| Since the pioneering work of Tarantola,waveform inversion has emerged as a tool for estimating velocity models of the subsurface using pre-stack seismic data.The waveform inversions have usually been performed in the time or frequency domain,but this can make it difficult to recover long-wavelength components of the velocity model due to the high nonlinearity of the objective function and the lack of low-frequency components in the field data.Instead,it has been recently suggested that Laplace-domain waveform inversion can circumvent these limitations.By using the zero-frequency component of the damped wavefield,the Laplace-domain waveform inversion can recover long-wavelength structures of the velocity model even if low-frequency components less than 5 Hz are unreliable or would be unusable in conventional inversions.Moreover,the Laplace-Fourier domain full-waveform inversion introduces the damping factor and inverts with the frequency.The Laplace-Fourier domain full-waveform inversion has a natural advantage over the multi-scale inversion strategy,which enriches the multi-scale inversion Strategy.In this paper,the main content of Laplace-Fourier domain scalar wave fullwave inversion method.The first part of the paper is the basic principle part of Laplace-Fourier domain full waveform inversion.Firstly,the basic theory of mathematics is introduced from the basic concept of mathematics.Then,the different optimization algorithms of solving nonlinear problems in mathematics are introduced.The principle of one-dimensional line search to determine the step size is introduced.The validity and accuracy of the forward method of the Laplace-Fourier domain wave equation are proved by the model test.The Laplace-Fourier domain full-wave inversion is carried out for the next step.Foundation.Finally,a theoretical framework of Laplace-Fourier domain full-waveform inversion is established.A logarithmic objective function is proposed for wave field matching of Laplace-Fourier domain full waveform inversion.The characteristics of logarithmic objective function are discussed,and the gradient operator,The basic flow of Laplace-Fourier domain full waveform inversion is introduced.The second part is the optimization part of Laplace-Fourier domain full-waveform inversion.Firstly,the advantages of full-waveform inversion of Laplace-Fourier domain are introduced.Then,the role of low-frequency information component in inversion is analyzed.The low-frequency information is less or less than 5Hz,so that the traditional full-waveform inversion has a strong dependence on the initial velocity model.Laplace-Fourier domain fullwaveform inversion method can reduce the low-frequency information component by introducing the attenuation factor Local minima,reduce the dependence of the inversion on the initial model and the low frequency information in the wave field,and improve the inversion accuracy.And test with the model description.We also discuss the multi-scale inversion strategy.The Laplace-Fourier domain full-waveform inversion introduces the damping wave field,which can be regarded as a layer stripping effect for a single damping factor.The depth of the wave can be controlled by controlling the weight of the wave field at different time by selecting the damping factor,so as to reduce the non-linearity caused by the crosstalk of different waveforms.Laplace-Fourier domain full-waveform inversion is carried out in Laplace-Fourier domain,and the frequencies of inversion can be freely selected.Different frequencies correspond to different scales.Therefore,Laplace-Fourier domain full-waveform inversion is natural in multi-scale inversion strategy And the multi-scale inversion strategy is enriched by the introduction of damping factor.Finally,the Laplace-Fourier domain full-waveform inversion algorithm is modeled.The Laplace-Fourier domain full-waveform inversion effect is mainly tested by the classical Marmousi model which highlights the ability of the full-waveform inversion of Laplace-Fourier domain to recover the low-frequency information component.At the same time,the LaplaceFourier domain full waveform And analyze the effect of multi-scale inversion. |
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