Post-critical Angles Research Articles

Approximate reflection and transmission coefficients at an interface between two contrasting viscoelastic anisotropic half-spaces

SUMMARY We present and compare two new methods to calculate the reflection and transmission coefficients of qP, qSV and qSH waves at a flat interface between two contrasting viscoelastic anisotropic half-spaces. The first method is a new version of the real ray tracing method that extends the traditional real ray tracing method to tackle the triplications of the qSV wave. This new version is valid for computing the reflection and transmission coefficients of the qP and qSV waves for all pre- and post-critical angles of incidence, which cannot be handled by the previous approach. The second method employs the real slowness direction and Snell's law to provide approximate reflection and transmission coefficients for such wave modes. This approximate method has attractive advantages which are not only the avoidance of the searching process inherent in the real ray tracing method but also offers competitive economical solutions to the first method. Many viscoelastic VTI models are utilized to validate the two methods. The numerical results show that (1) the ray angles, ray velocities, ray Q-factors, and the magnitudes of the reflection and transmission coefficients obtained by the two methods closely agree with each other, except in the vicinity of the critical angles and (2) the phase angles of the reflected and transmitted waves exhibit complexities due to the existence of some reverse phenomena between the approximate and the real ray tracing solutions. However, such phase discrepancies do not affect the magnitudes of the reflection and transmission coefficients. Moreover, the energy ratios of the different wave modes satisfy the energy conservation criterion at the interface.

Reflection coefficient frequency-dependent inversion of a planar interface with spherical waves: Using critical and post-critical angles

As a zero-order approximation of spherical wave theory, plane wave theory based on the Zoepprtizs equations and various linear approximations has been widely used in seismic exploration, such as prestack inversion, deconvolution and Q compensation. Since localized sources excited in field acquisition usually produce spherical wavefronts, the plane wave approximation fails to correctly utilize post-critical information and frequency-dependent characteristics have been completely neglected. Unlike conventional prestack inversion using multiple angles, the existence of frequency dependency makes the inversion using reflected record of a single angle possible. Therefore, the feasibility of frequency-dependent inversion of a planar interface using one angle is analyzed and investigated. A strategy of choosing frequency-band in frequency-domain inversion is given by fitting the geometric distribution of reflection coefficients on a complex plane. The sensitivity, resolution matrix and objective function are adopted to analyze the coupled relationship between different parameters and to determine the parameters to be inverted. Combining a global optimal algorithm, the effectiveness of frequency-dependent inversion is verified using synthetic data and cross-well seismic data. The velocity and density parameters are well-inverted, especially when the data of critical offset is included. Moreover, the inversion uniting frequency-dependency with multiple angles offers more accurate estimations for both density and velocity parameters.

Seismic analysis of a system of dam-massed foundation-reservoir under inclined excitation

One of the acceptable assumptions in engineering practice is vertical propagation of earthquake waves. When the source of earthquake is located very deep in the ground, this assumption is valid, but for sources located in shallow ground, it loses its viability. In this study, linear seismic analysis of a system of concrete dam-massed foundation-reservoir is performed under inclined earthquake excitation. Both P- and SV-type earthquakes are considered for the purpose of the seismic analysis. To consider the effects of inhomogeneous waves for the case of SV wave propagation, post-critical angles are also considered in the analysis. To investigate the effects of earthquake frequency content on the results, three different records with contents of low, intermediate, and high frequencies are selected. Results indicate that considering vertical propagation underestimates the obtained responses. For the case of SV-type earthquakes, post-critical angles must be looked at. Frequency content of the earthquake also has considerable effects on trend and absolute values of responses.

Amplitude variation with offset inversion using acoustic-elastic local solver

Conventional amplitude variation with offset (AVO) inversion analysis uses the Zoeppritz equations, which are based on a plane-wave approximation. However, because real seismic data are created by point sources, wave reflections are better modeled by spherical waves than by plane waves. Indeed, spherical reflection coefficients deviate from planar reflection coefficients near the critical and postcritical angles, which implies that the Zoeppritz equations are not applicable for angles close to critical reflection in AVO analysis. Elastic finite-difference simulations provide a solution to the limitations of the Zoeppritz approximation because they can handle near- and postcritical reflections. We have used a coupled acoustic-elastic local solver that approximates the wavefield with high accuracy within a locally perturbed elastic subdomain of the acoustic full domain. Using this acoustic-elastic local solver, the local wavefield generation and inversion are much faster than performing a full-domain elastic inversion. We use this technique to model wavefields and to demonstrate that the amplitude from within the local domain can be used as a constraint in the inversion to recover elastic material properties. Then, we focus on understanding how much the amplitude and phase contribute to the reconstruction accuracy of the elastic material parameters ([Formula: see text], [Formula: see text], and [Formula: see text]). Our results suggest that the combination of amplitude and phase in the inversion helps with the convergence. Finally, we analyze elastic parameter trade-offs in AVO inversion, from which we find that to recover accurate P-wave velocities we should invert for [Formula: see text] and [Formula: see text] simultaneously with fixed density.

