1.6.1. Main functions of the sector

1.7. Consumer sector (reception and computing complex)

1.7.1. Functions of the geodetic receiving and computing complex

1.7.2. Generalized block diagram of a geodetic satellite receiver

1.7.4. Selection of signals coming from various satellites

1.7.6. Principles of demodulation of received signals

1.7.7. Brief information about the operation of the GPS receiver control system

Section 2. Measurement and calculation methods used in satellite positioning systems

2.1. Absolute and relative methods of satellite measurements

2.2. Main types of differential methods

2.4. The principle of measuring pseudo-ranges and the practical use of this method

2.5. Simplified analysis of phase relationships in satellite ranging measurements

2.6. First, second and third differences based on phase measurements of carrier oscillations

2.6.1. First differences

2.6.2. Second differences

2.7. Integrated Doppler count

2.8. Principles for resolving ambiguities in phase measurements

2.8.1. Geometric method

2.8.3. Method for finding the most probable values ​​of an integer number of cycles

2.8.4. Non-trivial methods for resolving ambiguity

2.9. Detection of missed phase cycles

2.10. General scheme for processing observed data

Section 3. Coordinate and time systems used in satellite measurements

3.1. The role and significance of coordinate-time support for satellite positioning methods

3.1.2. Brief information about the time reference systems used in GPS and GLONASS

3.2. Coordinate systems typical for GPS and GLONASS

3.2.1. Stellar coordinate systems

3.2.2. Geodetic coordinate systems and their transformations

3.2.3. Transition to a common terrestrial coordinate system

3.2.4. Geocentric coordinate system PZ-90

3.2.5. Geocentric coordinate system WGS-84

3.3. Methods for converting coordinate systems for satellite GPS technology and transition parameters

3.4. Features of determining heights using satellite systems

4.3. Taking into account the influence of the external environment on the results of satellite measurements

4.3.1. Ionosphere influence

4.3.2. Troposphere influence

4.3.3. Multi-track

4.4.1. Errors due to instability of the clocks on the satellite and in the receiver

4.4.2. Errors caused by inaccurate knowledge of the reference point

4.5. Geometric factor

4.6. Reasons and methods for artificially reducing the accuracy of GPS measurements

Section 5. Design, organization and pre-processing of satellite measurements

5.1. Specifics of design and organization of satellite measurements

5.2. Pre-field planning in office conditions

5.2.1. Drawing up a technical project

5.4. Entering operating mode and monitoring the progress of measurements

5.5. End of observation session. Storage of collected information. Keeping a field journal

5.6. Specifics of reducing the results of satellite measurements with eccentric installation of receivers

Section 6. Processing of satellite measurements, reduction and adjustment of geodetic networks

6.1. Primary processing of satellite measurements performed in the receiver

6.2. Pre-processing of satellite measurements performed after the measurements are completed

6.3. Final processing of satellite measurements

6.3.1. Final processing of satellite measurements according to the program of the satellite receiver manufacturer

6.3.2. Final processing of satellite measurements using a specially developed program

6.4. Adjustment of geodetic networks created using satellite technology

6.4.1. Adjustment according to the program of the satellite receiver manufacturer

6.4.2. Adjustment using a specially developed program

6.4.3. Equating satellite measurements as trilateration networks

Section 7. Use of satellite technologies to build geodetic networks

7.1. Construction of a global geodetic reference network

7.2. Construction of continental geodetic reference networks

7.3. Construction of the state geodetic network of Russia based on satellite technologies

7.3.1. Fundamental Astronomical and Geodetic Network (FAGS)

7.3.2. High-precision geodetic network (HGN)

7.3.3. Satellite geodetic network 1st class (SGS-1)

7.4.3. On the need to coordinate work on the creation of state and city geodetic networks

7.4.4. Development of the project “Instructions for the creation and reconstruction of urban geodetic networks using GLONASS and GPS satellite systems”

Section 8. Special applications of satellite geodetic measurements for solving various geodetic problems

8.1. Solving geodynamic problems

8.2. Application of satellite technologies in applied geodesy

8.4. Carrying out aerial survey work using satellite coordinate determinations

8.5. The use of satellite technologies when performing topographic and various specialized surveys

8.6. Features of solving navigation problems using satellite receivers

8.6.1. Personal navigation systems

8.6.2. Vehicle navigation systems

Conclusion

Dictionary of English terms

Bibliography

Content

transport locations, ensures the effective further use of these points by all ministries and departments performing geodetic work.

3.2.5. Geocentric coordinate system WGS-84

The NAD 27 coordinate system was created in the USA in 1927 using data from astronomical, geodetic, gravimetric and leveling networks in North and Central America. The system is based on the Clark 1866 ellipsoid with parameters a = 6,378,206 m; / = = 1:295.0. The starting point is located in Meads Ranch, Kansas (B = +39°13"26.7"; L = -98°32"30.5"). This coordinate base served for almost 60 years and was replaced by the NAD 83 coordinate system in 1983. The WGS-84 geocentric coordinate system was initially obtained only using satellites, without connection with very long-baseline interferometry data, and is presented on the earth's surface in the form of a homogeneous global network with an accuracy of point coordinates of 1-2 m. The coordinate system has been refined several times and since 1994, the WGS-84 (G730) version has been used, which has a global consistency of the order of 10 cm.

When determining the parameters of the WGS-84 global coordinate system, the same fundamental constants were used:

Speed ​​of light;

- geocentric gravitational constant;

- angular velocity of the Earth's rotation.

