zotero/storage/AZ557D2L/.zotero-ft-cache

Journal of Applied Geophysics 42 Ž1999. 117–134

www.elsevier.nlrlocaterjappgeo

Gravity terrain corrections — an overview
D.A.G. Nowell )
St. John’s College, Cambridge CB2 1TP, UK Received 7 October 1997; accepted 8 July 1999

Abstract

Terrain corrections should normally be standardised out to a distance of 166.735 km and combined with the Bullard B

correction

for

the

curvature

of

the

Earth,

out

to

this

1

1 2

8

limit,

derived

from

a

simple

power-series.

Particular

care

must

be

taken with nearby topography which often forms a significant part of the total correction and changes rapidly with location.

This can either be surveyed in situ or calculated as rotational wedge segments of uniform slope out to a variable distance

from the point of observation. Increasing vertical separation between adjacent stations requires terrain corrections to be

extended further out than would otherwise be needed as they are height dependent. Distant topography beyond about 22 km

must be adjusted for the curvature of the Earth and can produce negative terrain corrections. It is important to take into

account bodies of water, and marine surveys must be converted to Bouguer gravity if they are to be combined with terrestrial

data. Methods for special cases, such as readings on the seabed, towers and under ground, require additional approaches in

their computation, especially when trying to detect deviations from Newtonian gravity. q 1999 Elsevier Science B.V. All

rights reserved.

Keywords: Bouguer gravity; Terrain correction; Marine; Height; Earth curvature

1. Introduction

The need to add a correction for the gravita-

tional attraction of the undulations of the terrain

about the plane through a gravity station was

first recognized by Hayford and Bowie Ž1912..

This terrain correction and a correction for the

curvature of the Bouguer slab on the Earth,

using tables by Cassinis et al. Ž1937., were

applied

out

to

1

1 2

8

or

a

distance

of

166.735

km

from the gravity station by Bullard Ž1936..

Lambert Ž1930. had discussed the origin of the

ideas about terrain corrections leading to the

) Present address: 2 Tudor Road, New Barnet, Herts EN5 5PA, UK.

work of Hayford and Bowie Ž1912., but failed to understand their significance and presented a closed-form solution of the curvature problem which did not address it from the exploration point of view. When Hammer Ž1939. modified the Hayford and Bowie Ž1912. system for high precision terrain corrections out to a distance of about 22 km from the gravity station, the method started to be widely applied in gravity surveys.
This paper refines the process of terrain correction for the effects of height, nearby terrain or buildings, the need to correct to the sea or the lake bed instead of to the water surface and for masses of water, as well as locations above and below ground level. Without considering all these factors, errors will occur in the final ter-

0926-9851r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 6 - 9 8 5 1 Ž 9 9 . 0 0 0 2 8 - 2

118

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

rain corrections for a gravity station which may be much more significant than errors of between 33 and 10 mGal in the measurement of gravity ŽRymer, 1989., or the short range precision of 10 mGal ŽValliant, 1991. and 3 to 5 mGal accuracy ŽLaCoste, 1991. for a gravity meter. Whatever the methods that are finally used in producing terrain corrections for a gravity survey, this paper aims to make the reader think carefully about how this is done, along with any likely errors and limitations of their chosen methods.

2. Standard gravity corrections

2.1. Corrections normally applied to graÕity data

The following corrections are normally applied to gravity data: Earth tides Žwith an amplitude of up to 300 mGal Ä0.3 mGal4.; instrumental drift; latitude; free air; topography.
With the free air correction ŽFAC., LaFehr Ž1991a. showed that the measured vertical gradients of gravity demonstrate considerable variation. As pointed out by LaFehr Ž1991a., this does not suggest that the normal free-air gradient of q0.3086 mGalrm ŽRobbins, 1981. is in error or needs to be locally adjusted. However, a far more precise height Ž h. and latitude Žf . dependent FAC is given by Lambert Ž1930. as:

FAC s Ž0.308,57 q 0.000,21 cos 2f . h

y 0.072Ž hr1000.2

Ž1.

and Heiskanen and Moritz Ž1967. as:

FAC s Ž0.308,768 y 0.001,434 sin2f . h

q 7.212 = 10y8h2.

Ž2.

Also, there should ideally be a correction for any departure from the expected level of the geoid, which is calculated from the Bouguer anomaly ŽTsuboi, 1983; p. 130. and thus would require a recursive solution to calculate. These anomalies should be taken into account when

the geoid height changes rapidly, as in moun-

tainous areas, because the indirect effect

ŽTalwani, 1998. due to the discrepancy between

the levels of the geoid and the reference ellip-

soid of the International Gravity Formula be-

comes significant. But, as normally these effects

change slowly with distance, it is justified sim-

ply to remove a regional field when gravity

modelling, since uncertainties about variations

in density are far more significant than such

very long wavelength anomalies.

As Bullard Ž1936. recognized, the correction

for the gravitational attraction of the topography

at a gravity station consists of three parts: the

Bouguer correction ŽBullard A., which approxi-

mates the topography to an infinite horizontal

thickness equal to the height of the station

above sea level or any other datum plane, the

Bouguer slab; the curvature of the Earth ŽBul-

lard B., which reduces the infinite Bouguer slab

to that of a spherical cap of the same thickness

with

a

surface

radius

of

166.735

km

Ž1

1 2

8.;

and

the terrain correction ŽBullard C., which takes

into account the undulations of the topography

above and below the curved surface of the Earth

at the height of the station ŽFig. 1a.. As Chapin

Ž1996. pointed out, these corrections to produce

the Bouguer anomaly are not gravity reductions,

in that the gravity value is not somehow moved

or ‘‘reduced’’ to a different location, as station

values remain fixed at the point of observation.

