Reflection Of Spherical Waves Research Articles

Time-domain model for spherical wave reflection in a flat surface with absorber character – Application to the SOPRA measurement method

This paper presents a developed time-domain model for the reflection of spherical waves in an absorber-like surface. The model is made to enable evaluation of measurement methods for assessing the sound absorptive properties of traffic noise barriers in direct sound field, it is thus called the Direct Field Absorption (DFA) model. In this study, the DFA model is applied to the in-situ SOPRA method for quick sound reflection index measurements on road noise barriers. The DFA model results in an impulse response (IR), based on a theoretically derived impedance of a certain absorber. The DFA IR is entered to the SOPRA formula to calculate the reflection index, RIQ, of the absorber, which is subsequently compared to the measured RIQ of a wall fitted with the absorber in question. If the results are similar, it is reasonable to assume that the SOPRA measurement results are valid. But if there are any significant differences between the DFA and SOPRA sound reflection indices, possible reasons for them should be examined.The first results are encouraging, showing that the DFA model can be a valuable tool to evaluate the results of reflection measurements. Furthermore, the DFA model could be useful for estimating the sound absorptive performance at lower frequencies of a noise barrier limited to its spatial extent in width and height, i.e., when it is physically impossible to measure. Further studies are necessary though, since the conclusions of this paper are based on only one kind of absorber.

Complex spherical-wave seismic inversion in anelastic media for the P-wave minimum quality factor

Attenuation always exists when seismic waves propagate in underground anelastic media, especially in hydrocarbon-bearing reservoirs. Quality factor Q or attenuation factor 1/ Q can be used to quantify the seismic wave attenuation and has become an important hydrocarbon indicator. The relationship between the plane-wave reflection coefficient ([Formula: see text]) in anelastic media and P- and S-wave quality factors has been widely used in the plane-wave seismic inversion to estimate the quality factors. The [Formula: see text] provides an adequate approximation for the deeper subsurface. However, for the shallow subsurface and anelastic wavefields excited by point sources, the [Formula: see text] is inaccurate and its meaning involves some fundamental difficulties. In view of this, a Q-dependent P-P spherical-wave reflection coefficient ([Formula: see text]) in anelastic media is used here. Considering that having too many parameters to be inverted will lead to unstable and inaccurate inversion results, we further derive an approximate anelastic [Formula: see text] and anelastic spherical-wave impedance ([Formula: see text]), which are frequency dependent and are the functions of P- and S-wave velocities, density, and P-wave minimum quality factor ([Formula: see text]). Finally, a complex spherical-wave seismic inversion approach in anelastic media for the P-wave minimum quality factor is developed. Using the Bayesian inversion approach and complex convolution model, we first estimate the multilayer [Formula: see text] from the complex seismic traces with different frequencies and incidence angles. Based on the inverted angle- and frequency-dependent [Formula: see text], the P- and S-wave velocities, density, and P-wave minimum quality factor are further estimated using a nonlinear inversion tool. Synthetic examples verify the feasibility and robustness of the complex spherical-wave seismic inversion approach in anelastic media. In the shallow subsurface, the spherical-wave inversion is superior to plane-wave inversion. A field example further demonstrates the accuracy and great potential of our approach in hydrocarbon-bearing reservoir prediction.

Plane-Wave Spectrum Analysis of Spherical Wave Absorption and Reflection by Metasurface Absorber

Plane-Wave Spectrum Analysis of Spherical Wave Absorption and Reflection by Metasurface Absorber

Spherical-wave elastic inversion in transversely isotropic media with a vertical symmetry axis

