Full-waveform inversion (FWI) is a powerful tool that can be used to invert for microseismic event locations and the source signature because it can exploit the complete waveform information. We have developed an algorithm to invert for a spatio-temporal source function that encapsulates microseismic events with spatially localized or distributed locations and source signatures. The algorithm does not require assumptions to be made about the number or type of sources; however, it does require that the velocity model is close to the true subsurface velocity. We reformulate the conventional FWI algorithm based on the [Formula: see text]-norm data-misfit function by adding sparsity constraints using a sparsity promoting [Formula: see text]-norm as an additional regularization term to get more focused and less noise-sensitive event locations. The Orthant-Wise Limited-memory quasi-Newton algorithm is used to solve the optimization problem. It inherits the advantageous (fast convergence) properties of the limited memory Broyden-Fletcher-Goldfarb-Shanno method and can easily overcome the nondifferentiability of [Formula: see text]-norm at null positions. We determine the performance of the algorithm on noise-free and noisy synthetic data from the SEG/EAGE overthrust model.

Attenuation of seismic waves is sensitive to the physical properties of the subsurface and has been observed in vertical seismic profiling (VSP) and reflection data. De et al. (1994) report measurements of the P- and S-wave quality factors from VSP surveys and sonic logs. Klimentos (1995) measures compressional and shear attenuation from sonic logs in sandstone formations with variable oil, water, and gas saturation and observes that [Formula: see text] and [Formula: see text] can be used for pore-fluid discrimination. Maultzsch et al. (2007) evaluate P-wave azimuthal attenuation anisotropy from 3D VSP data acquired over a fractured hydrocarbon reservoir and infer fracture directions from attenuation analysis. Attenuation anisotropy has also been observed in P-wave reflection data (Clark et al., 2009; Vasconcelos and Jenner, 2005). Seismic attenuation is most commonly measured using the spectral-ratio method. Zhu et al. (2007) extend the spectral-ratio method to anisotropic media and apply it to physical modeling data acquired for a transversely isotropic (TI) sample. Computing spectral ratios helps eliminate the source spectrum and can be used to obtain accurate effective and interval attenuation coefficients of P- and S-waves in layered anisotropic media (Behura and Tsvankin, 2009a; Shekar and Tsvankin, 2011).

Interval attenuation measurements provide valuable information for reservoir characterization and lithology discrimination. We extend the attenuation layer-stripping method of Behura and Tsvankin to mode-converted (PS) waves with the goal of estimating the S-wave interval attenuation coefficient. By identifying PP and PS events with shared ray segments and applying the PP þ PS ¼ SS method, we first perform kinematic construction of pure shear (SS) events in the target layer and overburden. Then, the modified spectral-ratio method is used to compute the effective shear-wave attenuation coefficient for the target reflection. Finally, application of the dynamic version of velocity-independent layer stripping to the constructed SS reflections yields the interval S-wave attenuation coefficient in the target layer. The attenuation coefficient estimated for a range of sourcereceiver offsets can be inverted for the interval attenuation parameters. The method is tested on multicomponent synthetic data generated with the anisotropic reflectivity method for layered VTI (transversely isotropic with a vertical symmetry axis) and orthorhombic media.

Seismic wave propagation in attenuative media can be efficiently modeled with ray-based methods. We present a methodology to generate reflection data from attenuative anisotropic media using the Kirchhoff scattering integral and summation of Gaussian beams. The Green's functions are computed in the reference elastic model by Gaussian-beam summation, and the influence of attenuation is incorporated as a perturbation along the central ray. The reflected P-wave is obtained by substituting the approximate Green's functions into the Kirchhoff scattering integral. Numerical examples for a transversely isotropic medium above a horizontal reflector and for a structurally complex acoustic model with a salt body confirm the accuracy of the method.