Orthogonal Body-fitted Grid Forward Modeling Under Undulating Ground Conditions

The forward modeling of seismic waves plays an irreplaceable role in the exploration of oil and gas resources.At present,seismic prospecting work is continually developing to complex areas,so it is of great practical significance to study the propagation characteristics and laws of seismic waves under complex surface.Among various seismic wave forward modeling methods,the finite difference method is one of the widely used methods.Because of its flexibility,adaptability,easy programming,simple operation and high computational efficiency in numerical simulation,it has attracted the attention of scholars and engineers.In this paper,the finite difference method in Cartesian coordinate system is used to simulate the surface fluctuation.Before the simulation,the wave equation form in Cartesian coordinate system with Cartesian coordinate components as control variables and its difference scheme in staggered grids are discussed in detail.Then the boundary conditions(absorbing boundary conditions and free boundary conditions)are discussed and the corresponding implementation methods are given.When the numerical simulation is carried out,the finite difference in conventional rectangular grids is found.The division method has a great influence on the boundary characterization of the undulating model,that is,the undulating interface will be divided into stepped strata,which will cause comer scattering at the inflection point of the step,resulting in the reduction of simulation accuracy and cannot achieve good simulation results.In view of the above problems,this paper introduces orthogonal body-fitted grids with good fit for the undulating surface to mesh the model,which is convenient for the development of numerical simulation.However,because orthogonal body-fitted grids are a kind of grids in curvilinear coordinates,the wave equation derived under the condition of conventional rectangular grids is no longer suitable for numerical simulation under orthogonal body-fitted grids.Therefore,we introduce the concept of tensor and use it as a control variable to derive the wave equation in curvilinear coordinates,and then obtain the tensor form of wave equation.Thereafter,the in-situ grid finite difference method is used to solve the discrete wave equation of this form,which avoids the complex interpolation operation needed in staggered grids.In the treatment of free boundary,because the grids are orthogonal,the treatment of free boundary is similar to that of horizontal free surface,and the stress mirror method is used to implement the treatment of free boundary.Complete matching layer method based on auxiliary differential equation is used to eliminate artificial scattering.Finally,numerical simulations of different models are carried out.Through the above research,we can find that:(1)the finite difference method under the conventional rectangular mesh can cause artificial numerical scattering in the forward simulation of the undulating surface model,which results in low simulation accuracy;(2)the orthogonal body-fitted mesh can do a good body-fitted subdivision of the undulating surface;(3)the forward simulation combined with the orthogonal body-fitted mesh generation technology can well implement the free boundary conditions.It eliminates comer scattering in conventional grids and improves the accuracy of simulation,and can clearly understand the propagation characteristics and laws of wave field under the fluctuation table.