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In velocity model building using full waveform inversion, diving waves are a source of the low-frequency information needed to update the low- to intermediate wavenumber components in the velocity model. Because full waveform inversion is a highly nonlinear optimization problem attempting to fit the information from the whole wavefield, it is prone to get stuck in local minimums and experience cycle skipping if the initial model does not contain accurate enough information about the low-wavenumbers to constrain the kinematics. Such initial velocity models will typically be estimated using alternative inversion methods that only apply information from parts of the wavefield in the forward modelling and recorded data. In migration velocity analysis, the unfocusing of the image, caused by inconsistency in the kinematics of reflected waves, is measured and is used to update the velocity model by applying differential semblance optimization. The same concept is applicable when using information from diving waves. For diving waves, little is known about the focusing behaviour, also known as imaging moveout, especially in the presence of azimuthal anisotropy, such as orthorhombic anisotropy, which is a good approximation for volcanic plains and sedimentary basins. Obtaining more knowledge about the focusing behaviour of diving waves for this type of anisotropy may help when choosing parameters to invert during inversion that yield a low trade-off between the parameters. Here, we derive approximate series-based equations describing the imaging moveout of diving qP waves in acoustic factorized orthorhombic anisotropic media. Due to the complexity of the analytical solutions for diving wave properties in such models, we find a vertical transverse isotropic type of simplification that manages to conserve the curvature and horizontal velocity of the diving qP wave. The accuracy of the simplified expressions is compared to the exact form and found to yield small errors. To verify the validity of the derived equations for imaging moveout, we compare the generated surfaces with images containing the shape of the unfocusing obtained using reverse time migration, showing a good fit. Of the anisotropic parameters defining the acoustic factorized orthorhombic model, the |$\epsilon _1$| and |$\epsilon _2$| parameters have the largest influence on the shape of the surface. The anelliptical parameters are shown only to induce minimal changes. Our method to obtain the series expressions describing the imaging moveout in the acoustic factorized orthorhombic model is also applicable to higher symmetry special cases of orthorhombic anisotropy by setting appropriate conditions. Here, this is demonstrated by additionally finding an approximate series expression describing the imaging moveout in acoustic factorized vertical transversely isotropic media, yielding a good match with the image obtained from reverse time migration. [ABSTRACT FROM AUTHOR]

SUMMARY: We define double (S1S2) and triple (PS1S2) singularity points and their degeneracy classes in triclinic anisotropic media. We derive equations for the group velocity image for all these cases. The degeneracy classes are defined by factorization of quadratic (double singularity point) and cubic (triple singularity point) forms with three variables. [ABSTRACT FROM AUTHOR]

We have derived an analytical approximate expression to estimate the delay in diving seismic waves due to thin layers of CO2. The expression is valid for high frequencies and can be used to estimate the delay in diving waves at seismic frequencies for large separations between the source and receiver (offset). The approximation may be used to assess CO2 detection limits using diving waves and to support survey planning for CO2 monitoring and full-waveform inversion (FWI) cycle skipping analysis. In this study, we analyse the diving-wave response to a thin layer of CO2 for band-limited data using acoustic finite-difference modelling, and compare the results against the analytical calculations. We find that the responses are offset-dependent and related to double- and single-leg interactions between the diving waves and the CO2. To test the methods, we created a synthetic representation of the 2010 subsurface conditions for the top CO2 layer at the Sleipner storage complex in the North Sea, by combining base and monitor post-stack seismic data with field velocity trends. Using the acoustic finite-difference method, we model pre-stack data that captures the complexity of field data and demonstrate the use of the diving-wave delay for CO2 migration monitoring and CO2 thin layer detection. [ABSTRACT FROM AUTHOR]

Accurate modelling of the frequency-dependence of seismic wave velocity related to fracture system and fluid content is crucial to the quantitative interpretation of seismic data in fractured reservoirs. Both mesoscale fractures and patchy saturation effects can cause significant velocity dispersion and attenuation in the seismic frequency band due to wave-induced fluid flow (WIFF) mechanism. Considering the coupled impact of 'mesoscale fractures' and 'patchy saturation', we derive expressions for the frequency-dependent anisotropy in partially saturated porous rock containing two fracture sets with different orientations, sizes and connectivities. Especially, we simplify the rock-physics model as an orthorhombic (ORT) media by assuming the mesoscale fractures to be orthogonal and give the explicit expressions for frequency-dependent elastic constants. Finally, we give the expressions for the frequency-dependent phase velocity in patchy saturated and fractured ORT media and investigate the effect of patchy saturation on P-wave velocity at different polar and azimuth angles. In this paper, we investigate the effects of fluid saturation and fluid pressure on frequency-dependent velocities and Thomsen anisotropy parameters. Also, the effect of the relative permeability is very noticeable. The relaxation frequency can be lower in partially saturated fractured rocks compared with the fully saturated case, which makes the rock have a larger stiffness. The non-monotonic relationships between frequency-dependent anisotropy and fluid saturation add complexity to seismic forward modelling and inversion in reservoirs with complex fracture patterns. [ABSTRACT FROM AUTHOR]

Application of the Floquet theorem and the matrix propagator method reduces the problem of the plane wave propagation in a periodically layered anisotropic media, to analysis of the properties of stationary envelopes of different wave modes propagating up- and downwards. We analyse the interchanging of stop- and pass-bands and their structure at low frequencies for a periodically layered medium with monoclinic symmetry. The analysis shows the effect of interaction between P , S 1 and S 2 wave multipliers for stop- and pass-band structure and gives insight into the wave propagation in vertically heterogeneous anisotropic media which is important in modelling and interpretation of seismic data. [ABSTRACT FROM AUTHOR]

Orthorhombic models are often used in the seismic industry nowadays to describe azimuthal and polar anisotropy and reasonably realistic in capturing the features of the earth interior. It is challenging to handle so many model parameters in the seismic data processing. In order to reduce the number of the parameters for P wave, the acoustic orthorhombic medium is proposed by setting all on‐axis S wave velocities to zero. However, due to the coupled behaviour for P and S waves in the orthorhombic model, the 'S wave artefacts' are still remained in the acoustic orthorhombic model, which kinematics needs to be defined and analysed. In this paper, we analyse the behaviour of S wave in acoustic orthorhombic media. By analysis of the slowness surface in acoustic orthorhombic media, we define the S waves (or S wave artefacts) that are more complicated in shape comparing to the one propagating in an acoustic transversely isotropic medium with a vertical symmetry axis. The kinematic properties of these waves are defined and analysed in both phase and group domain. The caustics, amplitude and the multi‐layered case for S wave in acoustic orthorhombic model are also discussed. It is shown that there are two waves propagating in this acoustic orthorhombic medium. One of these waves is similar to the one propagating in acoustic vertical symmetry axis media, whereas another one has a very complicated shape consisting of two crossing surfaces. [ABSTRACT FROM AUTHOR]

Compensation for geometrical spreading along the ray path is important in AVO (amplitude versus offset) analysis, since it contributes to the seismic amplitude preservation. The P-wave geometrical spreading factor is expressed by the reflection traveltime described by a non-hyperbolic moveout approximation using the traveltime parameters that can be estimated in velocity analysis. The generalized non-hyperbolic moveout approximation (GMA) for traveltime was proposed by Fomel and Stovas by using two reference rays. We develop a 3-D GMA-type approximation for the relative geometrical spreading in a homogeneous orthorhombic (ORT) medium using the same strategy. Two type of GMA form approximations are defined by different selection of reference rays: two reference rays with finite offsets and two horizontal reference rays in two corresponding vertical symmetry planes. One horizontal ray in between the vertical symmetry planes is selected to compute the cross-term coefficient in the approximation. In the numerical examples, we compare our proposed GMA form approximations with other type of approximations in a homogeneous transversely isotropic medium with a vertical symmetry axis (VTI), transversely isotropic medium with a horizontal symmetry axis (HTI), ORT and multilayered ORT models and show that our GMA-type approximations are superior among all approximations. [ABSTRACT FROM AUTHOR]

