Geodesy Notes: Datum & Geodetic Coordinate Systems

Datum & Geodetic Coordinate Systems

Definition of Datum

• Datum are the basis for all geodetic survey work, acting as reference points.
• Analogous to starting points when giving directions.
• Used by geodesists and surveyors to create maps, mark property boundaries, and plan infrastructure.
• A datum is a set of spatial information that acts as a foundation for other data.
• Without a datum, there would be nothing to securely support other spatial information, such as digital elevations, land use, or population.
• Two types of datum: horizontal and vertical.
• A geodetic datum or geodetic system is a coordinate system and a set of reference points used to locate places on the Earth to begin surveys and create maps.
• Geodetic datum defines the size and shape of the earth and the origin and orientation of the coordinate system used to map the earth.
• Modern geodetic datum range from flat-earth models used for plane surveying to complex models used for international applications which completely describe the size, shape, orientation, gravity field and angular velocity of the earth.

Local & Global Datum

• Local datum: an ellipsoid that has been established to fit the geoid (MSL) well over an area of local interest.
• Differences between the Geoid and the reference ellipsoid could be ignored, allowing accurate maps to be drawn.
• Global datum: an ellipsoid that best fits approximately to the Geoid that covers the whole Earth's shape.

Horizontal Datum

• Used for describing a point on the earth’s surface in latitude and longitude or other coordinate system.
• The horizontal datum is the model used to measure positions on the earth.
• A specific point on the earth can have substantially different coordinates, depending on the datum used to make measurement.
• There are hundreds of locally developed horizontal datum around the world, usually referenced to some convenient local reference point.
• Ellipsoids have varying position and orientations.
• An ellipsoid is positioned and oriented with respect to the local mean sea level (or Geoid) by adopting a latitude $\varphi$ and longitude $\lambda$ and ellipsoidal height $h$ of a so- called fundamental point and an azimuth to an additional point.
• The local horizontal datum is realized through a so- called triangulation network (or survey network).
• The angles in each triangle are measured in addition to at least one side of a triangle; the fundamental point is also a point in the triangulation network.
• Local horizontal datum have been established to fit the Geoid well over the area of local interest, which in the past was never larger than a continent.
• The most important global (or geocentric) spatial reference system for the GIS community is the International Terrestrial Reference System (ITRS).
• It 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 Earth north pole. The 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.
• 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.
• Constant re-measuring is needed because of the involvement of new control stations and ongoing geophysical processes (mainly tectonic plate motion) that deform the Earth’s crust at measurable global, regional and local scales.
• These deformations cause positional differences in time, and have resulted in more than one realization of the ITRS. Examples are the ITRF96 or the ITRF2000 datum.
• The World Geodetic System of 1984 (WGS84) datum has been refined on several occasions and is now aligned with the ITRF to within a few centimetres worldwide. The Global Positioning System (GPS) uses the WGS84 as its reference system.
• Global horizontal datum, such as the ITRF2000 or WGS84, are also called geocentric datum because they are geocentrically positioned with respect to the centre of mass of the Earth.
• To implement the ITRF in a region, a densification of control stations is needed to ensure that there are enough coordinated reference points available in the region.
• These control stations are equipped with permanently operating satellite positioning equipment (i.e. GPS receivers and auxiliary equipment) and communication links.
• Examples for (networks consisting of) such permanent tracking stations are the AGRS in the Netherlands and the SAPOS in Germany and also MyRTKnet in Malaysia.
• ITRF coordinates (X,Y and Z in metres) can be transformed into geographic coordinates ($\varphi$, $\lambda$, $h$) with respect to the GRS80 ellipsoid without the loss of accuracy.
• However, the ellipsoidal height $h$, obtained through this straightforward transformation, is has no physical meaning and contrary to our intuitive human perception of height. Therefore we use the height $H$, above the Geoid. Height $h$ above the geocentric ellipsoid, and height $H$ above the Geoid. The first is measured orthogonal to the ellipsoid, the second orthogonal to the Geoid.

