Propagator Matrices Research Articles

Computational implementation of the polarization propagator method

A computational implementation of the polarization propagator method is outlined. The program is constructed so that, in principle, there are no limitations either on the size of the basis set or on the size of the particle-hole excitation space. The lists of two-electron integrals and propagator matrices are both assumed to be too large to be kept in primary storage. A list of symbolic matrix elements is generated and the individual terms which contribute to the matrix elements are reordered so that the list of two-electron integrals can be processed sequentially. Timings for the various steps of the program are given for CO 2 as a test case. It is shown that a major part of the computational effort goes into the construction of the B(2) matrix and that the CPU and I/O times are comparable.

Calculation of complete theoretical seismograms in vertically varying media using collocation methods

Summary. We present a method for calculating complete theoretical seismograms in earth models whose velocity, density and attenuation profiles are arbitrary piecewise-continuous functions of depth only. A form of attenuation valid for low loss situations is included by allowing the seismic velocities to be complex, and frequency is also allowed to be complex to avoid wraparound problems in the time-domain seismograms. Solutions for the stressdisplacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. A seismic source is applied at the Earth’s surface and a radiation condition is applied at depth. The resulting two-point boundary value problem for the expansion coefficients is solved by a collocation technique which works best for those cases that other methods, e.g. propagator matrices, work most poorly, i.e. highly evanescent solutions. Solutions for the expansion coefficients are obtained in the depth, frequency and horizontal wavenumber domain. Phase velocity filtering may be effected at this point by restricting the portion of the frequency-wavenumber plane in which solutions are sought. The transformed strain tensor at depth is formed by taking linear combinations of the solutions. This strain tensor is transformed back into the space and time domain by successive application of a Bessel transform, a fast Fourier transform, and by multiplication of the time signal by a growing exponential to remove the exponential decay introduced by the use of complex frequency. The strain tensor is contracted with a seismic moment tensor, and a reciprocity relation for Green’s functions is used to obtain displacements at the Earth’s surface caused by a buried moment tensor source.

Evolution of the polarization in codirectional and contradirectional optical couplers

Jones and Mueller calculus for both forward- and backward-wave optical couplers is presented. By using Kronecker product algebra, the relationship between the system matrices of the amplitude, the coherency, and the polarization vectors is derived. The relationship between the corresponding transfer matrices is also established. Explicit expressions are given for the Mueller matrix of uniform two-mode devices supporting hermitian and skew-hermitian coupling. For uniform devices the eigenvalues and eigenvectors of the propagator matrices of the Jones and Stokes vectors are evaluated. Linearly tapered and chirped optical couplers are analyzed, and solutions are given for the transfer matrix of the modal amplitude vector.

Vertical displacements from a rectangular fault in layered elastic-gravitational media.

In previous papers (RUNDLE, 1980, 1981 a), the techniques needed for calculation of the displacements due to various internal sources of strain embedded in layered linear elastic-gravitational continua have been developed. The method involves straightforward solution of the coupled elastic-gravitational field equations, and makes use of a matrix technique for propagating the solution from one layer to the next. As in the free oscillation problem, the six quantities propagated include two displacement kernels, two normal traction kernels and a kernel each for the gravitational potential and its gradient minus a multiple of the vertical displacement. These propagated quantities lead to 6×6 matrices.The method was used to numerically compute the vertical displacements arising from a 30° dipping point double-couple thrust source in an elastic-gravitational layer over an elastic-gravitational half-space (RUNDLE, 1981 a). A result from this work is that most of the gravitational effect for dislocations arises from terms associated with the surface acceleration g, as opposed to the terms involved with the gravitational constant G0 which represents self-gravitation effects. The present paper exploits this result by noting that for fault dislocation sources, the equations can be decoupled, reducing the matrices in the problem to 4×4 rather than 6×6. The advantage gained is significant reduction in computation time.After derivation of the reduced propagator matrices from the more general ones given in RUNDLE (1980, 1981 a), vertical displacements due to a rectangular, dipping fault in an elastic-gravitational layer over an elastic-gravitational half space are computed and compared with corresponding elastic solutions. The character of the elastic-gravitational displacements differs substantially from the elastic displacements when moduli contrasts become large.

