Chapter2_Reference System bcbcccbcv.pptx

Umesh sir slide

1. Chapter 2: Reference System GE 601 UMESH BHURTYAL PASHCHIMANCHAL CAMPUS 6/16/2023 ‹#›

2. Content Reference system: 7 Hrs 2.1 Plane coordinate system, geodetic coordinate system, Geodetic, geocentric and astronomical latitude, Geodetic, geocentric and astronomical longitude, \Earth centered Earth fixed coordinate system, Laplace equation, deflection of vertical and Laplace station. 2.2 International terrestrial reference system (ITRF), WGS72, NAD83, WGS84 and Everest 1830 Ellipsoidal datum, Datum transformation from geodetic system to Cartesain and vice versa, determination of transformation parameters. 6/16/2023 ‹#›

3. Spatial Reference Spatial Reference = Datum + Projection + Coordinate system Datum – Horizontal and Vertical Datum Projection Coordinate System Coordinate systems are the central - mathematical - element of any geodetic reference system.  map projection is one of many methods used to represent the 3-dimensional surface of the earth or other round body on a 2-dimensional plane in cartography (mapmaking) 6/16/2023 ‹#›

4. Spatial Reference System Creating an unambiguous way of referencing location on the surface of the earth. 6/16/2023 ‹#›

5. Important of reference frame national and regional surveying and mapping works are all based on one single framework of geodetic control which is considered as a primary network of the country Applied in: Defense satellite lunching, missiles projecting , construction of major infrastructures of the country (dams, roads, sewerage system, irrigation, hydropower stations etc). 6/16/2023 ‹#›

6. Reference Surfaces A reference surface is a datum that provide a reference point to the Earth's surface Two main reference surfaces (or Earth figures) are used to approximate the shape of the Earth Geoid Ellipsoid the Geoid is the equipotential surface at mean sea level and is used for measuring heights represented on maps. starting point for measuring these heights are mean sea level points established at coastal places ellipsoid (also called spheroid) provides a relatively simple mathematical figure of the Earth used to measure locations, the latitude ( φ ) and longitude ( λ )of points of interest Some well known ellipsoids are the WGS84, GRS80, International 1924 (also known as Hayford), Krasovsky, Bessel, or the Clarke 1880 ellipsoid. 6/16/2023 ‹#›

7. Geoid The geoid is the shape that the ocean surface would have under the sole influence of gravity and the Earth’s rotation. when we say that the top of a hill is 130 metres above sea level, we actually consider it to be 130 metres above the geoid plumb line through any surface point is always perpendicular to Geoid the Geoid, is a reference surface for heights. 6/16/2023 ‹#›

8. The ellipsoid For the description of the horizontal coordinates (i.e. geographic coordinates) of points of interest ellipsoid is used ( oblate ellipsoid)  ellipsoid is formed when an ellipse is rotated about its minor axis provides a relatively simple figure which fits the Geoid to a first order approximation, An oblate ellipse, used to represent the Earth surface, defined by its the semi-major axis a and semi-minor axis b 6/16/2023 ‹#›

9. Defining Ellipsoid Flattening f Flattening f is dependent on both the semi-major axis a and the semi-minor axis b. The ellipsoid may also be defined by its semi-major axis a and eccentricity e, which is given by: 6/16/2023 ‹#›

10. Local and global ellipsoids Local ellipsoids have been established to fit the Geoid (mean sea level) well over an area of local interest, which in the past was never larger than a continent. differences between the Geoid and the reference ellipsoid could effectively be ignored, accurate maps can be drawn in the vicinity of the datum The Geoid, a globally best fitting ellipsoid for it, and a regionally best fitting ellipsoid for it, for a chosen region. 6/16/2023 ‹#›

11. The local horizontal datum Different local ellipsoids with varying position and orientation had to be adopted to best fit the local mean sea level in different countries or regions. Examples of local horizontal datums with their underlying ellipsoid and difference in position (datum shift) with respect to WGS84. North American Datum 1927 is used for the North American countries, the Tokyo Datum for Japan, the European Datum for the European countries, and the Amersfoort datum for the Netherlands 6/16/2023 ‹#›

