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Polynomial regression model for translation between GPS and local terrestrial network
Lü Zhongshu
Information institute of Photogrammetry and Remote
Sensing Yunnan Bureau of Surveying and Mapping,
Kunming 650034
P. R. China
Shi Kun
Department of Geomatics
Kunming University of Science
and Technology, Kunming
650093 P.R.China
shikun@263.net
Lü Zhongshu
Information institute of Photogrammetry and Remote
Sensing Yunnan Bureau of Surveying and Mapping,
Kunming 650034
P. R. China
Shi Kun
Department of Geomatics
Kunming University of Science
and Technology, Kunming
650093 P.R.China
shikun@263.net
Abstract
The polynomial regression model in which the precision information of satellite and local terrestrial networks is considered is applied to carry out translation between GPS and local terrestrial systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability, and determinability of parameters is applied to the translation model to select the the effective parameters. Finally, in order to test the model, an observed network in north of China has been calculated and analyzed with this model and translation parameters have been selected with determinability.
Introduction
The model selection for translation between GPS and terrestrial network has been an important problem in modern geodesy. As well known, it is important to select the optimum translation model within a limited area of a certain local terrestrial network for the departments such as engineering surveying, mineral, geology and so on. For the purpose, the problem has been studied widely and deeply.[2],[3],[4],[5].Traditional translation model, 7-parameters model, between the two systems mainly focused on the solution of seven translation parameters, i.e. three shifts Dx Dy Dz)of the two systems ;three components of rotation about X,Y,Z axes or Eular rotation elements(ex ey ez) and one scale element K. However, that this model is not suitable in a limited small area has been proved in reality. Considering the systematic error of local terrestrial network and the merits and demerits of existed main translation model between the two systems, the polynomial regression model is applied to the translation for the two systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability,.
Translation approach with polynomial regression model The realization of translation between GPS and terrestrial network is usually based upon the coordinate differences at the same point that has the coordinates of both satellite frame and terrestrial frame. According to Weierstrass[ ],the coordinate differences lR can be approximated by polynomial with certain order term, i.e. for the I point of data in translation model, considering the second order term of the polynomial, the coordinate differences between the two systems for the same vector can be expressed as follows,
(Rt) i=(Rg) i+(dR) i (1)
(dR) i=(dx dy dz) i (2)
and
In the formula,(Rt) i is the vector of i point in geocentric system or GPS frame;
(Rg) i is the vector of i point in geodetic system or local terrestrial frame;
(dR) i is the difference of the same vector of in the tow systems
And subscript k is the reference point of terrestrial network; m0,n0,l0 are the shifts of the original point; m1,n1,l1 are the scale elements of the directions of X,Y,Z axes; (m,n), (m,l), (n,l) are the rotation about X,Y,Z axes,…,m,n,…n,l,…l are the systematic errors that can not be distinguished.
Data Processing
Data processing in this work is based on the principle of adjustment of correlated observations[7]. For the i point in network, the observation equation can be established as following, (3)
In the formula,
X is the correction of the coordinate of i point;
Y is the parameters for translation between two systems;
A and B are the coefficients matrix relative to X and Y;
B is the approximate of geodetical latitude of the i point;
L is the approximate of geodetical longitude of the i point;
H is the approximate of geodetical height of the i point.
Suppose there are n common points in the network,the integrated form of the observation equation will be,
And suppose the weight matrix relative to the observation equation is
Here,
A=diag[Ai] i=1,2…n
B=[B]
Ps is the weight matrix of observation Vs;
PT is the weight matrix of observation VT;
Or, V=AX+BY-L P
According to the least square method, the solution of parameters can be achieved as following,
The Determinability of the Translation Parameters
In the formula (8), there usually are the appearance of overparameters because of limitations of small area for combination of two networks and fewer number common points in the area as well as the problem of precision of the points and the distribution of the points. In order to overcome the overparameters in a limited local area, based on the theory of reliability of Baarda [6], the determinability of parameters is put forward for the polynomial regression translation model to select the effective translation parameters. From formula [7],we have,
This is the extended Gauss-Markov model in adjustment. According to [7],we can express the prescribed minimum value of Y vector as following,
This formula gives us a p-dimensions prescribed minimum ellipsoid in which if the parameters are include, they will be thought immeasurable statistically and
d0 (S)is defined as the value of determinability with unit of s0
And d0(S)is the function of a0 and ·0 (a0 is the critical value and ·0 is the power of statistical test). Finally we obtain the value of determinability of single and multiple parameters as following,
Summary and Discussions
For the testing of the model, a calculation program has been worked out with Fortran and an observed network in the Southwest part of China has been calculated with it. The criteria for selection of parameters is the minimum one of values from determinability of
Table. 1: The determinability value and the precision of translation parameters
Table.2: Comparison of the mean value of mean square error of terrestrial coordinates before and after adjustment
shift m,n,l. Comparing the precision of translation parameters and coordinates after adjustment with that of before adjustment in table 1 and table 2,the conclusion can be drawn as following,
The polynomial regression model in which the precision information of satellite and local terrestrial networks is considered is applied to carry out translation between GPS and local terrestrial systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability, and determinability of parameters is applied to the translation model to select the the effective parameters. Finally, in order to test the model, an observed network in north of China has been calculated and analyzed with this model and translation parameters have been selected with determinability.
