GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 137
1.0 Introduction
Topographical effect is one of the most impor-
tant components in the solution of the Geodetic 
Boundary Value Problem (GBVP), and should be 
treated properly in the determination of a precise 
geoid. The classical solution of the geodetic BVP 
using Stokes’s formula for geoid determination 
assumes that there should be no masses outside 
the geoid. The input gravity anomalies should re-
fer to the geoid, which requires the actual Earth’s 
topography to be regularized in some way. The 
determination of the geoid as an internal geodetic 
boundary value problem is described by [Heck, 
1992; Moritz, 1980]. There are different reduction 
Gongola Basin Geoid Determination 
using Isostatic Models and Seismic Reflection Data 
and Geophysical Interpretation
© E. E. Epuh, J. B. Olaleye, O. G. Omogunloye, 2016
Department of Surveying and Geoinformatics, University of Lagos, Nigeria
Received September 9, 2016
Presented by the Editorial Board Member V. I. Starostenko
Применение формулы Стокса для вычисления ундуляций геоида требует отсутствия масс 
вне его. Как правило, постоянная плотность 2,67 г/см3 используется при определении геоида, 
что обусловливает ошибку в редуцированных гравитационных аномалиях (конденсация Гель-
мерта) и, следовательно, геоида. В работе использованы изостатические модели Эри—Хейска-
нена и Пратта—Хейфорда при определении геоида, рассматривая приближенные плоские и 
сферические модели. Рассчитан непрямой эффект изменения топографической латеральной 
плотности на геоиде в качестве аддитивной поправки для улучшения точности вычисленного 
геоида. Для расчета переменной плотности рассмотрена дополнительная информация о плот-
ности, полученная по сейсмическим и каротажным данным. По модели EGM 2008 рассчитаны 
геопотенциальные ундуляции геоида. Остаточный геоид получен путем вычитания локаль-
ного изостатического геоида из геопотенциального геоида. Также проведены исследования 
проводимости геоида и гравитационного поля в дополнение к изучению остаточного геоида.
Планарная и сферическая аппроксимации показали сходные характеристики, но с раз-
ными величинами в обеих моделях. Наши результаты свидетельствуют о том, что эффекты 
латерального изменения топографические плотности при определении геоида являются зна-
чительными и должны рассматриваться в рифтовых бассейнах. Геофизический анализ резуль-
татов определения геоида показывает, что северо-восточный  регион имеет положительные 
значения остаточного геоида, которые указывает на наличие интрузивных магматических 
пород высокой плотности, тогда как юго-западный имеет отрицательные величины остаточ-
ного геоида, что свидетельствует о доминирующем присутствии осадочных пород низкой 
плотности. Результаты также показывают, что радиальное распределение аномальной массы, 
полученное при использовании аномалий геоида/остаточного геоида, однозначно соответству-
ет полученному распределению с использованием данных сейсмического профилирования 
методом отраженных волн, которые обнаружили скопления углеводородов в юго-восточной 
зоне на территории проекта. Исследования проводимости гравитационного поля и геоида 
подтвердили результаты изучения остаточного геоида и сейсмических наблюдений.
Ключевые слова: ундуляции геоида, модель Эри—Хейсканена, модель, Пратта—Хейфорда, 
изостатическая остаточная аномалия силы тяжести, остаточная ундуляция геоида.
methods depending on the way in which these 
topographic masses are dealt with [Heiskanen, 
Moritz, 1967, p. 126—158]. One of such methods 
used is the topographic isostatic reduction. Recent 
studies along the line of geodetic isostacy include 
[Rummel et al., 1988; Engels et al., 1995; Tsoulis, 
2001, 2003a,b; Claessens, 2003; Kuhn, 2003; Wild, 
Heck, 2004a,b; Kaban et al., 2004; Heck, Wild, 
2005]. The basic concept of isostacy assumes that 
the outer masses of the earth down to a certain 
compensation depth are in hydrostatic equilibrium 
with the masses below [Kuhn, 2003]. According to 
topographic isostatic reduction, the topographic 
masses are not completely removed, but are shifted 
НАУЧНЫЕ СООБЩЕНИЯ
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
138 Геофизический журнал № 6, Т. 38, 2016
into the interior of the geoid [Heiskanen, Moritz, 
1967] Until now, the common practice in geoid 
determination has been the application of a crustal 
density of 2.67g/cm3 for the topographic masses 
[Vanicek, Kleusberg, 1987; Featherstone, 1992; 
Forsberg, Sideris, 1993; Abd-Elmotaal, 1999; Smith, 
Milbert, 1999; Featherstone et al., 2001]. However, 
the real density can differ from this value by 10 % 
or more [Martinec, 1993; Tziavos et al., 1996; Pa-
giatakis, Armenakis, 1998; Kuhn, 2000a,b; Huang 
et al., 2001]. While sedimentary rocks often have 
density values below 2.40 g/cm3 mafic igneous and 
plutonic rocks have density values above 3.0 g/cm3. 
Martinec [1993] showed theoretically that small 
lateral density variation of topographic masses may 
introduce errors on the geoid height of more than 
one decimeter.
