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

Abstract
It is not ignorable that there is base datum with not enough precision to support the network computation and adjustment of a constrained GPS network. With the help of modern geodesy techniques GPS and adjustment theory and computational techniques, we put forward a new method to carry out the adjustment of GPS network with not enough precision to support the computation Based on the theory and model, we have computed a small GPS network in southwest of China to test the theory and model Based upon the comparison, suggestions and recommends have been applied in this paper

Introduction
According to the case really occurred that most cities horizontal controlling network has been established with traditional way or advanced GPS techniques originally in China, and in order to conform to the standard issued by the Chinese Construction Department in 1997,’The Technical Regulations for the Urban Surveying with GPS’. It is reasonable to adapt the original datum to be the reference frame when we carry out the surveying engineering and designing surveying network with GPS techniques, and the original datum and existed large scale maps must be considered. How to make full use of the reasonable parts of the original reference system in a local city and develop the high accuracy information of data observed with GPS economically is the problem this paper deals with. And with time going, this problem will be more and more important in China.

Adjustment Model
The observations of GPS are composed of space vectors that are surveyed with different time intervals. They can be expressed as ( DX, DY,DZ), or differential coordinates of two arbitrary points on the ground. However, we may meet the case that there is base datum with not enough precision to support the network computation and adjustment of a constrained GPS network.

The ways to get the solution of base datum with not enough precision can be approached by following,

Strict adjustment model for the base datum with not enough precision
Adjustment strictly according to the principle of least square.i.e.adjustment base datum with not enough precision combined to the GPS observations. It is not a convenient and economic method to be use in real work;
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. It will be shown this is a better and economic way in constrained adjustment of GPS network.
- Strict adjustment model for the base datum with not enough precision
In following discussion we take it as acknowledge that base datum with not enough precision is a kind of observation with a prior-weight, and it is adjusted with general GPS vector observation. According to the principle of least square, we can write the indirect adjustment model as following,

V₁=A dY + CdX-l
V_s=dX^{^}+L_s ........ (1)
and
l=AY⁰+CX⁰+l
L_s=0 ....................(2)

In above formula, dyˆ are the parameter vectors of GPS network, dXˆ are the base datum parameter vector. And

Formula (1) will be,
V=BZ^+l ................(4)

We can then compose the normal equation based upon (4) as follows, B^TPBZ^+B^TPl=0 ...............(5)
Suppose,
N=B^TPB,..........W=B^TPl...............
We have,

Then we have,
Z^=-N^-1W ...........................(6)
and,
Q_zz=N^-1
Suppose,

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.

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