Plane Wave Reflection Coefficient Research Articles

Sub-bottom profiling by inverting ambient noise

In this paper a technique based on spectral factorization for restoring the phase of incoherent bottom sediment reflection coefficient measurements is presented, so that by Fourier transformation one can then obtain the minimum phase impulse response at each grazing angle. The method is developed and discussed in the context of another recently established technique for extracting the seabed’s plane wave reflection coefficient from ambient noise data measured on a moored or drifting vertical array (VLA). Limitations of the phase restoration method are discussed, and using modeled data, comparisons are made between the ‘‘true’’ impulse response derived from the known complex reflection coefficient and the result of applying spectral factorization to the absolute value of the reflection coefficient. For instance, in both cases one can see clear, matching arrivals from each layer boundary at angles above critical. Finally the method is demonstrated on experimental reflection loss inferred from ambient noise measurements at three moored VLA sites and one VLA drift track in the Mediterranean Sea. Convincing angular variation of the impulse response is shown for the moored sites. Sub-bottom profiles (impulse response vs position) are shown for the drift track demonstrating that one can survey with just a single directional receiver.

Geoacoustic inversion of a shallow fresh-water environment

A recent experiment was conducted at The University of Texas/Applied Research Laboratories test station located at Lake Travis, Austin, TX. Implosive (light bulb), explosive (firecracker), and tonal sources were recorded on a dual receiver system located on the bottom next to a range-independent underwater river channel. Inversion results of the broadband time series obtained over ranges less than 1.5 km were used to predict measured transmission loss at several tonal frequencies in the band from 250–1000 Hz. The average water depth was approximately 38 m along the channel during the experiment. Sound speed profiles were calculated from recorded temperature readings measured as a function of depth. Implosive source spectrums were measured and used to evaluate a model/data correlation cost function in a simulated annealing algorithm. Comparisons of inversion results using both a normal mode and a ray-based plane wave reflection coefficient forward model [Stotts et al., J. Acoust. Soc. Am. (submitted)] are discussed. Predicted transmission loss based on the inversion results are compared to the measured transmission loss. Differences between fluid and elastic layer bottom models will also be presented.

Thin-layer scaling effects on AVO modelling

In AVO modelling with log data, it is a general practice to upscale the log data for reasons such as computational efficiency. In this study, we investigate the AVO sensitivity to thin-layer effects by upscaling log data at different scales. A series of Kennett-based reflectivity modelling exercises are conducted using the resulting elastic 1D models. In the presence of thin layers, it is essential to incorporate interbed multiples (including converted wave modes) and therefore, primary-only Zoeppritz modelling could lead to misleading results. On one hand, very fine scale models involve a complex wavefield tuning that renders the theoretical Zoeppritz plane-wave reflection coefficients meaningless. On the other hand, wavefield complexities due to thin-layer multiple scattering also introduce uncertainties in the modelled AVO behaviour. In contrast, with increasing layer scales, Zoeppritz plane-wave reflection coefficients become more applicable, and interbed multiple and tuning effects also have less influence upon the AVO behaviour. At such scales, primary-only AVO modelling by ray tracing where reflection amplitudes are determined by Zoeppritz equations could be an efficient alternative to full-wavefield reflectivity modelling. In our synthetic examples, this occurs at layer scales (d) satisfying a relation to dominant frequency (?) of ?/d = 4 ~ 5. However, ? and d are interdependent, since the effective velocity used to calculate wavelength ? depends on scale d. To resolve this dilemma in practice, surface seismic data could be analysed and a Q sensitivity analysis may be necessary to determine the dominant wavelength at different target levels.

