PAGEOPH, Vol. 119 (1980/81) 0033-4553/81/002373~37501.50 + 0.20/0 �9 1981 Birkhiiuser Verlag, Basel

A New Method for Near-Topographic Correction in Gravity Surveys 1)

By E. KLINGELE 2)

A b s t r a c t - The paper is concerned with the problems of topographic reduction in gravity prospecting. Particular emphasis is placed on topographic conditions frequently encountered in mountainous areas such as in the Alps. New master curves are developed for several cases in which steep walls followed by inclined planes are located near the gravity sites. The computation methods are outlined and the precision as well as the rapidity are tested for various cases.

Key words: Gravity surveying; Topographic correction.

In a high precision gravity survey the topographic corrections for the near zones A 1

and B 1 are made in the field with the aid of topographic instruments (e.g. a levelling instrument, theodolite etc.). The surveyor has to make a number of measurements

depending on the slope of the topography (one or two for a gentle slope, four or more

for a steep slope). For a very rough topography the classical method suffers from the

disadvantage that it is very expensive. Generally, for economic reasons, the surveyor

chooses to place the gravimeter stations on roads or paths, the survey being done by

car. Consequently, the stations are located in a topographic environment characterized

by some important broken slopes (like walls and banks) and the determination of the

first four mean elevations could be very difficult.

The case previously described is illustrated in Fig. 1. Two simplifications can be

introduced. First, the slope of the ground can be approximated by planes (fluctuations

around the mean slope being of negligible effect). Second, the transverse slope (i.e. road

or path gradient) is very small (several per cent) and its effect is negligible.

These hypotheses indicate that the intersections of the horizontal and vertical planes

are horizontal or sub-horizontal. Therefore, the problem can be summarized: How can

a sector of ring involving walls and banks be replaced by a homogeneous sector of ring

having the same effect at the station?

The solution which we propose is to precompute a set of equivalent models for three

parameters; crthe angle between the horizon and the average slope of the topography, x

1) Contribution No. 301. Institute of Geophysics, Swiss Federal Institute of Technology, CH-8093 Zurich, Switzerland.

2) Swiss Geophysical Commission, ETH-H6nggerberg, CH-8093 Zurich, Switzerland.

E. Klingele

/

PAGEOPH, 374

Figure 1 Definition of the parameters x, y, and aused in the master curves and of the equivalent height H e of a sector

of ring.

He(M) t 15_[ / ALPHA - 40

R- 20m

H~ 4.7 (X-2.Y-5)

0 O.5 1.0 1.5 2.0 25 3.0 3.5 4.0 45 5.0

Figure 2

[M)

Master curves for cr = 40 degrees and showing the example given with x = 20 m, y = 5 m. These curves are valid for 0 < x < 5 m.

Vol. 119, 1980 /81 Near-Topographic Correction in Gravity Surveys 3 7 5

He(M)

O- x(M) 5 6 7 8 9 1011 12 15 14 15 16 17 18 19 20

Figure 3 Master curves for a= 40 degrees and valid for 5 m < x < 20 m.

the position of the station relative to the break in the slope and y the height of the wall.

The results are presented in the form of master curves giving an equivalent height, H e (Figs. 2 and 3). We think that the method of equivalent height is the most useful

method because the density is not taken into account and the resulting data can be

stocked and processed in the same form as the other results.

Computation method and presentation of results

The computation of the gravity effect has been done by numerical integration. The

step of integration has been chosen so as to maintain the precision and the time of computation within reasonable boundaries. The results are presented in form of

parametric curves, where y is constant, giving the equivalent height, H e, for a sector of ring (where 0 = 90 ~ as a function o f x (see Fig. 1). Each set of curves is computed for

a constant value of the slope c~ The number of sets available is 12, valid for

0 < a < 50 ~ For each 10 ~ increment there are two diagrams: one for 0 _< x < 5 m and

a second one for 5 m < x < 20 m. The values of 'H e' less than 0.2 m have not been

376 E. Klingele PAGEOPH,

computed because this height produces an effect less than 0.01 mgal with p = 3.0 g/cm 3. The example given in this paper (or= 40 ~ has been computed for R = 20 m but it is possible to extend this radius according to the necessity of the survey. It is also possible to compute the same type of master curves for square sectors such as those used for a digital topography in a computer-oriented system (KLINGELE and OLIVIER,

1979).

M e t h o d o f use

After choosing the site of the station, the surveyor has to determine the parameter x and y with a tape-measure and the slope orwith a clinometer. The next step consists of choosing the appropriate set of curves corresponding to the slope or (since near-station corrections are always of the same sign, irrespective of whether the topography is

positive or negative, or corresponds to the absolute value of the slope). The determination of ' H e' is now very simple, the intersection of the y curve with a vertical line coming from the x abscissa gives a point P, a horizontal line between this point and the vertical axis determines the required ' H e ' value.

In order to clarify these explanations an example is given (Fig. 2) with its solution

fory = 5 m; x = 2 m and or= 40 ~ The result is H e _~ 4.7 m.

