3.3 Orbit Determination
3.3.3 Orbit Representation
For many practical applications in satellite geodesy only a short part of the orbit is used, for instance that portion which can be directly observed from the participating
stations. In such cases simplified orbit representation techniques, which do not require orbital dynamics and the correct modeling of the acting forces, may be sufficient.
Instead, aspects of computer speed and memory requirements are well to the fore. The following procedures are frequently used:
− modeling of the deviations from a Keplerian orbit,
− polynomial representation, and
− “short-arc” representation with a simplified force-model.
In many cases a combination of the different representations is used.
3.3.3.1 Ephemeris Representation for Navigation Satellites
Here low memory requirements and efficient algorithms are of particular importance.
Two operational navigation systems have been in use: the TRANSIT system [6] until 1996, and currently the Global Positioning System (GPS) [7]. For both, the ephemeris representation is based on a mean Kepler ellipse with secular terms, e.g.d./dt, di/dt, anddω/dt. The time-dependent deviations of the predicted orbit from this reference ellipse are transmitted to the user. In the so-called TRANSIT broadcast ephemerides [6.2.2] the differences, in the three components of the orbital Cartesian system, were transmitted for every even minute (UTC) (cf. [3.2.1.3], Fig. 3.13). These are:
,E(t) correction in the direction of the motion,
,a(t) correction in the radial component, and (3.174) η(t) component perpendicular to the orbital plane.
If the satellite coordinates are required for epochs in between the even minutes, an appropriate interpolation algorithm, such as a polynomial interpolation [3.3.3.2] or a short-arc-solution [3.3.3.3], has to be applied.
For GPS ephemerides [7.1.5.2] the differences are given in the form of harmonic coefficients for modeling time-dependent sine and cosine correction terms, namely (cf. Fig. 7.12, p. 225):
Cus, Cuc amplitudes of the harmonic correction terms to the argument of latitude u= ν + ω, Cis, Cic amplitudes of the harmonic correction terms
to the inclination anglei, (3.175)
Crs, Crc amplitudes of the harmonic correction terms to the orbit radius.
This representation is continuous in time and thus suitable for real-time applications;
no interpolation is required. Each representation is, however, only valid for a limited time span, for example one or two hours. A smoothing algorithm may be necessary to remove the “jump” between adjacent portions of the orbit representation.
For GLONASS satellites the structure of the navigation message is different (Stew-art, Tsakari, 1998). Every 30 minutes the geocentric vector components for position and velocity as well as for the lunisolar acceleration are transmitted. The user has to apply adequate interpolation algorithms for intervening observation epochs (cf.
[7.7.1]).
3.3.3.2 Polynomial Approximation
The main advantages of this procedure are the simplicity of computation and the rather modest requirements for computer-time and memory. The main disadvantage is that polynomials are not suitable for the representation of trajectories longer than one or more revolutions. For this reason polynomials are not used for orbit predictions. On the other hand, polynomials can be successfully used for the representation of short arcs.
The computing time required for the representation of GPS orbits with polynomials is considerably less than for conventional orbit representation. This is why polynomials are preferable for real-time navigation, when small field computers are used. Linear approximation functions of the type
F (x) = a0ϕ(x) + a1ϕ1(x) + · · · + amϕm(x) (3.176) are common in practice. Different base functionsϕi(x) can be selected. Well known are power seriesϕi(x) = xi, i.e. the approximation with polynomials
F (x) = a0+ a1x + · · · + amxm. (3.177) Other common selections in satellite geodesy are trigonometric polynomials and Chebyshev polynomials.
For the approximation of orbits from the TRANSIT-type (broadcast ephemerides), the following trigonometric base functions have been used (Wells, 1974):
ϕ = {1, t, sin 2 ¯nt, cos 2 ¯nt} (3.178) where ¯n is the mean motion of the satellite. F (x) can be approximated by
P =
3 i=0
aiϕi (3.179)
and ˙F (x) by
P =˙
3 i=0
ai˙ϕi, (3.180)
with
˙ϕ = {0, 1, 2 ¯n cos ¯nt, −2 ¯n sin ¯nt} . (3.181) The first derivatives of the base functions are helpful when both the satellite coordinates X(t) and the velocities ˙X(t) are given. This was, for instance, the case with the
TRANSIT “precise ephemerides” for every minute, and it is the case for the GLO-NASS “broadcast ephemerides” [7.7.1].
