GEOLOGICA CARPATHICA, FEBRUARY 2007, 58, 1, 97—102
www.geologicacarpathica.sk
Geophysical indirect effect on interpretation of gravity data
in Slovakia
PETER VAJDA and JAROSLAVA PÁNISOVÁ
Geophysical Institute, Slovak Academy of Sciences, Dúbravská cesta 9, 845 28 Bratislava, Slovak Republic;
peter.vajda@savba.sk; jpanisova@yahoo.co.uk
(Manuscript received November 22, 2005; accepted in revised form June 22, 2006)
Abstract: This paper deals with the proper definition and correct compilation of the Bouguer gravity anomaly that
differs from the standard definition repeatedly advocated in the geophysical literature. The difference between the
new “correct” definition of the Bouguer anomaly and its “standard” definition has become known in geophysics as the
“geophysical indirect effect”. Here we investigate the magnitude and variation of the geophysical indirect effect as a
systematic error in gravity data inversion or interpretation in Slovakia. It is found that in Slovakia this effect is of the
order of 3 mGal with a spatial variability of about 0.5 mGal over some 100 km. The impact of the geophysical indirect
effect on the determination of the depth of a density interface is also estimated. In Slovakia, the impact of this
systematic error is of the order of about 300 meters for a density contrast of 300 kg/m
3
.
Key words: gravity data, Bouguer anomaly, geophysical indirect effect.
Introduction
When interpreting gravity data, we seek to determine the
density distribution in order to gather knowledge of sub-
surface geological structure. The inverse problem in
gravimetry is typically formulated in terms of relating the
anomalous density distribution to the parameters of the
anomalous field derived from the disturbing potential.
These parameters include the gravity anomaly, gravity
disturbance, geoidal undulation, and various derivatives
(horizontal, vertical, mixed) of the disturbing potential.
We focus in particular on the relation between anoma-
lous masses and anomalous gravity. In geophysical prac-
tice, the Bouguer gravity anomaly has been used most
extensively. Many geophysicists have already argued that
correct interpretation of gravity data requires that the
Bouguer anomaly is compiled using normal gravity eval-
uated at the station (which implies the use of ellipsoidal
heights as opposed to heights above sea level in the for-
mula for computing normal gravity), and using the refer-
ence ellipsoid (as opposed to sea level) as the lower
boundary of topographic masses in evaluating the topo-
graphic correction (Chapman & Bodine 1979; Vogel
1982; Jung & Rabinowitz 1988; Meurers 1992; Talwani
1998; Hackney & Featherstone 2003; Vajda et al. 2006;
and others). The failure to do so leads to a systematic error
that has become known in gravimetry as the “geophysical
indirect effect” (ibid).
Vajda et al. (2006) have recently demonstrated that if
we look for a rigorous match to the “gravitational effect”
(attraction) of anomalous density, we arrive at the quantity
compiled from observed gravity named by them the
“NETC gravity disturbance” (where NETC denotes ‘No el-
lipsoidal topography of constant density’, the definition
and compilation of which is reviewed in section 2) to dis-
tinguish it from the commonly used “Bouguer gravity
anomaly”. In other words, in the quest to determine the
anomalous density by means of gravity inversion or for-
ward and inverse modeling, the gravity data to be mod-
eled or to be inverted are strictly those gravity data
defined as the “NETC gravity disturbance”. The difference
between the NETC gravity disturbance and the Bouguer
gravity anomaly, which constitutes the geophysical indi-
rect effect (cf. ibid), is relatively small, of the order of
10 mGal. The issue we shall examine here is whether or
not this relatively small difference is significant to geo-
physical interpretations. Our aim is to numerically assess
the geophysical indirect effect in Slovakia, as well as to
estimate its impact on the interpreted anomalous density
distribution, especially its impact on the interpreted depth
of density interfaces.
