Study On Seismic Waveform Inversion Method In Data Domain

Seismic waveform inversion in data domain makes full use of the kinematics and dynamics information of the prestack seismic wavefield to reconstruct the parameters of the subsurface medium,which has the potential to reveal the structural and reservoir properties under complex geological background.Seismic waveform inversion in data domain transforms the PDEconstrained inverse problem into a large-scale data-matching optimization problem,which tries to reconstruct a wide and continuous wavenumber spectrum of the underground model,and recover the macro background velocity and high-wavenumber structures image simultaneously.Waveform inversion in data domain suffers from large computational burden and strong nonlinearity,which makes it is highly dependent on the accuracy of the initial model.In this paper,we have developed an efficient forward modeling method to reduce the computational cost.Under the framework of least squares inversion,the gradient of the seismic waveform inversion is calculated by the adjoint-state method,and the model parameters are updated by the nonlinear optimization method.In order to mitigate the ill-conditionedness the waveform inversion,we have analyzed the influencing factors of nonlinear waveform inversion from the linearization basis,the Born approximation,and lays a foundation for developing effective inversion strategies and methods.The forward modeling of seismic wavefield based on wave equation is the basis of waveform inversion in data domain.Based on regular grid and staggered grid finite difference method,the numerical simulation method of full waveform in time domain and frequency domain is carried out.The wave equation with the convolution perfect matching layer is also deduced,which effectively suppresses the boundary reflection caused by the artificial boundary truncation.The five-point method and the nine-point method of 2D frequency domain waveform forward modeling are introduced.In order to improve the efficiency of forward modeling,we develop an optimized FD scheme with high spatial and temporal accuracy fornumerical scalar wave modeling based on equivalent staggered grids.The final aim of optimized FD coefficients is to reduce phase velocity errors.We design an objective function for minimizing relative temporal and spatial dispersion errors of waves propagating in all directions.In addition,Newton method is applied to quickly solve this nonlinear problem.Dispersion analysis has demonstrated that space dispersion easily appears in the low velocity media,while time dispersion has more probability to occur in the high velocity media.Modeling tests indicate that the proposed method ensures high accuracy not only in the time domain but also in the space domain.For elastic waveform modeling,we have deduced analytical plane wave solutions in the wavenumber domain with eigenvalue decomposition method.Based on the elastic plane wave solutions,three new time–space domain dispersion relations of ESG elastic modelling are obtained,which are represented by three equations corresponding to P-,S-and converted-wave terms in the elastic equations,respectively.By using these new relations,we can study the dispersion errors of different spatial FD terms independently.The dispersion analysis showed that different spatial FD terms have different errors.It is therefore suggested that different FD coefficients to be used to approximate the three spatial derivative terms.Synthetic examples have demonstrated that this new optimal FD schemes have superior accuracy for elastic wave modelling compared to Taylor-series expansion and optimized space domain FD schemes.Conventional full waveform inversion(FWI)using least square distance between the observed and predicted seismograms suffers from local minima.Recently,optimal transport distance(OTD)has been introduced to FWI to compute the misfit between two seismograms.Instead of comparisons bin by bin,OTD allows to compare signal intensities across different coordinates.This measure has great potential to account for time and space shifts of events within seismograms.However,there are two main challenges in application of OTD to FWI.The first one is that the compared signals need to satisfy nonnegativity and mass conservation assumptions.The second one is that the computation of OTD between two seismograms is a computationally expensive problem.In this paper,a strategy is used to satisfy the two assumptions via decomposition and recombination of original seismic data.In addition,the computation of OTD based on dynamic formulation is formulated as a convex optimization problem.A primal-dual hybrid gradient method with linesearch has been developed to solve this large-scale optimization problem on GPU device.The advantages of the new method are that it is easy to implement and has high computational efficiency.Compared to LSD based FWI,the computation time of the proposed method will approximately increase by 11% in our case studies.A 1D time-shift signals case study has indicated that OTD is more effective in capturing time shift and makes the misfit function more convex.Two applications to synthetic data using transmissive and reflective recording geometries have demonstrated the effectiveness of OTD in mitigating cycle-skipping issues.We have also applied the proposed method to SEG2014 benchmark data,which has further demonstrated that OTD can mitigate local minima and provide reliable velocity estimations without using low frequency information in the recorded data.Least-squares reverse time migration(LSRTM),an effective tool for imaging the structures of the Earth from seismograms,can be characterized as a linearized waveform inversion problem.We have investigated the performance of three minimization functionals as the L2 norm,the hybrid L1/L2 norm,and the OTD for LSRTM.1D signal analysis has demonstrated that the OTD behaves like the L1 norm for two amplitude-varied signals.Unlike the L1 norm,the OTD does not suffer from the differentiability issue for null residuals.Numerical examples of the application of three misfit functions to LSRTM on synthetic data have demonstrated that,compared to the L2 norm,the hybrid L1/L2 norm and OTD can accelerate LSRTM and are less sensitive to non-Gaussian noise.For the field data application,the OTD produces the most reliable imaging results.The hybrid L1/L2 norm requires tedious trial-and-error tests for the judicious threshold parameter selection.Hence,the more automatic OTD is supposed to be recommended as a robust alternative to the customary L2 norm for time-domain LSRTM.Because of the high velocity contrast between salt and sediment,the relation between the waveform and velocity perturbation is strongly nonlinear.Therefore,salt inversion can easily get trapped in the local minima.Since the velocity of salt is nearly constant,we can make the most of this characteristic with total variation regularization to mitigate the local minima.In this paper,we develop an adaptive primal-dual hybrid gradient method to implement total variation regularization by projecting the solution onto a total variation norm constrained convex set,through which the total variation norm constraint is satisfied at every model iteration.The smooth background velocities are firs inverted and the perturbations are gradually obtained by successively relaxing the total variation norm constraints.Numerical experiment of the projection of the BP model onto the intersection of the total variation norm and box constraints has demonstrated the accuracy and efficiency of our adaptive primal dual hybrid gradient method.A workflow is designed to recover complex salt structures in the BP 2004 model and the 2D SEG/EAGE salt model,starting from a linear gradient model without using low-frequency data below 3 Hz.The salt inversion processes demonstrate that wavefield reconstruction inversion with a total variation norm and box constraints is able to overcome local minima and inverts the complex salt velocity layer by layer.