Research And Realization Of Multi-frequency Waveform Inversion In Two-dimensional Frequency-domain Full Waveform Inversion

Traveltime tomography was a main strategy in the past 20th century coping with seismic inversion.With the development of computer technology in recent years,the full waveform inversion(FWI)which takes a great amount of computation becomes hotspot of the researchers.The full waveform inversion is divided into time domain full waveform inversion and full waveform inversion in frequency domain.Full waveform inversion needs only a few frequency points to get the result with high accuracy,this huge advantage makes frequency domain full waveform inversion method becomes an indispensable tool in seismic exploration.With the deep research on the waveform inversion,a squentional problems are found and many places need to be improved to get a better result,among which the selection and invsersion strategy of frequency points is one of the significant section.On the full introduction of full waveform inversion,We start from the basic theory of frequency domain full waveform inversion,including forward hybrid grid method and PML boundary discretization absorption conditions,objective function and inversion of pretreatment gradient method,and the iterative search method on the step length of gradient are introduced.Then,the article discuss the process and details of the sequentional inversion of frequency domain full waveform inversion.After the preliminary study and comparison of the classical frequency domain inversion,the inversion strategy of multi frequency is discussed in detail.The article concentrates on the problems in the inversion where multi frequency conducts at the same time.One of the problem is that the objection of diffierent frequency varies,which lead to the result that low frequency component is ignored because of the relative low objective function value in the total objective function.As a result,simply sum of the objective function of diffierent frequency point always can not converge to a good result.In order to adjust the proportion of each frequency point to a reasonable one,under the revelation of Hu's article(2001),this paper propose a new multi frequency inversion strategy based on the balance of the gradient of the objective function after the detailed deduction,and We also take a deep discussion on the phenomenon that low frequency component in the objective function is neglected.Based on this discussion,we tried to get a new weight of each frequency.The inversion results of these methods are analyzed and compared.We conduct experiments on the simple three layer model and the classical Overthrust model using the mainstream of the frequency grouping strategy.Then we take the comparision of the experimental result on the two models of these referred inversion strategy to verify their effectiveness and the robustness,and find that although the sequentional frequency inversion strategy can get a result with higher resolution,yet the result also brings some fake information,the result of the other three multi frequency inversion strategy is relatively more smooth.On the other hand,this paper focus on the realization of the multi frequency simultaneous inversion.In order to achieve multi frequency simultaneous inversion,coping with the forward and inversion process,We take some changes on three process among the forward and the inversion.First,We change the form of the impedance matrix and the wave field value matrix in the forward process;Second,We change the form of the impedance matrix and the source value matrix to get the gradient value matrix in the inversion process;Third,We change the form of the impedance matrix and wave field matrix to get the Hessian value matrix.Taking advantage of the MUMPS tool to solve the linear equations,which can realize the goal to get the wave field value,gradient of the objective function and Hessian simultaneously.