Fraser and colleagues have developed a GIS 
based system to calculate terrain corrections using 
the real topographical rock density values [Fraser 
et al., 1998]. The results show that in the Skeena 
region of British Columbia Canada, the terrain 
corrections to gravity can change by a few mGals 
when real topographical density is used. Further, 
[Pagiatakis et al., 1999] showed that the effect of 
lateral density variations on the geoid can reach 
about 10 cm in the Skeena region and several mil-
limeters in New Brunswick, where the terrain is 
moderate. Also, [Sjöberg, 2004] showed that the 
total effect on the geoid from lateral density for 
the deepest lake on Earth (Lake Bajchal) and the 
highest mountain on Earth (Mt. Everest) can reach 
up to +1.5 cm and +1.78 cm respectively. These 
differences are the reason for using improved 
density model in geoid determination based on 
Stokes’ function [Kuhn, 2003]. Several studies have 
previously investigated the use of lateral varying 
topographic density models in gravimetric geoid 
determination (e. g. [Martinec et al., 1995; Marti, 
1997; Kuhtreiber et al., 1998; Pagiatakis et al., 1999; 
Tziavos, Featherstone, 2000; Haung et al., 2001; 
Hunegnaw, 2001; Kiamehr, 2006a,b]). In the Ni-
gerian geoid determination [Nwilo et al., 2007] 
modeled the local geoid of Lagos (Nigeria) based 
on geometrical interpolation approach. By using 
the orthometric height and ellipsoidal height, the 
empirical geoid height were computed. The surface 
interpolation utilized the kriging approach. Isioye 
and colleagues utilized a five to eight parameter 
model to fit the GPS/Leveling to the EGM 2008 
model to improve the determination of orthomet-
ric height observed from GPS in the study area 
(Portharcourt, Nigeria) [Isioye et al., 2011]. Ezeigbo 
and colleagues examined some factors that affect 
the accuracy of gravimetric geoid determination us-
ing mean gravity anomalies over geographically de-
fined grids [Ezeigbo et al., 2007]. Gravity anomaly 
data obtained from satellite altimetry mission were 
used in the evaluation of the Stokes’ and Vening 
Meinesz’s integral formulae. The result shows that 
the most significant parameters that affect the ac-
curacy of gravimetric geoid determination are the 
minimum spherical distance from the computation 
point, the size and distribution of the observed 
gravity anomaly data. Okiwelu and colleagues 
determined geoid undulation for Nigeria using 
the spherical harmonics expansion employed in 
the Earth Gravitational Model 2008 (EGM 2008) 
referenced to the WGS 84 (World Geodetic Sys-
tem 1984) [Okiwelu et al., 2011]. In their study, 
they found that the highest geoid undulations are 
centered over the North Central region of Nigeria 
with relatively lower values (16—20 m) confined to 
the Nigerian sedimentary basins (Bornu basin and 
Benue Trough). In this study, the Pratt—Hayford 
and Airy—Heiskanen isostatic models as well as 
density data derived from seismic and well log ob-
servations (Kolmani River-1 log) will be utilized in 
the determination of the geoid. The main advantage 
of using isostatic models in geoid determination is 
their small indirect effect, together with a smooth 
field of gravity anomalies [Kuhn, 2003[. Isostatic 
geoid are also long wavelength in nature and as 
such are dominated by the signature of deep-
mantle anomalies (e. g. [Hager, Clayton, 1989]) 
and/or subducted slabs (e. g. [Ricard et al., 1993]). 
The introduction of the additive lateral density 
variation indirect effect and the primary indirect 
effect terrain effects will improve the accuracy of 
the computed geoid and hence, the accuracy of 
the interpreted geophysical structures.
Variations in the height of the geoidal surface 
are related to density anomalous distributions 
within the Earth and the geoid undulations help 
to understand the internal structure of the earth. 
Fig. 1 [Lowrie, 2007] shows that positive geoid in-
dicates the presence of high density excess mass, 
while negative geoid indicate regions of mass defi-
ciency or low density mass deposits. The use of ge-
oid in geophysics has been presented by [Vanicek, 
Christou, 1994; Featherstone, 1997] in which; the 
relationship between the geoid and deep Earth 
mass density anomaly structure, strain and stress 
fields, tectonic forces, isostatic state of ocean litho-
sphere, Earth rotation, geophysical prospecting 
and ocean circulation are discussed. However, the 
importance of geoid in the determination of the 
anomalous density distribution has also been rec-
ognized by [Kaula, 1967; Chase, 1985; Lambeck, 
1988]. Brown in his work [Brown, 1983] recognized 
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 139
the correlation between the geoid and the deep-
Earth mass density anomalies; while [Christou et 
al., 1989] showed its correlation with near surface 
mass density anomalies. Also correlations between 
geoid and earth mantle convection is established 
by [Runcorn, 1967] and with plate tectonic features 
and seismic tomography by [Silver et al., 1988]. 
Featherstone used geoid to determine the lateral 
extent of known geological structures [Feather-
stone, 1997]. The geoid approach in this research, 
serves as a complementary way of studying the 
earth interior and mass density distribution in 
the Gongola basin. Most previous studies have 
relied heavily on geophysical approach; for ex-
ample [Okereke, 1988; Osazuwa et al., 1992; Ug-
bor, Okeke, 2010; Okiwelu et al., 2010; Okiwelu 
et al., 2011].
2.0 The Isostatic Models
2.1 Airy—Heiskanen Model (Planar and 
spherical approximation)
The model is applied under the following as-
sumptions [Rummel et al., 1998; Kuhn, 2003; Ilk, 
Witte, 2007] that the isostatic compensation takes 
place completely and locally, i.e. the compensation 
mass is directly under the regarded topographic 
mass which makes the compensation depth vari-
able (Fig. 2).
The root thickness t for planar approximation 
is obtained by [Heiskanen, Moritz, 1967; Grant, 
West, 1987] as:
 crpt H
ρ= Δρ , (1)
where tр — root thickness (planar), ρcr — crustal 
density (2.67 g/cm3), Δρ — the variable density 
contrast between the lower crust and the upper 
mantle, H — topographic height.
The root thickness tsp for spherical approxima-
tion is obtained by [Rummel et al., 1998; Ilk, Witte, 
2007] as
 ( ) 3 3 cr3sp 3(( ) )1 1 ( )
R H R
t R T
R T
⎡ ⎤+ − ρ⎢ ⎥= − − − − Δρ⎢ ⎥⎣ ⎦
, (2)
where R — radius of the earth (6371 km), T — nor-
mal thickness.
Usually it is applied assuming:
- a constant compensation depth (D) of 
100 km at which the hydrostatic equilibrium 
is achieved;
- that due to constant compensation depth, 
the condition of equilibrium leads to a later-
ally variable mass density. Fig. 3 illustrates 
the Pratt—Hayford model.
The variable density for the planar model is 
expressed as: 
 
pL cr
D
D H
ρ = ρ+ , (3)
pL
ρ  — density variation on land (planar), D — 
depth of compensation on land (100 km), H — 
topographic height on land.
For spherical approximation, 
pL
ρ is given as 
[Rummel et al., 1998; Ilk, Witte, 2007]:
 
spL cr
ρ = ρ −
 
( )( )
2
2
1
3
t
H D h DH H D
D R R
+ +⎡ ⎤+− + + ρ⎢ ⎥⎣ ⎦
, (4)
spL
ρ  — density variation on land (spherical), ρt 
— weathered tertiary density value, R — radius 
of the Earth.
