| Full-waveform inversion (FWI) is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms, and frequency-domain FWI is one of the important branches in FWI. The key ingredient of FWI is the efficient forward modeling method, which could greatly determine the accuracy and efficiency of FWI. In the real seismic data, Rayleigh waves are usually included, but combining surface waves and body waves in FWI is still a challenge because of their different physical behavior and nature. This convergence towards a local minimum can be attributed to surface waves and to the difficulty to fit both body and surface waves. So, considering free surface condition in frequency-domain seismic modeling is a fundamental work to solve this problem.Frequency-domain seismic modeling involves solving large scale linear sparse systems. Usually, these systems are highly sparse, nonsymmetrical and non-positive, which could be solved by direct-solver method and iterative solver, such as LU method and Conjugate gradient method. They also have their own advantages and disadvantages. So by comparing these two methods, we could more understand them and improve their shortcomings to increase computational efficiency.In this paper, based on frequency-domain elastic wave equations, first I perform two methods of seismic modeling in frequency domain—optimal nine-point finite-difference method and25-point finite-difference method. I compare and analyze these two methods meticulous from matrix property, grid dispersion, computational accuracy and efficiency. Then, I perform two methods to solve large scale linear sparse systems for seismic modeling—Conjugate gradient method and LU method. I also compare and analyze these two methods on several aspects, such as computational accuracy, memory requirement and CPU run time. What’s more, I discuss absorbing boundary condition and free surface condition in frequency-domain modeling, and I perform an implicit boundary condition to frequency-domain modeling and get Rayleigh waves. In order to evaluate the result, I calculate the dispersion energy of Rayleigh waves in f-v domain, and compare it with theoretical phase velocities. Finally, I design several media to analyze characteristics of Rayleigh waves and body waves.According to my works, I get several conclusions as follows: Firstly, optimal nine-point finite-difference method and25-point finite-difference method belong to the mixed-grid method, and they can be effectively applied in frequency-domain modeling. By using more grids around calculating points, the phase velocity dispersion of25-point method is significantly less than that of9-point method. According to my results, to keep the errors of phase velocity within1%,9-point method requires10grid points per shear wavelength, while25-point method only needs3.3grid points per shear wavelength. At the same time, phase velocity dispersion of25-point method is independent of Poisson’s ratio.Secondly, to keep the errors of phase velocity within1%, the results calculating by these two methods have the same accuracy. Although both the CPU time and memory requirement of9-point method are less than that of25-point method when grid size is the same,9-point method requires finer model split in order to achieve the accuracy requirements in case of the same model size.Thirdly, both Conjugate gradient method and LU method could effectively solve large scale linear sparse systems, but the non-zeros elements structure of sparse matrix has great influence to how efficiently a matrix can be factorized. By comparing these two methods, I find that Conjugate gradient method requires less memory, but spends more CPU time, while LU method requires larger memory but less time. And with the model becoming bigger and bigger, the gap of CPU time and memory requirement between these two methods become larger and larger.What’s more, the perfectly matched layer (PML) boundary condition could apply well in frequency domain modeling. Because the solution at each one frequency includes information for all times, so absorbing boundary in frequency domain is very critical. The boundary needs to set thicker than time domain modeling in order to get good results.Finally, applying an implicit boundary condition to frequency-domain modeling could simulate Rayleigh waves. Then I calculate the dispersion energy of Rayleigh waves in f-v domain using linear Radon transform, and compare it with theoretical phase velocities, which could demonstrate the correct of the Rayleigh waves. At the same time, I conclude that when25-point method is chosen, it requires more than20grid points per shear wavelength at the surface to get right result. |
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