Higher order Chebyshev polynomials are also frequently used for the approxi-mation of satellite orbits. Here we find for the satellite coordinates, velocities and accelerations (Kouba, 1983c):
X(t) = n
i=0
CXiTi(τ), ˙X(t) = n
i=1
CXiTi(τ), ¨X(t) = n
i=2
CXiTi(τ), (3.182)
with
τ = 2
,t(t − t0) and t ∈ t0; (t0+ ,t)!.
t0and,t are the starting epoch and the length of the fitting interval, n is the polynomial order,CXi are the adjusted Chebyshev coefficients for the satellite coordinatesx, y, z.
The Chebyshev polynomialsTiand their derivatives are determined recursively:
T0(τ) = 1, T1(τ) = τ,
Tn(τ) = 2τTn−1(τ) − Tn−2(τ); |τ| ≤ 1, n ≥ 2;
T1(τ) = dτ dt = ˙τ,
T2(τ) = 4τ ˙τ, (3.183)
Tn(τ) = 2n
n − 1τTn−1 (τ) − n
n − 2Tn−2 (τ); n ≥ 3, T2(τ) = 4( ˙τ)2,
T3(τ) = 24τ( ˙τ)2, Tn(τ) = 2n
n − 1( ˙τTn−1 (τ) + τTn−1 (τ)) − n
n − 2Tn−2 (τ), n ≥ 4.
For TRANSIT orbits a polynomial order of n = 8 to 10 proved to be successful.
For GPS orbits an order of 7 to 8 is sufficient. The advantage of the Chebyshev polynomials compared to other polynomial representations is that they give a much better approximation, even at the limits of the interval.
In addition to the previously mentioned representations of TRANSIT orbits (broad-cast and precise ephemerides), polynomial representations are suitable for fitting off-sets between the adjacent sections of the GPS broadcast orbits. These offoff-sets can reach several decimeters and thus may complicate continuous carrier phase solutions [7.1.5.2]. After a polynomial approximation the remaining offsets are below 2 cm.
In satellite laser ranging a smoothing of the orbit with polynomials aids the detec-tion of blunders and a first estimadetec-tion of the observadetec-tion accuracy [8.4.2]. Finally, large numbers of single observations can be condensed into normal points via smoothing functions. Such normal points play an important role in laser ranging, and they are also applied in the evaluation process of GPS observations.
A spatial smoothing of orbits with polynomials was proposed early on by Wolf (1967, 1970) in connection with the geometrical evaluation of satellite triangulations.
Through this procedure all observations could be referred to the same “smooth” orbital arc.
3.3.3.3 Simplified Short Arc Representation
In the short arc method forces are considered in the process of orbit approximation.
Here a short arc is defined as a portion of an orbit of less than one revolution. In this case it is usually sufficient to use a potential series expansion up to degree and order (10,10) for satellites in TRANSIT orbits (h ≈ 1000 km) and only (4,4) or (6,6) for GPS or GLONASS satellites (h ≈ 20 000 km).
Starting with initial conditions, for instance from a broadcast ephemeris, the orbital arc is determined using a method of numerical integration [3.3.2.2]. Depending on the quality of the approximate start values, several iteration steps may be necessary.
Compared with alternative approximation techniques, the short arc method requires a lot of computer time. It is, however, possible (Kouba, 1983b), to arrive at equally good results with a simplified and much faster short arc algorithm, when accelerations, caused by Earth’s gravity field, are introduced into a polynomial approximation. The accelerations acting on the satellite can be derived from the well known potential expansion up to degree and order N, and be calculated, for example every minute.
Equation (3.109) yields
where, λ are the geocentric latitude and longitude of the satellite. The components of acceleration at the satellite location, described in a Cartesian coordinate system, are obtained using the chain rule:
¨x = ∂V
The partial derivatives in (3.184) can be found in Anderle (1974) or Egge (1985).
Rewriting (3.184) gives (Kouba, 1983c,b):
¨x = ∂V
∂x + 2ω ˙y + ω2x, ¨y = ∂V
∂y − 2ω ˙x + ω2y, ¨z = ∂V
∂z. (3.185) ω is Earth’s rotation velocity; x, y, z are the satellite coordinates. The following
expressions also hold:
Finally, the partial derivatives required in equation (3.186) are given by:
∂V
The Legendre polynomialsPnm(sin ) and their derivatives Pnm (sin ) with respect to can be determined recursively (e.g. Tscherning et al., 1983; Egge, 1985). For potential coefficients CnmandSnm, which are given in fully normalized form, a de-normation is required:
Now, the satellite positions, velocities, and the accelerations (3.185) can be smoothed with a Chebyshev polynomial algorithm. For TRANSIT satellites the agree-ment with precise ephemerides was found to be at the 5 cm level (Kouba, 1983c;
Schenke, 1984) with a series expansion of the potential up to degree and order(10, 10).