Calculation Of Reflection Coefficients Research Articles

Errors and uncertainties in the ultrasonic pulse-echo reflectometry method for measuring acoustic impedance

Ultrasonic pulse-echo reflectometry provides a convenient method to measure the acoustic impedance of a layer of unknown material bonded to a substrate of another material whose acoustic properties are known. The amplitudes of echoes from the interface are used to calculate the interface reflection coefficient and, from this and a knowledge of the properties of the substrate material, the unknown impedance is obtained. The technique has potential for assessment of adhesive cure and measurement of mechanical moduli. The calculation is poorly conditioned in that small errors in raw echo data can result in unacceptably large errors in the estimate of the required impedance. This paper gives an analysis which demonstrates how bias and variance errors in raw data lead to bias and variance errors in the calculated reflection coefficient and how these lead to errors in impedance estimation which can be very much greater and which depend on the value of the coefficient of reflection at the interface. Experimental results are given which verify the analytical predictions of the error generation process. The effects of interface conditions such as transducer coupling film and paint coatings are considered quantitatively by a simulation of the wave system. The effects of substrate surface roughness are considered briefly in a series of experiments.

Secondary electron and positive ion emission from HTC superconductors by ion impact

We have measured the secondary electron emission (SEE) yield γ e and secondary positive ion yield γ + from high T c superconductors by ion impact at normal incidence, using a cylindrical Faraday cup, outside of which is surrounded by a shield, in vacuum of ∼10 −7 Torr. The bias V b was applied to the inner cylinder. For H + and Ar + impact on YBa 2Cu 3O 7 (YBCO), γ e and γ + reached constant values for V b > +25 and < − 90 V, respectively. The calculated reflection coefficient is much smaller than γ +, confirming that γ + is not due to the reflection of incident ions but secondary positive ions. It is found that γ e and γ + at saturation follow the electronic and nuclear stopping powers, respectively. It also appears that γ + is much larger than the calculated sputtering yield for H + impact.

Reflection coefficient calculation from marine high resolution seismic reflection (Chirp) data and application to an archaeological case study

Chirp sub-bottom profilers produce high-resolution images of the near-surface. An attribute of the sea-bed reflection in chirp data are fluctuations in polarity between adjacent traces. Two models are proposed and presented to explain this: the first incorporates changes in an acoustic impedance gradient at the sea bed; the second uses changes in the thickness of the uppermost sediment layer. Mixing of adjacent traces produces a consistent polarity for the sea-bed reflector. Reflection coefficients are calculated, using amplitude information derived from single-traces, and polarity information from trace mixing, with application to a marine archaeological case study. The reflection coefficient calculated for the top of a buried 18th century wooden wreck is -0.26.

Investigation of multislice RHEED calculations

A multislice program is developed to calculate RHEED intensities and compare them with the results of the standard algorithms for calculating RHEED intensities. In our approach the reflection coefficient for a range of k values is extracted from a single multislice calculation and the only errors are numerical and the error made in the multislice approximation. Two different methods for the calculation of reflection coefficients based on a reciprocal space approach have been compared. The accuracy of the multislice method is estimated for the one-beam case.

Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films

Fast computer algorithms for calculating the reflection coefficients of inhomogeneous films are developed. The applications of such algorithms are aimed at real-time monitoring and control of the growth of gradient index optical coatings. Inhomogeneous films with a refractive index varying along the direction normal to the surface are generally treated by dividing the film into a large number of homogeneous layers and using an iterative or transfer-matrix method to calculate reflectivities. We develop alternative methods. We show that the elements of the transfer matrix and the reflection coefficients can be written in terms of single, double, and higher-dimensional integrals over the films. In most cases it is sufficient to include only the first term, i.e., the single integrals, and the calculation of reflection coefficients is then between 1 and 2 orders of magnitude faster than that for the transfer-matrix and iteration methods. This gives the possibility of real-time determination of optical parameters of inhomogeneous layers for growth monitoring and opens up feedback control.

Imaging wooden artefacts using Chirp sources

This paper investigates the potential for imaging submerged and buried wooden artefacts in the marine environment using high-resolution seismic reflection techniques. Models to calculate theoretical reflection coefficients of 11 wood species buried in shallow marine sediments are developed. The models use estimates of acoustic impedance contrasts to predict the strength of reflections resulting from buried wooden artefacts. Traditionally, wooden wrecks are composed of oak, with lesser components of mahogany, pine and elm. Experimental results on wood samples from a sixteenth century oak wreck (the Mary Rose) are presented as a control on theory. Results indicate that wooden wrecks may be readily imaged by a suitable sub-bottom profiler operating under appropriate survey conditions. Particular reference is made to Chirp sonar, which transmits a frequency modulated pulse, providing high-resolution digital seismic data. Chirp data acquired over an eighteenth century buried oak wreck (the Invincible, Horsetail, East Solent) is presented. Calculations of reflection coefficients from reflection data acquired over the site support both the theoretically and experimentally derived data. © 1997 John Wiley & Sons, Ltd.

