ground track, and nominalcrossover points, respectively. As the nominal orbit andits ground track can be e�ectively described by a fewconstant parameters, a set of fundamental mathematicalrelations between the latitude, longitude, and ground-track crossing times can be easily established. ThereforeRowlands (1981) and Shum (1982) presented iterative

Journal of Geodesy (1997) 71: 749±767

positioning of actual single-satellite crossovers startingfrom nominal crossovers to utilize altimeter crossoverdi�erences in their orbit adjustment and determinationstudies. As also summarized in Schrama (1989) andSpoecker (1991), the general procedure of single-satellitecrossover locating has been based on the search for in-tersections between piecewise ground tracks segmentedby ascending and descending passes (i.e., by semirevolu-tions), and mainly consists of two steps of the predictionof nominal crossovers and the determination of actualcrossovers by interpolating discrete subsatellite pointsprojected from satellite trajectories. As well as the actualdetermination step, the analytical prediction step itselfneeds numerical iterations due to the inherent transcen-dental relation between the latitude and the di�erence ofcrossing timings as discussed by Colombo (1984). Such atwo-step procedure has also been adopted in ®nding dual-satellite crossovers. For a particular combination of twonominal orbits, Santee (1985) employed spherical trian-gular models to make an initial guess of simulated dual-satellite nominal crossovers followed by iterative re®ne-ments by updating inclinations. More or less generaldual-satellite crossover-®nding techniques, all based onthe semirevolutions search, have been also discussed byShum et al. (1990a), and by Moore and Ehlers (1993).

Given orbital ephemerides of actual satellites and aproper de®nition of ground tracks, the crossover locatingproblem is to determine the ground-track crossing time-tag pairs. Strictly speaking, the solutions are intersectionsof two unrelated curves. Depending on the relevant orbitgeometries, the number of dual-satellite crossovers can bea priori unknown. There can exist more than one inter-section (or no intersection) in a semirevolution ground-track pair and therefore one may lose some crossovers atthe prediction step. Even for the counting of crossovers inthe nominal single-satellite con®guration, we found someaspects overlooked in the previous studies. On the basisthat the theory to locate crossovers has not been surveyedin su�cient detail and that we need a more systematicapproach to the problem, this paper presents both ana-lytical and numerical aspects of locating crossovers in thecases of the general dual-satellite scenario which includethe single-satellite cases as a subset. We ®rst investigatethe fundamental geometric structure of ground tracks.With some relevant mathematical tools described in theAppendix, we establish the ground-track crossing con-dition, followed by numerical strategies for ®nding them.We will provide an answer to the predictability on thenumber of crossover solutions. With some treatments onthe issue of the nadir mapping of satellite ephemeridesonto the reference ellipsoid of revolution, actual appli-cations of the developed techniques to the precisely de-termined orbital ephemerides of the historic GEOSATand the currently orbiting ERS-1 and TOPEX/POSEI-DON altimeter satellites will be also presented.

2 Nominal ground tracks

The ground track of an earth-orbiting arti®cial satelliteis the time-trace of its subsatellite point: a nadir

mapping of the satellite's orbital trajectory onto aprescribed earth-surface model such as a referenceellipsoid of revolution (Torge 1991) or a sphericalharmonics geoid model (Shum 1982). While de®nitionsof the nadir mapping and the earth surface can bedependent on a speci®c problem and earth satellites areunder the in¯uence of a variety of perturbations, thenominal ground track is appropriate analytically toinvestigate the fundamental geometry of ground tracks.

In the rotating earth-®xed geocentric equatorialGreenwich-meridional reference frame (hereinafter, theearth-®xed frame), the orbital angular motion of a sat-ellite can be represented by the geocentric unit vector

r � rr� 1

r

x

y

z

8><>:9>=>; �

cos/ cos k

cos/ sin k

sin/

8><>:9>=>;

�cos s cos#ÿ cos i sin s sin#

cos s sin#� cos i sin s cos#

sin i sin s

8><>:9>=>; �1�

where fx; y; zg denotes the Cartesian components of thesatellite position vector r associated with its geocentricradius r, latitude /, and longitude k, s is the argument oflatitude and represents the satellite's orbital rotation, #is the longitude of the orbital ascending node measuredfrom the Greenwich meridian and represents therotation of the satellite's orbital plane with respect tothe rotating earth, and i is the inclination of the orbitalplane. For a nominal ground track, i becomes constantand the two rotations s and # become uniform variablesresulting in their linear dependence,

# � ke ÿ qs �2�where ke represents the longitudinal origin of thenominal ground track, and q denotes the angularvelocity ratio between the two uniform rotations. Withthe linear dependence we can consider s as a time-likecontinuously increasing independent variable (thealong-track location in the nominal ground track) and# as a dependent variable which is continuouslydecreasing, as the rotation of any orbital plane withrespect to the earth is always westward (i.e., q > 0).

Nominal ground tracks are thus on the three-di-mensional parametric space of �0� � i � 180�;0 < q <1; 0� � ke < 360��. Apart from the longitudi-nal origin ke, the manner in which a nominal groundtrack is laid down on the spherical earth is primarilydependent on i and q. While i determines the latitudinalbounds of a nominal orbit and its ground track, it is alsothe parameter to classify conveniently the longitudinaldirection of orbital motions. We say that an orbit isposigrade when 0� � i < 90�, is retrograde when90� < i � 180�, and is circumpolar when i � 90�. How-ever, such a simple classi®cation is valid only when or-bits are viewed from the inertial frame. When orbits areviewed from the earth-®xed frame, in which groundtracks are actually depicted, we need careful accountingfor the relative rotational motions between the orbital

750

planes and the earth. For a nominal orbit the relativerotation can be e�ectively quanti®ed by the parameter q.

To investigate the behavior of a nominal groundtrack, using Eqs. (1) and (2), we ®rst formulate the spin-axis component velocity of the unit vector r,

zr

� �0� /0 cos/ � sin i cos s � sgn�cos s�

������������������������������cos2 /ÿ cos2 i

p�3�

and the spin-axis component angular momentum,

xr

� � yr

� �0ÿ y

r

� � xr

� �0� k0 cos2 /

� cos iÿ q� q sin2 i 1ÿ cos2 sÿ � �4�

� cos iÿ q cos2 /

where the prime indicates di�erentiation with respect tos, and the sgn function takes 1, 0, or ÿ1 depending onthe sign of its argument. Equations (3) and (4) can becombined to yield

1

cos/d/dk� �

������������������������������cos2 /ÿ cos2 i

pcos iÿ q cos2 /

�5�

which represents the instantaneous slope of the nominalground track on the unit sphere. Note that for givenvalues of i and q, the slope is a double-valued functionof the latitude /.

The latitudinal motion of the nominal ground trackis bounded in cos/ � j cos i j and is

northward when cos s > 0

stationary when cos s � 0

southward when cos s < 0

8><>: �6�

as the sign of /0 in Eq. (3) will indicate. Meanwhile thelongitudinal motion can be somewhat complex. As thesign of k0 in Eq. (4) will indicate, the nominal groundtrack is

eastward when cos2 s < f cos2 / < qÿ1 cos iÿ �

stationary when cos2 s � f cos2 / � qÿ1 cos iÿ �

westward when cos2 s > f cos2 / > qÿ1 cos iÿ �

8>><>>:�7�

where

f � 1ÿ qÿ cos i

q sin2 i� �1ÿ q cos i� cos i

q�1ÿ cos i��1� cos i� �8�

is a constant which can be considered as a parametricfunction of i and q.

Based on Eqs. (7) and (8), in Fig. 1 we classify thelongitudinal direction of nominal ground tracks in thetwo-dimensional parametric space of �0 < q <1;ÿ1 � cos i � 1�. There are four regions separated byq � cos i � f � 1�, q cos i � 1 � f � 0�, cos i � 0 � f � 0�,and surrounded by cos i � ÿ1 � f � ÿ1�, cos i � 1� f � 1 when q < 1; f � ÿ1 when q > 1�, q � 0 (f �

ÿ1 when cos i < 0, f � 1 when cos i > 0), q � 1� f � ÿ cot2 i�. In the region q < cos i � 1, the associat-ed ground tracks are always eastward � f > 1�. In thetwo regions qÿ1 < cos i � 1 and ÿ1 � cos i < 0, the as-sociated ground tracks are always westward � f < 0�.Hereinafter these two regions are respectively referred toas the posigrade westward region and the retrogradewestward region, where the ``posigrade'' and ``retro-grade'' indicate the longitudinal direction with respect tothe inertial frame (the ``eastward'' and ``westward'' areused for the earth-®xed frame). In the remaining regionof 0 < cos i < min q; qÿ1

ÿ �(where 0 < f < 1), the asso-

ciated ground tracks change the longitudinal direction;eastward at higher latitudes of cos2 / < qÿ1 cos i andwestward at lower latitudes of cos2 / > qÿ1 cos i. Wewill refer to this region as the eastward/westward mixedregion or simply the mixed region. It can be said thatground tracks of all circumpolar orbits �cos i � 0� arewestward k0 � ÿq < 0� �. The point q � cos i � 1( f � 1=2 as a limit) corresponds to the geostationaryorbit whose ground track is motionless over a point onthe equator �/ � 0; k � ke�.

