Geodesy and Gravity Field

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This section covers the definitions and branches of geodesy, the shape of the earth, coordinate types, the geoid and the ellipsoid, the deflection of the vertical and the Laplace equation (condition).

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Definition and Branches • Geodesy (Helmert) is the science of measuring and mapping the earth's surface, including the determination of the earth's external gravity field and the variation of both with time.

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It provides the reference frames, datums and control on which all other surveying, mapping and navigation depend. • Branches: geometric (mathematical) geodesy — sizes, shapes and positions on the ellipsoid; physical geodesy — the gravity field, the geoid and heights; satellite (space) geodesy — GNSS, SLR, VLBI, DORIS and satellite altimetry/gravimetry; dynamic geodesy — earth rotation, tides and crustal motion.

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Plane surveying neglects curvature; geodesy does not.

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Shape of the Earth Surface Meaning Topographic (physical) surface The actual land and sea surface — irregular, the surface on which measurements are made but too complex for computation Geoid The equipotential (level) surface of the earth's gravity field that most nearly coincides with mean sea level, extended under the continents; everywhere perpendicular to the plumb line; smooth but irregular (undulating ±100 m from the ellipsoid) because of density variations; it is the natural vertical datum for orthometric heights Reference ellipsoid An ellipsoid of revolution obtained by rotating an ellipse about the minor axis — a simple mathematical surface for computation of positions; defined by a and f Sphere Sufficient for small-scale and approximate work; mean radius R ≈ 6 371 km • Typical ellipsoid dimensions (WGS84/GRS80): semi-major axis a = 6 378 137 m, semi-minor axis b ≈ 6 356 752 m, flattening f = 1/298.257 — the earth is flattened at the poles by about 21 km, so it is an oblate spheroid.

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Coordinates • Geodetic (ellipsoidal) coordinates — latitude φ (angle between the ellipsoidal normal and the equatorial plane), longitude λ and ellipsoidal height h measured along the normal. • Geocentric Cartesian (ECEF) coordinates X, Y, Z with the origin at the earth's centre of mass, Z along the mean rotation axis and X through the zero meridian. • Astronomical coordinates Φ, Λ — defined by the direction of the plumb line (gravity vertical), obtained by star observations. • Projected (map/grid) coordinates E, N; and local systems (north, east, up; or local plane coordinates for a project). • Heights: orthometric height H — above the geoid, measured along the curved plumb line (what levelling gives); ellipsoidal height h — above the ellipsoid along the normal (what GNSS gives); geoid undulation N = separation of geoid and ellipsoid, so h = H + N.

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Geoid models (EGM96, EGM2008, and national/regional models) supply N so that GNSS heights can be converted to usable levelled heights.

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Deflection of the Vertical and the Laplace Condition • The deflection of the vertical is the angle between the plumb line (normal to the geoid) and the ellipsoidal normal at a point.

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Its components are ξ = Φ − φ (in the meridian) and η = (Λ − λ) cos φ (in the prime vertical).

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It is usually a few seconds of arc, but reaches tens of seconds in mountainous regions such as the Himalaya. • Laplace equation (Laplace condition):

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A − α = η tan φ (plus small terms), connecting the astronomical azimuth A with the geodetic azimuth α.

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A station where both the astronomical azimuth and the deflection components are known is a Laplace station; such stations were inserted at intervals in classical triangulation to control the azimuth (orientation) error accumulating along the chains. • Because of the deflection, astronomical and geodetic latitudes/longitudes of the same point differ slightly, and levelled (orthometric) heights refer to the geoid while GNSS heights refer to the ellipsoid.

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This section covers basic ellipsoidal geometry, the calculation of coordinates on the ellipsoidal surface, the selection of an ellipsoid, normal sections, and meridional and prime vertical arcs.

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Ellipsoidal Parameters • The reference ellipsoid is generated by rotating an ellipse about its minor axis.

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Defining parameters: semi-major axis a (equatorial radius) and either the semi-minor axis b or the flattening f. • f = (a − b)/a; first eccentricity e² = (a² − b²)/a² = 2f − f²; second eccentricity e′² = (a² − b²)/b² = e²/(1 − e²); for the earth e² ≈ 0.00669.