The Bouguer slab below the gravity station

pulls downwards and increases the observed

value of gravity: hence this effect has to be

subtracted from readings. The formula for the

Bouguer Correction BA ŽBullard A. is:

BA s 2p Gr h

Ž3.

where G is the gravitational constant, r is the density of the surface layer and h is the thickness of the slab. The curvature correction ŽBullard B. out to 166.735 km consists of two parts ŽFig. 2.: the section of the spherical cap directly underlying the infinite slab which dominates up to elevations of 4150 m and pulls downwards

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

119

Fig. 1. Ža. The corrections for a gravity station ŽGS. above sea level on land. Žb. Key to the corrections for gravity stations in Ža. and Figs. 10–14: 1 s the Bouguer slab, which approximates the topography to an infinite horizontal thickness equal to the height of the station above sea level, corrected for by subtracting the Bouguer correction ŽBullard A. at the Bouguer density Žcross hatching.; 2 s the effects of topography above and below the horizontal plane at the height of the station, corrected for by adding the terrain correction ŽBullard C. at the Bouguer density Žvertical stripes.; 3 s the effect of water above the sea bed, corrected for by subtracting a correction at the density of the water Žhatching.; 4 s the effect of the lack of rock below the station due to the sea, corrected for by adding the marine correction equal to the difference in density between the Bouguer slab and the water Žhorizontal stripes.; 5 s the combined effects of the Bouguer slab Ž1. and topography Ž2.; 6 s the combined effects of topography Ž2. and water above the sea bed Ž3.. The curvature of the Earth ŽFig. 2. is omitted for simplicity and is corrected for by subtracting the Bullard B curvature correction at the Bouguer density.

increasing the observed value of gravity; and the truncation of the infinite Bouguer slab at 166.735 km which dominates at elevation above 4150 m and decreases the observed value of

gravity. Cogbill Ž1979. developed a more precise approximation than LaFehr Ž1991b. for the Bullard B correction BB Žat a density of 2670 kgrm3, with an Earth radius of 6371 km.,

Fig. 2. The Bullard B correction for the curvature of the Earth away from a gravity station ŽGS.: blacks the section of the spherical cap directly underlying the infinite slab, which pulls downwards increasing the observed value of gravity; vertical stripess the truncation of the infinite Bouguer slab at 166.735 km, which decreases the observed value of gravity, after LaFehr Ž1991b..

120

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

which is subtracted from readings, as a powerseries:

BB s Ah y Bh2 q Ch3 q Dh4

Ž4.

where h is the height of the station, A s 1.464,139 = 10y3, B s 3.533,047 = 10y7, C s 1.002,709 = 10y13 and D s 3.002,407 = 10y18 Žthe B term is not positive as misprinted in the version given by LaFehr, 1991b.. This approximation gives corrections to within 0.01 mGal and the exact formula quoted by LaFehr Ž1991b. can be used if needed. This effect near sea level for high-precision surveys is about 1.4 mGalrm Žthe typographical error in LaFehr, 1991a,b was corrected in an Erratum, 1992., and must be taken into account. Away from a latitude of 458 the correction varies linearly with height: at 4000 m above sea level on the equator the approximation is only out by 8.5 mGal ŽFig. 3.. The undulations of topography result in the upwards attraction of hills above the plane of the station and valleys below, which decrease the observed value of gravity, so these effects have to be added to readings.
As Chapin Ž1996. pointed out, the corrections to produce the Bouguer anomaly are not graÕity reductions, in that the gravity value is not somehow moved or ‘‘reduced’’ to a different location. The corrected value for a station remains

fixed at the height and point of observation and is never moved.
2.2. The Hammer method of terrain correction
The method Hammer Ž1939. used to make terrain corrections ŽBullard C. involved dividing the area surrounding the station into circular zones and compartments in which the height above or below the station datum could be estimated and the effect for a given compartment calculated. This is done by using a series of clear overlays showing the limits of the compartments over maps at suitable scales away from the station. If a compartment contains portions which are below and above the station level, the average of the height differences must be found and not the difference between the average height of the compartment and the station level. Manual terrain corrections are only subject to errors from the quality of the topographic maps used Žthis also applies to digital terrain models which are after all based on them., using too few compartments out to a given distance in an area of extreme topography and the ability of the interpreter correctly to average the height above or below a station in each compartment.

Fig. 3. The effect of latitude with increased height on the Bullard B correction, from LaFehr Ž1991b..

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

121

Hammer Ž1939. based his correction on the formula for the gravitational attraction Ž g . in the middle of a vertical hollow cylinder with its base at right angles to the horizontal plane through the station:

( g s 2p Gr R2 y R1 q Ž R12 q h2 .

(y

Ž

R

2 2

q

h2

.

Ž5.

where h is the height of the cylinder and R1 and R2 are the inner and outer radii. The cylinder is subdivided into compartments ŽTable 1. to make finding the height difference between the topography in the cylinder and the station easy. The corrections for the compartments in different cylinder or zones are simply added together to give the terrain correction. To obtain the zone spacing, Hammer Ž1939. used the following condition:

R2rR1 s Ž n q p .rŽ n y p .

Ž6.

where n is the number of compartments in the zone.

2.3. Computer methods of terrain correction

Bott Ž1959. showed that computers could be used to speed up large numbers of outer zone corrections with gridded elevation data and Nagy Ž1966. developed a formula for the gravitational attraction of a right rectangular prism. Blais and Ferland Ž1984. found that for distances greater than about 12 km, with a 1 km grid, the formula for a prism could be simplified to that for a vertical line mass, height Ž h., centred at a grid point:

g s Gr sŽ1rd y 1rdX .

Ž7.

where s is the cross sectional area, d is the horizontal distance from the station to the mid-
( dle of the prism and dX s Ž d2 q h2 . the dis-
tance between the station and the top of the prism.
With the advent of modern computers, terrain corrections can easily be calculated with a digital terrain model, e.g., Cogbill Ž1990. and Rollin

Table 1

Standard hammer zones with additional zones out to the

limit

of

the

Bullard

B

correction

1

1 2

8

Ž166.735

km.

and

108

Ž1110 km. with the drop below the tangential plane of a

station due to the curvature of the Earth

Zone
A B C D E F G H I J K L M

R1
0.0 2.0 16.6 53.3 170.1 390.1 894.9 1530 2615 4469 6653 9903 14,742

R2
2.0 16.6 53.3 170.1 390.1 894.9 1530 2615 4469 6653 9903 14,742 21,944

Re
– 4.25 25.2 83.0 238.4 546.9 1131 1933 3303 5349 7962 11,852 17,643

NC b
1– 4– 6– 6– 8– 8– 12 – 12 – 12 1 16 2 16 5 16 11 16 24

ŽN. 21,944 33,000 26,371 20 55 ŽO. 33,000 50,000 39,777 20 124 ŽP. 50,000 75,000 60,025 20 283 ŽQ. 75,000 110,000 89,218 20 625 ŽR. 110,000 166,735 132,610 20 1380

ÄS4 166,735 230,000 193,350 24 2930 ÄT4 230,000 315,000 265,910 24 5550 ÄU4 315,000 430,000 363,680 24 10,380 ÄV4 430,000 590,000 497,530 24 19,420 ÄW4 590,000 810,000 682,820 24 36,560 ÄX4 810,000 1,110,000 936,700 24 68,740

R1s the inner radius of the zone in metres. R2 s the outer radius of the zone in metres. Res the approximate radius of equality between the effect in the inner and outer parts of a zone in metres.
NC s the number of compartments in a given zone.
bs the drop below the tangential plane of a station at Re for a given zone in metres.

