CN109520522B - Control point stability determination method based on three-dimensional baseline - Google Patents

Abstract

The invention discloses a method for determining the stability of control points based on a three-dimensional baseline, which is characterized by comprising the following steps: (1) two-phase observation of control points in a GNSS three-dimensional control network, and each original observation of the entire network is performed. The data is adjusted by least square method, and the results such as spatial rectangular coordinates are obtained (2), a single baseline is judged; (3), all baselines are judged: all baselines are judged in step (2), if there are 2 The above baselines at the same endpoint are unstable, and the common point is determined to be unstable. Since the adjustment datum of the GNSS three-dimensional control network has no effect on the horizontal distance difference and the poor results in the vertical direction, this method is more reliable to determine the unstable point, and it is suitable for the situation where the network shape of two or more observations is inconsistent. .

Description

A method for determining the stability of control points based on 3D baseline

technical field

The invention relates to engineering surveying and mapping, in particular to a method for determining the stability of control points based on a three-dimensional baseline.

Background technique

In order to meet the requirements of water conservancy projects (such as long tunnels) and other traffic projects, two (three) construction plane control networks, CP I and CP II plane control networks are generally used to meet the requirements of engineering measurement accuracy (the plane control network is generally GNSS control network) , and re-test the control network regularly or irregularly to find out the possible coordinate displacement.

In the prior art, the absolute coordinate difference and the relative error of the coordinate difference between adjacent points of two-phase observations are generally used to analyze unstable points, or the difference in side length and azimuth angle are directly used to analyze unstable points ( It is called method 2), but the prior art has the following defects: method 1 cannot find all unstable points, method 2 gives a limit according to the error in the priori, and cannot accurately estimate the error of each side length, resulting in stability Judgment is unreliable. Both method 1 and method 2 require that the network shape and observation outline of two or more observations are consistent, and this condition is difficult to meet.

Therefore, it is necessary to develop a simple and reliable method for determining the stability of control points.

SUMMARY OF THE INVENTION

The purpose of the present invention is to solve the above-mentioned deficiencies of the background technology, and to provide a method for determining the stability of a control point based on a three-dimensional baseline.

The technical scheme of the present invention is: a method for determining the stability of a control point based on a three-dimensional baseline, which is characterized in that it includes the following steps:

(1) Carry out two-phase space rectangular coordinate observation for the control points in the GNSS three-dimensional control network, and perform least squares adjustment for each original observation data of the entire network;

(2) Judging a single baseline

2.1) Use the adjusted data to calculate the difference ΔP of the horizontal distance and the difference ΔU of the vertical distance between the two observations of each baseline in the Cartesian coordinate system of the NEU station center space;

利用

计算得到各基线第一期、第二期在水平方向上距离的方差

竖直方向上距离的方差

根据两期观测相互独立有水平距离较差之方差

竖直距离较差之方差

计算得到两期观测水平距离较差之中误差σ_ΔP、竖直距离较差之中误差σ_ΔU；2.2) Calculate the variance covariance matrix of the first and second phases of each baseline in the NEU station center space rectangular coordinate system using the adjusted data

use

Calculate the variance of the distance between the first and second periods of each baseline in the horizontal direction

The variance of the distance in the vertical direction

According to the two periods of observation are independent of each other, there is a variance of the horizontal distance difference

The variance of the vertical distance difference

The error σ _ΔP in the poor horizontal distance and the error σ _ΔU in the poor vertical distance of the two observations are obtained by calculation;

2.3) If the judgment formula |ΔP|≤2σ _ΔP and |ΔU|≤2σ _ΔU are satisfied, the baseline is determined to be stable; if not, the baseline is determined to be unstable;

(3) Judging all baselines:

Perform the judgment of step (2) on all the baselines. If more than two baselines with the same endpoint are unstable, the common point is determined to be unstable.

