Complete Wavefield Research Articles

Ground‐penetrating radar: Wave theory and numerical simulation in lossy anisotropic media

Subsurface georadar is a high‐resolution technique based on the propagation of high‐frequency radio waves. Modeling radio waves in a realistic medium requires the simulation of the complete wavefield and the correct description of the petrophysical properties, such as conductivity and dielectric relaxation. Here, the theory is developed for 2-D transverse magnetic (TM) waves, with a different relaxation function associated to each principal permittivity and conductivity component. In this way, the wave characteristics (e.g., wavefront and attenuation) are anisotropic and have a general frequency dependence. These characteristics are investigated through a plane‐wave analysis that gives the expressions of measurable quantities such as the quality factor and the energy velocity. The numerical solution for arbitrary heterogeneous media is obtained by a grid method that uses a time‐splitting algorithm to circumvent the stiffness of the differential equations. The modeling correctly reproduces the amplitude and the wavefront shape predicted by the plane‐wave analysis for homogeneous media, confirming, in this way, both the theoretical analysis and the numerical algorithm. Finally, the modeling is applied to the evaluation of the electromagnetic response of contaminant pools in a sand aquifer. The results indicate the degree of resolution (radar frequency) necessary to identify the pools and the differences between the anisotropic and isotropic radargrams versus the source‐receiver distance.

The radiation modes of a vertically varying half-space: a new representation of the complete Green's function in terms of modes

SUMMARY We prove that the complete Green’s function for an isotropic vertically varying halfspace can be written in a dyadic form as a sum on the usual normal modes plus an integral over radiation modes. The radiation modes form the basis on which to represent the body waves, which are not trapped in the structure. The expressions for their eigenfunctions, the so-called improper eigenfunctions, are found for both the SH and the P-SV wave operators. We prove that they are orthogonal to each other and to the normal-mode eigenfunctions. In a numerical example, we present improper eigenfunctions, and compare complete wavefield seismograms synthesized by the mode method and by the discrete wavenumber integration method.

Approximate hydrodynamic analysis of multi-column ocean structures

An approximate, semi-analytical technique is developed to calculate the wave-induced loads and added-mass and radiation damping coefficients for idealized ocean structures consisting of arrays of large-diameter, truncated, vertical circular cylinders. The approach involves replacing the complete wave field scattered/radiated by one cylinder in the vicinity of another cylinder by a plane wave together with a non-planar correction term and so is essentially a large-spacing approximation. Numerical results are presented which illustrate the accuracy of the approximate approach and provide estimates of the hydrodynamic characteristics of several practical array configurations.

Wave Propagation in Layered Soils: Theoretical Solution in Wavenumber Domain and Experimental Results of Hammer and Railway Traffic Excitation

A method of calculating the wave field of horizontally layered soils is presented. It combines numerical integration in the wavenumber domain with the matrix method for layered soils in the most simple way. The three-dimensional problem of a harmonic horizontal or vertical point load acting on the surface of the soil is investigated. The method yields the radial, transverse and vertical displacements as functions of frequency and distance, which describe the complete wave field due to a point load of any direction. Results are presented for the vertical displacement due to vertical excitation, as that is the situation which is usually of interest in soil prospecting and vibration analysis. The amplitudes and partly the phases (which yield the dispersion curves of the soil profile) of the complex transfer functions are given for some elementary layered soils, such as a soft layer on a stiffer half-space, a stiff layer on a softer half-space, and continuous variation of stiffness and damping. Comparison of the results with those for homogeneous half-spaces of different wave speeds leads to a general conclusion that usually only the soil material down to half a wavelength of the Rayleigh wave has an influence on the response of the surface, but some significant deviations from that rule are also shown. Experimental results for three different sites due to impulse, harmonic and railway traffic excitation are presented and compared with the theoretical results, and these demonstrate the strong influence of the soil profile on the distribution of wave amplitudes in distance and frequency.

