Madagascar Programming Reference Manual

Introduction

This section presents time-domain finite-difference modeling [1] written with the RSF library. The program is demonstrated with the C, C++ and Fortran 90 interfaces. The acoustic wave-equation

$$\Delta U - \frac{1}{v^2}\frac{\partial^2U}{\partial t^2} = f(t)\vert\Delta U - f(t)\vert v^2 = \frac{\partial^2U}{\partial t^2},$$

where $\Delta$ is the Laplacian symbol, $f(t)$ is the source wavelet, $v$ is the velocity, and $U$ is a scalar wavefield. A discrete time-step involves the following computations:

$$U_{i+1} = [\Delta U - f(t)]v^2\Delta t^2 + 2U_i - U_{i-1},$$

where $U_{i ? 1}$ , $U_i$ and $U_{i + 1}$ represent the propagating wavefield at various time steps. In this exercise we shall use a discrete Laplacian accurate up to the fourth order and the second derivative of time is accurate up to the second order.