Image-space wave-equation tomography in the generalized source domain

APPENDIX A

This appendix derives the perturbed one-way wave equation with respect to the slowness perturbation. Let us start with the one-way wave equation for the source wavefield as follows:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...,\omega) = \overline{f_s(\omega)\delta({\bf x}-{\bf x}_s)} \end{array} \right.,$$ (A-1)

We can rewrite the slowness and the source wavefield as follows:

$$s({\bf x}) = \widehat{s}({\bf x}) + \Delta s({\bf x})$$ (A-2)

$$D({\bf x},{\bf x}_s,\omega) = \widehat{D}({\bf x},{\bf x}_s,\omega) + \Delta D({\bf x},{\bf x}_s,\omega),$$ (A-3)

where $\widehat{s}({\bf x})$ and $\widehat{D}({\bf x},{\bf x}_s,\omega)$ are the background slowness and background wavefield, and $\Delta s({\bf x})$ and $\Delta D({\bf x},{\bf x}_s,\omega)$ are small perturbations in slowness and source wavefield, respectively. If $\Delta s({\bf x})$ is small, then the square root in the first equation of A-1 can be approximated using Taylor expansion as follows:

$$\sqrt{\omega ^2 s^2({\bf x})-\vert{\bf k}\vert ^2} \approx \sqrt{... ...)}{\sqrt{1-\frac{\vert{\bf k}\vert^2}{\omega^2 {\widehat{\bf s}}^2({\bf x})}}}.$$ (A-4)

Substituting Equations A-2, A-3 and A-4 into Equation A-1 and ignoring the second-order terms yield the following linearized one-way wave equation for the perturbed source wavefield:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...f x}_s,\omega) \\ \Delta D(x,y,z=0,{\bf x}_s,\omega) = 0 \end{array} \right. .$$ (A-5)

Similarly, we can also obtain the linearized one-way wave equation for the perturbed receiver wavefield as follows:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...bf x}_s,\omega) \\ \Delta U(x,y,z=0,{\bf x}_s,\omega) = 0 \end{array} \right..$$ (A-6)

Image-space wave-equation tomography in the generalized source domain