CN101210814A - Line engineering GPS control network height fitting method - Google Patents

Abstract

The invention relates to a method for GPS control measurement of a line project, in particular to an elevation fitting of the GPS control network for the line projects by using Lagrange interpolating function. The method comprises the following steps of: (1) carrying out adjustment calculation of a GPS control network to obtain the plane coordinates of all GPS control points and the geodetic height H; (2) selecting points for constructing the interpolation function; (3) projecting the on-line GPS points in order on a vertical plane to convert three-dimensional coordinates (x, y, h) of line points to two-dimensional coordinates (s, Zeta); (4) constructing a Lagrange interpolating function according to the point sequence (S<i>, Zeta<i>)i=0<n>; (5) calculating the elevation abnormal value (Zeta) according to the function; (6) calculating the elevation of the GPS control points according to the formula: h=H-Zeta; and (7) testing by 'addition-substraction-addition' method and giving accurate evaluation. The method can increase the accuracy of GPS control network elevation fitting of the line, and can replace the elevation level measurement partially to reduce the level measurement and increase the work efficiency in the line project.

Description

Line engineering GPS control network height fitting method

Technical field

The present invention relates to the method that line projects such as a kind of railway, highway, water conservancy, electric power conveying, oil and natural gas pipeline carry out the GPS control survey, particularly a kind of Lagrange's interpolation function that utilizes carries out line engineering GPS control network height fitting method.

Background technology

In recent years, country had strengthened the dynamics of investment to capital constructions such as railway, highway, water conservancy, electric power conveying, oil and natural gas pipelines.These engineerings are based on line project, have that wire, region span are big, the location change complicated, accuracy requirement is high, wait characteristics it is pressed for time, for vast mapping operations person has proposed new requirement, the GPS technology has become the main means of carrying out these engineering surveys.In the GPS control survey, because the physical mechanism of GPS measurement and positioning technology, the precision of its planimetric position can reach very high precision, and its vertical accuracy is lower than its plane precision.Main cause is: the geodetic height of the defined WGS-84 coordinate system of GPS only has geometric meaning and does not have the physics intension, and the existing geometric meaning of the defined normal height of the elevation system in the reality, the such physical significance of earth interior mass density skewness is arranged again, thereby produced height anomaly.Especially in the line projects such as railway, highway, water conservancy, electric power conveying, oil and natural gas pipeline, has wire, the region span is big, the location changes complicatedly, and the height anomaly value often changes from several meters and changes to tens meters, makes these engineerings can't directly use the GPS geodetic height.In recent years, vast mapping operations person tries to explore to adopt fitting method to replace a part of measurement of the level, to bring into play the effect of GPS fully, reduces labour intensity.Therefore, GPS control net height journey approximating method also correspondingly becomes one of heat subject that mapping educational circles tries to explore.

Summary of the invention

The purpose of this invention is to provide a kind of line engineering GPS control network height fitting method,, replace the measurement of the level of part elevation, thereby reduce the measurement of the level of line project, increase work efficiency so that improve circuit GPS control net height journey precision.

The object of the present invention is achieved like this, and line engineering GPS control network height fitting method the steps include:

(1) calculates by the GPS adjustment of control network, obtain the planimetric coordinates and the geodetic height H at whole GPS reference mark;

(2) choose Structure interpolating function point, select in the GPS reference mark at circuit two ends and the circuit GPS reference mark about every approximately 30km at interval, obtain its elevation h by measurement of the level, with these as Structure interpolating function point

(3) GPS on the circuit is pressed sequential projection on a vertical plane, make field location by three-dimensional (x, y, h) be converted into two dimension (s, ζ);

Wherein $s_i=\sqrt{\mathrm{\&Delta;}{x_i}^2+\mathrm{\&Delta;}{y_i}^2},$ ζ _i＝H _i-h _i，i＝0，...n；

In the formula, symbol is represented respectively:

s _iMileage for each point;

Δx _i＝x _i-x _i-1，Δy _i＝y _i-y _i-1；

ζ _iHeight anomaly value for each point;

H _iGeodetic height for each point;

h _iThe elevation of each point.