Vertical Datum

• A vertical datum is used for measuring the elevations of points on the earth’s surface.
• Vertical datum is either tidal based on sea levels, gravimetric based on a geoid, or geodetic based on the same ellipsoid models of the earth.
• In common usage, elevations are often cited in height above mean sea level; widely used tidal datum
• The local vertical datum (or height datum) is implemented through a levelling network.
• A levelling network consists of benchmarks, whose height above mean sea level has been determined through geodetic levelling (precise levelling).
• The implementation of the datum enables easy user access. The surveyors do not need to start from scratch (STAPS)every time they need to determine the height of a new point.
• They can use the benchmark of the levelling network that is closest to the point of interest.
• The use of satellite-based positioning equipment (e.g. GPS) to determine heights with respect to a reference ellipsoid (e.g. WGS84) is becoming more in use.
• These heights are known as the ellipsoidal heights (height $h$ above the ellipsoid).
• Ellipsoidal heights have to be adjusted before they can be compared to orthometric (mean sea level) heights.
• Geoid undulations ($N$) are used to adjust the ellipsoidal heights ($H = h - N$). Height $h$ above the reference ellipsoid and height $H$ above the Geoid for two points on the Earth surface. The ellipsoidal height is measured orthogonal to the ellipsoid. The orthometric height is measured orthogonal to the Geoid.

Function of Geodesy Datum

• Control the coordinate and bearing
• To ensure the consistency of the system
• Surveyor need to integrate the data sets
• Purpose applications with a border land between the border states
• Positioning use
• As a control to surveying work
• Ensure the consistency of regular coordinates

Geodetic Coordinate System

• Different kind of coordinates are used to position objects in a two- or three-dimensional space.
• Spatial coordinates (also known as global coordinates) are used to locate objects either on the Earth’s surface in a 3D space, or on the Earth’s reference surface (ellipsoid or sphere) in a 2D space.
• Specific examples are the geographic coordinates in a 2D or 3D space and the geocentric coordinates, also known as 3D Cartesian coordinates.
• Planar coordinates on the other hand are used to locate objects on the flat surface of the map in a 2D space. Examples are the 2D Cartesian coordinates and the 2D polar coordinates.

2D Geographic Coordinates ($\varphi$, $\lambda$)

• The most widely used global coordinate system consists of lines of geographic latitude (phi or $f$ or \j) and longitude (lambda or $l$).
• Lines of equal latitude are called parallels. They form circles on the surface of the ellipsoid.
• Lines of equal longitude are called meridians and they form ellipses (meridian ellipses) on the ellipsoid.
• Both lines form the graticule when projected onto a map plane.
• Note that the concept of geographic coordinates can also be applied to a sphere as the reference surface.
• Latitude and longitude represent the geographic coordinates ($f$,$l$) of a point $P'$ with respect to the selected reference surface.
• They are also called geodetic coordinates or ellipsoidal coordinates when an ellipsoid is used to approximate the shape of the Earth.
• Geographic coordinates are always given in angular units.

3D geographic coordinates ($\varphi$, $\lambda$, $h$)

• 3D geographic coordinates ($\varphi$, $\lambda$, $h$) 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. The latitude ($f$ ) and longitude ($l$) angles and the ellipsoidal height ($h$) represent the 3D gegraphic coordinate system.

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.

2D Cartesian Coordinates (X,Y)

• A flat map has only two dimensions: width (left to right) and length (bottom to top).
• Transforming the three dimensional Earth into a two- dimensional map is subject of map projections and coordinate transformations.
• Here, like in several other cartographic applications, two- dimensional Cartesian coordinates (x, y), also known as planar rectangular coordinates, are used to describe the location of any point in a map plane, unambiguously.
• The 2D Cartesian coordinate system is a system of intersecting perpendicular lines, which contains two principal axes, called the X- and Y-axis.
• The horizontal axis is usually referred to as the X-axis and the vertical the Y-axis (note that the X-axis is also sometimes called Easting and the Y-axis the Northing).
• The intersection of the X- and Y-axis forms the origin.