Gravity inversion/interpretation formulated in
terms of the attraction of anomalous masses
We shall use the following notation: the horizontal po-
sitions of points are represented by geographical coordi-
nates, latitude
φ and longitude λ, vertical position of a
point by ellipsoidal (geodetic) height h, which is reck-
oned from the reference ellipsoid as a height datum, not
by a height above sea level (in Slovakia: “normal height”)
H
N
, which is reckoned from “sea level” (in Slovakia:
“quasigeoid”) as a vertical datum. For brevity, the hori-
zontal position is often denoted by
Ω
φ, λ . Actual
(measured) gravity is denoted by g, normal gravity by
γ,
and quasigeoidal height (the separation between “sea lev-
el” and the reference ellipsoid) by N. In our developments,
the Newton volume integrals are evaluated using a spheri-
cal approximation (e.g. Vajda et al. 2004a) with a mean
98
VAJDA and PÁNISOVÁ
earth radius R. Thus, the reciprocal Euclidean distance be-
tween a computation point (h,
and any integration
point (h
’, ’ is (ibid, Eqs. [19] and [22]):
,
,
,
1
h
h
L
2
1
2
2
cos
2
ψ
h
R
h
R
h
R
h
R
,
(1)
where cos
ψ = sin φ sin φ’+ cos φ cos φ’cos (λ—λ’), ψ be-
ing the angular (spherical) distance between
Ω and Ω’.
The NETC gravity disturbance, at the observation
point, is defined as (Vajda et al. 2006):
(2)
The first term on the right hand side is the actual (ob-
served) gravity at the observation point (station), the sec-
ond term is the normal gravity at the observation point, and
the third term is the topographic correction to the gravity
disturbance, evaluated at the observation point, adopting
the reference ellipsoid (h = 0) as the lower boundary of topo-
graphic masses, the topographic surface (h = h
T
( )) as the
upper boundary, and constant density
0
for topographic
masses. G is the Newton gravitational constant,
0
stands
for the solid angle (representing integration over the entire
globe – full sphere), d
’=cos ’ d ’d ’ and the kernel
of the volume integral reads:
J(h, , h’, ’)
(3)
This topographic correction should strictly be evaluat-
ed over the whole globe. It represents the negative attrac-
tion of masses of constant density between the reference
ellipsoid and the topographic surface.
Let us now define (ibid) the gravitational effect of
anomalous masses ( A), i.e. the attraction of anomalous
density (
) within the whole earth (below the topo-
graphic surface), again assuming a spherical approxima-
tion (cf. Vajda et al. 2004a):
(4)
Equation (4) represents the Newton volume integral
(with the kernel J given by Eq. (3)) of the anomalous
density within the entire volume of the earth, bound by
the topographic surface. The background (reference) den-
sity model used to construct the anomalous density
from the real density is described in detail in (Vajda et al.
2004b, sec. 3.1; Vajda et al. 2006, sec. 3). It consists of a
model ‘normal’ density distribution inside the reference
ellipsoid that generates the normal gravitational field,
and of the constant density distribution between the ref-
erence ellipsoid and the topographic surface.
Vajda et al. (2006) give a rigorous proof that:
δg
NETC
h,
Ω) =
δA(h, Ω), (5)
which, in words, means that the attraction of anomalous
masses inside the earth (below the topographic surface) is
exactly equal to the anomalous gravity quantity defined
as the NETC gravity disturbance. Put another way: if we
are to solve for anomalous density using gravity data in-
version/interpretation, we have to (rigorously) match syn-
thetic gravity data, generated by our model of anomalous
density (via the Newton volume integral), with “observed”
NETC gravity disturbances (as opposed to “observed”
Bouguer gravity anomalies). Here “observed” means
“compiled from measured gravity”. As we have already
mentioned, the difference between the Bouguer gravity
anomaly and the NETC gravity disturbance is subtle, and
the aim here is to estimate the impact and significance of
the difference on gravity data interpretation, with particu-
lar focus on Slovakia.
Notice that using Eq. (5) the gravimetric inverse prob-
lem can be formulated, for instance, at the topographic
surface (h = h
T
). This means that in the case of forward and
inverse modeling, the matching of observed and synthetic
data can be done at the observation points (stations) and
elsewhere, also on the topographic surface. The gravimet-
ric inverse problem is solved in three steps, as follows:
1 – Compile the NETC gravity disturbance (
δg
NETC
)
from observed gravity using Eq. (2). For more details re-
garding the evaluation of normal gravity at the observa-
tion point (with a focus on Slovakia), refer to Vajda &
Pánisová (2005). For more details regarding the topo-
graphic correction to gravity disturbance, see Vajda et al.
(2004a) and references therein;
2 – Match the “observed” NETC gravity disturbance
with the gravitational effect of anomalous masses (cf.
Eqs. (4) and (5)) at the observation points (say on the to-
pography) as follows:
(6)
3 – Solve for anomalous density from the functional
on the right hand side of Eq. (6) using either direct inver-
sion methods or forward modeling (trial and error itera-
tive) methods (e.g. Blakely 1995).