12. The Global horizontal datum In contrast to local horizontal datum systems which apply only to a specific country or localized area of the Earth’s surface, global datum systems approximate the Geoid as a mean Earth ellipsoid. increasing demands for global surveying, activities are underway to establish  global reference surfaces 6/16/2023 ‹#›

13. WGS 84 The WGS 84 Coordinate System is a Conventional Terrestrial Reference System (CTRS). The definition of this coordinate system follows the criteria outlined in the International Earth Rotation Service (IERS) Technical Note It is geocentric, the center of mass being defined for the whole Earth including oceans and atmosphere. Its scale is that of the local Earth frame, in the meaning of a relativistic theory of gravitation. Its orientation was initially given by the Bureau International de l’Heure (BIH) orientation of 1984.0. Its time evolution in orientation will create no residual global rotation with regards to the crust. 6/16/2023 ‹#›

14. WGS 84 Origin = Earth’s center of mass Z-Axis = The direction of the IERS Reference Pole (IRP). This direction corresponds to the direction of the BIH Conventional Terrestrial Pole (CTP) (epoch 1984.0) with an uncertainty of 0.005”. X-Axis = Intersection of the IERS Reference Meridian (IRM) and the plane passing through the origin and normal to the Z-Axis. The IRM is coincident with the BIH Zero Meridian (epoch 1984.0) with an uncertainty of 0.005”. Y-Axis = Completes a right-handed, Earth-Centered Earth-Fixed (ECEF) orthogonal coordinate system. 6/16/2023 ‹#›

15. The global horizontal datum Global datum make geodetic results mutually comparable and to provide coherent results to other disciplines like astronomy and geophysics. The origin of the coordinate system is the geocentre. a three-dimensional coordinate system with a well-defined origin (the centre of mass of the Earth) and three orthogonal coordinate axes (X,Y,Z) Z -axis points towards a mean Earth north pole X -axis is oriented towards a mean Greenwich meridian and is orthogonal to the  Z -axis. The  Y -axis completes the right handed reference coordinate system (a) The International Terrestrial Reference System (ITRS), and; (b) the International Terrestrial Reference Frame (ITRF) visualized as a distributed set of ground control stations (represented by red points). The ITRS is realized through the International Terrestrial Reference Frame (ITRF), a distributed set of ground control stations that measure their position continuously using GPS. 6/16/2023 ‹#›

16. International Terrestrial Reference System (ITRS) This is a three-dimensional coordinate system with a well-defined origin (the centre of mass of the Earth) and three orthogonal coordinate axes (X, Y, Z). The Z-axis points towards a mean North Pole. The X-axis is oriented towards the mean Greenwich meridian and is orthogonal to the Z-axis. The Y -axis completes the right-handed reference coordinate system. Constant re-measuring is needed (addition of new control stations and ongoing geophysical processes (mainly tectonic plate motion)) deformations cause positional differences over time and have resulted in more than one realization of the ITRS Examples are the ITRF96 and the ITRF2000 . ITRF2000 or WGS84, are also called  geocentric datums 6/16/2023 ‹#›

17. ITRF 96 ITRF96 was established on 1 January 1997, which means that the measurements use data acquired up to 1996 to fix the geocentric coordinates (X, Y and Z in metres) and velocities (positional change in X, Y and Z in metres per year) of the different stations 6/16/2023 ‹#›

18. ITRF 2000  an improved frame in terms of quality, network and datum definition The ITRF2000 solution reflects the actual quality of space geodesy solutions, being free from any external constraints. It includes primary core stations observed by VLBI, LLR, SLR, GPS, and DORIS (usually used in previous ITRF versions) as well as regional GPS networks for its densification. 6/16/2023 ‹#›

