Spherical Wave Reflection Coefficient Research Articles

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.

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.

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.

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.

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.

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.

A novel expression of the spherical-wave reflection coefficient at a plane interface

Abstract The spherical-wave reflection coefficient (SRC) describes the reflection strength when seismic waves emanating from a point source impinge on an interface. In this study, the SRC at a plane interface between two infinite half-spaces is investigated. We derive an analytical equation of the SRC when kR → 0 (k is the wave number and R is the wave propagation distance). It only depends on the density ratio; it is independent of the velocity ratio and incidence angle. On the other hand, we find that the SRCs at different kR lie along an elliptical curve on the complex plane (the complex plane is a geometric representation of the complex numbers established by the real axis and perpendicular imaginary axis). Based on this feature, we construct a new analytical equation for the reflected spherical wave with high accuracy, which is applicable to both small and large kR. Using the elliptical distribution of the SRCs for a series of frequencies recorded at only one spatial location, the density and velocity ratios can be extracted. This study complements the spherical-wave reflection theory and provides a new basis for acoustic parameters inversion, particularly density inversion.

On the inversion of sound fields above a locally reacting ground for direct impedance deduction

A complex root-finding algorithm is typically used to deduce the acoustic impedance of a locally reacting ground by inverting the measured sound fields. However, there is an issue of uniquely determining the impedance from a measurement of an acoustic transfer function. The boundary loss factor F, which is a complex function, is the source of this ambiguity. It is associated with the spherical wave reflection coefficient Q for the reflected sound field. These two functions are dependent on a complex parameter known as the numerical distance w. The inversion of F leading to the multiple solutions of w can be identified as the root cause of the problem. To resolve this ambiguity, the zeroes and saddle points of F are determined for a given source/receiver geometry and a known acoustic impedance. They are used to establish the basins containing all plausible solutions. The topography of Q is further examined in the complex w-plane. A method for identifying the family of solutions and selecting the physically meaningful branch is proposed. Validation is provided by using numerical simulations as well as the experimentally data. The error and uncertainties in the deduced impedance are quantified.

AVO Analysis of Spherical Reflected P Waves

AbstractIn this work, the reflection coefficient formula of spherical reflected P wave was derived. According to the derived formula, the variations of reflection coefficient and phases of class I and class III AVO media versus angle were calculated, and compared with the result of plane wave Zoeppritz formula. The results show that for the class I AVO medium, spherical wave reflection coefficient near the critical angle is more accurate than the Zoeppritz's computation result, and is highly influenced by the interface depth, i.e. with depth increasing, the trend of spherical wave AVO is close to plane wave AVO. Finally, numerical simulation technology was applied to verify and analyze the spherical wave AVO gather. The results show that a larger variation occurs for the refraction caused by the near‐critical or super‐critical angle through the class I AVO medium. But for the class III AVO phenomenon, because there is no critical angle problem, the spherical wave simulation results has little difference with Zoeppritz's. These calculation and analysis results would provide a theoretical basis for the AVO analysis of large offset gather data in practice.

Recent advances in sound propagation above a non-locally reacting ground

In the absence of atmospheric effects, the sound fields above a locally reacting ground can be accurately predicted by the Weyl-van der Pol formula. The solution is based on an asymptotic analysis to yield two terms: a direct and a ground reflected wave terms. The reflected wave term can be written as a product of a spherical wave reflection coefficient and the sound reflected from a rigid ground. In the contrary, it is more challenging to derive a similar formula for the sound fields above non-locally reacting grounds. In the past, an approximation in the same form as the Weyl-van der Pol formula has been used which becomes inadequate for layered grounds. In this presentation, a brief review of the asymptotic analysis will be discussed. An overview of the analytical and numerical approaches will be presented for obtaining accurate prediction of sound fields above the non-locally reacting ground. It will be further demonstrated that the reflection coefficient can be split exactly into two terms - a plane wave reflection coefficient and a ground wave term involving the boundary loss factor. The correlation between the numerical distance and the location of the surface wave pole will be examined.

Numerical study of the influence of roughness on coherence of signals reflected by the seafloor

The context of this work is the study of broadband acoustic signals reflected by the seafloor and recorded on a 15‐hydrophone vertical array. Two different aspects of the coherence of these signals were examined: in the temporal domain with the maximum of the cross‐correlation coefficients and in the frequency domain with the phase of the cross‐spectra. In previous work, it was shown that these coherence parameters are sensitive to the geoacoustic nature of the seafloor which may open the door for a new method of probing the seabed properties on both large scales (i.e., deterministic) and small scales (i.e., statistical). The aspect examined in this present work is the influence of the seafloor roughness on the coherence. To do so, the time series are simulated by a three‐step process: 1) computation of the plane‐wave reflection coefficient, 2) computation of the spherical wave reflection coefficient by a numerical integration of the Sommerfeld integral, and 3) Inverse Fourier Transform. Using these simulations, with the measured data as a guide, the potential for detecting and extracting roughness estimates from the coherence in complex multilayered seabed environments is explored.