2.3 Spherical Harmonics Representation of 
Geoid
Spherical harmonics are often used to ap-
proximate the shape of the geoid. The geoid un-
dulations are evaluated from spherical harmonic 
coefficients at the surface of the ellipsoid, not tak-
ing into account the difference between height 
anomalies and geoid undulations [Heiskanen, 
Moritz, 1967, p. 325] nor the effect on the geoid 
of the downward continuation of gravity from the 
surface [Sjöberg, 1998a; Kaban et al., 2004]. Ge-
oid undulation (N) can be expressed in spherical 
harmonics as [Rapp et al., 1991]:
 (
2 0
cos
nN n
nm
n m
GM a
N C m
r r= =
⎛ ⎛ ⎞= λ +⎜ ⎜ ⎟⎜γ ⎝ ⎠⎝∑ ∑
 sin sinmnnmS m P
⎞+ λ ϕ⎟⎟⎠
. (5)
Fig. 1. A mass excess below the ellipsoid elevates the geoid 
above the ellipsoid. N is the geoid undulation [Lowrie, 2007].
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
140 Геофизический журнал № 6, Т. 38, 2016
The corresponding free air gravity anomaly 
can be obtained from the anomalous potential as 
[Heiskanen, Moritz, 1967, p. 97].
 (F 2
2 0
( 1) cos
nN n
nm
n m
GM a
g n C m
rr = =
⎛ ⎛ ⎞Δ = − λ +⎜ ⎜ ⎟⎜ ⎝ ⎠⎝∑ ∑
 sin sinmnnmS m P
⎞+ λ ϕ⎟⎟⎠
, (6)
ΔgF — free air anomaly.
The Earth Gravitational Model EGM 2008 [Pal-
vis et al., 2008] is complete to spherical harmonic 
degree and order 2159 and contains additional 
coefficients extending to degree 2190 and order 
2159. The harmonic degree implies that short 
wavelength anomalous features can be studied 
which is relevant in the determination of the near 
surface mineral depth and its mass density distri-
bution [Featherstone, 1997]. The free air anoma-
ly derived from the spherical harmonic model is 
stated as follows:
2.4 Isostatic Geoid Anomalies
2.4.1 Co-Geoid and Geoid undulations
The geoid undulation N is the vertical separa-
tion between the geoid and the ellipsoid. Fig. 4 
shows the geometric separate of the geoid and the 
co-geoid. In gravimetric geoid determination, the 
computed geoid does not match with the actual 
geoid. The difference between the actual geoid 
and the computed geoid is called the indirect ef-
fect.
The formula for computing co-geoid undu-
lation from a 3D density model Δρ(x, y, z) is ex-
pressed as [Turcotte, Schubert, 1982]:
 ( ) 0c
0
2 T t
m cr cr
H H
G
N z dz z dz
+
−
π= ρ −ρ + ργ ∫ ∫ , (7)
Nc — co-geoid, γ0 — normal gravity, G — universal 
gravitational constant, pm — mantle density, ρcr  — 
crustal density.
For Airy—Heiskanen compensation with depth 
of the normal T, surface topography H and the root 
Fig. 2. Airy—Heiskanen isostatic model in planar approximation [Ilk, Witte, 2007] 2.2 Pratt—Hayford model (Planar and spheri-
cal approximation).
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 141
t. From Eq. 1, taking downward continuation, the 
co-geoidal undulation is given for planar approxi-
mation [Crovetto et al., 2008] as:
 c 2cr m
0 m cr
2
G
N TH H
⎧ ⎫π ρ ρ= +⎨ ⎬γ ρ −ρ⎩ ⎭ . (8)
In the Airy—Heiskanen formula, we assume a 
perfect isostatic balance using T=30 km, density 
2.67 and density contrast ρm−ρcr=0.6 g/cm3. For 
spherical approximation with respect to Eq. 2, it 
is derived as:
 ( )c m cr
0
2
Gt
N t T
π= + ρ −ργ , (9)
сm — upper mantle density (3.27 g/cm
3).
In the use of Pratt—Hayford model for planar 
earth (Eq. 3), with depth compensation D and 
normal density ρcr, H — elevation, the co-geoid 
undulation associated with positive topography 
is derived from Eq. 7 as:
 c cr
0
G
N DH
π ρ= γ . (10)
For spherical approximation with respect to 
Eq. 4, it is derived from Eq. 7 as
 c 2 2cr L L
0
( )
G
N D H
π ⎡ ⎤= ρ −ρ +ρ⎣ ⎦λ . (11)
After computing the co-geoid undulation, the 
geoid undulation N is computed by adding a num-
ber of additive corrections (which sum up as the 
indirect effects) to the co-geoid
 сN N N= + δ , (12)
 PITE DWCpN N N NΔδ = δ + δ + δ , (13)
N — geoid undulation, δN — total indirect effect, 
which must use the same density model as the 
gravity anomalies, δNΔp — lateral isostatic density 
anomaly indirect effect on the geoid, δNDWC — in-
direct effect due to downward continuation. The 
contribution of this effect cancels in their sum.
2.4.2 Primary Indirect Terrain Effect
The primary indirect topographical effect 
(PITE) is the separation between the geoid and 
the co-geoid caused by the condensation of the 
topography and the atmosphere
Fig. 3. Pratt—Hayford compensating model (constant density) in planar approximation [Ilk, Witte, 2007].
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
142 Геофизический журнал № 6, Т. 38, 2016
 PITE
0
V
N
δδ = γ ,
δV — change of potential at the geoid which de-
pends on the reduction method used.
2.4.3 The Effect of Lateral Density Variation on 
the Geoid
Sjöberg [2004] showed that the total effect of 
the geoid due to the lateral density anomaly could 
be represented as a simple correction proportional 
to the lateral density anomaly and the elevation 
of the computation point square. The combined 
topographic effect on the geoid [Sjöberg, 2001; 
Kiahmehr, 2006b] including the zero and first de-
gree terms is well approximated by
 1 2comb
0
2 G
N H
π ρδ = γ , (14)
where G — gravitational constant, ρ=ρ(θ, λ) is the 
laterally variable topographic density at the co-
latitude (θ), and longitude (λ), H — height, γ0 — 
normal gravity.
The lateral density variations indirect effect on 
geoid determination was computed using Eq. 14. 
