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PRECISION ORBIT DETERMINATION (POD), Exercises of Relativity Theory

Duke UniversityRelativity Theory

The University of Texas at Austin. February 2001 ... The relativistic effects on GPS measurements can be summarized as follows. Due to the difference in the ...

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Geoscience Laser Altimeter System (GLAS)
Algorithm Theoretical Basis Document
Version 2.1
PRECISION ORBIT DETERMINATION (POD)
Prepared by:
H. J. Rim
B. E. Schutz
Center for Space Research
The University of Texas at Austin
February 2001
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Geoscience Laser Altimeter System (GLAS)

Algorithm Theoretical Basis Document

Version 2.

PRECISION ORBIT DETERMINATION (POD)

Prepared by: H. J. Rim B. E. Schutz Center for Space Research The University of Texas at Austin

February 2001

TABLE OF CONTENTS

  • 1.0 INTRODUCTION ....................................................................................................................
    • 1.1 B ACKGROUND ......................................................................................................................
    • 1.2 T HE POD P ROBLEM ..............................................................................................................
    • 1.3 GPS-BASED POD...................................................................................................................
      • 1.3.1 Historical Perspective ....................................................................................................
      • 1.3.2 GPS-based POD Strategies ............................................................................................
    • 1.4 O UTLINE ................................................................................................................................
  • 2.0 OBJECTIVE.............................................................................................................................
  • 3.0 ALGORITHM DESCRIPTION: ORBIT...............................................................................
    • 3.1 ICESAT/GLAS O RBIT D YNAMICS O VERVIEW ...................................................................
    • 3.2 E QUATIONS OF M OTION, T IME AND COORDINATE SYSTEMS .........................................
      • 3.2.1 Time System ....................................................................................................................
      • 3.2.2 Coordinate System........................................................................................................
    • 3.3 G RAVITATIONAL FORCES ..................................................................................................
      • 3.3.1 Geopotential .................................................................................................................
      • 3.3.2 Solid Earth Tides ..........................................................................................................
      • 3.3.3 Ocean Tides..................................................................................................................
      • 3.3.4 Rotational Deformation................................................................................................
      • 3.3.5 N-Body Perturbation ....................................................................................................
      • 3.3.6 General Relativity.........................................................................................................
    • 3.4 N ONGRAVITATIONAL FORCES ..........................................................................................
      • 3.4.1 Atmospheric Drag ........................................................................................................
      • 3.4.2 Solar Radiation Pressure .............................................................................................
      • 3.4.3 Earth Radiation Pressure .............................................................................................
      • 3.4.4 Thermal Radiation Perturbation ..................................................................................
      • 3.4.5 GPS Solar Radiation Pressure Models.........................................................................
      • 3.4.6 ICESat/GLAS "Box-Wing" Model ................................................................................
    • 3.5 E MPIRICAL FORCES ............................................................................................................
      • 3.5.1 Empirical Tangential Perturbation ..............................................................................
      • 3.5.2 Once-per Revolution RTN Perturbation.......................................................................
  • 4.0 ALGORITHM DESCRIPTION: MEASUREMENTS........................................................
    • 4.1 ICESAT/GLAS M EASUREMENTS O VERVIEW ..................................................................
    • 4.2 GPS M EASUREMENT M ODEL ............................................................................................
      • 4.2.1 Code Pseudorange Measurement.................................................................................
      • 4.2.2 Phase Pseudorange Measurement................................................................................
      • 4.2.3 Double-Differenced High-Low Phase Pseudorange Measurement..............................
      • 4.2.4 Corrections ...................................................................................................................
        • 4.2.4.1 Propagation Delay ................................................................................................
        • 4.2.4.2 Relativistic Effect .................................................................................................
        • 4.2.4.3 Phase Center Offset ..............................................................................................
        • 4.2.4.4 Ground Station Related Effects ............................................................................
      • 4.2.5 Measurement Model Partial Derivatives......................................................................
    • 4.3 SLR M EASUREMENT M ODEL ............................................................................................
      • 4.3.1 Range Model and Corrections......................................................................................
      • 4.3.2 Measurement Model Partial Derivatives......................................................................
  • 5.0 ALGORITHM DESCRIPTION: ESTIMATION................................................................
    • 5.1 LEAST SQUARES E STIMATION ..........................................................................................
    • 5.2 P ROBLEM FORMULATION FOR M ULTI -SATELLITE O RBIT D ETERMINATION ...............
    • 5.3 O UTPUT ...............................................................................................................................
  • 6.0 IMPLEMENTATION CONSIDERATIONS.......................................................................
    • 6.1 POD SOFTWARE SYSTEM ..................................................................................................
      • 6.1.1 Ancillary Inputs ............................................................................................................
    • 6.2 POD P RODUCTS ..................................................................................................................
    • 6.3 ICESAT/GLAS O RBIT AND ATTITUDE ..............................................................................
    • 6.4 POD ACCURACY ASSESSMENT .........................................................................................
    • 6.5 POD P ROCESSING STRATEGY ...........................................................................................
      • 6.5.1 Assumptions and Issues ................................................................................................
      • 6.5.2 GPS Data Preprocessing..............................................................................................
      • 6.5.3 GPS Orbit Determination.............................................................................................
      • 6.5.4 Estimation Strategy ......................................................................................................
    • 6.6 POD P LANS .........................................................................................................................
      • 6.6.1 Pre-Launch POD Activities ..........................................................................................
        • 6.6.1.1 Standards ..............................................................................................................
        • 6.6.1.2 Gravity Model Improvements...............................................................................
        • 6.6.1.3 Non-Gravitational Model Improvements..............................................................
        • 6.6.1.4 Measurement Model Developments .....................................................................
        • 6.6.1.5 Preparation for Operational POD .........................................................................
        • 6.6.1.6 Software Comparison ...........................................................................................
        • 6.6.1.7 POD Accuracy Assessment ..................................................................................
      • 6.6.2 Post-Launch POD Activities.........................................................................................
        • 6.6.2.1 Verification/Validation Period..............................................................................
        • 6.6.2.2 POD Product Validation.......................................................................................
        • 6.6.2.3 POD Reprocessing................................................................................................
    • 6.7 COMPUTATIONAL: CPU, M EMORY AND D ISK STORAGE ...............................................
  • 7.0 BIBLIOGRAPHY...................................................................................................................