Ž1990., or, as Herrera-Barrientos and Fernandez Ž1991. did, by fitting the topography with a set of Gaussian basis functions. An even more rapid method has been developed by Parker Ž1996. using Fourier methods, in which the attraction is calculated at points on the terrain surface by converting a power series for the topographic height into a series of convolutions. The cylindrical zone nearest to the station must be computed directly by integration to avoid convergence problems. However, terrain corrections

122

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

made with a digital terrain model are only as good as the data they are based on and depend on how representative the heights used are of the general topography. The digital terrain model is only as good as the original map and the sampling method used to obtain digital data for the terrain correction with a computer terrain model. Also, a digital terrain model has to be set up in the first place, and it may be simpler to make manual corrections for a small number of stations when this will take less time. They can also be done for a few sample stations to give a good idea of the terrain correction distribution and validate the computer calculations.
3. The inner terrain correction
Special consideration has to be given to the area nearest the station which can form a significant part of the total terrain correction, especially within the first 170 m where the correction used often to be approximated to a simple slope through the station ŽSandberg, 1958.. The latest method, developed by Lyman et al. Ž1997., uses a reflectorless laser rangefinding system to quickly survey the surface level round the station out to 50–100 m and calculate the terrain correction for this innermost zone ŽAiken and Cogbill, 1998.. Differences of up to 1.0 mGal were found between this method and calculating the correction from a 40 m terrain grid. The system can cope with grass, but not thicker vegetation or fog, as these block and dissipate the laser signals. A parabolic surface can be fitted to the mean elevations ŽRollin, 1990. through octant segments out to 895 m ŽTurnbull, 1980. and rotation of section through the station out to 895 m ŽTurnbull, 1984., though these methods can underestimate the effect by cutting out part of the average height of zones E and F with the fitted surface. A sloping wedge technique for calculating inner terrain corrections was developed by Barrows and Fett Ž1991., where the elevation and range of each change in slope away from a station are picked from a

map and calculated for a given arc around the

station. Oliver and Simard Ž1981. developed a

method

for

terrain

corrections

within

2

1 2

km

of

the station based on conic prisms, which can

smooth out the real topography, instead of

Hammer’s flat-topped compartments, which are

an average of the elevation. Blais and Ferland

Ž1984. found prisms with sloping tops should be

used for calculating corrections within 2.5 km

of a station, as there is only an average y0.09

mGal difference between this and the template

ŽHammer. method, compared to an erroneous

y2.90 mGal for calculations with the standard

flat topped prisms. Granser Ž1987. approxi-

mated the topography in a digital elevation

model to a single valued function, to calculate

terrain corrections between 50 m and about

1000 m from the station. This method involves

the conversion of the volume integral for the

gravity effect into a two-dimensional definite

integral which is solved using the Gauss–

Legendre quadrature formula. But the formulae

used to calculate this correction are very com-

plex, necessitate the use of a computer, and

assume that the points that form the given model

are representative of the general topography

around the station: if this is not so, errors will

occur. As Cogbill Ž1990. pointed out, the errors

in the location of a station and the digital eleva-

tion model relative to the actual topographic

surface within 250 m of the station will result in

errors. Corrections for this zone with a total

value of less than 300 mGal will result in errors

of less than 50 mGal, while larger corrections

can have errors in excess of 100 mGal. The

same effect can be given by using smaller Ham-

mer zones and compartments, though this also

requires more calculations than normal. Ham-

mer Ž1982. pointed out that the estimate of the

average elevation in each compartment requires

considerably more weight for the inner portion

of the topography. This problem is greatest for

the innermost zones, and so Hammer Ž1982.

subdivided zones B, C and D for very high

precision applications. The radius Ž re. at which the inner and outer halves of the correction in a

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

123

Fig. 4. The correction Ž g . for a quarter segment ŽQ. of a wedge with uniform slope u , above or below the station, with a radius R, rotated around the vertical axis through the observation point. g s1r2p Gr RŽ1ycos u . where G is the gravitational constant and p the density of the rock.

Hammer zone are equal varies slightly with
height and can only be calculated by iteration.
However, this radius Ž re. for any height can be approximated as:

re s Ž r2 q r1.r2

½( 5 y2 Ž r22 qr12r2. y Ž r2 qr1.r2

Ž8.

where r1 and r2 are the inner and outer radii of the zone.
A simple way of calculating the inner terrain
correction Ž g . is to use the formula for a wedge

of uniform slope Žu . above or below the station

of a given radius Ž R., rotated around the verti-

cal axis through the observation point:

g s 2p Gr RŽ1 y cos u . .

Ž9.

This can be adapted to calculate the effect for a quarter Ž908. segment ŽFig. 4.. This new method

is more realistic and flexible than the power law

approximation for a station on a uniform slope

out to zone C Ž53.3 m. ŽCampbell, 1980..

4. Buildings
Buildings near a gravity station will have an effect on readings, as their mass results in an upward attraction which decreases the observed value of gravity, and so a correction has to be added to readings. For detailed microgravity the effect of known sources have to be carefully modelled ŽPatterson et al., 1995. and the method used below can only serve as a rough estimate.
The effect of an average modern building was calculated as a Hammer zone compartment at given distances from the centre, and the resulting data plotted as a graph ŽFig. 5.. The

Fig. 5. The effect of a building, 27 m square, 100 m high, 25 = 106 kg mass and average density of 340 kgrm3, with distance from the centre of the building in mGal, calculated as a Hammer zone compartment of suitable dimensions and density.

