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Summary Considering reciprocity where the traveltime is an even function of the offset or horizontal-slowness, the fourth-order normal moveout (NMO) series are governed by the normal-incidence time and eight effective parameters: three second-order and five fourth-order. Local effective parameters are related to the individual layers, while the global effective parameters are related to the overburden multi-layer model. Local and global parameters are related by forward and inverse Dix-type transforms. The NMO formulae are different in the slowness and offset domains, but the eight parameters are the same in both cases. We suggest a new set of intuitive normalized effective parameters, classified into two “azimuthally isotropic” and six “azimuthally anisotropic” parameters. We provide feasible ranges for the normalized parameters, thus allowing their used for controlled inversion.

Summary Considering all types of pure-mode and converted waves, we derive the azimuthally-dependent normal moveout (NMO) series coefficients of near normal-incidence reflection waves in general anisotropic (triclinic) horizontally layered media, for a leading error term of order six. The NMO series can be either a function of the invariant horizontal-slowness (slowness domain) or the surface-offset (offset domain). The NMO series coefficients of different orders, also referred to as effective parameters, are associated with the corresponding azimuthally-dependent NMO velocity functions. We distinguish between local (single-layer) and global (overburden multilayer) effective parameters, where the local and global effective parameters are related by forward and inverse Dix-type transforms. We first consider the case in which the reciprocity assertion for incidence and reflected waves holds, i.e. pure-mode waves for general anisotropic horizontally-layered media, and converted waves for anisotropic horizontally-layered models sharing a common horizontal symmetry plane. Considering reciprocity, the odd-power coefficients of the NMO series cancel, and the remaining coefficients are zero-offset time, three second-order and five fourth-order effective parameters. Next we consider converted waves in general anisotropic media, where reciprocity no longer holds. Twelve additional parameters are required: two firstorder, four third-order and six fifth-order effective parameters.

We derive azimuthally dependent fourth-order effective velocity and moveout parameters for compressional waves propagating in a layered model that consists of orthorhombic layers. The subsurface layered medium is considered as a locally 1D model, where each layer can be characterized by orthorhombic parameters. The orthorhombic layers have a common vertical axis but different azimuthal orientations of horizontal axes. For a 1D vertically varying anisotropic model, the azimuth of the phase velocity is the same for all layers, while the azimuths of the ray velocity are generally different. We extend the existing studies on the moveout in an azimuthally anisotropic model, accounting for the azimuthal deviation between the phase and ray velocities. We compute the lag between the azimuth of the surface offset (source-receiver vector) and the phase velocity azimuth. An effective model that replaces the multi-layer model with a single azimuthally anisotropic layer is derived, whose moveout and offset azimuth are identical to those of the layer package up to the fourth-order terms. We verify the accuracy of the approximation for small to moderate reflection angles vs. exact analytical ray tracing. The proposed approximation is in particular important when analyzing residual moveouts measured along full-azimuth common image reflection angle gathers.

arametrically. The lengthwise (along the phase velocity azimuth) and transverse components of the lateral propagation and the traveltime are defined as functions of the horizontal slowness magnitude and azimuth. We compute the power series coefficients for infinitesimal slowness and for nearly critical slowness. With these coefficients, we design continuous parametric functions valid for the whole offset range, whose Tailor series expansions match the given coefficients. In addition to the zero offset time, we keep two “head” coefficients for small offsets and two “tail” coefficients for nearly critical slowness, per azimuth. One of the tail coefficients characterizes the propagation in the layer with the fastest horizontal velocity for the given azimuth. The other tail coefficient depends on the propagation and traveltime in the slow layers when the slowness reaches its critical value. We verify the accuracy of the approximation for all feasible reflection angles vs. exact analytical ray tracing.

In this work we study the hyperbolic moveout approximations for shear and converted waves in a layered orthorhombic medium with flat interfaces in the horizontal plane of symmetry, distinguishing two types of conversion - PS1 and PS2, with different shear polarizations. Note that the azimuthally dependent NMO velocity function for shear or converted waves is similar to that of compression waves - only the coefficients are different. Given a package of orthorhombic layers with different parameters, one can establish an equivalent effective model consisting of a single layer with the same vertical time as the original package, described by the fast and slow effective NMO velocities, the effective azimuth of the slow velocity, and the vertical compression velocity. The latter is normally obtained from non-seismic information such as check-shot or well logs. Note that the effective azimuths are different for models describing PP, PS1 and PS2 waves.

