Efficient Stochastic Orbit Modeling Techniques using Least Squares Estimators

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1 D N fficient tochastic Orbit Modeling Techniques using Least quares stimators A Jäggi, G Beutler, U Hugentobler Astronomical Institute, Universit of Berne, idlerstrasse 5, CH-3012 Berne, witzerland adrianjaeggi@aiubunibech Abstract Reduced-dnamic orbit determination for spaceborne GP receivers of low arth orbiting satellites is a successful method promising highest precision We review the principles of (reduced) dnamic orbit determination and develop the mathematical background for different efficient stochastic orbit parametrizations (eg, piecewise constant accelerations which provide not onl continuous but also differentiable orbits) using least squares methods imulated as well as real data from the CHAMP GP receiver show, to some extent, the equivalence of the different parametrizations and reveal the impressive performance of stochastic orbit modeling techniques Independent comparisons with orbits determined b other groups and validations with LR measurements show that our orbits are of high qualit Ke words Low arth orbiter, reduced-dnamic orbit determination, stochastic orbit parametrization 1 Introduction The idea of using spaceborne GP receivers for precise orbit determination (POD) of low arth orbiters (LOs) was demonstrated for the first time almost a decade ago (Bertiger, 1994) for dnamic and reduced-dnamic orbits of the TOPX/POIDON satellite This proof of concept stimulated an increasing number of follow-up missions of arth observing satellites to carr on-board GP receivers, not onl for POD, but also for purposes like atmosphere sounding (Kursinski, 1997) and gravit field recover For the latter purpose the analsis of CHAMP GP receiver and accelerometer data proved that it is possible to separate gravitational from nongravitational perturbations (Reigber, 2002) This article focuses on POD, namel on the promising approach of reduced-dnamic orbit determination and in particular on the stochastic orbit modeling part, which defines the degree of strength reduction of the dnamic laws b allowing for a stochastic component in the equations of motion The results for simulated and real GP data of the CHAMP GP receiver underline the applicabilit of the theoretical developments 2 Dnamic Orbit Determination The equation of motion of an arth orbiting satellite including all perturbations reads as! "$# (1) in the inertial frame with initial conditions &% ')( +*# % '( -,/ *:2+*$#;=< > 5?,=/@134A7 28 * where +* are the six! osculating " elements pertaining to and where are the unknown dnamical parameters describing the perturbing acceleration acting on the satellite (eg, gravit field coefficients, air-drag or radiation pressure parameters, ) Let us assume that an a priori orbit CB is available (eg, from a GP code solution) As B - must be a solution of the equation of motion (1) a priori values DF *HGJI, * / * 1 * 4 * A7 * 28 *=* $ *! " *LK for the unknown orbit parameters D are available, as well The actual orbit - ma therefore be linearized b a truncated Talor series - M CB - O!P R B - D T DF DF * #U where V XW I orbit parameters D! D\ " K I,/0! " K (2) ZY denotes the total number of unknown quation (2) allows it to improve the orbit if the partial derivatives of the a priori orbit wrt the unknown orbit parameters are known 21 Variational quations Let us assume that D is one of the parameters defining the initial values or the dnamics in q (1) and that the partial derivative of the orbit CB wrt the parameter D is designated b the function ]0^ B (3) The initial value problem associated with the partial derivative (3) is referred to as the sstem of variational equations in this article and ma be obtained

2 o ' D ' * ± Y * ± # Y 2 b taking the partial derivative of q (1) The result ma be written as ]0^ `_ B ]0^ _ ]0^ where the 3 a 3 matrices _ B and _ (4) are defined b b *c '=d &e *5f b c '=d 9e *f (5) Zg where denotes the i-th component of the total acceleration e in q (1) For D GhI,/02134A7 89* K q (4) is a linear, homogeneous, second order differential equation sstem with initial values ] ^ * #ji lk ]0^ +*# i mk, whereas for D GnI 5!!