An integrated imaging framework in visco-acoustic media based on the stereo-modeling method#xD;

Viscoacoustic full-waveform inversion (FWI) and viscoacoustic least-squares reverse time migration (LSRTM) are widely used for high-resolution seismic imaging. The resolution of both techniques largely depends on the accuracy of the seismic wave simulation. We use the fourth-order stereomodeling (STEM) method for solving the viscoacoustic wave equation in viscoacoustic FWI and LSRTM, thereby developing a high-precision imaging framework. For the fourth-order STEM spatial method, we discuss two time-matching schemes: STEM, which combines a fourth-order time scheme in the Newtonian system, and the symplectic STEM (SSM) method, which combines a fourth-order symplectic time scheme that we developed in the Birkhoffian system. We compare the various numerical properties of the STEM, SSM, and traditional finite-difference methods, such as the Lax-Wendroff correction (LWC) method, including numerical dispersion, accuracy, numerical errors, and efficiency. The results indicate that STEM and SSM have lower numerical dispersion with higher computational efficiency than LWC. Although STEM and SSM exhibit nearly identical accuracy, SSM maintains better energy stability than STEM due to its symplectic time scheme. However, STEM has higher computational efficiency, making it more suitable for conventional wavefield simulation. Furthermore, we derive the velocity and Q gradients for viscoacoustic FWI using the STEM method, namely STEM-FWI. We also derive the Born modeling operator, the adjoint equation, and the gradient for viscoacoustic LSRTM using the STEM method, namely STEM-LSRTM. The accurate velocity and Q models obtained from the two-step viscoacoustic STEM-FWI serve as the initial models for STEM-LSRTM, which can provide a high-resolution subsurface image. The results of the viscoacoustic overthrust model indicate that our QFWI-LSRTM framework achieves high accuracy in inversion and imaging with minor numerical dispersion. The framework, tested on field data, demonstrates its effectiveness.