Sensitivity of seismic waveforms to structure and lithology

This paper contains a numerical analysis of stability, order of accuracy and consistencyof the staggered grid finite difference scheme, used on the elastic wave-equation, as de-scribed by Levander (1988). 1 Finite Difference Analysis1.1 Analysis of Finite Difference SchemesA given partial differential equation can be solved using finite differences. As an example,consider the one way wave equation : ÆÆÆÆ(1)This hyperbolic differential equation, can be approximated in a finite grid, by replacing thepartial derivatives by finite differences. The place in the grid is defined by the discrete point inspace, , and the discrete point in time, . A point in the grid is referred to as : . The valueis thus approximated by the discrete point , where.One can choose many different forms of finite difference approximations. The set of ap-proximations chosen for the derivatives in the partial differential equation is termed the finitedifference scheme for the equation. A simple central difference approximation of equation 1 isgiven by :(2)where and are the sampling interval in space and time respectively.Equation 2 solved with respect togives :