= " K and q (4) is inhomogeneous with zero initial values Note that in the latter case the homogeneous part of q (4) is the same as for parameters D defining the initial values 3 Pseudo-tochastic Orbit Modeling Apart from the six initial conditions, accelerations are set up as the onl orbit parameters Without taking into account an a priori models for the nonconservative forces, piecewise constant stochastic accelerations partiall absorb these accelerations (eg, air-drag, radiation pressure, ) as well as accelerations due to a mismodeling in the gravitational attractions (eg, due to an imperfect gravit field model) The parametrization subsequentl used for the residual acceleration o in the inertial sstem reads as the sum of three accelerations in (r)adial, (a)longtrack and (c)ross-track directions in the satellite corotating sstem oqp osr is repre- where the acceleration osu sented b osu -,0*5f " xw!p - U ost G`I,=v K, Y, f "5z { ## u (6) -U (7) where u denotes the unit vector in one of the directions specified above and z -2 -{ L2 #? :0 ~} { ƒ > : (8) else Altogether a total number of W?$# orbit parameters, are *f " set up As opposed to the constant acceleration acting over the whole arc in q (7), the are referred to as pseudo-stochastic accelerations in this article, because the are characterized b an expectation value piecewise constant accelerations, f " of zero and an a priori weight f " Š) Œ Ž variance ˆ&, f " # X> : f " ˆ ˆ Š Œ Ž given b an a priori (9) where ˆ denotes the a priori root mean square error (RM) of unit weight The a priori weights in q (9) constrain the, estimates of the piecewise constant accelerations f ", not allowing them to deviate too much from zero Note that q (9) allows for an estimation of the parameter,0*5f " in q (7) As indicated in section 21, ever orbit parameter additionall set up requires in principle the simultaneous integration of one additional differential equation sstem (4) Facing the possibl large number of stochastic accelerations to be set up, it is mandator to develop efficient methods to calculate the partial derivatives wrt these parameters in order to avoid inefficienc due to a computational overload 31 Pseudo-tochastic Accelerations Let us now develop the mathematical background for in prede-, estimating constant accelerations f " termined directions u for -{ ƒ 1? 5! 2 For the sake of simplicit we drop the in- 13 dices, Y in the following and focus on one acceleration The contribution of this parameter,, in in q (1) is of the form for -{ Neglecting an possible velocit-dependent forces, the corresponding variational equation thus reads as ] r _ B ] r - :0 } { k : else Let us assume that the functions ]@ -š (10)?! W are the partial derivatives of the a priori orbit wrt the +* six parameters defining the six initial conditions at As these six functions ] form one complete sstem of homogeneous solutions of q (10) the solution of the inhomogeneous sstem (10) and its first time derivative can be obtained b the method of variation of constants and ma therefore be written as a linear combination of the homogeneous solutions ] r % '( - P œ ]@ % '( :ž< `> 5?R (11) where the coefficients œ - are functions of time Ÿ to! $ be determined Introducing the matrix notation ] Ÿ L! ª$ ] š? 5! W and vector notation k the solution ma be obtained b definite integrals Ÿ { A³ # ² A³ # A³ Ÿ { -A³ # µ0 A³ # A³9 (12)

3 , ¹ ¼¼¼¼ º ¼¼¼¼ k œ, r r,, Ê Ê Y where 5! œ # J ¹ º -{ :@ -{ :@ -{ :@» This implies that the solution ] r ] r reads as and where - :0 ƒ -{ P œ - ] ½:0 -{ ƒ P œ - # ] :0»ƒ (13) for the parameter Note that in the case»ƒ the coefficients œ - # constant in time 311 fficient olution for -{ ¾ (14) are Let us introduce an auxiliar problem and write the function (14) as a function of the solution of this auxiliar problem The parameter underling the auxiliar problem is a constant acceleration over the entire arc Note that, *f " can be identified with in q (7) The corresponding variational equation (see q (10)) reads as ]&À r _ B ]9À -ÂÁÃ r (15) As the difference Ä Å] r À Æ] r solves the homogeneous differential equation sstem Ä `_ B Ä in the designated time interval, its solution can be written as a linear combination of the functions ], therefore ] r % ')( `] À r % '( - P CÇ ] % '( -È:Ã< M> 5?R (16) valuating q (16) at time -{ and taking into account q (14), the coefficients Ç ma be obtained as a solution of the following linear sstem of algebraic equations: P Ç ] -{ # ½] À P Ç ] -{ 5# ]&À - { # - { #U (17) As the above equations form a linear sstem of six scalar equations for the six unknowns Ç it is possible to write the partial derivative (16) wrt the parame-, ter as a linear combination of the partial derivative wrt the parameter and the six partial derivatives ] Note that q (11) together with q (12) implies not onl continuit but also differentiabilit of at time { ] r»¾ 312 fficient olution for quations (11) and (14) impl that ] r is continuous and differentiable at time, as well, which means that ] r % ')( # ma be computed b evaluating q (16) at time The coefficients œ - # ma therefore be obtained as a solution of the following linear sstem of algebraic equations: P œ # ] # `] r # (18) P œ #q ] # ] r #È As the above equations again form a linear sstem it as a of six scalar equations for the six unknowns œ # is possible to write the partial derivative ] r linear combination of the six partial derivatives ] onl In conclusion not onl the position vector but also the velocit vector of the improved orbit remains continuous over the whole arc 32 Pseudo-tochastic Pulses Let us briefl outline the special case of instantaneous velocit changes É at times in predetermined directions # (Beutler, 1994) Focusing on one pulse É at time, the contribution of - É in in q (1) ma formall be written as É Ê # where denotes Dirac s delta function The corresponding variational equation thus reads as ]CËÌ `_ B ]CË Ê - # -È 3 (19) Using the same notation the solution of (19) ma be written again in the form of q (11) The coefficients ma be obtained in analog to q (12) b def- œ»¾ inite integrals which are given for b ± ² -A³ # Ÿ { -A³ # A³ # A³ Ÿ { # - # (20) Obviousl the œ»å are constant in time for Therefore the partial derivatives ] Ë - ma be written as a linear combination of the six partial derivatives wrt the initial conditions onl The drawback however is that ] Ë and therefore are no longer continuous The discontinuities lie at the epochs 4 imulation tudies We discuss two experiments using simulated GP phase zero-difference observations for the CHAMP satellite In both simulations the same phsical and mathematical models were used as in the real data

4 Í Ï simulated signal estimated signal along-track deviation accelerations pulses acceleration (nm/s^2) along-track deviation (mm) radial deviation (mm) time (min) Fig 1 Piecewise constant accelerations ever 15 minutes compensate for an unmodeled along-track signal The dotted curve denotes deviations wrt the true orbit (simulated data) time (min) Fig 2 Radial deviations wrt the true orbit for pseudostochastic accelerations resp pulses set up ever 6 minutes pochs of velocit changes are easil recognized on the dashed curve (simulated data) processing In addition the error-free GP data were affected b an artificial once per rev? along-track acceleration with an amplitude of > {Î m/s for the first simulation An orbit with identical initial values without being affected b the artificial signal served as a priori orbit To illustrate the feasibilit of our approach for this scenario, piecewise constant accelerations in along-track direction were estimated ever fifteen minutes For a selected time interval of about two revolution periods, Fig 1 shows that the estimated piecewise constant accelerations follow the artificial ( true ) signal and that the deviations from the true orbit in along-track direction are mostl below the one millimeter level High correlations with the time intervals of the accelerations can be recognized as well as a once per rev periodicit Both effects are caused b the unavoidable deficiencies of the parametrization The second simulation illustrates the applicabilit and, to some extent, the equivalence of both, pseudostochastic accelerations and pulses in a more realistic environment Again GP observations were simulated but as opposed to the first experiment the phase observations were affected b a white noise random error of 1 mm RM A dnamical orbit using the gravit field model IGN-1 (Reigber, 2002) up to degree/order 120 served as true orbit, whereas the same gravit field model, but truncated at degree/order 80, was used in the orbit improvement procedure To illustrate the properties of both parametrizations two different orbits were generated, one with the parametrization defined in q (6) (where stochastic accelerations were set up ever six minutes) and another with pulses instead of accelerations with the same spacing in time B optimizing the constraints in q (9), both orbits could be fitted equall well (postfit RM of about 102 mm) The differences wrt the true orbit ield almost the same RM values over one da, smaller than 2 mm in all three directions The largest differences occur in the radial component (caused b the large number of rather loosel constrained stochastic parameters, which reduce the dnamic to the extent that the resulting orbit tends to be more sensitive to the observation geometr (like a kinematic orbit) which penalizes the radial direction) Fig 2 shows the radial deviations wrt the true orbit over a selected time interval of about one revolution period where the differences for both parametrizations can be observed well The orbit generated with piecewise constant accelerations shows a