LAPLACE GRAVITY GRADIOMETER

GRADIOMETRE DE PESANTEUR LAPLACE

English Abstract

A gravity gradiometer which derives the vertical component T zz of the gravity
gradient tensor out of the two measured horizontal components T xx and T yy
based on
Laplace equation: T zz = - (T xx + T yy) comprises a single disc, and two
pairs of
radially-oriented horizontal axis accelerometers oppositely-mounted on such
disc
along respective X and Y axes. Each pair provides, respectively, T xx T x1 + T
X2 and
T yy=T y1, + T y2 where T X1 and T X2 are the respective outputs of the X axis
accelerometers and T y1, and T y2 are the respective outputs of the Y axis
accelerometers. The gravity gradiometer provides all five independent
components of the gravity gradient tensor based on the above Laplace equation
and tensor symmetry as follows: One of the above pairs of horizontal axis
accelerometers further comprises tangentially oriented horizontal
accelerometers capable of providing the T xy component of the gravity gradient
tensor, and an accelerometer module consisting of two horizontal
accelerometers aligned with the radially and tangentially oriented combination
is mounted at a location on the Z axis of the disc above or below the disc to
provide the T zx and T zy, components by substracting the values of T x and T
y ,
respectively, at that location from average values of T x and T y at the disc
plane.

Claims

CLAIMS
1. A gravity gradiometer for obtaining the vertical component T zz of the
gravity
gradient tensor out of the two horizontal components T xx and T yy based on
Laplace
equation T zz = - (T xx + T yy) comprising:
a. a single disc; and
b. two pairs of radially-oriented horizontal axis accelerometers oppositely-
mounted
on said disc along respective X and Y axes, each pair providing, respectively,
T xx = T x1 - T x2 and T yy = T y1 - T y2 where T x1 and T x2 are the
respective outputs of the X
axis accelerometers and T y1 and T y2 are the respective outputs of the Y axis
accelerometers.
2. A gravity gradiometer for obtaining all five independent components of the
gravity gradient tensor based on the Laplace equation T zz = - (T xx + T yy )
and tensor
symmetry, comprising:
a. a single disc; and
b. two pairs of radially-oriented horizontal axis accelerometers oppositely-
mounted
on said disc along respective X and Y axes, each pair providing respectively,
T xx = T x1 - T x2 and T yy = T y1 - T y2 where T x1, and T x2 are the
respective outputs of the X
axis accelerometers and T y1 and T y2 are the respective outputs of the Y axis
accelerometers;
wherein one of the pairs of horizontal axis accelerometers further comprises
tangentially
oriented horizontal accelerometers to provide by substraction the T xy
component of the
gravity gradient tensor, and further comprising an accelerometer module
consisting of
two horizontal accelerometers aligned with the radially and tangentially
oriented
combination mounted at a location on the Z axis of the disc above or below the
disc to
provide the T zx and T zy components by subtracting the values of T x and T y,
respectively,
at said location from average values of T x and T y at the disc plane.
3. A gravity gradiometer as defined in claim 1 or 2, wherein the disc is
rotatable
around the Z axis.

Description

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LAPLACE GRAVITY GRADIOMETER
This invention relates to a gravity gradiometer which obtains
components of the gravity gradient tensor using the Laplace equation and
tensor symmetry with horizontal accelerometers only.
Background of the invention
Existing commercial gravity gradiometers depend on measurement of
gravity gradients along axes inclined 45 degrees to the vertical (umbrella
configuration). Examples of these are the Bell gravity gradient survey system
GGSS of Bell-Textron of Buffalo, N.Y. which operates at room temperature,
and an experimental University of Maryland gradiometer which requires
cryogenic temperatures.
Gravity gradients at 45 degrees to the vertical are more difficult to
measure since compensation for a large gravity component (G cos 45 ~) is
required. This involves springs, which are subject to non-linearities,
hysteresis, fatigue, tares, inter-atomic slippage, etc. The above Bell GGSS
model uses three rotating discs, each populated with four tangentially-
oriented
single axis pendulous accelerometers.
Furthermore, gradiometer signals are measured in a relatively strong
aircraft motion acceleration noise field, with the vertical components
typically
several times higher than the horizontal components.
Statement of the invention
Applicant has found that there is no need to measure anomalous
gradients on a background of strong aircraft sub-vertical gravity
accelerations
directly, since the full tensor can be derived from horizontal gradients using
Laplace equation and the tensor symmetry.

