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Radio telescopes require a well defined coordinate systems for core calculations and to make accurate maps of the sky. The three core systems that have to be well defined are the celestial system (and epoch), the time system, and the earth-based system describing the location of the array antennas. Typically an array centre is defined, which defines the origin of a local cartesian coordinate system, and defines the latitude and longitude of the array. 

MWA tiles were surveyed by professional surveyors, who present the results in UTM coordinates "easting", "northing" and "height". This is a standard system for surveyors, but the directions for "easting" and "northing" don't correspond to local east and north in general. We need to convert the surveyed coords to a standard cartesian coordinate system used by radio telescopes.

The basic recipe for conversion is as follows:

  1. Convert the surveyed coords to latitude, longitude and height. This system needs a reference ellipsoid such as the WGS84 system, and the height is assumed to be relative to the reference ellipsoid (it is hard to find definitive information on what the height actually means).
  2. Convert the lat, lon and height to the X,Y,Z absolute coordinate system where the origin is the centre of the earth. This conversion needs to assume the same ellipsoid as the conversion from UTM coords. The X axis points out of the equator at zero degs longitude, the Z axis points through the north pole along the rotation axis of the earth, and the Y axis completes a right-handed coordinate system. (Y points out of the equator in the Indian ocean at longitude 90 degs.) It is possible to specify the location of the telescope antennas in this X,Y,Z system in many software packages, but the numbers are very large and non-intuitive, so a local coordinate system is often preferred.
  3. Convert the X,Y,Z coordinates of the antennas to a local cartesian coordinate system where the origin of the system is the defined centre of the array. The axes of the local system are east (defined at the array centre), north (defined at the array centre) and height or "up". It is also worth noting that some software confusingly defines another "local X,Y,Z" system where the global X,Y,Z system has been rotated around the Z axis such that X points out of the equator at the meridian of the array centre. I'm not sure if there is an official name to this system, and it doesn't help much because the coordinates of the antennas are still large numbers and non-intuitive because the local horizon is angled to all coordinate axes.
    1. To convert to local E,N,U units, a rotation on the sphere is required. Note that rotations on sphere do not commute, and we need to do the longitude rotation before the latitude rotation.


Converting surveyed coords to lat, lon and height

Converting from UTM coords to lat, lon and height uses well established formuae like Redfearn's formula. There are online tools that allow code to be checked against trusted sources. Example code is attached. Just reiterating, the height is assumed to be height above the reference ellipsoid.


Converting from lat, lon and height to X,Y,Z

This is analogous to finding the cartesian coords on a point on a sphere, except it is on the ellipsoid so there are some additional terms. Example python code based on WGS84 ellipsoid:

def Geodetic2XYZ(lat_rad,lon_rad,height_meters):
    EARTH_RAD_WGS84 = 6378137.0 # meters in the WGS84 (effectively same as GRS80)
    E_SQUARED = 6.69437999014e-3

    s_lat = math.sin(lat_rad)
    c_lat = math.cos(lat_rad)
    s_lon = math.sin(lon_rad)
    c_lon = math.cos(lon_rad)
    chi = math.sqrt(1.0 - E_SQUARED*s_lat*s_lat)

    X = (EARTH_RAD_WGS84/chi + height_meters)*c_lat*c_lon
    Y = (EARTH_RAD_WGS84/chi + height_meters)*c_lat*s_lon
    Z = (EARTH_RAD_WGS84*(1.0-E_SQUARED)/chi + height_meters)*s_lat
    return (X,Y,Z)

Converting from X,Y,Z to E,N,U

To do this, we first define the array centre. During the early days of MWA, a surveying mark was made on a rock outside the MWA donga, and this point was a well-defined and accurately measured reference point for the surveyors. Since this point is close to the hub of the MWA, it is as good a point as any to define as the centre of the array, so it was.




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