SUMMARY Although subsurface media are usually assumed to be isotropic, anisotropy is ubiquitous in crustal rocks and leads to the variation of seismic response with direction. Transversely isotropic media with a vertical symmetry axis (VTI media) are widely found in the real world, such as in textured shale reservoirs. Plane-wave reflection coefficients (PRCs) in VTI media have been widely exploited in amplitude variation with offset (AVO) inversion to estimate the elastic and anisotropy parameters of subsurface media. However, the PRCs in VTI media meet some fundamental problems, especially at near-critical or post-critical incidence angles where the spherical-wave effect is significant. To consider the wave front curvature, a complex spherical-wave reflection coefficient (SRC) in VTI media is derived. To better understand the spherical-wave seismic response in VTI media, we investigate the dependence of the complex SRC on frequency, reflector depth and Thomsen anisotropy parameters ($\varepsilon $ and $\delta $). Based on a complex convolution model, a spherical-wave AVO inversion approach in VTI media is proposed to estimate the vertical (symmetry-axis) compressional and shear wave velocities (P and S waves), density and Thomsen anisotropy parameters from observed seismic data with different incidence angle and frequency components. Synthetic data with Gaussian random noise are used to verify the robustness of the spherical-wave AVO inversion approach in VTI media. Field data examples show that the proposed approach can produce reasonable inversion results that match well with the well-logging data.

Frequency-dependent inversion based on spherical-wave reflection coefficient in elastic medium: Theory and methodology

Plane-wave theory based on the Zoeppritz's equations and its various approximations have been extensively utilized in seismic processing and interpretation. However, localized sources used in field acquisition usually generate spherical wave. As a zero-order approximation of spherical reflected wave, plane reflected wave varies with seismic offset, which can only use amplitude information of multiple pre-critical offsets to estimate elastic parameters. While, the change of spherical wavefront curvature across reflected interfaces causes the amplitude and phase of spherical reflected wave to vary with seismic frequency and offset. Therefore, in addition to conducting conventional amplitude variation versus with offset inversion using a single frequency of multiple offsets, spherical reflected wave makes frequency-dependent inversion using multiple low frequencies of a single offset possible. Herein, we derive the integral expression of spherical wave reflection coefficient from the general solution of wave equation and verify the consistency of displacements of reflected wave simulated by numerical integral and finite difference method in elastic medium. Subsequently, we analyze and investigate the feasibility of frequency-dependent inversion using spherical wave reflection coefficient of a single offset in synthetic data with a planar reflection interface. Combining a global optimal algorithm, low frequency-dependent inversion based on spherical wave reflection coefficient can offer accurate velocity and density parameters for further seismic interpretation.

Spherical-Wave Reflection from a Two-Layer Medium and Bistatic Microwave Sounding of the Lake Baikal Ice Cover

Spherical-Wave Reflection from a Two-Layer Medium and Bistatic Microwave Sounding of the Lake Baikal Ice Cover

Trans-dimensional Inversion in two spatial dimensions for geoacoustic parameters

We test a series of model parametrizations to invert a two-dimensional (2-D) seabed transect going from simple fixed-dimensional to more complex trans-dimensional configurations. Inverting spherical-wave reflection coefficient datasets independently as one-dimensional layered models for a 2-D section is less efficient than applying a more parsimonious 2-D parametrization while also carrying out full uncertainty quantification. We approach the problem by proposing different fixed and dynamically inferred parametrization schemes, and discuss implementation, computational cost and resulting accuracy. We demonstrate the application in geoacoustics using a dataset of 1711 source transmissions recorded on a 32-element linear hydrophone array with both source and array towed by an autonomous underwater vehicle along a 12 km transect on the Malta Plateau in the Mediterranean Sea.

Complex spherical-wave elastic inversion using amplitude and phase reflection information

Unlike the real-valued plane wave reflection coefficient (PRC) at the pre-critical incident angles, the frequency-and depth-dependent spherical-wave reflection coefficient (SRC) is more accurate and always a complex value, which contains more reflection amplitude and phase information. In near field, the imaginary part of complex SRC (phase) cannot be ignored, but it is rarely considered in seismic inversion. To promote the practical application of spherical-wave seismic inversion, a novel spherical-wave inversion strategy is implemented. The complex-valued spherical-wave synthetic seismograms can be obtained by using a simple harmonic superposition model. It is assumed that geophone can only record the real part of complex-valued seismogram. The imaginary part can be further obtained by the Hilbert transform operator. We also propose the concept of complex spherical-wave elastic impedance (EI) and the complex spherical-wave EI equation. Finally, a novel complex spherical-wave EI inversion approach is proposed, which can fully use the reflection information of amplitude, phase, and frequency. With the inverted complex spherical-wave EI, the velocities and density can be further extracted. Synthetic data and field data examples show that the elastic parameters can be reasonably estimated, which illustrate the potential of our spherical-wave inversion approach in practical applications.

