To describe accurately the propagation of elastic waves for characterizing and monitoring hydrocarbon reservoirs, as well as to obtain improved earth models, it is important to take into account seismic attenuation. We describe a method to estimate anelastic medium properties by a complete SH-waveform inversion. We use an optimization approach based on the iterative minimization of the mismatch between the seismic data and the computed response. To obtain a fast analytical imaging procedure, we include an asymptotic theory for attenuation in a linearized inverse scattering formulation. The forward modeling is solved by the Born approximation for a smooth and attenuative background medium. An asymptotic ray-tracing method is used to calculate traveltime, amplitude, and attenuation between source, receiver, and scattering points. The resulting method is computationally efficient and allows for a variety of data-acquisition geometries, including those with redundant or incomplete source-receiver coverage. Synthetic examples with realistic surface-to-surface geometry show an acceptable convergence in a few iterations when anomaly perturbations are less than 10% of the reference values and when associated diffracting structures are smaller than one-tenth of the predominant seismic wavelength. Through there remains the fundamental trade-off between density and shear modulus, the iterative asymptotic inversion is able to recover the elastic parameters (density and shear modulus) and the attenuation factor.

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