Orbit Determination from Single Pass Doppler Observations

336 IRE TRANSACTIONS ON MILITARY ELECTRONICS April-July

Orbit Determination from Single Pass Doppler Observations*R. B. PATTON, JR. t

Summary-This paper presents a method for determining the may be neglected without serious loss in accuracy. Thisorbit of a satellite by observing, in the course of a single pass, the reduces the problem to one of determining the six pa-Doppler shift in the frequency of a CW signal transmitted from theground and reflected by the satellite to one or more ground-basedareceivers at remote sites. The method is sufficiently general that, Doppler system with a ground-based transmitter. Ob-with minor modification, it may be applied to any type of satellite servations from such a system will be referred to in thisor ICBM tracking measurements. The computation consists of im- paper as passive data, as opposed to active data whichproving approximations for initial position and velocity components would be obtained from a system that included anby successive differential corrections which are obtained from a least air-borne transmitter as an integral part of the track-squares treatment of an over-determined system of condition equa-tions while imposing elliptic motion as a constraint. Methods for ob- ing instrumentation. From a data reduction point oftaining approximations for the initial position and velocity compon- view, there are several distinct advantages in having theents are likewise discussed. Results are presented for computations transmitter on the Earth's surface. Of primary interestwith typical input data. are the following.

INTRODUCTION 1) Both the transmitter frequency and drift may beY obse

...accurately measured if the signal source is ground-.srving the Doppler shift in a CW signal, an based whereas they become additional unknowns

orbit may be determined from a few minutes ofdata recorded in the course of a single pass of a f oreaysse th esan air-oretrans-.. . . ....................mitter. M1oreover, they are weakly determined

satellite. It is immaterial whether the source of the sig- when treated as unknowns in the latter system,nal is an air-borne transmitter in the satellite or ground- and in turn, they seriously degrade the accuracybased inistrumenitationi. IHowever, Ahile the latter re-

quiresa.oeoplxovralstuenai y with which the orbital paramneters are determined.quires a more complex over-all instrumlentation system, )Maueet yasnl eevraeisfiinit both sinmplifies and iiicreases the reliability of the 2.Mesieet bvasigercieraeisfiinibomputhnsimplifs anichareinvolvedinc therel ityf the for a reliable solution when the transmitter is car-computing methods which are involved inl the data re- ne. yh saelt.Onteohrad,sglr-

ried by the satellite. On the other hand, single re-duction process. The miiethod of solutioni, which is.

pre- ceiver solutions are frequently possible with pas-sented herein, has beeni developed for a system with a sive data so that reductions may still be obtainedground-based transmitter, but it may be readily ap- . .

' . . . ~~~~~evenwhen several receiving sites of a multi-re-plied to observationis from a system in which the signal ceiver system may fail to record data.source is carried in the satellite. Indeed, with minormodification, the method may be applied to any type ofsatellite or ICBM tracking measurements, since it con- plete elimination of air-borne equipment.sists essentially of standard least squares and differential DESCRIPTION OF OBSERVED DATAcorrection techniques.

Inasmuch as it was desired to develop a rapid, as well In order to arrive at a computing method, it wasas reliable, method of determining the orbit of a satel- necessary to consider the various forms in which thelite tracked by a Doppler system employing a minimum data may be recorded. The observations consist ofof instrumentation, emphasis was initially directed either Doppler frequency or the Doppler period, whichtoward the development of a solution for a set of data may be readily converted to frequency. In the sense inrecorded at a single receiver during a single pass of the which it is used here, Doppler frequency is defined to besatellite. To strengthen the geometry of the system, the frequency obtained by beatinig a local oscillatorand therefore the solution, the single receiver require- against the received signal reflected from the satellitement was relaxed, but the limitation of single pass meas- and then correcting the output for the bias introducedurements was considered essential. This results in a rela- by the difference in frequency between the transmittertively short time interval of observation which in turn and the local oscillator. If the Doppler frequency ispermits several simplifying assumptions. First, while plotted as a function of time, one obtains a curve of thethe Earth is treated geometrically as an ellipsoid, dy- form plotted in Fig. 1(a), usually referred to as an Snamically it is considered to be spherical. In addition, curve. The asymmetry is typical for a systema in whichdrag as well as ionospheric and atmospheric refraction the ground-based transmitter and receiver are separated

by an appreciable distance and the orbit is not sym-* Maucrp reevdb. h G I,Fbur 90 metrical with respect to the base line joining the twot U. S. Army Ballistic Res. Labs., Aberdeen Proving Ground, Md. instrumentation sites. If observations are recorded at

1960 Patton: Orbit Determination from Single Pass Doppler Observations 337

available observation rather than upon a selected fewct / of the total observations (which would be the case if,,rl / the computinig input were limited to a representative

numiiber of frequenicy mieasuremeints). Experience hasshowiv this to yield a very significanit gaini with regard

o / to the accuracy and conivergent properties of the solu-tioil.