124

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

building was assumed to be 100 m high and 27 m by 27 m in area with a density for building materials Žconcrete and steel. of 2300 kgrm3. Floor decks were 400 mm deep every 3.2 m and columns had sides 300 mm square with a spacing of 9 m between columns. The cladding round the block was assumed to be 0.1 m thick and the inner core of the building was equal to one outer side of cladding. The volume of the building materials was 10,825 m3 giving a total mass of about 25 = 106 kg which can be used as the basis for finding the effect of any large 20th century building by first calculating its volume. It should be noted that older large buildings often have an average density twice that of modern ones or more. An average 25 = 106 kg building at a distance of 30 m from its centre has a 50 mGal effect while at a distance of 100 m only 5.5 mGal, so a group of nearby buildings could have a significant effect. If the hypothetical 100 m high building had a basement 6.9 m deep, which is not uncommon in modern buildings, this would result an effect equal to half that of the building, given a density of 2670 kgrm3 for the surrounding rock, which would also have to be added to readings. As basements are out of sight, this could result in small errors, and so for many buildings it may be simpler to add one and a half times the building correction — or some smaller fraction, depending on local knowledge.
5. The outer limit of terrain corrections
There is the problem for a given survey of the distance at which to stop calculating the terrain correction. Hammer Ž1939. took 21.944 km, while modern surveys, such as the British Geological Survey ŽBGS., often go out to 50 km or further, and Bullard Ž1936. went out to 166.735 km. The effect of terrain beyond this distance was found to vary little throughout Czechoslovakia with a range of y112 to y109.5 mGal ŽPick, 1987. equivalent to an artificial southwestward gradient of about 7.6

mGalrkm. Sprenke Ž1989. devised a method for optimising the distance to which terrain corrections are made, based on a geostatistical analysis of the topography around a given area. This method cuts out unnecessary terrain corrections beyond a calculated limit based on the accuracy needed. Danes Ž1982. developed a method to estimate the terrain correction of an outer zone extending to infinity. These methods are based on the elevation of valley floors, summits and the station, and do not take into account the average elevation differences above and below the station. For example, a plain dissected by deep narrow gorges would give odd results, as the elevation of the terrain is not evenly distributed, with the average elevation at a level just below that of the plain and not half way down to the floor of the narrow gorges as these methods would imply. More importantly, Danes Ž1982. and Sprenke Ž1989. take no account of the effect of increasing station height, as terrain corrections at large distances are height dependent ŽHallinan, 1991. and significant for large height differences within a survey. This was illustrated by LaFehr Ž1998. for a range of station elevations and topographic relief. In mountainous areas, gravity investigations must not be confined to valleys, as errors of up to 10 mGal can be caused due to fault structures or low density sediments filling the floor of valleys ŽSteinhauser et al., 1990..
It should be remembered that the complete free-air and Bouguer correction assumes that the data are collected at a common datum. As the data are collected at the level of each observation, the formulae Žnot the data. are corrected for this incorrect assumption ŽChapin, 1996., so that the effects above Žor in special cases below. the datum are reduced in a uniform manner. The treatment of variable density should be left for the interpretation and not included as part of the standard data reduction ŽLaFehr, 1991a.. However, bodies of anomalous density outside the limits of the model will be unaccounted for, but these bodies can be estimated in the modelling process and if necessary their effects removed

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

125

from the data. Thus the resulting anomalies can be modelled using any competent program such as GRAVMAG ŽPedley, 1991.. 6. The effects of height
For a given height above or below a station, the terrain correction in a Hammer compartment will be less for zones further away from the station ŽFig. 6.. For a given Hammer zone the increase in terrain correction with height is not linear, and so errors in finding the average
Fig. 6. Increase in terrain correction in mGal with height in metres for a Hammer zone compartment Ž16th of a zone. in zone J Ž4469 m to 6653 m. and zone N Ž21,944 m to 32,667 m., density 2000 kgrm3.

height of a compartment will become more significant with an increased height difference between the height of the compartment and the station ŽFig. 6.. At heights larger than the dimension of the outer radius, the rate of change decreases for a given height difference with greater height, and as the height tends to infinity the correction reaches a limiting value as the zone forms an ever smaller part of the overall Bouguer slab.
At distances greater than about 22 km, the amount of terrain correction is height dependent ŽHallinan, 1991.. For an isolated mountain 2000 m high the Bouguer correction due to the Bouguer slab at a distance greater than 22 km is 10.2 mGal and even at 50 km is 4.5 mGal ŽFig. 7.. This must be removed by adding terrain corrections out to a large distance of up to 200 km as Hallinan Ž1991. did.
Extending terrain corrections out to the limit of Bullard B correction for the curvature of the Earth at 166.735 km means that at this point errors from nearby terrain are far more significant than errors resulting from distant topography ŽLaFehr, 1991a. and have a very long wavelength compared to the high-frequency components of nearby terrain. However, LaFehr Ž1991a, Fig. 2. concedes that topographic effects beyond 167 km may occasionally be worth taking into account. As this is beyond the Bullard B Bouguer cap, the topography between sea level and the level of the horizontal tangent from the station will result in an additional negative Bouguer correction, while topography above the tangent will result in a normal positive terrain correction, as will bodies of water Žcorrected for the difference with the Bouguer density. and valleys and voids below sea level. The difference between readings at sea level and a height of 4 km on an isolated island, due to the effect of a 4 km deep ocean beyond the Bullard B limit, is about 6.6 mGal given a density contrast of 1640 kgrm3. The selection of 166.735 km as the outer limit of the Bullard B correction was based on minimising the difference between the effect of the cap and that of

126

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

Fig. 7. Difference between Bouguer slab correction and gravitational attraction for a disk out to 21.944 km dashed line and 50 km solid line in mGal with increasing station height, density 2670 kgrm3.

an infinite horizontal slab for a significant range of elevations ŽLaFehr, 1991b. ŽFig. 8.. 7. The effect of the Earth’s curvature
A problem with distant terrain corrections for the effects of height would appear to be the curvature of the Earth, which is dealt with in the Hayford–Bowie system. At a distance of 22 km
Fig. 8. The varying corrections with height for reducing the Bouguer slab to spherical caps of differing radii, from LaFehr Ž1991b..

the surface of the Earth drops 38 m below the horizontal tangent of a station and can be approximated to:

b s r y cos u r

Ž10.

where r is the radius of the Earth Žtaken as an average 6371 km. and u the angle at the centre of the Earth due to the distance on the surface ŽFig. 9.:

u s 180Srp r .

Ž11.

This drop Ž b. can also be approximated more simply by:

b s S2r2 r

Ž12.