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Horizontal gradients can be measured more simply than gradients in the
vertical directions by such means as pendulum - based accelerometers such as
Bell VII or XI models of Bell-Textron of Buffalo, N.Y.). A pendulum is
stabilized by the force of gravity, and is, therefore, a reliable, sensitive
and
frequently used inertial element.
A first embodiment of the gravity gradiometer in accordance with the
present invention comprises two pairs of radially-oriented horizontal axis
accelerometers oppositely-mounted on a support , such as a single disc along
respective X and Y axes, each pair providing, respectively, TXX TXI + Txz and
Tri Ty, + Ty2 where Tx, and Tx2 are the outputs of the X axis accelerometers
and TY, and Ty2 are the outputs of the Y axis accelerometers.
A second embodiment of the gravity gradiometer in accordance with the
present invention provides all five independent components of the gravity
gradient tensor using the Laplace equation and tensor symmetry. In this
embodiment, one of the two pairs of horizontal axis accelerometers further
comprises tangentially oriented horizontal accelerometers providing the TXy
component of the gravity gradient tensor. In addition, an accelerometer module
consisting of two horizontal accelerometers aligned with the above radial and
tangential combination is mounted at a location on the Z axis of the disc
above
and/or below the disc to provide the T~ and TZY components by subtracting the
values of TX and Ty at that location respectively, from average values of TX
and
TY at the disc plane. Both embodiments include signal processing means for
combining the outputs of the accelerometers in order to derive corresponding
gravity gradients.
Brief Description of the Drawings
The invention will now be disclosed, by way of example, with reference
to the accompanying drawings in which:
Figure 1 illustrates an embodiment of a gravity gradiometer which
provides the vertical component TZZ of the gravity gradient from measurement

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of horizontal components Txx and TYY and
Figure 2 illustrates an embodiment of a gravity gradiometer providing
all five independent components of the gravity gradient tensor from
measurement of horizontal components.
Detailed Description of Preferred Embodiments
Before proceeding with the description of the preferred embodiments,
let us provide the following well known definitions:
Basic Formulas
Gravity Gradient is a second derivative of Gravity Potential T. It is
represented
by the second-order nine-component symmetric tensor T;~.
TXX TXY TXZ
TiJ TYx TYY TYz
TZX TZY TZZ
On and above the earth surface the value of its in-line
(diagonal) components conforms to a well known Laplace equation:
TXX+TYY+T~=0
from which follows:
Tzz = - (Txx + TYY)
Thus we can obtain vertical component out of the two horizontal components.
By virtue of gradient tensor symmetry
T;~ = T~;
it is clear that only five of the nine components are independent (which is a
well known theorem). Therefore, in order to describe fully the tensor it is
sufficient to measure two in-line (diagonal) components and three independent
cross-components. None of these has to be a vertical component.
Sensor Geometry
In a first embodiment of the invention illustrated in Figure 1, the vertical
gravity gradient TZZ is obtained through Laplace equation TZZ = - (Txx + TYY)
from two horizontal gradients Txx and TYY combining the outputs of two pairs 1

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and 2 of radially oriented horizontal axis accelerometers mounted on a
horizontal disc 6 along X and Y axes, respectively. Accelerometer pairs 1 and
2
provide T,~ and Tyy.
In a second embodiment of the invention illustrated in Figure 2, a11 five
independent components of the gravity gradient tensor can be obtained from
horizontal gradients based on Laplace equation and tensor symmetry. In this
embodiment, one of the pairs of radially-oriented horizontal accelerometers
(pair 1) is replaced with a pair 3 of radially and tangentially oriented
horizontal
accelerometers 3 providing the TXy component of the gravity gradient sensor.
In addition, an accelerometer module 4 consisting of two horizontal
accelerometers aligned with the above radial and tangential combination is
mounted at a location on the Z axis above or below the disc to provide the TZX
and TZY components by subtracting the values of TX and Ty respectively, at
that
location from average values of TX and TY at the disc plane.
As noted previously, both embodiments include signal processing
means for combining the outputs of the several accelerometers in order to
derive corresponding gravity gradients. The signal processing means, as those
skilled in this art will appreciate, may be of a standard off the-shelf
variety.
The accelerometer disc may be rotated around the vertical axis in order
to narrow the signal bandwidth and reduce noise thus increasing
signal-to-noise ratio, which is a standard industry practice, as for example
in
Bell GGSS gravity gradiometer.
The second embodiment can obtain the full tensor using a single
stationary or rotating horizontal disc configuration rather than three discs
in a
45 degree umbrella orientation such as employed in Bell GGSS, or three pairs
of spring accelerometers in the same configuration like in the above mentioned
University of Maryland cryogenic gradiometer. This simplification results in
lower noise as well as a decrease in complexity and cost.

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Although the invention has been disclosed, by way of example, with
reference to preferred embodiments, it is to be understood that other
alternatives are also envisaged within the scope of the following claims:

Representative Drawing