The main parameters of the global WGS-84 ellipsoid, obtained from satellite measurements on land and in the oceans, have the following values:

a = 6,378,137 m - semimajor axis of the earth's ellipsoid; / = 298.257 223 563 - the denominator of the compression of the earth's ellipsoid.

In addition to the WGS-84 global coordinate system, there are regional and national geocentric coordinate systems. The most famous of them is the European one, fixed on the earth's surface by the EUREF network.

3.3. Methods for converting coordinate systems for satellite GPS technology and transition parameters

There are two types of coordinate transformation when moving from one system to another:

Transformation of spatial rectangular or ellipsoidal coordinates of one coordinate system into another coordinate system

a dynatic system of the same type using precisely defined transition parameters;

Converting one coordinate system to another coordinate system of the same type using points whose coordinates are known in both systems.

In this case, three-dimensional, two-dimensional and one-dimensional transformation methods are distinguished.

Converting spatial rectangular or ellipsoidal coordinates of one coordinate system into another coordinate system of the same type according to fairly strict formulas using precisely defined transition parameters is a fairly simple task for the three-dimensional coordinate systems PZ90 and SK-42 and the associated two-dimensional topocentric systems (State coordinate system , local coordinate systems), as well as for the three-dimensional systems WGS-72 and WGS-84 and related two-dimensional topocentric systems (NAD-87 and others). Preliminary communication parameters of some coordinate systems are given in Table. 3.6.

					Table 3.6
	Preliminary		Coordinate systems
	conversion options
	AH, m
	A Y, m
	/i-10"6
	С0у

*data approximate

It should be noted that until recently there were no final values ​​for the connection parameters between the PZ-90 and WGS-84 coordinate systems. The works still provide approximate values ​​(see Table 3.6). The reason for this is that the parameters of each coordinate system are constantly being refined. Currently, the PZ-90 coordinate system has a rotation relative to the WGS-84 system around the Z axis by an amount of about 0.2", which corresponds to a shift in the longitudinal direction on the territory of Russia by 3-6 m. Such a rotation significantly exceeds the declared accuracy of the coordinate systems.

ordinates PZ-90 and WGS-84. A way out of this situation may be the adoption of a unified geocentric coordinate system for existing and future international and national satellite positioning systems. Some averaged ITRF implementation can be considered as such a system. All over the world, for the most precise tasks, for example for geodynamics problems, the system implemented in ITRF, created and supported by International service Earth Rotation (IERS) in accordance with Resolution No. 2 of the International Union of Geodesy and Geophysics, adopted in 1991 in Vienna.

Since international cooperation is based on the use of navigation systems regardless of nationality, precise communication between the two coordinate systems is necessary to take full advantage of their capabilities.

Adopted in August 2001, the state standard of the Russian Federation GOST R 51794-2001 “Global navigation radio navigation equipment satellite system and global positioning system. Coordinate systems. Methods for transforming coordinates of defined points" sets the following parameters for connecting coordinate systems (Table 3.7).

			Table 3.7
Options	Coordinate systems
transformation	SK-42 in PZ-90	SK-95 in PZ-90	PZ-90 to WGS-84
AH, m
AY, m
w-10-6
cog,

Currently, in Russia and abroad, the development of navigation and geodetic receivers operating on signals from GLONASS and GPS satellites is underway. It is known that to solve the coordinate problem and to take into account the influence of satellite receiver clock drift, the minimum number of satellites must be equal to four. In reality, the consumer is forced to receive signals from four GLONASS satellites and four GPS satellites, receiving two unrelated results. In the absence of four satellites in any of the systems, only one solution is obtained, and the three satellites of the other system cannot be used even to refine the definitions. Thus, completely

It is unlikely that an integrated system based on GLONASS and GPS satellites will be created in the near future.

Converting one coordinate system into another coordinate system of the same type using points whose coordinates are known in both systems, based on similarity theory, is today the most common method of coordinate conversion in practice.

Let's take a closer look at 3D transformation. Transformation parameters can be determined from solving the system of equations (3.31), which can be presented in the following form:

In this case, the linearized coordinate transformation model for one point can be presented in the following form:

							(3 -34 >
			Хoi = m0 RoXi +АХо.
The design matrix A and the parametric vector dP are determined
are divided by the following ratios:
			XQI - A X 0		Z 0 I - A Z 0	G0, - D G 0
A; = 0			Y0i -AY0	Z 0 / -AZ 0		X0i-AX0
	0 _1 Z0 / -AZ0			Y0i -AY0	X0i-AX0
				dZ dcox	do)Y dmz

When substituting values ​​from equations (3.35) into equation (3.34)

And (3.36) we obtain a system of linear equations for one point /. For

P points, the design matrix will look like:

	A = A 2

For three points whose coordinates are known in both systems, the design matrix can be represented by the following expression:

						Z0 1 - D20	Xt -AX0
				n. -DP Z0, -D20
				AZ„ -P.-DP
						Z 0 2 - A Z 0
				Y" - lg"
					G0 2 -DU0	X02 -AX0
						Z03 - AZ0
				Duo	Z m -D^o
				D 2 0

Two-dimensional transformation (transformation of one flat coordinate system into another similar coordinate system) using points whose coordinates are known in two systems, based on the theory of similarity, is a special case of three-dimensional transformation, but at the same time the most widespread geodetic problem in both classical and satellite geodesy. Converting coordinates to in this case is represented in the form of rotation and translation of the origin of coordinates (Fig. 3.8).