If the density of the topography at the computa-
tional point is
 crρ = ρ + Δρ , (15)
where ρcr is the standard density (2.67 g/cm
3), Δρ(ϕ, 
λ) is the lateral density anomaly with respect to 
the standard density. The total effect of Δρ on the 
geoid undulation becomes
 2
0
2 G
N HΔρ
π Δρδ = γ . (16)
For the Airy—Heiskanen model for geoid un-
dulation computation, the lateral density variation 
indirect effect on the geoid for planar approxima-
tion with respect to Eq. 1 is given as
 3cr(AH)
0
2 G
N H
tΔρ
π ρδ = γ . (17)
Based on the concept above, the indirect effect 
in spherical approximation with respect to Eq. 2, 
is derived as
 
2
(AH)
0
2 HG
NΔρ
π Δρδ = =γ
 
( )
( )
3 3
2cr
3
0
2 R H RG
H
R T
⎡ ⎤+ −π ρ ⎢ ⎥= γ ⎢ ⎥−⎣ ⎦
. (18)
From the Pratt—Hayford model for geoid un-
dulation computation, the lateral isostatic density 
variation indirect effect for planar approximation 
on the geoid with respect to Eq. 3, is given as [Ki-
amehr, 2006b]
 L crρ = ρ + Δρ , (15')
 
3
cr
(PH)
0
2 G H
N
D HΔρ
⎛ ⎞π ρδ = ⎜ ⎟⎜ ⎟γ +⎝ ⎠
. (19)
The indirect effect in spherical approximation 
with respect to (Eq. 4), is derived as
 
( ) 2cr L
(PH)
0
2 G H
NΔρ
π ρ −ρδ = γ . (20)
2.4.4 Geoid Undulations computations from 
isostatic models
Fig. 4. The Relationship between geoid and co-geoid [Kuhn, 2003]. 
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 143
From the above formulations for the co-geoid, 
primary indirect terrain effect and lateral density 
variation indirect effect, the final expression for 
geoid undulation obtained from the isostatic mod-
els are:
 2cr mAH Pl
0 m cr
2
G
N TH H−
⎡ ⎤⎧ ⎫π ρ ρ= + −⎢ ⎥⎨ ⎬γ ρ −ρ⎢ ⎥⎩ ⎭⎣ ⎦
 3cr
0 0
2
2 sin
2
GG Hd
H
t
⎡ ⎤ ⎡ ⎤π ρρ ψ⎛ ⎞− π +⎢ ⎥ ⎢ ⎥⎜ ⎟γ γ⎝ ⎠⎣ ⎦ ⎣ ⎦ , (21)
 ( )AH Sp m cr
0
2
Gt
N T−
⎡ ⎤π= + ρ −ρ −⎢ ⎥γ⎣ ⎦
 
0
2 sin
2
G Hd⎡ ⎤ρ ψ⎛ ⎞− π +⎢ ⎥⎜ ⎟γ ⎝ ⎠⎣ ⎦
 
( )
( )
3 3
2cr
3
0
2 R H RG
H
R T
⎡ ⎤⎡ ⎤+ −π ρ⎢ ⎥⎢ ⎥+ γ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦
, (22)
 crPH Pl
0 0
2 sin
2
G G Hd
N DH−
⎡ ⎤ ⎡ ⎤π ρ ρ ψ⎛ ⎞= − π +⎢ ⎥ ⎢ ⎥⎜ ⎟γ γ ⎝ ⎠⎣ ⎦ ⎣ ⎦
 
3
cr
0
2ѓ ПG H
D H
⎡ ⎤⎛ ⎞π+ ⎢ ⎥⎜ ⎟⎜ ⎟γ +⎢ ⎥⎝ ⎠⎣ ⎦
, (23)
 ( )2 2PH Sp cr L L
0
Gt
N D H−
⎡ ⎤π ⎡ ⎤= ρ −ρ +ρ −⎢ ⎥⎣ ⎦γ⎣ ⎦
 
0
2 sin
2
G Hd⎡ ⎤ρ ψ⎛ ⎞− π +⎢ ⎥⎜ ⎟γ ⎝ ⎠⎣ ⎦
 
( ) 2cr L
0
2 G H⎡ ⎤π ρ −ρ+ ⎢ ⎥γ⎢ ⎥⎣ ⎦
. (24)
2.5 Residual Geoid Undulations
In order to reveal the short wavelength geoi-
dal features which are assumed to reflect crustal 
and lithospheric structures, the long wavelength 
component of the geoid assumed to originate 
in the mantle is removed by the process termed 
detrending [Featherstone, 1997]. Residual geoid 
undulation is obtained as the difference between 
the geoid obtained from EGM 2008 model and 
the geoid undulations obtained from the isostatic 
models:
 ΔNI = NEGM 2008−NI, (25)
where ΔNI — residual undulation for the various 
isostatic models, NEGM 2008 — geoid undulation for 
EGM 2008, NI — geoid undulation for the various 
isostatic models.
2.6 Geoid and Gravity Admittance Evaluation
Gravity and geoid anomalies reflect lateral het-
erogeneities in the earth’s density structure. Be-
cause such anomalies often correlate with topog-
raphy, it became a standard approach to use the 
relationship between bathymetry and the gravity 
or geoid to gain information about the subsurface 
density structure and the style of isostatic com-
pensation of the topographic load assuming that 
compensation occurs on a regional basis. In the 
following, the geoid admittance (Geoid to Topo-
graphic Ratio) as a spectral function is calculated 
by [Keifer, Hager, 1991]:
 geoid
( )
( )
( )
N
A
H
λλ = λ , (26)
Ageoid(λ) — geoid admittance, N(λ) — geoid undu-
lation, H(λ) — topography.
In relating the geoid undulations to free air 
gravity anomalies in the spectral domain, the fol-
lowing expression is used [Keifer, Hager, 1991]
 0FA
2 ( )
( )
N
g
πγ λΔ λ = λ , (27)
 FAgravity
( )
( )
( )
g
A
H
Δ λλ = λ , (28)
where Agravity(λ) — gravity admittance, ΔgFA(λ) — 
free air anomaly.
In the case of the Airy—Heiskanen model, 
the topographic height is considered in terms of 
surface topography and mantle (dynamic topog-
raphy) to determine the deep earth mass density 
anomalies [Bowin, 1983] and near-surface mass 
density distribution [Christou et al., 1989].
3.0 Methodology
The following data were utilized in the gravi-
metric geoid determination.
1. Digital terrain model (DTM) for modeling 
the shape of topography.
2. Digital Density Model (DDM) for modeling 
the spatial distribution of density in the to-
pography and deeper masses. The informa-
tion from seismic and well log observations 
were used to derive the density model.
3. Isostatic models for analytically modeling 
the Earth’s outer masses.
4. EGM 2008 geoid undulation data.
5. Residual geoid obtained as difference be-
tween EGM 2008 and the local isostatic 
geoid. 
Fig. 5 shows the flowchart for the geoid model-
ing and its geophysical analysis.