19. ITRF 2020 ITRF2020 is the new realization of the International Terrestrial Reference System. Following the procedure already used for previous ITRF solutions, the ITRF2020 uses as input data time series of station positions and Earth Orientation Parameters (EOPs) provided by the Technique Centers of the four space geodetic techniques ( VLBI, SLR, GNSS and DORIS), as well as local ties at colocation sites. Based on completely reprocessed solutions of the four techniques, the ITRF2020 is expected to be an improved solution compared to ITF2014 6/16/2023 ‹#›

20. ITRF 2020 A number of innovations were introduced in the ITRF2020 processing, including: The time series of the four techniques were stacked all together, adding local ties and equating station velocities and seasonal signals at colocation sites; Annual and semi-annual terms were estimated for stations of the 4 techniques with sufficient time spans; Post-Seismic Deformation (PSD) models for stations subject to major earthquakes were determined by fitting GNSS/IGS data. The PSD models were then applied to the 3 other technique time series at earthquake colocation sites. 6/16/2023 ‹#›

21. Difference between ITRS and ITRF ITRF (International Terrestrial Reference Frame): The ITRF is a three-dimensional coordinate reference frame that provides a precise and consistent set of coordinates for points on the Earth's surface and in space. It is used as a common reference system by scientists, engineers, and surveyors worldwide. The ITRF is updated periodically to incorporate new measurements and improved modeling techniques. ITRS (International Terrestrial Reference System): The ITRS is a geocentric reference system that serves as the mathematical foundation for the ITRF. It defines the Earth's rotation and orientation parameters and provides a framework for relating the Earth-centered coordinates of the ITRF to the Earth's physical motion in space. The ITRS takes into account the motion of the Earth's tectonic plates, the effects of continental drift, and other geodynamic processes ITRS provides the theoretical framework for describing the Earth's rotation and orientation, while the ITRF is the practical realization of that framework, providing the actual coordinates for points on the Earth's surface and in space. 6/16/2023 ‹#›

22. Space techniques contributing to ITRS • very long baseline interferometry (VLBI) (high precision and long term stability) • satellite laser ranging (SLR) (long term stability and geo-centricity) • lunar laser ranging (LLR) (geo-centricity, long term stability, relativistic effects) • the French tracking system DORIS (excellent global station distribution) • Global Positioning System (densest global network, short term stability, high precision). Since the tracking network equipped with the instruments of these techniques is evolving and the period of data available increases with time, the ITRF is constantly being updated. 6/16/2023 ‹#›

23. VLBI VLBI provides the orientation of the ITRF relative to the celestial reference frame (i.e., the ‘distant stars’) and is also one of the two techniques currently used for accurately realizing the scale of the ITRF. SLR SLR is used to locate the center of mass of the Earth system and thereby defines the ITRF origin and contributes to the ITRF scale. GPS GPS contributes the large number of sites that define the ITRF (contributing to its density) and contributes to precise monitoring of polar motion. GPS, DORIS, and SLR are used to position space-orbiting platforms in the ITRF, and GPS is used to position instruments on the Earth’s land and sea surfaces (for example, tide gauges and buoys). None of the space geodesy techniques alone is capable of providing all the necessary parameters for ITRF definition (origin, scale, and orientation). 6/16/2023 ‹#›

24. The consequences of wrong Datum 6/16/2023 ‹#› Geodetic datum problems in air navigation were first encountered in Europe in the early 1970s during the development of multi-radar tracking systems for EUROCONTROL’s Maastricht Upper Airspace Centre (UAC), where plot data from radars located in Belgium, Germany and the Netherlands were processed to form a composite track display for air traffic controllers. Discrepancies in the radar tracks were found to be the result of incompatible coordinates.

25. Datum we use in Nepal A previous control survey of Nepal was made by Survey of India (SOI) during the period of 1964-1960. the geodetic reference system used was Indian Datum as defined by the following parameters Referenced Spheroid Everest (1830) Semi major axis a = 6377276.345 m Flattening f = 1/300.8017 Indian Datum Origin: Kalyanipur Latitude = 24 07’ 11.26” N Meridian component of the deflection of vertical = -0.29” Longitude = 77 39’ 17.57” E Prime vertical component of the deflection of vertical = +2.89” Geoid undulation N = 0 m 6/16/2023 Survey of India (SOI) established control points in Nepal for the topographical mapping in the scale 1 = 1 mile . Geodetic Survey Branch (GSB) established second, third and fourth order network of control points in the districts where cadastral survey has to be done.