Reflection and transmission coefficients of spherical waves at an interface separating two dissimilar viscoelastic solids

SUMMARY Spherical wave reflection and transmission (R/T) coefficients at an interface are not only of theoretical significance but also play an important role in the amplitude variation with offset (AVO) analysis of wide-angle reflection seismic data and cross-borehole surveys. For sources close to the interface the resulting wavefields cannot be adequately described in terms of a single incident plane wave. Rather, the spherical waves must be viewed as the superposition of an infinite number of plane waves. Moreover, the R/T coefficients for each individual plane wave in viscoelastic media have proven to be more complicated than expected due to the difficulty in selecting the correct vertical slowness. In such attenuating media the R/T coefficients cannot be properly determined by simply replacing the real elastic parameters with their complex viscoelastic counterparts. In this study, the reflection and transmission coefficients of spherical waves at a plane interface separating two dissimilar viscoelastic solids are rigorously investigated. The difficulty in selecting the vertical slowness is shown to be circumvented if the spherical wavefields are calculated from the plane wavefields using the Sommerfeld integral appropriate for the dissipative materials. However, some resulting phase curves of the complex spherical wave R/T coefficients tend to be of opposite sign to the corresponding phase curves of plane waves due to non-uniqueness of the latter for post-critical wave incidence. In this contribution we propose a new definition of spherical wave R/T coefficients for viscoelastic media which differs from the conventional one. Its advantages are that it is not explicitly expressed as a function of the R/T angles, it is valid for both P and S waves, yet it is consistent with the existing definitions of spherical wave R/T coefficients but is more robust. By way of examples we compute both spherical wave reflection coefficients (SWRC) and spherical wave transmission coefficients (SWTC) for two different viscoelastic models. Unlike plane waves, both the SWRC and the SWTC of converted PS waves are found to be non-zero at vertical incidence and may be drastically affected by the existence of longitudinal PS waves which are confirmed by full waveform calculations for the converted PS waves.

Seismic inversion using complex spherical-wave reflection coefficient at different offsets and frequencies

Compared with the plane-wave reflection coefficient, the spherical-wave reflection coefficient (SRC) can more accurately describe the reflected wavefield excited by a point source, especially in the case of low seismic frequency and short travel distance. However, unlike the widely used plane-wave amplitude-variation-with-offset/frequency (AVO/AVF) inversion, the practical application of spherical-wave AVO/AVF inversion in multilayer elastic media is still in the exploratory stage. One of the difficulties is how to fully use the amplitude and phase information of the complex-valued SRC and the spherical-wave response property of each frequency component to obtain the spherical-wave synthetic seismogram in multilayer elastic media. In view of this, we have developed a complex convolution model considering the amplitude and phase information of an SRC to obtain the complex synthetic seismogram of a certain frequency component. A simple harmonic superposition method is further developed. By superposing the complex synthetic seismograms of different frequency components, the synthetic seismogram of the full-frequency band can be obtained. In addition, a novel three-parameter SRC in terms of P- and S-wave moduli and density is derived. Based on the SRC and complex seismic traces with different offsets (or incidence angles) and frequency components, an inversion approach of complex spherical-wave amplitude and phase variation with offset and frequency is proposed. A noisy synthetic data example verifies the robustness of our complex spherical-wave inversion approach. Field data examples indicate that the P- and S-wave moduli estimated by the complex spherical-wave inversion approach can reasonably match the filtered well-logging data. Considering spherical waves rather than plane waves can improve the accuracy of seismic inversion results.