be the primary tracking data for the ICESat/GLAS POD, and SLR data will be used for POD validation.

1.2 The POD Problem The problem of determining an accurate ephemeris for an orbiting satellite involves estimating the position and velocity of the satellite from a sequence of observations, which are a function of the satellite position, and velocity. This is accomplished by integrating the equations of motion for the satellite from a reference epoch to each observation time to produce predicted observations. The predicted observations are differenced from the true observations to produce observation residuals. The components of the satellite state (satellite position and velocity and the estimated force and measurement model parameters) at the reference epoch are then adjusted to minimize the observation residuals in a least square sense. Thus, to solve the orbit determination problem, one needs the equations of motion describing the forces acting on the satellite, the observation-state relationship describing the relation of the observed parameters to the satellite state, and the least squares estimation algorithm used to obtain the estimate.

1.3 GPS-based POD Since the earliest concepts, which led to the development of the Global Positioning System (GPS), it has been recognized that this system could be used for tracking low Earth orbiting satellites. Compared to the conventional ground-based tracking systems, such as the satellite laser ranging or Doppler systems, the GPS

tracking system has the advantage of providing continuous tracking of a low satellite with high precision observations of the satellite motion with a minimal number of ground stations. The GPS tracking system for POD consists of a GPS flight receiver, a global GPS tracking network, and a ground data processing and control system.

1.3.1 Historical Perspective The GPS tracking system has demonstrated its capability of providing high precision POD products through the GPS flight experiment on TOPEX/Poseidon (T/P) [ Melbourne et al. , 1994]. Precise orbits computed from the GPS tracking data [ Yunck et al. , 1994; Christensen et al. , 1994; Schutz et al. , 1994] are estimated to have a radial orbit accuracy comparable to or better than the precise orbit ephemerides (POE) computed from the combined SLR and DORIS tracking data [ Tapley et al ., 1994] on T/P. When the reduced-dynamic orbit determination technique was employed with the GPS data, which includes process noise accelerations that absorb dynamic model errors after fixing all dynamic model parameters from the fully dynamic approach, there is evidence to suggest that the radial orbit accuracy is better than 3 cm [ Bertiger et al ., 1994]. While GPS receivers have flown on missions prior to T/P, such as Landsat-4 and -5, and Extreme Ultraviolet Explorer, the receivers were single frequency and had high level of ionospheric effects relative to the dual frequency T/P receiver. In addition, the satellite altitudes were 700 km and 500 km, respectively, and the geopotential models available for POD, as they are today, had large errors for

