Waveform Inversion in Riemannian Space

Waveform extrapolation employing forward modeling in Riemannian coordinates Sava and Fomel (2005); Shragge (2006) does not present a problem for general approach to waveform inversion developed herein because inversion does not take place in generalized coordinates. Rather, the calculated Green's functions are transformed back to global Cartesian grid through mapping pair

$$\begin{eqnarray} {\mathbf G}_0(\xi,{\mathbf s}) \approx & \mathbf{T}({\bf x};\xi... ... & \mathbf{T}^ \dag ({\bf x};\xi) {\mathbf G}_0(\xi,{\mathbf s}) \end{eqnarray}$$ (38) (39)

where $\mathbf{T}$ is a transformation matrix that interpolates from the Riemannian space defined by $\xi$ to global Cartesian space ${\bf x}$ that includes the transformation Jacobian. In practice, this is applied using weighted sinc interpolation. Thus, one may rewrite the adjoint of equation 27 in the following manner  

$$\Delta s({\bf x}) \approx -\omega^2 \sum_{{\mathbf s}} \sum_... ... \dag (\xi;{\mathbf r}) \Delta \Psi({\mathbf r},{\mathbf s}).$$ (40)