(4) according to point sequence { s _i, ζ _i} _I=0 ⁿStructure Lagrange's interpolation function:

$L_n\left(s\right)=\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&zeta;}}_k{l}_k^{\left(n\right)}\left(s\right)={\mathrm{\&zeta;}}_0{l}_0^{\left(n\right)}\left(s\right)+{\mathrm{\&zeta;}}_1{l}_1^{\left(n\right)}\left(s\right)+...+{\mathrm{\&zeta;}}_n{l}_n^{\left(n\right)}\left(s\right)$

Lagrange's interpolation basis function wherein:

$l_k^{\left(n\right)}\left(s\right)=\frac{(s-s_0)...(s-s_{k-1})(s-s_{k+1})...(s-s_n)}{(s_k-s_0)...(s_k-s_{k-1})(s_k-s_{k+1})...(s_k-s_n)}$

k＝0，1，2，…，n

(5) ask for the height anomaly value ζ of other point according to the function of structure;

(6) obtain the elevation at these GPS reference mark according to formula h=H-ζ;

(7) vertical accuracy that the GPS that takes the check of " adding-remove-add " method to obtain through interpolation fitting is ordered provides precision evaluation.

" adding-remove-add " method is taked in described check, and step is:

1), removes one by the method for above-mentioned structure interpolating function Structure interpolating function point, and this point obtained to ask for its spot elevation h again as unknown spot elevation _Inspection

2) the spot elevation difference of twice acquisition of comparison;

3) when difference during, this point is marked, adopt measurement of the level to encrypt translocation GPS reference mark greater than K (the K value is determined according to the accuracy class of engineering), as Structure interpolating function point

4) will increase newly Structure interpolating function pointAdd, re-construct interpolating function;

5) return the 1st) step;

6) when difference during less than K, whether incorporation of markings information detect all The structure interpolating function The pointThe interpolating function of structure all detects and finishes, and does not return the 1st) step;

7) Structure interpolating function pointThe interpolating function of structure all detects and finishes, and finishes.

Principle of the present invention is: the geodetic height that GPS surveyed is the height along normal direction to the reference ellipsoid face, is reference field with the mathematics curved surface promptly.It is a geometric sense, has clear and definite geometric meaning, does not have meaning physically.Road conduct along the line Structure interpolating function point, changing greatly at height anomaly, vertical accuracy requires when high, the density of laying is big, otherwise on the contrary.

Normal height is the height along vertical line to quasigeoid, is that the gravity equipotential surface of fluctuating is arranged is reference field with irregular, promptly object when this face moves, gravity do not do work (water can not flow on this face).Quasigeoid is an important physical reference surface describing the figure of the earth, also is the face of starting at of absolute elevation system, has strict physical significance, use so in engineering construction, have widely, and also be the general elevation system of China.

Gap between quasigeoid and the reference ellipsoid face is called height anomaly.There is relational expression as follows

ζ＝H-h

ζ is a height anomaly in the formula, and the expression quasigeoid is to the distance of reference ellipsoid face.H is a geodetic height, and h is normal high.

Because ζ is not a constant, thereby has caused in the actual engineering for the direct applied difficulty of GPS geodetic height.Carrying out after GPS measures, can obtain the geodetic height H of each point by the GPS three-dimensional adjustment, is the leveling conjuction point if in the net part GPS point is arranged, and the normal high h of promptly known these points then can try to achieve the height anomaly ζ of these points.Within the specific limits, height anomaly ζ is not a constant, but can think that variation is mild in this scope, and more available mathematical functions come match, try to achieve the function that can reflect that height anomaly changes in the GPS network control system scope, try to achieve the height anomaly of other each point in the net then by interpolation.

Description of drawings

The present invention is further described below in conjunction with the embodiment accompanying drawing.

Fig. 1 is a Lagrange's interpolation function mathematical model;

Fig. 2 is that the height anomaly fitted figure is seen.

Embodiment

As shown in Figure 1, establish y=f (x) and be the continuous function on interval [a, b], note is made f ∈ C[a, b].Known f goes up n+1 inequality point at [a, b]

a≤x ₀，x ₁，…，x _a-1，x _a≤b，x _i≠x _j(i≠j)

The value at place

L _n(x _i)＝f(x _i)，i＝0，1，2，…，n

Then have and be no more than n time lagrange polynomial

$L_n\left(x\right)=\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{y}_k{l}_k^{\left(n\right)}\left(x\right)=y_0{l}_0^{\left(n\right)}\left(x\right)+y_1{l}_1^{\left(n\right)}\left(x\right)+...+y_n{l}_n^{\left(n\right)}\left(x\right)$

Set up, wherein the Lagrange's interpolation basis function

$l_k^{\left(n\right)}\left(x\right)=\frac{(x-x_0)...(x-x_{k-1})(x-x_{k+1})...(x-x_n)}{(x_k-x_0)...(x_k-x_{k-1})(x_k-x_{k+1})...(x_k-x_n)}$

k＝0，1，2，…，n

By Lagrange's interpolation formula as can be known, L _n(x) be to be no more than n time polynomial expression.By its character as can be known, this function has uniqueness, and numerical evaluation has stability, is continuous function in interval of definition.