using double- and triple-differenced GPS carrier phase measurements. Kinematic solutions are more sensitive to geometrical factors, such as the direction of the GPS satellites and the GPS orbit accuracy, and they require the resolution of phase ambiguities. The dynamic orbit determination approach [ Tapley , 1973] requires precise models of the forces acting on user satellite. This technique has been applied to many successful satellite missions and has become the mainstream POD approach. Dynamic model errors are the limiting factor for this technique, such as the geopotential model errors and atmospheric drag model errors, depending on the dynamic environment of the user satellite. With the continuous, global, and high precision GPS tracking data, dynamic model parameters, such as geopotential parameters, can be tuned effectively to reduce the effects of dynamic model error in the context of dynamic approach. The dense tracking data also allows for the frequent estimation of empirical parameters to absorb the effects of unmodeled or mismodeled dynamic error. The reduced-dynamic approach [ Wu et al ., 1987] uses both geometric and dynamic information and weighs their relative strength by solving for local geometric position corrections using a process noise model to absorb dynamic model errors. Note that the adopted approach for ICESat/GLAS POD is the dynamic approach with gravity tuning and the reduced-dynamic solutions will be used for validation of the dynamic solutions.

1.4 Outline This document describes the algorithms for the precise orbit determination (POD) of ICESat/GLAS. Chapter 2 describes the objective for ICESat/GLAS POD algorithm. Chapter 3 summarizes the dynamic models, and Chapter 4 describes the measurement models for ICESat/GLAS. Chapter 5 describes the least squares estimation algorithm and the problem formulation for multi-satellite orbit determination problem. Chapter 6 summarizes the implementation considerations for ICESat/GLAS POD algorithms.

3.0 ALGORITHM DESCRIPTION: Orbit

3.1 ICESat/GLAS Orbit Dynamics Overview Mathematical models employed in the equations of motion to describe the motion of ICESat/GLAS can be divided into three categories: 1) the gravitational forces acting on ICESat/GLAS consist of Earth’s geopotential, solid earth tides, ocean tides, planetary third-body perturbations, and relativistic accelerations; 2) the non- gravitational forces consist of drag, solar radiation pressure, earth radiation pressure, and thermal radiation acceleration; and 3) empirical force models that are employed to accommodate unmodeled or mismodeled forces. In this chapter, the dynamic models are described along with the time and reference coordinate systems.

3.2 Equations of Motion, Time and Coordinate Systems The equations of motion of a near-Earth satellite can be described in an inertial reference frame as follows:

r &&^ = a g + ang + aemp (3.2.1) where r is the position vector of the center of mass of the satellite, a (^) g is the sum of the gravitational forces acting on the satellite, ang is the sum of the non-gravitational forces acting on the surfaces of the satellite, and a (^) emp is the unmodeled forces which

act on the satellite due to either a functionally incorrect or incomplete description of the various forces acting on the spacecraft or inaccurate values for the constant parameters which appear in the force model.

3.2.1 Time System Several time systems are required for the orbit determination problem. From the measurement systems, satellite laser ranging measurements are usually time- tagged in UTC (Coordinated Universal Time) and GPS measurements are time-tagged in GPS System Time (referred to here as GPS-ST). Although both UTC and GPS-ST are based on atomic time standards, UTC is loosely tied to the rotation of the Earth through the application of "leap seconds" to keep UT1 and UTC within a second. GPS-ST is continuous to avoid complications associated with a discontinuous time scale [ Milliken and Zoller , 1978]. Leap seconds are introduced on January 1 or July 1, as required. The relation between GPS-ST and UTC is

GPS-ST = UTC + n (3.2.2) where n is the number of leap seconds since January 6, 1980. For example, the relation between UTC and GPS-ST in mid-July, 1999, was GPS-ST = UTC + 13 sec. The independent variable of the near-Earth satellite equations of motion (Eq. 3.2.1) is typically TDT (Terrestrial Dynamical Time), which is an abstract, uniform time scale implicitly defined by equations of motion. This time scale is related to the TAI (International Atomic Time) by the relation

TDT = TAI + 3 2.184 s. (3.2.3) The planetary ephemerides are usually given in TDB (Barycentric Dynamical Time) scale, which is also an abstract, uniform time scale used as the independent variable for the ephemerides of the Moon, Sun, and planets. The transformation from the TDB time to the TDT time with sufficient accuracy for most application has been

for the continental mass on which tracking stations are affixed has been modeled based on the AM0-2 model [ Minster and Jordan , 1978; DeMets et al. , 1990; Watkins , 1990]. Yuan [1991] provides additional detailed discussion of time and coordinate systems in the satellite orbit determination problem.