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
144 Геофизический журнал № 6, Т. 38, 2016
3.1 The Study Area and Gravity Data Acquisition
Gongola basin of Northern Nigeria is one part 
of a series of Cretaceous and later rift basins in 
Central and West Africa whose origin is related 
to the opening of the South Atlantic [Obaje et al., 
2006]. Many authors have noted that the Benue 
rift (which includes Gongola basin) have many fea-
tures in common with other intra-continental East 
African rift such as Baikal rift and the Rio Graade 
rift, for example [Logatchev, 1993; Shemang et 
al., 2001; Ugbor et al., 2010]. These rift systems 
are associated with volcanism and regional up-
lift. The basin contains thick sediment accumula-
tions (mainly Cretaceous) in excess of 7 km in the 
North-Eastern part of the study area; deposited 
under varying environments. Seismic studies in 
the area have led the insight into the structure of 
the crust and mineral viability of the basin.
The gravitational data for this investigation is a 
set of Bouguer gravity and isostatic residual grav-
ity anomalies observed at 1813 shot points with 
station interval of 500 m within Gongola basin.
4.0 Results
The following results shown in Fig. 6—10 il-
Fig. 5. Geoid Modeling and Geophysical Analysis Flow Chart (Modified from [Ilk, Witter, 2007]).
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 145
lustrate the geoid and geophysical findings within 
the basin.
4.0. Discussion
4.1 Geoid and Residual Geoid Anomaly Maps
The spatial behavior of the geoid undulation 
for P-H and A-H and that of the EGM 2008 (Fig. 6) 
have similar characteristics with respect to the 
density variation in west-east direction; but dif-
fer in magnitude. However, the results from the 
P-H spherical approximations are better than the 
other results in the planar and spherical approxi-
mations in terms of producing the minimum sum 
of squares of the residual geoid values (Fig. 7).
The isostatic geoid undulations are found 
to have long wavelength features between 
4000<λ<8600 m. These features of the geoid are 
due to density variations in the lower mantle and 
resulting deformations of the core mantle bound-
ary and other boundaries in the mantle [Richards, 
Hager, 1984; Okiwelu et al., 2011]. The residual 
geoid undulations for the P-H isostatic models 
range between –7 m to +11 m in the spherical ap-
proximation model. The negative residual geoid 
undulation values correspond with the region of 
less dense intrusive rocks in the SE, while the posi-
tive residual geoid values correspond to regions 
of high density subsurface intrusive rock depos-
its zones of the project area. The spatial behavior 
of the P-H isostatic residual gravity anomaly, the 
P-H geoid undulation and the P-H residual geoid 
(as shown in Fig. 7, a, b) correlates and this helps 
in the definition of the basin’s subsurface density 
distribution. The large bias in the residual geoid un-
dulation is based on the use of spherical harmonics 
coefficients in the EGM 2008 geoid model. Gravity 
disturbances [Heiskanen, Moritz, 1967], rather than 
gravity anomalies, are computed from the spherical 
harmonics so as to account for the masses between 
the reference ellipsoid and geoid [Kaban et al., 
2004]. Geoid undulations are evaluated from the 
spherical harmonic coefficients at the surface of 
the ellipsoid, not taking into account the difference 
between height anomalies and geoid undulations 
[Heiskanen, Moritz, 1967] nor the effect on the 
geoid of the downward continuation of gravity from 
the surface [Sjöberg, 1998a]. Correlation exists in 
the geoid undulation between basement complex 
zones and the EGM 2008 geoid undulations. In 
comparison with the geology, there is a substantial 
agreement with respect to the sedimentary zones 
with large negative residual geoid undulations 
which is due to lateral density anomaly. This is 
due to the large density contrast which has a direct 
influence on the computed geoid. The area with 
high sedimentary thickness between 4 km and 7 km 
in the North-East has high density igneous intrusive 
rocks and as such, depicts positive residual geoid 
undulation values as shown in Fig. 7, c. The de-
Fig. 6. Pratt—Hayford, Airy—Heiskanen and EGM 2008 geoid undulations.
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
146 Геофизический журнал № 6, Т. 38, 2016
crease in geoid undulations from west to east within 
the Gongola sedimentary basin shows the presence 
of depression in the NE and SE zones which is 
Fig. 8. Correlation of Isostatic Residual Gravity Anomaly, Seismic Horizon H4 depth map and Residual Geoid map.
consistent with the evolution of the trough. The 
high geoid undulation transiting between the NE 
and SE (Fig. 6, a, b) shows the presence of a ridge 
Fig. 7. Correlation of the isostatic residual gravity, geoid and residual geoid maps.
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 147
Fig. 9. Gravity and geoid admittance maps.
within this segment of the basin. The evolution of 
the Benue Trough is closely linked with the open-
ing of the South Atlantic. Details on the sequence 
of events that led to its formation alongside other 
sedimentary basins in Nigeria are contained in the 
various literatures [King, 1950; Cratchley, Jones, 
1965; Wright, 1976; Benkhelil, 1982; Whiteman, 
1982]. It is a rift basin with plate dilation leading 
to the opening of the Gulf of Guinea [Benkhe-
lil, 1989; Fairhead, Binks, 1991]. Benkhelil [1989] 
suggested that the evolution trough could also be 
as a result of tension resulting in a rift or wrench 
related fault basin. Mesozoic to Cenezoic magma-
tism has accompanied the evolution of the tectonic 
rift as it is scattered all over and throughout in 
the trough [Coulon et al., 1996; Abubakar et al., 
2010]. A magmatic old rift was also suggested for 
the Gongola basin by [Shemang et al., 2001] while 
[Abubakar et al., 2010] suggested the evolution 
as a combination of mantle upwelling or rise of a 
mantle plume which resulted in crustal stretching 
and thinning and the emplacement of basic igne-
ous material within the basement and sediment 
which resulted in rifting. The reflection seismic 
time/depth structural maps also define the SE zone 
as a zone with favourable structural geometry for 
hydrocarbon accumulation. Fig. 8 shows the cor-
relation between the Pratt—Hayford isostatic re-
sidual gravity anomaly, the seismic depth map (at 
prospect horizon) and the Pratt—Hayford residual 
geoid undulation. The reflection seismic depth 
map corroborates the Pratt—Hayford geoid and 
residual geoid structural pattern and hence, the 
favourable locations for hydrocarbon accumulation 
(oval shapes in the maps) shown in the residual 
geoid are justified.
