##### Document Text Contents

Page 1

TERRAIN CORRECTIONS FOR GRAVIMETER STATIONS*

SIGMUND HAMMER**

ABSTRACT

In this paper the correction for the.gravitational attraction of the topography on a

gravity station is considered as consistmg of two parts; (I) the restricted but conven-

tional “Bouguer correction” which postulates as a convenient approximation that the

topography consists of an infinite horizontal plain, and (2) the “Terrain correction”

which is a supplementary correction taking into account the gravitational effect of the

undulations of the terrain about the plane through the gravity station. The paper illus-

trates the necessity of making terrain corrections if precise gravity surveys are desired

in hilly country and presents terrain correction tables with which this quantity may be

determined to a relative accuracy of one-tenth milligal. This accuracy is required to

fully utilize the high instrumental precision of modern gravimeters.

Applications of the gravity method of geophysical prospecting in

the search for small, local, geological structures require very precise

data. To supply this need gravimeters have been developed or im-

proved to the point where the probable error of the observed

gravity values is of the order of I/IO mg. or even less1~2~3,4 To fully

utilize this high instrumental precision it is obvious that corrections

for the several non-significant influences, which are present in the ob-

served gravity values, must be of the same order of accuracy. One of

the important non-significant influences is the gravitational attraction

of the topography in the vicinity of a gravity station. This so-called

“topographic correction” will be considered, by definition, in the

present paper to consist of two parts; (I) the “Bouguer correction”

which postulates as a convenient approximation, that the topography

consists of an infinite horizontal plain, and (2) the “Terrain correc-

tion” which evaluates the error in the Bouguer correction due to

undulations of the terrain. The purposes of this paper are (a) to

illustrate the necessity of making terrain corrections if precise

gravity surveys are desired in hilly country, and (b) to present

* Published by permission of the Gulf Research 8r Development Co., Pittsburgh’

Pa.

** Gulf Research & Development Co., Pittsburgh, Pa.

r E. A. Eckhardt, Geophysics I, pp. 292-293 (1936) (abstract).

’ D. C. Barton and W. T. White. Trans. Am. Geophys. Union, Part I, pp. 106-107

(1937).

s A. Graf. Zeits f. Geophys. 14, pp. 152-172 (1938).

’ H. Hedstrom, 4.T.M E. Tech. Publ. So. 953 (1938).

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 2

TERRAIN CORRECTZONS FOR GRa4 VIMETER STATIONS

tables with which the terrain correction may be determined

accuracy merited by the precision of modern gravimeters.

THE CORRECTION FOR TOPOGRAPHY

185

to the

A common procedure in gravity prospecting is to correct for the

attraction of the topography by use of an approximate value “B”

calculated from the simple Bouguer formula

B = anya(H - Ho) (I)

where I&II0 is the elevation of the gravity station above the level

datum

u is the density of the surface soil or rock

y is the gravitational constant.

This formula calculates the gravity effect of the matter between a

horizontal plane through the field station and the horizontal eleva-

I:IG. I. Schematic diagram of the Douguer correction, illustrating residual

~rnvitational effects due to undulating terrain.

tion-datum-plane on the assumption that the space between these

two planes out to an infinite radius is uniformly filled with matter if

the station is above the datum plane, or is uniformly empty if the

station is below this plane.5 (See Fig. I). Subtracting this quantity

(algebraically) from the observed gravity value amounts to removing

the assumed amount of matter in the infinite flat plate if the station is

above the datum plane or to completely filling up this space if the

station is below the datum plane so as to reduce, in effect, the surface

of the ground to the plane through the elevation datum.

6 liar an excellent brief stntemcnt of the nature of the Ilouguer reduction SW EIay

ford and Bowie, U.S.C. & G.S. Special Publication No. IO, p. 75 (1912). These authors

define the “Bouguer Correction” in the more general scnsc which is equivalent to our

“Topographic Correction.”

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 5

188 SIGMUWD HAMMER

evaluating it. By its very definition it is clear that the terrain correc-

tion depends upon the details of the topography. Therefore, a

topographic map, or the equivalent, of the vicinity of the gravity

station is required. The area about the station is divided into zones

FIG. 1. Zone chart for use in evaluating terrain corrections at gravity stations.

and compartments, for example, by laying a transparent terrain cor-

rection zone chart, such as that shown in Fig. 4, upon the map and

centering it at the gravity station. The average departure without

regard to sign of the topography in each compartment from the

plane through the station is then determined* and the terrain cor-

* Since the sign of the terrain correction (as defined in this paper) depends only

on the absolute value of the departure of an element of the topography from the plane

through the statron, and not on its sign (that is; whether it is above or below the plane)

it follows that the difference between the simple average elevation in each compartment

and the station elevation cannot be used, in principle, if the terrain within any one com-

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 6

rection corresponding to this average departure is evaluated for each

compartment by means of tables calculated for that purpose. Finally

these terrain corrections arc summed over all the zones in which there

are appreciable effects.

