| Seismic modeling based on wave equation plays an important role in designing the seismic geometry,seismic data processing and interpretation.To date,the first-order particle velocity-stress formulations have been widely applied to wave equation simulations.The staggered-grid method is an effective way to solve the first-order wave equation.However,the acoustic,elastic,viscoacoustic and viscoelastic wave simulations based on the staggered-grid suffer from the discretization errors of temporal derivatives and associated stability issues.I focus on these issues and propose new finite-difference stencils to suppress the temporal discretization errors for the acoustic and elastic modeling,and propose k-space methods to suppress the temporal dispersion errors for the viscoacoustic and viscoelastic wavefield simulations.Compared with conventional center-grid based finite-difference,the staggered-grid finite-difference has better accuracy and stability.However,when applying the conventional staggered-grid finite-difference to the acoustic and elastic cases,the accuracy is only(2M)th-order accuracy in space but only second-order accuracy in time.Recently developed temporal high-order staggered-grid finite-difference increases the modeling accuracy to(2M)th-order accuracy in both space and time.However,it adds too many extra grid points for finite-difference,and thus takes too much computational costs.To address the issues of both methods,I design a hybrid staggered-grid finitedifference methods,which are applied to acoustic and elastic cases,respectively.The hybrid staggered-grid methods maintain the high temporal accuracy and balance the computational cost and accuracy.Numerical examples and dispersion tests demonstrate that compared with conventional spatial high order methods,the hybrid staggered-grid method has higher temporal accuracy.Compared with conventional temporal high-order methods,hybrid staggered-grid method has less computational costs.Stability analyses reveal that the new hybrid staggered-grid finite-difference method has better performance than the conventional spatial or temporal high-order methods.Unlike the acoustic hybrid staggered-grid simulation,the new elastic hybrid staggered-grid finitedifference method obtains the finite-difference coefficients by P-or S-wave dispersion relations,respectively.And thus,it can ensure the accuracy of different wave types.The analyses show that the elastic hybrid staggered-grid can gain a good balance among accuracy and efficiency.To address the nonlinearity problem of the new hybrid staggered-grid method,I have proposed a split-step linear optimization method.The optimization method avoids the nonlinearity problem and obtains the global optimal finite-difference coefficients.Numerical cases and dispersion analyses validate the correctness of the split-step optimization method.To date,the fractional spatial derivative based viscoacoustic and viscoelastic wave equations have been widely applied to seismic modeling and imaging.The staggeredgrid pseudospectral method is often applied to solve these wave equations.Due to the temporal derivatives associated with the attenuation terms and time stepping terms,the temporal accuracy for the whole discrete viscoacoustic and viscoelastic wave equations is only first order.To tackle the issue,I have designed two k-space numerical solvers to completely compensate the discretization error in time.The numerical cases demonstrate that two k-space methods agree with the Green’s functions analytical solutions well.For heterogeneous media,the lowrank algorithm is applied to accelerate the wave simulation.Besides,the second k-space method adopts alternative wave propagator decomposition method,and reduces the number of computational expensive k-space operators drastically.Therefore,the computational efficiency is boosted greatly.Finally,the model tests also demonstrate that the new viscoacoustic and viscoelastic kspace methods are more stable than the conventional staggered-grid pseudospectral method. |
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