The general transformation equation is:
X( =X0 +mXcosa-m Ksina;
yj= Y0 +mXsina+m Kcosa.

In this case, four transformation parameters are used: X 0, Y 0, a, m. To determine these four parameters, it is enough to have two points whose coordinates are known in two systems. Using

If there are two points, the system of equations is solved using the least squares method to determine the parameters X0 and Y0, as well as

auxiliary parameters P and Q. Then the transformation parameters a and m are calculated using the formulas:

Combined transformation (transformation of a spatial coordinate system into another flat coordinate system) using points whose coordinates are known in two systems, based on similarity theory, is also a special case of three-dimensional transformation and also the most widespread geodetic task in satellite geodesy.

Rice. 3.8. Two-dimensional transformation of coordinate systems

The most critical and at the same time the most controversial parameter in two-dimensional and combined transformation is the scale factor m. On the one hand, GPS and GLONASS satellite systems are high-precision rangefinding systems, and the introduction of any scale factor into the results of their measurements requires serious justification. On the other hand, classical geodetic constructions are carried out, as a rule, with high metrological accuracy, which was and is currently provided by a fairly reliable system of technological methods and controls, which also makes the use of any scale factors very problematic. And, finally, on the third side, a formal transformation based on the theory of similarity of a rectangular coordinate system (spatial or flat) into another rectangular system created on the basis of one of the classical projections (UTM, Gauss-Kruger or others) for linear objects with a length of the order of tens kilometers or areal objects of the same size, especially those extended along the parallel, can lead to methodological transformation errors that exceed both the accuracy of satellite measurements and the accuracy of previously created classical geodetic constructions (Fig. 3.9).

One-dimensional transformation (transformation of one coordinate into another similar coordinate) using points whose coordinates are known in two systems, based on similarity theory, is a special case of three- and two-dimensional transformation and a fairly common geodetic problem in both classical and satellite geodesy. The transformation in this case is represented in the form of transformation of heights and transformation of base lines. Transforming heights will be discussed in the next subsection. The problem of transforming base lines can be solved quite strictly on the basis of knowledge of the exact value of the length of the base line measured by a satellite system on the physical surface of the Earth, the exact parameters of the coordinate system into which the base line is transformed, and the approximate coordinates of the ends of the line in this coordinate system, determined by one of the above methods.

	L->. YT.FTL"Y

Rice. 3.9. Distortions due to methodological incorrectness of transformation

So, for example, when one-dimensionally transforming lines measured by a GPS system into the SK-42 coordinate system, the classical reduction problem of higher geodesy is solved (Fig. 3.10).

In this case, the transition from the length of the line MN, measured on the physical surface of the Earth, to the length of the line M X N V reduced to the SK-42 coordinate system is carried out by three transformations:

1) introduction of corrections for the slope of the line, for example, according to the formula

AD H 2D 8Z)3 "

where h=H M - H N \ D - length of the line between points M and N;

2) reduction to the surface of the reference ellipsoid, for example, according to the formula

GOST R 51794-2008

Group E50

NATIONAL STANDARD OF THE RUSSIAN FEDERATION

Global navigation satellite systems

COORDINATE SYSTEMS

Methods for transforming coordinates of defined points

Global navigation satellite system and global positioning system. Coordinate systems. Methods of transformations for determinate points coordinate

OKS 07.040
OKSTU 6801

Date of introduction 2009-09-01

Preface

The goals and principles of standardization in the Russian Federation are established by Federal Law of December 27, 2002 N 184-FZ "On Technical Regulation", and the rules for applying national standards of the Russian Federation are GOST R 1.0-2004 "Standardization in the Russian Federation. Basic Provisions"

Standard information

1 DEVELOPED 29 by the Research Institute of the Ministry of Defense of the Russian Federation

2 INTRODUCED by the Technical Committee for Standardization TC 363 "Radio Navigation"

3 APPROVED AND ENTERED INTO EFFECT by Order of the Federal Agency for Technical Regulation and Metrology dated December 18, 2008 N 609-st

4 INSTEAD GOST R 51794-2001


Information about changes to this standard is published in the annually published information index " National standards", and the text of changes and amendments - in the monthly published information index "National Standards". In case of revision (replacement) or cancellation of this standard, the corresponding notification will be published in the monthly published information index "National Standards". The corresponding information, notice and texts are also posted V information system for general use - on the official website of the Federal Agency for technical regulation and metrology on the Internet

Amendments have been made, published in IUS No. 4, 2011, IUS No. 6, 2011, IUS No. 9, 2013

Amendments made by the database manufacturer

1 area of ​​use

1 area of ​​use

This standard applies to coordinate systems that are part of the geodetic parameter systems "Earth Parameters", "World Geodetic System" and the coordinate base of the Russian Federation, and establishes methods for transforming coordinates and their increments from one system to another, as well as the procedure for using numerical values ​​of elements transformation of coordinate systems when performing geodetic, navigation, cartographic work using the equipment of consumers of global navigation satellite systems.

2 Terms and definitions

The following terms with corresponding definitions are used in this standard:

2.1 semimajor axis of the ellipsoid : A parameter characterizing the size of the ellipsoid.

2.2 reference ellipsoid: An ellipsoid adopted for processing geodetic measurements and establishing a geodetic coordinate system.

2.3 geodetic coordinate system: A system of parameters, two of which (geodetic latitude and geodetic longitude) characterize the direction of the normal to the surface of the reference ellipsoid at a given point in space relative to the planes of its equator and prime meridian, and the third (geodesic height) represents the height of the point above the surface of the reference ellipsoid.

