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Report 23
Application of the Planetary Geodesy Methods (the Geoid Theory)
for the Reconstruction of the Earth’s Interior Structure in the Western
Antarctic
Rudolf Greku, Ksenya Bondar, Victor Usenko
Institute of Geological Sciences, National Academy of Sciences of Ukraine,
Kiev, 55B, Gonchara St., 01054, Kyiv, Ukraine, e-mail: satmar@svitonline.com
Abstract
The algorithm for calculation of differential geoid heights and anomalous harmonic densities inside the Earth is represented in the paper. Expansion of external gravitational potential in series of spherical functions is used. The application of geopotential topography and anomalous harmonic densities for studing of structural features of regional deep sections is illustrated. The structure of the Scotia Arc Region with a technique named by us as the gravimetric tomography method is considered along typical latitudinal and longitudinal cross-sections, and also on maps of the differential geoid for different ranges of spherical harmonics.
1. Theoretical background
Density of the Earth’s interior can be presented
as a sum of normal and
anomalous components:
,
where normal density 
determines
normal gravity potential of the Earth; disturbing potential is caused by anomalous
density .
The global gravity models
describe the disturbing potential of the Earth which depends on spherical
coordinates of the point on its surface. It can be represented
as Newton`s integral
,
where G – gravitational constant, l – distance from the investigated
point to the surface of the Earth, v – elementary volume. So, there is
a linear dependence between 
, where
N is a linear operator.
Hence, the inverse gravitational problem can be formulated in the following way:
the operator N-1 is unique if V is determined in the whole 3-dimensional
space [Moritz, 1990]. The kernel of the operator N-1 consists of
set of possible distributions
of density _0 inside closed contour S (within the Earth), which generate
ground external potential.
The uniquely defined inverse Newton`s operator
provides harmonic density .
Thus, the solution could be found in the way of summation of definite harmonic
density and the density of ground potential _0.
The densities of ground potential characterize features of spherical stratification inside the Earth. Local heterogeneity is described by anomalous harmonic density _h.
2. The algorithm for calculation of the disturbing potential (or geoid height)
The geoid height above reference ellipsoid is found from
the well-known formula
,
where 
km – radius of the Earth,
_nm and snm – normalized
coefficients of external spherical harmonics of gravity potential, _ – longitude
of the investigated point, _ – polar distance of the investigated point,
Pnm(cos_)
– Legendre polynomial, z – number of harmonics to the 360th inclusive.
The formula taken from [Shabanova, 1962] was used for calculation of normalized spherical functions:
The necessary set of formules for calculation of any spherical function can be received on the base of following ones
3.The algorithm for calculation of anomalous harmomic densities
The solution of
the inverse gravity problem is given by Moritz [1990] under condition that
distribution of density is a continuous function which can
be approximated
uniformly with a system of polynomials.
The density as a function of spherical coordinates can be expansed in series
of spherical harmonics
The coefficients fnm (=anm or bnm) are arbitrary and can be represented as polinomials
Having excluded a general solution, which
corresponds to the densities of ground potential, the expression for anomalous
harmonic densities as a series
of internal
spherical harmonics is received
,
where 
, Vnm – coefficients
of external gravity potential,
or 
,
M – mass of the Earth.
Thus, external spherical harmonics are used for determination of internal spherical harmonics. The internal spherical harmonics are used to receive distribution of positive and negative density inhomogeneities, which does not change external gravity potential as their total mass is equal to zero.
The final formula
is
,
Corresponding number of harmonics z is taken to calculate the density _h
on the depth (R-r).
4.Estimation of the depth of disturbing layer according to the number of harmonic
The obtained above density anomalies obviously are situated in different depths. However the mathematical methods used in physical geodesy though suppose but have no evident connection with the real geological structure of the Earth. Therefore a following question is remained: what layers of the Earth are responsible for disturbing gravitational potential.
In our method an assessment of the disturbing
layer’s depth is computed
by a known harmonic function in physical geodesy in the case of the potential
of the internal masses confined by a sphere
,
where r is the distance from the sphere surface down to the disturbing mass,
is the distance from the center of
the sphere up to the disturbing mass, R is the radius of the sphere,
is
the Legendre polynomial of
nth degree,
is the central angle
between R and .
Calculations was carries
out from nmin=2. At the right part of the expression the normalizing
coefficient (2n+1)1/2 was considered. If 
=0,
Pn . Under this condition for specified values of r, and
R corresponding
n was found.
Obviously, for large n the series 
converge weakly. That is why
restrictions are included into the procedure of calculation. Maximal degree
nmax is defined
under
condition
,
where coefficient 0.1 means 10% error from r.
Relationship between harmonic degrees
n and depths r of disturbing layers is shown on the bilogarithmic diagram (Fig.
1). Main boundaries of the lithosphere
are shown in accordance with the Bullard’s density model of the Earth.
The program allows to compute values of the disturbing gravity potential
in terms of heights of the geoid, values of harmonic anomalies in units of
g/cm3 and values
of upper cover depths of disturbing layers of the Earth. Spherical coefficients
of two geoid models OSU91A and EGM96 are used.
Figure 1. Relationship between harmonic degrees n and depths r of the disturbing layers of the Earth. Value n is the sum of harmonicas in a range from degree 2 up to n. Depth r corresponds to upper cover of the disturbing layer, which thickness is considered from the center of the Earth
5. Examples of density structure in the Scotia Arc region
Deep structure and
geodynamics of the Scotia Arc and adjacent provinces within limits of 48S-66S
and 80W-10W are submitted with the EGM96 gravity geoid model.
The distribution of density inhomogeneities of the Earth is displayed along
the Scotia Sea’s central 58S latitudinal vertical cross-section (Fig.
2) and on 100 km and 1 km lateral levels with the spatial resolution of 30
km also (Fig. 3).
The images of differential anomalies (relatively of homogeneous deep layers)
as three-dimensional surfaces show a detailed distribution of masses in the
upper layers of the lithosphere, geometry and sizes of density inhomogeneities,
their displacement in depth under influence of dynamic processes, and correlation
of subsurface bodies with the bottom topography also.
Figure 2. Maps of differential geoids topography corresponding disturbing layers with indicated depths of the upper cover. Lighter tint is more dense structure. The area of increased density is noted in depth of 100 km with epicenter 50S, 47W. It is a root part of the really Scotia body. On smaller depths (1 km) a structural differentiation of the Scotia Sea and correlation with the bottom relief are increased.
Figure 3. Vertical cross-section of the geopotential along 58S latitude.
The roots of the South Sandwich Island ridge (1) are sloped to the Scotia
Sea side and are immersed into depths 25-30 km. The SS Trench (2) is traced
up to depth of 20 km. A fragment (3) of a joint area of the Shackleton F.Z.
and the Western Scotia Ridge is observed at 57W.
References
Moritz H (1990) The Figure of the Earth.Theoretical Geodesy and
the Earth`s Interior. Wichmann, Karlsruhe.
Shabanova A I (1962) Calculation on ECVM «Ural-1» of characteristics
of gravity field applying gravity anomalies in spherical functions (in Russian)
J CNIIGAiK 145: 77-81.
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