CA2104180C - Laplace gravity gradiometer - Google Patents
Laplace gravity gradiometerInfo
- Publication number
- CA2104180C CA2104180C CA 2104180 CA2104180A CA2104180C CA 2104180 C CA2104180 C CA 2104180C CA 2104180 CA2104180 CA 2104180 CA 2104180 A CA2104180 A CA 2104180A CA 2104180 C CA2104180 C CA 2104180C
- Authority
- CA
- Canada
- Prior art keywords
- disc
- accelerometers
- horizontal
- gravity
- components
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired - Lifetime
Links
Classifications
-
- G—PHYSICS
- G01—MEASURING; TESTING
- G01V—GEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
- G01V7/00—Measuring gravitational fields or waves; Gravimetric prospecting or detecting
Landscapes
- Physics & Mathematics (AREA)
- Life Sciences & Earth Sciences (AREA)
- General Life Sciences & Earth Sciences (AREA)
- General Physics & Mathematics (AREA)
- Geophysics (AREA)
- Force Measurement Appropriate To Specific Purposes (AREA)
- Testing Of Balance (AREA)
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.
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.
Description
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.
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.
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 ' CA 02104180 1998-OS-11 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 -S-and 2 of radially oriented horizontal axis accelerometers mounted on a horizontal disc 6 along X and Y axes, respectively. Accelerometer pairs 1 and 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.
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:
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 ' CA 02104180 1998-OS-11 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 -S-and 2 of radially oriented horizontal axis accelerometers mounted on a horizontal disc 6 along X and Y axes, respectively. Accelerometer pairs 1 and 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.
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:
Claims (3)
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.
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.
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.
Priority Applications (4)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA 2104180 CA2104180C (en) | 1993-08-16 | 1993-08-16 | Laplace gravity gradiometer |
| AU74552/94A AU7455294A (en) | 1993-08-16 | 1994-08-16 | Laplace gravity gradiometer |
| PCT/CA1994/000442 WO1995005614A1 (en) | 1993-08-16 | 1994-08-16 | Laplace gravity gradiometer |
| US08/823,833 US5734104A (en) | 1993-08-16 | 1997-03-25 | Laplace gravity gradiometer |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA 2104180 CA2104180C (en) | 1993-08-16 | 1993-08-16 | Laplace gravity gradiometer |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CA2104180A1 CA2104180A1 (en) | 1995-02-17 |
| CA2104180C true CA2104180C (en) | 1999-08-10 |
Family
ID=4152154
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA 2104180 Expired - Lifetime CA2104180C (en) | 1993-08-16 | 1993-08-16 | Laplace gravity gradiometer |
Country Status (3)
| Country | Link |
|---|---|
| AU (1) | AU7455294A (en) |
| CA (1) | CA2104180C (en) |
| WO (1) | WO1995005614A1 (en) |
Families Citing this family (11)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US5969352A (en) * | 1997-01-03 | 1999-10-19 | Mds Inc. | Spray chamber with dryer |
| US5922951A (en) * | 1997-06-11 | 1999-07-13 | The Broken Hill Proprietary Company Ltd. | Gravity gradiometer |
| AU735871B2 (en) * | 1997-06-11 | 2001-07-19 | Fugro Finance Ag | Gravity gradiometer |
| US5962782A (en) * | 1997-06-11 | 1999-10-05 | The Broken Hill Proprietary Company Limited | Gravity gradiometer accelerometers |
| AUPR571301A0 (en) * | 2001-06-15 | 2001-07-12 | Bhp Billiton Innovation Pty Ltd | Airborne gravity gradiometers |
| JP4362766B2 (en) * | 2002-03-06 | 2009-11-11 | ロッキード マーティン コーポレーション | Gravity deviation meter |
| US7954375B2 (en) * | 2009-01-21 | 2011-06-07 | General Dyamics Advanced Information Systems, Inc. | Inclined axis gravity gradiometer |
| GB201003355D0 (en) * | 2010-03-01 | 2010-04-14 | Moir Christopher I | A method and apparatus for the measurement of earth's gravity and gravity gradient |
| CN104793258B (en) * | 2015-04-29 | 2017-05-17 | 中国地震局地球物理研究所 | Vertical gravity gradient measurement system |
| CN105005088B (en) * | 2015-04-29 | 2017-03-29 | 中国地震局地球物理研究所 | Vertically to gradiometry method |
| CN116774303B (en) * | 2023-06-26 | 2024-05-07 | 中国自然资源航空物探遥感中心 | Method and device for locating field source boundary and computer readable storage medium |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| DD269456A1 (en) * | 1988-02-24 | 1989-06-28 | Akad Wissenschaften Ddr | HEAVY GRADIOMETER FOR THE DETERMINATION OF ALL COMPONENTS OF THE HEAVY DUTY RADIATOR |
-
1993
- 1993-08-16 CA CA 2104180 patent/CA2104180C/en not_active Expired - Lifetime
-
1994
- 1994-08-16 WO PCT/CA1994/000442 patent/WO1995005614A1/en active Application Filing
- 1994-08-16 AU AU74552/94A patent/AU7455294A/en not_active Abandoned
Also Published As
| Publication number | Publication date |
|---|---|
| AU7455294A (en) | 1995-03-14 |
| WO1995005614A1 (en) | 1995-02-23 |
| CA2104180A1 (en) | 1995-02-17 |
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Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| EEER | Examination request | ||
| FZDC | Correction of dead application (reinstatement) |