4.1 Reference Coordinate Systems

• Appropriate, well defined and reproducable reference coordinate systems are necessary for the description of satellite motion, observables and models.
• The increasing accuracy of many satellite observation techniques requires the corresponding increase in the accuracy of reference systems.
• Reference coordinate systems in satellite geodesy are global and geocentric (Earth center of mass) by nature.
• Reference coordinate systems for terrestrial measurements are local and topocentric by nature.
• The relationship between global and local reference coordinate systems might be know with an accuracy compatible with the accuracy of the individual observations techniques: this is a goal.
• Some authors distinguish recently between:
  • A reference system: the conceptual idea, including the fundamental theory and standards.
  • A reference frame: the practical realization of the reference system through observations and a set of station coordinates (fiducial stations).

4.1.1 Reference Coordinate Systems in Satellite Geodesy

• Two systems are required:
  • A space-fixed, inertial reference system (CIS) for the description of satellite motion, where the Newton's laws of motion are valid.
  • An earth-fixed, terrestrial reference system (CTS) for the positions of the observations and for the description of results from satellite geodesy.
• A good approximation for a CIS is given by the quasi-inertial equatorial system of the Astronomy (named in that way due to the accelerations of the annual motion of the Sun around the Earth).
• Its definition is:
  • Origin: geocenter
  • The positive Z-axis oriented towards the north pole
  • The positive X-axis to the First Point of Aries ( ).
  • The Y-axis completes a right handed system
• The transformations between the associated spherical coordinates right ascension , declination and geocentric distance r with the cartesian coordinates X,Y,Z are:

  

• The CIS reference frame was given by the fundamental stars FK5 (accuracy of ) and now for the HIPPARCOS output catalogue ( ).
• A more precise definition (regarding to FK5) is given by means of radiotelescopes with the VLBI technique, but in the recent pass there were not direct links between the two reference frames.
• The CTS (also named Earth Centered Earth Fixed System, ECEF) can be realized through a set of Cartesian coordinates of fundamental stations within a global network, by means of the Conventional Terrestrial Pole (CTP) and a zero longitude on the equator (Greenwich Mean Observatory, GMO).
• The transformation from CIS to CTS is realized through a sequence of rotations that account for:
  • Precession
  • Nutation
  • Earth rotation including polar motion

  

• The Earth's axis of rotation and its equatorial plane are not fixed in space, but rotate with respect to an inertial system, basically due to the gravitational attraction of the Sun and the Moon on the equatorial bulge of the Earth.
• The total motion is composed of a mean secular component (precession: mean equator, mean equinox and mean positions) and a periodic component (nutation: true equator, true equinox and true positions): one consequence is that the Earth rotation axis describes an oscullating conic about the ecliptic pole .
• The mean positions can be transformed from the reference epoch (J2000) to the required observation epoch t by means of the precession rotation matrix:

  

  being the three rotation angles:

  

  and is counted in Julian centuries of 36525 days.

• The transformation from the mean equator and equinox to the instantaneous true equator and equinox for a given observation epoch:

  

  where is the obliquity of the ecliptic, is the nutation in obliquity and is the nutation in longitude counted in the ecliptic:

  

  being the mean longitude of the lunar ascending node and D the mean elongation of the moon from the Sun and ....

• Finally we obtain the true coordinates .
• For the transformation from an instantaneous space-fixed equatorial system to an conventional terrestrial reference system we need the apparent Greenwich sidereal time (GAST) and the pole coordinates (these are called Earth Rotation Parameters) -ERP- or Earth Orientation Parameters -EOP-) that cannot be described by theory: they must be observed: astronomical observations, SLR to the satellites and Moon, VLBI and GPS.
• So finally the matrix S that transform from the instantaneous space-fixed sytem to the CTS is:

  

  where

  

  and ( are small angles):

  

4.1.2 Ellipsoidal Reference Coordinate Systems

• For most practical applications ellipsoidal coordinate systems are preferred because:
  • They closely approximate the earth's surface.
  • They facilitate a separation of horizonal position and height.
• Usually a rotational ellipsoid is selected which is flattened at the poles and which is created by rotating the meridian ellipse about its minor axis b.
• The geometric parameters are semi-major axis a and flattening (or alternatively the first numerical eccentricity
• The global ellipsoidal system has:
  • The ellipsoidal geographic coordinates:
    • The ellipsoidal latitude
    • The ellipsoidal longitude
    • The ellipsoidal height h
  • The cartesian coordinate system associated to the ellipsoid:
    • Origin at the center of the ellipsoid O
    • -axis directed to the northern ellipsoidal pole (along the minor axis)
    • -axis directed to the ellipsoidal zero meridian
    • -axis completing the right-handed system
• The transformations are:

  

  where is the radius of curvature in the prime vertical

  

  The inverse problem must be solved iteratively as:

  

• A local ellipsoidal system (NEU) can be defined tied to the observation point P and to the ellipsoidal vertical as:
  • Origin at the observation point P
  • -axis in the direction of the ellipsoidal vertical
  • -axis directed to the north (geodetic meridian)
  • -axis directed to the east, completing a left-handed system.
  
•  
  
• The location of a second point in the local ellipsoidal system is given by:
  • The slant ranges s
  • The ellipsoidal azimuths
  • The ellipsoidal directions or horizontal angles
  • The ellipsoidal zenith angles
  They are related:

  

• The transformation of the coordinate differences from the local to the global ellypsoidal system:

  

  where

  

4.1.3 Ellipsoid, Geoid and Geodetic Datum

• The ellipsoid closely approximates the physical shape of the Earth, and it is smooth and convenient for mathematical operations: it is used as reference surface for horizontal coordinates in geodetic networks.
• The ellipsoid is less suitable as a reference surface for vertical coordinates (heights): instead the geoid is used.
• The geoid is defined as that level surface of the gravity field which best fits the mean sea level, and it is extended inside the solid body of the Earth.
• The vertical separation between the geoid and a particular reference ellipsoid is called geoid undulation N:

  

  being h the ellipsoidal height and H the orthometric height.

• The angle between the directions of the ellipsoidal normal and of the plumb line at point P is called the deflection of the vertical (or astrogeodetic deflections), that is divided into two components:

  

  being , obtained from astronomical observations and , from geodetic observations.

• The local ellipsoids are defined in such a way that the distributions of the known deflection of the vertical fulfilled some minimum condition in the adjustment: the best fitting ellipsoids.

• A geodeticum datum: the set of parameters that describe the relationship between a particular local ellipsoid and a global geodetic reference system. In general it is needed:
  • 3 translation components .
  • 3 rotations
  • 1 scale factor m
  
  
• Usually the rotation angles are very small:

  

• When the datum information is derived from satellite orbits, the potential coefficients of the Earth's gravity field as well as some fundamental constants (Earth rotation, velocity of light, geocentric gravitational constants) form part of the datum definition:

  Flattening f 1/298.257223563 Angular velocity

  Geoc. grav. constant GM 2nd. zonal harmonic

4.1.4 Three-dimensional Eccentricity Computation

The transformations between eccentricity observations in ellipsoidal coordinates ( , , ) to the cartesian eccentricities ( , , ):

where , are the radius of curvature in the meridian and in the prime vertical respectively:

The inverse formula is: