Geophysical Prospecting - 9 - Advances in Seismic Anisotropy, 2019

In this paper, we consider wave propagation in a periodically layered medium with orthorhombic symmetry. The weak‐contrast approximation is utilized to derive the low‐frequency dispersion in effective properties for P, S1 and S2 waves. We show that the dispersion term for all effective properties is controlled by the second‐order contrasts in elastic properties from the layers. We also compute the sensitivity matrices for second‐ and fourth‐order coefficients from eigenvalues of frequency‐dependent system matrix associated with kinematic parameters for individual wave modes.

The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine the Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth.

Degeneracies of the slowness surfaces of shear (and compressional) waves in low‐symmetry anisotropic media (such as orthorhombic), known as point singularities, pose difficulties during modelling and inversion, but can be potentially used in the latter as model parameter constraints. I analyse the quantity and spatial arrangement of point singularities in orthorhombic media, as well as their relation to the overall strength of velocity anisotropy. A classification scheme based on the number and spatial distribution of singularity directions is proposed. In normal orthorhombic models (where the principal shear moduli are smaller than the principal compressional moduli), point singularities can only be arranged in three distinct patterns, and media with the theoretical minimum (0) and maximum (16) number of singularities are not possible. In orthorhombic models resulting from embedding vertical fractures in transversely isotropic background, only two singularity distributions are possible, in contrast to what was previously thought. Although the total number of singularities is independent of the overall anisotropy strength, for general (non‐normal) orthorhombic models, different spatial distributions of singularities become more probable with increasing magnitude of anisotropy.

Although clay is composed of disconnected anisotropic clay platelets, many rock physics models treat the clay platelets in shale as interconnected. However, the clay matrix in shales can be modelled as anisotropic clay platelets embedded within a soft isotropic interplatelet region, allowing the influence of disconnected clay platelets on the elastic properties of the clay matrix to be analysed. In this model, properties of the interplatelet region are governed by its effective bulk and shear moduli, whereas the effective properties of the clay platelets are governed by their volume fraction, aspect ratio and elastic stiffness tensor. Together, these parameters implicitly account for variations in clay and fluid properties, as well as fluid saturation. Elastic stiffnesses of clay platelets are obtained from the literature, including both experimental measurements and first‐principles calculations of the full anisotropic (monoclinic or triclinic) elastic stiffness tensors of layered silicates. These published elastic stiffness tensors are used to compile a database of equivalent transverse isotropic elastic stiffness tensors, and other physical properties, for eight common varieties of layered silicates. Clay matrix anisotropy is then investigated by examining the influence of these different elastic stiffnesses, and of varying model parameters, upon the effective transverse isotropic elastic stiffness tensor of the clay matrix. The relationship between the different clay minerals and their associated anisotropy parameters is studied, and their impact on the resulting anisotropy of the clay matrix is analysed.

We consider a transversely isotropic medium that is long‐wave equivalent to a stack of thin, parallel, isotropic layers and is obtained using the Backus average. In such media, we analyse the relations among anisotropy parameters; Thomsen parameters, ε and δ, and a new parameter ϕ. We discuss the last parameter and show its essential properties; it equals 0 in the case of isotropy of equivalent medium and/or constant Lamé coefficient λ in layers. The second property occurs to make ϕ sensitive to variations of λ in thin‐bedded sequences. According to Gassmann, in isotropic media the variation of fluid content affects only the Lamé coefficient λ, not μ; thus, the sensitivity to changes of λ is an essential property in the context of possible detection of fluids. We show algebraically and numerically that ϕ is more sensitive to these variations than ε or δ. Nevertheless, each of these parameters is dependent on the changes of μ; to understand this influence, we exhibit comprehensive tables that illustrate the behaviour of anisotropy parameters with respect to specific variations of λ and μ. The changes of μ in layers can be presented by the Thomsen parameter γ that depends on them solely. Hence, knowing the values of elasticity coefficients of equivalent transversely isotropic medium, we may compute ϕ and γ, and based on the aforementioned tables, we predict the expected variation of λ; in this way, we propose a new method of possible fluid detection. Also, we show that the prior approach of possible detection of fluids, proposed by Berryman et al., may be unreliable in specific cases. To establish our results, we use the Monte Carlo method; for the range and chosen variations of Lamé coefficients λ and μ – relevant to sandstones – we generate these coefficients in thin layers and, after the averaging process, we obtain an equivalent transversely isotropic medium. We repeat that process numerous times to get many equivalent transversely isotropic media, and – for each of them  – we compute their anisotropy parameters. We illustrate ϕ, ε and δ in the form of cross‐plots that are relevant to the chosen variations of λ and μ. Additionally, we present a table with the computed ranges of anisotropy parameters that correspond to different variations of Lamé coefficients.

3D anisotropic waveform inversion could provide high‐resolution velocity models and improved event locations for microseismic surveys. Here we extend our previously developed 2D inversion methodology for microseismic borehole data to 3D transversely isotropic media with a vertical symmetry axis. This extension allows us to invert multicomponent data recorded in multiple boreholes and properly account for vertical and lateral heterogeneity. Synthetic examples illustrate the performance of the algorithm for layer‐cake and ‘hydraulically fractured’ (i.e. containing anomalies that simulate hydraulic fractures) models. In both cases, waveform inversion is able to reconstruct the areas which are sufficiently illuminated for the employed source‐receiver geometry. In addition, we evaluate the sensitivity of the algorithm to errors in the source locations and to band‐limited noise in the input displacements. We also present initial inversion results for a microseismic data set acquired during hydraulic fracturing in a shale reservoir.

In previous publications, we presented a waveform‐inversion algorithm for attenuation analysis in heterogeneous anisotropic media. However, waveform inversion requires an accurate estimate of the source wavelet, which is often difficult to obtain from field data. To address this problem, here we adopt a source‐independent waveform‐inversion algorithm that obviates the need for joint estimation of the source signal and attenuation coefficients. The key operations in that algorithm are the convolutions (1) of the observed wavefield with a reference trace from the modelled data and (2) of the modelled wavefield with a reference trace from the observed data. The influence of the source signature on attenuation estimation is mitigated by defining the objective function as the ℓ2‐norm of the difference between the two convolved data sets. The inversion gradients for the medium parameters are similar to those for conventional waveform‐inversion techniques, with the exception of the adjoint sources computed by convolution and cross‐correlation operations. To make the source‐independent inversion methodology more stable in the presence of velocity errors, we combine it with the local‐similarity technique. The proposed algorithm is validated using transmission tests for a homogeneous transversely isotropic model with a vertical symmetry axis that contains a Gaussian anomaly in the shear‐wave vertical attenuation coefficient. Then the method is applied to the inversion of reflection data for a modified transversely isotropic model from Hess. It should be noted that due to the increased nonlinearity of the inverse problem, the source‐independent algorithm requires a more accurate initial model to obtain inversion results comparable to those produced by conventional waveform inversion with the actual wavelet.

Transversely isotropic models with a tilted symmetry axis have become standard for imaging beneath dipping shale formations and in active tectonic areas. Here, we develop a methodology of wave‐equation‐based image‐domain tomography for acoustic tilted transversely isotropic media. We obtain the gradients of the objective function using an integral wave‐equation operator based on a separable dispersion relation that takes the symmetry‐axis tilt into account. In contrast to the more conventional differential solutions, the integral operator produces only the P‐wavefield without shear‐wave artefacts, which facilitates both imaging and velocity analysis. The model is parameterized by the P‐wave zero‐dip normal‐moveout velocity, the Thomsen parameter δ, anellipticity coefficient η and the symmetry‐axis tilt θ. Assuming that the symmetry axis is orthogonal to reflectors, we study the influence of parameter errors on energy focusing in extended (space‐lag) common‐image gathers. Distortions in the anellipticity coefficient η introduce weak linear defocusing regardless of reflector dip, whereas δ influences both the energy focusing and depth scale of the migrated section. These results, which are consistent with the properties of the P‐wave time‐domain reflection moveout in tilted transversely isotropic media, provide important insights for implementation of velocity model‐building in the image‐domain. Then the algorithm is tested on a modified anticline section of the BP 2007 benchmark model.