Numerical study of geoacoustic inversion in range-dependent ocean environments

Simulated acoustic data for two-dimensional waveguides were used to estimate the properties of a sea bed over which the depth of the water column varies with range. The simulated data were produced by a parabolic equation method. No approximate knowledge of the local sea-bed representation or parameter bounds was given prior to the inversions. In addition, little approximate information was given on the nature of the range dependence of the sea bed. The inversion methodology used a simulated annealing approach and a variation of the algorithm that generated the original acoustic data. For the purpose of consistency, a second method was employed, which used the same simulated annealing approach but with a range-dependent ray approach in which the sea-bed interaction is described by a complex plane wave reflection coefficient. Various combinations of frequencies and source-receiver positions commonly used in real acoustic measurements at sea were selected for the inversions. The goal of the inversion was to efficiently achieve a coarse description of the environment by several inversions; short-range data at low frequencies were employed for the purpose of efficiency. In the later stages of the inversion process, data from longer ranges and higher frequencies were included to provide additional details of the sea bed. Sea-bed parameters were assumed to vary linearly with range between geoacoustic profiles defined at specific ranges. Inversion results obtained prior to the reporting of the actual solutions are presented.

Acoustic near-field characteristics of a conical, premixed flame

The occurrence of self-excited pressure oscillations routinely plagues the development of combustion systems. These oscillations are often driven by interactions between the flame and acoustic perturbations. This study was performed to characterize the structure of the acoustic field in the near field of the flame and the manner in which it is influenced by oscillation frequency, combustor geometry, flame length and temperature ratio. The results of these calculations indicate that the acoustic velocity has primarily one- and two-dimensional features near the flame tip and base, respectively. The magnitude of the radial velocity components increases with temperature ratio across the flame, while their axial extent increases with frequency. However, the acoustic pressure has primarily one-dimensional characteristics. They also show that the acoustic field structure exhibits only moderate dependencies upon area expansion and flame temperature ratio for values typical of practical systems. Finally, they show that the local characteristics of the acoustic field, as well as the overall plane-wave reflection coefficient, exhibit a decreasing dependence upon the flame length as the area expansion ratio increases.

Free field measurements of the absorption coefficient for nonlocally reacting sound absorbing porous layers.

A simple asymptotic approximation with two parameters (the plane wave reflection coefficient and a correction factor) has been derived by Brekhovskikh and Godin [Acoustics of Layered Media II. Point Sources and Bounded Beams (Springer, New York, 1992)], for the spherical field reflected by nonlocally reacting surfaces. In the present work, an expression for the correction factor is obtained for the case of a homogeneous and isotropic porous layer. A free field method for evaluating reflection and absorption coefficients with this approximation is presented. The evaluation of the absorption coefficient at normal incidence is performed on a porous layer on a rigid backing, and compared to measurements performed using a two microphone technique and an impedance tube.

Acoustic intensity, impedance and reflection coefficient in the human ear canal.

The sound power per unit cross-sectional area was determined in human ear canals using a new method based on measuring the pressure distribution (P) along the length of variable cross-section acoustic waveguides. The technique provides the pressure/power reflection coefficients (R/R) as well as the acoustic intensity of the nonplanar incident wave (I+, the acoustic input to the ear) and the nonplanar outgoing wave (I-, the acoustic output of the ear). Results were compared to the classical acoustic impedance (Z) and associated plane-wave power reflection coefficient (R(Z)). Performance of the method was investigated theoretically using horn equation simulations and evaluated experimentally using pressure data recorded in nonuniform waveguides. The method was applied in normal-hearing young adults to determine ear-canal position- and frequency-dependence of I(+/-), R, and R(Z) using random phase broadband stimuli (1-15 kHz; approximately 75 dB SPL). Reflection coefficient (R) measurements at two different locations within individual human ear canals exhibited a position dependence averaging deltaR approximately 0.1 (over 6 mm distance)--a difference consistent with predictions of inviscid acoustics in nonuniform waveguides. Since this position dependence was relatively small, an "optimized" position-independent reflection coefficient was defined to facilitate practical application and intersubject comparisons.

Inversion of shallow water ocean acoustic data using a complex plane-wave reflection coefficient approach

A previously reported method of inversion in range-dependent environments using a complex plane-wave approach [Stotts et al., J. Acoust. Soc. Am. 109, 2334 (2001)] has been applied to data analysis. The method separates the propagation in the water column from the interaction with the seabed by first calculating and then storing the eigenray characteristics associated with the assumed known waveguide geometry. The inversion is then performed by calculating the plane-wave reflection coefficients at the predetermined angles and constructing the modeled field of the entire propagation for each perturbation of the seabed parameters. Results of applications of this model to data taken off the coast of Florida during the spring of 2001, will be presented. Consistent inversion results obtained from other models along with an examination of received time series, as well as sound speed profiles collected over the region, will also be discussed in the context of range-dependent inversion analysis. Comparisons of this method to a direct reflection coefficient inversion approach provide insight into the use of reflection coefficients as an inversion product.