Preferably, in the step 2.1), let the two ends of the baseline be the control points A and B, denoted as the baseline AB, and the spatial Cartesian coordinates after adjustment are respectively (X _A , Y _A , Z _A ) and (X _B , Y _B , Z _B ), convert the spatial Cartesian coordinates (X _A , Y _A , Z _A ) and (X _B , Y _B , Z _B ) of control points A and B into latitude and longitude coordinates (B _A , L _A , H _A ) and (B _B , L _B , H _B ), as well as the NEU station center space Cartesian coordinates (NA , E _A , U _A ) and (N _B , E _B , U _{B )} , so that the NEU station center space is at a right angle Under the coordinates:

The distance P _AB in the horizontal direction of the control points A and B satisfies:

Where: N _AB =N _B -N _A , E _AB =E _B -E _A ,

The distance U _AB in the vertical direction of the control points A and B satisfies:

U _{AB =} U _B - U _A ,

Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ), (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ after the first phase of baseline control points A and B ) is calculated to obtain P _AB ⁽¹⁾ , U _AB ⁽¹⁾ ;

After the second phase of baseline control points A, B coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ ) to calculate to get P _AB ⁽²⁾ and U _AB ⁽²⁾ , then we have

Poor baseline AB level distance ΔP _AB =P _AB ⁽²⁾ -P _AB ⁽¹⁾

Baseline AB vertical distance difference ΔU _AB =U _AB ⁽²⁾ -U _AB ⁽¹⁾ .

Further, in the step 2.2), under the space Cartesian coordinate system:

The variance covariance matrix of control points A and B is:

The coordinate difference between the control points A and B in the space rectangular coordinate system is as follows:

That is, dL=K ₀ L, where:

dL=(ΔX _AB ΔY _AB ΔZ _AB ) ^T

L=(X _A Y _A Z _A X _B Y _B Z _B ) ^T

Therefore, the variance covariance matrix between ΔX _AB , ΔY _AB and ΔZ _AB is as follows:

其中

可知：In the NEU station center space Cartesian coordinate system, the variance covariance matrix between N, E, and U: by

in

It is known that:

即

which is

Therefore, the variance-covariance matrix between N, E, U is as follows:

The latitude and longitude in the transformation matrix R is the average of the two points, that is

B=(B _A +B _B )/2, L=(L _A +L _B )/2;

Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ^{(1) )} of control points A and B through the baseline ) calculation to obtain the variance covariance matrix of the first phase of baseline AB in the NEU station center space rectangular coordinate system

Coordinates (X _A ⁽²⁾ , Y _A ⁽²⁾ , Z _A ⁽²⁾ ) and (X _B ⁽² , Y _B ⁽²⁾ , Z _B ⁽²⁾ ) of control points A and B through the baseline second period Calculate to get the variance covariance matrix of the second phase of baseline AB in the NEU station center space rectangular coordinate system

Further, in the described step 2.2), the distance P _AB on the horizontal direction of the control points A and B under the NEU station center space Cartesian coordinates satisfies:

Among them: N _AB =N _B -N _A , E _AB =E _B -E _A

Totally differentiate the functional expression to get:

即

which is

which is

Applying the covariance propagation law, we get:

从D_NEU中得到；in,

from D _NEU ;

K ₃ is obtained according to the coordinates of the control points A and B under the Cartesian coordinates of the NEU station center space,

For the vertical direction, we have

where cov(U _AB ,U _AB ) is obtained from D _NEU

基线AB第一期在竖直方向上距离的方差

Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ^{(1) )} of control points A and B through the baseline ) calculation to get the variance of the distance in the horizontal direction of the first period of baseline AB

The variance of the distance in the vertical direction in the first period of baseline AB

基线AB第二期在竖直方向上距离的方差

After the second phase of baseline control points A, B coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ ) calculation to obtain the variance of the distance in the horizontal direction of the second period of baseline AB

The variance of the distance in the vertical direction of the second period of baseline AB

Since the first and second period observations are independent of each other, there are:

代入式(7)、

代入式(8)中，得到基线AB水平距离较差之方差

基线AB竖直距离较差之方差

开方后取正得到基线AB水平距离较差之中误差

基线AB竖直距离较差之中误差

will be obtained

Substitute into formula (7),

Substitute into formula (8) to get the variance of the baseline AB horizontal distance difference

The variance of the vertical distance difference of the baseline AB

After square root, take the positive value to get the error in the difference between the baseline AB horizontal distance and the difference

Baseline AB vertical distance is poor middle error

Further, the conversion method from the spatial Cartesian coordinates of the control points to the geodetic coordinates is:

Wherein: N is the radius of the unitary circle; a is the long radius of the reference ellipsoid; b is the short radius of the reference ellipsoid; e is the first eccentricity of the reference ellipsoid; e' is the second eccentricity of the reference ellipsoid ;and

Further, the conversion method from the spatial rectangular coordinates of the control point to the spatial rectangular coordinates of the station center is:

Let point j be the coordinates in the Cartesian coordinate system of the station center with point o as the center as (N _oj , E _oj , U _oj ), point o takes any control point of the control network, and only one point is selected in one control network As point o, the coordinates of point j and point o in the space rectangular coordinate system are (X _j , Y _j , Z _j ), (X _o , Y _o , Z _o ), respectively, and point j and point o are in the geodetic coordinate system The coordinates below are (B _j , L _j , H _j ), (B _o , L _o , H _o ), then

Preferably, in step (1), a control point in the GNSS three-dimensional control network is selected as a fixed point for least squares adjustment processing.

