Xiaofeng Jia

Element-free method and its efficiency improvement in seismic modelling and reverse time migration

We present the theory of the element-free method (EFM) and its numerical applications in seismic modelling and reverse time migration. The absence of elements makes the method cheaper and more flexible than the finite element method (FEM). The shape function in the EFM only needs to satisfy the moving-least-squares (MLS) criterion, so it can be generated easily. Besides, it is convenient to deal with the local problems using weight functions and influence domains. Due to the MLS fitting method, the dependent variable and its derivative are both continuous and precise in the EFM. However, as a result of its heavy cost burden, this method seems difficult to be developed in seismic modelling and reverse time migration. The cost is mainly caused by improper storage of some large sparse matrices such as the mass matrix and the stiffness matrix, and improper operations (multiplication and inversion) on them. In this paper we compress the sparse matrices by the compressed sparse column (CSC) format and solve the linear equations instead of inverting sparse matrices, with the help of Intel linear sparse solver ‘PARDISO’. By these strategies, we have saved computer resources significantly. In the end, we show some applications of the improved method in seismic modelling and reverse time migration.

Accuracy improvement for super-wide angle one-way waves by wavefront reconstruction

Summary To overcome the angle limitation of regular one-way wave propagation methods, we propose to extend the capability of one-way propagators by a wavefront reconstruction method which combines and interpolates the two orthogonally propagated one-way wavefields. The proposed method has accurate super-wide angle (greater than 90o) propagation and can model turning waves. The method has the potential to be used in imaging steep subsalt reflectors and overhanging salt flanks. Numerical tests demonstrated the validity of the method.

An effective denoising strategy for wave equation migration based on propagation angles

We present an effective denoising strategy for two-way wave equation migration. Three dominant artifact types are analyzed and eliminated by an optimized imaging condition. We discuss a previously unsolved beam-like artifact, which is probably caused by the cross-correlation of downward transmitting and upward scattering waves from both the source and receiver side of a single seismic shot. This artifact has relatively strong crosscorrelation but carries no useful information from reflectors. The beam-like artifact widely exists in pre-stack imaging and has approximately the same amplitude as useful seismic signals. In most cases, coherent artifacts in the image are caused by directionally propagating energy. Based on propagation angles obtained by wavefield gradients, we identify the artifact energy and subtract its contribution in the imaging condition. By this process most artifacts can be accurately eliminated, including direct wave artifacts, scattering artifacts, and beamlike artifacts. This method is independent of the wavefield propagator and is easy to adapt to almost all current wave equation migration methods if needed. As this method deals with the physical artifact origins, little damage is caused to the seismic signal. Extra k-domain filtering can additionally enhance the stacking result image quality. This method succeeds in the super-wide-angle one-way migration and we can expect its success in other two-way wave equation migrations and especially in reverse time migration.

Imaging Steep Salt Flanks By Super-wide Angle One-way Method

The super-wide angle one-way method has been proposed to overcome the angle limitation of regular one-way wave propagation methods. In the method, a technique of wavefront reconstruction was used to combine the two orthogonally propagated one-way wavefields. This paper is focused on the accuracy improvement of the wave back propagation from the receiver array. The proposed method can model turning waves well and consequently can be employed in imaging overhanging salt flanks. Numerical tests demonstrated the validity of the method.

Staining algorithm for full-waveform inversion

Full waveform inversion (FWI) has the potential to provide high resolution velocity parameters. FWI is an iterative algorithm that modifies the velocity model each step. Each step of iteration consists of a reverse time migration process, which calculates the gradient of misfit function by correlating the forward wavefield of the actual sources and the backward wavefield of the receiver data residuals. If some structures such as those in subsalt shadow zones cannot be imaged well by reverse time migration, these areas can hardly be well reconstructed by full waveform inversion either. Staining algorithm is introduced to improve the signal-to-noise ratio of poorly illuminated subsurface structures. Staining algorithm has a significant advantage that it is amplitude-preserved for the locally recovered wavefield, which makes it feasible to be adopted to waveform inversion process. Combination of waveform inversion and staining algorithm is promising to get better resolution of velocity parameter in poorly illuminated areas.

A Dynamic Lattice Method for Seismic Wave Simulation in TI Media with Free Surface Topography

Based on a particle lattice model, a dynamic lattice method is proposed to simulate seismic wave propagation in transversely isotropic (TI) media with tilted symmetry axis (TTI media) in the presence of free surface topography. Different from other wave equation based numerical methods, the dynamic lattice approach calculates the micro-mechanical interactions between particles in the lattice instead of solving the wave equation. In the case of TI media, it is a challenge to find the correct particle lattice model which can reflect the anisotropic nature of TI media. Our study reveals the theoretical connection between TI medium and the particle lattice model, allowing us to model elastic seismic waves in TI media. On the other hand, the particle feature of this method makes it convenient to incorporate free surface topography. To achieve this, we only need to remove the particles above the surface topography from the lattice. The free surface boundary condition is automatically implemented through the interactions between particles near the free surface. Numerical results demonstrate the effect of our method in simulating elastic waves in TI media with free surface topography.

A Meshless Method and its Applications in Seismic Exploration

Wave equation method is one of the fundamental methods for seismic modeling and imaging. In this paper a meshless numerical method - the Element-Free Method (EFM) was applied to solve seismic wave equation and handle modeling and imaging problems. The theory of EFM consisted of two parts -the Moving Least Squares (MLS) criterion and the variational principle, in contrast with the Lagrange interpolation and the variational principle for the theory of the Finite Element Method (FEM). In EFM, it was necessary to calculate the Gauss quadrature on each Gauss point. Only the numbered nodes near to the Gauss point needed to be considered for the quadrature. These nodes were determined by the so-called influence domain of each node. The influence domain was a significant feature of EFM because it effected on the accuracy and cost of the method. At the same time, the absence of elements was also a key point of EFM, which yielded easier preprocessing and lower cost than those of FEM. In this paper, the scheme of EFM would be shown for full scalar or elastic wave equation. Based on the theory, a simple example was discussed in details to indicate the good effectivity of EFM. Furthermore, some synthetic examples will be shown to discover the good performance of EFM in seismic modeling and imaging problems. It is clear that complex structures can be modeled and imaged very well such as high-angle dip and high-velocity anoma1y even under complex surface conditions.