TERRAIN CORRECTION TABLES

,4 number of terrain correction tables have been published6,7-8

for geodetic and regional gravity work. However, the author knows

of no previously published tables which are sufficiently precise for

modern gravity prospecting needs. The tables presented in this paper

have been designed especially for this use after experience extending

over a number of years with terrain correction calculations.

The present tables were modelled after those by Hayfortl and

Bowie6*7 but are more precise and have been modified in principle to

evaluate, not the total topographic correction, but only the error in

the Bouguer correction as discussed above.* This modification permits

relatively larger compartments in the distant zones, for a given pre-

cision, and thus reduces the labor involved in the use of the tables.

However, to attain the precision desired the compartments in the

zones near the station are smaller in the present tables than in those

by Hayford and Bowie. Another important advantage of the moditica-

tion in the principle is that it is necessary to use the tables in moder-

~.________

parlment consists of elements both above and below the station. Thus, theoretically,

if the topography within one compartment consists of a hill above the station and a

valley below the station, the average departure must be determined as if the topogra-

phy consisted of two hills or two valleys. However, in practice, it turns out that the

terrain effect in such a compartment is ordinarily zero. Therefore, the mathematically

rigorous but cumbersome procedure outlined above may usually be replaced by the

much more rapid procedure of estimating the average elevations in all the compsrt-

ments (irrespective of Lbhether the topography in any one compartment rs above and

below the station) and then calculating the departures of these simple average eleva

tions from the station elevation. This would not be true, in general, in extremely

rugged topography of large relief but even then the complication may usually be

avoided by rotating the zone chart to a different azimuth. The azimuth must, of course,

be maintained while reading the elevations in all the compartments of any one zone.

’ Hayford and Bowie, lot. cit., pp. r3-53.

7 Hayford and Bowie, U.S.C. & G.S. Spec. Publ. 40, pp. 11-18 (1917).

* Bullard, Philos. Trans. Roy. Sot. London A235, pp. 4866491 (1936).

* Bullard’s tables (1~. cit.) are also of this type. His paper contains a detailed dis-

cussion of the advantages of this modification. A recent search m the literature reveals

the historically interesting fact that this n-ethod was described by F. R. Helmert in

1884 (“Thcorien der hoheren Geodasie,” Vol. II pp. 169-172) and was called “the

ordinary method” by F. A. Venning-Meinesz in ‘923 (“Observations de Pendule tlans

1~s Pays-Bas.” I’ubl. de la comm. g6od. nCer!andaise, p, 141).

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 10

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Page 11

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

TERRAIN CORRECTIONS FOR GRAVIMETER STATIONS*

SIGMUND HAMMER**

ABSTRACT

In this paper the correction for the.gravitational attraction of the topography on a

gravity station is considered as consistmg of two parts; (I) the restricted but conven-

tional “Bouguer correction” which postulates as a convenient approximation that the

topography consists of an infinite horizontal plain, and (2) the “Terrain correction”

which is a supplementary correction taking into account the gravitational effect of the

undulations of the terrain about the plane through the gravity station. The paper illus-

trates the necessity of making terrain corrections if precise gravity surveys are desired

in hilly country and presents terrain correction tables with which this quantity may be

determined to a relative accuracy of one-tenth milligal. This accuracy is required to

fully utilize the high instrumental precision of modern gravimeters.

Applications of the gravity method of geophysical prospecting in

the search for small, local, geological structures require very precise

data. To supply this need gravimeters have been developed or im-

proved to the point where the probable error of the observed

gravity values is of the order of I/IO mg. or even less1~2~3,4 To fully

utilize this high instrumental precision it is obvious that corrections

for the several non-significant influences, which are present in the ob-

served gravity values, must be of the same order of accuracy. One of

the important non-significant influences is the gravitational attraction

of the topography in the vicinity of a gravity station. This so-called

“topographic correction” will be considered, by definition, in the

present paper to consist of two parts; (I) the “Bouguer correction”

which postulates as a convenient approximation, that the topography

consists of an infinite horizontal plain, and (2) the “Terrain correc-

tion” which evaluates the error in the Bouguer correction due to

undulations of the terrain. The purposes of this paper are (a) to

illustrate the necessity of making terrain corrections if precise

gravity surveys are desired in hilly country, and (b) to present

* Published by permission of the Gulf Research 8r Development Co., Pittsburgh’

Pa.

** Gulf Research & Development Co., Pittsburgh, Pa.

r E. A. Eckhardt, Geophysics I, pp. 292-293 (1936) (abstract).

’ D. C. Barton and W. T. White. Trans. Am. Geophys. Union, Part I, pp. 106-107

(1937).

s A. Graf. Zeits f. Geophys. 14, pp. 152-172 (1938).

’ H. Hedstrom, 4.T.M E. Tech. Publ. So. 953 (1938).

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 2

TERRAIN CORRECTZONS FOR GRa4 VIMETER STATIONS

tables with which the terrain correction may be determined

accuracy merited by the precision of modern gravimeters.