A nominal ground track will eventually be parallel toits preceding trace with an o�set less than some presetlimit. To this extent all nominal ground tracks are re-peating, although the repeat interval may be as long aspossible. Hence we restrict our interest to the caseswhere q is a rational number, i.e.,

q � N=M �9�where M and N are positive coprime integers. With Eq.(9), which is a resonance condition between s and #, thenominal ground track becomes periodic and repeats itstrace after M orbital revolutions of the satellite, orequivalently, N nodal days. In order to visualize thetopological evolution of nominal ground tracks, in Fig. 2we geographically display 21 nominal ground tracks

Fig. 1. Longitudinal direction of nominal ground tracks classi®ed in�q; cos i� space

751

Fig. 2. Nominal ground tracks

752

(A to U) with the same value of ke � 180� but withvarying q and i as indicated by the corresponding 21points (A to U) in the �q; cos i� space shown in Fig. 1.The two arrows in each of the nominal ground tracksindicate the subsatellite point at the epoch �s � 0� andafter a quarter orbital revolution �s � p=2�. As thetracks are periodic, we can limit our interest in a repeatperiod �0 � s < 2Mp�.

The ®rst seven tracks (A±G) correspond toq � N=M � 1=3 but with varying i. A�i � 0� orcos i � 1� is an eastward equatorial track. In the periodof the three �M� orbital revolutions, the track makes two�M ÿ N� eastward equatorial revolutions with the nete�ects of one westward �ÿN� revolution of the orbitalnode with respect to the earth. As a degenerate case, thetrack is con®ned to the equator, and one can say that itsactual repeat period is not three �M� revolutions but oneand half, i.e., M=�M ÿ N�, revolutions. B�i � 60� orcos i � 1=2� is a typically inclined eastward groundtrack. One can say that the two longitudinal revolutionsof the track A have been separated with an equatorialsymmetry. Now the repeat period becomes three �M�orbital revolutions. There are three crossover points (atthe equator) where the two longitudinal revolutions in-tersect each other. C�i � 70:5� or cos i � 1=3� is at theborder �q � cos i� of the purely eastward region and theeastward/westward mixed region shown in Fig. 1. Thetrack crosses the equator vertically so that the anglebetween ascending and descending traces becomes 180�at the crossover points. With D�i � 85� orcos i � 0:087� we are now in the mixed region. The trackis eastward (resp. westward) above (resp. below)j/j � 59:2�, at which it becomes instantaneously meri-dional k0 � 0� �. There exist nine crossovers with theappearance of six nonequatorial crossovers whose lon-gitudes are aligned with those of equatorial crossovers.E�i � 90� or cos i � 0� is a circumpolar ground trackwhich consists of inclined straight lines as both the lat-itude and longitude become linear functions of s, i.e.,/0 � sgn�cos s� and k0 � ÿq. Each of the north andsouth poles becomes a collocation point of three cross-overs, and we therefore still have nine crossovers. WithF�i � 120� or cos i � ÿ1=2� we are now in the retro-grade westward region. While there still exist ninecrossover points, the longitudes of nonequatorialcrossover points are no more aligned with those of theequatorial crossover points. As the last example ofq � 1=3 cases, G�i � 180� or cos i � ÿ1� is a westwardequatorial track. Like the track A, the ground-trackcon®guration is again con®ned to the equator. In theperiod of the three �M� orbital revolutions the trackmakes four �M � N� westward equatorial revolutionsand one can say that its actual repeat period is 3/4, i.e.,M=�M � N�, revolutions. The equatorial tracks�cos i � �1� actually form degenerate cases, for whichthe ground-track motion becomes one dimensional.

The next six tracks (H±M) are examples of equa-torially asymmetric ground tracks characterized by theopposite parity of M and N . While H�q � 1=2; i � 55��closely corresponds to a ground track of the GlobalPositioning System (GPS) satellites (Leick 1995) which

generate no crossovers, its neighboring trackI�q � 11=20; i � 55�� has 160 crossovers showing thee�ect of a small change in the value of q.J�q � 1=2; i � 95�� is a highly inclined westward trackwhich shows four crossover points. The following threetracks (K, L, M) are for the same value of q � 2=3 butwith varying i. While both K�i � 70:5�� and L�i � 85��are located in the mixed region, they are quite di�erentas far as the number of crossovers is concerned; none inK and twelve in L. The track M�i � 120�� is anothertypical westward track having the same number ofcrossovers as L but with a quite di�erent topologicalstructure.

The next three tracks (N, O, P) correspond toq � 1=1. N�i � 60�� is a typical geosynchronous satelliteground track which forms a ®gure-eight curve. Thetrack librates both latitudinally and longitudinallyspending 12 h successively in each of its northern andsouthern loops. With q � 1=1, this geosynchronous be-havior is preserved between the geostationary case�i � 0�� and the circumpolar geosynchronous case�i � 90�� which is shown as the track O. As the trackbecomes purely retrograde westward �i > 90��, the lon-gitude no more librates but undergoes smooth rotationsas shown in the track P�i � 120��. Each of the threetracks has only one crossover point (as its epoch posi-tion).

The last ®ve tracks (Q±U) are for q � 2=1 but withvarying i. With the track Q�i � 30�� we are now in theposigrade westward region. The track generates nocrossover point. R�i � 60��, a cuspidal track, is a lim-iting case located at the border �q cos i � 1� of the mixedregion and the posigrade westward region. We see twocusps where the ground-track velocity instantaneouslybecomes zero. With the track S�i � 85��, we are again inthe mixed region showing a looping ground track withan equatorial asymmetry. The cusps of R have beenevolved into loops with the appearance of two cross-overs. Following another circumpolar track T with twocrossover points, the nominal ground track becomeswestward again as shown in the track U�i � 120��.

While all nominal ground tracks have longitudinalcyclic symmetry (i.e., the same pattern repeats M times),they also have either a symmetry or an asymmetry withrespect to the equator. The equatorial symmetry occurswhen M and N have the same parity (both odd) and theequatorial asymmetry occurs when M and N have theopposite parity. While one can count M equatorialcrossovers in the same parity cases, no equatorialcrossover point exists in the opposite parity cases due tothe asymmetry. In its repeat period, a nominal groundtrack undergoes M latitudinal librations such that itcrosses a zonal line, in j sin/j < sin i, 2M times (alongeach of its M ascending and M descending traces).Meanwhile the longitudinal behavior can be diverse. Incase of a retrograde westward track, there are M � Nlongitudinal revolutions which consist of M revolutionsdue to the satellite's orbital motion and N revolutionsdue to the net e�ects of the orbital plane's motion withrespect to the earth. Such a track crosses a meridionalline M � N times. Since the M � N longitudinal revo-

753

lutions interweave so that there exist M � N ÿ 1 cross-over latitudes and each of the crossover latitudes con-tains M crossover points, the number of crossovers isM�M � N ÿ 1�. In case of a purely eastward track, thereexist M ÿ N longitudinal revolutions which produceM ÿ N ÿ 1 crossover latitudes and consequentlyM�M ÿ N ÿ 1� crossovers. In case of a posigrade west-ward track, there exist N ÿM longitudinal revolutionswhich produce N ÿM ÿ 1 crossover latitudes and con-sequently M�N ÿM ÿ 1� crossovers. For the eastward/westward mixed tracks, as will be veri®ed later, thenumber of crossover latitudes varies betweenjjM ÿ N j ÿ 1j and M � N ÿ 1 depending on the values ofcos i and q, and consequently the number of crossoversvaries from M jjM ÿ N j ÿ 1j to M�M � N ÿ 1� with theincrement M .

Some previous studies have also presented formulasto count the number of single-satellite nominal cross-overs appearing in a repeat period. Tai and Fu (1986)claimed that the number equals M�M ÿ N � 1� ifcos i > 0 and equals M�M � N ÿ 1� if cos i < 0 (notethat M and N change their notational roles in Tai andFu). Spoecker (1991) claimed M2. Considering the caseq < 1, Cui and Lelgemann (1992) claimedM�M ÿ rN ÿ 1� where r � sgn�cos i�. Along with a signerror in Tai and Fu, and the complete ignorance of theearth rotation e�ects in Spoecker, these studies passedover the fact that the crossing geometry of a single-satellite nominal ground track is more signi®cantlyvariant in the two-dimensional parametric space of�q; cos i�. The reciprocal of q has been called theground-track parameter. Although some of its impor-tant roles have been recognized in the counting ofcrossover latitudes (King 1976; Farless 1985; Parke et al.1987), we claim that all of these previous studies missedthe existence of the eastward/westward mixed tracks andthe posigrade westward tracks.

In Fig. 3 we overlay a pair of nominal ground trackswith i1 � 60�; q1 � 1=3; ke1 � 180�� � and i2 � 95�;�q2 � 1=2; ke2 � 155��. In addition to the three cross-overs of the eastward track and the four crossovers ofthe westward track, one can count ten points where thetwo tracks intersect. These intersections, ordered from 1to 10 along the eastward track, are nominal examples ofthe dual-satellite crossover point. To fully classify dual-

satellite nominal ground-track con®gurations, one mayneed to survey a ®ve-dimensional parametric space ofi1; i2; q1 � N1=M1; q2 � N2=M2;� D � ke1 ÿ ke2�. Here Drepresents the mutual longitudinal origin of two groundtracks, and is introduced to reduce the number of pa-rameters (from six to ®ve) by using the fact that therelative geometry of two nominal ground tracks is in-variant upon a simultaneous rotation around a ®xedaxis. In Sect. 4 we will provide a more systematictreatment for the count of crossovers in both the single-and dual-satellite nominal ground-track con®gurations.

3 Ground-track intersection

Let r1 t1� � and r2 t2� � be two arbitrary orbital trajectoriesreferenced to the earth-®xed frame, where t1 and t2 aretime-tags (to be frequently omitted) used as independentvariables of the two trajectories. To the extent that thegeocentric nadir mapping is concerned, the ground-track crossing condition for the two trajectories can besimply written as

r1 t1� � � r2 t2� � �10�where r1 and r2 are geocentric unit vectors pointing r1and r2, respectively.