Goos–Hänchen induced normal moveout correction for wide-angle reflections

The Goos–Hanchen (GH) lateral shift has been theoretically simulated and observed in lab. GH lateral shift introduces additional traveltime and distance when the incidence angles are larger than the critical angle. For seismic wave, this GH shift is caused by the total reflection of an incident beam of P-wave from low to high impedance medium at near and post-critical angles. Because of its large influences on traveltime and lateral shift displacement, the GH shift should be corrected in normal moveout (NMO) correction for wide-angle reflections in seismic data processing. In this paper, we derive the partial derivatives of reflection coefficients (PP- and PSV-wave) with respect to circular frequency using the Zoeppritz equations. Then, the delay time and NMO correction term with the behavior of GH lateral shift is derived. The characteristics of delay time and GH induced time differences are analyzed. The results show that this GH shift could be either positive or negative and the delay on time has large influences on seismic reflections when the incidence angles are larger than the critical angles. The efficiency of GH induced NMO correction is tested using synthetic seismic data. The GH induced NMO correction should be done for wide-angle reflections during the progress of seismic data processing.

Novel wide-angle AVO attributes using rational function

Conventional AVO inversion employs Zoeppritz equations and various approximations to them to obtain the reflection coefficients of plane-waves, which are confined to a certain (small) angle range (mostly below $$40^{\circ }$$). However, near the critical angles (wide-angle), reflections at the post-critical angles provide much more potential for velocity and density inversion because of the large amplitudes and phases-shifted waveforms, while the Zoeppritz equations are not applicable anymore. Hence, there is a strong demand for the research into wide-angle AVO. With reflection coefficients at wide-angle corresponding to the features of rational function, we try to approximate the seismic data with vector fitting which is used to obtain the rational zero-pole and residual properties of wide-angle AVO. We apply this technique to classify AVO type and recognize the lithology. Our experiment shows that extending our research into wide-angle AVO is very promising in gathering richer data for a more accurate seismic analysis.

Frequency-Dependent Spherical-Wave Reflection in Acoustic Media: Analysis and Inversion

Spherical-wave reflectivity (SWR), which describes the seismic wave reflection in real subsurface media more accurately than plane-wave reflectivity (PWR), recently, again attracts geophysicists’ attention. The recent studies mainly focus on the amplitude variation with offset/angle (AVO/AVA) attributes of SWR. For a full understanding of the reflection mechanism of spherical wave, this paper systematically investigates the frequency-dependent characteristics of SWR in a two-layer acoustic medium model with a planar interface. Two methods are used to obtain SWR. The first method is through the calculation of classical Sommerfeld integral. The other is by 3D wave equation numerical modeling. To enhance computation efficiency, we propose to perform wave equation simulation in cylindrical coordinates, wherein we for the first time implement unsplit convolutional perfectly matched layer as the absorbing boundary. Both methods yield the same results, which demonstrate the validity and accuracy of the computation. From both the numerical tests and the theoretical demonstration, we find that the necessary condition when frequency dependence of SWR occurs is that the upper and lower media have different velocities. At the precritical small angle, the SWR exhibits complicated frequency-dependent characteristics for varying medium parameters. Especially when the impedance of upper medium equals that of lower one, the PWR is zero according to geometric seismics. Whereas the SWR is nonzero: the magnitude of SWR decreases with growing frequency, and approaches that of the corresponding PWR at high frequency; the phase of SWR increases with growing frequency, but approaches 90° or −90° at high frequency. At near- and post-critical angles, large difference exists between SWR and PWR, and the difference is particularly great at low frequencies. Finally, we propose a nonlinear inversion method to estimate physical parameters and interface depth of media by utilizing the frequency-dependent SWR at a precritical small angle. Tests show that good inversion results can be obtained when the input data contain both low and moderate frequencies.