Transverse isotropy with a vertical axis of symmetry is a common form of anisotropy in sedimentary basins, and it has a significant influence on the seismic amplitude variation with offset. Although exact solutions and approximations of the PP‐wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill‐posed. In this paper, we propose a seismic amplitude inversion method for the transversely isotropic media with a vertical axis of symmetry based on a modified approximation of the reflection coefficient. This new approximation consists of only three model parameters: attribute A, the impedance (vertical phase velocity multiplied by bulk density); attribute B, shear modulus proportional to an anellipticity parameter (Thomsen's parameter ε−δ); and attribute C, the approximate horizontal P‐wave phase velocity, which can be well estimated by using a Bayesian‐framework‐based inversion method. Using numerical tests we show that the derived approximation has similar accuracy to the existing linear approximation and much higher accuracy than isotropic approximations, especially at large angles of incidence and for strong anisotropy. The new inversion method is validated by using both synthetic data and field seismic data. We show that the inverted attributes are robust for shale‐gas reservoir characterization: the shale formation can be discriminated from surrounding formations by using the crossplot of the attributes A and C, and then the gas‐bearing shale can be identified through the combination of the attributes A and B. We then propose a rock‐physics‐based method and a stepwise‐inversion‐based method to estimate the P‐wave anisotropy parameter (Thomsen's parameter ε). The latter is more suitable when subsurface media are strongly heterogeneous. The stepwise inversion produces a stable and accurate Thomsen's parameter ε, which is proved by using both synthetic and field data.

Fractures in elastic media add compliance to a rock in the direction normal to the fracture strike. Therefore, elastic wave velocities in a fractured rock will vary as a function of the energy propagation direction relative to the orientation of the aligned fracture set. Anisotropic Thomson–Haskell matrix Rayleigh‐wave equations for a vertically transverse isotropic media can be used to model surface‐wave dispersion along the principal axes of a vertically fractured and transversely isotropic medium. Furthermore, a workflow combining first‐break analysis and azimuthal anisotropic Rayleigh‐wave inversion can be used to estimate P‐wave and S‐wave velocities, Thomsen's ε, and Thomsen's δ along the principal axes of the orthorhombic symmetry. In this work, linear slip theory is used to map our inversion results to the equivalent vertically fractured and transversely isotropic medium coefficients. We carried out this inversion on a synthetic example and a field example. The synthetic data example results show that joint estimation of S‐wave velocities with Thomsen's parameters ε and δ along normal and parallel to the vertical fracture set is reliable and, when mapped to the corresponding vertically fractured and transversely isotropic medium, provides insight into the fracture compliances. When the inversion was carried out on the field data, results indicated that the fractured rock is more compliant in the azimuth normal to the visible fracture set orientation and that the in situ normal fracture compliance to tangential fracture compliance ratio is less than half, which implies some cementation may have occurred along the fractures. Such an observation has significant implications when modelling the transport properties of the rock and its strength. Both synthetic and field examples show the potential of azimuthal anisotropic Rayleigh‐wave inversion as the method can be further expanded to a more general case where the vertical fracture set orientation is not known a priori.

Computation of complex‐valued traveltimes provides an efficient approach to describe the seismic wave attenuation for applications like attenuation tomography, inverse Q filtering and Kirchhoff migration with absorption compensation. Attenuating acoustic transverse isotropy can be used to describe the directional variation of velocity and attenuation of P‐waves in thin‐bedding geological structures. We present an approximate method to solve the acoustic eikonal equation for an attenuating transversely isotropic medium with a vertical symmetry axis. We take into account two similar parameterizations of an attenuating vertical symmetry axis medium. The first parameterization uses the normal moveout velocity, whereas the second parameterization uses the horizontal velocity. For each parameterization, we combine perturbation theory and the Shanks transform in different ways to derive analytic solutions. Numerical examples show that the analytic solutions derived from the second parameterization yield better accuracy. The Shanks transform solution with respect to only the anellipticity parameter from the second parameterization is demonstrated numerically to be the most accurate among all the analytic solutions.

