Dmitry Neklyudov

An iterative solver for the 3D Helmholtz equation

Abstract We develop a frequency–domain iterative solver for numerical simulation of acoustic waves in 3D heterogeneous media. It is based on the application of a unique preconditioner to the Helmholtz equation that ensures convergence for Krylov subspace iteration methods. Effective inversion of the preconditioner involves the Fast Fourier Transform (FFT) and numerical solution of a series of boundary value problems for ordinary differential equations. Matrix-by-vector multiplication for iterative inversion of the preconditioned matrix involves inversion of the preconditioner and pointwise multiplication of grid functions. Our solver has been verified by benchmarking against exact solutions and a time-domain solver.

Full-waveform inversion for macro velocity model reconstruction in look-ahead offset vertical seismic profile: numerical singular value decomposition-based analysis

Full-waveform inversion is currently considered as a potential tool for improving depth-velocity models for areas with complex geology. It is well-known that success of the inversion is very sensitive to the available low-frequency content of the data. In the paper we investigate this issue considering a look-ahead offset vertical seismic profile survey and applying singular value decomposition analysis of a linearized forward map as the main tool. We demonstrate with this technique the difference between the sequential full-waveform inversion strategy and the original time-domain approach proposed in the early 1980s.We emphasize the role of the lowest frequency in the data, which is necessary for reliable velocity model inversion in particular cases. Finally we show the existence of a trade-off between the lowest frequency and a regularization parameter of the inversion procedure. The presented approach may be adapted to answer general questions regarding the quality of data and acquisition system parameters required for feasible full-waveform inversion.

A Helmholtz iterative solver with semianalytical preconditioner for the frequency-domain full-waveform inversion

Summary A preconditioned iterative method for solving the Helmholtz equation in heterogeneous media is proposed. We consider it as a tool for frequency-domain fullwaveform inversion. Our method is based on a Krylov-type linear solver, similar to several other iterative approaches. The distinctive feature is the use of a right preconditioner, obtained as a solution of the complex damped Helmholtz equation in a 1D medium, where velocities vary only with depth. As a result, a matrix-by-vector multiplication of the preconditioned system may be efficiently evaluated using a fast 2D-Fourier transform in horizontal space coordinates, followed by the solution of a system of ordinary differential equations in the vertical space coordinate. To solve these equations, we treat the 1D background velocity as piecewise constant and search for the exact solution as a superposition of upgoing and downgoing waves. In our approach, we do not approximate derivatives by finite differences. The method has good dispersion properties in both lateral and vertical directions. We illustrate the properties of our method using a realistic 2D velocity model, and demonstrate propagation of waves without visible dispersion with fast convergence rates for a wide band of temporal frequencies. Finally, results of 2D fullwaveform inversion using the proposed forward modeling engine are presented for single-offset VSP data.

Frequency Domain Iterative Solver For Elasticity With Semi-analytical Preconditioner

A preconditioned iterative method for solving frequency domain elastic wave equations is presented. Our method is based on Krylov type linear solvers. Its distinctive feature is the use of a right preconditioner, obtained as a solution of the damped elastic wave equations in a vertically heterogeneous medium. We represent the actual differential operator as a perturbation of the preconditioner. As a result, a matrix-by-vector product of the preconditioned system is effectively evaluated via the fast Fourier transform in horizontal direction(s) followed by the solution of a number of systems of ordinary differential equations in the vertical direction. To solve these equations we introduce a piecewise constant 1D background medium and search for the exact solution in the 1D medium as a superposition of upgoing and downgoing Pand Swaves. The method has excellent dispersion properties because it does not use any finite-difference approximation of derivatives and converges reasonably fast.

Waveform Inversion of Surface Waves in F-K Domain in Locally 1D Formulation

Near-surface velocity variation is one of the main challenges for land data acquired in arid environment. Conventional methods have certain limitations and industry is constantly searching for better techniques for nearsurface characterization. One of the promising approaches is surface wave inversion since surface waves are naturally localized in the shallow part of the subsurface and they tend to be very sensitive to the S-wave velocity anomalies. The conventional method for surface wave analysis requires dispersion curve picking and therefore is quite timeand manpower consuming. An alternative approach that is recursively comparing amplitude spectrum of the data in FWI manner does not require any picking and can be applied automatically. In this work, we demonstrate the advantages of the new technique and compare it with the standard approach. For simplicity, we show all results for 1D model that is part of complex and quite realistic 2D model.