Let F denote the direction cosine between r1 and r2,i.e.,

F t1; t2� � � cosw t1; t2� � � r1 t1� � � r2 t2� � �11�where w represents the spherical angular distancebetween the two ground tracks. Locating ground-trackcrossovers can be considered as a two-dimensional root-®nding problem of a single scalar equation F t1; t2� � � 1,which replaces the vector equation in Eq. (10). Since Fyields its maximal value 1 at crossovers, the problem canbe solved by using the extremal condition

Ft1 � _r1 � r2 � 0; Ft2 � r1 � _r2 � 0 �12�where the subscripts t1 and t2 indicate partial di�eren-tiations. Given a t1; t2f g point close to a crossoversolution, the Newton-Raphson iteration scheme

t1t2

� � t1

t2

� �ÿ �r1 � r2 _r1 � _r2

_r1 � _r2 r1 � �r2

� �ÿ1 _r1 � r2r1 � _r2

� ��13�

may converge to the solution. The iteration fails whenthe determinant of the 2� 2 Hessian matrix, i.e., theGaussian curvature of F ,

G � Ft1t1Ft2t2 ÿ F 2t1t2 � �r1 � r2

ÿ �r1 � �r2ÿ �ÿ _r1 � _r2

ÿ �2 �14�vanishes.

Let K be the angle between _r1 and _r2, i.e.,

_r1 � _r2 � _r1k _r2�� �� cosK �15�

At a crossover point, r1 � r2, using the relation�r � r � ÿ _r

�� ��2, Eq. (14) yieldsGv � _r1

�� ��2 _r2�� ��2ÿ� _r1 � _r2�2 � _r1

�� ��2 _r2�� ��2sin2 K �16�Fig. 3. Overlay of two nominal ground tracks

754

where the subscript v, to be frequently omitted on theright-hand sides of equations, indicates evaluation at thecrossover point. We note that the iteration failure occurswhen the ground tracks become collinear�K � 0� or 180��.

A converged solution of Eq. (13) does not alwaysyield a crossover solution (where F � 1, a maximum ofF , or equivalently w � 0, a minimum of w) since othertypes of extrema also satisfy Eq. (12). In other words,Eq. (12) forms only a necessary condition for a cross-over. Figure 4 shows the variation of w in the two-di-mensional domain �0 � s1=2p < M1 � 3; 0 � s2=2p <M2 � 2� associated with the two nominal ground tracksshown in Fig. 3. Among the 14 minimal points of w inthe white area, 10 points correspond to crossovers whichare numbered from 1 to 10, consistent with Fig. 3. Theremaining four minima labeled by L are local minima.One may also count 14 maxima in the gray area. Thereactually exist 56 extremal points including 14 saddlepoints in each of the white and gray areas. The extremaof F can be classi®ed as

where H denotes the mean curvature of F given byH � �Ft1t1 � Ft2t2�=2. With the iteration scheme, Eq. (13),we need to check whether a converged solution is acrossover or not. With the symmetry in the numbers ofcenters and saddles, the symmetry in the numbers ofmaxima and minima, and the existence of local maxima,less than a quarter of converged solutions will becrossovers.

Now we present an alternative approach by repre-senting the orbital states of satellites in terms of spinors,which enable us to rewrite the ground-track crossingcondition in a more convenient form. At this point, theauthor encourages the reader to refer to the Appendix,which presents a summary on spinors, their algebra, andthe two-way transformations between the vector stateand the spinor state. Some equations in the Appendix,e.g., the spinor multiplication laws in Eq. (A1), will befrequently used in the following development.

Let fex; ey ; ezg be a right-handed orthogonal triad ofunit vectors and suppose it forms the earth-®xed frame.We consider the spinor decomposition of the two orbitalposition vectors

r1 � Ry1exR1; r2 � R

y2exR2 �18�

with

R1 � a1 � j b1ex � c1ey � d1ezÿ �

R2 � a2 � j b2ex � c2ey � d2ezÿ � �19�

where fa1; b1; c1; d1g and fa2; b2; c2; d2g denote the realcomponents of the spinors R1 and R2 respectively, j isthe unit imaginary number, and the superscript ydenotes the spinor conjugate.

With Eq. (18) the ground-track crossing conditionEq. (10) can be rewritten as

Ry1exR1

r1� R

y2exR2

r2�20�

where, for the magnitudes of the two vectors, we have

r1 � Ry1R1 � R1R

y1; r2 � R

y2R2 � R2R

y2 �21�

Premultiplying R1 and postmultiplying Ry2, Eq. (20)

yields

exR1Ry2 � R1R

y2ex �22�

on using Eq. (21).We introduce a new spinor

Q � R1Ry2 � A� j Bex � Cey � Dez

ÿ � �23�

Substituting Eq. (19) into the product in R1Ry2 in Eq. (23),

which is also given in Eq. (A5), one obtains

Crossovers�F � 1�Maxima�H < 0�h

Centers�G > 0�h Local Maxima�F < 1�Extrema�rF � 0�h Minima�H > 0�

Saddles�G < 0�

�17�

Fig. 4. Spherical angular distance (in degree) between two nominalground tracks

755

A � a1a2 � b1b2 � c1c2 � d1d2B � b1a2 ÿ a1b2 ÿ d1c2 � c1d2C � c1a2 � d1b2 ÿ a1c2 ÿ b1d2D � d1a2 ÿ c1b2 � b1c2 ÿ a1d2 �24�With Eq. (23), Eq. (22) yields

exR1Ry2 ÿ R1R

y2ex � exQÿ Qex � 2 Dey ÿ Cez

ÿ � � 0

�25�on using Eq. (A1). Note that the ground-track crossingcondition is now reduced to

C�t1; t2� � 0; D�t1; t2� � 0 �26�whereas A and B can be arbitrary.

We will refer to the two scalar equations in Eq. (26)as the spinor representation of the ground-track cross-ing condition. In the �t1; t2� domain the crossovers arelocated at the intersections of the two nonlinear curves;i.e., we are again working with a two-dimensional root-®nding problem for a set of the unknown crossover timepairs. The two equations in Eq. (26) can be solved withthe Newton- Raphson iteration scheme

t1t2

� � t1

t2

� �ÿ Ct1 Ct2

Dt1 Dt2

� �ÿ1 CD

� ��27�

with

Ct1 � _c1a2 � _d1b2 ÿ _a1c2 ÿ _b1d2

Ct2 � c1 _a2 � d1 _b2 ÿ a1 _c2 ÿ b1 _d2

Dt1 � _d1a2 ÿ _c1b2 � _b1c2 ÿ _a1d2

Dt2 � d1 _a2 ÿ c1 _b2 � b1 _c2 ÿ a1 _d2

�28�

obtained on di�erentiating the last two equations inEq. (24). It is quite noteworthy that the condition of Eq.(26) is necessary and su�cient, and therefore a convergedsolution of the spinor-version iteration scheme Eq. (27)always yields a crossover, in contrast to the vector-version iteration scheme of Eq. (13). Equation (27) failswhen the determinant of the 2� 2 Jacobian matrix,

J � Ct1Dt2 ÿ Dt1Ct2 �29�i.e., the Jacobian of the crossing condition Eq. (26),vanishes.

To further analyze the Jacobian, we take partialderivatives of Q in Eq. (23), i.e.,

Qt1 � _R1Ry2 � At1 � j Bt1 ex � Ct1 ey � Dt1 ez

ÿ �Qt2 � R1

_Ry2 � At2 � j Bt2 ex � Ct2 ey � Dt2 ez

ÿ � �30�

For the derivatives _R1 and _Ry2, from Eqs. (A44) and

(A45), we have

_R1 � 12F 1R1;

_Ry2 � 1

2Ry2Fy2 �31�

with

F 1 � _r1r1� j

g1r21

ez � �m1r1g1

ex

� �; F

y2 �

_r2r2ÿ j

g2r22

ez � �m2r2g2

ex

� ��32�

where g's are the magnitudes of the total angularmomenta,

g1 � g1g1 � r1 � _r1 � r21 r1 � _r1

g2 � g2g2 � r2 � _r2 � r22 r2 � _r2�33�

and �m's are the e�ective normal perturbations of thesatellites in the earth-®xed frame.

Substituting Eq. (31) into Eq. (30), we have

Qt1 � 12F 1R1R

y2 � 1

2F 1Q

Qt2 � R112Ry2Fy2 � 1

2Q Fy2

�34�

Now, substituting Eq. (32) into Eq. (34), in comparisonwith Eq. (30), we obtain

At1 �1

2

_r1r1

Aÿ g1r21

Dÿ �m1r1g1

B� �

Bt1 �1

2

_r1r1

B� g1r21

C � �m1r1g1

A� �

Ct1 �1

2

_r1r1

C ÿ g1r21

B� �m1r1g1

D� �

Dt1 �1

2

_r1r1

D� g1r21

Aÿ �m1r1g1

C� �

�35�

and

At2 �1

2

_r2r2

A� g2r22

D� �m2r2g2

B� �

Bt2 �1

2

_r2r2

B� g2r22

C ÿ �m2r2g2

A� �

Ct2 �1

2

_r2r2

C ÿ g2r22

B� �m2r2g2

D� �

Dt2 �1

2

_r2r2

Dÿ g2r22

Aÿ �m2r2g2

C� �

�36�

as similar to Eq. (A34).With Eqs. (35) and (36), at crossovers, where

C � D � 0, the Jacobian in Eq. (29) becomes

Jv � 1

2

g1r21

g2r22

AB �37�

which, as will be veri®ed, vanishes when the groundtracks become collinear in accordance with Gv in Eq.(16).