THE SOLtUTIONThe metlhod of solutioni consists of fittinig a comiiputed

TIME curve to a set of Doppler observationis by a least squares( ') technique in which a coiipatible set of approximiiationis

to the six orbital paramneters are derived anid theniimproved by repeatedly applying differentiai correc-

| / tioIIs until convergence is achieved. Kepleriani Imlotioni., is assumed. The equations of condition are derived fronm

the Taylor expansion about that poinlt which conlsistsof the approximate values of the six orbital paramiieters.All second and higher order terms of the expanisioni arenieglected. Convergence is greatly dependent upoIn theset of paramneters which are selected to describe theorbit. Not every set of orbital parameters will yield nor-mal equations which are sufficiently ind(lependenit to per-(1T)) IME mit a solution with this method. It has beeni found thatparamiieters conisisting of positioIn anid velocity coIIm-ponenits for a given timiie readilv yield a conivergenit solu-tioin, whereas a set comprising the semli-major axis, ec-

| * centricity, mean anonmaly at epoch, iniclination, rightU_ ascensioni of the ascenidinig niode, anid argumiient of perigee

proved to be altogether hopeless.As stated previously, the observations will consist

either of Doppler frequenicy as a funictioni of timne, orperiod measurements which imiay be coniverted to Dop-pler frequency. The desired conmputer iniput, on the

TIME other hand, is the total cycle count. Wheni a continuousrecord of observations is recorded for a substantial pe-riod of time, the total cycle count mav be obtained byFig. 1-Doppler frequency-time curves. Ithe simple process of summatioin over timne initervalsthat, in general, will be of several seconds duratioin. IfXN is the wavelength of the radiated signal anid N-j the

frequent intervals, such as one per second, Fig. l(a) total number of Doppler cycles resulting at the ith re-illustrates a typical set of data available for computer ceiver from the motion of the satellite between times tjinlput. However, in order to conserve power, it may be and tj+1, it follows that vij=XNij is a measure of thenecessary to use antenna systems consisting of three total change in path length froni the transmitter to thenarrow, fan-shaped beams which would provide con- satellite to the ith receiver betweeni times tj anid tj+i. LettiIIuous data at intermittent intervals as in Fig. l(b). gij be defined as the actual change in path length. It fol-A third possibility consists of discrete observations at lows from Fig. 2 that:regular initervals as shown- in Fig. 1(c). Such data could 9ijbe obtainled by a systeml which scanlned the sky with a gij (TS41 + RjSj1) (TSj+ RiSj), (1)narrow, faii-shaped beamn. where P is the tranlsmlittinlg site, Ri the locationl of: theAny of these sets of data may be used readily as input ith receiver, S,+i the position of the satellite at time

for the computing procedure. W;henever possible, this tj+1, and S1 the position of the satellite at time t?. In theinput consists of the total cycle count rather than the event that the observations consist only of discreteDoppler frequency, i.e., the area under the curves or measurements of frequency, the same definitions willarcs of curves presented in Fig. 1(a) and 1(b) . Hence, apply, but the time interval from tj to tj+i will be limitedthis method of solution is based, in a sense, upon every to one second.


wherez

t j+0(X~~~~j+Yi(X+,jZj+1) J (gi-/ dgt. dgi- dgij dtg dgii)(O--) -- , -, -- -,/ \xo yo Ozo dxo dye aso

'AV (AVV j) = (Vi- gij), (3)

Ay0AXT Azo

for all values of i and j. Hence, J is a matrix of order(i jX6), AV a matrix of order (i jX 1), and AX amatrix of order (6 X 1). Since there are six unknowns, aminimum of six equations are required for a solution. Inpractice, sufficient data are available to provide anover-determined system, thus permitting the leastsquares solution,

