Acoustic wavefield evolution as function of source location perturbation

The implementation

As we have seen earlier, the equations associated with the wavefield perturbation has a form similar to the wave equation. As a result, any of the methods typically used to solve the wave equation suffices for the perturbation partial differential equations. The most general and straightforward of these methods is the finite-difference approach. With proper space and time grid distribution this method provides acceptable solutions regardless of the complexity of the velocity model. Its only limitation is the relative slow execution speed.

To apply the source perturbation, I first solve the original wave equation for a particular point source as a background field for the perturbation step. In the process, we store the Laplacian evaluation as they are needed for the perturbation calculation. 2Using an initial condition of $D_x(x,z,t=0)$=0, we can solve for $D_x$ using equation (4) and add the solution to the original wavefield using equation 8, and thus, obtain a new wavefield shape approximating that for another source location. For higher-order accuracy, we also solve for $D_{xx}$ using equation (7) 2with a similar initial condition, $D_{xx}(x,z,t=0)$=0, and include it in the Taylor's series expansion with terms from the solutions of the original background source and $D_x$.

To avoid potential problems with the storage requirement especially in 3D, we can solve the wave equation and the corresponding perturbation equations simultaneously, and thus, use the already evaluated derivatives (Laplacian) directly. In this case, the cost of the perturbation finite difference application is similar to the cost of solving the wave equation. Thus, the cost of the first and second order expansions to obtain the wavefield for other sources is equivalent to two and three times, respectively, of the cost of solving the wave equation for a single source. However, the information obtained approximates the wavefield for infinite source possibilities in the vicinity of the original source.

Since the perturbation equations have the same form as the wave equation they adhere to the same Courant-Friedrichs-Lewy (CFL) condition [Courant et al. (1928)]. Thus, the time step is constrained by the grid spacing, which in turn relies on velocity, based on the following formula:

$$\frac{\min(\Delta x, \Delta y, \Delta z)}{\Delta t} > \sqrt{3} v$$ (15)

where $\Delta t$ is the time step interval and $\Delta x$, $\Delta y$, and $\Delta z$ are the grid spacing along the main axes and $v$ is the velocity.

Acoustic wavefield evolution as function of source location perturbation