Alford rotation, ray theory, and crossed-dipole geometry

Abstract

Abstract Two generalizations of Alford rotation have been proposed for processing 2 × 2-component data containing nonorthogonal split shear waves: singular value decomposition (SVD) and eigenvector-eigenvalue decomposition (EED). Using a simple crossed-dipole synthetic model, we demonstrate that the physical model behind the EED method is invalid. It incorrectly assumes that a vector source aligned with the particle motion of an anisotropic pure mode will excite only that one mode. Ray theory shows that a vector point-force source embedded in a homogeneous anisotropic medium instead excites all those modes with particle motions that are not perpendicular to the direction of the applied force, just as a vector point receiver detects all modes with polarizations that are not perpendicular to the receiver. Correctly generalized Alford rotation synthesizes vector sources and receivers such that each component is perpendicular to all but one of the pure modes of the medium. Although this ray-theory result does not allow for the possibility of a source or receiver on a free surface and so is not yet completely general, it does apply to the idealized homogeneous crossed-dipole geometry of our example. The new method, symmetric Alford diagonalization, differs from previous methods by becoming unstable when applied over excessively short time windows. This behavior is consistent with the physics of the problem: If nonorthogonal modes are allowed, then there is not enough information at a single time sample to determine a unique solution. Any method that can find a unique solution at a single time sample, including both the EED and SVD methods, does not respect the physics of the nonorthogonal problem. Although there appears to be no problem that recommends the EED method over standard Alford rotation for its solution, the SVD method is still applicable to the problem it was originally designed to solve: orthogonal modes with an unknown source or receiver orientation.