Appendix 1

The derivation of the Frechet derivative is now presented. For the 3-D Helmholtz equation we have:

 

$$(\nabla^2 + \omega^2 s(\bf {r}')^2 ) \tilde{P}(\bf {r}'\vert\bf {r}_s) \tilde{F}(\bf {r}',\bf {r}_s)$$ = (3.16)

where $\tilde{F}(\bf {r},\bf {r}_s)$ is the source term associated with a harmonically oscillating source at $\bf {r}_s$, and $\tilde{P}(\bf {r}'\vert\bf {r}_s)$ is the associated pressure field. The perturbed pressure field $\tilde P+\delta \tilde P$ due to the slowness perturbation $s+\delta s$is given by:

 

$$(\nabla^2 + \omega^2 (s(\bf {r}')+\delta s(\bf {r}'))^2 ) [\tilde{P}(\bf {r}'\vert\bf {r}_s)+\delta \tilde P] \tilde{F}(\bf {r}',\bf {r}_s)$$ = (3.17)

Subtracting equation 3.20 from  3.21 yields the following equation up to first-order in the perturbation terms:

 

$$(\nabla^2 + \omega^2 s(\bf {r}')^2 ) \delta \tilde P(\bf {r}_g,\bf {r}_s) -2\omega^2 s(\bf {r}') \delta s(\bf {r}') P(\bf {r}',\bf {r}_s).$$ = (3.18)

Neglecting higher-order terms is equivalent to the Born approximation, and is valid for small perturbations in the slowness parameters. Thus multiples are negligible in the perturbed pressure field.

Inverting the above equation by Green's theorem yields:

$$\delta \tilde{P}(\bf {r}_g\vert\bf {r}_s) 2 \omega^2 \int_{V'} s(\bf {r}')\tilde{P}(\bf {r}'\vert\bf {r}_s) \tilde G(\bf {r}'\vert\bf {r}_g) \delta s(\bf {r}') dx'dy'dz'.$$ = (3.19)

Setting $\delta s (\bf {r}') = \delta(\bf {r}-\bf {r}') \delta s(\bf {r})$ in the above equation yields:

$$\delta \tilde{P}(\bf {r}_g\vert\bf {r}_s) 2 \omega^2 s(\bf {r})\tilde{P}(\bf {r}\vert\bf {r}_s) \tilde G(\bf {r}\vert\bf {r}_g) \delta s(\bf {r}) .$$ = (3.20)

Dividing both sides by $\delta s(\bf {r})$ and replacing $\delta$ by $\partial$ gives the Frechet derivative for the pressure field:

$$\frac{\partial \tilde{P}(\bf {r}_g,\bf {r}_s) }{ \partial s(\bf {r})} 2 \omega^2 s(\bf {r})\tilde{P}(\bf {r}\vert\bf {r}_s) \tilde G(\bf {r}\vert\bf {r}_g).$$ = (3.21)