3.3 Gravitational Forces The gravitational forces can be expressed as: ag = Pgeo + Pst + Pot + Prd + Pn + Prel (3.3.1) where Pgeo = perturbations due to the geopotential of the Earth Pst = perturbations due to the solid Earth tides Pot = perturbations due to the ocean tides Prd = perturbations due to the rotational deformation Pn = perturbations due to the Sun, Moon and planets Prel = perturbations due to the general relativity

3.3.1 Geopotential The perturbing forces of the satellite due to the gravitational attraction of the Earth can be expressed as the gradient of the potential, (^) U , which satisfies the Laplace equation, ∇^2 U = 0:

∇ U = ∇( U (^) s + ∆ U (^) st + ∆ U (^) ot + ∆ U (^) rd ) = Pgeo + Pst + Pot + Prd (3.3.2) where U (^) s is the potential due to the solid-body mass distribution, ∆ U (^) st is the potential change due to solid-body tides, ∆ U (^) ot is the potential change due to the ocean tides, and ∆ U (^) rd is the potential change due to the rotational deformations.

The perturbing potential function for the solid-body mass distribution of the Earth, (^) U (^) s , is generally expressed in terms of a spherical harmonic expansion, referred to as the geopotential, in a body-fixed reference frame as [ Kaula , 1966; Heiskanen and Moritz, 1967]:

1 0 ( , , ) (sin ) cos sin e e l e^ l U (^) s r φ λ GMr GMr (^) l m ar Plm φ Clm m λ Slm m λ ∞ = =

= + ^   + 

where GM (^) e = the gravitational constant of the Earth ae =^ the mean equatorial radius of the Earth C (^) lm , S (^) lm = normalized spherical harmonic coefficients of degree l and order m Plm (sin ϕ) (^) = the normalized associated Legendre function of degree l and order m r , φ, λ = radial distance from the center of mass of the Earth, the geocentric latitude, and the longitude of the satellite To ensure that the origin of spherical coordinates coincides with the center of mass of the Earth, we define C 10 = C 11 = S 11 = 0.

The summation over k(l,m) means that each different l, m combination has a unique list of tidal frequencies, k , to sum over. The tidally induced variations in the Earth’s external potential can be expressed as variations in the spherical harmonic geopotential coefficients [ Eanes et al. 1983].

0 0

( 1) cos^ , 4 (2 ) sin^ ,

m k lm (^) e m k k k k C k H l^ m^ even a π δ l^ m^ odd

∆ = −  Θ^ −

− ∑  Θ^ −

0 0

( 1) sin^ , 4 (2 ) cos^ ,

m (^) k lm (^) e m k k k k S k H l^ m^ even a π δ l^ m^ odd

∆ = −  −^ Θ^ −

− ∑^  Θ^ − (3.3.5)

where δ 0 m is the Kronecker delta; ∆ C (^) lm and ∆ S (^) lm are the time-varying geopotential coefficients providing the spatial description of the luni-solar tidal effect.

3.3.3 Ocean Tides The oceanic tidal perturbations due to the attraction of the Sun and the Moon can be expressed as variations in the spherical harmonic geopotential coefficients. The temporal variation of the free space geopotential induced from the ocean tide deformation, ∆ U (^) ot , can be expressed as [ Eanes et al. , 1983] '^1 0 0

l l e^ l Uot π G ρ w ae (^) k l m l k^ ar ∞ − + = = +

∆ = +^ ^ 

× C (^) klm^ ±^ cos( Θ k ± m λ) + S (^) klm^ ±^ sin( Θ k ± m λ) Plm (sin φ) (3.3.6)

where ρ w is the mean density of sea water, k is the ocean tide constituent index, k (^) l '^ is the load Love number of degree l , C (^) klm^ ±^ and S (^) klm^ ±^ are the unnormalized prograde and retrograde tide coefficients, and Θ k is the Doodson argument for constituent k. The above variations in the Earth’s external potential due to the ocean tide can be expressed as variations in the spherical harmonic geopotential coefficients as follows [ Eanes et al. 1983].

∆ C lm = Flm ∑ k Aklm

∆ S lm = Flm ∑ k Bklm (3.3.7)

where Flm , Aklm , and Bklm are defined as

Flm = 4 π Mae^2 e^ ρ w^ ( l + m )! ( l - m )!(2 l +1)(2- δ 0 m )

1+ k (^) l ' 2 l +1 (3.3.8)

and A (^) klm B (^) klm = ( C^ klm

  • (^) + C (^) klm - (^) ) ( S (^) klm^ +^ - S (^) klm^ -^ ) cos Θ k + ( S^ klm

  • (^) + S (^) klm - (^) ) ( C (^) klm^ -^ - C (^) klm^ +^ ) sin Θ k (3.3.9)

3.3.4 Rotational Deformation Since the Earth is elastic and includes a significant fluid component, changes in the angular velocity vector will produce a variable centrifugal force, which consequently deforms the Earth. This deformation, which is called “rotational deformation”, can be expressed as the change of the centrifugal potential, U (^) c [ Lambeck , 1980] given by