2D Polar Coordinates ($\alpha$,$d$)

• Another possibility of defining a point in a plane is by polar coordinates ($\alpha$,$d$).
• This is the distance $d$ from the origin to the point concerned and the angle $\alpha$ between a fixed (or zero) direction and the direction to the point.
• The angle $\alpha$ is called azimuth or bearing and is measured in a clockwise direction.
• It is given in angular units while the distance $d$ is expressed in length units.

3D Cartesian Coordinates (X, Y, Z)

• The X axis is defined by the intersection of the plane and defined also by the prime meridian and the equatorial plane.
• The Y axis completes a right handed orthogonal system by plane 90° E of the X axis and its intersection with the equator.
• The Z axis points toward to the North Pole.

Geocentric & Topocentric Coordinate System

Geocentric Coordinate System

• Coordinate system defined using an ellipsoid
• Any point in the meridian will have the same longitude.
• This value is determined by the size of the angular between the normal line through the point and the plane of the equator.

Topocentric Coordinate System

• The framework is centered on the origin system which is located at a point on the topography surface.
• The system is known as local coordinate geodetic system.

Map Projection

• Maps are one of the world’s oldest types of document.
• For quite some time it was thought that our planet was flat, and during those days, a map simply was a miniature representation of a part of the world.
• Now that we know that the Earth’s surface is curved in a specific way, we know that a map is in fact a flattened representation of some part of the planet.
• The field of map projections concerns itself with the ways of translating the curved surface of the Earth into a flat map.
• A map projection is a mathematically described technique of how to represent the Earth’s curved surface on a flat map.
• To represent parts of the surface of the Earth on a flat paper map or on a computer screen, the curved horizontal reference surface must be mapped onto the 2D mapping plane.
• The reference surface for large-scale mapping is usually an oblate ellipsoid, and for small-scale mapping, a sphere.
• Mapping onto a 2D mapping plane means transforming each point on the reference surface with geographic coordinates ($\varphi$, $\lambda$) to a set of Cartesian coordinates (x, y) representing positions on the map plane.
• The actual mapping cannot usually be visualized as a true geometric projection, directly onto the mapping plane.
• This is mostly achieved through mapping equations.
• A forward mapping equation transforms the geographic coordinates ($f$, $l$) of a point on the curved reference surface to a set of planar Cartesian coordinates (x, y), representing the position of the same point on the map plane: $(x, y) = f (f, l)$
• The corresponding inverse mapping equation transforms mathematically the planar Cartesian coordinates (x,y) of a point on the map plane to a set of geographic coordinates ($f$, $l$) on the curved reference surface: $(f, l) = f (x, y)$

Classification of Map Projections

Map projections can be described in terms of their:

• class (cylindrical, conical or azimuthal),
• point of secancy (tangent or secant),
• aspect (normal, transverse or oblique), and
• distortion property (equivalent, equidistant or conformal).

Classes of Map Projection

• The three classes of map projections are cylindrical, conical and azimuthal.
• The Earth's reference surface projected on a map wrapped around the globe as a cylinder produces a cylindrical map projection.
• Projected on a map formed into a cone gives a conical map projection.
• When projected directly onto the mapping plane it produces an azimuthal (or zenithal or planar) map projection.

Point of Secancy

• The planar, conical, and cylindrical surfaces are all tangent surfaces; they touch the horizontal reference surface in one point (plane) or along a closed line (cone and cylinder) only.
• Another class of projections is obtained if the surfaces are chosen to be secant to (to intersect with) the horizontal reference surface.
• Then, the reference surface is intersected along one closed line (plane) or two closed lines (cone and cylinder).
• Secant map surfaces are used to reduce or average scale errors because the line(s) of intersection are not distorted on the map.

