Transmission Coefficient Matrices Research Articles

Azimuth‐dependent tuning of seismic waves reflected from fractured reservoirs

Reservoirs with thickness less than the seismic wavelength can still contain significant amounts of hydrocarbons. Such layers exhibit a tuning effect which involves the interference of reflected waves from the top and bottom of the reservoir. Natural fractures in such reservoirs can play an important role in determining fluid flow, which makes the density and orientation of fractures of great interest. In the presence of one or more sets of aligned vertical fractures, the amplitude of reflected waves at nonzero offset varies with azimuth; hence, the tuning effect will vary with azimuth. For wavelengths much greater than typical fracture spacing, equivalent medium theory allows such a vertically fractured layer to be modeled as a monoclinic layer with a plane of mirror symmetry parallel to the layer. The variation in reflection and transmission coefficients with incidence and azimuthal angle for a thin vertically fractured layer can be expressed in terms of the horizontal slowness, automatically accounting for the change of angle with azimuth for rays propagating through the layer and for the tuning effect which occurs for layers with thickness of the order of the wavelength. For low enough frequency (or equivalently, thin enough layers), approximate expressions for the reflection and transmission coefficient matrices and transmitted amplitudes are derived. These expressions demonstrate explicitly that all reflected pulses and all converted transmitted pulses have the same shape as the time derivative of the incident pulse, whereas for thicker layers, distinct reflections from the top and bottom of the layer are evident, particularly for small angles of incidence. When these reflections interfere, significant changes in pulse shape with azimuth are found which result from differences in the azimuthal variation of reflection coefficient from the top and bottom of the layer due to propagation effects in the layer.

General Optimization Approach To a Frequency-Domain Inversie Problem of a Stratified Bianisotropic Slab

An inverse problem for a stratified slab of a general bianisotropic medium is considered in the frequency domain. Considering obliquely impinging plane waves, imbedding equations and Green's functions equations for the reflection and transmission coefficient matrices and the internal fields are derived through the concept of wave-splitting. Two different optimization approaches, based on the imbedding method and the Green's functions method respectively, are applied to reconstruct the constitutive parameters as functions of the position using band-limited reflection and/or transmission data from different angles of illumination. Exact and explicit expressions for the gradients are derived, which result in a one order faster computation compared with numerical perturbations of the objective functionals. The reconstruction algorithm (based upon a conjugate gradient method) is tested with both clean and noisy data.

Electromagnetic reflection and transmission for a stratified bianisotropic slab

Electromagnetic reflection and transmission for a time-harmonic electromagnetic plane wave normally incident on a stratified bianisotropic slab are considered. A general scheme for obtaining the reflection and transmission coefficient matrices for a stratified slab of the general bianisotropic material is described and implemented numerically. >

Lg-Wave Modelling For the North Sea

SUMMARY Theoretical seismograms for inhomogencous 2-D media are computed via summation of surface wave modes and use of transmission coefficient matrices, in an attempt to explain the observed Lg-wave barrier within thc continental structure of the North Sea. Where other scientists have been able to trcat Lg-wave propagation in the North Sea for one frequency, many surface wave modes and a complicated structural model of the North Sea grabens, our method takes into account many frequencies, and is consequentiy limited to a few significant modes. and a relatively simple structural model.

Interaction of electromagnetic waves with general bianisotropic slabs

An efficient, elegant, and systematic formulation technique which, combining Fourier transform with matrix analysis methods, is suitable for problems related to radiation by dipole or other sources in the presence of an arbitrarily general stratified anisotropic medium has been recently developed. This technique is adapted further extended to allow the presence of general bianisotropic media described by four tensors with no limitations on their elements. Two specific applications pertaining to some canonical problems of fundamental importance are included to exemplify the method and demonstrate its usefulness: radiation by an arbitrarily oriented elementary electric dipole source located in the vicinity of a general bianisotropic slab, either grounded or ungrounded, leading to the expressions of the dyadic Green's function of the structure, and reflection and transmission of an arbitrarily polarized plane wave incident upon such a slab, leading to closed-form concise expressions for the reflection and transmission coefficient matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of elastic wave propagation in stratified fluid-filled porous media for interwell seismic applications