2.4 geodetic latitude: The angle between the normal to the surface of the reference ellipsoid passing through a given point and the plane of its equator.

2.5 geodetic longitude: The dihedral angle between the planes of the geodesic meridian of a given point and the prime geodesic meridian.

2.6 geodetic height: The height of a point above the surface of the reference ellipsoid.

2.7 geodetic meridian plane: A plane passing through the normal to the surface of the reference ellipsoid at a given point and parallel to its minor axis.

2.8 astronomical meridian plane: A plane passing through a plumb line at a given point and parallel to the Earth's axis of rotation.

2.9 plane of the prime meridian: The plane of the meridian from which longitudes are calculated.

2.10 geoid: An equipotential surface that coincides with the surface of the World Ocean in a state of complete rest and equilibrium and continues under the continents.

2.11 equipotential surface: A surface on which the potential has the same value.

2.12 Global Positioning System(Global Positioning System): Global navigation satellite system developed in the USA.

2.13 Earth's gravitational field; GPZ: Gravity field on the Earth's surface and in outer space, caused by the Earth's gravitational force and the centrifugal force resulting from the Earth's daily rotation.

2.14 quasigeoid: A mathematical surface that is close to the geoid and serves as a reference surface for establishing a system of normal heights.

2.15 space geodetic network; KGS: A network of geodetic points fixing a geocentric coordinate system, the position of which on the earth's surface is determined by observations artificial satellites Earth.

2.16 World geodetic system(World Geodetic System): A system of geodetic parameters developed in the United States.

2.17 model of the Earth's gravitational field: Mathematical description of the characteristics of the Earth's gravitational field.

2.18 normal height: The height of a point above the quasigeoid, determined by the geometric leveling method.

2.19 normal gravitational field of the Earth: The Earth's gravitational field, represented by the normal gravity potential.

2.20 common terrestrial ellipsoid; OSE: An ellipsoid whose surface is closest to the geoid as a whole, used to process geodetic measurements over the entire surface of the Earth in a common terrestrial (geocentric) coordinate system.

2.21 planetary model of the Earth's gravitational field: A model of the Earth's gravitational field, reflecting the gravitational characteristics of the Earth as a whole.

2.22 ellipsoid compression : A parameter characterizing the shape of the ellipsoid.

2.23 system of geodetic parameters of the Earth: A set of numerical parameters and accuracy characteristics of the fundamental geodetic constants of the earth's ellipsoid, the planetary model of the Earth's gravitational field, the geocentric coordinate system and the parameters of its connection with other coordinate systems.

2.24 fundamental geodesic constants: Mutually consistent geodetic constants that uniquely determine the shape of the general Earth ellipsoid and the normal gravitational field of the Earth.

2.25 elements of transformation of coordinate systems: Parameters used to convert coordinates from one coordinate system to another.

2.26 flat rectangular coordinates: Plane coordinates on the plane on which the surface of the reference ellipsoid is displayed according to a certain mathematical law.

3 Abbreviations and symbols

The following abbreviations and symbols are used in this standard:

3.1 GLONASS is a global navigation satellite system developed in the Russian Federation.

3.2 GPS is a global navigation satellite system developed in the USA.

3.3 GGS - state geodetic network.

3.4 GPZ - Earth's gravitational field.

3.5 KNS - space navigation system.

3.6 WGS; The World Geodetic System is a system of geodetic parameters developed in the USA.

3.7 OZE - common terrestrial ellipsoid.

3.8, , , - axes of the spatial rectangular coordinate system.

3.9 PZ; Earth Parameters is a system of geodetic parameters developed in the Russian Federation.

3.10 SC - coordinate system.

3.11 - semimajor axis of the general earth ellipsoid in the PZ system.

3.12 - semimajor axis of the global ellipsoid in the WGS system.

3.13 - semimajor axis of the Krasovsky ellipsoid.

3.14 - compression of the general earth ellipsoid in the PZ system.

3.15 - compression of the global ellipsoid in the WGS system.

3.16 - compression of the Krasovsky ellipsoid.

4 Systems of geodetic parameters

4.1 System of geodetic parameters "Earth Parameters"

The PP system includes: fundamental geodetic constants, OZE parameters, the PP coordinate system fixed by the coordinates of points of the space geodetic network, characteristics of the GPZ model and transformation elements between the PP system and the national reference coordinate systems of Russia. The numerical values ​​of the transformation elements between the PP system and the national reference coordinate systems of Russia and the order of their use when transforming coordinate systems are given in Appendices A, B.

Notes

1 for use for geodetic support of orbital flights and solving navigation problems, the geocentric coordinate system "Earth Parameters of 1990" (PZ-90) was given the status state system coordinates

2 By Order of the Government of the Russian Federation dated June 20, 2007 N 797-r, in order to improve the tactical and technical characteristics of the global navigation satellite system GLONASS, improve geodetic support for orbital flights and solve navigation problems, an updated version of the state geocentric coordinate system "Earth Parameters 1990" was adopted for use " (PZ-90.02).

3 The numerical values ​​of transformation elements between the PZ-90.02 and PZ-90 coordinate systems and the order of their use when transforming coordinate systems are given in Appendix D.


The theoretical definition of the PZ coordinate system is based on the following provisions:

a) the origin of the coordinate system is located at the center of mass of the Earth;

b) the axis is directed to the International Conditional Origin;

c) the axis lies in the plane of the prime astronomical meridian established by the International Time Bureau;

d) the axis complements the system to the right coordinate system.

The positions of points in the PP system can be obtained in the form of spatial rectangular or geodetic coordinates.



The center of the OSE coincides with the origin of the PZ coordinate system, the axis of rotation of the ellipsoid coincides with the axis, and the plane of the prime meridian coincides with the plane.

Note - The reference surface in the systems of geodetic parameters PZ-90 and PZ-90.02 is taken to be a general earth ellipsoid with a semi-major axis of 6378136 m and a compression of 1/298.25784.

4.2 System of geodetic parameters "World geodetic system"

The WGS parameter system includes: fundamental geodetic constants, a WGS coordinate system fixed by the coordinates of points of the space geodetic network, OSE parameters, characteristics of the GPZ model, transformation elements between the WGS geocentric coordinate system and various national coordinate systems.

The numerical values ​​of transformation elements between the PZ coordinate system and the WGS coordinate system, as well as the order of using transformation elements, are given in Appendices C and D.

Note - On January 1, 1987, the first version of the WGS-84 coordinate system was introduced. On January 2, 1994, a second version of the WGS-84 coordinate system was introduced, designated WGS-84(G730). On January 1, 1997, the third version of the WGS-84 coordinate system was introduced, designated WGS-84(G873). The fourth version of the WGS-84 coordinate system is currently in effect, designated WGS-84(G1150) and introduced on January 20, 2002. In the following designations for versions of the WGS-84 coordinate system, the letter "G" means "GPS", and "730", "873" and "1150" indicate the GPS week number corresponding to the date to which these versions of the WGS-84 coordinate system are assigned .

The theoretical definition of the WGS coordinate system is based on the provisions given in 4.1.

WGS point positions can be obtained as spatial rectangular or geodetic coordinates.

Geodetic coordinates refer to the OZE, the size and shape of which are determined by the values ​​of the semi-major axis and compression.

The center of the ellipsoid coincides with the origin of the WGS coordinate system, the axis of rotation of the ellipsoid coincides with the axis, and the plane of the prime meridian coincides with the plane.

Note - The reference surface in WGS is a global ellipsoid with a semi-major axis of 6378137 m and a compression of 1/298.257223563.

4.3 Reference coordinate systems of the Russian Federation

The coordinate base of the Russian Federation is represented by a reference coordinate system, implemented in the form of the GGS, which fixes the coordinate system on the territory of the country, and the state leveling network, which extends the system of normal heights (the Baltic system) throughout the entire territory of the country, the initial origin of which is the zero of the Kronstadt footpole.

The positions of the defined points relative to the coordinate base can be obtained in the form of spatial rectangular or geodetic coordinates, or in the form of flat rectangular coordinates and heights.

Geodetic coordinates in the reference coordinate system of the Russian Federation refer to the Krasovsky ellipsoid, the dimensions and shape of which are determined by the values ​​of the semi-major axis and compression.

The center of the Krasovsky ellipsoid coincides with the origin of the reference coordinate system, the axis of rotation of the ellipsoid is parallel to the axis of rotation of the Earth, and the plane of the prime meridian determines the position of the origin of longitude calculation.

Notes

1 In 1946, a unified reference coordinate system of 1942 (SK-42) was adopted for the entire territory of the USSR. The reference surface in SK-42 is Krasovsky's ellipsoid with a semimajor axis of 6378245 m and a compression of 1/298.3.

2 By Decree of the Government of the Russian Federation of July 28, 2000 N 568, a new reference system of geodetic coordinates of 1995 (SK-95) was adopted for use in geodetic and cartographic work. Krasovsky's ellipsoid is taken as the reference surface in SK-95.

5 Methods for transforming the coordinates of defined points

5.1 Converting geodetic coordinates to rectangular spatial coordinates and vice versa

The transformation of geodetic coordinates into rectangular spatial coordinates is carried out according to the formulas:

where , , are the rectangular spatial coordinates of the point;

, - geodetic latitude and longitude of the point, respectively, rad;

- geodetic height of the point, m;

- radius of curvature of the first vertical, m;

- eccentricity of the ellipsoid.

The values ​​of the radius of curvature of the first vertical and the square of the eccentricity of the ellipsoid are calculated, respectively, using the formulas:

where is the semimajor axis of the ellipsoid, m;

- compression of the ellipsoid.

To convert spatial rectangular coordinates to geodetic coordinates, iterations are required when calculating geodetic latitude.

To do this, use the following algorithm:

a) calculate the auxiliary quantity using the formula

b) analyze the value:

1) if 0, then

2) if 0, at

c) analyze the meaning:

1) if 0, then

2) in all other cases, calculations are performed as follows:

- find auxiliary quantities , , using the formulas:

Implement an iterative process using auxiliary quantities and:

If the value determined by formula (16) is less than the established tolerance value, then

, (17)

; (18)

If the value is equal to or greater than the specified tolerance value, then

and the calculations are repeated starting from formula (14).

When transforming coordinates, the value (10) is taken as the tolerance for terminating the iterative process. In this case, the error in calculating the geodetic height does not exceed 0.003 m.

5.2 Transformation of spatial rectangular coordinates

Users of GLONASS and GPS systems need to perform coordinate transformations from the PP system to the WGS system and vice versa, as well as from PP and WGS to the reference coordinate system of the Russian Federation. These coordinate transformations are performed using seven transformation elements, the accuracy of which determines the accuracy of the transformations.

