High precision dynamic orbit integration for

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ScienceDirect Advances in Space Research xxx (2017) xxx–xxx www.elsevier.com/locate/asr

High precision dynamic orbit integration for spaceborne gravimetry in view of GRACE Follow-on Matthias Ellmer ⇑, Torsten Mayer-Gu¨rr Graz University of Technology, Institute of Geodesy, Steyrergasse 30/III, 8010 Graz, Austria Received 13 February 2017; received in revised form 30 March 2017; accepted 17 April 2017

Abstract Future gravity missions like GRACE Follow-on and beyond will deliver low-low satellite-to-satellite ranging measurements of a much increased precision on the order of nanometers. This necessitates a re-evaluation of the processes used in gravity field determination with an eye to numerical stability and computational precision. This study investigates the computation of dynamic orbits, which are used for multiple purposes in gravity recovery. They are, for example, used in computing linearized observations for the low-low satellite-tosatellite tracking instruments. The precision at which the dynamic orbits are determined thus must surpass the precision of the ranging observations. Dynamic orbits for GRACE were computed both in a simple simulation, where the force model was reduced to a static potential of degree and order 60, and for real observational data. Encke’s method was employed while using a novel reference trajectory determined through rigorous optimization. This reference trajectory was parametrized with equinoctial elements to minimize errors resulting from imprecision in the reference motion. The differences in coordinates between successive iterations of orbit determination were used as a benchmark for the quality of the orbit solution. Using Encke’s method with equinoctial elements, the coordinate difference between iterations was reduced from on the order of tens of micrometers to some nanometers in the spectral range relevant to GRACE satellite-to-satellite tracking observations. The resulting dynamic orbits are self-consistent to below the expected precision of the GRACE Follow-on ranging instruments. Ó 2017 COSPAR. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/). Keywords: GRACE; GRACE Follow-on; Laser ranging instrument; Dynamic orbit; Encke method; ITSG-Grace2016

1. Introduction The Gravity Recovery and Climate Experiment (GRACE) satellite mission (Tapley et al., 2004) is a joint National Aeronautics and Space Administration (NASA) and German Aerospace Center (DLR) operation with the goal of mapping Earth’s gravitational potential. The main observation is low-low satellite to satellite tracking (ll-sst), where the range between the spacecraft is continuously ⇑ Corresponding author.

E-mail addresses: [email protected] (M. Ellmer), mayer-guerr@tugraz. at (T. Mayer-Gu¨rr).

monitored using a K-Band ranging (KBR) instrument. The successor mission to GRACE, Gravity Recovery and Climate Experiment Follow-on (GRACE-FO), is scheduled for a launch in late 2017 (Flechtner, 2016) to continue the record of climate data provided by GRACE. Additionally, GRACE-FO will fly a Laser Ranging Interferometer (LRI) instrument as a technology demonstrator, which is expected to provide measurements of increased precision. The computation of dynamic orbits is one of the key tasks to be performed in preparation of computing gravity field solutions from GRACE ll-sst observations. Dynamic orbits are computed strictly through integration of the accelerations acting on the spacecraft, with no direct

http://dx.doi.org/10.1016/j.asr.2017.04.015 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Ellmer, M., Mayer-Gu¨rr, T. High precision dynamic orbit integration for spaceborne gravimetry in view of GRACE Follow-on. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.04.015

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M. Ellmer, T. Mayer-Gu¨rr / Advances in Space Research xxx (2017) xxx–xxx

position observations considered. This gives a very smooth orbit that can be fitted to coarser observations like kinematic orbits from a Global Positioning System (GPS) instrument (Zehentner and Mayer-Gu¨rr, 2016). Dynamic orbits are then used multiple times throughout the processing chain. Positions and velocities from dynamic orbits are used to compute reference accelerations from background models: Gravity from a static potential, perturbing accelerations from the Atmosphere and Ocean De-aliasing Level1B (AOD1B) product, or air drag from an atmospheric model. Dynamic orbits are also used to compute observation equations, where they are used as a Taylor point for the linearization of Precise Orbit Determination (POD) or ll-sst observations. The precision at which the dynamic orbits are determined thus must surpass the precision of the ranging observations. The accelerations that are integrated to yield a dynamic orbit can be broadly categorized into two groups: Accelerations due to conservative forces stemming from a potential field, and accelerations due to non-conservative forces like air drag, solar radiation pressure, or Earth albedo. Conservative forces can not be measured directly, and must be derived from a background model using positions from dynamic orbits. Non-conservative forces can be measured by the accelerometer (ACC) installed on the GRACE spacecraft. As the non-conservative forces are measured directly, a change in the dynamic orbit positions does not affect them. For the conservative forces, however, a change in position does have an effect, as evaluating the underlying potential at a different position will yield a different acceleration. The use of dynamic orbit positions to determine the accelerations due to conservative forces thus means that dynamic orbit integration for GRACE must be an iterative procedure. Only when the accelerations derived from the integrated orbit equal those used to compute it can one say that an equilibrium state has been reached, and that the dynamic orbit is self-consistent. In other words, using a dynamic orbit as input into the integration routine must result in the same dynamic orbit as output. Any difference in the positions can be regarded as either a manifestation of insufficient convergence or as a defect in the orbit integration algorithm. This procedure, the integration and evaluation of dynamic orbits, is the focus of this paper. Section 2 discusses a general approach for determining dynamic orbits. Section 3 contains an overview of employed methods and the improvements over state-of-the-art integration procedures. Section 4 presents results from simulations and real data analysis. Section 5 gives analysis of the results and an outlook towards future work. 2. Background A dynamic orbit is determined through integration of the equation of motion €rðtÞ ¼ f ðt; rðtÞ; p; . . .Þ;

