## Sunday, September 18, 2011

Geodesy, the science of precise measurement of the Earth's shape and gravitational field, has at its root many of the methods employed by the local surveyor or cartographer, but is distinguished from these by the scale of interest: in general geodesy refers to the analysis of observations made over distances at which the curvature of the Earth becomes significant. Modern geodesy utilizes surface-to-surface and ground-based astronomical observations, and increasingly also data from signals transmitted and received by artificial Earth-orbiting satellites, to quantify the form of our planet.

__Geodesy and gravity__

Physical geodesy is that branch of the subject concerned with the measurement of the Earth's gravity field and the geoid, the surface of equal gravitational potential that in the oceans is close to mean sea level. On the planetary scale the geoid is approximated by an ellipsoid of revolution about its minor axis with a degree of flattening of around 1 part in 300 caused primarily by the Earth's rotation. Permanent deviations of the geoid from an ellipsoid occur as a result of non-uniform mass distribution within the Earth, and also temporarily over the oceans by up to 1 m in response to winds, tides, and ocean currents. Both types of perturbation may be of interest to Earth scientists, but to geodesists the term ‘geoid’ refers to the stable long-term features of the equipotential surface. Geoids are expressed in terms of the geoid–ellipsoid separation or geoid height, the perpendicular distance to the geoid from a given reference ellipsoid. Historically, different countries tended to use their own reference ellipsoids with differing origins and major and minor axes, chosen so as to minimize the geoid height in the region of interest, but the increasing use of space geodetic data makes it more appropriate nowadays to use a common global reference ellipsoid such as WGS-84 or GRS-80 with its origin at the centre of mass of the Earth. In this case, geoid heights may reach several tens of metres.

The geoid can be derived from gravity measurements at known positions over land or sea, or more directly by using precise radar altimetry readings taken from low-orbiting satellites over the oceans. The latter method has proved particularly efficient in determining the short-wavelength components of the geoid. A satellite orbiting at an altitude of 100 km and emitting microwave radar pulses with a beam width of 1° will illuminate patches of the sea surface with radii of about 3–5 km. The radar altimeter measurement is affected by the roughness of the ocean surface and by atmospheric refraction, and so the accuracy of the measurements is limited to approximately 0.2 m, comparable with the precision to which the satellite's position can be tracked. In contrast, obtaining a gravimetric geoid from terrestrial gravity measurements is much more complicated and error-prone. The spatial resolution of gravimetric geoids is limited to length-scales smaller than the area over which gravity observations are given, and the observations themselves must be subjected to elaborate corrections to remove the effects of matter situated above the points at which gravity was measured.

Terrestrial gravity measurements can be made in an absolute sense by measuring the time for a pellet to fall a fixed distance in an evacuated chamber, or in a relative sense by using mechanical force-balance instruments. Relative gravimeters can be of the static variety, in which the change in gravity from the nominal value displaces a mass on a spring by an amount proportional to the change, or the more accurate astatic variety in which the displacement for a given gravity perturbation is much larger. The mean value of the acceleration due to gravity at the Earth's surface is approximately 9.8 m s−2. Gravity anomalies are commonly expressed in units of gal (named after Galileo; 1 gal = 0.01 m s−2) or gravity units (1 gu = 10−6 m s−2 = 0.1 mgal). The precisions of modern astatic relative gravimeters are of the order of 0.1–0.01 mgal but are subject to long-term drift in excess of 0.05 mgal/month, whereas absolute gravimeters can reach accuracies of 0.001 mgal and do not suffer from temporal drift. Seaborne gravity observations are less accurate than those on land, first because of the pitch and roll of the ship, which cause unwanted accelerations of the gravimeter (although gyroscopic stabilizing platforms can limit these), and secondly because of the Eötvös effect. This effect is caused by the east–west component of the ship's velocity, which modifies the centrifugal acceleration due to the normal rotation of the Earth about its axis. An uncertainty in ship speed of 1 km hr−1 results in an error in the Eötvös correction of up to 4 mgal at the Equator and less at higher latitudes. With modern GPS navigation systems (see below) providing ship velocity, accuracies one order of magnitude better than this can easily be achieved. Airborne gravity measurements also require an Eötvös correction and are significantly degraded by vibration and acceleration of the aircraft. Another problem with airborne measurements is that the aircraft's altitude must be known to within 1 m to attain gravimetric precision of the order of 1 mgal.

