Eigendecomposition of the two-point correlation tensor for optimal characterization of mantle convection

The two-point correlation tensor provides complete information on mantle convection accurate up to second-order statistics. Unfortunately, the two-point spatial correlation tensor is in general a data-intensive quantity. In the case of mantle convection, a simplified representation of the two-point spatial correlation tensor can be obtained by using spherical symmetry. The two-point correlation can be expressed in terms of a planar correlation tensor, which reduces the correlation's dependence to only three independent variables: the radial locations of the two points and their angular separation. The eigendecomposition of the planar correlation tensor provides a rational methodology for further representing the second-order statistics contained within the two-point correlation in a compact manner. As an illustration, results on the planar correlation are presented for the thermal anomaly obtained from the tomographic model of Su, Woodward & Dziewonski (1994 ) and the corresponding velocity field obtained from a simple constant-viscosity convection model Zhang & Christensen 1993 ). The first 10 most energetic eigensolutions of the planar correlation, which constitute an almost three orders of magnitude reduction in the data, capture the two-point correlation to 97 per cent accuracy. Furthermore, the energetic eigenfunctions efficiently characterize the thermal and flow structures of the mantle. The signature of the transition zone is clearly evident in the most energetic temperature eigenfunction, which clearly shows a reversal of thermal fluctuations at a depth of around 830  km. In addition, a local peak in the thermal fluctuations can be observed around a depth of 600  km. In contrast, due to the simplicity of the convection model employed, the velocity eigenfunctions exhibit a simple cellular structure that extends over the entire depth of the mantle and do not exhibit transition-zone signatures.