Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

Identification of lateral discontinuities via multi‐offset phase analysis of surface wave data

Surface wave methods are based on the inversion of observed Rayleigh wave phase‐velocity dispersion curves. The goal is to estimate mainly the shear‐wave velocity profile of the investigated site. The model used for the interpretation is 1D, hence results obtained wherever lateral variations are present cannot be considered reliable. In this paper, we study four synthetic models, all with a lateral heterogeneity. When we process the entire corresponding seismograms with traditional f‐k approach, the resulting 1D profiles are representative of the subsurface properties averaged over the whole length of the receivers lines. These results show that classical analysis disregards evidences of sharp lateral velocity changes even when they show up in the raw seismograms. In our research, we implement and test over the same synthetic models, a novel robust automated method to check the appropriateness of 1D model assumption and locate the discontinuities. This new approach is a development of the recent multi‐offset phase analysis with the following further advantages: it does not need previous noise evaluation and more than one shot. Only once the discontinuities are clearly identified, we confidently perform classical f‐k dispersion curve extraction and inversion separately on both sides of the discontinuity. Thus the final results, obtained by putting side by side the 1D profiles, are correct 2D reconstructions of the discontinuous S‐wave distributions obtained without any additional ad‐hoc hypotheses.     

Unified multi‐depth‐level field decomposition

Wavefield decomposition forms an important ingredient of various geophysical methods. An example of wavefield decomposition is the decomposition into upgoing and downgoing wavefields and simultaneous decomposition into different wave/field types. The multi‐component field decomposition scheme makes use of the recordings of different field quantities (such as particle velocity and pressure). In practice, different recordings can be obscured by different sensor characteristics, requiring calibration with an unknown calibration factor. Not all field quantities required for multi‐component field decomposition might be available, or they can suffer from different noise levels. The multi‐depth‐level decomposition approach makes use of field quantities recorded at multiple depth levels, e.g., two horizontal boreholes closely separated from each other, a combination of a single receiver array combined with free‐surface boundary conditions, or acquisition geometries with a high‐density of vertical boreholes. We theoretically describe the multi‐depth‐level decomposition approach in a unified form, showing that it can be applied to different kinds of fields in dissipative, inhomogeneous, anisotropic media, e.g., acoustic, electromagnetic, elastodynamic, poroelastic, and seismoelectric fields. We express the one‐way fields at one depth level in terms of the observed fields at multiple depth levels, using extrapolation operators that are dependent on the medium parameters between the two depth levels. Lateral invariance at the depth level of decomposition allows us to carry out the multi‐depth‐level decomposition in the horizontal wavenumber–frequency domain. We illustrate the multi‐depth‐level decomposition scheme using two synthetic elastodynamic examples. The first example uses particle velocity recordings at two depth levels, whereas the second example combines recordings at one depth level with the Dirichlet free‐surface boundary condition of zero traction. Comparison with multi‐component decomposed fields shows a perfect match in both amplitude and phase for both cases. The multi‐depth‐level decomposition scheme is fully customizable to the desired acquisition geometry. The decomposition problem is in principle an inverse problem. Notches may occur at certain frequencies, causing the multi‐depth‐level composition matrix to become uninvertible, requiring additional notch filters. We can add multi‐depth‐level free‐surface boundary conditions as extra equations to the multi‐component composition matrix, thereby overdetermining this inverse problem. The combined multi‐component–multi‐depth‐level decomposition on a land data set clearly shows improvements in the decomposition results, compared with the performance of the multi‐component decomposition scheme.     

A seismic reflection from isotropic‐fractured fluid‐saturated layer

Average elastic properties of a fluid‐saturated fractured rock are discussed in association with the extremely slow and dispersive Krauklis wave propagation within individual fractures. The presence of the Krauklis wave increases P‐wave velocity dispersion and attenuation with decreasing frequency. Different laws (exponential, power, fractal, and gamma laws) of distribution of the fracture length within the rock show more velocity dispersion and attenuation of the P‐wave for greater fracture density, particularly at low seismic frequencies. The results exhibit a remarkable difference in the P‐wave reflection coefficient for frequency and angular dependency from the fractured layer in comparison with the homogeneous layer. The biggest variation in behaviour of the reflection coefficient versus incident angle is observed at low seismic frequencies. The proposed approach and results of calculations allow an interpretation of abnormal velocity dispersion, high attenuation, and special behaviour of reflection coefficients versus frequency and angle of incidence as the indicators of fractures.     