Introduction
The model selection for translation between GPS and terrestrial network has been an important problem in modern geodesy. As well known, it is important to select the optimum translation model within a limited area of a certain local terrestrial network for the departments such as engineering surveying, mineral, geology and so on. For the purpose, the problem has been studied widely and deeply.[2],[3],[4],[5].Traditional translation model, 7-parameters model, between the two systems mainly focused on the solution of seven translation parameters, i.e. three shifts Dx Dy Dz)of the two systems ;three components of rotation about X,Y,Z axes or Eular rotation elements(ex ey ez) and one scale element K. However, that this model is not suitable in a limited small area has been proved in reality. Considering the systematic error of local terrestrial network and the merits and demerits of existed main translation model between the two systems, the polynomial regression model is applied to the translation for the two systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability,.
Translation approach with polynomial regression model The realization of translation between GPS and terrestrial network is usually based upon the coordinate differences at the same point that has the coordinates of both satellite frame and terrestrial frame. According to Weierstrass[ ],the coordinate differences lR can be approximated by polynomial with certain order term, i.e. for the I point of data in translation model, considering the second order term of the polynomial, the coordinate differences between the two systems for the same vector can be expressed as follows,
(Rt) i=(Rg) i+(dR) i (1)
(dR) i=(dx dy dz) i (2)
and
In the formula,(Rt) i is the vector of i point in geocentric system or GPS frame;
(Rg) i is the vector of i point in geodetic system or local terrestrial frame;
(dR) i is the difference of the same vector of in the tow systems
And subscript k is the reference point of terrestrial network; m0,n0,l0 are the shifts of the original point; m1,n1,l1 are the scale elements of the directions of X,Y,Z axes; (m,n), (m,l), (n,l) are the rotation about X,Y,Z axes,…,m,n,…n,l,…l are the systematic errors that can not be distinguished.
Data Processing
Data processing in this work is based on the principle of adjustment of correlated observations[7]. For the i point in network, the observation equation can be established as following, (3)
In the formula,
X is the correction of the coordinate of i point;
Y is the parameters for translation between two systems;
A and B are the coefficients matrix relative to X and Y;
B is the approximate of geodetical latitude of the i point;
L is the approximate of geodetical longitude of the i point;
H is the approximate of geodetical height of the i point.
Suppose there are n common points in the network,the integrated form of the observation equation will be,
And suppose the weight matrix relative to the observation equation is
Here,
A=diag[Ai] i=1,2…n
B=[B]
Ps is the weight matrix of observation Vs;
PT is the weight matrix of observation VT;
Or, V=AX+BY-L P
According to the least square method, the solution of parameters can be achieved as following,
The Determinability of the Translation Parameters
In the formula (8), there usually are the appearance of overparameters because of limitations of small area for combination of two networks and fewer number common points in the area as well as the problem of precision of the points and the distribution of the points. In order to overcome the overparameters in a limited local area, based on the theory of reliability of Baarda [6], the determinability of parameters is put forward for the polynomial regression translation model to select the effective translation parameters. From formula [7],we have,
This is the extended Gauss-Markov model in adjustment. According to [7],we can express the prescribed minimum value of Y vector as following,
This formula gives us a p-dimensions prescribed minimum ellipsoid in which if the parameters are include, they will be thought immeasurable statistically and
d0 (S)is defined as the value of determinability with unit of s0
And d0(S)is the function of a0 and ·0 (a0 is the critical value and ·0 is the power of statistical test). Finally we obtain the value of determinability of single and multiple parameters as following,
Summary and Discussions
For the testing of the model, a calculation program has been worked out with Fortran and an observed network in the Southwest part of China has been calculated with it. The criteria for selection of parameters is the minimum one of values from determinability of
Table. 1: The determinability value and the precision of translation parameters
| Parameters | Determinability | Precision of parameters | Remarks | usd |
| ·X | 0.2051 | 0.1054 | a0=1.0% b=80% | 1.7278 |
| ·Y | 0.7744 | 0.4173 | ||
| ·Z | 1.3576 | 0.6575 | ||
| m | 0.3526 | 0.1740*10 | ||
| n | 1.1216 | 0.4908*10 | ||
| l | 0.3376 | 0.1700*10 |
Table.2: Comparison of the mean value of mean square error of terrestrial coordinates before and after adjustment
| mean value of mean square error % | B (m) | L (m) | H (m) |
| Before adjustment | 0.3483 | 0.3539 | 2.0067 |
| Adjustment with 7-parameters | 0.2397 | 0.3119 | 0.7861 |
| Adjustment with polynomial model | 0.2455 | 0.1727 | 0.5610 |
shift m,n,l. Comparing the precision of translation parameters and coordinates after adjustment with that of before adjustment in table 1 and table 2,the conclusion can be drawn as following,
- It is effectual and helpful to select translation parameters in polynomial regression model with the determinability of parameters between the two systems in a limited local area;
- In the case of coordinates with given precision within the network,we suggest that the translation parameters between the two systems should be chosen with the criteria of determinability.
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