The influence of the Goos–Hänchen effect on seismic data processing and AVO in attenuating media

The Goos–Hänchen (GH) effect induced by the total reflection of an incident beam of P-wave from a low-impedance medium to a high-impedance medium at near- and post-critical angles was theoretically simulated and discussed. For both PP- and PSV-waves, there may be large GH shifts (GHS) and penetration depths (PD) for both lossless and attenuating media. As the Q-factor increases, or the frequency of the seismic wave decreases, the GH effect is increased. However, in attenuating media, there may be non-zero GHS and PD at all non-zero incident angles, not just post-critical angles. GHS may be either positive or negative, while PD is positive only. Compared to the Q-factor in the incident medium, the Q-factor in the transmission medium may play a more dominant role in the determination of reflection coefficients, GHS, and PD. The GH-induced normal moveout (NMO) discrepancy of the PSV-wave may be larger than that of the PP-wave. Due to the GH effect, there may be an angle discrepancy (at fixed offset) between the GH-modified incident angle and the traditional incident angle. In addition, the GH effect at a given offset may produce two or three reflected waves, from different incident angles. These results suggest that, within their assumptions, the GH effect may lead to errors in NMO estimates and the vertical location of the reflector. Furthermore, there may be errors in offsets, incident angles, and reflection amplitudes, in the analyses of the amplitude variation with offset (AVO). These GH effects might be more important for seismic data at fixed offsets and shallow layers, and for sonic log data, which might fall into the post-critical angle regime. Therefore, there may be a necessity to take into account the GH effect in the interpretation of wide-angle reflection data in NMO and AVO analyses.

The Cause and Effects of Multilayer-Generated Guided-Waves

Seismic methods are commonly used by the petroleum industry to obtain reflections from geological boundaries that are prospective for oil and gas production. In the presence of nearsurface high-velocity layers (i.e. sea-floor carbonates or basalts), a significant portion of the primary seismic energy is scattered. This results in very poor reflection data as little energy remains to be transmitted further and subsequently reflected back to the surface. Such loss of energy is often characterised by the generation of guided-wave energy in low-velocity layers being sandwiched between two high-velocity layers. This results in the dominant horizontal propagation of seismic energy as guided-waves with leaky modes.Guided-wave data has been physically modelled to allow an indepth analysis of the problem. This paper shows that there is a relationship between the group velocity, dominant frequency and dominant wavelength of the guided-waves, to the thickness of the low-velocity layer. An empirical formula is presented to estimate the interval velocity of the low-velocity layer, for the case of horizontally layered media. The results of a three-component recording indicate that guided-wave propagation is of pseudo-Rayleigh wave particle motion. This implies that the generation of guided-waves is not solely due to total internal reflections within the waveguide, but also involves surface wave propagation along the surface of the underlying high-velocity layer. It is proposed that this surface wave refracts energy into the low-velocity layer at postcritical angles, thus providing additional energy for total internal reflections within the waveguide. Mode conversion from P- to SV-waves also occur at reflections beyond the critical angle, becoming more prominent as the offset increases (Ogilvie and Purnell, 1996).

Estimation of fracture height using microseismicity associated with hydraulic fracturing

The ratio of horizontal‐to‐vertical (H/V) particle velocity in background microseismic radiation associated with hydraulic fracturing is substantially higher in the dilatant, low‐velocity fractured zone than it is outside. This provides a useful diagnostic for determining the height of the fractured zone. Numerical synthesis of guided wave phenomena within the low‐velocity fractured zone accounts for much of the observed behavior, but measured H/V patterns are not totally consistent with either pure tensile or pure shear sources. A composite model containing both tensile‐compressional sources and asperity shear failures appears to satisfy the main observations better than either source type does alone. This composite is consistent with current models of earthquake aftershock sequences, which also have different mechanisms at the edges and in the interior of a fracture zone (tensile and shear, respectively). The H/V phenomenon is consistent with a predominance of energy with shear‐wave polarization traveling at postcritical angles, produced either directly by the source or by P-to-S conversion at the edges of the fracture zone. The H/V ratios are enhanced by increasing dilatancy, which decreases the velocity within the fracture zone.

Use of mode-converted waves in marine seismic data to investigate the lithology of the sub-bottom sediments in Isfjorden, Svalbard

The compression wavefield is efficiently converted to shear-wave energy at post-critical angles in areas of high impedance contrast at the sea floor. We have analysed mode-converted shear waves in a data set acquired with a hybrid marine/land geometry in Isfjorden, Svalbard. Through a kinematic 2D ray-tracing modellingVp/Vs ratios for part of the uppermost 5km of the crust are obtained. Low values (Vp/Vs=1.65) are tentatively associated with the section of Devonian sandstones which appears to attain a minimum thickness of 1.5km below 3 km depth about 10km west of Kapp Thorden.