ð1Þ

stating that the acceleration €rðtÞ experienced by the spacecraft is equal to the specific force exerted on it. This force f is described by the force function per unit mass, which encompasses all forces acting on the satellite: Those due to conservative fields, and those due to non-conservative sources. This is dependant on the time t, the location of evaluation rðtÞ, and the parameters of the forcegenerating functions p. The satellite orbit is divided into segments, or arcs, of length T ¼ tend tstart . These arcs are integrated independent of each other. Time t is normalized within an arc as t tstart s¼ : ð2Þ T The force can then be written as f ðsÞ ¼ f ðs; rðsÞ; p; . . .Þ:

ð3Þ

Integrating Eq. (1) yields the position and velocity of the spacecraft €rðsÞ ¼ f ðsÞ

ð4Þ

Z

s

f ðs0 Þds0 Z s rðsÞ ¼ r0 þ r_ 0 ðsT Þ þ T 2 ðs s0 Þf ðs0 Þds0 r_ ðsÞ ¼ r_ 0 þ T

ð5Þ

0

ð6Þ

0

depending on some initial values r_ 0 and r0 , for each epoch 1 . . . N (see e.g. Mayer-Gu¨rr, 2006). Based on this formulation, the variational equation approach (Beutler and Mervart, 2010; Montenbruck and Gill, 2000) can be used to compute dynamic orbits at high precision and for arc lengths of hours to days, or longer. This approach is based on a linearization of the integrated positions and velocities with regard to the sought force model parameters. It allows the computation of partial derivatives of the position and velocity with regard to the force model parameters through integration starting from some initial condition, and avoids explicit computation of all partial derivatives at each epoch. When setting up the variational equations, dynamic orbits appear as the linear term in a Taylor expansion of the integrated equations of motion (Eqs. (5) and (6)). Position and velocity of the satellite along its orbit arc are combined in the state vector rðsÞ yðsÞ ¼ : ð7Þ r_ ðsÞ The variational equations are then the partial derivatives of the satellite state with regard to the unknown force model parameters p and the unknown initial state of the orbit arc T y0 ¼ ½ r0 r_ 0 . These derivatives are grouped into the parameter sensitivity matrix 2 3 @rðsÞ @yðsÞ 4 @p 5 ¼ @ r_ ðsÞ SðsÞ ¼ ð8Þ @p @p

and the state transition matrix



@yðsÞ UðsÞ ¼ ¼ @y0

" @rðsÞ @r0

@ r_ ðsÞ @r0

@rðsÞ # @ r_ 0 @ r_ ðsÞ @ r_ 0

:

ð9Þ

S describes the reaction of the satellite orbit to changes in the force model parameters dp, for example a change in gravity field coefficients. U describes the dependence of the satellite orbit at each epoch along the arc on changes in the initial state dy0 . The linearized satellite state y at epoch s is then dp : ð10Þ yðsÞ ¼ yðsÞjp;y0 þ ½ SðsÞ UðsÞ dy0 Given a set of initial conditions y0 and the parameters p, the dynamic orbit is well-defined for the entire arc s 2 ½0; 1. Given initial states S 0 and U0 as well as the state transition matrix UðsÞ for all epochs s, the parameter sensitivity matrices SðsÞ; s 2 ð0; 1 can be computed through integration (Montenbruck and Gill, 2000) without numerically solving any differential equations. The resulting observation equations can then be used for orbit determination (compute the satellite state at each epoch) or gravity field determination (solve for gravity field parameters in p). 3. Methods Sections 3.1, 3.2, 3.3, 3.4, 3.5 describe a general method of dynamic orbit determination as applied to GRACE (cf. Mayer-Gu¨rr, 2006). Sections 3.6, 3.7, 3.8, 3.9 detail improvements to this methodology. The following text will use the convention that if a variable is given with a time s, it refers to that singular epoch. If the time argument is dropped, it shall refer to a vector containing the values T for a full arc, such that r ¼ ½ rð0Þ . . . rð1Þ . 3.1. Method of integration When computing a dynamic orbit, the equations of motion for the satellite must be solved. Classically, a variety of integration methods have been developed to perform this task numerically. Beutler and Mervart (2010) mention Euler’s algorithm, Runge-Kutta methods, and multi-step methods, among others. Such methods have in common that the state vector is propagated: The integral of the accelerations over one time interval is used to determine a new satellite state at the end of the interval by extrapolation. In some methods, the extrapolated state is then refined through an iterative procedure. Then, the integration interval is moved forward in time and the process is repeated. In the implementation described here, satellite state is not propagated from epoch to epoch, thereby avoiding extrapolation. The dynamic orbit is instead determined by taking advantage of the full knowledge of all accelerations along the arc. This allows one to determine the new state for each epoch by solving the definite integral of all accelerations along the arc (Beutler and Mervart, 2010), which can be efficiently achieved through numerical

3

integration techniques. In the implementation described here, the definite integral of the accelerations is computed using a sliding integration polynomial (Mayer-Gu¨rr, 2006, Sec. 4.1.1). The satellite states from an approximate orbit and the newly integrated positions are then used to estimate an initial state for the new dynamic orbit arc, thus fixing the new satellite state for the complete arc. Sections 3.2, 3.3, 3.4, 3.5 describe this algorithm in more detail. 3.2. Coarse approximation For GRACE, dynamic orbit determination begins with a coarse approximation of the orbit, e.g. a kinematic orbit from GPS observations. These are the approximate positions r , which deviate from the true positions r by some value . The approximate positions are used to evaluate the background models and compute the accelerations from conservative forces at each epoch: €rcons ðsÞ ¼ f ðs; r ðsÞ; pÞ

ð11Þ

The complete approximate accelerations €r are obtained by adding the measured accelerations from the GRACE accelerometer, yielding €r ðsÞ ¼ €rcons ðsÞ þ €rACC ðsÞ:

