Based on the linear slip fracture model, we first express stiffness matrix of tilted transversely isotropic (TTI) media in terms of the normal and tangential fracture weaknesses. Using perturbations in stiffness parameters for the case of an interface separating an isotropic medium and a TTI medium, we derive a linearized P-to-P reflection coefficient as a function of fracture weaknesses, in which titled fracture weaknesses involving effects of tilt angle and fracture weaknesses emerge. Following a Bayesian framework, we propose an inversion approach to use amplitude differences between seismic data along two azimuths to estimate the tangential fracture weakness and tilted normal and tangential fracture weaknesses based on the derived and simplified reflection coefficient. Synthetic tests confirm that the unknown parameter vector involving the tangential fracture weakness and tilted fracture weaknesses is estimated stably and reliably in the case of seismic data containing a moderate Gaussian noise. The inversion approach is also applied to a field data set acquired from a fractured carbonate reservoir, from which reasonable results of tilted fracture weaknesses are obtained. We conclude that the proposed inversion approach may provide additional proofs for fracture characterization, and it also make the estimation of tilt angle from observed seismic data for fractured reservoirs be available.
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