A finite-element analysis of reflection coefficient variability due to interface roughness

The effect of sea floor roughness on the reflection of sound diverging from a spherical source is investigated. An ensemble of water‐sediment interface realizations are modeled, each with height statistics matching those of a rough surface prepared in a laboratory experiment. Using the finite‐element method to solve the Helmholtz equation, time harmonic solutions from 40 to 60 kHz are calculated. From these frequency domain solutions, the band limited impulse response time domain solution is constructed via Fourier synthesis. The first two moments of the spherical wave reflection coefficient are determined as a function of grazing angle, and results are compared to measurements from the laboratory experiment. [Work supported by ONR, Ocean Acoustics.]

Deduction of ground impedance from measurements of excess attenuation spectra

An efficient numerical method of deducing ground-surface impedance from measurements of short-range propagation from a point source is proposed. The method involves finding the root of a complex equation and uses the fact that the classical approximation for the spherical wave-reflection coefficient near grazing incidence is an analytic function of the impedance. Examples of fitting measured excess-attenuation spectra are given. Methods for ensuring the uniqueness of the resulting impedance fits are discussed. Subsequent deductions of parameters in a two-parameter impedance model are also considered and exemplified.

Changes in acoustic impedance of marine sediment covered with liquid pollutants

Acoustic properties of porous material are slightly changed by a coating layer, which is thin compared to the acoustic wavelength. These changes affect mainly the spherical wave reflection coefficient at grazing incidence. The phase change of a surface wave component is observed and enables the easy detection of the thin layer by measuring the interference of direct and guided surface waves. This method is investigated for screening observations in order to monitor liquid pollutants on marine sediment. Since sediment is usually not plain but more or less irregularly grooved, the coating by liquid pollutants will occur in patches. Laboratory experiments are carried out with rough surfaces of sediment to investigate the sensitivity of detecting an average change in impedance. The results are compared with a numerical model incorporating Biot theory [Tooms et al., J. Acoust. Soc. Am. 93, 171–181 (1998)], which was successfully applied to model experimental data of plain sediment (Harms et al., 134th meeting of ASA). Since the rough surface produces a significant backscattering, the liquid patches are detectable in the backscattered signal as well. Experimental results of this contribution are presented and the possibility of using this method for monitoring purposes discussed. [Work supported by German Ministry of Technology BMFT.]

Reflection coefficient of marine sediment covered with liquid pollutants

Acoustic sounding of sediment properties is usually carried out with waves at more or less normal incidence. This method, however, fails to detect thin layers (d≪λ) of undissolved fluid covering the sea-bottom due to the small difference in impedance compared to water. But the spherical wave reflection coefficient is strongly affected at grazing incidence and allows for the detection of these layers. Laboratory measurements in model scale as well as model calculations are carried out in order to determine the capabilities of the new measuring method. Experiments were done in a water tank with a hydrophone as a spherical broadband sound source. Various liquid chemicals of density greater than water are poured on the sand in small amounts, forming layers of less than 3-mm thickness. Measured excess attenuation of the pure sediment is in very good agreement with simulations carried out with a fast field program (FFLAGS) that incorporates Biot theory [S. Tooms et al., J. Acoust. Soc. Am. 93, 173–181 (1993)]. The excess attenuation of the sediment coated by a thin layer differs significantly. In particular, the interference pattern is broadened and shifted toward lower frequencies. This behavior is explainable by FFLAGS as well. [Work supported by BMBF.]

Downwind propagation of sound in an atmosphere with a realistic sound-speed profile: A semianalytical ray model

A ray model is presented for downwind propagation of sound in a layered atmosphere, with a logarithmic or a power profile of the sound speed. The model employs a new method to compute ray paths, which takes advantage of the fact that ray paths are ordered in groups of four. The paths are obtained from an iterative solution of an integral equation. This solution requires a very small computation time, compared with computation times of numerical ray models and wave models. Ground reflections are taken into account by either the plane wave reflection coefficient or the spherical wave reflection coefficient (a pseudoreflection coefficient for a point source above a ground surface). At high frequency both coefficients yield good agreement with numerical solutions of the Helmholtz equation. An empirical reflection coefficient is introduced to obtain good agreement at low frequency. The ray model is used to compute effects of atmospheric turbulence on the received sound spectrum. It is found that these effects are rather small for downwind sound propagation. The ray model is used further to compute the flow of acoustical energy along the different rays through the atmosphere. It is found that much of the energy flows along the highest rays, in particular at high frequency.