By dot matrix { x _i, y _i} _I=0 ⁿCarry out interpolation, obtain interpolating function L _n(x), the value of independent variable x is composed successively to L _n(x), thereby obtain the functional value of other point, realize the purpose of interpolation fitting.

The step of interpolation fitting

(1) engineering distributing GPS reference mark, road along the line is observed, and obtains the planimetric coordinates and the geodetic height H of these points by compensating computation;

(2) leveling conjuction is carried out at the GPS reference mark or on leveling point the distributing GPS reference mark, obtain the elevation h of these points, as Structure interpolating function pointThese points should evenly distribute, and should suitably encrypt in the bigger location of topographic relief.

(3) with whole circuit in order projection on a vertical plane, make field location by three-dimensional (x, y, h) be converted into two dimension (s, ζ).

Wherein $s_i=\sqrt{\mathrm{\&Delta;}{x_i}^2+\mathrm{\&Delta;}{y_i}^2},$ ζi＝H _i-h _i，i＝0，...n。

(4) foundation Structure interpolating function pointSequence { s _i, ζ _i} _I=0 ⁿThe structure interpolating function:

$L_n\left(s\right)=\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&zeta;}}_k{l}_k^{\left(n\right)}\left(s\right)={\mathrm{\&zeta;}}_0{l}_0^{\left(n\right)}\left(s\right)+{\mathrm{\&zeta;}}_1{l}_1^{\left(n\right)}\left(s\right)+...+{\mathrm{\&zeta;}}_n{l}_n^{\left(n\right)}\left(s\right)$

Lagrange's interpolation basis function wherein:

$l_k^{\left(n\right)}\left(s\right)=\frac{(s-s_0)...(s-s_{k-1})(s-s_{k+1})...(s-s_n)}{(s_k-s_0)...(s_k-s_{k-1})(s_k-s_{k+1})...(s_k-s_n)}$

k＝0，1，2，…，n

(5) ask for the height anomaly value ζ at other GPS reference mark according to the function of structure.

(6) obtain the elevation at GPS reference mark according to formula h=H-ζ.

The vertical accuracy at the GPS reference mark that (7) check process interpolation fitting obtains, and make precision evaluation.

The inspection of fitting effect:

For fitting effect is checked, when carrying out The Fitting Calculation, can take " adding---removing---adds " method:

(1) when the structure interpolating function, at first selects all Structure interpolating function pointCarry out constructed fuction, and try to achieve the elevation h of other point _Inspection

(2) remove one of them Structure interpolating function pointAnd with this point as unknown spot elevation.Utilize remaining Structure interpolating function pointConstruct interpolating function, ask for the elevation h of all the other points _Inspection

(3) result of twice acquisition of comparison: the elevation difference of twice interpolation of all unknown point; Emphasis is checked as unknown point Structure interpolating function pointDifference with known elevation.

(4) remove the another one interpolation point, repeat (2), (3) step, obtain other one group of data.

(5) interpolation result is made evaluation.

Below by concrete experimental data the present invention is further elaborated.

The long 205km of certain Oil Transportation Pipeline Engineering, the distributing GPS reference mark is 25 altogether, by obtaining each point coordinate and geodetic height H after the compensating computation of GPS software; Obtained the elevation h of 14 points by leveling conjuction or alternate manner; Utilize ζ=H-h to obtain the height anomaly value of 14 known spot elevations.See Table one, table two.

Known spot elevation mileage, height anomaly value (unit: m) table one

Sequence number

Known spot elevation is called the roll

Mileage S

Height anomaly value ζ

1	YZ01	0	32.986
				2	YZ02	15971.3	33.657
3	YZ03	30539.6	33.649
				4	YZ04	39563.4	33.640
5	YZ05	49049.2	33.832
				6	YZ06	76964.2	34.890
7	YZ07	87475.4	35.250
				8	YZ08	116667.0	36.325
9	YZ09	150148.6	37.620
				10	YZ10	160168.5	37.833
11	YZ11	184033.5	38.009
				12	YZ12	192910.0	38.123
13	YZ13	197591.5	38.230
				14	YZ14	204482.7	38.582

Wait to ask spot elevation mileage, geodetic height (unit: m) table two

Sequence number	Wait to ask spot elevation to call the roll	Mileage S	Geodetic height H
				1	CM01	9073.5	1273.497
2	CM02	21883.5	1300.606
				3	CM03	61876.9	1343.972
4	CM04	68233.5	1338.123
				5	CM06	86033.3	1380.983
6	CM07	100411.0	1343.900
				7	CM08	108667.1	1329.380
8	CM10	131186.0	1318.847
				9	CM11	140754.4	1283.351
10	CM12	172024.3	1248.928
				11	CM13	189568.4	1284.048

With point sequence in the table one (S, ζ) conduct Structure interpolating function point, structure interpolating function L (x).