,AX = (J*j)-JJ*,Av, (4)x

where J* is the transpose of the Jacobian J. Finally,Feimproved values for the initial conditions are obtained

fromX+ AX,

The solution consists of improving a set of positionand velocity components which have been approxi- wheremated for a specific time. The latter will be defined as 0to, and as a matter of convenience, it is generally takenas the time at which tracking is initiated. Although the Yolocation of the coordinate system in which the positioni _ oand velocity vectors are defined is relatively immaterial Xfrom the standpoinit of convergence, a system fixed withrespect to the Earth's surface does provide two distinct 0advantages. First, the function gij is somewhat simpli- zofied since the coordinates of the instrumentation sitesremain fixed with respect to time. Of greater signifi- As a matter of convenience, no subscripts were intro-cance, however, is the fact that the method may be duced to indicate iteration; but at this point, the im-altered to accept other types of measurements for input proved values of X are used for the initial point and theby merely changing the definition of the function gjj and process is iterated until convergence is achieved.correspondingly, the expressions for its derivatives,with no additional modification to the balance of the EVALUATION OF AVprocedure which in fact, constitutes the major portion Since the function gij cannot be expressed readily inof the computing process. terms of X directly, the evaluation is obtained im-The set of initial approximations for the position anld plicitly. An ephemeris is computed for the assumed

velocity components are defined for time to as (x0, Yo, values of the position and velocity vectors at time to.zo, xo, 'o, s). The reference frame is the xyz-coordinate Thenl usinag (1), gjj may be evaluated for all values of isystem which iS defined in the following section. If and j.second and higher order terms are omitted from the In the process of this evaluation, it is expedient to useTaylor expansion about the point (x0, Yo, zo, xo, Y'o, so two rectangular coordinate systems in addition to thethe equations of condition may be written inI mlatrix Earth-bound system whose origin1 is at the transmittingform, site. Referring to Fig. 3, these coordinate systems are

AV = JfAX, (2) defined as follows.


z' right-hand direction and i is equal to 1, 2, or 3, accord-ing to whether the rotation is about the x, y, or z axis

xl respectively. The desired transfornmation follows,jF~~~~~~~PER IGEE)y*/ ~NORTH /\ XO |

/\ /C= == 1TRANSMITTER yYo' = R3(-6Jo)R2(4 - 900)ZO'

xo ~posin Ay o+ 0 (5)

/ = R3(-Oo)R2(k - 900)

xo posin Ayo + 0zo) pPcosA

xi /+ R3(-Oo)R2(q -900) i0 (6)( VERNALV(EQUINOX /I

Fig. 3-Coordinate systems.whereR3(-00)=the time derivative of R3(-6j) when Oj=0O,

1) The xyz-coordinate system is a right-hand rec- =j-the right ascension of the transmitting sitetangular system with the origin on the Earth's sur- at time tj,face at the transmitter. The x axis is positive s=the geodetic latitude of the transmittingsouth, the y axis is positive east and the z axis is site,normal to the Earth's surface at the transmitting p,the radius vector from the Earth's centersite. to a point on the Earth's sea level surface

2) The x'y'z'-coordinate system is a right-hand rec- at the latitude X,tangular system with the origin at the Earth's A=the difference between the geodetic andcenter. The x' axis lies in the plane of the equator geocentric latitudes at the transmittingand is positive in the direction of the vernal site.equinox, while the positive z' axis passes throughthe north pole. The y' axis is chosen so as to com- The next step in the computation involves the evalua-plete a right-hand system. tion of the following orbital parameters:

3) The x"y"z"-coordinate system is likewise a right- a=semi-major axis,hand rectangular system with the origin at the e-eccentricity,Earth's center. The x"y' plane lies in the orbital o-=mean anomaly at epoch,plane of the satellite, with the positive x" axis in i-inclination,the direction of perigee. The positive z" axis forms Q=right ascension of the ascending node,with the positive z' axis an angle equal to the in- - argument of perigee.clination of the orbital plane to the plane of theequator. The y" axis is chosen so as to complete a The evaluation of these orbital parameters is obtainedright-hand system. from the following equations:'