26. Problem with Indian Datum and SOI Coordinates computation and the adjustments was done based on co-ordinates of points made available from SOI converted into UTM rectangular system modified for Nepal large scale coordinates were not from rigorous primary geodetic network it is called the provisional coordinates Z. M. Wiedner former director of GSB stated that these coordinates are basically for cadastral purpose only Govt. of Nepal decided to adopt metric system in the country it was necessary to used the factor of conversion from foot to meter 6/16/2023 ‹#›

27. Problem with Indian Datum and SOI Coordinates The conversion factor and the value of semi-major axis was 1 Indian foot = 0 .3 04 79841 m a= 6 377 276.345 m Adopting the Everest Spheroid (1830) with the value of semi-major axis stated above using SOI controls GSB started to establish new control points. Subsequently latter it was found that the conversion factor adopted in India was: 1 Indian foot= 0.3047996 m Therefore the value of semi-major axis comes to be a= 6 377 301.243 m the major axis is incorrect by 24.898 m (phuyal et.al 1992 ) difference of the semi-major axis, there will be a different values of co-ordinates of the SOI points converted to the rectangular UTM system, 6/16/2023 ‹#›

28. Datum we use in Nepal The present control survey of Nepal was made by Directorate of Military Survey, Ministry of Defence UK (MoDUK) during the period of 1981-1984. The geodetic reference system used was Nepal Datum as defined by the following parameters Referenced Spheroid Everest (1830) Semi major axis a = 6377276.345 m Flattening f = 1/300.8017 Nepal Datum Origin: 12/157 Nagarkot Latitude = 27 41’ 31.04” N Meridian component of the deflection of vertical = -37.03” Longitude = 85 31’ 20.23” E Prime vertical component of the deflection of vertical = -21.57” Geoid undulation N = 0 m 6/16/2023 The deflections quoted are derived from an astronomic position observed by Czechoslovak Geodetic Institute. government of Nepal and MODUK established the first order geodetic control net of 68 points in the country.

29. Nepal Datum The Nepal datum represents a rigorous reference system. The net in properly oriented to the conventional origin (CIO) and the scale of the net is consistent with the international standards of length defined by the Doppler satellite observation. As stated in the Report submitted by MODUK, the geographical co-ordinates of first order points are of high standard and hence fulfill the requirement of a rigorous Geodetic datum in Nepal. 6/16/2023 ‹#›

30. UTM GRID ZONE for Nepal 6/16/2023 ‹#›

31. MUTM 6/16/2023 ‹#›

32. Comparing Everest 1830 and WGS 84 6/16/2023 ‹#›

33. Map of Nepal 6/16/2023 ‹#›

34. Latitude and Longitude The  latitude   ( φ)   o f a point P is the angle between the ellipsoidal normal through P' and the equatorial plane. Latitude is zero on the equator ( φ  = 0°), and increases towards the two poles to maximum values of  φ  = +90 (90°N) at the North Pole and  φ  = - 90° (90°S) at the South Pole. The  longitude   ( λ )  is the angle between the meridian ellipse which passes through Greenwich and the meridian ellipse containing the point in question. It is measured in the equatorial plane from the meridian of Greenwich  ( λ€ = 0°) either eastwards through  λ  = + 180° (180°E) or westwards through  λ  = -180° (180°W). They are also called geodetic coordinates or ellipsoidal coordinates 6/16/2023 ‹#›

35. Geodetic and Geocentric Latitude Definition of latitude for a geocentric and geodetic coordinate system 6/16/2023 ‹#›