The impact of surface‐wave attenuation on 3‐D seismic survey design

Three‐dimensional seismic survey design should provide an acquisition geometry that enables imaging and amplitude‐versus‐offset applications of target reflectors with sufficient data quality under given economical and operational constraints. However, in land or shallow‐water environments, surface waves are often dominant in the seismic data. The effectiveness of surface‐wave separation or attenuation significantly affects the quality of the final result. Therefore, the need for surface‐wave attenuation imposes additional constraints on the acquisition geometry. Recently, we have proposed a method for surface‐wave attenuation that can better deal with aliased seismic data than classic methods such as slowness/velocity‐based filtering. Here, we investigate how surface‐wave attenuation affects the selection of survey parameters and the resulting data quality. To quantify the latter, we introduce a measure that represents the estimated signal‐to‐noise ratio between the desired subsurface signal and the surface waves that are deemed to be noise. In a case study, we applied surface‐wave attenuation and signal‐to‐noise ratio estimation to several data sets with different survey parameters. The spatial sampling intervals of the basic subset are the survey parameters that affect the performance of surface‐wave attenuation methods the most. Finer spatial sampling will reduce aliasing and make surface‐wave attenuation easier, resulting in better data quality until no further improvement is obtained. We observed this behaviour as a main trend that levels off at increasingly denser sampling. With our method, this trend curve lies at a considerably higher signal‐to‐noise ratio than with a classic filtering method. This means that we can obtain a much better data quality for given survey effort or the same data quality as with a conventional method at a lower cost.     

Controlled source electromagnetic three‐dimensional grid‐modelling based on a complex resistivity structure of the seafloor: effects of acquisition parameters and geometry of multi‐layered resistors

A comprehensive controlled source electromagnetic (CSEM) modelling study, based on complex resistivity structures in a deep marine geological setting, is conducted. The study demonstrates the effects of acquisition parameters and multi‐layered resistors on CSEM responses. Three‐dimensional (3D) finite difference time domain (FDTD) grid‐modelling is used for CSEM sensitivity analysis. Interpolation of normalized CSEM responses provides attributes representing relative sensitivity of the modelled structures. Modelling results show that fine grid, 1 × 1 km receiver spacing, provides good correlations between CSEM responses and the modelled structures, irrespective of source orientation. The resolution of CSEM attributes decreases for receiver spacing >2 × 2 km, when using only in‐line data. Broadside data in the grid geometry increase data density by 100 – approximately 200% by filling in in‐line responses and improve the resolution of CSEM attributes. Optimized source orientation (i.e., oblique to the strike of an elongated resistor) improves the structural definition of the CSEM anomalies for coarse‐grid geometries (receiver spacing ≥3 × 3 km). The study also shows that a multi‐resistor anomaly is not simply the summation but a cumulative response with mutual interference between constituent resistors. The combined response of constituent resistors is approximately 50% higher than the cumulative response of the multi‐resistor for 0.5 Hz at 4000 m offset. A gradual inverse variation of offset and frequency allows differentiation of CSEM anomalies for multi‐layered resistors. Similar frequency‐offset variations for laterally persistent high‐resistivity facies show visual continuity with varying geometric expressions. 3D grid‐modelling is an effective and adequate tool for CSEM survey design and sensitivity analysis.     