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Ellipsoid a (m) 1/f Use Everest 1830 6 377 276.345 300.8017 India, Nepal (classical datum and MUTM/UTM sheets) Clarke 1866 6 378 206.4 294.978 North America (NAD27) GRS80 6 378 137 298.257222101 ITRF, NAD83 — the modern geodetic reference system WGS84 6 378 137 298.257223563 GPS broadcast ephemeris and global mapping Radii of Curvature and Arcs • The ellipsoid's curvature varies with latitude and direction: • Radius of curvature in the meridian, M = a(1 − e²)/(1 − e² sin²φ)3/2; • Radius of curvature in the prime vertical, N = a/(1 − e² sin²φ)1/2 (also the length of the normal from the surface to the minor axis); • N ≥ M at every latitude except at the poles, where both equal a/(1 − f) = a²/b; at the equator M = a(1 − e²) and N = a.

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The Gaussian mean radius R = √(MN), and the radius of the parallel of latitude = N cos φ. • Arc lengths: along a meridian, ds = M dφ; along a parallel, ds = N cos φ dλ.

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Consequently 1° of latitude increases from about 110.6 km at the equator to 111.7 km at the pole (because M increases with latitude), while 1° of longitude decreases from about 111.32 km at the equator to zero at the pole.

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Coordinates on the Ellipsoidal Surface • Geodetic to Cartesian:

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X = (N + h) cos φ cos λ, Y = (N + h) cos φ sin λ, Z = [N(1 − e²) + h] sin φ.

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The inverse (Cartesian to geodetic) is solved iteratively or by closed formulas (Bowring). • Geodesic: the shortest line between two points on the ellipsoid; it is a double-curved line, not plane.

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The direct (forward) problem computes the coordinates of the second point from the first point, the azimuth and the distance; the inverse problem computes the distance and the two azimuths from two sets of coordinates.

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Classical solutions:

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Puissant and Clarke for short lines, Bessel and Vincenty for long lines (Vincenty gives millimetre accuracy over any distance). • Normal section: the curve formed by cutting the ellipsoid with the plane containing the normal at one point and the other point.

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Because the normals at two points do not intersect (except on the same meridian or on the equator), the normal section from A to B differs from that from B to A — the two reciprocal normal sections — and the geodesic lies between them, about one-third of the way from the direct section.

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The separations are tiny (millimetres over tens of kilometres) but matter in precise geodesy; observed directions must be reduced to the geodesic.

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Selection of an Ellipsoid • A local (best-fitting) ellipsoid is chosen so that it fits the geoid closely over one country or region, which minimises deflections of the vertical and geoid undulations there — hence Everest 1830 for the Indian subcontinent and Nepal, Clarke 1866 for North America, Bessel 1841 for parts of Europe.

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Its centre does not coincide with the earth's centre of mass (a non-geocentric datum). • A global geocentric ellipsoid (GRS80, WGS84) fits the whole earth on average, has its centre at the geocentre, and is required for satellite positioning.

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Modern practice is to adopt a geocentric datum and publish transformation parameters to the older local datum; many countries, including Nepal, maintain the classical Everest-based system for cadastral records alongside WGS84/ITRF for GNSS work.

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This section covers the definition of datum, coordinate systems and reference frames — terrestrial, celestial and orbital, ITRS/ITRF/ECEF, local and global spheroids — coordinate and datum transformations, rotation and reflection matrices, and map projections including UTM and Nepal's MUTM.

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Datum, System and Frame • A geodetic datum is the set of parameters that fixes the size, shape, position and orientation of the reference surface used for coordinates: a horizontal datum = an ellipsoid (a, f) plus its position and orientation relative to the earth (origin point, orientation, and in classical datums the deflection and undulation adopted at the origin); a vertical datum = the surface (usually the geoid/mean sea level at a defined tide gauge) from which heights are reckoned. • A reference system is the theoretical definition (origin, axes, scale, units, constants); a reference frame is its realisation through the published coordinates (and velocities) of a set of stations.

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ITRS (the system) realised by ITRF2014/ITRF2020 (the frames), and the celestial ICRS realised by the ICRF from quasar positions. • Terrestrial (ECEF) system: origin at the geocentre, rotating with the earth, Z along the mean pole (IERS reference pole), X through the IERS reference meridian.