Aspect of Map Projection

• Projections can also be described in terms of the direction of the projection plane's orientation (whether cylinder, plane or cone) with respect to the globe. This is called the aspect of a map projection.
• The three possible aspects are normal, transverse and oblique.
• In a normal projection, the main orientation of the projection surface is parallel to the Earth's axis.
• A transverse projection has its main orientation perpendicular to the Earth's axis. Oblique projections are all other, non-parallel and non-perpendicular, cases.

Distortion Property

• The distortion properties of map are typically classified according to what is not distorted on the map:
• In a conformal (orthomorphic) map projection the angles between lines in the map are indentical to the angles between the original lines on the curved reference surface. This means that angles (with short sides) and shapes (of small areas) are shown correctly on the map.
• In an equal-area (equivalent) map projection the areas in the map are identical to the areas on the curved reference surface (taking into account the map scale), which means that areas are represented correctly on the map.
• In an equidistant map projection the length of particular lines in the map are the same as the length of the original lines on the curved reference surface (taking into account the map scale).

Coordinate Transformation

• Map and GIS users are mostly confronted in their work with transformations from one two-dimensional coordinate system to another.
• This includes the transformation of polar coordinates delivered by the surveyor into Cartesian map coordinates or the transformation from one 2D Cartesian (x, y) system of a specific map projection into another 2D Cartesian (x, y) system of a defined map projection. Integration of spatial data into one common coordinate system
• Datum transformations are also important, usually for mapping purposes at large and medium scale. An example, map and GIS users are often collecting spatial data in the field using satellite navigation technology and need to represent this data on published maps on a local horizontal datum.
• The principle of changing from one unknown projection into a known projection using a 2D Cartesian transformation. A number of 2D control points are required to determine the relation between both systems. The principle of changing from one into another projection using the mapping equations.

Datum Transformation

• Datum transformations are transformations from a 3D coordinate system (i.e. horizontal datum) into another 3D coordinate system.
• The transformation parameters to take us from one datum system to another datum system are estimated on the basis of a set of selected points whose coordinates are known in both datum systems. Datum shift between two geodetic datum. Apart from different ellipsoids, the centres or the rotation axes of the ellipsoids do not coincide.
• 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 ($\varphi$, $l$) in datum A.
• Next, the datum transformation takes us from the geographic coordinates ($\varphi$, $l$) in datum A to the geographic coordinates ($\varphi$, $l$) in datum B.
• Finally, the forward equation of projection B is used to take us from the geographic coordinates ($\varphi$,$l$) in datum B to the map coordinates (x’,y') of projection B.
• A height coordinate ($h$ or $H$) may be added to the geographic coordinates. The principle of changing from one into another projection combined with a datum transformation from datum A to datum B.

3 Parameters of Datum Transformation

• The shift on X axis ($DX$)
• The shift on Y axis($DY$)
• The shift on Z axis($DZ$)

7 Parameters of Datum Transformation

• The shift on X axis ($DX$)
• The shift on Y axis($DY$)
• The shift on Z axis($DZ$)
• X axis rotation ($Rx$)
• Y axis rotation ($Ry$)
• Zaxis rotation ($Rz$)
• A scale correction ($Sc$)

Datum Transformations Via Geocentric Coordinates

• Datum transformations via the geocentric coordinates (x, y, z) are 3D similarity transformations.
• Essentially, these are transformations between two orthogonal 3D Cartesian spatial reference frames together with some elementary tools from adjustment theory.
• The three most applied methods for a datum transformation via the 3-dimensional geocentric coordinates are:
  • The geocentric translation,
  • The Helmert 7-parameter transformations
  • The Molodensky-Badekas 10- parameter transformation.

Datum Transformations Via Geographic Coordinates

• Datum transformations via the geographic coordinates directly relate the ellipsoidal latitude ($\varphi$) and longitude ($l$), and possibly also the ellipsoidal height ($h$), of both datum systems.
• The applied methods for a datum transformation via the 3-dimensional geographic coordinates are:
  • the geographic offsets,
  • the Molodensky and Abridged Molodensky transformation,
  • the multiple regression transformation.

