Question 2 of 7: Computations of Positions on the Ellipsoid

Question 2: Computations of Positions on the Ellipsoid (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Relative positioning of a new point from a known point, carried out either two-dimensionally on the reference ellipsoid or three-dimensionally in the geocentric Cartesian frame.

Find. (a) the “given / observed / wanted” of the direct problem in each case; (b) the required observation reductions, classified as physical or geometrical; (c) a comparison of the computational complexity.

(a) The direct problem. In the 2-D (ellipsoidal) direct problem the given is the geodetic latitude and longitude \((\varphi_1,\lambda_1)\) of the known point plus the ellipsoid parameters \((a,f)\); the observed quantities are the geodesic (ellipsoidal) distance \(s\) and the geodetic (forward) azimuth \(\alpha_{12}\) to the new point; the wanted is the latitude and longitude \((\varphi_2,\lambda_2)\) of the new point (and the back-azimuth \(\alpha_{21}\)). In the 3-D (spatial) direct problem the given is the geocentric Cartesian position \((X_1,Y_1,Z_1)\) of the known point; the observed are the spatial distance, the astronomic/geodetic azimuth, and the vertical (zenith) angle — equivalently the components of the baseline vector \((\Delta X,\Delta Y,\Delta Z)\); the wanted is the Cartesian position \((X_2,Y_2,Z_2)\), afterward converted to \((\varphi_2,\lambda_2,h_2)\). The two are equivalent because the ellipsoidal coordinates and the Cartesian coordinates are related by an exact, invertible transformation.

(b) Required reductions and their type. For the 2-D problem the raw field observations must be reduced from the terrain to the ellipsoid: (i) correcting directions/azimuths for the deflection of the vertical and applying the Laplace correction — physical (they depend on the gravity field); (ii) reducing the measured slope distance for atmospheric refraction and for the station height above the ellipsoid, and reducing the geodesic to the ellipsoid — the atmospheric part is physical while the height (chord-to-geodesic, sea-level) reduction is geometrical; (iii) the skew-normal and height-of-target corrections — geometrical. For the 3-D problem the same raw observations are used, but the vertical reference is handled explicitly: the zenith angle must be corrected for atmospheric refraction and the direction for the deflection of the vertical (both physical), after which the baseline is formed purely by geometrical vector algebra. In short, both problems share the same physical (gravity/atmosphere) corrections; the 2-D problem carries extra geometrical reductions to force the data onto a 2-D surface.

(c) Complexity. The 2-D direct problem requires solving the direct geodetic problem on the ellipsoid — integrating along a geodesic (Bessel, Rainsford, Vincenty or Gauss mid-latitude formulae), which involves series expansions or iteration and is analytically intricate, though only latitude, longitude and azimuth propagate. The 3-D problem is conceptually simpler: once the observations are reduced, positioning is straightforward vector addition \(\mathbf{X}_2=\mathbf{X}_1+\Delta\mathbf{X}\) in Cartesian space, with a routine conversion to \((\varphi,\lambda,h)\). The price of that simplicity is that the 3-D approach must keep the observations in the local astronomic (plumb-line) frame and so needs the deflections of the vertical explicitly, whereas the 2-D approach also needs the deflections plus station ellipsoidal heights (orthometric height + geoid undulation) for its distance reductions, and in addition pays for it with elaborate geodesic mathematics. This is why they are equivalent when done correctly: the same corrected data, organized two different ways.