3-4 Geometric reductions, Reduction of horizontal angles

This video lecture describes the geometric reduction of directions, azimuths, and horizontal angles.

After removing the effects of the Earth's gravity field from the vertical and horizontal directions, the impact of the differing geometry between the plane and the ellipsoid must be considered. The first reduction addresses the skewness of the normals to the ellipsoid at the points of observation, where the target point has a geodetic height h. In this case, the plane tangent to the normal at the first point will not be tangent to the normal at the second point. However, what we require is the azimuth between the projections of both points on the surface of the ellipsoid. Therefore, the physically reduced azimuth must be further adjusted to account for the position of the normal and the height of the target point.

The second geometric reduction converts the reduced azimuth, adjusted for the height of the target, to the geodetic azimuth, which is defined as the angle between the local meridian and the tangent to the geodesic passing through the two points.

Astronomical azimuths can be corrected by applying four reduction terms: two related to physical reductions and two to geometric corrections. Horizontal angles can be reduced to the angles between two geodesics on the surface of the ellipsoid, either by adjusting the two azimuths of the pointing directions or by subtracting the corresponding corrections and applying the result to the measured angle on the Earth's surface.

For more details, refer to:

Bomford, B. G. (1952) Geodesy, Clarendon Press, Oxford, UK
Vaníček, P. and Krakiwsky, E. J. (1986) Geodesy: The Concepts, 2nd Edition, Elsevier Science Publishers, Netherlands