Alford rotation, ray theory, and crossed-dipole geometry

Two generalizations of Alford rotation have been proposed for processing 2 x 2-component data containing nonorthogonal split shear waves: singular val ue decomposition (SVD) and eigenvector-eigenvalue decomposition (EED). Usin g a simple crossed-dipole synthetic model, we demonstrate that the physical model behind the EED method is invalid. It incorrectly assumes that a vect or 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 sour ce embedded in a homogeneous anisotropic medium instead excites all those m odes with particle motions that are not perpendicular to the direction of t he applied force, just as a vector point receiver detects all modes with po larizations that are not perpendicular to the receiver. Correctly generaliz ed 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 sou rce 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 exam ple. The new method, symmetric Alford diagonalization, differs from previou s methods by becoming unstable when applied over excessively short time win dows. This behavior is consistent with the physics of the problem: If nonor thogonal modes are allowed, then there is not enough information at a singl e time sample to determine a unique solution. Any method that can find a un ique solution at a single time sample, including both the EED and SVD metho ds, does not respect the physics of the nonorthogonal problem. Although the re appears to be no problem that recommends the EED method over standard Al ford rotation for its solution, the SVD method is still applicable to the p roblem it was originally designed to solve: orthogonal modes with an unknow n source or receiver orientation.