In the previous studies, many authors considered that, the conditions at which
the directions from one station to another was determined with the highest accuracy
from synchronous optical observations (Batrakove, 1969;
Lambeck, 1968; Zhongolovitch, 1970;
Allan and Weiss, 1980; Liao, 1985).
Also, GPS was used for determined the acceleration and torsion displacements
data of a Yonghe bridge tower (Kaloop and Hui, 2009).
In the previous study, GPS system was employed for solving the problem of mobile
machine localization (Yuan et al., 2010). The
system error of this kind of system was determined and corrected by lidar sensors.
In this case, lidar scan-matching was proposed to correct the error of GPS to
localize a mobile machine accurately.
Since, the laser measurements of the range to satellites became more accurate
and wide spread. Laser optical devices were proposed for the surface detection
on defect features concerning the past low speed, smaller size in detection
and less inconvenience in the testing or operation (Chu
et al., 2011). Also, laser optical observation was considered for
illuminating artificial satellites and other space objects from H-SLR station
with determined the accuracy of the calculations (Ibrahim
et al., 2011). It was of interest to consider more general problem
in which, besides the usual synchronous optical observations of two positions
of the satellite from two stations, one measurement of the range was available
(Gili et al., 2000; Hakli,
2004; Martin and Jahn, 2000; Eckl
et al., 2001). Furthermore (Ekuma, 2007)
considered the model, which describing the laser light and optical objects,
for studying the infinite nature of most distant bodies. In addition, laser-satellite
optical observations in SLR station were achieved for investigating and studying
the effect of the satellite signatures on the data fitting accuracy (Hanna
et al., 2011).
The analytical expressions for the error of the distance should be presented depending on errors of measurements. In addition, the problem of conditions, which give the minimum of this error, was taken in our consideration.
SYSTEM OF COORDINATES
For the sake of simplicity, we introduced the frame of reference in the following way seen in Fig. 1.
||The system of coordinates M1: Station at the origin,
M2: Station was laid on the Z-axis S1 and S2:
Two successive positions of the satellite between the two stations
The origin was placed at the first station M1, the Z-axis was passed through the second station M2. The XZ plane was passed through the first observed satellite S1 and Y-axis was completed the right hand system, with X and Z-axis.
In this frame of reference, the error of Z-coordinate of M2 station was equivalent to the error of the distance.
having origin at Mi station and the end at Sj satellite
and ρij were topocentric distance (from the station to the satellite),
θij, φij were angular coordinates of the satellite
measured as seen from Fig. 1.
THE EQUATIONS OF CONDITION
From Fig. 1, we have:
where, (xj, yj, zj) and (Xi, Yi, Zi); (j, i = 1, 2) are the coordinates of the satellite and station, respectively. Differentiation of (1), gives the equations of condition:
We must put in Eq. 2:
Besides, it must be taken into account that this problem could allow to determine only relative positions of satellites. In addition, the position of M1 station must be taken as known, this gives:
Now, with Eq. 4, we have nine unknowns, three coordinates for every position of the satellite and three coordinates of M2 station. However, it is possible to write only eight Eq. 2 types and it is necessary to add one more equation, which is given by the range measurements to these eight equations. Let us consider this measured distance to correspond to indices i = j = 1, such supposition do not break the generality, because every measured distance can be taken for ρ11 at the proper choice of the frame of reference. For the range measurement, we have the equation of condition, taking into account the conditions of Eq. 3 and 4:
Solving Eq. 1-5 with respect to the correction
dZ2 as an example, we obtain after sufficiently troublesome computations,
the following expression:
Let us suppose that ,
dθij are independent random values distributed according to
the normal laws with the equal dispersion σ20 (σ20
is the square of the mean quadratic error of one measurement). The random value
dρ11 is also considered to be independent and normal and its
dispersion is taken to be χ2 σ20,
where χ a coefficient is transforms the dimensions and the values of the
angular errors into the dimensions and the values of the range error. The dispersion
of dZ2 is determined by the formula:
The mean square error of dZ2 can be obtained by computing the square
root from the both parts of Eq. 7. Let us analyze now the
expression (7), the second item of the right hand side of (7) does not depend
on φ and the first item as can easily seen has minimum at sin2
φ = 1 which corresponds to the condition:
It means that planes of synchronous observations must be perpendicular one to another. At sin2 φ = 1, then we have:
where, the angles φ21 and (θ21-θ11) may be considered as naturally independent variables. The expression between brackets in the right hand side of Eq. 8 has minimum at cos2 θ21 = cos2 (θ21-θ11) = 0. It corresponds to sin2 θ21 = 1 (i.e., when sin2 θ21, is maximum).
In this case, we have the solution with physical meaning:
According to this solution, the first satellite must Zlace at the point M2.
In this case ρ21 naturally is equal to zero. Finally, we obtained
the following minimum dispersions of dZ2:
The received conditions from Eq. 9, gave very small error in determining the distance between two stations. It was evident that we could speak only about the approximation to these ideal conditions, which were allowed by the visibility conditions and by the requirement of the negligibility of differential refraction effect, which influence the determination of the position noticeably at small angles above the horizon.