18-Geom-A3 Geodesy and Positioning · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, May 2015 — 3 hours, closed book (approved Casio/Sharp calculators only). EIGHT numbered questions; six constitute a complete paper and each is of equal value. Most answers are required in essay format — clarity and organization are marked. All eight questions are solved below for completeness.
Reference texts: Vaníček & Krakiwsky, Geodesy: The Concepts (2nd ed., North-Holland); Torge & Müller, Geodesy (4th ed., de Gruyter); Hofmann-Wellenhof, Lichtenegger & Wasle, GNSS — Global Navigation Satellite Systems (Springer, 2008); Snyder, Map Projections — A Working Manual (USGS PP 1395); Heiskanen & Moritz, Physical Geodesy (Freeman, 1967); Natural Resources Canada / Canadian Geodetic Survey references for NAD83(CSRS), CGVD2013 and CGG2013. Canadian datums and regulators throughout (NRCan; Ontario CORS network).
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. Eleven short-definition terms spanning reference frames, satellite geodesy, orbital mechanics and map projections.
Find. A concise 2–3 sentence description of each, with a sketch where useful.
(a) IERS — International Earth Rotation and Reference Systems Service. The international body that defines, realizes and maintains the International Terrestrial Reference System/Frame (ITRS/ITRF) and the International Celestial Reference Frame (ICRF), and publishes the Earth-orientation parameters (polar motion, UT1−UTC, nutation) that link them. It also issues the IERS Conventions used by all precise geodetic computations.
(b) VLBI — Very Long Baseline Interferometry. A radio-astronomy technique in which two or more widely separated antennas record the signal from a distant quasar; the time delay between arrivals gives the baseline vector between the antennas with millimetre precision. VLBI uniquely defines the frame’s scale and orientation and ties the terrestrial frame to the inertial (celestial) frame, and it is essential for measuring Earth rotation.
(c) SLR — Satellite Laser Ranging. A ground station fires short laser pulses at retro-reflector-equipped satellites (e.g. LAGEOS) and times the two-way travel to measure the range to millimetre accuracy. Because it senses the dynamics of the orbit about the geocentre, SLR is the primary technique for defining the origin (Earth’s centre of mass) and the scale of the ITRF.
(d) CBN — Canadian Base Network. A national network of stable, precisely monumented geodetic control stations across Canada, observed by GPS and integrated with the Canadian Active Control System (CACS) to realize NAD83(CSRS). The CBN provides passive, re-observable access to the Canadian Spatial Reference System .
(e) Beidou. China’s Global Navigation Satellite System (BDS), the fourth global GNSS alongside GPS, GLONASS and Galileo. It uses a mixed constellation of MEO, IGSO and GEO satellites, provides global positioning, navigation and timing, and (like the others) is interoperable with modern multi-GNSS receivers.
(f) Satellite orbit inclination. The angle \(i\) between the satellite’s orbital plane and the Earth’s equatorial plane, measured at the ascending node. It is one of the six Keplerian orbital elements and controls the range of latitudes the ground track covers (e.g. GPS orbits are inclined at about 55°).
(g) Right ascension of the ascending node (RAAN). The angle \(\Omega\), measured in the equatorial plane from the vernal equinox (the reference \(x\)-axis of the inertial frame) eastward to the ascending node — the point where the satellite crosses the equator going north. Together with the inclination it orients the orbital plane in inertial space.
(h) Orbital coordinate system. A coordinate frame attached to the orbital plane, in which the satellite’s motion is simplest: the origin is at the geocentre (focus of the ellipse), one axis points to perigee and another lies in the orbital plane perpendicular to it, with the third completing the right-handed set normal to the plane. Positions computed here (from the Keplerian elements) are then rotated into the inertial and terrestrial frames.
(i) Deflection of the vertical. The small angle between the true vertical (the plumb line / direction of gravity) and the normal to the reference ellipsoid at a point, usually resolved into a north–south component \(\xi\) and an east–west component \(\eta\). It reflects the local gravity field’s departure from the ellipsoidal model and must be accounted for when combining astronomic and geodetic directions.
(j) Conformal map. A map projection that preserves angles (and therefore the shape of infinitesimally small features): the scale factor at any point is the same in every direction, so meridians and parallels intersect at right angles as on the ellipsoid. UTM, MTM and Lambert Conformal Conic are conformal — the property that makes them suitable for survey and engineering work, where correct angles matter.
(k) Tissot’s indicatrix. A graphical device showing a projection’s distortion: an infinitesimal circle on the ellipsoid maps to an ellipse (the indicatrix) on the projection, whose axes give the maximum and minimum scale factors and their directions at that point. For a conformal map the indicatrix is always a circle (equal scale in all directions) whose size varies with position; a non-circular indicatrix signals angular distortion.