As we have already stressed, in practice the role of the
left hand side in Eq. (6) is typically and commonly
played by the Bouguer gravity anomaly, be it planar or
spherical, complete or incomplete etc. (e.g. Hayford &
Bowie 1912; Bullard 1936; Heiskanen & Moritz 1967;
Vaníček & Krakiwsky 1986; LaFehr 1991; Chapin 1996;
Talwani 1998; Arafin 2004; Vaníček et al. 2004). For our
purposes we consider an “ideal” spherical complete Bou-
guer anomaly (SCBA). “Ideal” means that the topograph-
ic correction needed to construct the SCBA is considered
as ideally evaluated over the whole globe. Of course, we
acknowledge that in practice the topographic correction
for the Bouguer anomaly is generally computed only to
the Hayford-Bowie radius (1
o
29
’58” or 167 km). The ef-
0
)
,
,
,
(
— G
2
0
d
dh
h
R
h
h
J
T
h
0
)
,
,
,
(
1
h
h
h
L
.
cos
=
3
1
L
h
R
h
R
=
,
h
A
δ
=
,
h
g
NETC
δ
;
)
,
,
,
(
)
,
(
0
2
d
dh
h
R
h
h
J
h
G
T
h
R
δρ
0
.
)
,
,
,
(
)
,
(
2
d
dh
h
R
h
h
J
h
G
T
h
R
δρ
,
Ω
h
δ
g
NETC
,
,
Ω
Ω
h
h
g
99
GEOPHYSICAL INDIRECT EFFECT ON INTERPRETATION OF GRAVITY DATA IN SLOVAKIA
fect of the truncation error in computing the topographic
correction to the Bouguer anomaly has been discussed in
detail most recently by Mikuška et al. (2006), and previ-
ously by Talwani (1998), LaFehr (1991), Kotake & Hagi-
wara (1987), and Pick et al. (1973). This systematic error
may become significant when interpreting gravity data
on a regional or global scale and on a local scale in
mountainous areas (Mikuška et al. 2006). This effect is,
however, outside the scope of our investigations. We
shall focus on the “ideal” SCBA. Let us at last define the
SCBA (cf. Vaníček et al. 1999, Eq. [38]; Vaníček et al.
2004):
(7)
Notice that compared to the NETC gravity disturbance
(Eq. (2)), when the SCBA is compiled (using Eq. (7)) from
observed gravity, normal gravity is not evaluated at the
station. This is because normal height (height above sea
level) is used in the formula for normal gravity computa-
tion instead of the ellipsoidal (geodetic) height. Further,
sea level (geoid or quasigeoid) is used as the lower bound-
ary of topographic masses in the topographic correction
needed to compile the SCBA. This contrasts to the case in
which the ellipsoid is used as the lower topographic
boundary in the topographic correction for computing the
NETC gravity disturbance.
Geophysical indirect effect
The Geophysical Indirect Effect (GIE) is defined, in its
most general form, as the systematic deviation of the
(“ideal”) SCBA from the gravitational effect of the subsur-
face anomalous masses:
)
,
(h
GIE
,
h
A
δ
)
,
(h
g
SCB
. (8)
Realizing that the gravitational effect of anomalous
masses is rigorously equal to the NETC gravity distur-
bance (Vajda et al. 2006), we get:
=
)
,
(h
GIE
)
,
(h
g
NETC
δ
)
,
(h
g
SCB
. (9)
By substituting for the two anomalous gravity quanti-
ties from Eqs. (2) and (7), we get:
=
)
,
(h
GIE
)
,
(
)
,
(
h
H
h
N
γ
γ
.
)
,
,
,
(
—
0
2
0
0
N
d
dh
h
R
h
h
J
G
(10)
Equation (10) represents the most general form of the
geophysical indirect effect (Vajda et al. 2006). Notice that
the GIE is a spatial rather than a surface quantity, that is,
its value varies not only with horizontal position, but also
with the altitude of the observation station. The GIE con-
sists of two terms,
=
)
,
(h
GIE
)
,
(
1
h
GIE
)
,
(
+
2
h
GIE
.