Elements of transformation between the PZ and WGS coordinate systems are given in Appendices C, D.

The conversion of coordinates from the WGS system to the coordinates of the reference system of the Russian Federation is carried out by sequentially transforming the coordinates first into the PZ system, and then into the coordinates of the reference system.

The transformation of spatial rectangular coordinates is performed according to the formula

where, , - linear elements of transformation of coordinate systems when moving from system A to system B, m;

, , - angular elements of transformation of coordinate systems when moving from system A to system B, rad;

- a large-scale element of transformation of coordinate systems when moving from system A to system B.

The inverse transformation of rectangular coordinates is performed according to the formula

5.3 Conversion of geodetic coordinates

The transformation of geodetic coordinates from system A to system B is performed according to the formulas:

where , - geodetic latitude and longitude, expressed in units of plane angle;

- geodetic height, m;

, , - corrections to the geodetic coordinates of the point.

Amendments to geodetic coordinates are determined using the following formulas:

where , - corrections to geodetic latitude, longitude, ...;

- correction to geodetic height, m;

, - geodetic latitude and longitude, rad;

- geodetic height, m;

, , - linear elements of transformation of coordinate systems during the transition from system A to system B, m;

, , - angular elements of transformation of coordinate systems when moving from system A to system B, ...;

- a large-scale element of transformation of coordinate systems when moving from system A to system B;

Radius of curvature of the meridian section;
- radius of curvature of the first vertical;

Major semi-axes of ellipsoids in coordinate systems B and A, respectively;

, - squares of eccentricities of ellipsoids in coordinate systems B and A, respectively;

- number of arc seconds in 1 radian [(206264.806)].

When converting geodetic coordinates from system A to system B, the values ​​of geodetic coordinates in system A are used in formula (22), and when converting back - in system B, and the sign of the corrections , , in formula (22) is reversed.

Formulas (23) provide the calculation of corrections to geodetic coordinates with an error not exceeding 0.3 m (in linear measure), and to achieve an error of no more than 0.001 m, perform the second iteration, i.e. take into account the values ​​of corrections to geodetic coordinates using formulas (22) and repeat the calculations using formulas (23).

Wherein

Formulas (22), (23) and the accuracy characteristics of transformations using these formulas are valid up to latitudes of 89°.

5.4 Converting geodetic coordinates to plane rectangular coordinates and vice versa

To obtain flat rectangular coordinates in the Gauss-Kruger projection adopted on the territory of the Russian Federation, geodetic coordinates on the Krasovsky ellipsoid are used.

Flat rectangular coordinates with an error of no more than 0.001 m are calculated using the formulas

where , - flat rectangular coordinates (abscissa and ordinate) of the determined point in the Gauss-Kruger projection, m;

- geodetic latitude of the determined point, rad;

- the distance from the determined point to the axial meridian of the zone, expressed in radian measure and calculated by the formula

Geodetic longitude of the determined point, ...°;

The integer part of the expression enclosed in square brackets.

The transformation of flat rectangular coordinates in the Gauss-Kruger projection on the Krasovsky ellipsoid into geodetic coordinates is carried out according to the formulas

where , - geodetic latitude and longitude of the determined point, rad;

- geodetic latitude of a point, the abscissa of which is equal to the abscissa of the point being determined, and the ordinate is equal to zero, rad;

- number of the six-degree zone in the Gauss-Kruger projection, calculated by the formula

The integer part of the expression enclosed in square brackets;

- ordinate of the determined point in the Gauss-Kruger projection, m.

The values ​​of , and are calculated using the following formulas:

where is an auxiliary quantity calculated by the formula

Auxiliary quantity calculated by the formula

Abscissa and ordinate of the determined point in the Gauss-Kruger projection, m.

Error of coordinate transformation according to formulas (25); (26) and (32)-(36) is no more than 0.001 m.

5.5 Converting increments of spatial rectangular coordinates from system to system

The transformation of increments of spatial rectangular coordinates from coordinate system A to system B is carried out according to the formula

The inverse transformation of increments of spatial rectangular coordinates from system B to system A is performed according to the formula

In formulas (37) and (38), the angular transformation elements , , are expressed in radians.

5.6 Relationship between geodetic and normal heights

Geodetic and normal heights are related by the relation:

where is the geodetic height of the determined point, m;

- normal height of the determined point, m;

- height of the quasigeoid above the ellipsoid at the determined point, m.

The heights of the quasi-geoid above the reference ellipsoid of the systems of geodetic parameters GZ and WGS are calculated using GZ models, which are integral part systems of geodetic parameters.

When recalculating the heights of a quasi-geoid from coordinate system A to coordinate system B, use the formula

where is the height of the quasigeoid above the OSE, m;

- height of the quasigeoid above the Krasovsky ellipsoid, m;

- correction to geodetic height, calculated using formula (23), m.