Modal coupling in acoustic waveguides: planar discontinuities

A new derivation for a scattering matrix for reflection and transmission of higher order modes at the planar junction of two waveguides is presented. The derivation is extended to include finite junction wall impedance and offset waveguides. The resulting matrix equations are analyzed and the physical significance of the matrices is explained. As an example of the theory, analytical expressions for the coupling coefficients at a size change in a rectangular duct are developed and the resulting reflection and transmission coefficients are computed. The results should be of interest to the HVAC noise control community. The paper also shows the effects of modal truncation on the accuracy and convergence of the solution. It is shown that the proper selection of the ratio of the number of modes on either side of the discontinuity is related to the ratio of the characteristic sizes of the waveguides. Finally it is shown that at least one higher mode should be included for reasonable accuracy in the computation of plane wave reflection and transmission coefficients except at the very lowest frequencies.

The Kirchhoff-Helmholtz Integral Pair

The Kirchhoff–Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff–Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff–Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff–Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff–Helmholtz integral.

THE KIRCHHOFF–HELMHOLTZ INTEGRAL PAIR

The Kirchhoff–Helmholtz integral models the reflected acoustic wavefield by an integration along the reflector over the incident field multiplied by the specular plane-wave reflection coefficient. Based on the structural relationships between the reflector and the reflection-traveltime surface, we design an asymptotic inverse Kirchhoff–Helmholtz integral. Analogously to the forward integral, the proposed inverse consists of an integration along the reflection-traveltime surface over the recorded reflected field. We show that the new inverse integral asymptotically recovers the input to the standard Kirchhoff–Helmholtz integral, that is, the reflector position and the reflection coefficients along it. A simple numerical example demonstrates the inverse relationship between the proposed and the standard Kirchhoff–Helmholtz integrals. In this way, a new technique for kinematic (positioning) and dynamic (amplitude) wavefield inversion becomes available. This is realized by means of an integral operation that is most naturally related to its counterpart Kirchhoff–Helmholtz integral.

Geoacoustic inversion in range-dependent environments: A plane-wave reflection coefficient approach

One of the most challenging aspects of geoacoustic inversion in ocean waveguides involves range-dependent problems, such as in shallow water or continental shelf environments. Inversion calculations may require tens or hundreds of thousands of calls to an acoustic propagation model. When the forward propagation model is range dependent, the computation time can become extremely large. If, however; the range dependence is known, an approximation is introduced that significantly simplifies the problem. Using the measured bathymetry and range-dependent sound-speed profile, the geometrical optics approximation is used to compute the eigenrays between sources and receivers. Assuming a rigid bottom boundary condition, the medusa algorithm is used for this purpose. This range-dependent ray theory calculation separates the propagation in the water column from that in the seabed. The modeled broadband pressure fields are obtained by computing the plane-wave reflection coefficient (PWRC) at the angles and frequencies as needed in the eigenray expansion. Namely, each perturbation of the seabed does not require a new range-dependent calculation of eigenrays, only a recalculation of the PWRC. Examples are shown that demonstrate the accuracy, efficiency, and limitations of the approach. [Research supported by SPAWAR and ONR.]

Exact and approximate formulas for P‐SV reflection and transmission coefficients for a nonwelded contact interface

