Peculiarity of the Geoid determination at the Antarctic Region
The gravitation effect of topographic masses is very important for the gravity anomaly calculations and geoid heights determination. The new approach for the computation of the topographicisostatic corrections and indirect effect of topography in spherical approximation is developed and tested in An...
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irk-123456789-1293972018-01-20T03:06:02Z Peculiarity of the Geoid determination at the Antarctic Region Marchenko, D.O. Maksymchuk, V.U. Геолого-геофізичні дослідження The gravitation effect of topographic masses is very important for the gravity anomaly calculations and geoid heights determination. The new approach for the computation of the topographicisostatic corrections and indirect effect of topography in spherical approximation is developed and tested in Antarctic region. При изучении гравитационного поля Земли в тех случаях, когда вычисляются как гравитационные аномалии, так и высоты геоида, следует учитывать гравитационный эффект топографических масс. Разработана методика вычисления топографо-изостатических поправок и поправок на рельеф в сферической аппроксимации с переменной плотностью. При вивченні гравітаційного поля Землі у випадку обчислення як гравітаційних аномалій, так і висот геоїда повинен бути врахований гравітаційний ефект топографічних мас. Розроблено методику обчислення топографо-ізостатичних поправок та поправок на рельєф у сферичній апроксимації з перемінною густиною. 2012 Article Peculiarity of the Geoid determination at the Antarctic Region / D.O. Marchenko, V.U. Maksymchuk // Український антарктичний журнал. — 2011-2012. — № 10-11. — С. 106-113. — Бібліогр.: 7 назв. — рос. 1727-7485 http://dspace.nbuv.gov.ua/handle/123456789/129397 528.21/22 ru Український антарктичний журнал Національний антарктичний науковий центр МОН України |
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Геолого-геофізичні дослідження Геолого-геофізичні дослідження |
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Геолого-геофізичні дослідження Геолого-геофізичні дослідження Marchenko, D.O. Maksymchuk, V.U. Peculiarity of the Geoid determination at the Antarctic Region Український антарктичний журнал |
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The gravitation effect of topographic masses is very important for the gravity anomaly calculations and geoid heights determination. The new approach for the computation of the topographicisostatic corrections and indirect effect of topography in spherical approximation is developed and tested in Antarctic region. |
| format |
Article |
| author |
Marchenko, D.O. Maksymchuk, V.U. |
| author_facet |
Marchenko, D.O. Maksymchuk, V.U. |
| author_sort |
Marchenko, D.O. |
| title |
Peculiarity of the Geoid determination at the Antarctic Region |
| title_short |
Peculiarity of the Geoid determination at the Antarctic Region |
| title_full |
Peculiarity of the Geoid determination at the Antarctic Region |
| title_fullStr |
Peculiarity of the Geoid determination at the Antarctic Region |
| title_full_unstemmed |
Peculiarity of the Geoid determination at the Antarctic Region |
| title_sort |
peculiarity of the geoid determination at the antarctic region |
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Національний антарктичний науковий центр МОН України |
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2012 |
| topic_facet |
Геолого-геофізичні дослідження |
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http://dspace.nbuv.gov.ua/handle/123456789/129397 |
| citation_txt |
Peculiarity of the Geoid determination at the Antarctic Region / D.O. Marchenko, V.U. Maksymchuk // Український антарктичний журнал. — 2011-2012. — № 10-11. — С. 106-113. — Бібліогр.: 7 назв. — рос. |
| series |
Український антарктичний журнал |
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AT marchenkodo peculiarityofthegeoiddeterminationattheantarcticregion AT maksymchukvu peculiarityofthegeoiddeterminationattheantarcticregion |
| first_indexed |
2025-07-09T11:18:48Z |
| last_indexed |
2025-07-09T11:18:48Z |
| _version_ |
1837167995506393088 |
| fulltext |
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
106
УДК 528.21/22
PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC
REGION
D.O. Marchenko, V.U. Maksymchuk
Carpathian branch of the Subbotine Institute of Geophysics NAS Ukraine, Lviv, Naukova 3B Str.,
dmitriy.marchenko@gmail.com
Summary. The gravitation effect of topographic masses is very important for the gravity anomaly
calculations and geoid heights determination. The new approach for the computation of the topographic-
isostatic corrections and indirect effect of topography in spherical approximation is developed and tested in
Antarctic region.
Key words: gravity, terrain effect, Moho, density, topographyc-isostatic corrections.
Реферат. При вивченні гравітаційного поля Землі у випадку обчислення як гравітаційних аномалій,
так і висот геоїда повинен бути врахований гравітаційний ефект топографічних мас. Розроблено
методику обчислення топографо-ізостатичних поправок та поправок на рельєф у сферичній
апроксимації з перемінною густиною.
