Tensor operator

Abstract: Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm (θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R 3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm (θ, φ) to the spherical harmonics of R 4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

Abstract: There are many occasions on which the magnetotelluric impedance tensor is affected by local galvanic distortion (channelling) of electric currents arising from induction in a conductive structure which is approximately two-dimensional (2-D) on a regional scale. Even though the inductive behavior is 2-D, the resulting impedance tensor can be shown to have three-dimensional (3-D) behavior. Conventional procedures for rotating the impedance tensor such as minimizing the mean square modulus of the diagonal elements do not in general recover the principal axes of induction and thus do not recover the correct principal impedances but rather linear combinations of them. This paper presents a decomposition of the impedance tensor which separates the effects of 3-D channeling from those of 2-D induction. Where the impedance tensor is actually the result of regional 1-D or 2-D induction coupled with local frequency independent telluric distortion, the method correctly recovers the principal axes of induction and, except for a static shift (multiplication by a frequency independent real constant), the two principal impedances. Also obtained are two parameters (twist and shear), which partially describe the effects of telluric distortion. It is shown that the tensor operator which describes the telluric distortions can always be factored into the product of three tensor sub-operators (twist, shear, local anisotropy) and a scalar. This product factorization allows assimilation of local anisotropy, if present, into the regional anisotropy. The method of decomposition is given in the paper along with a discussion of the improvements obtained over the conventional method and an example with real data.

Abstract: 1 Preliminaries.- 2 Spherical Harmonics.- 3 Differentiation and Integration over the Sphere.- 4 Approximation Theory.- 5 Numerical Quadrature.- 6 Applications: Spectral Methods.

Abstract: Addition theorems are described for spherical vector wave functions, under both rotations and translations of the coordinate system. These functions are the characteristic solutions in spherical coordinates of the vector wave equation, such as occurs in electromagnetic problems. The vector wave function addition theorems are based on corresponding theorems for the spherical scalar wave functions. The latter are reviewed and discussed. (auth)