| Seismic is one of the most important geophysical methods in mineral and energy resources exploration. There is one wave which is spread only along the free-surface and produced under the intervene effection of P-wave and SV-wave near the free-surface named Rayleigh-wave. Its existence was strictly proved and its propagation property was studied first by Lord Rayleigh. Since its energy is very strong, in earthquake, it is the most damaging wave. And in other kinds of seismic prospecting methods because of its strong energy and short time consuming, it is a strong noise which is need to be filter in data processing. But there are two important properties for Rayleigh-wave, its dispersing property is related with the dynamic property of elastic meida and its different penetrating depth along frequency. We can use these two characteristics to detect the aimed objects underground and study the inner structured of the earth.Numerical simulation is one of most important research method of seismic and seismology. As same as other kinds of geophysical forward methods, seismic numerical modeling is under the basic assuming in which the model and relatively physical parameter of underground structure is known. And then all kinds of seismic wave components are simulated by computer program which can be used to study the regulation of the seismic wave spread in different kinds of media. Along with the development of seismic prospecting in complex territory such as mountainous region, seismic numerical simulation method with topography is wide attended by geophysical specialist. Seismic data is very complex especially when the surface is not a plane surface. There are a low signore-to-noise ratio and some complex noisy waves in it. The problem of static correction is very difficult to implement. Many typical theory of elastic wave propagating in horizontal layer earth model is not credible when it is used in the earth model with topography surface, so seismic numerical simulation become the main research implementation. There are two key problems need to be solved when modeling topography surface. The first is how to realize the topography in numerical method and the second is how to realize the free-surface condition in numerical arithmetic. This research has been carried on for many years by several scholars. There are many kinds of seismic numerical modeling methods, but the staggered-grid finite-difference method is the most useful technique up to now, especially in full wave modeling. In this paper, staggered-grid finite-difference method was used to modeling elastic wave propagating in harmonic half space model and layer half space model with topography. And the main work of this paper can be concluded as follows.(1) Some researching history of Rayleigh-wave researching and seismic numerical simulation researching was given in in this paper first. And then the 3D P-SV elastic wave formulation expressed by the relation of velocity and stress was given.(2) The staggered-grid finite-difference format of the 3D P-SV elastic wave formulation was given including its stability condition.(3) The perfectly matched layer method was introduced to filter the artificial reflections, and its effect has been analysed.(4) The AEA method which was first given by XU to solve the numerical realizing problem of 2D plane free-surface in his paper in 2007 was used with step approximate method to solve the topography free-surface numerical problem. At first the AEA scheme for 2D vertical free-surface and 3D horizontal and vertical free-surface was given. And then according to the AEA scheme had given before, the new scheme adapted to 2D and 3D topography free-surface was given.(5) 2D and 3D half space model with topography was modeling with staggered-grid finite-difference method and the new scheme for topography free-surface realizing. To validate the feasibility of the new scheme, two models with incline free-surface which was variety in incline direction was introduced. And the numerical results were compared with the analytical results of half space model. Later, we find that along with the increase of the angle of incline surface the error between numerical results and analytical results increased. And also the error increased along with the distance of source and receiver. To get a fine numerical result fine grid was needed, 80 grids per minimal wavelength in this paper was given.(6) Through analyzing the numerical simulation results of 2D and 3D half space, we find that there are great changes in elastic wave field when topography free-surface was introduced. Arid it is more, obviously for valley than apophysis.(7) At last the dispersing curve of 2D and 3D layer model with plane and topography free-surface were given by the method which XIA had used in his paper first. And it was concluded that the topography free-surface was not well for creating dispersing curve after analyzing the dispersing curve.
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