The vertical deflection (deflection of the plumb line, astro-geodetic deflection) at a point on the earth is a measure of how far the direction of the local gravity field has been shifted by local anomalies such as nearby mountains. (Here "gravity" means apparent gravity—true gravity "reduced" by the earth's spin). They are widely used in geodesy, for surveying networks and for geophysical purposes.

"Has been shifted"... shifted from what? Suppose the earth were perfectly spherical and homogenous, and stationary in space (not spinning). A plumb bob would always point to the center of the earth. Now say the same homogenous earth starts spinning; the plumb bob no longer points to the center of the earth (except at the pole and the equator). But it does always line up with a point on the earth's axis, the same point for a plumb bob anywhere on that parallel of latitude. The vertical deflection is everywhere zero, and the "sea level" surface is a slightly-squashed sphere with no lumps—its curvature at any point is given by the formulas for an ellipse.

In reality the sea level surface is lumpy; surveyors and mapmakers try to approximate it with an ellipsoid, and also try to measure how good the approximation is—they want to know how far sea level is above or below the ellipsoid they've chosen, and how unparallel it is. The former is geoid height and the latter is vertical deflection.

The vertical deflection (abbrev. VD or ξ,η) is the local difference between the true zenith (plumb line) and the line that would be perpendicular to the surface of the reference ellipsoid chosen to approximate the earth's sea-level surface. VDs are caused by mountains and by geological irregularities of the subsurface, and amount to angles of 10″ (flat areas) or 20-50″ (alpine terrain)[citation needed]. 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 between astronomic and geodetic latitude; the latter is usually calculated by geodetic network coordinates. The value of η is the difference between the corresponding longitudes.

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

## Determination of vertical deflections

The deflections are connected with the local and regional undulation 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 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.

## Application of deflection data

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.

## Example: Hawaii

On their website NGS gives the NAD83 latitude and longitude for the water tank north of the airport at Hilo, on the big island of Hawaii. We can plug that lat-lon into another part of the website and learn Xi is +12.67 seconds and Eta is +48.32 sec. So a plumb bob is deflected 50 sec in azimuth 255 degrees.

105.5 km away on the other side of the island, at the lat-lon for the Keahole Pt lighthouse, Xi comes out -12.42 and Eta is -63.47: the plumb bob is deflected 65 seconds in azimuth 79 degrees.

88 km southeast of there at another survey station Xi is -56.68 and Eta is +32.89, so the plumb bob is deflected 65.5 sec in azimuth 330 degrees.

So anyone trying to determine the "actual" latitude and longitude in Hawaii in, say, 1950 would have a tough time. When they surveyed those three points they got their positions relative to each other reasonably accurately, but the latitude was 11 seconds more than NAD83 and the longitude was 10 seconds more. (The NGS data sheets give the old lat-lon (labelled "OLD HI") as well as the current NAD83.)

Few places have 4000-meter mountains 50 km from deep ocean. But any large mountain range will pull a plumb bob 10-20 arc seconds toward it, or more; determining the overall shape of the Earth was no simple matter in the days before artificial satellites.