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Because the plates move, ITRF coordinates carry an epoch and a velocity.

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WGS84, the GPS system, now agrees with ITRF at the centimetre level.

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Celestial systems are non-rotating and used for satellite orbits and star positions; orbital systems describe a satellite's position by its Keplerian elements.

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Coordinate and Datum Transformation • Coordinate transformation changes the representation within the same datum: geodetic (φ, λ, h) ↔ Cartesian (X, Y, Z) ↔ projected (E, N) ↔ local (n, e, u). • Datum transformation changes from one datum to another: the 3-parameter (Molodensky) transformation applies three translations ΔX, ΔY, ΔZ; the 7-parameter Helmert (similarity) transformation adds three rotations (εx, εy, εz) and a scale factor:

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X2 = T + (1 + s)·R·X1; a 14-parameter version adds the rates of change for time-dependent frames; and grid-based methods (distortion grids) model local distortions of old networks.

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Parameters are derived by least squares from points with coordinates in both datums. • Rotation matrices about the axes:

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Rx(θ), Ry(θ), Rz(θ) contain cos θ and ± sin θ terms; they are orthogonal (R−1 = RT), preserve lengths and angles, and have determinant +1.

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A reflection matrix (e.g., diag(1, 1, −1)) reverses the handedness of the system and has determinant −1; it is needed when converting between left-handed and right-handed systems (for example some local survey systems where the axes are swapped).

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Map Projections • A map projection transforms the curved ellipsoidal surface to a plane.

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Every projection distorts shape, area, distance or direction; only the sphere-to-plane distortion pattern can be chosen, not avoided. • Classification by developable surface: cylindrical, conical and azimuthal (planar); by aspect: normal, transverse and oblique; by property: conformal (preserves angles and local shape — Mercator, Transverse Mercator, Lambert conformal conic, stereographic; used for topographic mapping and navigation), equal-area (equivalent) (preserves area — Albers, Lambert azimuthal; used for statistical and thematic maps), equidistant (true distances from one point or along certain lines) and azimuthal (true directions from the centre).

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Projection Definition and use UTM (Universal Transverse Mercator) Conformal transverse Mercator in 60 zones each 6° wide, from 80° S to 84° N; central-meridian scale factor 0.9996; false easting 500 000 m, false northing 0 in the north and 10 000 000 m in the south;

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Nepal lies in zones 44 and 45 (central meridians 81° E and 87° E) Projection Definition and use MUTM (Modified UTM) — Nepal Transverse Mercator with 3°-wide zones, central meridians at 81°, 84° and 87° E, scale factor 0.9999 and false easting 500 000 m, on the Everest 1830 ellipsoid — used by the Survey Department for topographic and cadastral mapping, since narrower zones and a scale factor nearer unity reduce distortion in Nepal's east-west extent Lambert conformal conic Conformal conic with one or two standard parallels — suited to regions extending east-west in mid latitudes Polyconic / Cassini Used in older cadastral series in the subcontinent • Other essentials: the grid convergence (angle between grid north and true north), the scale factor which varies across a zone (less than 1 at the central meridian, greater than 1 at the zone edges), and the reduction of observed distances and directions to the grid (t − T correction) in precise work.

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This section covers Newton's laws, gravity force and potential, theoretical, measured and normal gravity, gravity anomalies, undulations and heights, and level surfaces and plumb lines.

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Gravitation and Gravity • Newton's law of universal gravitation:

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F = G·m1m2/r², with G = 6.674 × 10−11 m³/(kg·s²).

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The gravitational acceleration at a point is the force per unit mass. • Gravity g is the resultant of the gravitational attraction of the earth's masses and the centrifugal acceleration due to the earth's rotation (which acts outward, perpendicular to the rotation axis, and is greatest at the equator).

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Therefore g increases from about 9.78 m/s² at the equator to 9.83 m/s² at the poles — the combined effect of flattening and rotation. • Units:

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1 gal = 1 cm/s² = 0.01 m/s²;

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1 mGal = 10−5 m/s² — the practical unit of gravimetry (μGal for precise work).

