One of the main challenges in anisotropic velocity analysis and imaging is simultaneous estimation of velocity gradients and anisotropic parameters from reflection data. Approximating the subsurface by a factorized VTI (transversely isotropic with a vertical symmetry axis) medium provides a convenient way of building vertically and laterally heterogeneous anisotropic models for prestack depth migration.

Nonhydrostatic stress, an often-ignored source of seismic anisotropy, is universally present in the subsurface and may be as common as intrinsic or fracture-induced anisotropy. Nonhydrostatic stress, applied to an initially transversely isotropic solid with vertical symmetry axis (VTI), results in an effective medium having almost orthorhombic symmetry (provided that one of the principal stresses is aligned with the symmetry axis). The symmetry planes observed in this orthorhombic medium are aligned with the orientations of the principal stresses, and anisotropic parameters ((1,2) , δ (1,2,3) , and γ (1,2)) can reveal information about the stress magnitudes. Thus, time-lapse monitoring of changes in anisotropy potentially can provide information on temporal variations in the stress field. We use nonlinear elasticity theory to relate the anisotropic parameters to the magnitudes of the principal stresses and verify these relationships in a physical modeling study. Under the assumption of weak background and stress-induced anisotropy, each effective anisotropic parameter reduces to the sum of the corresponding Thomsen parameter for the unstressed VTI background and the corresponding parameter associated with the nonhydrostatic stress. The stress-related anisotropic parameters depend only on the differences between the magnitudes of principal stresses; therefore, principal stresses can influence anisotropic parameters only if their magnitudes differ in the symmetry plane in which the anisotropic parameters are defined. We test these predictions on a physical modeling data set acquired on a block of Berea Sandstone exhibiting intrinsic VTI anisotropy. Uniaxial stress, applied normal to the VTI symmetry axis, i.e., horizontally, produces an effective medium that is close to orthorhombic. We use two different methods to estimate the anisotropic parameters and study their variation as a function of stress. The first method utilizes conventional measurements of transmission velocities along the principal axes of the sample. The second method uses PP and PS reflection data acquired along seven different azimuths on the surface of the block. In accordance with theoretical predictions, the anisotropic parameters in the vertical plane normal to the stress are almost insensitive to the magnitude of the stress. In contrast, anisotropic parameters in the vertical plane of the applied stress increase approximately in a linear fashion with increasing stress. Except for the parameter δ (1) , comparison of the measured values of anisotropic parameters with theoretical predictions shows satisfactory agreement. Despite some documented discrepancies, we believe that nonlinear elasticity may provide a suitable framework for estimating pore pressure and 3D stresses from seismic data.

Conventional semblance velocity analysis is equivalent to modeling prestack seismic data with events that have hyperbolic moveout but no amplitude variation with offset (AVO). As a result of its assumption that amplitude is independent of offset, this method might be expected to perform poorly for events with strong AVO—especially for events with polarity reversals at large offset, such as reflections from tops of some class 1 and class 2 sands. We find that substantial amplitude variation and even phase change with offset do not compromise the conventional semblance measure greatly. Polarity reversal, however, causes conventional semblance to fail. The semblance method can be extended to take into account data with events that have amplitude variation, expressed by AVO intercept and gradient (i.e., the Shuey approximation). However, because of the extra degrees of freedom introduced in AVO‐sensitive semblance, resolution of the estimated velocities is decreased. This is because the data can be modeled acceptably with a range of combined erroneous velocity and AVO behavior. To address this problem, in addition to using the Shuey equation to describe the amplitude variation, we constrain the AVO parameters (intercept and gradient) to be related linearly within each semblance window. With this constraint we can preserve velocity resolution and improve the quality of velocity analysis in the presence of amplitude and even polarity variation with offset. Results from numerical tests suggest that the modified semblance is accurate in the presence of polarity reversals. Tests also indicate, however, that in the presence of noise, the signal peak in conventional semblance has better standout than does that in the modified semblance measures.