The first step in this phase of the computation con- =(oj+ yt2+(o)(7sists of computing values for the initial position andV(o)+(y)'f(z),(7velocity components in the x'y'z'-coordinate system. Vo - V/(X 1)2 + (y,1)2 + (0t')', (8)This may be achieved by one rotation and one transla- r0i0 =xO'xo' +I yo'yo' + zo'zo', (9)tion which are independent of time, and a second rota- =Rtion which varies with time. Let the notation Ri(a) = gR'indicate the matrix performing a rotation through anangle ae about the ith axis of the frame of reference such 'eie yD.B afne,BlitcRs as,Aedethat the angle is positive when the rotation is in the Proving Ground, Md.


where g is the imean gravitational constant and R is the whereradius of the Earth, which is assumed to be spherical in I sgn L.zo' + y(xo'k- y k)/ ( 1-sgn [xs0 + y(Xo ko-yok1)l~~10the developmnient of the equations, 2

a ro (10) = (-1)qcos-1N1, (31)2ju- rov0 where

(roko)2 ( ro \1/2 1-sgn N2

Eo = tan-1 Having obtained values for a, e, o, i, Q, and w, gij mayV/,ua(61 _ r0r be evaluated for each time of observation. The first stepL\ a/a in the computing procedure consists of solving for the

'r eccentric anomaly Ej in Kepler's equation,7r ( r \+ t1- sgn(1--- (12)2 { ( a) Ej-e sin Ej = nj +o-. (32)

where The position of the satellite as a function of time is de-s /n(1 ) i ro) termined in the x"y"z"-coordinate system.

sgn 1- I-- if I1 > 07

sgn (1--)--1 if (1--)ro 0,\fj = 2 tantla /[ tan( 29] (33)(g a t/i \\1 a) 0a a rj = a(l - ecosEj); (34)

n = /4/ (13) xj' = r1cosfj; (35)

o- = Eo- e sin Eo - n/o, (14) y" = rj sinfj. (36)

Yo'zo'- zo'n', (15) z," is zero according to the definition of the x"y"z"-coordinate system. A transformatioii to the x'y'z'

2 - 0Z0, (16) coordinates can be achieved by three rotations as/II3 = XOf01 - YO'X'j (17) follows:hi =s/il2 +- 1122 +-h12 (18) f xi' xi-hl=, (1)Y''j= R3(-Q)R,(-i)R3(-W) y;" . (37)k1= 1 (19)

112k.2 =--,7(20) Finally the position in the xyz-coordinate system mayh be obtainied by two additional rotations anid a tranis-h3 lationl.

k3 =-7 (21) X. xj) p1. sin AT = \1h12 +-h22 (22) Yj = R2(900 - O)R3(6j) yj' - 0 (38)

h2 Lz1J Kl') pO cosAN,.= - -1T (23) Referring to (1), gjj may be then expressed in terms of

hi the satellite's positions at time ti and tj+,.N2 = (24)

gij = VXj+12 + yj+12 + zj+12i= cos-1 k3, where 0 i r, (25)= a(cos E.,- e), (26) V,(x1-j+=x)2 + (yS - yi) (Z1+1 -Z)9= a(sink.o)V1- e2, (27) - VXj + Y}2 + Zj2- -_ i2(9

rO N =xotNl+ yO'N2,~~where (xi, yI, zi) is the surveyed positionl of the ith re-

roXk N (yo'k3 - zk2) + N72(zo'k - xo'k3), (29) ceiver. Finally, the residuals A\v21 may be determined=~~~~~~x(ro N) 2-(o N)V (30 from

a7=(-1)Pcos 1 x-- y(ro X k 3)/i zi-i-(0


EVALUATION OF J second in each velocity componient has beeii founid suf-Having evaluated the vector A*V, there remains the ficient to secure convergence. The comiputing process

problem of determiining the Jacobian J. The niecessary has occasionally converged with larger initial errors, butdifferentiation may be carried out numerically, but the the figures presented are initended to specify limits with-computing time will be reduced and the accuracy in- in which convergence imiay reasonably be assured.creased if the derivatives are evaluated from analytical Clearly, a supporting con putation to furnish moder-expressions. Recalling that for all values of i and j ately accurate initial approximations is essential to the

successful application of the comiputing imiethod. Several/ gio, g-i, * approaches to this phase of the problem have beeni con-

J-Jl Y . , ) sidered, but one in particular is preferred. It inivolves0,I YOI ZOI -~O~0) ZO using an analog computer to fit a computed curve to

let an observed Doppler frequency-time curve. In the fit-J = J1J2J3J4, ting procedure, the boundary values of a second-order

where differential equation are systematically varied until thecomputed S curve matches the observed S curve. The