General Treatment of Reflection of Spherical Electromagnetic Waves from Spherical, Uneven Antarctic Surface and Its Implications for the Mystery Events Detected by ANITA

General Treatment of Reflection of Spherical Electromagnetic Waves from Spherical, Uneven Antarctic Surface and Its Implications for the Mystery Events Detected by ANITA

On the reflection of time-domain acoustic spherical waves by a sinusoidal diffraction grating

This work reports on some results obtained from numerical simulations of time-domain acoustic wave propagation in the presence of a periodically rough interface. Emphasis is put on the structure of the reflected signals in the presence of a sinusoidal grating. More specifically, we investigate the effect of the frequency bandwidth of the emitted signal and the effect of the incident wavefront sphericity on the signals reflected from the rough interface and associated with the different diffraction orders.

Acoustic characterisation of towing tanks

Large water tanks known as towing tanks, wave basins and cavitation tunnels, are used to determine the performance of ships, boats and submarines at model-scale. They may also be used for noise measurements but suffer from the effects of reverberation, which makes it difficult to capture the free field conditions necessary for complete characterisation of the noise. This paper describes an adapted Image Source Model and its use to understand these reverberation effects. It extends the conventional Image Source Model by the use of spherical wave reflection coefficients at the tank boundaries, in addition to the inclusion of volumetric absorption. Validation of the Image Source Model is undertaken by comparison with published transfer function measurements made in the depressurized wave basin at the Maritime Research Institute Netherlands (MARIN). The relative contribution to the tank response from the direct field and free surface is examined. The tank response is found to vary with three frequency-scales (periodicities) resulting from interactions with the free surface and axial modes.

General treatment of reflection of spherical electromagnetic waves from a spherical surface and its implications for the ANITA anomalous polarity events

We develop a general formalism to treat reflection of spherical electromagnetic waves or, more precisely, dipole radiation from a spherical surface. Our main objective is interpretation of radio wave signals produced by cosmic ray interactions with Earth’s atmosphere which are observed by the Antarctica based ANITA detector after reflection off the ice surface. The incident wave is decomposed into plane waves and each plane wave is reflected off the surface using the standard Fresnel formalism. For each plane wave the reflected wave is assumed to be locally a plane wave. This is a very reasonable assumption and there are no uncontrolled approximations in our treatment of the reflection phenomenon. The surface roughness effects are also included by using a simple model. We apply our formalism to the radiation produced by the balloon-borne HiCal radio-frequency (RF) transmitter. Our final results for the reflected power are found to be in good agreement with data for all elevation angles. We also study the properties of reflected radio pulses in order to study their phase relationship with direct pulses. We find that for some roughness models the pulse shape can be somewhat distorted. However this is a rather small effect and is unable to provide an explanation for the observed mystery events by ANITA.

Frequency-dependent spherical-wave nonlinear AVO inversion in elastic media

SUMMARY The plane-wave reflection coefficient (PRC) plays a remarkable role in conventional amplitude variation with offset (AVO) analysis and inversion. Compared with the widely exploited PRC that breaks down at the near- and supercritical incidence angles, the spherical-wave reflection coefficient (SRC) can overcome the influence of wide-angle reflection and give an accurate description of the actual seismic wave reflection phenomenon based on spherical-wave fronts. However, SRC is not widely used in AVO inversion due to its nonlinearity and computational complexity. In our study, the characteristics of frequency–depth-dependent monochromatic SRC are discussed and a novel three-parameter SRC is derived. Compared with the conventional six-parameter SRC, the novel three-parameter SRC improves the stability of spherical-wave AVO inversion. In addition, the concept of SRC within the Fresnel zone is proposed, and the accuracy of SRC within the Fresnel zone in the deep subsurface is tested. Finally, a nonlinear spherical-wave AVO inversion method for elastic media is proposed, which can make full use of all frequency components of wavelet. The robustness of the proposed method is verified by the application on synthetic seismogram with white Gaussian noise. The feasibility and practicability of this method are verified by comparing the spherical-wave AVO inversion results with the filtered well logs at the known well location.