Ключові слова: сила тяжіння, територіальний ефект, Мохо, топографо-ізостатичні поправки.
Реферат. При изучении гравитационного поля Земли в тех случаях, когда вычисляются как
гравитационные аномалии, так и высоты геоида, следует учитывать гравитационный эффект
топографических масс. Разработана методика вычисления топографо-изостатических поправок и
поправок на рельеф в сферической аппроксимации с переменной плотностью.
Ключевые слова: сила притяжения, территориальный эффект, Мохо, топографо-изостатические
поправки.
Introduction
Computation of topographic corrections to measured values of functionals of the disturbing
potential is one among key problems of the physical geodesy. In the frame of the classical
approach to the determination of the disturbing potential based on the famous Stokes integral
formula, introducing of such corrections provides the basic condition of absence of attracting
masses outside the geoid. In the frames of advanced approaches based on application of the
remove-restore procedure, introducing of topographic corrections leads to essential smoothing of a
residual field due to elimination of very high frequencies. As a result, we may hope on more stable
determination of the disturbing potential.
In general, the fundamental requirement of absence of zero-degree harmonic in the disturbing
potential (Heiskanen and Moritz, 1967; Moritz, 1980; Neyman, 1979) can be violated by
application of topographic mass potential TV in the remove-restore procedure. This essential
disadvantage may be eliminated by additional introducing the isostatic compensation potential IV
in accordance with one of known compensation model based on equal mass principle (Moritz,
1990). As a result, we come to the topographic / isostatic correction
ITTI VVV . (1)
УКРАЇНСЬКИЙ АНТАРКТИЧНИЙ
ЖУРНАЛ
УАЖ, № 10-11, 106-113 (2011/2012)
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
107
At arbitrary point Р at the Earth’s surface the potential of masses located between the geoid
surface
),(11 rr , (2)
and the Earth’s physical surface
),(22 rr , (3)
may be defined by the integral (Heiskanen and Moritz, 1967):
2
0 0
22
1
sin)(
r
r
cT ddrd
l
rGPV , (4)
where, in general, the density c of the Earth’s crust is considered as
),,( rcc . (5)
In the above expressions G is the gravitational constant, l is spatial distance between the point
Р and the mass element ddrdrdm c sin2 :
cos222 rrrrl PP , (6)
Pr is the geocentric distance of the point Р, is the geocentric angular distance between the point
P and the mass element dm :
)cos(sinsincoscoscos PPP , (7)
PP , are spherical coordinates (polar angle and longitude, respectively) of the point P, and
, are spherical coordinates of the mass element dm .
The potential of compensating masses located between the surfaces
),(11 tt , ),(22 tt , (8)
may be defined in similar way by the integral
2
0 0
22
1
sin)(
t
t
I ddrd
l
rGPV , (9)
with a variable density
),,( r . (10)
With the definitions (4) and (9), total topographic / isostatic correction is expressed by the
sum
)()()( PVPVPV ITTI , (11)
and any linear geodetic functional (Moritz, 1980; Neyman, 1979) of the potential (11) can be
written in the form
)()()( I
P
T
P
TI
P VLVLVL . (12)
So, by applying the traditional spherical approximation (Heiskanen and Moritz, 1967), we get
the attraction of topographic and compensating masses:
ITI
P
T
P
TI
P
TI AAV
r
V
r
V
r
A
, (13)
which is nothing else but the effect of the potential (11) into gravity disturbances:
)()( PAPg TITI . (14)
In similar way we get the effects of the potential (11) into gravity anomalies
P
TI
TI
P
TI
P
TI
TI r
PVPg
r
PV
r
PVPg )(2)()(2)()(
, (15)
and into geoid heights as well:
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
108
P
TI
TI
PVPN
)()( , (16)
where P is normal gravity at the point P.
By analogy with the traditional disturbing potential T, we must require that zero-degree
harmonic is absent in the potential (11). This requirement is achieved if the sum of topographic
and compensating masses is equal to zero. As a result, we come to the condition
0sin
2
0 0
22
2
1
2
1
dddrrdrr
t
t
r
r
c
, (17)
which must be taken into account in computations of topographic-isostatic corrections.