Subduction zone dip angles and flow driven by plate motion

Kinematic models of the large scale flow in the mantle accompanying the observed plate motions are calculated by neglecting thermal buoyancy forces. The large scale flow is therefore determined by the mass flux imposed by the moving plates. The energy and momentum equations decouple, and with the assumption of a radially symmetric Newtonian viscosity, the flow accompanying the plate motions can be obtained using harmonic analysis and propagator matrices. The resulting flow models predict remarkably well the observed dips of subducted slabs if the flow extends into the lower mantle. The plates drag along a thick boundary layer which should be included in models of the heating of subducted slabs.

Asymptotic results for elastodynamic propagator matrices in plane-stratified and spherically-stratified earth models

Summary. Asymptotic results, applicable at high frequencies, are established for elastodynamic propagator or transfer matrices. We are concerned here with a single layer', by which we shall mean a region of a spherically- symmetric earth model in which physical properties vary smoothly with radius, or a region of a plane-stratified earth model in which physical properties vary smoothly with depth. Results are obtained for the spheroidal and toroidal oscillations of a solid spherical layer, the spheroidal oscillations of a spherical fluid layer, P-SV and SH waves in a plane solid layer and P waves in a plane fluid layer. Each of these results is given to first order in 0-l where w is the angular frequency.

Propagator and Green Matrices for Body Force and Dislocation

Summary The use of propagator and matrical elements matrices allows one to derive simultaneously the solutions to the problems of the excitation of a stratified medium by body forces and dislocations.

Propagator matrix formulation for earth currents of deep internal origin

The problem of the distribution of the poloidal electric field in the mantle, originating from the core-mantle boundary, is formulated in terms of propagator matrices suitable for calculations for a wide variety of piece-wise continuous models of the mantle. The elements of the propagator matrix for some interesting variations of the electric conductivity are presented.

The connection between elastodynamic representation theorems and propagator matrices

Abstract Two independent powerful methods in elastodynamics have been developed over the last few years: these are the use of representation theorems and propagator matrix techniques. The connection between these two approaches is derived in this paper. This is then used to show the equivalence of approximate techniques in elastic wave scattering based on representation theorems and a recent method based on matrix theory.

Propagator matrices for some geothermal problems

In problems of linear flow of heat in inhomogeneous media, the governing equation is a second order ordinary differential equation with variable coefficients. When transformed into a set of first order ordinary differential equations with variable coefficients, the problem becomes amenable to an elegant method of propagator matrices. In this paper the propagator matrices for some steady and unsteady heat conduction problems (including a case of heat generation by an irreversible first order reaction) having conductivity and heat generation functions as piecewise continuous, have been described.

Propagator matrix formulation of heat transfer in spherically stratified media

In the paper we have transformed the steady and unsteady conductive heat transfer differential equation in spherical coordinates into a system of first order differential equations and processed them by method of propagator matrices to extrapolate the known surface heat flux and temperature to any desired depth. The elements of propagator matrices have been summarised for various piecewise continuous conductivity and rate of heat generation functions to approximate inhomogeneities in the earth. In the analysis the rate of heat generation is either assumed to depend linearly upon temperature or correspond to first order irreversible chemical reactions.

Propagator matrices in elastic wave and vibration problems

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerica methods for their computation. When the dispersion relation is somem th order minor of the integral matrix it is possible to deal withm th minor propagators so that the dispersion relation is a single element of them th minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations forSH and forP-SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations forP-SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley-Hamilton theorem and the Lagrange-Sylvester, interpolation formula, are derived.

To: “Propagator matrices in elastic wave and vibration problems,” by Freeman Gilbert and George E. Backus, April, 1966, p. 326–332

In the article entitled “Propagator matrices in elastic wave and vibration problems,” by Freeman Gilbert and George E. Backus, April, 1966, p. 326–332, the second of equations (2.10) on page 327 should read differently.