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Increasing interest in ever more detailed reservoir evaluation includes identification and mapping of small scale inhomogeneities. On a macroscopic scale, such inhomogeneities manifest themselves as anisotropy to seismic wave propagation. As a consequence, there is a need for seismic inversion algorithms which can reconstruct physical properties of anisotropic materials. An important type of small scale inhomogeneities for reservoir evaluation are that caused by fractures since fractures control the fluid flow of hydrocarbons. At the macroscopic scale, the fractured rock formations behave as azimuthally anisotropic materials. They affect seismic amplitudes, in particular seismic amplitude variations with azimuths (AVAZ). So, the problem of reconstructing physical properties of fractured rock formations can be posed as that of determining elastic properties of an azimuthally anisotropic inclusion present in a smooth background medium. If the background medium is assumed azimuthally isotropic and depth dependent only, the AVAZ can be cast in terms of a Fourier series of azimuths, φ,

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