Coordinate Geometry (COGO)
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What is the difference between a 3D Local NEU (north, east, up) Inverse
and a 3D ECEF (x, y, z) Inverse.
The major difference between these two inverse types is the coordinate system origin.
For the 3D Local NEU Inverse, the origin is the AT
station. This origin is located at a position defined by its latitude,
longitude and ellipsoidal height. If you were to measure an azimuth,
zenith angle and chord (slope) distance from an AT station, you could
compute a Local NEU vector from the results by applying the formulas:
Local North = chord distance * sin(zen) * cos(az)
Local East = chord distance * sin(zen) * sin(az)
Local Up = chord distance * cos(zen)
The 3D ECEF origin is based on the center of the ellipsoidal model being
used (NAD 83, WGS 84, etc.). GPS observations (dx, dy, dz) are based
on this inverse type.
For any two stations (an AT and TO station pair), the magnitude of a
Local NEU Inverse results and the GPS dXYZ Inverse results are identical. That is:
is numerically the same as:
dN squared + dE squared + dUP squared
dX squared + dY squared + dZ squared.
When computing a 3D Astro Geodetic Inverse, how do the deflection of the vertical components (optionally provided for each geodetic station) affect the results?
For the 3D Astro Geodetic Inverse, 3D Geodetic Traverse or 3D Geodetic
Network Adjustment, the deflection of vertical components (when known)
are used to correct terrestrial observations measured in the field to
geodetic observations used within the geodetic mathematical model.
Astronomic observations are measured in reference to the direction of
gravity at the AT station. Geodetic observations are in reference to
the ellipsoidal normal at the same AT station. The difference
between the direction of gravity and the ellipsoidal normal can be as
little as a few seconds to several seconds depending on the location of
your survey project. The longer your measurements, the greater these
differences will have on the computed coordinates of the TO station.
When computing the 3D Astro Geodetic Inverse, all computations are based
on the reference ellipsoid (using latitude, longitude and height). The
resulting Geodetic observations (azimuth, zenith, chord) are then
corrected to Astronomic observations and reported. These corrections
are based on the provided deflection of the vertical data. If these data
are not provided, no correction is made and hence the computed Geodetic
observations and Astronomic observations are assumed to be the same.
All inverse computation results are mark to mark observations.
They can be utilized in the field if instrument and target heights are zero OR
they are identical (and not very large - less than a couple meters). The latter is
only a very close approximation.
Network Adjustment and Coordinate Transformation
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