Appendix A (mandatory). Elements of transformation between the refined coordinate system of the Earth Parameters and the reference coordinate systems of the Russian Federation

Appendix A
(required)

Conversion of coordinates from the 1942 reference coordinate system to the PZ-90.02 system

23.93 m; 0;
-141.03 m; -0.35;
-79.98 m; -0.79;
-130.97 m; 0.00;
-81.74 m; -0.13;
(-0.22)·10;

Conversion of coordinates from the PZ-90.02 coordinate system to the 1995 reference coordinate system

Appendix B (mandatory). Elements of transformation between the Earth Parameters coordinate system and the reference coordinate systems of the Russian Federation

Appendix B
(required)

Converting coordinates from the 1942 reference coordinate system to the PZ-90 system

25 m; 0;
-141 m; -0.35;
-80 m; -0.66;
0;

Converting coordinates from the PZ-90 coordinate system to the 1942 reference coordinate system

Converting coordinates from the 1995 reference coordinate system to the PZ-90 system

25.90 m;
-130.94 m;
-81.76 m;

Conversion of coordinates from the PZ-90 coordinate system to the 1995 reference coordinate system

Appendix B (mandatory). Elements of transformation between the refined coordinate system of the Earth Parameters and the coordinate system of the World Geodetic System

Appendix B
(required)

Converting coordinates from the PZ-90.02 coordinate system to the WGS-84 system

0.36 m; 0;
+0.08 m; 0;
+0.18 m; 0;
0;

Converting coordinates from the WGS-84 coordinate system to the PZ-90.02 system

Appendix D (mandatory). Elements of transformation between the Earth Parameters coordinate system and the World Geodetic System coordinate system

Appendix D
(required)

Converting coordinates from the PZ-90 coordinate system to the WGS-84 system

1.10 m; 0;
-0.30 m; 0;
-0.90 m; -0.20±0.01;
(-0.12)·10;

Converting coordinates from the WGS-84 coordinate system to the PZ-90 system

Appendix D (mandatory). Transformation elements between the refined coordinate system PZ-90.02 and the coordinate system PZ-90

Appendix D
(required)

Converting coordinates from the PZ-90.02 coordinate system to the PZ-90 system

1.07 m; 0;
+0.03 m; 0;
-0.02 m; +0.13;
(+0.22) ·10;

Converting coordinates from the PZ-90 coordinate system to the PZ-90.02 system

Electronic document text
prepared by Kodeks JSC and verified against:
official publication
M.: Standartinform, 2009

To move from one coordinate system to another, there are fundamentally 2 types of transformations:

- coordinate transformation using officially published transformation parameters, also called global transformation methods, since they specify the algorithm for transition between coordinate systems as a whole, throughout the entire space of action of these coordinate systems, for example, between WGS-84 and SK-95, ITRF and SK-95, PZ-90 and WGS-84, etc.;

- transformation of coordinates using transformation parameters calculated using a limited set of control points located on the local territory, the coordinates of which are known in both of these CS, also called local conversion methods, since they specify a coordinate recalculation algorithm that operates only in relation to the local territory on which the control points are located.

Classic three-dimensional coordinate transformation methods used primarily for global transformations between spatial three-dimensional rectangular or ellipsoidal (geodesic) coordinate systems are the Helmert method and the Molodensky method, respectively.

Conversion from one spatial (3D) rectangular coordinate system X,Y,Z(SK-1) to another spatial system of rectangular coordinates (SK-2) according Helmert consists of performing three operations:

Transferring the beginning of CK1 to the beginning of CK2 by shifting along the axes XYZ by magnitudes T X, T Y, T Z, corresponding to the difference in coordinates of the origins of coordinate systems 1 and 2 (or, similarly, by the value of the coordinate values ​​of the final coordinate system SK-2 in the original SK-1);

Rotate around each of the coordinate axes by amounts w X , w Y , w Z ,;

Scaling (introducing a multiplier m, characterizing the change in the scale of the final SC-2 relative to the scale of the initial SC-1).

Thus, the Helmert transformation is specified by the 7 above parameters, which is why it is often called the 7-parameter transformation, or the Euclidean similarity transformation, and the transformation parameters included in it are called the Helmert parameters.

For the 7-parameter Helmert transform, the formula is used

Where [ X, Y, Z]SK1- coordinates of the point in the original coordinate system;

Where, [ X, Y, Z]SK2- coordinates of the point in the final coordinate system;

T X, T Y, T Z- the displacement value of the origin of coordinate system 1 along the corresponding axes to the origin of coordinate system CK2;

w X , w Y , w Z- rotation around each of the axes of the coordinate system;

m- scale factor taking into account the different scales of these SCs, its value is usually<10 -6 и дается в единицах 6-го знака после запятой.

Method Molodensky used to convert between two spatial geodetic coordinate systems B, L, H(i.e., eliminating the need to change to rectangular XYZ coordinates).

To transform coordinates using the Molodensky method, use the formula

. (5)

,

.

Classical 3D method The calculation of the 7-parameter transformation is sometimes carried out in two modifications: Bursa-Wolf and Molodensky-Badekas.

The difference between the modifications is that in the Bursch-Wolf transformation, the center of rotation is the origin of the original coordinate system A and the 7 above-described parameters of the Helmert transformation are used - CLASSIC.

and in the Molodensky-Badekas modification, the center of rotation is the “center of gravity” (a point on the work site that has average coordinates) of the control points in the original coordinate system A, therefore, in this modification of the classical three-dimensional transformation, 3 more parameters are added to the 7 Helmert parameters (coordinates of the center of rotation X 0 , Y 0 , Z 0). In LGO it is implemented like this

The scheme of coordinate transformations when performing geodetic work using GNSS technologies is given below

12. Free adjustment, varieties of minimally limited adjustment, limited adjustment, limited adjustment with simultaneous estimation of transformation parameters.