__Positions and reference frames__

Fundamental to positional geodesy is the reference frame in which the resulting coordinates are expressed. Early marine navigators could compute their astronomic latitude relatively easily, by observing the highest and lowest angles above the horizon attained by the Sun and other stars. Astronomic longitude is a slightly trickier problem because an accurate determination relies on the ability to ascertain the local time with respect to the time at the origin of longitude (by modern convention the Greenwich meridian). Until accurate chronometers were developed in the eighteenth century, this could be achieved only by means of complex calculations based on the eclipses of stars and planetary bodies. Although astronomical measurements can give independent positions at any point on the Earth, their disadvantage is that the observations are taken with respect to the local vertical (the direction of the gravitational force, perpendicular to the geoid). Because the geoid is not a perfect ellipsoid and undulates gently, the directions of the vertical in two places can in fact be parallel, resulting in identical astronomical coordinates for the two locations. Correction must be made for this deflection of the vertical to give geodetic coordinates (latitude, longitude, and height above the ellipsoid); these coordinates, being relative to a given ellipsoid, are unique. The parameters of the ellipsoid used must then be specified together with the coordinates.

Because, as described above, many countries used their own ellipsoidal parameters (major and minor axis radii and location of the centre) to achieve as close as possible a match between the geoid and ellipsoid in their region, ellipsoidal coordinates are not universal. An alternative way of describing the position of a point is by using Cartesian geocentric coordinates with respect to three mutually perpendicular axes with their origin at the centre of mass of the Earth. The Z-axis is chosen to coincide with the Earth's rotation axis, the X-axis intersects the Equator at the Greenwich meridian, and the Y-axis is at 90°E. The rotation axis changes continuously in a way that is not entirely predictable in the short term as a result of day-to-day and seasonal redistribution of mass within the atmosphere and oceans; it also changes more predictably in the longer term as a result of the Chandler wobble with a period of 14 months and as a result of nutation of the Earth (small oscillations of its axis) with a period of 18.6 years. In practice, therefore, the Z-axis is defined by the mean rotation axis over a fixed interval. The X- and Y-axes are no longer defined directly by observations at Greenwich, but in an implied sense through the choice of coordinates for several sites around the globe.

When dealing with astronomical and satellite measurements it is necessary to consider not just the Earth-fixed reference frame referred to above but also a suitable celestial reference frame in which the distant stars are (approximately) fixed and the equations of motion of satellites can be written without including any external rotational terms (an inertial frame). In practice such a frame is not perfectly realizable, because even the most distant stars exhibit small proper motions with respect to each other, and because it is not yet possible to model accurately all the resistive forces acting on near-Earth satellites. According to the problem in question, it may be more appropriate to consider an Earth-centred or Sun-centred (barycentric, fixed to the centre of mass of the Solar System) reference frame. In either case, the system is defined with respect to the ecliptic, the plane of the Earth's orbit around the Sun, and the Earth's Equator. The line of intersection of these two planes marks the equinoxes, the points at which the Sun apparently crosses the Equator as it passes from one hemisphere to the other. The vernal equinox (also known as the ascending node), where the Sun passes from the southern to the northern hemisphere, is chosen as the fundamental axis of the reference frame. Like the Earth's rotation axis, the actual Equator and ecliptic vary with time, and so the celestial reference frame is defined using mean values of these entities, observed over a fixed period.