ð12Þ

Using integration polynomials (Mayer-Gu¨rr, 2006), the definite integrals from Eqs. (5) and (6) are computed Z s €r ðs0 Þds0 ðsÞ ¼ T ð13Þ r_ int 0 Z s 2 rint ðs s0 Þ€r ðs0 Þds0 ð14Þ ðsÞ ¼ T 0

for the complete arc, giving integrated positions and velocities. This yields the integrated equations of motion _ 0 þ r_ int r_ dyn ðsÞ ¼ r ðsÞ rdyn ðsÞ

¼ r0 þ sT r_ 0 þ

ð15Þ rint ðsÞ;

ð16Þ

where, rdyn and r_ dyn are the dynamic orbit computed from the approximate orbit r and r_ . To determine the initial state, the approximate state transition matrix is computed as 1 sT Ur ðsÞ UðsÞ ¼ ¼ ð17Þ r_ ðsÞ 0 1 U by taking the partial derivative of Eqs. (15) and (16) with regard to the initial state y0 . It proves sufficient to use only positions for a stable estimate of y0 , allowing one to neglect the velocity component of the approximate and dynamic orbit. A least-squares estimate ^y0 for the initial state that best fits the approximate positions r can then be computed !

¼ r : by rearranging Eq. (16) and setting rdyn r rint ¼ U r y0 :

ð18Þ

The first complete approximate dynamic orbit is then y0 þ yint : ^y ¼ U^

ð19Þ


4


The shape of the approximate orbit is smooth as defined by the integrated accelerations. Its position is fixed to be close to the initial approximate orbit through the adjusted initial state ^ y0 . 3.3. Refinement The first approximation for the dynamic orbit as determined in Section 3.2 has a flaw, the evaluation of the forces from background models was performed at the approximate positions r , not the true positions of the satellite r. This leads to a deviation in the derived accelerations from the true accelerations, and thus also a deviation of the computed positions rdyn — the orbit is not self-consistent. A strategy to deal with this problem in the context of a boundary value problem for orbit integration is described in detail in Mayer-Gu¨rr (2006, Section 4.2.4.3). The equivalent formulations for the initial value problem are outlined here for completeness. For simplicity of notation, two operators for both the integration of velocities and positions Z s jr_ ðsÞ ¼ ðÞds0 ð20Þ 0 Z s jr ðsÞ ¼ ðs s0 ÞðÞds0 ð21Þ 0

With a matrix 2 rf ðr ðs1 ÞÞ 6 T¼6 4 0

3

0 ..

7 7; 5

.

ð29Þ

rf ðr ðsN ÞÞ

reordering Eq. (28) and inserting into Eq. (27) gives rdyn rdyn ¼ K r Tðr r Þ: ð30Þ Given correct implementation and convergence, then !

rdyn ¼ r. Inserting Eq. (24) into Eq. (30) gives r y0 K r€r ¼ K r Tðr r Þ: rU

ð31Þ

Subtracting the approximate position from both sides and reordering gives r y0 þ K r€r r ; r r K r Tðr r Þ ¼ U ð32Þ which can be solved for the coordinate difference Dr ¼ r r : r y0 þ K r€r r ½I K r T ðr r Þ ¼ U 1 Dr ¼ ðr r Þ ¼ ½I K r T ½U r r : r y0 þ K r€

ð33Þ ð34Þ

Dr is an estimate of the linearization error made due to evaluation of the potential at r instead of r. The estimate of the spacecraft position at each epoch can then be updated:

are introduced. As these operations can be computed through polynomial integration, they can be discretised and represented as matrices K r ; K r_ (Mayer-Gu¨rr, 2006 eq. 4.71). This allows one to write

r ¼ r þ Dr

r_ int r ¼ K r_ €

is In Eq. (17), the approximate state transition matrix U computed as the derivative of position and velocity with regard to the initial state y0 . In this first approximation, int it is neglected that the terms r_ int a and ra also depend on the position along the arc — and thus the initial state — as they are computed from the accelerations €r as derived from the force models at r . To compute the complete state transition matrix for the velocity U, one must take the derivative of Eq. (6) with regard to the initial state y0 : Z s @rðsÞ @ ðr0 þ sT r_ 0 Þ @f ðs0 Þ 0 ¼ þ T2 ðs s0 Þ ds : ð36Þ @y0 @y0 @y0 0

rint

ð22Þ

¼ K r€r ;

ð23Þ

with rint and r_ int vectors of all integrated positions and velocities along the arc. Eq. (16) can be written once with the approximate positions r and once with the true position r as input: r y0 þ K r€r rdyn ¼U r y0 þ K r€r rdyn ¼ U

ð24Þ ð25Þ

Taking the difference of Eqs. (24) and (25) yields ¼ K r ð€r €r Þ: rdyn rdyn

ð26Þ

Remembering that the accelerations €r are computed as a function of the force f ðrÞ, Eq. (26) can also be written as r

dyn

rdyn

¼ K r ½f ðrÞ f ðr Þ:

ð27Þ

The Taylor expansion of f ðrÞ at the approximate position r is, up to the linear term: f ðrÞ ¼ f ðr Þ þ rf jr ðr r Þ:

ð28Þ

Here, rf is the gradient of the force, or the Marussi tensor.