smooth behaviour, whereas epochs with velocit changes can be identified quite easil on the other orbit Nevertheless, the differences are ver small (below a few millimeters in the vicinit of the epochs of the stochastic parameters) and do not significantl affect the overall orbit qualit Consequentl, no significant gain or loss has to be expected using one or the other orbit parametrization with a reasonabl large number of stochastic parameters, when processing real data The real errors are much higher due to the non-gravitational forces, which were neglected in this simulation, and therefore exceed the subtleties reported above 5 CHAMP Orbit Test Campaign Using GP final orbits and high-rate clocks (Bock, 2002) from the COD analsis center (Center for Orbit Determination in urope) real CHAMP GP zero-difference phase observations were processed to stud the impact of constrained stochastic parameters on the orbit The orbit qualit was assessed b internal (eg, formal accuracies along the orbit) and external indicators (eg, comparisons with orbits from

5 Ú 5 Table 1 RM of plain differences to an orbit from CR resp RM of LR residuals (da 141) Constraint RM wrt CR RM of LR residuals (m/sð ) (cm) (cm) Ñ Ò$Ó3Ô ÕCÖ ÓÃÒ$Ó3Ô ÕCÖ Ñ Ò$Ó3Ô ÕC ÓÃÒ$Ó3Ô ÕC radial accurac (mm) e-8 m/s^2 1e-8 m/s^2 5e-9 m/s^2 1e-9 m/s^2 other groups, LR residuals) For this purpose we used the time span of the CHAMP orbit test campaign from Ma 20 to 30, 2001 (das ), which was selected as the test period within the IG LO Pilot Project As the orbits from the Center of pace Research (CR) at Austin were found to be the best of the twelve contributing centers (Boomkamp, 2002) the serve here as benchmark orbits 51 Internal ualit Assessment In order to assess the orbit qualit, the full parametrization (defined in q (6)) with stochastic parameters set up ever six minutes was used to generate CHAMP orbits for both, accelerations and pulses with different (but in each direction equal) constraints To obtain best results, the gravit field model IGN-1 was used and the attitude measurements from the star sensors were introduced The dependenc of the postfit RM on the constraints was found to be equal for both parametrizations, obviousl allowing a better fit for looser constraints However this does not necessaril impl that such orbits are reall better On the contrar, it turns out that setting up a large number of loosel constrained stochastic parameters ma result in worse orbits, which ma be verified b an external comparison Table 1 shows for da 141 the RM of plain 1-d differences (no Helmert transformation applied) to an orbit from CR as well as the RM of LR residuals for four different tpes of constraints (in each direction equal) when setting up stochastic accelerations ever six minutes According to this comparison an optimal solution exists in the range considered? and most probabl the constraints of Ø > {Ù m/s are quite close to the optimum Note that almost identical results can be obtained if pulses are set up It is instructive to analze the formal errors along the orbit as an additional indicator of qualit Appling the general law of error propagation together with the variance-covariance information allows the computation of formal errors in the sstem co-rotating with the satellite (ie, in radial, along-track and cross-track direction) Fig 3 shows as an example the errors in radial direction for the four different constraints from Table 1 Apart from the common characteristics (eg, the increased error level around 3h and 10h caused b time (h) Fig 3 Formal accuracies of orbit positions in radial direction for differentl constrained pseudo-stochastic accelerations set up ever 6 minutes using the gravit field model IGN-1 (da 141) a reduced number of successfull tracked satellites), Fig 3 implies that good solutions in Table 1 turn out to have low formal accuracies along the orbit as well The accuracies in the other directions show a similar behaviour, whereas the highest error level can be found in the along-track direction indicating that the orbits (as opposed to the simulations in section 4) are all governed b the dnamical laws due to the relativel tight constraints Nevertheless the? most kinematic solution with constraints of Ø > {Û m/s alread suffers considerabl from the limit of the CHAMP BlackJack receiver at that time to track onl up to 8 satellites simultaneousl 52 xternal ualit Assessment Table 2 shows the dail RM of plain 1-d differences wrt CR orbits (middle column) for the time span of the CHAMP orbit comparison test campaign Over the eleven das an RM of 529 cm wrt the orbits from CR (corresponding to a 3-d RM of 916 cm) is