The complete ordered ray expansion-I. Calculation of synthetic seismograms

SUMMARY As high-quality broad-band data become more widely available, efficient methods for modelling and inverting such data are increasingly required. In this paper we describe a novel method for calculating seismograms, referred to as the Complete Ordered Ray Expansion or CORE. This technique is based on a ray-generation algorithm involving the symbolic manipulation of complete wavefield expressions from reflectivity theory, truncated to produce a finite ray series. CORE represents an extremely flexible tool for the generation of body wave synthetics, and for the interpretation of observed seismograms. It also provides a framework for waveform inversion that exploits the natural division between smooth structure and discontinuities. We describe the ray-generation algorithm, and show examples of the effect of varying degrees of truncation on the ray series. Although CORE uses asymptotic ray theory to generate the seismograms from the ray series, the resulting traces show surprisingly good agreement with full reflectivity synthetics. We show comparisons of broad-band CORE and reflectivity synthetics, as well as a comparison with observed data. Our method is particularly useful in modelling the complete broad-band wave train at teleseismic distances, where the computing time for reflectivity becomes excessively large. The computational procedure is readily extended to a laterally heterogeneous model, where it forms the basis for a fully automated procedure for phase association and waveform inversion.

Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at a cylindrical inclusion : Stange, S; Friederich, W Geophys JV111, N3, Dec 1992, P470–482

Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at a cylindrical inclusion : Stange, S; Friederich, W Geophys JV111, N3, Dec 1992, P470–482

Point–source/point–receiver ultrasonics: A versatile materials characterization tool

The essential elements of a point–source/point–receiver ultrasonic system include a broad bandwidth, small‐aperture source generating elastic waves with a broad angular spectrum and a receiver with similar characteristics to detect them. By moving the source or receiver and stacking the received waveforms, one generates a scan image that provides a view of the complete elastic wave field in a test specimen. Wave arrivals are related to the speeds of propagation of various wave modes in various directions and the signal amplitudes reflect the propagational characteristics of the material. An interpretation of the scan images requires an understanding of the propagation of transient elastic waves in a bounded structure. Recent developments on both the forward and inverse problems are summarized and the full solution of the forward problem for the response of a general axisymmetric anisotropic viscoelastic plate is reported. The computed results are compared to measured waveforms in viscoelastic and transver...

Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at a cylindrical inclusion

We present an exact treatment of wave propagation across a cylindrical inclusion embedded in a layered waveguide, which may serve as a test case allowing an estimation of the accuracy and validity range of approximation methods applied to surface wave propagation. Since, in contrast to a plane vertical interface, a cylindrical inclusion is of finite spatial extent, the test case presented here is especially well suited to assess the quality of approaches based on scattering theory. The wavefield is represented by Love and Rayleigh type modes. A 3-D orthonormality relation is derived, expressed as an integral over the cylinder surface which allows a direct and unique computation of basis solutions. After solving a linear system of equations these basis solutions are superimposed to form the complete wavefield both within and outside the cylinder. It is shown that exact continuity of displacements and tractions at the interface can not be achieved with propagating modes only. Non-propagating modes, which have complex wavenumbers and hence decay in the propagation direction, have to be included in the modal series. In contrast to propagating modes, complex modes have vanishing energy flux. Numerical results are presented for a layered halfspace, where only propagating modes were used, and for a layered waveguide for various sizes of the cylindrical inclusion. Astonishingly, vertical and horizontal displacements behave very differently. For instance, the scattered vertical displacement generated by an incoming Rayleigh fundamental mode is clearly dominated by the mode itself, while the scattered horizontal displacement is severely influenced by the excited higher Rayleigh modes. This might explain the difficulties met when interpreting horizontal components of surface wave data within a single-mode concept.

Guided wave propagation across sharp lateral heterogeneities: the complete wavefield at plane vertical discontinuities

SUMMARY 3-D wave propagation in a waveguide composed of laterally homogeneous partitions separated by vertical interfaces is treated in an exact manner. In each laterally homogeneous subregion, the wavefield is represented by Love- and Rayleigh-type modes, combined in such a way that displacements and tractions are continuous at the vertical discontinuities. In order to achieve exact continuity at the interfaces, near-field modes which exponentially decay in the propagation direction and are associated to fully complex wavenumbers, must be included in the modal representation. The expansion coefficients of the mode series are computed directly by exploiting orthogonality relations between modes of different type, order and propagation direction. Since the boundary conditions at the discontinuities are satisfied exactly, the resulting expansion of the wavefield is valid even on the interfaces themselves. Numerical results are presented for a single vertical discontinuity and a lamella.

Multicomponent Seismic Reflection Profiling–Some Scale Model Experiments

Reflection experiments have been carried out on simple two-dimensional laboratory models to demonstrate the advantages of multicomponent profiling over conventional scalar wavefield recording. The 2 ´ 2 matrix of observations were taken with a biaxial piezoelectric receiver (vertical and horizontal receivers) and two piezoelectric source types ? vertical force (P wave) and horizontal force (S wave) ? to record the complete vector wavefield. The four component data were processed using a controlled direction reception (CDR) filter, which transforms the data into four sections for interpretation: PP, PS, SP and SS, including all mode conversions. In these sections not only is the specified linearly polarised wave type enhanced but coherent noise such as Rayleigh waves (elliptic polarisation) and incoherent background noise (isotropically polarised) is suppressed. With a single component source and a single component receiver, recording and interpretation of the various wavefields is difficult. Using vector wavefield recording and processing, it is easy to separate and identify compressional-wave reflections, shear-wave reflections, and all mode converted waves (PS reflections and SP reflections). The availability of S wave sections, in addition to conventional P wave sections, lends new insight to seismic interpretation. It improves the accuracy in prediction/seismic mapping and yields better vertical and horizontal resolution of the structure. In addition, the multicomponent sections contain diagnostic information on lithology, porosity, fluid content and anisotropy of the medium.

Broadband acoustic‐field simulations from standard ray theory

The complete wave field over a small region around 1000 km from a pulsed source is reconstructed in two ways. First, all the rays from the source to a vertical array of receivers at 1000 km are found, along with their travel times, number of caustics, arrival angles, and intensities. The pattern of wave fronts in a space at a given time is then reconstructed on a closely spaced grid surrounding 1000 km by treating these rays in an appropriate way. Second, the parabolic equation method is used at multiple frequencies to synthesize a pulse. The two fields are compared. Finally, the effect of internal waves is simulated by use of the first method, introducing random fluctuations on the travel times and arrival angles of each ray. [Work supported by ONR, Code 1125OA.]

Response of an oceanic wave guide to an explosive point source using leaking modes

AbstractA new method for computing transient wave fields in a stack of liquid layers overlying a solid elastic layered half-space is presented. By taking the frequency, ω, and wavenumber, k, simultaneously as complex variables and choosing appropriate paths of integration in the ω-plane, the integration with respect to k in the well-known integrals representing the exact solution is performed exactly using Cauchy residue theory. The remaining integration (with respect to ω) is then performed approximately by use of the Fast Fourier transform and by other numerical integration methods. The theoretical results derived show that the complete wave field, including all possible body waves, can be represented simply as a superposition of modes. The method is easy to apply and computationally efficient, and is valid for both near- and far-field solutions. The same method is also applicable to axisymmetric borehole problems. The application of the method is illustrated in detail with a numerical example.

