Petr Petrov

Three dimensional finite‐difference modeling of elastic wave propagation in the Laplace‐Fourier domain

We have developed new heterogeneous 3D second and fourth order staggered-grid finite-difference schemes for modeling seismic wave propagation in the Laplace-Fourier domain. Recent interest in full waveform inversion in the Laplace-Fourier domain has been the motivation for the development for these types of wave-field simulators. Our approach is based on the principles of an integral equation approximation technique for the velocity-stress formulation in the Cartesian coordinate system. The fourth-order scheme is obtained by the combination of integral identities for the two elementary volumes – “small” and “large” around nodes where the wave variables are defined. The final matrix formulation for the fourth or second-order scheme is performed only for velocity components and the resulting linear system is solved by Krylov iterative methods. We have applied these simulators for the investigation of the wave fields of the SEG/EAGE model in the LaplaceFourier domain along with other test models. Our eventual goal is to embed it in an inversion scheme for joint seismicelectromagnetic imaging.

Multistep inversion workflow for 3D long-offset damped elastic waves in the Fourier domain

ABSTRACT We present a new workflow for imaging damped three‐dimensional elastic wavefields in the Fourier domain. The workflow employs a multiscale imaging approach, in which offset lengths are laddered, where frequency content and damping of the data are changed cyclically. Thus, the inversion process is launched using short‐offset and low‐frequency data to recover the long spatial wavelength of the image at a shallow depth. Increasing frequency and offset length leads to the recovery of the fine‐scale features of the model at greater depths. For the fixed offset, we employ (in the imaging process) a few discrete frequencies with a set of Laplace damping parameters. The forward problem is solved with a finite‐difference frequency‐domain method based on a massively parallel iterative solver. The inversion code is based upon the solution of a least squares optimisation problem and is solved using a nonlinear gradient method. It is fully parallelised for distributed memory computational platforms. Our full‐waveform inversion workflow is applied to the 3D Marmousi‐2 and SEG/EAGE Salt models with long‐offset data. The maximum inverted frequencies are 6 Hz for the Marmousi model and 2 Hz for the SEG/EAGE Salt model. The detailed structures are imaged successfully up to the depth approximately equal to one‐third of the maximum offset length at a resolution consistent with the inverted frequencies.