achieved olving for a seven parameter Helmert transformation shows that our orbits agree with the CR orbits on a RM level better than 5 cm However both orbit tpes show a mean shift in the earth-fixed z-direction of 332 cm which ma be due to slightl different reference frames the used GP orbits and clocks refer to Table 2 shows the dail RM of LR residuals (right column) for the time span of the CHAMP orbit comparison test campaign A total of 1837 LR measurements (normal points) of 16 stations were used for this validation Due to the sparse LR tracking the differences of LR measurements (corrected for the tropospheric dela) and our orbit positions give onl snapshots of the 1-d orbit accurac in selected directions during the short tracking passes For the eleven

6 á 6 Table 2 RM of plain differences to orbits from CR resp RM of LR residuals (das ) An overall RM of 529 cm resp 337 cm results Da of ear RM wrt CR RM of LR residuals (cm) (cm) das an RM of 337 cm over all LR residuals is obtained without showing an significant LR bias 53 Comparison with Accelerometer Data If a perfect gravit field could be used the estimated accelerations should directl represent the non-conservative forces acting on the satellite Fig 4 shows for a selected time interval of about three revolution periods how the piecewise constant accelerations in along-track direction agree with the measured accelerations (bias and scale applied) from the TAR accelerometer, when the gravit field model IGN- 1 is used Despite the crude approximation with constant accelerations over six minutes the correlation (778%) is quite remarkable whereas the agreement in the other directions is less significant (correlation of 361% resp 347% in radial resp crosstrack direction) At least in along-track direction reliable bias and drift parameters ÞÝ { ma be derived ßÝ b { such comparisons (-3518 Ü and Ü $à da for the time span of the orbit comparison campaign) for internal validation purposes 6 ummar We developed the mathematical background for a stochastic orbit modeling in the environment of least squares estimators and presented two possible pseudo-stochastic orbit parametrizations (piecewise constant accelerations resp instantaneous velocit changes) which ma be set up in an efficient wa Due to the densit and high accurac of GP phase tracking data both parametrizations are well suited for reduced-dnamic LO POD and ma even be considered to some extent as equivalent as shown with simulations The processing of CHAMP zero-difference phase data (das in 2001) showed that the estimated reduced-dnamic orbits ma be validated with LR residuals on a level of 337 cm RM without an significant LR bias and that the agree on along-track acceleration (nm/s^2) time (min) measured estimated Fig 4 Piecewise constant accelerations ever six minutes in along-track direction compared with measurements from the TAR accelerometer (bias and scale applied) using the gravit field model IGN-1 (da 141) an RM level of better than 5 cm with orbits from CR The additional comparison with accelerometer data showed a high correlation (778%) between the measured and the estimated piecewise constant accelerations in along-track direction References Bertiger, W, Y Bar-ever, Christensen, Davis, J Guinn, B Haines, R Ibanez-Meier, J Jee, Lichten, W Melbourne, R Muellerschoen, T Munson, Y Vigue, Wu, T Yunck, B chutz, P Abusali, H Rim, M Watkins, P Willis (1994) GP Precise Tracking of TOPX/POIDON: Results and Implication Journal of Geophsical Research, 99(C12): Beutler, G, Brockmann, W Gurtner, U Hugentobler, L Mervart, M Rothacher (1994) xtended orbit modeling techniques at the COD processing center of the international GP service for geodnamics (IG): theor and initial results Manuscripta Geodetica, 19: Bock, H, U Hugentobler, TA pringer, G Beutler (2002) fficient Precise Orbit Determination of LO atellites using GP Advanced pace Research, 30/2, pp Boomkamp, H (2002) The CHAMP Orbit Comparison Campaign In: First CHAMP Mission Results for Gravit, Magnetic and Atmospheric tudies, edited b C Reigber et al, pp 53-58, pringer, Berlin, IBN Kursinski, R, GA Hajj, JT chofield, RP Linfield, KR Hard (1997) Observing arth s Atmosphere with Radio Occultation Measurements Using the Global Positioning stem Journal of Geophsical Research, 102(D19): Reigber, Ch, G Balmino, P chwintzer, R Biancale, A Bode, J-M Lemoine, R Koenig, Loer, H Neumaer, J-C Mart, F Barthelmes, F Perosanz, Y Zhu (2002) A High ualit Global Gravit Field Model from CHAMP GP Tracking Data and Accelerometr (IGN-1) Geophsical Research Letters, 29(14),101029/2002GL015064

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