The first term:
,
1
h
GIE
)
,
(
)
,
(
h
H
h
N
γ
γ
(11)
accounts for the change of normal gravity along the ver-
tical displacement (between the telluroid and the topo-
graphic surface, if the observation point lies on the
topographic surface (ibid)). It can be approximated (Vaj-
da et al. 2006, sec. 9; Vajda & Pánisová 2005, sec. 5) as:
1
GIE
(12)
where
γ
0
(
φ) is the normal gravity on the reference ellip-
soid, the vertical gradient of normal gravity is given in
[mGal/m], and the geoidal height is given in [m]. In Slo-
vakia, this approximation is good to 90
µGal, while the
Fig. 1. The
GIE
1
term, Eq. (11), in Slovakia [mGal].
)
,
(
h
g
SCB
)
,
(h
g
)
,
(
N
H
h
γ
.
)
,
,
,
(
— G
)
(
2
0
0
T
h
N
d
dh
h
R
h
h
J
)
(
),
(
0
0
N
h
N
)
(
]
/
[
3086
.
0
N
m
mGal
≅
100
VAJDA and PÁNISOVÁ
effect itself is of the order of 10 mGal (Vajda & Pánisová
2005, sec. 5). Notice that with this approximation, this term
becomes a surface quantity, i.e. it only varies with horizontal
location and no longer varies with the altitude of the obser-
vation station. This effect computed for Slovakia is displayed
in Fig. 1.
The second term:
,
2
h
GIE
0
2
0
0
)
,
,
,
(
N
d
dh
h
R
h
h
J
— G
ρ
(13)
is the gravitational effect of masses of constant density
between the reference ellipsoid and the sea level (geoid or
quasigeoid). It is computed by numerical integration of
the volume integral, while the integration must be carried
out over the whole globe. Alternatively, it can be approxi-
mated by the gravitational effect of the Bouguer shell
(Blakely 1995, Secs 3.2.1 and 3.2.2) with a thickness
equal to the (quasi-)geoidal height at the horizontal posi-
tion of the observation point:
,
2
h
GIE
)
(
G
3
4
2
3
3
0
h
R
R
N
R
ρ
π
.
(14)
Notice that the second term of the GIE, as well as its Bou-
guer shell approximation, is a spatial quantity, i.e. it varies
not only with horizontal location, but also with altitude. This
effect, the second term of the GIE, computed for Slovakia us-
ing the Bouguer shell approximation for stations on the to-
pographic surface, is displayed in Fig. 2. The accuracy of the
Bouguer shell approximation of
GIE
2
will be subject to fu-
ture investigation. The variability of
GIE
2
, as computed us-
ing the Bouguer shell approximation, with the altitude of the
station is (for altitudes up to 8 km) three orders of magnitude
smaller than the effect itself. Hence,
GIE
2
with the
Bouguer
shell approximation can be further approximated as:
2
GIE
, (15)
which becomes a surface quantity. Assuming that the den-
sity of masses between sea level and the reference ellip-
soid is constant (
ρ
0
= 2670 kg/m
3
), this leads to the
following approximation for the whole geophysical indi-
rect effect as a surface quantity (cf. Eqs. (12) and (15)):
GIE
(16)
This formula gives a simple estimate of the GIE. A sepa-
ration between sea level and the ellipsoid of 100 m pro-
duces a GIE of about 8 mGal.
If the second term of the GIE is neglected, then the GIE
is referred to as the “Free-Air GIE”. If the second term of
the GIE is approximated by the gravitational effect of the
Bouguer plate/shell, then the GIE is referred to as the
“Bouguer GIE”. The GIE for Slovakia, computed using the
Bouguer shell approximation and evaluated on the topo-
graphic surface, is displayed in Fig. 3.
The geophysical indirect effect in Slovakia has an am-
plitude of about 3 mGals. Spatially, it varies on a regional
scale by a few tenths of a mGal. It appears that for local
studies in Slovakia, the GIE can be neglected as a trend of
no interest. However, in regional studies of larger extent,
this may not be the case.
Let us give a rough estimate of the impact of the GIE on
the determination of the depth to a density interface. The
error d in the interpreted depth d of a density interface
(with a density contrast
ρ) stemming from neglecting the
GIE can be estimated using the gravitational effect of a
Bouguer plate (e.g. Blakely 1995, sec. 10.3.1) as:
. (17)
Fig. 2. The
,
2
T
GIE
h
h
term in Slovakia [mGal], evaluated with the Bouguer shell approximation for stations on the topographic
surface, Eq. (14).