The beneficial effects of the present invention are:

1. Since the adjustment datum of the GNSS three-dimensional control network has no effect on the horizontal distance difference and the poor results in the vertical direction, this method is more reliable for determining unstable points.

2. According to the variance covariance matrix between any two control points, the variance of the corresponding horizontal distance difference and the variance of the corresponding vertical distance difference of the two observations are obtained. The theory of this method is strict.

3. It is suitable for the situation that the network shape and observation outline of two or more observations are consistent or inconsistent, such as the destruction of individual control points of the construction control network, and the analysis of unstable points is not affected.

Description of drawings

Figure 1 is a schematic diagram of the relationship between the space rectangular coordinates and the space rectangular coordinates of the station center

Figure 2 is a schematic diagram of judging common points

3 is a schematic diagram of a control point network in an embodiment

Detailed ways

The following specific examples will further illustrate the present invention in detail.

As shown in Figures 1-2, the method for determining the stability of a control point based on a three-dimensional baseline provided by the present invention includes the following steps.

(1) Carry out two-phase space rectangular coordinate observation of the control points in the GNSS three-dimensional control network, and perform least squares adjustment for each original observation data of the entire network,

(2) Judging a single baseline

2.1) Use the adjusted data to calculate the difference ΔP of the horizontal distance and the difference ΔU of the vertical distance between the two observations of each baseline in the Cartesian coordinate system of the NEU station center space;

Take the baseline AB formed by two control points A and B as an example:

The two ends of the base are the control points A and B, the spatial Cartesian coordinates after adjustment are sum (X _B , Y _B , Z _B ), respectively, and the spatial Cartesian coordinates of the control points A and B (X _A , Y _A , Z _{A )} ) and (X _B , Y _B , Z _B ) are converted into longitude and latitude coordinates (B _A , L _A , H _A ) and (B _B , L _B , H _B ), and the NEU station center space Cartesian coordinates (N _A , E _A ,U _A ) and (N _B ,E _B ,U _B ),

The conversion method from the spatial Cartesian coordinates of the control points to the geodetic coordinates is as follows:

Wherein: N is the radius of the unitary circle; a is the long radius of the reference ellipsoid; b is the short radius of the reference ellipsoid; e is the first eccentricity of the reference ellipsoid; e' is the second eccentricity of the reference ellipsoid ;and

For the WGS4 reference ellipsoid, a=6378137m, b=6356752.314m.

The conversion method from the spatial Cartesian coordinates of the control points to the geodetic coordinates is:

Let point j be the coordinates in the Cartesian coordinate system of the station center with point o as the center as (N _oj , E _oj , U _oj ), point o takes any control point of the control network, and only one point is selected in one control network As point o, the coordinates of point j and point o in the space rectangular coordinate system are (X _j , Y _j , Z _j ), (X _o , Y _o , Z _o ), respectively, and point j and point o are in the geodetic coordinate system The coordinates below are (B _j , L _j , H _j ), (B _o , L _o , H _o ), then

Converting the spatial rectangular coordinates of the control point into longitude and latitude coordinates and the spatial rectangular coordinates of the NEU station center belongs to the conventional prior art.

Under the Cartesian coordinates of the NEU station center space:

The distance P _AB in the horizontal direction of the control points A and B satisfies:

Where: N _AB =N _B -N _A , E _AB =E _B -E _A ,

The distance U _AB in the vertical direction of the control points A and B satisfies:

U _{AB =} U _B - U _A ,

Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ), (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ after the first phase of baseline control points A and B ) is calculated to obtain P _AB ⁽¹⁾ , U _AB ⁽¹⁾ ;

After the second phase of baseline control points A, B coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ ) to calculate to get P _AB ⁽²⁾ and U _AB ⁽²⁾ , then we have

ΔP _AB =P _AB ⁽²⁾ -P _AB ⁽¹⁾

ΔU _AB =U _AB ⁽²⁾ −U _AB ⁽¹⁾ .