ð35Þ

3.4. State transition matrix

Applying the chain rule to the derivative of the force function yields @f ðs0 Þ @f ðs0 Þ @rðs0 Þ ¼ @y0 @rðs0 Þ @y0

ð37Þ

and thus @rðsÞ @ ðr0 þ sT r_ 0 Þ ¼ @y0 @y0 |ffl{zffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} Ur

r U

Z

s @f ðs0 Þ@rðs0 Þ 0 þ T2 ðs s0 Þ ds : @rðs0 Þ @y 0 |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflffl}|fflffl{zfflffl}|fflffl{zffl0ffl} Kr

T

ð38Þ

Ur



5

This linear equation system can be solved for the complete state transition matrix U, giving

yielding the desired result without relying on an inverse containing K r_ .

r: Ur ¼ ½I K r T 1 U

3.5. Final estimate

ð39Þ

Both Eqs. (34) and (39) have the form 1

l ¼ ½I K r T x;

ð40Þ

and can thus be solved in a similar fashion. The inverse contains three position components per epoch, and thus has size 3N 3N , which makes direct inversion of the complete matrix costly even for moderate arc lengths of a few hours. One can however make use of the special structure of the matrices involved. The matrix of gravity tensors T is block-diagonal with one block of size 3 3 per epoch. The integration matrix K r is only populated on or below the main block diagonal, as integration up to time s only depends on values before s. This structure allows for the solution of the equation system to be computed per epoch by an iterative solver, such as BICGSTAB (van der Vorst, 1992), which is magnitudes faster and more efficient than direct inversion. To determine Ur_ , one can set up a similar equation system r_ Ur_ ¼ ½I K r_ T U 1

ð41Þ

which can be solved using the same iterative algorithm. However, one must implement it a second time as the integration matrix for velocity K r_ is different from the one for the position K r used before. Instead Ur_ can be derived using algorithms that are already implemented. First the state transition matrix for the accelerations U€r must be computed by taking the derivative of Eq. (1) with regard to the initial state: @€rðsÞ @f ðsÞ ¼ : @y0 @y0

ð42Þ

Applying the chain rule yields @€rðsÞ @f ðsÞ @rðsÞ ¼ @y0 @rðsÞ @y0

ð43Þ

in which one can identify U€r ¼ TUr :

ð44Þ

To compute Ur_ , the same derivative is taken of Eq. (5), again using the chain rule: Z s @ r_ ðsÞ @ r_ 0 @f ðsÞ 0 ¼ þT ds ð45Þ @y0 @y0 @y0 0 Z s @ r_ 0 @f ðsÞ @rðsÞ 0 ¼ þT ds : ð46Þ @y0 0 @rðsÞ @y0 Here, one can identify Z T r_ þ T Tðs0 ÞUr ðs0 Þds0 Ur_ ¼ U 0

r_ þ K r TUr ¼U r_ þ K r U€r ; ¼U

All previously integrated positions are based on the forces evaluated at the approximate positions f ðr Þ (compare Eq. (11)). Using the linearization of the force function from Eq. (28) and the coordinate difference Dr , an updated estimate for the accelerations due to the forces experienced by the spacecraft can be computed. These are the corrected accelerations €rc ¼ €r þ TDr

which can again be integrated to corrected velocities and positions with rc r_ int c ¼ K r_ € rint c

ð49Þ

¼ K r€rc :

ð50Þ

Repeating the steps from Eq. (18) — but now also knowing the complete state transition matrix Ur — a new estimate for the initial state ^y0 can be determined from the system r rint c ¼ U r y0 :

ð51Þ

The estimated initial state is then used to compute the final positions and velocities, the final dynamic orbit, for the spacecraft: r_ ¼ Ur_ ^y0 þ r_ int c

ð52Þ

r ¼ Ur ^y0 þ

ð53Þ

rint c :

Due to the linearizations performed, all steps from Eq. (11) to Eq. (52) must be repeated, using the newly computed positions as input or approximate positions to the algorithm, until convergence is achieved. Dynamic orbit computation following this schema is thus an inherently iterative process. 3.6. Quality of convergence For real observations the true position of the spacecraft is unknown, denying the obvious path for a quality check on orbit determination. The iterative nature of the dynamic orbit determination provides an approach for an alternative quality check. The correction applied to the spacecraft positions at iteration n to give the new positions at iteration n þ 1 is Dr (cf. Eq. (35)). Given correct implementation and data, Dr must grow smaller with each iteration, as the approximate orbit approaches the true orbit. It is thus expected that for a perfect dynamic orbit determination process, the difference between subsequent orbits must vanish: lim Dr ¼ 0:

n!1

ð47Þ

ð48Þ

ð54Þ

In practice, constraints such as the limited precision of computations or flawed algorithms limit the achievable repeatability of the orbit integration, precluding the differ-


6


ences in Eq. (54) from disappearing. Instead, after some number of iterations, the differences stop to grow smaller, even as more iterations of computation are performed. This effect can be used to test the quality of orbit determination strategies both on their own merit and amongst different implementations. The smaller the differences become, the better the algorithm is designed and implemented – given the same input data. 3.7. Improved algorithm A numerically challenging section in this procedure is the actual integration of the computed accelerations to positions and velocities in Eq. (13) and (14). As orbit arcs get longer, the numeric values being integrated grow larger, leading to a loss of precision in the least significant digits. One approach to regain precision is to split the integral into two parts, a numerically large part that can be solved analytically, and a smaller part that must be solved numerically: €rðsÞ ¼ f ðsÞ ¼ f 0 ðsÞ þ Df ðsÞ

ð55Þ

Here, f 0 ðsÞ is the acceleration due to the reference force which can be computed analytically and Df ðsÞ is the acceleration due to the disturbing force which must be computed numerically. This is a general formulation of the well-known Encke method for perturbed orbit propagation (Encke, 1852), which was also independently developed by Bond (1849). For compactness, the following derivations are given for the position only, the formulations for velocity and acceleration follow the same structure. The acceleration due to the reference motion is, along with its integrals, equivalent to Eqs. (4)–(6): Z s rref ðsÞ ¼ rref;0 þ r_ ref;0 ðsT Þ þ T 2 ðs s0 Þf 0 ðs0 Þds0 : ð56Þ 0