The analytical solution for seismic wave propagation associated with a point force in a fluid-filled porous medium is developed. The point force solution is applied to solve the boundary value problem of seismic wave propagation in a stratified poroelastic medium. The coupled Biot vector wave equations are expressed in cylindrical coordinates and expanded in Fourier series with respect to azimuth. The resulting equations are transformed to the wave-number domain using the Hankel transform method. Following this analysis, one set of three coupled partial differential equations associated with fast compressional, slow compressional, and vertically polarized shear waves is derived. The unknowns are the radial and vertical displacements associated with the solid frame motion as well as the fluid pore pressure. A separate partial differential equation associated with waves whose particle motion is polarized in horizontal planes (SH waves) is derived as well. The general solution of the three coupled differential equations is obtained by the Kupradze method. This solution of the Biot’s motion equations in the wave-number domain leads directly to closed form expressions for the vector wave displacement and the pressure produced by a point force in a poroelastic unbounded medium. In order to develop the solution of a point force in the presence of a stratified porous medium, the displacement-stress matrix, the pressure, and the vertical component of the displacement of the fluid in its relative motion versus the solid for different regions are expressed in terms of upgoing and downgoing waves and unknown wave coefficients. The wave coefficients are determined by applying boundary conditions of continuity of displacements, pressure and stresses across each layer interface, and the radiation conditions at infinity. To determine the unknown wave coefficients, a method that consists in expressing the kernels of the Hankel transform integrals in terms of factorization of upgoing and downgoing wave amplitudes in each layer is used. These factorizations are based on the generalized reflection and transmission coefficient matrices, which are formed recursively, from one layer boundary to the next, including all of the reflection/conversion/transmission properties of the layered medium. Their factorization method allows the field within each layer above or below the source to be determined once the field in the medium containing the source is known. The final equations provide a complete description of the field throughout the layered medium. The particular form of the equations makes possible the simultaneous evaluation of the response at a number of detector locations for a number of different source positions in a borehole for interwell seismic applications. Numerical model results demonstrate the validity of this theoretical development for predicting spectral responses associated with porosity and permeability effects. The seismic pressure response of a thin gas-saturated porous layer was analyzed. The results inferred that the gas-saturated porous layer strongly attenuates the waveforms observed by detectors within the layer. Alternatively, large multiple reflections and converted waves from the layer are observed by detectors in the water-saturated porous formation.

A generalized reflection-transmission coefficient matrix and discrete wavenumber method for synthetic seismograms

Expressions for displacements on the surface of a layered half-space due to point force are given in terms of generalized reflection and transmission coefficient matrices (Kennett, 1980) and the discrete wavenumber summation method (Bouchon, 1981). The Bouchon method with complex frequencies yields accurate near-field dynamic and static solutions. The algorithm is extended to include simultaneous evaluation of multiple sources at different depths. This feature is the same as in Olson's finite element discrete Fourier Bessel code (DWFE) (Olson, 1982). As numerical examples, we calculate some layered half-space problems. The results agree with synthetics generated with the Cagniard-de Hoop technique, P-SV modes, and DWFE codes. For a 10-layered crust upper mantle model with a bandwidth of 0 to 10 Hz, this technique requires one-tenth the time of the DWFE calculation. In the presence of velocity gradients, where finer layering is required, the DWFE code is more efficient.

A method of full wave analysis with improved stability

A new version of a previously published method of full wave analysis is presented. The purpose of the procedure is to avoid the numerical instability which in some cases is encountered when the original version is used. In the old method the reflection and inverse transmission coefficient matrices were recursively calculated starting from the top of the layer. It is found that better stability is obtained, if the reflection coefficients are calculated as before, but the transmission coefficients are computed starting from the bottom of the layer and proceeding upwards in the direction of the incident wave.

A simple method for obtaining reflection and transmission coefficients and fields for an electromagnetic wave in a horizontally stratified ionosphere

A simple method for obtaining the reflection and transmission coefficients and fields for an electromagnetic wave propagating vertically in a horizontally stratified ionosphere is introduced. The ionosphere is divided into a great number of thin layers, and the boundary conditions of the electric and magnetic fields are applied at all layer interfaces. A recursive formalism is derived, which gives the altitude dependence of the reflection and inverse transmission coefficient matrices starting from the top of the layer. It is then shown how these results can be used in calculating the height dependence of a wave corresponding to any incident polarization. Test results are also presented in order to demonstrate the applicability of the procedure.

