International Journal of Geosciences, 2012, 3, 918-929
http://dx.doi.org/10.4236/ijg.2012.325094 Published Online October 2012 (http://www.SciRP.org/journal/ijg)
Reformulation of the Vening-Meinesz Moritz Inverse
Problem of Isostasy for Isostatic Gravity Disturbances
Robert Tenzer1, Mohammad Bagherbandi2,3
1National School of Surveying, University of Otago, Dunedin, New Zealand
2Division of Geodesy and Geoinformatics, Royal Institute of Technology (KTH), Stockholm, Sweden
3Department of Industrial Development, IT and Land Management, University of Gävle, Gävle, Sweden
Email: email@example.com, firstname.lastname@example.org
Received July 30, 2012; revised August 31, 2012; accepted September 30, 2012
The isostatic gravity anomalies have been traditionally used to solve the inverse problems of isostasy. Since gravity
measurements are nowadays carried out together with GPS positioning, the utilization of gravity disturbances in various
regional gravimetric applications becomes possible. In global studies, the gravity disturbances can be computed using
global geopotential models which are currently available to a relatively high accuracy and resolution. In this study we
facilitate the definition of the isostatic gravity disturbances in the Vening-Meinesz Moritz inverse problem of isostasy
for finding the Moho depths. We further utilize uniform mathematical formalism in the gravimetric forward modelling
based on methods for a spherical harmonic analysis and synthesis of gravity field. We then apply both mathematical
procedures to determine globally the Moho depths using the isostatic gravity disturbances. The results of gravimetric
inversion are finally compared with the global crustal seismic model CRUST2.0; the RMS fit of the gravimetric Moho
model with CRUST2.0 is 5.3 km. This is considerably better than the RMS fit of 7.0 km obtained after using the
isostatic gravity anomalies.
Keywords: Crust; Gravity; Isostasy; Moho Interface; Spherical Harmonics
The functional models of solving the inverse problem of
isostasy have been traditionally formulated by means of
the isostatic gravity anomalies (cf. Heiskanen and Moritz
, p. 133). According to these definitions the topog-
raphic mass surplus and the ocean mass deficiency are
compensated either by a variable crustal thickness or
density. The isostatic equilibrium is described in terms of
the isostatic gravity anomalies which should theoretically
be equal zero, provided that the refined Bouguer gravity
anomalies are isostatically fully rebalanced by the corre-
sponding gravitational attraction of compensating masses.
The Pratt-Hayford isostatic model is based on adopting a
constant depth of compensation while considering a
variable density contrast (Pratt , Hayford , Hayford
and Bowie ). In the Airy-Heiskanen isostatic model a
constant density contrast is assumed while a depth of
compensation is variable (Airy , Heiskanen and Ven-
ing Meinesz ). Both these classical isostatic models are
based on a local compensation scheme. Vening-Meinesz
 modified the Airy-Heiskanen theory of isostasy with
applying the regional instead of local compensation.
Moritz  utilized the Vening-Meinesz inverse problem
of isostasy for finding the Moho depths. Sjöberg  fur-
ther generalized this concept for finding the Moho depths
and density contrast. Sjöberg and Bagherbandi 
computed the Moho depths based on solving Moritz’s
generalization of the Vening-Meinesz inverse problem of
isostasy (VMM isostatic model). Bagherbandi and Sjö-
berg  demonstrated that the VMM Moho depths bet-
ter agree with the Moho data taken from the global
crustal seismic model CRUST2.0 (Bassin et al. )
than those obtained based on solving the Airy-Heiskanen
The results of regional and global studies have shown
often existing significant disagreement between the isos-
tatic and seismic Moho depths. Several different reasons
explaining this misfit were proposed and also confirmed
by numerical experiments. Kaban et al. , for instance,
demonstrated that the isostatic compensation does not
take place only within the Earth’s crust but essentially
also within the lithospheric mantle. This finding was later
confirmed by Kaban et al.  and Tenzer et al. [15,16].