4.2 Analysis of Geoid and Gravity Admittance 
Maps
Short wavelength geoid provides information 
about the near surface features while long wave-
length geoid with low degree (n=6, 7) provides in-
formation about the mass anomalies in the crust 
mantle zone [Bowin, 1985]. Their isostatic model 
gravity/geoid admittance were used to determine 
the mass anomalies in the crust mantle zone. The 
gravity admittance map (Fig. 9, a) from the Pratt 
Hayford isostatic model shows a uniform mass 
density distribution that corresponds with the 
result obtained from the mass density structure 
of EGM 2008 gravity admittance (Fig. 9, b). The 
mass anomalies from EGM 2008 geoid admittance 
corroborate with the gravity admittance structures 
from the isostatic model. The mass anomalies vary 
between 60 and 82 m/km in the northeast while it 
varies between 38 m/km and 60 m/km in the south-
east of the project area. Consequently, it could be 
inferred that the northeast zone has high density 
mass deposits while the southeast has low density 
mass deposits. The northeast is therefore a favour-
able location for solid minerals while the southeast 
is a favourable location for hydrocarbon deposits. 
The gravity/geoid admittance structures corrobo-
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
148 Геофизический журнал № 6, Т. 38, 2016
rate the findings established in the geoid/residual 
geoid and seismic reflection studies.
Conclusion
Gongola basin geoid is primarily affected by 
the lateral density variation from the crust/mantle 
discontinuities. Based on this, the lateral density 
variation indirectly affected the accuracy of the 
geoid significantly by about 3 cm.
The geoid and residual geoid undulations have 
shown in this research to corroborate the findings 
of the reflection seismic data in the determination 
of the lateral extent of known geological structures. 
This has also shown that the geoid can be used as 
a complementary method in the definition of the 
location and radial distribution of the geological 
structures for mineral and hydrocarbon exploration.
Abd-Elmotaal H., 1999. Comparison among different ge-
oid solutions for the Egyptian south-western desert 
using FFT technique boll. Boll. Geofis. Teor. Appl. 
40(3-4), 563—569.
Abd-Elmotaal H., Kuhtreiber N., 2003. Geoid determina-
tion using adapted reference field, seismic Moho 
depths and variable density contrast. J. Geodesy 77, 
77—85.
Abubakar Y. I., Umegu M. N., Ojo S. B., 2010. Evolution 
of Gongola Basin Upper Benue Trough Northeastern 
Nigeria. Asian J. Earth Sci. 3, 62—72.
Anderson E. G., 1979. The effect of topography on solu-
tion of Stokes’ problem. Unisurv Report S-14, Uni-
References
versity of New South Wales, Kensington, Australia, 
252 p.
Benkhelil J., 1989. The Origin and evolution of the cre-
taceous Benue Trough (Nigeria). J. Afr. Earth Sci. 
8, 251—282.
Bird D. E., 2001. Shear Margins: Continent-Ocean trans-
forms and fracture zone boundaries. The Leading 
Edge 20(2), 150—159.
Bomford G., 1982. Geodesy. 5th edn. Oxford: University 
Press.
Carter J. D., Barber W., Tait A. E., Jones J. P., 1963. The 
Geology of part of Adamawa, Bauchi and Borno 
Gongola Basin Geoid Determination 
using Isostatic Models and Seismic Reflection Data 
and Geophysical Interpretation
© E. E. Epuh, J. B. Olaleye, O. G. Omogunloye, 2016
The application of Stokes’ formula to create geoid undulation requires no masses outside the geoid. Usually, 
a constant density of 2.67 g/cm3 is used in the determination of the geoid which introduces error in the reduced 
gravity anomalies (Helmert’s condensation) and consequently the geoid. In this paper, isostatic models of Airy—
Heiskanen and Pratt—Hayford were utilized in the determination of the geoid by considering the planar and 
spherical approximation models the indirect effect of the topographic lateral density variation on the geoid was 
computed as additive correction for the improvement of the accuracy of the computed geoid. Additional density 
information deduced from seismic and well log information were considered for the variable density computation. 
The geopotential geoid undulations were computed from the EGM 2008 model. The residual geoid was obtained 
by subtracting the local isostatic geoid from the geopential geoid. Geoid and gravity admittance studies were also 
carried out to complement the results from the residual geoid.
The planar and spherical approximation results showed similar characteristics; but a change in magnitude in 
both models. Our results suggest that the effects of topographic lateral density variations in geoid determination 
are significant and should be considered in rift basins. The geophysical analysis of the geoid results show that the 
north-east has positive residual geoid which indicates the presence of high density intrusive igneous rocks, while 
the south-east has negative residual geoid which indicates the dominant presence of low density sedimentary 
rocks. The results also show that the radial distribution of the anomalous mass obtained using the geoid/residual 
geoid anomaly uniquely matched that obtained using the seismic reflection data which inferred the presence of 
hydrocarbon accumulation in the southeast zone of the project area. The gravity and geoid admittance studies 
corroborated the residual geoid and seismic reflection results.
Key word: geoid undulation, Airy—Heiskanen model, Pratt—Hayford model, isostatic residual gravity anomaly, 
residual geoid undulation.
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 149
Provinces in North Eastern Nigeria. Bull. Geol. Surv. 
Nigeria 30, 1—108.
Claessens S. A., 2003. A synthetic Earth model. Analysis, 
implementation, validation and application. Delft 
University Press. 61 p.
Cook F. A., Brown L. D., Oliver J. E., 1981. The Late 
Precambrian-Early. Paleozoic continental edge in 
the Appalachian Orogen. Am. J. Sci. 281, 993—1008.
Coulon C., Vida P., Dupuy C., Baudin P., Pupoff M., 
Maluski H., Hermite D., 1996. The Mesozoic to Early 
Cenezoic Magmatism of the Benue Trough (Nige-
ria): Geochemical Evidence for the Involvement of 
the St. Helena plume. J. Petrol. 37, 1341—1358.
Cratchley C. R., Jones G. P., 1965. An interpretation of 
the geology and gravity anomalies of the Benue 
Valley of Nigeria. Overseas geological survey. Geo-
physical Paper (1). 26 p.
Crovetto C. B., Introcaso A., 2008. Alternative gravimet-
ric methodology for isostatic analyses: an example 
of Bolivian Andes. Boletin del Instituto de Fisiografia 
y Geologia 78(1-2), 1—11.
Engels J., Grafarend E. W., Sorcik P., 1995. The gravi-
tational field of topographic isostatic masses and 
the hypothesis of mass compensation. Part I and II. 
Technical Report Department of Geodesy Univer-
sity, Stuttgart.
Featherstone W. E., 1992. A GPS controlled gravimetric 
determination of the geoid of the British Isles. D. Phil 
Thesis, Oxford University.