Here, the initial values of the reference motion T yref;0 ¼ ½ rref;0 r_ ref;0 appear. The complete integral of both the reference acceleration and the accelerations due to the disturbing forces gives the true motion Z s rðsÞ ¼ r0 þ r_ 0 ðsT Þ þ T 2 ðs s0 Þ½f 0 ðs0 Þ þ Df ðs0 Þds0 ; ð57Þ 0

in which the original initial values y0 appear. The difference between the true motion and the reference motion is given by the Encke vectors D€r; D_r, and Dr. Using Eqs. (55)–(57) and the equivalent formulations for the velocities, they are D€rðsÞ ¼€rðsÞ €rref ðsÞ ¼ Df ðsÞ

Z

ð58Þ s

Df ðs0 Þds0 ð59Þ 0 Z s DrðsÞ ¼ rðsÞ rref ðsÞ ¼ Dr0 þ D_r0 ðsT Þ þ T 2 ðs s0 ÞDf ðs0 Þds0 D_rðsÞ ¼ r_ ðsÞ r_ ref ðsÞ ¼ D_r0 þ T

0

ð60Þ

with D_r0 ¼ r_ 0 r_ ref;0 and Dr0 ¼ r0 rref;0 . This formulation is similar to the original problem treated in the previous sections, except that the initial values y0 are replaced by T the differential initial values Dy0 ¼ ½ Dr0 D_r0 and the full force f ðsÞ is replaced by the disturbing force Df ðsÞ. This system can be solved with minor adjustments to the previously presented algorithm. The steps are as follows: 1. Select a reference force f 0 with an associated reference trajectory. Compute rref and r_ ref for the entire arc. Compute the disturbing forces Df at the approximate position r according to Eq. (55). 2. Following Eqs. (22) and (23), compute the integrated Encke position and velocity Drint r : ¼ K r D€

ð61Þ

r as in Eq. (17), solve the system 3. With U r rref Drint ¼ Ur Dy0

ð62Þ

to compute an estimate of the differential state D^ y0 . 4. Similar to Eq. (34), the estimated coordinate difference to the true position is 1 ð63Þ Dr ¼ ½I K r T U y0 þ Drint r D^ þ rref r : 5. Compute Ur ; Ur_ , and U€r according to Eqs. (39), (44) and (47). 6. Following Eq. (48), use Dr to correct the accelerations due to the disturbing forces D€rc ¼ D€r þ TDr :

ð64Þ

7. Integrate the corrected accelerations as in Eqs. (49) and (50) with D_rint rc ; c ¼ K r_ D€

ð65Þ

Drint c

ð66Þ

¼ K r D€rc ;

and, similar to Eq. (51), compute a new estimate D^ y0 of the differential state from r rref Drint c ¼ Ur Dy0 :

ð67Þ

8. Compute the final dynamic orbit as in Eqs. (52) and (53) with r_ ¼ r_ ref þ Ur_ D^y0 þ D_rint c

ð68Þ

Drint c

ð69Þ

r ¼ rref þ Ur D^y0 þ

This is a general formulation of the reduced initial value approach to dynamic orbit determination. 3.8. Reference motion The formulations in Section 3.7 are fully independent of the choice of reference force f 0 . If, for example, one were to choose f 0 ðsÞ ¼ 0 and Dy0 ¼ y0 , the approach would simplify into the same apparatus as presented in Sections 3.2, 3.3, 3.4 and 3.5. In his work on the perturbations of planets, Encke (1852) suggests to compute the perturbed



orbit relative to a Keplerian reference motion that is identical to that of the perturbed orbit at the first epoch (Encke, 1852; Encke, 1857). The ellipse described by this type of reference motion is termed osculating. The reference force would then be the central term of the gravitational attraction f 0 ðsÞ ¼ GM

rðsÞ jrðsÞj

3

;

ð70Þ

jDrj ; jrj

N X 2 jDrðsi Þj ! min :

ð74Þ

ð71Þ

Knowing that the approximate position is the sum of the reference position as a function of its initial values and the Encke vector

ð72Þ

r ¼ rref ðyref;0 Þ þ Dr;

This approach has the undesirable property that the reference motion and the perturbed orbit will diverge significantly after only a short period of integration, leading to the loss of any numerical advantages attributed to the method. This separation is commonly quantified in the Encke ratio ¼

temporal modification of the Kepler parameters due to higher order terms or external perturbations. Instead, the choice of the initial parameters of the reference orbit yref;0 is reconsidered. The Encke ratio is small if the Encke vector Dr is small. The goal must thus be to minimize the magnitude of the Encke vectors over the whole arc

i¼1

with the initial values of the reference motion equaling those of the approximate position: rref;0 ¼ r ð0Þ r_ ref;0 ¼ r_ ð0Þ

7

ð73Þ

the ratio of the magnitude of the Encke vector in relation to the magnitude of the position vector. If the Encke ratio grows large, the increase in numerical precision associated with the Encke method is lost. Lundberg et al. (2000) states that the general recommendation is to keep < 1%. The general approach to dealing with the problem of large is rectification, meaning the interruption of the integration at a certain epoch and continuing from there using a newly defined reference trajectory which will again provide a small . In essence, this means restarting the integrator, with possibly negative effects on precision of the orbit arc (Milani and Nobili, 1987). The first efforts to reduce the Encke ratio for long arc orbit determination, or orbit integration in general, were based on the premise of considering Earth’s oblateness in the reference force (Kyner and Bennett, 1966; Escobal, 1966). Liu and Hu, 1997 focuses on considering higher order terms of Earth’s potential and secular terms in perturbing bodies. Lundberg et al. (2000) develops a long arc model that allows general variations in all six orbital elements, mentioning successful results with Encke ratios on the order of 10–20%. These studies have in common that they all consider medium to high orbiting laser ranging satellites like the Laser Geodynamics Satellite (LAGEOS) (Lundberg et al., 1990; Lundberg et al., 2000; Liu and Hu, 1997) or the Satellite de Taille Adapteé avec Re´flecteurs Laser pour les Etudes de la Terre (STARLETTE) (Lundberg et al., 2000). The arc lengths in these works are on the order of years, not hours as is usual in GRACE processing. For the GRACE case of a low-earth orbiter with moderate arc lengths of at most 24 h, a simpler solution presents itself: A mean static Kepler ellipse, with no