ŽParker, 1995.. The exact drop at any given point would be difficult to calculate as the radius of the Earth varies with latitude and with the direction away from the station.
If, as the BGS does, a curved Bouguer cap is used for the Bouguer correction Žpers. comm. K.E. Rollin., then the curvature of the Earth is taken into account by the BGS. A small positive elevation difference above the station at distances greater than about 20 km can produce a negative terrain correction ŽSazhina and Grushinsky, 1971; Rollin, 1990., as can happen with the system devised by Hayford and Bowie Ž1912.. This results from the mass of a hill, below the tangential plane of the station, pulling downwards and increasing the observed value

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

127

Fig. 9. The geometrical framework for calculating the effects of curvature on the surface of a sphere, where: r s the radius of the Earth; bs the drop below the tangent from the station to a point on the Earth’s surface; Ss the distance along the surface of the Earth between the station and a given point; Ds the direct distance, through the Earth, between the station and a given point; u s the angle at the centre of a sphere formed by a straight line distance on the surface; T s the distance along the tangent between a given point and the station; ps the difference between the drop b and the radius of the Earth r.
of gravity, so this effect has to be subtracted from the readings, as it was part of the Bouguer slab removed in the Bullard B correction, while the portion of larger hills above the plane of the station will result in a normal positive terrain correction. The combined effects of these two corrections for terrain above the Bouguer cap are set out in tables by Sazhina and Grushinsky Ž1971.. Topography below the height of the station is corrected for in the normal way, after adjusting it for the drop due to the Earth’s curvature, as valleys below the height of the station start at the top of the Bouguer cap after the Bullard B correction has been made.
As the drop below the horizontal plane of the station will be different for each of the two radii of a Hammer zone, the correction for the drop Ž b. would result in a sloping compartment at the correct level relative to the station, which is far more complex to calculate. To get round this the average drop for the compartment can be calculated at the radius of equal weighting Ž re.

ŽTable 1.. The curvature adjusted terrain correction for a Hammer compartment can be approximated by calculating the difference between the combined thickness of the height plus drop less the effect of a compartment equal to the drop only. Another effect of increased curvature is that the local vertical of a point at a distance away from the station will be out by u from that of the vertical at the station, so that the top of a column or radius above this point will be further away and one below nearer to the station. Also, the direct distance Ž D. between the station and the point will be less than that on the surface of the sphere ŽFig. 9. and can be calculated with:

(½ 5 Ds Ž rycos u r .2q Žsin u r .2 .

Ž13.

However, apart from the drop Ž b. below the horizontal tangent from the station, these effects are small, because topographic variations out to distances that influence corrections are small compared with the radius and circumference of the Earth. Even at 108 Ž1111.95 km., the difference between the surface and direct distances is only 1.41 km or 0.127% and the top of a 5000 m high column is only 868 m further away: this is less than a thousandth of the total distance. With radial line elements ŽTalwani, 1973., the effect of distant terrain on a spherical Earth can be directly evaluated using a computer program.

8. Terrain corrections to the sea bed
As Bullard Ž1936. recognized, it is important that terrain corrections are taken down to the sea bed or lake bottom, as has been done on 1:250,000 Bouguer anomaly maps of New Zealand ŽReilly, 1972., while Steinhauser et al. Ž1990. realized that not only lakes had to be taken into account but glaciers with a density of between 790 kgrm3 and 880 kgrm3 for the ice. The effect of not including the void below sea level is even more significant than just the difference between the density of the sea water Ž1030 kgrm3. and that used for the terrain

128

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

Fig. 10. The corrections for a gravity station ŽGS. above sea level on land, down to the sea bed, see Fig. 1b for key.
correction ŽFig. 10.. This arises from the increasing curvature of the Hammer compartment correction graph ŽFig. 6. with height, so that the increase in thickness of the compartment will produce a larger correction than that for the layer of water alone, even if it had the same density as that of the terrain correction. This could be overcome by taking the difference in terrain correction in a compartment between the thickness down to sea level and the sea bed and dividing by the ratio of densities to get the correction for the water. But this ignores the fact that a given depth of water does not have that effect in a Hammer compartment. By calculating the effect of the water separately to that of the terrain the terrain correction density can be easily changed as the density of the body of water is fixed.

To show that taking terrain corrections down to the sea bed has a significant effect, gravity data from a survey of a coastal part of eastern Iceland ŽNowell, 1994. were examined and set out in Table 2. For a station on the coast, the depth of the sea off Iceland was up to 100 m or more deeper in Zone M. The effect of height can be illustrated by a low level station at the head of a fjord ŽLJ 20. where the terrain correction down to the sea bed had little effect Ž4 mGal. while a station further inland but at a height of over 600 m ŽLJ 11. has a net effect of 25 mGal. Coastal stations near sea level ŽSW6, SW8, SW10. have small effects of up to 45 mGal, while a fictitious station ŽM1. at 700 m on the hill just inland above them has a large overall net effect of 309 mGal with the water only accounting for 6 mGal. The importance of taking the terrain corrections down to the sea bed will be much greater for areas where the depth of water increases much faster than off Iceland, such as small mid-ocean islands.
9. Stations on the sea surface
Normally, free air gravity surveys are acquired at sea, but a Bouguer correction can be made for gravity readings taken on the sea

Table 2 Contribution to the total terrain correction from taking the terrain correction down to the sea bed, accounting for the effect of water. Taken from a gravity survey of Eastern Iceland ŽNowell, 1994.

Location

Grid Ref

H

TC

SB

W

NE

PD

SW6 SW8 SW10 M1 LJ11 LJ

393 754 461 778 444 825 439 799 239 883 244 860

q15.0

q6195

q36

y6

q30

0.48

q16.3

q2422

q57

y12

q45

1.86

q15.4

q3047

q8

0

q8

0.26

q700.0

q11,108

q315

y6

q309

2.78

q615.2

q9058

q26

y1

q25

0.28

q33.1

q6162

q5

y1

q4

0.06

Grid Ref s grid reference on 1:50,000 U.S. Army Map of Iceland.
H s height of station in metres above sea level. TC s terrain correction down to sea bed in mGal given a density of 2600 kgrm3. SB s effect due to going down to the sea bed instead of sea level in mGal. W s effect of water above sea bed given a density of 1030 kgrm3 in mGal. NE s net effect of taking the terrain correction down to sea bed in mGal.
PD s percentage of terrain correction due to going down to the sea bed instead of sea level.