Featherstone W. E., Kirby J. F., Kearsley A. H. W., Gil-
liland J. R., Johnston G. M., Steed J., Forsberg R., 
Sideris M. G., 2001. The AUSgeoid98 geoid model 
of Australia: data treatment, computations and 
comparisons with GPS-levelling data. J. Geodesy 
75, 313—330.
Forsberg R., 1984. A study of terrain reductions, den-
sity anomalies and geophysical inversion methods 
in gravity field modeling. Department of Geodetic 
Science and Surveying. Report 355. The Ohio State 
University, Columbus.
Forsberg R., Sideris M. G., 1993. Geoid computations 
by multi-band spherical FFT approach. Manuscr. 
Geod. 18, 82—90.
Fraser D., Pagiatakis S. D., Goodacre A. K., 1998. In-
situ rock density and terrain corrections to gravity 
observations. Proc. of the 12th Annual Symposium on 
Geographic Information Systems, Toronto 6—9 April 
1998. P. 357—360.
Heck B., 1992. Some remarks on the determination of 
the geoid in the framework of the internal geodetic 
boundary value problem. In: First continental work-
shop on the geoid in Europe. Research Institute of 
Geodesy, Topography and Cartography, Prague. 
P. 458—471.
Heck B., Wild F., 2005. Topographic and isostatic re-
ductions in Satellite Gravity Gradiometry based on 
a generalized condensation model. In: A Window 
on the Future of Geodesy, International Association 
of Geodesy Symposia. Vol. 128. Berlin, Heidelberg, 
New York: Springer, P. 294—299.
Heiskanen W. A., Moritz H., 1967. Physical geodesy. 
San Francisco: W. H. Freeman. Institute of Physi-
cal Geodesy. 364 p.
Heiskanen W. A., Vening Meinesz F. A., 1958. The Earth 
and its gravity field. New York: McGraw-Hill Book 
Company, Inc. 470 p.
Huang J., Vanicek P., Pagiatakis S., Brink W., 2001. Ef-
fect of topographical mass density variation on grav-
ity and geoid in the Canadian Rocky Mountains. J. 
Geodesy 74, 805—815.
Hunegnaw A., 2001. The effect of lateral density varia-
tion on local geoid determination. Proc. IAG 2001 
Scientific Assembly, Budapest, Hungary.
Ilk K. H., Witte B., 2007. The Use of Topographic-Iso-
static Mass Information in Geodetic Applications. 
Inaugural Lecture. Institute of Geodesy and Geoin-
formation, University of Bonn, Germany.
Isioye O. A., Olaleye J. B., Youngu R., Aleem K. F., 2011. 
Modelling orthometric Heights from GPS-Leveling 
Observations an Global Gravity Model (EGM 2008) 
for Rivers State, Nigeria. Nigerian Journal of Survey-
ing and Geoinformatics  3(2), 56—69.
Kaban M. K., Schwintzer P., Reigber C. H., 2004. A new 
isostatic model of the lithosphere and gravity field. 
J. Geodesy 78, 368—385.
Kiamehr R., 2006a. A strategy for determining the re-
gional geoid in developing countries by combining 
limited ground data with satellite-based global geo-
potential and topographical models: a case study of 
Iran. J. Geodyn. 79(10-11), 602—612.
Kiamehr R., 2006b. The impact of lateral density varia-
tion model in the determination of precise gravimet-
ric geoid in mountainous areas: a case study of Iran. 
Geophys. J. Int. 67, 521—527. doi:10.1111/j.1365-
246X.2006.03143.x.
Kiamehr R., Sjöberg L. E., 2005. Effect of the SRTM glob-
al DEM in the determination of a high-resolution 
geoid model of Iran. J. Geodyn. 79(9), 540—551.
Kuhn M., 2000. Density modeling for geoid determina-
tion. GGG2000, July 31-August 4, 2000, Alberta , 
Canada.
E. E. EPUH, J. B. OLALEYE, O. G. OMOGUNLOYE
150 Геофизический журнал № 6, Т. 38, 2016
Kuhn M., 2003. Geoid Determination with density hy-
potheses from isostatic models and geological in-
formation. J. Geodesy 77, 50—65/
Kuhtreiber N., 1998. Precise geoid determination using 
a density variation model. Phys. Chem. Earth 23(1), 
59—63.
Logatcher N. A., 1993. History of Geodynamics of Baikal 
Rift (East Siberia): A Review. Bulletin due center de 
la recherche exploration production 17, 353—370.
Martinec Z., 1998. Boundary-value problems for gravi-
metric determination of a precise geoid. Lecture 
Notes in Earth Sciences, no. 73. Berlin, Heidelberg, 
New York: Springer.
Martinec Z., 1993. Effect of lateral density variations of 
topographical masses in view of improving geoid 
model accuracy over Canada. Final rep. under DSS 
contract No. 23244-2-4356/01-SS, Geodetic Survey 
of Ottawa.
Martinec Z., Vanicek P., 1994. The indirect effect of 
Stokes-Helmert’s technique for a spherical approxi-
mation of the geoid. Manuscr. Geod. 18, 417—421.
Martinec Z., Vanicek P., Mainville A., Veronneau M., 
1995. The effect of lake water on geoidal height. 
Manuscr. Geod. 20, 199—203.
Moritz H., 1965. The boundary value problem of physi-
cal geodesy. Publication of the Isostatic Institute of 
the IAG, no. 50, Helsinki.
Moritz H., 1990. The figure of the Earth: theoretical ge-
odesy and the Earth’s interior. Karlsruche: Herbert 
Wichmann, 292 p.
Nettleton l. L., 1971. Elementary gravity and magnetic 
for geologists and seismologists. Society of Explora-
tion Geophysicists (SEG), Tulsa, Oklahoma, USA, 
121 p.
Nwilo P. C., Opaluwa Y. D., Adejare Q. A., Ayodele E. G., 
Ayeni A. M., 2009. Local Geoid Modeling of Lagos 
island Area using Geometrical Interpolation Meth-
od. Nigeria Journal of Surveying and Geoinformatics 
2(2), 68—82.
Obaje N. G., Attah D. O., Opeloye S. A., Moumouni A., 
2006. Geochemical Evaluation of the Hydrocarbon 
Prospects of Sedimentary Basins in Northern Nige-
ria. Geochem. J. 40, 227—243.
Okereke C. S., 1988. Contrasting modes of rifting: The 
Benue Trough and Cameroon volcanic lines, West 
Africa. Tectonics 7(4), 775—784.
