DEPTH DETERMINATION BY MEANS OF GRAVIMETRICAL
SOUNDING*
by Franc Sumi
With 5 figures
By the gravimetrical interpretation it must be kept in mind that
a certain mass distribution causes a definite gravimetrical anomaly.
Conversely, any gravimetrical anomaly can be caused by various mass
distributions. For a quantitative gravimetrical interpretation it is therefore
necessary to get an assumption on the probable shape and size of the
body, in ordier to be able to determine the position and the depth from
gravity anomalies.
There are several well known methods of gravimetrical interpretation
in the geophysical literature, by means of Bouguer anomalies, residual
values, second derivatives etc. Possibilities of a quantitative interpretation
and a depth estimation are described as well. In my paper: Analyse der
Interpretation der gravimetrischen Messungen bei Erdöluntersuchungen,
the possibility of a gravimetrical sounding along a cross profile above a
two-dimensional body is mentioned. The principles and advantages of
this method as well as its practical application will be discussed here.
By comparing the approximate equation for the residual value R (1)
and the second derivative g^" (2) of the gravity on a profUe above a
two-dimensional body
it can be seen, that these equations are very similar. The only difference
is the coefficient 2ir^ and the fact that the interval r in the equation (1)
must be large, whereas in the equation (2) it is small, g,, is the gravity at
the examined point, gr is the average value at the same point, defined as
the arithmetical mean of gravity at two points at a distance r to the
right and to the left from the point g^.
* Presented at the 7th eAEG Meeting, The Hague, December, 1954.
303
It should be noted that the equation (2) differs Uttle from
Mr Baranov's excellent equation (3) for the second derivative, calculated i
for a two-dimensional case, |
where besides the average gravity value gf at a distance r, the average
value g2r at a distance 2r is also considered.
When applying equation (2) for the second derivative, a cor-
responding interval r must be chosen. By two-dimensional bodies
comparisons are made between theoretical values of the second derivative,
computed directly from the known position of the body, and the second
derivative computed by the use of the equation (2) for various intervals r.
In the case of a horizontal cylinder, the interval r in the equation (2)
must be leas than K of the depth to the centre of the cyhnder, in order
to keep the error in the range of less than 10 "/o of the actual second
derivative value. This is particularly important when second derivatives
are used for the quantitative interpretation.
Similarly to the equations (1) and (2) for the residual value and the
second derivative, we can develop an analogous equation for the vertical
gravity gradient g^' at the same point
The equation (4) is vahd only for a certain interval r and depends on
the position, the shape and the depth of the two-dimensional body. The
interval r for a point directly above the centre of a cylinder should be
equal to its depth. Other values of r correspond to other two-dimensional
bodies.
The similarity between the equations (1), (2) and (4) was already
mentioned. When one of these equations for various values of r is used,
the functions (1), (2) and (4) are proportional to the second derivative,
iE r is small compared to the depth, to the first derivative by a certain r
and to the residual value by a large r. When successive increased values
of r are used in equation (4), the function go' reaches its maximum by
Гтах and depends on the shape and the depth of the body. Conversely,
by a geologically supposed shape and an already obtained Гтах it is
possible to determine the depth of the centre of the body. Due to the
analogy with the electrical resistivity method, where by way of the
expansion of electrodes the depth penetration is increased, the present
gravimetrical method is called gravimetrical sounding.
In order to simplify the calculation and the interpretation, the
equation (4) may be written as follows
304
where gL and gR represent the gravity values to the left and to the right
of g|) at a distance r. Gl and Gr are the average horizontal gradients
on a profue to the left and to the right from g,,, between points go and gL
and between gu and g„. In order to be able to apply the already known
equations for horizontal gradients of bodies of various shapes, the
sounding is not done by means of the average gradients Gl and Gr as in
equations (4) and (5), but by the use of the actual gradients Gi. and Gr
at a distance o to the left and to the right from the point g„. These
gradients can be obtained either by close gravimeter measurements on
the profile or by the torsion balance. In most cases sufficient accuracy in
sounding is obtained by modem gravimeters. The torsion balance should
be applied only for bodies near the surface.