ρ
G
π
2
GIE
.
)
(
4
0
N
G
]
[
)
(
]
/
[
0863
.
0
)
(
4
3086
.
0
0
m
N
m
mGal
N
G
d
101
GEOPHYSICAL INDIRECT EFFECT ON INTERPRETATION OF GRAVITY DATA IN SLOVAKIA
Naturally, the smaller the density contrast across the in-
terface, the greater the impact of the GIE on the interpreted
depth of the interface. The estimated error in the interpret-
ed depth of a density interface of, for example,
ρ = 300 kg/m
3
due to neglecting the GIE (as displayed
in Fig. 3) is portrayed in Fig. 4. In Slovakia, the magnitude
of this error is about 300 m.
Using the approximation of Eq. (16) we can estimate
this impact as:
(18)
The above simple estimate tells us that if the GIE is ne-
glected, for each meter of separation between sea level and
Fig. 4. The impact of the GIE on gravimetric interpretation in Slovakia. This map shows a coarse estimate of the error [m], due to ne-
glecting the GIE, in the interpreted depth of a density interface with a contrast of 300 kg/m
3
(chosen as an example), cf. Eq. (17).
Fig. 3. The geophysical indirect effect
GIE
(h = h
T
, ) in Slovakia [mGal], evaluated on the topographic surface using the Bouguer shell
approximation, Eqs. (11) and (14).
the reference ellipsoid there is an error of about 7 meters in
the interpreted depth of a density interface with a density
contrast of 300 kg/m
3
. The above estimate based on the
gravitational effect of a Bouguer plate is coarse. Better in-
sight would be gained by adding the GIE to the input Bou-
guer anomalies in a specific gravity interpretation, and then
examining the change imposed on the interpreted model.
Historically, the lack of ellipsoidal (geodetic) heights at
observation points meant that the separation between sea
level (geoid or quasigeoid) and the reference ellipsoid was
ignored in gravity data inversion or interpretation. Nowa-
days, in the era of GPS positioning and with the wide-
spread availability of geoid/quasigeoid models, ellipsoidal
heights are readily available. This means that it is now pos-
d
.
N
m
kg
m
kg
]
/
[
]
/
[
073
2
3
3
102
VAJDA and PÁNISOVÁ
sible to compute the GIE and to add it to the Bouguer
anomaly, thereby leading to improved gravimetric interpre-
tation and inversion.
Discussion and conclusions
The geophysical indirect effect, as a systematic error in
gravity data inversion or interpretation, originates in: 1) the
failure to use ellipsoidal heights instead of heights above
sea level in evaluating the normal gravity used for compil-
ing the Bouguer gravity anomaly; and 2) the failure to com-
pute the topographic correction using the reference
ellipsoid instead of sea level as the lower boundary of the
topographic masses. The issue we have pursued here was
the size and shape of the geophysical indirect effect as the
source of a systematic error, and its impact on gravity data
interpretation.
In Slovakia, the GIE is small (Fig. 3). It has a magnitude
of 3 mGal and varies by about 0.5 mGal/100 km. We also
give a simple estimate of the impact of this systematic er-
ror in interpretation of density distribution, namely in
terms of an error in the computed depth of a density inter-
face (Eq. (18)). In Fig. 4, we portray the error of the deter-
mination of the depth of a density interface with a density
contrast of 300 kg/m
3
(taken as an example) caused by the
GIE in Slovakia. This error has a magnitude of about
300 meters. We conclude that in most applications in Slo-
vakia, the GIE as a systematic error may be considered in-
significant and neglected. However, in other areas with
higher mountains, or on larger regional scales, this may
not be the case. On the other hand, since this effect is a
systematic error, and since it is possible to evaluate it nu-
merically, it can be added to Bouguer gravity anomalies
for the sake of an improved gravimetric interpretation. The
addition of the GIE to the Bouguer gravity anomaly trans-
forms the Bouguer gravity anomaly, rigorously speaking,
to the NETC gravity disturbance (Vajda et al. 2006).
Acknowledgments: This work was carried out with partial
support from the Science and Technology Assistance Agen-
cy of the Slovak Republic under contract No. APVT-
51-002804, and the VEGA Grant Agency Projects No. 2/
3057/23 and 2/3004/23. We are grateful for the comments
of the reviewers Bruno Meurers and Miroslav Bielik and es-
pecially for the detailed comments by Ron Hackney.
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