ABSTRACT I derive the kinematic properties of single-mode P, S1, and S2 waves as well as converted PS1, PS2, and S1S2 waves in elastic orthorhombic media including vertical velocity, two normal moveout velocities defined in vertical symmetry planes, and three anelliptic parameters (two of them are defined in vertical symmetry plane and one parameter is the cross-term one). I show that the azimuthal dependence of normal moveout velocity and anellipticity is different in phase and group domains. The effects on-vertical-axis singularity and on-vertical-axis triplication are considered for pure-mode S1 and S2 waves and converted-mode S1S2 waves. The conditions and properties of on-vertical-axis triplication are defined in terms of kinematic parameters. The results are illustrated in four homogeneous orthorhombic models and one multilayered orthorhombic model with no variation in azimuthal orientation for all the layers. [ABSTRACT FROM AUTHOR]

The presence of triplications (caustics) can be a serious problem in seismic data processing and analysis. The traveltime curve becomes multi-valued and the geometrical spreading correction factor tends to zero due to energy focusing. We analyse the conditions for the qSV-wave triplications in a homogeneous transversely isotropic medium with vertical symmetry axis. The proposed technique can easily be extended to the case of horizontally layered vertical symmetry axis medium. We show that the triplications of the qSV-wave in a multilayered medium imply certain algebra. We illustrate this algebra on a two-layer vertical symmetry axis model. [ABSTRACT FROM AUTHOR]

An anisotropic medium with vertical symmetry axis (VTI) often presents a good model for describing real rocks. Propagation of quasi-P- and quasi-SV-waves in such media requires an expression of the vertical phase slowness, a complicated function of the horizontal phase slowness and the medium parameters. For converted-wave phase-shift migration methods, it is desired to have slowness expressions that are simple and accurate at wide angles of propagation. Taylor-series representations of the squared vertical slowness for quasi-P- and quasi-SV-waves result in new wide-angle phase-slowness approximations based on truncated series and continued-fraction representations. Slowness approximations that are exact for both vertical propagation and at a horizontal slowness corresponding to horizontally traveling qP-waves are derived. The approximation for quasi-SV-waves can be used in phase-shift migration in media where the quasi-SV wavefront contains triplications. These approximations are tested on several models and compared to previously published approximations. The numerical tests suggest that the new continued-fraction approximations are more accurate. They can be used in phase-shift migration algorithms, which are more efficient for large angles than the existing approximations. [ABSTRACT FROM AUTHOR]

ABSTRACT We consider a layered heterogeneous viscoelastic transversely isotropic medium with a vertical symmetry axis (a viscoelastic TIV medium) and parameters that depend on depth only. This takes into account intrinsic attenuation, anisotropy and thin layering. The seismic wavefield is decomposed into up- and downgoing waves scaled by the vertical energy flux. This scaling gives important symmetry relationships for both reflection and transmission (R/T) responses. For a stack of homogeneous layers, the exact reflection response can be computed in a numerically stable way by a simple layer-recursive algorithm. We derive exact plane-wave R/T coefficients and several linear and quadratic approximations between two viscoelastic TIV media, as functions of the real-valued horizontal slowness. The approximations are valid for pre- and post-critical values of horizontal slowness provided that the proper complex square roots are used when computing the vertical slowness. Numerical examples demonstrate that the quadratic approximations can be used for large differences in medium parameters, while the linear approximations can be used for small differences. For weak anisotropy it is sufficient to use an isotropic background medium, while for strong anisotropy it is necessary to use a weak TIV or TIV background medium. We also extend the O'Doherty–Anstey formula to the P- and SV-wave transmission responses of a stack of viscoelastic TIV layers, taking into account intrinsic attenuation, anisotropy and thin layering. [ABSTRACT FROM AUTHOR]