<>+ 9

(~ 1 measurement ~ 2m X 2 measurements [ ] 4 measurements .~ 20 rn

Figure 4 Disposition of the mean elevations in the test computation.

1 m

easu

rem

ent

SL

a V

(mga

l)

AV

(r

egal

)

10

0,09

4 0,

086

20

0.18

5 0.

150

30

0.27

6 0.

197

40

0.37

5 0.

231

50

0.48

3 0.

250

60

0,60

8 0.

257

V (m

gal)

A

V

(reg

al)

00-

0.

135

| 0.

100

I 0.

205

1 0.

126

I 0,

284

| 0.

140

I 0.

376

| 0.

143

] 0

49

2~

.0.1

41

I

1075

73

8 42

9 28

6 24

9 15

9 16

0 97

10

7 61

73

40

V (

mga

l)

AV

6

Tab

le 1

2 m

easu

rem

ents

4

mea

sure

men

ts

Mas

ter

curv

es

8

(reg

al)

0.04

9 0.

041

13

0.13

4 0.

099

| 28

3 0.

234

0.15

5 |

196

0.34

1 0.

197

| 13

7 0.

459

0.22

6 |

97

0.59

4 0.

243

[ 69

V (

mga

l)

A V

(r

egal

)

0.00

8 0

0.03

7 0.

002

0.08

3 0.

004

0.14

5 0,

001

0,23

4 0.

001

0.35

1 0

SL

= V

alue

of t

he s

lope

in d

egre

es.

V =

Gra

vity

eff

ect

(in

mga

l) o

btai

ned

wit

h th

e m

enti

oned

sch

eme

for

a ha

lf c

ircl

e an

d w

ith

A =

2.6

7 g/

cm 3.

A

V =

Dif

fere

nce

betw

een

V a

nd V

e.

e =

Err

or i

n pe

r ce

nt c

ompu

ted

by (

V--

Ve)

/V e.

V

~ = E

xact

val

ue c

ompu

ted

by S

andb

erg'

s fo

rmul

a.

0 6 5 0.69

0.

43

0

Exa

ct

valu

e V

e (m

gal)

0.00

8 0.

035

0.07

9 0.

144

0.23

3 0.

351

Z

"7,

o O

o~

C3

e~

t~

378 E. Klingele PAGEOPH,

Precision and rapidity of the method

In order to test the precision of our method, we have computed the exact value for x=y- - - -0 with a slope a varying between 10 and 60 degrees. The results may be compared with those we can obtain with the classical method where the altitudes have been measured in the field. For this case we have computed three examples

corresponding to 1, 2 and 4 measurements in each sector consistent with Fig. 4. The exact value has been calculated by the analytical method according to the solution given

by SANDBERG (1958):

where

Ag = 2?pR(n- 2 cos 0 �9 K(sin 0))

K(s in 6) =

y = Universal gravitational constant p = Density of the ground R = External radius

~/2

f �9 l - K ' s i n 2~, 0

= First-order elliptic integral

K* = sin 0

Comparison with Table 1 shows that for radius R equal to 20 m the error in the

classical method lies between 40 and 1075%. For the examples computed by our new method the maximum error is equal to 6%.

It is rather difficult to estimate the rapidity of the new method. Indeed it depends on the speed with which the surveyor can install his topographic instrument. In spite of this factor, and taking account of our own experience based on more than a thousand stations measured with this method, we are able to estimate that the new method is about eight times faster in the field. It is relevant to note that a linear interpolation is precise enough between two slope values differing by 10 degrees. Using it, the maximum error is about 4.5/agal for p = 2.67 g/cm 3. If we would dispose of a more complete set of curves, for example each 5 degrees, this error would decrease to 1.5/tgal for the coupledzones A 1 and B 1 and for the same density.

Acknowledgments

This paper was critically reviewed by my colleagues H.-G. Kahle, Institute of Geodesy and Photogrammetry, W. Lowrie and L. Rybach, both of the Institute of Geophysics. Their helpful suggestions for the improvement of the manuscript are

gratefully acknowledged.

Vol. 119, 1980/81 Near-Topographic Correction in Gravity Surveys 3 7 9

REFERENCES

KLINGELE, E. (1972), Contribution ~ l'dtude gravimdtrique de la Suisse romande et des rdgions avoisinantes. Mat. pour la Geol. de la Suisse, S6rie Geophysique No. 15 (Kfimmedy & Frey, Bern).

KLINGELE, E. (1974), Report to the Swiss ~lational Funds on the Project 'The New Swiss Gravity Map'. (Unpublished.)

KLINGELE, E., and OLrVmR, R. (1979), Topographic Models used in Switzerland for Terrain Correction in Gravity. EGS Meeting, Vienna.

SANDBERG, CH. (1958), Terrain Corrections for an Inclined Plane in Gravity Computations, Geophysics 23, 701-711.

SPIEGEL, R. M. (1968), Mathematical Handbook o f Formulae and Tables. Schaum's Reviews (McGraw-Hill Comp. Inc., New York), 271 pp.

(Received 27th June 1980)