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

129

Fig. 11. The corrections for a gravity station ŽGS. on the sea surface, see Fig. 1b for key.

surface to account for the missing rock between the sea bed and sea level ŽFig. 11.. This was done by Lefort et al. Ž1999. to contour the local variations in Moho depth over the continental margin between France and Ireland by processing and filtering the resulting marine Bouguer anomaly map. However, they gave no details of exactly how these Bouguer corrections were made. If marine and land based Bouguer corrected gravity data are to be combined, this marine Bouguer correction must be made in order to avoid errors in the maritime part of any model and especially at the transition between the data sets along the coastline. The lack of mass below the station results in a decrease in the observed value of gravity, so the marine Bouguer correction has to be added to readings. The effects of the terrain of the sea bed and any nearby land also have to be taken into account. The surface of any nearby land is above the station and will result in an upward attraction, which decreases the observed value of gravity, so this effect also has to be added to readings. As it would be complex to add a Bouguer slab and then remove the effects due to sea bed terrain, a combined marine Bouguer correction can be added. The marine Bouguer correction consists of a correction for the disk of water directly below the station and Hammer zones down to the sea bed away from the station. The density used is the difference between the density of the Bouguer slab and the water. The formula for the gravitational attraction Ž g . of the cylinder around the station on the sea surface down to the sea bed is

( g s 2p Gr R y Ž R2 q h2 . q h

Ž14.

where h is the depth of water and R the radius of the cylinder. The Hammer zones down to the sea bed are calculated in the normal way with h as the depth of water in each compartment. The effects of the disk and Hammer compartments are added together to get the marine correction. The effect of any nearby land can be accounted for by adding the normal terrain correction with heights relative to sea level and with the density of the Bouguer slab. A rapid and efficient Fourier technique for these corrections for marine gravity surveys in shallow water has been developed by Parker Ž1995.. The depth of water and the height of any land will affect how far out the marine and terrain corrections have to be taken: the deeper the water or the higher the land, the further out corrections will have to go.
10. Stations on the sea bed
The corrections for gravity readings on the sea bed are more complex than those on the sea surface ŽFig. 12.. As the value of gravity increases towards the centre of the Earth within the crust, the FAC of normally 0.3086 mGalrm below sea level is subtracted from readings. The effect of the water above the station pulls upwards and decreases the observed value of gravity, so these effects have to be added to readings if you are normalising to sea level for a marine Bouguer correction. Sea bed below the level of the station will have a lack of mass due to the missing rock and result in a decrease in the observed value of gravity, and so this effect will
Fig. 12. The corrections for a gravity station ŽGS. on the sea bed, see Fig. 1b for key.

130

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

have to be added to readings. Sea bed and land above the station will also result in an upward attraction from the rock above the station and decrease the observed value of gravity, so these effects have to be added to readings as a terrain correction. As a result the sea bed correction for stations on the sea bed has to be divided into three. First, the effect of the water above the station is taken into account by correcting for a cylinder of water above the station and the water in the Hammer compartments either down to the sea bed, shallower than the station, or to the depth of the station in deeper water. Second, the sea bed below the station is corrected for by calculating the effect in Hammer compartments with a density equal to the difference between that of the Bouguer density and the water, where the thickness of the compartment is the depth between the level of the station and the sea bed below. Third, any sea bed and land above the level of the station are taken into account by finding the height difference between the topography and the station and calculating the effect for each Hammer compartment with the Bouguer density. The effects of all these Hammer compartments and the cylinder of water above the station are added together to give the sea bed correction, which is added to readings. To standardise methods the marine Bouguer correction should be carried out to the same distance away from the station as land surveys, so that they can be compared.
11. Stations above and below ground level
Corrections for gravity stations above ground level, such as up a tower or on an airborne survey, consist of two parts: a Bouguer correction up to the level of the station; and a terrain correction at the Bouguer density, above and below the level of the station ŽFig. 13.. The free-air correction is added up to the level of the station as normal. The terrain correction, at the Bouguer density and to the level of the station, consists of a correction for the void below the

Fig. 13. The corrections for a gravity station ŽGS. above ground and sea level, see Fig. 1b for key.
station, which is approximated to a cylinder directly below the station, along with Hammer zones down to ground level, as well as Hammer zones for any topography above the level of the station. The terrain correction in effect cancels out that part of the Bouguer slab above the level of the ground, and where applicable it is taken down to the sea or lake bed with any water corrected for. Also, unless the flight line is below the highest topography, the terrain effect in airborne data will be less than those for conventional ground observations ŽHammer, 1983. and falls with increasing altitude.
A correction for the tower on which a reading is taken can be based on the gravitational attraction of a right rectangular prism ŽNagy, 1966. or cylinder of given length ŽFormula 14.. The correction for material above a station is positive and below a station is negative. With increased height above sea level it may again be necessary to extend the terrain corrections outwards until there is little remaining effect. Kuo et al. Ž1969., as had Hammer Ž1938., made these corrections for readings in tall buildings and showed that these vary with height, but failed to extend them beyond Hammer zone M Ž; 22 km. and did not account for the effects of surrounding buildings, even though the effect of the basement still needed a correction for readings on the highest floors.
Corrections for gravity stations below ground level are more complex, depending on whether the station is above or below sea level. The corrections for a gravity station below ground

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

131

level but above sea level consist of adding a normal FAC and subtracting a Bouguer correction to the level of the station, before adding a terrain correction ŽFig. 14.. The terrain correction, at the Bouguer density, consists of a correction for the cylinder of rock directly above the station along with Hammer zones, away from the station, down to this level, or up to it in the case of topography below the station.
The correction for a gravity station below the ground and below sea level consists of subtracting a FAC to account for the increase in gravity towards the centre of the Earth, within the crust, before adding a terrain correction, at the Bouguer density, for all the rock above the station which is pulling upwards and decreasing gravity at that point. If the compartments at larger distances are still thick and giving significant corrections, the terrain correction will have to be extended outwards to a greater distance, possibly beyond the limit of the Bullard B correction. In the special case of water being above the plane of the station, an additional correction for the water pulling upwards and decreasing the observed gravity at that point is made using the terrain correction method with the density of the water.
Any voids below ground can be corrected for by calculating the effect of rectangular prisms ŽNagy, 1966. with the density of the rock Žnormally the Bouguer slab density. relative to the station. This can also be done with a vertical line element ŽTalwani, 1973. of cross sectional area Ž s. at horizontal distance Ž r . from the

