An optimised one-way wave migration method for complex media with arbitrarily varying velocity based on eigen-decomposition

The classical one-way generalised screen propagator (GSP) and Fourier finite-difference (FFD) schemes have limitations in imaging large angles in complex media with substantial lateral variations in wave velocity. Some improvements to the classical one-way wave scheme have been proposed with optimised methods. However, the performance of these methods in imaging complex media remains unsatisfying. To overcome this issue, a new strategy for wavefield extrapolation based on the eigenvalue and eigenvector decomposition of the Helmholtz operator is presented herein. In this method, the square root operator is calculated after the decomposition of the Helmholtz operator at the product of the eigenvalues and eigenvectors. Then, Euler transformation is applied using the best polynomial approximation of the trigonometric function based on the infinite norm, and the propagator for one-way wave migration is calculated. According to this scheme, a one-way operator can be computed more accurately with a lower-order expansion. The imaging performance of this scheme was compared with that of the classical GSP, FFD and the recently developed full-wave-equation depth migration (FWDM) schemes. The impulse responses in media with arbitrary velocity inhomogeneity demonstrate that the proposed migration scheme performs better at large angles than the classical GSP scheme. The wavefronts calculated in the dipping and salt dome models illustrate that this scheme can provide a precise wavefield calculation. The applications of the Marmousi model further demonstrate that the proposed approach can achieve better-migrated results in imaging small-scale and complex structures, especially in media with steep-dipping faults.

A Numerical Method for SolvingTwo-Dimensional Nonlinear Parabolic ProblemsBased on a Preconditioning Operator

This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods.

A computing method for bending problem of thin plate on Pasternak foundation

The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

On Helmholtz Equations and Counterexamples to Strichartz Estimates in Hyperbolic Space

In this paper, we study nonlinear Helmholtz equations (NLH)$$\begin{equation} -\Delta_{\mathbb{H}^N} u - \frac{(N-1)^2}{4} u -\lambda^2 u = \Gamma|u|^{p-2}u \quad\textrm{in}\ \mathbb{H}^N, \;N\geq 2, \end{equation}$$where $\Delta _{\mathbb{H}^N}$ denotes the Laplace–Beltrami operator in the hyperbolic space $\mathbb{H}^N$ and $\Gamma \in L^\infty (\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $\lambda>0$ and $p>2$. The oscillatory behaviour and decay rate of radial solutions is analyzed, with extensions to Cartan–Hadamard manifolds and Damek–Ricci spaces. Our results rely on a new limiting absorption principle for the Helmholtz operator in $\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.

Cross Orthogonality Between Eigenfunctions and Conjugate Eigenfunctions of a Class of Modified Helmholtz Operator for Pekeris Waveguide

The eigenfunctions of the modified Helmholtz operator have no orthogonality in a bounded domain with a perfectly matched layer (PML), which makes it difficult to calculate the coordinates under two different local bases when the marching algorithm is applied. In this paper, we derive the conjugate eigenfunctions of the operator and discuss the cross orthogonality between the eigenfunctions and their conjugate eigenfunctions. On the other hand, we derive a simple formula for calculating the coordinates under the local base. The numerical results indicate that this method is effective.