ð75Þ

Eq. (75) can be solved in a least squares sense, treating the Encke vector Dr as if it represented the residuals of the least squares fit. This finds the initial values of the reference T ellipse ^yref;0 ¼ ^rref;0 ^r_ ref;0 that lead to the minimal

square sum DrT Dr, fulfilling the condition in Eq. (74). The differential initial values are then # " ^rref;0 r0 Dy0 ¼ y0 ^yref;0 ¼ : ð76Þ ^r_ ref;0 r_ 0

As DrT Dr is minimized, the solution ^yref;0 also minimizes the Encke ratio over the whole arc. Fig. 1 illustrates this optimized best-fit reference ellipse. 3.9. Parametrization of reference motion Numerical tests show that a parametrization of the reference ellipse using the traditional Kepler parameters r ¼ ½ a e I x X M T is not precise enough in double precision arithmetic. It is desirable to avoid porting the complete algorithm to quadruple precision arithmetic, as this would result in a significant performance penalty, with slowdowns on the order of a factor of 5–10 (Bailey and Borwein, 2015). Instead, an implementation of the reference motion T using the equinoctial elements re ¼ ½ a h k p q k as given in Broucke and Cefola (1972) is chosen. Danielson et al. (1995, Section 2.1) gives a general introduction to the derivation and use of the equinoctial elements. Here, all computations are performed in double precision arithmetic except for the computation of k, the fastmoving variable giving the position of the satellite along

Fig. 1. Osculating ellipse (in pink) and best-fit ellipse (in green) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).


8


the orbit arc. k is computed and stored in quadruple precision. As k is not used in any expensive operations, the impact on overall performance is negligible. Danielson et al. (1995) also gives formulations to compute position, velocity, and acceleration from the equinoctial elements using no trigonometric functions.1 With this parametrization and implementation, the conversion from orbital elements to Cartesian coordinates shows higher stability. The results presented in Sections 4.3 and 4.4 illustrate this statement. 4. Results The results presented in Sections 4.1, 4.2 and 4.3 are based on a simulated orbit for a single spacecraft moving in the GOCO05s static gravitational potential (MayerGu¨rr et al., 2015), expanded to degree and order 60. Where applicable, the Marussi tensor T was expanded to degree and order 10. No further conservative or nonconservative forces were applied. All orbits were determined for an arc length of 24 h at a sampling of 5 s resulting in 17,280 epochs. The initial approximate orbit was computed using an in-house integration polynomial based orbit propagator. Sections 4.4, 4.5 then give results from the processing of real GRACE data in the scope of the ITSG-Grace2016 gravity field solution. When absolute coordinate differences are shown, they are given for the along-track axis only, as it shows, in accordance to theory (Huang and Innanen, 1983), the largest errors. 4.1. Encke ratio Fig. 2 shows the Encke ratio for two cases of initial orbital elements for the reference ellipse. In the first configuration ( ), no differential initial values were applied, i.e. Dy0 ¼ 0. This is the classical Encke case of an osculating reference ellipse, which is congruent to the approximate

orbit at the first epoch. In the second case ( ), the Encke ratio is determined using a best-fit reference ellipse as described in Section 3.8. While the Encke ratio for the osculating orbit is 0% at the initial epoch, it quickly grows to larger than 1%, reaching 20% by the end of the arc. This corresponds to an Encke vector of 1240 km. The Encke ratio for the best-fit orbit never increases much beyond 0.1%, staying well within the bounds given by Lundberg et al. (2000). In absolute terms, the deviation from the reference motion is at most 8.2 km. 4.2. Convergence As detailed in Section 3.6, the convergence of the dynamic orbit solution is a good indicator for the correctness of the algorithm. To test for the number of iterations needed for convergence to occur, the simulated orbit was deteriorated with gaussian white noise of rr ¼ 50 m and rr_ ¼ 0:5m=s respectively. Using this noisy orbit as a first approximation, Fig. 3 shows the convergence of the dynamic orbit solution over 100 iterations of integration and correction, for five different configurations of dynamic orbit determination. The configuration with no reference motion ( ) can be easily identified as the one with the worst convergence. Two configurations using an osculating reference ellipse and Kepler elements ( ) or equinoctial elements ( ) also show comparatively large position differences between iterations. The last two configurations use a best-fit orbit and Kepler elements ( ) or equinoctial elements ( ), respectively. These show the smallest differences. Where at first the changes in positions between iterations are on the order of tens of meters, they quickly shrink to values smaller than a millimeter. No significant reduction in differences can be observed after the fourth iteration, leading to the conclusion that all methods have converged. The results in the next section will show the differences after a generous number of 80 iterations, picked at random from the available results. 4.3. Absolute differences

Fig. 2. Osculating ellipse (in pink) and best-fit ellipse (in green). Encke ratio over one orbit arc for osculating reference orbit (in pink) and best-fit orbit (in green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 1 Beware however of wrong partial derivatives of the equinoctial element a with regard to Cartesian position and velocity as given by Danielson et al. (1995, Section 2.1.6, Eqs. (2) and (4)). Comparison with Broucke and Cefola (1972) gives partials in Danielson et al. (1995)’s the correct @r @ r_ 1 notation: @a ¼ 1a r r_ 3t2 and @a ¼ 2a r_ GM j3r t These partials rj3 are needed when computing the best-fit Kepler ellipse from Eq. (75).