Several authors argued that the isostatic balance is also
partially affected by the changing rigidity, glacial iso-
static adjustment, plate motion, ocean lithosphere cooling,
and other geophysical processes. Moreover, large portion
of the isostatic mass balance is attributed to variable li-
opyright © 2012 SciRes. IJG
R. TENZER, M. BAGHERBANDI 919
thospheric density structures which are usually not taken
into consideration in computing the isostatic gravity
anomalies. Therefore, the models for gravimetric recov-
ery of the Moho parameters should incorporate all known
information on subsurface density structures. One exam-
ple can be given in Greenland and Antarctica where the
application of the ice density contrast stripping correction
to gravity data is essential for a realistic interpretation of
gravimetric results. Another significant gravitational con-
tribution to be modelled and subsequently subtracted
from gravity data is due to large sedimentary basins.
Braitenberg et al.  and Wienecke et al.  demon-
strated that the misfit of the isostatic assumption of the
Moho interface to the long-wavelength part of gravity
field is explained by large sedimentary basins and rigid-
ity variations of the crustal plate.
The gravity anomalies have been primarily used in re-
gional and global studies investigating the Earth’s inner
structures. Vajda et al. , however, argued that the
definition of gravity disturbances in the context of these
studies is theoretically more appropriate. Moreover,
modern gravity observation techniques (such as air-born
gravimetry) nowadays incorporate the GPS positioning
systems. Therefore, it is expected that the gravity distur-
bances will become the most often used gravity data type
in all gravimetric applications. This is due to the fact that
GPS observations provide the geodetic heights above the
reference ellipsoid surface, while the definition of gravity
anomalies requires topographic heights with respect to
sea level. Tenzer et al. [20-22] utilized the definition of
gravity disturbances in the forward modeling of gravita-
tional field generated by major known crustal density
structures. Following this concept, here we define the
VMM inverse problem of isostasy by means of the
isostatic gravity disturbances.
2. Refined Gravity Disturbances
To solve the gravimetric problem of isostasy for finding
the Moho parameters, the gravitational contributions of
all known density contrasts within the Earth’s crust
should be modeled and subsequently removed from grav-
ity data. Moreover, the inhomogeneous density structures
within the mantle lithosphere and sub-lithosphere mantle
should be taken into consideration provided that reliable
data of global mantle density structures are availableial mantle
density structure is restricted by the lack of reliable glob-
al data. A possible way how to estimate the maximum
degree of long-wavelength spherical harmonics which
should be removed from the refined gravity field was
given by Eckhardt . The principle of this procedure is
based on finding the representative depth of gravity sig-
nal attributed to each spherical harmonic degree term.
The spherical harmonics which have the depth below a
certain limit (chosen, for instance, as the maximum Mo-
ho depth) are then removed from the gravity field. Non-
etheless, the complete subtraction of the mantle gravity
signal using this procedure is still questionable due to the
fact that there is hardly any unique spectral distinction
between the long-wavelength gravity signal from the
mantle and the expected higher-frequency signal of the
Moho geometry. Tenzer et al.  demonstrated the
presence of significant correlation (>0.6) between the
mantle gravity signal and the Moho geometry at the me-
dium gravity spectrum (between 60 and 90 of spherical
harmonics degree terms). On the other hand, the gravity
signal of the core-mantle boundary and deep mantle
structures could almost completely be subtracted from
the refined gravity data as it is mainly attributed to the
long-wavelength part of gravity spectra.
