Evaluation of Isotropic Covariance Functions of Torsion Balance Observations

Abstract

Torsion balance observations in spherical approximation may be expressed as second-order partial derivatives of the anomalous (gravity) potential T:

$$ T_{13} = \frac{{\partial ^2 T}} {{\partial x_1 \partial x_3 }}, T_{23} = \frac{{\partial ^2 T}} {{\partial x_2 \partial x_3 }}, T_{12} = \frac{{\partial ^2 T}} {{\partial x_1 \partial x_2 }}, T_\Delta = \frac{{\partial ^2 T}} {{\partial x_1^2 }} - \frac{{\partial ^2 T}} {{\partial x_2^2 }} $$

where x ₁, x ₂, and x ₃ are local coordinates with x ₁ “east,” x ₂ “north,” and x ₃ “up.” Auto- and cross-covariances for these quantities derived from an isotropic covariance function for the anomalous potential will depend on the directions between the observation points. However, the expressions for the covariances may be derived in a simple manner from isotropic covariance functions of torsion balance measurements. These functions are obtained by transforming the torsion balance observations in the points to local (orthogonal) horizontal coordinate systems with first axis in the direction to the other observation point. If the azimuth of the direction from one point to the other point is α, then the result of this transformation may be obtained by rotating the vectors

$$ \left[ {\begin{array}{*{20}c} {T_{13} } \\ {T_{23} } \\ \end{array} } \right] and \left[ {\begin{array}{*{20}c} {T_\Delta } \\ {2T_{12} } \\ \end{array} } \right] $$

the angles α − 90° and 2(α − 90°) respectively.

The reverse rotations applied on the 2×2 matrices of covariances of these quantities will produce all the direction dependent covariances of the original quantities.