Fig. 4 shows the differences of the in-track coordinate component for one complete day after 80 iterations. The exact numeric values of course change from iteration to iteration, but the spectral behavior of the differences is rather consistent. In Fig. 4a, the configuration using no reference motion ( ) can easily be identified as the one with the largest errors. Both configurations using the osculating reference ellipse also show comparatively large errors ( and ), where the differences for the configurations employing the best-fit ellipse can not be seen at this scale ( and ). Fig. 4b shows a magnification of the bestfit cases, where it becomes clear that the differences for the configuration of a best-fit reference motion using equinoctial elements ( ) are smaller than those for a best-fit reference motion using Kepler elements ( ).



9

Fig. 3. Simulated data. RMS of 3D coordinate difference Dr for a complete arc after each iteration.

Fig. 4. Simulated data. In-Track coordinate differences Dralong between iterations in both the spatial and spectral domain after convergence. The vertical gray lines in Fig. 4c denote multiples of the orbital frequency, with the leftmost line showing one cycle per revolution. Colors denoting the configurations are the same as in Fig. 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


10


The differences between the configurations become most clear when observing the Power Spectral Density (PSD) of the coordinate differences. The PSDs were computed using Welch’s method with a segment length of 6 h and plotted in Fig. 4c. The best-performing configuration of a best-fit orbit using equinoctial elements ( ) shows white noise behavior at frequencies higher than 2 cycles per orbital revolution. The magnitude of the differences in the highfrequency part of the spectrum is of the same magnitude as the numerical resolution of a double precision floating point number at orbital altitude, meaning that the machine precision is completely exhausted here. At longer wavelengths, some residual error can be observed. The best-fit configuration using Kepler elements ( ) has reduced fidelity in the higher frequencies, due to the inferior stability of the Keplerian reference trajectory. The configurations based on the osculating reference orbit ( and ) show much larger deviations at lower frequencies, with the error at one cycle per revolution being two orders of magnitude larger than those using the best-fit orbit. At shorter wavelengths, the configuration employing equinoctial elements ( ) displays differences smaller than those achieved by using a best-fit orbit with Kepler elements ( ), but they never approach the low error level of the best-fit equinoctial solution ( ). The osculating configuration using Kepler elements ( ) is never better than the best-fit configuration using Kepler elements ( ). At very high frequencies close to the Nyquist frequency, this configuration shows no improvement over using no reference motion at all ( ). Not shown in Fig. 4, the coordinate differences in the cross-track and radial axes show similar spectral behavior. The magnitude of the differences is however smaller by approximately two orders of magnitude.

based on a simple static potential, these orbits include a number of background models (see Table 1 for details) and observations. Fig. 5 shows that both the osculating and ) show and best-fit Keplerian configurations ( no improvement over the implementation without a reference trajectory ( ). The advantages of the best-fit reference trajectory become clear when comparing the configurations using equinoctial elements. Here, the configuration using a best-fit trajectory ( ) is superior to the configuration using an osculating trajectory ( ) in the same way as could be seen in the simulated data. Overall, the impact of this improved methodology is much larger for real data than could be expected from the simpler simulations of Section 4.3. 4.5. Propagation to ranging measurement As mentioned in Section 1, the final dynamic orbits are used at multiple steps in gravity field determination. In one step, a linearization of the GRACE ranging measurements is computed from the dynamic orbits of both satellites, rA and rB . The impact of the dynamic orbit noise on a ranging measurement derived from it can be computed from the ranging equation qðsÞ ¼ jrB ðsÞ rA ðsÞj ¼ sðsÞ:

ð77Þ

With pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ Dx2 þ Dy 2 þ Dz2 ;

ð78Þ

simple error propagation gives " 2 2 # 2 Dx Dy Dz 2 2 2 rx þ ry þ r2z : rq ¼ 2 s s s

ð79Þ

With x being the in-track axis, one can approximate that 4.4. Application to GRACE The dynamic orbit computation was also performed with real data in the context of the new ITSG-Grace2016 gravity field solution (Mayer-Gu¨rr et al., 2016; Klinger et al., 2016). Where the results in the previous sections were

Dx 1 s

ð80Þ

and that Dy Dz 0: s s

ð81Þ

Table 1 Background models used in ITSG-Grace2016. Effect

Model

Earth rotation Third body forces Solid earth tides Pole tides Ocean tides Ocean pole tides Atmospheric tides Dealiasing Relativistic effects

IERS 2010 (Petit and Luzum, 2010) JPL DE421 (Folkner et al., 2008) IERS 2010 IERS 2010 EOT11aa Desai, 2002b Atmospheric Tide Loading Calculator (van Dam and Ray, 2010) AOD1B RL05 (Flechtner et al., 2014) IERS 2010

a The EOT11a model was produced by DGFI based on multi-mission altimeter data and distributed via OpenADB (http://openadb.dgfi.badw.de). More details on the product are available in Savcenko et al. (2012). b Using coefficients published in IERS 2010 conventions.