Okiwelu A., Okwueze C., Okereke C., Osazuwa I., 2010. 
Crustal Structure and Tectonics of the Calabar 
Flank, West Africa, based on Residual Gravity In-
terpretation. Eur. J. Soc. Sci. Res. 42(2), 195—203.
Okiwelu A. A., Okwueze E. E., Ude I. O., 2011. Determi-
nation of Nigerian Geoidal Undulation from Spheri-
cal Harmonic Analysis. Appl. Phys. Res. 3(1), 8—14.
Osazuwa I. B., Adeniyi O. O., Ojo J. B., 1992. A gravity 
study of the older granites suite in the Zaria Area 
of Kaduna State. Nigerian Journal of Mining and 
Geology 28(2), 231—236.
Pagiatakis S. D., Armenakis C., 1998. Gravimetric ge-
oid modelling with GPS. Int. Geoid Serv. Bull. 8, 
105—112
Pagiatakis S. D., Fraser D., McEwen K., Goodacre A. K., 
Veronneau M., 1999. Topographic mass density and 
gravimetric geoid modeling. Boll. Geofis. Teor. Appl. 
40, 189—194.
Pratt J. H., 1855. On the attraction of the Himalaya 
Mountains, and of the elevated regions beyond them 
upon the plumb-line in India. Phil. Trans. Roy. Soc. 
London 145, 53—100.
Rummel R., Rapp R. H., Sunkel H., Tscherning C. C., 
1988. Comparisons of global topographic-isostatic 
models to the Earth’s observed gravity field. Reports 
of the Department of Geodetic Science and Survey-
ing No. 388, Ohio State University, Columbus, Ohio.
Shemang E. M., Ajayi C. O., Jacoby W. R., 2001. A mag-
netic failed rift beneath the Gongola arm of the 
Upper Benue Trough, Nigeria? J. Geodyn. 32(3), 
355—371.
Sideris M., 1996. International tests of the new GSFC/
DMA Geopotential Models. Gravity, Geoid and Ma-
rine Geodesy. International Association of Geodesy 
Symposia (Tokyo) 117, 478—485.
Sjöberg L. E., 2004. The effect on the geoid of lateral top-
ographic density variations. J. Geodesy 78, 34—39.
Sjöberg L. E., 1998a. The exterior Airy/Heiskanen topo-
graphic-isostatic gravity potential, anomaly and the 
effect of analytical continuation in Stokes’ formula. 
J. Geodesy 72, 654—662.
Sjöberg L. E., 2001. Topographic and atmospheric cor-
rections of the gravimetric geoid determination with 
special emphasis of the effects of degrees zero and 
one. J. Geodesy 75, 283—290.
Sjöberg L. E., 1998b. On the Pratt and Airy models of iso-
static geoid undulations. J. Geodyn. 26(1), 137 —147.
Sun W., 2002. A formula for gravimetric terrain correc-
tions using powers of topographic height. J. Geodesy 
76(8), 399—406.
Sun W., Murata I., 1994. Divergence of high-degree 
harmonic solutions of a rotating elliptical Earth. 
Geophys. J. Int. 118, 269—271.
Sunkel H., 1985. An isostatic Earth model. Rep. 367, De-
GONGOLA BASIN GEOID DETERMINATION USING ISOSTATIC MODELS AND SEISMIC...
Геофизический журнал № 6, Т. 38, 2016 151
partment of Geodetic Science and Surveying, The 
Ohio State University, Columbus.
Tsoulis D., 2001. A comparison between the Airy-Heis-
kanen and the Pratt-Hayford isostatic models for the 
computation of potential harmonic coefficients. J. 
Geodesy 74(9), 637—643.
Tsoulis D., Tziavos I. N., 2003. A comparison of some 
existing methods for the computation of terrain 
corrections in local gravity field modelling. Grav-
ity and Geoid 2002. Ziti-Publishing Thessaloniki, 
P. 156—16.
Tziavos I. N., Featherstone W. E., 2000. First results of 
using digital density data in gravimetric geoids 
computation in Australia. IAG Symposia, GGG2000. 
Vol. 123. Berlin, Heidelberg: Springer Verlag, 
P. 335—340.
Tziavos I. N., Sideris M. G., Sunkel H., 1996. The effect 
of surface density variation on terrain modeling- a 
case study in Austria. Proc. EGS Society General 
Assembly. The Hague, The Netherlands, May, 1996. 
Report of the Finnish Geodetic Institute. P. 99—110.
Turcotte D. L., Schubert G., 1982. Geodynamics. Applica-
tions of continuum physics to geological problems. 
New York: John Wiley & Sons Ed., 450 p.
Ugbor D. O., Okeke F. N., 2010. Geophysical investi-
gation in the lower Benue Trough of Nigeria using 
gravity method. Int. J. Phys. Sci. 5(11), 1757—1769.
Vanicek P., Huang J., Brink W., Novak P., 1998. Prelimi-
nary investigation of the effect of topographic mass 
density variations on gravity and geoid in the Cana-
dian Rocky Mountains. Progress report on contract 
«Theoretical and practical refinements of precise 
geoid determination methods». Geodetic Survey 
Division, Natural Resource Canada, Ottawa.
Vanicek P., Kleusberg A., 1987. The Canadian geoid-
stokesian approach. Manuscr. Geod. 12, 86—98.
Vanicek P., Martinec Z., 1994. The Stokes-Helmert 
scheme for the evaluation of precise geoid. Manu-
scr. Geod. 19, 119—128.
Vanicek P., Sun W., Ong P., Martinec Z., Najafi M., Vaj-
da P., ter Horst B., 1996. Downward continuation of 
Helmert’s gravity. J Geodesy 71, 21—34.
Wichiencharoen C., 1982. The direct effects on the com-
putation of geoid undulation. Rep. 336, Department 
of Geodetic Science and Surveying, The Ohio State 
University, Columbus.
Wild F., Heck B., 2004a. A comparison of different 
isostatic models applied to Satellite Gravity Gra-
diometry. Gravity, Geoid and Space Missions. Inter-
national Association of Geodesy Symposia. Vol. 129. 
Berlin, Heidelberg, New York: Springer, P. 230—235.
Wild F., Heck B., 2004b. Effects of topographic and iso-
static masses in Satellite Gravity Gradiometry. The 
Geoid and Oceanography. Proceedings of the Second 
International GOCE User Workshop Frascati/Italy, 
March 8-10, 2004. CD-ROM.