For orbital computations, simultaneous observations are needed to be carried
out by the individual observing stations (Batrakov, 1961;
Degnan, 1994; Cech et al.,
1998). If the different observational data, obtained from the individual
stations, are not carried out at the same time we would deal with functions
which either is not known at every value of the independent variable within
an interval or their expressions are so complicated that the evaluation of the
function is prohibitive.
So far, we are dealing, in fact, with data fitting. This can be subjected to
the methods of interpolation. Mathematically, we are concerned with a function
y = f(x), whose values xi, (i = 0, 1, 2,
, n) of the independent
variable x and it is required to adopt a function p(x), to replace the function
f(x) which assumes the same values as f(x) for xi and from which
other values can be easily calculated to a desired degree of accuracy. In dealing
with SLR data, the interpolating polynomials may not be suitable for use as
an approximation. Particularly, when fitting large number of data points using
high degree polynomial, the fitting may suffer from noticeable error due to
the nature of approximation (Johnson and Riess, 1982).
For this reason, either the Spline or overlapping techniques are used as an
In this article, we present two techniques for fitting SLR data, the Spline
technique and overlapping technique. We apply these two techniques on the SLR
data of 7 artificial satellites taken from Helwan SLR station.
SPLINE TECHNIQUE FOR MATCHING CONTINUOUS PROGRESSIVELY INTERVALS
Spline technique is defined as an artificially constructed sequence of polynomials
matched one to another at the end points of the intervals at which the polynomials
are defined (Nmadu et al., 2009; Hama-Salh,
2010). This technique is used in order to join lower degree polynomials
which leads to a good fitting (De Boor, 1978; Nurnberger,
Starting with observational measurements yi = 1, 2, ..., n taken at time ti = 1, 2, ..., n. These data are divided into groups. Two consecutive groups, yij, i = 1, 2 and j = 0, 1, ..., ni are selected. The closed time intervals [ti0, tin] are mapped into the closed intervals [-1, 1]i, using the following Xtransformation:
Two independent Chebyshev polynomials (Taiwo and Abubakar,
2011; Zhang et al., 2012) are constructed
for yij as functions of variable x are given in the following form:
where, fik(x) are the Chebyshev polynomials as given by Hanna
The functions yi, i = 1, 2 are two independent approximating polynomials representing two successive intervals of one set of the data. These two constructed polynomials are, in general, disjoined. The value of the polynomial at the first data point of the second subinterval may not be matched with the value of the polynomial at the end data point of the first subinterval. The combined adjustment method, given below, is to construct the proper Spline.
In order to join the two constructed interpolating polynomials yi,
i = 1, 2 for the two successive intervals, the coefficients
have to be determined for the constraint (Hanna et al.,
and then the coefficients can be obtained in the form:
are the coefficients of the 2nd polynomial (Hanna et
al., 2004) as a result of applying the proper Spline.
THE OVERLAPPING TECHNIQUE
The overlapping method is based on using the overlapped intervals of observations.
The reason that it has not been used in orbit computations is connected probably
with the fact that the statistical characteristics of the normal points obtained
by the overlapping method have not yet been studied and their strict use is
difficult (Bjorck and Yuan, 1999). We have not the aim
to investigate these characteristics completely but we shall restrict ourselves
to a more particular problem. This problem is how to form a normal point corresponding
to the all-available measurements (for example, laser measurements of the range
during one passage of an artificial Earth's satellite in the zone of visibility
of a tracking station by using normal points obtained with overlapping subintervals.
Let the interval of time L, covering one passage of the satellite over the station, be divided into a number of subintervals Li ( i = 1, 2,
, m) successively matched to one another. Assume that we have a sufficiently good preliminary orbit of a satellite. The discrepancy between the observed and the calculated ranges are small and can be smoothed by a polynomial of a low degree on both of the subintervals Li and the whole interval L. The measurements are considered to be independent and of equal accuracy.
Let the first intermediate normal point be formed as an arithmetic mean of the differences between the observed and computed ranges at the united interval containing L1 and L2. The second arithmetic mean is formed by an analogous way at the united interval including L2 and L3 and so on until the united interval Lm-1 and Lm. We suppose also that in a quite similar way the normal point at the united interval L1 and Lm has been determined.
Analysis of the variations within and between the overlapping intervals will
be found and discussed by Hanna (2002) and Samwel
et al. (2005).
COMPARATIVE STUDIES ON THE OVERLAP AND SPLINE TECHNIQUES
In the present study, both Spline and overlapping techniques are applied for
30 samples of ranging data for satellites AJISAI (Aj), TOPEX (TP) and BEACON-C
(BC) (Ibrahim et al., 2011). The number of data
points in these samples range from 153 to 1482. Table 1 summarized
the results obtained.
The name of satellite and the number of data points are represented in the first column. Second column shows the polynomial degree and the corresponding Standard Deviation (SD) obtained in case of fitting total number of data. Third and forth columns represent the polynomial degree and the corresponding standard deviation obtained using Spline technique and overlapping technique, respectively.
As represented in Table 1, the average polynomial used for fitting the whole data is of 8th degree while it is of 4th degree using both spline and overlapping technique. In addition the average standard deviation reached in case of fitting the whole data is about 882.7 mm while it is of about 1.2 mm for spline technique and 3.97 mm for overlapping technique which clarify that the used polynomial and the corresponding standard deviation of the two techniques, spline and overlap techniques, are much better than that obtained when fitting the total number of data.
