Multiparameter Inversion From Wave Equations By Frequency Domain Full Waveform Inversion

With the development of technologies for seismic exploration, researches onseismic forward and inverse problems for complex mediun have increased. Waveequations which can precisely model the propagation properties of seismic wave incomplex medium usually contain more than one physical parameters describing theproperties of the complicated medium, and the accurate inversion of these parametersis theoretically and practically significant to the description of rock features andreservoir prediction. Nowadays, full waveform inversion (FWI) is an efficient methodto obtain parameters from wave equations, and the most commonly invertedparameter is velocity where in the inversion process the other parameters are set to beknown but they are unknown actually. This problem can be avoided if theseparameters are inverted simultaneously. However, the intrinsic difficulties ofmultiparameter inversion make it not easy to implement the simultaneous inversion ofmore than one parameter. Considering the importance and difficulties ofmultiparameter inversion, wave equations, such as Stokes equation and visco-acousticequation, which have taken the viscous property of the medium into considerationwhen the wave propagation is modeled, are used to test the multiparameter inversion.The two wave equations are chosen because they can not only describe the wavepropagation accurately but also not too complex. Besides, the parameters obtained areimportant and helpful for the description of tectonic distribution and the prediction ofoil and gas.Before the multiparameter inversion is started, the forward modeling of Stokesequation and2D acoustic equation is analyzed first. Since frequency domain FWI isused in the multiparameter inversion, and the observed seismic data are recorded inthe time domain, time domain seismic records should be transformed into the frequency domain by fast Fourier Transformation (FFT) in frequency domain FWI.Based on that, formulas of finite-difference time domain (FDTD) and finite-differencefrequency domain (FDFD) about Stokes equation and2D acoustic equation are given.Wavefields directly computed in frequency domain are compared with that obtainedby transforming the time domain wavefields to frequency domain by FFT. Then theformulas are tested on synthetic models to obtain the time domain and frequencydomain wavefields, and comparisons about the transformations of the wavefieldsbetween the domains are given. The comparison results of2D acoustic wavefieldsshow that, for low frequency, wavefields directly computed in frequency domain arenearly the same as that transformed from the time domain by FFT while for highfrequency there are some differences in the values.After the forward modeling of Stokes equation is analyzed, multiparameterinversion from Stokes equation by frequency domain FWI is researched. Thesensitivity analysis of the parameters to the misfit function are first given before themultiparameter inversion is implemented, and the results show that the sensitivities ofdensity and velocity to misfit function are higher to that of viscosity coefficient tomisfit function. Then mono-parameter inversion (i.e. velocity, density, viscositycoefficient) is implemented and the influences of some relative factors are analyzed.According to the sensitivity analysis, different inversion strategies and step lengthselection methods are compared in the simultaneous inversion of velocity andviscosity coefficient, and the results show that simultaneous inversion strategy andparabolic step length computing method are the best, which are used in thesimultaneous inversion of density and velocity as well as density and viscositycoefficient. In the three parameter inversion (i.e. density, velocity and viscositycoefficient), different inversion strategies are compared, and the results show that thestrategy constrained by the Gardner formula can obtain acceptable inversion results.With the same Gardner constraint, influences of different factors are compared andanalyzed.The gradient is scaled by the diagonal approximate Hessian matrix in themultiparameter inversion from Stokes equation by frequency domain FWI. When this method is extended to the inversion of2D acoustic equation, it seems to be tootime-consuming because the computation grids increases and the computationstorages are large with the dimension extended from one to two. To find a properoptimization method, commonly used optimization methods, such as gradientmethods and Newtonian methods are tested on the same synthetic model. Thereconstructed models by these methods are compared and analyzed from the aspectsof relative error and computation time, and the results show that proper optimizationmethod should be chosen according to the problem to be solved. Then a newquasi-Newton method named memoryless quasi-Newton (MLQN) method is appliedin frequency domain FWI to invert velocity from surface seismic data for the firsttime. This method can attain acceptable results with low computational cost and smallmemory storage requirements. To test the efficiency of the MLQN method in FWI,two synthetic models, a modified Marmousi model and a modified overthrust model,are examined from the surface seismic data with and without white Gaussian noise.For comparison, the conjugate gradient (CG) method is carried out for the samevelocity models with the same parameters. The inverted velocities by the two methodsare compared based on the aspects of memory storage requirements, computation timefor each iteration, and error. By keeping the memory storage requirements andcomputation time in each iteration similar, the reconstructed velocity models obtainedusing the MLQN method are closer to the true velocity models than those obtainedusing the CG method, especially for the noise-added data. The numerical tests showthat the MLQN method is feasible and reliable in FWI.In this thesis, MLQN method is applied in the multiparameter inversion from2Dvisco-acoustic wave equation by frequency domain FWI. Sensitivity analysis of theparameters to the misfit function is given from the aspects of partial derivationwavefields and the variation of the misfit function with the variation of the parameters,and the results show that density and velocity are more sensitive to the misfit functionthan Q. In the mono-parameter inversion (i.e. velocity, density, Q), different seismicinformation is used in the inversion and their influences to the rebuilt models aregiven. As for the two-parameter inversion (i.e. density and velocity, velocity and Q, density and Q), the results show that for parameter couples with Q, acceptablereconstructed models can obtained once appropriate normed Q-1is selected. In thethree parameter inversion (i.e. density, velocity and Q), two inversion strategies arecompared: the first inversion strategy is that the three parameters are inverted at thesame time; the second inversion strategy is that in the first stage, density and velocityare inverted with Q being the initial model and in the second stage, with the initialdensity and velocity model being the models obtained in the first stage, the threeparameters are inverted simultaneously. The results show that the inverted Q model isin better accordance with the true Q models when the second inversion strategy isused.The simultaneous multiparameter inversion from Stokes equation andvisco-acoustic equation by frequency domain FWI can not only obtain more than oneparameters, supplying more information for the description of rock property andprediction of reservoirs, but also lay foundations for the research on themultiparameter inversion from more complicated equations. Besides, the research ofmultiparameter inversion from seismic data can also supply reference to the samequestion in other fields.