Design of waveguide-to-orotron-resonator transition with modified Bethe theory

The modified Bethe theory is applied to the waveguide-to-resonator transition, which is used for output coupling in an orotron oscillator designed for operation at 94 GHz. The finite thickness of the coupling hole between output waveguide and resonator is taken into account. The approach results in an equivalent circuit, which describes the reflection-type resonator, with respect to the output waveguide. Its equivalent circuit parameters are given in closed-form expressions. The calculated reflection coefficients are compared to experimental results. The assumed single-mode operation in the resonator is demonstrated by field-profile measurements.

Investigation of frequency-angular resonances of sound reflection coefficients from the bottom

Features of the behavior of exact frequency-angular dependencies of the reflection coefficients V(fd,tet) (f—frequency, d—thickness of a sediment layer, tet—angle of slid) from a structure and the acoustic properties of various types of the marine bottom are investigated. Angular and frequency resonances of reflection coefficients and change of position of resonance depending on a ratio of acoustic parameters of sediment layers and half-space are revealed. For exact calculations of a reflection coefficients, a matrix propagator was used. Resonant formalism was expanded on the layered elastic bottom case, in terms of the Breit–Wigner variables. The comparisons of resonance and exact calculations of reflection coefficients for two types of resonances show good agreement of the results. Calculations of exact and resonant reflection coefficients for real parameters of the shallow sea bottom was carried out. The relation between resonance position and width and parameters of environment was ascertained. [Work supported by RFFI Project N 96-02-18944.]

Energy band structure of Bi 12SiO 20 and Bi 12GeO 20 single crystals

The energy band structures of the Bi 12SiO 20 and Bi 12GeO 20 single crystals were calculated using a local pseudopotential method. An iteration procedure was carried out by use of increasing number of plane waves (up to 1500) until the required energy eigenvalue stability was less than 0.02 Ry. An agreement between the calculated reflection coefficient and the experimental optical reflection data was obtained. Moreover, a striking similarity of the Bi 12SiO 20 and Bi 12GeO 20 energy band structures was found. The local pseudopotential approach is suggested to be more effective than any nonlocal one in order to describe other complex single crystals.

Reflection coefficients of D, T ions with Maxwell energy distribution

The reflection coefficients of D, T ions in fusion plasma incident to the first wall of a fusion reactor are necessary for us to understand and evaluate the energy balance, fuel recycling in the fusion reactor operation. Due to different energy and different direction of D, T ions in fusion plasma, the calculation of reflection coefficients of D and T ions in fusion plasma is generally a heavy task. In this paper we will present a new method, which is based on the scaling property of ion transport, for calculating the reflection coefficients of Maxwell D, T ions of fusion plasma. A systematic and comprehensive calculation for D, T reflection coefficients has been completed for getting the scaling formulas, and some reflection coefficients of Maxwell D, T ions with different temperature are given.

The one-dimensional elastic wave equation: A finite-difference formulation for animated computer applications to full waveform propagation

A FORTRAN computer program for modeling full waveform propagation through a layered homogeneous one-dimensional medium is presented. The output synthetic seismograms are generated using a finite-difference approximation to the wave equation, as opposed to the more conventional approach of simply convolving the calculated reflection coefficient wavetrains with representative wavelets. The strength of the program in wavefront propagation is screen animated and can be captured on hard copy. The software is user-friendly, and was developed primarily as an instructional tool. Each of the finite-difference input parameters is explained in detail for those unfamiliar with the finite-difference theory.

Effect of Segmentation of Conductive Coatings on the Radio Wave Reflection from Solar Control Sheet Glass

Solar control glass often becomes an obstacle to television broadcast wave reception inside a room, because coated conductive films can reflect incident electromagnetic waves with frequencies lower than that of the plasma oscillation of the films. Results of calculation of reflection coefficients from theoretical equations showed that the radio wave reflection from solar control glass decreases by dividing the conductive film into discontinuous segments with a periodic array. The segmental effect has been examined experimentally. Experimental results for the system of periodic conductive segments fixed on soda lime silicate glass agreed very well with the theoretical results when the edge effect of the array was very small. It was found also the power reflection coefficients for aluminium foil segments of 25mm square were less than 0.05 below 200MHz.