station between two levels Ž Z1. near and Ž Z2. further above or below the plain of the station:
½ ( ( 5 gsGr s 1r Ž . r 2qZ12 y1r Ž r 2qZ22 . .
Ž15.
The correction for a void above a station is negative and for a void below a station is positive. In a gravity survey of two mines, these corrections for the shaft tended to cancel out ŽGibb and Thomas, 1980. but were greatest at the top and bottom of the shafts — up to q70 mGal and y50 mGal, respectively — with a maximum total void correction of y430 mGal midway down a shaft due to the effects of galleries at that level which did not cancel. Gibb and Thomas Ž1980. found that corrections for the galleries in which the readings were made decayed rapidly with distance from the station: about 85% of the correction occurred within 10 m of a station.
As corrections both above and below ground level vary with height, they should have been made in tower and mine experiments searching for deviations from Newtonian gravity ŽFishbach and Talmadge, 1992.. Since this seems not to have been done, these results will be erroneous until they are recalculated to take the full terrain correction into account. Even with submarine measurements of gravity in vertical profiles to depths of 5000 m to calculate the gravitational constant, terrain corrections were only made in a 60 km square region ŽZumberge et al., 1991..

Fig. 14. The corrections for a gravity station ŽGS. below ground level but above sea level, see Fig. 1b for key. The correction for a void above ŽVA. a station is negative and for a void below ŽVB. a station is positive.

12. Summary and conclusions

The effect of topography at a gravity station

is removed with a Bouguer correction ŽBullard

A., a correction for the curvature of the Earth

ŽBullard B. and a terrain correction ŽBullard C..

With modern computers, full corrections out to

a distance of 166.735 km, the Bullard B limit of

1

1 2

8,

should

become

standard.

This

arbitrary

dis-

tance can then be used to judge uniformly

132

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

whether it is worth taking into account the effects of terrain beyond this cutoff. Also, this would eliminate one possible source of mismatch between surveys and reduce the introduction of processing error in the final Bouguer anomalies for little additional cost ŽLaFehr, 1998.. This will be especially true if future gravity instruments ŽChapin, 1998. improve and have higher accuracies down to the mGal level or better. In addition, this higher resolution combined with gravity gradiometry ŽBell, 1997, 1998. will greatly enhance geological interpretations made with gravity data ŽPawlowski, 1998..
In order to get a precise correction, many factors have to be taken into account: otherwise, errors of many mGal or only a few mGal will result, depending on which factors are ignored. The most significant effect is that of increasing station height. This must be accounted for by applying the Bullard B correction for the curvature of the Earth and extending terrain corrections adjusted for Earth curvature outwards, until they have no discernible effect. If this is not done, high stations may be out by many mGal relative to lower stations in an area. The elevations for distant terrain correction can be clustered together being calculated for each station ŽHallinan, 1991.. Special attention must be given to the ground nearest the station as this can have a significant effect, and a simple correction using the formula

g s 1r2p Gr RŽ1 y cos u .

Ž16.

can be applied for the sloping quarter segments around the station. A building near a station can result in effects of a few mGal which can be calculated and removed, assuming that the building has a uniform average density. Also terrain corrections must be applied down to the sea or lake bed, otherwise errors of over 1 mGal could result with stations near deeper water than about 100 m for Hammer zone M off Iceland. Stations in special locations such as the sea surface or bed and above and below ground level can have gravity corrections calculated if all the effects on a station are considered.

It should be hoped that in the future, published Bouguer gravity maps and calculated models will state in much more detail how the corrections to the data Žreadings. were made so that the final result does not appear to be a geophysical conjuring trick.
Acknowledgements
Dr. Allen Cogbill is thanked for his informative discussions on this topic and sending me a copy of part of his PhD thesis. Keith Rollin of the BGS Regional Geophysics group at Keyworth is thanked for providing information on how the BGS correct gravity data. Dr. Bill Addis and James Lee are thanked for their help in calculating the mass of an average tower block. Dr. Peter Styles and Dr. David Chapin are thanked for critically reviewing earlier drafts of this paper.
References
Aiken, C.L., Cogbill, A.H., 1998. Recent developments in digital gravity data acquisition on land. The Leading Edge 17, 93–97.
Barrows, L.J., Fett, J.D., 1991. A sloping wedge technique for calculating gravity terrain corrections. Geophysics 56, 1061–1063.
Bell, R.E., 1997. Gravity gradiometry resurfaces. The Leading Edge 16, 55–59.
Bell, R.E., 1998. Gravity gradiometry. Scientific American, June, 56–61.
Blais, J.A.R., Ferland, R., 1984. Optimization in gravimetric terrain corrections. Canadian Journal of Earth Sciences 21, 505–515.
Bott, M.H.P., 1959. The use of electronic digital computers for the evaluation of gravimetric terrain corrections. Geophysical Prospecting 7, 46–54.
Bullard, E.C., 1936. Gravity measurements in East Africa. Philosophical Transactions of the Royal Society, London 235, 445–534.
Campbell, D.L., 1980. Gravity terrain corrections for stations on a uniform slope — a power law approximation. Geophysics 45, 109–112.
Cassinis, G., Dore, P., Ballarin, S., 1937. Tavole fonda-

