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The vertical deflections (deflections of the plumb line, astro-geodetic deflections) are important parameters of the local gravity field. They are widely used in geodesy, for surveying networks and for geophysical purposes.

The vertical deflection (abbrev. VD or ξ,η) is the local difference between the true zenith (plumb line) and its theoretical vertical direction on a global ellipsoid. VDs are caused by mountains and by geological irregularities of the subsurface, and amount to 10" (flat areas) or 20-50" (alpine terrain). At the axis of valleys the values are rather small, whereas the maxima occur at steep mountain slopes.

The deflection of the vertical is a difference vector and therefore has two components: a north-south component ξ and an east-west component η. The value of ξ is the difference of geographic and ellipsoidal latitude; the latter is usually calculated by geodetic network coordinates. The value of η corresponds to the difference of the adequate longitudes.

Earth's ellipsoid, geoid, and two types of vertical deflection

The deflections are connected with the local and regional undulations of the geoid — and also with gravity anomalies — for they are functionals of 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 astro-geodetic levelling, a geoid determination technique. As a vertical deflection describes the difference between the geoidal and ellipsoidal normals, it represents the horizontal gradient of the undulations of the geoid, i.e. the separation between geoid and reference ellipsoid. Given a starting value for the geoid undulation at one point, determining geoid undulations for an area becomes a matter for simple integration.

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 of the Großglockner (3798 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.

Vertical deflections are principally used in a threefold matter:

1. For precise calculation of survey networks. The geodetic theodolites and levelling instruments are oriented with respect to the true vertical, but its deflection exceeds the geodetic measuring accuracy by a factor of 5 to 50. Therefore the data have to be corrected exactly with respect to the global ellipsoid. Without these reductions, the surveys may be distorted by some centimeters or even decimeters per km.
2. For the geoid determination (mean sea level) and for exact transformation of elevations. The global geoidal undulations amount to 50-100 m, and their regional values to 10-50 m. They are adequate to the integrals of VD components ξ,η and therefore can be calculated with cm accuracy over distances of many kilometers.
3. For GPS surveys. The satellites measurements refer to a pure geometrical system (usually the WGS84 ellipsoid), whereas the terrestrial heights refer to the geoid. We need accurate geoid data to combine the different types of measurements.
4. For geophysics. Because VD deflection data are affected by the physical structure of the Earth's crust and mantle, geodesists are engaged in models to improve our knowledge of the earth's interior. Additionally and similar to applied geophysics, the VD data can support the future exploration of raw materials, oil, gas or ores.

• Vertical
• Zenith
• Gravity anomaly