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Because events in image gathers generated after prestack depth migration are sensitive to the velocity field, they are often used in migration velocity analysis for isotropic media. Here, we present an analytic and numerical study of P-wave image gathers in transversely isotropic media with a vertical symmetry axis (VTI) and establish the conditions for flattening such events and positioning them at the true reflector depth. Application of the weak-anisotropy approximation leads to concise expressions for reflections in image gathers from homogeneous and factorized v(z) media in terms of the VTI parameters and the vertical velocity gradient k z. Flattening events in image gathers for any reflector dip requires accurate values of the zero-dip NMO velocity at the surface [V nmo (z = 0)], the gradient k z , and the anellipticity coefficient η. For a fixed error in V nmo and k z , the magnitude of residual moveout of events in image gathers decreases with dip, while the moveout caused by an error in η initially increases for moderate dips but then decreases as dips approach 90 •. Flat events in image gathers in VTI media, however, do not guarantee the correct depth scale of the model because reflector depth depends on the vertical migration velocity. For factorized v(x, z) media with a linear velocity variation in both the x-and z-directions, the moveout on image gathers is controlled by V nmo (x = z = 0), k z , η, and a combination of the horizontal velocity gradient k x and the Thomsen parameter δ (specifically, k x √ 1 + 2δ). If too large a value of any of these four quantities is used in migration, reflections in the image gathers curve downward (i.e., they are undercorrected; the inferred depth increases with offset), while a negative error results in overcorrection. Lateral heterogeneity tends to increase the sensitivity of moveout of events in image gathers to the parameter η, and errors in η may lead to measurable residual moveout of horizontal events in v(x, z) media even for offset-to-depth ratios close to unity. These results provide a basis for extending to VTI media conventional velocity analysis methods operating with image gathers. Although P-wave traveltimes alone cannot be used to separate anisotropy from lateral heterogeneity (i.e., k x is coupled to δ), moveout of events in image gathers does constrain the vertical gradient k z. Hence, it may be possible to build VTI velocity models in depth by supplementing reflection data with minimal a priori information, such as the vertical velocity at the top of the factorized VTI layer.

Nonhydrostatic stress, an often-ignored source of seismic anisotropy, is universally present in the subsurface and may be as common as intrinsic or fracture-induced anisotropy. Nonhydrostatic stress, applied to an initially transversely isotropic solid with vertical symmetry axis (VTI), results in an effective medium having almost orthorhombic symmetry (provided that one of the principal stresses is aligned with the symmetry axis). The symmetry planes observed in this orthorhombic medium are aligned with the orientations of the principal stresses, and anisotropic parameters ( (1,2) , δ (1,2,3) , and γ (1,2) ) can reveal information about the stress magnitudes. Thus, time-lapse monitoring of changes in anisotropy potentially can provide information on temporal variations in the stress field.We use nonlinear elasticity theory to relate the anisotropic parameters to the magnitudes of the principal stresses and verify these relationships in a physical modeling study. Under the assumption of weak background and stress-induced anisotropy, each effective anisotropic parameter reduces to the sum of the corresponding Thomsen parameter for the unstressed VTI background and the corresponding parameter associated with the nonhydrostatic stress. The stress-related anisotropic parameters depend only on the differences between the magnitudes of principal stresses; therefore, principal stresses can influence anisotropic parameters only if their magnitudes differ in the symmetry plane in which the anisotropic parameters are defined.We test these predictions on a physical modeling data set acquired on a block of Berea Sandstone exhibiting intrinsic VTI anisotropy. Uniaxial stress, applied normal to the VTI symmetry axis, i.e., horizontally, produces an effective medium that is close to orthorhombic. We use two different methods to estimate the anisotropic parameters and study their variation as a function of stress. The first method utilizes conventional measurements of transmission velocities along the principal axes of the sample. The second method uses PP and PS reflection data acquired along seven different azimuths on the surface of the block.In accordance with theoretical predictions, the anisotropic parameters in the vertical plane normal to the stress are almost insensitive to the magnitude of the stress. In contrast, anisotropic parameters in the vertical plane of the applied stress increase approximately in a linear fashion with increasing stress. Except for the parameter δ (1) , comparison of the measured values of anisotropic parameters with theoretical predictions shows satisfactory agreement.Despite some documented discrepancies, we believe that nonlinear elasticity may provide a suitable framework for estimating pore pressure and 3D stresses from seismic data.

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