1. Des Éléments Importants des Systèmes de Référence et de la Géodésie au CERN Mark Jones EN\MEF-SU

2. Outline Introduction CERN Coordinate System (CCS) Altitudes Geoid Models CERN Geodetic reference frames Z  H Transformation Conclusions

3. The Survey Team Large Scale Metrology Section Metrology Measurement Alignment Monitoring As-built surveys First Surveyors at CERN in 1954 Our 60 th Anniversary this year too! Surveying is the application of Geodesy

4. Geodesy Geodesy is the science concerned with the Shape, Size, and the Gravity Field of the Earth (International Association of Geodesy) One of the oldest sciences Includes temporal variations 1 st Geodesist Eratosthenes, 200 BC ~5950 km (6371 km) Aswan Alexandrie Distance

5. Surveying Determine point positions Different types of Observations Directions / Angles / Azimuths Distances Redundant Observations Identify errors Optimisation algorithms (Least Squares) Simplify calculations as much as possible Done by hand for many hundreds of years! a b c 11 33 22 Pt 1 Pt 3 Pt 2

6. Surveying Different types of instruments Directions (and distances) Theodolite / Camera / Total Station / Laser Tracker / Laser Scanner Distances Invar wires / EDM / Digital Scales Height differences Levels

7. Measured positions 2D + 1 Reference system Horizontal / Planimetric positions Latitude, , and Longitude, Eastings, E, and Northings, N, (or X, Y) in a mapping plane Altitudes, H Heights above Mean Sea Level

8. Mapping

9. CERN Reference System A Reference System covering the whole of the CERN site First version established at the start of the PS Ring construction at CERN Defines the relative location all things at CERN Sites Buildings Tunnels Accelerators Experiments

10. CERN Reference System -1955 P0P0 P1P1 d 

11. CERN Reference System -1959 X Y P1P1 P0P0 P3P3

12. CERN Reference System -1962 X Y P2P2 P1P1 (X, Y) = (1000, 1000)

13. CERN Reference System -1966 P2P2 P1P1 (X, Y) = (2000, 2000) X Y

14. Altitude (Orthometric Height) Height above Mean Sea Level Mean Sea Level Represents 70% of the Earth’s surface! Traditionally determined by Tide Gauges An equipotential surface of the gravity field Equipotential Surface is modelled by a reference surface, Geoid The surface we choose depends on the accuracy required The accuracy required will also define the area over which a given surface is valid

15. CERN Vertical Reference –1954-1970 A horizontal plane (or different planes) OK for a small area Larger area means lower accuracy Easy for surveyors A Flat Earth Challenging for physicists!

16. CERN Vertical Reference –1954-1969 PS Horizontal Plane Altitude 433.660 m ISR Horizontal Plane Altitude 445.460 m

17. CERN Reference System -1970 CERN Coordinate System (CCS) A Reference Frame with a 3D Cartesian Coordinate System Principal Point, pillar P 0 X and Y-axes directions unchanged Z-axis coincident with local vertical P2P2 P1P1 (X, Y) = (2000, 2000) X Y P0P0

18. CCS –Principal Point Z-coordinate of PS Ring 2433.66000 m P 0 XY-Coordinates (m) (2000.00000, 2097.79265) Z-coordinate 2433.66000 m

19. Vertical Reference –a Sphere Sphere more complicated than a plane Higher accuracy over larger areas, Still easily defined mathematically

20. Z-Coordinates and Altitudes Z CCS = H + 2000

21. Z-Coordinates and Altitudes Z CCS ≠ H + 2000

22. Z-Coordinates and Altitudes Z-coordinate of PS Ring 2433.66000 m Z-coordinate of P 0 2433.66000 m Altitude (H) of PS Ring 433.66000 m Altitude (H) of P 0 433.65921 m Z CCS ≠ H + 2000

23. Z XY-Plane Altitude H = 10 000 m Z = 10 000 m H = 10 000 m Z = 0 m

25. CERN Reference System -1983 CERN Coordinate System (CCS) Unchanged New Vertical Reference Surface Increase in area covered by LEP (LHC) Higher precision model required

26. Biaxial Ellipsoid Model Ellipsoid of Revolution Ellipse rotated around one of its axes Mathematics not too complicated Closer match to the Earth’s shape and gravity field Positioned locally for an even better match Geodetic Reference Ellipsoid, GRS-80