36. Different Latitudes The geodetic latitude   ( φ)   o f a point P is the angle between the ellipsoidal normal through P' and the equatorial plane. The  geocentric latitude  ( φ ' ) is the angle between the equatorial plane and a line from the center of the ellipsoid (used to represent the Earth). The  astronomic latitude   ( Φ ) is the angle between the equatorial plane and the normal to the Geoid (i.e. a plumb line). Three different latitudes: the geodetic (or geographic) latitude ( φ  ), the astronomic latitude ( Φ ) and the geocentric latitude ( φ€ ' ).   only difference from coordinate systems in geographic and geocentric formats is their definition of latitude. 6/16/2023 ‹#›

37. Use of geodetic latitudes Geodetic latitude is widely used in various applications that require accurate location measurements and mapping on Earth's surface. Navigation and GPS: Geodetic latitude is essential for navigation systems and GPS (Global Positioning System) devices. Cartography and Mapping: Geodetic latitude is crucial for creating accurate maps and charts. It forms the basis for defining grid systems, such as latitude and longitude, that are used to represent Earth's surface on flat maps. Geographical and Earth Sciences: Geodetic latitude is fundamental to geographic information systems (GIS) and other geographical research Civil Engineering and Infrastructure: Geodetic latitude plays a crucial role in civil engineering projects, including the design and construction of roads, bridges, tunnels, and buildings. Used in location measurements, mapping, and spatial analysis on Earth's surface. 6/16/2023 ‹#›

38. Use of geocentric latitude Geocentric latitude is primarily used in geodesy, which is the science of measuring and understanding Earth's shape, size, and gravitational field. Satellite Tracking:Geocentric latitude is employed in tracking and predicting the positions of satellites orbiting the Earth Geodetic Reference Systems: International Terrestrial Reference System (ITRS) and the World Geodetic System (WGS) Geodetic Coordinate Transformations: transforming coordinates between different reference frames or coordinate systems, geocentric latitude serves as an intermediate step. geodetic latitude and longitude >>> geocentric Cartesian coordinates (x, y, z) using geocentric latitude as an intermediary parameter. Gravitational Studies: Geocentric latitude plays a role in gravitational studies, such as modeling the Earth's gravitational field and determining the geoid (a representation of Earth's shape considering gravitational equipotential surfaces). Geocentric latitude is used in calculating gravity anomalies and understanding the Earth's gravitational forces. 6/16/2023 ‹#›

39. Use of astronomical latitude Astronomical latitude, also known as celestial latitude, is primarily used in the field of astronomy and celestial navigation. Celestial Navigation: Astronomical latitude is crucial in celestial navigation, which involves determining the position of a vessel or observer on Earth using celestial bodies. Astronomical Observations: Astronomical latitude is used to calculate the position of celestial objects in the sky relative to an observer's location on Earth. Astrometry: Astrometry is the precise measurement of the positions and motions of celestial objects. Astronomical latitude is employed in astrometry to accurately determine the declination (angular distance north or south of the celestial equator) of celestial objects. Timekeeping and Earth's Orientation: Astronomical latitude plays a role in determining and monitoring Earth's orientation in space. Celestial Mechanics: In the field of celestial mechanics, astronomical latitude is employed in orbital calculations and the study of the motion of celestial bodies. 6/16/2023 ‹#›

40. Geodetic, geocentric and astronomical longitudes For all systems, the longitude of a point is defined as the given angle between the meridian of origin and the meridian passing through that point. 6/16/2023 ‹#›

41. 3D geographic coordinates are obtained by introducing the ellipsoidal height h to the system The ellipsoidal height (h) of a point is the vertical distance of the point in question above the ellipsoid. It is measured in distance units along the ellipsoidal normal from the point to the ellipsoid surface. 3D geographic coordinates can be used to define a position on the surface of the Earth (point P in figure below). 6/16/2023 ‹#›

42. Geocentric coordinates (X,Y,Z) An alternative method of defining a 3D position on the surface of the Earth is by means of geocentric coordinates (x,y,z), also known as 3D Cartesian coordinates. The system has its origin at the mass-centre of the Earth with the X- and Y-axes in the plane of the equator. The X-axis passes through the meridian of Greenwich, and the Z-axis coincides with the Earth's axis of rotation. The three axes are mutually orthogonal and form a right-handed system. Geocentric coordinates can be used to define a position on the surface of the Earth (point P in figure below). 6/16/2023 ‹#›

