This study attempts to validate a mathematical formalism of introducing attenuation into Schoenberg's linear slip model. This formalism is based on replacing the real-valued weaknesses by complex-valued ones. During an ultrasonic experiment, performed at a central frequency of 100 kHz on a plate-stack model with 1-mm-thick Plexiglas™ plates, the velocity and attenuation ͑inverse of the quality factor Q͒ of P-, SH-, and SV-waves are measured in directions from 25°to 90°with the symmetry axis for dry and oil-saturated models and loading uniaxial pressures of 2 and 4 MPa. The velocity and attenuation data are fitted by the derived theoretical functions. The values of the real and imaginary parts of the complex-valued weaknesses are estimated. The real parts of the weaknesses, which have a clear physical meaning ͑they affect the weakening of the material͒, are three times larger for the dry model than for the oil-saturated one. The imaginary parts of the weaknesses are responsible for attenuation; their values are an order of magnitude smaller than the real parts. The derived expressions for angle-dependent velocities and attenuations can be used to distinguish between dry and oil-saturated fractures. In particular, the P-wave attenuation function in the symmetry-axis direction ͑normal to fracture planes͒ is different in dry and saturated media. The experiment shows that the platestack model is inhomogeneous because of the nonuniform pressure distribution, which degrades the experimental results and creates difficulties in the inversion for the complex-valued weaknesses -particularly in joint inversion of P-and S-wave data.

This is a study on anisotropy of seismic attenuation in a transversely isotropic (TI) model, which is a long-wavelength equivalent of an isotropic medium with embedded parallel fractures. The model is based on Schoenberg's linear-slip theory. Attenuation is introduced by means of a complex-valued stiffness matrix, which includes complex-valued normal and tangential weaknesses. To study the peculiarities of seismic attenuation versus wave-propagation direction in TI media, numerical modeling was performed. The modelinput data were the complex-valued weaknesses found from the laboratory ultrasonic experiment made with a Plexiglas plate-stack model, oil-saturated (wet) and air-filled (dry). The laboratory experiment and the numerical modeling have shown that in the vicinity of the symmetry axis, in the wet model, P-wave attenuation is close to S-wave attenuation, while in the dry model, P-wave attenuation is much greater than S-wave attenuation. Moreover, the fluid fill affects the P-wave attenuation pattern. In the dry (air-saturated) model, the attenuation pattern in the vicinity of the symmetry axis exhibits steeper slope and curvature than in the wet (oil-saturated) model. To define the slope or the curvature, a QVO gradient was introduced, which was found to be proportional to the symmetry-axis Q S /Q P -ratio, which explains the differences between dry and wet models. Thus, depending on the Q S /Q P -ratio, the QVO gradient can serve as an indicator of the type of fluid in fractures, because the QVO gradient is greater in gas-saturated than in liquid-saturated rocks. The analysis of P-wave attenuation anisotropy in seismic reflection and vertical seismic profiling data can be useful in seismic exploration for distinguishing gas from water in fractures.

S U M M A R Y There are three types of surfaces which are used for studying wave propagation in anisotropic media: normal surfaces, slowness surfaces and wave surfaces. Normal surfaces and slowness surfaces have been researched in detail. Wave surfaces are the most complicated and comparatively poorly known compared with the other two.Areas of complicated geometrical structure of the wave surfaces are located in the vicinity of conical acoustic axes. There is an elliptical hole on the quick shear wave surface and complicated folds and cusps on the slow shear wave surface. Decomposition of the slow shear wave surface into smooth sheets is used for the study of its geometrical structure. Complexity of shear wave surfaces can be expressed by the number of waves corresponding to a fixed ray. An original approach to the calculation of wave normals depending on ray direction is presented.

The simplest effective-medium model of fractured rocks known as the Linear Slip (LS) model of Schoenberg (1980) represents a single fracture set in an isotropic background rock. In the LS model, the stiffness C 13 is not independent as in the overall transversely isotropic (TI) model, but it is related to other stiffnesses by the equation, C 11 C 33-C 2 13 = 2C 66 (C33+C13). We have studied a physical sense of this constraint on the C 13 and found out that in terms of the TI elastic compliance tensor S it leads to the equality 12 = 13 , where 13 and 12 are the two different horizontal Poisson's ratios. In contrast to the Linear Slip model, in the overall TI model, one of these Poisson's ratios, 13 , is always greater than the other one, 12 , that is validated by numerous static and dynamic laboratory measurements of these Poisson's ratios in VTI-type rocks. Thus we have revealed a contradiction and inconsistency in the constraint on the C 13 for the LS model. Moreover, the restriction on C 13 for the LS model doesn't work for the overall TI medium in which there are physical constraints on the C 13 , namely, C 13_min < C 13 < C 13_max (e.g., Yan et al., 2013). We have revealed that mathematical expression for its lower bound, C 13_min , coincides with that for the constraint on the C 13 for the LS model. This means that the restriction on C 13 for the LS model, C 13 = C 13_min , does not satisfy the physical constraint for the overall TI model that is inequality C 13 > C 13_min. The LS model is not a universal model for real rocks. It may work successfully only in several special cases, or under certain conditions, for example, when the normal fracture weakness Δ N =0 (fluid-saturated cracks), or in the case of Δ N = Δ T (dry cracks). Also we have revealed that the LS model may suit better for sandstones and carbonates than for shales.