Diffraction and conversion of elastic waves at a corrugated interface

Numerical modeling is used to investigate the effect of small‐scale irregularities of a reflecting boundary on elastic wave reflections. The scattered wave field is computed by using a discretized form of boundary integral equations and a plane‐wave decomposition of seismic wave fields. For various values of incidence angle of the P wave, we compute the distribution of diffracted energy for both P waves and S waves as a function of reflection angle. We show that corrugations with mean wavelength of the order of, or smaller than, the seismic wavelength have little effect on the reflected P wave. However, the pattern of P‐to‐S conversion is very different from that with a plane boundary. Scattered S waves appear at postcritical angles for any angle of incidence of the P wave. The amplitude of these nongeometrical shear waves decreases rapidly with decreasing amplitude of the corrugations, or when the mean wavelength of the corrugations becomes larger than the dominant seismic wavelength. The local geometry of the irregularities has a negligible effect on the scattered S waves. By analogy with perturbation theory, we propose interpreting the postcritically scattered S waves as the contribution to the shear wave field of converted inhomogeneous P waves diffracted along the boundary.

An experimental study of the penetration of a water–sediment interface by a parametric beam

A recent theoretical study [D. J. Wingham, J. Acoust. Soc. Am. 76, 1192–1200 (1984)] presented calculations of the farfield secondary pressure in sediment due to a parametric array incident on a water–sediment interface, assuming the array to be completely truncated at the interface. An experimental investigation of this theory is undertaken and the measured secondary pressure contours throughout the sediment are compared with the theoretical estimates. The theory is shown to provide good estimates of the secondary field. The measured secondary pressure variation as a function of beam incidence angle at various ranges is presented and it is apparent that only when the array is truncated in its primary nearfield does the performance of the parametric array differ significantly from a conventional array. It is concluded that the anomalous behavior of parametric arrays, at postcritical angles, relies on virtual sources which are both close to the interface and suitably phased to radiate into the sediment.

A theoretical study of the penetration of a water sediment interface by a parametric beam

A theoretical investigation of the farfield secondary radiation present in sediment due to a parametric array incident on a water–sediment boundary is conducted, based on the assumption that the interface completely truncates the parametric array in the nearfield of its primary radiation. The field is found to be due to two apertures, one coincident with the transducer, the second with the truncation. At subcritical angles of primary beam incidence the field is similar to that produced by a conventional beam. At postcritical angles of primary beam incidence the presence of the truncation aperture results in a steeper and deeper penetration of the sediment than in the conventional case. It is concluded that the presence of virtual sources near the interface is responsible for the postcritical penetration of sediment by a parametric beam.

Inversion for sediment structure using multichannel data

The problem of inferring a velocity or impedance structure for a layered sediment has traditionally been posed in terms of a restricted data set, such as a normal incidence reflection seismogram or a refraction travel time curve. Data acquisition systems used for petroleum exploration record 96 or more channels in a linear array, and thus obtain a virtually unaliased representation of the wavefield. The data contains reflections from all angles of incidence, including postcritical angles, all normal modes, leaky modes, and head waves. A key tool in dealing with this kind of data is the Radon transform, which transforms the data set into a plane-wave representation. A raw data set u(x,t) is transformed into v(p,τ), where p, which parameterizes the plane waves, is the x component of slowness, and τ is a reduced (intercept) time. The ensemble of plane-wave seismograms is the basis for inversion of the data to obtain a one-dimensional velocity/impedance model. Modeling the lossy elastic transient plane-wave response is easily done by one of the modifications of the Thomson-Haskell matrix method. The algorithm can be vectorized to run very rapidly on a machine with an array processor. The object of the inversion is to obtain a model whose plane-wave decomposition agrees exactly with that of the observed wavefield. In the absence of noise, one plane-wave seismogram suffices to determine the model. An ensemble for a range of wave parameters overdetermines the model. In the presence of noise, the problem has the character of an ill-posed, overdetermined least squares problem. In order to carry through the solution by means of least squares, it is necessary to go through a construction procedure to make the trial model linearly close to the correct one in some parameter subspace. Several construction procedures proceed by stripping from the top down, although none have been reduced to a fully automatic alogrithm, and require some human intervention in the iterations.