Also, it is found that the average polynomial degrees and average standard
deviations of both Spline and overlapping techniques are comparable with slight
merit for spline technique over the overlap technique.
|| Results of fitting SLR data. The standard deviation and the
corresponding degree of the used polynomials are given
The average standard deviation using Spline technique is 1.2 mm with polynomial
of 4th degree, while it is about 3.97 mm with polynomial of 4th degree using
For more clarity, Fig. 1 represents the results obtained
for one of the samples under study. The data taken in the present case consist
of 552 data points for satellite Beacon-C (6503201) which was observed in 4/9/2000
from Helwan SLR station. Figure 1a represents the range and
best fitting for the whole number of data. Figure 1b and c
represent the range and best fitting of the second subinterval, using Spline
technique and overlapping technique, respectively.
As it is clarified in Fig. 1, the polynomial, used for fitting the whole data is of 10th degree with corresponding standard deviation of 75.84 mm while the polynomials used for fitting data using both Spline and overlapping technique are of the 4th degree and the corresponding average standard deviations are about 0.15 mm for Spline technique and about 1.04 mm for overlapping technique.
||The range and best fitting of satellite Beacon-C for (a) Fitting
whole data, (b) Fitting using Spline technique and (c) Fitting using overlapping
||The range and best fitting of satellite TOPEX for (a) Fitting
data using Spline technique and (b) Fitting data using overlapping technique
However, it is found that about 11/30 cases have slight merit for overlap technique over the Spline technique as represented in Fig. 2 which be explained by the existence of gaps within the data points. So, generally, the distribution of the data points in each interval may adversely affect the degree of the used polynomial and the corresponding standard deviation obtained.
For more clarification, Fig. 2 represents the results obtained for one of the samples under study. The data taken in the present case consist of 985 data points for satellite Beacon-C (6503201) which was observed in 10/9/2000 from Helwan SLR station. Figure 2a represents the range and best fitting of the second subinterval, using Spline technique, Fig. 2b represents the range and best fitting of the second subinterval, using the overlapping technique.
As it is clarified from Fig. 2, the data is fitted using polynomial of 4th degree with corresponding average standard deviation of about 6.89 mm using spline technique and of 0.89 mm, using the overlapping technique which clarify how adversely the distribution of data can affect on the fitting process.
SATELLITE SIGNATURE EFFECT
According to the Satellite Laser Ranging (SLR) technique, the incident laser
beam on a retro reflector onboard an artificial satellite is reflected in a
path that is parallel to the initial incident one but in an opposite direction.
The size of the geodetic satellites has become one of the limiting factors of
the laser ranging data precision and is known as satellite signature effect.
The most visible geodetic satellites are AJISAI (perigee height of 1485 km),
TOPEX (perigee height of 1350 km) and BEACONC (perigee height of 927 km).
Also, the number and type of the retro reflectors as well as their orbital trajectory
vary significantly from satellite to another, leading to different expected
optical responses in which AJISAI has the most favorable one (Villoresi
et al., 2008). Besides, the spread of retro reflectors due to multiple
reflectors on the satellites is now recognized as key error factor and known
as target effect (Appleby, 1992). Furthermore, it is
known that the large array of corner cubes reflectors on the satellites, made
them possible to obtain a fairly large amount of range data. However, the varying
size of the array causes a severe target signature effect (Neubert,
1994; Otsubo et al., 1999). These previous
satellite signature effect mentioned above have a great influence on the precision
of the SLR data obtained. AJISAI (AJ), TOPEX (TP) and BEACON-C (BC) are taken
as examples of high signature satellites. ERS-2 (ES) and STARLETTE (ST) satellites
are taken as examples for low signature satellites.
The characteristics of satellites under study are represented in Table 2. Column 1 represents the satellite name, while the corresponding ID is indicated in column 2. Column 3, 4 and 5 represent the retro-reflector array shape, diameter and reflectors of each satellite. Finally, column 6 represents the satellite altitude in kilometers.
In the present study, a comparison between the SLR data precisions for both low and high satellite signatures is performed. The precision of the SLR data is expressed by the used polynomial degree for data fitting and the corresponding standard deviation as represented in Table 3.
As a result of the high degree of satellite signature of the Ajisai, Topex and Beacon-C in comparable with the satellite signatures of ERS-2 and Starlette, it becomes clear that the average precision of the fitting of the data taken for the satellites STARLETTE and ERS-2 are much better than those obtained for the satellites AJISAI, TOPEX and BEACON-C. The average precision of the high signature satellites is about 470.45 mm while it is 21.43 mm for the low signature satellite.
|| Characteristics of the used satellites
|| Polynomial degree and standard deviation of low and high
In the present study, two different methods are used for analyzing SLR data. These are the Spline and overlapping techniques. It is concluded that the polynomial degree and the corresponding standard deviation of the two techniques are much better than that obtained when fitting the total number of data. In addition, it is found that the average polynomial degrees and the corresponding average standard deviations of both Spline and overlapping techniques are comparable with slight advantage for Spline technique over the overlap technique.
However, it is found that about 11/30 samples under study represent that the overlap technique is better than the spline technique which may refers to the distribution of the data points inside the intervals.
Furthermore, the effect of the satellite signatures on the SLR data precision is examined. It is found that the precision of the data fitting for the low signatures satellites are much better than those obtained for high signature satellites.