We present new and exact mathematical formulas for the P–SV particle displacement reflection and transmission coefficients for elastic plane waves incident upon a nonwelded contact (or linear slip) interface separating two solid half‐spaces. We represent the nonwelded contact at the interface with displacement discontinuity boundary conditions, that is, the traction is continuous across the interface, but the particle displacement is not: the discontinuity in the displacement is proportional to the traction. The formulas are derived by algebraically solving these boundary conditions. The formulas can be expressed in the form of those for the welded contact case, with some terms being modified due to nonwelded contact, plus additional terms due purely to nonwelded contact. Such formulas are useful because they provide insight into the nature of the coefficients and can be used to derive approximate formulas which have applications in the inversion and interpretation of seismic data. The exact formulas can also be applied to the case of a viscous nonwelded interface by simply replacing one of their parameters (the tangential specific compliance) with a modified parameter which includes the specific viscosity. We also apply the exact formulas to the specific case in which the incidence and transmission media are identical (e.g., a joint, fault, or fracture in a single homogeneous medium). In this case, reflected elastic waves are produced, unlike the case of welded contact, in which no reflections are produced because of the lack of an impedance contrast. This effect may partially explain occasional reports in the literature of anomalously large seismic amplitudes observed in areas where impedance contrasts are small. We present some numerical examples illustrating the effects of nonwelded contact on reflection and transmission coefficients. We show that energy is conserved at all incidence angles at a nonwelded interface between two different elastic solids. We also present some examples of applications of the exact formulas: (1) we derive approximate formulas for a weakly nonwelded interface between two identical solids, (2) we use them to study the sensitivity of the coefficients to the normal and parallel specific compliances (measures of the amount of nonweldedness), finding that they are generally more sensitive to the normal than the parallel compliance (except for the SS reflection), that the sensitivities vary considerably with incidence angle, and that the PP and SS transmission coefficients are basically insensitive to the compliances for a weakly nonwelded interface, and (3) we further approximate the formulas in example 1 to the case of small propagation angles, resulting in formulas that could be useful in amplitude versus offset studies.

Application of the Gelfand–Levitan method to geoacoustic inversion in shallow water

The Gelfand–Levitan inverse technique is an exact inversion method originating in quantum scattering theory [I. M. Gelfand and B. M. Levitan, Am. Math. Soc. Transl. 1, 253–304 (1955)]. At a single frequency, the required input data are the Fourier transform of the plane wave reflection coefficient as a function of incident vertical wave number. The method was originally employed in underwater acoustics for a deep-water environment where a simple relationship exists between the reflection coefficient and the depth-dependent Green’s function, which is obtained by Hankel transforming measurements of the point source pressure field [A. A. Merab, Sc.D. thesis, MIT/WHOI Joint Program, Woods Hole, MA (January 1987)]. In shallow water, the Green’s function is characterized by poles corresponding to the eigenvalues of the perfectly trapped modes in the waveguide. These poles complicate the relationship between the Green’s function and the reflection coefficient, and therefore the application of the Gelfand–Levitan technique. However, through application of the Darboux transform to the governing wave equation and the reflection coefficient, the Gelfand–Levitan method can be employed for shallow-water geoacoustic inversion [J. R. McLaughlin and S. Wang, in Mathematical and Numerical Aspects of Wave Propagation, edited by J. A. DeSanto (SIAM, Philadelphia, 1998), pp. 232–236]. In this talk, the Darboux transform will be illustrated and examples of geoacoustic inversion for synthetic shallow-water acoustic data will be presented. [Work supported by ONR.]

Comparison between the dispersion curves calculated in complex frequency and the minima of the reflection coefficients for an embedded layer

Analytical solutions of Lamb functions for symmetric and antisymmetric elastodynamic modes propagating within a solid layer embedded in an infinite medium are presented. Alternative theoretical analyses of such modes are performed, first in terms of the usual approach of harmonic heterogeneous plane waves (real frequency and complex slowness) and then in terms of transient homogeneous plane waves (complex frequency and real slowness). An example structure of a 0.1-mm-thick "alpha case" (an oxygen-rich phase of titanium that is relatively stiff) plate embedded in titanium is used for the study. A large difference between the usual dispersion curves calculated in real frequency and complex slowness and those calculated in complex frequency and real slowness is shown. Thus the choice between a spatial and a temporal parameter to describe the imaginary part of the guided waves is shown to be significant. The minima and the zeros of the longitudinal and shear plane-wave reflection coefficients are calculated and are compared with the dispersion curves. It is found that they do not match with the dispersion curves for complex slowness, but they do agree quite well with the dispersion curves for complex frequency. This implies that the complex frequency approach is better suited for the comparison of the modal properties with near-field reflection measurements.