Utilizing shear‐wave (S‐wave) data acquired with compressional waves (P‐waves) is becoming more common as joint imaging and inversion techniques improve. Interest in S‐waves radiated from vertical sources and buried explosives exploits conversion to P‐waves as primary reflections (SP‐waves) for reducing acquisition costs and for application to legacy data. However, recent investigations overstate the extent of SP‐wave illumination and show isotropic processing results with narrow bandwidth frequency and wavenumber data. I demonstrate that illumination with SP‐waves is limited in general to near vertical polar angles up to around 30° or 35° for VP/VS of 2 or 3, respectively. At greater angles, S‐waves are typically in the P‐wave evanescent range and cannot excite SP‐wave reflections. Contrary to recent claims, these sources for P‐wave do not radiate SH‐waves polarized in horizontal planes in all azimuths. I show these properties for isotropic media with radiation expressions for amplitude derived in vector slowness coordinates. Also, I extend these expressions to transversely isotropic media with a vertical symmetry axis to show agreement with synthetic seismic data that only quasi SV‐waves are radiated and become more narrowly focused towards 45°. Furthermore, in orthorhombic media, synthetic data show that fast S1‐ and slow S2‐waves polarized parallel and perpendicular to fractures may appear as SV‐ and SH‐waves. For the partially saturated fracture model studied here, S1‐wave radiation has broader azimuthal illumination than slow S2‐waves, which are more narrowly focused in azimuth. These produce SP‐wave splitting signatures on vertical component reflection data that are nearly identical to PS‐wave signatures on radial horizontal component data. Separating these fast and slow SP‐waves is an additional processing challenge.

The collision of the Indian and Eurasian plates, to the east of the eastern Himalayan syntaxes, forms the Sanjiang lateral collision zone in the southeast margin of the Tibetan Plateau, where there are intense crustal deformation, active faults, earthquakes, as well as a metallogenic belt. Given the lack of adequate seismic data, shear‐wave splitting in this area has not been studied. With seismic data from a temporary seismic linear array, as well as permanent seismic stations, this paper adopts the identification on microseismic event to pick more events and obtains shear‐wave splitting parameters from local earthquakes. From the west to the east, the study area can be divided into three subzones. The “fast” polarization (i.e. the polarization of the fast shear wave) varies gradually from NNW to NS to NNE in these three subzones. The time delay of the slow shear wave (i.e. the time difference between the two split shear waves) also increases in the same direction, indicating the presence of seismic anisotropy above 25 km in the crust. Both shear‐wave splitting parameters are closely related to stress, faults and tectonics. The scatter and the “dual” (i.e. two) dominant orientations of the fast polarizations at several stations indicate strong distortions caused by nearby faults or deep tectonics. The anisotropic parameters are found to be related to some degree to the metallogenic belt. It is worth to further analyse the link between the anisotropic pattern and the metallogenic area, which suggests that shear‐wave splitting could be applied to study metallogeny. This paper demonstrates that the identification on microseismic event is a useful tool in detecting shear‐wave splitting details and exploring its tectonic implications.

Seismic anisotropy is a unique observational tool for remotely studying deformation and stress within the Earth. Effects of anisotropy can be seen in seismic data; they are due to mineral alignment, fractures or layering. Seismic anisotropy is linked to local stress and strain, allowing modern geophysics to derive geomechanical properties from seismic data for supporting well planning and fracking. For unravelling anisotropic properties of the crust, the teleseismic receiver functions methodology has started to be widely applied recently due to its ability in retrieving the three‐dimensional characteristics of the media sampled by the waves. The applicability of this technique is tested here by a field test carried out around the Kontinental Tiefbohrung site in southeastern Germany. We compare our results to previous investigations of the metamorphic rock pile of the Zone Erbendorf‐Vohenstrauss, drilled down to 9 km depth, which sampled an alternating sequence of paragneiss and amphibolite, in which a strong foliation has been produced by ductile deformation. The application of the receiver functions reveals the presence of two distinct anisotropic layers within the metamorphic rock pile at 0–4 km and below 6 km depth, with up to 8% anisotropy; the depth of these two layers corresponds to the location of mica‐rich paragneiss which show intense foliation, and finally proves the relation between the signal in the receiver functions, rock texture and presence of cracks. We have now the capability of providing insights from passive seismic data on geomechanical properties of the rocks, useful for geological exploration and engineering purposes, which will help influencing expensive drilling decisions thanks to future application of this seismic technique.