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.