2.2)

a. Calculate the variance covariance matrix between ΔX _AB , ΔY _AB and ΔZ _AB in the first and second period spatial Cartesian coordinates

The spatial Cartesian coordinates of control points A and B are (X _A , Y _A , Z _A ) and (X _B , Y _B , Z _B ), respectively, and the variance covariance matrix D _XYZ between these two control points is:

The coordinate difference between these two control points is as follows:

That is, dL=K ₀ L, where:

dL=(ΔX _AB ΔY _AB ΔZ _AB ) ^T

L=(X _A Y _A Z _A X _B Y _B Z _B ) ^T

Therefore, the variance covariance matrix between ΔX _AB , ΔY _AB and ΔZ _AB is as follows:

Set the coordinates of the first phase control points A and B (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ⁽¹⁾ ) Substituting into the above equations (1)-(3), we get

b. Calculate the variance covariance matrix in the first phase and the second phase NEU station center space rectangular coordinate system

In the Cartesian coordinate system of the NEU station center space

其中

可知：Depend on

in

It is known that:

即

which is

Therefore, the variance-covariance matrix between N, E, U is as follows:

The latitude and longitude in the transformation matrix R is the average of the two points, that is

B=(B _A +B _B )/2, L=(L _A +L _B )/2;

得到

Through the first phase of the baseline control points A, B geodetic coordinates and

get

得到

Through the second phase of the baseline control points A, B geodetic coordinates and

get

c. Calculate the error σ _ΔP in the difference in the horizontal distance and the error σ _ΔU in the difference in the vertical distance between the two observations:

The distance P _AB of the control points A and B in the horizontal direction in the Cartesian coordinates of the NEU station center satisfies:

Among them: N _AB =N _B -N _A , E _AB =E _B -E _A

Totally differentiate the functional expression to get:

即

which is

which is

Applying the covariance propagation law, we get:

从D_NEU中得到；in,

from D _NEU ;

K ₃ is obtained according to the coordinates of the control points A and B under the Cartesian coordinates of the NEU station center space,

For the vertical direction, we have

where cov(U _AB ,U _AB ) is obtained from D _NEU

得到

The spatial Cartesian coordinates of the NEU station center through the control points A and B of the first phase of the baseline and

get

得到

The spatial Cartesian coordinates of the NEU station center through the control points A and B of the first phase of the baseline and

get

Since the first and second period observations are independent of each other, there are:

代入式(7)、

代入式(8)中，得到

开方后取正得到

will be obtained

Substitute into formula (7),

Substitute into formula (8), we get

After opening the square, take it right and get it

且竖直方向上的较差有

则说明ΔP_AB和ΔU_AB主要由观测误差造成，基线AB稳定，即控制点A、B稳定；否则，认为这两个控制点中至少有一点不稳定，则该基线不稳定。2.3) If the horizontal distance difference between the two observations is

And the difference in the vertical direction is

It means that ΔP _AB and ΔU _AB are mainly caused by observation errors, and the baseline AB is stable, that is, the control points A and B are stable; otherwise, it is considered that at least one of the two control points is unstable, and the baseline is unstable.

(3) Check the difference in the horizontal distance and the difference in the vertical direction of the two observations of all the baselines. If there are more than 2 baselines (AC and BC) with the same end point that fail to pass the inspection, determine their common point ( C) Instability, see Figure 2.

The following is an application example of this scheme

This example calculates a control network composed of four synchronously observed control points. The selected data are four control points in a water diversion project and a control network composed of 6 GNSS baselines. The control point network is shown in the figure. 3 shown.