Fig. 1. A point arrangement for a gravimetrical sounding
1. si. Razpored točk za gravimetrično sondiranje
The final equations for the gravimetrical sounding is
when gravimeter data are used, and
for a torsion balance.
The arrangement of the points g^ to g4 can be seen on Fig. 1. Gl and
Gr are the components of the horizontal gradients in the direction jf
the profile at the points, which are at a distance q to the left and to the
right from the point g^.
Equations (6) and (7) show, that the possible regional gradient is
automaticaHy eliminated. This is particularly important in the case of
close measurements with the torsion balance on the hilly country, where
terrain corrections need to be determined only in the vicinity of stations.
Cartographic corrections are almost equal for all stations and can be
considered as a regional gradient.
In the electrical method bodies of good or poor conductivity are
located by resistivity mapping. Their depth is then determined by way
of resistivity sounding. Similarly we map gravimetricaliy after equation
Geologija 6 — 20
305
(6) with small and constant Л on the cross profile above a two- ;
dimensional body. The direction of the cross profile is perpendicular to i
the geologically supposed or gravimetrically determined strike. When
mapping by the use of the equation (6) a regional gradient is eliminated, ;
so that the maxima he directly above the bodies and are not displaced
as on an original gravimetrical profile. The maxima are gravimetrically
sounded by the use of the equation (6) or (7). Similarly to the electrical
resistivity method,  a gravimetrical sounding curve is designed. The
vertical axis shows the expanding values o, and the horizontal one the
functions f (o) (6).
On Fig. 2. the theoretical case of a horizontal cylinder is shown.
Besides a gravity anomaly g, caused by a cylindrical mass, a stron;J
regional effect is superposed and the maximum of g is displaced because
Fig. 2. Gravimetrical mapping and sounding above a horizontal cylinder
2. si. Gravimetrično kartiranje in sondiranje nad horizontalnim valjem
of them. By applying a small value g, the gravimetrical mapping gives
a curve g" (the approximate second derivative) with a maximum just,
above the cylinder. The sounding curve at this point is shown on Fig. 2. |
The function f (o) has its maximum by Qmax and depends on the shaped
and the depth of the body.
For bodies of simple geometrical shapes the values о„глх which depend ■
on the depth, can be calculated.
At the point above a sphere, the function f (o) has a maximum for j
a value 1
(8)
306 i
determined after the same equation as Xmax for the horizontal;
gradient above a sphere. {)шах obviously does not depend on the radius!
and the density of the sphere.
^rnax for a horizontal cylinder can be obtained in the same way (Fig. 2.)
In the case of a cylinder Omax does not depend on the radius and
the density.
Fig. 3. Diagrams for gravimetrical sounding above a thin plate and a gentle |
anticline
3. si. Diagram za interpretacijo tanke plošče in blage antiklinale z metodo
gravimetričnega sondiranja
The relation (9) can be used for the gravimetrical sounding above
a steep anticline. Equations for a thin plate and a flat triangle are
derived for anticlines of gentle slopeness.
The position of the centre of a thin plate can be determined by
gravimetrical mapping. A sounding curve at this point gives a value Ошах
corresponding to f (i))max, and Qy^ where the function f {q) has a value
of one half of f (i))max- In the sounding curve, two f (0)1/2
responding o    are obtained. In all the equations only oy^ which are
307
greater than Omax are considered. The depth d to the centre and thej
width 2u of the plate can be calculated from ¿>max and Qy^. \
The diagram a on Fig. 3. is the graphical representation of the j
equation (10). From the diagram b the depth d can be taken. In the same
diagrams the correnponding curve for a flat triangle are drawn. In the
case of a flat triangle i>max is evidently i
The diagram a (Fig. 3.) shows, that аД the triangles of the same
value of ^max are within the circle of a radius Qm^x }/з.
Both diagrams in Fig. 3. are strictly valid only, when the plate is
infinitely thin and the triangle infinitely flat. When sounding after
equations (6) and (7) above a thick plate or triangle, an error in the
depth determination occurs, and depends upon (I) the relations between
the width (2u) and the thickness (h) of the plate or the triangle, and
upon (II) the relation between the depth to the centre of the plate or the
triangle and the thickness. In the case of a plate the above mentioned
relations may vary in wide limits, but the error remains neghgible;
it is not so in the case of a triangle.
