The Gravity Gradiometer instrument

Theory of FTG Measurement (by John Brett, Bell Geospace Inc.)

Introduction

Gravity Gradiometers have existed for over 100 years but until recently were only available on stationary platforms in very quiet environments. Through the US military, an initiative was undertaken in the 1970s to make a gravity gradiometer that would work on a moving platform. The only result from that initiative was a system developed by Bell Aerospace (now Lockheed Martin) and that system is now being used commercially by Bell Geospace, Inc. to provide high resolution, 3D Gravity Gradients for the oil & gas industry. This paper will discuss some of the problems associated with getting this system to work on a moving platform and how this system solved those problems.

The Invention

The Rotating Accelerometer Gravity-Gradiometer is an instrument which employs two pairs of opposing accelerometers mounted orthogonally on a continuously rotating platen. This configuration solves the two most important FTG measurement problems. First, in order to not be influenced by the vehicle accelerations, the scale factor of the opposing accelerometers must be precisely matched. Second, to eliminate the red noise (low frequency noise) of the individual accelerometers, the measured gradient signal must be shifted to a higher frequency. The rotating accelerometer scheme accomplishes both of these goals. The scale factor difference is modulated by the rotation frequency, which can be separated from the gradient measurement and used to adjust the scale factor of each pair. The gradient measurement is also modulated by twice the rotation frequency and therefore can be easily separated from the low frequency red noise. While the concept is relatively simple the engineering problems associated with making the instrument accurate to one part in 10^11 are formidable. Another engineering problem is the stabilization of the entire assembly. Since, any rotation rate is a true gradient and 10^-9 radians per second squared is equal to one Eotvos, the intended accuracy, stabilization needs to be very precise. In the gravity-gradiometer, now employed commercially by Bell Geospace, there are three assemblies of four rotating accelerometers (GGIs) These GGIs together with an advanced Gravity Measurement Assembly (GMA) are mounted on a single gyro stabilized assembly. This arrangement provides for continuous measurement of all five independent gravity-gradient tensor elements and the total gravity field.

Noise Reduction

Earlier, I indicated that the measurement of gravity gradient by opposing pairs of accelerometers eliminates the effect of the host vehicle acceleration. That statement would be true if the accelerometers were perfect instruments. However, in the real world such instruments are not perfectly linear. Although the non-linear coefficients are small, (less than 1 part in 10^6) they can cause noise due to host vehicle accelerations within the desired bandwidth. Unlike gravity measurements, this noise is not a direct measurement of host vehicle accelerations but instead are the various products of acceleration and the accelerometer non-linear coefficients. Therefore, if the coefficients are known and the host vehicle acceleration is accurately measured, the induced noise can be determined and eliminated. A post mission scheme is utilized in which the non-linear coefficients are determined for each 20-hour segment of any given survey. This is accomplished by multiplying the recorded accelerations of the correct order to the assumed coefficients. A technique that regresses on the value of each coefficient until the noise is minimized is then employed. To insure that this process does not distort the measured gradient elements, the noise is sampled from frequencies just beyond the useful gradient frequency band. The process is called High Rate Post Mission Compensation or HRPMC and has proven effective for host vehicle accelerations approaching 0.1g standard deviation. Two additional elements which can induce host vehicle acceleration noise are: (1) any misalignment of the combination of accelerometers within each GGI with respect to the plane of rotation and (2) any scale factor difference between the two accelerometer pairs. Both of these offsets are corrected just prior to each survey while still at sea. The misalignment is adjusted by offsetting one accelerometer in each GGI until host vehicle acceleration output is minimized. The scale factor of the accelerometer pairs in each GGI is adjusted by modulating the GGI rotation rate and adjusting one of the scale factors until the observed modulating frequency output is minimized. Gravity-Gradient measurements are very sensitive to near masses, which disturb the gravity field. Such masses include host vehicle structure and stores. Since such masses move with the host vehicle, it is necessary to calibrate and remove their influence from the measured data. The calibration is accomplished in a specially designed survey pattern. Any residual self-gradient can be removed by observing gradient data, which is fixed to the host vehicle motion. BGI gradiometer surveys are always conducted in an orthogonal pattern, resulting in many crossing points. These crossing points are used to remove bias drift in each gradiometer output and in the gravimeter data. It is at this point in the process that residual ship self-gradient is also removed. The inner element of the gyro-stabilized platform is continuously rotated at a constant rate. Such rotation aids in the separation of the residual self-gradient. This entire process is called Low Rate Post Mission Compensation LRPMC).