11

ranging accuracy with regard to the KBR instrument (Flechtner, 2012). 5. Discussion

Fig. 5. Real data. In-Track coordinate differences after convergence for real GRACE orbits. Colors denoting the configurations are the same as in Figs. 3 and 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

_ Fig. 6. Real data. Propagation of orbit noise to range rate measurement q. Solid black is a noise model for the GRACE KBR and ACC instrument noise, Dashed black is a noise model for the GRACE-FO LRI and ACC instruments. The other colors are the same as in Figs. 3–5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

This gives pffiffiffi rq ¼ 2 r x :

ð82Þ

As the standard deviation is given once for each frequency f, one can differentiate the error PSD in the frequency domain to compute the range rate error rq_ ðf Þ ¼ 2pf rq ðf Þ:

ð83Þ

The resulting PSD is displayed in Fig. 6. Here, it becomes clear that all configurations lead to a range-rate error that is below the precision of the current KBR instrument on GRACE (solid black line). The accuracy of the configuration using no reference motion ( ) as well as the accuracy of both the Kepler configurations ( and ) are above the performance estimate for the GRACE-FO LRI instrument (dashed black line) in the branch of the error spectrum that is dominated by the ranging instrument noise. The configuration using an osculating reference ellipse and equinoctial elements ( ) is slightly below the noise level of the LRI instrument. The configuration using a best-fit ellipse and equinoctial elements ( ) is well below the noise level of the LRI over the entire spectrum. The KBR noise spectrum in Fig. 6 is based on spectral analysis of real GRACE ll-sst residuals. The performance estimate for the LRI is based on real GRACE ACC performance data, a description of the noise characteristics of the LRI (Heinzel et al., 2012), and an optimistic assumption of a 50-fold improvement of the

In view of GRACE-FO processing, an improved method for dynamic orbit determination was developed. Encke’s method was applied, thus directly minimizing the effect of the central term of Earth’s potential on the numerical integration of accelerations. Encke’s method was then refined in two ways: Firstly, the osculating reference ellipse proposed by Encke was substituted by a rigorously optimized best-fit reference ellipse. This step reduces the error in dynamic orbits at low frequencies. It was shown that through this choice, complex higher-order or time-variable parametrizations of the reference ellipse could be avoided. Secondly, the parametrization of the reference ellipse was transformed from Kepler elements to equinoctial elements. This improved the stability of the reference motion at high frequencies. The resulting orbits reach machine precision at frequencies above two cycles per orbital revolution. The precision of the original and improved orbits were put into relation with GRACE and GRACE-FO ll-sst observations through error propagation to the range rate domain. It was shown that the improved orbits are selfconsistent to below the expected precision of the GRACE-FO LRI instrument. This represents an improvement of several orders of magnitude over previous results. The improvement is important as the dynamic orbits are used as a Taylor point in the linearization of the observation equations for gravity field recovery from GRACE (Mayer-Gu¨rr, 2006), so any extraneous errors that stem from the processing chain, not from the observations, should be avoided. The most significant aspect of this work is the reduction of the in-track error by several orders of magnitude, as it is the component with the largest influence on the GRACE ranging measurement. It is conceivable to reduce this error even more using an ensemble approach, treating each iteration of the orbit determination after convergence as a separate realization of the dynamic orbit. These realizations could then be used to compute an orbit of best agreement, possibly reducing the error at each epoch. The presented assessment of the effectiveness of the equinoctial best-fit reference ellipse is specific to the GRACE orbital configuration and chosen arc length. For this case, precession of the orbital plane is negligible for the integration period, ensuring a small Encke ratio. For satellites in similar orbital configurations, such as the European Space Agency’s GOCE mission (Drinkwater et al., 2003) or the SWARM constellation (Friis-Christensen et al., 2006), the method is directly applicable. The methodology can also be applied to satellites in other orbits, however the effect of nodal precession must possibly be considered. The nodal precession of the orbital plane is dependant on the inclination of the satellite (Brouwer, 1959), meaning that the Encke ratio would change at a


12


different rate for a satellite at a different inclination. A coprecessing reference ellipse would better approximate such an orbit, keeping the Encke ratio at a smaller value for longer periods. Jezewski (1983a) and Jezewski (1983b) give analytical solutions for such a reference motion, where the rotation of the ellipse is due to the oblateness of the Earth. This method improves on the processing of the data only, resulting in an orbit that is more compatible with the observations used to create it. This means that the method is not able to compensate for deficiencies in the input data. Given noisy accelerometer data or imperfect background models, applying this method would not lead to orbits that are necessarily closer to reality, but merely fit the noisy observations better. Spectral analysis of converged orbit differences show that there is still some extraneous oscillation in the orbits at frequencies at or below one cycle per revolution. This can be attributed to the static nature of the employed ellipse. For applications in GRACE processing, the presented results are however satisfactory, as the critical part of the error spectrum is located at higher frequencies, where precision greater than that of the LRI instrument was achieved (cf. Fig. 6). The applicability of the often suggested method of rectification of the reference orbit to the GRACE case is uncertain. Continuous orbit arcs are highly desirable in the context of integrating the variational equations and dynamic orbits. Milani and Nobili (1987) give an algorithm for a rectification method which can be used without restarting the integration procedure. In the method presented in this text, the state transition matrix of the variational equations is modified through corrections computed from the Marussi tensor. It is not immediately clear how Milani and Nobili’s (1987) method can be applied to the presented modified Encke approach given this circumstance. Acknowledgements This work is funded by the Austrian Research Promotion Agency (FFG) in the frame of the Austrian Space Applications Programme Phase 11 (Project 847971). The GRACE Level 1B dataset (Case et al., 2010) was obtained from the NASA EOSDIS Physical Oceanography Distributed Active Archive Center (PO.DAAC) at the Jet Propulsion Laboratory, Pasadena, CA ([dataset] JPL, 2001). The GRACE AOD1B dataset (Flechtner et al., 2014) was also obtained from PO.DAAC ([dataset] GFZ, 2012). We would like to thank the German Space Operations Center (GSOC) of the DLR for providing continuously and nearly 100% of the raw telemetry data of the twin GRACE satellites. References Bailey, D., Borwein, J., 2015. High-precision arithmetic in mathematical physics. Mathematics 3, 337–367. http://dx.doi.org/10.3390/ math3020337.

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