Gravimetric estimation of the E'otv'os components
📝 Abstract
The elements of the E"otv"os matrix are useful for various geodetic applications, such as the interpolation of the elements of the deflection of the vertical, the determination of gravity anomalies and the determination of geoid heights. A torsion balance instrument is customarily used for the determination of the E"otv"os components. In this work, we show that it is possible to estimate the E"otv"os components, at a point on the physical surface of the Earth, using gravity measurements at three nearby points, comprising a very small network. In the first part, we present the method in detail, while in the second part we demonstrate a numerical example. We conclude that this method is able to estimate the elements of the E"otv"os matrix with satisfactory accuracy.
💡 Analysis
The elements of the E"otv"os matrix are useful for various geodetic applications, such as the interpolation of the elements of the deflection of the vertical, the determination of gravity anomalies and the determination of geoid heights. A torsion balance instrument is customarily used for the determination of the E"otv"os components. In this work, we show that it is possible to estimate the E"otv"os components, at a point on the physical surface of the Earth, using gravity measurements at three nearby points, comprising a very small network. In the first part, we present the method in detail, while in the second part we demonstrate a numerical example. We conclude that this method is able to estimate the elements of the E"otv"os matrix with satisfactory accuracy.
📄 Content
Gravimetric estimation of the Eötvös components
G. Manoussakis(1), R. Korakitis(1) and P. Milas(2)
(1)Dionysos Satellite Observatory and (2)Higher Geodesy Laboratory, Department of Surveying, National Technical University of Athens, Iroon Polytechniou 9, Zografos 157 80, Greece, tel. +30– 2107722693, fax +30–2107722670.
Abstract
The elements of the Eötvös matrix are useful for various geodetic applications, such as the interpolation of the elements of the deflection of the vertical, the determination of gravity anomalies and the determination of geoid heights. A torsion balance instrument is customarily used for the determination of the Eötvös components. In this work, we show that it is possible to estimate the Eötvös components at a point on the Earth’s physical surface using gravity measurements at three nearby points, comprising a very small network. In the first part, we present the method in detail, while in the second part we demonstrate a numerical example. We conclude that this method is able to estimate the elements of the Eötvös matrix with satisfactory accuracy.
1 Introduction The Eötvös matrix is the second order derivative of the Earth’s gravity potential at a point P and is significant to several applications. For example, it plays an important role in the “Geodetic Singularity Problem”: if the determinant of the Eötvös matrix at point P is equal to zero, then it is rank deficient and this classifies point P as a singular point. This means that it is not possible to replace (pseudo)differentials of unholonomic coordinate systems, which are related to moving local astronomical frames, with differentials of holonomic coordinate systems (Grafarend, 1971, Livieratos, 1976). Another application of the Eötvös matrix is the determination of the deflection of the vertical at points on the Earth’s physical surface (Völgyesi, 1993). The elements of the Eötvös matrix which are involved are Wxx, Wxy and Wyy. A third application of the Eötvös matrix is the determination of the geoid undulation by an alternative solution for the astrogeodetic leveling (Völgyesi, 2001), and to determine the gravity anomaly with the help of the elements Wxz and Wyz (Völgyesi et al., 2005). The elements of the Eötvös matrix (except Wzz) are customarily determined by torsion balance measurements at point P. The appreciation of the Eötvös matrix is lately increased, since a large number of torsion balance measurements are carried out around the world (Völgyesi, 2015), in order to detect lateral underground mass inhomogeneities and geological fault structures. The aim of the present work is to develop a method for the estimation of the Eötvös components using a gravimeter instead of a torsion balance instrument. The components will be estimated at a chosen point S on the Earth’s physical surface, using gravity measurements at S and three nearby points, comprising a very small network. The proposed method will be described in detail in the next sections.
2 Methodology
2.1 Estimation of the values of the Eötvös components except Wxy
Let S be a point on the Earth’s physical surface with known geodetic coordinates,
gravity value and geometric height. The Earth’s gravity potential is expressed in a
local Cartesian system (x, y, z), which is centered at point S (point of interest), the z –
axis is perpendicular to the equipotential surface passing through point S pointing
outwards, the x – axis is tangent to the equipotential surface passing through point S
pointing North and the y – axis is tangent to the aforementioned equipotential surface
pointing East.
In addition, let A, B, C be three points in the neighborhood of point S (within a few
meters) with known local Cartesian coordinates and gravity values. Point A is taken
on the x – axis (y = 0), point B on the y – axis (x = 0) and point C on the z – axis (x =
y = 0), as in Figure 1.
Figure 1 The value of Wzz can be directly computed from the gravity measurements at points S and C (see (2.4)). For the other four Eötvös components at point S we proceed as follows:
A parametric vector equation for the actual equipotential surface of point S around
this point, expressed in the local Cartesian system, is
(2.1)
and the tangent vectors of the equipotential surface are
))
,
(
,
,
(
)
,
(
)
,
(
:
:
3
2
y
x
z
y
x
y
x
s
y
x
D
s
z x x W W s x s ,0,1 (2.2)
z y y W W s y s ,1,0 (2.3)
The value of Wzz at point S is obtained by:
C S C zz zz z g g
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