We consider the two rotating satellite-®xed orbitalframes (relative to the rotating earth-®xed frame);

r1g1 � r1

g1

8<:9=; � 1

r1Ry1

ex

ey

ez

8<:9=;R1;

r2g2 � r2

g2

8<:9=; � 1

r2Ry2

ex

ey

ez

8<:9=;R2

�38�The spinor algebra enables us to write

756

g1 � g2 � g1g2h i � 1

r1r2Ry1ezR1R

y2ezR2

D E� 1

r1r2ezR1R

y2ezR2R

y1

D E�39�

� 1

r1r2ezQezQ

yD E� 1

r1r2A2 ÿ B2 ÿ C2 � D2ÿ �

on using Eq. (23). In Eq. (39), the bracket denotes thescalar operator in which the product of two spinors iscommutative as applied to R

y1 and ezR1R

y2ezR2. In similar

ways, one can obtain the remaining eight directioncosines between the two orbital frames to write

which generalizes Eq. (A18). With the relation

Q Qy � R1R

y2R2R

y1 � r1r2 � A2 � B2 � C2 � D2 �41�

Eq. (40) yields r1 � r2, and

g1 � r1g1

� �v

� 1

A2 � B2

A2 ÿ B2 2ABÿ2AB A2 ÿ B2

� �g2 � r2

g2

� �v

�42�at crossovers �C � D � 0�.

With the relations

_r1 � g1r21

g1 � r1; _r2 � g2r22

g2 � r2 �43�

Eq. (42) states that, at crossovers, the angle between g1and g2 is same as that of _r1 and _r2. (The ground-trackcrossing angle is essentially formed by two osculatingorbital planes instantaneously passing through the cross-over point. When observed from the earth-®xed frame,the orientations of the orbital planes can be e�ectivelyrepresented by the unit vectors g1 and g2.) We can write

cosKv � A2 ÿ B2

A2 � B2; sinKv � 2AB

A2 � B2�44�

so that the correct quadrant of the crossing angle Kv isde®ned in �0�; 360��.

With Eq. (44), Eq. (37) yields

Jv � 1

4

g1r1

g2r2sinKv �45�

Meanwhile, with Eq. (43), Eq. (16) can be written as

Gv � g1r21

g2r22sinKv

� �2

�46�

We therefore obtain a simple relation

Gv � 4Jv

r1r2

� �2

�47�

between the Jacobian (appearing in the spinor-versioniteration scheme) and the Gaussian curvature (in thevector-version) at crossovers.

4 Number of crossovers

The ground-track crossing condition derived in theprevious section is exact and general as far as thegeocentric nadir mapping is concerned. Given orbitalephemerides of two satellites in their ®nite missionperiods, the crossover locating problem is now reducedto solve the crossing condition. How many crossovers

exist? And when do they (pairs of time-tags) occur?Strictly speaking, the number of solutions can be apriori unknown as highly perturbed, irregularlymaneuvered satellites can generate signi®cantly com-plicated ground-track patterns, and there is nothingspecial about any crossover point from either satel-lite's point of view. Let us consider instead thenumber of crossovers in nominal ground tracks. Inpractice, as most of geodetic satellites have near-circular orbits, nominal ground tracks often havetopologically equivalent structure to true groundtracks. Nevertheless, no satisfactory answer has beengiven to the question regarding the number of cross-over solutions between two nominal ground tracks(even for single-satellite nominal ground tracks asdiscussed before).

We ®rst consider the single-satellite nominal ground-track crossing by representing the spinor Q in terms ofEuler angles which are less useful than spinor compo-nents for numerical purposes but superior in providingsome qualitative insights. Substituting Eq. (A48) intoEq. (24), with some trigonometry one can derive

A � ��������r1r2p ÿ sin sÿ cos i� sin#ÿ � cos sÿ cos iÿ cos#ÿ� �

B � ��������r1r2p � sin s� sin i� sin#ÿ � cos s� sin iÿ cos#ÿ� �

C � ��������r1r2p � cos s� sin i� sin#ÿ ÿ sin s� sin iÿ cos#ÿ� �

D � ��������r1r2p � cos sÿ cos i� sin#ÿ � sin sÿ cos iÿ cos#ÿ� �

�48�with

s� � 12 s1 � s2� �; i� � 1

2 i1 � i2� �;#ÿ � 1

2 #1 ÿ #2� � �49�

From the last two equalities in Eq. (48), with i1 � i2 � i,q1 � q2 � q, ke1 � ke2 � ke, and #ÿ � ÿqsÿ, the ground-track crossing condition C � 0 and D � 0 yield

cos s� sin qsÿ sin i � 0 �50�

r1g1 � r1

g1

8<:9=; � 1

r1r2

A2 � B2 ÿ C2 ÿ D2 2�BC � AD� 2�BDÿ AC�2�BC ÿ AD� A2 ÿ B2 � C2 ÿ D2 2�CD� AB�2�BD� AC� 2�CDÿ AB� A2 ÿ B2 ÿ C2 � D2

24 35 r2g2 � r2

g2

8<:9=; �40�

757

and

cos sÿ sin qsÿ cos i � sin sÿ cos qsÿ �51�Meanwhile we restrict our interests to a repeat period

which corresponds to the two-dimensional domain0 � s2< s1 < 2Mp 0 < sÿ<Mp;� sÿ � s� < 2Mpÿ sÿ�shown in Fig. 5. Note that sin sÿ 6� 0 and sin qsÿ 6� 0 asthe trivial s1 � s2 sÿ � 0� � is excluded, and that Eqs. (50)and (51) show the uncoupling of s� and sÿ. While Eq.(50) is reduced to cos s� � 0 and readily yields 2M so-lutions for 0 � s� < 2Mp, Eq. (51) can be written as

�1ÿ cos i� sin �M � N� sÿM

h i� �1� cos i� sin �M ÿ N� sÿ

M

h i�52�

since q � N=M .Let L denote the number of solutions in Eq. (52) for

0 < sÿ < Mp. We immediately note L � M � N ÿ 1 andL � jjM ÿ N j ÿ 1j, respectively, for the limiting cases ofcos i � ÿ1 and cos i � 1. To investigate the variation ofL in ÿ1 < cos i < 1, we decompose Eq. (51) into

cos sÿ � cos qsÿ � 0 �53�and

cos i � K sÿ� � � sin sÿ cos qsÿcos sÿ sin qsÿ

�54�

where the function K is introduced for convenience.Note that Eq. (53) yields at most one solutionsÿ � Mp=2 which exists when both M and N are odd.This peculiar solution corresponds to crossovers at theequator, whereas Eq. (54) yields nonequatorial cross-overs (if they exist). For an example case of q � 5=9,Fig. 6 presents the sÿ; cos i� � solution loci of Eqs. (53)and (54) showing that L varies from 13 to 3 as the value

of cos i varies from ÿ1 to 1. Note that the changes of Loccur at the ®ve maximal points of the function K.

Now consider the di�erentiation of K in Eq. (54),

K 0 sÿ� � � sin qsÿ cos qsÿ ÿ q sin sÿ cos sÿcos sÿ sin qsÿ� �2 �55�

where the prime indicates di�erentiation with respect tosÿ. Let s�ÿ be an extremal solution of K such that itsatis®es K 0 s�ÿ

ÿ � � 0, or equivalently

cos qs�ÿcos s�ÿ

� qsin s�ÿsin qs�ÿ

�56�

on using the numerator in Eq. (55).Let K� � K s�ÿ

ÿ �denote an extremum of the function

K. Using Eqs. (54) and (56), one can verify

K� � qsin s�ÿsin qs�ÿ

� �2

� qÿ1cos qs�ÿcos s�ÿ

� �2

� 0 �57�

and

K� ÿ q� � K� ÿ qÿ1ÿ � � cos2 s�ÿ ÿ cos2 qs�ÿ

cos s�ÿ sin qs�ÿ

� �2

� 0 �58�

which indicate the range of the extremun K�. In otherwords, the function K is monotone when K < 0 andwhen min q; qÿ1

ÿ �< K < max q; qÿ1

ÿ �. As shown in Fig.

6, the value of L changes at the extrema. Varying theinclination, the value of L can change only in the mixedregion in the �q; cos i� space shown in Fig.1. Insummary, we have

L � M � N ÿ 1 in ÿ1 � cos i < 0kM ÿ N j ÿ 1j � L � M � N ÿ 1 in 0 < cos i < min q; qÿ1

ÿ �L � kM ÿ N j ÿ 1j in min q; qÿ1

ÿ �< cos i � 1

8<:�59�

which veri®es the discussion presented in Sect. 2. Thereexist 2ML crossover solutions (the combination of the2M solutions of s� and the L solutions of sÿ) in the two-dimensional domain of 0 < sÿ < Mp; 0 � s� < 2Mp� �.

Fig. 5. The s1; s2� � domain for single-satellite nominal crossoversolutions

Fig. 6. Loci of single-satellite nominal crossover solutions insÿ; cos i� � plane for q � 5=9

758

Since we are working in its semidomain 0 < sÿ�< Mp; sÿ � s� < 2Mpÿ sÿ�, we say that the numberof single-satellite nominal crossovers is ML.

As we have seen in the two-dimensional parametricspace of single-satellite cases (which forms a subspace ofthe ®ve-dimensional parametric space of dual-satellitecases), changes may occur in the number of crossoversolutions in the mixed region 0 < cos i < min q; qÿ1

ÿ �.

These changes correspond to ``bifurcations''(Guckenheimer and Holmes 1986, p. 117): the termoriginally used to describe the splitting of equilibriumsolutions in a family of di�erential equations at para-metric points or regions where the Jacobian derivative ofthe di�erential equations has a zero eigenvalue. In ourcase, the splitting of crossover solutions occurs wherethe Jacobian of the ground-track crossing conditionEq. (26) is singular. We know that the Jacobian vanisheswhen the ground-track crossing angle becomes 0� or180�. In other words, as it can be quite naturally un-derstood, births and deaths of crossover solutions canoccur at parametric regions where the two ground trackscan be collinear. Such regions will be referred to as bi-furcating regions.