Seismic wave propagation in a very permeable water‐saturated surface layer

According to Biot's (1956a, b) model, the presence of water plays an important role in the propagation of seismic waves in at least three different ways: (1) in an infinite medium, water saturation induces an attenuation that can be accounted for by a complex formulation of wave velocities, as in viscoelastic media; (2) at the boundaries of the saturated medium, pore pressure and water flux determine specific continuity conditions; and (3) there is a second compressional wave, called the P2 wave. In this paper, we discuss the latter two effects. Biot's model is presented first, with homogenization theory used to provide the numerical values of the different coefficients used in the model. In an infinite medium, the model is of practical interest when the frequency ƒ is about the same order of magnitude as a characteristic frequency noted ƒc, which depends on the properties of the constituents. This limits the application of Biot's model to a few particular fields in geophysics. In a layered medium, Biot's model has a wider scope in that it provides a tool for modeling fluid‐solid interaction at the boundaries of the saturated medium. This is illustrated in our paper for the case of a very permeable water‐saturated surface layer over an elastic half‐space. Two examples are given; in the first example (rigid sands) we discuss the physics of the strongly attenuated P2 wave predicted by Biot, the amplitude of which becomes significant when the ƒ/ƒc ratio is about equal to or greater than 0.1. In the second example (soft unconsolidated sediments) the P2 wave is negligible, but the calculation of the complete wave field is required when the ƒ/ƒc ratio is about 0.01. There is no adequate equivalent single phase model that gives a correct estimation of the amplitude of the ground motion. In this case, we argue that the P2 wave is not important in itself, but Biot's model allows the description of the fluid‐solid interaction at the water table; continuity of effective stress and pore pressure can be explicitly formulated.

Diffraction of elastic waves by cracks or cavities using the discrete wavenumber method

A new method to calculate the diffraction of elastic waves by cracks or cavities is presented. The method yields the complete diffracted wave field and is valid at any frequency. It is based on a discretization of the inclusion surface and on the assumption of a periodic repetition of the diffracting inclusion. Body forces are applied at the discretized points along the inclusion surface and the discrete wavenumber method is used to evaluate the resulting radiation. The application of the boundary conditions then leads to a linear system of equations, which is solved by matrix inversion. Examples of calculation and comparisons with results from other techniques are presented in the case of plane incident waves as well as in the case of a line source. The presence of a free surface is also considered.

Global matrix formulation of wave phenomena in plane layered media

A direct global matrix formulation for computing the multilayer Green's impulse response function in media with arbitrary depth-dependent elastic or fluid properties is reniewed, after recent work by Schmidt. The global Green's function matrix for solving the resulting linear system of elastodynamic equations is formally analogous to the global stiffness matrix in the direct Finite Element Method and proves both computationally efficient and numerically stable. This analogy and the matrix solution are discussed, as embodied in the general applications code, “SAFARI” ( ∗ SA clant FA st Field Program for R ange- I ndependent Environments ∗ ), developed by one of us (H.Schmidt). Cros s-disciplinary applications of this direct global matrix algorithm are considered for three pertinent areas of current research; namely underwater acoustic propagation, ultrasonic scattering, and crustal seismic modeling. Results obtained show that the SAFARI model correctly reproduces the complete wavefield response in plane-layered, welded, homogeneous media. Arbitrary source/receiver numbers, frequencies, locations, and geometries can also be considered over much larger-scale multilayered media models than previously feasible with extant propagator matrix-based codes. The physical simplicity, CPU savings, and flexible performance of the direct global matrix algorithm suggests that many more full wavefield phenomena can now be simulated without excessively complicated, specialized, or expensive modeling schemes.

Propagation of propeller tone noise through a fuselage boundary layer

In earlier experimental and analytical studies, it was found that the boundary layer on an aircraft could provide significant shielding from propeller noise at typical transport airplane cruise Mach numbers. In this paper a new three-dimensional theory is described that treats the combined effects of refraction and scattering by the fuselage and boundary layer. The complete wave field is solved by matching analytical expressions for the incident and scattered waves in the outer flow to a numerical solution in the boundary layer flow. The model for the incident waves is a near-field frequency-domain propeller source theory developed previously for free field studies. Calculations for an advanced turboprop (Prop-Fan) model flight test at 0.8 Mach number show a much smaller than expected pressure amplification at the noise directivity peak, strong boundary layer shielding in the forward quadrant, and shadowing around the fuselage. Results are presented showing the difference between fuselage surface and free-space noise predictions as a function of frequency and Mach number. Comparison of calculated and measured effects obtained in a Prop-Fan model flight test show good agreement, particularly near and aft of the plane of rotation at high cruise Mach number.