Table 1 WGS84 space Cartesian coordinates (single-point positioning) obtained after the first phase of GNSS survey

Table 2 WGS84 space Cartesian coordinates (single-point positioning) obtained after the second phase of GNSS survey

The original data of the two-phase observation are shown in Table 1-2 above, and the baseline calculation is performed on the two-phase observation data to ensure that the baseline meets the accuracy requirements. According to the existing method 1 in the background technology, the control point GPS2802 is used as a fixed point, and three-dimensional constraint adjustment (least square method adjustment processing) is performed, and the coordinates of the control point after adjustment are as follows:

Table 3 WGS84 space Cartesian coordinates of the point with the same name after adjustment

From the space rectangular coordinates in Table 3, the geodetic coordinates and station center coordinates of the control point can be calculated as shown in Table 4-5 below:

Table 4 Geodetic coordinates of control points

Table 5 Control point station center coordinates

In order to further explore the stability of other control points, for the baseline between any two points in the entire control network, the variance covariance matrix is shown in Table 6 below:

Table 6 Variance covariance matrix D _XYZ of all baselines in the control network

Calculate the horizontal distance and height difference in the vertical direction obtained from the two periods of observation, as well as the poor median error of the horizontal distance and the poor median error of the vertical height difference.

Table 7 Baseline horizontal distance poor judgment

Table 8 Poor judgment of baseline height difference

It can be seen from Table 7-8 that the baseline changes from point GPS2881 to point GPS2802, GPS2605, and GPS2629 are not significant, which is a stable baseline. Therefore, this point GPS2881 is determined to be a stable point. Similarly, points GPS2605 and GPS2629 can be determined to be stable points. Finally, it is concluded that all control points in the control network are stable points.

Claims (7)

1. a control point stability determination method based on three-dimensional baseline, is characterized in that, comprises the following steps: (1) Carry out two-phase space rectangular coordinate observation for the control points in the GNSS three-dimensional control network, and perform least squares adjustment for the original observation data of each phase of the entire network; (2) Judging a single baseline 2.1) Use the adjusted data to calculate the difference ΔP of the horizontal distance and the difference ΔU of the vertical distance between the two observations of each baseline in the Cartesian coordinate system of the NEU station center space; 2.2) Calculate the variance covariance matrix of the first and second phases of each baseline in the NEU station center space rectangular coordinate system using the adjusted data

use

Calculate the variance of the distance between the first and second periods of each baseline in the horizontal direction

The variance of the distance in the vertical direction

According to the two periods of observation are independent of each other, there is a variance of the horizontal distance difference

The variance of the vertical distance difference

The error σ _ΔP in the poor horizontal distance and the error σ _ΔU in the poor vertical distance of the two observations are obtained by calculation; 2.3) If the judgment formula |ΔP|≤2σ _ΔP and |ΔU|≤2σ _ΔU are satisfied, the baseline is determined to be stable; if not, the baseline is determined to be unstable; (3) Judging all baselines Perform the judgment of step (2) on all the baselines. If more than two baselines with the same endpoint are unstable, the common point is determined to be unstable. 2. the control point stability determination method based on three-dimensional baseline as claimed in claim 1, is characterized in that, in described step 2.1), make baseline both ends be control points A, B, be denoted as baseline AB, after adjustment The spatial Cartesian coordinates of the control point A and B are (X _A , Y _A , Z _A ) and (X _B , Y _B , Z _B ), respectively, and the spatial Cartesian coordinates (X _A , Y _A , Z _A ) and ( X _B , Y _B , Z _B ) are converted into latitude and longitude coordinates (B _A , L _A , H _A ) and (B _B , L _B , H _B ), and the NEU station center space Cartesian coordinates (N _A , E _A , U _A ) and (N _B , E _B , U _B ), thus under the NEU station center space Cartesian coordinate system: The distance P _AB in the horizontal direction of the control points A and B satisfies:

Where: N _AB =N _B -N _A , E _AB =E _B -E _A , The distance U _AB in the vertical direction of the control points A and B satisfies: U _AB =U _B -U _A , Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ), (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ^{(1) )} of control points A and B through the first phase of the baseline ) is calculated to obtain P _AB ⁽¹⁾ , U _AB ⁽¹⁾ ; Coordinates (X _A ⁽²⁾ , Y _A ⁽²⁾ , Z _A ⁽²⁾ ) and (X _B ⁽²⁾ , Y _B ⁽²⁾ , Z _B ^{(2) )} of control points A and B through the baseline second period ) to calculate to get P _AB ⁽²⁾ and U _AB ⁽²⁾ , then we have Poor baseline AB level distance ΔP _AB =P _AB ⁽²⁾ -P _AB ⁽¹⁾ Baseline AB vertical distance difference ΔU _AB =U _AB ⁽²⁾ -U _AB ⁽¹⁾ . 3. the control point stability determination method based on three-dimensional baseline as claimed in claim 2, is characterized in that, in described step 2.2), under the space Cartesian coordinate system: The variance covariance matrix of control points A and B is:

The coordinate difference between the control points A and B in the space rectangular coordinate system is as follows:

That is, dL'=K ₀ L', where: dL'=(ΔX _AB ΔY _AB ΔZ _AB ) ^T

L′=(X _A Y _A Z _A X _B Y _B Z _B ) ^T Therefore, the variance covariance matrix between ΔX _AB , ΔY _AB and ΔZ _AB is as follows:

In the NEU station center space Cartesian coordinate system, the variance covariance matrix between N, E, and U: by

in

It is known that:

which is

Therefore, the variance-covariance matrix between N, E, U is as follows:

The latitude and longitude in the transformation matrix R is the average of the two points, that is B=(B _A +B _B )/2, L=(L _A +L _B )/2; Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ^{(1) )} of control points A and B through the baseline ) calculation to obtain the variance covariance matrix of the first phase of baseline AB in the NEU station center space rectangular coordinate system

Coordinates (X _A ⁽²⁾ , Y _A ⁽²⁾ , Z _A ⁽²⁾ ) and (X _B ⁽²⁾ , Y _B ⁽²⁾ , Z _B ^{(2) )} of control points A and B through the baseline second period ) calculation to obtain the variance covariance matrix of the second phase of baseline AB in the NEU station center space rectangular coordinate system

4. the control point stability determination method based on three-dimensional baseline as claimed in claim 3, is characterized in that, in described step 2.2), the distance P on control point A, B horizontal direction under NEU station center space Cartesian coordinate system _AB satisfies:

Among them: N _AB =N _B -N _A , E _AB =E _B -E _A Totally differentiate the functional expression to get:

which is

which is

Applying the covariance propagation law, we get:

in,

from D _NEU , K ₃ is obtained according to the coordinates of the control points A and B in the Cartesian coordinate system of the NEU station center space, For the vertical direction, we have

where cov(U _AB , U _AB ) is obtained from D _NEU , Coordinates (X _A ⁽¹⁾ , Y _A ⁽¹⁾ , Z _A ⁽¹⁾ ) and (X _B ⁽¹⁾ , Y _B ⁽¹⁾ , Z _B ^{(1) )} of control points A and B through the baseline ) calculation to get the variance of the distance in the horizontal direction of the first period of baseline AB

The variance of the distance in the vertical direction in the first period of baseline AB

Coordinates (X _A ⁽²⁾ , Y _A ⁽²⁾ , Z _A ⁽²⁾ ) and (X _B ⁽²⁾ , Y _B ⁽²⁾ , Z _B ^{(2) )} of control points A and B through the baseline second period ) calculation to obtain the variance of the distance in the horizontal direction of the second period of baseline AB

The variance of the distance in the vertical direction of the second period of baseline AB

Since the first and second period observations are independent of each other, there are:

will be obtained

Substitute into formula (7),

Substitute into formula (8) to get the variance of the baseline AB horizontal distance difference

The variance of the vertical distance difference of the baseline AB

After square root, take positive to get the error of the horizontal distance difference of baseline AB

The vertical distance of the baseline AB is poor and the error

5. the control point stability determination method based on three-dimensional baseline as claimed in claim 2 is characterized in that, the conversion method of the space rectangular coordinate of control point to geodetic coordinate is:

Wherein: N' is the radius of the unitary circle; a is the long radius of the reference ellipsoid; b is the short radius of the reference ellipsoid; e is the first eccentricity of the reference ellipsoid; e' is the second eccentricity of the reference ellipsoid Rate; and

6. the control point stability determination method based on three-dimensional baseline as claimed in claim 2, is characterized in that, the conversion method of the space rectangular coordinate of control point to station center space rectangular coordinate is: Let point j be the coordinates in the Cartesian coordinate system of the station center centered on point o as (N _oj , E _oj , U _oj ), point o takes any control point of the control network, and only one point is selected in one control network As point o, the coordinates of point j and point o in the space rectangular coordinate system are (X _j , Y _j , Z _j ), (X _o , Y _o , Z _o ), respectively, and point j and point o are in the geodetic coordinate system The coordinates below are (B _j , L _j , H _j ), (B _o , L _o , H _o ), then

7. The method for judging the stability of a control point based on a three-dimensional baseline as claimed in claim 1, wherein in step (1), a control point in the GNSS three-dimensional control network is selected as a fixed point for least squares adjustment.

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