Reflection coefficients of P-SV waves in weak anisotropic media

We introduce the Thomsen anisotropic parameters into the approximate linear reflection coefficient equation for P-SV wave in weakly anisotropic HTI media. From this we get a new, more effective, and practical reflection coefficient equation. We performed forward modeling to AVO attributes, obtaining excellent results. The combined AVO attribute analysis of PP and PS reflection data can greatly reduce ambiguity, obtain better petrophysical parameters, and improve parameter accuracy.

Scattering of Water Waves by Dual Symmetric Inclined Floating Porous Barriers Using the DBEM

The scattering of normally incident water waves by two surface-piercing inclined perforated barriers in water with a uniform finite depth is investigated within the framework of linear water wave theory.Considering that thin barriers are zero-thickness,a novel numerical method involving the the coupling of the dual boundary element method(DBEM)with damping layers is applied.In order to effectively damp out the reflected waves,two damping layers,instead of pseudoboundaries are implemented near the two side boundaries of the computational domain.Thus,the modified linearized free surface boundary conditions are formulated and used for solving both the ordinary boundary integral equation as well as the hypersingular boundary integral equation for degenerate boundaries.The newly developed numerical method is validated against analytical methods using the matched eigenfunction expansion method for the special case of two vertical barriers or the inclined angle to the vertical being zero.The influence of the length of the two damping layers has been discussed.Moreover,these findings are also validated against previous results for several cases.After validation,the numerical results for the reflection coefficient,transmission coefficient and dissipation coefficient are obtained by varying the inclination angle and porosity-effect parameter.The effects of both the inclination angle and the porosity on the amplitudes of wave forces acting on both the front and rear barriers are also investigated.It is found that the effect of the inclination angle mainly shifts the location of the extremal values of the reflection and the transmission coefficients.Additionally,a moderate value of the porosity-parameter is quite effective at dissipating wave energy and mitigating the wave loads on dual barriers.

不均匀海底地形影响下柔性板与斜向来浪的相互作用

The present work analyzes the interaction of oblique waves by a porous flexible breakwater in the presence of a step-type bottom.The physical models for both scattering and trapping cases are considered and developed within the framework of small amplitude water-wave theory.Darcy’s law is used to model the wave interaction with the porous medium.It is assumed that the varying bottom extends over a finite interval,connected by a finite length of uniform bottom near an impermeable wall,and a semi-infinite length of bottom in the open water region.The boundary value problem is solved using the eigenfunction expansion method in the uniform bottom regions,while a modified mild-slope equation(MMSE)is used for the region with the varying bottom.Additionally,a mass-conserving jump condition is employed to handle the solution at slope discontinuities in the bottom.A system of equations is derived by matching the solutions at interfaces.The reflection coefficient and force on the breakwater and impermeable wall are plotted and analyzed for various parameters,such as the length of the varying bottom,depth ratio,angle of incidence,and flexural rigidity.It is observed that moderate values of flexural rigidity and depth ratio significantly contribute to an optimum reflection coefficient and reduce the wave force on the wall and breakwater.Remarkably,the outcomes of this study are assumed to be applicable in the construction of this type of breakwater in coastal regions.

Wave Reflection by Rectangular Breakwaters for Coastal Protection

In this study,we focus on the numerical modelling of the interaction between waves and submerged structures in the presence of a uniform flow current.Both the same and opposite senses of wave propagation are considered.The main objective is an understanding of the effect of the current and various geometrical parameters on the reflection coefficient.The wave used in the study is based on potential theory,and the submerged structures consist of two rectangular breakwaters positioned at a fixed distance from each other and attached to the bottom of a wave flume.The numerical modeling approach employed in this work relies on the Boundary Element Method(BEM).The results are compared with experimental data to validate the approach.The findings of the study demonstrate that the double rectangular breakwater configuration exhibits superior wave attenuation abilities if compared to a single rectangular breakwater,particularly at low wavenumbers.Furthermore,the study reveals that wave mitigation is more pronounced when the current and wave propagation are coplanar,whereas it is less effective in the case of opposing current.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.