Elliptical anisotropy—Its significance and meaning

Nonspherical wavefronts in anisotropic media are often assumed to be approximately elliptical. However, in transversely isotropic media only the wavefront of SH waves is always an oblate ellipsoid. The wavefront of SV waves is never an ellipsoid, and the wavefront of P waves is an oblate ellipsoid if and only if the expression [Formula: see text] vanishes. This cannot happen if the anisotropy is due to lamellation (periodic layering with a spatial period small in comparison to the wave length). The occurrence of elliptical wavefronts cannot be detected on the basis of surface observations of times alone, since the complete wave field can be transformed into one with spherical wavefronts everywhere by simple stretching of layers. Neither arrival times nor apparent slownesses (and thus Snell’s law) are affected by this transformation. All concepts and algorithms applicable to spherical wavefronts are applicable to elliptical wavefronts, in particular the determination of a velocity as the zero‐offset limit of the stacking velocity. Since the resulting velocity is invariant under the stretching transformation, it can represent only a velocity that is itself an invariant, viz., the velocity at right angles to the direction of stretching. However, amplitude observations of SH waves give an independent indication of elliptical anisotropy. Although the wavefronts of P and SV waves can never be ellipsoids if the anisotropy is the result of lamellation, pieces of the wavefront can be represented with sufficient accuracy by an ellipsoid. This representation allows a simple determination of the ratio “zero‐offset limit of stacking velocity/vertical velocity.” Constraints on the parameters of the thin layers that constitute a lamellated medium can be translated into constraints of the above velocity ratio. For P waves this ratio is centered around unity for a wide range of constituent parameters.

Space-time ray method for seismic wave fields

The space-time ray method can be applied to the evaluation and continuation (extrapolation) of the complete seismic wave field in laterally inhomogeneous media with curved interfaces. The wave field propagates along certain space-time curves, called space-time rays. Their space projections correspond to standard rays. Examples of possible applications of the space-time ray method, where the standard ray method fails, are as follows: a) The propagation of seismic waves in slightly dissipative media, b) The computation of seismic wave fields generated by seismic sources with direction-dependent source-time variations. c) Downward continuation of the seismic wave field (actual seismograms) measured at the Earth's surface.

Computation of wave fields in inhomogeneous media -- Gaussian beam approach

An asymptotic procedure for the computation of wave fields in two-dimensional laterally inhomogeneous media is proposed. It is based on the simulation of the wave field by a system of Gaussian beams. Each beam is continued independently through an arbitrary inhomogeneous structure. The complete wave field at a receiver is then obtained as an integral superposition of all Gaussian beams arriving in some neighbourhood of the receiver. The corresponding integral formula is valid even in various singular regions where the ray method fails (the vicinity of caustic, critical point, etc.). Numerical examples are given.

Direct Fourier Synthesis of Waves: Application to Acoustic Source Radiation

In numerous situations of current interest in aeroacoustics the sound field propagates in a nonhomogeneous medium. This is typically the case of jet noise radiation. A widely used model in that situation consists of a thin vortex sheet separating two uniformly moving fluids. This model may be treated as a layered medium, and acoustic source radiation may be analyzed by Fourier transform techniques. The inversion of the Fourier solution is however difficult and requires far-field approximations. This difficulty is here avoided by constructing the wave field by direct Fourier synthesis. This method was shown, in a previous work to be fast, reliable, and superior to the standard approximations. It is applied here to source radiation in the vicinity of a vortex sheet separating uniformly moving streams. Complete wave field maps are given for subsonic and supersonic flow combinations, providing new insights on jet noise radiation and refraction in open wind tunnels.