Adjustment and accuracy estimation for the base data with not enough precision in city’s GPS networks

By the law of reversal matrix, we can obtain the result as follows,

Q₁₁=N^-1₁₁+N^-1₁₁N^-1₁₂N^-1₂₂N^-1>21N^-1₁₁
Q₁₂=-N^-1₁₁N₁₂N^-1₂₂N^-1₂₂Q^T₂₁
Q₂₂=N^-1₂₂=(N₂₂-N₂₁N^-1₁₁N₁₂)^-1 ..............

Suppose there is a function of parameters as following,

j^{^}=j^{^}(Z^{^}) ..............dj^{^}=f^T_YdY^{^}+f^T_YdX^{^} ...................... (7)

The covariance will be,

available approximation adjustment model

Supposing the precision of the base datum is enough in adjustment of GPS network, estimating the contribution of the base datum with not enough precision when evaluation of the computation the final results. According to the principle of indirective adjustment model, we can have,

V=AdY-l,P₁ ......................(9)
and,
A^Tp₁AdY^{^}+A^Tp₁l=0
NdY^{^}+W=0...............(10)
we have,
dY^{^}=-N^-1W ...............(11)
QY^{^}Y^{^}=N^-1

Suppose there is a function of parameters as following,
j^{^}=j^{^}(y^{^})...............dj^{^}=f^Tdy^{^}+f^T_ldl................(12)
The covariance will be,

Example
A small GPS network in southwest of China in which there are two base datum points with not enough precision. And their covariance matrix is there are four independent observations l1,l2,l3,l4 and their covariance are 2,1,2,2. We are going to make adjustment with available approximation adjustment model and strict adjustment model for the base datum with not enough precision.

(1), strict adjustment model
By formula (1), we have

(2), available approximation adjustment model

Summary and suggestions

• Compare the two jj Q from the two ways, available approximation adjustment model is better than strict adjustment model for the base datum with not enough precision. And the most important thing is that the former does not change the base datum. But the later will change the base datum and this will destroy the uniform of the original local frame;
• The quantity of computation for the available approximation adjustment model is much less and simple than strict adjustment model;
• The unit standard deviation(usd) in the two models is different. We should pay enough attention to this fact. They are
  
• The matching of prior-weight must be considered before adjustment in two models for estimation of precision. And we suggest surveying engineer apply the available approximation adjustment model in real work.

References

• Petre Venicek, E J.Krakiwsky,Geodesy:The Concepts,North-Holland Publihsing Company, Amsterdam 1982.
• D.E.Wells et al,GPS Guidance,translated into Chiness by WTUSM 1988.4
• Sjoberg I E On the Quasigeoid to Geoid Seperation,Manuscripta Geodetica 1995.20.182-192.
• Elkins T.A.Vertical Gradient of Gravity on Axis for Hollow and Solid Cylinders, Geophysics, 1996.31(40). 816-820
• T.Vicenty,On the Use of GPS Vectors in Densification Adjustment,Surveying and Mapping,Vol 47,No.2
• 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,CanadaSurveying, 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

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