( 0gi ,g.. j (* j41) differential equation has been derived on the basis of a*pp1, l=+, V -1 ( ) circular orbit and a nonrotating Earth. The initial ap-

proximiations for the positionl anid velocity comiponents,/J2-( ~~~j+l1) j+l xXy y (42) which serve as input for the prinmary computationi, are

a, e, c, WI Q, i then computed from the results obtained by the curve, e, a, co fitting procedure. A variation of this miiethod conlsists ofo, I

'-IIPI (43) plotting a frequency-time curve for the observed data

'\X0, ye', ZO, X, Ye, Z' / and then matching this to the appropriate member of aX0', ye', z', x0',y0', '0 large family of S curves previously computed for vari-

J4 = . . (44) ous circular orbits. Both methods have been successfullyx0, Ye, so, xe, zX), Qoemployed.The orders of the above matrices are (I.jX6) for J, The following definitions will be useful in the deriva-

and (6X6) for J2, J3, and J4. A distinict advantage of tion of the differential equation used in the fittingthis method of evaluatinig J results fromii the fact that procedure.the function gij appears onily in J,. Hence, if the solution RT=the radius vector from the Earth'sis applied to other types of measurements, only J, needs center to the transmitting site.revision for the appropriate evaluation of J. This is rj_the radius vector from the Earth'strivial compared to the effort required to derive the de- center to the position of the satelliterivatives conitained in J2 and J3. The expressions for at time tj.many of the elemenits of these Jacobian matrices are gj1_the angle betweeni RT and rj.rather long and involved, and therefore will not be pre- (XT", YT", Z") _the position of the transmitting site.sented here, but the complete results are reported in [Il. (xj", yj", 0) -the position of the satellite at time tj.

pj=the distance from the transmittingINITIAL APPROXIMATIONS site to the satellite at timne tj.

Of primiary importance to the success of the method pi the distance from the ith receiver tothat has been presented, is the capability of establishing the satellite at time ti.

. '.. . . ~~~~~~~H_the altitude of the satellite above thlea compatible set of initial approximations which are suf- Earth' surface.ficiently close to the actual values to permit convergence Earth's surface.of the computing process. It has been determined thatthe computation will converge with input consisting of To simplify the problem, certaini assumiptions have beendata from a single receiver systemn, if the base line from madethe transmitter to the receiver is not excessively short,and the initial approximation to (xe, Ye, Ze, x0, ye, se) is 1) the satellite nmoves in a circular Keplerian orbit,moderately accurate. For example, with base lines of the 2) dj is relatively small throughout the period oforder of 400 miles and either continuous or intermittent observation,observations over time intervals of approximately five 3) the Earth is not rotating.minutes, convergence may reasonably be expected whenthe error in each coordinate of the initial estimate is not A nlumlber of useful relationlships mlay be derived as ain excess of 50 to 75 miles anld the velocity componenlts result of these assumptionls.are correct to within 2 to 1 mile per second. For a two- |rreceiver systeml, onl the other hand, an accuracy of 150 =r=R+Hto 200 mliles inl each coordinate and 1 to 2 miles per where H is constanlt.


v = nr = V(AV')2 + (5v")2. where z is an estimate of the altitude based on experi-/R \ / RV2 \ ence. The former is always an adequate approximation

(R H) ( _ Iwhile the latter may usually be estimated to within 100\R + H / H / to 200 miles by merely observing the general character-Xt/I-In2Xjll; y " = - n2yj. istics of the S curve. In the event that the original esti-

Cos:J#j 1. mate of z0 proves to be so poor that it prevents con-vergence, a series of values for z may be tried without

RT = constant. becoming involved in excessive computation. EstimatesThe reference framne for this derivation is the x"y"z"- for xo and yo are obtained by solving the equations

coordinate system which has been defined previously. 2+ yO2 + Z2 = p02It follows from the definition of pj that (O)))((X(x-x)2 + (yo y,)2 + sZ,) I = p,o2. (49)

pi = (Xj" - XTI")2 + (yj" - YT")2 + (ZT' )2. (45)Differentiating pj and pij with respect to time yields, forDifferentiating twice with respect to time yields j = 0 and 3o = 0,

'2 + pj*j (X1)2 + (yj/)2 + computing procedure for the desired set of initial

= n2(RT.-r), approximations.= rRcos COMPUTATIONAL RESULTS

= v2( R Numerous convergent solutions have been obtainedR +H with both simulated and actual field data serving as