Non-local boundary conditions for high-order parabolic equation algorithms

Numerical parabolic equation (PE) solvers for underwater sound propagation usually approximate the subbottom radiation condition by appending an absorbing layer to the computational mesh. An alternative procedure uses a non-local boundary condition (NLBC) applied along (or below) the sea–bottom interface to transform the semi-infinite PE configuration exactly into an equivalent bounded domain. In this paper, we analyze NLBCs that are expressed in terms of plane-wave reflection coefficients and describe their implementation in the context of high-order Padé-based PE algorithms. In particular, we develop non-local boundary conditions to account for the coherent losses due to scattering by a statistically-rough sea surface. Several numerical examples are presented to illustrate the numerical implementation of NLBCs for high-order PEs.

Coupled mode theory for resonant excitation of waveguiding structures

Resonant coupling of light beams via high-index media or gratings to planar waveguiding structures are of interest for both applications and from a theoretical point of view. Coupled Mode theory (CMT) can give an accurate description of the coupling process in terms of relatively simple expressions involving often a large number of coupling parameters. In this paper it is shown, using time reversal and energy conservation how these parameters are interrelated. The evaluation of the remaining independent parameters is shown to be possible using a few reflection and transmission coefficients for incoming plane waves, including in the calculations, if present, the effect of the grating. Further, it is proved that under certain condition a grating coupler may show exactly 100% reflection. Analytical expressions for the reflected and transmitted beams and the amplitude distribution of the excited mode are given for the case of incoming Gaussian beams. A few applications of the theory and considerations on its applicability are presented.

The dependence of target strength of the northern right whale (Eubalaena glacialis) on the acoustic properties of blubber

Whale detection and tracking using active sonar is a subject of recent study [J. H. Miller, D. C. Potter, T. Weber, and J. Felix, J. Acoust. Soc. Am. 105, 992 (1999)]. In that previous work, the measured target strength of a northern right whale (Eubalaena glacialis) was found to be less than that measured for a humpback whale (Megaptera novaeangliae) of the same size and aspect (about 0 and 5 dB, respectively). The difference was explained by modeling the thicker blubber of the right whale as a lossy layer in a plane wave reflection coefficient. However, values for the acoustic properties of the blubber were estimated from tissue properties of other mammals [R. C. Chivers and R. J. Parry, J. Acoust. Soc. Am. 63 (1978)]. In April, 1999, a 60-ton northern right whale named Staccato was found dead near Cape Cod. Blubber samples were acquired by the National Marine Fisheries Service and acoustic properties determined using a multi-sensor core logger in URI Marine Geomechanics Laboratory. Sound speed and density data have been used to refine the plane wave reflection coefficient model and corroborate the earlier work. [Work supported by NMFS.]

Erratum: “Plane-wave reflection and transmission coefficients for a three-layered elastic medium” [J. Acoust. Soc. Am. 97, 954–961 (1995)

Erratum: “Plane-wave reflection and transmission coefficients for a three-layered elastic medium” [J. Acoust. Soc. Am. <b>97</b>, 954–961 (1995)

Plane wave reflection coefficient estimation by use of spatial parametric signal modeling

The plane wave reflection coefficient of an absorbing infinite plane can be estimated from pressure distributions on two lines parallel to the surface. The measured pressures are decomposed into incoming and outgoing plane waves in the wave-number domain after the Hankel transform is applied to the spatial data. In practice, only a finite number of pressure measurements can be made on the measurement lines. As a result, errors resulting from aliasing and truncation are induced in the wave-number spectra and subsequently the reflection coefficient. The aliasing results from insufficiently small sample spacing, and can be easily controlled. The truncation error results from the limited size of the measurement aperture, which may be difficult to expand for practical reasons. Therefore, extrapolation techniques based on parametric signal modeling have been developed to ‘‘virtually’’ expand the measurement aperture, thus minimizing truncation effects. However, some restrictions, particularly on spatial sampling rate, must be considered in order to obtain good estimates of the reflection coefficient. Prony series modeling is one procedure that can be used to model and extrapolate the spatial sound pressure data. Reflection coefficients estimated by using the Prony series approach will be used to illustrate this procedure.