Gravimetry (derived from Latin gravis, meaning heavy, and Greek μετρω, meaning to measure) is the empirical determination of the acceleration of gravity and its derivative, the gravity gradient. This is accomplished by observing the behavior of a test mass. Two centuries ago such observations could be made only on the surface of the Earth. Today observations are collected also from ships, airplanes, and satellites. Data cover the entire planet.

The unit of the acceleration of gravity is ms-2 in the SI system. The gravity gradient has unit s-2. In geodesy and geophysics a historical remnant from the CGS system has survived, i.e., the unit 1 Gal = 1 cm s-2, named in honor of Galileo Galilei. In practical observations the derived quantities milligal (mGal) = 10-5 ms-2 and microgal (µGal) = 10-8 ms-2 are encountered.

The gravity vector is formed by the gravitational acceleration (caused by the masses of the Earth) and centrifugal acceleration (caused by the rotation of the Earth). The gravity vector defines the local direction of the vertical. Observations from platforms in space are not affected by the rotation of the planet. External forces on the test mass (e.g., gravitation due to the Sun, Moon, and planets; tidal and loading effects; variations in the direction of the Earth axis) are small and may be corrected for by models or supplemental observations.

The test mass may be mounted on an arm balanced by a spring or by torsional forces, or it may be in free fall, or it may be a physical pendulum. The latter two involves repeated time measurements of the motions. In satellite gravimetry the entire satellite in orbit may be considered as the test mass. Orbital changes with time are analyzed to derive gravitational acceleration. Relative position changes between twin satellites and three-axis accelerometers are other approaches for dedicated gravimetric missions.

The foundation of observational gravimetry dates back to the 17th century when Galileo Galilei experimented with pendulums and the free fall of solid objects. The properties deduced formed the base for Christian Huygens' theory of the mathematical and physical pendulums. Jean Richer discovered that surface gravity changed with latitude and Isaac Newton realized that free fall was the equivalent phenomenon of planetary motion. Johan Kepler discovered empirical laws of planetary motion based on observations collected by Tycho Brahe. This led to Newton's law of gravitation in 1687. Different applications of hydrostatic equilibrium allowed Huygens and Newton to conclude that the Earth must be flattened at the poles. In the 18th century theoretical works by P. Bouguer, C. MacLaurin, and L. Euler set the foundation for A. C. Clairaut (1743) to formulate the application of gravimetry to geodetic problems.

In Norway, gravimetric observations have been collected both for scientific purposes and in support of national mapping and geophysical exploration. We primarily address the scientific perspectives in this paper but mention other applications when appropriate.

The first technological milestone of observational gravimetry is Johann Bohnenberger's (1811) principle for a reversible pendulum. If a physical pendulum is constructed to swing around one of two rotational points such that the oscillation period is equal in both situations, it becomes unnecessary to know the location of the center of mass. Henry Kater constructed the first instrument in England in 1818. The brass physical pendulum had a test mass of 1 kg and an adjustable mass of 32 g. Knife edges at either end of the 1 m long pendulum allowed the pendulum to be reversed. Adjustment of the small mass produced equal oscillation periods. Kater (1818) was able to obtain results in London with an uncertainty of ±0.0004 ms-2 (±40 mGal) (Torge, 1989). This served as a reference site and invariable pendulums were made to determine relative gravity differences between London and other locations (Kater, 1819). Expeditions were sent to remote regions of the Earth. The data collected would enter into Clairaut's theorem (1743) to estimate the flattening of the Earth.

Edward Sabine was a British naval officer and natural scientist. An expedition in the North Atlantic with The Griper allowed him to perform the first gravity observations in Norway in 1823 using invariable pendulums by Kater. The oscillation period was determined in London before and after the expedition. Sabine (1825) arrived at Hammerfest on 4 June 1823 and established a temporary observatory at Fuglenes. Small wooden sheds were covered on the outside with canvas and soil to reduce the effect of wind on the instruments. The drift of the pendulum clock was controlled astronomically by observing the Sun and stars with a Dollond transit instrument. Kater's pendulums were mounted in a separate shed 9 m above sea level. The oscillation periods of two pendulums were determined by 27 observation series between 9 and 22 June 1823.

Following visits and successful observations in Greenland and Spitzbergen (now known as Svalbard), The Griper arrived at Trondheim on 8 October 1823. Sabine set up his instruments in a house located north of the city and made a temporary astronomical observatory in the garden. A room in the ground floor of the house had its floor removed to establish independent foundations for Kater's pendulum and the pendulum clock. A trigonometric determination established that the pendulum instrument was 37 m above mean sea level. Oscillation periods were determined by 31 observation series between 16 October and 1 November 1823.