This paper discusses the application of imaging and characterization in the local angle domain (LAD) in a shale reservoir. The system delivers high-quality images of the reservoir and geomechanical characterization of rocks with the precision needed to steer horizontal drilling, detect sweet spots, locate geobodies resistant to fracturing, or image geobodies of irregular shape. This technique is particularly attractive in Poland, where conventional seismic imaging in complex tectonics has frequently resulted in dry wells. Using this method, and combined with well data, image horizons were accurately tied to well markers, and stress/fracture orientations were highly correlated in the vicinity of the wells.

Orthorhombic models comprise subsurface anisotropy caused by vertical azimuthally-aligned fractures and layering, or by two orthogonal sets of vertical fractures, with or without layering. In this paper we derive new relations for hyperbolic and non-hyperbolic moveout approximations for pure compression waves, considering a 1D model that consists of a set of orthorhombic layers. The layers have a common vertical axis but different orientations of horizontal orthorhombic axes. For 1D models, the azimuth of the phase velocity is the same for all layers, while the azimuths of the ray velocity are generally different. We extend the existing studies on moveout in an orthorhombic model, accounting for the azimuthal deviation between the phase and ray velocities. We then formulate the azimuthally-dependent NMO velocity for a package of layers. Finally, we compare the derived full quartic moveout term with its acoustic approximation, and verify the accuracy of the approximation vs. exact analytical ray tracing.

We demonstrate how we solve the two-point ray tracing problem in the presence of complex geological structures. For a point diffractor (PD) single-ray ray fan, we define surface-to-subsurface mapping (and vice versa) which maps the subsurface angles (take-off dips and azimuths) of individual traced rays to their surface locations. This mapping provides us with an interpolated ray for any subsurface angle or any surface location. We also introduce the reliability factor, which is based on a high-order interpolation technique in which we integrate several physical parameters calculated along the rays. This factor serves as a way of measuring the quality of the surface-to-subsurface mapping, and allows us to obtain a full ray (wave) field representation at any location, similar to wave equation methods. The same methodology is used for Common Reflection Point (CRP) specular ray pairs which are traced from a given subsurface location, with various opening angles and azimuths and a given reflector directivity, to the surface.

We consider a case where full-azimuth reflection angle gathers were generated using a background VTI depth model. Residual moveouts (RMO) which were automatically picked on these 3D gathers along major horizons indicate considerable periodic azimuthal variations. Our aim is to use the azimuthally dependent RMOs to convert the background VTI model into interval orthorhombic layer parameters. Our method is based on a newly derived generalized Dix-based theory, assuming a locally varying 1D orthorhombic model, where at each location the vertical axis is the same for all layers but the azimuthal orientations are different. An effective model for such a layered structure represents a single layer with identical vertical time, effective compression vertical velocity, effective Thomsen parameters and an effective orientation. The NMO velocity and the surface offset azimuth of the effective model coincide with the parameters of the original package of layers for any azimuth of the phase velocity. Our approach starts with a Fourier-based conversion of the RMOs into azimuthally dependent NMO velocities, which are then inverted into three effective parameters. Finally, we apply the proposed generalized Dix-based inversion approach to estimate the interval orthorhombic parameters within each layer.

The paper overviews some results of 5D regularization on model and real data in order of missing information reconstruction. Besides there is description of used Orthogonal Matching Pursuit (OMP) algorithm, which recalculate all the expansion coefficients after a new component has been added to the expansion

We present a multi dimensional interpolation method for the regularization of seismic data. The method operates in frequency slices where in each slice the data is represented by a Fourier expansion. The algorithm operates iteratively where in each step one Fourier component is selected from an overly redundant space. The coefficients of all previously selected Fourier components are recalculated at every step and in this the present algorithm differs from most algorithms presented in the geophysical literature. The regularization method is tested in two examples of pre stack data.