A methodology for improving gravity-gradient measurement data exploits the constraints imposed upon the components of the gravity-gradient tensor by the conditions of integrability needed for reconstruction of the gravitational potential. These constraints are derived from the basic equation for the gravitational potential and from mathematical identities that apply to the gravitational potential and its partial derivatives with respect to spatial coordinates.

Consider the gravitational potentialφ in a Cartesian coordinate system {x1,x2,x3}. The ith component of gravitational acceleration is given by

(where i = 1, 2, or 3) and the (α,β) component of the gravity-gradient tensor is given by

where α = 1, 2, or 3 and β = 1, 2, or 3). The aforementioned constraints are such that the components of the gravity-gradient tensor are not independent of each other. In particular, it is easily shown that the gravity-gradient tensor is symmetrical and has a zero trace; that is,

Γαβ = Γβα and Γ11+ Γ2233 = 0.

Hence, if one measures all the components of the gravity-gradient tensor at all points of interest within a region of space in which one seeks to characterize the gravitational field, one obtains redundant information. One could utilize the constraints to select a minimum (that is, nonredundant) set of measurements from which the gravitational potential could be reconstructed. Alternatively, one could exploit the redundancy to reduce errors from noisy measurements.

A convenient example is that of the selection of a minimum set of measurements to characterize the gravitational field at n3 points (where n is an integer) in a cube. Without the benefit of such a selection, it would be necessary to make 9n3 measurements because the gravity-gradient tensor has 9 components at each point. It has been shown that when the constraints are applied to the measurement points in an appropriately chosen sequence, the number of measurements needed to compute all 9n3 components is only n3 + n2 + 3n .

The problem of utilizing the redundancy to reduce errors in noisy measurements is an optimization problem: Given a set of noisy values of the components of the gravity-gradient tensor at the measurement points, one seeks a set of corrected values — a set that is optimum in that it minimizes some measure of error (e.g., the sum of squares of the differences between the corrected and noisy measurement values) while taking account of the fact that the constraints must apply to the exact values. The problem as thus posed leads to a vector equation that can be solved to obtain the corrected values.

This work was done by Igor Kulikov and Michail Zak of Caltech for NASA's Jet Propulsion Laboratory. NPO-30536

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Reducing Errors by Use of Redundancy in Gravity Measurements

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This article first appeared in the June, 2004 issue of NASA Tech Briefs Magazine.

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