Spherical approximation of the topographic / isostatic potential
Now we will apply the next traditional assumptions:
the figure of the geoid is the sphere of mean Earth’s radius
constRr 1 ; (18)
the topographic masses have constant density
constc . (19)
Also we will use the well-known Airy-Heiskanen model with local isostatic compensation
(Moritz, 1990) that leads to next additional assumptions:
the compensation masses have constant density, which is equal to density jump at the
crust-mantle boundary (Moho boundary):
const ; (20)
without the topographic and compensating masses, the (normal) Earth’s crust have the
(normal) thickness
0D , (21)
and bounded by the geocentric radii
constrRDRtconst 11 . (22)
We use these conditions for the deriving of the Moho boundary and the integration over
geocentric distance. Next, more adequate conditions will be considered for the developing of the
corresponding algorithm, which allows to use measured values of densities and Moho depths.
Moho boundary from zero mass condition
With the above assumptions, the zero mass condition (17) may be expressed especially for the
case of local compensation as
0
22
22
t
DR
r
R
c drrdrr . (23)
Considering (23) as a third degree equation regarding 2t , we get
3
33
2
3
2 )( RrDRt c
. (24)
This expression gives the geocentric distance of Moho boundary, obtained in the frame of
Airy-Heiskanen model in the spherical approximation. Note here, that by representing 2t and 2r
in the forms
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
109
dDRt 2 , hRr 2 , (25)
substituting (25) into (24) and applying Taylor linearization, we get the expression
h
DR
Rd c
2
, (26)
which coincides exactly with that used in (Rummel et al., 1988; Tsoulis, 2001) for
investigations of spectral properties of topographic / isostatic potential.
Integration over geocentric distance
Next, we replace the integration over , by the integration over spherical distance and
azimuth in (4), (11). With the assumptions (18) – (22) the potentials become
2
0 0
22 sin)(
r
R
cT ddrd
l
rGPV , (27)
2
0 0
22 sin)(
t
DR
I ddrd
l
rGPV . (28)
After the integration over the variable r, we come to
0
2
0
2
2
2
)( ddvrGPV r
R
P
cT
, (29)
0
2
0
2
2
2
)( ddvrGPV t
DR
P
I
, (30)
where
sin)cosln()1cos3()cos3(),( 2 xxxvv , (31)
cos21 2 xx
r
l
P
, (32)
Pr
rx . (33)
The function (31) has no any singularity within the interval ],0[ . With 1x it follows
immediately from (31):
0),()0,( xvxv . (34)
In the case 1x the function (31) gives
sin
2
sin)1cos3(2
2
sin2
2
sin2ln)1cos3(),1( 22
v , (35)
and, therefore
0),1( v , (36)
0),1(lim)0,1(
0
vv . (37)
By differentiating (29), (30) with respect to Pr , we get in accordance with (13):
0
2
0
2)( ddarGPA r
RPcT
, (38)
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
110
0
2
0
2)( ddarGPA t
DRPI
, (39)
where
),(sin),(
3
xvxxaa
. (40)
Again, the function (40) has no singularity within the interval ],0[ . With 1x we get
directly:
0),()0,( xaxa . (41)
With 1x the function (40) transforms to
),1(
2
cos),1(
va , (42)
and taking into account (36) and (37) we see that
0),1( a , (43)
1)0,1( a . (44)
As a result, we see that the function (40) has the jump at )0,1( x . The functions (31)
and (40) are shown in the Figure 1. for 1x .
),1( ,),1( av
30 60 90 120 150 180
-1
0
1
2
Fig. 1. Functions ),1( v (simple curve) and ),1( a (bold curve).
Integration over spherical template compartments
Now we divide the spherical Earth’s surface onto blocks in such a way that the geocentric
distance (3) of the Earth’s physical surface and the densities c , may be taken as a constant
within each block. If k-th block is bounded by the azimuths 21 , and the spherical distances
21, , we can write
constc ),( , constr ),(2 , ],[ 21 , ],[ 21 . (45)
Also we have from (24) and (45):
const ),( , constt ),(2 , ],[ 21 , ],[ 21 . (46)
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
111
These assumptions allow performing the integration within a block in (29), (30), (38) and
(39). As a result, the following expressions are valid for a block:
2
1
2
2
)(
2
r
R
P
c
k
T wrGPV , (47)
2
1
2
2
)(
2
t
DR
Pk
I wrGPV
, (48)
2
1
2)(
r
RPc
k
T brGPA , (49)
2
1
2)(
t
DRP
k
I brGPA
. (50)
In these formulas
12 , (51)
)cosln(cossincos)cos(
3
2),( 22
xxxww , (52)
),(),( 2 xwxxbb , (53)
and the quantities x and are defined by the expressions (32), (33) respectively.
The functions (52), (53) have not a singularity at 0 :
)12)(1(1
3
1),(lim)0,(
0
xxxxwxw
, (54)
)1(1
3
1),(lim)0,( 2
0
xxxxbxb
. (55)
Moreover, these functions have non-singular first order derivatives with respect to :
0),(),(
00
xbxw . (56)
Figure 2 shows the functions (52), (53) with 1x .
