1. The Forward Problem

Many types of wave motion can be described mathematically by the equation u_{tt} = \nabla \cdot (c^2\nabla u)+f. We use a more compact notation for the partial derivatives to save space:

u_t = \frac{\partial u}{\partial t}, \quad u_{tt} = \frac{\partial^2 u}{\partial t^2}

Let’s implement a solver for the 1D scalar acoustic wave equation with absorbing boundary conditions. Let u(x,t) be the displacement at time t and at space location x, which is the wavefield. The displacement function u is governed by the following mathematical model.

\begin{split} \frac{1}{c(x)^2} u_{tt}(x,t) - u_{xx}(x,t) & = f(x,t), \\ \frac{1}{c(0)}u_t(0,t)-u_x(0,t) & = 0, \\ \frac{1}{c(1)}u_t(1,t)+u_x(1,t) & = 0, \\ u(x,t) & = 0 \quad\text{for}\quad t \le 0, \end{split}

where the middle two equations are the absorbing boundary conditions, the last equation gives initial conditions, x \in [0,1], and t \in [0,T]. The model velocity is given by the function c(x).

In our notation, we write that solving this PDE is equivalent to applying a nonlinear operator \mathcal{F} to a model parameter m, where m(x) = \frac{1}{c(x)^2} for the scalar acoustics problem.

We then write that \mathcal{F}[m] = u.

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