__Terrestrial geodetic measurements__

Geodetic positioning by surface-to-surface observations has in some respects been superseded by space geodesy (in particular GPS, see below), but for small- and medium-scale positioning it often remains the best method. Also, many geophysical studies, in particular those of Earth rotation and lithospheric deformation, benefit from the longer history of terrestrial geodetic measurements. The first true geodetic measurements, dating back to the time of ancient Greece, were astronomical in nature. Eratosthenes (a contemporary of Archimedes) derived a remarkably accurate value for the Earth's circumference by comparing the zenith angle of the noonday sun at Alexandria with that at Aswan. Advances in timekeeping, telescopes, star catalogues, and the modelling of atmospheric refraction of light mean that astronomical positions this century are accurate to within 0.3≤ of latitude or longitude, an absolute positional accuracy of 10 m, which is insufficient for survey purposes. The difference in position between two nearby points can, however, be measured more precisely: the azimuth from one to the other can be measured to within 0.5≤ of arc, corresponding to 25 mm over a 10-km baseline. Many modern survey networks have such azimuth observations to provide orientation control.

Triangulation, first proposed by the sixteenth-century Dutch geodesist Gemma Frisius, was until the development of satellite geodesy the main method by which regional geodetic networks were measured. In its simplest form, triangulation relies on the fact that if the angles at the vertices of a set of abutting triangles are known, and also the length and orientation of one side of a triangle, then the positions of all the other vertices can be calculated using simple trigonometric relations. This method obviously requires that the vertices are intervisible, which is why triangulation pillars are usually sited on the tops of mountains or on tall buildings. The angle measurements are made with a theodolite, which is essentially a telescope free to rotate in the horizontal and vertical planes and with graduated circles so that the angle can be read off to within up to 0.1≤. Atmospheric refraction of light is a significant source of error, and to counter this several sets of measurements are taken, often at night when the temperature structure of the atmosphere is more stable.

Historically, the main advantage of triangulation over traverse surveys (in which successive bearings and distances are used to proceed from the known point to each unknown point in turn) was that there is a need for only one distance measurement in the entire network, although many networks had more than one to counteract errors. Distance measurements were previously made with Invar tape, designed to have a low coefficient of thermal expansion, but this method entailed laborious corrections for gravitational sag of the tape and the underlying topography between the two ends of the baseline. Since the 1950s electronic distance measurement (EDM) systems, using microwave radar or visible light, have made it relatively easy to measure the distance between intervisible points. Atmospheric refraction errors are mitigated by using two frequencies of light, because at optical frequencies the atmosphere is dispersive (i.e. it delays different signals by an amount depending on their wavelength, which can later be compensated). In this way the distance-dependent errors of EDM can be limited to about 1 ppm. Using EDM, trilateration, which is similar to triangulation but uses distance measurements instead of angles, becomes feasible.

The vertical component of position can be measured either by vertical triangulation (trigonometric levelling), which is severely limited by atmospheric refraction errors because the density of the atmosphere changes rapidly upwards, or more accurately by spirit levelling. The difference in height between successive points a few tens of metres apart is measured using a telescopic level and graduated rod and the process is repeated along a traverse or loop. Accuracies of up to 0.1 mm per kilometre of traverse can be achieved.

At the length-scales involved in geodesy, ordinary plane trigonometry cannot be used and the computations must be performed for a curved reference surface. This is a major reason why an appropriate ellipsoid must be chosen to approximate the geoid in the region. All these terrestrial methods are affected by the deflection of the vertical explained above, and this must be estimated either independently or, as part of the complete geodetic problem, together with the positions of the points. Because trigonometric and spirit levelling give the orthometric height (relative to the geoid), not the ellipsoidal height, the difference between the two (i.e. the geoid height or geoid–ellipsoid separation) must also be computed.

__Space geodesy and satellite positioning.__

Soon after the discovery of extraterrestrial radio sources in the 1930s, it was realized that these signals could be used to determine the distances between radio telescopes. Use of this technique of very-long-baseline interferometry (VLBI) for geodesy did not come to full fruition until the late 1960s, when precise atomic clocks made it possible to record and time-tag independently the signals received at widely separated points. The time difference between the arrivals of a wavefront at each telescope is proportional to the component in the direction towards the radio star of the baseline joing the two telescope. Measurements are made in the S- and X-bands (2–8 GHz). At these frequencies, signal delay caused by charged particles in the ionosphere is a problem which can be overcome by measuring at more than one frequency because the effect is dispersive. More significant is the delay caused by the troposphere, in particular tropospheric water vapour, which is non-dispersive and must be estimated using models of pressure, temperature, and humidity variation with altitude.