Rv4 computer input. Since the latter are of major interest,- cos/Cs, the discussion will be restricted to results obtained from

g real data. This method of solution was developed spe-Rv4 cifically for the DOPLOC system, which is reported

upon by deBey.2 The initerim version of this Dopplersystem complex consists of a 50-kw continuous wave

It follows that illuminator transmitter station located at Fort Sill,Rv4 Okla., and two DOPLOC receiving stations, one at__-pj2 White Sands Missile Range (WSMR) and a second at

..______

(47) Forrest City, Ark. To conserve power, the antenna sys-pi tem has been limited to three narrow, fan-shaped beams

which provide continuous data at intermittent intervalsA similar expression may be derived for

tht

j. Recalling as illustrated in Fig. l(b). The present results were ob-the definition for gui, we conclude that tained from observations which were recorded during a

A - pj2 A - j.2 period when the WSMR receiver was inoperable and,g1i + (48) therefore, are derived from data recorded by a single re-

Pi Pii ceiver in the course of a single pass over the instrumen-where tation site. Included with the DOPLOC results are

Rv4 orbital parameters which were determined and pub-A = lished by the National Space Surveillance Control Cen-

ter (Space Track). As an aid in comparing the two setsThe initial step in the fitting process on the analog of determinations, the Space Track parameters have

computer consists of approximating A, po, Pio, Po, and been converted to the epoch times of the DOPLOC re-Puo, then computing and plotting that curve for gij ductions.which satisfies (48) while observing the constraints The initial successful solution with field data from thegis =vj0 and gt0 =Z3t. The five parameters are adjusted DOPLOC system was achieved for Revolution 9937 ofsystematically until a sufficiently good fit is obtained. Sputnik III. M4easurements were recorded for 28 sec-.The final value for A may be used to determine the onds in the south antenna beam of the system, 7 secondsvelocity from which the altitude may be estimated on in the center beam, and 12 seconds in the north beam,the basis of a circular orbit, but this has not beers par-ticularly successful for highly eccentric orbits. The pre- 2L .dBy Takn nsaeb OLC"ti su,pferred procedure iS to assume that so-=0 and zo = z, 332.


with two gaps in the data of 75 seconds each. Thus, agreement in the values for a, e, i, and Q, particularlyobservations were recorded for a total of 47 seconds for the latter two. This is characteristic of the single-within a time interval of 3 miniutes and 17 seconds. On pass solution when the eccenitricity is small anid thethe first pass through the comiputinig imiachinie, the computationial iniput is limnited to Doppler frequency.computation coniverged in three iterationis to initial Since the orbit is almost circular, a anid w are less sig-positioni anid velocity componienits that were equivalenit nificanit thani the other parameters anid likewise, areto the following orbital parameters: mnore difficult for either system to determine accurately.

a = 4149 miles, However, as a result of the small eccentricity, the sumof co and o is a rather good approximation to the angular

e= 0.0153, distance along the orbit from the nodal point to theo- = 288.04, position of the satellite at epoch time and as such, pro-i-65.370 vides a basis of comparison between the two systems.In the DOPLOC solution, (co+o-) =32.660 while theQ= 178.240, Space Track determination yields a value of 35.730, aco-104.620. difference of 3.070 between the two sets of results. To