Application of 3D vertical seismic profile multi‐component data to tight gas sands‡

Due to the complicated geophysical character of tight gas sands in the Sulige gasfield of China, conventional surface seismic has faced great challenges in reservoir delineation. In order to improve this situation, a large‐scale 3D‐3C vertical seismic profiling (VSP) survey (more than 15 000 shots) was conducted simultaneously with 3D‐3C surface seismic data acquisition in this area in 2005. This paper presents a case study on the delineation of tight gas sands by use of multi‐component 3D VSP technology. Two imaging volumes (PP compressional wave; PSv converted wave) were generated with 3D‐3C VSP data processing. By comparison, the dominant frequencies of the 3D VSP images were 10–15 Hz higher than that of surface seismic images. Delineation of the tight gas sands is achieved by using the multi‐component information in the VSP data leading to reduce uncertainties in data interpretation. We performed a routine data interpretation on these images and developed a new attribute titled ‘Centroid Frequency Ratio of PSv and PP Waves’ for indication of the tight gas sands. The results demonstrated that the new attribute was sensitive to this type of reservoir. By combining geologic, drilling and log data, a comprehensive evaluation based on the 3D VSP data was conducted and a new well location for drilling was proposed. The major results in this paper tell us that successful application of 3D‐3C VSP technologies are only accomplished through a synthesis of many disciplines. We need detailed analysis to evaluate each step in planning, acquisition, processing and interpretation to achieve our objectives. High resolution, successful processing of multi‐component information, combination of PP and PSv volumes to extract useful attributes, receiver depth information and offset/ azimuth‐dependent anisotropy in the 3D VSP data are the major accomplishments derived from our attention to detail in the above steps.     

Elastic full waveform inversion of multi‐component OBC seismic data

Elastic full waveform inversion of seismic reflection data represents a data‐driven form of analysis leading to quantification of sub‐surface parameters in depth. In previous studies attention has been given to P‐wave data recorded in the marine environment, using either acoustic or elastic inversion schemes. In this paper we exploit both P‐waves and mode‐converted S‐waves in the marine environment in the inversion for both P‐ and S‐wave velocities by using wide‐angle, multi‐component, ocean‐bottom cable seismic data. An elastic waveform inversion scheme operating in the time domain was used, allowing accurate modelling of the full wavefield, including the elastic amplitude variation with offset response of reflected arrivals and mode‐converted events. A series of one‐ and two‐dimensional synthetic examples are presented, demonstrating the ability to invert for and thereby to quantify both P‐ and S‐wave velocities for different velocity models. In particular, for more realistic low velocity models, including a typically soft seabed, an effective strategy for inversion is proposed to exploit both P‐ and mode‐converted PS‐waves. Whilst P‐wave events are exploited for inversion for P‐wave velocity, examples show the contribution of both P‐ and PS‐waves to the successful recovery of S‐wave velocity.     

Wave dispersion in high‐rise buildings due to soil–structure interaction

Nonparametric techniques for estimation of wave dispersion in buildings by seismic interferometry are applied to a simple model of a soil–structure interaction (SSI) system with coupled horizontal and rocking response. The system consists of a viscously damped shear beam, representing a building, on a rigid foundation embedded in a half‐space. The analysis shows that (i) wave propagation through the system is dispersive. The dispersion is characterized by lower phase velocity (softening) in the band containing the fundamental system mode of vibration, and little change in the higher frequency bands, relative to the building shear wave velocity. This mirrors its well‐known effect on the frequencies of vibration, i.e. reduction for the fundamental mode and no significant change for the higher modes of vibration, in agreement with the duality of the wave and vibrational nature of structural response. Nevertheless, the phase velocity identified from broader band impulse response functions is very close to the superstructure shear wave velocity, as found by an earlier study of the same model. The analysis reveals that (ii) the reason for this apparent paradox is that the latter estimates are biased towards the higher values, representative of the higher frequencies in the band, where the response is less affected by SSI. It is also discussed that (iii) bending flexibility and soil flexibility produce similar effects on the phase velocities and frequencies of vibration of a building. Copyright © 2014 John Wiley & Sons, Ltd.     