We now return to the ground-track slope equation(5). One can write the ground-track collinear conditionby

1

cos/d/dk

� �1

� 1

cos/d/dk

� �2

�60�

which yields�������������������������������cos2 /ÿ cos2 i2

pcos i1 ÿ q1 cos2 /ÿ �

� ��������������������������������cos2 /ÿ cos2 i1

pcos i2 ÿ q2 cos2 /ÿ � �61�

where, in the dual sign, ``plus'' is for the 0� crossingangle cases (i.e., both tracks are north-going or south-going) and ``minus'' is for the 180� crossing angle cases(i.e., one track is north-going and the other south-going). It is noteworthy that D is absent in Eq. (61), andwe can therefore work with a four-dimensional para-metric space of i1; i2; q1; q2� � to search the bifurcatingregions. As the roles of two tracks can be exchanged, wecan set q2 � q1. Taking squares of both sides, Eq. (61)can be written as

X cos6 /� Y cos4 /� Z cos2 / � 0

with max cos2 i1; cos2 i2� � cos2 / � 1 �62�

and

X = q21 ÿ q22Y = 2 q2 cos i2 ÿ q1 cos i1� �

� q22 cos2 i1 ÿ q21 cos

2 i2ÿ �

Z = cos2 i1 ÿ cos2 i2�2 cos i1 cos i2 q1 cos i2 ÿ q2 cos i1� �

8>>>>>><>>>>>>:�63�

Equation (62) is a cubic equation in cos2 /. Excludingthe peculiar triple solution cos2 / � 0, which can occur

when both ground tracks are circumpolar, it is essen-tially a quadratic equation and may yield 0, 1, or 2 realsolutions depending on the parametric values. Theexistence of a solution (or solutions) implies that thecorresponding parametric point is in bifurcating regionswhere the number of crossovers can be changed as theparameter values vary.

In Fig. 7 we display eleven i1; i2� � sections of thei1; i2; q1; q2� � space by varying the values of q1; q2� �. Theeleven points (A±K) in the q1; q2� � section at the rightlower corner depict the approximate values of q1; q2� �corresponding to the eleven i1; i2� � sections. In each ofthe i1; i2� � sections, bifurcation regions are striped(horizontally, vertically, or in both ways). The hori-zontally striped regions correspond to the existence ofthe 0� crossing angle solution, and the vertically stripedregions correspond to the existence of the 180� crossingangle solution. While both roots exist in the regionswhere the horizontal stripes and vertical stripes cross, noreal root exists in the white areas, which will be calledregular regions. The lines labeled a±g are

a. sin i2 ÿ i1� � � q1 sin i2 ÿ q2 sin i1b. sin i2 � i1� � � q1 sin i2 � q2 sin i1c. cos i1 � � cos i2d. cos i1 � ÿ cos i2e. q2 cos i2 � 1

f. q1 cos i1 � 1

g. Y 2 ÿ 4XZ � 0

�64�

at which the characteristics of the parametric space canbe signi®cantly changed.

In each of the eleven sections, there essentially existfour regular regions (left, right, lower, and upper whiteareas) which are separated by the diagonal and antidi-agonal bifurcating regions. We focus our discussion onthe section A q1 � N1=M1 � 1=3; q2 � N2=M2 � 1=2� �which is typical for two satellites below the geosyn-chronous altitude (i.e., q < 1). In each of the four reg-ular regions, the number of crossovers is invariant. Sinceone can consider the four limiting points i1� 0�;�i2 � 90��, i1�180�; i2 �90�� �, i1 � 90�; i2 � 0�� �, i1 ��90�; i2 � 180�� as representative points of the four reg-ular regions, the corresponding numbers of crossoversare readily summarized as 2M2 M1 ÿ N1j j; 2M2 M1 � N1� �,2M1 M2 ÿ N2j j; 2M1 M2 � N2� �, respectively. For in-stance, with i1 � 0�; i2 � 90�� � each of the M2 revolu-tions of the circumpolar ground track crosses with eachof the M1 ÿ N1j j revolutions of the eastward equatorialtrack twice (one along its ascending trace and one alongits descending trace). In the bifurcating regions, thenumber of crossovers cannot be easily guessed and maydepend on the value of the remaining parameter D. Infact, the ground-track crossing can be highly sensitive tosmall changes in any of the ®ve parameters. For actualground tracks, such a small change can be easily intro-duced as the parameters experience perturbed varia-tions.

759

5 Applications

We now aim to ®nd crossovers associated with ephemer-ides of actual satellites. Those satellites are GEOSAT(which ¯ew fromMarch 1985 to December 1989), ERS-1(which has been orbiting since 22 April 1991), and

TOPEX/POSEIDON (which has been orbiting since 10August 1992). For each of the three satellites, afterchecking their repeatability using the ephemerides to bedescribed, we selected a complete repeat cycle:17.050458355 days from 02:55:43 10 April 1987 in theGEOSAT Exact Repeat Mission (ERM), 34.999940729

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

760

days from 03:45:56 22 February 1993 in the ERS-1 Phase-C mission, and 9.915647338 days from 04:29:45 2 March1994 in the TOPEX/POSEIDON mission. The threerepeat missions have, respectively, iG � 108�; qG ��NG=MG �17=244�, iE � 98:5�; qE � NE=ME � 35=501� �,and iT � 66�; qT � NT=MT � 10=127� � where the sub-scripts denote mission initials. We can say that theGEOSAT and ERS-1 ground tracks are retrogradewestward, and the TOPEX/POSEIDON ground track isposigrade eastward.

Orbital trajectories of actual satellites are often pro-vided in terms of tabulated ephemerides. In addition tothe TOPEX/POSEIDON ephemerides (Tapley et al.1994), the GEOSAT-ERM and ERS-1 Phase-C ephe-merides used were consistently generated using theUTOPIA orbit determination software system at theCenter for Space Research at the University of Texas atAustin. Each of the three satellites' orbital trajectorieswas given in the form of a discretely tabulated time-series of earth-®xed geocentric equatorial Cartesianstates at 10-s intervals. For usage of such tabulatedstates in the search of ground-track crossovers, we needan e�cient and accurate interpolator to compute thestate at any instant time in the table. For the state in-terpolation, we adopted the Hermite interpolationscheme (Ferziger 1981, p. 12). To ®nd the location of aspeci®c time in the state tables, the routine ``hunt'' inPress et al. (1922, pp. 111±113) is used.

The spherical earth so far used to describe groundtracks can be replaced by a more realistic earth-surfacemodel. Let the subsatellite point ro � xo; yo; zof g be anormal projection of the satellite's position r � fx; y; zgonto a generalized earth-surface model E ro� � � 0. Thesurface normal projection to minimize the satellite's al-titude above the surface can be formulated by

Minimize P ro; g� � � r ÿ roj j2� gE ro� � �65�where g is the Lagrange multiplier. A proper earth-surface model is the geoid, especially for oceanographicapplications of satellite altimetry, as employed by Shum(1982) by utilizing a spherical harmonics geopotentialmodel. In practice, as it is widely adopted in satellitegeodesy, the geoid is well approximated by a referenceellipsoid of revolution (Torge 1991, pp. 44±49) with itssemimajor axis ae and semiminor axis be, such that

E � x2o � y2o � kz2o ÿ a2e with k � a2eb2e

�66�

Albeit for the ellipsoidal nadir mapping, closed-formformulas exist with some complexities (Grafarend andLohse 1991), we prefer the sequences

xo � x1� g

; yo � y1� g

; zo � z1� kg

E � xoxo � yoyo � kzozo ÿ a2e

xgo �

ÿxo1� g

; ygo �

ÿyo1� g

; zgo �

ÿkzo1� kg

Eg � 2 xoxgo � yoyg

o � kzozgo

ÿ �

g gÿ EEg

Iterate if jEj isn't small enoughxto �

_x1� g

; yto �

_y1� g

; zto �

_z1� kg

Et � 2 xoxto � yoyt

o � kzozto

ÿ �_g � ÿ Et

Eg

_xo � xto � _gxg

o; _yo � yto � _gyg

o; _zo � zto � _gzg

o

starting from a proper positive value of g. Thesuperscript t and g appearing in the above equationsimply partial di�erentiations. Although the ground-track crossing condition derived before was based on themapping of satellite states to a spherical-earth model, itis noteworthy that a generalized surface mapping can bere¯ected simply by using the nadir-mapped ground-track states instead of the satellite's orbital states inconstructing the crossing condition and solving it to ®ndcrossovers. Here, for the computation of the threesatellites' ground tracks, we use the ellipsoidal nadirmapping of satellite states on the TOPEX/POSEIDONstandard reference ellipsoid of revolution withae � 6378136:3m and be � 6356751:6m (the reciprocal¯attening of 298.257).

To ®nd actual crossovers, one can pursue an ap-proach speci®c to a particular combination of groundtracks. However, in general we have to map out the full�t1; t2� domain. ``There are no good, general methods forsolving systems of more than one nonlinear equation'' asstated in Press et al. (1992, p. 372). Here, we pursue agrid-search scheme by setting initial guesses of solutionson a regular grid of the corresponding �t1; t2� domain;the grid scheme can be outlined as

Loop for �t1k � t1i; t1k � t1f ; t1k � t1k � Dt1�Loop for �t2k � t2i; t2k � t2f ; t2k � t2k � Dt2�0. Initialize iteration: ft1; t2g � ft1k; t2kg;1. Compute a1; b1; c1; d1; _a1; _b1; _c1; _d1 at t1 and

a2; b2; c2; d2; _a2; _b2; _c2; _d2 at t2;2. Compute Ct1 ;Dt1 ;Ct2 ;Dt2 using Eq. (28);3. Compute J using Eq. (29);4. If J is tiny, abort iteration;5. Update ft1; t2g using Eq. (27);6. If jt1 ÿ t1kj > pDt1 or jt2 ÿ t2kj > pDt2, abort

iteration;7. Compute C;D using Eq. (24);8. If C and D are not su�ciently small, go to 1;9. If t1i � t1 � t1f and t2i � t2 � t2f , store ft1; t2g as

a crossover time-tag pair;End Loop

End Loop

Sort the stored crossover time-tag pairs and removeduplicated pairs.