summarize, when limited to single-pass, single-receiverFor comparison, the orbital parameters reported In observations, the DOPLOC systemn provides an excel-Space Track Bulletin No. 230 for 1958 Delta II (Sput- l dermationsothe Orien of theorilnik III) were used to compute the value of the parame- lane determination of the shapeof the orbit,ters for the same epoch time as that of the solution. The and af ooodetermination of th e orie ofp

~~~~~~~~~anda fair-to-poor determination of the orientation ofresults are as follows: the ellipse within the orbital plane.a = 4111 miles, Occasionally, excellent results have been obtained fore 0.0130, both oc and co; but in general, the interim DOPLOC

257.780 systeimi with its present limitations fails to provide coIn-sistently good evaluations of these two quantities.i 65.060, Therefore, in presenting the remaining DOPLOC reduc-Q = 178.220, tions, o- and co have been eliminated from further coni-0-137 95 sideration. Results have been indicated in Table I for

*o=137.95g. six revolutions of Discoverer XI, includinig nutber 172In compariing the DOPLOC and Space Track solu- which was the last known revolution of this satellite.

tions, it will be noted that there is reasonably good As a matter of interest, the position determined by the

TABLE ICOMPARISON OF DOPLOC AND SPACE TRACK RESULTS FOR DISCOVERER XI

Revolution a Total Amount Interval ofNumber (miles) e (degrees) (degrees) of Data Observation

____________

(seconds) (seconds)DOPLOC 30 4186 0.0295 80.15 215.92 35 123Space Track 4189 0.0291 80.01 215.84Difference

-3 0.0004 0.14 0.08DOPLOC 124 4115 0.0198 80.31 207.50 44 138Space Track 4135 0.0203 80.10 207.27Difference

-20 -0.0005 0.21 0.23DOPLOC 140 4138 0.0198 80.44 205.22 55 141Space Track 4121 0.0178 80.10 205.78Difference 17 0.0020 0.34 -0.56DOPLOC 156 4143 0.0300 80.79 204.50 25 117Space Track 4108 0.0148 80.10 204.30Difference

_35 0.0152 0.69 0.20DOPLOC 165 4037 0.0111 79.99 203.63 49 165Space Track 4099 0.0128 80.10 203.50Differenlce -62 -0.0017 -0. I11 0.13DOPLOC 172 4093 0.0189 80.44 203.18 35 90Space Track 4091 0.0111 80.10 202.84lDifference 2 0.0078 0.34 0.34


initerim DOPLOC system for this pass indicated anl tained for actual field data, and further that such solu-altitude of 82 miles as the satellite crossed the base line tions were completely independent of other measuring55 miles west of Forrest City. To provide a basis for systems. Results from the latter were used onily for corn-evaluation of the DOPLOC results, orbital parameters, parison and did not enter any phase of the computationsobtained by convertinig Space Track determinations to leading to these solutions.the appropriate epoch times, have been included in the In conclusion, the method is general and therefore,table. Table I also contains a listing of the amount of need not be confinied to either Doppler observations ordata available for each reduction in additioni to the total Keplerian orbits. For more complex orbits, it may betime interval within which the observationis were col- desirable to replace anialytical with numunerical differen-lected. tiation and it will, of course, be necessary to modify the

computation for the ephemeris; otherwise the solutionCONCLUSION will be fundamentally the same. Insofar as the use of

This method of solution has been shown to be both other types of observations are concerned , only Ji andpractical and useful by numerous successful applica- gij need be modified to allow the method to be appliedtions with real as well as simulated data. Although to observations from any satellite or ICBM trackingresults have been presented for passive data only, the system, provided the limitation of Keplerian orbits iscomputing procedure has been altered slightly and ap- retained.plied to active data with considerable success. Com- ACKNOWLEDGMENTputing times are reasonable sinice convergent solutionshave required from 20 to 40 minutes on the ORDVAC The author is indebted to Dr. Boris Garfinkel forwith the coding in floating decimal, whereas more mod- many helpful suggestions.ern machines would perform the same computationi in2 to 4 minutes. This miethod allows the determiiniatioi of BIBLIOGRAPHYa relatively accurate set of orbital parameters with as [1] R. B. Patton, Jr., "A Method of Solution for the Determinationof Satellite Orbital Parameters from DOPLOC Measurements,"little as 1.5 to 3 minutes of intermittenit observations Ballistic Res. Labs., Aberdeen Proving Ground, Md., Memo.from a single receiver when the signal source is a ground- Rept. No. 1237; September, 1959.[2] W. H. Guier and G. C. Weiffenbach, "Theoretical Analysis ofbased tranismitter. While multi-receiver systems provide Doppler Radio Signals from Earth Satellites," Appl. Phys. Lab.,niumerous distinct advanitages, it has been shown that Silver Spring, Md., Bumblebee Rept. No. 276; April, 1958.[3] R. C. Davis, "Techniques for the Statistical Analysis of Con-it is possible to obtain, routinely, quite satisfactorv re- tinuous-Wave Doppler Data," U. S. Naval Ord. Test Station,sults with observationis from a single receiver. China Lake, Calif., NAVORD Rept. 1312; April, 1951.[41 F. R. Moulton, "An Introduction to Celestial Mechanics," Mac-It should be emphasized that solutions have been ob- millan & Co., Ltd., New York, N. Y.; 1902.

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Orbit Determination from Single Pass Doppler Observations

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