As seen from the functional model in Section 3, the
errors in computed gravity data propagate proportionally
Copyright © 2012 SciRes. IJG
R. TENZER, M. BAGHERBANDI 927
to the Moho depth errors. The expected relative uncer-
tainties in gravity data thus cause the errors of about 10%
in the estimated Moho depths. Čadek and Martinec 
estimated the uncertainties of Moho depths in their glob-
al crust thickness model to be about 20% (5 km) for the
oceanic crust and of about 10% (3 km) for the continen-
tal crust. The results of more recent seismic and gravity
studies, however, revealed that these error estimates are
too optimistic. Grad et al.  demonstrated that the
Moho uncertainties (estimated based on processing the
seismic data) under the Europe regionally exceed 10 km
with the average error of more than 4 km. Much large
Moho uncertainties are obviously expected over large
parts of the world where seismic data are absent or insuf-
ficient. A significant contribution to these Moho uncer-
tainties is expected to be explained by several geophysi-
cal phenomena which are not accounted for in the isos-
tatic functional model. Since the geophysical processes
which contribute to isostatic imbalance have a different
regional character, the global model which accounts for
all these contributions (such as the glacial isostatic ad-
justment or oceanic lithosphere cooling) is difficult to
establish and solve numerically. A possible method how
to overcome to some extent these practical limitations
was proposed and applied by Bagherbandi and Sjöberg
. They combine the gravity and seismic data in order
to model and subsequently account for the differences
between the isostatic and seismic Moho models. The
application of this method is out of the scope of this
7. Summary and Conclusions
We have redefined the Vening-Meinesz Moritz inverse
problem of isostasy for the isostatic gravity disturbances
while adopting the double layer model for defining the
Moho density interface. The definition of the isostatic
gravity disturbances was based on the refined gravity
disturbances which have a maximum correlation with the
Moho geometry. With reference to results of the correla-
tion analysis by Tenzer et al. , the gravity distur-
bances corrected for the gravitational contributions of
topography and density contrasts due to ocean, ice and
sediments hold this condition. The application of the ad-
ditional stripping gravity correction accounting for ano-
malous density structures within the crystalline crust was
not taken into consideration as the currently available
global data of this crustal component are still unreliable.
Moreover, the mantle density structures were not mod-
eled too due to the same reason. The methods for a
spherical harmonic analysis and synthesis were used for
computing the refined gravity disturbances. A spectral
representation was also used in definition of the observa-
tion equation for solving the VMM model. The devel-
oped computational schemes were then applied to com-
pute the isostatic gravity disturbances and to determine
the Moho depths. The numerical experiment was carried
out globally on a 1 × 1 arc-deg grid. The results were
validated using the CRUST2.0 Moho depths.
The global map in Figure 6 revealed the signature of
gravity signal of which pattern corresponds with a spatial
geometry of the Moho interface. The negative gravity
values are prevailing over continents, while oceanic areas
are dominated by the positive gravity values. The gravity
minima agree with locations of the maximum continental
crustal thickness under Himalayas, Tibet Plateau and
Andes. The corresponding gravity maxima are seen es-
pecially in central Pacific Ocean. The contrast between
the oceanic and continental lithospheric plate boundaries
is marked along continental margins by the absolute
gravity minima. The signature of the mantle lithosphere
structures is also presented especially along the mid-
The Moho depths computed using the isostatic gravity
disturbances based on solving the VMM model have a
better agreement with the CRUST2.0 seismic model than
those computed using the isostatic gravity anomalies.
The RMS fit of the VMM Moho depths with CRUST2.0
for the isostatic gravity disturbances was found to be 5.3
km. This RMS fit is about 24% better than the corre-
sponding RMS fit of 7.0 km obtained when using the
isostatic gravity anomalies. The facilitation of the isos-
tatic gravity disturbances in the VMM model improved
significantly the systematic bias otherwise found when
using the isostatic gravity anomalies. The mean of dif-
ferences between the VMM and CRUST2.0 Moho depths
obtained after using the isostatic gravity disturbances and
gravity anomalies was found to be 0.1 and –3.4 km re-
spectively. The systematic bias was thus almost com-
pletely eliminated. This numerical improvement is quite
remarkable as the differences between the gravity ano-
malies and disturbances are globally mostly within 30
mGal. A possible explanation for such improvement
might be due to the fact that the differences between
these two gravity data types have a long-wavelength
character. The respective changes in Moho results are
thus more substantial than those attributed to the changes
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