孔隙率对矩形沟槽上垂直多孔屏障的波浪散射效应影响

The effect of porosity on surface wave scattering by a vertical porous barrier over a rectangular trench is studied here under the assumption of linearized theory of water waves.The fluid region is divided into four subregions depending on the position of the barrier and the trench.Using the Havelock’s expansion of water wave potential in different regions along with suitable matching conditions at the interface of different regions,the problem is formulated in terms of three integral equations.Considering the edge conditions at the submerged end of the barrier and at the edges of the trench,these integral equations are solved using multi-term Galerkin approximation technique taking orthogonal Chebyshev’s polynomials and ultra-spherical Gegenbauer polynomial as its basis function and also simple polynomial as basis function.Using the solutions of the integral equations,the reflection coefficient,transmission coefficient,energy dissipation coefficient and horizontal wave force are determined and depicted graphically.It was observed that the rate of convergence of the Galerkin method in computing the reflection coefficient,considering special functions as basis function is more than the simple polynomial as basis function.The change of porous parameter of the barrier and variation of trench width and height significantly contribute to the change in the scattering coefficients and the hydrodynamic force.The present results are likely to play a crucial role in the analysis of surface wave propagation in oceans involving porous barrier over submarine trench.

Improving seismic interpretation: a high-contrast approximation to the reflection coefficient of a plane longitudinal wave

Linearized approximations of reflection and transmission coefficients set a foundation for amplitude versus offset（AVO） analysis and inversion in exploration geophysics.However,the weak properties contrast hypothesis of those linearized approximate equations leads to big errors when the two media across the interface vary dramatically.To extend the application of AVO analysis and inversion to high contrast between the properties of the two layers,we derive a novel nonlinearized high-contrast approximation of the PP-wave reflection coefficient,which establishes the direct relationship between PPwave reflection coefficient and P-wave velocities,S-wave velocities and densities across the interface.（A PP wave is a reflected compressional wave from an incident compressional wave（P-wave）.） This novel approximation is derived from the exact reflection coefficient equation with Taylor expansion for the incident angle.Model tests demonstrate that,compared with the reflection coefficients of the linearized approximations,the reflection coefficients of the novel nonlinearized approximate equation agree with those of the exact PP equation better for a high contrast interface with a moderate incident angle.Furthermore,we introduce a nonlinear direct inversion method utilizing the novel reflection coefficient equation as forward solver,to implement the direct inversion for the six parameters including P-wave velocities,S-wave velocities,and densities in the upper and lower layers across the interface.This nonlinear inversion algorithm is able to estimate the inverse of the nonlinear function in terms of model parameters directly rather than in a conventional optimization way.Three examples verified the feasibility and suitability of this novel approximation for a high contrast interface,and we still could estimate the six parameters across the interface reasonably when the parameters in both media across the interface vary about 50%.

Ultrasonic characterization of modified Cr_2O_3 coatings by reflection coefficient spectroscopy

Pores,microcracks and density of plasma sprayed Cr2O3 coatings before and after high-intensity pulsed ion beam(HIPIB) irradiation were investigated using the ultrasonic reflection coefficient spectroscopy(URCS).The URCS was analyzed based on an acoustic transmission model for the multi-layered structure.The longitudinal velocity in the coatings was calculated from the experimental URCS,and the attenuation coefficient expression was deduced by comparing the experimental and numerical fitting amplitude spectral lines.The longitudinal velocity of as-sprayed Cr2O3 coating is 2 002 m/s,and increases to 2 099 and 2 148 m/s after being irradiated by HIPIB with 1 and 5 shots.Correspondingly,the factor A changes from 0.046 to 0.026 and 0.020 and n from 1.702 to 1.658 and 1.649 in the attenuation coefficient expression of α=Af n.It is observed that the surface morphology of Cr2O3 coatings changes from rough and porous to smooth and uniform with the increase of shot number,which accords with the ultrasonic analyses reasonably.The URCS seems to provide a convenient and nondestructive method to characterize surface modification of the plasma sprayed coatings.

Forward Modeling of the Relationship Between Reflection Coefficient and Incident Angle of the P Wave in a Coal Seam

Although the Zoeppritz equation is suitable for a single interface in a thick deposit, it has some limitations for composite reflection waves from both the floor and the roof of coal seams. Based on the ray model, the relationship of the overall reflection coefficient of composite reflection P waves, from coal seam versus incidence angle (AVO), is dis- cussed. The result shows that: 1) the overall reflection coefficient of composite reflection waves from coal seams is a negative value and is determined mainly by the lithology of roof and floor, which is different from the reflection coeffi- cient of a single interface; 2) if the incidence angle ranges from 0° to 6°, the reflection coefficient of composite waves of a coal seam does not change with the incidence angle and 3) if the incidence angle ranges from 6–60° , the reflection coefficient increases monotonically.