),1( ),,1( bw
30 60 90 120 150 180
-0.5
0.0
0.5
1.0
1.5
2.0
Fig. 2. Functions ),1( w (simple curve) and ),1( b (bold curve).
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
112
As a result, we can get potentials of topographic and compensating masses as well as
corresponding attractions by the summation over all compartments
k
k
TT PVPV )()( , (57)
k
k
II PVPV )()( , (58)
k
k
TT PAPA )()( , (59)
k
k
II PAPA )()( . (60)
Derived expressions realize application of spherical mathematical template for computation
of 3 dimensional integrals for topographic and isostatic potentials and corresponding attractions. In
principle, such approach is well known in gravimetry and physical geodesy. For example, classical
handbook by W.A.Heiskanen and Н.Moritz (1967) contains detailed consideration of application
of a mathematical template for computation of topographic / isostatic corrections in plane
approximation.
Conclusions
Both above considered approaches to computation of topographic / isostatic corrections are
based on application of mathematical templates (spherical or planar) constructed around a point of
interest. Closed expressions for the potential and the attraction obtained for each compartment of a
template provide good practical flexibility of the discussed technique. In fact, because the
requirements on constant densities c and are essential strictly just within a compartment,
we see the possibility to use observed density data without any modifications of above derived
expressions (47) – (50) or (82) – (85). Moreover, seismically observed Moho depth data can be
used also within each compartment.
Using the ETOPO1 digital terrain model (Fig. 3 and 4) we get two sets of the terrain
reductions, one based on topographic surface data, second one – on the combination of the ice
shield data and topography (Fig. 3–6 see at the color paste between pages 112 and 113).
From the Fig. 5 we can really fill how big the differences between the ice shield and
topography at the Antarctic continent.
As a result two sets of the Fay anomaly were calculated. In the next step on the frame of well-
known remove-restore technic the EGM2008 global gravity model was removed from the resulted
sets of Fay anomalies. Using the sequential multipole analysis the residual gravity field was
approximated by the sets of multipoles of different degree and order. But this is a topic of another
paper.
With gridded data we can provide simple and effective interpolation of corresponding values
onto centers of a template compartments. Usually, regular height data are available in the form of
digital terrain models (DTM) with various resolutions. Much probably that data of other types are
not available as corresponding grids. In this case general prediction techniques should be applied
to irregular data for a grid creation. Next, we should note that choice of a template steps over
azimuth and spherical (or horizontal) distance seems as practically important problem.
Indeed, very small steps will produce essential computation time expenses whereas too large
steps will violate the conditions (45), (46) or (80), (81). In this view we should note that template
steps must be agreed with smallest size of used grids. In addition, it may be necessary to perform
grid densification in small vicinity of a computation point. It is obvious that application of the
spherical template lead to more consistent results.
Finally, we should note that some modification of classical mathematical template was
proposed by R.Forsberg (1984), who used division onto rectangular blocks in accordance with
D.O. Marchenko: PARCULARITY OF THE GEOID DETERMINATION AT THE ANTARCTIC REGION
113
DTM structure. However, potentials of rectangular prism were derived in (Forsberg, 1984) only as
plane approximations of the potentials (4), (9). In addition, singularities may appear for some
positions of computation point regarding DTM nodes. With this respect we can conclude that the
technique based on rectangular prism potentials has not essential advantages in comparison with
the techniques based on classical mathematical templates.
References
Forsberg, R. (1984) A study of terrain reductions, density anomalies and geophysical inversion
methods in gravity field modeling. Reports of the Department of Geodetic Science and Surveying, Ohio
State University, Report No. 355.
Heiskanen, W.A., and Moritz, H. (1967) Physical Geodesy. W.H. Freeman, San Francisco.
Moritz, H. (1980) Advanced Physical Geodesy, H. Wichmann, Karlsruhe, 1980.
Moritz, H. (1990) The Figure of the Earth. Theoretical Geodesy and Earth’s Interior, Wichmann,
Karlsruhe, 1990.
Neyman, Yu.M. (1979) Variational Method of Physical Geodesy. Nedra: Moscow. (In Russian).
Rummel, R., Rapp, R.H., Sünkel, 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 Surveying, Ohio State University, Report No. 388.
Tsoulis, D. (2001) A comparison between the Airy/Heiskanen and the Pratt/Hayford isostatic
models for the computation of potential harmonic coefficients // Journal of Geodesy, No. 74, p. 637–
643.
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