Gradient calculation for waveform inversion of microseismic data in VTI media
Jarillo Michel, O. and Tsvankin, I., 2014. Gradient calculation for waveform inversion of microseismic data in VTI media. Journal of Seismic Exploration, 23: 201-217. In microseismic data processing, source locations and origin times are usually obtained using kinematic techniques, whereas moment-tensor estimates are typically based on linear inversion of P- and S-wave amplitudes. Waveform inversion (WI) can potentially provide more accurate source parameters along with an improved velocity model by incorporating information contained in the entire trace including the coda. Here, we address one of the key issues in implementing WI for microseismic surveys - efficient calculation of the gradient of the objective function with respect to the model parameters. Application of the adjoint-state method helps obtain closed-form expressions for the gradient with respect to the source location, origin time, and moment tensor. Computation of the forward and adjoint wavefields is performed with a finite-difference algorithm that handles elastic VTI (transversely isotropic with a vertical symmetry axis) models. Numerical examples illustrate the properties of the gradient for multicomponent data recorded by a vertical receiver array placed in homogeneous and horizontally layered VTI media.
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