Upon his return to England, Sabine made extensive observations at the reference site in London, e.g., to determine the effect of temperature variations. He reduced his results to local mean sea level and derived the length of a 2 s pendulum in Trondheim to be 39.17456 in. and in Hammerfest 39.19519 in. (Sabine, 1825). We have converted these values into gravity in SI units and listed them in Table 2 (column 2).

Sabine (1825) derived a reciprocal flattening of the Earth of 289, based on his own observations supplemented by those of Henry Kater in England and French data on the continent, a total of 25 stations between the Equator and Spitzbergen (80∘ N). Hansteen (1838) reanalyzed these data and added other stations observed by Rümker, Bessel, and the French Uranie expedition to the Southern Hemisphere. He derived a reciprocal flattening of 292 ± 3 (mean error), based on 33 stations with latitudes from 52∘ S to 80∘ N. Contemporaneously Bessel derived 299 ± 5 from geodetic arc measurements. The modern value is 298.25.

The first wave of international gravity observations faded out after 1830. Attempts to improve the observational precision were unsuccessful until Robert von Sterneck (1887) in Vienna miniaturized the pendulum instrument in the 1880s. His 25 cm pendulum was able to obtain a precision of typically ±0.0002 ms-2 (±20 mGal) by comparing oscillation times between sites. The best observing conditions gave ±0.00005 ms-2 (±5 mGal) (Torge, 1989). Sterneck provided a reference value for Vienna when delivering an instrument.

These activities were initiated as part of a multidecadal international project that began as “Mittel-Europäische Gradmessung” (Torge, 2005, 2012). Norway was among the 13 countries that joined the project in 1862 to improve the Earth ellipsoid and investigate its deviations from the real shape of the planet's equipotential surface. Other countries continued to join and the collaboration eventually developed into “Internationale Erdmessung”, which in 1919 became the International Association of Geodesy (IAG), a component of the International Union of Geodesy and Geophysics (IUGG). The Norwegian contributions to the project were achieved by close collaboration between the University of Oslo and the Geographical Survey of Norway. New geodetic baselines were established in 1864 and a meridian arc from Oslo to Levanger was measured during the next several years. The orientation of the arc was determined by astronomical observations for 1868–1888 in a dozen selected stations, which also revealed deflections of the vertical. The longitude differences between Oslo, Stockholm, and Copenhagen were determined astronomically in 1865 by telegraphic time transfer (Fearnley et al., 1890; Pettersen, 2007) and between Oslo and Bergen in 1880 (Fearnley 1884). The first tide gauges were mounted in the 1880s to provide scale zero for a leveled height system. This refers to a potential surface in the gravity field and thus required gravity observations as well.

Physics professor O. E. Schiøtz at the University of Oslo acquired an instrument with four individual pendulums (production no. 19–22) from Sterneck in 1892 (Fig. 1), funded by the Norwegian Commission for the International Arc Measurements. Another instrument with two pendulums (production no. 33–34) was acquired for Fridtjof Nansen's Fram expedition into the Arctic Ocean. A national reference gravity station was established at the Oslo University Observatory.

When delivering the instruments, Sterneck provided observational results for the reference site in Vienna, which had a stated gravity value of g = 9.80866 ms-2. Sterneck repeatedly measured the oscillation periods of each half-second pendulum at this site, and an average value for each pendulum was provided with seven decimals. A measurement of the oscillation period T at another observing site would provide a gravity value g by gobservedgVienna=TViennaTobserved2. Schiøtz (1893, 1894, 1895, 1901a) made repeated observations at the reference site in the Oslo University Observatory with both pendulum instruments. Table 1 lists gravity values from individual observing sessions, which are plotted in Fig. 2. The average is 9.81955 ± 0.00008 ms-2.

Between 1892 and 1903, Schiøtz (1901b, 1908) determined the acceleration of gravity at 42 sites in Norway (Fig. 3). The average value derived for the four pendulums at each site had standard deviations between ±0.00005 and ±0.00018 ms-2.

Pendulum observations could not provide gravity values at sea because the motions of the ship affected the results. The polar vessel Fram was constructed to withstand the forces of freezing-in with the sea ice at high latitudes. A multidisciplinary science expedition headed by professor Fridtjof Nansen was carried out in 1893–1896. Fram was taken into the ice north of Russian Siberia and remained frozen in as the currents of the Arctic Ocean transported the surface ice on a north-westerly trajectory north of Svalbard. The ship was released in 1896 and returned to Norway.