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Gravity Potential, Level Surfaces and Plumb Lines • The gravity potential W = V + Φ, where V is the gravitational potential of the masses and Φ the centrifugal potential; g = grad W in magnitude and direction (the vector of gravity is perpendicular to the surfaces W = constant and points along the inward normal). • Level (equipotential) surfaces W = constant are the surfaces of equal potential; the geoid is the particular one with W = W0 ≈ mean sea level.

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Plumb lines are the curved lines orthogonal to all level surfaces; their curvature and the non-parallelism of level surfaces (they converge towards the poles because gravity increases) mean that the height difference between two level surfaces depends on latitude — this is why precise levelling requires the orthometric correction and why 'height' must be defined carefully. • Geopotential number C = W0 − WP (the potential difference from the geoid, in geopotential units) is the rigorous measure of height; from it come the orthometric height H = C/ḡ (mean gravity along the plumb line), the normal height (with normal gravity) and the dynamic height (C divided by a constant normal gravity).

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Theoretical, Measured and Normal Gravity • Measured (observed) gravity g is obtained with absolute or relative gravimeters (3.5) at the earth's surface, after tidal, drift, instrument-height and other corrections. • Normal (theoretical) gravity γ is the gravity of the normal (level) ellipsoid — a rotating equipotential ellipsoid of the same mass and angular velocity as the earth.

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It is computed from the closed Somigliana formula or the series γ = γe(1 + β sin²φ − β1 sin²2φ), with GRS80 values γe = 9.780 327 m/s² at the equator and γp = 9.832 186 m/s² at the pole. • Gravity disturbance δg = gP − γP (both at the same point); gravity anomaly Δg = gP (reduced to the geoid) − γQ (normal gravity on the ellipsoid).

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Reduction / anomaly Formula and meaning Free-air correction +0.3086 mGal per metre of height — accounts only for the height above the geoid, ignoring the masses between; gives the free-air anomaly, which is small on average and used in geoid determination Bouguer correction Removes the attraction of the plate of rock between the station and the geoid:

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2πGρh = 0.1119 mGal/m for ρ = 2 670 kg/m³ (subtracted); with the terrain correction (always positive, since both hills above and valleys below reduce the measured gravity) it gives the complete Bouguer anomaly, used in geophysical exploration and to map subsurface density Isostatic anomaly Bouguer anomaly further corrected for the compensating mass deficiency (root) beneath mountains — nearly zero where the topography is in isostatic equilibrium Undulation and Heights • The geoid undulation N follows from the disturbing potential T = W − U (actual minus normal potential) by Bruns' formula N = T/γ, and is computed from global gravity anomalies by Stokes' integral (or, in modern practice, from a global model such as EGM2008 combined with local gravity data and GNSS/levelling points).

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The Vening Meinesz formulae give the deflection components from the same data. • Practical use:

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H = h − N converts GNSS ellipsoidal heights into orthometric heights, so a good national geoid model turns GNSS into a levelling tool; the accuracy of the derived heights is that of the geoid model.

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This section covers Laplace's equation in spherical coordinates, spherical harmonics, Clairaut's formula, gravimeters, and isostatic and non-isostatic gravity reductions.

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Laplace's Equation and Spherical Harmonics • Outside the attracting masses the gravitational potential V satisfies Laplace's equation ∇²V = 0 (it is a harmonic function); inside the masses it satisfies Poisson's equation ∇²V = −4πGρ.

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This is the mathematical foundation of physical geodesy: the external field can be determined from boundary values on a surface. • In spherical coordinates (r, θ, λ) the solution is expanded in a series of spherical harmonics: • V(r, φ, λ) = (GM/r) Σn (a/r)n Σm [C̄nm cos mλ + S̄nm sin mλ] P̄nm(sin φ), where n is the degree and m the order, P̄nm are the normalised associated Legendre functions and C̄, S̄ are the Stokes coefficients describing the earth's mass distribution. • Harmonics are called zonal (m = 0 — depend on latitude only; the most important is C̄20, related to the dynamic form factor J2, which expresses the earth's flattening), sectorial (m = n) and tesseral (0 < m < n).

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A model of maximum degree N has a spatial resolution of about 20 000 km/N;

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EGM2008 reaches degree 2 190 (≈ 9 km), and the satellite missions CHAMP, GRACE/GRACE-FO and GOCE have determined the long- and medium-wavelength field, including its variation in time (ice-mass and groundwater change).