In this outline, t1i and t1f denote the initial and ®naltimes for the ®rst satellite, t2i and t2f denote the initialand ®nal times for the second satellite, t1k and t2k denote

761

initial guesses of crossover solutions on the grid knots,Dt1 and Dt2 denote the grid-knot steps, and p denotes aparameter to determine the extent of duplicated surveyby controlling the size of a box, which is centered at theinitial grid knot ft1k; t2kg and brackets the solutionsearch. For the computation of spinor states, we needinterpolation of the vector states using ephemerides ta-bles, their nadir mappings, and vector-to-spinor ground-track state transformations. The outline is written for anillustrative purpose of a general dual-satellite crossoverlocating problem, and may not be an optimal scheme fora speci®c problem.

There are six ground-track con®gurations for thethree repeat cycles of GEOSAT-ERM, ERS-1, andTOPEX/POSEIDON; three single-satellite cases andthree dual-satellite cases. With the Hermite state inter-polator, the ellipsoidal nadir mapping, and the two-waytransformations between the vector and spinor states atour disposal, we applied the grid-search scheme to allthe six cases. We have deployed the initial guesses ofcrossover solutions on a 1=8� 1=8-revolution grid re-gardless of single- or dual-satellite ground-track sce-narios. In other words, the grid-knot step size is 754.69,754.49, and 843.22 s, respectively for GEOSAT-ERM,ERS-1, and TOPEX/POSEIDON. We used p � 1:0 sothat the search at a grid knot covers a 1=4� 1=4-re-volution box.

The number of single-satellite crossovers numericallyfound are 63440, 268035, and 14732, respectively for thethree cycles. These numbers agree exactly with thenumber of nominal single-satellite crossovers MG�MG�NG ÿ 1�; ME�ME � NE ÿ 1�; MT�MT ÿ NT ÿ 1� of thethree repeat missions. The number of dual-satellitecrossovers found are 261522, 117234, 57638, respec-tively, for the three combinations of GEOSAT-ERMand ERS-1, ERS-1 and TOPEX/POSEIDON, andGEOSAT-ERM and TOPEX/POSEIDON. The ®rsttwo numbers exactly match with 2ME�MG � NG� and2ME�MT ÿ NT�, indicating that they are located at theregular regions. However the number of crossovers be-tween the GEOSAT-ERM and TOPEX/POSEIDONcannot be easily explained. Returning to Fig. 7 andseeing the section J�q1 � 17=244; q2 � 10=127�, whichis topologically equivalent to the section A, one mayapproximately locate the parametric point�i1 � 108�; i2 � 66�� corresponding to the dual groundtracks of the GEOSAT-ERM and TOPEX/POSEI-DON. The point is located in the vertically striped bi-furcating region along the diagonal. It is locatedbetween the lower regular region and the right regularregion and the number 57638 is between2MG�MT ÿ NT� � 57096 and 2MT�MG � NG� � 66294.For the combination of the GEOSAT-ERM and TO-PEX/POSEIDON, Eq. (62) yields a real solutioncos2 / � 0:785, indicating that crossovers near 27:5�S or27:5�N have small acute angle and therefore a slightperturbation can easily generate or destroy crossovers.These results support the bifurcation analysis presentedin the previous section.

6 Conclusions

Techniques to locate satellite ground-track crossovershave been frequently presented as an auxiliary proce-dure of other main topics related to the applications ofsatellite altimetry. In this paper, we have studied theissue as an independent topic. The principal new aspectsinvestigated (or presented) herein are: (1) classi®cationof idealized nominal ground tracks in the two-dimensional parametric space of �q; cos i� (2) classi®ca-tion of dual-satellite nominal ground-track bifurcationsin the four-dimensional parametric space of �i1; i2; q1; q2�in conjunction with the number of crossover solutions,and (3) numerical searching schemes of actual cross-overs in both vector and spinor forms. Indeed, some ofthese interesting aspects seem not to have been surveyedbefore.

In a problem related to satellite orbits, a proper se-lection of variables often leads to simpli®cation of theproblem, as we have used the spinor representation ofthe satellite state to establish the ground-track crossingcondition. In satellite altimetry, a more conventionalcrossover locating method is based on usage of thegeodetic latitude and longitude, see, e.g., Rowlands(1981) or Wisse et al. (1995, pp. 72±76). The use ofgeodetic coordinates originated from the fact that thesatellite-altimeter geophysical data record (GDR) al-ready contains them along with the altimeter-rangemeasurement, its time-tag, and other environmental/geophysical corrections. However, when one pursues thedirect locating of crossovers from satellite-state tables,the vectors or spinors will be more adequate. Once theunknown crossover time-tags are obtained, it is a matterof data hunting and interpolation to get the corre-sponding altimeter data records. One may also preferthe vector version of the iteration scheme to the spinorversion, as it can be more understandable. The mainadvantage of using spinors is that a converged solutionof iteration always yields a crossover. The main dis-advantage of using spinors is that, as the satellite statesare usually given by Cartesian vectors, one needs con-versions between the vector and spinor state. There is atrade-o� unless the satellite states are initially given byspinors. However, since no trigonometrics are involvedin the state conversions, the increase of computationallabor produced by using the spinor- language should notbe overestimated. The selection of variables will behighly dependent on one's object.

Acknowledgements. Thanks are due to B. Tapley, B. Schutz, andR. Eanes for their encouragement and interest on this subject, toJ. Ries for providing ephemerides of the three altimeter satellites,to S. Bettadpur for reading a part of the paper in its ®rst draft, andto Y. Hahn in Jet Propulsion Laboratory and M. Anzenhofer inGFZ (GeoForschungsZentum, Potsdam) for ®nding some refer-ences. Sincere appreciation is due to anonymous reviewers whosecomments were helpful in improving the paper.

762

Appendix

Satellite state in spinor space

Let fex; ey ; ezg and feu; ev; ewg be right-handed orthogo-nal triads of geocentric unit vectors which form therotating earth-®xed frame and the non- rotating inertialframe, respectively. In the spinor algebra, which forms asubalgebra of the so-called Cli�ord algebra [geometricalgebra in Hestenes (1983)], such a triad corresponds tothe Pauli spin matrices (Goldstein 1980, p.156) andyields the spinor multiplication laws (Cartan 1981, p.44;Landau and Lifshitz 1977, p.202), e.g.,

exex � ey ey � ezez � 1

exey � jez � ÿey ex (in cyclic order)�A1�

where j is the unit imaginary number.Let �v1 � b1ex � c1ey � d1ez and �v2 � b2ex � c2ey

�d2ez be two arbitrary vectors. With the multiplicationlaws of Eq. (A1), one can readily verify the relation

v1v2 � v1 � v2 � j v1 � v2 �A2�between the spinor product, the scalar product, and thevector product of the two vectors. Equation (A2) can bealso represented by

v1 � v2 � 12�v1v2 � v2v1�

v1 � v2 � ÿ12j�v1v2 ÿ v2v1� �A3�

The spinor space is a combination of the scalar andvector spaces such that a spinor R � a� j v and itsconjugate R

y � aÿ j v, where v � bex � cey � dez, arede®ned by the four real-valued scalar componentsfa; b; c; dg. It is equivalent to the two-dimensionalcomplex vector space appearing in quantum mechanicsin conjunction with the wave functions (Greiner 1994, p.306). We will frequently denote the scalar part of R withhRi, i.e.,hRi � a � hRyi � 1

2�R� Ry� �A4�

where the bracket h i will be called the scalar operator.Let R1 � a1 � j v1 and R2 � a2 � j v2 be two arbitrary

spinors. With Eq. (A2), one obtains

R1Ry2 � �a1 � j v1��a2 ÿ j v2�� a1a2 � v1 � v2 � j�a2v1 ÿ a1v2 � v1 � v2� �A5�

which can be further expanded in terms of the realcomponents fa1; b1; c1; d1g and fa2; b2; c2; d2g to yieldthe equations given in Eq. (24) in the main text.

The products of the two spinors are in general notcommutative, e.g., R1R

y2 6� R

y2R1. But they are commuta-

tive inside the scalar operator, i.e., with Eq. (A5), we have

hR1Ry2i � hRy2R1i � a1a2 � v1 � v2� a1a2 � b1b2 � c1c2 � d1d2 �A6�

For the norm of the spinor R, we use a positive scalar

jRj � �R Ry�1=2 � �RyR�1=2 � �a2 � b2 � c2 � d2�1=2

�A7�The spinor R can be decomposed into jRjR where R is aunimodular spinor (i.e., jRj � 1�.

The spinor algebra is equivalent to the quaternionalgebra, which has long been known as the ideal tool todescribe rotational motions of rigid bodies. In practice,the four components of a unimodular spinor correspondto the Euler parameters (Goldstein 1980, p. 155). Forinstance, the earth's rigid body rotation can be e�ec-tively represented by

ex

ey

ez

8<:9=; � Ey

eu

ev

ew

8<:9=;E � xe �

eu

ev

ew

8<:9=; �A8�

where E is a unimodular spinor which accounts for theprecession, nutation, spin axis rotation and polarmotion of the earth (Lambeck 1988), and xe is thecorresponding angular velocity vector.