In the case of a vertical dike of infinite depth extent, the point above
the centre can be located by way of gravimetrical mapping. From the
sounding curve at this point the depth d and the width 2 u of the upper
edge (Fig. 4.) can be determined by means of Omax when the function f {q)
reaches its maximum, and o when the value of the function f (o) is one
half of the maximal, oi/j greater than ^шах is considered only.
Relations between Omax, Qy^- d and 2u are as follows ^
From the equation (12) it can be seen that the upper edge of the
dike lies within the semicircle with the centre above the dike and the
radius Omax-
In the case of an inclined dike (Fig. 4.) the method of gra\àmetrical
sounding has the advantage that the sounding curve at the point above
the centre of the upper edge of the dike shows identical ¿«max and q\¡^ for
308
vertical and for inclined dikes. Equations (12) and (13) are therefore!
extended to inclined dikes. '
In order to determine the dip angle i of a dike, a series of sounding]
curves on the profile above a dike must be calculated, and a point with \
an absolute maximal value of the function f (o) and the corresponding i
í>Max must be found. The point with the absolute maximum f (о)мах lies ;
on the side from the point O, on which a dike is inclined.
Fig. 4. Gravimetrical sounding above a vertical and an inclined dike of infinite
depth extent
4. si. Gravimetrično sondiranje nad vertikalnim in nagnjenim dajkom, ki vpada
zelo globoko
The dip angle i is
In Fig. 4. a theoretical example of gravimetrical sounding above a \
vertical and an inclined dike adopted for the torsion balance is shown.;
The sounding curves a and b are calculated at the point O, just above-
the centre of the upper edge of the dike. The dotted curve a is obtained!
in the case of a vertical dike, the solid curve b in the case of an'
309!
inclined dike. The values Qma.x and are equal for both curves. The
depth a and the width 2 u can be detemined by the use of equations (12)
and (13). In order to find the dip angle i, sounding curves for a series
of points on the profile above a dike are calculated and a gravimetrical
sounding curve c at the point M with a maximal value f (р)мах and cor-
responding ipMax is found. The dip angle i is determined by the use of
the equation (14). Using equations (6) and (7) in gravimetrical sounding,
the regional effect is eliminated.
When two bodies of approximately equal size and depth are
relativly close together^, gravimetrical maxima above them are displaced
and the apparent depth d' obtained by gravimetrical sounding differs
from the actual depth d. The case of two parallel horizontal cylinders
Fig. 5. Corrections for gravimetrical sounding in the case of two parallel
cylinders
5. si. Korekcije pri gravimetričnem sondiranju v primeru dveh vzporednih
valjev
at different distances is analysed. By gravimetrical mapping on the
profile, perpendicular to the strike, the position of both cylinders can
be determined, when they do not lie to close. The distance D between
the centres of the cylinders must be greater than the depth d. When
sounding directly above a cylinder by the use of the equations (6) or
(7), the regional effect of gravity is eliminated, but the gravimetrical
effect of the other cylinder remains however and the obtained apparent
depth d' differs from the actual depth d. The diagram on Fig. 5. shows
the relations between the quotients d'/d and D/d in the case, (I) when
the sounding point lies directly above the cylinder (x 0), (II) when
the sounding points is displaced for 10 and 20 % from the depth d in
the direction of the other cylinder (x = + 01 d, x = + 02 d), and finally
(III) when the sounding point is displaced for the same distance in the
opposite direction (x = — 01 d, x = — 02 d). It can be seen, that the error
for distances D, greater than 3 d is negligible. For distances D smaller
than 3 d, the correction can be taken from the diagram on Fig. 5.