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Clairaut's Theorem • Clairaut's theorem connects the geometric flattening of the earth with the variation of gravity with latitude.

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Writing normal gravity as γ = γe(1 + β sin²φ), the theorem states f + β = (5/2)m, where β is the gravity flattening (γp − γe)/γe and m = ω²a/γe is the ratio of centrifugal to gravitational acceleration at the equator. • Its importance is that the flattening of the earth can be determined from gravity measurements alone, without geometric (arc) measurements — historically a key result of physical geodesy.

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Gravimeters and Gravity Measurement Instrument Principle and use Absolute gravimeters Measure g directly: the free-fall (ballistic) type times a corner-cube falling in a vacuum with a laser interferometer and atomic clock, reaching a few μGal; older pendulum instruments.

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Used for national reference stations and calibration baselines Relative gravimeters Measure differences of gravity using a mass on a spring:

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LaCoste-Romberg (metal zero-length spring) and Scintrex CG-5/CG-6 (fused quartz) instruments, with a reading precision of 1–10 μGal; fast and portable, but require correction for instrument drift (repeat readings at a base station) and earth tides, and periodic calibration Superconducting gravimeters A levitated niobium sphere; sub-μGal stability for continuous recording of tides and long-term changes Airborne, shipborne and satellite gravimetry Gravity over inaccessible or oceanic areas (with strong filtering); satellite missions CHAMP, GRACE/GRACE-FO (inter-satellite ranging — time-variable gravity) and GOCE (gradiometry — high-resolution static field) • Corrections to gravity observations: instrument drift, earth and ocean tides, instrument height, latitude (normal gravity), free-air, Bouguer and terrain (3.4), and atmospheric effects.

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Gravity networks are adjusted by least squares onto absolute base stations.

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Isostasy and Gravity Reductions • Isostasy is the state of mass balance in which the topography is supported by a compensating mass deficiency at depth, so that below a certain depth of compensation the pressure is equal everywhere — the crust 'floats' on the denser mantle. • Airy-Heiskanen model: the crust has constant density and mountains have roots proportional to their height (for a crust-mantle density contrast of ≈ 600 kg/m³, the root is roughly 4.5 times the topographic height — the Himalaya therefore has a crustal root of the order of 70 km).

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Pratt-Hayford model: the depth of compensation is constant and the density varies — high mountains are underlain by lower-density material.

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Vening Meinesz extended Airy's model to regional (flexural) compensation of an elastic crust. • Isostatic reduction removes both the topographic masses and their compensation, giving the isostatic anomaly — which is small where the region is in isostatic equilibrium and reveals uncompensated loads (active tectonics, sedimentary basins, ore bodies) where it is not.

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Large positive free-air anomalies over mountains and strongly negative Bouguer anomalies are the classic evidence of isostatic compensation (first noticed in the survey of India near the Himalaya). • Non-isostatic reductions — free-air, Bouguer (simple and complete), and condensation methods such as Helmert's — are used for geoid determination and for geophysical interpretation, each removing a different part of the topographic effect.

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This section covers the celestial sphere and celestial coordinate systems, the spherical (astronomical) triangle and Napier's rules, sidereal and universal time, local and standard time with conversions, the motions of heavenly bodies, dependent and independent coordinate systems, and the star almanac.

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The Celestial Sphere • The celestial sphere is an imaginary sphere of infinite radius centred on the observer (or the earth) on which the stars appear projected.

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Its main references: the celestial poles (extension of the earth's axis), the celestial equator, the ecliptic (the sun's apparent annual path, inclined at the obliquity ε ≈ 23.44°), the vernal equinox (First Point of Aries, ♈) where the sun crosses the equator northwards, the observer's zenith and nadir, the horizon, the observer's meridian, the prime vertical, hour circles (great circles through the poles) and vertical circles (through the zenith).