With the S of each point in the table two, bring function L (x) respectively into, thereby obtain the height anomaly value of all the other each points, and try to achieve the elevation of each point, see Table three.

Height anomaly reckoner (unit: m) table three

Sequence number	Wait to ask a roll-call	Mileage S	Height anomaly value ζ	Geodetic height H	Elevation h
						1	CM01	9073.50	33.3544	1273.4970	1240.1426
2	CM02	21883.50	33.7189	1300.6060	1266.8871
						3	CM03	61876.90	34.3083	1343.9720	1309.6637
4	CM04	68233.50	34.5630	1338.1230	1303.5600
						5	CM06	86033.30	35.2017	1380.9830	1345.7813
6	CM07	100411.00	35.6946	1343.9000	1308.2054
						7	CM08	108667.10	36.0036	1329.3800	1293.3764
8	CM10	131186.00	36.9351	1318.8470	1281.9119
						9	CM11	140754.40	37.3141	1283.3510	1246.0369
10	CM12	172024.30	37.9401	1248.9280	1210.9879
						11	CM13	189568.40	38.0702	1284.0480	1245.9778

Claims (2)

1. line engineering GPS control network height fitting method the steps include:

(1) calculates by the GPS adjustment of control network, obtain the planimetric coordinates and the geodetic height H at whole GPS reference mark;

(2) choose Structure interpolating function point, select in the GPS reference mark at circuit two ends and the circuit GPS reference mark about every approximately 30km at interval, obtain its elevation h by measurement of the level, with these as Structure interpolating function point

(3) GPS on the circuit is pressed sequential projection on a vertical plane, make field location by three-dimensional (x, y, h) be converted into two dimension (s, ζ);

Wherein $s_i=\sqrt{\mathrm{\&Delta;}{x_i}^2+\mathrm{\&Delta;}{y_i}^2},$ ζ _i＝H _i-h _i，i＝0，...n；

In the formula, symbol is represented respectively:

s _iMileage for each point;

Δx _i＝x _i-x _i-1，Δy _i＝y _i-y _i-1；

ζ _iHeight anomaly value for each point;

H _iGeodetic height for each point;

h _iElevation for each point;

(4) according to point sequence { s _i, ζ _i} _I=0 ⁿStructure Lagrange's interpolation function:

$L_n\left(s\right)=\underset{k=0}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&zeta;}}_k{l}_k^{\left(n\right)}\left(s\right)={\mathrm{\&zeta;}}_0{l}_0^{\left(n\right)}\left(s\right)+{\mathrm{\&zeta;}}_1{l}_1^{\left(n\right)}\left(s\right)+...+{\mathrm{\&zeta;}}_n{l}_n^{\left(n\right)}\left(s\right)$

Lagrange's interpolation basis function wherein:

$l_k^{\left(n\right)}\left(s\right)=\frac{(s-s_0)...(s-s_{k-1})(s-s_{k+1})...(s-s_n)}{(s_k-s_0)...(s_k-s_{k-1})(s_k-s_{k+1})...(s_k-s_n)}$

k＝0，1，2，…，n

(5) ask for the height anomaly value ζ of other point according to the function of structure;

(6) obtain the elevation at these GPS reference mark according to formula h=H-ζ;

(7) vertical accuracy that the GPS that takes the check of " adding-remove-add " method to obtain through interpolation fitting is ordered provides precision evaluation.

2. line engineering GPS control network height fitting method according to claim 1 is characterized in that: described taking " adds-remove-add " method, and step is:

1), removes one by the method for above-mentioned structure interpolating function Structure interpolating function point, and this point obtained to ask for its spot elevation h again as unknown spot elevation _Inspection

2) the spot elevation difference of twice acquisition of comparison;

3) when difference during, this point is marked, adopt measurement of the level to encrypt translocation GPS reference mark greater than K (the K value is determined according to the accuracy class of engineering), as Structure interpolating function point

4) will increase newly Structure interpolating function pointAdd, re-construct interpolating function;

5) return the 1st) step;

6) when difference during less than K, whether incorporation of markings information detect all The structure interpolating function The pointThe interpolating function of structure all detects and finishes, and does not return the 1st) step;

7) Structure interpolating function pointThe interpolating function of structure all detects and finishes, and finishes.