The procedure for mathematical processing of satellite measurements:

Ø processing of GNSS measurements and calculation of baselines,

Ø calculation of residuals of closed figures,

Ø assessment of measurement accuracy based on figure residuals,

Ø network equalization,

Ø accuracy assessment based on adjustment results

Tools for mathematical processing of satellite measurements– special commercial software for processing satellite measurements

Equalization Concepts

In general, the development of GGS through GNSS measurements involves determining the coordinates of a large number of stations with a limited number of GNSS receivers. The observations carried out in the project are divided into sessions consisting of observations at individual stations (points). The following methods for adjusting satellite observations have been developed and used:

· adjustment of observations made on one station (for the case of absolute (point) positioning);

· treatment one baseline and subsequent integration of the baselines into a network,

· combined adjustment of all received observations of a separate session ( adjustment of observations of many stations of one session), And

· combining solutions many sessions into a rigorous, all-encompassing network solution,

· combining satellite and traditional geodetic measurements.

Equalization one station(point positioning, “one-point” solution) provides absolute station coordinates in the WGS-84 (or PZ-90) system. If only code measurements are processed, then due to low accuracy these results are usually of little interest for geodetic applications, but they often meet the requirements of some geophysics, GIS and remote sensing applications. A typical area of ​​this application is navigation.

Concept single baseline very widely used in satellite data processing software. In joint adjustment, observations from two simultaneously operating receivers are processed, primarily in the form of double differences. The result is the components of the baseline vector and the corresponding covariance matrix K XYZ

Individual baselines are used as input data in network equalization program. Processing of observations in the network breaks down into primary adjustment(baseline solution) and secondary adjustment(equalization of baseline vectors).

Most manufacturers offer programs with their receivers that use the baseline concept. These programs are useful for small projects, field data verification, and real-time applications.

IN adjustment of many stations of one session all data that was observed simultaneously by all receivers participating in the session is jointly processed. In this case, the results of the solution are R-1 independent vectors and a covariance matrix of size 3( R- 13( R- 1). Depending on the software available, results can also be given in sets of 3 R coordinates and a covariance matrix of size 3 R´3 R. The covariance matrix is ​​also block-diagonal, in which the size of the non-zero diagonal blocks is a function of the number of receivers R. Therefore, it is a strict adjustment of observations using all reciprocal stochastic relations. For geodetic purposes, such a “multipoint” adjustment has conceptual advantages over the baseline method, since the full potential of SRNS accuracy is used.

Multiple session solutions can be combined into equalization of many sessions or, more precisely, in solution for many stations and many sessions. This is a common technique when large networks are split into pieces due to a limited number of receivers. The main condition in such an adjustment is that each session is connected to at least one other session through one or more common stations at which observations were made in both sessions. Increasing the number of common stations increases the stability and reliability of the entire network.

Combining satellite and traditional types of measurements is necessary for the transition from global coordinates of satellite network points to the state reference system SK-95 and to the Baltic system of normal heights.

Adjustment of geodetic networks built using satellite technologies, is a necessary stage in the technology of geodetic work. The objectives of equalization are:

· coordination of the totality of all measurements in the network,

· minimization and filtering of random measurement errors,

· identification and rejection of rough measurements, elimination of systematic errors,

· obtaining a set of adjusted coordinates and the corresponding elements of the baselines with an assessment of accuracy in the form of errors or covariance matrices,

· transforming coordinates into the required coordinate system,

· converting geodetic heights to normal heights above a quasi-geoid.

Thus, the main goal of adjustment is to increase accuracy and present the results in the required coordinate system with an accuracy assessment.

There are free, minimally limited and limited (non-free) adjustment.

IN free adjustment All points of the network are considered unknown, and the position of the network relative to the geocenter is known with the same accuracy as the coordinates of the starting point of the network. In this case, the coefficient matrix of the system of correction equations (plan matrix) and, therefore, the normal matrix will have a rank defect equal to three. However, the use of the matrix pseudo-inversion apparatus used in some programs allows for adjustment. Its results reflect the internal accuracy of the network, which is not distorted by errors in the original data.

When fixing the coordinates of one point, we get minimally limited adjustment, in which the normal matrix turns out to be non-degenerate. To achieve meaningful control, a vector network should not contain vectors whose ends are not connected to at least two stations.

Free and minimally constrained adjustment are used to solve the first three adjustment problems (harmonizing the totality of all measurements in the network, minimizing and filtering random measurement errors, identifying and rejecting rough measurements, eliminating systematic measurement errors).

When fixing more than three coordinates - limited adjustment. In this case, additional restrictions will be imposed in relation to the minimum necessary.

Limited adjustment performed after successful completion of minimally limited adjustment to include the newly constructed network in the existing network, in its coordinate system, including the elevation system. To do this, the new network must be connected to at least two stations of the existing network.

A special problem is the joint adjustment of satellite and conventional geodetic measurements. Its essence is that traditional geodetic measurements (angle measurements, leveling, astronomical determinations, etc.) are performed using a level, that is, the geoid is used as a reference surface. Baseline measurements are made in the system of axes of a common earth ellipsoid. To correctly reduce data to one particular system, it is necessary to know the heights of the geoid above the ellipsoid with appropriate accuracy.

With limited adjustment, the following can be inserted as additional unknowns into the parametric equations: connection parameters between coordinate and elevation systems.

The combination of satellite and traditional measurements is carried out with limited adjustment. Mathematical models for spatial coordinates are based on the Helmert method (local transformation using the method of similarity of coordinates in Cartesian). In this transformation, the scale factor is the same in all directions, as a result of which the shape of the network is preserved, i.e. The angles are not distorted, but the lengths of the lines and the positions of the points may change.