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Coordinate system Coordinates and character Horizon system Altitude (or zenith distance) and azimuth — simple to observe but dependent on the observer's position and on time Equatorial (hour-angle) system Declination δ and hour angle H — δ is independent of the observer, H changes with time and longitude (semi-dependent) Equatorial (right-ascension) system Declination δ and right ascension α (measured eastwards from the vernal equinox) — independent of observer and time; used in star catalogues and almanacs Ecliptic and galactic systems Celestial latitude/longitude referred to the ecliptic; galactic coordinates for astronomy The Astronomical Triangle and Napier's Rules • The astronomical (PZS) triangle is formed on the celestial sphere by the pole P, the zenith Z and the star S.

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Its sides are the co-latitude (90° − φ), the polar distance (90° − δ) and the zenith distance (90° − altitude); the angles are the hour angle H at the pole, the azimuth A at the zenith and the parallactic angle at the star. • It is solved by spherical trigonometry: the cosine rule cos a = cos b cos c + sin b sin c cos A and the sine rule sin a/sin A = sin b/sin B = sin c/sin C.

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For example, sin(altitude) = sin φ sin δ + cos φ cos δ cos H. • Napier's rules of circular parts apply to right-angled spherical triangles: writing the five circular parts (the two sides adjacent to the right angle, and the complements of the other two angles and of the hypotenuse) in a circle, the sine of any middle part equals the product of the tangents of the adjacent parts, and also the product of the cosines of the opposite parts. • Field astronomy used these relations to determine azimuth (by observation of the sun or Polaris — the classical check on triangulation, and the basis of Laplace stations), latitude (from circum-meridian altitudes or Polaris) and longitude/time — before GNSS made them routine, though astronomical azimuth is still used for deflection studies and in tunnels/mines.

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Time Systems Time Definition Sidereal time Based on the rotation of the earth with respect to the vernal equinox: the sidereal day = 23 h 56 m 04 s of mean solar time; the local sidereal time equals the hour angle of the vernal equinox, and also the right ascension of a star on the observer's meridian Apparent (true) solar time Hour angle of the true sun + 12 h — what a sundial shows; irregular because the earth's orbit is elliptical and the ecliptic is inclined Mean solar time Hour angle of a fictitious mean sun moving uniformly along the equator; the equation of time = apparent − mean solar time, varying between about −14 and +16 minutes through the year Universal Time (UT) Mean solar time at the Greenwich meridian;

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UT1 follows the earth's actual rotation, while UTC is atomic time kept within 0.9 s of UT1 by inserting leap seconds;

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TAI is international atomic time and GPS time runs without leap seconds (currently a fixed number of seconds ahead of UTC) Local and standard time Local mean time = UT + λ/15 (longitude east positive, in hours); standard time is the local mean time of a chosen standard meridian for the whole country — Nepal uses 86° 15′ E, giving UTC + 5 h 45 min;

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India uses 82°30′ E (UTC + 5:30) Motions of the Heavenly Bodies • Precession — the slow conical motion of the earth's axis caused by the attraction of the sun and moon on the equatorial bulge, with a period of about 26 000 years; it moves the vernal equinox westwards (≈ 50″ per year), so star coordinates must be quoted for an epoch (J2000.0). • Nutation — a small periodic 'nodding' superimposed on precession, principal period 18.6 years, caused mainly by the moon's orbital motion. • Polar motion — the wandering of the instantaneous rotation pole relative to the crust by a few metres, with the Chandler wobble (≈ 435 days) and an annual component; monitored by the IERS and needed for precise reference frames. • Aberration — apparent displacement of a star due to the finite speed of light combined with the observer's motion (annual aberration up to 20.5″, diurnal aberration up to 0.3″). • Parallax — apparent displacement due to a change of the observer's position: annual (stellar) parallax from the earth's orbit, and diurnal (geocentric) parallax, significant for the moon and sun. • Atmospheric refraction — bends light downwards so that bodies appear higher than they really are; about 34′ at the horizon (so the sun is geometrically below the horizon when it appears to rise) and about 1′ at 45° altitude; it must be corrected in all altitude observations.

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Proper motion is the star's own motion across the sky. • Star almanac: the Star Almanac for Land Surveyors / Astronomical Almanac tabulate the apparent places of the sun and bright stars — Greenwich hour angle (GHA), declination, right ascension and the equation of time — from which the surveyor computes azimuth, latitude or time for the moment of observation, after applying corrections for refraction, parallax, aberration and semi-diameter.