In the preface to volume II of the scientific report from the Fram expedition, Fridtjof Nansen (1901) wrote:

When I planned the expedition, I considered it not impossible that we might meet with unknown land in high latitudes; and as in such a case it would be of great importance to be able to take pendulum observations, Prof. O. E. Schiøtz kindly undertook to equip us for this purpose. […] We met with no land in the North Polar Basin, and thus the ordinary conditions for making pendulum observations did not exist. But Scott-Hansen thought that the strong ship frozen firmly into the drifting ice, or the ice itself, might possibly afford a sufficiently solid base for the pendulum apparatus, and decided to make some observations as an experiment. Thus the first series of pendulum observations, which, to my knowledge, have ever been made over the sea, were made over the deep North Polar Basin. We had some doubt as to the value of the observations taken under such extraordinary circumstances; but thanks to Prof. Schiøtz's able elaboration and discussion of the material, it now appears that these observations afford perhaps some of the most important results of the expedition.

Sigurd Scott-Hansen made one observation in Russian Siberia in 1893 and several observations onboard Fram during the polar expedition (Schiøtz 1901a, b), between latitudes 79 and 86∘ N. Three observations in June 1895 were made on the ice, outside of the vessel. It appears that some of the observations were affected by vibrational noise caused by screw ice.

In 1946 Gunnar Nørgaard of the Danish Geodetic Institute employed two instruments of his own design to connect gravity values in Copenhagen and Oslo. He derived g = 9.819362 ms-2 (in the Potsdam system), referring to mean sea level in Oslo. The Geographical Survey of Norway acquired its own Nørgaard gravimeter in 1947. This represents the introduction of spring gravimeters to Norway. Gravity differences between Oslo and reference stations in Denmark, England, and Sweden were determined with standard deviations of ±0.00001 ms-2 (±1 mGal). The value derived for Oslo, referring to mean sea level, was g = 9.819378 ms-2 in the Potsdam system (Trovaag and Jelstrup, 1950). By comparing the gravity difference between Oslo and England, they concluded that the absolute gravity value for Potsdam deviated by 13 mGal. The precision and ease of operation led to spring gravimeters being the choice instrument in Norway for the second half of the 20th century.

A Worden gravimeter was acquired in 1953. It was transported on a SAS polar flight between Oslo and Anchorage, Alaska, in February 1957 (Sømod, 1957b). This was the first transfer of gravity values from the European to the Americas. Comparisons at pendulum stations in Oslo and Anchorage revealed a difference of 0.5 mGal.

During the summer of 1956 the Worden (and Nørgaard) gravimeter was employed at 101 leveling stations from Oslo at latitude 59∘ 55′ via Bodø to Hammerfest at latitude 70∘ 40′ (Sømod, 1957a). The distance between individual stations was 20–25 km. The purpose was to validate the gravity differences between the pendulum stations observed with the Cambridge pendulum in 1955. The Worden gravimeter was within 0.2 mGal of the pendulum results.

Three validated control stations and the improved instrument precision now allowed for a new national gravity net to be established. Annual gravity campaigns were conducted throughout the 1950s. Between 300 and 800 stations were observed each year, mainly coinciding with leveling markers. The new network consisted of 5200 stations. In 1969 the Geographical Survey of Norway began improving and expanding this network by LaCoste & Romberg instruments. Figure 5 shows photographs of the suite of spring gravimeters. Observations were made in closed loops to determine and correct for the daily drift of the springs. Across Norway, 36 first-order stations were established, often at airports and near highways. They served as reference stations for more than 200 second-order gravity stations, all of which were accessible by car and separated by less than 80 km. This network was complete by 1972 (Harsson, 1973, 1978a), and selected stations became the Norwegian contribution to the new global gravity system, IGSN 71. A densification to one station per 100 km2 followed, and then further densification in selected regions. The observations continued well into the 1990s. The current database of the Norwegian Mapping Authority (institutional name change in 1986) contains 11 800 stations. These data are used to generate both regional Nordic geoid models and global geoid models.

Norwegian jurisdiction in the Arctic region allowed access to islands in the high north. Following a major volcanic eruption in 1970, a gravimetric reference station was established at Jan Mayen in 1973. A network of stations southwest of the Beerenberg volcano was repeatedly monitored in 1976 and 1979 without detecting changes (Harsson, 1978b).