There is a complete analogy between the orbital an-gular motion of a satellite and the rotational motion of arigid body. Let s � ueu � vev � wew represent the orbitalposition vector of a satellite in the geocentric inertialframe. Let S � a� j�beu � cev � dew� be a spinor basedon the triad feu; ev; ewg. As it can be directly veri®edusing the multiplication laws of Eq. (A1), which is alsoapplicable to feu; ev; ewg, the spinor algebra enables usto write

s � SyeuS �A9�

with

u � a2 � b2 ÿ c2 ÿ d2; v � 2�bc� ad�; w � 2�bdÿ ac��A10�

Let s be the magnitude of s. We also have

s� sj j � u2 � v2 � w2ÿ �1=2� S

�� ��2� S Sy � S

yS

� a2 � b2 � c2 � d2�A11�

On di�erentiating Eq. (A9), we have

_s � _SyeuS � S

yeu

_S �A12�which can be expanded to relate the scalar components

of _s � _ueu � _vev � _wew and those of _S � _a� j� _beu � _cev

� _dew�. One can actually write

u _uv _vw _ws _s

26643775 �

a b ÿc ÿdd c b aÿc d ÿa ba b c d

26643775

a 2 _ab 2 _bc 2 _cd 2 _d

26643775 �A13�

on using the equations in (A10), (A11), and theirdi�erentiations. As the six components of s and _sconstitute the orbital state in the vector space, the eight

components of S and _S constitute the orbital state in thespinor space. While Eq. (A13) stands for the spinor-to-

763

vector state transformation, we need more informationto write its inverse transformation (note that s and _s areauxiliaries and the 4� 4 matrix is inherently singular).

As the vector s can be decomposed into its dilation sand rotation s, i.e., s � ss, where s is the unit vector

pointing s, the spinor S can be also decomposed into its

dilation S�� �� and rotation S, i.e., S � S

�� ��S. As shown in

Hestenes (1983), the unimodular spinor S can be de-

composed into the Eulerian rotations, i.e.,

S � exp� jew12s� exp� jeu

12i� exp� jew

12X� �A14�

where the three Euler angles X; i; s (Goldstein 1980, p.146) correspond, respectively, to the longitude of theascending node measured from the equinox, the incli-nation, and the argument of latitude of the orbit. Withthe relation S

�� �� � s1=2, we can write

S � s1=2�cos 12s� jew sin12s��cos 12i� jeu sin

12i�

� �cos 12X� jew sin12X� �A15�

which can be expanded to yield the spinor components

a � s1=2 cos 12�X� s� cos 12i; b � s1=2 cos 12�Xÿ s� sin 12i

c � s1=2 sin 12�Xÿ s� sin 1

2i; d � s1=2 sin 12�X� s� cos 12i

�A16�showing their correspondence to the Euler parameters(Goldstein 1980, p. 155) and to the Cayley-Kleinparameters (Whittaker 1937, p. 12).

Let h be the orbital angular momentum vector of thesatellite, i.e.,

h � hh � s� _s � ss� � _ss� s _s� � s2s� _s �A17�where the scalar h and the unit vector h represent themagnitude and direction of h, respectively. Now weintroduce a new triad fs; h� s; hg, which consists of theradial, transversal, and normal unit vectors of therotating satellite-®xed orbital frame. The spinor algebraenables us to write the transformation

where the four-parametric representation of the 3� 3transformation matrix can be also found in classicaldynamics textbooks, e.g., Goldstein (1980, p. 153). Wedenote the nine elements of the transformation matrixby T11; T12; . . . ; T33. These can be easily obtained fromthe vector state fs; _sg, as they correspond to theCartesian components of s; h� �s; sh. Note that thetransverse unit vector h� s is pointing towards thevelocity of the radial unit vector

_s � hs2

h� s �A19�

which itself is not a unit vector. We have

j _sj � hs2

�A20�

Based on Shepperd (1978), an e�cient means ofcomputing the four components of S from the trans-formation matrix is to write

with T � T11 � T22 � T33: the trace of the transformationmatrix. Several ways exist to obtain the four unknowns,e.g.,

a � ������������T � s4

r; b � T23 ÿ T32

4a

c � T31 ÿ T134a

; d � T12 ÿ T214a

�A22�

where the sign of a cannot be determined but can beselected freely assuming that we are working with a one-to-one correspondence, or isomorphism, between s andthe pair �S. One may say that S is a double-valuedfunction of the transformation matrix. While thesolution given in Eq. (A22) is singular when a � 0, wecan write three more solutions by changing the compo-nent to be used as the divisor. At least one nonzerocomponent exists as long as s 6� 0. For numericalpurposes, the optimal choice among the four possibil-ities corresponds to the largest among absolute values ofa; b; c; d, whose magnitude order is same as that ofT ; T11; T22; T33.

sh� s

h

8<:9=; � 1

sSy

eu

ev

ew

8<:9=;S � 1

s

a2 � b2 ÿ c2 ÿ d2 2�bc� ad� 2�bdÿ ac�2�bcÿ ad� a2 ÿ b2 � c2 ÿ d2 2�cd� ab�2�bd� ac� 2�cdÿ ab� a2 ÿ b2 ÿ c2 � d2

24 35 eu

ev

ew

8<:9=; �A18�

4a2 � 2T ÿ T � s4b2 � 2T11 ÿ T � s4c2 � 2T22 ÿ T � s4d2 � 2T33 ÿ T � s

8>><>>: and4ad � T12 ÿ T21; 4bc � T12 � T214ab � T23 ÿ T32; 4cd � T23 � T324ac � T31 ÿ T13; 4bd � T31 � T13

8<: �A21�

764

Now, to obtain the components of the time derivative_S, we consider the Eulerian angular velocity vector ofthe orbit expressed as (Hestenes 1983)

xs � dsdt

h� didt

f � dXdt

ew �A23�

with

h � sin i sinX eu ÿ sin i cosX ev � cos i ew �A24�and

f � cosX eu � sinX ev �A25�where f represents the unit vector toward the ascendingorbital node. For the time derivatives of the Eulerangles, we have the Gauss variation of parameterequations (Battin 1987, p.488)

dXdt� �n

shsin ssin i

;didt� �n

shcos s;

dsdt� h

s2ÿ �n

shsin s cos isin i

�A26�where �n represents the normal component of the satelliteacceleration.

Substituting Eqs. (A24±A26) into Eq.(23), with somemanipulation, one can obtain

xs � 1

sSy h

s2ew � �n

sh

eu

� �S �A27�

or

xs � hs2� �n

sh

�A28�

on using the ®rst equality in Eq. (A18). Meanwhile, thevelocity vector _s can be decomposed into (Hestenes andLounesto 1983)

_s � _ss� xs � s � _ss

sÿ 12 j�xssÿ s xs� � 1

2 �Wys� sW �

�A29�with

W � _ss� j xs �A30�

Comparing Eqs. (A12) and (A29), we obtain the timederivative,

_S � 12 S W �A31�

which can be also written as

_S � 12G S �A32�

with

G � 1

sS W S

y � _ss� j

hs2

ew � �nsh

eu

� ��A33�

on using Eqs. (A27) and (A30). Now, with Eq. (A33),one can expand Eq. (A32) to yield the four componentsof _S,

_a � 12

_ss aÿ h

s2 dÿ �n sh b

ÿ �; _b � 1

2_ss b� h

s2 c� �n sh a

ÿ �_c � 1

2_ss cÿ h

s2 b� �n sh d

ÿ �; _d � 1

2_ss d� h

s2 aÿ �n sh c

ÿ � �A34�To obtain the right-hand sides in Eq. (A34) we need

to know �n. However, having the same instantaneousposition and velocity as the true orbit, we can set �n � 0,which relates the spinor state to the osculating Keplerianorbit. From Eq. (A34) one can extract

c _a� d _bÿ a _cÿ b _d � 0 �A35�and

ÿb _a� a _b� d _cÿ c _d � �ns2

h�A36�

The relation in Eq. (A35) is called the spinor gaugecondition (Hestenes and Lounesto 1983) and alsoappears in the Kustaanheimo-Stiefel transformation(Stiefel and Scheifele 1971; Deprit et al. 1994). Thetwo relations of Eqs. (A35) and (A36) compensate thetwo redundancies in the spinor state.

So far we have established the two-way transforma-tion between the vector and spinor states observed in theinertial frame. The transformation is general andequivalently applicable to states for the earth-®xedframe. To clarify, let us ®rst denote the orbital positionand velocity vectors of a satellite observed in the earth-

®xed frame by r � xex � yey � zez � RyexR and _r � _xex

� _yey � _zez � _RyexR� R

yex

_R, where R � a� j bex � ceyÿ

�dez� is a spinor based on the rotating earth-®xed triad

ex; ey ; ez�

. Note that r � s but their Cartesian compo-nents are di�erent, i.e., fx; y; zg 6� fu; v;wg, as they arereferenced to the earth-®xed frame and the inertialframe, respectively. Also note that _r 6� _s as the timederivatives are observed in the two di�erent referenceframes. The di�erence is the e�ects of the earth rotation,i.e.,

_r � _sÿ xe � s �A37�The total angular velocity (to be denoted by xr) of theorbit observed in the earth-®xed frame combines theorbital angular velocity and the earth-rotation e�ect inthe sense of taking their di�erence, i.e.,

xr � xs ÿ xe �A38�and therefore the total angular momentum g � r � _r isdue to the orbital angular momentum h � s� _s and theearth-rotation e�ect, i.e.,

g � hÿ r � xe � r � hÿ r2xe � r � xe� �r �A39�on using Eq. (A37).