Single-parameter expression for ions reflection from surfaces

Based on a detailed analysis of the Boltzmann equation for ion transport in solids, it has been shown that for low-energy ions incident on a heavy element target the distribution function of the ions can be determined by one single parameter, called the scaled transport cross-section, which was defined earlier [1]. This means that the transport quantities of different ion-target-energy combinations should be similar only when their scaled transport cross-section is the same. To test this significant conclusion, we undertook a set of systematic and extensive calculations of reflection coefficients using the improved bipartition model of ion transport. The systematic calculations include 3410 ion-target-energy combinations, namely H, D, T, He, Li, B, C, N, O, Ne ions with energy ranges from 10 eV to 1 MeV normally incident to C, Al, Cu, Mo, Ag, W, Au, Pb, U targets. The only restrictions isM1/M2<1/6. The calculations verify that particle and energy reflection coefficients present an excellent one-to-one correspondence to the scaled transport cross-section. Furthermore, based on the calculations, universal expressions for both particle reflection coefficients and energy reflection coefficients for normal ion incidence have been obtained by fitting the numerical data. By comparing the results calculated by the universal expressions with experimental and Monte Carlo data, it is shown that the expression can describe reflection coefficients well.

Displacement Structures of Covariance Matrices, Lossless Systems, and Numerical Algorithm Design

Low displacement rank theory underlies many fast algorithms designed for structured covariance matrices. Some of these have gained notoriety for their numerical instability problems, particularly fast least-squares algorithms. Recent studies have shown that instability is not inherent to fast algorithms, but rather comes from the violation of backward consistency constraints. This paper thus details the connection between covariance matrices of a given displacement inertia and lossless rational matrices, as well as the role of this connection in numerically consistent algorithms. This basic connection allows displacement structures to be parametrized via a sequence of rotation angles obtained from a lossless system. The utility of this approach is that, irrespective of errors in the rotation parameter set, they remain consistent with a positive definite matrix of a prescribed displacement inertia. This property in turn may be rephrased as meaningful forms of backward consistency in numerical algorithms. The rotation parameters then take the form of Givens or Jacobi angles applied to data, in contrast to classical approaches which directly manipulate dyadic decompositions of the displacement structure. The concepts are illustrated in popular signal processing applications. In particular, these connections lend clear insight into the stable computation of reflection coefficients of Toeplitz systems, and also serve to resolve the numerical instability problem of fast least-squares algorithms.

Analysis of aperture-coupled microstrip antenna and array with an airgap

The tunable characteristic of an aperture-coupled microstrip antenna with an airgap between the substrate and the ground plane is investigated. Numerical results on input impedance, SWR bandwidth, resonant frequency and far-field radiation patterns are given. Computed results of input SWR against frequency are compared with experimental data. The scan characteristics of an infinite array of aperture-coupled microstrip antennas with an airgap are also studied using a full-wave analysis. The periodic spectral Green&apos;s functions are derived by the Floquet theorem. The calculated reflection coefficients versus scan angle in an arbitrary scan plane and against frequency when scanning at the broadside direction are presented.

Generation of shear and compression waves in an inhomogeneous elastic medium

This paper considers an inhomogeneous solid sediment, whose properties vary continuously with depth, lying between two homogeneous media—an upper fluid layer representing the ocean and a semi-infinite, homogeneous, solid substrate. The problem considered is that of determining the reflection coefficient of a plane wave incident on the sediment from above. It is assumed that the shear modulus in the sediment is small compared with the bulk modulus. This results in a relatively simple pair of coupled equations governing the generation of shear and compression waves. The equations clearly demonstrate how a density gradient in the sediment results in the continuous generation of shear waves within the bulk of the medium, and not just at the interfaces between adjacent media. With a suitably chosen density variation in the sediment, the equations can be solved analytically. Exact solutions are derived for the isovelocity case, and are used to investigate the effect of a continuous density variation within the sediment, on the reflection of an incoming plane wave from the upper layer. These exact solutions are also compared with a simpler method for calculating reflection coefficient, in which density gradient terms are ignored and solutions for homogeneous media are matched simply by ensuring that the density is given its correct value at each interface. The shape of the density profile in the sediment has a significant effect on reflection coefficient if wavelength is comparable to the thickness of the sediment. The influence of the density profile decreases rapidly at higher frequencies, and the simple impedance matching technique is found to be adequate for wavelengths which are small compared with sediment thickness.

Eclogite-facies shear zones—deep crustal reflectors?