43. Geocentric coordinates (X,Y,Z) It should be noted that the rotational axis of the Earth changes its position over time (referred to as polar motion). To compensate for this, the mean position of the pole in the year 1903 (based on observations between 1900 and 1905) has been used to define the so-called 'Conventional International Origin' (CIO). 6/16/2023 ‹#›

44. Laplace Equation The Laplace equation expresses the relationship between astronomic azimuth, geodetic azimuth and the astronomic longitude and geodetic longitude. the astronomic latitude is normally observed at each Laplace station Deflection-of-the-Vertical The Laplace correction is used to relate a geodetic azimuth to an astronomical azimuth. Such a correction is made at a point called a Laplace Station, where the deflection-of-the-vertical components are known. Laplace equations are used to connect the physical world with a mathematical representation. 6/16/2023 ‹#›

45. Laplace equation The difference between the ellipsoid normal and the vertical plumb line is called deflection-of-the-vertical Let geodetic azimuth α G • Astronomical azimuth α A • Let geodetic longitude λ G • Astronomical longitude λ A α A - α G = ( λ A - λ G )sin φ This is called Laplace Azimuth Equation 6/16/2023 ‹#›

46. The relationships between astronomical latitude, longitude, and azimuth and geodetic latitude, longitude, and azimuth are Given Equations can be used two ways— if the geodetic latitude and longitude and the astronomic latitude and longitude are known, then deflections-of-the-vertical can be computed at that point. If the deflection-of-the-vertical components are known, then geodetic latitude, longitude, and azimuth can be computed from astronomic latitude, longitude, and azimuth and vice versa. 6/16/2023 ‹#›

47. Datum transformation 6/16/2023 ‹#› Datum shift between two geodetic datums. Apart from different ellipsoids, the centres or the rotation axes of the ellipsoids do not coincide. when the source projection is based upon a different horizontal datum than the target projection Datum transformation is required Datum transformations   are transformations from a 3D coordinate system (i.e. horizontal datum) into another 3D coordinate system. spatial data with different underlying horizontal datums needs a  datum transformation for proper alignment Datum transformation  is part of  Coordinate transformation

48. Datum Transformation The inverse mapping equation of projection A is used first to take us from the map coordinates ( x , y ) of projection A to the geographic coordinates ( φ , λ ) in datum A. Next, the datum transformation takes us from the geographic coordinates ( φ , λ ) in datum A to the geographic coordinates ( φ , λ ) in datum B. Finally, the forward equation of projection B is used to take us from the geographic coordinates ( φ , λ ) in datum B to the map coordinates ( x’ , y' ) of projection B. 6/16/2023 ‹#› The principle of changing from one into another projection combined with a datum transformation from datum A to datum B.

49. GCS to CCS and CCS to GCS 6/16/2023 ‹#› These formulae can be used to convert geographic coordinates, latitude (  Φ  ), longitude (  λ  ), and height ( h ), into Cartesian coordinates ( X, Y, Z  ) and vice versa N is radius of curvature in the prime vertical

50. Example: Convert the given geographic (or geodetic) to geocentric (or 3D Cartesian) coordinates 6/16/2023 ‹#› φ   (ITRF)   =  48° 46' 59.6564" N λ  (ITRF) =  9° 10' 30.6113" E h (ITRF) =  330.397m X (ITRF) =  4,156,939.96m Y (ITRF) =  671,428.74m Z (ITRF) =  4,774,958.21m

51. Transformation Parameters additions to linear coordinates in the transformation from source to target denoted by Δ X, Δ Y & Δ Z in the case of Cartesian coordinates and Δ x, Δy in the case of grid coordinates arise from the movement of the origin from the source datum to the target datum, as illustrated in Figures Translation parameters do not affect conformality (preservation of shape) 6/16/2023 ‹#› Change of origin giving rise to translation parameters for grid coordinates Change of origin giving rise to translation parameters for Cartesian coordinates.