Substituting Eq. (A28) into Eq. (A38) and usingEq. (A39), we can write

765

xr � gr2� �m

rg

�A40�

with

�mrg� �n

shÿ r � xe �A41�

where �m can be called the normal component acceler-ation of the satellite e�ective in the rotating earth-®xedframe.

Note that the two two-way transformationsu; v;w; _u; _v; _w� � $ ÿ

a; b; c; d; _a; _b; _c; _d�and x; y; z; _x; _y; _z� �

$ a; b; c; d; _a; _b; _c; _dÿ �

are equivalent under Eq. (A41)which is nothing but r � xr � r � xs ÿ r � xe. We canwrite the counterparts of Eqs. (A30)±(A36) simply byusing new variables. Let

V �_rr� j xr �A42�

then,

_R � 12 R V �A43�

and

_R � 12 F R �A44�

with

F � 1

rR V R

y � _rr� j

gr2

ez � �mrg

ex

� ��A45�

and so on.The vector state r; _r

� is independent of the e�ective

normal acceleration �m which changes _R (but, not R).From Eq. (A41), we note that �m is caused by �n and theradial component of xe (i.e., r � xe). For �n one may re-¯ect the e�ects of any perturbation, e.g., the earth'soblateness with

�n � ÿ3J2le

r2re

r

� �2zrcos i �A46�

where le is the gravitational parameter of the earth, re isthe mean radius of the equator, and J2 is the secondzonal harmonic of the geopotential. Again, for thepurpose of state transformation, we can work with anosculating Keplerian orbit, i.e., we can set �n � 0. Like �n,r � xe does not change r; _r

� and R. However, the

component of xe normal to r changes R and _R as well as_r (but, not r). We consider a simpli®ed earth-rotationmodel uniformly rotating along the ew-axis, i.e.,

xe � xoew � xoez �A47�where xo represents the constant rate of the Greenwichmean sidereal time to be denoted by h. Then, as thecounterpart of Eq. (A16), the components of R can beexpressed as

a � r1=2 cos 12 �#� s� cos 12 i; b � r1=2 cos 12 �#ÿ s� sin 12 i

c � r1=2 sin 12 �#ÿ s� sin 1

2 i; d � r1=2 sin 12 �#� s� cos 12 i

�A48�

where

# � Xÿ h �A49�is the argument of ascending orbital node measuredfrom the Greenwich meridian.

References

Battin RH (1987) An introduction to the mathematical methods ofastrodynamics. American Institute of Aeronautics and Astro-nautics, New York

Born GH, Tapley BD, Santee ML (1986) Orbit determinationusing dual crossing arc altimetry. Acta Astronaut 13±4:157±163

Cartan E (1981) The theory of spinors. Dover, New YorkCloutier JR (1983) A technique for reducing low-frequency, time-

dependent errors present in network-type surveys. J GeophysRes 88:659±663

Colombo OC (1984) Altimetry, orbits and tides. NASA TechnicalMemorandum 86180, Goddard Space Flight Center

Cui C, Lelgemann D (1992) On cross-over di�erences of the radialorbital perturbations as functions of force model parameters.In: Geodesy and physics of the earth, International Union ofGeodesy and Geophysics. Springer, Berlin Heidelberg NewYork

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 TOPEXmission design. AAS 85±301:13±36

Ferziger JH (1981) Numerical methods for engineering application.Wiley, New York

Gaspar P, Ogor F, Le Traon P-Y, Zanife O-Z (1994) Estimatingthe sea state bias of the TOPEX and POSEIDON altimetersfrom crossover di�erences. J Geophys Res 99:24981±24994

Goldstein H (1980) Classical mechanics (2nd edn) Addison-Wesley,Reading Mass

Grafarend EW, Lohse P (1991) The minimal distance mapping ofthe topographic surface onto the (reference) ellipsoid of revo-lution. Manuscr Geod 16:92±110

Greiner W (1994) Quantum mechanics, an introduction (3rd edn)Springer, Berlin Heidelberg New York

Guckenheimer G, Holmes P (1986) Nonlinear oscillations, dy-namical systems, and bifurcations of vector ®elds. Springer,Berlin Heidelberg New York

Hagar H (1977) Computation of circular orbit ground trace in-tersections. Engineering Memorandum 315±29, Jet PropulsionLaboratory

Hestenes D (1983) Celestial mechanics with geometric algebra.Celestial Mech 30:151±170

Hestenes D, Lounesto P (1983) Geometry of spinor regularization.Celestial Mech 30:171±179

Hsu SK, Geli L (1995) The e�ect of introducing continuity con-ditions in the constrained sinusoidal crossover adjustmentmethod to reduce satellite orbit errors. Geophys Res Lett22:949±952

KimMC (1993) Determination of high-resolution mean sea surfaceand marine gravity ®eld using satellite altimetry. Ph.D Disser-tation. Dept Aerospace Engineering and Engineering Mechan-ics, The University of Texas at Austin

King JC (1976) Quantization and symmetry in periodic coveragepatterns with applications to earth observation. J Astronaut Sci24:347±363

Klokocnik J, Wagner CA (1994) A test of GEM-T2 from GEO-SAT crossovers using latitude lumped coe�cients. Bull Geod68:100±108

Kozel BJ (1995) Dual-satellite altimeter crossover measurementsfor precise orbit determination. Ph.D. Dissertation. DeptAerospace Engineering and Engineering Mechanics, The Uni-versity of Texas at Austin

766

Lambeck K (1988) Geophysical geodesy, the slow deformations ofthe earth. Oxford University Press, Oxford

Landau LD, Lifshitz EM (1977) Quantum mechanics (non-rela-tivistic theory) (3rd edn) Pergamon Press, Oxford

Leick A (1995) GPS satellite surveying (2nd edn) Wiley, New YorkMa XC, Shum CK, Eanes RJ, Tapley BD (1994) Determination of

ocean tides from the ®rst year of TOPEX/POSEIDON altimetermeasurements. J Geophys Res 99:24809±24820

Moore P, Ehlers S (1993) Orbital re®nement of ERS-1 using dualcrossover arc techniques with TOPEX/POSEIDON. ManuscrGeod 18:249±262

Moore P, Rothwell DA (1990) A study of gravitational and non-gravitational modeling errors in crossover di�erences. ManuscrGeod 15:187±206

Nerem RS (1995) Global mean sea level variations from TOPEX/POSEIDON altimeter data. Science 268:708±711

Parke ME, Stewart RH, Farless DL (1987) On the choice of orbitsfor an altimetric satellite to study ocean circulation and tides. JGeophys Res 92:11693±11707

Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992)Numerical recipes in FORTRAN (2nd edn) Cambridge Uni-versity Press, Cambridge

Rowlands D (1981) The adjustment of SEASAT altimeter data ona global basis for geoid and sea surface height determination.Report No. 325, Dept Geodetic Science and Surveying, TheOhio State University

Sandwell DT, Milbert DG, Douglas BC (1986) Global nondy-namic orbit improvement for altimeter satellites. J Geophys Res91:9447±9451

Santee ML (1985) Orbit correction using dual satellite crossing arctheory. CSR-85-4, Center for Space Research, The University ofTexas at Austin

Schrama EJO (1989) The role of orbit errors in processing of sat-ellite altimeter data. Ph.D. Thesis, Delft University Press, Delft

Schutz BE, Tapley BD, Shum CK (1982) Evaluation of the SEA-SAT altimeter time tag bias. J Geophys Res 87:3239±3245

Shepperd SW (1978) Quatenion from rotation matrix. J GuidanceControl 1:223±224

Shum CK (1982) Altimeter methods in satellite geodesy. Ph.D.Dissertation. The University of Texas at Austin

Shum CK, Zhang BH, Schutz BE, Tapley BD (1990a) Altimetercrossover methods for precision orbit determination and themapping of geophysical parameters. J Astronaut Sci 38:355±368

Shum CK, Werner RA, Sandwell DT, Zhang BH, Nerem RS,Tapley BD (1990b) Variations of global mesoscale eddy energyobserved from GEOSAT. J Geophys Res 95:17865±17876

Spoecker NT (1991) Crossover generation. Technical Report ERS-D-ADP-32601, DGFI Dept 1, Munich, Germany

Stiefel EL, Scheifele G (1971) Linear and regular celestial me-chanics. Springer, Berlin Heidelberg New York

Stottlemyer BA (1996) Improvement of the Venusian topographymodel using dual satellite altimeter crossover data from Ma-gellan and the Pioneer Venus orbiter. Technical Memorandum(CSR-TM-94-04), Center for Space Research, The University ofTexas at Austin

Tai CK, Fu LL (1986) On crossover adjustment in satellitealtimetry and its oceanographic implications. J Geophys Res91:2549±2554

Tapley BD, Ries JC, Davis GW, Eanes RJ, Schutz BE, Shum CK,Watkins MM, Marshall JA, Nerem RS, Putney BH, KloskoSM, Luthcke SB, Pavlis D, Williamson RG, Zelensky NP(1994) Precision orbit determination for TOPEX/POSEIDON.J Geophys Res 99:24383±24404

Torge W (1991) Geodesy, an introduction (2nd edn) Walter deGruyter, Berlin

Whittaker ET (1937) A treatise on the analytical dynamics ofparticles and rigid bodies; with an introduction to the problemof three bodies (4th edn) Cambridge Univ Press, Cambridge

Wisse E, Naeije MC, Scharroo R, Smith AJE, Vossepoel FC,Wakker KF (1995) Processing of ERS-1 and TOPEX/POSEI-DON altimeter measurements. BCRS report 95-02, Delft Uni-versity of Technology, Faculty of Aerospace Engineering,Section Space Research and Technology

767