Experimental study of reflection coefficient and wave forces acting on perforated caisson

The reflection coefficient and the total horizontal forces of regular waves acting on theperforated caisson are experimentally investigated. The empirical relationship between reflection coefficient and the ratio of the total horizontal forces acting on the perforated caisson to those on solid vertical walls with the relative chamber width, relative water depth and porosity of perforated wall, etc. are given. Moreover, the results of the ratio of the total horizontal forces are also compared with formulas given by Chinese Harbour Design Criteria and Takahashi, which may be useful for the practical engineering application.

Seismic reflection and transmission coefficients of a single layer sandwiched between two dissimilar poroelastic solids

The seismic reflection and transmission characteristics of a single layer sandwiched between two dissimilar poroelastic solids saturated with two immiscible viscous fluids are investigated. The sandwiched layer is modeled as a porous solid with finite thickness. The propagation of waves is represented with potential functions. The displacements of particles in different phases of the aggregate are defined in terms of these potential functions. Due to the presence of viscosity in pore fluids, the reflected and transmitted waves are inhomogeneous in nature, i.e., with different directions of propagation and attenuation. The closed-form analytical expressions for reflection and transmission coefficients are derived theoretically for appropriate boundary conditions. These expressions are calculated as a non-singular system of linear algebraic equations and depend on the various parameters involved in this non-singular system. Hence,numerical examples are studied to determine the effects of various properties of the sandwich layer on reflection and transmission coefficients. The essential features of layer thickness, incident direction, wave frequency, liquidsaturation and capillary pressure of the porous layer on reflection and transmission coefficients are depicted graphically and discussed. The analysis shows that reflection and transmission coefficients are strongly associated with incident direction and various properties of the porous layer.

Calculation of Wave Reflection Coefficient for Perforated-Pipe Breakwater

The hollow-pipe perforated breakwater is of low reflection. In this paper the functions of reflection coefficients of both regular and random waves are theoretically derived, based on the concept of linear superimposition of reflected and incident waves and with the total flow rate continuity of integral form instead of the non-continuity of the boundary condition, and based on the concept of linear wave spectrum theory. Comparisons between theoretical results presented here and measurements of model tests show reasonable agreement.

Analysis of Directional Spectra and Reflection Coefficients in Incident and Reflected Wave Field

In this paper, the modified Bayesian method for the analysis of directional wave spectra and reflection coefficients is verified by numerical and physical simulation of waves. The results show that the method can basically separate the incident and reflected directional spectra. In addition, the effect of the type of wave gage arrays, the number of measured wave properties, and the distance between the wave gage array and the reflection line on the resolution of the method are investigated. Some suggestions are proposed for practical application.

Hydrodynamic Coefficient Investigation on a Partial Permeable Stepped Breakwater Under Regular Waves

Traditional breakwater takes the advantage of high protection performance and has been widely used.However,it contributes to high wave reflection in the seaside direction and poor water exchange capacity between open seawater and an inside harbor.Consequently,a partially permeable stepped breakwater(PPSB)is proposed to ensure safety and good water exchange capacity for an inside harbor,and a 3-D computational fluid dynamics(CFD)mathematical model was used to investigate the hydrodynamic coefficients using Reynolds-Averaged Navier-Stokes equations,Re-Normalization Group(RNG)k-εequations,and the VOF technique.A series of experiments are conducted to measure the wave heights for validating the mathematical model,and a series of dimensionless parameters considering wave and PPSB effects were presented to assess their relationships with hydrodynamic coefficients,respectively.With the increase in the reciprocal value of PPSB slope,incident wave steepness and permeable ratio below still water level(SWL),the wave reflection coefficient decreases.The wave transmission coefficient decreases with an increase in the reciprocal value of the PPSB slope and incident wave steepness;however,it increases with the increase in the permeable ratio below SWL.With increases in the reciprocal value of the PPSB slope,permeable ratio below SWL and incident wave steepness for relatively high wave period scenarios,the wave energy dissipation coefficient increases;however,it decreases slightly with increases in the incident wave steepness for the smallest wave period scenarios.Furthermore,simple prediction formulas are conducted for predicting the hydrodynamic coefficients and they are well validated with the related data.