52. Scaling Parameter These are multiplying factors applied to distances in the directions of the axes. In the often occurring case of conformal transformations, there is only one scaling parameter and it is applied to distances in any direction. A scaling parameter can be expressed as a scale factor L or a scale change Δ L (where L=1+ Δ S). Δ S is often given in parts per million (ppm). define rotation of axes in a grid plane or in 3-dimensions Rotation parameters do not affect conformality 6/16/2023 ‹#› Rotation Parameter Rotation convention for a position vector in the Oxy plane. Rotation conventions for position vectors in the OYZ plane, the OZX plane and the OXY plane

53. Transformation models Several mathematical models have been developed which describe the functional relationship between pairs of three dimensional coordinates. The two most commonly used mathematical models to transform positions between the reference systems are Bursa-Wolf (Bursa, 1962, Wolf, 1963) and Molodensky (Molodensky et.al., 1962) These are the standard models due to their extensive use around the world over a number of years. The only difference between these two methods is Molodensky uses local origin about which the transformation is performed where as Bursa-Wolf method uses reference system origin. Bursa–Wolf transformation more popular than Molodensky transformation 6/16/2023 ‹#›

54. Bursa-Wolf The Bursa-Wolf method assumes a similarity three dimensioned relationship between two consistent sets of Cartesian coordinate through seven parameters: If U, V and W represent the Cartesian components of a station in reference frame 1 say Everest and X, Y, Z represent the Cartesian component of same stations in reference frame number 2 say WGS-84, the transformation can be expressed as: …………….(1) 6/16/2023 ‹#› The approximation is very good as long as the three rotation angles are rather small, on the order of two or three seconds of arc. When rotation angles exceed these values, the results begin to strain normally expected levels of accuracy. ppm is also "mm per km" since there are 1 million millimetres in 1 kilometre.

55. Bursa-wolf where R represents a 3 x 3 rotation matrix and defined a If all three angles are small the above rotation matrix can be written in its simplified form by setting sine of an angle equal to the angle itself, cosine of the angle equal to 1 and the product of sines equal to zero. This approximation is valid only for small angles. Thus after simplification the above matrix will appear as ……..(2) The transformation equation (1) can now be written as 6/16/2023 ‹#› Note: the rotation parameters ( R ) must be converted from arc-seconds to radians before being used in this equation.

56. Method of estimation of transformation parameters A point physically identifiable on the surface of Earth which has been assigned coordinates in at least two separate systems of coordinates is termed as collocated station. The Cartesian coordinates of sufficient number of collocated stations (U, V, W, X, Y, Z) can be used as observations in a least square adjustment for the seven transformation parameters. The model in symbolical form can be written as: 6/16/2023 ‹#› By arranging the above equation into this form will result in

57. Method of estimation of transformation parameters… These three equations represent the functional relationship between any two closely oriented, closely scaled, ortho normal Cartesian coordinates systems. Since the observations (6 Cartesian components per station) have systematic and other errors with them, the usual combined least squares procedure of minimizing the weighted sum of residuals squared is followed. Finally transformation parameters can be computed. 6/16/2023 ‹#›

58. Datum Transformation parameters of Nepal 6/16/2023 ‹#› Source Manandhar, N. : Geoid Studies of Nepal, Thesis submitted for Master of Engineering, School of Geomatic Engineering, University of New South Wales, Sydney. This transformation parameter has been used to transform the topographic database of 1: 25000, and 1: 50,000 series 3-dimentional coordinates based on WGS-84 were taken as the controls established by ENTMP. This network consists of 29 primary stations and 72 secondary stations. A total of 11 primary GPS points are common stations in first order geodetic network of Nepal. From these 11 stations only 9 stations are used for the derivation of the transformation parameter. The Bursa-Wolf method was used to estimate the transformation parameter. Transformation Parameter from Nepal Datum to WGS 84