Calibration of Gradient Data

As earlier indicated, an angular rate produces a true acceleration gradient. Centrifugal acceleration = r x W^2. Differentiating with respect to r produces W^2, a true acceleration gradient. While this fact requires that the GGI instruments be precisely isolated from angular rates it also results in a simple calibration mechanism. By rotating the inner element of the platform at two precise rates we can calibrate each gradient instrument scale factor and separate the bias terms. The rotation rate can be easily accomplished by providing a precise torque to the output axis of the vertical rate-sensing gyro.

High Resolution Gravity

Integrating the gradient data for the higher frequencies and adding that to the low frequency gravity measurements develops a gravity measurement, which has the equivalent spacial resolution of the gradient signature and without the influence of host vehicle accelerations. We call this result, "Enhanced Gravity" or TZe.


Navigation

The navigation system utilizes the combination of DGPS data and the inertial sensor data (gyros and accelerometers contained within the FTG platform), in a 24 state Kalman filter. This combination produces exceptionally accurate navigation data, which is used for both position location and to provide Eotvos corrections for the gravity data. The fact that the FTG platform inner element is continuously rotated further improves the inertial data quality.


Summary

The Full Tensor Gravity-Gradiometer employed by Bell Geospace provides for high resolution gravity measurements on ships at sea. High survey speeds and rough sea conditions are easily tolerated without loss of data resolution. In addition the direct measurement of all five independent gradient tensor components provides significant additional data for interpretation and inversion. Future tests are planned to demonstrate similar utilization in airborne vehicles.

The FTG system measures changes to the gravity vector components, the gradients or spatial rates of change, in the gravity field. Unlike a conventional gravimeter, which measures only the magnitude of the gravity field, the FTG system acquires data from all directions. The gravity tensor consists of nine components which can be mathematically arranged in a 3x3 matrix of partial derivatives. Of these nine tensor components, four are redundant, leaving only five independent components.

Examples

The following examples aim at enhancing the different FTG components uses. Even if the vertical derivative Tzz is the more meaningful component, you will learn how to see and interpret the horizontal derivatives of the vertical component Tzx and Tzy, and even the horizontal components derivatives Txx and Tyy.

In each FTG response :

Tzz gradient data measures up-down changes in up-down gravity. Tzz represents the difference between the near and far response. It highlights all edges and is the easiest gradient to interpret directly. Geologic structure is usually evident in the data when large mass anomalies, such as salt, are present. Tzz gradient data combines Txx and Tyy gradients. It highlights all edges and is useful for understanding the approximate shape of the dominant mass anomaly.

Theoretical cube Theoretical ellipse Real salt dome
Tzz
 Tzz
 Tzz

Txx gradient data measures east-west changes in east-west gravity, whereas Tyy measures north-south changes in the north-south gravity. Txx and Tyy emphasizes north-south and east-west trending edges.

Txx
 Txx
 Txx
Tyy
 Tyy
 Tyy

Txz and Tyz gradient data delineates respectively the north-south and the east-west mass anomaly axis. It also helps show north-south and east-west trending edges. Notice that a positive mass anomaly gives rise to both positive and negative gradients values with the center of mass defined by the axes of inflection.

Txz
 Txz
 Txz
Tyz
 Tyz
 Tyz

You can click on the pictures below to have an overview of the 5 components for each example used in this page.

Theoretical cube Theoretical ellipse Real salt dome