27. 23/Mai/2001Mark Jones EST/SU -Séminaire Technique Topography of the Earth The ellipsoid doesn’t take into account the topography The Earth is irregular in shape The gravity field is affected by these irregularities

28. 23/Mai/2001Mark Jones EST/SU -Séminaire Technique Mountains affect the Gravity Field An equipotential surface of the gravity field Direction of the gravity vector Geoid Mass

29. Geoid Model –CERN Geoid 1985 Calculated differences between an ellipsoid and the Mean Sea Level equipotential of the gravity field Geoidal Undulations Institut d’Astronomie, BERN University A grid of data points Modelled by a polynomial surface Hyperbolic Paraboloid CG1985

30. CERN Reference System -2000 CERN Coordinate System (CCS) Unchanged Geodetic Reference Ellipsoid Unchanged New Geoid Model Assure direction of CNGS beamline Best recent model required

31. Geoid Model –CERN Geoid 2000 Calculated differences between an ellipsoid and the Mean Sea Level equipotential of the gravity field Geoidal Undulations Office Fédéral de Topographie, CH A grid of data points Interpolated between grid points CG2000

32. Vertical Reference Surfaces at CERN Geoid model, CG2000 Grid of points (1 km spacing) Cubic spline interpolation Geoid Model, CG1985 Hyperbolic paraboloid Spherical Model Cartesian Z-coordinate but how do we transform Z  H

33. Z  H Transformation Need to determine the relationship between the CCS Cartesian system and the Geoid Model Geoid model is tied to the Geodetic Reference Ellipsoid Need to establish the local position and orientation of the Reference Ellipsoid with respect to the CCS

34. Geodetic reference ellipsoid 23/Mai/2001Mark Jones EST/SU -Séminaire Technique Parameters: 2 radii Geodetic reference ellipsoid established locally to better model the geoid Position and orientation established by 7 parameters :  0, 0  latitude, longitude h 0  geodetic height  0  azimuth  0,  0  deflections of the vertical N 0  geoidal undulation

35. CERN Reference Ellipsoids Sphere Both radii equal Mean Earth Radius defined by the IUGG (International Union of Geodesy and Geophysics) R = 6371 km Reference Ellipsoid GRS-80 adopted by the IUGG a = 6 378 137 m, equatorial radius b = 6 356 752 m, polar radius

36. Geodetic Coordinates Latitude, , Longitude,, geodetic height, h P Geodetic reference ellipsoid Geodetic reference frame  h ZGZG XGXG YGYG

37. Geodetic Coordinates –P 0 Fix Latitude,  0 = 51.3692 grad Longitude, 0 = 6.72124 grad geodetic height, h 0 = 433.66000 m   h0h0 ZGZG XGXG YGYG P0P0

38. Fix  0 = 0.0000 grad  0 = 0.00000 m Geoid 23/Mai/2001Mark Jones EST/SU -Séminaire Technique h0h0 P0P0 p0p0 Horizontal plane Plan XY G ZGZG  Geoid

39. CCS and Geodetic Reference Frame 23/Mai/2001Mark Jones EST/SU -Séminaire Technique h0h0 P0P0 p0p0 CCS XY-plane Plan XY G ZGZG  Horizontal plane CCS Z-Axis Vertical Fix  0 = 0.0000 grad  0 = 0.0000 grad

40. CCS and Geodetic Reference Frame   h0h0 ZGZG XGXG YGYG P0P0  CCS = 37.77864 Grad X CCS Y CCS Z CCS  CCS

41. CERN Geodetic Reference Frame Provides the link between different coordinate systems (1D, 2D & 3D) CCS (3D) Altitudes (1D) Latitude and Longitude (2D) Mapping Planes (2D) Global Geocentric Reference Frames Relies upon a model for the shape of the Earth and the Gravity Field

42. Z and H 23/Mai/2001Mark Jones EST/SU -Séminaire Technique h0h0 P0P0 p0p0 CCS XY-plane Plan XY G ZGZG  P HPHP ZPZP CCS Z-Axis NPNP

43. Conclusions Z CCS ≠ H + 2000 Three different vertical reference surfaces Implies three Z  H Transformations Care is needed to use the right transformation!

44. Conclusions Things aren’t quite as simple as they used to be … … and things get more complicated as the required precision increases Changes in the gravity field Tides, atmospheric pressure, water tables, plate tectonics … More precise determination of the gravity field

45. Conclusions Fortunately we no longer calculate things by hand! We have developed a database and various software applications to help

46. Thank you for your attention!