Frequency-dependent spherical-wave reflection coefficient inversion in acoustic media: Theory to practice

As an approximation of the spherical-wave reflection coefficient (SRC), the plane-wave reflection coefficient does not fully describe the reflection phenomenon of a seismic wave generated by a point source. The applications of SRC to improve analyses of seismic data have also been studied. However, most of the studies focus on the time-domain SRC and its benefit to using the long-offset information instead of the dependency of SRC on frequency. Consequently, we have investigated and accounted for the frequency-dependent spherical-wave reflection coefficient (FSRC) and analyzed the feasibility of this type of inversion. Our inversion strategy requires a single incident angle using reflection data for inverting the density and velocity ratios, which is distinctly different from conventional inversion methods using amplitude variation with offset. Hence, this investigation provides an alternative approach for estimating media properties in some contexts, especially when the range of aperture of the reflection angles is limited. We apply the FSRC theory to the inversion of noisy synthetic and field data using a heuristic algorithm. The multirealization results of the inversion strategy are consistent with the feasibility analysis and demonstrate the potential of the outlined method for practical application.

On the three-dimensional sound fields from a moving monopole source above a non-locally reacting ground.

This study derives an accurate analytic solution for sound fields due to a moving point monopole source above a non-locally reacting ground. Based on the property of the invariant phase function, the acoustic pressure and normal particle velocity are matched across the air/ground interface in the frequency-wave number domain. Expressions for the integral form of the sound fields in three different frames are used to analyze and compare the original physical frame in the reception time geometry, the Lorentz frame, and the emission time frame. For ease of interpretation, the final form of the asymptotic solution is expressed in the geometry of the emission time frame. The accuracy of the asymptotic solution is validated using the direct numerical integration technique. The three-dimensional point source solution derived from the current study is compared with that of the two-dimensional line source solution. Explicit comparisons of the cylindrical and spherical wave reflection coefficients are available, allowing a better understanding of the impact of the ground wave term in the vicinity of a non-locally reflecting plane. Finally, contour plots are presented that demonstrate the need to employ the asymptotic formula for the non-locally reacting ground instead of approximating it with a locally reacting property.

On the reflection, diffraction and refraction of a spherical wave of parallel polarization by a sphere of electrically long radius

A novel mathematical approach is proposed for the analysis of the reflection, diffraction and refraction of spherical waves from curved material interfaces. As a paradigm, this approach is applied to the radiation of a vertical electric dipole over an electrically homogenous sphere. The material of the sphere can be dielectric or conducting. The radius of the sphere is assumed to be much greater than the wavelength in its less dense exterior. The formulation is developed for any height of the dipole above the sphere and for any radial distance of the point of observation from the center of the sphere. Novel expansions are applied to the integral representations of the resulting fields. It is shown that contributions to the value of the fields can be produced either from stationary-phase points of the integrands that are generated by such expansions or from any of the two strings of poles of these integrands. Contributions from rapidly converging residue series, as opposed to those coming from stationary-phase points, correspond to wave trajectories containing a curved path in addition to rectilinear paths. For the sake of demonstration, approximate analytical formulas for computation are derived for a number of select cases. The method yields leading-order approximations for the reflected and refracted fields. The affinity with the reflection from a planar interface is analysed.

Reverberation time and audibility in phased geometrical acoustics using plane or spherical wave reflection coefficients.

Room acoustics parameters are typically predicted using some form of geometrical acoustics for large rooms. For smaller rooms, phased geometrical acoustics improves results for lower frequencies. The use of a spherical wave reflection coefficient improves the results further, yet the exact impact on room acoustics parameters is not fully known. This work predicts the reverberation time in medium-sized rooms (27 m3 < V < 300 m3) using phased geometrical acoustics. The difference between the use of plane and spherical wave reflection modeling is analyzed for a variety of boundary conditions, including non-uniform distribution of absorption. Since calculated differences are greater than the conventional just-noticeable-difference of 5% for reverberation time, laboratory listening tests are performed to confirm audibility of the modeled differences. Two narrow band noise stimuli (octave bands with central frequency 125 and 250 Hz) with a duration of 1 s were used for comparisons of 18 acoustic scenarios by means of a three-alternative forced choice method. More than half of the listeners could hear the differences in all 36 cases. Statistically significant results (chi-squared test was used) were found in two-thirds of the cases, corresponding to those with longer reverberation times.

Reverberation time and audibility in phased geometrical acoustics using plane or spherical wave reflection coefficients

Reverberation time and audibility in phased geometrical acoustics using plane or spherical wave reflection coefficients