Frequency‐dependent anisotropy of porous rocks with aligned fractures

Naturally fractured reservoirs are becoming increasingly important for oil and gas exploration in many areas of the world. Because fractures may control the permeability of a reservoir, it is important to be able to find and characterize fractured zones. In fractured reservoirs, the wave‐induced fluid flow between pores and fractures can cause significant dispersion and attenuation of seismic waves. For waves propagating normal to the fractures, this effect has been quantified in earlier studies. Here we extend normal incidence results to oblique incidence using known expressions for the stiffness tensors in the low‐ and high‐frequency limits. This allows us to quantify frequency‐dependent anisotropy due to the wave‐induced flow between pores and fractures and gives a simple recipe for computing phase velocities and attenuation factors of quasi‐P and SV waves as functions of frequency and angle. These frequency and angle dependencies are concisely expressed through dimensionless velocity anisotropy and attenuation anisotropy parameters. It is found that, although at low frequencies, the medium is close to elliptical (which is to be expected as a dry medium containing a distribution of penny‐shaped cracks is known to be close to elliptical); at high frequencies, the coupling between P‐wave and SV‐wave results in anisotropy due to the non‐vanishing excess tangential compliance.     

Comparison of P‐wave attenuation models of wave‐induced flow

Wave‐induced oscillatory fluid flow in the vicinity of inclusions embedded in porous rocks is one of the main causes for P‐wave dispersion and attenuation at seismic frequencies. Hence, the P‐wave velocity depends on wave frequency, porosity, saturation, and other rock parameters. Several analytical models quantify this wave‐induced flow attenuation and result in characteristic velocity–saturation relations. Here, we compare some of these models by analyzing their low‐ and high‐frequency asymptotic behaviours and by applying them to measured velocity–saturation relations. Specifically, the Biot–Rayleigh model considering spherical inclusions embedded in an isotropic rock matrix is compared with White's and Johnson's models of patchy saturation. The modeling of laboratory data for tight sandstone and limestone indicates that, by selecting appropriate inclusion size, the Biot‐Rayleigh predictions are close to the measured values, particularly for intermediate and high water saturations.     

Observation of shear‐wave splitting in the multicomponent node data from Atlantis field,Gulf of Mexico

In 2005, a multicomponent ocean bottom node data set was collected by BP and BHP Billiton in the Atlantis field in the Gulf of Mexico. Our results are based on data from a few sparse nodes with millions of shots that were analysed as common receiver azimuthal gathers. A first‐order look at P‐wave arrivals on a common receiver gather at a constant offset reveals variation of P‐wave arrival time as a function of azimuth indicating the presence of azimuthal anisotropy at the top few layers. This prompted us to investigate shear arrivals on the horizontal component data. After preliminary processing, including a static correction, the data were optimally rotated to radial (R) and transverse (T) components. The R component shows azimuthal variation of traveltime indicating variation of velocity with azimuth; the corresponding T component shows azimuthal variation of amplitude and phase (polarity reversal). The observed shear‐wave (S‐wave) splitting, previously observed azimuthal P‐wave velocity variation and azimuthal P‐wave amplitude variation, all indicate the occurrence of anisotropy in the shallow (just below the seafloor) subsea sediment in the area. From the radial component azimuthal gather, we analysed the PP‐ and PS‐wave amplitude variation for the first few layers and determined corresponding anisotropy parameter and V_P/V_S values. Since fracture at this depth is not likely to occur, we attribute the observed azimuthal anisotropy to the presence of microcracks and grain boundary orientation due to stress. The evidence of anisotropy is ubiquitous in this data set and thus it argues strongly in favour of considering anisotropy in depth imaging for obtaining realistic subsurface images, at the least.     

Surface‐wave inversion for a P‐velocity profile with a constant depth gradient of the squared slowness