THE CORRECTION FOR TOPOGRAPHY

185

to the

A common procedure in gravity prospecting is to correct for the

attraction of the topography by use of an approximate value “B”

calculated from the simple Bouguer formula

B = anya(H - Ho) (I)

where I&II0 is the elevation of the gravity station above the level

datum

u is the density of the surface soil or rock

y is the gravitational constant.

This formula calculates the gravity effect of the matter between a

horizontal plane through the field station and the horizontal eleva-

I:IG. I. Schematic diagram of the Douguer correction, illustrating residual

~rnvitational effects due to undulating terrain.

tion-datum-plane on the assumption that the space between these

two planes out to an infinite radius is uniformly filled with matter if

the station is above the datum plane, or is uniformly empty if the

station is below this plane.5 (See Fig. I). Subtracting this quantity

(algebraically) from the observed gravity value amounts to removing

the assumed amount of matter in the infinite flat plate if the station is

above the datum plane or to completely filling up this space if the

station is below the datum plane so as to reduce, in effect, the surface

of the ground to the plane through the elevation datum.

6 liar an excellent brief stntemcnt of the nature of the Ilouguer reduction SW EIay

ford and Bowie, U.S.C. & G.S. Special Publication No. IO, p. 75 (1912). These authors

define the “Bouguer Correction” in the more general scnsc which is equivalent to our

“Topographic Correction.”

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 5

188 SIGMUWD HAMMER

evaluating it. By its very definition it is clear that the terrain correc-

tion depends upon the details of the topography. Therefore, a

topographic map, or the equivalent, of the vicinity of the gravity

station is required. The area about the station is divided into zones

FIG. 1. Zone chart for use in evaluating terrain corrections at gravity stations.

and compartments, for example, by laying a transparent terrain cor-

rection zone chart, such as that shown in Fig. 4, upon the map and

centering it at the gravity station. The average departure without

regard to sign of the topography in each compartment from the

plane through the station is then determined* and the terrain cor-

* Since the sign of the terrain correction (as defined in this paper) depends only

on the absolute value of the departure of an element of the topography from the plane

through the statron, and not on its sign (that is; whether it is above or below the plane)

it follows that the difference between the simple average elevation in each compartment

and the station elevation cannot be used, in principle, if the terrain within any one com-

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 6

rection corresponding to this average departure is evaluated for each

compartment by means of tables calculated for that purpose. Finally

these terrain corrections arc summed over all the zones in which there

are appreciable effects.

TERRAIN CORRECTION TABLES

,4 number of terrain correction tables have been published6,7-8

for geodetic and regional gravity work. However, the author knows

of no previously published tables which are sufficiently precise for

modern gravity prospecting needs. The tables presented in this paper

have been designed especially for this use after experience extending

over a number of years with terrain correction calculations.

The present tables were modelled after those by Hayfortl and

Bowie6*7 but are more precise and have been modified in principle to

evaluate, not the total topographic correction, but only the error in

the Bouguer correction as discussed above.* This modification permits

relatively larger compartments in the distant zones, for a given pre-

cision, and thus reduces the labor involved in the use of the tables.

However, to attain the precision desired the compartments in the

zones near the station are smaller in the present tables than in those

by Hayford and Bowie. Another important advantage of the moditica-

tion in the principle is that it is necessary to use the tables in moder-

~.________

parlment consists of elements both above and below the station. Thus, theoretically,

if the topography within one compartment consists of a hill above the station and a

valley below the station, the average departure must be determined as if the topogra-

phy consisted of two hills or two valleys. However, in practice, it turns out that the

terrain effect in such a compartment is ordinarily zero. Therefore, the mathematically

rigorous but cumbersome procedure outlined above may usually be replaced by the

much more rapid procedure of estimating the average elevations in all the compsrt-

ments (irrespective of Lbhether the topography in any one compartment rs above and

below the station) and then calculating the departures of these simple average eleva

tions from the station elevation. This would not be true, in general, in extremely

rugged topography of large relief but even then the complication may usually be

avoided by rotating the zone chart to a different azimuth. The azimuth must, of course,

be maintained while reading the elevations in all the compartments of any one zone.

’ Hayford and Bowie, lot. cit., pp. r3-53.

7 Hayford and Bowie, U.S.C. & G.S. Spec. Publ. 40, pp. 11-18 (1917).

* Bullard, Philos. Trans. Roy. Sot. London A235, pp. 4866491 (1936).

* Bullard’s tables (1~. cit.) are also of this type. His paper contains a detailed dis-

cussion of the advantages of this modification. A recent search m the literature reveals

the historically interesting fact that this n-ethod was described by F. R. Helmert in

1884 (“Thcorien der hoheren Geodasie,” Vol. II pp. 169-172) and was called “the

ordinary method” by F. A. Venning-Meinesz in ‘923 (“Observations de Pendule tlans

1~s Pays-Bas.” I’ubl. de la comm. g6od. nCer!andaise, p, 141).

Downloaded 26 Aug 2012 to 186.30.73.226. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

Page 10

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Page 11

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