Vertical deflection

The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anomalies such as nearby mountains. They are widely used in geodesy, for surveying networks and for geophysical purposes.
The vertical deflection are the angular components between the true zenith-nadir curve (plumb line) tangent line and the normal vector to the surface of the reference ellipsoid (chosen to approximate the Earth's sea-level surface). VDs are caused by mountains and by underground geological irregularities and can amount to angles of 10″ in flat areas or 20–50″ in mountainous terrain).
The deflection of the vertical has a north-south component "ξ" (xi) and an east-west component "η" (eta). The value of "ξ" is the difference between the "astronomic latitude" and the "geodetic latitude" (taking north latitudes to be positive and south latitudes to be negative); the latter is usually calculated by geodetic network coordinates. The value of "η is" the product of cosine of latitude and the difference between the "astronomic longitude" and the longitude (taking east longitudes to be positive and west longitudes to be negative). When a new mapping datum replaces the old, with new geodetic latitudes and longitudes on a new ellipsoid, the calculated vertical deflections will change, too.
Determination.
The deflections reflect the undulation of the geoid and gravity anomalies, for they depend on the gravity field and its inhomogeneities.
VDs are usually determined astronomically. The "true zenith" is observed astronomically with respect to the stars, and the "ellipsoidal zenith" (theoretical vertical) by geodetic network computation, which always takes place on a reference ellipsoid. Additionally, the very local variations of the VD can be computed from gravimetric survey data and by means of digital terrain models (DTM), using a theory originally developed by Vening-Meinesz.
VDs are used in astrogeodetic levelling: as a VD describes the difference between the geoidal and ellipsoidal normal direction, it represents the horizontal spatial gradient of the geoid undulations of the geoid (i.e., the separation between geoid and reference ellipsoid). 
In practice, the deflections are observed at special points with spacings of 20 or 50 kilometers. The densification is done by a combination of DTM models and areal gravimetry. Precise VD observations have accuracies of ±0.2″ (on high mountains ±0.5″), calculated values of about 1–2″.
The maximal VD of Central Europe seems to be a point near the Großglockner (3,798 m), the highest peak of the Austrian Alps. The approx. values are ξ = +50″ and η = −30″. In the Himalaya region, very asymmetric peaks may have VDs up to 100″ (0.03°). In the rather flat area between Vienna and Hungary the values are less than 15", but scatter by ±10″ for irregular rock densities in the subsurface.
More recently, a combination of digital camera and tiltmeter have also been used, see zenith camera.
Application.
Vertical deflections are principally used in four matters:
Historical implications.
VD were used to measure Earth's density in the Schiehallion experiment.
VD is the reason why modern prime meridian passes more than 100 m to the east of the historical astronomic prime meridian in Greenwich.
The meridian arc measurement made by Nicolas-Louis de Lacaille north of Cape Town in 1752 (de Lacaille's arc measurement) was affected by VD. The resulting discrepancy with Northern Hemisphere measurements was not explained until a visit to the area by George Everest in 1820; Maclear's arc measurement resurvey ultimately confirmed Everest's conjecture.
Apparent errors in the meridian arc of Delambre and Méchain determination, which affected the original definition of the metre, were later found to be caused by VD.