On Ulcinjsko Polje (IJlcinj Plain) in Montenegro, a bedrock of
Cretaceous limestone is overlain by flysh and Tertiary beds. The density
difference between Hmestone-flysh is 0'3 and between limestone-Tertiary
310^
0'6. Gravimetrical measurements were carried out to locate the probable
structures. The gravimetrical map on Fig. 6. shows a strong regional
gradient in the north-west direction. The anticlines are therefore masked
and can be recognized only by way of windings of the isogams. The
general strike can be taken from the gravimetrical map. It is in good
agreement with the geologically supposed NW—SE strike, which is cha-^
Fig. 6. Gravimetrical map of Ulcinjsko Polje. Soimding curve
6. si. Gravimetrične meritve na Ulcinjskem Polju. Krivulja sondiranja
racteristic for the eastern Adriatic coast. The gravimetrical sounding
curve at the point C on the profile A—B reaches its maximum by
Omax = 370 m (Fig. 6.). Geologically, steep anticlines are assumed, so that
the equation (9) for a horizontal cylinder may be used for the deter-
mination of the depth. The depth d = 640 m to the centre of the
anticline is obtained. Considering the geologically assumed shape of the
anticHne and the density difference, the detph to the top of the anti-
311Š
cline can be estimated. Three parallel structures occur in the same field. |
The distance D from the point C to the southwestern structure is
about 2200 m. Since the quotient D/d is greater than 3, the error in
depth determination caused by the gravimetrical effect of a paralleli
structure can be neglected. Results obtained at bore holes agree with :
the gravimetrical data.
I
REFERENCES \
Bar ano V, V., 1953, Calcule du gradient vertical du champ de gravite;
ou du champ magnétique mesuré à la surface du sol. Geophysical Prospecting,:
Vol. 1, No. 3.
E Ik in s., A. A., 1951, The second derivative method of gravity inter-
pretation. Geophysics, Vol. XVI, No. 1.
S a X 0 V , S., N y g a a r d , K., 1953, Residual anomalies and depth |
estimation. Geophysics, Vol. XVIII, No. 4. \
Sumi, F., 1954, Analiza interpretacije gravimetriskih merenja pri-1
menjenih u istraživanju nafte. With summary in German. Vcsnik Zavoda za i
geološlca i geofizička istraživanja NR Srbije, knj.XI. i
DOLOČEVANJE GLOBINE S POMOCJO GRAVIMETRICNEGA
SONDIRANJA*
Povzetek
Pri interpretaciji gravimetričnih anomalij moramo upoštevati dej-
stvo, da iz znanega razporeda mas pod zemljo lahko izračunamo ano-
malno gravimetrično polje, obratno pa določenemu anomalnemu polju
ustreza neskončno mnogo kombinacij razporedov mas, tako da za
interpretacijo moramo privzeti neke predpostavke. Pri interpretaciji gra-
vimetričnih podatkov uporabljamo višje odvode težnostnega potenciala,
moremo pa, podobno kot pri geoelektričnem sondiranju, določiti globino
telesa, ki povzroča težnostno anomalijo. Krivuljo gravimetričnega son-
diranja (6) nanesemo v diagram kot funkcijo razdalje o (1. sL). Ko kri-
vuljo interpretiramo, moramo vsaj približno vedeti, kakšno obliko ima
telo, ki povzroča anomalijo. Cesto zadostuje, če vemo, ali so dimenzije
telesa v vse smeri istega reda velikosti, ali pa je telo razpotegnjeno v neki
smeri. Pri raziskavah na nafto so antiklinalne in sinklinalne strukture
običajno razpotegnjene v neki smeri; pri raziskavah rudišč imajo rudna
telesa popolnoma nepravilno obliko, ali pa obliko plošče. Da bi mogli
interpretirati vse možne oblike teles, moramo obdelati limitne primere
oblik: od krogle (8), preko horizontalnega valja (9), (2. si.), do horizon-
talne plošče (10) in blage antiklinale (11), (3. si.); razen horizontalne
plošče pa še vertikalno (12 in 13) in nagnjeno (14), (4. si.). Ce leži več
teles v bližini, moramo upoštevati tudi to (5. sL).
Uporabo metode gravimetričnega sondiranja smo prikazali na pri-
meru raziskav na Ulcinjskem Polju (6. sL), kjer smo določili položaj in
globino strmih antiklinalnih struktur.
* Predavanje na VII. kongresu EAEG v Haagu, decembra 1954.
312