Elastic reflection traveltime inversion with pure-wave propagators

Elastic full-waveform inversion (EFWI) updates high-resolution model parameters by minimizing the misfit function between the observed and modeled data. EFWI possesses strong nonlinearity and is likely to converge to a local minimum when the inversion begins with inaccurate initial models. Elastic reflection waveform inversion (ERWI) recovers the low-wavenumber components of P- and S-wave velocities along the "rabbit ear" wave paths to provide initial velocity models for EFWI. However, every iteration of ERWI requires six times as many forward calculations with elastic-wave equations which can be computationally expensive. Hence, we have developed a pure-wave reflection waveform inversion (PRWI) approach, which sequentially inverts low-wavenumber components of P- and S-wave velocity models. In our PRWI, we decompose elastic-wave operators into background and perturbed pure-wave parts and derive PRWI gradients using pure-wave operators. Both the background and perturbed wavefields in PRWI gradients are vector wavefields with single wave mode. PRWI can remove the high-wavenumber noise caused by S-wave stress decomposition, and reduce the computational cost of ERWI by almost 70%. Under the framework of PRWI, we have further developed the pure-wave reflection traveltime inversion (PRTI) approach to alleviate the issue of cycle skipping caused by waveform mismatch. In order to ensure the recovery of low-wavenumber components, we mute out the contribution of wavefields with small opening angles to PRTI gradients. Numerical examples have demonstrated that our PRTI method can provide good initial velocity models for EFWI efficiently.

Elastic reflection traveltime inversion with decoupled wave equation

Elastic full-waveform inversion (FWI) updates high-resolution model parameters by minimizing the residuals of multicomponent seismic records between the field and model data. FWI suffers from the potential to converge to local minima and more serious nonlinearity than acoustic FWI mainly due to the absence of low frequencies in seismograms and the extended model domain (P- and S-velocities). Reflection waveform inversion can relax the nonlinearity by relying on the tomographic components, which can be used to update the low-wavenumber components of the model. Hence, we have developed an elastic reflection traveltime inversion (ERTI) approach to update the low-wavenumber component of the velocity models for the P- and S-waves. In our ERTI algorithm, we took the P- and S-wave impedance perturbations as elastic reflectivity to generate reflections and a weighted crosscorrelation as the misfit function. Moreover, considering the higher wavenumbers (lower velocity value) of the S-wave velocity compared with the P-wave case, optimizing the low-wavenumber components for the S-wave velocity is even more crucial in preventing the elastic FWI from converging to local minima. We have evaluated an equivalent decoupled velocity-stress wave equation to ERTI to reduce the coupling effects of different wave modes and to improve the inversion result of ERTI, especially for the S-wave velocity. The subsequent application on the Sigsbee2A model demonstrates that our ERTI method with the decoupled wave equation can efficiently update the low-wavenumber parts of the model and improve the precision of the S-wave velocity.

Elastic reflection-based waveform inversion with a nonlinear approach

Full-waveform inversion (FWI) is a highly nonlinear problem due to the complex reflectivity of the earth, and this nonlinearity only increases under the more expensive elastic assumption. In elastic media, we need a good initial P-wave velocity and even better initial S-wave velocity models with accurate representation of the low model wavenumbers for FWI to converge. However, inverting for the low-wavenumber components of P- and S-wave velocities using reflection waveform inversion (RWI) with an objective to fit the reflection shape, rather than produce reflections, may mitigate the limitations of FWI. Because FWI, performing as a migration operator, is preferred of the high-wavenumber updates along reflectors. We have developed an elastic RWI that inverts for the low-wavenumber and perturbation components of the P- and S-wave velocities. To generate the full elastic reflection wavefields, we derive an equivalent stress source made up by the inverted model perturbations and incident wavefields. We update the perturbation and propagation parts of the velocity models in a nested fashion. Applications on the synthetic isotropic models and field data indicate that our method can efficiently update the low- and high-wavenumber parts of the models.