Surface waves are often used to estimate a near‐surface shear‐velocity profile. The inverse problem is solved for the locally one‐dimensional problem of a set of homogeneous horizontal elastic layers. The result is a set of shear velocities, one for each layer. To obtain a P‐wave velocity profile, the P‐guided waves should be included in the inversion scheme. As an alternative to a multi‐layered model, we consider a simple smooth acoustic constant‐density velocity model, which has a negative constant vertical depth gradient of the squared P‐wave slowness and is bounded by a free surface at the top and a homogeneous half‐space at the bottom. The exact solution involves Airy functions and provides an analytical expression for the dispersion equation. If the ratio is sufficiently small, the dispersion curves can be picked from the seismic data and inverted for the continuous P‐wave velocity profile. The potential advantages of our model are its low computational cost and the fact that the result can serve as a smooth starting model for full‐waveform inversion. For the latter, a smooth initial model is often preferred over a rough one. We test the inversion approach on synthetic elastic data computed for a single‐layer P‐wave model and on field data, both with a small ratio. We find that a single‐layer model can recover either the shallow or deeper part of the profile but not both, when compared with the result of a multi‐layer inversion that we use as a reference. An extension of our analytic model to two layers above a homogeneous half‐space, each with a constant vertical gradient of the squared P‐wave slowness and connected in a continuous manner, improves the fit of the picked dispersion curves. The resulting profile resembles a smooth approximation of the multi‐layered one but contains, of course, less detail. As it turns out, our method does not degrade as gracefully as, for instance, diving‐wave tomography, and we can only hope to fit a subset of the dispersion curves. Therefore, the applicability of the method is limited to cases where the ratio is small and the profile is sufficiently simple. A further extension of the two‐layer model to more layers, each with a constant depth gradient of the squared slowness, might improve the fit of the modal structure but at an increased cost.     

Laboratory measurements of guided‐wave propagation within a fluid‐saturated fracture

A fluid‐saturated flat channel between solids, such as a fracture, is known to support guided waves—sometimes called Krauklis waves. At low frequencies, Krauklis waves can have very low velocity and large attenuation and are very dispersive. Because they propagate primarily within the fluid channel formed by a fracture, Krauklis waves can potentially be used for geological fracture characterization in the field. Using an analogue fracture consisting of a pair of flat slender plates with a mediating fluid layer—a trilayer model—we conducted laboratory measurements of the velocity and attenuation of Krauklis waves. Unlike previous experiments using ultrasonic waves, these experiments used frequencies well below 1 kHz, resulting in extremely low velocity and large attenuation of the waves. The mechanical compliance of the fracture was varied by modifying the stiffness of the fluid seal of the physical fracture model, and proppant (fracture‐filling high‐permeability sand) was also introduced into the fracture to examine its impact on wave propagation. A theoretical frequency equation for the trilayer model was derived using the poroelastic linear‐slip interface model, and its solutions were compared to the experimental results.     

3‐D shallow‐water seismic survey evaluation and design using the focal‐beam method: a case study offshore Abu Dhabi

For 3‐D shallow‐water seismic surveys offshore Abu Dhabi, imaging the target reflectors requires high resolution. Characterization and monitoring of hydrocarbon reservoirs by seismic amplitude‐versus‐offset techniques demands high pre‐stack amplitude fidelity. In this region, however, it still was not clear how the survey parameters should be chosen to satisfy the required data quality. To answer this question, we applied the focal‐beam method to survey evaluation and design. This subsurface‐ and target‐oriented approach enables quantitative analysis of attributes such as the best achievable resolution and pre‐stack amplitude fidelity at a fixed grid point in the subsurface for a given acquisition geometry at the surface. This method offers an efficient way to optimize the acquisition geometry for maximum resolution and minimum amplitude‐versus‐offset imprint. We applied it to several acquisition geometries in order to understand the effects of survey parameters such as the four spatial sampling intervals and apertures of the template geometry. The results led to a good understanding of the relationship between the survey parameters and the resulting data quality and identification of the survey parameters for reflection imaging and amplitude‐versus‐offset applications.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.     

An adaptive free‐surface expression for three‐dimensional finite‐difference frequency‐domain modelling of elastic wave

Finite‐difference frequency‐domain modelling of seismic wave propagation is attractive for its efficient solution of multisource problems, and this is crucial for full‐waveform inversion and seismic imaging, especially in the three‐dimensional seismic problem. However, implementing the free surface in the finite‐difference method is nontrivial. Based on an average medium method and the limit theorem, we present an adaptive free‐surface expression to describe the behaviour of wavefields at the free surface, and no extra work for the free‐surface boundary condition is needed. Essentially, the proposed free‐surface expression is a modification of density and constitutive relation at the free surface. In comparison with a direct difference approximate method of the free‐surface boundary condition, this adaptive free‐surface expression can produce more accurate and stable results for a broad range of Poisson's ratio. In addition, this expression has a good performance in handling the lateral variation of Poisson's ratio adaptively and without instability.     