A reference station at Spitzbergen was established in 1978 and was extended to a network of gravity stations during the 1980s. In collaboration with international partners the Norwegian Mapping Authority were also engaged in the temporal variations of the gravity field and monitoring of various kinds of surface loads as measured by relative gravimeters. Long time series for the analysis of tidal effects at Spitzbergen (80∘ N) were collected in 1969 (Melchior et al., 1970) and again in 1996, the latter with a LaCoste & Romberg gravimeter equipped with an electronic feedback system (Bos et al., 2002).

A superconducting gravimeter has recorded the temporal changes in gravity at Ny-Ålesund, Svalbard, since 1999 (Sato el. al., 2001). Episodic calibrations have been made by absolute gravimeters. A 9-year time series reveals seasonal variability and long-term trends (Omang and Kierulf, 2011), interpreted as elastic effects from current glacier mass loss superposed on a viscoelastic response due to glacial isostatic adjustment since the last ice age.

A 80 km2 water reservoir for a hydroelectric plant in southern Norway was created by the construction of several dams in the 1980s. The accumulation of 3 billion tonnes of water created a regional load that was monitored by several geodetic and geophysical observing techniques, including gravimetry (Harsson and Bungum, 1992; Jentzsch and Koss, 1997). An elastic subsidence of 3 cm was recorded, with a bulge forming a few kilometers away from the lake.

Slow viscoelastic deformations due to postglacial land uplift in Fennoscandia were monitored under the auspices of the Nordic Geodetic Commission. Longitudinal arcs along the parallels at 56, 61, 63 and 65∘ N have been repeatedly observed by relative gravimeters for several decades. The best coverage is at 63∘ N, which has been repeated eight times since 1966 (Mäkinen et al., 2005).

Relative gravimetry on the ocean floor has been extensively employed by Statoil and collaborating partners since 1998 to monitor the vertical subsidence of offshore oil and gas production facilities in the North Sea (Eiken et al., 2008; Sasagawa et al., 2008; Zumberge et al., 2008). A recent application is to monitor the injection of CO2 into the ocean bed (Alnes et al., 2011). A unique instrument package has been developed. The relative gravimeters are routinely calibrated on land using vertical calibration lines established by an absolute gravimeter. The standard deviation of the survey observations has improved from ±20 µGal in 1998 to ±3 µGal at present.

The first absolute measurement of gravity in Norway was made in Hammerfest in 1976 (Cannizzo et al., 1978). A JILA instrument of the Finnish Geodetic Institute observed in Stavanger, Trysil, and Tromsø in 1991–1992. The improved instrument version FG5 was employed in 1993, 1995, and 1998 in a collaboration between the Norwegian Mapping Authority and institutions in Germany and USA (e.g., Klopping et al., 1995). These instruments are all based on methods of free-falling test masses.

The Norwegian University of Life Sciences acquired its own FG5 instrument in 2004 (Fig. 6). It has performed annual campaigns to extend the time series and expand the network of absolute stations in Norway. A measurement precision of ±2 µGal allowed testing of global ocean loading models along the Norwegian coast and to develop regional models for ocean loading corrections of gravity observations (Lysaker et al., 2008; Breili 2009a, b). A time series at an inland mountain station revealed seasonal variability in gravity due to changing precipitation, groundwater levels, and local as well as regional snow loads (Breili and Pettersen, 2009).

When corrected for short period and seasonal variability, the time series of absolute gravity values reveals secular changes specific to each observing site in Norway (Ophaug et al., 2016). Postglacial land uplift is a viscoelastic rebound from the melting of the Fennoscandian ice sheet that began about 104 years ago. This effect is smallest along the west coast of Norway and increases towards the inner parts of the Gulf of Bothnia, the likely location of the thickest part of the ice sheet. A comparison of gravity change versus changes in geometric height provides insight into geodynamic processes of the isostatic adjustment (Pettersen, 2011).

The extensive observing program with FG5-226 has produced redundant observations for many sites. A national network of 16 stations is the most accurate reference for gravity in Norway (Breili et al., 2010).

Gravity anomalies are essential data input for geoid determinations by Stokes' formula. An integration is required over the entire surface of the Earth. Thus data are required across the globe, both on land and at sea, which can be facilitated by the use of moving platforms.

Absolute gravity data for the sites in Table 2 (column 5) are found in Breili et al. (2010), Table 3, 10.1080/00291951.2010.481125, and in Ophaug et al. (2016), supplemental data, 10.1016/j.jog.2016.09.001. The values in Table 2 (column 5) were derived by correcting for the height of the observing station above sea level.