Strongly foliated eclogite-facies rocks in 30–150 m thick shear zones of Caledonian age occur within a Grenvillian garnet granulite-facies gabbro-anorthosite terrain in the Bergen Arcs of Norway. The predominant eclogite-facies mineral assemblages in the shear zones are omphacite + garnet + zoisite + kyanite in gabbroic anorthosite and omphacite + garnet in gabbro. Eclogite-facies rocks in shear zones are generally fine-grained; alternating omphacite/garnet- and kyanite/clinozoisite-rich layers define gneissic layering. A strong shape preferred orientation of omphacite, kyanite, and white mica (phengitic muscovite and/or paragonite) define the foliation. The anorthositic eclogites show omphacite b-axis maxima approximately normal to the foliation and c-axis girdles within the foliation plane. P-wave velocities ( V p ) determined at confining pressures to 600 MPa for samples from eclogite-facies shear zones range from 8.3 to 8.5 km s −1 and anisotropy ranges from 1 to 7%. The few samples with more pronounced anisotropy tend to be approximately transversely isotropic with minimum velocities for propagation directions normal to foliation and maximum velocities for propagation directions parallel to foliation. The fast propagation direction lies within the c-axis girdles (parallel to foliation) and the slow propagation direction is parallel to the b-axis concentration (normal to foliation) in samples for which omphacite crystallographic preferred orientation was determined. V p for the granulite-facies protoliths average about 7.5 km s −1. High calculated reflection coefficients for these shear zones, 0.04–0.14, indicate that they are excellent candidates for deep crustal reflectors in portions of crust that experienced high-pressure conditions but escaped thermal reactivation.

Two-Dimensional Synthetic Seismic and Log Cross Sections from Stratigraphic Forward Models

In an effort to fully use deterministic stratigraphic forward-modeling techniques in subsurface stratigraphic analysis, we developed a computer interface to routinely create synthetic logs and one-dimensional and two-dimensional (2-D) seismic responses from 2-D stratigraphic simulations. Each 2-D stratigraphic model can contain up to 200 timelines defining age-equivalent stratigraphic layers with laterally variable lithofacies and depths. Synthetic gamma ray, density and velocity responses are calculated for the simulated lithofacies using user-specified rock and fluid properties. Vertically incident synthetic seismograms are created using calculated reflection coefficients and user-defined input wavelets. Because stratigraphic simulations provide chronostratigraphy as a nown, log correlations and the interpolated seismic geometries follow timelines exactly. The power of this technique as an aid in sequence stratigraphic interpretations is shown from three examples: (1) a simulation of an idealized clastic system assuming constant clastic input and sinusoidal fourth-order and third-order sea level variations, (2) a detailed simulation of one third-order carbonate depositional sequence (lower-middle San Andres Formation) from the Northwest shelf, Permian basin, and (3) simulations of the Permian mixed clastic-carbonate infill of the Midland basin. Some general conclusions from these examples include the following: (1) seismic and well log-defined topsets, foresets, bottomsets, and related event terminations can be directly related to relative sea level fluctuations, (2) Exxon-type sequence boundaries (i.e., unconformities) are not necessarily seismically imageable, and their identification on well logs is not straightforward, and (3) lateral variations in amplitude related to lithofacies variations can be modeled. Synthetic logs and seismic sections from stratigraphic forward models such as these may be useful in constraining interpretations of subsurface data and thus aiding the prediction of reservoir and seal.

The nonlinear wave effects by a submerged horizontal plate to the vertical wall

Wave pressure reduction due to a submerged horizontal plate placed before a vertical wall was investigated both numerically as well as experimentally. Pressure-reduction due to the submerged plate for short period waves was clearly evident. For all the case studies, the optimal submergence depth was found to be equal to d = 0.20h rather than 0.45h, where h is the total water depth. However, at this depth, d = 0.20/;, the pressurereduction effect will vary according to wave conditions, the width of the plate, (l), and the distance of the plate away from the vertical wall (w). Furthermore, it was found that the wider the plates, the longer the waves that will be affected. Calculated reflection coefficients were found to be in satisfactory agreements with experiments for nonbreaking waves.

Acoustic propagation in anisotropic periodically multilayered media: A method to solve numerical instabilities

Acoustic propagation through thick composites has become a subject of intensive study due to their application to nondestructive evaluation. The anisotropic multilayered media are now usually studied by the propagator matrix formalism. Though this formalism is very convenient, it leads to numerical instabilities for thick composites at high frequencies. These numerical instabilities come from the combination of very high exponential terms which reduces the dynamics of the calculation. A very interesting case is the one of anisotropic periodically multilayered media. The method developed in this paper uses Floquet waves which correspond to the modes of an infinite periodically multilayered medium. They are linear combinations of the real waves propagating in each layer of the medium. The Floquet wave numbers are the eigenvalues of the propagation matrix of one period of the medium. The anisotropic periodically multilayered medium can then be considered as a dummy medium in which the Floquet waves propagate. High exponential terms can be avoided through a judicious choice of reference of the Floquet waves’ amplitudes. This method enabled us to calculate reflection coefficients up until 40 MHz, of thick composites of carbone/epoxy placed in water. Furthermore, it has permitted us to not have a limitation for a single layer of any given material, at any given frequency.