中国东北地区噪声层析成像

中国东北地区是中国唯一的深震孕育区,获取该区的壳幔结构,对于研究板块俯冲、深震以及火山活动等有重要的科学意义.本文利用该区159个固定台站2011年1月至2012年6月和27个流动台站2011年1月至2011年6月间的垂向波形连续记录,计算了台站间的预估格林函数,并采用基于连续小波变换的时频分析方法,测量了双台路径上瑞雷波的群速度和相速度频散曲线.通过质量控制和筛选,最终得到了2204条路径上周期5~40 s的群/相速度频散曲线.检测板测试表明,反演结果可以达到2°×2°的分辨.利用Ditmar & Yanovskaya反演方法,我们得到了研究区（105°E—135°E,39°N—52°N）周期8~30 s的瑞雷波的群速度和相速度分布图.不同周期的群/相速度分布图,反映了不同深度S波速度的横向变化情况.研究结果显示：中国东北地区的地壳上地幔S波速度结构存在横向非均匀性.短周期（如8 s）的群/相速度分布与地表地质构造具有明显的相关性,具体来说,山区显示为高速,沉积盆地显示为低速;随着周期的增大（如15 s,22 s）,地形的控制作用相对减弱;较长周期（如30 s）的群/相速度分布与地壳厚度密切相关.     

Quantitative imaging of the Permo‐Mesozoic complex and its basement by frequency domain waveform tomography of wide‐aperture seismic data from the Polish Basin

In this study we present the workflow and results of 2D frequency domain waveform tomography applied to the global‐offset seismic data acquired in central Poland along a 50‐km long profile during the GRUNDY 2003 experiment. The waveform tomography method allows full exploitation of the wide‐aperture content of these data and produces in a semi‐automatic way both the detailed P‐wave velocity model and the structural image (i.e., perturbations in respect to the starting model). Thirteen frequencies ranging from 4 to 16 Hz were inverted sequentially, gradually introducing higher wavenumbers and more details into the velocity models. Although the data were characterised by relatively large shot spacings (1.5 km), we obtained clear images both of the Mesozoic and Permian sedimentary cover. Velocity patterns indicated facies changes within the Jurassic and Zechstein strata. A high velocity layer (ca. 5500 m/s) was found near the base of Triassic (Scythian), which made the imaging of a deeper layer difficult. Nevertheless, we were able to delineate the base of the Permian (i.e., base of the Rotliegend), which was not possible to derive from conventional common‐depth‐point processing, as well as some deeper events, attributed to the Carboniferous. The sub‐Permian events formed a syn‐form which favoured our previous interpretation of a depression filled with Upper Carboniferous molasse. The validity of the waveform tomography‐derived model was confirmed by well‐log data. Forward ray‐tracing modelling and synthetic seismograms calculations provided another justification for the key structures present in the waveform tomography model.     

Converted‐wave beam migration with sparse sources or receivers

Gaussian beam depth migration overcomes the single‐wavefront limitation of most implementations of Kirchhoff migration and provides a cost‐effective alternative to full‐wavefield imaging methods such as reverse‐time migration. Common‐offset beam migration was originally derived to exploit symmetries available in marine towed‐streamer acquisition. However, sparse acquisition geometries, such as cross‐spread and ocean bottom, do not easily accommodate requirements for common‐offset, common‐azimuth (or common‐offset‐vector) migration. Seismic data interpolation or regularization can be used to mitigate this problem by forming well‐populated common‐offset‐vector volumes. This procedure is computationally intensive and can, in the case of converted‐wave imaging with sparse receivers, compromise the final image resolution. As an alternative, we introduce a common‐shot (or common‐receiver) beam migration implementation, which allows migration of datasets rich in azimuth, without any regularization pre‐processing required. Using analytic, synthetic, and field data examples, we demonstrate that converted‐wave imaging of ocean‐bottom‐node data benefits from this formulation, particularly in the shallow subsurface where regularization for common‐offset‐vector migration is both necessary and difficult.