Experimental Study on Reflection Coefficient of Curved Perforated Plate

A set of experiments is carried out in a towing tank to study the effects of the curvature of perforated plates on the wave reflection coefficient （Cr）. The curvature of a perforated plate can be changed by rotating a reference perforated plate about its origin point according to the parabolic equation y=-x2 A plunger-type wave maker is used to generate regular waves. The reflection coefficients are calculated using Goda and Suzuki’s （1976） method. The results are compared with those of vertical or sloped passive wave absorbers. The comparison shows that a perforated plate with a curved profile is highly efficient in terms of reducing the wave reflection coefficient. A correlation is established to estimate the reflection coefficient of curved perforated plates as a function of both flow and geometry characteristics.

A method of Robust low-angle target height and compound reflection coefficient joint estimation

It is always a challenging issue for radar systems to estimate the height of a low-angle target in the multipath propagation environment.The highly deterministic maximum likelihood estimator has a high accuracy,but the errors of the ground reflection coefficient and the reflecting surface height have serious influence on the method.In this paper,a robust es-timation method with less computation burden is proposed based on the compound reflection coefficient multipath model for low-angle targets.The compound reflection coefficient is es-timated from the received data of the array and then a one-di-mension generalized steering vector is constructed to estimate the target height.The algorithm is robust to the reflecting sur-face height error and the ground reflection coefficient error.Fi-nally,the experiment and simulation results demonstrate the validity of the proposed method.

Wave Reflection Coefficient Spectrum

The wave reflection coefficient frequency spectrum and directional spectrum for concrete face slope breakwaters and rubble mound breakwaters are investigated through physical model tests in the present study. The reflection coefficients of oblique irregular waves are analyzed by the Modified Two-Point Method (MTPM) proposed by the authors. The results show that the wave reflection coefficient decreases with increasing wave frequency and incident angle or decreasing structure slope. The reflection coefficient frequency spectrum and its variation with Iribarren number are given in this paper. The paper also suggests an empirical 3-dimensional reflection coefficient spectrum, i.e. reflection coefficient directional spectrum, which can be used to illustrate quantitatively the variation of reflection coefficient with the incident angle and the Iribarren number for oblique irregular waves.

Analysis of reflection coefficient for coated optical waveguide devices

From an ordinary condition,using a full three-dimensional model theory and an infinite perturbation expansion method,an exact solution of the reflection coefficient for the coated narrow stripe-geometry optical waveguide devices has been derived.All six components and the vector property of the electromagnetic field have been considered.The results are suitable for the symmetric and asymmetric waveguides.

Research on Thin Layer Structure Identification and Sedimentary Facies of Middle and Deep Layers Based on Reflection Coefficient Inversion—By Taking Dongying Formation of CFD Oilfield in Bohai Offshore as an Example

The sand layer B of Dongying Formation of CFD oilfield in Bohai offshore belongs to the middle deep layer of buried hill overlap deposit. Its reservoir distribution has the characteristics of large burial depth, thin thickness and rapidly lateral change. Because of low resolution of seismic data and overlying sand layer. It is difficult to identify and interpret the structure of sand layer accurately. The uncertainty of structure and reservoir restricts the fine development of B sand layer. In order to identify the top surface of reservoir effectively. The seismic data are processed by using the reflection coefficient inversion method. The results show that the inversion resolution of reflection coefficient is significantly higher than that of original data. The top surface of sand layer B and its overlying sand layer can be well identified and traced. Carrying out structural interpretation of B sand layer based on reflection coefficient inversion data and the microstructure and the formation tip extinction point are implemented. Based on the constraint of new interpretation level, the sedimentary facies plane distribution of B sand layer is described and make prediction of dominant reservoir development area in detail combining with sedimentary paleogeomorphology, along layer attribute section and limited drilling data. The research shows that the study area is mainly from the northwest material sources, the slope belt in the northwest is close to the lake shoreline with a gentle slope and shallow water depositional environment, which is located on the main transport and deposition channels. The shallow water gentle slope landform is suitable for forming large-area sand bar deposition, mainly composed of underwater distributary channel and debouch bars facies, which is the dominant reservoir development area. The research conclusion guides the deployment and implementation of the development well location effectively.

Algorithm and Design Methodology to Develop a PIFA with Optimized Reflection Coefficient for the ISM 868 MHz Band

This paper presents a methodological approach to design a printed Inverted F antenna for the ISM 868 MHz band. For this design, the ground plane dimensions were kept fixed and the meandered radiating arm was modified to obtain the best compromise integration/performances. This approach was then generalized to design meandered printed inverted F antennas.