Conversions From Geographic To Geocentric Coordinates And Visa Versa

• The principle of a datum transformation via the geocentric coordinates. The datum transformation is combined with conversions between the 3D geographic coordinates and geocentric coordinates in both datum systems.
• The conversion from the latitude and longitude coordinates into the geocentric coordinates is rather straightforward and turns ellipsoidal latitude ($f$) and longitude ($l$), and possibly also the ellipsoidal height ($h$), into X,Y and Z, using 3 direct equations.
• If the ellipsoidal semi-major axis is a, semi-minor axis b, and inverse flattening 1/f, then:
  • $X = (N + h) \cos f \cos l$
  • $Y = (N + h) \cos f \sin l$
  • $Z = [(1-e^2)N + h] \sin f$
  • $N = \frac{a}{\sqrt{1 - e^2 \sin^2 f}}$

Geocentric To Geographic Conversion

• The inverse equations for the reverse conversion are more complicated and require either an iterative calculation of the latitude and ellipsoidal height, or it makes use of approximating equations like those of Bowring.
• These last have millimetre precision for 'Earth-bound' points, i.e. points that are at most 10 km away from the ellipsoidal surface (any point on the Earth surface)

2D Cartesian Coordinate Transformations

• 2D Cartesian coordinate transformations can be used to transform 2D Cartesian coordinates (x, y) from one 2D Cartesian coordinate system to another 2D Cartesian coordinate system. The three primary transformation methods are:
  • the conformal transformation,
  • the affine transformation, and
  • the polynomial transformation.

The Conformal Transformation

• A conformal transformation is a linear (or first-order) transformation and relates two 2D Cartesian coordinate systems through a rotation, a uniform scale change, followed by a translation. The rotation is defined by one rotation angle (a), and the scale change by one scale factor (s). The translation is defined by two origin shift parameters (xo, yo). The equation is:
  • $X' = s X \cos(a) - s Y \sin(a) + x_o$
  • $Y' = s X \sin(a) + s Y \cos(a) + y_o$
• The simplified equation is:
  • $X' = aX - bY + x_o$
  • $Y' = bX + aY + y_o$
• where a=s cos(a) and b=s sin(a). The transformation parameters (or coefficients) are a,b,xo,yo.

The Affine Transformation

• An affine transformation is a linear (or first-order) transformation and relates two 2D Cartesian coordinate systems through a rotation, a scale change in x- and y-direction, followed by a translation.
• The transformation function is expressed with 6 parameters: one rotation angle (a), two scale factors, a scale factor in the x-direction (sx) and a scale factor in the y-direction (sy), and two origin shifts (xo, yo). The equation is:
  • $X' = s<em>x X \cos(a) - s</em>y Y \sin(a) + x_o$
  • $Y' = s<em>x X \sin(a) + s</em>y Y \cos(a) + y_o$
• The simplified equation is:
  • $X' = aX - bY + x_o$
  • $Y' = cX + dY + y_o$
• where the transformation parameters (or coefficients) are a,b,c,d,xo,yo.

The Polynomial Transformation

• A polynomial transformation is a non-linear transformation and relates two 2D Cartesian coordinate systems through a translation, a rotation and a variable scale change. The transformation function can have an infinite number of terms. The equation is:
  • $X' = x<em>o + a</em>1X+ a<em>2Y+ a</em>3XY + a<em>4X^2+ a</em>5Y^2+ a<em>6X^2Y+ a</em>7XY^2+ a_8X^3+……..$
  • $Y' = y<em>o+ b</em>1X+ b<em>2Y+ b</em>3XY + b<em>4X^2+ b</em>5Y^2+ b<em>6X^2Y+ b</em>7XY^2+ b_8X^3+……..$
• Polynomial transformations are sometimes used to georeference uncorrected satellite imagery or aerial photographs or to match vector data layers that don't fit exactly by stretching or rubber sheeting them over the most accurate data layer.