Polynomial regression model for translation between GPS and local terrestrial network


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 DeterminabilityPrecision of parametersRemarksusd
·X 0.2051 0.1054a0=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.
References
  • Petr, Vanicek, Geodesy: The concepts, North-Holland Publishing Company, New York,1982.
  • E.J.Krakiwsky, Mathmatical Model for the Combination of Terrestrial and Satellite Network, Canada Surveying,28(5),1974
  • T.Vincenty, Methods of Adjustment Space Systems Data and Terrestrial Measurements, Bull.Geod.56, 1982.
  • Helmut Wolf, Das Lage und Hohen Problem In grossen geodatischen, ZFV.Hefts.1985
  • Forstner, Reliability and Discernability of Extended Gauss- Markov Models, DGK,Reine A,Nr.98,Munchen 1983.
  • Deren Li, The Theory of Error and Reliability, Publishing House of Chinese Surveying and Mapping, 1988.
  • W.Jordan and O.Eggert, Handbuch der vermessungskund, Bd, 1962.
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