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

133

mentali per la riduzione dei valori osservati della
gravit`a. Publicazione dell’ Istituto di Geodesia, n 13.
Chapin, D.A., 1996. The theory of the Bouguer gravity anomaly: a tutorial. The Leading Edge 15, 361–363.
Chapin, D.A., 1998. Gravity instruments: past, present, future. The Leading Edge 17, 100–112.
Cogbill, A.H., 1979. The relationship between crustal structure and seismicity in the Western Great Basin. Unpublished PhD thesis, Northwestern University, Evanston, IL, 289 pp.
Cogbill, A.H., 1990. Gravity terrain corrections calculated using digital elevation models. Geophysics 55, 102– 106.
Danes, Z.F., 1982. An analytic method for the determination of distant terrain corrections. Geophysics 47, 1453–1455.
Erratum, 1992. Geophysics, Vol. 57, 370. Fishbach, E., Talmadge, C., 1992. Six years of the fifth
force. Nature 356, 207–214. Gibb, R.A., Thomas, M.D., 1980. Density determination
of basic volcanic rocks of the Yellowknife supergroup by gravity measurements in mine shafts — Yellowknife, Northwest Territories. Geophysics 45, 18–31. Granser, H., 1987. Topographic reduction of gravity measurements by numerical integration of boundary integrals. Geophysical Prospecting 35, 71–82. Hallinan, S.E., 1991. Gravity studies of the Guayabo Caldera and the Miravalles Geothermal Field, Costa Rica. Unpublished PhD thesis, The Open University, Milton Keynes, 381 pp. Hammer, S., 1938. Investigation of the vertical gradient of gravity. Transactions of the American Geophysical Union, 19th annual meeting, Part 1, 72–82. Hammer, S., 1939. Terrain corrections for gravimeter stations. Geophysics 4, 184–194. Hammer, S., 1982. Critique of terrain corrections for gravity stations. Geophysics 47, 839–840. Hammer, S., 1983. A note on airborne gravity terrain corrections. Geophysics 48, 396–399. Hayford, J.F., Bowie, W., 1912. The Effect of Topography and Isostatic Compensation upon the Intensity of Gravity. U.S. Coast and Geodetic Survey Special Publication No. 10, Washington, DC. Heiskanen, W.A., Moritz, H., 1967. Physical Geodesy. Freeman, San Francisco, 364 pp. Herrera-Barrientos, J., Fernandez, R., 1991. Gravity terrain using Gaussian surfaces. Geophysics 56, 724–730. Kuo, J.T., Ottaviani, M., Singh, S.K., 1969. Variations of vertical gravity gradient in New York City and Alpine, New Jersey. Geophysics 34, 235–248. LaCoste, L., 1991. A new calibration method for gravity meters. Geophysics 56, 701–704. LaFehr, T.R., 1991a. Standardization in gravity reduction. Geophysics 56, 1170–1178.

LaFehr, T.R., 1991b. An exact solution for the gravity curvature ŽBullard B. correction. Geophysics 56, 1178– 1184.
LaFehr, 1998. On Talwani’s ‘‘Errors in the total Bouguer reduction’’. Geophysics 63, 1131–1136.
Lambert, W.D., 1930. The reduction of observed values of
gravity to sea level. Bulletin Ge´ode´sique 26, 107–181.
Lefort, J.P., Agarwal, B.N.P., Jaffal, M., 1999. A tentative chronology of the Moho undulations in the Celtic sea region. Geodynamics 27, 161–174.
Lyman, G.D., Aiken, C.L., Cogbill, A., Balde, M., Lide, C., 1997. Terrain mapping by reflectorless laser rangefinding systems for inner zone gravity terrain corrections. Expanded Abstracts, 1997 SEG annual meeting, November 2–7, Dallas, TX.
Nagy, D., 1966. The gravitational attraction of a right rectangular prism. Geophysics 31, 362–371.
Nowell, D.A.G., 1994. Gravity studies of two silicic volcanic complexes. Unpublished M Phil thesis, The Open University, Milton Keynes, 193 pp.
Oliver, R.J., Simard, R.G., 1981. Improvement of the conic prism model for terrain correction in rugged topography. Geophysics 46, 1054–1056.
Parker, R.L., 1995. Improved Fourier terrain correction: Part I. Geophysics 60, 1007–1017.
Parker, R.L., 1996. Improved Fourier terrain correction: Part II. Geophysics 61, 365–372.
Patterson, D.A., Davey, J.C., Cooper, A.H., Ferris, J.K., 1995. The investigation of dissolution subsidence incorporating microgravity geophysics at Ripon, Yorkshire. The Quarterly Journal of Engineering Geology 28, 83–94.
Pawlowski, B., 1998. Gravity gradiometry in resource exploration. The Leading Edge 17, 51–52.
Pedley, R., 1991. GRAVMAG — User Manual ŽInteractive 2.5D Gravity and Magnetic Modelling Program.. Integrated Geophysical Services, British Geological Survey, Keyworth, England, 83 pp.
Pick, M., 1987. On the calculation of the gravity terrain corrections in Czechoslovakia. Studia Geophysica et Geodetica 31, 131–194.
Reilly, W.I., 1972. New Zealand gravity map series. New Zealand Journal of Geology and Geophysics 15, 3–15.
Robbins, S.L., 1981. Reexamination of the values used as constants in calculating rock density from borehole gravity data. Geophysics 46, 208–210.
Rollin, K.E., 1990. Terrain corrections for gravity stations using a Digital Terrain Model. British Geological Survey Technical Report WKr89r8.
Rymer, H., 1989. A contribution to precision microgravity data analysis using Lacoste and Romberg gravity meters. Geophysical Journal 97, 311–322.
Sandberg, C.H., 1958. Terrain corrections for an inclined plane in gravity computations. Geophysics 23, 701–711.

134

D.A.G. Nowellr Journal of Applied Geophysics 42 (1999) 117–134

Sazhina, N., Grushinsky, N., 1971. Gravity Prospecting. Moscow, 491 pp.
Sprenke, K.F., 1989. Efficient terrain corrections: a geostatistical analysis. Geophysics 54, 1622–1628.
Steinhauser, P., Meurers, B., Ruess, D., 1990. Gravity investigations in mountainous areas. Exploration Geophysics 21, 161–168.
Talwani, M., 1973. Computer usage in the computation of gravity anomalies. In: Bolt, B.A. ŽEd.., Methods in Computational Physics: Geophysics. Academic Press, New York, pp. 343–389.
Talwani, M., 1998. Errors in the total Bouguer reduction. Geophysics 63, 1125–1130.
Tsuboi, C., 1983. Gravity. George Allen and Unwin, London, 254 pp.

Turnbull, G., 1980. PIT80: Pocket calculator Inner zone Terrain corrections. Institute of Geological Sciences, Applied Geophysics Unit Computer Report No. 44 Žunpublished..
Turnbull, G., 1984. GRITZ84: Gravity Inner Terrain Zones 1984 ŽCasio Basic. British Geological Survey, Regional Geophysics Research Group Computer Program Report No. 52 Žunpublished..
Valliant, H.D., 1991. Gravity meter calibration at La Caste and Romberg. Geophysics 56, 705–711.
Zumberge, M.A., Hildebrand, J.A., Stevenson, M.J., Parker, R.L., Chave, A.D., Ander, M.E., Spiess, F.N., 1991. Submarine measurement of the newtonian gravitational constant. Physical Review Letters 67, 3051– 3054.