Reciprocity Relations in Light Propagation through a Multilayer Birefringent System

A method of calculating 2×2 reflection and transmission-coefficient matrices for a multilayer birefringent system, developed for use in a plane stratified magnetoplasma, is applied to an optical system. The diagonal matrix elements refer to the direct reflection and transmission coefficients of the two characteristic or normal modes, and off-diagonal elements refer to inter-mode coupling between them. For waves normally incident on the plane stratified system, in the positive or negative z directions, corresponding reflection and transmission-coefficient matrices, R± and T±, are defined in terms of suitably normalized wave fields. A reciprocity theorem is established wherein it is shown that the matrices R+ and R− are symmetric, and the matrix T+ is the transpose of T−, provided that the dielectric tensors of the layers composing the system are either (a) all symmetric, or (b) all gyrotropic (magnetoactive), having a common principal axis that is generally the external magnetic-field direction. Optically inactive, isotropic layers may be included in either class, but care must be taken in defining characteristic modes for such layers. Some applications are discussed, including the feasibility of constructing light valves.

A Reciprocity Theorem for Magnetoionic Modes

An arbitrary plane slab is considered within a magneto plasma whose parameters vary in a direction perpendicular to the (horizontal) plane of stratification, the plane of the slab faces. 2 × 2 reflection and transmission coefficient matrices are defined for the slab in terms of up‐going waves incident from below, Ru and Tu, and in terms of downgoing waves incident from above, RD and TD. The matrix elements compare amplitudes of the wave fields of the characteristic magnetoionic modes, so normalized that in regions of negligible absorption the squares of the amplitudes are proportional to the mean Poynting vector. The theorem that is proved states that the matrix Tu is the transpose of the matrix TD and that the matrices Ru and RD are symmetric. Some applications are discussed.

Coupling processes in the D- and E-regions at low and very low frequencies—I Frequencies less than critical: a study of the daytime ionosphere

Use is made of a generalized numerical method of thin film optics to derive 2 × 2 reflection and transmission coefficient matrices of horizontal ionospheric slabs, the top ends of which become progressively higher. The matrix elements are expressed in terms of characteristic magneto-ionic modes, off-diagonal elements representing inter-mode coupling terms. Propagation processes in the frequency range 10–200 kHz are followed by presenting the various matrix elements graphically as functions of the ionospheric slab height. In the VLF range direct (gradient) reflections of the ordinary wave and the Z-trace reflection processes are competitive, but in the LF range the Z-trace is the main process reflecting ordinary wave energy. The ‘coupling echo’ whereby the upgoing ordinary wave is converted to downgoing extraordinary at the X = 1 level (when Z ≈ Z c ), is almost non-existent, but there is a strong ‘ Z coupling echo’ produced by the reflection at the X = 1 + Y level of an extraordinary wave generated near the X = 1 level by an upgoing ordinary wave. The same process taken backwards, with upgoing extraordinary returning as ordinary is, by reciprocity, of equal magnitude. The ‘inverse Z-transmission’ process, with upgoing extraordinary being coupled into upgoing ordinary at the X = 1 level is very weak. It is shown that a wave of arbitrary polarization, incident on the ionosphere from below, will tend to maintain its initial polarization well into the ionosphere, even though the characteristic local wave-polarization may have changed considerably. In the upper LF range, where the X = 1 + Y level is virtually the only reflection region, a number of simple relations are shown to exist between the various transmission and reflection coefficient matrix elements. The problem of nomenclature of characteristic magneto-ionic modes at frequencies ‘less than critical’ is discussed.

Plasma-Filled Waveguide with Axial Magnetization. II. Scattering by a Section of Finite Length

The problem of mode conversion and scattering by a finite-length section of magnetoplasma-filled waveguide is formulated in matrix notation. This formulation becomes increasingly exact as the dimensions of the matrices (i.e., the number of modes considered) increases. The desired reflection- and transmission-coefficient matrices are readily evaluated with a digital computer. Computations of TE010-mode reflection coefficients indicate that good accuracy can be obtained with a fairly small (∼10) number of modes. Large errors result from considering only the incident (TE010) mode, however.